Mathematical Methods in the Applied Sciences 2011 Vol34 Issue1

Research Article

Received 26 May 2009 Published online 19 May 2010 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1325MOS subject classification: 74 A 30

Dynamic steady-state crack propagationin quasi-crystals

Enrico Radia and Paolo Maria Marianob∗†

Communicated by M. Grinfeld

The steady propagation of planar cracks in quasi-crystalline bodies with velocity lower than the one of bulk elasticmacroscopic waves is under scrutiny. Closed-form solutions to the balance laws are provided. Unusual Mach numberlimits are determined. Numerical experiments describing peculiar aspects of the crack propagation in quasi-crystalsare performed by varying parametrically the coupling coefficient between macroscopic deformation and substructuralevents. In this way, classes of quasi-crystals are then compared. Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: quasi-crystals; crack propagation; mechanics of complex bodies

1. Introduction

Quasi-crystals are a special class of quasi-periodic alloys characterized by atomic clusters displaying incompatible symmetrieswith periodic tiling of atoms in space: icosahedral symmetry in three dimensions, pentagonal symmetry in two dimensions (seedescriptions in [1--3]). Quasi-periodicity in space is assured by atomic rearrangements which create and annihilate atomic clusterswith symmetry different from the prevailing one—in this sense, the phase of the matter changes locally.

The representation of the generic material element—the essential starting point of any continuum model of condensed matter—can be done by picturing it as a cluster of atoms not fixed once and for all. Since in traditional continuum mechanics every materialelement is represented just by a point, its inner dimensions are not represented in what is chosen as ambient space. So the possibleatomic rearrangements within material elements have to be described by inner degrees of freedom, represented in an appropriatespace. They are called phason degrees of freedom, and are collected point by point in the values of a vector field defined over themacroscopic region occupied by the body under scrutiny. A differentiable, orientation preserving map describes the macroscopicdeformation. The mechanics of quasi-crystals at continuum level (long-wavelength approximation) admits, in this sense, a multiscaleand multifield description.

Peculiar actions appear: they are different from the standard macroscopic stress, determined only by crowding and shearing ofmaterial elements, and are called phason stresses. They are due to changes in the spatial inhomogeneity of the phason degrees offreedom. Phason stresses influence the macroscopic mechanical behavior in various aspects such as wave propagation, equilibriumand possible evolution of defects [4]. A paradigmatic example is one of the cracks propagating in quasi-crystals: phason stressesinfluence the force driving the crack tip. Molecular dynamics based simulations in [5--7] and analyses of field equations at acontinuum level in [8--10] furnish precise indications in this sense (see also [11--13]). In particular, a result in [9] is indicative: Considerthe quasi-static evolution of a stooping crack in a quasi-crystal specimen. The crack path determined by taking into account phasoneffects is perturbed with respect to the result obtained by neglecting phason effects.

Here, fracture mechanics in quasi-crystals is tackled once again. A planar crack propagating steadily in a three-dimensionalquasi-crystalline body, with a speed lower than the bulk wave velocity, is considered. Diffusion associated with phason degrees offreedom (the so-called phason diffusion), a common phenomenon in quasi-crystals, is neglected. The investigation is developedin a linear constitutive setting. Closed-form solution to the balance equations is obtained under general loading conditions. Sucha solution is applied to a concrete case, where it is shown that the amplitude of the phason stress increases with the incrementof the coupling between gross deformation and phason degrees of freedom. In the linear setting treated here, such a coupling isgoverned by a single coefficient. Its experimental evaluation is uncertain. Its potentially admissible range is bounded from above by

aDi.S.M.I., Università di Modena e Reggio Emilia, via Amendola 2, I-42122 Reggio Emilia, ItalybDICeA, University of Florence, via Santa Marta 3, I-50139 Firenze, Italy∗Correspondence to: Paolo Maria Mariano, DICeA, University of Florence, via Santa Marta 3, I-50139 Firenze, Italy.†E-mail: [email protected]

Contract/grant sponsor: GNFM-INDAMContract/grant sponsor: MIUR-PRIN08

Copyright © 2010 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2011, 34 1–23

1

E. RADI AND P. M. MARIANO

the value preventing the propagation of shear waves. Below such a threshold, there is a range of incompatible values with Rayleighwave propagation. The recognition of the existence of such a range is a result obtained here.

It is also shown that macroscopic and phason stresses display square root singularities at the crack tip. Stress intensity factorsare determined. The energy release rate is evaluated for subsonic crack propagation.

Finally, Yoffe’s solution for fixed finite-length cracks traveling in a linear elastic simple body is also extended to the presentsetting. It is interpreted as a tool for analyzing the ‘rigid body’-type evolution of a linear continuous distribution of dislocations.

The analysis developed here is an evolution of a proposal in [14--16] dealing with anisotropic elasticity of simple bodies. It isbased on the Stroh formalism [17, 18] (see also [19--21] for a comprehensive review). A similar approach has been adopted for theanalysis of crack propagation in a piezoelectric media in [22--26] and in magneto-electro-elastic solids [27].

The constitutive tensors involved are isotropic and the solution procedure furnishes a degenerate eigenvalue problem: a funda-mental matrix admits two double eigenvalues and these identical roots have a single eigenvector associated with them. Strohformalism has been modified in agreement with the generalization proposed in [28--31] for orthotropic elastic media for the caseof degenerate eigenvalue problem.

As the velocity of the crack tip goes to zero, the eigenvalue problem associated with the description of the crack propagationadmits a single eigenvalue with algebraic multiplicity equal to 3. The static case, then, requires analysis a part (see [32] for therelevant results).‡

Some notations. For n×n matrices A and B, the notation AB indicates the standard row-by-column product determining a n×nmatrix. If b is a n-dimensional vector, the product Bb furnishes a n-dimensional vector. For notational convenience, the symbol〈〈ak〉〉 is adopted to denote a diagonal matrix, for example diag(a1, a2, a3, a4). The operators Re and Im extract from a given operatorwith complex entries (or simply from a complex number) the real and imaginary parts. An overbar denotes complex conjugate inthe sense of complex numbers. As usual, the notation (·) := (··) means that (·) is defined by (··). R3∗ is the dual version of R3 andis (of course) isomorphic to R3 itself. B indicates an open, connected set in the three-dimensional ambient physical space R3 withsurface-like boundary �B oriented by the outward unit normal n everywhere but a finite number of corners and edges (essentiallyit sufficies to assume that B has the Lipshitz boundary). For F a linear operator, FT indicates the transpose.

2. Mechanics of quasi-crystals: essential elements

The non-linear setting for the mechanics of quasi-crystals discussed in [4, 34] and its linearization (see [35--37]) are summarizedbelow with the aim of introducing the reader to the analyses in the ensuing sections. Quasi-crystals are essentially consideredas a prominent example of complex bodies, those bodies with gross mechanical behavior prominently influenced by chances inthe material texture, an influence exerted through peculiar interactions. For quasi-crystals, the microstructural changes are localatomic rearrangements: jumps of atoms between neighboring places and/or collective atomic modes generated by the flipping ofcrisscrossing topological alterations needed to maintain matching rules [35, 37, 38]. Creation and annihilation of clusters of atoms,with symmetry differing from the prevailing one, determine the characteristic quasi-periodicity of quasi-crystals. Internal degreesof freedom, the ones exploited by the atoms to rearrange themselves, are then attributed to material elements in the continuum

modeling. They are collected at every x in a vector � belonging to an isomorphic copy of R3, indicated by R3

.Although quasi-periodicity is a global property, the field description (a local one) in terms of the phason field x �−→�(x) (see

[4, 34, 35, 37]) accounts at macroscopic level for the mechanism generating quasi-periodicity rather than the periodicity itself thatre-appears in the structure of the constitutive equations. The word phason recalls that the crystalline phase changes locally.

2.1. Placements and morphological descriptors

The ambient space R3 hosts a quasi-crystalline body in its macroscopic reference configuration B. Every point x in B indicatesthe place of a generic material element. Other macroscopic configurations are achieved by means of deformations: orientationpreserving differentiable bijections

x �−→y :=y(x)∈R3, x ∈B.

Really, it is convenient to consider macroscopic configurations different from the reference one as placed in an isomorphic copy ofR3, giving in this way a true exclusive role to B, a role becoming clear when covariance and existence theorems of minimizers forthe elastic energy are investigated (see relevant comments in [4, 39]). Such a distinction is not essential here and is then skipped,it is only mentioned to clarify the connection with the complete version of the theory summarized in the ensuing notes, a theorypresented in [4].

The spatial derivative of a deformation is indicated by F :=Dy(x) and is, at every x, a linear operator from the tangent space to B

at x to the tangent space to the actual place Ba :=y(B) of the body at y(x). The assumption that x �−→y be orientation preservingimplies det F>0 at every x, as it is well known. The natural measure of deformation (referred to B) is the tensor E :=1 / 2(FTF− I),where FTF is the left Cauchy–Green tensor. Essentially, E is half of the relative difference between the pull-back on B of the naturalmetric in Ba and the original metric in the reference place itself—meaning relative that the difference between the two metrics is

‡Use is made of appropriate modifications of techniques discussed in [15, 16, 22, 30, 31, 33].

2



multiplied by the inverse of the metric in B. Once the time t is introduced, a macroscopic motion in a time interval [0, d] is given by

(x, t) �−→y :=y(x, t)∈R3, x ∈B, t ∈ [0, d].

Sufficient smoothness in time t is presumed. The macroscopic velocity is defined by y := (d / dt)y(x, t) in the referential description(that is as a field over the tube B×[0, d]). The field

(x, t) �−→� :=�(x, t)∈ R3

, x ∈B, t ∈ [0, d]

indicates the phason degrees of freedom in time. Besides differentiability in space, sufficient smoothness in time is presumed. The

time rate of change of � in referential description is � := (d / dt)�(x, t)∈ R3

, the spatial derivative of (x, t) �−→� is indicated by N :=D�(x).

It is a linear operator from the tangent space to B at x to the space R3

, where the phason degrees of freedom are represented.

2.2. Linearized macroscopic kinematics

Consider the displacement field

(x, t) �−→u :=u(x, t)=y(x, t)−x, (x, t)∈B×[0, d].

The spatial derivative of (x, t) �−→u is indicated here by W :=Du(x, t). It follows that y(x, t)= u(x, t) and F = I+W , with I the second-rankidentity tensor. As usual, the condition

|W|�1

at every point x and instant t, defines the infinitesimal deformation regime. The measure of deformation E reduces to its linearizedpart � :=symW . Moreover, B and Ba can be also ‘confused’ within the limits of the infinitesimal deformation setting. No distinctionis also made between x �−→� and y �−→�a :=�◦y−1, with �a is the Eulerian (actual) representation of �.

2.3. Consequences of a d’Alembert–Lagrange-like principle

In addition to standard bulk forces and stresses due to (i) interaction of the body with the external environment and (ii) crowdingand shearing of material elements, peculiar actions connected with microscopic changes in the material texture arise. They aredefined by the power that they develop in the rate of the phason degrees of freedom.

Appropriate integral balances can be obtained by requiring the invariance of the overall power of all external actions overa generic part of the body under changes in observers, which alter isometrically the ambient space. Consequent alterations in

the space R3

collecting the phason degrees of freedom are induced by a differentiable homomorphism (see [9] for an extensivediscussion of non-standard changes in observers). An independent integral balance of phason actions does not appear naturally.The invariance procedure furnishes integral balances associated with the Killing fields of the metric in space. Different local balancescan be derived from them: they include the independent local (so not global) balance of phason actions. The result presented in [4]corrects the integral balances assumed in [35]. A discussion on these aspects can be found in the last section of [4]. In this way, therepresentation of actions and constitutive structures following from the second law of thermodynamics can be tackled separately.

However, for the purposes of this paper, the matter can be treated from a concise point of view by merging the abstractrepresentation of actions and the prescription of their constitutive structure into an unique principle. As discussed in [4], appropriateto this case is a d’Alembert–Lagrange-like principle expressed by

�

(∫B×[0,t]

Ldx∧ dt

)+∫B×[0,t]

zv ·��dx∧ dt =0, (1)

where

L :=L(x, y, y, F, N)= 1

2�|y|2 −e(x, F, N)+w(y)

with e the elastic potential and w the potential of external actions. No peculiar (phason) kinetic energy is associated here withthe phason degrees of freedom. It seems that scattering data do not support the introduction of a phason kinetic energy. Someauthors, however, introduce it on the basis of different considerations. The matter is a bit controversial (see [10, 34--36, 40]).

Also, the Lagrangian density does not depend on � alone, rather only on its gradient, as suggested also by data. The circumstanceprecludes the presence of a conservative self-action, a type of action appearing in general in complex bodies (see details in [41]).A dissipative self-action, however, appears and is associated with the diffusive nature of phason modes (see e.g. [35]). It is indicatedby

zv =zv(x, F,�, N, �)∈ R3∗

and is such that

zv · ��0 (2)


3


for any choice of �, with equality sign holding only when �=0. The inequality defines the dissipative nature of zv . A solution to(2) is

zv =c�,

with c the value at x and t of a positive-definite function

c :=c(x, F,�, N, �)

such that

c(x, F,�, N, 0)=0

(see [40] for estimates of c).Under C1-regularity of L and C2-regularity of transplacement and phason fields, Euler–Lagrange equations for (1) are given by

Div P+b=�y, DivS=c�,

where P :=−�FL is the first Piola–Kirchhoff stress, b :=�yL the vector of body forces, � the mass density in the reference place,S :=−�NL the so-called phason stress, due to the interactions between neighboring material elements with different amount ofphason degrees of freedom. The elastic potential e is further constrained by the requirement to be invariant under the evaluationof different observers, at least the ones mentioned previously, which involve isometric changes in the ambient space R3 and the

action of SO(3) on R3

. Another possible requirement is the invariance of the Lagrangian density or the whole integrand in (1) underthe class of changes in observers already mentioned. In this case, it follows that

skw(�F eFT +�⊗zv +�NeTN)=0

(details are in [4]). So there is a lack of symmetry in Cauchy’s stress

� := (det F)−1�F eFT

as a consequence of the presence of phason actions.§

Piola transform determines the actual counterparts of the other actions in Eulerian representation

ba := (det F)−1b, zva := (det F)−1zv, Sa := (det F)−1SFT,

where the index a recalls that they are defined in the actual place Ba. The balance equations in the Eulerian representation thenread

div�+ba =�a�, divSa =ca�a, (3)

where � is the actual velocity, �a := (det F)−1�, ca := (det F)−1c and �a is the rate of the map �a :=�◦y−1, that is the actual repre-sentation of the morphological descriptor field.

In the analysis presented in the ensuing sections, phason diffusion is neglected (the second balance (3) then reads divSa =0).One reason is to make easier the analysis of the crack dynamics under scrutiny. Another reason is that such a diffusion is associatedwith a self-action which does not contribute to the evolution equation of the crack tip directly (see the results in [9]); it contributesindirectly because it influences the fields around the crack. To be more precise, the J-integral¶ at a point x, belonging to a crack

§Symmetry of � can be recovered for special choices of the constitutive structures, as the ones adopted further on in this paper. Symmetry of � can be

recovered also if it is presumed that changes in observers in the ambient space have no connection with the ones in the phason space R3

. The question isrelevant for general complex bodies and is a strict consequence of the notion of observer that can be considered appropriate for the mechanics of complexbodies. Here, the point of view is the one discussed by one of us in [4, 9]. In fact, the description of the mechanics of quasi-crystals in terms of ambient andphason spaces is only an ideal representation of phenomena occurring in the real physical space, as all mathematical models are. Deformations and atomicrearrangements in quasi-crystals occur in the physical space. If a change in observer is considered as a change of atlas (coordinate system) in the physicalspace, in the mathematical representation at hands it is necessary to consider then all geometrical environments used to represent morphology and motionof a quasi-crystalline body. Moreover, since, there is an interplay between gross deformation and phason degrees of freedom, the physical change in observer

must be described not only in the mathematical ambient space but also in the space of the phason degrees of freedom, namely R3

. These remarks apply to the

general model-building framework of the mechanics of complex bodies. There, R3

is replaced, in general, by a finite-dimensional differentiable manifold M.

The link between the changes in the atlas in the ambient space and in M (which is R3

in the case treated here), a link described by a differentiablehomomorphism, represents the fact that microstructural changes (atomic rearrangements in the case of quasi-crystals) occur in the physical space. Analyticaldetails about such a description of changes in observers are skipped here. They can be found in [9] for the general format of the mechanics of complex bodies,and in [4] for the case of quasi-crystals. The question is mentioned here just to underline that only precise physical circumstances—they must be specified timeto time—may suggest to not consider between changes in atlas just mentioned. When the link cannot be considered, independently of constitutive choicesCauchy stress � results symmetric. The circumstance, however, is special. Balance equations for quasi-crystals proposed in [35] seem to agree with this specialcase. The point is further discussed in [4].

¶The present expression of J is a special case of the J-integral determined in [9, 41] within the general model-building framework of the mechanics of complexbodies, involving manifold-valued geometrical descriptors of the material microstructures.

4



tip, is in fact given here by

J(x) :=n(x) · lim�→0

∫�D�(x)

((e+ 1

2�|y|2

)I−FTP−NTS

)n d�,

where n is the direction of crack propagation at x, and D�(x) is a disk, centered at x, and belonging to the orthogonal plane to thetangent to the tip at x. The J-integral determines the driving force at the crack tip in addition to the limit value at the tip of thesurface energy and the local contribution of the tip own energy time the local tip scalar curvature (see e.g. [9]). No direct role isplayed in the expression of J by the self-action zv =c�. The term c� influences the J-integral only indirectly, through the balanceequations. In particular, c� determines dispersion of elastic waves in dynamics and decaying phason modes.

The analysis developed here is restricted to a purely elastic range, at a fixed temperature that allows brittle phase. Also, infinitesimaldeformation setting is considered, so that the distinction between referential and actual representations of the actions becomevolatile.

2.4. Linearized constitutive setting

In the linearized kinematical setting, the elastic potential can be a quadratic form‖ in W and N. It reads

e(W, N)= 12 (CW) ·W + 1

2 (KN) ·N+(K′N) ·W.

C, K and K′ are fourth-rank constitutive tensors. C is the standard elastic tensor and K′ describes the coupling between grossdeformation and phason degrees of freedom, the latter pertaining to K. Explicit expressions of such tensors are here taken from[35]∗∗

Cijhk = ��ij�kl +(�ik�jl +�il�jk),

Kijkl = k1�ik�jl +k2(�ij�kl −�il�jk),

K′ijkl = k3(�i1 −�i2)(�ij�kl −�ik�jl +�il�jk),

where in the last equation no summation over repeated indices is assumed, � and are standard Lamé constants, k1 and k2 describethe pure phason contribution to the energy and k3 is the so-called coupling coefficient.

The stress measures � and S (remind the elimination of the index ’a’) are then the derivatives of e with respect to W and Nrespectively, namely

�=CW +K′N, S=K′TW +KN,

where the exponent T over fourth-rank tensors indicates major transposition. More explicitly, such constitutive equations read

� = 2SymW +(�div u)I+k3(2 Sym N− I div�)R, (4)

Sa = k3R(2 Sym W − I div u)+k1N−k2(N∗− I div�), (5)

where R :=e1 ⊗e1 −e2 ⊗e2, with e1 and e2 two vectors of the vector bases e1,e2,e3, associated with �i1 and �i2.

2.5. Boundary conditions

Dirichlet’s-type boundary conditions are natural for the displacement field on some part �Bu of the body boundary. They can bealso prescribed in terms of � because it is possible to imagine, for example, that no phason degrees of freedom pertain to thematerial elements placed along the boundary of B.

In principle, standard loading devices allow one to assign the macroscopic tractions Pn. No loading device seems available forthe prescription of non-zero phason tractions Sn.

In terms of Sn, the natural boundary condition is then only

Sn=0.

This problem is common to all complex bodies admitting multifield description of their morphology and interactions.To avoid it, a surface energy can be considered over the boundary (like the body would be coated in some way) and appropriate

boundary phason tractions can be constructed through it. The construction is, however, not completely natural and contains thedegree of freedom associated with the constitutive choice of the explicit expression of such a surface energy. Details about therelated formalism can be found in [4].

‖In non-linear deformation setting, e cannot be convex in F =W + I (I the unit tensor) for reason of objectivity, whereas it can be convex in N.∗∗The theory presented here is different from the material reviewed in [35] on the explicit characterization of the foundational aspects leading to and justifying

balance equations. The results in [4] rectify in a sense the form of the integral balances presented in [35].


5


x2

vt

σ0 τ0

x

y

Figure 1. Semi-infinite rectilinear crack propagating at constant speed v, loaded by normal and shear standard stresses, �0 and0 applied on a portion of length � of the crack surface.

3. Steady-state crack propagation in the infinitesimal deformation setting

3.1. Preliminary issues: Stroh formalism

The whole framework summarized above is restricted from now on to an intrinsically two-dimensional problem. A planar crackpropagating with constant speed v in an infinite quasi-crystalline body is considered (see Figure 1). The crack remains planar duringits evolution. Crack margins are in contact. Deformations are infinitesimal.

Reference is made to two Cartesian coordinate systems: (Oxyz) and (Oxyz). The former is fixed, the latter moves with the cracktip. Crack kinking and curving are excluded during the evolution by homogeneity and type of boundary conditions imposed. It isthen possible to select the x1-axis to be coincident with the direction of propagation. The x-axis is also selected along the x1-axis.The x3-axis is taken along the straight crack front. Since the crack is straight, it is possible to restrict the analysis to the plane x3 =0.Symmetry suggests also to consider essentially planar the distribution of phason degrees of freedom.

The problem under scrutiny is then two-dimensional.Steady-state crack propagation is under analysis. It means that for any differentiable field (x1, x2, t) �−→a(x1, x2, t), the condition

a(x1, x2, t) :=a(x−vt, y)

holds, so that

a,1 =a,x, a,2 =a,y, a :=−va,1,

where the comma denotes partial derivative, the indices 1 and 2 indicate, respectively x1 and x2, the superposed dot denotes timederivative, as usual.

It is convenient to define

p := (�11,�21,S11,S21), q := (�12,�22,S12,S22),

r := (u1,1, u2,1,�1,1,�2,1), s := (u1,2, u2,2,�1,2,�2,2).

In this way, the linear constitutive Equations (4) and (5) can be then be written as

p=Ar+Bs, q=BTr+Cs,

where

A=

⎡⎢⎢⎢⎢⎢⎢⎣

2+� 0 k3 0

0 0 k3

k3 0 k1 0

0 k3 0 k1

⎤⎥⎥⎥⎥⎥⎥⎦

, B=

⎡⎢⎢⎢⎢⎢⎢⎣

0 � 0 k3

0 −k3 0

0 −k3 0 k2

k3 0 −k2 0

⎤⎥⎥⎥⎥⎥⎥⎦

, C=

⎡⎢⎢⎢⎢⎢⎢⎣

0 −k3 0

0 2+� 0 −k3

−k3 0 k1 0

0 −k3 0 k1

⎤⎥⎥⎥⎥⎥⎥⎦

.

