Viscoelastic materials with a double porosity structure

UPCommons Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints This is a post-peer-review, pre-copy edit version of an article published in Comptes rendus mécanique . The final authenticated version is available online at: https://doi.org/10.1016/j.crme.2018.12.004. Published paper : Iesan, D.; Quintanilla, R. Viscoelastic materials with a double porosity structure. "Comptes rendus mécanique", 25 Gener 2019, p. 1-17. doi:10.1016/j.crme.2018.12.004 URL d'aquest document a UPCommons E-prints: https://upcommons.upc.edu/handle/2117/127597 Viscoelastic materials with a double porosity structure D. Ie¸san1, R. Quintanilla2 1“Octav Mayer” Institute of Mathematics (Romanian Academy), Bd. Carol I, nr. 8, 700506, Ia¸si, Romania, [email protected]o 2Dep. Matem`atiques, ESEIAAT, Universitat Polit`ecnica de Catalunya, Colom, 11. Terrassa (08222). Barcelona, Spain, [email protected] Abstract This paper in concerned with the linear theory of materials with memory which possess a double porosity structure. First, the formulation of the initial-boundary- value problem is presented. Then, a uniqueness result is established. The semigroup theory of linear operators is used to prove existence and continuous dependence of solutions. A minimum principle for the dynamical theory is also derived. Key words: Viscoelastic porous materials; Uniqueness and existence results; Reciprocity; Minimum principle. 1 Introduction The mechanics of solids with a double porosity structure is of interest in geophysics and in mechanics of bone. In the recent years the deformation of these materials has been a subject of intensive study (see, e.g., Cowin, 1993; Berryman and Wang, 2000; Khalili and Selvadurari, 2003; Straughan, 2013, 2016; Ie¸san and Quintanilla, 2014; Svanadze, 2018, and references therein). In the first studies of the so-called double porosity model, the authors used Darcy’s law and deduced the equations for the displacement vector and the pressures associated with the porous structure. In the equilibrium theory the fluid pressures become independent of the displacement vector field. By using the theory of materials with voids, Ie¸san and Quintanilla (2014) have derived a theory of thermoelasticity for materials with a double porosity structure. In this theory the porosities are coupled with the displacement field even in the static case. The theory of elastic materials with voids has been established Preprint submitted to Elsevier 18 December 2018 by Nunziato and Cowin (1979) for the behavior of porous solids in which the skeletal or matrix materials are elastic and the interstices are void of material. Recently, some papers have been devoted to the rate theory of viscoelastic materials with double porosity (see, e.g., Svanadze, 2014). The present paper is concerned with the linear theory of materials with memory which possess a double porosity structure. The history of motion is important for rheological materials and in the dissipation and relaxation