A nonstandard method of modelling of thermoelastic periodic composites

A NONSTANDARD METHOD OF MODELLING THERMOELASTIC PERIODIC COMPOSITES CZESVAW Dept of Mechanics,

pacult)

WOiNlAK

of Mathematics. Informatics and Mechamcs. PKiN. IX Floor. 0%901 Warsra~a. Poland ((‘ommunicatcd

OF

University

of Warsa\\.

hq Z. OLESIAK)

Abstract The purpose of the paper is to derive from the equations of the nonlinear thermoelasticlty a certain class of homogenized models of periodic composite materials. The maln feature of the proposed approach IS a possihllity of modelling local stresses and heat fluxes with varloua approximations. The approach is based on the methods of nonstandard analysis. The hnearized equations and an example of application to laminated materials arc dIscussed.

INTRODUCTION

The list of papers on modelling of bodies with “micro”-periodic material structures (such as laminated bodies) is rather extensive and will not be discussed here. Among various approaches to the formulation of homogenized models of periodic material structures we mention those based on certain asymptotic theorems, [I,21 or on certain physical assumptions. [3.4]. The purpose of the paper is to propose an alternative approach to the formulation of homogenized models of “micro”-periodic thermoelastic composites, based on some theorems of the nonstandard analysis [S]. The main feature of the proposed method is a possibility of modelling of mean and local stresses and heat fluxes in “micro”periodic bodies with various approximations. The homogenized models of composites are derived in the paper from the general equations of the nonlinear thermoelasticity theory by means of the methods of nonstandard analysis in mechanics. [6], combined with the method of internal constraints [7,8]. That is why the proposed approach has been called a nonstandard method of modelling. I. BASIC

NOTATIONS

AND

FORMLAE

Let (J. t)~ R3 x R be inertial coordinates in the Gallilean space-time, (to, t ,) be a known time interval and B, stands for a region in R” occupied by a thermoelastic body in its undeformed state. In B,, we introduce two systems of material coordinates. namely: I” the Cartesian coordinates which coincide in B, with spatial coordinates y = 0.‘) and will be denoted by z = (.Y’),YE B,t, 2 the curvilinear reference coordinates X = (X’), given by means of a mapping .Y = k.(X), X ERR. where R, is a regular region in R” and functions x(.) = (J?(.)) are assumed to be single-valued and continuously __. ~ differentiable, except possibly at some points. lines or surfaces. and such that K(!&) = B,. We shall use the symbols: J(X) = detVk-(X) and A(X) = (VK(X))~’ (i.e. A”i(X)h-‘, p(X) = 0;). which hold for every XER,. The positive number 0, will represent the (constant) absolute temperature of the undeformed body. The position vectors y and the temperature 0 (at a time instant t, where fc(f,, t,)) will be related to the reference material coordinates X’ in B,, and represented by ?’ = K(X) + u(X, f), 0 = 0” + O(X.t); respectively.

Here ~(.,t) is the displacement

t Indices i, J, k and B,,, respectively. I-or [,(X, I) = ?f(X. t)/?X”, Symbol R(‘“-” stands stands for Kronecker

XERR.

field from B,, and O(., 1) is the temperature

r. /j, 7, ri run over I, 2, 3 and are related to the material coordinate systems (xl). (X’) In an arbitrary smooth field /‘(X,0, XE~&, tt(tO.tl). we define vf(X.1) E [f,,(X.r)], ?,,f(X. t) = clf(X, I);(:[. If M is an arbitrary matrix then MT stands for the transpose of M. for a set of all symmetric 3 x 3 matrices, elements of which are real numbers. Symbol 6; delta.

4x4

(. WO%hMh

increment field, defined at a time instant t. The Lagrangian strain tensor and the temperature gradient, both related to the curvilinear reference coordinate system in B,,, will be denoted by L(X, t) = (f&X, t)) and g(X, t) = (g,(X, t)), respectively, where

ux, 2) = ; [vK’-(x)vu(x,1) + VU’(X,f)VK(X)

+

Vu’(X, C)V#(X, t,],

g(X, t) = V@X, 1).

(1.1)

Hence the components of the Lagrangian strain tensor and temperature gradient, related to the material Cartesian coordinates in B,, are given by AT(X)L(X, t)A(X) and AT(X)g(X, t), respectively. Throughout the paper we shall assume that there are known: 1” mass density pO(X), X E QR, in the undeformed body B,, 2” free energy function 40 = (p[X, AT(X)L(X, t)A(X), Q(X, t)], X E!&, 3” heat flux density function H,[X, t), AT(X)g(X, t)] = (Ha[X, 0(X, t), AT(X)g(X, t)]}, X E QR, related to the undeformed body BO. We also assume that all fields depend on X = (X”)EQ~ and t g(to, t,) as independent variables. For the first Piola-Kirchhoff stress tensor, the heat flux density (both related to the element dI/(X) = dX’dX2dX3 of !J,) and the specific entropy, we obtain the known formulae

p/(X,

t) =

-AL

dH(X, t)’

rp = cp(X. A “@‘)I@, t)A(X), 0(X, t)),

(1.2)

which hold in QzR,and for t ~(t,, ti). Moreover, let h[X, u(X, t), t] = {b’[X, u(X, t), f}, r[X, u(X, t), 0(X, t), t] be the body forces and the heat absorption, respectively, in the deformed body (at a time instant t). Then the known equations

