Shakedown analysis of composites

Pergamon Mechanics Research Communications, Vol. 26, No. 3, pp. 309-318, 1999

Copyright © 1999 Elsevier Scieace Ltd Printed in the USA. All fights reserved

0093-6413/99/S--see front matter

PII S0093-6413(99)00029-4

SHAKEDOWN ANALYSIS OF COMPOSITES

D. Weichert, A. Hachemi and F. Schwabe Institute of General Mechanics, RWTH-Aachen Templergraben 64, 52056 Aachen, Germany

(Received 16 December 1998," accepted for print 15 March 1999)

Introduction

One of the advantages of composite materials compared to conventional materials stems from the different mechanical characteristics of each component of a composite. This may be used to design materials for specific technological purposes, where in some sense controversial material properties are required. As example one may quote metal-matrix-composites (MMC's) exhibiting at the same time high hardness and fracture toughness [1]. To predict failure of this type of materials, it is important to understand the complex processes on the microstructural level leading to failure, including in particular inelastic effects, and to link them to the macroscopic material properties. In this paper, failure of metal-matrix-composites under variables loads due to accumulated plastic deformations is investigated. For this, shakedown-analysis [2] is carried out on the micro- level. Then, by use of the technique of homogenisation [3], the results are linked to the overall material properties on the macro-level. The methodology is analogous to that given by Suquet [3] for the case of limit analysis of heterogeneous media (see also [4, 5]).

General considerations and definitions

The considered composite material is composed of inclusions, embedded according to a regular pattern as shown in FIG. 1 in an elastic-perfectly plastic metal matrix. The macroscopic behaviour of this heterogeneous material is observed on the scale (Xl, x2) and the microscopic behaviour on the scale (Yl, Y2).

309

310 D. WEICHERT, A. HACHEM/and F. SCHWABE

x21

Heterogeneous material

Matrix RVE (unit cell) e O : O O •

i i i , ~ :

! * Inclusion Yl O .+..\.. v . . i .

: e • e : e e

,~xt I

FIG. 1. Composite with square RVE.

Stresses and strains on macro- and micro-level are defined by X(x), E(x), a(y) and ~(y), respectively, linked through [6]

1 f~ tydV (1) X= < o > =V v~

'I dV. (2) E = < ~ > = V x0

Here, V denotes the volume of the (periodic) representative volume element (RVE) as shown in FIG. 1. We assume that at each point of the RVE, there exists in the space of stresses a surface delimiting the non empty set P(y) of all physically admissible stress states

a(y) ~ P(y), Vy • V (3)

P(y) is defined by means of a yield function g ( y , o)

P(y) = {a / ~.qZ-(y, a) < O, Vy e V}. (4)

It is assumed that the yield function ~7-(y, a) is of von Mises type

J-(y, a) = (3/2 cy D • o D ) I / 2 _ t~F(y ) _< 0 (5)

where cr~ (y) denotes the yield stress and a D the deviatoric part o fo .

SHAKEDOWN ANALYSIS OF COMPOSITES 311

Restricting to geometrically linear theory, the total rate of strains ~ can be split into an elastic and a plastic part

= ~e + ~p

with

~ e = A : 6 and ~p_~ O~ar - c3o

where A stands for the elasticity tensor; ~. is a non-negative scalar factor.

(6)

(7)

The convexity of ~r (y, o) and the validity of the normality rule is expressed by the maximum plastic work inequality

( e - ~) : ~P > 0 (8)

where ~ is any safe state of stresses defined by

~(y) = {~ / ~r(y, ~) < 0, Vy e v } . (9)

Static shakedown theorem

The macroscopic admissible domain pm of stresses ]~ is defined by

pm= {~/~,= < o > , o(y) ~ P(y), Vy ~ V}. (10)

To determine pm shakedown-analysis is carded out on the level of the RVE. For this, we introduce the notion of a "reference representative volume element (RVE(E)) '' differing from the actual one only by the fact that the material is supposed to behave purely elastically (FIG. 2).

1

a V J a(y) ~(y) cP(y)

/ V

Elastic-plastic RVE

Y~I ! ! . . . . )'1

Purely elastic RVE CE)

FIG. 2. Actual and reference representative volume element.