phenomena it plays a central role. In the classical theory of viscoelastic materials, Day (1971) proved that the work done in every closed strain path starting from zero is invariant under time-reversal if and only if the stress relaxation function is symmetric. Gurtin (1972) derived an extension of Day’s result within the context of the thermodynamics of materials with memory. In the first part of this paper we use the results established by Day (1971) and Gurtin (1972) to present the constitutive equations of a viscoelastic material with a double porosity structure. Then, by using the method given by Gurtin et al. (1979) we derive a uniqueness theorem for the initial-boundary-value problem. In the second part of the paper we consider the dynamic theory with Dirichlet boundary conditions. We use the semigroup theory of linear operators to obtain existence and continuous dependence of solutions. With a view toward a presentation of a minimum principle we first establish a reciprocity relation. Then we present a minimum principle of Reiss type. 2 Porous viscoelastics solids Let us denote by Bthe reference configuration occupied by a body at time t0. We refer to motion of the continuum to a fixed system of rectangular Cartesian axes Oxi, (i= 1,2,3). The conventions adopted with regard to tensor indices are as follows: Latin indices (unless otherwise specification) are understood to range over the integers (1,2,3) whereas Greek indices are confined to the range (1,2). The usual summation convention applies to all indices. Moreover, subscripts preceded by a comma denote partial differentiation with respect to the corresponding material coordinate and a superposed dot denotes the material derivative with respect to the time t. Following Gurtin (1972a), we write ∇(n)f(x, t) for the n-th gradient of fwith respect to xholding tfixed and f(n)(x, t) for the n-th derivative of fwith respect to tholding xfixed. We say that fis of class CM,N on B×(0, t0) if fis continuous on B×(0, t0) and the functions ∇(m)f(n),m∈ {0,1, . . . , M}, n∈ {0,1, . . . , N},m+n≤max(M, N), exist and are continuous on B×(0, t0). 2 In what follows we consider the linear theory of viscoelastic materials with a double porosity structure. Let uibe the components of the displacement field. The components of the infinitesimal strain field are given by eij =1 2(ui,j +uj,i).(1) We denote by ν1the volume fraction field corresponding to pores and by ν2 the volume fraction field corresponding to fissures. Let us denote by ν∗ 1and ν∗ 2the volume fractions in the reference configuration. Let ϕbe the change in volume fraction ν1from the reference value ν∗ 1, and let ψbe the change in volume fraction ν2from the reference value ν∗ 2. The equations of motion can be expressed as (Nunziato and Cowin, 1979) tji,j +Fi=ρ¨ui, σj,j +ξ+G=κ1¨ϕ, τj,j +ζ+L=κ2¨ ψ. (2) Here we have used the following notations: tij is the stress tensor, σjand τjare equilibrated stress vectors, ξand ζare the intrinsic equilibrated body forces, Fiis the body force, Gis