1 Introduction
The behavior of chiral elastic bodies is a topic of current interest both from a theoretical and an experimental point of view. The chiral elastic solid was used as model for various crystalline materials, carbon nanotubes, bones, honeycomb structures, as well as composites with inclusions. Eringen [1] introduced the theory of microstretch continua as a generalization of Cosserat theory. In this theory, the material particles undergo a uniform microdilatation and a rigid microrotation. The intended applications of the theory are to porous media filled with gas, composite materials reinforced with chopped elastic fibers, bones and other materials with microstructure. In the case of a microstretch continuum, the change of microdeformation tensor, from the constant value in the natural state, has the form \(\varphi \delta _{ij} + \varepsilon _{jik} \psi _{k}\), where \(\varphi \) is the microdilatation function, \(\psi _{k}\) is the microrotation vector, \(\delta _{ij}\) is the Kronecker delta and \(\varepsilon _{jik}\) is the alternating symbol. In the absence of microrotations, the equations of the linear theory of microstretch elastic solids coincide with the equations of the elastic materials with voids introduced by Cowin and Nunziato [2] to study the deformation of porous solids. In the theory of elastic materials with voids, the volume fraction field is a function strictly positive and less than 1. Considering the multiple possibilities of using the solid microstretch model, in this paper we will use the theory of microstretch continua to investigate the deformation of porous bodies. In what follows, the subscripts preceded by a comma denote partial differentiation with respect to the corresponding coordinate. The function \(\varphi \) is dimensionless, and the functions \(\varphi _{,j}\) and \(u_{i,rs}\) have the same dimensions. If the functions \(\varphi \) and \(\varphi _{,j}\) are considered as independent constituent variables, then the second-order partial derivatives of the displacement vector have to be included in the set of independent constituent variables. Thus, the introduction 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., [7‐9] 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, 13‐18]). 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.
Advertisement
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.
2 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 \(\partial B.\) The boundary \(\partial B\) 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 \(\varphi \) be the microstretch function (microdilatation function). The strain tensors are defined byIn the case of chiral porous thermoelastic bodies, the constitutive equations are given by (see, e.g., [1, 8, 22])where \(\tau _{ij}\) is the stress tensor, \(\mu _{ijk}\) is the dipolar stress tensor, \(\sigma _{i}\) is the microstretch stress vector, g is the intrinsic body force, T is the temperature measured from the constant absolute temperature of reference state, \(\delta _{ij}\) is Kronecker delta, \(\varepsilon _{ijk}\) is the alternating symbol, and \(\lambda , \mu , \alpha _{s}(s=1,2,\ldots ,5)\), \(\beta _{1}, \beta _{2}, \beta ,\) \(b,d,a_{0}, \xi \) 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 \(e_{ij}, \kappa _{ijk},\varphi \) and \(\varphi _{,k}\). The equations of equilibrium, in the absence of body loads, are given byThe 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.
$$\begin{aligned} e_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i}),\;\; \kappa _{ijk} = u_{k,ij}. \end{aligned}$$
(1)
$$\begin{aligned} \tau _{ij}&= \lambda e_{rr}\delta _{ij} + 2\mu e_{ij} + \textrm{d}\varphi \delta _{ij} + f (\varepsilon _{ikm}\kappa _{jkm}+\varepsilon _{jkm}\kappa _{ikm}) - b T \delta _{ij},\nonumber \\ \mu _{ijk}&= \frac{1}{2}\alpha _{1}(\kappa _{rri}\delta _{jk} + 2\kappa _{krr}\delta _{ij} + \kappa _{rrj}\delta _{ik})\nonumber \\&\quad +\alpha _{2}(\kappa _{irr}\delta _{jk} + \kappa _{jrr}\delta _{ik}) +2\alpha _{3}\kappa _{rrk}\delta _{ij} + \beta _{1}\delta _{ij}\varphi _{,k}+ \beta _{2}(\delta _{ik}\varphi _{,j} + \delta _{jk}\varphi _{,i})\nonumber \\&\quad + 2\alpha _{4} \kappa _{ijk} + \alpha _{5}(\kappa _{kji} + \kappa _{kij}) +f(\varepsilon _{iks}e_{js} + \varepsilon _{jks}e_{is}),\nonumber \\ \sigma _{i}&=\beta _{1}\kappa _{rri} + 2\beta _{2}\kappa _{irr} + a_{0}\varphi _{,i}, \;\; g = de_{rr} + \xi \varphi - \beta T, \end{aligned}$$
(2)
$$\begin{aligned} \tau _{ji,j} - \mu _{kji,kj}=0, \;\; \sigma _{j,j} - g= 0. \end{aligned}$$
(3)
Toupin [3, 4] introduced the functions \(P_{i}, R_{i}\) and \(Q_{i}\) defined bywhere \(n_{j}\) are the components of the outward unit normal of \(\partial B, D_{i}\) are the components of the surface gradient, \(D_{i} = (\delta _{ik} - n_{i}n_{k})\partial /\partial x_{k}, s_{i}\) are the components of the unit vector tangent to C and \(<g>\) denotes the difference of limits of g from both sides of C.
$$\begin{aligned}&P_{i} = (\tau _{ki} - \mu _{ski,s})n_{k} - D_{j}(n_{r}\mu _{rji}) + n_{s}n_{p}\mu _{spi}(D_{k}n_{k}),\nonumber \\&R_{i} = \mu _{rsi}n_{r}n_{s}, Q_{i} = <\mu _{pji}n_{p}n_{q}>\varepsilon _{jrq}s_{r}\;\; \text {on}\; \partial B, \end{aligned}$$
(4)
Advertisement
In the case of traction problem, the boundary conditions are [1, 24]where \({\widetilde{P}}_{i}, {\widetilde{R}}_{i}, {\widetilde{\sigma }}\) and \({\widetilde{Q}}_{i}\) are prescribed functions.
$$\begin{aligned} P_{i} = {\widetilde{P}}_{i}, R_{i} = {\widetilde{R}}_{i}, \sigma _{i}n_{i} = {\widetilde{\sigma }}\;\; \text {on}\; \partial B \backslash C, Q_{i} = {\widetilde{Q}}_{i}\;\; \text {on}\; C, \end{aligned}$$
(5)
We assume that the region B from here on refers to the interior of a right cylinder of length h with the cross section \(\Sigma \) and the lateral boundary \(\Pi \). Let \(\Gamma \) be the boundary of \(\Sigma \). The coordinate system consists of the orthonormal basis \(\{\textbf{e}_{1}, \textbf{e}_{2}, \textbf{e}_{3}\}\) 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 \(\Sigma _{1}\) and \(\Sigma _{2}\), respectively, the cross section located at \(x_{3} = 0\) and \(x_{3}=h\). Let \(\Gamma _{\alpha }\) be the boundary of \(\Sigma _{\alpha }\), \((\alpha =1,2)\). 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 \(\Pi \). 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 conditionsLet \(\textbf{F} = (F_{1}, F_{2}, F_{3})\) and \(\textbf{M} = (M_{1}, M_{2}, M_{3})\) be prescribed vectors representing the resultant force and resultant moment about O of the tractions acting on \(\Sigma _{1}\). On \(\Sigma _{2}\), there are tractions applied so as to satisfy the equilibrium conditions of the cylinder. We have introduced the mechanical loads \(\textbf{F}\) and \(\textbf{M}\) in order to compare the effects of thermal field with those produced by the resultants \(\textbf{F}\) and \(\textbf{M}\). On the end \(\Sigma _{1}\), we have the boundary conditionsFrom (4), we find thatwhere \((n_{1}, n_{2},0)\) are the direction cosines of the exterior normal to \(\Pi .\)
$$\begin{aligned} T = T_{0}(x_{1}, x_{2}) + x_{3}T_{1}(x_{1},x_{2}), \end{aligned}$$
(6)
$$\begin{aligned} P_{i} = 0, R_{i} = 0, \sigma _{\alpha }n_{\alpha } = 0\;\; \text {on}\; \Pi . \end{aligned}$$
(7)
$$\begin{aligned}&\int _{\Sigma _{1}}P_{\alpha }\textrm{d}a + \int _{\Gamma _{1}}Q_{\alpha } \textrm{d}s = F_{\alpha }, \end{aligned}$$
(8)
$$\begin{aligned}&\int _{\Sigma _{1}} P_{3}\textrm{d}a + \int _{\Gamma _{1}}Q_{3}\textrm{d}s = F_{3},\end{aligned}$$
(9)
$$\begin{aligned}&\int _{\Sigma _{1}}(x_{\alpha }P_{3} + R_{\alpha })\textrm{d}a + \int _{\Gamma _{1}}x_{\alpha }Q_{3}\textrm{d}s = \varepsilon _{\beta \alpha 3}M_{\beta },\end{aligned}$$
(10)
$$\begin{aligned}&\int _{\Sigma _{1}} \varepsilon _{\alpha \beta 3}x_{\alpha }P_{\beta }\textrm{d}a + \int _{\Gamma _{1}}\varepsilon _{a\beta 3}x_{\alpha }Q_{\beta } \textrm{d}s = M_{3}. \end{aligned}$$
(11)
$$\begin{aligned}&P_{i} = -\tau _{3i} +2\mu _{\alpha 3i,\alpha } + \mu _{33i,3}, R_{i} = \mu _{33i}\;\; \text {on}\; \Sigma _{1},\nonumber \\&Q_{i} = -2\mu _{\alpha 3i}n_{\alpha }\;\;\text {on}\; \Gamma _{1}, \end{aligned}$$
(12)
3 Two-dimensional problems
In what follows, we will reduce the problem formulated in Sect. 2 to the study of some two-dimensional problems. In this section, we present some results concerning the generalized plane strain of a chiral thermoelastic cylinder. We assume now that a body force \({\mathscr {F}}_{j}\) and a microstretch force density l are prescribed on B. Let us suppose that \({\mathscr {F}}_{j}, l, {\widetilde{P}}_{i}, {\widetilde{R}}_{i}\) and \({\widetilde{\sigma }}\) are independent of the axial coordinate \(x_{3}\). We define the state of generalized plane strain of the cylinder B to be that state in which the displacement vector, the microdilatation function and the temperature are independent of the axial coordinate,From (1), we obtain \(e_{33} = 0, \kappa _{j3i} = 0\) andThe constitutive equations (2) reduce toandThe equations of equilibrium in the presence of body loads can be written asIt follows from (4) that in the plane problems the functions \(P_{i}\) and \(R_{i}\) have the following form on the lateral surfaceThe conditions on the surface \(\Pi \) becomeThe two-dimensional problem of thermoelasticity consists in finding the functions \(u_{j}\) and \(\varphi \) satisfying the Eqs. (14), (15) and (17) on \(\Sigma _{1}\), and the boundary conditions (19) on \(\Gamma _{1}\). We denote by \(({\mathscr {H}})\) the problem characterized by the Eqs. (14), (15), (17) and the boundary conditions (19). Using the results established by Hlavacek and Hlavacek [25], we can easily deduce the following proposition.
