A Constitutive Formulation for the Linear Thermoelastic Behavior of Arbitrary Fiber-Reinforced Composites

Abstract

The linear thermoelastic behavior of a composite material reinforced by two independent and inextensible fiber families has been analyzed theoretically. The composite material is assumed to be anisotropic, compressible, dependent on temperature gradient, and showing linear elastic behavior. Basic principles and axioms of modern continuum mechanics and equations belonging to kinematics and deformation geometries of fibers have provided guidance and have been determining in the process of this study. The matrix material is supposed to be made of elastic material involving an artificial anisotropy due to fibers reinforcing by arbitrary distributions. As a result of thermodynamic constraints, it has been determined that the free energy function is dependent on a symmetric tensor and two vectors whereas the heat flux vector function is dependent on a symmetric tensor and three vectors. The free energy and heat flux vector functions have been represented by a power series expansion, and the type and the number of terms taken into consideration in this series expansion have determined the linearity of the medium. The linear constitutive equations of the stress and heat flux vector are substituted in the Cauchy equation of motion and in the equation of conservation of energy to obtain the field equations.

1. Introduction

Generally, composite materials are separated into natural composites and artificial composites. While natural and artificial composites have functional similarities, they differ greatly in terms of methods of production and purposes of use. Natural composites are not the result of a manufacturing and production method implemented by humans for a certain purpose. Having extremely fine and complex subsystems, structures of this kind comprise known natural structure elements by combination in a certain distribution at a time and on grounds determined according to a universal program of sometimes microscopic and other times macroscopic level. Artificial composites appear as a product of a certain manufacturing process produced by intellectual capabilities of the human mind to create a material with superior characteristics for a certain application purpose. Dealing with microlevel study for natural and artificial composite structure elements is a subject of micromechanics [1].

Generally, studies of composite materials are divided into two main branches, namely, micromechanical and macromechanical analyses. The micromechanical analysis aims to uncover certain mechanical characteristics relating to the general behavior of composite materials using the physical and mechanical properties of matrix and reinforcement materials as a starting point. Micromechanical methods can be separated into three, that is, the energy method, elasticity method, and material mechanics method [2]. Composites are broadly used in civilian and military aircraft, aerospace technologies, automotive industry, sea vehicles primarily in ships, pneumatic vessels, power transmission axles, and orthopedic devices [3].

In our previous study [4], viscoelastic composites of a single fiber family have been studied assuming that the medium has a discontinuity surface. In our study [5], it has been assumed that a viscoelastic medium with two different inextensible fiber families does not have a discontinuity surface. Again, in the studies in [6, 7], the exposure of a viscoelastic medium to the effect of electrical and magnetic fields in addition to its reinforcement by a single fiber family has been researched in the form of separate studies. Furthermore, in his study [8], Usal has examined the electromechanical behavior of a piezoelectric viscoelastic medium with two fiber families. Since temperature was assumed to be constant in all of the previous studies, temperature change has not been taken into consideration. In this study, constitutive equations have been obtained that indicate the stress and heat distribution determining the thermoelastic behavior of a composite material reinforced by two arbitrary independent and inextensible fiber families. Since the temperature is not constant, a temperature gradient has been included in the calculations as an independent constitutive variable.

Researchers like those in [9–11] have made progress in the studies they have conducted on the formulation of thermoelastic effect on a variety of materials. A study by Nowacki fills a large gap in thermoelasticity and its applications [12].

The thermal properties can play a significant role in affecting the design and manufacture of composite structures in their industrial applications [13]. The subject of thermoelastic behavior of composites has been studied by a number of different researchers [14–21]. Fiber-reinforced composite materials belong to a very important class of materials which are often employed in a wide variety of industrial applications. Typically, these composite materials consist of a fabric structure where the fibers are continuously arranged in a matrix material, and, at the macroscopic level, these composite materials exhibit strong directional dependencies. The vehicle tyres furnishe a typical example of technological application of such man-made composites [21].

