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On linear thermoelasticity of composite materials
InI I. EQVK SC. Vol ?I. No. Printed in Great Britain.
5. pp. 44348.
W?~7?!S/83/0S044M6$03.0010 @ 1983 Pergamon Press Ltd.
1983
ON LINEAR THERMOELASTICITY MATERIALS
OF COMPOSITE
HORIA I. ENE INCREST, Department of Mathematics, Bd. Pkii 220. 79622Bucharest, Romania (Communicated by E. Sd0.S) Abstract-We establish the equations of the linear thermo-elasticity for a composite material, using the homogenization method. The macroscopic coefficients are deduced and it is proved that the macroscopic energy equation contains new corrective coefficients.
1. INTRODUCTION
1.1 Generalities In the general framework of the homogenization method[l, 21 we consider the problem of linear thermoelasticity. The periodic structure of the composite material is associated with a small parameter c. The asymptotic process, e + 0, implies that the number of periods is very large. All mechanical and thermal properties are different in the matrix and in the inclusions, and their magnitude is of order one. 1.2 General equations We consider a parallelipipedic period Y of the space of the variables yi (i = 1, 2, 3) formed by two parts Y, and Yz, separated by a smooth boundary I. We also denote by Yi (i = 1,2) the union of the Yi parts of all periods and assume that the Yi is connected. If R is the domain of the “composite material” in the space of the variables xi, we introduce the small parameter E and the domains fl,i defined by R,i = {Xi X E R, XE CYi} (i = 1, 2) In R,i we have the equations of balance of momentum and energy[5, 61
(1.1) (1.2) and the constitutive equation U.3) with (1.4) where o;j and e;j are the respective linear stress and strain tensors, 0’ is the temperature, uf are the components of the displacement vector, fi are the body force components, r is the heat supply and r, is the absolute reference temperature. The stiffness tensor c&,, and the strain-temperature tensor /3 are symmetric tensors Tj
444
H. I. ENE
The coefficients are k&-the thermal conductivity tensor, cf the specific heat at constant deformation and p’ the mass density. We look for Y-periodic coefficients in the variable y = X/C
c rjkh(X)
=
Cijkh
k:(x) = kij (:);
(:);
p’(X)
P7jtx)
=
p
=
(:);
Pij
(:)
Cc(X)
=
C (:)s
The boundary conditions on I are [UC]= 0 [U:jtlj] [VI
= 0
(1.5)
= 0
1.3 Two-scale asymptotic process In order to study the asymptotic process E+ 0 we consider the classical expansions [ 1, 21 u’(x, t) = UO(X,t) + EU’(X,y, t) t . .. B'(x,t)= e"(x, y,t>t &'(x,y,t)t . . eTjE
fTij(U’)
= f?yj(X, Y, t) + EfTij(X.
Y, t)
.
(1.6)
+...
where all functions are considered to be Y periodic with respect to the variable y. The dependence in x is obtained directly and through the variable y. The derivatives must be considered as d -*L+!L
dXi axi E Jyi and then
f?ij(
U) =
U) t
fZij,(
k
eijy
(U)
From (1.4) we have
&x,
Y, t> =
GjAu",
+
fGjy(U’)
e:j(x, Y, t) = eij,(u’> + eijJu’>
and from (1.3) U; = Ui(X,
Y,
t)t EU!j(X, Y* t) +. . .
(1.7)
with U7j(x, Y, t) = Cijkh(Y)e$,
Y, t) - Pij(Y>e”(x7 Yv t)
Uij(Xt Y, t) = cijkh(Y)e:j(%
Y, t)-
2. MACROSCOPIC
BALANCE
fiij(Y)e’(%
Yt t)
OF MOMENTUM
In order to obtain the “macroscopic equation” we use (1.6) and (1.7) in (1.1) and (1S) and we
On linear thermoelasticity of composite materials
445
identified the successive powers of e. At order CC’we have
a&O
(2.1)
a.vi
and at order lo
ad. a& $+$-P(y)$J=-fi(X). J
(2.2)
I
The mean operator (2.3) applied to (2.2) give us the macroscopic equation of balance of momentum
~_gE&_fi
(2.4)
J
where in the calculation we see from the Y-periodicity that:
Remark 2.1. The eqn (2.4) is the classical “homogeneized equation” for the linear elasticity[2].
