Strain gradient theory of porous solids with initial stresses and initial heat flux

In this paper we present a strain gradient theory of thermoelastic porous 
solids with initial stresses and initial heat flux. First, we establish the equations 
governing the infinitesimal deformations superposed on large deformations. Then, we 
derive a linear theory of prestressed porous bodies with initial heat flux. The theory is capable to describe the deformation of chiral materials. A reciprocity relation and a uniqueness result with no definiteness assumption on the elastic constitutive coefficients are presented.


1.
Introduction. In recent years considerable attention has been given to the mechanical behaviour of porous elastic solids (see, e.g., [9,16,33], and references therein). In [10,32] Nunziato and Cowin established a theory of elastic materials with voids for the treatment of porous solids. This theory introduces an additional degree of kinematical freedom. There has been very much written in the last years on the theories of elastic solids with inner structure in which the deformation is described not only by the usual vector displacement field, but by other vector or tensor fields as well. The origin of the theories of continua with microstructure goes back to the papers of Mindlin [27], Eringen and Suhubi [12] and Green and Rivlin [18]. Mindlin [27] formulated a theory of an elastic solid which has some properties of a crystal lattice as a result of the inclusion of the idea of the unit cell. Mindlin begins with the general concept of an elastic continuum each material point of which is itself a deformable medium. In the theory developed by Mindlin each material point is constrained to deform homogeneously. In this theory the degrees of freedom for each material point are twelve: three translations, u i , and nine microdeformations, χ ij . A special class of bodies with microstructure [13] is characterized by a microdeformation tensor of the form ψδ ij , where ψ is the microstretch function (or microdilatation function) and δ ij is the Kronecker delta. In this case the material points undergo a uniform microdilatation (a breathing motion). The linear equations which describe the behavior of an elastic body with this kind of microstructure coincide with the equations of the linear theory of elastic materials with voids established by Cowin and Nunziato [10] (cf. Eringen [13]). In what follows we shall refer to this model as a porous elastic continuum.
The linear theory of elastic materials with voids is the simplest theory of elastic bodies that takes into account the microstructure of the material. The theory of elastic solids with microstructure is characterized by constitutive functions which depend on the deformation gradient, microdeformation and microdeformation gradient. Rymarz [35], Brulin and Hjalmars [5], and Hjalmars [19] have shown that in order to obtain a consistent grade level for the microstructure theory it is necessary to add the second-order displacement gradient to the above independent constitutive variables. The equations of motion, constitutive equations, and the boundary conditions of the theory of nonsimple elastic bodies of grade 2 were first established by Toupin ([37,38]). The linear theory has been developed by Mindlin [27] and Mindlin and Eshel [28]. In the first part of this paper we use the results established by Toupin [38] and Nunziato and Cowin [32] to derive a non-linear theory of porous thermoelastic solids of grade 2. This work is motivated by the recent interest in the study of non-simple solids (see, e.g, [1,2,3,4,15,36]) and the use of the continuum with microstructure as model for engineering materials (see, e.g., [11,30]).
We establish a strain gradient theory of thermoelastic porous solids with initial stresses and initial heat flux. The theory of initially stressed elastic bodies is of considerable interest both from the mathematical and the technical point of view. It provides a natural extension of the classical theory of elasticity, as the initial configuration needs no longer be unstressed. The theory also establishes a basis for discussing stability of finite deformations (see Knops and Wilkes [23]). The theory of initially stressed bodies has given impetus to theoretical research into the equations of elastic bodies for which little or no information is known concerning the elasticities (see Knops and Payne [24], Sections 2.1, 4.4, 8.3 and Knops [25]). We note the recent interest in the study of prestressed bodies with inner structure (see, e.g., [39] and references therein). By using the non-linear theory of classical thermoelasticity , Ieşan [20] established a linear theory of thermoelasticity with initial stresses and initial heat flux. Some theorems in this theory have been established by Chirita [8] and Navarro and Quintanilla [29]. Martinez and Quintanilla [26] have established the theory of small thermoelastic deformations superposed on a large deformation at non-uniform temperature in the context of the nonsimple materials of grade 2. An existence result and continuous dependence of solutions upon the initial state and supply terms have been also presented. This paper is structured as follows. In Section 2 we establish a non-linear theory of thermoelastic solids in which the independent constitutive variables are the deformation gradient, the second-order displacement gradient, volume fraction field, the gradient of volume fraction field, temperature and temperature gradient. Section 3 presents the equations governing the infinitesimal thermoelastic deformations superposed on large deformations at non-uniform temperature. In Section 4 we establish a linear theory of prestressed porous bodies with initial heat flux. Section 5 is devoted to reciprocity and uniqueness results in the strain gradient theory of prestressed thermoelastic porous solids.