Also, the balance equations read now

p,1 +q,2 =�v2Dr,1,

where D :=diag(1, 1, 0, 0). Use of constitutive equations then implies

(A−�v2D)r,1 +2(SymB)r,2 +Cs,2 =0,

after the exploitation of the relation s,1 = r,2, a relation justified by Schwartz’s theorem.

6



Since the matrix Q :=A−�v2D is not singular, the balance equations can be further rewritten as(r

s

),1

+[

2Q−1(SymB) Q−1C

−I 0

](r

s

),2

=(

0

0

), (6)

where I is the 4×4 identity matrix.

3.2. Degenerate eigenvalue problem

The spectrum of the matrix of coefficients in the equations of motion is given by the eigenvalue problem††

[2Q−1(SymB)−�kI Q−1C

−I −�kI

]⎛⎝h(k)

g(k)

⎞⎠=

(0

0

)(7)

with h(k) and g(k) are four-dimensional vectors. It reduces to

[C−�k2 SymB+�2kQ]g(k) =0, h(k) =−�kg(k).

The eigenvalues �k are the roots of the characteristic equation

det(C−�k2 SymB+�2kQ)=0.

By defining

� := k3

k1,

the characteristic equations can be rewritten as

(1+�2k )2((1+�2

k )(2+�−�k3)−�2k�v2)((1+�2

k )(−�k3)−�2k�v2)=0.

Purely imaginary roots with positive imaginary parts and related conjugates are available. By introducing Mach numbers

m21 = �v2

2+�−�k3, m2

2 = �v2

−�k3,

and recalling that i denotes (as usual) the imaginary unit, the roots with non-negative imaginary parts are given by

�1 = i√1−m2

1

, �2 = i√1−m2

2

, �3 = i.

The algebraic multiplicity of the root �3 is 2, whereas the eigenvalues �1 and �2 are distinct for v>0. Moreover, for subsonic crackpropagation, the crack tip speed v is smaller than each of the speeds of the elastic waves in the bulk in the x1-direction; consequentlyboth Mach numbers m1 and m2 are smaller than 1. For a stationary crack, namely for v =0, the four roots �k , k =1, 2, 3, 4, arecoincident.‡‡

The eigenvectors h(k) and g(k), corresponding to �k , k =1, 2, 3, are given by non-trivial solutions of (7) for each �k , namely

g(1) = (im21, −m2

1

√1−m2

1, −i�(4−3m21), �(4−m2

1)√

1−m21),

g(2) = (im22

√1−m2

2, −m22, −i�(4−m2

2)√

1−m22, �(4−3m2

2)),

g(3) = (0, 0, −i, 1),

h(1) =⎛⎝ m2

1√1−m2

1

, im21, −�

4−3m21√

1−m21

, −i�(4−m21)

⎞⎠ ,

h(2) =⎛⎝m2

2, im2

2√1−m2

2

, −�(4−m22), −i�

4−3m22√

1−m22

⎞⎠ ,

h(3) = (0, 0, −1, −i).

††Solutions of negative exponential type are in general considered in the Stroh formalism. However, when the problem is degenerate (as the one treated here),other paths have to be followed (see [21]).

‡‡The stationary case is treated in [32].


7


The eigenvectors h(k) have been obtained by making use of the relation h(k) =−�kg(k). The eigenvalue problem (7) is degeneratebecause the geometric multiplicity of the repeated eigenvalues �3 and �3 is 1. There exists only one eigenvector for each of theseeigenvalues. For them, the geometric multiplicity is less than the corresponding algebraic multiplicity, which is 2. A generalizedeight-dimensional eigenvector (h(4), g(4)) can be then defined for the repeated eigenvalue �3. It is linearly independent of the otherthree eigenvectors and is a solution to [

Q−12(SymB)−�3I Q−1C

−I −�3I

]⎛⎝h(4)

g(4)

⎞⎠=

⎛⎝h(3)

g(3)

⎞⎠ (8)

(see [21, 42]). As a consequence, it follows that

g(4) =(

2(�+)+�v2

4�(�+), i

2(�+)−�v2

4�(�+),

(�v2)2

8�k3(�+), 0

),

h(4) =(

−i2(�+)+�v2

4�(�+),

2(�+)−�v2

4�(�+), i

(1− (�v2)2

8�k3(�+)

), −1

).

For k =1, 2, 3, 4, the eigenvectors h(k) and g(k) can be considered respectively the columns of the matrices

H :=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

m21√

1−m21

m22 0 −i

2(�+)+�v2

4�(�+)

im21 i

m22√

1−m22

02(�+)−�v2

4�(�+)

−�4−3m2

1√1−m2

1

−�(4−m22) −1 i

(1− (�v2)2

8�k3(�+)

)

−i�(4−m21) −i�

4−3m22√

1−m22

−i −1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

and

G :=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

im21 im2

2

√1−m2

2 02(�+)+�v2

4�(�+)

−m21

√1−m2

1 −m22 0 i

2(�+)−�v2

4�(�+)

−i�(4−3m21) −i�(4−m2

2)√

1−m22 −i

(�v2)2

8�k3(�+)

�(4−m21)√

1−m21 �(4−3m2

2) 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Equations (7), for k =1, 2, 3, (8) and their complex-conjugate equivalent relations can be then rewritten as[Q−12(SymB) Q−1C

−I 0

][H H

G G

]=[

H H

G G

][S+N 0

0 S+N

], (9)

where S and N are the semi-simple nilpotent matrices defined by

S :=〈〈�k〉〉, N :=

⎡⎢⎢⎢⎢⎢⎣

0 0 0 0

0 0 0 0

0 0 0 1

0 0 0 0

⎤⎥⎥⎥⎥⎥⎦ ,

in tune with the Jordan decomposition theorem. Note that �3 =�4 in S. Equation (9) implies that H=−G(S+N)Consider a four-dimensional differentiable vector field (x1, x2) �−→b(x1, x2), defined by(

r

s

)=[

H H

G G

](b

b

)=2Re

(Hb

Gb

).8



By introducing such a relation into (6) and using (9), it follows that

(b

b

),1

+[

S+N 0

0 S+N

](b

b

),2

=(

0

0

)

which is equivalent to

b,1 +(S+N)b,2 =0 (10)

together with its complex-conjugate. Componentwise, (10) reads

bk,1 +�kbk,2 =0 for k =1, 2, 4 (11)

and

b3,1 +�3b3,2 =−b4,2. (12)

By introducing the complex variable

zk =x1 + ix2

√1−m2

k , k =1, 2, 3, 4 (13)

with the convention m3 =m4 =0, which implies z3 =z4 =x1 + ix2 =: z, Equations (11) and (12) become respectively

�bk

�zk=0 for k =1, 2, 4

and

�b3

�z3=−1

2b4,2.

It follows that the complex function (x1, x2) �−→bk(x1, x2), k =1, 2, 4, must be analytic in the complex variable zk—actually bk =ak(zk)—then

�b3

�z3=−1

2b′

4(z)�z

�x2=− i

2b′

4(z),

where the prime denotes derivative with respect to z. Integration of the last relation yields

b3(x1, x2)=a3(z)− i

2zb′

4(z), (14)

with a3(z), an arbitrary analytic function of z.In terms of the vector b, the expressions of the displacement, phason degrees of freedom and stress fields then read

r = 2Re(Hb), s=2Re(Gb),

p = 2Re(Jb), q=2Re(Kb),

where J :=AH+BG and K :=BTH+CG.It is possible to write

b(z)=a(z)− i

2zNa′(z), (15)

(where ak(z)=bk(z) for k =1, 2, 4. a3(z) is the value of the function introduced in (14)), then

r(z) = 2Re(H(a− izNa′)(z)), s(z)=2Re(G(a− izNa′)(z)),

p(z) = 2Re(J(a− izNa′)(z)

), q(z)=2Re(K(a− izNa′)(z)).

The formal expressions derived in this section become explicit once a is determined for a specific boundary value problem.Two explicit cases are treated in the ensuing section to clarify the point. Moreover, an application to the surface wave problem isconsidered in Appendix B.


9


4. Semi-infinite straight crack loaded along its margins

4.1. Crack tip fields

Assume that the crack be loaded on both the margins with a uniform distribution of normal and shear standard stresses indicatedrespectively by �0 and 0. Phason stresses are prescribed to vanish along the crack margins. �0 and 0 are distributed along aportion of length �, measured by starting from the tip. Figure 1 represents the situation just described.

It is presumed that both macroscopic and phason stresses vanish at infinity. The assumption about the standard Cauchy stress israther natural. As regards the phason stress, it is possible to imagine that at infinity there is a boundary layer where phason degreesof freedom are absent. The assumption has a constitutive nature.

Since p and q vanish at infinity by assumption, vector functions b and a vanish at infinity too.Standard and phason tractions are continuous across x1-axis:

q+(x1)−q−(x1)=0. (16)

In fact, no external loads are applied far from [−�, 0] along x1-axis, the margins of the crack are in contact, and they are loaded withthe same load along [−�, 0].

Displacement and phason fields are continuous across x1-axis ahead of the crack tip, namely

r+(x1)−r−(x1)=0 for x1>0.

The assumed asymptotic behavior of the stress fields at infinity and the linear constitutive setting impose that the functionz �−→a(z) vanish at infinity.

The continuity of standard and phason stresses across the margins of the crack implies that the function

z �−→ j(z) :=(

K(

a− i

2zNa′

))(z)−

(K(

a+ i

2zNa′

))(z)

be analytic in the whole complex plane. By the Liouville theorem, it is then constant, so it vanishes as a consequence of theasymptotic behavior z �−→a(z). It follows that(

K(

a− i

2zNa′

))(z)=

(K(

a+ i

2zNa′

))(z). (17)

Moreover, the continuity condition for displacement and phason fields implies that

H(

a+− i

2x1Na′+

)(x1)−H

(a++ i

2x1Na′+

)(x1)=H

(a−− i

2x1Na′−

)(x1)−H

(a−+ i

2x1Na′−

)(x1) (18)

for x1>0. In the previous formula, a+ and a− are the standard limit values at the x1-axis defined by a±(x1) := limx2→0± a±(x1 + i2x2).

An analogous definition holds for a′±.The relation (17) implies (

a±+ i

2x1Na′±

)(x1)=

(K

−1K(

a±− i

2x1Na′±

))(x1).

The insertion of this last relation in (18) and the assumption that the matrix

Re(iHK−1)

be not singular imply (a+− i

2x1Na′+

)(x1)−

(a−− i

2x1Na′−

)(x1)=0 for x1>0. (19)

Finally, indicating by q0 the vector collecting the applied loads along [−�, 0], the loading conditions imply that

K(

a+− i

2x1Na′+

)(x1)+K

(a−+ i

2x1Na′−

)(x1)=

{q0 for −��x1�0

0 for x1<−�,

along the margins of the crack, so that

(a+− i

2x1Na′+

)(x1)+

(a−− i

2x1Na′−

)(x1)=

{K−1q0 for −��x1�0

0 for x1<−�.(20)1

0



Equations (19) and (20) define an inhomogeneous Riemann–Hilbert problem for the analytic vector function z �−→ (a−(i / 2)zNa′)(z).Solution is given by (

a− i

2zNa′

)(z)= 1

L(z)K−1q0 (21)

(see [43, 44]), where L(z) is a 4×4 matrix and is defined by

L(z) :=⟨⟨

1

2ilog

√zk + i

√�

√zk − i

√�

−√

�

zk

⟩⟩.

Since N is nilpotent (that is N2 =0), it follows that

Na′(z)= 1

2 (z+�)

(�

z

)3/2NK−1q0. (22)

The introduction of (21) and (22) in (15) then yields

b(z)= 1

(L(z)− x2

2(z+�)

(�

z

)3/2N

)K−1q0.

The displacement field and the field of phason degrees of freedom, collected in a vector d := (u1, u2,�1,�2), can be obtained bydirect integration of the function z �−→ r(z) with respect to x1. It follows that

d(z)= 2

Re

(H

(�(zk)−x2

(1

2ilog

√z+ i

√�√

z− i√

�−√

�

z

)N

)K−1q0

),

where �(z) is a 4×4 matrix defined by

�(z) :=⟨⟨

zk +�

2ilog

√zk + i

√�

√zk − i

√�

−√

�zk

⟩⟩.

4.2. Energy release rate

The evaluation of the energy release rate during a crack propagation can be obtained by specializing abstract results obtained inthe general model-building framework of the mechanics of complex bodies (see [9, 41]).

Traditional expressions of the energy release rate in simple bodies (see [45, 46]) can be also used: quasi-crystalline structurescan be considered, in fact, as incommensurate projections over the three-dimensional ambient space of six-dimensional standardperiodic crystalline structures. To fix ideas on a special case, consider a periodic lattice made of square cells living in a plane, take aline with irrational inclination with respect to the lattice, and consider a strip of infinite extension and finite thickness—a thicknessincluding some atomic layers. The strip is symmetric across the line itself. A one-dimensional quasi-crystal in the plane can beobtained by projecting over the inclined line the atoms contained in the selected strip. Higher-dimensional quasi-crystals can beobtained in an analogous way.

The energy release rate can be obtained by adapting common formulas in fracture mechanics of simple bodies to the higher-dimensional setting, then ‘projecting’ in an appropriate way them into the physical ambient space.

In the present setting, the energy release rate G then reads

G=−

2lim�0

√�q(�) · lim

�0

√�(r(�exp(i ))−r(�exp(−i ))),

where � is a radial coordinate in a polar coordinate system centered at the tip. With the notations introduced previously, it is alsopossible to write

G=4�

q0 ·Re(iHK−1)q0.

The 4×4 matrix Y := iHK−1 is Hermitian (see the proof in Appendix A)—it means Y= YT—so that its real part is symmetric. G isthen positive when Re(Y) is positive definite.

4.3. Remotely loaded crack with free crack surface

Consider q0 =0. Formally (see once more [43, 44]), the solution reads

b(z)= 12 L(z)K−1a0,

where a0 ∈R4 is a constant, and

L(z) :=〈〈(2 zk)−1/2〉〉


11


is a 4×4 matrix. Consequently, the traction vector q ahead of the crack tip at x2 =0 and for x1>0 is given by

q= (2 x1)−1/2a0.

Assume that the stress ahead of the crack tip can be written in terms of stress intensity factors, all collected in the list k :=(KII, KI, TII, TI), where KI and KII are the intensity factors of the macroscopic stress, whereas TI and TII are the ones of the phasonstress. Such an assumption is motivated by the remark that the mechanics of quasi-crystals can be considered to some extent as themechanics of simple bodies in a space with dimension higher that the one of the physical space, modulo appropriate projections.

The primary consequence is that

a0 =k

and

q= (2 x1)−1/2k.

Then, it follows that

Na′(z)=− 1

4z√

2 zNK−1k,

and also

b(z) = 1

2

1√2

(⟨⟨1√zk

⟩⟩+ x2

2z√

zN)

K−1k,

d(z) =√

2

Re

((⟨⟨√zk⟩⟩− x2

2√

zN)

K−1k)

.

The energy release rate is then given by

G= 12 k ·Re(iHK−1)k.

5. Yoffe’s traveling crack: a scheme for the evolution of linear continuous distributionsof dislocations

In the 1951 paper [47], Yoffe described the steady state propagation in an infinite, planar, elastic body of a straight crack withconstant length 2�. Yoffe’s analysis is developed in an infinitesimal deformation regime. Loads are placed at infinity. The geometryof the case under analysis is explained in Figure 2: a straight crack is depicted, the frame of reference traveling with the crack itselfis centered at the middle point of the crack span with the x1-axis aligned with the crack.

The assumption that a traveling crack maintains constant its length is somewhat artificial. Yoffe’s solution, however, has physicalinterest. In fact, a linear crack (as it is the one under analysis here) can be considered as a continuous distribution of dislocationsalong the interval of the x1-axis where the crack is placed (see [48]). For this reason, analyzing a traveling linear crack with constantlength is tantamount to consider the ‘rigid body’-type evolution of a linear continuous distribution of dislocations—the dislocationsare orthogonal to the plane considered.

Yoffe’s solution is extended here to the mechanics of quasi-crystals in linear deformation setting. The aim is to furnish a tool foranalyzing continuous distributions of straight dislocations, at least in some aspects of their mechanical behavior.

With reference to Figure 2, loading conditions are prescribed at infinity, namely

p∞ := (�∞11,�∞

21, 0, 0), q∞ := (�∞12,�∞

22, 0, 0).

σ11∞

σ22∞

σ12∞

x1

x2

vt

xy

Figure 2. Yoffe’s crack propagating at constant speed v, loaded by remotely applied standard stress �∞ .

12



Figure 3. Contours of normalized phonon and phason stress fields under Mode I loading conditions, namely for 0 =0, �=5, and m2 =0.6.

The presence of a remote loading implies

b(z)=b∞+a(z)− 1

2izNa′(z).

It then follows

r(z) = r∞+2Re

(H(

a− i

2zNa′

)(z)

), s(z)=s∞+2Re

(G(

a− i

2zNa′

)(z)

),

p(z) = p∞+2Re

(J(

a− i

2zNa′

)(z)

), q(z)=q∞+2Re

(K(

a− i

2zNa′

)(z)

),

where

r∞ :=2Re(Hb∞), s∞ :=2Re(Gb∞), p∞ :=2Re(Jb∞), q∞ :=2Re(Kb∞).

The conditions fulfilled along the crack margins by the displacement field, the phason degrees of freedom, and the standard andphason stresses are then used as in the previous sections. The first result is

r+(x1)= r−(x1) for |x1|>�.

Then, by taking into account that the matrix Re(iHK−1) is not singular, it is possible to write(a+− i

2x1Na′+

)(x1)−

(a−− i

2x1Na′−

)(x1)=0 for |x1|>�.


13



The absence of tractions along the crack surfaces implies(a+− i

2x1Na′+

)(x1)+

(a−− i

2x1Na′−

)(x1)=−K−1q∞ for |x1|<�.

The two conditions define an inhomogeneous Riemann–Hilbert problem for the analytical vector function z �−→ (a−(i / 2)zNa′)(z).The following solution vanishing at infinity [43, 49] is ammissible:(

a− i

2zNa′

)(z)= 1

2(L(z)−I)K−1q∞, (23)

where I is once more the 4×4 unit matrix and

z �−→L(z) :=⟨⟨

zk√z2

k −�2

⟩⟩

is a 4×4 matrix-valued function that generates a square root singularity at the crack tip for both the standard and phason stressfields.

As in the previous example, since N is nilpotent, it follows that

Na′(z)=− �2

2(z2 −�2)3/2NK−1q∞. (24)

The formulas (23) and (24) then imply

a(z)− i

2zNa′(z)= 1

2

(L(z)−I+ �2x2

(z2 −�2)3/2N

)K−1q∞.

14



Figure 5. Contours of normalized phonon and phason stress fields under Mode II loading conditions, namely for �0 =0, �=5, and m2 =0.6.

The explicit expression of the vector d :=(u1, u2,�1,�2), collecting the in-plane components of standard displacements and phasonfield, follows by direct integration of the map z �−→ r(z) with respect to x1. The result is

d=〈〈zk〉〉Re(H(2a∞−K−1q∞)+Re

(H

(�(

zk)− zx2√

z2 −�2N

)K−1q∞

),

where

�(zk) :=⟨⟨√

z2k −�2

⟩⟩

is a 4×4 matrix at every zk . On the basis of the previous results, the energy release rate G is given by

G= �

2q∞·Re(iHK−1)q∞.

5.1. Displacement and phason fields at the tip

A polar coordinate system (�,�), with the origin at the traveling crack tip, is useful for the ensuing developments. In this frame ofreference, the complex variable (13) is expressed by

zk =�+�

(cos�+ i

√1−m2

k sin�

)for k =1, 2, 3, 4.


15



When � is small with respect to � — the half-length of the crack — q and d are

q =√

�

2�Re

(K(⟨⟨

1

�k (�)

⟩⟩+ 1

2sin�

(cos

3

2�− i sin

3

2�

)N)

K−1q∞)

,

d =√

2��Re

(H(

〈〈�k(�)〉〉− 1

2sin�

(cos

�

2− i sin

�

2

)N)

K−1q∞)

,

where

�k(�) :=√

cos�+ i√

1−m2k sin�.

6. Special cases

Once values of the constitutive coefficients are selected, insertion of them in the closed-form solutions to the problems discussedhere permits the specialization of the results. The experimental determination of values for the constitutive coefficients is discussedvariously in the pertinent literature. Different values are proposed (see e.g. [38, 40, 50--55]). Values used in [50, 52, 55] are adoptedhere. They read

�=85 GPa, =65 GPa, k1 =0.044 GPa, k2 =−0.0396 GPa.

16



0-0.0010

-0.0005

0

0.0005

0.0010

S r1/

2 /KI

1 /r1/

2 K

I

-0.2

0.2

0.4

0.6

0.8

0 45 90

45 90

45 90

135

0

r r1/

2 /KI

r1/2 /K

I

0-50

-40

-30

0

-20

-10

180

135 180

135 180

(a)

(c) (d)

(e)

0-0.0004

-0.0002

0

0.0002

0.0004

S r r1/

2 /KI

2 /r1/

2 K

I

0

-40

-30

0

-10

-20

-50

0-0.5

0.5

1.0

1.5

2.0 = 5

u 1/r

1/2 K

I u 2

/r1/

2 KI

m2 = 0.6

m2 = 0.2

m2 = 0.8

0

45 90

45 90

45 90

135 180

135 180

135 180

(b)

(f)

Figure 7. Angular variations of standard, phason stresses and displacement and phason fields around the crack tip under remote Mode I loading conditions,for �=5, and for different values of m2.

Uncertainty is associated with the experimental determination of the coupling coefficient k3. In particular, values �=5 and�=35 are considered here for comparison purposes. The latter value is close to the limit value at which shear wavesdo not propagate (see the top straight line in Figure 9 indicating this value: about 38 in the special conditions adoptedhere).

The dynamics of a semi-infinite straight crack is considered in the figures. Yoffe’s problem is viewed in a sense as a special case.

• Figure 3 collects the standard and phason stresses around the crack tip in Mode I loading conditions (0 =0). For m2 =0.6 and�=5 the magnitude of the phason stress is ten times smaller than the one of the standard stresses.

• In analogous conditions and with �=35 (high coupling, close to the limit of no-propagation of shear waves), the amplitudesof both the standard and phason stresses are comparable (Figure 4).