T&(X, t) + J(X)p,(X)h’[X, u(X, t), t] = .I(x)p~(x)c7,&u’(x,t), h&,(X, t) + J(XMX)rCX, 4X, t), 11 = J(X)W, t)MX, t),

(1.3)

are assumed to hold in QR and for t E (to, t J Equations (l.l)-( 1.3), constitute the governing relations of the nonlinear thermoelasticity theory and are the starting point of the analysis. 2. “MICRO’‘-PERIODIC THERMOELASTIC COMPOSITES Let (cl, r2, t3) be a triplet of positive numbers and <, = (dJ
@.= 1, 2, 3}.

If for every Z E R,, such that Z + t, E R, for some ME ( 1,2,3),

PO(Z)= PO@+ 4aL

cp(Z, E, 0) = cp(Z + t,, 6 @I,

conditions

Hh(Z, 0, h) = Hb(Z + t,, 0, h) (2.1)

hold for every E ER c3 ’ 3’ 0 E R h E R3 then we shall say that the body has a periodic material structure. For such a body the domain of definition of functions pO(.), cp(., E, H), H’(., 8, h) can be uniquely extended to R3. In the sequel we shall deal with periodic material structures and with functions pe(.), cp(.,E, O), hh(., 0, h) defined on R3. Let lR be the smallest characteristic length dimension of region OR in R3, define c=Jt;“58s,,a n d assume that a body with periodic material structure satisfies conditions: 4 << lR, K(X + 5,) - K(X) N VK(X)<,, X, X + &en,, CI= 1, 2, 3, where “cc” and “5” stand

A nonstandard

method

of modelling

of thermoelastic

periodic

for intuitive relations “is much smaller” and “is nearly equal”, respectively. confine ourselves to the problems of thermoelasticity in which conditions

Vu(X, t)

‘v

vqx, t) =

Vu(X + 5,, q,

V&X

4x5

composites

c 5,, t),

Let us also

c( = 1, 2, 3,

(2.2)

hold in Sz, and for every t E&, tl). Then the thermoelastic body under consideration will be referred to as a “micro”-periodic thermoelastic composite body. It has to be emphasized that a term “periodic” is related here to the reference (curvilinear) coordinates X = (X’) in undeformed body Bo. Let us also observe that conditions (2.2) can be verified only after obtaining the solution to the problem under consideration. The main purpose of the paper is to derive from eqns (l.l)-( 1.3) such homogenized models of a “micro’‘-periodic thermoelastic composite body, which enable us to describe stresses and heat fluxes in every periodicity cell. Unfortunately, the physical concept of a “micro’‘-periodic body formulated above, involving intuitive relations << (“is much smaller”) and 1: (“is nearly equal”), does not lead to any well defined class of bodies. However, this concept can be strictly precised if the nonstandard model of analysis is taken as a mathematical tool of mechanics, [5,6]. In the next Section, using the methods of nonstandard analysis, we are to show the way leading from eqns (f.l)-(1.3) to equations describing various homogenized models of a well defined “micro’‘-periodic thermoclastic body.

3. NONSTANDARD MODELLING AND HOMOGENIZATION STATEMENT Using the nonstandard approach to the mathematical modelling in mechanics, [6], we shall embed all sets and relations introduced in Sections 1 and 2 in a certain full mathematical structure W and then we shall pass to a nonstandard model *9X of %!I, [S]. It means that all mathematical entities introduced in Sections 1 and 2 have to be interpreted now as entities of *YJ. Let us specify in *W a class of problems by assuming that: lo in the place of entities Be, Qt,, K(.), tic), r(.) we introduce standard entities, which will be denoted by *B,, *!&, *KY(.),*PC), *r(.), respectively, 2” in the place of entities p,,(.), cp(.,E,8), Hb(*,t),h) we introduce entities PO(.), Cp(.,E,O), &(., 0, h), respectively, defined by the nonstandard stretching of variables

f%dX) = *p&m,

$(X, E, 6)

EE

*&5X,

E, 8),

q)(X, 8,h) = *H&5X, 0, Iz),

(3.1)

where ii is a fixed infinite positive number and *pot), *q(.), *Hb(.) are standard functions, Taking into account eqns (2.1) and setting E, = t$Jc;j, we obtain now the following periodicity conditions

ik(Z) = PO@+ &a), Ab(Z, 0, h) =

Cp(Z,E, 0) = $Z

rrgz -t c,, 8, h);

f e,, E, O),

CI= 1, 2, 3;

ZE*R3,

(3.2)

with the infinitely small periods E”, where E= E P/t&. Hence vectors 5, in eqns (2.1) have to be replaced by vectors E, = &&5. At the same time, conditions of Section 2, involving the intuitive relations “is much smaller” and “is nearly equal”, have now well defined meaning?. Functions PO(.), tp(, E, O), Rf,(-, 6, h), where E E *Rg3 ’ 3), h E *R3, 0 E *R are parameters, are periodic with infinitesimal periods .sa = 5”/& Thus, we have specified the well defined class of what will be called nonstandard micro-periodic composite bodies. From eqns (1.1). (1.3), t Relation G(CC8, where OL,/I E *R + , holds iff a/P E M, , where M 1is a set of all infinitely small numbers. Relation a = /I, where now a, fleMO, holds iff a - /IEM~; here MO is a set of all finite numbers. Notation of the nonstandard analysis used throughout the paper is based on that used in [S].