312 D. WEICHERT, A. HACHEMJ and F. SCHWABE

Proposition : I f there exists a safety factor ~ > l, a time-independent self equilibrated residual

stresses tensor ~(R) and a sanctuary of elasticity [7] ~m c pm

p m = { ~ / ~ = < ~ > , ~(y) ~ ~(y), Vy ~ V} (11)

then the periodic heterogeneous material shakes down. Here, the safe state of stresses ~ is defined as usual (see e.g. [8, 9])

~(y, t) = c t a (E) (y, t) + N(a)(y) (12)

where if(E) is the stress field which would occur in the RVE (E) under the same loading as the actual RVE. This field satisfies the following relations :

(i) for given macroscopic stresses ~,

d ivo(E)=0 inV ; O~E).n=~.n oncOV (13)

(ii) for given macroscopic strains

divtJ (E)=0 inV • u (~)=E.y onSV ; <a(E)>=~2 0 4 )

where n is the outward normal vector to aV.

The field of the residual stresses ~(R) satisfies

d ive (R)=0 inV ; ~(R).n=0 oncW ; <~(R)>=0 . (15)

Proof: For the proof we show that the average of the microscopic dissipation is bounded, which means that the plastic strains in the whole RVE are limited in an energetic sense. For that, we introduce the positive definite form

W ( t ) = 2 < ( a ( R ) -- ~-(R)) : A : (O (R) - ~ ( R ) ) > . ( 1 6 )

Using the identity

o(y, t) = o'(E)(y, t) + G(R)(y, t)

one obtains the following decomposition of the actual strain field

~(U) =E(E)+IE(R)+~P

= A : o (E) + A : fir (R) + E p.

(17)

(18)


The first term of the r.h.s in eqn. (18) constitutes the strain field in the RVE (E) when exposed to the same boundary conditions as the actual one. This field is derives from the displacement field u {E) of the RVE rE) and therefore fulfils the conditions of compatibility. As the total strain field g is also compatible the last two terms result from a displacement u (R) called "residual displacement"

~(E) = grads (u(E)), g i n = E(R) + IE p = grads (U(R)) (19) u = u (E) + u (R) ( 2 0 )

where u denotes the actual displacement in the RVE compatible with g.

Derivation of W with respect to time gives

With

one gets

W(t) = - <(o(m - @~)) : ~P> + <(o(m - ~{m) : (~ _ ~(E))>.

< ( o ( R ) _ ~(R)) : (~ _ ~(E))> = <~o(R) _ ~[(R)> : < ~ _ ~(E)> = 0

(21)

(22)

Integration with respect to time leads to

I i < o ~p> cc 1 ~(R)>. dt < _ a _ I ~ < ~ ( R ) : A :

This proves the boundedness of the macroscopic dissipation.

(27)

(Z < [-~V(t)]. (26) - o t - I

X;V(t) = - < ( o - ~) : ~P>. (23)

As W(t) _> 0, one can conclude that

W(0) > W(t) for t > 0 and X~V(t) ~ 0, W(t) ~ const, for t ~ oo. (24)

The averaging procedure for additive functions allows to compute the macroscopic dissipation as the average of the microscopic one

D = <d> = < o : ~P>. (25)

We can deduce from (8) and (23) that

< o : ~ P > < ~ < ( o - ~ ) : ~P> - a - I

314 D. WEICHERT, A. HACHEMI and F. SCHWABE

Discretised formulation

The present approach is based on a finite element discretisation with test functions for the displacement fields. Then, to calculate the elastic stresses orE) in the reference RVE (E), we use the virtual work principle

I v {oCE)}{6~ CE)} dV= I~¢} {p}{~Su(E) } dS + IV){f}{6u~E)} dV. (28)

Analogously, the field of residual stress is defined by

I {~(R)(y)} {~i~(E)(y)} dV = 0. (29) v)

By using the well-known Gauss-Legendre technique, this integral can be calculated for each finite element. The integration has to be carried out over all Gaussian points NGE with their weighting factors w i in the considered element "e"

I NGE NK {~(R)(y)} {61~tE)(y)} dV = ~ w i [ ~ [Bk(Yi) ] {~u~}] T {~(R)(yi) }.

ve ) i= 1 k = 1 (30)

Here, [Bk(Y)] and {6u~} denote the k-th matrix of the derivatives of the shape functions and the

vector of virtual displacement of the k-th node of the element "e", respectively. NK denotes the total number of nodes of each element. By summation of the contributions of all elements and by variation of the virtual node-displacements with regard to the boundary conditions, one finally gets the linear system of equations (see also [ 10])

NG Y. [c3 {#I m} = [c] {~m} = {0}, t31) i=l

where NG denotes the total number of Gaussian points of the RVE ~E), [C] is a constant matrix,

uniquely defined by the dis.:etised system and the boundary conditions and {sCR~} is the global residual stress vector of the discretised RVE ~E).