17


0-0.0015

-0.0010

-0.0005

0

1 /r1/

2 K

II

0 45 90

45 90

45 90

135

r1/2 /K

II

r r1/

2 /KII

0-40

-30

10

0

-20

-10

180

135 180

135 180

(a)

(c) (d)

(e)

0-0.0015

-0.0010

-0.0005

0

2 /r1/

2 K

II

0

10

20

40

30

0

0

u 2/r

1/2 K

II

u 1/r

1/2 K

II

45 90

45 90

45 90

135 180

135 180

135 180

(b)

(f)

=5

-8

0

0.4

0.6

m2 = 0.8

m2 = 0.6

m2 = 0.20.2

-0.2

-0.4

-0.6

Sr1/

2

KII

S rr1/

2

KII

-1.0

1.5

1.0

0.5

-0.5

0

Figure 8. Angular variations of standard, phason stresses and displacement and phason fields around the crack tip under remote Mode II loading conditions,for �=5, and for different values of m2.

• In Mode II conditions (�0 =0), for m2 =0.6 and �=5, the magnitude of the phason stress is one hundred times lesser than theone of the standard stresses (Figure 5). With increased coupling the amplitude of the phason stress increases too and becomesten times lesser than the one of the standard stresses at �=35 (Figure 6).

• The angular plot of standard stresses is reminiscent of the shapes obtained in linear fracture dynamics in simple bodies (seeFigure 7(a),(b) and [45], also Figure 8(a),(b))

• The magnitude of the phason degrees of freedom decreases when Mach number increases in Mode I conditions and for�=5 (Figure 7(e),(f)). The same behavior is recognized in Mode II conditions only with respect to the phason field component�1, whereas �2 increases with Mach number (Figure 8(e),(f)). The same monotonous behavior is shown by the phason stress�r-component in Mode II conditions and with �=5.

• Under the limit value for � beyond which the shear waves do not propagate, there is a range of values in the plane �−m2—the region delimited by the curves in Figure 9—indicating non-admissibility conditions for crack propagation. Two possibleinterpretations of the result can be as follows: (1) In that region—that is with that combinations of coupling coefficient and

18



Figure 9. Energetically favorable (G>0) and non-favorable (G<0) regimes for crack propagation in quasi-crystals in the �−m2 plane.

Mach number—crack propagation could be admissible only if the mechanical behavior is not purely linear elastic. (2) Somepairs (�, m2) of values are not physically admissible. Experiments can corroborate one interpretation or the other, or, else, theycan falsify both.

Appendix A: Orthogonality relations

The orthogonality relations between the actual and the generalized eigenvectors are displayed here. Hermitian character of thematrix Y := iHK−1 is also shown.

Define the two 8×8 matrices

J :=⎡⎣2(SymB) C

C 0

⎤⎦ , M :=

⎡⎣Q−12(SymB) Q−1C

−I 0

⎤⎦ , (A1)

where M is the matrix of coefficients of the system (7) and J=JT. The exponent T indicates matrix transposition. Remind that Cand Q are also symmetric. Multiplication of M by J furnishes

JM=⎡⎣2(SymB)Q−12(SymB)−C 2(SymB)Q−1C

CQ−12(SymB) CQ−1C

⎤⎦= (JM)T =MTJ. (A2)

Denote by v(k) the eight-dimensional eigenvectors constituted by h(k) and g(k), namely

v(k) :=

⎛⎜⎝h(k)

g(k)

⎞⎟⎠ for k =1, 2, 3, 4,

where v(4) is the generalized eigenvector of the degenerate matrix M, the other values of k correspond to standard eigenvectors.From the definition of v(k), it follows that

JMv(k) = �kJv(k), JMv(k) = �kJv(k) for k =1, 2, 3, (A3)

JMv(4) = �3Jv(4) +Jv(3), JMv(4) = �3Jv(4) +Jv(3). (A4)

By using (A2), it is immediate to realize that the vectors Jv(k) and Jv(k), k =1, 2, 3, 4, are the eigenvectors and the generalizedeigenvectors of the degenerate matrix MT. The relation (A3), (A4) and the symmetry of the matrices J and JM, for h, k =1, 2, 3,imply

�kJv(k) ·v(h) = JMv(k) ·v(h) =v(k) ·JMv(h) =�hv(k) ·Jv(h) =�hJv(k) ·v(h),

�kJv(k) · v(h) = JMv(k) · v(h) =v(k) ·JMv(h) = �hv(k) ·Jv(h) =�hJv(k) · v(h).


19


Since the six eigenvalues �k and �k , for k =1, 2, 3, are not equal, the following orthogonality relations must hold:

Jv(k) ·v(h) = Jv(k) · v(h) =0 for h, k =1, 2, 3, h �=k,

Jv(k) · v(h) = Jv(k) ·v(h) =0 for h, k =1, 2, 3. (A5)

By using (A4)1, one can also find that

JMv(4) ·v(3) =v(4) ·JMv(3) =�3v(4) ·Jv(3) =�3Jv(4) ·v(3) = (JMv(4) −Jv(3)) ·v(3), (A6)

and

�3Jv(4) ·v(k) = (JMv(4) −Jv(3)) ·v(k) =JMv(4) ·v(k) =v(4) ·JMv(k) =�kv(4) ·Jv(k) =�kJv(4) ·v(k), (A7)

for k =1, 2.The use of (A4) then implies

�3Jv(4) · v(k) = �kJv(4) · v(k) for k =1, 2, 3, (A8)

�3Jv(4) · v(4) = �3Jv(4) · v(4). (A9)

From (A6) it follows that

Jv(3) ·v(3) =0, Jv(3) · v(3) =0.

Moreover, (A7), (A8) and (A9) imply

Jv(4) ·v(k) = Jv(k) ·v(4) =0 for k =1, 2,

Jv(4) · v(k) = Jv(4) ·v(k) =0 for k =1, 2, 3, 4,

(A10)

so that from (A5) and (A10) it follows that

UTJU=[

T 0

0 T

], (A11)

where

T :=

⎡⎢⎢⎢⎢⎢⎣

J11 0 0 0

0 J22 0 0

0 0 0 J34

0 0 J34 J44

⎤⎥⎥⎥⎥⎥⎦

with Jhk :=Jv(k) ·v(k). The matrix U collects eigenvectors, namely

U := [v(1) v(2) v(3) v(4) v(1) v(2) v(3) v(4)]=[

H H

G G

]. (A12)

The introduction of (A12) and (A1)a in (A11) and the use of the definition K :=BTH+CG yields

HTK+KTH=HTK+KTH=0, HTK+KTH=0, (A13)

namely

⎡⎣HT KT

HT KT

⎤⎦[K K

H H

]=[

T 0

0 T

],2

0



so that [K K

H H

]−1

=[

T−1 0

0 T−1

]⎡⎣HT KT

HT KT

⎤⎦

where

T−1 :=

⎡⎢⎢⎢⎢⎢⎣

1 / J11 0 0 0

0 1 / J22 0 0

0 0 −J44 / J234 1 / J34

0 0 1 / J34 0

⎤⎥⎥⎥⎥⎥⎦ .

The orthogonality relation (A13)a implies that the matrix Y := iHK−1 is Hermitian. Relations similar to (A13 ) have been alsoderived by Stroh (see [17, 18]) for anisotropic standard linear elasticity, with reference to a non-degenerate matrix. Such a result byStroh has been extended by Ting to the degenerate case in [20].

Appendix B: Rayleigh waves in quasi-crystals

Stroh formalism adopted in the previous sections to analyze the evolution of a planar crack can be used straight away to analyzethe evolution of surface waves in quasi-crystals. The approach is summarized here briefly.

Consider a quasi-crystalline body occupying the half plane x2 =0 in the Cartesian frame of reference (0x1x2x3). Assume that thepotential bh(x1, x2), h=1, 2, 3, 4, appearing in (10), be of the form

bh(x1, x2)= fh(x2) exp(ikx1),

where k is a positive number. Substitution into (10) implies

ikf(x2)+(S+N)f′(x2)=0

that is

ikfh(x2)+�hf′h(x2)=0, h=1, 2, 4,

ikf3(x2)+�3f′3(x2)=−f′4(x2).

Solution is expressed by

f(x2)=〈〈exp(−k√

1−m2hx2)〉〉(c− ikx2Nc),

where m3 =m4 and c is a constant vector to be determined by means of the boundary conditions.By imposing that q=0 at x2 =0, since q=2Re(Kb), as derived in the main body of this work, it follows that

2Re(Kf(0) exp(ikx1))=0.

Moreover, since f(0)=c, from the explicit expression of f above, the condition

Kcexp(ikx1)+Kc exp(−ikx1)=0

is satisfied at every x1. The arbitrariness of x1 implies that

Kc=0,

which is the eigenvalue problem for the matrix K, a problem admitting non-trivial solutions only when

detK=0.

Such a characteristic equation is satisfied for values of the Mach number plotted in Figure 9)—where it is indicated by mlim—as afunction of the coupling parameter � between the gross deformation and phason activity.

Acknowledgements

This work has been developed within the programs of the research group in ‘Theoretical Mechanics’ of the ‘Centro di RicercaMatematica Ennio De Giorgi’ of the Scuola Normale Superiore at Pisa. The support of the GNFM-INDAM is acknowledged. ERacknowledges also Italian MIUR for financial support under the contract ‘Multiscale modelling, numerical and experimental analysisof complex materials and structures with novel applications’-PRIN08.


21


References1. Steurer W. Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals. Zeitschrift

für Kristallographie 2004; 219:391--446.2. Steurer W, Deloudi S. Fascinating quasicrystals. Acta Crystals 2008; A64:1--11.3. Schmicker D, van Smaalen S. Dynamical behavior of aperiodic intergrowth crystals. International Journal of Modern Physics B 1996; 10:2049--2080.4. Mariano PM. Mechanics of quasi-periodic alloys. Journal of Nonlinear Science 2006; 16:45--77.5. Mikulla R, Stadler J, Krul F, Trebin H-R, Gumbsch P. Crack propagation in quasicrystals. Physical Review Letters 1998; 81:3163--3166.6. Rösch F, Rudhart C, Roth J, Trebin H-R, Gumbsh P. Dynamic fracture of icosahedral model quasicrystals: a molecular dynamic study. Physical

Reviews B 2005; 72:014128-1--014128-9.7. Rudhart C, Trebin H-R, Gumbsh P. Crack propagation in perfectly ordered and random tiling quasicrystals. Journal of Non-Crystalline Solids 2004;

334 & 335:453--456.8. Fan T-Y, Mei Y-W. Elastic theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials. Advances in Applied

Mechanics 2004; 57:325--343.9. Mariano PM. Cracks in complex bodies: covariance of tip balances. Journal of Nonlinear Science 2008; 18:99--141.

10. Zhu A-Y, Fan T-Y. Dynamic crack propagation in decagonal Al–Ni–Co quasicrystals. Journal of Physics: Condensed Matter 2008; 20:295217-1--295217-9.

11. Li X-F, Fan T-Y, Sun Y-F. A decagonal quasicrystal with a Griffith crack. Philosophical Magazine 1999; 79:1943--1952.12. Wang X, Pan E. Analytical solutions for some defect problem in 1D hexagonal and 2D octagonal quasicrystals. PRAMANA—Journal of Physics

2008; 70:911--933.13. Guo Y-C, Fan T-Y. A mode II Griffith crack in decagonal quasicrystals. Applied Mathematical Mechanics 2001; 22:1311--1317.14. Piva A, Viola E. Crack propagation in an orthotropic medium. Engineering Fracture Mechanics 1988; 29:535--548.15. Viola E, Piva A, Radi E. Crack propagation in an orthotropic medium under general loading. Engineering Fracture Mechanics 1989; 34:1155--1174.16. Piva A, Radi E. Elastodynamic local fields for a crack running in an orthotropic medium. Journal of Applied Mechanics 1991; 58:982--987.17. Stroh AN. Dislocations and cracks in anisotropic elasticity. Philosophical Magazine 1958; 3:625--646.18. Stroh AN. Steady state problems in anisotropic elasticity. Journal of Mathematical Physics 1962; 41:77--103.19. Gao Y, Zhao YT, Zhao BS. Boundary value problems of holomorphic vector functions in 1D QCs. Physica B: Condensed Matter 2007; 394:56--61.20. Ting TCT. Anisotropic elasticity. Theory and applications. Oxford University Press: Oxford, 1996.21. Tanuma K. Stroh formalism and Rayleigh waves. Journal of Elasticity 2007; 89:5--154.22. Suo Z, Kuo CM, Barnett DM, Willis JR. Fracture mechanics for piezoelectric ceramics. Journal of the Mechanics and Physics of Solids 1992;

40:739--765.23. Wang BL, Mai YW. Crack tip field in piezoelectric/piezomagnetic media. European Journal of Mechanics A 2003; 22:591--602.24. Piva A, Tornabene F, Viola E. Crack propagation in a four-parameter piezoelectric medium. European Journal of Mechanics A 2006; 25:230--249.25. Piva A, Tornabene F, Viola E. Subsonic Griffith crack propagation in piezoelectric media. European Journal of Mechanics A 2007; 26:442--459.26. Boldrini C, Viola E. Crack energy density of a piezoelectric material under general electro-mechanical loading. Theoretical and Applied Fracture

Mechanics 2008; 49:321--333.27. Gao CF, Kessler H, Balke H. Crack problems in magnetoelectroelastic solids. Part I: exact solution of a crack. International Journal of Engineering

Science 2003; 41:969--981.28. Ni L, Nemat-Nasser S. A general duality principle in elasticity. Mechanics of Materials 1996; 24:87--123.29. Ting TCT, Hwu C. Sextic formalism in anisotropic elasticity for almost non-semisimple matrix N. International Journal of Solids and Structures

1998; 24:65--76.30. Gentilini C, Piva A, Viola E. On crack propagation in orthotropic media for degenerate states. European Journal of Mechanics A 2004; 23:247--258.31. Piva A, Viola E, Tornabene F. Crack propagation in an orthotropic medium with coupled elastodynamic properties. Mechanics Research

Communications 2005; 32:153--159.32. Radi E, Mariano PM. Stationary straigth cracks in quasicrystals. International Journal of Fracture 2010; in print.33. Wang YM, Ting TCT. The Stroh formalism for anisotropic materials that possess an almost extraordinary degenerate matrix N. International

Journal of Solids and Structures 1997; 34:401--413.34. Lubensky TC, Ramaswamy S, Toner J. Hydrodynamics of icosahedral quasicrystals. Physical Reviews B 1985; 32:7444--7452.35. Hu C, Wang R, Ding D-H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Reports on Progress in Physics

2000; 63:1--39.36. De P, Pelcovits RA. Linear elasticity theory of pentagonal quasicrystals. Physical Reviews B 1987; 35:8609--8620.37. Jeong H-C, Steinhardt PJ. Finite-temperature elasticity phase transition in decagonal quasicrystals. Physical Reviews B 1993; 48:9394--9403.38. de Boissieu M, Boudard M, Hennion B, Bellissent R, Kycia S, Goldman A, Janot C, Audier M. Diffuse scattering and phason elasticity in the

AlPdMn icosahedral phase. Physical Review Letters 1995; 75:89--92.39. Mariano PM, Modica G. Ground states in complex bodies. ESAIM: Control, Optimisation and Calculus of Variations 2009; 15:377--402.40. Rochal SB, Lorman VL. Minimal model of the phonon-phason dynamics in icosahedral quasicrystals and its application to the problem of

internal friction in the i -AlPbMn alloy. Physical Reviews B 2002; 66:144204-1--144204-9.41. Mariano PM. Multifield theories in mechanics of solids. Advances in Applied Mechanics 2002; 38:1--93.42. Hirsh MW, Smale S. Differential Equations, Dynamic Systems and Linear Algebra. Academic Press: New York, 1974.43. Muskhelishvili NI. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff: Groningen, 1953.44. Gakhov FD. Boundary Value Problems. Pergamon Press: New York, 1996.45. Freund LB. Dynamic Fracture Mechanics. Cambridge University Press: Cambridge, 1990.46. Broberg KB. Cracks and Fracture. Cambridge University Press: Cambridge, 1999.47. Yoffe EH. The moving Griffith crack. Philosophical Magazine 1951; 7:739--750.48. Landau LD, Lifshitz EM. Course of Theoretical Physics. vol. 7. Theory of Elasticity (3rd edn). Pergamon Press: Oxford,1986.49. Sih G. Liebowitz CH. Fracture: An Advanced Treatise. Academic Press: New York, 1968.50. Amazit Y, Fischer M, Perrin B, Zarembowitch A. Ultrasonic investigations of large single AlPdMn icosahedral quasicrystals. Proceedings of the

5th International Conference on Quasicrystals, Janot C, Massieri RS (eds). World Scientific: Singapore, 1995; 584--587.51. Fan TY, Wang XF, Lin W, Zhu AY. Elasto-hydrodynamics of quasicrystals. Philosophical Magazine 2009; 89:501--512.

22



52. Létoublon A, de Boissieu M, Boudard M, Mancini L, Castaldi J, Hennion B, Caudron R, Bellissent R. Phason elastic constants of the icosahedralAl-Pd-Mn phase derived from diffuse scattering experiments. Philosophical Mag Letters 2001; 81:273--283.

53. Tanaka K, Mitarai Y, Koiwa M. Elastic constants of Al-based icosahedral quasicrystals. Philosophical Magazine 1996; 73:1715--1723.54. Zhu AY, Fan TY, Guo LH. Elastic field for a straight dislocation in an icosahedral quasicrystal. Journal of Physics: Condensed Matter 2007;

23:236216-1--236216-8.55. Walz C. Zur Hydrodynamik in Quasikristallen. Thesis, University of Stuttgart, 2003.


23

Research Article

Received 26 May 2009 Published online 19 May 2010 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1325MOS subject classification: 74 A 30

Dynamic steady-state crack propagationin quasi-crystals

Enrico Radia and Paolo Maria Marianob∗†

Communicated by M. Grinfeld

The steady propagation of planar cracks in quasi-crystalline bodies with velocity lower than the one of bulk elasticmacroscopic waves is under scrutiny. Closed-form solutions to the balance laws are provided. Unusual Mach numberlimits are determined. Numerical experiments describing peculiar aspects of the crack propagation in quasi-crystalsare performed by varying parametrically the coupling coefficient between macroscopic deformation and substructuralevents. In this way, classes of quasi-crystals are then compared. Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: quasi-crystals; crack propagation; mechanics of complex bodies

1. Introduction

Quasi-crystals are a special class of quasi-periodic alloys characterized by atomic clusters displaying incompatible symmetrieswith periodic tiling of atoms in space: icosahedral symmetry in three dimensions, pentagonal symmetry in two dimensions (seedescriptions in [1--3]). Quasi-periodicity in space is assured by atomic rearrangements which create and annihilate atomic clusterswith symmetry different from the prevailing one—in this sense, the phase of the matter changes locally.

The representation of the generic material element—the essential starting point of any continuum model of condensed matter—can be done by picturing it as a cluster of atoms not fixed once and for all. Since in traditional continuum mechanics every materialelement is represented just by a point, its inner dimensions are not represented in what is chosen as ambient space. So the possibleatomic rearrangements within material elements have to be described by inner degrees of freedom, represented in an appropriatespace. They are called phason degrees of freedom, and are collected point by point in the values of a vector field defined over themacroscopic region occupied by the body under scrutiny. A differentiable, orientation preserving map describes the macroscopicdeformation. The mechanics of quasi-crystals at continuum level (long-wavelength approximation) admits, in this sense, a multiscaleand multifield description.

Peculiar actions appear: they are different from the standard macroscopic stress, determined only by crowding and shearing ofmaterial elements, and are called phason stresses. They are due to changes in the spatial inhomogeneity of the phason degrees offreedom. Phason stresses influence the macroscopic mechanical behavior in various aspects such as wave propagation, equilibriumand possible evolution of defects [4]. A paradigmatic example is one of the cracks propagating in quasi-crystals: phason stressesinfluence the force driving the crack tip. Molecular dynamics based simulations in [5--7] and analyses of field equations at acontinuum level in [8--10] furnish precise indications in this sense (see also [11--13]). In particular, a result in [9] is indicative: Considerthe quasi-static evolution of a stooping crack in a quasi-crystal specimen. The crack path determined by taking into account phasoneffects is perturbed with respect to the result obtained by neglecting phason effects.

Here, fracture mechanics in quasi-crystals is tackled once again. A planar crack propagating steadily in a three-dimensionalquasi-crystalline body, with a speed lower than the bulk wave velocity, is considered. Diffusion associated with phason degrees offreedom (the so-called phason diffusion), a common phenomenon in quasi-crystals, is neglected. The investigation is developedin a linear constitutive setting. Closed-form solution to the balance equations is obtained under general loading conditions. Sucha solution is applied to a concrete case, where it is shown that the amplitude of the phason stress increases with the incrementof the coupling between gross deformation and phason degrees of freedom. In the linear setting treated here, such a coupling isgoverned by a single coefficient. Its experimental evaluation is uncertain. Its potentially admissible range is bounded from above by

aDi.S.M.I., Università di Modena e Reggio Emilia, via Amendola 2, I-42122 Reggio Emilia, ItalybDICeA, University of Florence, via Santa Marta 3, I-50139 Firenze, Italy∗Correspondence to: Paolo Maria Mariano, DICeA, University of Florence, via Santa Marta 3, I-50139 Firenze, Italy.†E-mail: [email protected]

Contract/grant sponsor: GNFM-INDAMContract/grant sponsor: MIUR-PRIN08


1


the value preventing the propagation of shear waves. Below such a threshold, there is a range of incompatible values with Rayleighwave propagation. The recognition of the existence of such a range is a result obtained here.

It is also shown that macroscopic and phason stresses display square root singularities at the crack tip. Stress intensity factorsare determined. The energy release rate is evaluated for subsonic crack propagation.

Finally, Yoffe’s solution for fixed finite-length cracks traveling in a linear elastic simple body is also extended to the presentsetting. It is interpreted as a tool for analyzing the ‘rigid body’-type evolution of a linear continuous distribution of dislocations.

The analysis developed here is an evolution of a proposal in [14--16] dealing with anisotropic elasticity of simple bodies. It isbased on the Stroh formalism [17, 18] (see also [19--21] for a comprehensive review). A similar approach has been adopted for theanalysis of crack propagation in a piezoelectric media in [22--26] and in magneto-electro-elastic solids [27].

The constitutive tensors involved are isotropic and the solution procedure furnishes a degenerate eigenvalue problem: a funda-mental matrix admits two double eigenvalues and these identical roots have a single eigenvector associated with them. Strohformalism has been modified in agreement with the generalization proposed in [28--31] for orthotropic elastic media for the caseof degenerate eigenvalue problem.

As the velocity of the crack tip goes to zero, the eigenvalue problem associated with the description of the crack propagationadmits a single eigenvalue with algebraic multiplicity equal to 3. The static case, then, requires analysis a part (see [32] for therelevant results).‡

Some notations. For n×n matrices A and B, the notation AB indicates the standard row-by-column product determining a n×nmatrix. If b is a n-dimensional vector, the product Bb furnishes a n-dimensional vector. For notational convenience, the symbol〈〈ak〉〉 is adopted to denote a diagonal matrix, for example diag(a1, a2, a3, a4). The operators Re and Im extract from a given operatorwith complex entries (or simply from a complex number) the real and imaginary parts. An overbar denotes complex conjugate inthe sense of complex numbers. As usual, the notation (·) := (··) means that (·) is defined by (··). R3∗ is the dual version of R3 andis (of course) isomorphic to R3 itself. B indicates an open, connected set in the three-dimensional ambient physical space R3 withsurface-like boundary �B oriented by the outward unit normal n everywhere but a finite number of corners and edges (essentiallyit sufficies to assume that B has the Lipshitz boundary). For F a linear operator, FT indicates the transpose.