$$\begin{aligned} u_{j}= u_{j}(x_{1}, x_{2}), \varphi = \varphi (x_{1}, x_{2}), T = T(x_{1}, x_{2}), (x_{1}, x_{2})\in \Sigma _{1}. \end{aligned}$$
(13)
$$\begin{aligned} e_{\alpha \beta } = \frac{1}{2}(u_{\alpha ,\beta } + u_{\beta ,\alpha }), 2e_{\alpha 3} = u_{3,\alpha }, \kappa _{\alpha \beta j} = u_{j,\alpha \beta }. \end{aligned}$$
(14)
$$\begin{aligned} \tau _{\alpha \beta }&= \lambda e_{\rho \rho }\delta _{\alpha \beta } + 2\mu e_{\alpha \beta } + \textrm{d}\varphi \delta _{\alpha \beta }+f(\varepsilon _{\alpha \rho 3}\kappa _{\beta \rho 3} + \varepsilon _{\beta \rho 3}\kappa _{\alpha \rho 3}) - b T\delta _{\alpha \beta },\nonumber \\ \tau _{\alpha 3}&= 2\mu e_{\alpha 3} + f\varepsilon _{\rho \beta 3}\kappa _{\alpha \rho \beta },\nonumber \\ \mu _{\alpha \beta \gamma }&= \frac{1}{2}\alpha _{1}(\kappa _{\rho \rho \alpha }\delta _{\beta \gamma } +2\kappa _{\gamma \rho \rho }\delta _{\alpha \beta } + \kappa _{\rho \rho \beta }\delta _{\alpha \gamma }) + \alpha _{2}(\kappa _{\alpha \rho \rho }\delta _{\beta \gamma } + \kappa _{\beta \rho \rho }\delta _{\alpha \gamma })\nonumber \\&\quad + 2\alpha _{3}\kappa _{\rho \rho \gamma }\delta _{\alpha \beta } +2\alpha _{4}\kappa _{\alpha \beta \gamma } + \alpha _{5}(\kappa _{\gamma \beta \alpha } + \kappa _{\gamma \alpha \beta }) + \beta _{1}\delta _{\alpha \beta }\varphi _{,\gamma }\nonumber \\&\quad +2\beta _{2}(\delta _{\alpha \gamma } \varphi _{,\beta } + \delta _{\beta \gamma }\varphi _{,\alpha }) +f(\varepsilon _{\alpha \gamma 3}e_{\beta 3} + \varepsilon _{\beta \gamma 3}e_{\alpha 3}),\nonumber \\ \mu _{\alpha \beta 3}&= 2\alpha _{3}\kappa _{\rho \rho 3}\delta _{\alpha \beta } + 2\alpha _{4}\kappa _{\alpha \beta 3} + f(\varepsilon _{\rho \alpha 3}e_{\beta \rho } + \varepsilon _{\rho \beta 3}e_{\alpha \rho }), \nonumber \\ \sigma _{\alpha }&= \beta _{1}\kappa _{\rho \rho \alpha } + 2\beta _{2}\kappa _{\alpha \rho \rho }+a_{0}\varphi _{,\alpha },\;\; g = de_{\rho \rho } + \xi \varphi -\beta T, \end{aligned}$$
(15)
$$\begin{aligned} \tau _{33}&= \lambda e_{\rho \rho }+\textrm{d}\varphi - b T,\nonumber \\ \mu _{3\alpha \beta }&= \frac{1}{2}\alpha _{1}\kappa _{\rho \rho 3}\delta _{\alpha \beta } +\alpha _{5}\kappa _{\beta \alpha 3} + f \varepsilon _{\beta \rho 3}e_{\alpha \rho },\nonumber \\ \mu _{3\alpha 3}&= \frac{1}{2}\alpha _{1}\kappa _{\rho \rho \alpha } + \alpha _{2}\kappa _{\alpha \rho \rho } + f \varepsilon _{\rho \alpha 3}e_{3\rho } + \beta _{2}\varphi _{,\alpha },\nonumber \\ \mu _{33\alpha }&= \alpha _{1}\kappa _{\alpha \rho \rho } + 2\alpha _{3}\kappa _{\rho \rho \alpha } + \beta _{1}\varphi _{,\alpha } + 2f \varepsilon _{3\alpha \rho }e_{3\rho },\nonumber \\ \mu _{333}&= (\alpha _{1}+2\alpha _{3})\kappa _{\rho \rho 3}, \sigma _{3} = \beta _{1}\kappa _{\rho \rho 3}. \end{aligned}$$
(16)
$$\begin{aligned} \tau _{\alpha i,\alpha } - \mu _{\alpha \beta i, \alpha \beta } + {\mathscr {F}}_{i} = 0, \sigma _{\alpha ,\alpha } - g+l =0\;\;\text {on}\; \Sigma _{1}. \end{aligned}$$
(17)
$$\begin{aligned} P_{i}&= (\tau _{\beta i} - \mu _{\rho \beta i,\rho })n_{\beta } - D_{\rho }(n_{\beta }\mu _{\beta \rho i}) + n_{\beta }n_{\alpha }\mu _{\beta \alpha i}(D_{\rho }n_{\rho }),\nonumber \\ R_{i}&= \mu _{\rho \alpha i}n_{\rho }n_{\alpha }. \end{aligned}$$
(18)
$$\begin{aligned} P_{i} = {\widetilde{P}}_{i}, R_{i} = {\widetilde{R}}_{i}, \sigma _{\alpha }n_{\alpha } = {\widetilde{\sigma }}\;\;\text {on}\; \Gamma _{1}. \end{aligned}$$
(19)
Theorem 1
The necessary and sufficient conditions for the existence of a solution to the problem \(({\mathscr {H}})\) are given byIn view of (14) and (15), the equations of equilibrium can be written in the formThe functions \(\tau _{33}, \mu _{3\alpha \beta }, \mu _{3\alpha 3}, \mu _{33\alpha },\mu _{333}\) and \(\sigma _{3}\) can be calculated after the determination of displacements and microdilatation.
$$\begin{aligned} \int _{\Sigma _{1}} {\mathscr {F}}_{j}\textrm{d}a + \int _{\Gamma _{1}}{\widetilde{P}}_{j} \textrm{d}s&= 0,\nonumber \\ \int _{\Sigma _{1}} \varepsilon _{3\alpha \beta }x_{\alpha } {\mathscr {F}}_{\beta } \textrm{d}a + \int _{\Gamma _{1}} \varepsilon _{3\alpha \beta }(x_{\alpha }{\widetilde{P}}_{\beta } + n_{\alpha }{\widetilde{R}}_{\beta })\textrm{d}s&= 0. \end{aligned}$$
(20)
$$\begin{aligned}&\mu \Delta u_{\alpha } + (\lambda +\mu )u_{\beta ,\beta \alpha } - 2(\alpha _{3}+\alpha _{4})\Delta \Delta u_{\alpha }- 2(\alpha _{1}+\alpha _{2}+\alpha _{5})\Delta u_{\beta ,\beta \alpha } \nonumber \\&\qquad + 2 f \varepsilon _{\alpha \beta 3} \Delta u_{3,\beta } + \textrm{d}\varphi _{,\alpha }- (\beta _{1}+2\beta _{2})\Delta \varphi _{,\alpha } - b T_{,\alpha } + {\mathscr {F}}_{\alpha } = 0,\nonumber \\&\mu \Delta u_{3} - 2(\alpha _{3} + \alpha _{4})\Delta \Delta u_{3} + 2f \varepsilon _{\rho \nu 3}\Delta u_{\nu ,\rho } + {\mathscr {F}}_{3} = 0,\nonumber \\ \qquad&(\beta _{1} + 2\beta _{2})\Delta u_{\rho ,\rho } + a_{0}\Delta \varphi - d u_{\rho ,\rho } - \xi \varphi + \beta T + l =0. \end{aligned}$$
(21)
4 Plane temperature field
In this section, we present the solution of the problem under the following assumptionsIt 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 formwhere \(u_{j}^{(k)}, \varphi ^{(k)}, w_{j}\) and \(\psi \) 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 \(u_{j}^{(k)}\) and microdilatation function \(\varphi ^{(k)}\). The body loads and the boundary data associated with the problems \(A^{(k)}\) will be precised in what follows. We introduce the notationsIt follows from (15) that the constitutive equations implyWe shall use the notationsWe denote by \(({\mathcal {T}})\) the plane thermoelastic problem which corresponds to the temperature \(T_{0}\) and is characterized by the displacement vector \(w_{j}\) and the microdilatation \(\psi .\) Thus, the constitutive equations in the problem \(({\mathcal {T}})\) arewhereWe shall use the notationsFrom (1) and (23), we obtainIn view of (2) and (29), we find that the functions \(\tau _{ij},\mu _{ijk},\sigma _{i}\) and g are given byHere we have used the relations (25)–(27).
$$\begin{aligned} T = T_{0}(x_{1},x_{2}), \;\;\;\textbf{F} = F_{3} \textbf{e}_{3}, \;\;\; \textbf{M} = M_{j}\textbf{e}_{j}. \end{aligned}$$
(22)
$$\begin{aligned} u_{\alpha }&=-\frac{1}{2}a_{\alpha }x_{3}^{2} + \varepsilon _{3\beta \alpha }a_{4}x_{\beta }x_{3} + \sum _{k=1}^{4}a_{k}u_{\alpha }^{(k)} + w_{\alpha }(x_{1},x_{2}),\nonumber \\ u_{3}&= (a_{1}x_{1} + a_{2}x_{2} + a_{3})x_{3} + \sum _{k=1}^{4}a_{k}u_{3}^{(k)}+w_{3}(x_{1},x_{2}),\nonumber \\ \varphi&= \sum _{k=1}^{4}a_{k}\varphi ^{(k)}+\psi , \end{aligned}$$
(23)
$$\begin{aligned} e_{\alpha \beta }^{(k)} = \frac{1}{2}(u_{\alpha ,\beta }^{(k)} + u_{\beta ,\alpha }^{(k)}), 2e_{\alpha 3}^{(k)} = u_{3,\alpha }^{(k)}, \kappa _{\alpha \beta j}^{(k)} = u_{j,\alpha \beta }^{(k)}. \end{aligned}$$
(24)
$$\begin{aligned} \tau _{\alpha \beta }^{(k)}&= \lambda e_{\rho \rho }^{(k)}\delta _{\alpha \beta } + 2\mu e_{a\beta }^{(k)} + \textrm{d}\varphi ^{(k)}\delta _{\alpha \beta } + f(\varepsilon _{\alpha \rho 3}\kappa _{\beta \rho 3}^{(k)} + \varepsilon _{\beta \rho 3}\kappa _{\alpha \rho 3}^{(k)}),\nonumber \\ \tau _{\alpha 3}^{(k)}&= 2\mu e_{\alpha 3}^{(k)} + f \varepsilon _{\rho \beta 3}\kappa _{\alpha \rho \beta }^{(k)},\nonumber \\ \mu _{\alpha \beta \gamma }^{(k)}&= \frac{1}{2}\alpha _{1}(\kappa _{\rho \rho \alpha }^{(k)} \delta _{\beta \gamma } + 2\kappa _{\gamma \rho \rho }^{(k)} \delta _{\alpha \beta }+ \kappa _{\rho \rho \beta }^{(k)}\delta _{\alpha \gamma })\nonumber \\&\quad +\alpha _{2}(\kappa _{\alpha \rho \rho }^{(k)}\delta _{\beta \gamma } + \kappa _{\beta \rho \rho }^{(k)}\delta _{\alpha \gamma }) + 2\alpha _{3}\kappa _{\rho \rho \gamma }^{(k)}\delta _{\alpha \beta }\nonumber \\&\quad +2\alpha _{4}\kappa _{\alpha \beta \gamma }^{(k)} + \alpha _{5}(\kappa _{\gamma \beta \alpha }^{(k)} + \kappa _{\gamma \alpha \beta }^{(k)}) + \beta _{1}\delta _{\alpha \beta }\varphi _{,\gamma }^{(k)}\nonumber \\&\quad +\beta _{2}(\delta _{\alpha \gamma }\varphi _{,\beta }^{(k)} + \delta _{\beta \gamma }\varphi _{,\alpha }^{(k)}) + f(\varepsilon _{\alpha \gamma 3}e_{\beta 3}^{(k)} + \varepsilon _{\beta \gamma 3}e_{\alpha 3}^{(k)}),\nonumber \\ \mu _{\alpha \beta 3}^{(k)}&= 2\alpha _{3}\kappa _{\rho \rho 3}^{(k)}\delta _{\alpha \beta } + 2\alpha _{4}\kappa _{\alpha \beta 3}^{(k)} +f(\varepsilon _{\rho \alpha 3}e_{\beta \rho }^{(k)} + \varepsilon _{\rho \beta 3}e_{\alpha \rho }^{(k)}),\nonumber \\ \sigma _{\alpha }^{(k)}&= \beta _{1}\kappa _{\rho \rho \alpha }^{(k)} + 2\beta _{2}\kappa _{\alpha \rho \rho }^{(k)} + a_{0}\varphi _{,\alpha }^{(k)}, \;\; g^{(k)} = de_{\rho \rho }^{(k)} + \xi \varphi ^{(k)}. \end{aligned}$$
