ON COMPLEX VARIABLE METHOD IN FINITE ELASTICITY

ADE AKINOLA

Abstract. We highlight the alternative presentation of the Cauchy-Riemannconditions for the analyticity of a complex variable function and considerplane equilibrium problem for an elastic transversely isotropic layer, in fi-nite deformation. We state the fundamental problems and consider tractionboundary value problem, as an example of fundamental problem-one. Asimple solution of ”Lame’s problem” for an infinite layer is obtained. Theprofile of the deformed contour is given; and this depends on the order ofthe term used in the power series specification for the complex potentialand on the material constants of the medium.

AMS Mathematics Subject Classification: 73G05.Key words and phrases: Analyticity, complex potentials, fundamental prob-lems, finite deformation, anisotropic parameter.

1. Introduction

In the last three or so decades now, in continuum mechanics, specifically insolid mechanics, it is modern to apply finite element method to solving prob-lems. This is particularly so in the cases where analytical solution is not readilyattainable. It is true that robust computational methods and correspondingpackages abound everywhere today. However, it is no less true that, when an-alytical method for a given problem exists, then the analytical solution is moreinformative than any numerical solution. So, sustained efforts is always on-goingto unravel and develope suitable analytical method of solution in continuum me-chanics. A ready avenue to obtaining analytical solutions in plane problems isto invoke the complex variable method. An example in this direction is the so-called theory of first and the second fundamental problems [1] in plane problemsof elasticity.

In fact, the universal compatibility equation of Levy-Maurice, that providedthe theoretical basis and hence opened the floodgate of the photoelastic method

Applied Mathematics, 2009, 1, 1-16

of analysis in solid mechanics, has revealed the potency of plane problems inelasticity. In the same vein, the pioneering works of Kolosov, Muskhelishvili andStevenson [2] have endowed the plane problem of elasticity with the effectivetools of complex analysis. Later, this tool of complex analysis has come to beextended to the investigation of phenomena in the theory of finite elasticity [3,4].Ogden and Isherwood [5] considered some finite plane strain problems using thestress function method. In the present work, a complex potential approach isadopted.

Specifically, if an expression could be shown to satisfy a Laplace equation,hence analytic in the appropriate sense, then it can be expressed in a pertinentconvergent series; which often admits some form of analytical solution and otherindepth analysis. In fact, on the basis of the Cauchy integral theorem/formular,by the known value of an analytic function on a contour of a region we canestablish that function in the whole plane region. The treasure in this for theplane problem of elasticity is that if we can find or specify, for instance, theresultant force vector and the resultant moment vector on the contour of aplane region then, by the Cauchy integral formular, any plane problem could beassumed solved. Consequently, most problems are reduced to finding the twoanalytic functions in the domain where solutions are sought.

The question then arises, how do we find these analytical functions in the caseof finite deformations? Moreover, an explicit and tractable universal expressionfor the energy function is not often known in finite elasticity. Here, on the basisof the semi-linear material of John [4,6], we give the general relations for thestatic, plane, finite deformation problem of elasticity for a transversely isotropicbody.

Using the parameter of anisotropy, we carry out the anisotropic expansionof pertinent state variables and reduce the initial-boundary value problem to arecurrence system of Laplace and Poisson equations. For the first system wegive the complex potentials and the fundamental problems, which are nonlineardue to the consideration of finite deformations. An example traction boundaryvalue problem is then considered for an infinite transversely isotropic layer, onthe basis of the fundamental problem-one.

PROBLEM SETTING

Analyticity of a Complex Variable Expression

We recall that the analyticity of a complex variable function f(z) = u(x, y)+iv(x, y), i ≡

√−1, in the finite plane C, is guaranteed by the Cauchy-Riemann

equations:

∂u

∂x=∂v

∂u,

∂u

∂y= −∂v

∂x. (1.1)

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An equivalent expression to (1.1), through the complex variable z = x + iyand its conjugate z = x− iy, is also given in the literature as

∂f

∂z= 0. (1.2)

In this paper, we shall adopt the presentation in (1.2).