the extrinsic equilibrated body force associated to pores, Lis the equilibrated body force associated to fissures, ρis the reference mass density, and κ1and κ2are coefficients of inertia (cf. Nunziato and Cowin, 1979). In the classical theory of viscoelastic materials, Day (1971) proved that the work done in every closed strain path starting from zero is invariant under time-reversal if and only if the stress relaxation function is symmetric. Gurtin (1972b) extended this result to the thermodynamics of materials with memory. By using the results of Day (1971) and Nunziato and Cowin (1979) we are led to the following constitutive equations of centrosymmetric viscoelastic materials with a double porosity structure tij(x, t) = Zt −∞ [Cijkl(x, t −s)˙ekl(x, s) + Bij(x, t −s) ˙ϕ(x, s) +Dij(x, t −s)˙ ψ(x, s)]ds, σi(x, t) = Zt −∞ [αij(x, t −s) ˙ϕ,j(x, s) + βij(x, t −s)˙ ψ,j(x, s)]ds, τi(x, t) = Zt −∞ [βji(x, t −s) ˙ϕ,j(x, s) + γij(x, t −s)˙ ψ,j(x, s)]ds, (3) ξ(x, t) = −Zt −∞ [Bij(x, t −s)˙eij(x, s) + α1(x, t −s) ˙ϕ(x, s) +α3(x, t −s)˙ ψ(x, s)]ds, ζ(x, t) = −Zt −∞ [Dij(x, t −s)˙eij(x, s) + α3(x, t −s) ˙ϕ(x, s) +α2(x, t −s)˙ ψ(x, s)]ds. The relaxation functions are twice continuously differentiable and have the 3 following symmetries Cijrs =Crsij =Cjirs, Bij =Bji, Dij =Dji, αij =αji, γij =γji.(4) By an admissible process we mean and ordered array of functions π= (ui, ϕ, ψ, eij, tij, σi, τi, ξ, ζ), defined on B×(−∞,∞) with the properties: (i) ui, ϕ and ψare of class C2; (ii) ui,˙ui,¨ui, ϕ, ˙ϕ, ¨ϕ, ψ, ˙ ψ, ¨ ψ, eij, ϕ,i and ψ,i are continuous on B×(−∞,∞); (iii) eij =eji,tij =tji; (iv) tij, σjand τkare functions of class C1,0on B×(−∞,∞); (v) tij, tij,i,σk, σj,j, τj, τi,i,ξand ζare continuous on B×(−∞,∞). To the above equations we must adjoin the initial data and boundary conditions. The initial data consists of the functions π∗= (u∗ i, ϕ∗, ψ∗, e∗ ij, t∗ ij, σ∗ i, τ∗ i, ξ∗, ζ∗) defined on B×(−∞, t∗). In what follows we shall consider t∗= 0.We consider processes that correspond to initial data π∗, π(r)=π∗,(5) where π(r)= (ui, ϕ, ψ, eij, tij, σi, τi, ξ, ζ) is the restriction of the admissible process πto B×(−∞,0). Clearly, if πis an admissible process that satisfies the condition (5), then ui, ϕ and ψsatisfy the initial conditions ui(x,0) = lim t→0u∗ i(x, t)≡u0 i(x),˙ui(x,0) = lim t→0˙u∗ i(x, t)≡v0 i(x), ϕ(x,0) = lim t→0ϕ∗(x, t)≡ϕ0(x),˙ϕ(x,0) = lim t→0˙ϕ∗(x, t)≡ϕ0 1(x),(6) ψ(x,0) = lim t→0ψ∗(x, t)≡ψ0(x),˙ ψ(x,0) = lim t→0 ˙ ψ∗(x, t)≡ψ0 1(x),x∈B. We consider the following boundary conditions ui=e uion S1×I, tjinj=e tion S2×I, ϕ=e ϕon S3×I, σjnj=e σon S4×I, (7) ψ=e ψon S5×I, τjnj=e τon S6×I, where I= (0,∞), Sk,(k= 1,2,...,6), are subsets of the boundary ∂B so that S1∪S2=S3∪S4=S5∪S6, S1∩S2=S3∩S4=S5∩S6=∅, and e ui,e ϕ, e ψ, e ti,e σand e τare prescribed functions. Throughout this paper we assume that: (i) Fi, G and Lare continuous on B×I; (ii) e ui,e ϕand e ψare continuous on S1×I, S3×I and S5×I, respectively; (iii) e ti,e σand e τare continuous in time and piecewise regular on S2×I,S4×Iand S6×I, respectively; (iv) ρis