For n independent loads, the load domain Q~ represents a n-dimensional polyhedron, defined by

~ = ] _ ~ / . ~ = ~ k 2 ° [ k' /Zk ~ [~ ' /~1 I (32) i= l

where _~ is the vector of generalized loads, /.z k are scalar multipliers with upper and lower

bounds ju~ and/.t~, respectively. _ ~ are n fixed and independent generalized loads (macroscopic


stresses, macroscopic strains or combinations). Now we are able to present the discretised formulation of the static shakedown theorem for the determination of the macroscopic admissible

domain p m of stresses ,Y.

with the subsidiary conditions

{XSD = m a x cx (33)

[ e l {~(R)} = {01

-(R) °r ) < 0 ,_~'-(~ a(i E) ( ~ j ) + (I i ,

(34)

(35)

(E) --(R) ~ = ~ < ( I i > + < O i >

Vi ¢ [1,NG] and Vj ~ [1,2n].

(36)

This is a problem of mathematical programming, with ct as objective function to be optimised

with respect to ~(R) and with relations (34) and (35) as linear and nonlinear constraints, respectively. The resolution of this problem is carried out by using the code LANCELOT [11] which is based on an augmented Lagrangian method. LANCELOT automatically transforms inequality constraints (35) into equations. This technique is extensively used in simplex-like methods for large scale linear and nonlinear programs [12]. The constrained maximisation problem (33)-(35) is solved by finding approximate maximisers of the augmented Lagrangian

function O, for a carefully constructed sequence of Lagrange multiplier estimates 2 i, constraint

scaling factors sii and penalty parameter fl,

with

1 m • (x, 2, s, ,6) = f(x) + 2~ Ai bi(x) + X bi(x) 2

i=l 2"~i =1 $ii (37)

f(x) = ct (38) --(g)

b p ( x ) = C t ~ q p = 1 ..... mr; q = 1 ..... m 2 (39) C~'[~ O.(E) + ~(R) \

b r (x ) = ,1 ~,~ r r ' OF) r = m! + 1 ..... m. (40)

Here, m i is the number of degrees of freedom, m 2 = NG.NSK and m = m n + 2n.NG, where NSK

is the dimension of stress vector at each Gaussian point. The number of optimisation variables x

which corresponds to ct and ~(R) is equal to 1 + m 2. The first-order necessary conditions for a

feasible point x (k) = (cx (k), ~(R)(k)) o f the iteration k to solve the problem (37), require that there are

Lagrangian multipliers, 2 (t0, for which the projected gradient of the Lagrangian function at x (k)

and 2 (k) and the general constraints (39)-(40) at x (k) vanish. For k = 0 we set cttk) = ct E and

~(R)(k) = {0} where ct z denotes the elastic limit factor of the RVE (E). One may then assess the


convergence of the augmented Lagrangian method by the size of the projected gradient and constraints at x (k) and A (k). The optimisation will be terminated if the conditions :

and I Ix(k)_ p ( x ~ k ) _ Vx~(X(k), 2(k), slk), fl(k))) [[ < c/

[I b(x%11_< ec

(41)

(42)

hold for some appropriate small convergence tolerances 8 t and ec, where P denotes the projection operator (for more detail about the algorithm of optimisation we refer to [11 ]).

