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1、arXiv:math-ph/0405055v1 21 May 2004 Saint-Venants principle in dynamical porous thermoelastic media with memory for heat fl ux Gerardo Iovane,Francesca Passarella Department of Information Engineering and Applied Mathematics (DIIMA), University of Salerno, 84084 Fisciano (Sa), Italy Abstract In the
2、present paper, we study a linear thermoelastic porous material with a constitutive equation for heat fl ux with memory.An approximated theory of thermodynamics is presented for this model and a maximal pseudo free energy is determined. We use this energy to study the spatial behaviour of the thermod
3、ynamic processes in porous materials. We obtain the domain of infl uence theorem and establish the spatial decay estimates inside of the domain of infl uence.Further, we prove a uniqueness theorem valid for fi nite or infi nite body. The body is free of any kind of a priori assumptions concerning th
4、e behaviour of solutions at infi nity. 1Introduction The SaintVenants principle has a central role in more theoretical and applied questions of elasticity. An important review of research on the spatial behaviour of solutions for statical and dynamical problems was given by Horgan and Knowles 1 and
5、Horgan 2, 3. Relevant information on the spatial behaviour of the solutions for the dynamical problems of elasticity are given by the domain of infl uence theorem as it is presented by Gurtin 4. A further study in this connection was made by Chirit a 5 for the linear theory of thermoelasticity, wher
6、e a bounded solid is subjected to the action of nonzero boundary loads only on the end face. Recently, Fabrizio, Lazzari and Munoz Rivera 6 have studied a linear thermoelastic material which exhibits a constitutive equation for heat fl ux with memory. Some existence, uniqueness and asymptotic theore
7、ms have been established in connection with this approach. Further, Chirit a and Lazzari 7 have studied the spatial behaviour of the thermoelastic processes. The hereditary eff ects are taken into account in a dissipative boundary condition by Ciarletta 8 and Bartilomo and Passarella 9. E-mail addre
8、ss: iovanediima.unisa.it, passarelladiima.unisa.it 1 On the other hand, Ie san 10 has developed a linear theory of thermoelastic materials with voids, by generalizing some ideas of Cowin and Nunziato 11.In this theory the bulk density is written as the product of two fi elds, the matrix material den
9、sity fi eld and the volume fraction fi eld. This representation introduces an additional degree of kinematic freedom and it is compatible with the theory of granular materials developed by Goodman and Cowin 12. The spatial and temporal behaviour in linear thermoelasticity of materials with voids has
10、 been studied by Chirit a and Scalia in 13. Further, it is shown by Iovane and Passsarella in 14 the spatial exponential decay for the boundaryfi nal value problem associated to elastic porous materials with a memory eff ect for the intrinsic equilibrated body forces. In the present paper, we study
11、a linear thermoelastic porous material which exhibits eff ects of fading memory in the constitutive equation for the heat fl ux. Such model exhibits fi nite speeds for the propagation of thermal disturbances and hence leads to accurate modeling of the transient thermal behaviour. By applying the the
12、rmodynamical laws, we can prove the existence of a pseudofree energy potential such that the internal dissipation vanishes despite of the dissipative character of the model. Then, we use this energy for studying the spatial behaviour of thermodynamical processes. The paper is organized as follows. I