2. Basic equations. In this section we present a non-linear theory of thermoelastic porous solids of grade 2. We consider a body that at time t 0 occupies the region B of the Euclidean three-dimensional space and is bounded by the smooth surface ∂B. Let (t 0 , t 1 ) be a given interval of time The motion of the body is referred to a fixed system of rectangular Cartesian axes and to the reference configuration. The coordinates of a typical point in the reference configuration are X K . The coordinates of this particle at time t are denoted by x i , where If the deformation is possible in a real material, then we have We assume the continuous differentiability of x i with respect to each of the variables X K and t as many times as required. The concept of a distributed body asserts that the mass density ρ at time t has the decomposition where γ is the density of the matrix material and ν is the volume fraction field. Throughout this paper, a superposed dot denotes the material time derivative. Partial derivatives are denoted by comma preceding a subscript: differentiation with respect to x i is indicated by small Latin indices and differentiation with respect to X K by Latin capitals, thus ∂f /∂X K = f ,K and ∂f /∂x i = f ,i . The usual Cartesian summation convention is used. Let B be at rest relative to the considered system of reference. We consider an arbitrary region P of the continuum at time t bounded by the surface ∂P, and we suppose that P is the corresponding region at time t 0 , bounded by the surface ∂P . Following [18,32,38], the conservation of energy, for every regular region P of B and every time t, can be expressed as where dv and da are elements of volume and area in the reference configuration, ρ 0 is the mass density in the reference configuration, e is the internal energy per unit mass, f i is the body force per unit mass, l is the extrinsic equilibrated body force per unit mass, S is the heat supply per unit mass, T i is the stress vector associated with the surface ∂P and measured per unit area of ∂P , M ji is the hypertraction associated with the surface surface ∂P and measured per unit area of ∂P , H is the equilibrated stress associated with the surface ∂P and measured per unit undeformed area, Q is the heat flux across the surface ∂P and measured per unit area of ∂P, κ is the equilibrated coefficient of inertia, and v i =ẋ i . We use the method given by Green and Rivlin [18] to obtain the equations of motion from the balance of energy and the invariance requirements under superposed rigid body motions. First, we consider motions of the body which differ from those given by (1) only by superposed uniform rigid body translational velocities, the continuum occupying the same position at time t. We assume that ρ 0 ,ė, f i , l, S, T i , M ji , H, Q and κ are unaltered by such rigid body velocities. Then, from (4) we obtain the balance of momentum. From this law we get and the local form of the balance of momentum, Here, T Ki is the first Piola-Kirchhoff stress tensor and N K are the components of the unit outward normal vector to the surface ∂P . If we use (5) and (6) then the relation (4) reduces to If we apply this equation to a region which in the reference state was a tetrahedron bounded by coordinate planes through the point X and by a plane whose unit normal is N K , we obtain where M Kji is the hyperstress tensor, H K is the equilibrated stress and Q K is the heat flux, associated with surfaces in the deformed body which were originally coordinate planes perpendicular to the X K -axes through the point X, measured per unit area of these planes. If we use (8) in (7) and apply the resulting equation to an arbitrary region, then we find the local form of the conservation of energy where g is defined by The function g is called the intrinsic equilibrated body force (cf. [4], [5]). This function is specified by a constitutive equation. We introduce the notations The equation (9) becomes Let us denote We consider a motion of the body which differs from the given motion only by a superposed uniform rigid body angular velocity, and we assume that ρ 0 ,ė, T Ki , M KLi , H K , g, Q K and S are not affected by such motion. Then, the equation (12) implies that We postulate the entropy production inequality in the form for every part P of B and every time. Here η is the entropy per unit mass, and T is the absolute temperature, which is assumed to be positive. If we introduce the Helmholtz free-energy ψ by then the equation (12) can be presented in the form A thermoelastic material is defined as one for which the following constitutive equations hold The response functions are assumed to be sufficiently smooth. For a given deformation, v i,j andν in (8) may be chosen arbitrarily so that, on the basis of the constitutive equations (18) we find If we use (19) in (15) then we obtain the following local form of the second law of thermodynamics From (17), (18) and (20) we get where σ = ρ 0 ψ. Following Toupin [38] and Mindlin [27], the skew symmetric part of M KLi , with respect to K and L, makes no condition to the rate of work over any closed surface in the body, or over the boundary. We shall assume that M KLi = M LKi and that there is no kinematical constraint. From the inequality (21) we get [14] (11) and (22) we find that In view of (22) the energy equation (17) reduces to The functions σ and Q K must be invariant under Euclidean displacements. It can be shown (see [32], [38]) that the functions σ and Q K are expressible in the form Here δ KL is the Kronecker delta. From (22), (24), (26) and (27) we obtain