486

C. WOiNIAK

treated as relations governing equations

of *YJI (i.e. in the nonstandard model of analysis), it follows that the of microperiodic bodies can be assumed in the form of equations

f&(X, t) = ~[V*K’(X)VU(X, T;(X,

t) + Vu”(X, I)V*K(X)

+ Vu’(X

,t)Vu(X.

r)].

sx(X.

r) =- rl,,tx.

r 1.

aL “(“x t) C*“fs(X) + uf/l(X, l)l>

t) = *J(x)&(X)

M ’ hyR = *J(X)ETb(X, Q(X,t),*AT(X)g(X,t))*Ay(X), &j

r/(X,t) = --.

@=

d&X, t)’

rgx, *A’(X)L(X,

(3.3)

t)*A(X), 0(X, t)),

and conditions

s

T”,‘u,,~d V = *J&[*b’(X, s *RR *IlR

h& s *Rx

dV=

which have to hold for shall tacitly assume that Define E = J??&$, 1Y”I -=c0.52”, c( = 1,2,3}. oscillating with periods

s *nfi

*Jp,J*r(X,

u, t) - +Q&+

u, 0, t) - (t), + O)C;,ry]id V,

every u E *C:(Q,), [E *C,(!AJ, such all functions in (3.3), (3.4) satisfy the 1” E E’/c, A., E (6~I.‘,6~,?2,6~A3), a Let P(.): R3 + R, p E { 1,. , m} u {I,. R”

s

l’(Z) = P(Z + A,),

d V,

1”(Z)dZ1dZZdZ3

= 0;

(3.4)

that uI*AI1x= 0, I$,,~~, = 0. We desired regularity conditions. = 1, 2, 3 and C:= [ YER~/ , M} be continuous functions,

2 = 1, 2, 3,

c

and having continuous derivatives except at finite number of smooth every periodicity cell. Now define in structure *YJI internal functions means of

l;(z) = E*l'

04 ,

Z

E

surfaces or lines in If(.): *R3 -+ *R, by

*R”.

(3.5)

Here and throughout the whole paper p E { 1,. . , m} u {r,. . , M}. Functions 1:(.) are microperiodic (i.e. 1:(Z) = 1f(Z + c,), E, = (~:E’,~:E~,~ZE~), a = 1, 2, 3), they attain only infinitesimal values, but their derivatives V/f(Z) = V*P(Z/c) attain also standard values. Using the method of internal constraints to eqns (3.3), (3.4), [7,8], we postulate functions P(.): R3 + R and then we impose on fields u(., t), O(.,t), t E *(to, tl), the following restrictions? ui(x,

t,

=

*wi(x,

t,

+

*4ndX,

t)13x),

U(X, t) = *9(x, t) + *yA(x, t)l,A(X); x

E

*l&, t E *(to, tl),

(3.6)

where *wi(., t), *qai(‘, t), *9c, t), *yA(., t), t E *(to, tl), are (sufficiently regular) unknown standard functions. Functions w(., t), 9(., t) describe what will be called macro-displacement and macro-temperature fields, respectively. Functions qai(‘, t), yA(., t), which will be called micromorphic parameters describe, from the quantitative point of view, the effects due to t Indices a, b and A, B run over 1, ,m and I, _,M, respectively, problem; summation convention holds with respect to a, b and A, B.

where m, A4 are known

in every speckal

A nonstandard

method of modelling of thermoelastic

periodic composites

4x7

the micro-periodic material structure of the body. At the same time functions la(.). II = I,. , IH and I”(.), A = I,. . , M, which have to be postulated a priori in every special problem under consideration, describe the expected qualitative character of these effects and will be called the shape functions?-. Using the method of constraints we have to assume that the integral conditions (3.4) have to be satisfied now by every

V,(X) = *V”Jx) + *vui(x)~Z(x), i(X) = *r&f)

Let

LIS

+ *iA(x)l

XE*R,,

define (in !JJ1and hence in *%1) the following

strain

(3.7)

measures:

Setting 7 = (7,. , yu) c R”, using denotations (3.8) and those introduced in Sections 1 and 2, WC formulate now the following. Homogenization stutenzent. Let relations (3.3)-(3.6) hold in *!N. Then in !JJ1 there exist uniquely defined: 1” positive number fio, 2” differentiable real valued function (I’,= 4(E, F, G, :j, A), 3’ differentiable real valued functions &(g,x,~), Gt&($,g, y), such that the following relations hold in ‘3J1:

T~IK

l) +

P”J(X)hk(X,w(X, t), t) = ,G”J(X)+?,Wk(X, t),

Sik(X. 1) = 0.