2. Mechanics of quasi-crystals: essential elements

The non-linear setting for the mechanics of quasi-crystals discussed in [4, 34] and its linearization (see [35--37]) are summarizedbelow with the aim of introducing the reader to the analyses in the ensuing sections. Quasi-crystals are essentially consideredas a prominent example of complex bodies, those bodies with gross mechanical behavior prominently influenced by chances inthe material texture, an influence exerted through peculiar interactions. For quasi-crystals, the microstructural changes are localatomic rearrangements: jumps of atoms between neighboring places and/or collective atomic modes generated by the flipping ofcrisscrossing topological alterations needed to maintain matching rules [35, 37, 38]. Creation and annihilation of clusters of atoms,with symmetry differing from the prevailing one, determine the characteristic quasi-periodicity of quasi-crystals. Internal degreesof freedom, the ones exploited by the atoms to rearrange themselves, are then attributed to material elements in the continuum

modeling. They are collected at every x in a vector � belonging to an isomorphic copy of R3, indicated by R3

.Although quasi-periodicity is a global property, the field description (a local one) in terms of the phason field x �−→�(x) (see

[4, 34, 35, 37]) accounts at macroscopic level for the mechanism generating quasi-periodicity rather than the periodicity itself thatre-appears in the structure of the constitutive equations. The word phason recalls that the crystalline phase changes locally.

2.1. Placements and morphological descriptors

The ambient space R3 hosts a quasi-crystalline body in its macroscopic reference configuration B. Every point x in B indicatesthe place of a generic material element. Other macroscopic configurations are achieved by means of deformations: orientationpreserving differentiable bijections

x �−→y :=y(x)∈R3, x ∈B.

Really, it is convenient to consider macroscopic configurations different from the reference one as placed in an isomorphic copy ofR3, giving in this way a true exclusive role to B, a role becoming clear when covariance and existence theorems of minimizers forthe elastic energy are investigated (see relevant comments in [4, 39]). Such a distinction is not essential here and is then skipped,it is only mentioned to clarify the connection with the complete version of the theory summarized in the ensuing notes, a theorypresented in [4].

The spatial derivative of a deformation is indicated by F :=Dy(x) and is, at every x, a linear operator from the tangent space to B

at x to the tangent space to the actual place Ba :=y(B) of the body at y(x). The assumption that x �−→y be orientation preservingimplies det F>0 at every x, as it is well known. The natural measure of deformation (referred to B) is the tensor E :=1 / 2(FTF− I),where FTF is the left Cauchy–Green tensor. Essentially, E is half of the relative difference between the pull-back on B of the naturalmetric in Ba and the original metric in the reference place itself—meaning relative that the difference between the two metrics is

‡Use is made of appropriate modifications of techniques discussed in [15, 16, 22, 30, 31, 33].

2



multiplied by the inverse of the metric in B. Once the time t is introduced, a macroscopic motion in a time interval [0, d] is given by

(x, t) �−→y :=y(x, t)∈R3, x ∈B, t ∈ [0, d].

Sufficient smoothness in time t is presumed. The macroscopic velocity is defined by y := (d / dt)y(x, t) in the referential description(that is as a field over the tube B×[0, d]). The field

(x, t) �−→� :=�(x, t)∈ R3

, x ∈B, t ∈ [0, d]

indicates the phason degrees of freedom in time. Besides differentiability in space, sufficient smoothness in time is presumed. The

time rate of change of � in referential description is � := (d / dt)�(x, t)∈ R3

, the spatial derivative of (x, t) �−→� is indicated by N :=D�(x).

It is a linear operator from the tangent space to B at x to the space R3

, where the phason degrees of freedom are represented.

2.2. Linearized macroscopic kinematics

Consider the displacement field

(x, t) �−→u :=u(x, t)=y(x, t)−x, (x, t)∈B×[0, d].

The spatial derivative of (x, t) �−→u is indicated here by W :=Du(x, t). It follows that y(x, t)= u(x, t) and F = I+W , with I the second-rankidentity tensor. As usual, the condition

|W|�1

at every point x and instant t, defines the infinitesimal deformation regime. The measure of deformation E reduces to its linearizedpart � :=symW . Moreover, B and Ba can be also ‘confused’ within the limits of the infinitesimal deformation setting. No distinctionis also made between x �−→� and y �−→�a :=�◦y−1, with �a is the Eulerian (actual) representation of �.

2.3. Consequences of a d’Alembert–Lagrange-like principle

In addition to standard bulk forces and stresses due to (i) interaction of the body with the external environment and (ii) crowdingand shearing of material elements, peculiar actions connected with microscopic changes in the material texture arise. They aredefined by the power that they develop in the rate of the phason degrees of freedom.

Appropriate integral balances can be obtained by requiring the invariance of the overall power of all external actions overa generic part of the body under changes in observers, which alter isometrically the ambient space. Consequent alterations in

the space R3

collecting the phason degrees of freedom are induced by a differentiable homomorphism (see [9] for an extensivediscussion of non-standard changes in observers). An independent integral balance of phason actions does not appear naturally.The invariance procedure furnishes integral balances associated with the Killing fields of the metric in space. Different local balancescan be derived from them: they include the independent local (so not global) balance of phason actions. The result presented in [4]corrects the integral balances assumed in [35]. A discussion on these aspects can be found in the last section of [4]. In this way, therepresentation of actions and constitutive structures following from the second law of thermodynamics can be tackled separately.

However, for the purposes of this paper, the matter can be treated from a concise point of view by merging the abstractrepresentation of actions and the prescription of their constitutive structure into an unique principle. As discussed in [4], appropriateto this case is a d’Alembert–Lagrange-like principle expressed by

�

(∫B×[0,t]

Ldx∧ dt

)+∫B×[0,t]

zv ·��dx∧ dt =0, (1)

where

L :=L(x, y, y, F, N)= 1

2�|y|2 −e(x, F, N)+w(y)

with e the elastic potential and w the potential of external actions. No peculiar (phason) kinetic energy is associated here withthe phason degrees of freedom. It seems that scattering data do not support the introduction of a phason kinetic energy. Someauthors, however, introduce it on the basis of different considerations. The matter is a bit controversial (see [10, 34--36, 40]).

Also, the Lagrangian density does not depend on � alone, rather only on its gradient, as suggested also by data. The circumstanceprecludes the presence of a conservative self-action, a type of action appearing in general in complex bodies (see details in [41]).A dissipative self-action, however, appears and is associated with the diffusive nature of phason modes (see e.g. [35]). It is indicatedby

zv =zv(x, F,�, N, �)∈ R3∗

and is such that

zv · ��0 (2)


3


for any choice of �, with equality sign holding only when �=0. The inequality defines the dissipative nature of zv . A solution to(2) is

zv =c�,

with c the value at x and t of a positive-definite function

c :=c(x, F,�, N, �)

such that

c(x, F,�, N, 0)=0

(see [40] for estimates of c).Under C1-regularity of L and C2-regularity of transplacement and phason fields, Euler–Lagrange equations for (1) are given by

Div P+b=�y, DivS=c�,

where P :=−�FL is the first Piola–Kirchhoff stress, b :=�yL the vector of body forces, � the mass density in the reference place,S :=−�NL the so-called phason stress, due to the interactions between neighboring material elements with different amount ofphason degrees of freedom. The elastic potential e is further constrained by the requirement to be invariant under the evaluationof different observers, at least the ones mentioned previously, which involve isometric changes in the ambient space R3 and the

action of SO(3) on R3

. Another possible requirement is the invariance of the Lagrangian density or the whole integrand in (1) underthe class of changes in observers already mentioned. In this case, it follows that

skw(�F eFT +�⊗zv +�NeTN)=0

(details are in [4]). So there is a lack of symmetry in Cauchy’s stress

� := (det F)−1�F eFT

as a consequence of the presence of phason actions.§

Piola transform determines the actual counterparts of the other actions in Eulerian representation

ba := (det F)−1b, zva := (det F)−1zv, Sa := (det F)−1SFT,

where the index a recalls that they are defined in the actual place Ba. The balance equations in the Eulerian representation thenread

div�+ba =�a�, divSa =ca�a, (3)

where � is the actual velocity, �a := (det F)−1�, ca := (det F)−1c and �a is the rate of the map �a :=�◦y−1, that is the actual repre-sentation of the morphological descriptor field.

In the analysis presented in the ensuing sections, phason diffusion is neglected (the second balance (3) then reads divSa =0).One reason is to make easier the analysis of the crack dynamics under scrutiny. Another reason is that such a diffusion is associatedwith a self-action which does not contribute to the evolution equation of the crack tip directly (see the results in [9]); it contributesindirectly because it influences the fields around the crack. To be more precise, the J-integral¶ at a point x, belonging to a crack

§Symmetry of � can be recovered for special choices of the constitutive structures, as the ones adopted further on in this paper. Symmetry of � can be

recovered also if it is presumed that changes in observers in the ambient space have no connection with the ones in the phason space R3

. The question isrelevant for general complex bodies and is a strict consequence of the notion of observer that can be considered appropriate for the mechanics of complexbodies. Here, the point of view is the one discussed by one of us in [4, 9]. In fact, the description of the mechanics of quasi-crystals in terms of ambient andphason spaces is only an ideal representation of phenomena occurring in the real physical space, as all mathematical models are. Deformations and atomicrearrangements in quasi-crystals occur in the physical space. If a change in observer is considered as a change of atlas (coordinate system) in the physicalspace, in the mathematical representation at hands it is necessary to consider then all geometrical environments used to represent morphology and motionof a quasi-crystalline body. Moreover, since, there is an interplay between gross deformation and phason degrees of freedom, the physical change in observer

must be described not only in the mathematical ambient space but also in the space of the phason degrees of freedom, namely R3

. These remarks apply to the

general model-building framework of the mechanics of complex bodies. There, R3

is replaced, in general, by a finite-dimensional differentiable manifold M.

The link between the changes in the atlas in the ambient space and in M (which is R3

in the case treated here), a link described by a differentiablehomomorphism, represents the fact that microstructural changes (atomic rearrangements in the case of quasi-crystals) occur in the physical space. Analyticaldetails about such a description of changes in observers are skipped here. They can be found in [9] for the general format of the mechanics of complex bodies,and in [4] for the case of quasi-crystals. The question is mentioned here just to underline that only precise physical circumstances—they must be specified timeto time—may suggest to not consider between changes in atlas just mentioned. When the link cannot be considered, independently of constitutive choicesCauchy stress � results symmetric. The circumstance, however, is special. Balance equations for quasi-crystals proposed in [35] seem to agree with this specialcase. The point is further discussed in [4].

¶The present expression of J is a special case of the J-integral determined in [9, 41] within the general model-building framework of the mechanics of complexbodies, involving manifold-valued geometrical descriptors of the material microstructures.

4



tip, is in fact given here by

J(x) :=n(x) · lim�→0

∫�D�(x)

((e+ 1

2�|y|2

)I−FTP−NTS

)n d�,

where n is the direction of crack propagation at x, and D�(x) is a disk, centered at x, and belonging to the orthogonal plane to thetangent to the tip at x. The J-integral determines the driving force at the crack tip in addition to the limit value at the tip of thesurface energy and the local contribution of the tip own energy time the local tip scalar curvature (see e.g. [9]). No direct role isplayed in the expression of J by the self-action zv =c�. The term c� influences the J-integral only indirectly, through the balanceequations. In particular, c� determines dispersion of elastic waves in dynamics and decaying phason modes.

The analysis developed here is restricted to a purely elastic range, at a fixed temperature that allows brittle phase. Also, infinitesimaldeformation setting is considered, so that the distinction between referential and actual representations of the actions becomevolatile.

2.4. Linearized constitutive setting

In the linearized kinematical setting, the elastic potential can be a quadratic form‖ in W and N. It reads

e(W, N)= 12 (CW) ·W + 1

2 (KN) ·N+(K′N) ·W.

C, K and K′ are fourth-rank constitutive tensors. C is the standard elastic tensor and K′ describes the coupling between grossdeformation and phason degrees of freedom, the latter pertaining to K. Explicit expressions of such tensors are here taken from[35]∗∗

Cijhk = ��ij�kl +(�ik�jl +�il�jk),

Kijkl = k1�ik�jl +k2(�ij�kl −�il�jk),

K′ijkl = k3(�i1 −�i2)(�ij�kl −�ik�jl +�il�jk),

where in the last equation no summation over repeated indices is assumed, � and are standard Lamé constants, k1 and k2 describethe pure phason contribution to the energy and k3 is the so-called coupling coefficient.

The stress measures � and S (remind the elimination of the index ’a’) are then the derivatives of e with respect to W and Nrespectively, namely

�=CW +K′N, S=K′TW +KN,

where the exponent T over fourth-rank tensors indicates major transposition. More explicitly, such constitutive equations read

� = 2SymW +(�div u)I+k3(2 Sym N− I div�)R, (4)

Sa = k3R(2 Sym W − I div u)+k1N−k2(N∗− I div�), (5)

where R :=e1 ⊗e1 −e2 ⊗e2, with e1 and e2 two vectors of the vector bases e1,e2,e3, associated with �i1 and �i2.

2.5. Boundary conditions

Dirichlet’s-type boundary conditions are natural for the displacement field on some part �Bu of the body boundary. They can bealso prescribed in terms of � because it is possible to imagine, for example, that no phason degrees of freedom pertain to thematerial elements placed along the boundary of B.

In principle, standard loading devices allow one to assign the macroscopic tractions Pn. No loading device seems available forthe prescription of non-zero phason tractions Sn.

In terms of Sn, the natural boundary condition is then only

Sn=0.

This problem is common to all complex bodies admitting multifield description of their morphology and interactions.To avoid it, a surface energy can be considered over the boundary (like the body would be coated in some way) and appropriate

boundary phason tractions can be constructed through it. The construction is, however, not completely natural and contains thedegree of freedom associated with the constitutive choice of the explicit expression of such a surface energy. Details about therelated formalism can be found in [4].

‖In non-linear deformation setting, e cannot be convex in F =W + I (I the unit tensor) for reason of objectivity, whereas it can be convex in N.∗∗The theory presented here is different from the material reviewed in [35] on the explicit characterization of the foundational aspects leading to and justifying

balance equations. The results in [4] rectify in a sense the form of the integral balances presented in [35].


5


x2

vt

σ0 τ0

x

y

Figure 1. Semi-infinite rectilinear crack propagating at constant speed v, loaded by normal and shear standard stresses, �0 and0 applied on a portion of length � of the crack surface.

3. Steady-state crack propagation in the infinitesimal deformation setting

3.1. Preliminary issues: Stroh formalism

The whole framework summarized above is restricted from now on to an intrinsically two-dimensional problem. A planar crackpropagating with constant speed v in an infinite quasi-crystalline body is considered (see Figure 1). The crack remains planar duringits evolution. Crack margins are in contact. Deformations are infinitesimal.

Reference is made to two Cartesian coordinate systems: (Oxyz) and (Oxyz). The former is fixed, the latter moves with the cracktip. Crack kinking and curving are excluded during the evolution by homogeneity and type of boundary conditions imposed. It isthen possible to select the x1-axis to be coincident with the direction of propagation. The x-axis is also selected along the x1-axis.The x3-axis is taken along the straight crack front. Since the crack is straight, it is possible to restrict the analysis to the plane x3 =0.Symmetry suggests also to consider essentially planar the distribution of phason degrees of freedom.

The problem under scrutiny is then two-dimensional.Steady-state crack propagation is under analysis. It means that for any differentiable field (x1, x2, t) �−→a(x1, x2, t), the condition

a(x1, x2, t) :=a(x−vt, y)

holds, so that

a,1 =a,x, a,2 =a,y, a :=−va,1,

where the comma denotes partial derivative, the indices 1 and 2 indicate, respectively x1 and x2, the superposed dot denotes timederivative, as usual.

It is convenient to define

p := (�11,�21,S11,S21), q := (�12,�22,S12,S22),

r := (u1,1, u2,1,�1,1,�2,1), s := (u1,2, u2,2,�1,2,�2,2).

In this way, the linear constitutive Equations (4) and (5) can be then be written as

p=Ar+Bs, q=BTr+Cs,

where

A=

⎡⎢⎢⎢⎢⎢⎢⎣

2+� 0 k3 0

0 0 k3

k3 0 k1 0

0 k3 0 k1

⎤⎥⎥⎥⎥⎥⎥⎦

, B=

⎡⎢⎢⎢⎢⎢⎢⎣

0 � 0 k3

0 −k3 0

0 −k3 0 k2

k3 0 −k2 0

⎤⎥⎥⎥⎥⎥⎥⎦

, C=

⎡⎢⎢⎢⎢⎢⎢⎣

0 −k3 0

0 2+� 0 −k3

−k3 0 k1 0

0 −k3 0 k1

⎤⎥⎥⎥⎥⎥⎥⎦

.

Also, the balance equations read now

p,1 +q,2 =�v2Dr,1,

where D :=diag(1, 1, 0, 0). Use of constitutive equations then implies

(A−�v2D)r,1 +2(SymB)r,2 +Cs,2 =0,

after the exploitation of the relation s,1 = r,2, a relation justified by Schwartz’s theorem.

6



Since the matrix Q :=A−�v2D is not singular, the balance equations can be further rewritten as(r

s

),1

+[

2Q−1(SymB) Q−1C

−I 0

](r

s

),2

=(

0

0

), (6)

where I is the 4×4 identity matrix.

3.2. Degenerate eigenvalue problem

The spectrum of the matrix of coefficients in the equations of motion is given by the eigenvalue problem††

[2Q−1(SymB)−�kI Q−1C

−I −�kI

]⎛⎝h(k)

g(k)

⎞⎠=

(0

0

)(7)

with h(k) and g(k) are four-dimensional vectors. It reduces to

[C−�k2 SymB+�2kQ]g(k) =0, h(k) =−�kg(k).

The eigenvalues �k are the roots of the characteristic equation

det(C−�k2 SymB+�2kQ)=0.

By defining

� := k3

k1,

the characteristic equations can be rewritten as

(1+�2k )2((1+�2

k )(2+�−�k3)−�2k�v2)((1+�2

k )(−�k3)−�2k�v2)=0.

Purely imaginary roots with positive imaginary parts and related conjugates are available. By introducing Mach numbers

m21 = �v2

2+�−�k3, m2

2 = �v2

−�k3,

and recalling that i denotes (as usual) the imaginary unit, the roots with non-negative imaginary parts are given by

�1 = i√1−m2

1

, �2 = i√1−m2

2

, �3 = i.

The algebraic multiplicity of the root �3 is 2, whereas the eigenvalues �1 and �2 are distinct for v>0. Moreover, for subsonic crackpropagation, the crack tip speed v is smaller than each of the speeds of the elastic waves in the bulk in the x1-direction; consequentlyboth Mach numbers m1 and m2 are smaller than 1. For a stationary crack, namely for v =0, the four roots �k , k =1, 2, 3, 4, arecoincident.‡‡

The eigenvectors h(k) and g(k), corresponding to �k , k =1, 2, 3, are given by non-trivial solutions of (7) for each �k , namely

g(1) = (im21, −m2

1

√1−m2

1, −i�(4−3m21), �(4−m2

1)√

1−m21),

g(2) = (im22

√1−m2

2, −m22, −i�(4−m2

2)√

1−m22, �(4−3m2

2)),

g(3) = (0, 0, −i, 1),

h(1) =⎛⎝ m2

1√1−m2

1

, im21, −�

4−3m21√

1−m21

, −i�(4−m21)

⎞⎠ ,

h(2) =⎛⎝m2

2, im2

2√1−m2

2

, −�(4−m22), −i�

4−3m22√

1−m22

⎞⎠ ,

h(3) = (0, 0, −1, −i).

††Solutions of negative exponential type are in general considered in the Stroh formalism. However, when the problem is degenerate (as the one treated here),other paths have to be followed (see [21]).

‡‡The stationary case is treated in [32].


7


The eigenvectors h(k) have been obtained by making use of the relation h(k) =−�kg(k). The eigenvalue problem (7) is degeneratebecause the geometric multiplicity of the repeated eigenvalues �3 and �3 is 1. There exists only one eigenvector for each of theseeigenvalues. For them, the geometric multiplicity is less than the corresponding algebraic multiplicity, which is 2. A generalizedeight-dimensional eigenvector (h(4), g(4)) can be then defined for the repeated eigenvalue �3. It is linearly independent of the otherthree eigenvectors and is a solution to [

Q−12(SymB)−�3I Q−1C

−I −�3I

]⎛⎝h(4)

g(4)

⎞⎠=

⎛⎝h(3)

g(3)

⎞⎠ (8)

(see [21, 42]). As a consequence, it follows that

g(4) =(

2(�+)+�v2

4�(�+), i

2(�+)−�v2

4�(�+),

(�v2)2

8�k3(�+), 0

),

h(4) =(

−i2(�+)+�v2

4�(�+),

2(�+)−�v2

4�(�+), i

(1− (�v2)2

8�k3(�+)

), −1

).

For k =1, 2, 3, 4, the eigenvectors h(k) and g(k) can be considered respectively the columns of the matrices

H :=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

m21√

1−m21

m22 0 −i

2(�+)+�v2

4�(�+)

im21 i

m22√

1−m22

02(�+)−�v2

4�(�+)

−�4−3m2

1√1−m2

1

−�(4−m22) −1 i

(1− (�v2)2

8�k3(�+)

)

−i�(4−m21) −i�

4−3m22√

1−m22

−i −1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

and

G :=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

im21 im2

2

√1−m2

2 02(�+)+�v2

4�(�+)

−m21

√1−m2

1 −m22 0 i

2(�+)−�v2

4�(�+)

−i�(4−3m21) −i�(4−m2

2)√

1−m22 −i

(�v2)2

8�k3(�+)

�(4−m21)√

1−m21 �(4−3m2

2) 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Equations (7), for k =1, 2, 3, (8) and their complex-conjugate equivalent relations can be then rewritten as[Q−12(SymB) Q−1C

−I 0

][H H

G G

]=[

H H

G G

][S+N 0

0 S+N

], (9)

where S and N are the semi-simple nilpotent matrices defined by

S :=〈〈�k〉〉, N :=

⎡⎢⎢⎢⎢⎢⎣

0 0 0 0

0 0 0 0

0 0 0 1

0 0 0 0

⎤⎥⎥⎥⎥⎥⎦ ,

in tune with the Jordan decomposition theorem. Note that �3 =�4 in S. Equation (9) implies that H=−G(S+N)Consider a four-dimensional differentiable vector field (x1, x2) �−→b(x1, x2), defined by(

r

s

)=[

H H

G G

](b

b

)=2Re

(Hb

Gb

).8



By introducing such a relation into (6) and using (9), it follows that

(b

b

),1

+[

S+N 0

0 S+N

](b

b

),2

=(

0

0

)

which is equivalent to

b,1 +(S+N)b,2 =0 (10)

together with its complex-conjugate. Componentwise, (10) reads

bk,1 +�kbk,2 =0 for k =1, 2, 4 (11)

and

b3,1 +�3b3,2 =−b4,2. (12)

By introducing the complex variable

zk =x1 + ix2

√1−m2

k , k =1, 2, 3, 4 (13)

with the convention m3 =m4 =0, which implies z3 =z4 =x1 + ix2 =: z, Equations (11) and (12) become respectively

�bk

�zk=0 for k =1, 2, 4

and

�b3

�z3=−1

2b4,2.