DORIN IEŞAN
We note that if S Ki and M KLi have the form (28) then the equation (14) is identically satisfied. It follows from (11) and (28) that By (22) and (26), As in classical thermoelasticity (see, e.g., Carlson [7]) the inequality (23) implies that The basic equations consists of the equations of motion (6) and (10), the equation of energy (25), the constitutive equations (26), (28)- (30), and the geometrical equations (27). To the field equations we must adjoin boundary conditions and initial conditions. We suppose that B is a bounded region with Lipschitz boundary ∂B. The boundary ∂B consists in the union of a finite number of smooth surfaces, smooth curves (edges) and points. Let C be the union of the edges. We introduce the surface gradient D K defined by D K = (δ KL − N K N L )∂/∂X L , and the notation Df = f ,K N K . Following Toupin [38] and Mindlin [32] we can write where Here, < f > denotes the difference in values of f as a given point on an edge is approached from either side, s K are the components of the unit vector tangent to C, and ε KLM is the alternating symbol.
In this section we derive the equations governing the infinitesimal thermoelastic deformations superposed on nonlinear deformations of non-simple porous bodies. In the case of simple materials, the theory of a thermoelastic body, deformed from a state of zero stress and which is subsequently subjected to small perturbations, was investigated in various works (see, e.g., [17,21,22,23] and references therein).We consider three states of the body: the reference configuration B and two present configurations B and B . In this section we wish to look at perturbations of the motion described by the equations of Section 2. Following Knops and Wilkes [23], we call B the primary state and B the secondary state. We shall designate as incremental those quantities associated with a difference of motion between the secondary and primary states. If the point X in the reference configuration B moves to x in the primary state and to y in the secondary, then u i = y i − x i is the incremental displacement. We have with det(∂y i /∂X K ) > 0.The thermomechanical quantities associated with the secondary state will be denoted with an asterisk. Now let We assume that u i = εu i , ϕ = εϕ , θ = εθ , where ε is a constant small enough for squares and higher powers to be neglected, and u i , ϕ and θ are independent of ε. The problem consists in establishing the equations, boundary conditions and initial conditions for u i , ϕ and θ, when the functions x i (X K , t), ν(X K , t) and T (X K , t), (X K , t) ∈ B × (t 0 , t 1 ), and the loadings corresponding to primary and secondary states are known. We refer all quantities to the configuration B. We have The equations of motion of the secondary deformation are In the secondary motion the balance of equilibrated forces is The energy equation can be written in the form In view of the constitutive equations, we have We consider the following boundary conditions for the secondary deformation where y i , d * i , ν * , T * , P * i , R * i , H * , Q * and Γ * i are given. In the secondary motion we have the following initial conditions where y 0 i , ξ 0 i , ν 0 , ζ 0 and η 0 are prescribed functions. The entropy production inequality implies that Let us derive the equations, boundary conditions and initial conditions for the functions u i , ϕ and θ. From (27) and (43) we find that where If we take into account (42) and (48), then to a second-order approximation we obtain Clearly, Let us note that (31) implies We introduce the notations From (11), (42) and (53) we get From (6), (39), (53) and (54) we find the equations Similarly, from (10), (37), (40) and (53) we obtain
We introduce the notations It follows from (28), (42), (49), (53) and (58) that The functions s Ki and m KLi can be written in the form The basic equations for the functions u i , ϕ and θ consist of the equations of motion (55) and (56), the equation of energy (57), the constitutive equations (59) and (60), and the geometrical equations (48). From (34) and (44) we obtain the following boundary conditions

and we have used the notations
The relations (35) and (45) lead to the following initial conditions for any ε KL , γ KLM , ϕ, ϕ ,K , θ and θ ,K .

Linear theory of materials with initial stresses and initial heat flux.