&AX,

t) + i%J(X)r(X, 4X, t), 9(X, t), t) = fioJ(X)[Oo + 9(X, t)]c?,ij(X, t),

,&x3 I) = 0.

(3.9)

where

((X, t) =

&(X, t) = &($(X,

t), A’(2)V!j(X, t), y(X, t))A:(X)J(X),

g$x. t) = G;:;(9(X, t), A“(X)VS(X, t), y(X, t))A:(X)J(X).

(3.10)

Equations (3.X)--(3.10) have to be satisfied in C& and for t g(to, t,). Scalar density PO, vector densities $,(g,g,~), G$6(9,g,y) (related to the volume element of B,), and function @(E, F, G, 9, A), are independent of material coordinates X = (X’) E QR. t Also functions I:(%) E ~*P‘(z/i:) and L;(Z) = tP’(Z/<). respectively. will be called the shape functions.

PE

{ 1,. , m} v (I.. , M},

defined on *K”

and K”.

4X8

C. WOiNIAK

Hence eqns (3.8))(3.10) represent in W a certain homogenized model of a microperiodic thermoelastic body which in *%R was described by eqns (3.3). The form of the model depends on constraints (3.6) i.e. on the form and number of the shape functions I@(.):R3 + R, which have to be postulated a priori. It has to be emphasized that the homogenization statement deals exclusively with the governing relations of the nonlinear thermoelasticity theory (in *fm and in %R) and it does not concern formulations and solutions of the boundary-value problems. However, taking into account different sequences of shape functions I”(.), a = 1,. ,m, and I”(.), A = I,. , M, we can obtain more or less accurate homogenized models for the special physical problem we are to describe and solve?. The proof of the homogenization statement can be obtained by the direct calculations. To this aid let us substitute the RHS of eqns (3.7) into the integral formulae (3.4). Then we arrive at the conditions which involve integrals of the form

s

aW*B(-W W),

dI’(X) = dX’ dX2 dX3,

*RX

where a(.) is an internal nonstandard function and */I(.) is a standard JC,l = c1e2e3, C,(X) : = { YE R,I 1a - X”I < OW, CI= 1, 2, 3) and

(a(.))(X)

1 = (c,I c,(x) a( Y)d I’( Y), s

function.

Let ea = t”/&,

dV(Y) = dY’ dY2dY3.

(3.11)

Moreover, let {C,(Z), ZEA}, where A:= {YE*!&/ Y= n’~~, n’.*N, CY= 1,2,3}, be an internal fine partition of *R, (cf. [SJ, p. 71). Taking into account the known theorems of the nonstandard integral calculus (cf. [S], pp. 71&79), we obtain

s

a(X)*DW)dWV =

c

s

%EA

‘QR

=

a(X)*BWd V-9

C,(Z)

4X)dW)*B(Z) = 1 (49>(Z)*W)lC,l. SJ,,,, ’

ZEA

P

Now assume that for every function (a(.))(Z), ZEA, under consideration there exists its standard part ‘(E(.))(X), XER, (cf. [S], pp. 57 and 115) which is an integrable function. Then

s

4W*B(x)dVW = 1 *C”(a(.)>l(Z)*P(Z)lC,l &A *al

o(a(~)>WP(WW). Using relations of the form (3.12), under assumption hold and making use of notation1

that the desired regularity

Do= o(Po(.)>

(3.12)

conditions

(3.13)

t The situation is analogous to that of the finite element method, where we also deal with different choices of the shape functions. $ If a(Z), Z E *R3, is a periodic function with periods 6’”along Z”, a = 1, 2, 3, then by means of (a())() = const. we shall define (a()) = (a())(X) for every X.

489

and 7$(X, t) 52 (I( TE(., l))(X),

$(X,

t) 5 (‘( T$y, f)/:.,(.))(x),

&(X,f)

g$x,

f) 5= ‘)(kjK(*,f,&(.))(X),

= “(h”,(.,t))(X),

q(X, t) = “(&(.)~(., r))(X)fi,

(3.14)

‘,

we derive from (3.4), (3.6), (3.7) system of equations (3.9). Now let us substitute of eqns (3.3) for TF(X, t), &(X, t), $X, f) into de~nitioIls (3.14). Setting in *!W

the RHS

define @(E, I;, c;, 9, A) = “(&(.)q(.,

*A“@(., *E, “F, *G)*A. “3));;

l.