Geometry of Deformation

Let Ω be a transverse isotropic medium in three dimensional euclidean spaceE3. We look at the equilibrum state of Ω in plane finite deformation. Thedeformation of Ω is given by specifying the position vector ~X of a particle priorto deformation in the initial (reference) configuration Ωo with the boundary Σo

and its orientation normal unit vector ~N and the position vector ~x in the currentconfiguration Ω with the boundary Σ and its orientation normal vector ~n :

~X = X1 ~e1 +X2 ~e2 +X3 ~e3, (1.3)

~x = x1 ~e1 + x2 ~e2 + x3 ~e3, (1.4)

such that for plane deformation:

xα = xα(X1, X2), x3 = lX3, α = 1, 2, (1.5)

where Xm, xm are the material coordinates in Ωo and Ω respectively; ~em is theorthonormal basis; m = 1, 2, 3; l is any real constant.

Let the deformation gradient be the tensor-gradient of the position vector ~xin Ω taken with respect to the initial configuration Ωo. That is, applying theoperator of gradient-vector in the reference configuration,

o

∇≡ ~em∂

∂Xm , on theposition vector ~x in the current configuration, we obtain the tensor-gradient (ordeformation gradient):

A ≡o

∇~x = ~eα ⊗ ~eβ∂xβ

∂Xα+ ~e3 ⊗ ~e3l. (1.6)

We also consider the rotation tensor of the medium:

R = V−1 · A = I cosχ+ (1 − cosχ)~e3 ⊗ ~e3 − ~e3 × I sinχ (1.7)

where, V, such that V2 = A · AT , is the symmetric left stretch tensor, arisingfrom the polar decomposition of the deformation gradient, A = V · R; I =~e1 ⊗ ~e1 + ~e2 ⊗ ~e2 + ~e3 ⊗ ~e3 is the unit tensor in E3, ~e3 × I is the cross productof ~e3 and I and

cosχ =1q

(∂x1

∂X1+∂x2

∂X2

), sinχ =

1q

(∂x2

∂X1− ∂x1

∂X2

),

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q =

√(∂x1

∂X1+∂x2

∂X2

)2

+(∂x2

∂X1− ∂x1

∂X2

)2

. (1.8)

For any vectors (or tensors, as may be appropriate) φ and ψ, here and elsewhere,we denote their dot product, double dot product and cross product respectivelyby φ · ψ, φ · ·ψ and φ× ψ.

Static Equation for Transversely Isotropic Material

We look at the equilibrium of Ω in a state of plane finite deformation. Forthis, we first introduce the energy function for an isotropic semi-linear materialin finite deformation, proposed by F. John [6]:

W = µS2 + 1/2λS21 , (1.9)

where S1 and S2 are the invariants S1 = I · ·(V − I), S2 = (V− I) · ·(V − I). λand µ are the Lame constants. Then, on the basis of (1.9) we consider an energyfunction [7,8] for a transversely isotropic semi-linear material:

W = λ2S2 + 1/2λ1S21 + λ0S0, (1.10)

where S0 = ~c ·V2 ·~c is an additional invariant of deformation, due to anisotropy.~c is the unit vector characterizing the direction of anisotropy (i.e., the axis oftransverse isotropy). λ0, λ1, λ2 are the material constants. In the case of arandomly unidirectional fibre reinforced composite or a lamina composite thematerial constants represent the effective moduli [8,9,10]

λ2 = 〈µ〉, λ1 = 〈λ〉+

⟨λ

(λ+2µ)

⟩2

⟨1

(λ+2µ)

⟩ −⟨

λ2

(λ+ 2µ)

⟩, λ3 =

1〈 1

µ 〉, λo = 2(λ3−λ2).

(1.11)We note that in the case of degeneracy into isotropy, the energy function (1.10)automatically reduces to the energy (1.9) and, accordingly, for the effective mod-uli λ3, λ2, λ1, we have

λ3 = λ2 = µ, λ1 = λ, λo = 0. (1.11)′

For any finite function ϕ(~ξ) ∈ Ω, 〈ϕ〉 denotes its geometric average over Ω〈ϕ〉 = 1

|Ω|∫ΩϕdΩ, where |Ω| denotes the volume of Ω.

