On the thermal stresses in chiral porous elastic beams

This paper is concerned with the strain gradient theory of porous thermoelastic solids. We study the deformation of isotropic chiral cylinders subjected to a temperature field that is linear in the axial coordinate. It is shown that the solution can be reduced to the study of two-dimensional problems. The results are used to investigate the deformation of a circular cylinder subjected to a uniform temperature variation. In contrast to the case of achiral materials, the thermal field in chiral cylinders produces torsional effects.

of the microdilatation function requires a strain gradient theory. Toupin [3,4] and Mindlin [5] have introduced the strain gradient theory of elasticity. Interest in the study of this theory is stimulated by the fact that it is suitable to investigate problems related to the effects of size and nanotechnology. The deformation of chiral elastic materials cannot be described within classical elasticity [6]. In various papers, the authors have studied the behavior of chiral materials by using the linear theory of gradient elasticity (see, e.g., [7][8][9] and references therein). The strain gradient elasticity has been used to investigate the behavior of carbon nanotubes (see, e.g., [10,11]). This paper is concerned with a theory of thermoelasticity for isotropic microstretch continua, without microrotations, where the second-order displacement gradient is added to the classical set of independent constitutive variables. We have considered this theory since the chirality behavior in strain gradient theory is controlled by a single material parameter, in contrast to the three additional material parameters required in Cosserat theory. We study the equilibrium problem for a cylinder which, in the absence of body forces and lateral loading, is subjected to prescribed surface tractions on bases and to a thermal field that is linear in the axial coordinates. The origin of this problem goes back to the work of Boley and Weiner [12] that is devoted to classical thermoelasticity. The deformation of achiral porous elastic solids has been investigated in various papers (see, e.g., [2,[13][14][15][16][17][18]). In this paper, we focus our attention to the case of homogeneous and isotropic chiral porous elastic materials. Since the cancellous bone is considered as a porous body [19] as well as a chiral material [20], it seems that the linear theory of gradient elasticity of porous solids is adequate to describe the mechanical behavior of bones. This paper is concerned with uncoupled system in the sense that the temperature field can be found by solving the heat flow problem associated with the heat conduction and energy equation. We shall treat the temperature field as having already been determined [21]. The temperature is a prescribed function that is independent of time.
The paper is structured as follows. First, we present the basic equations of chiral porous thermoelastic solids and formulate the problem of deformation of the right cylinders. We have introduced mechanical loads on the ends in order to compare the effects of thermal field with those produced by the resultants of the tractions that act on the bases. Then, we investigate the generalized plane strain problem. In the following section, we establish the solution of the problem when the temperature distribution is independent of the axial coordinate. It is shown that this thermal field produces extension, bending and torsion. The next section deals with the deformation of cylinders subjected to a temperature field that is linear in the axial coordinate. We present a method to reduce the three-dimensional problem to the study of plane problems. The results are used to study the deformation of a circular cylinder subjected to a uniform temperature variation. In contrast to the case of centrosymmetric materials, the thermal field in chiral cylinders produces torsional effects.