continuous and strictly positive on B. We say that π= (ui, ϕ, ψ, eij, tij, σi, τi, ξ, ζ) is a viscoelastic process corresponding to the body loads (Fi, G, L) if πis an admissible process that satisfies the equations (1)-(3). By a solution of the problem we mean a viscoelastic process corresponding to the body loads (Fi, G, L) that satisfies the initial history condition (5) and the boundary conditions (7). Let us present an alternative form of the constitutive equations. We introduce the convolution (f∗g)(x, t) = Zt 0f(x, t −s)g(x, s)ds, 4 where fand gare functions on B×Ithat are continuous in time. We denote sij(t) = Z∞ 0[˙ Cijmn(t+s)emn(−s) + ˙ Bij(t+s)ϕ(−s) +˙ Dij(t+s)ψ(−s)]ds, πi(t) = Z∞ 0[ ˙αij(t+s)ϕ,j(−s) + ˙ βij(t+s)ψ,j(−s)]ds, χi(t) = Z∞ 0[˙ βji(t+s)ϕ,j(−s) + ˙γij(t+s)ψ,j(−s)]ds, (8) ν(t) = −Z∞ 0[˙ Bij(t+s)eij(−s) + ˙α1(t+s)ϕ(−s) + ˙α3(t+s)ψ(−s)]ds, ϑ(t) = −Z∞ 0[˙ Dij(t+s)eij(−s) + ˙α3(t+s)ϕ(−s) + ˙α2(t+s)ψ(−s)]ds, where, for convenience, we have expressed the argument x. Since the process π∗is prescribed it follows that sij, πi, χi, v and ϑare given functions. With the help of (8) we can express the constitutive equations (3) in the form tij =sij +d dt(Cijkl ∗ekl +Bij ∗ϕ+Dij ∗ψ), σi=πi+d dt(αij ∗ϕ,j +βij ∗ψ,j), τi=χi+d dt(βji ∗ϕ,j +γij ∗ψ,j),(9) ξ=ν−d dt(Bij ∗eij +α1∗ϕ+α3∗ψ), ζ=ϑ−d dt(Dij ∗eij +α3∗ϕ+α2∗ψ). 3 Uniqueness Uniqueness results in the classical viscoelasticity have been presented in various works (see, e.g., Leitman and Fisher, 1973 and references therein). A uniqueness theorem in the case of viscoelastic materials with voids has been presented by Ciarletta and Scalia (1971). In this section we use the results of Gurtin et al. (1979) to derive a uniqueness result for the problem formulated in Section 2. Let π= (ui, ϕ, ψ, eij, tij, σi, τi, ξ, ζ) be a viscoelastic process corresponding to the body loads (Fi, G, L). In view of the equations of motion (2), we find that tij ˙eij +σi˙ϕ,i +τi˙ ψ,i −ξ˙ϕ−ζ˙ ψ= (tji ˙ui+σj˙ϕ+τk˙ ψ),j +Fi˙ui+G˙ϕ+L˙ ψ−ρ˙ui¨ui−κ1˙ϕ¨ϕ−κ2˙ ψ¨ ψ. 5 By using the divergence theorem we get ZB(tij ˙eij +σi˙ϕ,i +τi˙ ψ,i −ξ˙ϕ−ζ˙ ψ)dv =Z∂B(tji ˙ui+σj˙ϕ+τj˙ ψ)njda (10) +ZB(Fi˙ui+G˙ϕ+L˙ ψ)dv −1 2 d dt ZB(ρ˙ui˙ui+κ1˙ϕ2+κ2˙ ψ2)dv. We define the functions fij,ηand χby fij(x, t1, t2) = eij(x, t1)−eij(x, t2), η(x, t1, t2) = ϕ(x, t1)−ϕ(x, t2), χ(x, t1, t2) = ψ(x, t1)−ψ(x, t2),x∈B, t1, t2∈I. (11) Let us introduce the notations W(t1, t2;q) = 1 2Cijmn(q)fij(t1, t2)fmn(t1, t2) + Bij(q)fij(t1, t2)η(t1, t2) +Dij(q)fij(t1, t2)χ(t1, t2) + 1 2αij(q)η,i(t1, t2)η,j(t1, t2) +βij(q)η,i(t1, t2)χ,j(t1, t2) + 1 2γij(q)χ,i(t1, t2)χ,j(t1, t2) +1 2α1(q)η2(t1, t2) + α3(q)η(t1, t2)χ(t1, t2) + 1 2α2(q)χ2(t1, t2), V(t1, t2;q) = 1 2˙ Cijmn(q)fij(t1, t2)fmn(t1, t2) + ˙ Bij(q)fij(t1, t2)η(t1, t2) +˙ Dij(q)fij(t1, t2)χ(t1, t2) + 1 2˙αij(q)η,i(t1, t2)η,j(t1, t2) +˙ βij(q)η,i(t1, t2)χ,j(t1, t2) + 1 2˙γij(q)χ,i(t1, t2)χ,j(t1, t2) (12) +1 2˙α1(q)η2(t1, t2) + ˙α3(q)η(t1, t2)χ(t1, t2) + 1 2˙α2(q)χ2(t1, t2), U(t1, t2;q) = 1 2¨ Cijmn(q)fij(t1, t2)fmn(t1, t2) + ¨ Bij(q)fij(t1, t2)η(t1, t2) +¨ Dij(q)fij(t1, t2)χ(t1, t2) + 1 2¨αij(q)η,i(t1, t2)η,j(t1, t2) + ¨ βij(q)η,i(t1, t2)χ,j(t1, t2) +1 2¨γij(q)χ,i(t1, t2)χ,j(t1, t2) + 1 2¨α1(q)η2(t1, t2) + ¨α3(q)η(t1, t2)χ(t1, t2) +1 2¨α2(q)χ2(t1, t2), t1, t2, q ∈I, where we have suppressed the argument x. Lemma 1. Let π= (ui, ϕ, ψ, eij, tij, σi, τi, ξ, ζ)be an admissible process that 6 corresponds to null initial history and satisfies the equations (3). Then Zt 0(tij ˙eij +σi˙ϕ,i +τi˙ ψ,i −ξ˙ϕ−ζ˙ ψ)dv =W(t, 0; t) −Zt 0V(s, 0; s)ds −Zt 0V(t, s;t−s)ds (13) +1 2Zt 0Zt 0U(r, s;|r−s|)drds. Proof. Since πcorresponds to null initial history, we have ui(x, t)=0, ϕ(x, t)=0, ψ(x, t) = 0,x∈B, t ∈(−∞,0].(14) In this case, from (11) we find that fij(x, t, 0) = eij(x, t), η(x, t, 0) = ϕ(x, t), γ(x, t, 0) = ψ(x, t), t ∈I. (15) Moreover, the constitutive equations (3) become tij =Cijmn(0)emn(t) + Bij(0)ϕ(t) + Dij(0)ψ(t) +Zt 0[˙ Cijmn(t−s)emn(s) + ˙ Bij(t−s)ϕ(s) + ˙ Dij(t−s)ψ(s)], σi=αij(0)ϕ,j(t) + βij(0)ψ,j(t) + Zt 0[ ˙αij(t−s)ϕ,j(s) + ˙ βij(t−s)ψ,j(s)]ds, τi=βji(0)ϕ,j(t) + γij(0)ψ,j(t) + Zt 0[˙ βji(t−s)ϕ,j(s) + ˙γij(t−s)ψ,j(s)]ds, (16) ξ=−Bij(0)eij(t)−α1(0)ϕ(t)−α3(0)ψ(t) −Zt 0[˙ Bij(t−s)eij(s) + ˙α1(t−s)ϕ(s) + ˙α3(t−s)ψ(s)]ds, ζ=−Dij(0)eij(t)−α3(0)ϕ(t)−α2(0)ψ(t)−Zt 0[˙ Dij(t−s)eij(s) + ˙α3(t−s)ϕ(s) + ˙α2(t−s)ψ(s)]ds. By using (14), we can write Zt 0(tij ˙eij +σi˙ϕ,i +τi˙ ψ,i −ξ˙ϕ−ζ˙ ψ)ds =tijeij +σiϕ,i +τiψ,i −ξϕ −ζψ −Zt 0(˙ tijeij + ˙σiϕ,i + ˙τiψ,i −˙ ξϕ −˙ ζψ)ds. (17) It follows from (4), (12), (15) and (16) that tijeij +σiϕ,i +τiψ,i −ξϕ −ζψ = 2W(t, 0; 0) + J(t),(18) 7 where J(t) = Zt 0{˙ Cijmn(t−s)eij(t)emn(s) + ˙ Bij(t−s)[eij(t)ϕ(s) + eij(s)ϕ(t)] +˙ Dij(t−s)[eij(t)ψ(s) + eij(s)ψ(t)] + ˙αij(t−s)ϕ,j(s)ϕ,i(t) +˙ βij(t−s)[ψ,j(s)ϕ,i(t) + ψ,j(t)ϕ,i(s)] + ˙γij(t−s)ψ,j(s)ψ,i(t) (19) + ˙α1(t−s)ϕ(s)ϕ(t) + ˙α3(t−s)[ψ(s)ϕ(t) + ψ(t)ϕ(s)] + ˙α2(t−s)ψ(s)ψ(t)}ds. In a similar way we obtain ˙ tijeij + ˙σiϕ,i + ˙τiψ,i −˙ ξϕ −˙ ζψ =d dtW(t, 0; 0) + 2V(t, 0; 0) + P(t),(20) where P(t) = Zt 0{¨ Cijmn(t−s)emn(s)eij(t) +¨ Bij(t−s)[ϕ(s)eij(t) + ϕ(t)eij(s)] + ¨ Dij(t−s)[ψ(s)eij(t) +ψ(t)eij(s)] + ¨αij(t−s)ϕ,j(s)ϕ,i(t) + ¨ βij(t−s)[ψ,j(s)ϕ,i(t) (21) +ψ,j(t)ϕ,i(s)] + ¨γij(t−s)ψ,j(s)ψ,i(t) + ¨α1(t−s)ϕ(s)ϕ(t) + ¨α3(t−s)[ψ(s)ϕ(t) + ψ(t)ϕ(s)] + ¨α2(t−s)ψ(s)ψ(t)}. By (17), (18) and (20) we get Zt 0(tij ˙eij +αi˙ϕ,i+τi˙ ψ,i−ξ˙ϕ−ζ˙ ψ)ds =W(t, 0; 0)−2Zt 0V(s, 0; 0)ds+Q(t),(22) where Q(t) = J(t)−Zt 0P(τ)dτ, t ∈I. (23) 8 Therefore, we can obtain the system        B∗C∗D∗ E∗F∗H∗ J∗K∗M∗               u ϕ ψ        =        m n1 n2        ,(41) where B∗ i(u) = ui−ρ−1[(Cijkl(0) + Z∞ 0 ˙ Cijkl(s)e−sds)uk,l],j,B∗= (B∗ i), C∗ i(ϕ) = −ρ−1[(Bij(0) + Z∞ 0 ˙ Bij(s)e−sds)ϕ],j,C∗= (C∗ i), D∗ i(ψ) = −ρ−1[(Dij(0) + Z∞ 0 ˙ Dij(s)e−sds)ψ],j,D∗= (D∗ i), E∗(u) = −κ−1 1[(Bij(0) + Z∞ 0 ˙ Bij(s)e−sds)ui,j], F∗ϕ=ϕ−κ−1 1[(αij(0)+Z∞ 0˙αij(s)e−sds)ϕ,i],j +κ−1 1[(α1(0)+Z∞ 0˙α1(s)e−sds)ϕ], H∗ψ=−κ−1 1[(βij(0) + Z∞ 0 ˙ βij(s)e−sds)ψ,j],i +κ−1 1[(α3(0) + Z∞ 0˙α3(s)e−sds)ψ], J∗(u) = −κ−1 2[(Dij(0) + Z∞ 0 ˙ Dij(s)e−sds)ui,j], K∗ϕ=−κ−1 2[(βij(0) + Z∞ 0 ˙ βij(s)e−sds)ϕ,i],j +κ−1 2[(α3(0) + Z∞ 0˙α3(s)e−sds)ϕ], M∗ψ=ψ−κ−1 