(25)
$$\begin{aligned} \tau _{33}^{(k)}&= \lambda e_{\rho \rho }^{(k)} + \textrm{d}\varphi ^{(k)},\;\; \mu _{3\alpha \beta }^{(k)} = \frac{1}{2}\alpha _{1}\kappa _{\rho \rho 3}^{(k)}\delta _{\alpha \beta } + \alpha _{5}\kappa _{\beta \alpha 3}^{(k)} + f\varepsilon _{\beta \rho 3}e_{\alpha \rho }^{(k)},\nonumber \\ \mu _{3\alpha 3}^{(k)}&= \frac{1}{2}\alpha _{1}\kappa _{\rho \rho \alpha }^{(k)} + \alpha _{2}\kappa _{\alpha \rho \rho }^{(k)} + f \varepsilon _{\rho \alpha 3}e_{3\rho }^{(k)} + \beta _{2}\varphi _{,\alpha }^{(k)},\nonumber \\ \mu _{33\alpha }^{(k)}&= \alpha _{1}\kappa _{\alpha \rho \rho }^{(k)} + 2\alpha _{3}\kappa _{\rho \rho \alpha }^{(k)} + \beta _{1}\varphi _{,\alpha }^{(k)} + 2f \varepsilon _{3\alpha \rho }e_{3\rho }^{(k)},\nonumber \\ \mu _{333}^{(k)}&= (\alpha _{1}+2\alpha _{3})\kappa _{\rho \rho 3}^{(k)},\; \sigma _{3}^{(k)} = \beta _{1}\kappa _{\rho \rho 3}^{(k)}. \end{aligned}$$
(26)
$$\begin{aligned} t_{\alpha \beta }&= \lambda \eta _{\rho \rho }\delta _{\alpha \beta } + 2\mu \eta _{\alpha \beta } + \textrm{d}\psi \delta _{\alpha \beta } + f(\varepsilon _{\alpha \rho 3}\xi _{\beta \rho 3}+\varepsilon _{\beta \rho 3}\xi _{\alpha \rho 3}) - b T_{0} \delta _{\alpha \beta },\nonumber \\ t_{\alpha 3}&= 2\mu \eta _{\alpha 3} + f\varepsilon _{\rho \beta 3}\xi _{\alpha \rho \beta },\nonumber \\ m_{\alpha \beta \gamma }&= \frac{1}{2}\alpha _{1}(\xi _{\rho \rho \alpha }\delta _{\beta \gamma } + 2\xi _{\gamma \rho \rho }\delta _{\alpha \beta } + \xi _{\rho \rho \beta }\delta _{\alpha \gamma }) +\alpha _{2}(\xi _{\alpha \rho \rho }\delta _{\beta \gamma } + \xi _{\beta \rho \rho }\delta _{\alpha \gamma }) \nonumber \\ {}&\quad + 2\alpha _{3}\xi _{\rho \rho \gamma }\delta _{\alpha \beta } + 2\alpha _{4}\xi _{\alpha \beta \gamma } + \alpha _{5}(\xi _{\gamma \beta \alpha } + \xi _{\gamma \alpha \beta }) + \beta _{1}\delta _{\alpha \beta }\psi _{,\gamma },\nonumber \\&\quad +2\beta _{2}(\delta _{\alpha \gamma }\psi _{,\beta } + \delta _{\beta \gamma }\psi _{,\alpha }) + f(\varepsilon _{\alpha \gamma 3}\eta _{\beta 3} + \varepsilon _{\beta \gamma 3}\eta _{\alpha 3}),\nonumber \\ m_{\alpha \beta 3}&= 2\alpha _{3}\xi _{\rho \rho 3}\delta _{\alpha \beta } + 2\alpha _{4}\xi _{\alpha \beta 3} + f(\varepsilon _{\rho \alpha 3}\eta _{\beta \rho } + \varepsilon _{\rho \beta 3}\eta _{\alpha \rho }), \nonumber \\ \pi _{\alpha }&= \beta _{1}\xi _{\rho \rho \alpha } + 2\beta _{2}\xi _{\alpha \rho \rho } + a_{0}\psi _{,\alpha },\;\; \gamma = \textrm{d}\eta _{\rho \rho } +\xi \psi - \beta T_{0}, \end{aligned}$$
(27)
$$\begin{aligned} \eta _{\alpha \beta } = \frac{1}{2}(w_{\alpha ,\beta } + w_{\beta ,\alpha }), 2\eta _{\alpha 3} = w_{3,\alpha }, \xi _{\alpha \beta j} = w_{j,\alpha \beta }. \end{aligned}$$
(28)
$$\begin{aligned} t_{33}&= \lambda \eta _{\rho \rho } +\textrm{d}\psi - b T_{0},\nonumber \\ m_{3\alpha \beta }&= \frac{1}{2}\alpha _{1}\xi _{\rho \rho 3}\delta _{\alpha \beta } + \alpha _{5}\xi _{\beta \alpha 3} + f \xi _{\beta \rho 3}\eta _{\alpha \rho },\nonumber \\ m_{3\alpha 3}&= \frac{1}{2}\alpha _{1}\xi _{\rho \rho \alpha } + \alpha _{2}\xi _{\alpha \rho \rho } + f \xi _{\rho \alpha 3}\eta _{3\rho } + \beta _{2}\psi _{,\alpha },\nonumber \\ m_{33\alpha }&= \alpha _{1}\xi _{\alpha \rho \rho } + 2\alpha _{3}\xi _{\rho \rho \alpha } + \beta _{1}\psi _{,\alpha } + 2f \varepsilon _{3\alpha \rho }\eta _{3\rho },\nonumber \\ m_{333}&= (\alpha _{1}+2\alpha _{3})\xi _{\rho \rho 3}\xi _{\rho \rho 3}, \pi _{3} = \beta _{1}\xi _{\rho \rho 3}. \end{aligned}$$
(29)
$$\begin{aligned} e_{\alpha \beta }= & {} \eta _{\alpha \beta } + \sum _{k=1}^{4}a_{k}e_{\alpha \beta }^{(k)}, e_{\alpha 3} = \frac{1}{2}\varepsilon _{3\beta \alpha }x_{\beta }a_{4}\nonumber \\{} & {} +\eta _{\alpha 3} + \sum _{k=1}^{4}a_{k}e_{\alpha 3}^{(k)},e_{33} = a_{1}x_{1}+a_{2}x_{2}+a_{3},\nonumber \\ \kappa _{\alpha \beta \gamma }= & {} \xi _{\alpha \beta \gamma } + \sum _{k=1}^{4}a_{k}\kappa _{\alpha \beta \gamma }^{(k)}, \kappa _{\alpha \beta 3} = \xi _{\alpha \beta 3} + \sum _{k=1}^{4}a_{k}\kappa _{\alpha \beta 3}^{(k)},\nonumber \\ \kappa _{\beta 3\alpha }= & {} \varepsilon _{3\beta \alpha }a_{4},\kappa _{\alpha 33} = -\kappa _{33\alpha } = a_{4},\kappa _{333} = 0. \end{aligned}$$
(30)
$$\begin{aligned} \tau _{\alpha \beta }&= [(a_{1}x_{1}+a_{2}x_{2}+a_{3}) - 2f a_{4}]\delta _{\alpha \beta } + \sum _{k=1}^{4}a_{k}\tau _{\alpha \beta }^{(k)} +t_{\alpha \beta },\nonumber \\ \tau _{\alpha 3}&= \mu a_{4}\varepsilon _{3\beta \alpha }x_{\beta } + 2f \varepsilon _{\alpha \rho 3}a_{\rho } + \sum _{k=1}^{4}a_{k}\tau _{\alpha 3}^{(k)} + t_{\alpha 3},\nonumber \\ \tau _{33}&= (\lambda +2\mu )(a_{1}x_{1}+a_{2}x_{2}+a_{3})+4fa_{4}+t_{33} +\sum _{k=1}^{4}a_{k}\tau _{33}^{(k)},\nonumber \\ \mu _{111}&=2(\alpha _{2}-\alpha _{3})a_{1} + \sum _{k=1}^{4}a_{k}\mu _{111}^{(k)} + m_{111},\nonumber \\ \mu _{222}&= 2(\alpha _{2}-\alpha _{3})a_{2} + \sum _{k=1}^{4}a_{k}\mu _{222}^{(k)}+m_{222},\nonumber \\ \mu _{221}&= (\alpha _{1}-2\alpha _{3})a_{1} - fa_{4}x_{1} + \sum _{k=1}^{4}a_{k}\mu _{221}^{(k)} + m_{221},\nonumber \\ \mu _{112}&= (\alpha _{1} - 2\alpha _{3})a_{2} - f a_{4}x_{2} + \sum _{k=1}^{4}a_{k}\mu _{112}^{(k)}+m_{112},\nonumber \\ \mu _{121}&= \frac{1}{2}(2\alpha _{2}-\alpha _{3})a_{2} + \frac{1}{2}fa_{4}x_{2}+ \sum _{k=1}^{4}a_{k}\mu _{121}^{(k)} + m_{121},\nonumber \\ \mu _{122}&=\frac{1}{2}(2\alpha _{2}-\alpha _{1})a_{1} +\frac{1}{2}fa_{4}x_{1} + \sum _{k=1}^{4}a_{k}\mu _{122}^{(k)} + m_{122},\nonumber \\ \mu _{\alpha 33}&= \frac{1}{2}(2\alpha _{2} - \alpha _{1}+4\alpha _{4})a_{4} - \frac{1}{2}fa_{4}x_{\alpha }+\sum _{k=1}^{4}a_{k}\mu _{\alpha 33}^{(k)} + m_{\alpha 33},\nonumber \\ \mu _{33\alpha }&=(\alpha _{1}-2\alpha _{3} - 2\alpha _{4} + 2\alpha _{5})a_{\alpha }+fa_{4}x_{\alpha }+\sum _{k=1}^{4}a_{k}\mu _{33\alpha }^{(k)} + m_{33\alpha },\nonumber \\ \mu _{\alpha 3\beta }&= \varepsilon _{3\alpha \beta } f(a_{1}x_{1}+a_{2}x_{2}+a_{3}) + \varepsilon _{3\alpha \beta }(2\alpha _{4}-\alpha _{5})a_{4}\nonumber \\&\quad +\sum _{k=1}^{4}a_{k}\mu _{\alpha 3\beta }^{(k)} +m_{\alpha 3\beta },\nonumber \\ \mu _{\alpha \beta 3}&=\sum _{k=1}^{4}a_{k}\mu _{\alpha \beta 3}^{(k)}+m_{\alpha \beta 3},\mu _{333} = \sum _{k=1}^{4}a_{k}\mu _{333}^{(k)} + m_{333},\nonumber \\ \sigma _{\alpha }&= (2\beta _{2} - \beta _{1})a_{\alpha } +\sum _{k=1}^{4}a_{k}\sigma _{\alpha }^{(k)}+\pi _{\alpha },\sigma _{3} = \sum _{k=1}^{4}a_{k}\sigma _{3}^{(k)} +\pi _{3},\nonumber \\ g&= d(a_{1}x_{1}+a_{2}x_{2}+a_{3})+\sum _{k=1}^{4}a_{k}g^{(k)} + \gamma . \end{aligned}$$
(31)
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 \(\tau _{\alpha j}^{(k)}, \mu _{\alpha \beta i}^{(k)}, \sigma _{\alpha }^{(k)}\) and \(g^{(k)}\) \((k=1,2,3,4)\) satisfy the following equationswhere we have used the notationsThe equilibrium equations (3) reduce toLet us introduce the functionsWe require that the boundary conditions (7) be satisfied for any constants \(a_{k}\). Thus, we find that the functions \(\tau _{\alpha j}^{(k)}, \mu _{\alpha \beta i}^{(k)}\) and \(\sigma ^{(k)}\) have to satisfy the following boundary conditionswhereWe conclude that the problem \(A^{(k)}\) consists in finding the functions \(u_{j}^{(k)}\) and \(\varphi ^{(k)}\) satisfying the Eqs. (24), (25) and (32) on \(\Sigma _{1}\), 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 \(\Sigma \) and the constitutive constants. The boundary conditions (7) reduce toThe problem \(({\mathcal {T}})\) consists in finding the functions \(w_{j}\) and \(\psi \) which satisfy the Eqs. (27), (28) and (34) on \(\Sigma _{1}\), and the boundary conditions (38) on \(\Gamma _{1}\).
