Effective properties of cosserat composites with periodic microstructure

Pergamon Mechanics Research Communtcations, Vol. 28. No. 3, pp. 265-270.2001

Copyright 0 2001 Elsevier Science Ltd

Printed in the USA. All rights reserved

0093~6413/01/$-5ee front matter

PII: SOO93-6413(01)00172-O

EFFECTIVE PROPERTIES OF COSSERAT COMPOSITES WITH PERIODIC MICROSTRUCTURE

Xi Ywn*, Yoshihim Tom&a**

*kdate school of !kience & Technology, Kobe University, Kobe 657-850 1, Japan

**Faculty of l%gkkng, Kobe University, Kobe 657-8501, Japan

fRecerved 31 October 2000: accepted jbr pnnt 15 Apnl2001)

l.htroductioa

In the last three decades, numbers of homogenization theories have been developed in order to predict the macroscopic behavior of heterogeneous materials from their microsuucture. However, most of these

theories are developed in the field of the classical comimnnn mechanics and give results independeut of the

scale of the microstructure. Despite the conspicuous development of the generalized continuum or nonlocal

theories recently, works concerning the homogenization analysis of heterogeneous nonlocal continuum, e.g.

[ 1,2], is rather rare and most chara&m of such kind of heterogeneous medium are not clear yet.

The aim of this work is to develop a homogenization method for heterogeneous Cosserat materials. Such

materials [3-51 are charactaized by rotational degrees of tkedom 0, which are independent of the translational motion u. They belong to nonlocal contintmm and have found much application, e.g. [6-81, now.

We 19, lo] once developed a homogenization method for this type of material by asymptotic homogenization

approach. In this paper, a homogenization method is derived after the discussion of properties of the Cosserat composite with periodic microstructure on the micromechanics sense. Then some scale-dependent behaviors

of porous media are discussed.

Throughout the paper, a bold character denotes a tensor. Dot means fmt order contraction and : the

double contraction. Q stands for tensor product and V is the Nabla operator.

2.Micro~olar cornno& with wiodic microstn~cture

2.1. Basic field equations

A composite SIl with a periodic microstructme can be de&ted by the smallest repeatable element, i.e.,

repmsentative volume element (RVE) or unit cell Y (see Fig. 1). Here we suppose that the constituents of the

RVE are micropolar materials and their spatial distribution and mechanical properties are given. The static

elastic problem with no body force and body couple is considered in this paper. The equations governing the

265

266 X. YUAN and Y. TOMITA

equilibrium stress and defmtion fields in such a linear isotropic micropolar elastic’composite material are

theu r31

E=VBu-eocp, tc=V@q& (la)

v.t=o, V.u+e:t=O. (lb)

t=D’:a or t=&(t?)II+j18+jl,ar,

~=D’:K ~=&(K)I,+YK+~Kr, (lc)

where $ K are strain and curvature strain tensors, u and u are displacement and rotation vectors, t and u are stress and

couple stress tensors. Besides, 12 denotes unit tensors of the

second order. The elastic moduli il and p+h=2C are

readily recognized as Lame coeflicients, while pb=2G

denotes the Cosserat shear modulus which couples the

skew-symmetric stress-strain components. For plane deformation problem, an intrinsic length scale I, and a

coupling factor Nare commonly introduced as

I,‘=y/2C, N2 =G/(C+G). (2)

With N=O, equation (lc) gives the same stress-strain

relation as the classical elastic theory.

FIG. 1 Composite with periodic microslructure

and unit cell

Assume that the microscopic scale (the sire of tit cell Y) is small enough compared to the macroscopic

scale (the size of a) that the heterogeneities can be ‘smeared-out’ and n can be homogenized to behaves as a

homogeneous body. Then imposing some boundary displacement, force, or couple force will result in a homogeneous strain E and a homogeneous curvature K throughout Y. These homogeneous strain and

curvature generate a homogeneous stress T and a homogeneous couple stress M and the homogenized (or

effective) constitutive relations express the relations between E, K and T, M.