Preliminaries
In this section, we present the equations of equilibrium in the context of the strain gradient theory of porous thermoelastic materials. Mindlin [5] presented three forms of the linear theory of gradient elasticity. In what follows, we use the strain measures introduced in the first form of the theory. We consider a body that in the undeformed state occupies the bounded region B with Lispchitz boundary ∂ B. The boundary ∂ B consists of the union of a finite number of smooth surfaces, smooth curves (edges) and points (corners). We denote by C the union of the edges. Throughout this paper, a rectangular Cartesian coordinate system Ox j , ( j = 1, 2, 3), is used. We shall employ the usual summation and differentiation conventions.
We assume that B is occupied by a homogeneous and isotropic chiral thermoelastic solid. Let u i be the components of the displacement vector, and let ϕ be the microstretch function (microdilatation function). The strain tensors are defined by In the case of chiral porous thermoelastic bodies, the constitutive equations are given by (see, e.g., [1,8,22]) where τ i j is the stress tensor, μ i jk is the dipolar stress tensor, σ i is the microstretch stress vector, g is the intrinsic body force, T is the temperature measured from the constant absolute temperature of reference state, δ i j is Kronecker delta, ε i jk is the alternating symbol, and λ, μ, α s (s = 1, 2, . . . , 5), β 1 , β 2 , β, b, d, a 0 , ξ and f are constitutive constants. In the case of achiral materials, the coefficient f is equal to zero. In what follows, we assume that the elastic potential is a positive definite quadratic form in the variables e i j , κ i jk , ϕ and ϕ ,k . The equations of equilibrium, in the absence of body loads, are given by The equilibrium theory of linear elastic heat conductors has been studied in various books (see., e.g., [13,22,23]). We investigate the effects of temperature variation of the deformation of cylinders. Toupin [3,4] introduced the functions P i , R i and Q i defined by where n j are the components of the outward unit normal of ∂ B, D i are the components of the surface gradient, D i = (δ ik − n i n k )∂/∂ x k , s i are the components of the unit vector tangent to C and < g > denotes the difference of limits of g from both sides of C.
In the case of traction problem, the boundary conditions are [1,24] where P i , R i , σ and Q i are prescribed functions. We assume that the region B from here on refers to the interior of a right cylinder of length h with the cross section and the lateral boundary . Let be the boundary of . The coordinate system consists of the orthonormal basis {e 1 , e 2 , e 3 } and the origin O. We choose the system Ox j such that x 3 -axis is parallel to the generator of the cylinder and the x 1 Ox 2 plane contains one of the terminal cross sections. We denote by 1 and 2 , respectively, the cross section located at x 3 = 0 and x 3 = h. Let α be the boundary of α , (α = 1, 2). Throughout this paper, the Greek subscripts range over the integers (1,2).
In what follows, we assume that the thermal field is linear in the axial coordinate, where the functions T 0 and T 1 are prescribed. We suppose that the lateral surface is smooth. This assumption implies that the functions Q i are equal to zero on . The cylinder is supposed to be free of lateral loading and subjected to appropriate stress resultants over its ends. On the lateral surface of the cylinder, we have the conditions Let From (4), we find that where (n 1 , n 2 , 0) are the direction cosines of the exterior normal to . The problem consists in finding the functions u i and ϕ satisfying the equations (1)-(3) on B, the conditions (7) on and the conditions (8)-(11) on the ends, when the temperature T , the constitutive coefficients and the constants F j and M j are prescribed.

Theorem 1 The necessary and sufficient conditions for the existence of a solution to the problem (H ) are given by
In view of (14) and (15), the equations of equilibrium can be written in the form The functions τ 33 , μ 3αβ , μ 3α3 , μ 33α , μ 333 and σ 3 can be calculated after the determination of displacements and microdilatation.