2[(γij(0)+Z∞ 0˙γij(s)e−sds)ψ,j],i+κ−1 2[(α2(0)+Z∞ 0˙α2(s)e−sds)ψ], mi=u0 i+v0 i+ρ−1Z∞ 0Zs 0eτ−s˙ Cijkl(s)z0 k,l(τ)+ ˙ Bij(s)k0(τ)+ ˙ Dij(s)l0(τ),jdτds, n1=ϕ0+φ0+κ−1 1Z∞ 0Zs 0eτ−s˙αij(s)l0 ,i(τ) + ˙ βij(s)k0 ,i(τ),jdτ −κ−1 1Z∞ 0Zs 0eτ−s˙ Bij(s)z0 i,j(τ) + ˙α1(s)l0(τ) + ˙α3(s)k0(τ)dτds, n2=ψ0+χ0+κ−1 2Z∞ 0Zs 0eτ−s˙ βij(s)l0 ,i(τ) + ˙γij(s)k0 ,i(τ),jdτds −κ−1 1Z∞ 0Zs 0eτ−s˙ Dij(s)z0 i,j(τ) + ˙α3(s)l0(τ) + ˙α2(s)k0(τ)dτ In order to study the system (40) we consider the bilinear form defined on W1,2 0×W1,2 0×W1,2 0 R[(u, ϕ, ψ),ˆ u,ˆϕ, ˆ ψ)] = (42) =<(B∗u+C∗ϕ+D∗ψ, E∗u+F∗ϕ+H∗ψ, J∗u+K∗ϕ+M∗ψ),(ρˆu, κ1ˆϕ, κ2ˆ ψ > where this product is taken in L2×L2×L2. It is clear that this product is bounded. On the other side we see that R[(u, ϕ, ψ),(u, ϕ, ψ)] = (43) 15 ZB(ρuiui+κ1ϕ2+κ2ψ2)dv +ZBZ∞ 0e−s[Cijkl(s)ui,juk,l+2Bij(s)ui,jϕ+2Dij(s)ψ+α1(s)ϕ2+α2(s)ψ2+2α3(s)ϕψ]dsdv +ZBZ∞ 0e−s[αij(s)ϕ,iϕ,j + 2βij(s)ϕ,iψ,j +γ(s)φ,iψ,j]dsdv. Therefore, in view of (30) and (31) Ris a coercive bilinear form. On the other side it is clear that (m, n1, n2)∈W−1,2×W−1,2×W−1,2. Lax-Milgram theorem implies the existence of a solution to our problem. We also see the existence of (v, φ, χ) in W1,2 0×W1,2 0×W1,2 0and we can prove that z(s), l(s), k(s) belongs in the corresponding space. We have proved that Theorem 2. The operator Adefined previously generates a contractive semigroup in Z. As a consequence, we obtain that Theorem 3. Let us assume that Fi, G, L ∈C1([0,∞, L2]) ∩C0([0,∞, W1,2 0]),and U0∈ D. Then, there exists a unique solution to the problem determined by the system (27) and the conditions (28), (29) such that U(t)∈C1([0,∞,Z]). 5 Minimum principle In this section we use the results of Reiss (1978) and Reiss and Haug (1978) to derive a minimum principle for the porous viscoelastic materials. First, we give an alternative characterization of the problem formulated in Section 2 and derive a reciprocity relation. We introduce the functions jand lon [0,∞) by l(t) = 1, j = (l∗l)(t) = t, t ∈[0,∞).(44) Let Hi, S and Tbe functions on B×[0,∞) defined by Hi=j∗Fi+ρ(tv0 i+u0 i), s =j∗G+κ1(tϕ0 1+ϕ0), T=j∗L+κ2(tψ0 1+ψ0).(45) Following Gurtin (1972a) we have 16 Lemma 4. Let ui, ϕ, ψ ∈C0,2,tij, σi, τi∈C1,0, and ξ, ζ ∈C0. Then ui, ϕ, ψ, tij, σi, τi, ξ and ζsatisfy the equations (2) and the initial conditions (6) if and only if j∗tki,k +Hi=ρui, j ∗(σi,i +ξ) + S=κ1ϕ, j ∗(τi,i +ζ) + T=κ2ψ, (46) on B×[0,∞). The next proposition is an immediate consequence of Lemma 4. Theorem 4. Let πbe an admissible process. Then πis a solution of the problem if and only if πsatisfies the equations (1), (9), the initial history condition (5), and the boundary conditions (7). We consider two external data systems L(α)={F(α) i, G(α), L(α),e u(α) i,e ϕ(α), e ψ(α),e t(α) i,e σ(α),e τ(α), π∗(α)}, (α= 1,2), and denote by π(α)= (u(α) i, ϕ(α), ψ(α), e(α) ij , t(α) ij , σ(α) i, τ(α) i, ξ(α), ζ(α)) a solution corresponding to L(α).We introduce the notations t(α) i=t(α) ji nj, σ(α)=σ(α) jnj, τ(α)=τ(α) jnj, H(α) i=j∗F(α) i+ρ(tv0(α) i+u0(α) i), S(α)=j∗G(α)+κ1(tϕ0(α) 