$$\begin{aligned} \tau _{\alpha j,\alpha }^{(k)} - \mu _{\alpha \beta j,\alpha \beta }^{(k)} + {\mathscr {F}}_{j}^{(k)} = 0,\;\; \sigma _{\alpha ,\alpha }^{(k)} - g^{(\kappa )} + l^{(k)} = 0\;\; \text {on}\; \Sigma _{1}, \end{aligned}$$
(32)
$$\begin{aligned} {\mathscr {F}}_{j}^{(\rho )} = \lambda \delta _{j\rho },\;\; {\mathscr {F}}_{j}^{(3)} = 0,\;\; {\mathscr {F}}_{j}^{(4)}=0,\;\, l^{(\rho )}=-\textrm{d}x_{\rho },\;\; l^{(3)} = -d,\; l^{(4)} = 0. \end{aligned}$$
(33)
$$\begin{aligned} t_{\alpha i,\alpha } - m_{\alpha \beta i,\alpha } = 0,\;\, \pi _{\alpha ,\alpha } - \gamma = 0\;\;\text {on}\;\; \Sigma _{1}. \end{aligned}$$
(34)
$$\begin{aligned} P_{i}^{(k)}&= (\tau _{\beta i}^{(k)} - \mu _{\rho \beta i,\rho }^{(k)}) n_{\beta } - D_{\rho }(n_{\beta }\mu _{\beta \rho i}^{(k)})+n_{\beta }n_{\alpha }\mu _{\beta \alpha i}^{(k)}(D_{\rho }n_{\rho }),\nonumber \\ R_{i}^{(k)}&= \mu _{\rho \alpha i}^{(k)} n_{\rho }n_{\alpha },\nonumber \\ P_{i}^{*}&= (t_{\beta i} - m_{\rho \beta i,\rho })n_{\beta } - D_{\rho }(n_{\beta }m_{\beta \rho i})+n_{\beta }n_{\alpha }m_{\beta \alpha i}(D_{\rho }n_{\rho }),\nonumber \\ R_{i}^{*}&= m_{\rho \alpha i}n_{\rho }n_{\alpha },\;\; \text {on}\;\; \Gamma _{1}. \end{aligned}$$
(35)
$$\begin{aligned} P_{i}^{(k)} = {\widetilde{P}}_{i}^{(k)},\;\; R_{i}^{(k)} = {\widetilde{R}}_{i}^{(k)},\;\; \sigma _{\alpha }^{(k)}n_{\alpha } = {\widetilde{\sigma }}^{(k)}\; \text {on}\; \Gamma _{1}, \end{aligned}$$
(36)
$$\begin{aligned} {\widetilde{P}}_{1}^{(1)}&= -\lambda x_{1}n_{1} + (\alpha _{1} - 2\alpha _{2})\varepsilon _{3\alpha \nu }(n_{1}n_{2})_{,\nu }n_{\alpha },\nonumber \\ {\widetilde{P}}_{2}^{(1)}&= -\lambda x_{1}n_{2}+ \frac{1}{2}(\alpha _{1} - 2\alpha _{2})\varepsilon _{3\alpha \rho }(n_{1}^{2} - n_{2}^{2})_{,\alpha }n_{\rho }, P_{3}^{(1)} =2f n_{2},\nonumber \\ {\widetilde{R}}_{1}^{(1)}&= 2\alpha _{3} - \alpha _{1}+(\alpha _{1} - 2\alpha _{2})n_{1}^{2}, {\widetilde{R}}_{2}^{(1)}=(\alpha _{1}-2\alpha _{2})n_{1}n_{2},\nonumber \\ {\widetilde{R}}_{3}^{(1)}&= 0, {\widetilde{\sigma }}^{(1)} = (\beta _{1}-2\beta _{2})n_{1},\nonumber \\ {\widetilde{P}}_{1}^{(2)}&= -\lambda x_{2}n_{1} + \frac{1}{2}(\alpha _{1}-2\alpha _{2})\varepsilon _{3\alpha \nu }(n_{1}^{2}-n_{2}^{2})_{,\alpha }n_{\nu },\nonumber \\ {\widetilde{P}}_{2}^{(2)}&= -\lambda x_{2}n_{2} + (\alpha _{1}-2\alpha _{2})\varepsilon _{3\alpha \nu }(n_{1}n_{2})_{,\nu }n_{\alpha }, {\widetilde{P}}_{3}^{(2)} = -2 f n_{1},\nonumber \\ {\widetilde{R}}_{1}^{(2)}&= (\alpha _{1}-2\alpha _{2})n_{1}n_{2}, {\widetilde{R}}_{2}^{(2)} = 2\alpha _{3}-\alpha _{1}+ (\alpha _{1}-2\alpha _{2})n_{2}^{2},\nonumber \\ {\widetilde{R}}_{3}^{(2)}&= 0, {\widetilde{\sigma }}^{(2)} = (\beta _{1}-2\beta _{2})n_{2},\nonumber \\ {\widetilde{P}}_{\alpha }^{(3)}&= -\lambda n_{\alpha }, {\widetilde{P}}_{3}^{(3)} = 0, {\widetilde{R}}_{j}^{(3)} = 0, {\widetilde{\sigma }}^{(3)} =0,\nonumber \\ {\widetilde{P}}_{1}^{(4)}&= \frac{1}{2}f[5n_{1} + D_{1}(x_{2}n_{2})+D_{2}(x_{2}n_{1}-2x_{1}n_{2})-2(x_{2}n_{1}n_{2} - x_{1}n_{2}^{2})(D_{\rho }n_{\rho })],\nonumber \\ {\widetilde{P}}_{2}^{(4)}&= \frac{1}{2}f[5n_{2} + D_{1}(x_{1}n_{2} - 2x_{2}n_{1}) + D_{2}(x_{1}n_{1}) - 2(x_{1}n_{1}n_{2} - x_{2}n_{1}^{2})(D_{\rho }n_{\rho })],\nonumber \\ {\widetilde{P}}_{3}^{(4)}&= \mu \varepsilon _{3\beta \nu }x_{\nu }n_{\beta }, {\widetilde{R}}_{1}^{(4)} = f(x_{1}n_{2}^{2} - x_{2}n_{1}n_{2}),\nonumber \\ {\widetilde{R}}_{2}^{(4)}&= f(x_{2}n_{1}^{2} - x_{1}n_{1}n_{2}), {\widetilde{R}}_{3}^{(4)}=0, {\widetilde{\sigma }}^{(4)}=0. \end{aligned}$$
(37)
$$\begin{aligned} P_{i}^{*} = 0,\;\; R_{i}^{*} = 0, \; \pi _{\alpha }n_{\alpha } = 0\;\; \text {on}\; \Gamma _{1}. \end{aligned}$$
(38)
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 obtainIn view of (7) and (31), we find from (39) thatThus, the conditions (8), with \(F_{\alpha } = 0\), 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 notationswhereThe 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 thatThe functions \(t_{3j}\) 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.
$$\begin{aligned} \int _{\Sigma _{1}} P_{\alpha }\textrm{d}a + \int _{\Gamma _{1}}Q_{\alpha }\textrm{d}s= & {} - \int _{\Gamma _{1}}(x_{\alpha } {\widetilde{P}}_{3} + n_{\alpha }{\widetilde{R}}_{3})\textrm{d}s \nonumber \\ {}{} & {} -\int _{\Sigma _{1}}[x_{\alpha }(\tau _{33,3} - \mu _{333,33})+2\mu _{\alpha 33,3} - \mu _{33\alpha ,3}]\textrm{d}s,\nonumber \\ \int _{\Sigma _{1}}P_{3}\textrm{d}a + \int _{\Gamma _{1}}Q_{3}\textrm{d}s= & {} - \int _{\Sigma _{1}}(\tau _{33} - \mu _{333,33})\textrm{d}a,\nonumber \\ \int _{\Sigma _{1}} (x_{\alpha }P_{3} +R_{\alpha })\textrm{d}a + \int _{\Gamma _{1}}x_{\alpha }Q_{3}\textrm{d}s= & {} -\int _{\Sigma _{1}}[x_{\alpha }(\tau _{33} - \mu _{333,3}) +2\mu _{\alpha 33} - \mu _{33\alpha }]\textrm{d}a,\nonumber \\ \int _{\Sigma _{1}}\varepsilon _{\alpha \beta 3}x_{\alpha }P_{\beta }\textrm{d}s + \int _{\Gamma _{1}}\varepsilon _{\alpha \beta 3}x_{\alpha }Q_{\beta }\textrm{d}s= & {} \int _{\Sigma _{1}}[\varepsilon _{\alpha \beta 3}x_{\alpha } (\mu _{33\beta ,3} - \tau _{3\beta })-2\varepsilon _{\alpha \beta 3}\mu _{\alpha 3\beta }]\textrm{d}a. \end{aligned}$$
(39)
$$\begin{aligned} \int _{\Sigma _{1}}P_{\alpha }\textrm{d}a + \int _{\Gamma _{1}}Q_{\alpha }\textrm{d}s=0. \end{aligned}$$
$$\begin{aligned}&\sum _{k=1}^{4}D_{\alpha k}a_{k} = \varepsilon _{3\alpha \beta }(M_{\beta } +M_{\beta }^{*}),\nonumber \\&\sum _{k=1}^{4}D_{3k}a_{k} = -F_{3} - F_{3}^{*},\nonumber \\&\sum _{k=1}^{4}D_{4k}a_{k} = -M_{3} - M_{3}^{*}. \end{aligned}$$
(40)
$$\begin{aligned}&D_{\alpha k} = \int _{\Sigma _{1}}(x_{\alpha }S_{33}^{(k)} + 2N_{\alpha 33}^{(k)} - N_{33\alpha }^{(k)})\textrm{d}a,\nonumber \\&D_{3k} = \int _{\Sigma _{1}} S_{33}^{(k)}\textrm{d}a, D_{4k} = \int _{\Sigma _{1}}\varepsilon _{3\alpha \beta }(x_{\alpha }S_{3\beta }^{(k)} +2N_{\alpha 3\beta }^{(k)})\textrm{d}a, \nonumber \\ {}&F_{3}^{*} = \int _{\Sigma _{1}}t_{33}\textrm{d}a, M_{\alpha }^{*} = \varepsilon _{3\alpha \beta }\int _{\Sigma _{1}}(x_{\beta }t_{33} + 2m_{\beta 33} - m_{33\beta })\textrm{d}a,\nonumber \\&M_{3}^{*} =\int _{\Sigma _{1}}\varepsilon _{3\alpha \beta }(x_{\alpha }t_{3\beta } +2m_{\alpha 3\beta })\textrm{d}a, \end{aligned}$$
(41)
$$\begin{aligned} S_{33}^{(\rho )}&=(\lambda +2\mu )x_{\rho }+\tau _{33}^{(\rho )}, S_{33}^{(3)}=\lambda +2\mu +\tau _{33}^{(3)},\nonumber \\ S_{33}^{(4)}&= 4f +\tau _{33}^{(4)}, S_{3\alpha }^{(\rho )} =2f \varepsilon _{\alpha \rho 3} + \tau _{\alpha 3}^{(\rho )},\nonumber \\ S_{3\alpha }^{(3)}&= \tau _{\alpha 3}^{(3)}, S_{3\alpha }^{(4)}= \mu \varepsilon _{3\beta \alpha }x_{\beta } + \tau _{\alpha 3}^{(4)},\nonumber \\ N_{\alpha 33}^{(i)}&= \frac{1}{2}(2\alpha _{2} - \alpha _{1} + 4\alpha _{4})\delta _{i\alpha }+\mu _{\alpha 33}^{(i)}, (i=1,2,3),\nonumber \\ N_{\alpha 33}^{(4)}&= -\frac{1}{2}fx_{\alpha }+\mu _{\alpha 33}^{(4)}, N_{33\alpha }^{(i)} = (\alpha _{1} -2\alpha _{3}-2\alpha _{4} + \alpha _{5})\delta _{i\alpha }\nonumber \\ {}&\quad +\mu _{33\alpha }^{(i)},\;\; N_{33\alpha }^{(4)} = fx_{\alpha }+\mu _{33\alpha }^{(4)}, N_{\alpha 3\beta }^{(\rho )} = \varepsilon _{3\alpha \beta }fx_{\rho } + \mu _{\alpha 3\beta }^{(\rho )},\nonumber \\ N_{\alpha 3\beta }^{(3)}&= \varepsilon _{3\alpha \beta }f + \mu _{\alpha 3\beta }^{(3)}, N_{\alpha 3\beta }^{(4)} = \varepsilon _{3\alpha \beta }(2\alpha _{4}-\alpha _{5})+\mu _{\alpha 3\beta }^{(4)}. \end{aligned}$$
(42)
$$\begin{aligned} \text{ det }(D_{mn})\ne 0. \end{aligned}$$
(43)
$$\begin{aligned} D_{mn} = D_{nm}. \end{aligned}$$
5 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 datawhere \(T_{1}\) and \(F_{\alpha }\) are prescribed.