13、n Section 2, we present the thermodynamical restric- tions imposed by the thermodynamical laws on the model in question; then, we introduce a maximal pseudofree energy. In Section 3, we deduce a result for describing the domain of infl uence by using an appropriate time-weighted surface measure. Thi
14、s domain at instant t 0, T is the set Dtof all points of continuum for which r t, where r represents the distance from the bounded support DTof the (external) given data in the time interval 0, T and is a constant characteristic to the thermodynamic coeffi cients. We obtain, into the inner of the do
15、main of infl uence, a spatial decay of exponential type. The decay rate, characterized by a factor independent of time, is exp( r), where is a positive parameter independent of time and defi ning the time-weighted surface power measure. In Section 4, we prove that, into the inner of the domain of in
16、fl uence Dt, an energetic measure associated with the thermodynamic process tends to zero with a decay rate equal to ? 1 r t ? . 2Basic equations. Thermodynamic restrictions We consider a porous thermoelastic body that occupies a (regular) region B of the physical space IR3 in an assigned reference
17、confi guration. By identifying IR3with the associated vector space, an orthonormal system of reference is introduced, so that vectors and tensors will have components denoted by Latin subscripts ranging over 1, 2, 3. The letters in boldface stand for the tensor L of an order p 1, and Lij. . . k(p su
18、bscripts) are the components of the tensor L. Summation over repeated subscripts and other typical conventions for diff erential operations are implied, such as a superposed dot or a comma followed by a subscript to denote partial derivative with respect to time or with respect to the corresponding
19、coordinate. 2 We denote U u, , , where u is the displacement vector fi elds, is the change in volume fraction starting from the reference confi guration, is the temperature variation from the uniform reference temperature 0( 0). Further, is the bulk mass density and is the equilibrated inertia in th
20、e reference state. We restrict our attention to the linear theory of thermoelasticity in which the eff ect of fading memory is contained in the thermal gradient . The local balance equations become 10 for the present problem Sji,j+ fi= ui, hj,j+ g + = , qj,j+ r = ,on B (0,). (1) In the previous equa
21、tions: S and f are the stress tensor and body force, respectively; h, g and are the equilibrated stress vector, intrinsic and extrinsic equilibrated body force, respectively; , q and r are the rate at which heat is absorbed for a unit of volume, the heat fl ux vector and the (extrinsic) heat supply,
22、 respectively. Let E be the strain fi eld associated with u, tbe the history up to time t for the thermal gradient and t be the integrated history of the thermal gradient, i. e. Eij= 1 2(ui,j + uj,i),(2) t ,j(s) = ,j(t s), t ,j(s) = Z t ts ,j()d,s 0,+).(3) As it was shown by Ie san in 10, the consti
23、tutive equations are Sij= CijrsErs+ Dijs,s+ Bij + Mij, hi= DrsiErs+ Ais,s+ bi + ai, g = BrsErs bs,s m, = 0Mrs Ers 0as ,s 0m + c, (4) where the material coeffi cients of eqs. (4) are supposed continuous and bounded fi elds on B (the set B is the closure of B). Moreover, they satisfy the symmetry rela
24、tions Cijrs= Crsij= Cjirs,Dijr= Djir,Aij= Aji, Bij= Bji,Mij= Mji. (5) 3 Further, we consider for the heat fl ux the following constitutive equation as suggested by Fabrizio, Lazzari and Munoz Rivera in 6 qi(t) = 0 Z 0 Kij(s)t ,j(s)ds, (6) with the relaxation thermal conductivity K.We are assuming Ki