Now, for hyperelasticity, we take the Frechet derivative [4,11] of the energywith respect to deformation gradient A and obtain the Piola stress tensor P, towhich it is energy conjugate:

P =∂W

∂A= 2λ2A + (λ1S1 − 2λ2)R + 2λ0~c⊗ ~c ·A. (1.12)

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In the absence of body force, we have the equilibrium equation

o

∇ ·P = 0

and the accompanying boundary condition

~fdΣ = ~N · PdΣo,

where dΣ is the element of the boundary surface area in the current configurationon which a prescribed force per unit surface area ~f acts while dΣo is the elementof the boundary surface area in the reference configuration, with the outer unitnormal vector ~N = (N1, N2).

The component form of these relations is:

∂P 11

∂X1+∂P 21

∂X2= 0,

P 12

∂X1+∂P 22

∂X2= 0,

∂P 33

∂X3= 0 (1.13)

and

f1dΣdΣo

= N1P11 +N2P

21, f2dΣdΣo

= N1P12 +N2P

22, (1.13)′

where by the deformation (1.5), we have dΣ = ldsdX3 and dΣo = dSdX3 andthe surface area ratio dΣ

dΣocan be expressed through the corresponding arclengths

ds and dS as l dsdS .

2. Complex Variable Formulation

Complex Variable Representation of Equilibrium Equation

We now look at Ξ, a two-dimension cross-sectional region of the originalmedium Ω, as a subspace of the complex plane C. In place of the materialcoordinates X1, X2 and x1, x2, in the reference configuration and the currentconfiguration respectively, we introduce the complex variables:

Z = X1 + iX2 ∈ Ξo and z = x1 + ix2 ∈ Ξ, (2.1)

where i =√−1 is the imaginary unit. Then,

∂

∂Z=

12

(∂

∂X1− i

∂

∂X2

),

∂

∂Z=

12

(∂

∂X1+ i

∂

∂X2

)(2.2)

and

2∂z

∂Z=

(∂x1

∂X1+∂x2

∂X2

)+ i

(∂x2

∂X1− ∂x1

∂X2

),

or, in view of (1.6),

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∂z

∂Z=

12q exp (iχ) and exp (iχ) =

∂z

∂Z/

∣∣∣∣∂z

∂Z

∣∣∣∣ . (2.3)

The Piola stress tensor (1.12), the equilibrium equations (1.13) and the bound-ary conditions (1.13)′ respectively become

P 11 + iP 12 = φ(q)exp (iχ) + 2iλ2∂z

∂X2+ 2λo

(q exp (iχ) + i

∂z

∂X2

),

P 22 − iP 21 = φ(q)exp (iχ) − 2λ2∂z

∂X1, (2.4)

P 33 = P 33(X1, X2),

∂Φ(Z,Z)∂Z

= −2λ0∂2z

∂X1∂X1(2.5)

and

ifds

dS− dZ

dSΦ(Z,Z) + 4λ2

dz

dS= 2λ0

∂z

∂X1iN1, (2.5)′

where N = N1 + iN2; f = f1 + if2 is the complex form of the boundary force perunit present length; ds and dS are the arc elements in the current and referenceconfigurations respectively, and the yet to be established function

Φ(Z,Z) ≡ φ(q)exp (iχ), φ(q) = (λ1 + 2λ2)(q − 2) + 2λ2 + λ1(l − 1). (2.6)

Anisotropic Expansion of State Variables

We note that if Ξ were to be an isotropic medium, then the right-hand sidein (2.5) and (2.5)′ would vanish. This implies that λ0 is a true parameter ofanisotropy. So, we non-dimensionalize it and expand the dependent variables interms of

β =λ0

λ1 + 2λ2< 1. (2.7)

That is,

z = z0 + βz1 + β2z2 + β3z3 + ...,

Φ = Φ0(Z,Z) + βΦ1(Z,Z) + β2Φ2(Z,Z) + β3Φ3(Z,Z) + ..., (2.8)

~f = ~f0 + β ~f1 + β2 ~f2 + β3 ~f3 + ....