3. MACROSCOPIC
THERMAL CONDUCTIVITY
TENSOR
Using (1.6) in (1.2) we have
[+~+!$+$+‘(~+$Jt”-l,
(&+f&)[kjtY) -To/j,(y)
(%t
,$t..
.)-c(y)
($t
2/g+..
.) =-r(x)
(3.1)
We shall see that 19’does not depend on y. From (3.1) at order E-* we have
(3.2) This equation holds in the hole Y in the sense of distributions (we have used the boundary conditions (1.5) written for O”,of course). Then from the Y-periodicity we obtain O”= f3’(x,t). Now, in the same way at order 6-l we obtain (3.3)
or
(3.3’)
446
H.I.ENE
This classical equation give us[l, 21
[kj(Y) (g+$)]-=k’ij$
(3.4)
k!i=kijt[ki\(,)g]
(3.5)
B'(x. y,t)= w'(y)$pc(x. t) I
(3.6)
where wi is the solution of the problem Find wi E H;,,(y) with 9’ = 0 satisfying
(3.7) Remark 3.1. The “macroscopic thermal conductivity tensor” is obtained in classical way L2.31. 4.MACROSCOPICCONSTITUTIVEEQUATION
In order to obtain the macroscopic constitutive equation we must return to the eqn (2.1) named the “local equation” [l, 21. Using (1.7), the eqn (2.1) takes the form
-$
=
B’!$.
[Cijkh(Y)ekhy(U')] ekh,
I
(4.1)
1
The variational formulation of (4.1) is
Find r~’ E Hi,,(y) satisfying
= ekdu@l2 I Cijkh(Y)ekhr(U’)eijy(_u)dY I v
k
I
UidY
Now we proceed with the calculation. We define Wk” and Q the solutions of the problems (4.3) respectively (4.4): Find W”” E H,!,,,(Y) with p”’ = 0 satisfying
= 1.2 I Cijm,e,,,(Wkh)eijr(~)dy y
Tidy (WC E
H,L(Y)
(4.3)
with 0 = 0 satisfying
Then the solution of (4.2) is g ‘(x, y. t) = &h,(!@)ykh - 6”@ abstraction of a function depending on x and t.
(4.5)
On linear thermoelasticity of composite materials
441
Using (4.5) we have succesively
em&‘) = ekh&O)emn,(Wkh) - f)“e,,,(@)
u;=Cijmn[SmkSnh +
%ny(~kh)lekJ&O)
- eO[Pij + cijft2fteiiJO)l and applying the mean operator (2.3) to the last equation we obtain the macroscopic constitutive equation (4.6) with cikh
cijkh
=
P Fj
=
(4.7)
+ [Cijmnemy(Wkh)l-
Pij +
[Cijf7&&0)1-
(4.8)
Remark 4.1. The macroscopic constitutive eqn (4.6) is of the same type than the microscopic one, but the macroscopic coefficients are different from the simply mean value. In fact (4.7) is the macroscopic stiffness tensor obtained first in the classical linear elasticity[2]. The macroscopic strain-temperature tensor p!j (4.8) depends also of the microscopic stiffness. 5. MACROSCOPIC
ENERGY EQUATION
From (3.1) at order 6’ we have the equation
- TOPij(Y)!g _
C(Y) g
= - r(x).
(5.1)
If we take the mean value of (5.1) we have succesively (eqns (3.9, (3.6) and (4.5) are used)
r!j
=
bij
+ yij Y =
=
@ij •+ [pkhekhy(b?l-
(5.2)
[Pij(Ykij,(O)l-
(5.3) Then, the macroscopic energy equation is
(5.4) Remark 5.1. The coefficient 71 from (5.4) is equal with /3: given by (4.8). For that it is necessary to use (4.3) and (4.4) in the difference ri:! - /3: and the value is zero.
