A Generalization of the Cowin and Nunziato's Domain

BULLETIN OF THE TRANSILVANIA UNIVERSITY OF BRAŞOV

A GENERALIZATION OF THE COWIN AND NUNZIATO'S DOMAIN

M. MARIN*, O. FLOREA*

Abstract. The domain of influence, proposed by Cowin and Nunziato, is extended to cover the thermoelasticity of micropolar bodies with voids. We prove

that for a finite time t > 0 the displacement field ui , the microrotation vector

i , the temperature  and the change in volume fraction  generate no disturbance outside a bounded domain B .

1. Introduction

It is remarkable to note that the theory of materials with voids or vacuous pores was first proposed by Nunziato and Cowin [8]. In this theory the authors introduce an additional degree of freedom in order to develope the mechanical behavior of a body in which the skeletal material is elastic and interstices are voids of material. The intended applications of the theory are to geological materials like rocks and soil and to manufactured porous materials. The linear theory of elastic materials with voids was developed by Cowin and Nunziato in [3]. Here the uniqueness and weak stability of solutions are also derived. Iesan in [4] has established the equations of thermoelasticity of materials with voids. An extension of these results to cover the theory of micropolar materials with voids was been made in our studies [6], [7]. In the present paper we first consider the basic equations and conditions of the mixed initial-boundary value problem in the context of thermoelasticity of micropolar bodies with voids. Next we define the domain of influence Bt of the data at time t associated with the problem. We adopt the method used in [1] and [5] to establish a domain of influence theorem. The main result asserts that in the context of theory considered, the solutions of the mixed initial-boundary value problem vanishes outside Bt , for a finite time t > 0 .

2. Basic equations

An anisotropic elastic material is considered. Assume a such body that occupies a properly regular region B of three-dimensional Euclidian space R3 bounded by a piecewise smooth surface B and we denote the closure of B by B . We use a fixed system of rectangular Cartesian axes Oxi ,(i = 1,2,3) and adopt Cartesian tensor notation. A

* Faculty of Mathematics and Computer Science, Transilvania University of Brasov * 158 A Generalization Of The Cowin And Nunziato's Domain

superposed dot stands for the material time derivate while a comma followed by a subscript denotes partial derivatives with respect to the spatial coordinates. Einstein summation on repeated indices is also used. Also, the spatial argument and the time argument of a function will be omitted when there is no likehood of confusion. The basic equations from thermoelasticity of micropolar bodies with voids are, [7] tij, j  Fi = u˙˙i , mij, j   ijkt jk  M i = Iij˙˙ j , (1) hi,i  g  L = ˙˙, (2)

T0˙ = qi,i  r. (3) The equations (1) are the motion equations, (2) is the balance of the equilibrated forces and (3) is the energy equation. We complete the above equations with - the constitutive equations tij = Cijmn mn  Bijmn mn  Bij  Dijk ,k  ij, mij = Bmnij mn  Cijmn mn  Cij  Eijk ,k ij, hi = Aij , j  Dmni mn  Emni mn  di  ai, (4) g = Bij ij  Cij ij   di ,i  m,

 = ij ij ij ij  m  ai ,i  a, qi = kij, j ; - the kinetic relations

 ij = u j,i   jikk ,  ij =  j,i , = T T0 ,  =  0. (5) In the above equations we have used the following notations:  -the constant mass density;  -the specific entropy; T0 -the constant absolute temperature of the body in its reference state; Iij -coefficients of micro inertia;  -the equilibrated inertia; ui -the components of displacement vector; i -the components of micro rotation vector;  -the volume distribution function which in the reference state is 0 ;  -the change in volume fraction measured from the reference state;  -the temperature variation measured from the reference temperature T0 ;  ij , ij -kinematic characteristics of the strain; tij -the components of the stress tensor; mij -the components of the couple stress tensor; hi -the components of the equilibrated stress vector; qi -the components of the heat flux vector; Fi -the components of the body forces; M i -the components of the body couple; r -the heat supply per unit time; g -the intrinsic equilibrated force; L -the extrinsic equilibrated body force; Aijmn , Bijmn ,...,kij -the characteristic functions of the material, and they obey the symmetry relations