Let y denote the position of a point in the unit cell. The local strain field s and K can be split into the

overall strain E, K and a perturbation terms s+ and K*, which account for the presence of heterogmeities.

Then it obtains that:

a=E+e* , K=K+K* (3)

Correspondingly, the displacement and microrotation field u and cp can be split into U, 0 and periodic

fields u* and cp* respectively, where

K=VQO, K*=v@g*

E=V@U-a, E*=v@U*-(p* (4)

Here all the deformation fields, 6, K, t, p , u+ and cp* , conforming themselves to the periodic arrangement

of the cells, are supposed periodic at the microscopic scale, which means

g’69=g’(y+d), (5)

where g denotes an arbitrary deformation parameter, d is the oonstant vector that de&mines the period of the

strucmre.

2.2. Properues of overall stress and strain tensors

The defiitions of overall strain and curvature strain tensors have been given in Eq. (4). ln addition, we

de6netheoverallstressandcoupleslressas

T=< t>, M=< u>, (6)

where cgz denotes the average of any variable g over the volume of unit cell Y, i.e.

COSSERAT COMPOSITIES WITH PERIODIC MICROSTRUCTURE 267

<gr (Y( r - ‘j gdY.

One of the most impohnt propehs of the abovaddined overall strain and stress tensors is: LemmaI:~tand~btselfequilibratadstreesandcouplestresefi~donYwhichfu161l(lb)asrlsaodKthe compatible strain and curvature fields given by (4) and satisfy (1 a), then if boundary condition (5) is satisfied, the following relation holds:

<t:s+p:K>I T:E+M:K (8)

In another word, the average of the microscopic internal work is precisely the macroscopic work of effective stress <t>, <cu. This conclusion plays a fhiamental role in the discussion of effective properties of micropolar composite materials.

To prove (8), we fintly consider the internal work of perturbation strains 6+ and I(*. After the

idrodudon of(4), ai~l considering the equilibrium equation (2), it can be obtained that:

=< t:V@u+>+<p:VOc)*-e:t.g*>

=c V.(t.u*)>-<V.t.u*>+<V.(p.q*)>-<(V.p+e:t).p*>

=< V.(t.u*)>+<V.(p.~*)>

(9)

It then can be obtained hm Gauss’s theorem and the periodicity of t, 1 and u*,qP that the above equation gives zero value. Then Eq. (8) can be easily obtained atIer the introduction of Eq. (4) into Eq. (9).

2.3 Solution of perhhed defommtion field and e&ctive rrmcIoBcopic prapaties

The local stress and strain fields induced at the microscopic unit cell Y by overall strain E, Kor overall stress T, M can be obtained by solving the following equilibrium and periodic boundary conditions:

v.t(y) = 0, V.p(y)+e:t(y)=O

t(y)=D’(y):(s’(y)+E), fi)=D’(y):(~*(y)+K) (10)

QYEY, U*,K*,t,JL tllZ&lel’iodiC

Once the problem (10) is solved, the macroscopic stress <t> and couple stress <p can then be computed and the affective stiffness D of the composite is detezmhed through the relation

D:[+ [;j>=< D_$[;:;+]% (11)

Itcaubeprovedthat D sharothosamesymmehyas D,=,namely D,, =D, [lo].

For the convenience of finite element calculation, we transfer tho boundary condition problem (10) iota its variational form:

I, (a’ (V) : D, : b(v)+ 8 (v l ) : D, : 6 c(v)w = 0 (12)

wherev={u,~}, V={U,@}, 8(v) = {VJBu-e: (p,V@o}. Tho solutionofEq. (12)yiolds:

j+tI: D,,, : S s(v)dY = I, D,,, : ba(v)dY (13)

v”=fu,~)+~:p(v,@) (14)

When the overall (macroscopic) deformation field U, @ is given, the microscopic deformation field v*

268 X. YUAN and Y. TOMJTA

can then obtained fkom Eqs. (13), (14).

The above homogenization method is applied to in~eatigate the elastic plane strain problem of porous mater&. The shapes of the voids inside the matrix are supposed ellipse and the geome&y of the unit cell is given in Fig. 2. As shown in the figure, the unit cell has its height and width of 2L and contains an elliptic void of aspect ratio a& at its center. The geometry of the voids is also xpresented by equivalent radius R, where R%b.