Putting (2.8) in (2.5) and (2.5)′ we obtain for the equilibrium equation and theboundary condition, respectively,

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∞∑

m=0

βmFm = 0, (2.9)

∞∑

m=0

βmPm = 0, (2.9)′

where

Fm ≡ ∂Φm

∂Z+ 2(λ1 + 2λ2)

∂2zm−1

∂X1∂X1,

Pm ≡ ifmds

dS− dZ

dSΦm(Z,Z) + 4λ2

dzm

dS− 2λ0

∂zm−1

∂X1iN1,

for m = 0, 1, 2, ..., z1 = 0.Now, setting every coefficient of the powers of β to zero in equations (2.9)

and (2.9)′ we obtain a recurrence system of equations, i.e., Fm = 0 and Pm =0;m = 0, 1, 2, . . . . The first equation in the recurrence system due to (2.9) ishomogeneous (i.e. zero right hand side), while each of the subsequent ones isnonhomogeneous with right hand sides depending recursively on the solution ofthe previous equation.

Thus, this much is the effect of anisotropy on the medium: and has in no wayinfluenced the fact or exposed the issue of finite deformation. The effect of finitedeformation is exposed in what follows.

Complex Potential and Nonlinear Fundamental Problems

We now deduce the Fundamental Problems, analogous to the Kolosov - Muskhe-lishvili first and second fundamental problems of infinitesimal elasticity, given interms of two complex potentials [1-4,12].

Let

z0 ≡ v = v1 + iv2, Φ0 ≡ F (Z,Z) and f0 ≡ h = h1 + ih2. (2.10)

Then, the first equation in the recurrence system due to (2.9) is:

∂F (Z,Z)∂Z

= 0, (2.11)

with the corresponding boundary condition from recurrence system (2.9)′

ihds

dS=dZ

dSF (Z,Z) − 4λ2

dv

dS, (2.11)′

where h ≡ f0 is the specified force per unit length of the current boundary con-tour. By the Cauchy-Riemann equations, i.e., (1.2), (2.11) implies that F (Z,Z)is an analytic function of the variable Z only, in the finite domain. It can thenbe written as a uniformly convergent series in the form

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F (Z,Z) ≡ F (Z) =∞∑

m=0

αmZm,

where αm are constants.Now, F (Z) is related to the material positions v(Z,Z) in the current plane

in a pertinent manner. In fact, setting l = 1 in (1.6) for convenience and using(2.3), (2.6) and (2.10), we write

F (Z) = φ0(q0) exp (iχ0), q0 = 2∣∣∣∣∂v

∂Z

∣∣∣∣ , exp (iχ0) =∂v

∂Z/

∣∣∣∣∂v

∂Z

∣∣∣∣ . (2.12)

Then,

F (Z) = 2(λ1 + 2λ2)∂v

∂Z− 2(λ1 + λ2)

F (Z)|F (Z)|

and

(λ1 + 2λ2)∂v

∂Z= λ2ϕ

′2(Z) + (λ1 + λ2)ϕ′(Z)ϕ′(Z)

, (2.13)

whereF (Z) ≡ 2λ2ϕ

′2(Z). (2.14)Following Goursat or Muskhelishvili [1,2], we integrate (2.13) and differentiate

the result with respect to Z to give

(λ1 + 2λ2)v(Z,Z) = λ2

∫ϕ′2(Z)dZ − (λ1 + λ2)

[ϕ(Z)ϕ′(Z)

+ ψ(Z)

](2.15)

and

−λ1 + 2λ2

λ1 + λ2

∂v

∂Z=ϕ(Z)ϕ′′(Z)ϕ′(Z)ϕ′(Z)

− ψ′(Z). (2.16)

Here, ψ(Z) is another analytic function that emerges as a result of integration,in the form of ψ(Z). Thus, in place of F (Z) we now have two analytic functions,called complex potentials, through which we can define any particle positionv(Z,Z) on the current plane Ξ.

Now, on any arc length S on the contour Γo of the reference configuration wehave:

dv

dS=∂v

∂Z

dZ

dS+∂v

∂Z

dZ

dS.

Putting (2.15), (2.16) in this, we obtain

−i(λ1 + 2λ2)dv

dS=

[λ2ϕ

′2(Z) + (λ1 + λ2)ϕ′(Z)ϕ′(Z)

]N

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−(λ1 + λ2)

[ϕ(Z)ϕ′′(Z)ϕ′(Z)ϕ′(Z)

+ ψ′(Z)

]N, (2.17)

where N = N1 + iN2 = −idZdS ; N = N1 − iN2 = idZ

dS .This now enables us to obtain boundary conditions pertinent for equilibrium.