25、jare continuous and bounded fi elds on B; moreover, Kij(s) L2(0,) and they satisfy the relations Kij(s) = Kji(s),Kij() = 0.(7) By taking into account t ,j(0) = 0 and d dt t ,j(s) = ,j(t) t ,j(s), d ds t ,j(s) = t ,j(s), (8) the constitutive equation (6) can be written as qi(t) = 0 Z 0 Kij(s)t ,j(s)d
26、s. (9) In what follows, we study the restriction imposed by the fundamental laws of thermodynam- ics in terms of cyclic processes. With this aim, we recall the laws of the thermodynamics as considered in 6, 15, 16: First Law of Thermodynamics: For every cyclic process the following equality holds I
27、n (t) + Srs(t) Ers(t) + hs(t) ,s(t) g(t) (t) o dt = 0.(10) Second Law of Thermodynamics: For every cyclic process the following equality holds 1 2 0 I (t)(0 (t) + qs(t),s(t)dt 0,(11) and the equality sign holds if and only if the process is reversible. The relations (6, 11) imply Z d 0 ?Z 0 Kij(s)t
28、,j(s)ds ? ,i(t)dt 0(12) for any cycle on 0, d), and the equality sign holds if and only if tis a constant history. If we put ,i(t) = kicos(t)+kisin(t) into the relation (12) with d = 2 , then we obtain Z 0 (kiKij(s)kj+kiKij(s)kj)cos(s)ds+ Z 0 (kiKij(s)kjkiKij(s)kj)sin(s)ds 0, (13) 4 where is strictl
29、y positive and ki, kiare the continuous functions on B. Now, we introduce the Fourier transform, sine and cosine transforms of function f L2(, ), i. e. fF() = Z f()exp(i)d, fS() = Z 0 f()sin()d,fC() = Z 0 f()cos()d. (14) We remark that the Fourier inversion formula imply f() = r 2 Z 0 fS()sin()d;(15
30、) and the Plancherels theorem of the Fourier transform gives Z f()g()d = r 1 2 Z fF()gF ()d,(16) with f, g L2(-, ) and with g F is the complex conjugate of gF. We can easily prove by eqs. (8, 14) that tC ,i = tS ,i , tS ,i = 1 ,i(t) + tC ,i .(17) By putting ki= ki=kiin the relation (13) as Chirita a
31、nd Lazzari in 7, we obtain kiKC ijkj 0,k (k1,k2,k3) 6= 0, 0.(18) Thus, the relations (17, 18) imply that KS is a negative defi nite tensor. By using eqs. (9, 15, 16), we have qi(t) = r 2 0 Z 0 KS ij() tS ,j ()d,(19) and Kij() = r 2 Z 0 KS ij()sin()d. (20) If we integrate eq. (20) with the respect an
32、d we take into account the Riemann Lebesgue lemma and the hypotheses Kij() = 0, then it follows Kij(0) = r 2 Z 0 1 KS ij()d. (21) 5 The relations (7, 21) and the negative defi niteness of the tensor KSimply that K(0) is a symmetric and positive defi nite tensor, i.e. kiKij(0)kj 0,k (k1,k2,k3) 6= 0,
33、0.(22) Consequently, it follows that Kij(0)kikj KMkiki,k =k1,k2,k3,(23) where KM(x) 0 is the largest characteristic eigenvalue of K(x, 0). In what follows, we will assume that the bulk mass density , the equilibrated inertia and the constant heat c are strictly positive, continuous and bounded fi el
34、ds on B, so that 0 0 inf xB ,0 0 inf xB ,0 0. In particular, in the case E Eij, ,i, , from (4, 29) we get Sij= Sij(E) + Mij,hi=hi(E) + ai,g = g(E) m.(33) Taking into account eqs. (29, 30, 32, 33), we deduce W W(E) = 1 2(Sij Mij)Eij+ (hi ai),i ( g + m), W = Sij Eij+ hi ,i g + 0 0 , (34) and 2W M ?E i
35、jEij+ ,i,i+ 2?. (35) By setting F =S(E) in the relations (31,32), we have 2W(S(E) M ? Sij(E)Sij(E) + 1 hi(E)hi(E) + g2(E) ? .(36) Therefore, we conclude thanks to (31, 36) and thanks to Cauchy-Schwarzs inequality that ? Sji(E)Sji(E) + 1 hi(E)hi(E) + g2(E) ?2 = h 2F(E,S(E) i2 4W W(S(E) 2W M ? Sji(E)S
36、ji(E) + 1 hi(E)hi(E) + g2(E) ? . Consequently, it follows ? ? ?S(E) ? ? ? 2 =Sij(E)Sij(E) + 1 hi(E)hi(E) + g2(E) 2MW . (37) Thanks to the inequality for second-order tensors L and G (Lij+ Gij)(Lij+ Gij) (1 + )LijLij+ (1 + 1 )GijGij, 0, 7 and thanks to eqs. (33, 37), we get SijSij+ 1 hihi + g2 (1 + )
37、 ? ? ?S(E) ? ? ? 2 + (1 + 1 ) ? MijMij+ 1 aiai + m2 ? 2 (1 + )2W + (1 + 1 )M c 0 2, 0, (38) with sup xB M,M sup xB 0 c ? MijMij+ 1 aiai + m2 ? .(39) The Laws of thermodynamics imply the existence of the thermodynamical internal potential energy e and the entropy such that e(t)=(t) + Sji(t) Eji(t) +