. Consistent with definition of the material constants of isotropic elastic micropolar media given in Eqs. (lc) and (2), here we define the characteristic length of the homogenized materialas

l- D 3131 (15) FIG.2

x - 2(4212 + 4u,) ’ Geometxy and typical ftite element

and the homogenized value of the Cosserat shear moduli C and meshofunitc8ll

Gas:

Cn=D~za+D~nl, GH=DIZZ-DI~~ (16)

The unit cell problem Eq. (13) is solved by finik: element method. Triangle element is used and typical finite element mesh of the unit cell is also given in Fig.2. To emphasize,the d.i&rence of current method with classical homogenization approach, we focus our attention on the analysis of some size effect phenomenon, i.e., the dependence of the above-def~ned elastic constants on the size of unit cell L. The influence of the geometxy of unit cell, relative void radius R/L. and aspect ratio a/b, are also analyzd. In the following part, subscripts m and H denote matrix and homogenized values, respectively.

0.55- 0 50 LY 150 2M) 0 ml 160 200

L/I,

(4 (b)

FIG3

Dependenceof~modulusC~cmtherelativasiztoflmitcell~and(a) aspectratioofvoid,(b) relative radius of void RiL

COSSERAT COMPOSITIES WITH PERIODIC MICROSTRUCTURE 269

Tbu value of homogenized Cosserat elastic modulus CH is shown in Fig.3. It indicates that with the imxease of the size of unit cell L, CH deaxases and converges to some m, while the classical rwults (when coupling factor N=O) show no such size dependence. Furthermore, the values d above constauts are dependent on the geometry of the unit cell. When aspect ratio s/b or relative radius R/L of the intmml void becomea Iazger, the constants become smaller. Fmthermorc, large value of a/b and WL also give rise tc quicker change of CH. In other words, it shows more cbvicus size dependeace in this case.

R/L.=0.4 -db=l El -_.___ a/b=2 -.-_._.- a/b=3

50 If/4”. 150 2

(4

-I

!oO

FIG4

a/b=2 - RhO.1 -_---_ R/L=,,.2 _-_._.- R/L*.3

Depednw of homogenized module Gff on the relative size of unit cell L&and (a) aspect ratio of void, (b) relative radius of void R/L

0 50 ml 150 200

l-4 (a)

,,. _,__,_____________________ ._.__._. __ ,_._ ___.__.__._.__ ._._. _ 0 50 loo 150 2M)

L/L

0

FIG5

Dependence of homogenized charactexistic length 1~ on the relative size of unit cell L&, and (a) aspect ratio of void, (b) relative radius of void R/L

Fig.4 gives ihe value of homogenized Cosserat elastic modulus Gx+ It shows that with the increase of the size of unit cell L, CH decrea~ and converges to zero rapidly. Although it could be also seen the intluence of the~ratioa/bandrelativer~~wLoftheintanalvoid,itisqnitcli#leand~i~~le.

Fig.5 shows the changes of the homogenized chamcbistic length 1~. It is a much inbrebg parameter because it greatly settles the nonlocal behavior of composite. The conclusion given iu the figure is: larger the value of a/b or/and R/L, larger the value of 1~. That means it tends tc show nonlocal properties. At the same

270 X. YUAN and Y. TOMITA

time, if the size of unit cell is quite Small, say at the same order of the &mcte&ic length 1, the

&rack&c length of the cumposite 1~ is much smaller.

4. CoMwion:

Some propertks of Come& composite witb periodic micro&ucture are d&used. It is proved that the

composite can be made equal to a homogenow Cowerat medium with the same value of strain energy.

Add&a@, a homogenization theoawn is developed. The numericel implementation of the theorem is also

prea&d aad used to analyze some scaladent deformation behaviors of porous materials. The

calc&ed resulta show that the homogenized elastic constants depend obviously on tbe size of unit cell and gumuhy oftbe i&anal void.

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