By (1.13)’ we have

[ϕ′2(Z) − ϕ′(Z)

ϕ′(Z)

]N −

[ϕ(Z)ϕ′′(Z)ϕ′(Z)ϕ′(Z)

+ ψ′(Z)

]N =

λ1 + 2λ2

2λ2(λ1 + λ2)hds

dS. (2.18)

We note that on any contour Γo of the cross-section of the body, for anyfunction θ(Z,Z):

∫

Γo

θ(Z,Z)NdS =∫

Γo

(N1 − iN2)θ(Z,Z)dS =∫

Ξo

(∂

∂X1− i

∂

∂X2

)θ(Z,Z)dΞo

= 2∫

Ξo

∂θ(Z,Z)∂Z

dΞo

and ∫

Γo

θ(Z,Z)NdS =∫

Ξo

∂θ(Z,Z)∂Z

dΞo. (2.19)

Then, by (2.13), (2.14), (2.16), (2.19) the resultant force vectorH =∫Γh(x1, x2)ds

and the resultant moment M =∫Γ(x1h2−x2h1)ds = i

2

∫Γ(vh−vh)ds will vanish.

Now, let

Z = r exp(iθ),∂Z

∂θ= iZ,

∂Z

∂θ= −iZ,

∫ϕ′2(Z)dZ = I(Z), v′(Z,Z) =

∂v

∂θ, ho(Z,Z) =

k

2λ2

ds

dS(h1 + ih2), (2.20)

c ≡ λ2

λ1 + λ2= 1 − 2νo, k ≡ λ1 + 2λ2

λ1 + λ2= 1 + c = 2(1 − νo), νo ≡ 1

2λ1

λ1 + λ2.

Then, (2.18) becomes

I(Z) − ϕ(Z)ϕ′(Z)

+ ψ(Z) =ik

2λ2H(Z,Z) (2.21)

and by (2.17)

cI(Z) +ϕ(Z)ϕ′(Z)

− ψ(Z) = kv(Z,Z). (2.22)

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Thus, we have established two nonlinear fundamental boundary-value prob-lems, given respectively by (2.21) and (2.22), for plane finite deformation. Theseare analogous to those already known in the literature for the case of linear elas-ticity [1,2].

(i)Fundamental Problem I: Find the state of elastic equilibrium of a bodywhen the external load applied on the boundary Γ is specified by (2.21), i.e.,H(Z,Z) is given on the contour Γ.

(ii)Fundamental Problem II: find the state of elastic equilibrium for a bodywhose contour deforms from Γo into Γ, given by the position function (2.22),i.e., v(Z,Z) is given on the contour Γ.

This implies that the solution of plane problems of finite elasticity impingeson knowing ϕ(Z) and ψ(Z). These can be established by their values on theappropriate boundary contours, on the basis of the Cauchy integral theorem,due to their analyticity property.

Physical Components of Stress

It is desirable to express the physical components of stress in the currentconfiguration, since, in practice, it is in this form that they are often used. Forthis purpose, we make use of the expression connecting the Cauchy stress T andthe Piola stress P, noting that 0 < J ≡ | ∂xα

∂Xβ | = ∂v∂Z

∂v

∂Z− ∂v

∂Z

∂v∂Z :

T =1J

AT · P =1J

(∂xα

∂XβP βγ~eα ⊗ ~eγ + P 33~e3 ⊗ ~e3

);α, β, γ = 1, 2

and

t1 + t2 = ~e1 ·T · ~e1 + ~e2 · T · ~e2 =1J

∂xα

∂XβPαβ ,

t1 + t2 + 4λ2 =1Jq0φ(q0),

t2 − t1 − 2it12 = ~e2 ·T · ~e2 − ~e1 · T · ~e1 − i(~e1 ·T · ~e2 + ~e2 · T · ~e1)

=1J

[∂x2

∂XαPα2 − ∂x1

∂Xα− i(

∂x1

∂XαPα2 +

∂x2

∂XαPα1)

]

=1Jφ(q0) exp(iχ0)

(∂v

∂X1+ i

∂v

∂X2

)= −

4J

φ(q0)q0

∂v

∂Z

∂v

∂Z.