38、hj(t) ,j(t) g(t) (t),(40) (t) 1 2 0 (t)(0 (t) + qj(t),j(t).(41) We introduce the pseudofree energy potential (t) = e(t) 0(t).(42) Moreover, we defi ne the maximal pseudofree potential energy Msuch that M(t) = (t)(t) 0 + Sji(t) Eji(t) + hj(t) ,j(t) g(t) (t) qj(t),j(t) 0 .(43) Then, the relations (40-
39、43) imply (t) M(t).(44) By taking into account the relations (4, 19, 342), a maximal pseudofree energy potential is M(t) = W (t) + c2(t) 20 1 Z 0 KS ij(x,) tS ,i (x,)tS ,j (x,)+ + KS ij(x,) tC ,i (x,)tC ,j (x,)d. (45) Since W is a positive defi nite quadratic form, , and c are strictly positive, KSi
40、s the negative defi nite tensor, we deduce that the functional Mis a norm. As in 7, we can prove by eq. (19) that |q(t)|2= 20 Z 0 qi(t) KS ij(x,) tS ,j(x,)d (46) 0 ? 2 Z 0 1 qi(t)qj(t) KS ij()d ?1/2? 2 Z 0 KS ij() tC ,i ()tS ,j()d ?1/2 . The relations (21, 25, 26, 45, 46) imply |q(t)|2 K00c ? 2M(t)
41、2W (t) c2(t) 0 ? .(47) 8 3A time-weighted surface power measure Throughout this work by an admissible process we mean an ordered array u, E, S, , , h, , k, q with the following properties i.ui, C2,2(B 0, +), C1,1(B 0, +); ii.Eij= Eji, i=,i, ki= t ,i C0,1(B 0, +); iii.Sij= Sji, hi, qi C1,0(B 0, +), a
42、nd which meets the equations of motion (1), the geometrical equations (2, 3), the consti- tutive equations (4, 6) and the following initial conditions ui(0) = u0 i, ui(0) = u0 i, (0) = 0, (0) = 0,(0) = 0.(48) Let s, h and q be the surface tractions, the surface equilibrated stress and the heat fl ux
43、, so si(t) = Sji(t)nj,h(t) = hj(t)nj,q(t) = qj(t)nj.(49) We denote by fi, si, u0 i, u 0 i, l, h, 0, 0, r, q, 0, 0 ,j the external data and assume that all functions are prescribed continuous functions. Now, we introduce the support DTof the external data and the body supplies on the time interval 0,
44、 T, i. e. the set of all x B such that: (1) if x B, then u0 i 6= 0 or u0 i 6= 0 or 06= 0 or 06= 0 or 06= 0 or 0 ,i(s) 6= 0 for some s (,0 (50) or fi(s) 6= 0 or (s) 6= 0 or r(s) 6= 0 for some s 0,T; (2) if x B, then si(s) ui(s) 6= 0 or h(s) (s) 6= 0 or q(s)(s) 6= 0 for some s 0,T. In what follows, we
45、 will assume that DTis a bounded set. We introduce a nonempty set D T such that DT D T B and (a) if DTB 6= , then we choose D T to be the smallest bounded regular region in B that includes DT; in particular, we set D T = DTif DTit is a regular region; (b) if 6= DT B, then we choose D T to be the sma
46、llest regular subsurface of B that includes DT; in particular, we set D T = DTif DTis a regular subsurface of B; (c) if DT= , then we choose D T to be an arbitrary nonempty regular subsurface of B. On this basis, we introduce the set Dr, by Dr= x B : D T (r) 6= , 9 where (r) is the open ball with ra
47、dius r and center at x. Further, we shall use the notation Brfor the part of B contained in B Drand we set B(r1, r2) = Br2 Br1, r1 r2. We denote by Srthe subsurface of Brcontained into the inner of B and whose outward unit normal vector is forwarded to the exterior of Dr. We can observe that the dat
48、a are null on Br, Sr. We defi ne the following time-weighted surface power function I(r, t) I(r,t) = Z t 0 Z Sr essi(s) ui(s) + h(s) (s) 1 0 q(s)(s)dads.(51) for a fi xed positive parameter and for any r 0, t 0, T The following theorems establish a set of properties for the surface power function I.
49、 These theorems will be useful in the study of the spatial behaviour of the thermoelastic processes. Lemma 1: Let be a thermoelastic process and b DTbe the bounded support of the external data on the time interval 0, T. Moreover, let I(r, t) be the time-weighted surface power function associated with and K be the kine