It follows that the complex function (x1, x2) �−→bk(x1, x2), k =1, 2, 4, must be analytic in the complex variable zk—actually bk =ak(zk)—then

�b3

�z3=−1

2b′

4(z)�z

�x2=− i

2b′

4(z),

where the prime denotes derivative with respect to z. Integration of the last relation yields

b3(x1, x2)=a3(z)− i

2zb′

4(z), (14)

with a3(z), an arbitrary analytic function of z.In terms of the vector b, the expressions of the displacement, phason degrees of freedom and stress fields then read

r = 2Re(Hb), s=2Re(Gb),

p = 2Re(Jb), q=2Re(Kb),

where J :=AH+BG and K :=BTH+CG.It is possible to write

b(z)=a(z)− i

2zNa′(z), (15)

(where ak(z)=bk(z) for k =1, 2, 4. a3(z) is the value of the function introduced in (14)), then

r(z) = 2Re(H(a− izNa′)(z)), s(z)=2Re(G(a− izNa′)(z)),

p(z) = 2Re(J(a− izNa′)(z)

), q(z)=2Re(K(a− izNa′)(z)).

The formal expressions derived in this section become explicit once a is determined for a specific boundary value problem.Two explicit cases are treated in the ensuing section to clarify the point. Moreover, an application to the surface wave problem isconsidered in Appendix B.


9


4. Semi-infinite straight crack loaded along its margins

4.1. Crack tip fields

Assume that the crack be loaded on both the margins with a uniform distribution of normal and shear standard stresses indicatedrespectively by �0 and 0. Phason stresses are prescribed to vanish along the crack margins. �0 and 0 are distributed along aportion of length �, measured by starting from the tip. Figure 1 represents the situation just described.

It is presumed that both macroscopic and phason stresses vanish at infinity. The assumption about the standard Cauchy stress israther natural. As regards the phason stress, it is possible to imagine that at infinity there is a boundary layer where phason degreesof freedom are absent. The assumption has a constitutive nature.

Since p and q vanish at infinity by assumption, vector functions b and a vanish at infinity too.Standard and phason tractions are continuous across x1-axis:

q+(x1)−q−(x1)=0. (16)

In fact, no external loads are applied far from [−�, 0] along x1-axis, the margins of the crack are in contact, and they are loaded withthe same load along [−�, 0].

Displacement and phason fields are continuous across x1-axis ahead of the crack tip, namely

r+(x1)−r−(x1)=0 for x1>0.

The assumed asymptotic behavior of the stress fields at infinity and the linear constitutive setting impose that the functionz �−→a(z) vanish at infinity.

The continuity of standard and phason stresses across the margins of the crack implies that the function

z �−→ j(z) :=(

K(

a− i

2zNa′

))(z)−

(K(

a+ i

2zNa′

))(z)

be analytic in the whole complex plane. By the Liouville theorem, it is then constant, so it vanishes as a consequence of theasymptotic behavior z �−→a(z). It follows that(

K(

a− i

2zNa′

))(z)=

(K(

a+ i

2zNa′

))(z). (17)

Moreover, the continuity condition for displacement and phason fields implies that

H(

a+− i

2x1Na′+

)(x1)−H

(a++ i

2x1Na′+

)(x1)=H

(a−− i

2x1Na′−

)(x1)−H

(a−+ i

2x1Na′−

)(x1) (18)

for x1>0. In the previous formula, a+ and a− are the standard limit values at the x1-axis defined by a±(x1) := limx2→0± a±(x1 + i2x2).

An analogous definition holds for a′±.The relation (17) implies (

a±+ i

2x1Na′±

)(x1)=

(K

−1K(

a±− i

2x1Na′±

))(x1).

The insertion of this last relation in (18) and the assumption that the matrix

Re(iHK−1)

be not singular imply (a+− i

2x1Na′+

)(x1)−

(a−− i

2x1Na′−

)(x1)=0 for x1>0. (19)

Finally, indicating by q0 the vector collecting the applied loads along [−�, 0], the loading conditions imply that

K(

a+− i

2x1Na′+

)(x1)+K

(a−+ i

2x1Na′−

)(x1)=

{q0 for −��x1�0

0 for x1<−�,

along the margins of the crack, so that

(a+− i

2x1Na′+

)(x1)+

(a−− i

2x1Na′−

)(x1)=

{K−1q0 for −��x1�0

0 for x1<−�.(20)1

0



Equations (19) and (20) define an inhomogeneous Riemann–Hilbert problem for the analytic vector function z �−→ (a−(i / 2)zNa′)(z).Solution is given by (

a− i

2zNa′

)(z)= 1

L(z)K−1q0 (21)

(see [43, 44]), where L(z) is a 4×4 matrix and is defined by

L(z) :=⟨⟨

1

2ilog

√zk + i

√�

√zk − i

√�

−√

�

zk

⟩⟩.

Since N is nilpotent (that is N2 =0), it follows that

Na′(z)= 1

2 (z+�)

(�

z

)3/2NK−1q0. (22)

The introduction of (21) and (22) in (15) then yields

b(z)= 1

(L(z)− x2

2(z+�)

(�

z

)3/2N

)K−1q0.

The displacement field and the field of phason degrees of freedom, collected in a vector d := (u1, u2,�1,�2), can be obtained bydirect integration of the function z �−→ r(z) with respect to x1. It follows that

d(z)= 2

Re

(H

(�(zk)−x2

(1

2ilog

√z+ i

√�√

z− i√

�−√

�

z

)N

)K−1q0

),

where �(z) is a 4×4 matrix defined by

�(z) :=⟨⟨

zk +�

2ilog

√zk + i

√�

√zk − i

√�

−√

�zk

⟩⟩.

4.2. Energy release rate

The evaluation of the energy release rate during a crack propagation can be obtained by specializing abstract results obtained inthe general model-building framework of the mechanics of complex bodies (see [9, 41]).

Traditional expressions of the energy release rate in simple bodies (see [45, 46]) can be also used: quasi-crystalline structurescan be considered, in fact, as incommensurate projections over the three-dimensional ambient space of six-dimensional standardperiodic crystalline structures. To fix ideas on a special case, consider a periodic lattice made of square cells living in a plane, take aline with irrational inclination with respect to the lattice, and consider a strip of infinite extension and finite thickness—a thicknessincluding some atomic layers. The strip is symmetric across the line itself. A one-dimensional quasi-crystal in the plane can beobtained by projecting over the inclined line the atoms contained in the selected strip. Higher-dimensional quasi-crystals can beobtained in an analogous way.

The energy release rate can be obtained by adapting common formulas in fracture mechanics of simple bodies to the higher-dimensional setting, then ‘projecting’ in an appropriate way them into the physical ambient space.

In the present setting, the energy release rate G then reads

G=−

2lim�0

√�q(�) · lim

�0

√�(r(�exp(i ))−r(�exp(−i ))),

where � is a radial coordinate in a polar coordinate system centered at the tip. With the notations introduced previously, it is alsopossible to write

G=4�

q0 ·Re(iHK−1)q0.

The 4×4 matrix Y := iHK−1 is Hermitian (see the proof in Appendix A)—it means Y= YT—so that its real part is symmetric. G isthen positive when Re(Y) is positive definite.

4.3. Remotely loaded crack with free crack surface

Consider q0 =0. Formally (see once more [43, 44]), the solution reads

b(z)= 12 L(z)K−1a0,

where a0 ∈R4 is a constant, and

L(z) :=〈〈(2 zk)−1/2〉〉


11


is a 4×4 matrix. Consequently, the traction vector q ahead of the crack tip at x2 =0 and for x1>0 is given by

q= (2 x1)−1/2a0.

Assume that the stress ahead of the crack tip can be written in terms of stress intensity factors, all collected in the list k :=(KII, KI, TII, TI), where KI and KII are the intensity factors of the macroscopic stress, whereas TI and TII are the ones of the phasonstress. Such an assumption is motivated by the remark that the mechanics of quasi-crystals can be considered to some extent as themechanics of simple bodies in a space with dimension higher that the one of the physical space, modulo appropriate projections.

The primary consequence is that

a0 =k

and

q= (2 x1)−1/2k.

Then, it follows that

Na′(z)=− 1

4z√

2 zNK−1k,

and also

b(z) = 1

2

1√2

(⟨⟨1√zk

⟩⟩+ x2

2z√

zN)

K−1k,

d(z) =√

2

Re

((⟨⟨√zk⟩⟩− x2

2√

zN)

K−1k)

.

The energy release rate is then given by

G= 12 k ·Re(iHK−1)k.

5. Yoffe’s traveling crack: a scheme for the evolution of linear continuous distributionsof dislocations

In the 1951 paper [47], Yoffe described the steady state propagation in an infinite, planar, elastic body of a straight crack withconstant length 2�. Yoffe’s analysis is developed in an infinitesimal deformation regime. Loads are placed at infinity. The geometryof the case under analysis is explained in Figure 2: a straight crack is depicted, the frame of reference traveling with the crack itselfis centered at the middle point of the crack span with the x1-axis aligned with the crack.

The assumption that a traveling crack maintains constant its length is somewhat artificial. Yoffe’s solution, however, has physicalinterest. In fact, a linear crack (as it is the one under analysis here) can be considered as a continuous distribution of dislocationsalong the interval of the x1-axis where the crack is placed (see [48]). For this reason, analyzing a traveling linear crack with constantlength is tantamount to consider the ‘rigid body’-type evolution of a linear continuous distribution of dislocations—the dislocationsare orthogonal to the plane considered.

Yoffe’s solution is extended here to the mechanics of quasi-crystals in linear deformation setting. The aim is to furnish a tool foranalyzing continuous distributions of straight dislocations, at least in some aspects of their mechanical behavior.

With reference to Figure 2, loading conditions are prescribed at infinity, namely

p∞ := (�∞11,�∞

21, 0, 0), q∞ := (�∞12,�∞

22, 0, 0).

σ11∞

σ22∞

σ12∞

x1

x2

vt

xy

Figure 2. Yoffe’s crack propagating at constant speed v, loaded by remotely applied standard stress �∞ .

12




The presence of a remote loading implies

b(z)=b∞+a(z)− 1

2izNa′(z).

It then follows

r(z) = r∞+2Re

(H(

a− i

2zNa′

)(z)

), s(z)=s∞+2Re

(G(

a− i

2zNa′

)(z)

),

p(z) = p∞+2Re

(J(

a− i

2zNa′

)(z)

), q(z)=q∞+2Re

(K(

a− i

2zNa′

)(z)

),

where

r∞ :=2Re(Hb∞), s∞ :=2Re(Gb∞), p∞ :=2Re(Jb∞), q∞ :=2Re(Kb∞).

The conditions fulfilled along the crack margins by the displacement field, the phason degrees of freedom, and the standard andphason stresses are then used as in the previous sections. The first result is

r+(x1)= r−(x1) for |x1|>�.

Then, by taking into account that the matrix Re(iHK−1) is not singular, it is possible to write(a+− i

2x1Na′+

)(x1)−

(a−− i

2x1Na′−

)(x1)=0 for |x1|>�.


13



The absence of tractions along the crack surfaces implies(a+− i

2x1Na′+

)(x1)+

(a−− i

2x1Na′−

)(x1)=−K−1q∞ for |x1|<�.

The two conditions define an inhomogeneous Riemann–Hilbert problem for the analytical vector function z �−→ (a−(i / 2)zNa′)(z).The following solution vanishing at infinity [43, 49] is ammissible:(

a− i

2zNa′

)(z)= 1

2(L(z)−I)K−1q∞, (23)

where I is once more the 4×4 unit matrix and

z �−→L(z) :=⟨⟨

zk√z2

k −�2

⟩⟩

is a 4×4 matrix-valued function that generates a square root singularity at the crack tip for both the standard and phason stressfields.

As in the previous example, since N is nilpotent, it follows that

Na′(z)=− �2

2(z2 −�2)3/2NK−1q∞. (24)

The formulas (23) and (24) then imply

a(z)− i

2zNa′(z)= 1

2

(L(z)−I+ �2x2

(z2 −�2)3/2N

)K−1q∞.

14




The explicit expression of the vector d :=(u1, u2,�1,�2), collecting the in-plane components of standard displacements and phasonfield, follows by direct integration of the map z �−→ r(z) with respect to x1. The result is

d=〈〈zk〉〉Re(H(2a∞−K−1q∞)+Re

(H

(�(

zk)− zx2√

z2 −�2N

)K−1q∞

),

where

�(zk) :=⟨⟨√

z2k −�2

⟩⟩

is a 4×4 matrix at every zk . On the basis of the previous results, the energy release rate G is given by

G= �

2q∞·Re(iHK−1)q∞.

5.1. Displacement and phason fields at the tip

A polar coordinate system (�,�), with the origin at the traveling crack tip, is useful for the ensuing developments. In this frame ofreference, the complex variable (13) is expressed by

zk =�+�

(cos�+ i

√1−m2

k sin�

)for k =1, 2, 3, 4.


15



When � is small with respect to � — the half-length of the crack — q and d are

q =√

�

2�Re

(K(⟨⟨

1

�k (�)

⟩⟩+ 1

2sin�

(cos

3

2�− i sin

3

2�

)N)

K−1q∞)

,

d =√

2��Re

(H(

〈〈�k(�)〉〉− 1

2sin�

(cos

�

2− i sin

�

2

)N)

K−1q∞)

,

where

�k(�) :=√

cos�+ i√

1−m2k sin�.

6. Special cases

Once values of the constitutive coefficients are selected, insertion of them in the closed-form solutions to the problems discussedhere permits the specialization of the results. The experimental determination of values for the constitutive coefficients is discussedvariously in the pertinent literature. Different values are proposed (see e.g. [38, 40, 50--55]). Values used in [50, 52, 55] are adoptedhere. They read

�=85 GPa, =65 GPa, k1 =0.044 GPa, k2 =−0.0396 GPa.

16



0-0.0010

-0.0005

0

0.0005

0.0010

S r1/

2 /KI

1 /r1/

2 K

I

-0.2

0.2

0.4

0.6

0.8

0 45 90

45 90

45 90

135

0

r r1/

2 /KI

r1/2 /K

I

0-50

-40

-30

0

-20

-10

180

135 180

135 180

(a)

(c) (d)

(e)

0-0.0004

-0.0002

0

0.0002

0.0004

S r r1/

2 /KI

2 /r1/

2 K

I

0

-40

-30

0

-10

-20

-50

0-0.5

0.5

1.0

1.5

2.0 = 5

u 1/r

1/2 K

I u 2

/r1/

2 KI

m2 = 0.6

m2 = 0.2

m2 = 0.8

0

45 90

45 90

45 90

135 180

135 180

135 180

(b)

(f)

Figure 7. Angular variations of standard, phason stresses and displacement and phason fields around the crack tip under remote Mode I loading conditions,for �=5, and for different values of m2.

Uncertainty is associated with the experimental determination of the coupling coefficient k3. In particular, values �=5 and�=35 are considered here for comparison purposes. The latter value is close to the limit value at which shear wavesdo not propagate (see the top straight line in Figure 9 indicating this value: about 38 in the special conditions adoptedhere).

The dynamics of a semi-infinite straight crack is considered in the figures. Yoffe’s problem is viewed in a sense as a special case.

• Figure 3 collects the standard and phason stresses around the crack tip in Mode I loading conditions (0 =0). For m2 =0.6 and�=5 the magnitude of the phason stress is ten times smaller than the one of the standard stresses.

• In analogous conditions and with �=35 (high coupling, close to the limit of no-propagation of shear waves), the amplitudesof both the standard and phason stresses are comparable (Figure 4).


17


0-0.0015

-0.0010

-0.0005

0

1 /r1/

2 K

II

0 45 90

45 90

45 90

135

r1/2 /K

II

r r1/

2 /KII

0-40

-30

10

0

-20

-10

180

135 180

135 180

(a)

(c) (d)

(e)

0-0.0015

-0.0010

-0.0005

0

2 /r1/

2 K

II

0

10

20

40

30

0

0

u 2/r

1/2 K

II

u 1/r

1/2 K

II

45 90

45 90

45 90

135 180

135 180

135 180

(b)

(f)

=5

-8

0

0.4

0.6

m2 = 0.8

m2 = 0.6

m2 = 0.20.2

-0.2

-0.4

-0.6

Sr1/

2

KII

S rr1/

2

KII

-1.0

1.5

1.0

0.5

-0.5

0

Figure 8. Angular variations of standard, phason stresses and displacement and phason fields around the crack tip under remote Mode II loading conditions,for �=5, and for different values of m2.

• In Mode II conditions (�0 =0), for m2 =0.6 and �=5, the magnitude of the phason stress is one hundred times lesser than theone of the standard stresses (Figure 5). With increased coupling the amplitude of the phason stress increases too and becomesten times lesser than the one of the standard stresses at �=35 (Figure 6).

• The angular plot of standard stresses is reminiscent of the shapes obtained in linear fracture dynamics in simple bodies (seeFigure 7(a),(b) and [45], also Figure 8(a),(b))

• The magnitude of the phason degrees of freedom decreases when Mach number increases in Mode I conditions and for�=5 (Figure 7(e),(f)). The same behavior is recognized in Mode II conditions only with respect to the phason field component�1, whereas �2 increases with Mach number (Figure 8(e),(f)). The same monotonous behavior is shown by the phason stress�r-component in Mode II conditions and with �=5.

• Under the limit value for � beyond which the shear waves do not propagate, there is a range of values in the plane �−m2—the region delimited by the curves in Figure 9—indicating non-admissibility conditions for crack propagation. Two possibleinterpretations of the result can be as follows: (1) In that region—that is with that combinations of coupling coefficient and

18



Figure 9. Energetically favorable (G>0) and non-favorable (G<0) regimes for crack propagation in quasi-crystals in the �−m2 plane.

Mach number—crack propagation could be admissible only if the mechanical behavior is not purely linear elastic. (2) Somepairs (�, m2) of values are not physically admissible. Experiments can corroborate one interpretation or the other, or, else, theycan falsify both.

Appendix A: Orthogonality relations

The orthogonality relations between the actual and the generalized eigenvectors are displayed here. Hermitian character of thematrix Y := iHK−1 is also shown.

Define the two 8×8 matrices

J :=⎡⎣2(SymB) C

C 0

⎤⎦ , M :=

⎡⎣Q−12(SymB) Q−1C

−I 0

⎤⎦ , (A1)

where M is the matrix of coefficients of the system (7) and J=JT. The exponent T indicates matrix transposition. Remind that Cand Q are also symmetric. Multiplication of M by J furnishes

JM=⎡⎣2(SymB)Q−12(SymB)−C 2(SymB)Q−1C

CQ−12(SymB) CQ−1C

⎤⎦= (JM)T =MTJ. (A2)

Denote by v(k) the eight-dimensional eigenvectors constituted by h(k) and g(k), namely

v(k) :=

⎛⎜⎝h(k)

g(k)

⎞⎟⎠ for k =1, 2, 3, 4,

where v(4) is the generalized eigenvector of the degenerate matrix M, the other values of k correspond to standard eigenvectors.From the definition of v(k), it follows that

JMv(k) = �kJv(k), JMv(k) = �kJv(k) for k =1, 2, 3, (A3)

JMv(4) = �3Jv(4) +Jv(3), JMv(4) = �3Jv(4) +Jv(3). (A4)

By using (A2), it is immediate to realize that the vectors Jv(k) and Jv(k), k =1, 2, 3, 4, are the eigenvectors and the generalizedeigenvectors of the degenerate matrix MT. The relation (A3), (A4) and the symmetry of the matrices J and JM, for h, k =1, 2, 3,imply

�kJv(k) ·v(h) = JMv(k) ·v(h) =v(k) ·JMv(h) =�hv(k) ·Jv(h) =�hJv(k) ·v(h),

�kJv(k) · v(h) = JMv(k) · v(h) =v(k) ·JMv(h) = �hv(k) ·Jv(h) =�hJv(k) · v(h).


19


Since the six eigenvalues �k and �k , for k =1, 2, 3, are not equal, the following orthogonality relations must hold:

Jv(k) ·v(h) = Jv(k) · v(h) =0 for h, k =1, 2, 3, h �=k,

Jv(k) · v(h) = Jv(k) ·v(h) =0 for h, k =1, 2, 3. (A5)

By using (A4)1, one can also find that

JMv(4) ·v(3) =v(4) ·JMv(3) =�3v(4) ·Jv(3) =�3Jv(4) ·v(3) = (JMv(4) −Jv(3)) ·v(3), (A6)

and

�3Jv(4) ·v(k) = (JMv(4) −Jv(3)) ·v(k) =JMv(4) ·v(k) =v(4) ·JMv(k) =�kv(4) ·Jv(k) =�kJv(4) ·v(k), (A7)

for k =1, 2.The use of (A4) then implies

�3Jv(4) · v(k) = �kJv(4) · v(k) for k =1, 2, 3, (A8)

�3Jv(4) · v(4) = �3Jv(4) · v(4). (A9)

From (A6) it follows that

Jv(3) ·v(3) =0, Jv(3) · v(3) =0.

Moreover, (A7), (A8) and (A9) imply

Jv(4) ·v(k) = Jv(k) ·v(4) =0 for k =1, 2,

Jv(4) · v(k) = Jv(4) ·v(k) =0 for k =1, 2, 3, 4,

(A10)

so that from (A5) and (A10) it follows that

UTJU=[

T 0

0 T

], (A11)

where

T :=

⎡⎢⎢⎢⎢⎢⎣

J11 0 0 0

0 J22 0 0

0 0 0 J34

0 0 J34 J44

⎤⎥⎥⎥⎥⎥⎦

with Jhk :=Jv(k) ·v(k). The matrix U collects eigenvectors, namely

U := [v(1) v(2) v(3) v(4) v(1) v(2) v(3) v(4)]=[

H H

G G

]. (A12)

The introduction of (A12) and (A1)a in (A11) and the use of the definition K :=BTH+CG yields

HTK+KTH=HTK+KTH=0, HTK+KTH=0, (A13)

namely

⎡⎣HT KT

HT KT

⎤⎦[K K

H H

]=[

T 0

0 T

],2

0



so that [K K

H H

]−1

=[

T−1 0

0 T−1

]⎡⎣HT KT

HT KT

⎤⎦

where

T−1 :=

⎡⎢⎢⎢⎢⎢⎣

1 / J11 0 0 0

0 1 / J22 0 0

0 0 −J44 / J234 1 / J34

0 0 1 / J34 0

⎤⎥⎥⎥⎥⎥⎦ .