(3.15)

Let us also define

Then, taking into account definitions (3.15) (3.16) and bearing in mind that the terms involving shape functions /I’(.), ,UE ( I,. . , PI] u (I,. . , A4 I (but not their derivatives!) are in~nitesimal and can be neglected if we pass from *%I1to W, cf. [6]. from (3.14) and (3.3) we obtain eqns (3.10). This ends the proof of the homogenization statment. Equations (3.8)- (3.10) represent a homogenized model of the (rlonstand~~rd) thermoelastic microperiodic body. 4. HOMOGENIZED

THERMOELASTICITY

WITH

MICROMORPHIC

PARAMETERS

Now we shall transform eqns (3.8), (3.10) to more convenient form, in which all scalar and vector densities are related to the volume element dx,d.x,rl.x, of the undeformed body B,. Setting

PJ(X. t) E ~~(X)~~(X. t1.1 ‘(X), i;:,(X,t) = i;zR(X, r).i ‘(X),

wyx, t) 5% &(X)wk(X, t), h”( x, .) = A;(X)hk(X,.),

we transform

S;;Z(X,f) =

~~iX)S~k(x, t)J l(x),

‘&X, r) = &X,f)JJ &(X, I) =

‘(X),

A;(X)ql;(X,I),

eqns (3.9) to the form?

kY,(X,t) +

&r[X,

w(X. f), 9(X,t), t]

= fi,[O, -t 9(X, t)]C,fj(X, t).

‘c&X, t) = 0. 3 The vertical line m eqns (4.1) stands for the covariant X ~52~. of the curvilinear coordinate system in R,,.

(4.1) diffcrcntiation

in the metric Gap(X) E ti~z(X)~!B(X)6,,.

490

Similarly,

(‘. WO%NIAK

eqns (3.10) will take the form

fj(X,t) = -~

a@ &9(X, t)

Kgx, t) = Eigqx, t), A’(X)VS(X, r), y(X, t))AP(X), g:(X, t) =

G;f:(S(X, t), A“(X)VS(X, t), y(X, t))A;(X).

(4.2)

Equations (3.Q (4.1), (4.2) will be called the governing equations of the (nonlinear) homogenized thermoelasticity with micromorphic parameters. Substituting the RHS of eqns (4.2) into eqns (4.1) and using (3.8) we obtain the system of equations for macrodisplacement fields w(., t), macro-temperature fields 9(., t), micromorphic kinematic parameters qy(., t) and micromorphic thermal parameters yn(., t), t E (to, tl). Let us observe that from (4.1), (4.2), (3.8) we obtain for micromorphic parameters q,(X, t), y,JX, t) two systems of equations [which are not differential equations for q,(X, t), y”(X, t)!] [SF + wNly(X, t)]

si-3 +c&(X, qac “(“xt) = 0, 27

ah

’

3

G$(S(X,t), A“(X)VS(X, t), y(X, t))Aq(X) = 0,

(4.3)

respectively. Hence the boundary and initial conditions for eqns (3.8), (4.1), (4.2) will be formulated only for macro-fields w’(., t), 9(., t), i.e. they have a form analogous to that of the well known conditions of the classical thermoelasticity. So far, eqns (3.8), (4.1), (4.2) were represented by a certain homogenized model with micromorphic parameters of the nonstandard (i.e. defined in structure *%R) thermoelastic microperiodic body. Now we are to interrelate this model with the “micro’‘-periodic thermoelastic body (described in structure %R), the idea of which has been outlined in Section 2. To this aid we take into account subsets C(X): = { YcCIR( IY” - X”j -c OSt”, Tx= 1,2,3} defi ne d now for every X E Sz, such that C(X) c QK (cf. Section 2). Every C(X) constitutes a periodicity cell for any periodic function ti: R3 + R, where ll/(Z) = $(.Z + <,), c( = 1, 2, 3, holds for every ZE R”. Setting ICI = <‘<*t3, define a constant

<$(.)>c = &

s

c(x) $(Y)d k’(Y)>

dV(Y) = dY’dY2dY3,

(4.4)

which is independent of the choice of point X. Moreover, let $: *R3 -+ *R be a (nonstandard) function defined by q(Z) = *$(GZ), Z6*R3 (cf. eqns (3.1)). Then, taking into account definition (3.1 l), equality 5” = r% and footnote $ on page 488, we obtain by the (nonstandard) stretching of variables Z = 6Y, that

“<$C,>=

IG(Y)dUY) “(&j e C,(X)

(4.5)

Now define l!(Z) = tP(Z/& ,u = a, A, where 1;(.): R3 + R are oscillating with periods l”, i.e.

@a= [‘i(Z +

5,x

u = 1,. .,m,

s

&Z)dk’(Z) (‘;(X)

A = I,.

= 0,

., M.

Functions

A nonstandard

method

of m~delli~g

of thermoel~stic

periodic

composites

491

and satisfy in *W conditions

UZ) = *lg,,(cliz),

ZE*R”

Hence the derivatives I&(Z) are obtained from IT,#(Z) by the (nonstandard) stretching of variables. Using the aforementioned stretching of variables and formula (4..5), we obtain the followitlg ~~~~~~u~~?Fz~~s~u~~~n~F~r. Constant PO and functions &E, F, G, 9, A), .f7b(S, n’g,,), Ggp(S, Arg, y), which determine the material properties of a homogenized model of the thermoelastic microperiodic (nonstandard) body are uniquely determined by the material properties of the “micro”-periodic (standard) body by means of