On the contour Γ we have,

tn + tns = 2λ2ϕ′2(Z)

∂v

∂s

∂Z

∂s− 2λ2 (2.23)

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where, tn, tns are the normal and tangential physical components of the Cauchystress, with respect to the contour Γ.

In the cases of simple specification of potentials, it is possible to obtain exactsolutions.

3. Example: An Exact Solution

Using the semi-inverse method, we can consider examples on the application ofthe exposed theory of the complex variable method to large deformation theoryof elasticity, with respect to two cases of exact boundary value problems. Thefirst case is the so-called simple solution, whereby the two complex potentials,ϕ(Z) and ψ(Z), are carefully specified from the onset. The second case, namedmain (or mixed) solution is implemented by specifying one of the potentials, sayϕ(Z), while the second, ψ(Z) is deduced from the problem vis-a-vis ϕ(Z). By(2.22) and (2.23) we then obtain the displacement and the physical stress fieldon every contour Γ of the current configuration, Ξ.

Specifically, we consider here a simple problem, by the fundamental problem-one.

Simple Solution for an Infinite Plane - ”Lame’s Problem”

Consider the equilibrium of an infinite transversely-isotropic plane; in thecurrent configuration having a unit circular hole, centered at the origin. Theaxis of transversal-isotropy is in the radial direction. We are then interestedin the displacement and stress field developed in the medium as a result of anapplied ”dead load”, σ1, on the boundary of the hole.

By the semi-inverse method, in the case of an infinite domain, let

ϕ(Z) = Z, ψ(Z) =b

Zm, (3.1)

where b is a real number and m is a positive integer.Now, by putting (3.1) in (2.22), we obtain the displacement field

v(Z,Z) = Z − k′Z−m

= r exp(iθ) − kr−m exp(imθ), Z ≡ r exp(iθ), k′ ≡ b

k.

(3.2)This implies that (3.1) transforms a circle of radius r = const. into a closed

curve R ≡ |v| = const. :

R2 = vv = r2 − 2k′r1−m cos(m− 1)θ + (k′r−m)2. (3.3)

With respect to stress, consider the following:

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Figure 1. Block partitioning for L

∂Z

∂θ= iZ,

∂v

∂θ= i(Z −mk′Z

−m),

∂v

∂θ= −i(Z −mk′Z−m) and

∂v

∂θ

∂Z

∂θ= r2 −mk′r1−mσ1−m,

Rθ ≡ ∂v

∂θ

∂v

∂θ= r2 − 2mk′r1−m cos(m− 1)θ + (mk′r−m)2. (3.4)

By (2.23) we obtain

tn + itns

= 2λ2[−1 +R−2θ (r2 −mk′r1−mσ1−m)]

= 2λ2−1 +R−2θ r2 −mk′r1−m[cos(1 −m)θ + i sin(1 −m)θ]

= 2mk′λ2R−2θ [r1−m cos(1 −m)θ −mk′r−2m] − ir1−m sin(1 −m)θ.

This gives

tn = 2mk′λ2R−2θ [r1−m cos(1 −m)θ −mk′r−2m],

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Figure 2. Block partitioning for L

tns = −2mk′λ2R−2θ r1−m sin(1 −m)θ. (3.5)

When m = 1, from (3.2), (3.3) and (3.5) we obtain

v(Z,Z) = Z − k′

Z, R = (r − k′

r)1/2, tn = 2k′λ2(r2 − k′)−1, tns = 0. (3.6)

That is, the only non-vanishing components for the displacement and stressare the normal ones. What is more, the expression (3.6)1 for the particle posi-tion in the current configuration, where m = 1, is similar to the correspondingsolutions (6.8), (6.24) obtained in [5]. When m > 1 in (3.2), the nonlinear effectbecomes obvious. On the other hand, the situation m = 1 corresponds to Lame’sproblem of classical elasticity:

un(r) = −k′

r; tn(r) =

2k′λ2

r2 − k′,

the only unknown constant k′ is found from the loading condition, on the bound-ary: tn(r)|r=1 = σ1. Then, k′ = σ1

2λ2+σ1, and the displacement will have the

expression:

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Figure 3. Block partitioning for L

un = − σ1

2λ2 + σ1

1r.