The orthogonality relation (A13)a implies that the matrix Y := iHK−1 is Hermitian. Relations similar to (A13 ) have been alsoderived by Stroh (see [17, 18]) for anisotropic standard linear elasticity, with reference to a non-degenerate matrix. Such a result byStroh has been extended by Ting to the degenerate case in [20].

Appendix B: Rayleigh waves in quasi-crystals

Stroh formalism adopted in the previous sections to analyze the evolution of a planar crack can be used straight away to analyzethe evolution of surface waves in quasi-crystals. The approach is summarized here briefly.

Consider a quasi-crystalline body occupying the half plane x2 =0 in the Cartesian frame of reference (0x1x2x3). Assume that thepotential bh(x1, x2), h=1, 2, 3, 4, appearing in (10), be of the form

bh(x1, x2)= fh(x2) exp(ikx1),

where k is a positive number. Substitution into (10) implies

ikf(x2)+(S+N)f′(x2)=0

that is

ikfh(x2)+�hf′h(x2)=0, h=1, 2, 4,

ikf3(x2)+�3f′3(x2)=−f′4(x2).

Solution is expressed by

f(x2)=〈〈exp(−k√

1−m2hx2)〉〉(c− ikx2Nc),

where m3 =m4 and c is a constant vector to be determined by means of the boundary conditions.By imposing that q=0 at x2 =0, since q=2Re(Kb), as derived in the main body of this work, it follows that

2Re(Kf(0) exp(ikx1))=0.

Moreover, since f(0)=c, from the explicit expression of f above, the condition

Kcexp(ikx1)+Kc exp(−ikx1)=0

is satisfied at every x1. The arbitrariness of x1 implies that

Kc=0,

which is the eigenvalue problem for the matrix K, a problem admitting non-trivial solutions only when

detK=0.

Such a characteristic equation is satisfied for values of the Mach number plotted in Figure 9)—where it is indicated by mlim—as afunction of the coupling parameter � between the gross deformation and phason activity.

Acknowledgements

This work has been developed within the programs of the research group in ‘Theoretical Mechanics’ of the ‘Centro di RicercaMatematica Ennio De Giorgi’ of the Scuola Normale Superiore at Pisa. The support of the GNFM-INDAM is acknowledged. ERacknowledges also Italian MIUR for financial support under the contract ‘Multiscale modelling, numerical and experimental analysisof complex materials and structures with novel applications’-PRIN08.


21


References1. Steurer W. Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals. Zeitschrift

für Kristallographie 2004; 219:391--446.2. Steurer W, Deloudi S. Fascinating quasicrystals. Acta Crystals 2008; A64:1--11.3. Schmicker D, van Smaalen S. Dynamical behavior of aperiodic intergrowth crystals. International Journal of Modern Physics B 1996; 10:2049--2080.4. Mariano PM. Mechanics of quasi-periodic alloys. Journal of Nonlinear Science 2006; 16:45--77.5. Mikulla R, Stadler J, Krul F, Trebin H-R, Gumbsch P. Crack propagation in quasicrystals. Physical Review Letters 1998; 81:3163--3166.6. Rösch F, Rudhart C, Roth J, Trebin H-R, Gumbsh P. Dynamic fracture of icosahedral model quasicrystals: a molecular dynamic study. Physical

Reviews B 2005; 72:014128-1--014128-9.7. Rudhart C, Trebin H-R, Gumbsh P. Crack propagation in perfectly ordered and random tiling quasicrystals. Journal of Non-Crystalline Solids 2004;

334 & 335:453--456.8. Fan T-Y, Mei Y-W. Elastic theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials. Advances in Applied

Mechanics 2004; 57:325--343.9. Mariano PM. Cracks in complex bodies: covariance of tip balances. Journal of Nonlinear Science 2008; 18:99--141.

10. Zhu A-Y, Fan T-Y. Dynamic crack propagation in decagonal Al–Ni–Co quasicrystals. Journal of Physics: Condensed Matter 2008; 20:295217-1--295217-9.

11. Li X-F, Fan T-Y, Sun Y-F. A decagonal quasicrystal with a Griffith crack. Philosophical Magazine 1999; 79:1943--1952.12. Wang X, Pan E. Analytical solutions for some defect problem in 1D hexagonal and 2D octagonal quasicrystals. PRAMANA—Journal of Physics

2008; 70:911--933.13. Guo Y-C, Fan T-Y. A mode II Griffith crack in decagonal quasicrystals. Applied Mathematical Mechanics 2001; 22:1311--1317.14. Piva A, Viola E. Crack propagation in an orthotropic medium. Engineering Fracture Mechanics 1988; 29:535--548.15. Viola E, Piva A, Radi E. Crack propagation in an orthotropic medium under general loading. Engineering Fracture Mechanics 1989; 34:1155--1174.16. Piva A, Radi E. Elastodynamic local fields for a crack running in an orthotropic medium. Journal of Applied Mechanics 1991; 58:982--987.17. Stroh AN. Dislocations and cracks in anisotropic elasticity. Philosophical Magazine 1958; 3:625--646.18. Stroh AN. Steady state problems in anisotropic elasticity. Journal of Mathematical Physics 1962; 41:77--103.19. Gao Y, Zhao YT, Zhao BS. Boundary value problems of holomorphic vector functions in 1D QCs. Physica B: Condensed Matter 2007; 394:56--61.20. Ting TCT. Anisotropic elasticity. Theory and applications. Oxford University Press: Oxford, 1996.21. Tanuma K. Stroh formalism and Rayleigh waves. Journal of Elasticity 2007; 89:5--154.22. Suo Z, Kuo CM, Barnett DM, Willis JR. Fracture mechanics for piezoelectric ceramics. Journal of the Mechanics and Physics of Solids 1992;

40:739--765.23. Wang BL, Mai YW. Crack tip field in piezoelectric/piezomagnetic media. European Journal of Mechanics A 2003; 22:591--602.24. Piva A, Tornabene F, Viola E. Crack propagation in a four-parameter piezoelectric medium. European Journal of Mechanics A 2006; 25:230--249.25. Piva A, Tornabene F, Viola E. Subsonic Griffith crack propagation in piezoelectric media. European Journal of Mechanics A 2007; 26:442--459.26. Boldrini C, Viola E. Crack energy density of a piezoelectric material under general electro-mechanical loading. Theoretical and Applied Fracture

Mechanics 2008; 49:321--333.27. Gao CF, Kessler H, Balke H. Crack problems in magnetoelectroelastic solids. Part I: exact solution of a crack. International Journal of Engineering

Science 2003; 41:969--981.28. Ni L, Nemat-Nasser S. A general duality principle in elasticity. Mechanics of Materials 1996; 24:87--123.29. Ting TCT, Hwu C. Sextic formalism in anisotropic elasticity for almost non-semisimple matrix N. International Journal of Solids and Structures

1998; 24:65--76.30. Gentilini C, Piva A, Viola E. On crack propagation in orthotropic media for degenerate states. European Journal of Mechanics A 2004; 23:247--258.31. Piva A, Viola E, Tornabene F. Crack propagation in an orthotropic medium with coupled elastodynamic properties. Mechanics Research

Communications 2005; 32:153--159.32. Radi E, Mariano PM. Stationary straigth cracks in quasicrystals. International Journal of Fracture 2010; in print.33. Wang YM, Ting TCT. The Stroh formalism for anisotropic materials that possess an almost extraordinary degenerate matrix N. International

Journal of Solids and Structures 1997; 34:401--413.34. Lubensky TC, Ramaswamy S, Toner J. Hydrodynamics of icosahedral quasicrystals. Physical Reviews B 1985; 32:7444--7452.35. Hu C, Wang R, Ding D-H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Reports on Progress in Physics

2000; 63:1--39.36. De P, Pelcovits RA. Linear elasticity theory of pentagonal quasicrystals. Physical Reviews B 1987; 35:8609--8620.37. Jeong H-C, Steinhardt PJ. Finite-temperature elasticity phase transition in decagonal quasicrystals. Physical Reviews B 1993; 48:9394--9403.38. de Boissieu M, Boudard M, Hennion B, Bellissent R, Kycia S, Goldman A, Janot C, Audier M. Diffuse scattering and phason elasticity in the

AlPdMn icosahedral phase. Physical Review Letters 1995; 75:89--92.39. Mariano PM, Modica G. Ground states in complex bodies. ESAIM: Control, Optimisation and Calculus of Variations 2009; 15:377--402.40. Rochal SB, Lorman VL. Minimal model of the phonon-phason dynamics in icosahedral quasicrystals and its application to the problem of

internal friction in the i -AlPbMn alloy. Physical Reviews B 2002; 66:144204-1--144204-9.41. Mariano PM. Multifield theories in mechanics of solids. Advances in Applied Mechanics 2002; 38:1--93.42. Hirsh MW, Smale S. Differential Equations, Dynamic Systems and Linear Algebra. Academic Press: New York, 1974.43. Muskhelishvili NI. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff: Groningen, 1953.44. Gakhov FD. Boundary Value Problems. Pergamon Press: New York, 1996.45. Freund LB. Dynamic Fracture Mechanics. Cambridge University Press: Cambridge, 1990.46. Broberg KB. Cracks and Fracture. Cambridge University Press: Cambridge, 1999.47. Yoffe EH. The moving Griffith crack. Philosophical Magazine 1951; 7:739--750.48. Landau LD, Lifshitz EM. Course of Theoretical Physics. vol. 7. Theory of Elasticity (3rd edn). Pergamon Press: Oxford,1986.49. Sih G. Liebowitz CH. Fracture: An Advanced Treatise. Academic Press: New York, 1968.50. Amazit Y, Fischer M, Perrin B, Zarembowitch A. Ultrasonic investigations of large single AlPdMn icosahedral quasicrystals. Proceedings of the

5th International Conference on Quasicrystals, Janot C, Massieri RS (eds). World Scientific: Singapore, 1995; 584--587.51. Fan TY, Wang XF, Lin W, Zhu AY. Elasto-hydrodynamics of quasicrystals. Philosophical Magazine 2009; 89:501--512.

22



52. Létoublon A, de Boissieu M, Boudard M, Mancini L, Castaldi J, Hennion B, Caudron R, Bellissent R. Phason elastic constants of the icosahedralAl-Pd-Mn phase derived from diffuse scattering experiments. Philosophical Mag Letters 2001; 81:273--283.

53. Tanaka K, Mitarai Y, Koiwa M. Elastic constants of Al-based icosahedral quasicrystals. Philosophical Magazine 1996; 73:1715--1723.54. Zhu AY, Fan TY, Guo LH. Elastic field for a straight dislocation in an icosahedral quasicrystal. Journal of Physics: Condensed Matter 2007;

23:236216-1--236216-8.55. Walz C. Zur Hydrodynamik in Quasikristallen. Thesis, University of Stuttgart, 2003.


23

Research Article

Received 4 January 2010 Published online 7 May 2010 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1327MOS subject classification: 34 B 10; 34 B 15

A numerical solution to nonlinear multi-pointboundary value problems in the reproducingkernel space

Yingzhen Lin∗† and Minggen Cui

Communicated by S. Georgiev

In this paper, a new numerical algorithm is provided to solve nonlinear multi-point boundary value problems in a veryfavorable reproducing kernel space, which satisfies all complex boundary conditions. Its reproducing kernel functionis discussed in detail. The theorem proves that the approximate solution and its first- and second-order derivativesall converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solvingnonlinear multi-point boundary value problems. Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: nonlinear problems; multi-point boundary value conditions; reproducing kernel space

1. Introduction

In Reference [1], authors have given a method for solving the problem of second-order multi-point boundary value (1). We shallprovide other method in this paper. ⎧⎪⎨

⎪⎩u′′(x)+f (x)+g(u, u′)=0, 0�x�1,

u(0)=�, u(1)=m∑

i=1�iu(�i)+�.

(1)

where �i ∈ (0, 1), i=1, 2,. . . , m. We may assume that �=0, �=0, since the boundary conditions u(0)=�, u(1)=∑mi=1 �iu(�i)+� can be

reduced to u(0)=0, u(1)=∑mi=1 �iu(�i).

They arise in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams and plate deflection theory.In recent years, much attention has been given to solving the multi-point boundary value problems [1--3]. Several numerical andanalytical techniques are being developed for solving these problems [1, 4--8]. In this paper, we shall give a numerical method thatis different from Reference [1] dependent on reproducing kernel theory. A reproducing kernel that satisfies all boundary conditionswill be obtained. It will be used as an approximate solution of the equation.

2. A constructive method for the reproducing kernel space H0[0, 1]

In Reference [9], Hilbert reproducing kernel space is given by

H[0, 1]={u(x)|u′(x) is absolutely continuous, u(0)=0, u′′(x)∈L2[0, 1]}.

The reproducing kernel function of H[0, 1] is Ry(x).The subspace H0[0, 1] of H[0, 1] is defined by:

H0[0, 1]={u(x)∈H[0, 1], u(1)−m∑

i=1�iu(�i)=0}. (2)

It is a closed subspace of H[0, 1].

Department of Mathematics, Harbin Institute of Technology at Weihai, Shandong, 264209, People’s Republic of China∗Correspondence to: Yingzhen Lin, Department of Mathematics, Harbin Institute of Technology at Weihai, Shandong, 264209, People’s Republic of China.†E-mail: [email protected]

44


Y. LIN AND M. CUI

It is very important to obtain the representation of reproducing kernel in H0(�), since it is a base of our algorithm. Therefore, ourwork starts from a lemma first.

Lemma 2.1R1(x)−∑m

i=1 �iR�i (x) �∈H0[0, 1].

ProofOtherwise, ∀u(x)∈H[0, 1],

0=⟨

u(·), R1(·)−m∑

i=1�iR�i (·)

⟩=〈u(·), R1(·)〉−

m∑i=1

�i〈u(·), R�i (·)〉=u(1)−m∑

i=1�iu(�i)

then u(x)∈H0[0, 1], which is contradictory. �

According to Lemma 2.1, we can get A=R1(1)−2∑m

i=1 �iR�i (1)+∑mi,j=1 �i�jR�j (�i) �=0. Now we consider a function

Ky(x)=Ry(x)−R1(x)[Ry(1)−∑m

i=1 �iRy(�i)]−Ry(1)∑m

i=1 �iR�i (x)+∑mi,j=1 �i�jR�j (x)Ry(�i)

A. (3)

One can check that Ky(x)∈H0[0, 1] and it is the reproducing kernel of H0[0, 1] carefully.

3. Numerical algorithm

In this section, we shall explain how to obtain an approximate solution of Equation (1). Let an operator L : H0[0, 1]→L2[0, 1],(Lu)(x)=u′′(x). Operator L is bounded clearly. Rewriting Equation (1) as:

(Lu)(x)=−f (x)−g(u, u′). (4)

Let us choose a countable dense subset S={x1, x2,. . .}⊂ [0, 1] and define �i(x) by �i(x)def= Lxi Kxi (x)=�2Kxi (x) / �x2

i . {�i(x)}∞i=1 is acomplete system of the space H0[0, 1] (see Reference [10]).

Applying Gram–Schmidt process, we obtain an orthogonal basis {�i(x)}∞i=1 of H0[0, 1], such that �i(x)=∑ik=1 �ik�k(x),

Theorem 3.1If u(x) is the solution of Equation (4), then

u(x)=∞∑

i=1

i∑k=1

�ik[−f (xk, )−g(u(xk), u′(xk))]�i(x). (5)

ProofAccording to {�i(x)}∞i=1 is orthogonal basis of H0[0, 1], we have

u(x) =∞∑

i=1

i∑k=1

〈u(x), �i(x)〉�i(x)=∞∑

i=1

i∑k=1

�ik〈u(x),�k(x)〉�i(x)

=∞∑

i=1

i∑k=1

�ik〈u, (LKxk )(x)〉�i(x)=∞∑

i=1

i∑k=1

�ik[L〈u, Rxk 〉](x)�i(x)

=∞∑

i=1

i∑k=1

�ik(Lu)(xk)�i(x)=∞∑

i=1

i∑k=1

�ik[−f (xk)−g(u(xk), u′(xk))]�i(x).

�

Let us denote g(u(xk), u′(xk)) by �k , and it follows that (5) would be

u(x)=∞∑

i=1

i∑k=1

�ik[−f (xk, )−�k]�i(x). (6)

In this way, converts solving Equation (4) to search �k . In order to obtain �k , we truncate the series of the left-hand side of (6),obtain

un(x)=n∑

i=1

i∑k=1

�ik[−f (xk, )−�k]�i(x, t). (7)


45

Y. LIN AND M. CUI

We can get �k based on the minimum-point of function

J(�1,�2,. . . ,�n)=n∑

k=1[g(un(xk), u′

n(xk))−�k]2.

Consequently, the approximate solution of Equation (4) be obtained by

un(x)=n∑

i=1

i∑k=1

�ik[−f (xk, )−�k]�i(x), (8)

un(x) in (8) is the truncated of Fourier series of u(x). Hence, un(x)→u(x) in H0[0, 1]. Last, we shall give the algorithm of obtaining �k .

1. Pick initial values �0k .

2. Substitute �0k into (8) and compute un(x).

3. Calculate J(�01,�0

2,. . . ,�0n).

4. If J(�01,�0

2,. . . ,�0n)<10−20, then computation terminate; otherwise, substitution of un(x) into (7) yields new �1

k .


2,. . . ,�1n).

6. If J(�11,�1

2,. . . ,�1n)<J(�0

1,�02,. . . ,�0

n), then replace �0k with �1

k and return to 2; otherwise, give up �1k . Taking �0

k as initial values, we

can obtain the minimal value points of J(�1,�2,. . . ,�n) using Mathematics, that is new �1k and replace �0

k with �1k , return to 2.

Theorem 3.2The approximate solutions un(x) and u′

n(x), u′′n(x) uniformly converge to exact solutions u(x) and u′(x), u′′(x), respectively.

ProofFrom un(x)→u(x) in H0[0, 1], and

|u(k)n (x)−u(k)(x)|=|〈(u(k)

n −u(k))(y), Kx(y)〉|=|〈(un −u(y),dkKx(y)

dyk〉|�‖un −u‖‖ dkKx(y)

dyk‖�Mk‖un −u‖,

where Mk are constants, k =0, 1, 2, proof of the theorem is completed. �

4. Numerical examples

In this section, numerical examples are studied to demonstrate the accuracy of the present method. The results obtained by themethod are compared with analytical solution of each example and are found to be in good agreement with each other.

ExampleFor nonlinear second-order differential equation:

u′′+xu′+(1−x)u+x(1−x)2u2 = f, x ∈ (0, 1). (9)

In Reference [1], F. Z. Geng has obtained a approximate solution under fourth-point boundary value conditions u(0)=0, u(1)=∑4i=1(1 / (1+ i))u(i / 5)+0.600831. The algorithm error is shown by Figure 3 [1]. The maximum absolute error is 0.00001. Yet, when

we apply our method in this paper and take the number of nodes n=10, 30, 50, the errors are shown in Figure 1. This illustratesthat the accuracy of approximate solution will be getting better and better as n increases.

Finally, we re-calculate the approximate solution for (9) with ten-point boundary value conditions u(0)=0, u(1)=∑10i=1(1 / 1+ i)

u(i / 11)−0.434465. The errors are shown in Figure 2. The errors of derivatives are |u′−u′50|=0.0000116744 and |u′′−u′′

50|=0.000018421.

Figure 1. Absolute errors |u−u10|, |u−u30|, |u−u50|, respectively.

46


Y. LIN AND M. CUI


5. Conclusion

In this paper, we construct a novel reproducing kernel space and give the way to express reproducing kernel function Ky(x). Basedon these, a numerical algorithm is presented for solving nonlinear multi-point boundary value problems. The convergence is provedstrictly, and the experimental results illustrate the effectiveness and superiority of the algorithm.

References1. Geng F-Z, Cui M-G. Solving nonlinear multi-point boundary value problems by combining homotopy perturbation and iteration methods.

International Jounal of Nonlinear Sciences and Numerical Simulation 2009; 10(5):597--600.2. Denche M, Marhoune AL. A three-point boundary value problem with an integral condition for parabolic equations with the bessel operator.

Applied Mathematics Letters 2000; 13:85--89.3. Momani SM. Some problems in non-Newtonian fluid mechanics. Ph.D. Thesis, Walse University: U.K., 1991: 1--15.4. Dehghan M. Numerical techniques for a parabolic equation subject to an overspecified boundary condition. Applied Mathematics and

Computation 2002; 132:299--313.5. Geng F, Cui M. Solving singular nonlinear two-point boundary value problems in the reproducing kernel space. Journal of the Korean

Mathematical Society 2008; 45:631--644.6. Geng F, Cui M. Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. Applied Mathematics

and Computation 2007; 192:389--398.7. Chawla MM, Katti CP. Finite difference methods for two-point boundary-value problems involving higher order differential equation. BIT 1979;

19:27--33.8. Ma TF, Silva J. Iteration solution for a beam equation with nonlinear boundary conditions of third order. Applied Mathematics and Computation

2004; 159(1):11--18.9. Cui M, Lin Y. Nonlinear numerical analysis in reproducing kernel hilbert space. Nova Science Publisher: New York, 2009; 1--153.

10. Zhou Y, Cui M, Lin Y. Numerical algorithm for parabolic problems with non-classical conditions. Journal of Computational and Applied Mathematics2009; 230:770--780.


47

Research Article

Received 4 January 2010 Published online 7 May 2010 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1327MOS subject classification: 34 B 10; 34 B 15

A numerical solution to nonlinear multi-pointboundary value problems in the reproducingkernel space

Yingzhen Lin∗† and Minggen Cui

Communicated by S. Georgiev

In this paper, a new numerical algorithm is provided to solve nonlinear multi-point boundary value problems in a veryfavorable reproducing kernel space, which satisfies all complex boundary conditions. Its reproducing kernel functionis discussed in detail. The theorem proves that the approximate solution and its first- and second-order derivativesall converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solvingnonlinear multi-point boundary value problems. Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: nonlinear problems; multi-point boundary value conditions; reproducing kernel space

1. Introduction

In Reference [1], authors have given a method for solving the problem of second-order multi-point boundary value (1). We shallprovide other method in this paper. ⎧⎪⎨

⎪⎩u′′(x)+f (x)+g(u, u′)=0, 0�x�1,

u(0)=�, u(1)=m∑

i=1�iu(�i)+�.

(1)

where �i ∈ (0, 1), i=1, 2,. . . , m. We may assume that �=0, �=0, since the boundary conditions u(0)=�, u(1)=∑mi=1 �iu(�i)+� can be

reduced to u(0)=0, u(1)=∑mi=1 �iu(�i).