Aijmn = Amnij , Cijmn = Cmnij , kij = k ji . (6) The entropy inequality implies kij,i, j  0. (7) Bulletin of the Transilvania University of Brasov Vol. 13(48) - 2006 159

To the system of field equations (1)-(5) we adjoin the following initial conditions 0 1 0 1 ui (x,0) = ui (x), u˙ i (x,0) = ui (x), i (x,0) = i (x), ˙ i (x,0) = i (x), (8)  (x,0) =  0 (x),  (x,0) =  0 (x), ˙ (x,0) =  1(x), x  B, and the following prescribed boundary conditions c ui = ui on B1 [0,t0 ), ti  tij n j = ti on B1 [0,t0 ), c i = i on B2 [0,t0 ), mi  mij n j = mi on B2 [0,t0 ), (9) c  =  on B3 [0,t0 ), h  hini = h on B3 [0,t0 ), c  =  on B4 [0,t0 ), q  qini = q on B4 [0,t0 ), c c c c where B1,B2 ,B3 and B4 with respective complements B1 ,B2 ,B3 and B4 are subsets of B , ni are the components of the unit outward normal to B , t0 is some instant 0 1 0 1 0 0 1 that may be infinite, ui , ui , i , i , ,  ,  , ui , ti , i , mi ,  , , q and h are prescribed functions in their domains. Introducing (5) and (4) into equations (1), (2) and (3), we obtained the following system of equations

u˙˙i = (Aijmn mn  Bijmn mn  Bij  Dijk ,k  ij ), j  Fi ,

Iij˙˙ j = (Bmnij mn  Cijmn mn  Cij  Eijk ) ,k ij ), j 

  ijk (Ajkmn mn  B jkmn mn  B jk  D jkm ,m   jk )  M i ,

˙˙ = (Dmni mn  Emni mn  di  Aij ) , j  ai ), j  L  (10)

 Bij ij  Cij ij   di ,i  m, ˙ 1 1 a = (kij, j ),i  r  ij˙ij ij˙ij  m˙  ai˙ ,i . T0 T0

By a solution of the mixed initial boundary value problem of the theory of thermo elasticity of bodies with voids in the cylinder 0 = B[0,t0 ) we mean an ordered array

(ui ,i ,, ) which satisfies the system (10) for all (x,t)0 , the boundary conditions (9) and the initial conditions (8).

3. Main result

We begin this section with the definition of the domain of influence. Next, we establish a domain of influence inequality, which is a counterpart of the inequality established in [5]. Finally, we shall prove a domain influence theorem in the context of thermo elasticity of bodies with voids. In all what follows we shall use the following assumptions on the material properties i)   0, I ij  0, k  0, T0  0, a  0; 160 A Generalization Of The Cowin And Nunziato's Domain

ii) Aijmn xij xmn  2Bijmn xij ymn  Cijmn yij ymn  2Bij xij  2Dijl xij zk  2  2Cij yij  2Eijk yij zk  2di zi    Aij zi z j  2  (xij xij  yij yij  zi zi   ), for all xij , yij , zi , ; iii) kiji j  ii , for all i . These assumptions are in agreement with the usual restrictions imposed in the mechanics of continua. The assumption iii) represent a considerable strengthening of the consequence (7) of the entropy production inequality.