448
H. 1. ENE
Than, the macroscopic energy equation is (5.4’) This fact is in accord with classical thermoelastical results. 6. FINAL REMARKS
6.1 All considerations hold in the case of uncoupled thermoelasticity. In this case the eqn (1.2) does not contain the second term in the 1.h.s. Then the macroscopic energy equation takes the form:
This is an example of the convergence theorem proved by G. Pass and D. Poli$evski[4]. 6.2 If we introduce the dimensional quantities like in [6], the coupling constant in the eqn (5.4) take the form EC2
3$E( 1 - 2V) where E is the Young’s modulus, v the Poisson’s ratio and (Y is the coefficient of volume expansion. For a composite material (glass/epoxy or born/epoxy) this constant is of order 1. Then in the study of dynamic problem of thermoelasticity it is necessary to take into consideration the coupled theory. Acknowledgement--I am very indebted to Eugen Sbos for interesting discussions and suggestions REFERENCES [l] A. BENSOUSSAN, 1. L. LIONS and G. PAPANICOLAU, Asympfolic Analysis for Periodic Sfructure. NorthHolland, Amsterdam (1978). [2] E. SANCHEZ-PALENCIA. Topics in Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127, Springer, Berlin (1980). [3] H. I. ENE and E. SANCHEZ-PALENCIA, Int. .I. Engng Sci. 20,623 (1982). [4] D. POLISEVSKI and G. PASA. Nume. Fun& Anal. and Optimiz. 3(l), 95 (1981). [5] R. M. CHRISTENSEN, Mechanics of Composite Materials. Wiley, New York (1979). [6] P. CHADWICK. Progress in Solid Mechanics (Edited by IN Suedden and R. Hill). Vol. 1. Chap. 6 May. North-Holland (1960). (Receiced 5 May 1982)
5. pp. 44348.
W?~7?!S/83/0S044M6$03.0010 @ 1983 Pergamon Press Ltd.
1983
ON LINEAR THERMOELASTICITY MATERIALS
OF COMPOSITE
HORIA I. ENE INCREST, Department of Mathematics, Bd. Pkii 220. 79622Bucharest, Romania (Communicated by E. Sd0.S) Abstract-We establish the equations of the linear thermo-elasticity for a composite material, using the homogenization method. The macroscopic coefficients are deduced and it is proved that the macroscopic energy equation contains new corrective coefficients.
1. INTRODUCTION
1.1 Generalities In the general framework of the homogenization method[l, 21 we consider the problem of linear thermoelasticity. The periodic structure of the composite material is associated with a small parameter c. The asymptotic process, e + 0, implies that the number of periods is very large. All mechanical and thermal properties are different in the matrix and in the inclusions, and their magnitude is of order one. 1.2 General equations We consider a parallelipipedic period Y of the space of the variables yi (i = 1, 2, 3) formed by two parts Y, and Yz, separated by a smooth boundary I. We also denote by Yi (i = 1,2) the union of the Yi parts of all periods and assume that the Yi is connected. If R is the domain of the “composite material” in the space of the variables xi, we introduce the small parameter E and the domains fl,i defined by R,i = {Xi X E R, XE CYi} (i = 1, 2) In R,i we have the equations of balance of momentum and energy[5, 61
(1.1) (1.2) and the constitutive equation U.3) with (1.4) where o;j and e;j are the respective linear stress and strain tensors, 0’ is the temperature, uf are the components of the displacement vector, fi are the body force components, r is the heat supply and r, is the absolute reference temperature. The stiffness tensor c&,, and the strain-temperature tensor /3 are symmetric tensors Tj
444
H. I. ENE
The coefficients are k&-the thermal conductivity tensor, cf the specific heat at constant deformation and p’ the mass density. We look for Y-periodic coefficients in the variable y = X/C
c rjkh(X)
=
Cijkh
k:(x) = kij (:);
(:);
p’(X)
P7jtx)
=
p
=
(:);
Pij
(:)
Cc(X)
=
C (:)s
The boundary conditions on I are [UC]= 0 [U:jtlj] [VI
= 0
(1.5)
= 0
1.3 Two-scale asymptotic process In order to study the asymptotic process E+ 0 we consider the classical expansions [ 1, 21 u’(x, t) = UO(X,t) + EU’(X,y, t) t . .. B'(x,t)= e"(x, y,t>t &'(x,y,t)t . . eTjE
fTij(U’)
= f?yj(X, Y, t) + EfTij(X.