(4.6) where

The proof of the equivalence statement can be obtained directly from definitions (3.13), (3.1 S), (3.16), from interrelations I&(Y) = *I$,(&Y), P = hi,A, u = 1,. . . , m, A = 1, _. . , M, and from formula (4.5). The equivalence statement makes it possible to derive the governing equations (3.8) (4.1), (4.2) of the homogenized thermoelasticity with micromorphic parameters directly from the equations of thermoelasticity of “‘micro”-periodic bodies (1.2), (1.3), (2.1). provided that there are known shape functions I!(.): R3 + R, pc { 1,. . , , m} u {I,. . . , M), oscillating with periods [“, continuous in R3 and having continuous derivatives except at a finite number of smooth surfaces in every periodicity cell C(Z), ZE R3. Remark. All obtained results are valid if functions determining the material properties of the ‘~micro”-periodic body are independent either on variables X2, X” or on variable X3. Then formula (4.4) holds under condition that either C(X):= {Y’ E RI /Y’ - X1/ < OS<‘), or C(X):= {(Y’, Y*)ER~\ IYK ~ X”/
u(X, r) = Ql(X. t) + p&X, t)&(x),

0(X, t) = 3(X, t) + ya(X, t)l;‘(X),

x~~R,~~@“,t,l.

(4.8)

492

C‘. WOiNIAK

Now let us take into account the know formulae for Cauchy stress T’j(X, L)and heat llux h’(X, t) in the deformed body

7+(X,6 =

detCV~(X) + Vu(X, t)] - ‘[ti!Jx)

c uf=(X,t)] T%(x, t).

h’(X, t) = detCVrc(X) + Vu(X, t)] _ r [x-(,(X) + nf,( X, [)1&(X, r),

(4.9)

where T$X, t), &(X, C)are given by eqns (1.2), .Z. Then, using formulae (4.8), we calculate stresses and heat fluxes for XEQ, and at an arbitrary time instant t, t~(t,,t,). The RHS of eqns (4.9) depend on gradients Vu(X, t) =

vwfx,4 -I” %(X,W~(X> + VYAX,t)rg(x?,

vqx, t) = vqx,

t) + ?.‘,j(X,t)vq(x)

+ Vy,(X, t)/,“(X).

(4.10)

The obtained results have a physical meaning only if conditions (2.2) hold; from eqns (4.10) we conclude that such situation takes place if values of Vw(X, t), qJX, t), yA(X, t), Vq,(X, t), Vy,(X, t), VS(X, t) in every periodicity ceil C(Z), C(Z) c QR, can be approximately treated as constant. It has to be emphasized that for “micro’‘-periodic material structures, for which the homogenized thermoelasticity with micromorphic parameters can be applied (i.e. for sufficiently small values of periods t”, provided that all other entities hold constant, cf. Section 2), the values IF(X) of the shape functions are very small [since l:(X) + 0 as < -+ 0] and hence all underlined terms in eqns (4.8), (4.10) often can be neglected. On the other hand, terms in eqns (4.10) involving gradients of the shape functions are not small, even for a very fine periodic structures (since the maximal values of gradients V&X) = VP(X/<) are not small and hold constant if I_” + 0). Thus we conclude that the micromorphic parameters qa(X, t), yA(X,t) play an essential role mainly in modelling of local stresses and heat fluxes in “micro”-periodic bodies.

5.

LINEARIZED

EQUATIONS

WITH

MICROMORPHIC

PARAMETERS

We shall linearize eqns (3.8), (4.1), (4.2) of the homogenized thermoelasticity with mi~romorphic parameters with respect to the incremental fields w(, r), 9(, r), qa(.,tf, y,,,(,.t), r~(t,,t,f. From eqns (3.8) we obtain the following linearized strain measures?

Hence (4.7) yields

Define

t For the sake of simplicity we use the same symbols the general strain measures (3.8),,, and (4.7).

E,,, O,,, for the linearized

strain measures

(5.1) and for

A nonstandard

method

of modeI~in~ of the~oel~stic

periodic

where all indices run over 1, 2, 3. A free energy density function function. H,, = Hk) will be assumed now in the known form

where K;(X)

The linearized

is a tensor

of a thermal

form of constitutive

conductivity.

equations

composites

493

tp, s pop and a heat flux

Then formulae

(4.6), (4.4) lead tot

(4.2) will be now given by

?O(X, t) = A~~(X)[(~jkf~E~~(X,

t) + (C’jk’l;,,>q,&X,t)]

-I- A”f(X)( ~j~,~~X, t),

SYX, t) = A~~(X)[(Cijk’l~,pl~,y>qbd(X, + (Cijk’~;,p)E,a(X,

fj(X,t) = -(s)P-‘O,‘t’J(X,t) + W&>P

t)

t)] f A;~(X)(B”‘I&)S(X,

- A~~(X)~(Bij)~-‘E,p(x,t)

- ‘q,pw,

KL

gA(X,t) = A~~(X)C(K~I4l,)g,(X, t) -t (Kf&&J(X, t)]. t In linearized formulations we neglect subscript clement of B,. At the Same time we define ($) formula (4.4).

t),

(5.2)

“0” tacitly assuming that every density is related to the volume = (G(.)), for every function $c), which is periodic in R3, cf.