For a sufficiently small  > 0 , let W (z) be a smooth nondecreasing function, vanishing in (,0] and equal to one in [,) and for 0  s  t ,  R  r  G(x, s) = W   t  s (11)    for some fixed positive R and t, where r =| x  x0 |, x0 is an arbitrary fixed point, c is a positive constant to be determined later. G(x, s) is a smooth function on B[0,t], vanishing outside  where

 =  S[x0 , R  c(t  s)]. s[0,t]

The sphere S(x0 , R) is defined as 3 S(x0 , R) = {x  R :| x  x0 |< R}. (12) Let U (x, s) be the function defined as 1 U (x, s) = u˙ u˙  I ˙ ˙  ˙ 2  a 2  A    2B    2 i i ij ij ij ijmn ij mn ijmn ij mn (13)  Cijmn ij mn  2Bij ij  2Dijk ,k  ij  2Cij ij  2  2Eijk ,k  ij  2di ,i  Aij ,i , j   (x, s) We also define the function K(x, s) 1 K(x, s) = [u˙ u˙  I ˙ ˙  ˙ 2  a 2        2   ](x, s). (14) 2 i i ij i i ij ij ij ij ,i ,i Taking into account the assumptions i) and ii) from (13) and (14) we deduce K(x, s)  U (x, s). (15) The next theorem is a necessary step to prove the main result.

Theorem 1. Let (ui ,i ,, ) be a solution to the system of equations (10) with the initial conditions (8) and the boundary conditions (9). Then for any R > 0,t > 0 and x0  B , we have that Bulletin of the Transilvania University of Brasov Vol. 13(48) - 2006 161

1 t U (x,t)dV  k   dV  U (x,0)dV  D[ x ,R] 0 D[ x ,Rc(ts)] ij ,i , j D[x ,Rct] 0 T0 0 0 t 1  [F u˙  M ˙  L˙  r ]dVds  0 D[ x ,Rc(ts)] i i i i 0 T0 (16) t 1  [tiu˙ i  mi˙ i  h˙  q ]dSds, 0 D[x ,Rc(ts)] 0 T0

where D(x0 , R) = {x  B :| x  x0 |< R}, D(x0 , R) = {x  B :| x  x0 |< R}.

Proof. Multiplying the equation (10) 1 by Gu˙ i , it results 1 d G (u˙ iu˙ i ) = GFiu˙ i  (Gtij u˙ i ), j  G, j tij u˙ i  2 dt (17)

 G(Aijmn mn  Bijmn mn  Bij  Dijk ,k   ij )u˙ i, j .

Multiplying the equation (10) 2 by G˙ i , it results 1 d G (I ˙ ˙ ) = GM ˙  (Gm ˙ )  G m ˙  2 dt ij i j i i ij i , j , j ij i (18)  G(Bmnij mn  Cijmn mn  Cij  Eijk ,k  ij )˙ i, j

  ijk (Ajkmn mn  B jkmn mn  B jk  D jkm ,m   jk )˙ i .

Multiplying the equation (10) 3 by G˙ , we get 1 d G (˙ 2 ) = GL˙  (Gh ˙ )  G h ˙  2 dt i ,i ,i i (19)  G(Aij , j˙ ,i  Dmni mn˙ ,i  Emni mn˙ ,i  di˙ ,i  ai˙ ,i ) 

 G(Bij ij˙  Cij ij˙  ˙  di ,i˙  m˙ ).

At last, multiplying the equation (10) 4 by G , we are led to

1 d 2 1 1 G (a ) = Gr  [(Gqi ),i  G,iqi ]  2 dt T0 T0 (20) 1  Gkij,i, j  G( ij˙ ij   ij˙ ij  m˙  ai˙ ,i ). T0 Additing equations (17), (18), (19) and (20) together, it results 162 A Generalization Of The Cowin And Nunziato's Domain

1 d G u˙ u˙  I ˙ ˙  ˙ 2  a 2   GF u˙  GM ˙  GL˙  2 dt i i ij i j i i i i 1  1   Gr  Gt u˙  m ˙  h ˙  q   T  ij i ij i j T j  0  0 , j

 GAijmn mn˙ij  Bijmn  mn˙ ij  ˙ mn ij  Cijmn mn˙ ij  (21)

 Bij ˙ ij   ij˙  Cij ˙ ij   ij  Dijk  ij˙ ,k  ˙ ij ,k 

 Eijk  ij˙ ,k  ˙ ij ,k  di ˙ ,i  ˙ ,i  Aij ,i˙ , j  ˙  1 1  G, j tij u˙ i  G, j mij˙ i  G,i hi˙  G,i qi  Gkij,i, j T0 T0 The relation (21) may be restated as follows 1 d G u˙ u˙  I ˙ ˙  ˙ 2  a 2  A    2B    2 dt i i ij i j ijmn mn ij ijmn mn ij