Plane temperature field
In this section, we present the solution of the problem under the following assumptions It is known that in the classical thermoelasticity, a uniform thermal field produces bending, extension and a plane deformation. We try to solve the problem by combining a solution of Saint-Venant's type with a solution of the plane problem. We seek the solution in the form where u (k) j , ϕ (k) , w j and ψ are unknown functions which are independent of x 3 , and a k are unknown constants. We denote by A (k) (k = 1, 2, 3, 4), the isothermal (T = 0) plane strain problems characterized by the displacement vector u (k) j and microdilatation function ϕ (k) . The body loads and the boundary data associated with the problems A (k) will be precised in what follows. We introduce the notations It follows from (15) that the constitutive equations imply We shall use the notations We denote by (T ) the plane thermoelastic problem which corresponds to the temperature T 0 and is characterized by the displacement vector w j and the microdilatation ψ. Thus, the constitutive equations in the problem (T ) where We shall use the notations From (1) and (23), we obtain α3 , e 33 = a 1 x 1 + a 2 x 2 + a 3 , In view of (2) and (29), we find that the functions τ i j , μ i jk , σ i and g are given by Here we have used the relations (25)-(27). Let us impose that the equations of equilibrium (3) and the boundary conditions (7) be satisfied by the functions (31). If we require that the coefficients of the constants a k that appear in the equilibrium equations be equal to zero, then we find that the functions τ α and g (k) (k = 1, 2, 3, 4) satisfy the following equations where we have used the notations The equilibrium equations (3) reduce to t αi,α − m αβi,α = 0, π α,α − γ = 0 on 1 .
Let us introduce the functions We require that the boundary conditions (7) be satisfied for any constants a k . Thus, we find that the functions τ αβi and σ (k) have to satisfy the following boundary conditions where We conclude that the problem A (k) consists in finding the functions u (k) j and ϕ (k) satisfying the Eqs. (24), (25) and (32) on 1 , and the boundary conditions (36). The necessary and sufficient conditions for the existence of the solution are satisfied for each problem A (k) . The solutions of these problems depend only on the cross section and the constitutive constants. The boundary conditions (7) reduce to The problem (T ) consists in finding the functions w j and ψ which satisfy the Eqs. (27), (28) and (34) on 1 , and the boundary conditions (38) on 1 . Let us determine now the constants a j , ( j = 1, 2, 3, 4). By using the divergence theorem and the relations (3), (5) and (12), we obtain In view of (7) and (31), we find from (39) that Thus, the conditions (8), with F α = 0, are identically satisfied. It follows from (31) and (39) that the conditions (9)-(11) reduce to the following system, for the constants a k In (40), we have used the notations where S (ρ) The constants D mn (m, n = 1, 2, 3, 4) can be calculated after the solving of the problems A (k) (k = 1, 2, 3, 4). As in classical elasticity, the positive definiteness of the potential energy implies that [17,26].
Thus, the constants a k are determined by the system (40). By using the reciprocal theorem, we find that The functions t 3 j and m j33 that appear in F * 3 and M * 3 depend on the temperature T 0 . We conclude that a plane temperature field produces axial extension, bending and torsion.