1+ϕ0(α)), T(α)=j∗H(α)+κ2(tψ0(α) 1+ψ0(α)), s(α) i=s(α) ji nj, π(α)=π(α) knk, χ(α)=χ(α) knk,(47) f H(α) i=j∗s(α) ki,k,e S(α)=j∗(π(α) k,k +ν(α)), e T(α)=j∗(χ(α) k,k +ϑ(α)), Γαβ =ZB[(H(α) i+f H(α) i)∗u(β) i+ (S(α)+e S(α))∗ϕ(β) + (T(α)+e T(α))∗ψ(β)]dv +Z∂B j∗[(t(α) i−s(α) i)∗u(β) i + (σ(α)−π(α))∗ϕ(β)+ (τ(α)−χ(α))∗ψ(β)]da. Lemma 5. If the body is subjected to two external data systems, then the corresponding solutions π(α),(α= 1,2), satisfy the reciprocity relation Γ12 = Γ21.(48) Proof. If we denote Jαβ =l∗[(t(α) ij −s(α) ij )∗e(β) ij + (σ(α) i−π(α) i)∗ϕ(β) ,i + (τ(α) i−χ(α) i)∗ψ(β) ,i −(ξ(α)−ν(α))∗ϕ(β)−(ζ(α)−ϑ(α))∗ψ(β)],(49) 17 then from (9) we find that Jαβ =Cijkl ∗e(α) kl ∗e(β) ij +Bij ∗(ϕ(α)∗e(β) ij +ϕ(β)∗e(α) ij ) +Dij ∗(ψ(α)∗e(β) ij +ψ(β)∗e(α) ij ) + αij ∗ϕ(α) ,j ∗ϕ(β) ,i +βij ∗(ψ(α) ,j ∗ϕ(β) ,i +ψ(β) ,j ∗ϕ(α) ,i ) + γij ∗ψ(α) ,j ∗ψ(β) ,i (50) +α1∗ϕ(α)∗ϕ(β)+α3∗(ψ(α)∗ϕ(β)+ψ(β)∗ϕ(α)) + α2∗ψ(α)∗ψ(β). In view of (4) and (50) we obtain J12 =J21.(51) On the other hand, by (49), (1) and (46) we obtain l∗Jαβ =j∗[(t(α) ki −s(α) ki )∗u(β) i+ (σ(α) k−π(α) k)∗ϕ(β)+ (τ(α) k−χ(α) k)∗ψ(β)],k −ρu(α) i∗u(β) i−κ1ϕ(α)∗ϕ(β)−κ2ψ(α)∗ψ(β)+H(α) i∗u(β) i+S(α)∗ϕ(β)(52) +T(α)∗ψ(β)+j∗[s(α) ki,k ∗u(β) i+ (π(α) k,k +ν(α))∗ϕ(β)+ (χ(α) k,k +ϑ(α))∗ψ(β)]. If we integrate this relation over B, and use (47), then we get ZBl∗Jαβdv =Z∂B j∗[(t(α) i−s(α) i)∗u(β) i+ (σ(α)−π(α))∗ϕ(β) + (τ(α)−χ(α))∗ψ(β)]da +ZB[(H(α) i+f H(α) i)∗u(β) i(53) + (S(α)+e S(α))∗ϕ(β)+ (T(α)+e T(α))∗ψ(β)−ρu(α) i∗u(β) i −κ1ϕ(α)∗ϕ(β)−κ2ψ(α)∗ψ(β)]dv. From (51) and (53) we obtain (48). We say that fhas a Laplace transform f(or Lf) if there exist a real number p0≥0 such that for every p∈[p0,∞) the integral f(x, p) = Z∞ 0e−ptf(x, t)dt, converges uniformly on B. We assume that (A1)Fi, G, L, e ui,e ϕ, e ψ, e ti,e σ, e τand the constitutive functions possess Laplace transforms; (A2)Cijmnfijfmn + 2Bijfijv+ 2Dijfijw+αijgigj+ 2βijhjgi+γijhjhi+α1v2 +α1v2+ 2α3vw +α2w2≥0, for any fij, gi, hi, v and wwith fij =fji. 18 The assumption (A2) is similar to that used by Edelstein and Gurtin (1964) and Reiss and Haug (1978) in classical viscolelasticity. We shall write F[n](x, t) for the nth derivative of Fwith respect to tholding xfixed. Following Reiss (1978) we introduce the set Mof admissible weight functions. We say that g∈ M if gis a function on [0,∞) with the following properties Z∞ 0Z∞ 0g[k](t+s)dtds exists for k > 0,(54) g(t) = Zt 0Γ(p)e−ptdp, t ∈[0,∞),(55) where Γ is a continuous and positive function on [0,∞) and has a finite limit at infinity. For example, g(t) = (t+a)−n, n > 2, a > 0 with Γ(t) = [tn−1exp(−at)]/(n−1)! is a weight function (Reiss, 1978). We denote l∗f=b f, where lis defined by (44). We say that his bounded at infinity if lim t→∞ h(x, t) exists for each x∈B. We shall assume that the functions used to formulate the problem are bounded at infinity. We say that π= (ui, ϕ, ψ, eij, tij, σi, τi, ξ, ζ) is a kinematically admissible process if πis an admissible process that satisfies the equations (1), (3), the initial history conditions (5) and the boundary conditions