$$\begin{aligned} T = x_{3}T_{1}(x_{1},x_{2}),\;\; \textbf{F}=F_{\alpha }\textbf{e}_{\alpha },\;\; \textbf{M} = \textbf{0}, \end{aligned}$$
(44)
We try to solve the problem assuming thatwhere \((u_{j}^{(k)},\varphi ^{(k)})\) is the solution of problem \(A^{(k)} (k=1,2,3,4)\), \(U_{j}, \Phi , v_{j}\) and \(\chi \) 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 \(\chi \). The strain measures in this problem are defined byWe denote by \(E_{\alpha j}\) and \(K_{\alpha \beta j}\) the strain measures in the plane strain problem associated with the displacement vector \(U_{j}\) and microdilatation \(\Phi \),In view of (1) and (45)–(47), we obtainLet us introduce the notationsClearly, \(s_{ij}\) is the stress tensor, \(\nu _{ijk}\) is the dipolar stress tensor, \(\pi _{j}\) is the microstretch stress vector and p is the intrinsic body force in an isothermal plane problem corresponding to the strain measures \(\gamma _{ij}\) and \(\zeta _{ijk}\). We now consider a thermoelastic plane problem associated with the thermal field \(T_{1}\), displacement vector \(U_{j}\) and microdilatation function \(\Phi \). In this problem, we denote the stress tensor, the dipolar stress tensor, the microstretch stress vector and the intrinsic body force by \(T_{ij}, M_{ijk}, H_{\alpha }\) and L, respectively. Thus, we haveFrom the constitutive equations (2) and the relations (48)–(50), we find that the stress tensor \(\tau _{ij}\) is given bywhereThe functions \(\mu _{ijk}\), \(\sigma _{j}\) and g can be expressed asIn these relations, we have used the following notationsIf we take into account (32), (33), (51) and (53), then the equilibrium equations (3) reduce to the equationsandIn equations (55) and (56), we have used the notationsFollowing (4) and (18), we define the functionsWith the help of relations (36), (37), (51), (53) and (58), we see that the conditions on the lateral surface (7) reduce toandwhereWe denote by \(({\mathcal {P}}_{1})\) the isothermal plane problem which consist in finding the functions \(v_{j}\) and \(\chi \) that satisfy the geometrical equations (46), the constitutive equations (49), the equilibrium equations (55) and the boundary conditions (59). Let us denote by \(({\mathcal {P}}_{2})\) 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 \(({\mathcal {P}}_{2})\) are satisfied for any thermal field. The necessary and sufficient conditions to solve the problem \(({\mathcal {P}}_{1})\) areBy using the divergence theorem, we find thatIt follows from (51), (57), (61) and (63) thatBy using the equilibrium equations, we findThe condition \(P_{3} = {\widetilde{P}}_{3}\) on the lateral boundary can be expressed in the formso that the relation (65) implies thatLet us note the identityFrom (65) and (68), we getIn view of (7), (51), (53) and (69), we find thatso that the first two conditions from (61) are satisfied. With the help of (40), (57) and (60), we obtainThe last two conditions from (62) reduce toLet us impose the conditions (8). In view of (12), we getBy using (67), we obtainWith the help of (7), (40), (51), (53) and (71), we see that the conditions (8) reduce toThe system (70), (72) uniquely determines the constants \(b_{k} (k=1,2,3,4)\).
$$\begin{aligned} u_{\alpha }&= - \frac{1}{2}c_{\alpha }x_{3}^{2} - \frac{1}{6}b_{\alpha }x_{3}^{3} + \varepsilon _{3\beta \alpha }(c_{4}x_{3} + \frac{1}{2}b_{4}x_{3}^{2})x_{\beta } \nonumber \\ {}&\quad +\sum _{k=1}^{4}(c_{k} + b_{k}x_{3})u_{\alpha }^{(k)} + x_{3}U_{\alpha } + v_{\alpha },\nonumber \\ u_{3}&= (c_{1}x_{1} + c_{2}x_{2}+c_{3})x_{3} + \frac{1}{2}(b_{1}x_{1} + b_{2}x_{2} + b_{3})x_{3}^{2}\nonumber \\&\quad +\sum _{k=1}^{4}(c_{k} + b_{k}x_{3})u_{3}^{(k)} + x_{3}U_{3} + v_{3},\nonumber \\ \varphi&= \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\varphi ^{(k)} + x_{3}\Phi +\chi , \end{aligned}$$
(45)
$$\begin{aligned} \gamma _{\alpha \beta } = \frac{1}{2}(v_{\alpha ,\beta } + v_{\beta ,\alpha }), 2\gamma _{\alpha 3} = v_{3,\alpha }, \zeta _{\alpha \beta j} = v_{j,\alpha \beta }. \end{aligned}$$
(46)
$$\begin{aligned} E_{\alpha \beta } = \frac{1}{2}(U_{\alpha ,\beta } + U_{\beta ,\alpha }), 2E_{\alpha 3} = U_{3,\alpha }, K_{\alpha \beta j} = U_{j,\alpha \beta }. \end{aligned}$$
(47)
$$\begin{aligned} e_{\alpha \beta }&= \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})e_{\alpha \beta }^{(k)} + x_{3}E_{\alpha \beta } +\gamma _{\alpha \beta },\nonumber \\ e_{\alpha 3}&= \frac{1}{2}\varepsilon _{3\beta \alpha }(c_{4} + b_{4}x_{3})x_{\beta } + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3}) e_{\alpha 3}^{(k)} + \gamma _{\alpha 3}+x_{3}E_{\alpha 3}\nonumber \\&\quad +\frac{1}{2}\sum _{k=1}^{4}b_{k}u_{\alpha }^{(k)} +\frac{1}{2}U_{\alpha },\nonumber \\ e_{33}&= c_{1}x_{1} + c_{2}x_{2} + c_{3} + (b_{1}x_{1} + b_{2}x_{2}+b_{3}) + \sum _{k=1}^{4}b_{k}u_{3}^{(k)} +U_{3},\nonumber \\ \kappa _{\alpha \beta \gamma }&= \sum _{k=1}^{4}(c_{k} + b_{k}x_{3}) \kappa _{\alpha \beta \gamma }^{(k)} + \zeta _{\alpha \beta \gamma }+x_{3}K_{\alpha \beta \gamma },\nonumber \\ \kappa _{\alpha \beta 3}&= \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\kappa _{\alpha \beta 3}^{(k)} + \zeta _{\alpha \beta 3} +x_{3}K_{\alpha \beta 3},\nonumber \\ \kappa _{\beta 3\alpha }&= \varepsilon _{3\beta \alpha }(c_{4} + b_{4}x_{3}) + \sum _{k=1}^{4}b_{k}u_{\alpha ,\beta }^{(k)} + U_{\alpha ,\beta },\nonumber \\ \kappa _{\alpha 33}&= c_{\alpha } + b_{\alpha }x_{3} + \frac{1}{2}\sum _{k=1}^{4}b_{k}e_{3\alpha }^{(k)} + \frac{1}{2}E_{\alpha 3},\nonumber \\ \kappa _{33\alpha }&= -c_{k}-b_{\alpha }x_{3} + \varepsilon _{3\beta \alpha }b_{4}x_{\beta },\;\; \kappa _{333} = b_{1}x_{1} + b_{2}x_{2} + b_{3}. \end{aligned}$$
(48)
$$\begin{aligned} s_{\alpha \beta }&= \lambda \gamma _{\rho \rho }\delta _{\alpha \beta } + 2\mu \gamma _{\alpha \beta } + f(\varepsilon _{\alpha \rho 3}\zeta _{\beta \rho 3} + \varepsilon _{\beta \rho 3}\zeta _{\alpha \rho 3}) + \textrm{d}\chi \delta _{\alpha \beta }, \nonumber \\ s_{\alpha 3}&= 2\mu \gamma _{\alpha 3} +f \varepsilon _{\rho \beta 3}\zeta _{\alpha \rho 3}, s_{33} = \lambda \gamma _{\rho \rho } + \textrm{d}\chi ,\nonumber \\ \nu _{\alpha \beta \gamma }&= \frac{1}{2}\alpha _{1}(\zeta _{\rho \rho \alpha }\delta _{\beta \gamma } + 2\zeta _{\gamma \rho \rho }\delta _{\alpha \beta } + \zeta _{\rho \rho \beta }\delta _{\alpha \gamma })+ \alpha _{2}(\zeta _{\alpha \rho \rho }\delta _{\beta \gamma } + \zeta _{\beta \rho \rho }\delta _{\alpha \gamma }) \nonumber \\&\quad + 2\alpha _{3}\zeta _{\rho \rho \gamma }\delta _{\alpha \beta }+2\alpha _{4}\zeta _{\alpha \beta \gamma } + \alpha _{5}(\zeta _{\gamma \beta \alpha } + \zeta _{\gamma \alpha \beta }) + f(\varepsilon _{\alpha \gamma 3}\gamma _{\beta 3} + \varepsilon _{\beta \gamma 3}\gamma _{\alpha 3})\nonumber \\&\quad +\beta _{1}\delta _{\alpha \beta }\chi _{,\gamma } + 2\beta _{2}(\delta _{\alpha \gamma }\chi _{,\beta } + \delta _{\beta \gamma }\chi _{,\alpha }),\nonumber \\ \nu _{\alpha \beta 3}&= 2\alpha _{3}\zeta _{\rho \rho 3}\delta _{\alpha \beta } + 2\alpha _{4}\zeta _{\alpha \beta 3} + f(\varepsilon _{\rho \alpha 3}\gamma _{\beta \rho } + \varepsilon _{\rho \beta 3}\gamma _{\alpha \rho }),\nonumber \\ \nu _{3\alpha \beta }&= \frac{1}{2}\alpha _{1}\zeta _{\rho \rho 3}\delta _{\alpha \beta } + \alpha _{5}\zeta _{\beta \alpha 3} + f\varepsilon _{\beta \rho 3}\gamma _{\alpha \rho },\nonumber \\ \nu _{3\alpha 3}&= \frac{1}{2}\alpha _{1}\zeta _{\rho \rho \alpha } + \alpha _{2}\zeta _{\alpha \rho \rho } + f \varepsilon _{\rho \alpha 3}\gamma _{3\rho } + \beta _{2}\chi _{,\alpha }, \nonumber \\ \nu _{33\alpha }&= \alpha _{1}\zeta _{\alpha \rho \rho }+2\alpha _{3}\zeta _{\rho \rho \alpha }+2f \varepsilon _{3\alpha \rho }\gamma _{3\rho }+\beta _{1}\chi _{,\alpha },\nonumber \\ \nu _{333}&= (\alpha _{1}+2\alpha _{3})\zeta _{\rho \rho 3}, h_{\alpha } = \beta _{1}\zeta _{\rho \rho \alpha } +2\beta _{2}\zeta _{\alpha \rho \rho } + a_{0}\chi _{,\alpha },\nonumber \\ h_{3}&= \beta _{1}\zeta _{\rho \rho 3}, p= \textrm{d}\gamma _{\rho \rho } + \xi \chi . \end{aligned}$$