Y, t)
.
(1.6)
+...
where all functions are considered to be Y periodic with respect to the variable y. The dependence in x is obtained directly and through the variable y. The derivatives must be considered as d -*L+!L
dXi axi E Jyi and then
f?ij(
U) =
U) t
fZij,(
k
eijy
(U)
From (1.4) we have
&x,
Y, t> =
GjAu",
+
fGjy(U’)
e:j(x, Y, t) = eij,(u’> + eijJu’>
and from (1.3) U; = Ui(X,
Y,
t)t EU!j(X, Y* t) +. . .
(1.7)
with U7j(x, Y, t) = Cijkh(Y)e$,
Y, t) - Pij(Y>e”(x7 Yv t)
Uij(Xt Y, t) = cijkh(Y)e:j(%
Y, t)-
2. MACROSCOPIC
BALANCE
fiij(Y)e’(%
Yt t)
OF MOMENTUM
In order to obtain the “macroscopic equation” we use (1.6) and (1.7) in (1.1) and (1S) and we
On linear thermoelasticity of composite materials
445
identified the successive powers of e. At order CC’we have
a&O
(2.1)
a.vi
and at order lo
ad. a& $+$-P(y)$J=-fi(X). J
(2.2)
I
The mean operator (2.3) applied to (2.2) give us the macroscopic equation of balance of momentum
~_gE&_fi
(2.4)
J
where in the calculation we see from the Y-periodicity that:
Remark 2.1. The eqn (2.4) is the classical “homogeneized equation” for the linear elasticity[2].
3. MACROSCOPIC
THERMAL CONDUCTIVITY
TENSOR
Using (1.6) in (1.2) we have
[+~+!$+$+‘(~+$Jt”-l,
(&+f&)[kjtY) -To/j,(y)
(%t
,$t..
.)-c(y)
($t
2/g+..
.) =-r(x)
(3.1)
We shall see that 19’does not depend on y. From (3.1) at order E-* we have
(3.2) This equation holds in the hole Y in the sense of distributions (we have used the boundary conditions (1.5) written for O”,of course). Then from the Y-periodicity we obtain O”= f3’(x,t). Now, in the same way at order 6-l we obtain (3.3)
or
(3.3’)
446
H.I.ENE
This classical equation give us[l, 21
[kj(Y) (g+$)]-=k’ij$
(3.4)
k!i=kijt[ki\(,)g]
(3.5)
B'(x. y,t)= w'(y)$pc(x. t) I
(3.6)
where wi is the solution of the problem Find wi E H;,,(y) with 9’ = 0 satisfying
(3.7) Remark 3.1. The “macroscopic thermal conductivity tensor” is obtained in classical way L2.31. 4.MACROSCOPICCONSTITUTIVEEQUATION
In order to obtain the macroscopic constitutive equation we must return to the eqn (2.1) named the “local equation” [l, 21. Using (1.7), the eqn (2.1) takes the form
-$
=
B’!$.