ui=e ui, ϕ =e ϕ, ψ =e ψon ∂B ×I. (56) We denote by Ω the set of all kinematically admissible process πsuch that π and grad πpossess Laplace transforms. Theorem 5. Assume that hypotheses (A1)and (A2)hold. Let Λg{·} be the functional on Ωdefined by Λg{π}=ZBZ∞ 0Z∞ 0g(t+s){(b Cijmn ∗emn)(x, t)eij(x, s) + 2( b Bij ∗ϕ)(x, t)eij(x, s) + 2(c Dij ∗ψ)(x, t)eij(x, s) + (b αij ∗ϕ,j)(x, t)ϕ,i(x, s) + 2( b βij ∗ψ,j)(x, t)ϕ,i(x, s) (57) + (b γij ∗ψ,j)(x, t)ψ,i(x, s)+(b α1∗ϕ)(x, t)ϕ(x, s) + 2(b α3∗ψ)(x, t)ϕ(x, s)+(b α2∗ψ)(x, t)ψ(x, s) +ρui(x, t)ui(x, s) + κ1ϕ(x, t)ϕ(x, s) + κ2ψ(x, t)ψ(x, s) −2(Hi+j∗ski,k)(x, t)ui(x, s)−2[S+j∗(πk,k +ν)](x, t)ϕ(x, s) −2[T+j∗(χk,k +ϑ)](x, t)ψ(x, s)}dtdsdvx, for every π∈Ω.If πis a solution of the Dirichlet problem, then Λg{π} ≤ Λg{e π},(58) for every e π∈Ω. 19 Proof. We consider π, e π∈Ω and define π0=e π−π. Clearly, π0={u0 i, ϕ0, ψ0, e0 ij, t0 ij, σ0 i, τ0 i, ξ0, ζ0}is an admissible process that satisfies the equation (1), (3), null initial history and the boundary conditions u0 i= 0, ϕ0= 0, ψ0= 0 on ∂B ×I. (59) With the help of (54) we find that ZBZ∞ 0Z∞ 0g(t+s)Φ(x, t)Ψ(x, s)dtdsdvx=ZBZ∞ 0Γ(p)Φ(x, p)Ψ(x, p)dpdvx. (60) Clearly, we have (Ll)(p) = p−1,(Lj)(p) = p−2,L(Φ ∗Ψ) = Φ Ψ.(61) Let us calculate Λg{π+π0}. In view of (60) and (61) we get ZBZ∞ 0Z∞ 0g(t+s){(b Cijmn ∗emn +b Bij ∗ϕ+c Dij ∗ψ)(x, t)e0 ij(x, s) + ( b Cijmn ∗e0 mn +b Bij ∗ϕ0+c Dij ∗ψ0)(x, t)eij(x, s) + (b αij ∗ϕ,j +b βij ∗ψ,j)(x, t)ϕ0 ,i(x, s)+(b αij ∗ϕ0 ,j +b βij ∗ψ0 ,j)(x, t)ϕ,i(x, s) + ( b βji ∗ϕ,j +b γij ∗ψ,j)(x, t)ψ0 ,i(x, s)+(b βji ∗ϕ0 ,j +b γij ∗ψ0 ,j)(x, t)ψ,i(x, s) + ( b Bij ∗eij +b α1∗ϕ+b α3∗ψ)(x, t)ϕ0(x, s) (62) + ( b Bij ∗e0 ij +b α1∗ϕ0+b α3∗ψ0)(x, t)ϕ(x, s)+(c Dij ∗eij +b α3∗ϕ +b α2∗ψ)(x, t)ψ0(x, s)+(c Dij ∗e0 ij +b α3∗ϕ0+b α2∗ψ0)(x, t)ψ(x, s)}dtdsdvx = 2 ZB 0Z∞ 0p−1Γ(p)Π(p)dpdvx, where Π(p) = Cijmn(p)emn(p)e0 ij(p) + Bij(p)(ϕe0 ij +ϕ0eij)(p) +Dij(p)(ψe0 ij +ψ0eij)(p) + αij(p)ϕ0 ,j(p)ϕ,i(p) (63) +βij(p)(ψ,jϕ0 ,i +ψ0 ,jϕ,i)(p) + γij(p)ψ,j(p)ψ0 ,i(p) +α1(p)ϕ(p)ϕ0(p) + α3(ψϕ0+ψ0ϕ)(p) + α2(p)ψ(p)ψ0(p). With the help of (49) and (50) we obtain Π(p) = p−1[(tij −sij)e0 ij + (σi−πi)ϕ0 ,i + (τi−χi)ψ0 ,i −(ξ−ν)ϕ0−(ζ−ϑ)ψ0].(64) Suppose that πis a solution. Then, in view of (1) and (46) we get p−1Π(p) = p−2[(tki −ski)u0 i+ (σk−πk)ϕ0+ (τk−χk)ψ0],k −ρuiu0 i−κ1ϕϕ0−κ2ψψ0+ (Hi+p2ski,k)u0 i(65) + [S+p2(πk,k +ν)]ϕ0+ [T+p2(χk,k +ϑ)]ψ0. 20 From (57), (59), (61), (62) and (65) we find that Λ{e π}= Λ{π}+ZBZ∞ 0Z∞ 0g(t+s){(b Cijmn ∗e0 mn)(x, t)e0 ij(x, s) + 2( b Bij ∗ϕ0 ,j)(x, t)ϕ0 ,i(x, s) + 2( b βij ∗ψ0 ,j)(x, t)ϕ0 ,i(x, s) (66) + (b γij ∗ψ0 ,j)(x, t)ψ0 ,i(x, s)+(b α1∗ϕ0)(x, t)ϕ0(x, s) + 2(b α3∗ψ0)(x, t)ϕ0(x, s)+(b α2∗ψ0)(x, t)ψ0(x, s)}. It follows from (62), (66) and the hypothesis (A2) that (58) holds. Minimum principles for other boundary conditions can be also derived (see Reiss and Haug, 1978). Variational principles for linear theories of viscoelastic materials have been investigated in various papers (see, e.g., Leitman and Fisher, 1973; Luo and Hua, 2007). 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