(49)
$$\begin{aligned} T_{\alpha \beta }&= \lambda E_{\rho \rho }\delta _{\alpha \beta } + 2\mu E_{\alpha \beta }+\textrm{d}\Phi \delta _{\alpha \beta } + f(\varepsilon _{\alpha \rho 3} K_{\beta \rho 3} + \varepsilon _{\beta \rho 3}K_{\alpha \rho 3}) - b T_{1}\delta _{\alpha \beta },\nonumber \\ T_{\alpha 3}&= 2\mu E_{\alpha 3} + f\varepsilon _{\rho \beta 3}K_{\alpha \rho \beta },\nonumber \\ M_{\alpha \beta \gamma }&= \frac{1}{2}\alpha _{1}(K_{\rho \rho \alpha }\delta _{\beta \gamma } + 2K_{\gamma \rho \rho }\delta _{\alpha \beta }+K_{\rho \rho \beta }\delta _{\alpha \gamma })\nonumber \\&\quad + \alpha _{2}(K_{\alpha \rho \rho }\delta _{\beta \gamma } +K_{\beta \rho \rho }\delta _{\alpha \gamma }) + 2\alpha _{3}K_{\rho \rho \gamma }\delta _{\alpha \beta }+ 2\alpha _{4}K_{\alpha \beta \gamma } + \alpha _{5}(K_{\gamma \beta \alpha }+K_{\gamma \alpha \beta })\nonumber \\&\quad +\beta _{1}\delta _{\alpha \beta }\Phi _{,\gamma } +2\beta _{2}(\delta _{\alpha \gamma }\Phi _{,\beta } + \delta _{\beta \gamma }\Phi _{,\alpha }) + f(\varepsilon _{\alpha \gamma 3}E_{\beta 3}+\varepsilon _{\beta \gamma 3}E_{\alpha 3}),\nonumber \\ M_{\alpha \beta 3}&= 2\alpha _{3}K_{\rho \rho 3}\delta _{\alpha \beta } + 2\alpha _{4}K_{\alpha \beta 3}+f(\varepsilon _{\rho \alpha 3}E_{\beta \rho } + \varepsilon _{\rho \beta 3}E_{\alpha 3}),\nonumber \\ H_{\alpha }&= \beta _{1}K_{\rho \rho \alpha } + 2\beta _{2}K_{\alpha \rho \rho } + a_{0}\Phi _{,\alpha }, H_{3} = \beta _{1}K_{\rho \rho 3},\nonumber \\ L&= dE_{\rho \rho } +\xi \Phi - \beta T_{1}, T_{33} = \lambda E_{\rho \rho } + \textrm{d}\Phi - b T_{1},\nonumber \\ M_{3\alpha \beta }&= \frac{1}{2}\alpha _{1}K_{\rho \rho 3}\delta _{\alpha \beta } + \alpha _{5}K_{\beta \alpha 3}+f\varepsilon _{\beta \rho 3}E_{\alpha \rho },\nonumber \\ M_{3\alpha 3}&= \frac{1}{2}\alpha _{1}K_{\rho \rho \alpha } + \alpha _{2}K_{\alpha \rho \rho } + f\varepsilon _{\rho \alpha 3} E_{3\rho }+ \beta _{2}\Phi _{,\alpha }\nonumber \\ M_{33\alpha }&= \alpha _{1}K_{\alpha \rho \rho } + 2\alpha _{3}K_{\rho \rho \alpha }+\beta _{1}\Phi _{,\alpha }+2f \varepsilon _{3\alpha \rho }E_{3\rho },\;\; M_{333} = (\alpha _{1}+2\alpha _{3})K_{\rho \rho 3}. \end{aligned}$$
(50)
$$\begin{aligned} \tau _{\alpha \beta }&=s_{\alpha \beta } + x_{3}T_{a\beta }+\{\lambda [c_{1}x_{1}+c_{2}x_{2} + c_{3} + (b_{1}x_{1} + b_{2}x_{2}+b_{3})x_{3}]\nonumber \\&\quad -2f (c_{4} + b_{4}x_{3})\} \delta _{\alpha \beta } + \sum _{k=1}^{4}(c_{k}+b_{k}x_{3})\tau _{\alpha \beta }^{(k)} + G_{\alpha \beta },\nonumber \\ \tau _{\alpha 3}&= s_{\alpha 3} + x_{3}T_{\alpha 3} + 2f \varepsilon _{\alpha \rho 3}(c_{\rho } + b_{\rho }x_{3}) + \mu \varepsilon _{3\beta \alpha }(c_{4}+b_{4}x_{3}) x_{\beta } \nonumber \\ {}&\quad + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3}) \tau _{\alpha 3}^{(k)} + G_{\alpha 3},\nonumber \\ \tau _{33}&= s_{33}+x_{3}T_{33} + (\lambda +2\mu )[c_{1}x_{1} + c_{2}x_{2} + c_{3}+ (b_{1}x_{1} + b_{2}x_{2}+b_{3})x_{3}]\nonumber \\&\quad +4f (c_{4} +b_{4}x_{3}) +\sum _{k=1}^{4}(c_{k}+b_{k}x_{3}) \tau _{33}^{(k)} + G_{33}, \end{aligned}$$
(51)
$$\begin{aligned} G_{a\beta }&= f[\varepsilon _{3\rho \alpha }U_{\rho ,\beta } + \varepsilon _{3\rho \beta }U_{\rho ,\alpha } + \sum _{k=1}^{4}b_{k} (\varepsilon _{3\rho \alpha }u_{\rho ,\beta }^{(k)} + \varepsilon _{3\rho \beta }u_{\rho ,\alpha }^{(k)})],\nonumber \\ G_{\alpha 3}&= \mu (U_{\alpha } + \sum _{k=1}^{4}b_{k}u_{\alpha }^{(k)}) + f\varepsilon _{\alpha \rho 3}(U_{3,\rho } + \sum _{k=1}^{4}b_{k}u_{3,\rho }^{(k)}) -fb_{4}x_{\alpha },\nonumber \\ G_{33}&= (\lambda +2\mu )(U_{3} + \sum _{k=1}^{4}b_{k}u_{3}^{(k)}) + 2f \varepsilon _{3\alpha \beta }(U_{\alpha ,\beta }+\sum _{k=1}^{4}b_{k}u_{\alpha ,\beta }^{(k)}). \end{aligned}$$
(52)
$$\begin{aligned} \mu _{111}&= \nu _{111} + x_{3}M_{111} + 2(\alpha _{2} - \alpha _{3})(c_{1}+b_{1}x_{3}) + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3}) \mu _{111}^{(k)}+J_{111},\nonumber \\ \mu _{222}&= \nu _{222}+x_{3}M_{222} + 2(\alpha _{2}-\alpha _{3})(c_{2}+b_{2}x_{3})+\sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{222}^{(k)} + J_{222},\nonumber \\ \mu _{221}&= \nu _{221} + x_{3}M_{221}+(\alpha _{1}-2\alpha _{3})(c_{1}+b_{1}x_{3}) - f(c_{4} +b_{4}x_{3})x_{1}\nonumber \\&\quad +\sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{221}^{(k)} + J_{221},\nonumber \\ \mu _{112}&= \nu _{112} + x_{3}M_{112} + (\alpha _{1}-2\alpha _{3})(c_{2} +b_{2}x_{3}) - f(c_{4} +b_{4}x_{3})x_{2}\nonumber \\&\quad +\sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{112}^{(k)} + J_{112}, \nonumber \\\mu _{121}&= \nu _{121} + x_{3}M_{121} + \frac{1}{2}(2\alpha _{2} -\alpha _{1})(c_{2}+b_{2}x_{3})+\frac{1}{2} f(c_{4} + b_{4}x_{3})x_{2} \nonumber \\&\quad + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{121}^{(k)} + J_{121},\nonumber \\ \mu _{122}&= \nu _{122} + x_{3}M_{122} + \frac{1}{2}(2\alpha _{2} -\alpha _{1})(c_{1}+b_{1}x_{3}) +\frac{1}{2}f(c_{4} + b_{4}x_{3})x_{1}\nonumber \\ {}&\quad +\sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{122}^{(k)} + J_{122},\nonumber \\\mu _{\rho 33}&= \nu _{\rho 33}+x_{3}M_{\rho 33} + \frac{1}{2}(2\alpha _{2}-\alpha _{1}+4\alpha _{4})(c_{\rho } + b_{\rho }x_{3})\nonumber \\&\quad -\frac{1}{2}f (c_{4} +b_{4}x_{3})x_{\rho } + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{\rho 33}^{(k)} + J_{\rho 33},\nonumber \\ \mu _{33\rho }&= \nu _{33\rho } + x_{3}M_{33\rho } + (\alpha _{1}-2\alpha _{3} - 2\alpha _{4} + 2\alpha _{5})(c_{\rho }+b_{\rho }x_{3})\nonumber \\ {}&\quad + f(c_{4} + b_{4}x_{3})x_{\rho } + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{33\rho }^{(k)} + J_{33\rho }, \nonumber \\ \mu _{\alpha 3\beta }&= \nu _{\alpha 3\beta } + x_{3}M_{\alpha 3\beta } + (2\alpha _{4} - \alpha _{5})(c_{4} + b_{4}x_{3})\varepsilon _{\alpha \beta 3}\nonumber \\&\quad +f[c_{1}x_{1} + c_{2}x_{2} + c_{3} + (b_{1}x_{1} + b_{2}x_{2} + b_{3})x_{3}]\varepsilon _{\alpha \beta 3} \nonumber \\ {}&\quad +\sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{\alpha 3\beta }^{(k)} + J_{\alpha 3\beta },\nonumber \\ \mu _{\alpha \beta 3}&= \nu _{\alpha \beta 3} + x_{3}M_{\alpha \beta 3} + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\mu _{\alpha \beta 3}^{(k)}+J_{\alpha \beta 3}, \nonumber \\\mu _{333}&= \nu _{333} + x_{3}M_{333} + (\alpha _{1}+2\alpha _{3})\sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\kappa _{\alpha \alpha 3}^{(k)} + J_{333}, \nonumber \\\sigma _{\alpha }&= h_{\alpha } + x_{3}H_{\alpha } + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\sigma _{\alpha }^{(k)} + (2\beta _{2} - \beta _{1})(c_{\alpha } +b_{\alpha }x_{3}) \nonumber \\ {}&\quad +\beta _{1}\varepsilon _{3\beta \alpha }b_{4}x_{\beta } + \beta _{2}(\sum _{k=1}^{4}b_{k} e_{\alpha 3}^{(k)} + E_{\alpha 3}),\nonumber \\ \sigma _{3}&= h_{3}+x_{3}H_{3} + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3})\sigma _{3}^{(k)} + (b_{1}+2\beta _{2})(b_{1}x_{1} + b_{2}x_{2} +b_{3})\nonumber \\&\quad + 2\beta _{2}(\sum _{k=1}^{4}b_{k} u_{\rho ,\rho }^{(k)} + U_{\rho ,\rho }) + a_{0}(\sum _{k=1}^{4}b_{k} \varphi ^{(k)} + \Phi ),\nonumber \\ g&= p+x_{3}L + \sum _{k=1}^{4}(c_{k} + b_{k}x_{3}) g^{(k)} + d[c_{1}x_{1} + c_{2}x_{2} + c_{3} \nonumber \\ {}&\quad + (b_{1}x_{1} + b_{2}x_{2}+b_{3})x_{3} + \sum _{k=1}^{4}b_{k}u_{3}^{(k)} + U]. \end{aligned}$$
(53)