Due to some technological requirements, it is aspired that specific construction elements have rather elastic properties, provided that they have high durability in certain directions. Fiber-reinforced composite materials are produced by sticking fibers in a polymeric matrix which is elastic but with low strength. These fibers are manufactured from high-strength graphite or bor. They can be easily bent due to the very small size of their cross-section and it can be assumed that these fibers show a continuous distribution in a medium. Assuming inextensibility of the fibers is a reasonable approach since the rigidity of the fibers is very high compared to the rigidity of the matrix. Inextensibility of the fibers is broadly accepted in practice for formulation purposes. Thus, fiber families are assumed to be inextensible (𝜆²_𝑎=𝐶_𝐾𝐿𝐴_𝐾𝐴_𝐿=1, 𝜆²_𝑧=𝐶_𝐾𝐿𝑍_𝐾𝑍_𝐿=1) [22]. On the other hand, the composite material taken into consideration in this work is assumed to be compressible and shows linear elastic behavior. In a class of engineered fiber composites for structural load-bearing components in civil or aerospace applications, an assumption of linear elastic behavior is suitable and this class of composites belongs to a compressible material response.

2. Kinematics of Fibers Deformation

It is assumed that an element from two different continuous fiber families is placed on each point of the composite material. Before deformation and after deformation, these fiber families are represented by continuous unit vectors 𝐀(𝐗), 𝐙(𝐗), 𝐚(𝐱), and 𝐳(𝐱), respectively. The fibers deform along with the material; that is, fibers do not have a relative motion with respect to the material in which they are embedded. Relationships given below are true for an A-fiber family [23, 24] 𝑎_𝑘𝑑𝑙=𝑥_𝑘,𝐾𝐴_𝐾𝑑𝐿,𝑎_𝑘𝑑𝑙𝑑𝐿=𝑥_𝑘,𝐾𝐴_𝐾. (2.1) Rates of extension of fiber family 𝐴 can be defined as follows:𝜆_𝑎≡𝑑𝑙𝑑𝐿_𝐴. (2.2) If expression (2.2) is substituted into (2.1), the following expression is obtained: 𝑎_𝑘=𝜆_𝑎⁻¹𝑥_𝑘,𝐾𝐴_𝐾. (2.3) Deformation geometry of the fiber family 𝐴 is expressed by relationship (2.3). Because vectors 𝐀 and 𝐚 here represent unitized vectors of the fiber family 𝐴, operations are true|𝐀|=|𝐚|=1,𝑎_𝑘𝑎_𝑘=1=𝜆_𝑎⁻²𝑥_𝑘,𝐾𝑥_𝑘,𝐿𝐴_𝐾𝐴_𝐿=𝜆_𝑎⁻²𝐶_𝐾𝐿𝐴_𝐾𝐴_𝐿. (2.4) Accordingly, the form is found to be 𝜆²_𝑎=𝐶_𝐾𝐿𝐴_𝐾𝐴_𝐿. (2.5) Relationships that are true for the 𝑍-fiber family can be expressed as follows [23, 24]: 𝑧_𝑘=𝜆_𝑧⁻¹𝑥_𝑘,𝐾𝑍_𝐾,𝜆_𝑧≡𝑑𝑙𝑑𝐿_𝑍,𝜆²_𝑧=𝐶_𝐾𝐿𝑍_𝐾𝑍_𝐿, (2.6) where 𝑑𝐿 and 𝑑𝑙 are, respectively, arc length of fiber before and after deformation, 𝐴_𝐾 and 𝑍_𝐾 are fiber unit vector components before deformation, 𝑎_𝑘 and 𝑧_𝑘 are fiber unit vector components after deformation, 𝑥_𝑘,𝐾=𝜕𝑥_𝑘/𝜕𝑋_𝐾 is deformation gradient, 𝜆_𝑎 and 𝜆_𝑧 are rates of extension of fiber families, and 𝐶_𝐾𝐿=𝑥_𝑘,𝐾𝑥_𝑘,𝐿 is Green deformation tensor.

3. Thermomechanic Balance Equations

Balance equations, mass, linear momentum, angular momentum, energy balances, and entropy inequality have been summarized in [22, 25]. We have the following:

Conservation of mass:𝜌̇+𝜌𝑣_𝑘,𝑘()=𝜌=0,𝜌𝐱,𝑡⁰(𝐗)𝐽(𝐱,𝑡)conservationofmassinmaterialrepresentation, (3.1) balance of linear mometum: 𝜌𝑣̇_𝑝=𝜌f_𝑝+𝑡_{𝑟𝑝,𝑟}, (3.2) balance of moment of momentum: 𝜀_𝑘𝑟𝑝𝑡_𝑟𝑝=0,𝑡_𝑟𝑝=𝑡_𝑝𝑟, (3.3) conservation of energy:𝜌𝜀̇=𝑡_𝑘𝑙𝑑_𝑘𝑙+𝑞_𝑘,𝑘+𝜌ℎ, (3.4) Clausius-Duhem inequality:̇1𝜌𝜃𝜂−∇⋅𝐪+𝜃𝐪⋅∇𝜃−𝜌h⩾0. (3.5) Here, 𝑣 stands for the velocity field in a continuous medium, 𝜌₀ for mass density before deformation, 𝜌 for mass density after deformation, 𝐽≡det[𝑥_𝑘,𝐾]=𝜌₀/𝜌(𝐱,𝑡) for jacobian, 𝑣̇ for acceleration, 𝑡_𝑙𝑘 for stress tensor, f_𝑘 for the mechanical volumetric force per unit of mass, 𝜀 for internal energy density per unit of mass, 𝑞_𝑘 for heat flux vector, ℎ for heat source per unit of mass, 𝜂 for entropy density per unit of mass, 𝜃(𝐗,𝑡) for the absolute temperature of a material point 𝐗 at a moment 𝑡, and 𝜀_𝑖𝑗𝑘 for permutation tensor.

4. Thermodynamic Constraints and Modeling Constitutive Equations

Taking (𝜌ℎ) from the local energy (3.4) and substituting it in the entropy inequality (3.5) will give us the following:−𝜌𝜀̇−𝜃𝜂̇+𝑡_𝑘𝑙𝑑_𝑙𝑘+1𝜃𝑞_𝑘𝜃_,𝑘≥0. (4.1) Since the material derivative of the entropy density in this expression cannot be controlled inside a thermodynamic process, a defined Legendre transformation like the one provided below can be used to transform the derivative of these values into the controllable value 𝜃𝜓≡𝜀−𝜃𝜂. (4.2) As a result, the entropy inequality is transformed as follows, expressed in new terms: −𝜌𝜓̇+𝜃̇𝜂+𝑡_𝑘𝑙𝑑_𝑙𝑘+1𝜃𝑞_𝑘𝜃_,𝑘≥0. (4.3) Entropy inequality is obtained as follows in the material form [26]: −Σ̇+𝜌₀̇+1𝜃𝜂2𝑇_𝐾𝐿𝐶̇_𝐾𝐿+1𝜃𝜃,_𝐾𝑄_𝐾≥0. (4.4) Terms relating to the new values have been provided below:Σ≡𝜌₀𝐶̇𝜓,_𝐾𝐿=2d_𝑘l𝑥_𝑘,𝐾𝑥_l,𝐿⟹𝑑_𝑘𝑙=12𝐶̇_𝐾𝐿𝑋_𝐾,𝑘𝑋_𝐿,𝑙𝑇_𝐾𝐿,≡𝐽𝑋_𝐾,𝑘𝑋_𝐿,𝑙𝑡_𝑘𝑙⟹𝑡_𝑘𝑙=𝐽⁻¹𝑥_𝑘,𝐾𝑥_𝑙,𝐿𝑇_𝐾𝐿𝑄_𝐾,≡𝐽𝑋_𝐾,𝑘𝑞_𝑘⟹𝑞_𝑘=𝐽⁻¹𝑥_𝑘,𝐾𝑄_𝐾𝐺_𝐾,≡𝜃_,𝐾=𝑥_𝑘,𝐾𝜃_,𝑘⟹𝑔_𝑘≡𝜃_,𝑘=𝑋_𝐾,𝑘𝜃_,𝐾. (4.5)(4.6)(4.7)(4.8)(4.9) Here, Σ stands for thermodynamic stress potential, 𝜓 for generalized free energy density, 𝑑_𝑘𝑙 for deformation (strain) rate tensor, 𝑋_𝐾,𝑘=𝜕𝑋_𝐾/𝜕𝑥_𝑘 for the deformation gradient of the reverse motion, 𝑇_𝐾𝐿 for the stress tensor on material coordinates, 𝑄_𝐾 for the heat flux vector on material coordinates, and 𝐺_𝐾 for the temperature gradient on material coordinates.

To be able to use the inequality (4.4), we need to know the independent variables on which the thermodynamic potential Σ depends. Arguments of Σ and the variables they depend on have been found using constitutive axioms based on the selected material. According to the axioms of causality and determinism [22, 25], our stress potential, as a response functional at a material point 𝑋 at a time 𝑡, can be written as follows:Σ(𝐗,𝑡)=Σ𝐱𝐗^′,𝑡^′,𝜃𝐗^′,𝑡^′,𝐗𝑋∈𝑉−∞<𝑡^′≤𝑡. (4.10) Here, 𝑡^′ is any point in time between now and the past. 𝐗^′ stands for all material points other than 𝐗.