In this section we present the strain gradient theory of elastic materials with initial stresses and initial heat. Following Green [17], we now assume that the primary configuration B is identical with the reference configuration B.We suppose that B is subjected to an initial stress, and is at non-uniform temperature T . We consider that the distribution of voids in B is non-uniform so that we assume that ν = ν, where ν is a prescribed function. We then assume that the configuration B is obtained from B (or B ) by an infinitesimal deformation. We introduce the notations: u i is the displacement vector, ϕ is the change in the volume fraction field, θ is the temperature variation. We have the derivatives being evaluated at E KL = 0, G KLM = 0, ν = ν and T = T . The functions P KL , Π KLM and H K define the initial stresses in the body B. The functions Q K are the components of the initial heat flux. The work of preceding section can be applied to this special case and yields a strain gradient theory of thermoelastic porous solids which are under initial stresses and initial heat flux. We assume that the body is in equilibrium in the state B. We introduce the notations In this case, from (48) we find that If we denote then from (59), (60) and (67) we obtain the following constitutive equations In view of (67) and (69) the equations (55) can be written in the form where F M = δ iM F i . We can write the field equations by using small indices. Thus, the basic equations of the strain gradient theory of prestressed thermoelastic porous solids consist of the equations of motion the equations of energy the constitutive equations and the geometrical equations on B × (t 0 , t 1 ). From (51) we get The coefficients from (74) are defined by (50), and are evaluated at E KL = 0, G KLM = 0, ν = ν and T = T . From (52) we get To the system of equations (72)-(75) we adjoin the initial conditions (64) and the boundary conditions where n i = δ iK N K , and P i , R i and L i are given by In view of (75) and (76) we can express the equations (74) in the form where a ijrs = A ijrs + P sj δ ri , b ijpqr = B ijpqr + Π pqj δ ir , c ijkpqr = C ijkpqr .
It follows from (58), (76) and (81) that The coefficients from (80) depend on the initial stresses and the temperature T . If the body is isotropic then we have P ij = pδ ij , Π ijk = 0, where p is a specified function of x j . When the body has a constant reference temperature then Q i = 0, R ijpq = 0, J ipqr = 0, C i = 0, L ij = 0 and E i = 0. The strain gradient theory of elasticity is an adequate tool to describe the deformation of chiral solids (see Papanicolopulos [34] and references therein). By using the results of [28] and [34] we find that in the case of isotropic chiral elastic solids with a constant reference temperature, the constitutive equations (74) become where the coefficients are prescribed functions of x k . The behaviour of chiral materials has received in recent years a widespread attention. Examples of chiral materials include auxetic materials, bones, carbon nanotubes, as well as as some porous composites.

5.
Uniqueness. In this section we use the method of [6] to establish reciprocity and uniqueness results in the strain gradient theory of prestressed thermoelastic porous solids with a constant reference temperature T 0 . The field equations consist of the equations (72), (73) and the following constitutive equations τ ij = a ijrs u r,s + b ijpqr u r,pq + b ij ϕ + ζ ijk ϕ ,k − β ij θ, We assume that the coefficients in (83) are prescribed functions which are continuously differentiable on B and satisfy the relations (76) and (82). We first establish a reciprocity relation which involves two processes at different times. In what follows we suppose that the underlying time interval is I = (0, ∞). If f is a continuous function on B × I, then we denote by f the function defined by We introduce the notation G = ρ 0 s + T 0 γ 0 .
i ) is a solution corresponding to D (α) . We introduce the notations ,j (x, s)]dv, for all r, s ∈ I. A general reciprocity relation is expressed by the following theorem.
Theorem 5.1. Assume that the conductivity tensor k ij is symmetric and the relations (76) and (82) hold. Let for all r, s ∈ I. Then Proof. We introduce the notations In (91), for convenience, we have suppressed the argument x. With the help of constitutive equations (83) we obtain On the basis of (82) and (92) we conclude that for all r, s ∈ I. On the other hand, in view of (72), (86) and (91) we obtain ,j (s).
If we integrate the above relation on B and use the divergence theorem, (87) and (88) we find In view of (32), (87) and (88) Let the constitutive coefficients be as in Theorem 1 and let for all r, s ∈ I. Then Proof. In view of (90) we get We apply the relation (98) to the process A (1) = A. From (88), (89) and (97) we obtain In a similar way we get  The relation (100) can be transformed in a similar way. If we take into account (98) then we conclude that the relation (97) holds.
Theorem 2 forms the basis of the following uniqueness result. (iv) the symmetry relations (76) and (82) hold. Then the boundary-initial-value problem has at most one solution.
Theorem 1 implies the following reciprocal theorem.