$$\begin{aligned} J_{111}&= - (\alpha _{1}+2\alpha _{3})b_{4}x_{2} + (\alpha _{1}+2\alpha _{2})(U_{3,1} + \sum _{k=1}^{4}b_{k}u_{3,1}^{(k)}), \nonumber \\ J_{222}&= (\alpha _{1}+2\alpha _{3})b_{4}x_{1} + (\alpha _{1} + 2\alpha _{2})(U_{3,2} + \sum _{k=1}^{4} b_{k}u_{3,2}^{(k)}), \nonumber \\ J_{221}&= -2\alpha _{3}b_{4}x_{2} + \alpha _{1}(U_{3,1} + \sum _{k=1}^{4}b_{k}u_{3,1}^{(k)}) - f(U_{2} + \sum _{k=1}^{4}b_{k}u_{2}^{(k)}), \nonumber \\J_{112}&= \alpha _{1}(U_{3,2} + \sum _{k=1}^{4} b_{k}u_{3,2}^{(k)}) + 2\alpha _{3}(b_{4} x_{1}+\sum _{k=1}^{4}b_{k}u_{1}^{(k)}) \nonumber \\ {}&\quad + f(U_{1} + \sum _{k=1}^{4} b_{k}u_{1}^{(k)}),\nonumber \\ J_{121}&= \frac{1}{2}\alpha _{1}b_{4}x_{1} + \alpha _{2}(U_{3,2} + \sum _{k=1}^{4} b_{k}u_{3,2}^{(k)}) -\frac{1}{2}f (U_{1} + \sum _{k=1}^{4}b_{k}u_{1}^{(k)}),\nonumber \\ J_{122}&= -\frac{1}{2}\alpha _{1}b_{4}x_{2} + \alpha _{2}(U_{3,1} + \sum _{k=1}^{4}b_{k}u_{3,1}^{(k)}) + \frac{1}{2}f (U_{2} + \sum _{k=1}^{4} b_{k}u_{2}^{(k)}), \nonumber \\ J_{\rho 33}&= -\frac{1}{2}(\alpha _{1} + 2\alpha _{5})\varepsilon _{3\rho \beta }b_{4}x_{\beta } + (\alpha _{2} + 2\alpha _{4} + \alpha _{5}) (U_{3,\rho }\nonumber \\&\quad + \sum _{k=1}^{4}b_{k}u_{3,\rho }^{(k)}) + \frac{1}{2}f \varepsilon _{3\rho \beta }(U_{\beta }+\sum _{k=1}^{4} b_{k}u_{\beta }^{(k)}),\nonumber \\ J_{33\rho }&= - 2(\alpha _{3}+\alpha _{4})\varepsilon _{3\rho \beta } b_{4}x_{\beta } + (\alpha _{1}+2\alpha _{5}) (U_{3,\rho }+\sum _{k=1}^{4} b_{k}u_{3,\rho }^{(k)}) \nonumber \\ {}&\quad + f \varepsilon _{3\rho \beta }(U_{\beta } + \sum _{k=1}^{4} b_{k}u_{\beta }^{(k)}), \nonumber \\J_{\alpha 3\beta }&= 2\alpha _{4}(U_{\beta ,\alpha }+\sum _{k=1}^{4} b_{k}u_{\beta ,\alpha }^{(k)}) + \alpha _{5}(U_{\alpha ,\beta } +\sum _{k=1}^{4} b_{k}u_{\alpha ,\beta }^{(k)}) \nonumber \\ {}&\quad + \delta _{\alpha \beta }[\frac{1}{2}(\alpha _{1}+2\alpha _{2})(b_{1}x_{1} + b_{2}x_{2}+b_{3}) + \alpha _{2}(U_{\rho ,\rho } + \sum _{k=1}^{4} b_{k}u_{\rho ,\rho }^{(k)})] \nonumber \\ {}&\quad + f(U_{3} + \sum _{k=1}^{4} b_{k}u_{3}^{(k)})\varepsilon _{a\beta 3},\nonumber \\ J_{\alpha \beta 3}&= \delta _{\alpha \beta }[(\alpha _{1}+2\alpha _{3})(b_{1}x_{1} + b_{2}x_{2} + b_{3}) +\alpha _{1}(U_{\rho ,\rho } + \sum _{k=1}^{4}b_{k}u_{\rho ,\rho }^{(k)})\nonumber \\&\quad + \beta _{1}(\Phi + \sum _{k=1}^{4}b_{k}\varphi ^{(k)})] + 2\alpha _{5}(E_{\alpha \beta } + \sum _{k=1}^{4}b_{k}e_{\alpha \beta }^{(k)}),\nonumber \\ J_{333}&= 2(\alpha _{1} + \alpha _{2}+\alpha _{3} + \alpha _{4}+\alpha _{5})(b_{1}x_{1} + b_{2}x_{2}+b_{3}) + (\alpha _{1} + 2\alpha _{2})U_{\rho ,\rho } \nonumber \\&\quad +(\beta _{1}+2\beta _{2})(\Phi +\sum _{k=1}^{4} b_{k}\varphi ^{(k)}). \end{aligned}$$
(54)
$$\begin{aligned} s_{\beta j, \beta } - \nu _{\alpha \beta j, \alpha \beta } + \Psi _{j} = 0, \;\; h_{\alpha ,\alpha } - p +\mathcal {H} = 0\;\; \text {on}\; \Sigma _{1}, \end{aligned}$$
(55)
$$\begin{aligned} T_{\alpha j,\alpha } - M_{\alpha \beta j, \alpha \beta } = 0,\;\; H_{\alpha ,\alpha }-L = 0\;\; \text {on}\; \Sigma _{1}. \end{aligned}$$
(56)
$$\begin{aligned} \Psi _{\alpha }= & {} G_{\beta \alpha ,\beta } +T_{\alpha 3} - 2M_{3\rho \alpha ,\rho } - \sum _{k=1}^{4}(2\mu _{3\rho \alpha ,\rho }^{(k)} - \tau _{\alpha 3}^{(k)})b_{k}\nonumber \\{} & {} +4f \varepsilon _{3\alpha \beta }b_{\beta } + \mu \varepsilon _{3\rho \alpha }b_{4}x_{\rho } - J_{\rho \eta \alpha ,\rho \eta }, \nonumber \\\Psi _{3}= & {} G_{\alpha 3,\alpha } + T_{33} - 2M_{\alpha 33,\alpha } - J_{\alpha \beta 3,\alpha \beta }\nonumber \\{} & {} +(\lambda +2\mu )(b_{1}x_{1} + b_{2}x_{2} + b_{3}) + 6f b_{4} - \sum _{k=1}^{4} (2\mu _{\alpha 33,\alpha }^{(k)} - \tau _{33}^{(k)})b_{k},\nonumber \\ \mathcal {H}= & {} H_{3}+\beta _{2}(E_{\alpha 3,\alpha } + \sum _{k=1}^{4} b_{k}e_{\alpha 3,\alpha }^{(k)}) -d (U_{3} + \sum _{k=1}^{4}b_{k}u_{3}^{(k)}) + \sum _{k=1}^{4} b_{k}\sigma _{3}^{(k)}. \end{aligned}$$
(57)
$$\begin{aligned}&\Pi _{j}^{(1)} = (s_{\beta j} - \nu _{\rho \beta j,\rho })n_{\beta } -D_{\beta }(n_{\rho }\nu _{\rho \beta j}) + (D_{\alpha }n_{\alpha })n_{\rho }n_{\eta }\nu _{\rho \eta j}, \nonumber \\ {}&\Lambda _{k}^{(1)} = \nu _{\rho \beta j}n_{\rho }n_{\beta }, \Lambda _{j}^{(2)} = M_{\rho \beta j}n_{\rho }n_{\beta },\nonumber \\&\Pi _{j}^{(2)} = (T_{\beta j} - M_{\rho \beta j,\rho })n_{\beta } - D_{\beta }(n_{\rho }M_{\rho \beta j}) + (D_{\nu }n_{\nu }) n_{\rho }n_{\eta }M_{\rho \eta j}. \end{aligned}$$
(58)
$$\begin{aligned} \Pi _{j}^{(1)} = B_{j}, \Lambda _{j}^{(1)} = C_{j}, h_{\alpha }n_{\alpha } = h\;\; \text {on}\; \Gamma _{1}, \end{aligned}$$
(59)
$$\begin{aligned} \Pi _{j}^{(2)} =0, \Lambda _{j}^{(2)} = 0, H_{\alpha }n_{\alpha } = 0\; \text {on}\; \Gamma _{1}, \end{aligned}$$
(60)
$$\begin{aligned} B_{\alpha }&= 2n_{\rho }(M_{3\rho \alpha } + \sum _{k=1}^{4} b_{k}\mu _{3\rho \alpha }^{(k)}) - 2\varepsilon _{\alpha \rho 3}n_{\rho }[f(b_{1}x_{1} + b_{2}x_{2} + b_{3}) \nonumber \\ {}&\quad +(2\alpha _{4} - \alpha _{5})b_{4}] - B_{\alpha }^{*}, \nonumber \\ B_{3}&= 2n_{\rho }[M_{\rho 33} + \frac{1}{2}(2\alpha _{2} - \alpha _{1}+4\alpha _{4})b_{\rho } - \frac{1}{2}f b_{4}x_{\rho }] \nonumber \\ {}&\quad +\sum _{k=1}^{4} b_{k}\mu _{\rho 33}^{(k)} - B_{3}^{*},\; C_{j} = -J_{\alpha \beta j}n_{\alpha }n_{\beta },\nonumber \\ h&= [\beta _{1}\varepsilon _{3\alpha \beta } b_{4}x_{\beta } - \beta _{2}(E_{\alpha 3} +\sum _{k=1}^{4} b_{k}e_{\alpha 3}^{(k)})]n_{\alpha }, \nonumber \\ B_{j}^{*}&= (G_{\beta j} - J_{\rho \beta j,\rho })n_{\beta } - D_{\nu }(n_{\rho }J_{\rho \nu j}) + (D_{\alpha }n_{\alpha })n_{\rho }n_{\nu }J_{\rho \nu j}. \end{aligned}$$
(61)
$$\begin{aligned}&\int _{\Sigma _{1}}\Psi _{j}\textrm{d}a +\int _{\Gamma _{1}}B_{j}\textrm{d}s = 0,\nonumber \\&\int _{\Sigma _{1}} \varepsilon _{3\alpha \beta }x_{\alpha }\Psi _{\beta }\textrm{d}a + \int _{\Gamma _{1}}\varepsilon _{3\alpha \beta } (x_{\alpha }B_{\beta } + n_{\alpha }C_{\beta })\textrm{d}s=0. \end{aligned}$$
(62)
$$\begin{aligned}&\int _{\Sigma _{1}} (G_{\beta j,\beta } - J_{\rho \eta j, \rho \eta })\textrm{d}a + \int _{\Gamma _{1}} B_{j}^{*}\textrm{d}s = 0,\nonumber \\&\int _{\Sigma _{1}}\varepsilon _{3\alpha \beta }x_{\alpha }(G_{\rho \beta ,\rho } - J_{\nu \rho \beta , \nu \rho })\textrm{d}a + \int _{\Gamma _{1}} \varepsilon _{3\alpha \beta }(x_{\alpha }B^{*}_{\beta } + n_{\alpha }C_{\beta })\textrm{d}s=0. \end{aligned}$$
(63)
$$\begin{aligned} \int _{\Sigma _{1}}\Psi _{\alpha }\textrm{d}a + \int _{\Gamma _{1}} B_{\alpha }\textrm{d}s = \int _{\Sigma _{1}} \tau _{\alpha 3,3}\textrm{d}a. \end{aligned}$$
(64)
$$\begin{aligned} \tau _{\alpha 3}&= \tau _{\alpha 3} + x_{\alpha }(\tau _{j3,j} -\mu _{rs3,rs}) = [x_{\alpha }(\tau _{\beta 3} - \mu _{\beta \nu 3,\nu })]_{,\beta }\nonumber \\&\quad + x_{\alpha }(\tau _{33,3} - 2\mu _{3\beta 3,3\beta } - \mu _{333,33}) + \mu _{\alpha \nu 3,\nu }. \end{aligned}$$
(65)
$$\begin{aligned} (\tau _{\beta 3} - \mu _{\beta \nu 3,\nu }) n_{\beta } = {\widetilde{P}}_{3} + [2\mu _{3\beta 3,3} - (\mu _{\alpha \beta 3}n_{\alpha }n_{\gamma } - \mu _{\rho \gamma 3}n_{\rho }n_{\beta })_{,\gamma }]n_{\beta }, \end{aligned}$$
(66)
$$\begin{aligned} \int _{\Sigma _{1}}\tau _{\alpha 3}\textrm{d}a&= \int _{\Gamma _{1}}[\mu _{\alpha \nu 3,\nu } + x_{\alpha }(\tau _{33,3} - 2\mu _{3\beta 3,3\beta } - \mu _{333,33})] \textrm{d}a\nonumber \\&\quad +\int _{\Gamma _{1}} x_{\alpha } \{{\widetilde{P}}_{3} + (\mu _{\rho \gamma 3}n_{\rho }n_{\beta } - \mu _{\alpha \beta 3}n_{\alpha }n_{\gamma })_{,\gamma }n_{\beta } + 2\mu _{3\beta 3,3}n_{\beta }\}\textrm{d}s. \end{aligned}$$
(67)
$$\begin{aligned} \int _{\Sigma _{1}}x_{\alpha }(\mu _{\rho \gamma 3}n_{\rho }n_{\beta } - \mu _{\alpha \beta 3}n_{\alpha }n_{\gamma })_{,\gamma } n_{\beta } \textrm{d}s = \int _{\Gamma _{1}}n_{\alpha } {\widetilde{R}}_{3}\textrm{d}s - \int _{\Sigma _{1}} \mu _{\alpha \nu 3,\nu }\textrm{d}a. \end{aligned}$$
(68)