Using the results of causality, determinism, objectivity, smooth neighborhood, and admissibility axioms [22, 25], the arguments on which Σ depends in a composite with two fiber families exposed to mechanical loading and temperature change can be expressed as follows:Σ𝑋_𝐾,𝑡=Σ𝐶_𝐾𝐿𝑋_𝐾,𝑡,𝐺_𝐾𝑋_𝐾,𝑡,𝐴_𝐾𝑋_𝐾,𝑍_𝐾𝑋_𝐾,𝜃𝑋_𝐾,𝑡,𝑋_𝐾. (4.11) Assuming that the materials are homogenous, 𝐗 will be eliminated from among the arguments given in the expression (4.11) on which Σ depends. Because the fiber vectors 𝐴_𝐾 and 𝑍_𝐾 do not depend on time, the following expression is obtained by taking the material derivative of expression (4.11).Σ̇=𝜕Σ𝜕𝐶_𝐾𝐿𝐶̇_𝐾𝐿+𝜕Σ𝜕𝜃_,𝐾𝐺̇+𝜕Σ̇𝜕𝜃𝜃. (4.12) Substituting this expression in (4.4) gives us the following inequality:12𝑇_𝐾𝐿−2𝜕Σ𝜕𝐶_𝐾𝐿𝐶̇_𝐾𝐿−𝜌₀𝜂+1𝜌₀𝜕Σ̇𝜕𝜃𝜃−𝜕Σ𝜕𝐺_𝐾𝐺̇_𝐾+1𝜃𝐺_𝐾𝑄_𝐾≥0. (4.13) Since we are able to arbitrarily replace the arguments in inequality (4.13) from 𝜃 to 𝜃̇, from 𝐶_𝐾𝐿 to 𝐶̇_𝐾𝐿, and from 𝐺_𝐾 to 𝐺̇_𝐾, for the inequality (4.13) to be satisfied, the coefficients of 𝜃̇, 𝐶̇_𝐾𝐿 and 𝐺̇_𝐾 will be zero. The coefficient of 𝐺_𝐾 cannot be zero as due to 𝐺_𝐾’s presence in the arguments of Σ, it cannot be arbitrarily replaced. Equalizing the coefficients of 𝐶̇_𝐾𝐿, 𝜃̇, and 𝐺̇_𝐾 to zero will give us the following expressions:𝑇_𝐾𝐿=2𝜕Σ𝜕𝐶_𝐾𝐿1,𝜂=−𝜌₀𝜕Σ𝜕𝜃,𝜕Σ𝜕𝐺_𝐾=0. (4.14)(4.15)(4.16) It is understood from expression (4.16) that the stress potential does not depend on 𝐺_𝐾. Therefore, arguments on which the stress potential depends are expressed as follows:Σ=Σ𝐶_𝐾𝐿,𝐴_𝐾,𝑍_𝐾,𝜃. (4.17) Thus, inequality (4.13) is reduced to the following form:1𝜃𝐺_𝐾𝑄_𝐾≥0. (4.18) For the heat flux vector, expression (4.18) gives the Clausius-Duhem inequality and the following expression indicates the arguments on which the heat flux vector depends𝑄_𝐾=𝑄_𝐾𝐶_𝐾𝐿,𝐴_𝐾,𝑍_𝐾,𝐺_𝐾,𝜃. (4.19) In consideration of expression (4.19), inequality (4.18) is written down as follows:𝐺_𝐾𝑄_𝐾𝐶_𝐾𝐿,𝐴_𝐾,𝑍_𝐾,𝐺_𝐾,𝜃≥0or𝐐(𝐂,𝐀,𝐙,𝐆,𝜃,𝐗)⋅𝐆≥0. (4.20) In inequality (4.20), when 𝐺_𝐾=0, 𝑄_𝐾 must also be equal to zero. Accordingly, maintaining the order of independent constitutive variables in expression (4.20), the following expression should be written:𝑄_𝐾𝐶_𝐾𝐿,𝐴_𝐾,𝑍_𝐾,0,𝜃=0. (4.21) On the other hand, internal energy density 𝜀 can be written as follows from the expressions (4.2), (4.5), and (4.15):1𝜀=𝜌₀Σ−𝜕Σ𝜃𝜕𝜃. (4.22) From the constitutive equations offered by expressions (4.14) and (4.19), it is understood that the stress is derived from the stress potential Σ, while the heat flux vector appears as a vectorial form with known arguments independent of the stress potential. Thus, the explicit forms of Σ and 𝑄_𝐾, which appear as constitutive functions with definite arguments, should be determined. However, constraints imposed on the constitutive functions of the material in question by the material symmetry axiom should firstly be revised.