This gives pertinent constraint on the character and magnitude of the admissibleboundary load σ1.

The variation of the radial displacement against the dimensionless boundarystress σd ≡ σ1

2λ2, for a concentric cylindrical composite layer of unit internal

radius, is shown in figure 1. The transverse isotropy is in the radial direction;above the horizontal-axis (i.e. the radial displacement axis) the cross-section ofthe medium is in compression while below, it is under internal pressure. Forσd > −1 we obtain a stress-displacement relation similar to the Lame’s problemof classical elasticity: the theoretical dashed curve below the asymptotic lineσd = −1 is not countenanced by the small deformation theory.

Based on equations (3.1) and (3.2), the specific type of curve described by(3.3) depends on the numerical values of k′ andm. So also is the stress field givenby (3.5). Consequently, for the particle positions given by (3.3) , we identify thefollowing situations: (i) m = 1 gives a circle, as also obtained in the linear theory,(ii) m = 2 corresponds to a Pascal curve, (iii) m = 3 describes an ellipse, (iv) m

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= 4 approximates a triangle etc. Figure 2. shows the shape of these curves inthe current configuration for different combinations of values of m and k′.

5. Conclusion

The application of the complex variable method to obtain the fundamentalproblems one and two for plane equilibrium problem, in the case of anisotropicfinite elasticity, has been carried out. The discussion here-in shows that therelevance and potency of the complex variable method in plane problems remainas valid as ever, and as when it was first introduced by Kolosov-Muskhelishvili-Stevensen [2,3] at about the begining of this century; even, in the case of finitedeformations and anisotropy. All the relations obtained here, due to finite defor-mation, also remain true for the case of isotropy/homogeneity, when the effectiveconstants take the corresponding expressions given in (1.11)’. In fact, (2.11) isvalid independent of (1.11)’; since material constants do not feature there.

References

1. A. A. El-Bary, The first and the second fundamental problems for an elastic infinite platewith holes, Korean J. Comput. & Appl. Math. Vol. 8(2001), No.3, 666-675.

2. N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, P.Noordhoff Ltd Groningen-Holland, 1953.

3. A. E. Green and W. Zerna, Theoretical Elasticity, Oxford University Press, London, 1968.4. A. I. Lurie, Nonlinear Elasticity, Nauka Publishers, Moscow, 1980 (in Russian).5. R. W. Ogden and D. A. Isherwood, Solution of Some Finite Plane-Strain Problems for

Compressible Elastic Solids, Q. J. Mech. Appl Math. 31(2), (1978) 219 -249.6. F. John, Plane Strain Problem for a Perfectly Elastic Material of Harmonic Type, Com-

mun. Pure and Appl. Math. 13(2), (1978) 239-296.7. A. Akinola, On Interacting Longitudinal-Shear Waves in Large Elastic Deformation of a

Composite Laminate, International Journal of Nonlinear Mechanics, 34(2), (1999) 405-414.8. A. Akinola, Complex Potentials in Plane Problems of Finite Elasticity, Congress of the

International Union of Theoretical and Applied Mechanics, ICTAM2000 ISSN 0073-5264,No950 (2000) pp116.

9. R. M. Christensen, Mechanics of Composite Materials, John Wiely and Sons, 1978.10. B. E. Pobedria B.E. Mechanics of Composite Materials, Moscow State University Press,

Moscow, 1984 (in Russian).11. J. T. Oden and J. N. Reddy, Variational Methods in Theoretical Mechanics. 2nd Edition,

Springer-Verlag, NY, 1983.12. N. Sujunshkaliev, On Solution of Plane Boundary Value Problem of Nonlinear Elasticity,

Dok. Acad. Nauk UzSSR, 1, pp12-16 (1985) (in Russian).

Ade Akinola obtained his Ph.D. in 1985, in the Department of Theory of Elasticity,Moscow State University, Moscow. Since 1986 he has been in the Faculty of the ObafemiAwolowo University, Ile-Ife, Nigeria. He currently teaches applied mathematics in the De-partment of Mathematics of this University. His present area of interest is finite deformation

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of elastic material and composite mechanics. He has widely published papers in this areain local, national and international journals.

Mathematics Department, Obafemi Awolowo University, ILE-IFE, Nigeria

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