Numerical Application

To validate the presented method, we consider a square unit cell assuming perfect bonding between inclusion and metal matrix. The unit cell is discretised in space by two-dimensional finite elements under the assumption of plane stress. Geometry and the adopted mesh are shown in FIG. 3a. Here, triangular isoparametric elements with six nodes are used. For symmetry reasons, only a quarter of the unit cell is considered using 206 elements and 457 nodes. To satisfy the boundary and symmetry conditions, the unit cell is subjected to biaxial uniform displacements at the edges. The following loading cases are studied :

(i) The displacements u 1 and u 2 increase proportionally, which corresponds to the limit analysis problem:

0 0 U 1 = ~ U 1 and u 2 = g u 2 with 0 _< g < p.+ ;

(ii) The displacements uj and u 2 vary independently, which corresponds to the shakedown analysis problem:

0 0 + < + U 1 =]-t 1U 1 a n d u 2 = ~ 2 u 2 wi th O<gl < g l and 0 _ < ~ 2 _ ~ 2 .

The admissible macroscopic domain against failure is obtained for different values of mechanical

properties of inclusions embedded in a metal matrix with Young's modulus E = 2.1.105 MPa, Poisson's ratio v = 0.3 and yield stress o F = 280 MPa and different ratios of diameter d of the inclusions and the length L of the unit cell. The domain of admissible macroscopic stress for the homogeneous unit cell (without inclusion) is shown in FIG. 3b, where the bounds of the elastic, limit and shakedown domains are represented normalised by the yield stress of the matrix. The bounds of elastic (curves 1), limit (curves 2) and shakedown (curves 3) macroscopic stress domains are identical due to the fact, that in this case, the state of stress in the RVE is homogeneous. FIG. 4 shows the admissible macroscopic domain by considering the influence of the inclusion. The results are obtained with the following values of material properties :


FIG. 4a FIG. 4b

Young's modulus E (MPa) 14.07.104

FIG. 4i

7.14.104

Poisson's ratio v

0.26 0.23

Yield stress

o F (MPa) 233.3 186.6

FIG. 4c 0.21"104 0.20 140.0

FIG. 4d 14.07"104 0.26 233.3 FIG. 4e 7.14.104 0.23 186.6

FIG. 4f 0.21.10 4 0.20 140.0 FIG. 4g 14.07"104 0.26 233.3 FIG. 4h 7.14" 104 0.23 186.6

0.21 "10 4 0.20 140.0

d/L

0.1 0.1 0.1

0.2

0.2

0.2 0.3 0.3 0.3

u i

U2

~ d ~

.t---

L - - -

i T

= L -b !

..p

(a)

FIG. 3.

!i i*~ ,2,j3

I _ _ ~ / ~ < ~ . , ~ .

i i i i ; ;

(b)

(a) Geometry and discretisation of unit cell (b) Macroscopic admissible stresses domain.

i , ~ i", !

--I/ . . . . ; s

! ! [ L I (a) (b)

~'i[O F

J J

2: ; i

L~O F

(c)

(Figure continued)


(Figure continued)

(d)

! ! i ~~ ~ i

(0

- ' . . . .

//: r

(g) lh) (i)

(1) Bounds of elastic domains, (2) Bounds of limit domains, (3) Bounds of shakedown domains FIG. 4. Macroscopic admissible stresses domain of metal matrix composite.

References

1. H. Betas, A. Melander, D. Weichert, N. Asnafi, C. Broeckmann and A. Gross-Weege, Comput. Mat. Sci. 11, 166. (1998).

2. E. Melan, Sitzber. Akad. Wiss., Wien, Abt. 145, 195. (1938). 3. P. Suquet, C.R. Acad. Sci. 296, 1355. (1983). 4. A. Taliercio, Int. J. Plast. 8, 741. (1992). 5. P. Temin-Gendron and P. Laurent-Gengoux, Comput. Struct. 45, 947. (1992). 6. R. Hill, J. Mech. Phys. Solids 11,357. (1963). 7. B. Nayroles and D. Weichert, C.R. Acad. Sci. 316, 1493. (1993). 8. A. Hachemi and D. Weichert, Comput. Methods Appl. Mech. Engng. 160, 57. (1998). 9. D. Weichert and A. Hachemi, Int. J. Plast. 14, 891. (1998). 10. J. Gross-Weege, Int. J. Mech. Sci 39, 417. (1997). 11. A.R. Conn, N.I.M. Gould and Ph.L. Toint, LANCELOT: A fortran package for large-scale

nonlinar optimization (Release A), Springer -Verlag, Berlin (1992). 12. P.E. Gill, W. Murray and M.H. Wright, Practical optimization, Academic press, London and

New York (1981).

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Shakedown analysis of composites