 Cijmn mn ij  2Bij ij  2Cij ij  2Dijk  ij ,k  2  2Eijk  ij ,k  2di ,i  Aij ,i , j     (22)  1   1   G F u˙  M ˙  L˙  r   Gt u˙  m ˙  h ˙  q    i i i i T   ij i ij i j T j   0    0 , j 1 1 G, j tij u˙ i  G, j mij˙ i  G,i hi˙  G,i qi  kij,i, j T0 T0 That is 1 ˙ 1 1 GU  kij,i, j = G(Fiu˙ i  M i˙ i  L˙  r )  2 T0 T0 (23) 1 1  G(tij u˙ i  mij˙ i  h j˙  q j ), j  G, j (tij u˙ i  mij˙ i  h j˙  q j ). T0 T0 Integrating both sides of equations (23) over B[0,t] and by using the divergence theorem and the boundary conditions (9), we deduce 1 t GU (x,t)dV  Gkij,i, j dVds = GU (x,0)dV  B 0 B B T0 t 1  G(tiu˙ i  mi˙ i  h˙  q )dVds  0 B T0 (24) t 1  G(F u˙  M ˙  L˙  r )dVds  0 B i i i i T0 t t 1  G˙ U (x, s)dVds  G (t u˙  m ˙  h ˙  q  )dVds 0 B 0 B , j ij i ij i j j T0 Bulletin of the Transilvania University of Brasov Vol. 13(48) - 2006 163

Taking into account the definition (11) of the function G, we find that 1  G, j tij u˙ i  G,l mij˙ i  G,i hi˙  G,i qi  T0

1 x j 1 x j 1 x j 1 x j  W ' tij u˙ i  W ' mij˙ i  W ' hi˙  W ' qi  c r c r c r cT0 r 1 1 (25) W ' A  x  B  x  B x  D  x   x u˙  c  r ijmn mn j ijmn mn j ij j ijk ,k j ij j i

 Bmnij mn x j  Cijmn mn x j  Cijx j  Eijk ,k x j   ijx j ˙ i  1  Dmni mn xi  Emni mn xi  Aij , j xi  dixi  aixi ˙  kij, jxi ] T0 where dW W ' =  .  dr We now make use of arithmetic-geometric mean inequality 1 a 2 ab  (  b 2 p 2 ) (26) 2 p 2 to the last terms of relation (25) and by choosing suitable parameters p we can find c such that

1 ' | G, jtiju˙ i  G, j mij˙ i  G,ihi˙  G,iqi | W K(x, s), (27) T0 and that t ˙ t 1 GU (x, s)dVds  (G, j tij u˙ i  G, j mij˙ i  G,i hi˙  G,i qi )dVds  0 B 0 B T 0 (28) t '  W (x, s)[K(x, s) U (x, s)]dVds  0. 0 B By using the inequality (28) in equation (24), it results 1 t GU (x,t)dV  Gk   dVds  GU (x,0)dV  B 0 B ij ,i , j B T0 t 1  G(Fiu˙ i  M i˙ i  L˙  r )dVds  (29) 0 B 2  T0 t 1  G(tiu˙ i  mi˙ i  h˙  q )dVds. 0 B T0 Letting   0 into relation (29), G tends bounded to the characteristic function of  and we get the inequality (16). Based on the above estimations, we can now prove the main result of our study: the domain of influence theorem. 164 A Generalization Of The Cowin And Nunziato's Domain

Let B(t) be the set of points x  B such that: 0 1 0 1 0 1 0 (1) x  B  ui  0 or ui  0 or  i  0 or i  0 or   0 or   0 or   0 or  [0,t] such that Fi (x, )  0 or M i (x, )  0 or L(x, )  0 or r(x, )  0;

(2) x B1   [0,t] such that ui (x, )  0, c (3) x B1   [0,t] such that ti (x, )  0,

(4) x B2   [0,t] such that i (x, )  0, c (5) x B2   [0,t] such that mi (x, )  0,

(6) x B2   [0,t] such that  (x, )  0, c (7) x B2   [0,t] such that h(x, )  0,

(8) x B3   [0,t] such that  (x, )  0, c (9) x B3   [0,t] such that q(x, )  0. The domain of influence of the data at instant t is defined as

Bt = {x0  B : B(t)  S (x0 ,ct)  }, (30) where  is the empty set.