C’. WOiNIAK

494

The linearized form ofeqns (4. l), after taking into account F(X, t) = Por(X, 0, 0, t), will be represented by

Taq,j(X,f)

syx,

+

F(X. t)

the notation

/?(X, t) s p,,h”( A’, (1,r).

= ~jrtdrwU(X. f),

1) = 0,

F/,(X, 1) + qx, t) = j2&?,f(X,

t).

g”(X, t) = 0.

(5.3)

Equations (5.1), (5.2), (5.3) constitute the general linearized form of the governing relations of thermoelasticity with micromorphic parameters. Now let us pass to the isotropic case, setting Cijkf(X) = 6ij6kfi(X) + (dikSj’ + S”&‘)fl(X). where A(X), p(X) are Lame’s moduli: A(X) = 1.(X + t,), p(X) = p(X + 5,), z = 1,2,3. Similarly, let B”(X) = fi(X)S’j, KY(X) = k(X)@, where b(X) = /I(X + &J, k(X) = k(X + <,). GI= 1, 2, 3. Setting Gap(X) = .4”i(X)ABjX)Sij, we obtain from (5.2) ps(X,

t) = G""(X)G'"(X)l(i>E,~(X, t) + (Al;,,)q&Y,

+ [G"'(X)G""(X) + W;,A#‘~

t)l

+ Gah(X)GBy(X)] [(,u)E#‘, [)I +

t)

G”“O’KB)~(X,th

Saa(X, t) = Gap(X)Gy”(X)[(3.1;I,~~~,~)q,,(X, t) + (A1T,II)Ey6(X, t)] + [Gay(X) +

G”“(W?X)l

W~;,&,>Y,&‘> t)

f W&J&,(X, Gl + @(XKB~~,~)W, 0, ,W(X, t) = 0, ’ (s)W, t) +

w&J%&f.

G”“(X)C(B)E,p(X, t) t)l?

&X, t) = G”pW)C t) + W&A4K

Equations (5.1), (5.3), (5.4) are the governing relations with micromorphic parameters for the case of isotropic

6. RECTILINEAR

PERIODICITY

t)l,

of the linearized materials.

thermoelasticity

OF COMPOSITES

Now assume that the “micro’‘-periodic material structure of the composite body in the undeformed state B, is related to the Cartesian material coordinates x = (xi). In this case curvilinear reference material coordinates X = (X”) coincide with Cartesian material coordinates x = (xi). Then K = id and hence f12, = Be, A:(X) = Sz, r&X) = 6:. Using the aforementioned equalities we can easily pass from eqns (3.8), (4.1), (4.2) to the equations which describe the rectilinear periodicity case, i.e. the case in which the “micro’‘-periodic structure of II,, is related to the Cartesian coordinates. In the linear case from eqns (5.1) we obtain

Ei,Xx,t) = i Cwi,AX, t) + wj,i(x, t)l

(6.1)

A nonstandard

method

of modelling

of thermoclastic

periodic

composites

and qllcr(.x,t) = ~Sky,~(x,t). At the same time eqns (5.3) can be transformed

495

to the form

7yj,j(x,t) + lTi(X,t) = /3dtiY,Wi(X,t) soi(x, t, = O,

(6.2) and eqns (5.4) yield

h”i(r,1) = (k)9,i(x,t) SA(~, f) =

+ (kl&)y,,,(.x,f),

(kl$,i)lJ,i(X,t) +

(kl~,,/~~,)y,(x,

t).

(6.3)

Equations (6.1))(6.3) are the governing relations of the linearized thermoelasticity with micromorphic parameters for the case of isotropic materials and the rectilinear periodicity. From eqns (6.1) (6.3) we obtain the system of equations for macro-displacement fields w(., t), macro-temperature fields 9(., r), displacement micromorphic parameters y,(., 1) and thermal micromorphic parameters yA(., t), t~(t,,t,). Bearing in mind that 9 = (,I), this system will take the form

Eliminating from eqns (6.4) all micromorphic parameters qOi, ;‘” (which is always possible), WC arrive at the system of equations for w~(.;Q(.;). The mixed boundary value problems for this system can be formulated either in terms of wi(., t), 9(., t) and “mean” fields T;(.. t), Ki(., t) [cf. eqns (6.3)r.J or in term of fields Ui(X, t) = Wi(X, t) + yJ.w, t)Qx),

0(x, t) = 9(x, t) + yn(x, t)$(x),

(6.5)

496

C. WOPNIAK

and TAX,

t) = I(X)bijU&X,

hi(X, t) =

k(X)e.i(Xt

t) +

/L(X)[“i,jx3

l) +

uj,i(x3

t)l + P(xP(x, f)6tj*

[)3

16.6)

which describe the “micro”-periodic,i.e. nonhomogeneous body. After obtaining solution to the boundary-value problem and after calculating qoi(x,t), yA(x,f) from eqns (6.4),,,, formulae (6.5) can be used to determine the unknown fields in the “micro’‘-periodic body (cf. the comments at the end of Section 4).