Deformation due to a temperature that is linear in the axial coordinate
This section is concerned with the deformation of the beam subjected to the following external data where T 1 and F α are prescribed. We try to solve the problem assuming that where (u 1, 2, 3, 4), U j , , v j and χ are unknown functions of x 1 and x 2 , and c k (k = 1, 2, 3, 4) are unknown constants. Let us consider a plane strain in which the components of the displacement vector are v j and the microdilatation function is χ . The strain measures in this problem are defined by We denote by E α j and K αβ j the strain measures in the plane strain problem associated with the displacement vector U j and microdilatation , In view of (1) and (45)-(47), we obtain Let us introduce the notations s αβ = λγ ρρ δ αβ + 2μγ αβ + f (ε αρ3 ζ βρ3 + ε βρ3 ζ αρ3 ) + dχδ αβ , s α3 = 2μγ α3 + f ε ρβ3 ζ αρ3 , s 33 = λγ ρρ + dχ, ν αβγ = 1 2 α 1 (ζ ρρα δ βγ + 2ζ γρρ δ αβ + ζ ρρβ δ αγ ) + α 2 (ζ αρρ δ βγ + ζ βρρ δ αγ ) + 2α 3 ζ ρργ δ αβ + 2α 4 ζ αβγ + α 5 (ζ γβα + ζ γ αβ ) + f (ε αγ 3 γ β3 + ε βγ 3 γ α3 ) + β 1 δ αβ χ ,γ + 2β 2 (δ αγ χ ,β + δ βγ χ ,α ), ν αβ3 = 2α 3 ζ ρρ3 δ αβ + 2α 4 ζ αβ3 + f (ε ρα3 γ βρ + ε ρβ3 γ αρ ), Clearly, s i j is the stress tensor, ν i jk is the dipolar stress tensor, π j is the microstretch stress vector and p is the intrinsic body force in an isothermal plane problem corresponding to the strain measures γ i j and ζ i jk . We now consider a thermoelastic plane problem associated with the thermal field T 1 , displacement vector U j and microdilatation function . In this problem, we denote the stress tensor, the dipolar stress tensor, the microstretch stress vector and the intrinsic body force by T i j , M i jk , H α and L, respectively. Thus, we have From the constitutive equations (2) and the relations (48)-(50), we find that the stress tensor τ i j is given by where The functions μ i jk , σ j and g can be expressed as In these relations, we have used the following notations 3,1 ), 2 ), αβ ), If we take into account (32), (33), (51) and (53), then the equilibrium equations (3) reduce to the equations and In equations (55) and (56), we have used the notations Following (4) and (18), we define the functions (1) j = (s β j − ν ρβ j,ρ )n β − D β (n ρ ν ρβ j ) + (D α n α )n ρ n η ν ρηj , (1) k = ν ρβ j n ρ n β , (2) With the help of relations (36), (37), (51), (53) and (58), we see that the conditions on the lateral surface (7) reduce to (1) and (2) where α3 )]n α , We denote by (P 1 ) the isothermal plane problem which consist in finding the functions v j and χ that satisfy the geometrical equations (46), the constitutive equations (49), the equilibrium equations (55) and the boundary conditions (59). Let us denote by (P 2 ) the thermoelastic plane problem associated with the temperature T 1 and characterized by the geometrical equations (47), the constitutive equations (50), the equilibrium equations (56) and the boundary conditions (60). Clearly, the necessary and sufficient conditions to solve the problem (P 2 ) are satisfied for any thermal field. The necessary and sufficient conditions to solve the problem (P 1 ) are By using the divergence theorem, we find that It follows from (51), (57), (61) and (63) that By using the equilibrium equations, we find The condition P 3 = P 3 on the lateral boundary can be expressed in the form so that the relation (65) implies that Let us note the identity 1 x α (μ ργ 3 n ρ n β − μ αβ3 n α n γ ) ,γ n β ds = From (65) and (68), we get In view of (7), (51), (53) and (69), we find that 1 τ α3,α da = 0, so that the first two conditions from (61) are satisfied. With the help of (40), (57) and (60), we obtain The last two conditions from (62) reduce to Let us impose the conditions (8). In view of (12), we get By using (67), we obtain With the help of (7), (40), (51), (53) and (71), we see that the conditions (8) reduce to The equilibrium equations (34) and the boundary conditions (38) are satisfied if the constants E 1 and E 2 are given by where The positive definiteness of the elastic potential implies that D = 0, Thus, the solution of the problem (T ) is given by (76). It follows from (40) and (77) that In a similar way, we can study the problems A (3) and A (4) . The solution of the problem A (3) is where and D is given by (79). The problem A (4) has the following solution where In view of symmetry of D mn , we obtain D 3α = 0 and D 4α = 0. Thus, with the help of (82) and (83) we find that the system (40) reduces to D αβ a β = 0, D 33 a 3 + D 34 a 4 = −F * 3 , D 43 a 3 + D 44 a 4 = 0.
It follows from (42) that this system uniquely determines the constants a k . We find that where q −1 = D 33 D 44 − D 2 34 .
Let us note that we have solved the problem without using the solutions of the problems A (1) and A (2) . In view of (23), (76), (81), (82) and (85), we see that the solution of the problem is given by u α = ε 3βα a 4 x β x 3 + a 3 S 1 x α + a 4 K 1 x α + w α , u 3 = a 3 x 3 , ϕ = a 3 S 2 + a 4 K 2 + ψ.
From (86), we conclude that the thermal field produces torsion, extension and a variation of microdilatation.
In the case of an achiral material, we have f = 0 and (83) implies that D 34 = 0. Then, from (85) we obtain a 4 = 0. In this case, the thermal field does not produce torsion.

Conclusions
The results presented in this paper can be summarized as follows: a. In the context of the strain gradient theory of porous thermoelastic solids, we study the deformation of isotropic chiral cylinders subjected to a temperature field that is linear in the axial coordinate. b. We introduce the problem of generalized plane deformation of a chiral cylinder, and we express the equilibrium equations in terms of displacements and microdilatation function. c. We establish the solution of the problem when the temperature distribution is independent of the axial coordinate. It is shown that the thermal field produces extension, bending and torsion. d. We study the thermoelastic deformation of cylinders subjected to a thermal field which is linear in the axial coordinate. We present a method to reduce the three-dimensional problem to the study of plane problems. e. We use the results to study the deformation of a circular cylinder subjected to a uniform temperature variation. In contrast to the case of achiral materials, the thermal field in chiral cylinders produces torsional effects.
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