Theorem 2. Let (ui ,i ,, ) be a solution to the system of equations (10) with the initial conditions (8) and the boundary conditions (9). Then we have ui = 0,  = 0,  = 0, on {B \ Bt }[0,t].

Proof. For any x0  B \ Bt and  [0,t], by using the inequality (16) with t =  and R = c(t  ), we obtain 1  U (x, )dV  k   dVds  D[ x ,c(t )] 0 D[x ,c(ts)] ij ,i , j 0 T0 0  1  U (x,0)dV  (Fiu˙ i  M i˙ i  L˙  r )dVds  (31) D[x ,ct)] 0 D[x ,c(ts)] 0 0 T0  1  (tiu˙ i  mi˙ i  h˙  q )dSds. 0 D[x ,c(ts)] 0 T0

Since x0  B \ Bt , we have x  D(x0 ,ct)  x  B(t) and hence U (x,0)dV = 0. (32) D[ x0,ct]

Moreover, since D[x0 ,c(t  s)]  D(x0 ,ct), we have  1 (Fiu˙ i  M i˙ i  L˙  r )dVds = 0, (33) 0 D[x ,c(ts)] 0 T0 Bulletin of the Transilvania University of Brasov Vol. 13(48) - 2006 165

 1 (tiu˙ i  mi˙ i  h˙  q )dVds = 0. (34) 0 D[ x ,c(ts)] 0 T0 Taking into account the assumption iii) and the relations (32), (33) and (34) we obtain U (x, )dV  0, (35) D[ x0,c(t )] and with the aid of inequality (15), we get K(x, )dV  0, (36) D[ x0,c(t )] From the definition of K, it results u˙ i (x0 , ) = 0, u˙ i (x0 , ) = 0, (x0 , ) = 0,  (x0 , ) = 0, for any (x0 , ){B \ Bt }[0,t].

Finally, since ui (x0 ,0) = 0, i (x0 ,0) = 0 for any x0  B \ Bt , we deduce ui (x0 , ) = 0, i (x0 , ) = 0, (x0 , ) = 0,  (x0 , ) = 0, for any (x0 , ){B \ Bt }[0,t] and the proof of Theorem 2 is complete.

References

1. Carbonaro,B., Russo,R., J. Elasticity, 14, 163-174 (1984) 2. Chandrasekharaiah,D.S., J. Elasticity, 18, 173-179 (1987) 3. Cowin, S.C., Nunziato, J.W., J. Elasticity, 13, 125-147 (1983) 4. Iesan, D., J. Elasticity, 15, 215-224 (1985) 5. Ignaczak,J., Carbonaro,B., J. Thermal Stresses, 9, 79-91 (1986) 6. Marin, M., C.R. Acad. Sci. Paris, t. 231, Serie II b, 1995, 475-480 7. Marin, M., J. Comp. Appl. Math., 67 (1996), 235-247 8. Nunziato,J.W., Cowin, S.C., Arch. Rat. Mech. Anal., 72, 175- 201 (1979)

O generalizare a domeniului lui Cowin si Nunziato

Rezumat: Domeniul de influenta, propus de Cowin si Nunziato este extins pentru a acoperitermoelasticitatea corpurilor micropolare cu gauri.

Demonstram ca pentru un timp finit t  0 campul de deplasare ui , vectorul de

microrotatie i , temperatura  si schimbarea in fractiunea de volum  nu genereaza perturbari in afara domeniul marginit B.