7. EXAMPLE: LAMINATED MEDIUM As an illustrative example of the general considerations of the paper, we shall take into account multilayered (laminated) body, made of an isotropic linear-thermoelastic composite material. We assume that the material properties of the body are periodic (in the undeformed state B,) in the direction of the Cartesian x,-coordinate. Let 6 stand for a (constant) thickness of a layer and assume that every layer is composed of two homogeneous isotropic sublayers with thicknesses S,, 6,, where 6, + 6, = 6. Setting m = M = 1, 6 = [, I&x,) = /j(x,) = Ii( x1 E R, we obtain from (6.5)

ui(x* f, = wi(x7 t, +

@x, t) =

qitx,f)1&(xl)9

9(x, t) + 24%Wx,),

(7.1)

where I,(.): R --+R is a certain (continuous) shape function, oscillating with period 6. Let d stands for an arbitrary value of x,-coordinate, such that a coordinate plane x1 = d separates two homogeneous sublayers and x,-axis coincides with an outer normal to the sublayer of the thickness 6,. Then shape function la(.) in every periodicity cell (d - IS,, d + 6,) can be assumed in the form

ml)

=

I

if

x, e[d

- 6,,d],

if

x1 E [d,d + S,].

(7.21

Let I/I(.):R + R stand for any periodic function A(.), p(.), j?(.), k(.), s(.), p(.) in brackets (.) of eqns (6.4). Moreover, let $r, tiz be the constant values of $(x1) in parts (d - 6,,d). (d, d + a,), respectively, of an arbitrary periodicity cell (d - 6,, d + 6,). Then from formula (4.4) and after introducing more convenient notation, we obtain

$ = (*l,,lL%l) = ;*1 +flL2, 1 2

(*l&K)

=

0,

<*w&x)

= 0,

K = 2,3,

k = 1,2,3,

(7.3)

A nonstandard

where $t and $2 stand equations (6.4) yield (I +

fi)wk,ki(x,

t,

+

fiWi,kAX*

method

of modelling

for ir, pr,,..,pl

[)

+

[P1(j!qk,ktx,

of thermoe~astic

497

composites

and &, p2,. . ,p2, respectively.

Then governing

t)

+ CaIqI.i(X, t) + [p]qi.*(X, t) +

f&x, t)

periodic

= -- [k]9, ,(x, f)

Dg,i(X, f, +

gi(x,

t,

=

C~*atwi(x,

‘1.

(7.4)

After obtaining solutions to the boundary value problem for eqns (7.4), we can calculate displacements ui(x, 1) and temperature 0(x, t) in the nonhomogeneous “micro’‘-periodic composite body from eqns (6.5) and then stresses IT;-j(X, t) and heat fluxes hi(x, t) from eqns (6.6). It has to be remembered that the material moduli L(x), . . , k(x) in eqns (6.6) depend on the .x,-coordinate only and are constant in every sublayer. If the body under consideration constitutes a thick infinite plate bounded by planes .x1 = 0, x, = L. and the boundary conditions [formulated in term of (6.3, (6.6)] are independent of x2? x3, then in the static case we arrive at one dimensional problems which have elementary solutions. It can be shown that, after neglecting the heat adsorption r(,x, t), the heat conduction problems for the homogenized model with micromorphic parameter v(x,t) lead to solutions which coincide with the exact solutions of the linear theory of heat conduction, [lo). Similarly, after neglecting body forces bi(x, I), the solutions of the one dimensional isothermal problems for the homogenized model with micromorphic parameters qi(x, 1) coincide with the exact solutions of the linear theory of elasticity, [lo]. CONCLUDING

REMARKS

The homogenized models of thermoelastic “micro’‘-periodic composite bodies, which were derived and discussed throughout the paper, can be treated as a basis of a theory and a starting point for various applications of thermoelasticity with mi~romorphi~ parameters. The main features of the proposed models are: 1“ they can be applied to linear as well as to nonlinear problems, including the case of large deformations; 2” they can be applied also to the curvilinear composite material structures: 3” they describe not only mean but also local stresses and heat periodicity cell of a composite by means of eqns (4.9) and (4.10); 4” they can be formulated in various forms, given by eqns (4.8) involving more or less micromorphic parameters qa(X, t), u = 1,. . ,m, yA(X, t), A = I,. , M, and hence describe problems under consideration with various approximations; 5” in every problem they can be based on the physical premises, which enable us to choose the proper form of shape functions I$(.) in (4.8); 6’ their analytical structure is not more involved than that of the governing relations of th~rnloelasticity of homogeneous bodies [mi~romorphi~ parameters are determined by the nondifferential equations and hence can be easily eliminated from governing equations (X8), (4.1) (4.2)]. From the example given in Section 7 we also conclude, that for some special problems the models with micromorphic parameters lead to the solutions which coincide with the “exact” solutions. The proposed nonstandard method of modelling of composites with the use of micromorphic parameters can also be applied to various non-elastic periodic material structures, for example elastic-plastic, [9]. For the examples of application of the proposed method, cf. the forthcoming papers [lo 161; a special case of the theory of thermoelasticity with micromorphic parameters has been discussed in [17].

49x

C‘. WOiNIAK REFERENCES

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