They arise in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams and plate deflection theory.In recent years, much attention has been given to solving the multi-point boundary value problems [1--3]. Several numerical andanalytical techniques are being developed for solving these problems [1, 4--8]. In this paper, we shall give a numerical method thatis different from Reference [1] dependent on reproducing kernel theory. A reproducing kernel that satisfies all boundary conditionswill be obtained. It will be used as an approximate solution of the equation.

2. A constructive method for the reproducing kernel space H0[0, 1]

In Reference [9], Hilbert reproducing kernel space is given by

H[0, 1]={u(x)|u′(x) is absolutely continuous, u(0)=0, u′′(x)∈L2[0, 1]}.

The reproducing kernel function of H[0, 1] is Ry(x).The subspace H0[0, 1] of H[0, 1] is defined by:

H0[0, 1]={u(x)∈H[0, 1], u(1)−m∑

i=1�iu(�i)=0}. (2)

It is a closed subspace of H[0, 1].

Department of Mathematics, Harbin Institute of Technology at Weihai, Shandong, 264209, People’s Republic of China∗Correspondence to: Yingzhen Lin, Department of Mathematics, Harbin Institute of Technology at Weihai, Shandong, 264209, People’s Republic of China.†E-mail: [email protected]

44


Y. LIN AND M. CUI

It is very important to obtain the representation of reproducing kernel in H0(�), since it is a base of our algorithm. Therefore, ourwork starts from a lemma first.

Lemma 2.1R1(x)−∑m

i=1 �iR�i (x) �∈H0[0, 1].

ProofOtherwise, ∀u(x)∈H[0, 1],

0=⟨

u(·), R1(·)−m∑

i=1�iR�i (·)

⟩=〈u(·), R1(·)〉−

m∑i=1

�i〈u(·), R�i (·)〉=u(1)−m∑

i=1�iu(�i)

then u(x)∈H0[0, 1], which is contradictory. �

According to Lemma 2.1, we can get A=R1(1)−2∑m

i=1 �iR�i (1)+∑mi,j=1 �i�jR�j (�i) �=0. Now we consider a function

Ky(x)=Ry(x)−R1(x)[Ry(1)−∑m

i=1 �iRy(�i)]−Ry(1)∑m

i=1 �iR�i (x)+∑mi,j=1 �i�jR�j (x)Ry(�i)

A. (3)

One can check that Ky(x)∈H0[0, 1] and it is the reproducing kernel of H0[0, 1] carefully.

3. Numerical algorithm

In this section, we shall explain how to obtain an approximate solution of Equation (1). Let an operator L : H0[0, 1]→L2[0, 1],(Lu)(x)=u′′(x). Operator L is bounded clearly. Rewriting Equation (1) as:

(Lu)(x)=−f (x)−g(u, u′). (4)

Let us choose a countable dense subset S={x1, x2,. . .}⊂ [0, 1] and define �i(x) by �i(x)def= Lxi Kxi (x)=�2Kxi (x) / �x2

i . {�i(x)}∞i=1 is acomplete system of the space H0[0, 1] (see Reference [10]).

Applying Gram–Schmidt process, we obtain an orthogonal basis {�i(x)}∞i=1 of H0[0, 1], such that �i(x)=∑ik=1 �ik�k(x),

Theorem 3.1If u(x) is the solution of Equation (4), then

u(x)=∞∑

i=1

i∑k=1

�ik[−f (xk, )−g(u(xk), u′(xk))]�i(x). (5)

ProofAccording to {�i(x)}∞i=1 is orthogonal basis of H0[0, 1], we have

u(x) =∞∑

i=1

i∑k=1

〈u(x), �i(x)〉�i(x)=∞∑

i=1

i∑k=1

�ik〈u(x),�k(x)〉�i(x)

=∞∑

i=1

i∑k=1

�ik〈u, (LKxk )(x)〉�i(x)=∞∑

i=1

i∑k=1

�ik[L〈u, Rxk 〉](x)�i(x)

=∞∑

i=1

i∑k=1

�ik(Lu)(xk)�i(x)=∞∑

i=1

i∑k=1

�ik[−f (xk)−g(u(xk), u′(xk))]�i(x).

�

Let us denote g(u(xk), u′(xk)) by �k , and it follows that (5) would be

u(x)=∞∑

i=1

i∑k=1

�ik[−f (xk, )−�k]�i(x). (6)

In this way, converts solving Equation (4) to search �k . In order to obtain �k , we truncate the series of the left-hand side of (6),obtain

un(x)=n∑

i=1

i∑k=1

�ik[−f (xk, )−�k]�i(x, t). (7)


45

Y. LIN AND M. CUI

We can get �k based on the minimum-point of function

J(�1,�2,. . . ,�n)=n∑

k=1[g(un(xk), u′

n(xk))−�k]2.

Consequently, the approximate solution of Equation (4) be obtained by

un(x)=n∑

i=1

i∑k=1

�ik[−f (xk, )−�k]�i(x), (8)

un(x) in (8) is the truncated of Fourier series of u(x). Hence, un(x)→u(x) in H0[0, 1]. Last, we shall give the algorithm of obtaining �k .

1. Pick initial values �0k .

2. Substitute �0k into (8) and compute un(x).


2,. . . ,�0n).

4. If J(�01,�0

2,. . . ,�0n)<10−20, then computation terminate; otherwise, substitution of un(x) into (7) yields new �1

k .


2,. . . ,�1n).

6. If J(�11,�1

2,. . . ,�1n)<J(�0

1,�02,. . . ,�0

n), then replace �0k with �1

k and return to 2; otherwise, give up �1k . Taking �0

k as initial values, we

can obtain the minimal value points of J(�1,�2,. . . ,�n) using Mathematics, that is new �1k and replace �0

k with �1k , return to 2.

Theorem 3.2The approximate solutions un(x) and u′

n(x), u′′n(x) uniformly converge to exact solutions u(x) and u′(x), u′′(x), respectively.

ProofFrom un(x)→u(x) in H0[0, 1], and

|u(k)n (x)−u(k)(x)|=|〈(u(k)

n −u(k))(y), Kx(y)〉|=|〈(un −u(y),dkKx(y)

dyk〉|�‖un −u‖‖ dkKx(y)

dyk‖�Mk‖un −u‖,

where Mk are constants, k =0, 1, 2, proof of the theorem is completed. �


In this section, numerical examples are studied to demonstrate the accuracy of the present method. The results obtained by themethod are compared with analytical solution of each example and are found to be in good agreement with each other.

ExampleFor nonlinear second-order differential equation:

u′′+xu′+(1−x)u+x(1−x)2u2 = f, x ∈ (0, 1). (9)

In Reference [1], F. Z. Geng has obtained a approximate solution under fourth-point boundary value conditions u(0)=0, u(1)=∑4i=1(1 / (1+ i))u(i / 5)+0.600831. The algorithm error is shown by Figure 3 [1]. The maximum absolute error is 0.00001. Yet, when

we apply our method in this paper and take the number of nodes n=10, 30, 50, the errors are shown in Figure 1. This illustratesthat the accuracy of approximate solution will be getting better and better as n increases.

Finally, we re-calculate the approximate solution for (9) with ten-point boundary value conditions u(0)=0, u(1)=∑10i=1(1 / 1+ i)

u(i / 11)−0.434465. The errors are shown in Figure 2. The errors of derivatives are |u′−u′50|=0.0000116744 and |u′′−u′′

50|=0.000018421.


46


Y. LIN AND M. CUI


5. Conclusion

In this paper, we construct a novel reproducing kernel space and give the way to express reproducing kernel function Ky(x). Basedon these, a numerical algorithm is presented for solving nonlinear multi-point boundary value problems. The convergence is provedstrictly, and the experimental results illustrate the effectiveness and superiority of the algorithm.

References1. Geng F-Z, Cui M-G. Solving nonlinear multi-point boundary value problems by combining homotopy perturbation and iteration methods.

International Jounal of Nonlinear Sciences and Numerical Simulation 2009; 10(5):597--600.2. Denche M, Marhoune AL. A three-point boundary value problem with an integral condition for parabolic equations with the bessel operator.

Applied Mathematics Letters 2000; 13:85--89.3. Momani SM. Some problems in non-Newtonian fluid mechanics. Ph.D. Thesis, Walse University: U.K., 1991: 1--15.4. Dehghan M. Numerical techniques for a parabolic equation subject to an overspecified boundary condition. Applied Mathematics and

Computation 2002; 132:299--313.5. Geng F, Cui M. Solving singular nonlinear two-point boundary value problems in the reproducing kernel space. Journal of the Korean

Mathematical Society 2008; 45:631--644.6. Geng F, Cui M. Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. Applied Mathematics

and Computation 2007; 192:389--398.7. Chawla MM, Katti CP. Finite difference methods for two-point boundary-value problems involving higher order differential equation. BIT 1979;

19:27--33.8. Ma TF, Silva J. Iteration solution for a beam equation with nonlinear boundary conditions of third order. Applied Mathematics and Computation

2004; 159(1):11--18.9. Cui M, Lin Y. Nonlinear numerical analysis in reproducing kernel hilbert space. Nova Science Publisher: New York, 2009; 1--153.

10. Zhou Y, Cui M, Lin Y. Numerical algorithm for parabolic problems with non-classical conditions. Journal of Computational and Applied Mathematics2009; 230:770--780.


47

Research Article

Received 12 November 2008 Published online 25 May 2010 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1331MOS subject classification: 65 C 20; 60 H 35; 65 N 12

Numerical solution of random differentialinitial value problems: Multistep methods

J. C. Cortésa∗†, L. Jódara and L. Villafuerteb

Communicated by T. E. Simos

This paper deals with the construction of numerical methods of random initial value problems. Random linear multistepmethods are presented and sufficient conditions for their mean square convergence are established. Main statisticalproperties of the approximations processes are computed in several illustrative examples. Copyright © 2010 John Wiley& Sons, Ltd.

Keywords: random initial value problem; mean square calculus; random linear multistep scheme

1. Introduction

Mathematical models are useful to describe reality up to a certain point. Parameters and functions involved in mathematical modelsare subject to uncertainty because errors of measurements, complexity, ignorance and so on [1, 2]. For these reasons, it is moreconvenient dealing with random models instead of deterministic ones, where parameters are random variables and functions arestochastic processes. As a consequence, in the last years an important number of mathematical models have been proposed bymeans of random differential equations of the form

X(t) = f (X(t), t), t ∈T = [t0, te],

X(t0) = X0,

}(1)

where X0 is a random variable, and the unknown X(t) as well as the right-hand side f (X, t) are stochastic processes defined on thesame probability space (�,F, P). Explicit solutions of linear and quadratic random differential equations as well as applications aregiven in [3, 4]. In particular, conditions on the random data are established in order to have explicit formulae for the exact solution.These formulae coincide with the well-known solutions for the deterministic case. However, as in the deterministic case, in generalthe exact solution stochastic process is not always available. This motivates the development of random numerical methods toapproximate the process solution of (1). In [5, 6], a random Euler method to construct numerical solution of (1) is proposed.

In this paper, we extend [5, 6] by considering random multistep methods that allow us to improve Euler’s approximations.A discrete numerical method for computing mean square (m.s.) approximations of random diffusion models as well as its meansquare consistency and stability has been proposed in [7]. As an essential difference with respect to stochastic differential equationsregarding uncertainty through the white noise, see [8--12], our approach allows the consideration of other types of randomness. In[13, 14] for instance, m.s. stability of some difference schemes for stochastic differential equations is studied but without studyingstatistical moments of the approximations.

This paper is organized as follows. Section 2 deals with some preliminaries of the m.s. calculus addressed to clarify the presentationof concepts and results used later. Random multistep methods are presented in Section 3. Section 4 deals with the m.s. consistency

aInstituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, SpainbFacultad de Ingeniería, Universidad Autónoma de Chiapas, Tutxla Gutiérrez, Chiapas, Mexico∗Correspondence to: J. C. Cortés, Instituto Universitario de Matemática Multidisciplinar, Building 8G, Access C, 2nd floor, Universidad Politécnica de Valencia,

Valencia PO46022, Spain.†E-mail: [email protected]

Contract/grant sponsor: Spanish M.E.C.Contract/grant sponsor: FEDER; contract/grant numbers: MTM2009-08587, TRA2007–68006–C02–02Contract/grant sponsor: Universidad Politécnica de Valencia; contract/grant number: PAID-06-09 (ref. 2588)Contract/grant sponsor: Mexican Conacyt


63

J. C. CORTÉS, L. JÓDAR AND L. VILLAFUERTE

and the m.s. stability of random multistep methods. Section 5 deals with the m.s. convergence of random linear multistep methods.The last section includes several illustrative examples.

2. Preliminaries

We are interested in second-order random variables (2-r.v.’s) Y , having a density function fY ,

E[Y2]=∫ ∞

−∞y2fY (y) dy<∞,

where E denotes the expectation operator, and it allows us the introduction of the Banach space L2 of all the 2-r.v.’s endowed withthe norm

‖Y‖=√

E[Y2],

see [15, Chapter 4]. This norm does not satisfy the Banach inequality ‖XY‖�‖X‖‖Y‖. In fact let X =Y =U1/2, where U is a uniform

r.v. defined on [0,1]; thus ‖XY‖= (E[U2])1/2 =√

33 , and ‖X‖‖Y‖=E[U]= 1

2 . Hence ‖XY‖>‖X‖‖Y‖.A sequence of 2-r.v.’s {Yn} converges in m.s. to a 2-r.v. Y as n→∞, and it will be denoted by limn→∞ Yn =Y , if

limn→∞‖Yn −Y‖2 = lim

n→∞E[(Yn −Y)2]=0. (2)

A stochastic process Y(t) defined on the same probability space (�,F, P) is said to be a second-order stochastic process (2-s.p.) iffor each t, Y(t) is a 2-r.v.

For the sake of clarity in the presentation, we recall an important result [15, p. 88] that will be crucial in the next sections.

Lemma 2.1Let {Xn} and {Yn} be two sequences of 2-r.v.’s m.s. convergent to the 2-r.v’s X and Y , respectively, i.e.

limn→∞ Xn =X and lim

n→∞ Yn =Y.

Then

limn→∞E[XnYn]=E[XY]. (3)

One of the most practical advantages of the m.s. convergence is the following consequence of Lemma 2.1: if a sequence {Xn} of2-r.v.’s is m.s. convergent to the 2-r.v. X , then its expectation E[Xn] converges to E[X] and its variance Var[Xn] converges to Var[X].

Now we recall some definitions and properties that will play a prominent role in the last section where several illustrative examplesare given. The expectation of �X = (X1,. . . , Xn)T is the deterministic vector E[�X]= (E[X1],. . . , E[Xn])T. The covariance matrix of �X is definedby the following square matrix of size n :

��X =E[(�X −E[�X])(�X −E[�X])T]= (vij)n×n, (4)

where the (i, j)-entry is given by

vij =E[(Xi −E[Xi])(Xj −E[Xj])]=E[XiXj]−E[Xi]E[Xj], 1�i, j�n. (5)

Note that vii denoted by Var[Xi] is the variance of the r.v. Xi , 1�i�n. In the particular case that n=1 and �X =X1 =X , (4) is the varianceof the scalar 2-r.v. X .

Given two random vector of second order, say �X = (X1,. . . , Xn)T and �Y = (Y1,. . . , Yn)T with n�0, one defines their cross-covariancematrix by

��X,�Y =E[(�X −E[�X])(�Y −E[�Y])T]= (�ij)n×n, (6)

where

�ij =E[(Xi −E[Xi])(Yj −E[Yj])]=E[XiYj]−E[Xi]E[Yj], 1�i, j�n. (7)

In the particular case that n=1, �X =X1 =X and �Y =Y1 =Y , expression (6) is called the covariance between the 2-r.v.’s X , Y , and itwill denoted in the following by Cov[X, Y]. It can be expanded as Cov[X, Y]=E[X Y]−E[X] E[Y]. From the above definitions, one gets��X,�X =��X . Also, note that the following property holds:

��Y,�X = (��X,�Y )T. (8)

64



Finally, we introduce a property that will be used later. It provides the covariance matrix of two linear combination of randomvectors in terms of the covariance matrices of each pair of the random vectors:

��X,�Y =n∑

j=1

m∑k=1

ajbk��Xj,�Ykwhere �X =

n∑j=1

aj �Xj, �Y =m∑

k=1bk �Yk, (9)

where {aj : 1�j�n} and {bk : 1�k�m} are real numbers.

The meaning of X(t) in (1) is the m.s limit in L2 of the expression

X(t+�t)−X(t)

�tas �t →0. (10)

Definition 2.2Let S be a bounded set in L2, an interval T ⊆R and h>0, we say that f : S×T →L2 is randomly bounded time uniformly continuousin S if

limh→0

�(S, h)=0, (11)

where

�(S, h)= supX∈S⊂L2

sup|t−t∗|�h

‖f (X, t)−f (X, t∗)‖, t, t∗ ∈T. (12)

Example 2.3Let f (X, t)=AX +B(t), where 0�t�1, A is a 2-r.v. and B(t) is the standard Brownian motion process.

Note that by properties of the standard Brownian motion, see [15, p. 63], it follows that

‖f (X, t)−f (X, t∗)‖=‖B(t)−B(t∗)‖=|t−t∗|1/2.

Hence f (X, t) is randomly bounded time uniformly continuous in any bounded set S⊂L2. The next lemma will be used extensivelyfor establishing the m.s. convergence of random multistep methods.

Lemma 2.4Let � be a 2-r.v. and let � be a real number and suppose that ‖�‖�|�|. Then there exists a 2-r.v. � such that �= �� with ‖�‖�1.

ProofIf � �=0, take �=� / �, then ‖�‖= (1 / |�|)‖�‖�1. The result is trivial when �=0. �

3. Random linear multistep methods

Considering the discretization tn = t0 +nh, n=0, 1, 2,. . ., h= (te −t0) / N for the random initial differential problem (1), a random linearmultistep method of order k has the general form

k∑j=0

�jXn+j =hk∑

j=0�jfn+j , (13)

where the starting 2-r.v.’s X0, X1,. . . , Xk−1 are given and fm = f (Xm, tm), being Xm, for m=1, 2,. . . , an approximation of the exactsolution stochastic process X(t) at t = tm, and �j , �j , j=0,. . . , k, are real numbers satisfying

�k �=0, |�0|+|�0|>0.

Note that if �k is zero, then the corresponding method (13) is explicit and is implicit otherwise.By a similar way as in the deterministic case, if for

F(X)=h�k

�kf (X, tn+k)+c, (14)

where

c= 1

�k(h[�k−1fn+k−1 +·· ·+�0fn]−�k−1Xn+k−1 −·· ·−�0Xn), (15)

there exists a constant K : 0�K<1, such that

‖F(Y)−F(X)‖�K‖Y −X‖ ∀X, Y ∈L2, (16)

then the random equation (13) has a unique m.s. solution.


65


Let us continue by defining the concept of m.s. convergence of method (13). A random linear multistep method (13) is said tobe m.s. convergent if, for each problem (1) having m.s. solution X(t) on [t0, te],

limh→0

Xn =X(t), t = tn (17)

holds for all t ∈ [t0, te] and for all solutions {Xn} of the scheme (13) with starting values Xi =Xi(h), i=0, 1,. . . , k−1, satisfying

limh→0

Xi(h)=X0, i=0, 1,. . . , k−1. (18)

In order to study the error en =Xn −X(tn), n=1, 2,. . . , of the method given by (13), it is convenient to introduce the operator L[X(t); h]and the polynomials �() and () associated with (13) defined by

L[X(t); h] =k∑

j=0�jX(t+ jh)−h

k∑j=0

�j X(t+ jh), (19)

�() =k∑

j=0�j

j , ()=k∑

j=0�j

j , (20)

respectively. Observe that the above operator acts on any m.s. differentiable stochastic process.

4. m.s. consistency and stability

The above definition of m.s. convergence has some consequences that will allow us to introduce the random concepts of m.s.square consistency and m.s stability. Let us begin by assuming that the method (13) is m.s. convergent, that is,

limh→0

Xn =X(t), t = tn, nh= tn −t0, (21)

or equivalently

X(t)=Xn+j +�n,j(h), limh→0

�n,j(h)=0, j =0, 1,. . . , k. (22)

By a simple algebraic manipulation of (22) and considering (13), one gets

k∑j=0

�j =hk∑

j=0�jfn+j +

k∑j=0

�j�n,j(h). (23)

Now considering (22) and taking lim as h→0 in (23), we deduce

k∑j=0

�j =0, (24)

where we have assumed that the solution s.p. X(t) is not trivial.On the other hand, we have

limh→0

Xn+j −Xn

jh= X(t), j =1,. . . , k, t = tn,

which means

Xn+j −Xn = jhX(t)+ jh�n,j(h), j =1,. . . , k, (25)

where limh→0 �n,j(h)=0. From (25) one gets

k∑j=0

�jXn+j −(

k∑j=0

�j

)Xn =h

(k∑

j=0j�j

)X(t)+h

k∑j=0

j�j�n,j(h). (26)

From (13) and (24), expression (26) reads as

k∑j=0

�jfn+j =(

k∑j=0

j�j

)X(t)+

k∑j=0

j�j�n,j(h). (27)

66



Table I. Approximations of Example 4.2.

tn =nh yn yn −exp(nh)

0 1.0000 0.00000.1 1.1052 0.00000.2 1.2207 −0.00070.3 1.3462 −0.00360.4 1.4786 −0.01330.5 1.6065 −0.04230.6 1.6944 −0.12770.7 1.6344 −0.37930.8 1.6370 −0.58850.9 −0.7351 −3.19471.0 −6.5429 −9.2612

As f is m.s. Lipschitz, it follows

limh→0

fn+j = f (X(t), t)= X(t), j =0, 1,. . . , k;

thus, taking l.i.m as h→0 in (27), we find

k∑j=0

�j =k∑

j=0j�j . (28)

This motivates the following

Definition 4.1We say that scheme (13) is m.s. consistent if it satisfies conditions (24) and (28).

Hence, we have established that if the scheme (13) is m.s. convergent then it is m.s. consistent, but we shall show in the nextexample that the reciprocal result is not true.

Example 4.2Let us consider the random initial value problem

X(t)=X(t), X(0)=X0, (29)

where X0 is an exponential 2-r.v. with parameter �=1, then E[X0]=1.

The following scheme of order k =2

− 12 Xn+2 +2Xn+1 − 3

2 Xn =hXn, tn =nh, n=0, 1, 2,. . . (30)

satisfies conditions (24) and (28), that is, (30) is m.s. consistent; however, it is not m.s. convergent. In fact, if the multistep method(30) were m.s. convergent then by Lemma 2.1, the deterministic scheme

− 12 yn+2 +2yn+1 − 3

2 yn =hyn, yn =E[Xn] (31)

would be also convergent. It is easy to check that (31) is unstable because at least one the roots of the polynomial �() associatedwith (31) has modulus greater than one. To see this fact, let us compute the deterministic starting values y0 and y1 from theexpectation of the theoretical solution process X(t)=X0 et of (29), see [15, Chapter 6], that is, y0 =E[X0]=1, y1 =E[X1]=ehE[X0]=eh.Table I shows an evident divergence of scheme (31). Previous example exhibits that if the multistep method (13) is m.s. convergent,then every root of the polynomial �() given by (20) has to be modulus less than one.