$$\begin{aligned} \int _{\Sigma _{1}}\tau _{\alpha 3}\textrm{d}a = \int _{\Sigma _{1}} [x_{\alpha }(\tau _{33,3} - \mu _{333,33}) + 2\mu _{\alpha 33,3}]\textrm{d}a + \int _{\Gamma _{1}} (x_{\alpha }{\widetilde{P}}_{3} +n_{\alpha }{\widetilde{R}}_{3})\textrm{d}s. \end{aligned}$$
(69)
$$\begin{aligned} \int _{\Sigma _{1}} \tau _{\alpha 3,\alpha }\textrm{d}a = 0, \end{aligned}$$
$$\begin{aligned}&\int _{\Sigma _{1}} \Psi _{3}\textrm{d}a + \int _{\Gamma _{1}}B_{3}\textrm{d}s = \int _{\Sigma _{1}}T_{33}\textrm{d}a + \sum _{k=1}^{4}D_{3k}b_{k},\\&\int _{\Sigma _{1}}\varepsilon _{3\alpha \beta }x_{\alpha }\Psi _{\beta }\textrm{d}a + \int _{\Gamma _{1}} \varepsilon _{3\alpha \beta } (x_{\alpha }B_{\beta } + n_{\alpha }C_{\beta })\textrm{d}s\\ {}&= \int _{\Sigma _{1}}\varepsilon _{3\alpha \beta }x_{\alpha }T_{\beta 3}\textrm{d}a + \sum _{k=1}^{4}D_{4k}b_{k}. \end{aligned}$$
$$\begin{aligned} \sum _{k=1}^{4}D_{3k}b_{k} = -\int _{\Sigma _{1}} T_{33}\textrm{d}a, \sum _{k=1}^{4}D_{4k}b_{k} = -\int _{\Sigma _{1}}\varepsilon _{3\alpha \beta }x_{\alpha }T_{\beta 3}\textrm{d}a. \end{aligned}$$
(70)
$$\begin{aligned} \int _{\Sigma _{1}} P_{\alpha }\textrm{d}a + \int _{\Gamma _{1}}Q_{s}\textrm{d}s = -\int _{\Sigma _{1}} (\tau _{3\alpha } - \mu _{33\alpha ,3})\textrm{d}a. \end{aligned}$$
$$\begin{aligned} \int _{\Sigma _{1}}P_{\alpha }\textrm{d}a + \int _{\Gamma _{1}}Q_{\alpha }\textrm{d}s&= -\int _{\Gamma _{1}}(x_{\alpha }{\widetilde{P}}_{3} + n_{\alpha }{\widetilde{R}}_{3})\textrm{d}s - \int _{\Sigma _{1}} [x_{\alpha }(\tau _{33,3} - \mu _{333,33})\nonumber \\ {}&\quad +2\mu _{\alpha 33,3} - \mu _{33\alpha ,3}]\textrm{d}a. \end{aligned}$$
(71)
$$\begin{aligned} \sum _{k=1}^{4}D_{\alpha k}b_{k} = -F_{\alpha } - \int _{\Sigma _{1}}(x_{\alpha }T_{33} +2M_{\alpha 33} - M_{33\alpha })\textrm{d}a. \end{aligned}$$
(72)
If we take into account the relations (38), (40), (51) and (53), we find that the conditions (9)–(11) take the formwhere we have used the notationsThe constant \(c_{k}\) are determined by the system (73). First, we have to find the solutions of the plane problems \(A^{(k)}\) \((k=1,2,3,4)\). Then, we can determine the constants \(D_{mn}\) from (41) and the solution of the problem \(({\mathcal {P}}_{2})\). Next, we determine the constants \(b_{k}\) from the system (70), (72). Now, we can solve the problem \(({\mathcal {P}}_{1})\) and to find from (73) the constants \(c_{k}\) \((k=1,2,3,4)\). The above results can be used to study the case when the temperature field is a polynomial in the axial coordinate.
$$\begin{aligned}&\sum _{k=1}^{4}D_{\alpha k}c_{k} = -\varepsilon _{3\alpha \beta }(M_{\beta } + M_{\beta }^{0}),\nonumber \\&\sum _{k=1}^{4} D_{3k}c_{k} = -F_{3} - F_{3}^{0}, \sum _{k=1}^{4} D_{4k}c_{k} = -M_{3} - M_{3}^{0}, \end{aligned}$$
(73)
$$\begin{aligned} F_{3}^{0}&= \int _{\Sigma _{1}} [s_{33} + G_{33} - M_{333} - (\alpha _{1}+2\alpha _{3})\sum _{k=1}^{4}b_{k}\kappa _{333}^{(k)}]\textrm{d}a, \nonumber \\M_{\alpha }^{0}&= \varepsilon _{3\alpha \beta }\int _{\Sigma _{1}}\{ x_{\beta }[s_{33} + G_{33} - M_{333} - (\alpha _{1}+2\alpha _{3})\sum _{k=1}^{4}b_{k}\kappa _{333}^{(k)}] \nonumber \\ {}&\quad + 2\nu _{\beta 33} +2J_{\beta 33} -\nu _{33\beta } - J_{33\beta }\}\textrm{d}a,\nonumber \\ M_{3}^{0}&= \int _{\Sigma _{1}}\{\varepsilon _{\alpha \beta 3}[s_{\beta 3} + G_{\beta 3} - M_{33\beta } - (\alpha _{1}-2\alpha _{3} -2\alpha _{4}+2\alpha _{5}) b_{\beta }\nonumber \\&\quad - f b_{4}x_{\beta } - \sum _{k=1}^{4}b_{k}\mu _{33\beta }^{(k)}] + 2\varepsilon _{\alpha \beta 3}(\nu _{\alpha 3\beta } + J_{\alpha 3\beta })\}\textrm{d}a. \end{aligned}$$
(74)
6 Application
In this section, we use the solution (23) to investigate the thermoelastic deformation of a chiral circular cylinder subjected to a uniform temperature. We consider a homogeneous cylinder that occupies the domain \(B = \{(x_{1}, x_{2}, x_{3}): x_{1}^{2} + x_{2}^{2}<a^{2}, 0<x_{3}<h\}\), \((a>0)\). We assume that the mechanical loads are absent. In this case, we havewhere \(T^{*}\) is a prescribed constant. We seek the solution of the problem \(({\mathcal {T}})\) in the formwhere \(E_{1}\) and \(E_{2}\) are arbitrary constants. From (28), (27) and (76), we obtainThe equilibrium equations (34) and the boundary conditions (38) are satisfied if the constants \(E_{1}\) and \(E_{2}\) are given bywhereThe positive definiteness of the elastic potential implies that \(D\ne 0\), Thus, the solution of the problem \(({\mathcal {T}})\) is given by (76). It follows from (40) and (77) thatIn a similar way, we can study the problems \(A^{(3)}\) and \(A^{(4)}\). The solution of the problem \(A^{(3)}\) iswhereand D is given by (79). The problem \(A^{(4)}\) has the following solutionwhereIt follows from (81), (82) and (41) thatIn view of symmetry of \(D_{mn}\), we obtain \(D_{3\alpha } = 0\) and \(D_{4\alpha } = 0.\) Thus, with the help of (82) and (83) we find that the system (40) reduces toIt follows from (42) that this system uniquely determines the constants \(a_{k}\). We find thatwhereLet us note that we have solved the problem without using the solutions of the problems \(A^{(1)}\) and \(A^{(2)}\).
$$\begin{aligned} T = T^{*},\;\; F_{j} = 0,\;\;\, M_{j} = 0, \end{aligned}$$
(75)
$$\begin{aligned} w_{\alpha } = E_{1}x_{\alpha },\;\, w_{3} = 0,\;\, \psi = E_{2}, \end{aligned}$$
(76)
$$\begin{aligned}&\eta _{\alpha \beta } = E_{1}\delta _{\alpha \beta }, \eta _{\alpha 3} =0, \xi _{ijk} = 0, \nonumber \\&t_{\alpha \beta } = [2(\lambda +\mu )E_{1} + d E_{2} - b T^{*}]\delta _{\alpha \beta }, t_{\alpha 3} = 0, m_{\alpha \beta j} = 0, \nonumber \\ {}&\pi _{\alpha } = 0, \gamma = 2d E_{1}+\xi E_{2} - \beta T^{*}, m_{3\alpha \beta }= f E_{1}\varepsilon _{\beta \alpha 3}, m_{3\alpha 3} = 0,\nonumber \\ {}&m_{33j} = 0, P_{\alpha }^{*} = [2(\lambda +\mu )E_{1} + dE_{2}- bT^{*}]n_{\alpha }, P_{3}^{*} = 0,R_{i}^{*} = 0. \end{aligned}$$
(77)
$$\begin{aligned} E_{1} = (\beta d - b \xi )T^{*}/(2D), E_{2} = [db - \beta (\lambda +\mu )]T^{*}/D, \end{aligned}$$
(78)
$$\begin{aligned} D = d^{2}-\xi (\lambda +\mu ). \end{aligned}$$
(79)
$$\begin{aligned} F_{3}^{*} = \pi a^{2}(2\lambda E_{1} + d E_{2}- bT^{*}), M_{j}^{*} = 0. \end{aligned}$$
(80)
$$\begin{aligned} u_{\alpha }^{(3)} = S_{1}x_{\alpha }, u_{3}^{(3)} = 0, \varphi ^{(3)} = S_{2}, \end{aligned}$$
(81)
$$\begin{aligned} S_{1} = (\lambda \xi -d^{2})/(2D), \;\; S_{2} = \textrm{d}\mu / D, \end{aligned}$$
$$\begin{aligned} u_{\alpha }^{(4)} = K_{1}x_{\alpha }, u_{3}^{(4)} = 0, \varphi ^{(3)} = K_{2}, \end{aligned}$$
(82)
$$\begin{aligned} K_{1} = -f \xi / D,\;\; K_{2} = 2bf/D. \end{aligned}$$
$$\begin{aligned} \begin{aligned}&D_{\alpha 3} = 0, D_{\alpha 4} = 0, D_{33} = \pi \mu a^{2}[3d^{2} - (3\lambda +2\mu )\xi ]/D, \\ {}&D_{44} = \mu \frac{\pi a^{4}}{2} + 4(2a_{4} - \alpha _{5}-K_{1}f)\pi a^{2},\\ {}&D_{34} = D_{43} = 2\pi a^{2}f [3d^{2} - (3\lambda +2\mu )\xi ]/D. \end{aligned} \end{aligned}$$
(83)
$$\begin{aligned} D_{\alpha \beta }a_{\beta } = 0, D_{33}a_{3} + D_{34}a_{4} = -F_{3}^{*}, D_{43}a_{3} + D_{44}a_{4} = 0. \end{aligned}$$
(84)
$$\begin{aligned} a_{1} = a_{2} = 0, a_{3} = - D_{44}F_{3}^{*}q, a_{4} = D_{34}F_{3}^{*}q, \end{aligned}$$
(85)
$$\begin{aligned} q^{-1} = D_{33}D_{44} - D_{34}^{2}. \end{aligned}$$
In view of (23), (76), (81), (82) and (85), we see that the solution of the problem is given byFrom (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.
$$\begin{aligned}&u_{\alpha } = \varepsilon _{3\beta \alpha }a_{4}x_{\beta }x_{3} + a_{3}S_{1}x_{\alpha } + a_{4}K_{1}x_{\alpha } + w_{\alpha },\nonumber \\&u_{3} = a_{3}x_{3}, \varphi = a_{3}S_{2} + a_{4}K_{2}+\psi . \end{aligned}$$
(86)
7 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.
Funding
Open access funding provided by Universitá degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.