[Cijkh(Y)ekhy(U')] ekh,
I
(4.1)
1
The variational formulation of (4.1) is
Find r~’ E Hi,,(y) satisfying
= ekdu@l2 I Cijkh(Y)ekhr(U’)eijy(_u)dY I v
k
I
UidY
Now we proceed with the calculation. We define Wk” and Q the solutions of the problems (4.3) respectively (4.4): Find W”” E H,!,,,(Y) with p”’ = 0 satisfying
= 1.2 I Cijm,e,,,(Wkh)eijr(~)dy y
Tidy (WC E
H,L(Y)
(4.3)
with 0 = 0 satisfying
Then the solution of (4.2) is g ‘(x, y. t) = &h,(!@)ykh - 6”@ abstraction of a function depending on x and t.
(4.5)
On linear thermoelasticity of composite materials
441
Using (4.5) we have succesively
em&‘) = ekh&O)emn,(Wkh) - f)“e,,,(@)
u;=Cijmn[SmkSnh +
%ny(~kh)lekJ&O)
- eO[Pij + cijft2fteiiJO)l and applying the mean operator (2.3) to the last equation we obtain the macroscopic constitutive equation (4.6) with cikh
cijkh
=
P Fj
=
(4.7)
+ [Cijmnemy(Wkh)l-
Pij +
[Cijf7&&0)1-
(4.8)
Remark 4.1. The macroscopic constitutive eqn (4.6) is of the same type than the microscopic one, but the macroscopic coefficients are different from the simply mean value. In fact (4.7) is the macroscopic stiffness tensor obtained first in the classical linear elasticity[2]. The macroscopic strain-temperature tensor p!j (4.8) depends also of the microscopic stiffness. 5. MACROSCOPIC
ENERGY EQUATION
From (3.1) at order 6’ we have the equation
- TOPij(Y)!g _
C(Y) g
= - r(x).
(5.1)
If we take the mean value of (5.1) we have succesively (eqns (3.9, (3.6) and (4.5) are used)
r!j
=
bij
+ yij Y =
=
@ij •+ [pkhekhy(b?l-
(5.2)
[Pij(Ykij,(O)l-
(5.3) Then, the macroscopic energy equation is
(5.4) Remark 5.1. The coefficient 71 from (5.4) is equal with /3: given by (4.8). For that it is necessary to use (4.3) and (4.4) in the difference ri:! - /3: and the value is zero.
448
H. 1. ENE
Than, the macroscopic energy equation is (5.4’) This fact is in accord with classical thermoelastical results. 6. FINAL REMARKS
6.1 All considerations hold in the case of uncoupled thermoelasticity. In this case the eqn (1.2) does not contain the second term in the 1.h.s. Then the macroscopic energy equation takes the form:
This is an example of the convergence theorem proved by G. Pass and D. Poli$evski[4]. 6.2 If we introduce the dimensional quantities like in [6], the coupling constant in the eqn (5.4) take the form EC2
3$E( 1 - 2V) where E is the Young’s modulus, v the Poisson’s ratio and (Y is the coefficient of volume expansion. For a composite material (glass/epoxy or born/epoxy) this constant is of order 1. Then in the study of dynamic problem of thermoelasticity it is necessary to take into consideration the coupled theory. Acknowledgement--I am very indebted to Eugen Sbos for interesting discussions and suggestions REFERENCES [l] A. BENSOUSSAN, 1. L. LIONS and G. PAPANICOLAU, Asympfolic Analysis for Periodic Sfructure. NorthHolland, Amsterdam (1978). [2] E. SANCHEZ-PALENCIA. Topics in Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127, Springer, Berlin (1980). [3] H. I. ENE and E. SANCHEZ-PALENCIA, Int. .I. Engng Sci. 20,623 (1982). [4] D. POLISEVSKI and G. PASA. Nume. Fun& Anal. and Optimiz. 3(l), 95 (1981). [5] R. M. CHRISTENSEN, Mechanics of Composite Materials. Wiley, New York (1979). [6] P. CHADWICK. Progress in Solid Mechanics (Edited by IN Suedden and R. Hill). Vol. 1. Chap. 6 May. North-Holland (1960). (Receiced 5 May 1982)