Definition 4.3A random multistep linear method (13) is said to be m.s. stable if the modulus of no root of the associate polynomial �() given by(20) exceeds one and the roots of modulus one be simple.

5. On the m.s. convergence of multistep methods

The aim of this section is to establish sufficient conditions to guarantee the m.s. convergence of random linear multistep methodof the form (13), that is, we are interested in proving

limh→0

en =0, en =Xn −X(tn), (32)


67


where tn = t0 +nh, n=0, 1, 2,. . . , Xn is the m.s. solution of scheme (13) and X(tn) is the m.s. solution of (1) at t = tn. In order to getthis goal, let us assume the following hypotheses:

A1. f is randomly bounded time uniformly continuous, that is, f satisfies definition 2.2.A2. f satisfies the m.s. Lipschitz condition, that is,

‖f (X, t)−f (Y, t)‖�K(t)‖X −Y‖ where

∫ t1

t0

K(t) dt<∞.

A3. The scheme (13) is m.s. consistent and m.s. stable.A4. The starting values Xi =Xi(h), i=0, 1, 2,. . . , k−1, satisfy

limh→0

�(h)= limh→0

maxi=0,1,. . .,k−1

‖Xi(h)−X(t0 + ih)‖=0. (33)

Before starting the m.s. convergence proof, we require to demonstrate auxiliary results that will be useful to get a bound of them.s. error. Let us start by defining the deterministic coefficients l , l =0, 1, 2,. . . , as follows:

1

�k +�k−1+·· ·+�0k= 0 + 1+·· · . (34)

As in the deterministic case, a consequence of the m.s. stability is that, see [16, p. 242],

�= supl=0,1,2,. . .

| l|<∞. (35)

Multiplying both sides of (34) by �k +�k−1+·· ·+�0k and comparing coefficients in the resulting expansion in powers of , itturns out

�k l +�k−1 l−1 +·· ·+�0 l−k ={

1 if l =0,

0 if l =1, 2,. . . ,(36)

where it is assumed that l =0 for l<0.Consider the random linear non-homogeneous difference equation

k∑j=0

�jZj+m =hk∑

j=0�jgj+m +�m, m=0, 1, 2,. . . , (37)

where

gp = f (Xp, tp)−f (X(tp), tp), p=0, 1, 2,. . . ,

Zp and �m are 2-r.v.’s, k is a fixed positive integer and �j , �j are real numbers.

Lemma 5.1Let �() be the polynomial given by (20) that satisfies the stability condition and B∗, � and � be non-negative constants such that

k∑j=0

|�j|�B∗, |�k|��, ‖�m‖��, m=0, 1, 2,. . . , (38)

and let h be a positive number such that 0�h<|�k|(K�)−1 where K is the Lipschitz constant of f . Then every solution s.p. of (37) forwhich

‖Zi‖�Z, Z>0, i=0, 1,. . . , k−1, (39)

satisfies

‖Zn‖�K∗ enhL∗, n=0, 1,. . . , N, (40)

where

L∗ = �∗B∗K, K∗ =�∗(N�+AZk), (41)

A =k∑

j=0|�j|, �∗ = �

1−h|�k|−1�K. (42)

The proof of this result follows an analogous way as in the deterministic case, see [16, p. 243] but as in general the Banach inequality‖XY‖�‖X‖‖Y‖ does not hold for X , Y 2-r.v.’s, we have adapted the scheme handled in the above reference to the form (37), wherethere are not two random factors involved.

68



Now we are ready to prove the main result, which is the m.s. convergence of random multistep method (13), that is, condition(32). In order to do that, one requires first to obtain a bound of ‖L(X(t); h)‖, where L denotes the difference operator defined by(19). To begin with let us introduce

�()= sup|t−t∗|�

‖f (X(t), t)−f (X(t∗), t∗)‖, >0, (43)

then, by considering (1) into (43) and lemma 2.4, one gets

X(tn + ih)= X(tn)+ �i�(ih), i=0, 1,. . . , k, (44)

where �i is a 2-r.v.

�i =

⎧⎪⎨⎪⎩

X(tn + ih)− X(tn)

�(ih)if i=1,. . . , k, h>0

0 if i=0.

(45)

Note that by (45) and definition (43) one gets ‖�i‖�1, i=0, 1,. . . , k. Furthermore, by the fundamental theorem of m.s. integral calculus(see, [15, p. 104]) it follows that

X(tn + ih)=X(tn)+∫ tn+ih

tn

X(u) du, i=0, 1,. . . , k. (46)

This expression can be written in the equivalent form

X(tn + ih)=X(tn)+h

∫ i

0X(t+ph) dp, t = tn, (47)

as is easy to check by introducing the change of variable u= tn +ph, with 0�p�i, p a real number.Following the same argument that for (44), we again write

X(tn +ph)= X(tn)+ �p�(ph), 0�p�i, (48)

and by (47), one gets

X(tn + ih)=X(tn)+ ihX(tn)+h

∫ i

0�p�( ph) dp. (49)

On the other hand by considering (49) and (44), the operator L defined by (19) can be expressed at t = tn as follows:

L[X(t); h]=k∑

j=0�j

(X(t)+ jhX(t)+h

∫ j

0�p�( ph) dp

)−h

k∑j=0

�j(X(t)+ �j�( jh)). (50)

Since L is assumed to be m.s. consistent, from (24), (28) and (50), one gets

L[X(t); h]=h

{(k∑

j=0�j

)(∫ j

0�p�( ph) dp

)−

k∑j=0

�j�j�( jh)

}. (51)

Note that by the convexity inequality for m.s. Riemann integrals (see, [15, p. 102]), we can write∥∥∥∥∥k∑

j=0�j

(∫ j

0�p�( ph) dp

)∥∥∥∥∥�k∑

j=0|�j|

∫ j

0‖�p‖�( ph) dp�

k∑j=0

|�j|∫ j

0�( ph) dp��(kh)

k∑j=0

j|�j|, (52)

where we have used that ‖�p‖�1 and � is an increasing function. Then by lemma 2.4, one can assume the existence of a 2-r.v. �such that

k∑j=0

�j

(∫ j

0�p�(ph) dp

)= ��(kh)

k∑j=0

j|�j|, (53)

where ‖�‖�1. In analogous way, one gets∥∥∥∥∥k∑

j=0�j�j�(jh)

∥∥∥∥∥�k∑

j=0|�j|‖�j‖�(jh)��(kh)

k∑j=0

|�j|, (54)

and

k∑j=0

�j�j�(jh)= ��(kh)k∑

j=0|�j|, (55)

where ‖�‖�1.


69


Now we are ready for bounding the operator L. In fact, considering (53) and (55), expression (51) can be written as follows:

L[X(t); h]=h�(kh)

{�

k∑j=0

j|�j|− �k∑

j=0|�j|}

. (56)

Then taking norms one gets

‖L[X(t); h]‖�Ch�(kh), (57)

where

C =k∑

j=0(j|�j|+|�j|).

With the purpose to obtain an equation in terms of the errors, by subtracting L[X(tn); h] from the corresponding relationship of thelinear multistep method

k∑j=0

�jXn+j −hk∑

j=0�jfn+j =0,

satisfied by Xn, then writing en =Xn −X(tn), gn = f (Xn, tn)−f (X(tn), tn), n=0, 1,. . ., considering (52), (19) and Lemma 2.4, it follows that

k∑j=0

�jen+j −hk∑

j=0�jgn+j =C��(kh)h,

where ‖�‖�1. Next, applying Lemma 5.1 with Zn =en, �n =C�n�(kh)h, N= (tn −t0) / h, |�j|�� for j=0, 1,. . . , k, it follows that

‖en‖��∗[(tn −t0)C�(kh)+Ak�(h)] e(tn−t0)L∗, (58)

where �∗ and L∗ are given by (41) and (42), respectively. Note that by condition A2 one gets

�(kh) = sup|t−t∗|�kh

‖f (X(t), t)−f (X(t∗, t∗))‖

� sup|t−t∗|�kh

‖f (X(t), t)−f (X(t), t∗))‖+ sup|t−t∗|�kh

‖f (X(t), t∗)−f (X(t∗), t∗))‖

� sup|t−t∗|�kh

‖f (X(t), t)−f (X(t), t∗))‖+K(t∗) sup|t−t∗|�kh

‖X(t)−X(t∗)‖, (59)

then by condition A1, first term of the right-hand side of (59) tends to zero and by m.s. continuity of X(t), second term also tends tozero as h→0. Therefore, �(kh)→0 as h→0. On the other hand, by (33) one gets �(h)→0 as h→0. Thus, from (58) it follows that

limh→0

en =0.

Hence, the following result has been established.

Theorem 5.2Consider the random differential initial value problem (1). Let us assume that f satisfies hypotheses A1, A2 and the scheme givenby (19) satisfies hypotheses A3 and A4. Then the random linear multistep method defined by (13) is m.s. convergent.

If the right-hand side of (1) has m.s continuous derivatives of order p, then one can establish using the fundamental theorem ofm.s. calculus, see [15, p. 104], that for sufficiently accurate random starting values, the m.s. error en =Xn −X(tn) is of order O(hp).


This section contains two stochastic models where random multistep methods are applied. Both are selected as test examples inorder to measure the accuracy of random multistep methods because the solution stochastic processes are available. As the matrixcase of random multistep method can be proved following the same idea of the scalar case, we use a matrix random multistepmethod to approximate a system of random differential equations in the second example.

Example 6.1Consider the random initial value problem

X(t)=AX(t)+B(t), X(0)=X0, 0�t�1, (60)

70



where A is a beta 2-r.v. with parameters �=2 and �=1, A∼Be(2, 1); B(t) is the standard Brownian motion and the initial condition X0follows an exponential distribution with parameter �=1. Let us assume that for each t ∈ [0, 1], A, X0 and B(t) are pairwise independent2-r.v.’s. From [4], the theoretical solution is given by

X(t)=eAtX0 +∫ t

0eA(t−s)B(s) ds, 0�t�1, (61)

where the integral has to be interpreted in the m.s. Riemann sense. As E[B(t)]=0 and E[X0]=1, the expectation of X(t) has the form

E[X(t)]=E[eAt]E[X0]+∫ t

0E[eA(t−s)]E[B(s)] ds=

∫ 1

02A eAt dA= 2+2et(−1+t)

t2. (62)

Now, taking into account that E[(X0)2]=2, E[B(s)B(r)]=min(r, s), one gets

E[(X(t))2] = E[e2At]E[(X0)2]+2E[X0]

∫ t

0E[eA(2t−s)]E[B(s)] ds+

∫ t

0

∫ t

0E[eA(2t−r−s)]E[B(s)B(r)] dr ds

= 2

∫ 1

02A e2At dA+

∫ t

0

∫ s

0

(∫ 1

02A eA(2t−s−r) dA

)r dr ds+

∫ t

0

∫ t

s

(∫ 1

02AeA(2t−s−r) dA

)s dr ds. (63)

To approximate the theoretical solution s.p. (61), we will use the random Adams–Bashforth method, which is given by

Xn+2 −Xn+1 = h

2[3fn+1 −fn]. (64)

In this case, the above method reads as

Xn+2 =[

3

2hA+1

]Xn+1 − h

2AXn + h

2[3B(tn+1)−B(tn)], n�0,

X0 = X(0), X1 = (1+Ah)X0,

where the random starting values were obtained from the random Euler method by considering that B(0)=0 from the definition ofthe standard Brownian motion. It is straightforward to check that the random scheme (64) satisfies condition A3. Note that X0 andX1 satisfy condition A4. Note by Example 2.3 that f (X, t)=AX +B(t) satisfies condition A1. Considering that A has bounded realizationsbecause it is a beta 2-r.v., it is easy to check condition A2. In Figures 1 and 2, the numerical and theoretical expectation as well asnumerical and theoretical variance are compared. They show that the approximations are closer to theoretical values even for notso small values of h.

Figures 3 and 4 show a comparison between the random Euler method and the random Adams method. Observe that for theEuler method, we need to take smaller values of h than the random Adams–Bashforth method for getting a good approximationof the first moments.

Example 6.2Consider the problem of determining the effect on an earthbound structure of the earthquake-type disturbance. Let us assume thatthe structure is at rest at t =0, and let X(t)>0, t�0, be the relative horizontal displacement of the roof with respect to the ground.Then, based upon an idealized linear model, the relative displacement X(t) is governed by

X(t)+2�0X(t)+�20X(t)=−Y(t), t�0,

X(0)=0, X(0)=0,

}(65)

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

t

Exp

ecta

tion

E[X(t)]h=1/20h=1/10

Figure 1. Expectations of X(t) and Xn for Example 6.1.


71


0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

5.5

t

Var

ianc

e

Var[X(t)]h=1/10h=1/20

Figure 2. Variances of X(t) and Xn for Example 6.1.

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

t

Exp

ecta

tion

E[X(t)]AdamsEuler

Figure 3. Expectations of X(t) and Xn for Example 6.1 for h= 110 .

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

5.5

t

Var

ianc

e

Var[X(t)]AdamsEuler

Figure 4. Variances of X(t) and Xn for Example 6.1 for h= 120 .

where Y(t) is the 2-s.p. given by

Y(t)=m∑

j=1taj e−�j t cos(�jt+�j), t�0, (66)

72



and aj , �j , �j and are positive real numbers and �j are pairwise independent 2-r.v.’s uniformly distributed on [0, 2�]. The m.s.solution of the problem (65) has the form, see [15, p. 165],

X(t)=−∫ t

0h(t−z)Y(z) dz, (67)

with <1, where h(t) is the impulse response

h(t)= 1

�0e−�0t sin(�0t), t�0, �0 =�0

√1−2. (68)

The expectation and the auto-correlation functions of the 2-s.p. Y(t) are given by

E[Y(t)]=0, t�0, (69)

and

�Y (u, v)=E[Y(u)Y(v)]= 1

2

m∑j=1

uva2j e−�j(u+v) cos((u−v)�j), u, v�0. (70)

Hence from Equations (67)–(70), it follows that

E[X(t)]=0, t�0,

and

Var[X(t)]=E[X(t)X(t)]=∫ t

0

∫ t

0h(t−u)h(t−v)�Y (u, v)du dv, t�0. (71)

In order to apply the Mid point random scheme, which is given by

�Xn+2 = �Xn +2hf (�Xn+1, tn+1), n=0, 1, 2,. . . , (72)

to the problem (65) in the interval [0, 1], we shall transform this second-order differential equation into a system of first-orderdifferential equations. Let X1(t)=X(t), X2(t)= X(t), then the vector–matrix form of Equation (65) is

�X(t)=A(t)�X(t)+ �G(t), (73)

where

�X =[

X1

X2

], A(t)=A=

[0 1

−�20 −2�0

], �G(t)=

[0

−Y(t)

](74)

and (�X(0))T = (�0)T = [0, 0]. Note that for obtaining an approximation using the Mid point scheme, previously one needs to apply aone-step method, in our case we choose the random Euler method that has the form

�Xn+1 = �Xn +hf (�Xn, tn)= �Xn +h(A�Xn + �G(tn)), n=0, 1, 2,. . . . (75)

Thus, taking into account that �X0 = �X(0)=�0 and �G(t0)= �G(0)=�0 (note that from (66), Y(0)=0) in (75), one obtains that �X1 =�0. Now,as E[Y(t)]=0, from (74) one gets E[�G(t)]=�0, and from (72)–(73) it follows that

E[�Xn+2]=E[�Xn]+2h(AE[�Xn+1]+E[�G(tn+1)])=E[�Xn]+2hAE[�Xn+1], n=0, 1, 2,. . . . (76)

As a consequence of (76) and E[�X0]=E[�X1]=�0, one gets E[�Xn]=�0 for all n=0, 1, 2,. . . . Now we are interested in computing thecovariance matrix of the nth approximation given by the random vector �Xn. In order to reach this goal, first one requires to computethe cross-covariance matrix of �Xn and �G(tm), denoted by ��Xn,�G(tm). Taking into account that E[�G(t)]=�0, from (6)–(7) and (72)–(73)one gets

��Xn,�G(tm) = E[�Xn(�G(tm))T]

= E[{�Xn−2 +2hA�Xn−1 +2h�G(tn−1)}(�G(tm))T]

= ��Xn−2 ,�G(tm) +2hA��Xn−1 ,�G(tm) +2h��G(tn−1),�G(tm), n=2, 3,. . . , m=0, 1,. . . (77)


73


0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

t

Var

ianc

e

Mid Point h=1/10Mid Point h=1/20Euler h=1/20Euler h=1/40Theoretical variance

Figure 5. Variances of Xn for Example 6.2, using the random Mid point and Euler methods.

This recurrence starts from the cross-covariance matrix ��G(tn),�G(tm) of the data �G(t) together with the square null matrices of size 2:

��X0 ,�G(tm) =��X1 ,�G(tm) =0 since �X0 = �X1 =�0. Then from (6)–(7), (9) and (72)–(73), the cross-covariance of �Xn and �Xm are computed as

follows:

��Xn,�Xm= E[�Xn(�Xm)T]

= ��Xn−2 ,�Xm−2+2h��Xn−2 ,�Xm−1

(A)T+2h��Xn−2 ,�G(tm−1) +2hA��Xn−1 ,�Xm−2+4h2A��Xn−1 ,�Xm−1

(A)T

+4h2A��Xn−1 ,�G(tm−1) +2h��G(tn−1),�Xm−2+4h2��G(tn−1),�Xm−1

(A)T+4h2��G(tn−1),�G(tm−1), n, m=2, 3,. . . , (78)

where the terms ��G(tn−1),�Xm−2and ��G(tn−1),�Xm−1

can be computed directly from (77) taking into account that by property (8) onegets

��G(tn−1),�Xm−2= (��Xm−2 ,�G(tn−1))

T, ��G(tn−1),�Xm−1= (��Xm−1 ,�G(tn−1))

T.

Note that if ��Xn=��Xn,�Xn

= (Vij)2×2 denotes the square matrix of size 2 given by (78), then V11 is the approximating variance of

the solution s.p. of Equation (65) obtained from the random Mid point method. Figure 5 compares the approximate variance usingthe random Mid point and the Euler method for different values of h and the theoretical variance, Var[X(tn)], given by (71), takinginto (65) with the values =0.05, a0 =1, �0 =1, and �j =1, aj =1, �j = ( 1

2 )j , 1�j�20=m, in Y(t), given by (66). It illustrates that thenumerical values from the Mid point method are closer to the theoretical variance than the Euler method when h decreases.

7. Conclusions

This paper deals with the construction and the numerical analysis of m.s. solutions of random differential equations by means ofrandom multistep numerical methods. These methods provide higher accuracy order than the random Euler method introduced in[5, 6]. Several examples illustrate this fact as well as numerical simulations enclosed in the paper.

The paper also shows that in analogous way to the deterministic case, for random initial value problems, random multistepnumerical methods can be used and analyzed versus the sample approach based upon taking realizations of the problem andfurther management of them. The sample approach presents serious difficulties overcoming the lack of information about howmany realizations must be taken into account in order to get reliable results, see [15, Appendix A], [17], for details. Moreover,the sample approach imposes restrictive conditions on the trajectories associated with the random differential equations, such asdifferentiability that often does not take place in applications, as long as m.s. approach allows us to overcome these difficulties bytaking advantage of its powerful methods and properties.

Acknowledgements

Thanks to the anonymous reviewer whose comments greatly enhanced the paper. This work has been partially supported bythe Spanish M.E.C. and FEDER grants MTM2009-08587 and TRA2007–68006–C02–02, the Universidad Politécnica de Valencia grantPAID-06-09 (ref. 2588) and Mexican Conacyt.

74



References1. MacCluer CR. Industrial Mathematics. Prentice-Hall: NJ, 2000.2. Chilès JP, Delfiner P. Geostatistics. J. Wiley-Interscience: New York, 1999.3. Cortés JC, Jódar L, Villafuerte L. Random linear–quadratic mathematical models: computing explicit solutions and applications. Mathematics

and Computers in Simulation 79(7):2076--2090. DOI: 10.1016/j.matcom.2008.11.008.4. Villafuerte L, Braumann CA, Cortés JC, Jódar L. Random differential operational calculus: theory and applications. Computers and Mathematics

with Applications 2010; 59(1):115--125. DOI: 10.1016/j.camwa.2009.08.061.5. Cortés JC, Jódar L, Villafuerte L. Mean square solutions of random differential equations: facts and posibilities. Computers and Mathematics with

Applications 2007; 53(7):1098--1106. DOI: 10.1016/j.camwa.2006.05.030.6. Cortés JC, Jódar L, Villafuerte L. Numerical solution of random differential equations: a mean square approach. Mathematical and Computer

Modelling 2007; 45(7–8):757--765. DOI: 10.1016/j.mcm.2006.07.017.7. Cortés JC, Jódar L, Villafuerte L, Villanueva RJ. Computing mean square approximations of random diffusion models with source term. Computers

and Mathematics in Simulation 2007; 76(1–3):44--48. DOI: 10.1016/j.matcom.2007.01.020.8. Brugnano L, Burrage K, Burrage PM. Adams-type methods for the numerical solution of stochastic ordinary differential equations. BIT Numerical

Mathematics 2000; 40(3):451--470. DOI: 10.1023/A:1022363612387.9. Denk G, Schäffler S. Adams methods for the efficient solution of stochastic differential equations with additive noise. Computing 1997;

59(2):153--167. DOI: 10.1007/BF02684477.10. Ewald BD, Témam R. Numerical analysis of stochastic schemes in geophysics. SIAM Journal on Numerical Analysis 2005; 42(6):2257--2276.

DOI: 10.1137/S0036142902418333.11. Kloeden PE, Platen E. Numerical Solution of Stochastic Equations. Springer: New York, 1999.12. Talay D, Tubaro L. Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Analysis and

Applications 1990; 8(4):483--509.13. Ryashko LB, Schurz H. Mean square stability analysis of some linear stochastic systems. Dynamic Systems and Applications 1997; 6(2):165--190.14. Schurz H. The invariance of asymptotic laws of linear stochastic systems under distretization. Zeitschrift für Angewandte Mathematik und Mechanik

1999; 79(6):375--382. DOI: 10.1002/(SICI)1521–4001.15. Soong TT. Random Differential Equations in Science and Engineering. Academic Press: New York, 1973.16. Henrici P. Discrete Variable Methods in Ordinary Differential Equations. Wiley: New York, 1962.17. Boyce WE. In Probabilistic Methods in Applied Mathematics I, Bharucha-Reid AT (ed.). Academic Press: New York, 1968.


75

Documents

Mathematical Methods in the Applied Sciences 2011 Vol34 Issue1