Let the symmetry group of the material be the full orthogonal group (isotropic material) or any of its subgroups (anisotropic material). Let 𝑆=[𝑆_𝐾𝐿] be any arbitrary matrix representing the orthogonal transformation of material coordinates (or rigid configurations of the material medium according to the reference coordinate frame) and pertaining to the symmetry group of the medium. According to the material symmetry axiom, constitutive functionals under each transformation 𝑋^′_𝐾=𝑆_𝐾𝐿𝑋_𝐿,𝑋_𝐿=𝑆^𝑇_𝐿𝐾𝑋^′_𝐾,𝑆⁻¹=𝑆^𝑇 (4.23) established using the orthogonal matrix 𝑆 should remain form invariant. Mathematically, this indicates the validity of the transformationsΣ𝑆𝐶𝑆^𝑇,𝑆𝐴,𝑆𝑍,𝜃=Σ𝐶,𝐴,𝑍𝑄,𝜃,𝑆𝐶𝑆^𝑇,𝑆𝐴,𝑆𝑍,𝑆𝐺,𝜃=𝑄𝐶,𝐴,𝑍,𝐺,𝜃. (4.24)(4.25) The following conditions should be satisfied since the fiber families are assumed to be inextensible [22, 27]:𝜆²_𝑎=𝐶_𝐾𝐿𝐴_𝐾𝐴_𝐿=1,𝜆²_𝑧=𝐶_𝐾𝐿𝑍_𝐾𝑍_𝐿=1. (4.26) Thus, the constitutive equation for the stress is obtained as follows in spatial and material coordinates:𝑡_𝑘𝑙=Γ_𝑎𝑎_𝑘𝑎_𝑙+Γ_𝑧𝑧_𝑘𝑧_𝑙+2𝑥_𝑘,𝐾𝑥_𝑙.𝐿𝜕Σ𝜕𝐶_𝐾𝐿,𝑇_𝐾𝐿=Γ_𝑎𝐴_𝐾𝐴_𝐿+Γ_𝑧𝑍_𝐾𝑍_𝐿+2𝜕Σ𝜕𝐶_𝐾𝐿. (4.27) In these expressions, Γ_𝑎 andΓ_𝑧 are Lagrange coefficients and are defined by field equations and boundary conditions.

In this study, the matrix material has been considered as an anisotropic medium. In the scope of this approach, the stress potential and heat flux vector functions are expanded in power series in terms of the components of arguments on which they depend, giving us the thermoelastic behavior of the composite medium. The reference position of the medium has been selected at a uniform temperature 𝑇₀ in a stress-free natural condition, and it has been assumed that the medium moves away from that position by small displacements and deformations and small changes in temperature. By referring to small changes in temperature, we mean 𝜃=𝑇₀+𝑇, 𝑇₀>0, |𝑇|≪𝑇₀ [22]. The type and the number of terms taken in the series expansion have been determined based on the linearity condition of the medium. Moreover, because the matrix material remains insensitive to change of direction along the fibers, expressions of vector fields representing the fiber distribution through outer products in even numbers as arguments should be considered. The linear constitutive equation of stress has been obtained by taking the derivative of the stress potential according to its deformation tensor. Field equations have been obtained by substituting the linear constitutive equations of the stress and heat flux vector in the Cauchy motion equation and in the equation of conservation of energy.

5. Determination of Stress Constitutive Equation in Linear Thermoelasticity

Since the relation 𝐶_𝐾𝐿=𝛿_𝐾𝐿+2𝐸_𝐾𝐿 exists between the Green deformation tensor and strain tensor, 𝐸_𝐾𝐿 can be expressed as 𝐸_𝐾𝐿≅𝐸_𝐾𝐿≡(1/2)(𝑈_𝐾,𝐿+𝑈_𝐿,𝐾)in a linear theory, and the arguments of the stress potential given by expression (4.17) can be written down as follows:Σ=Σ𝐸_𝐾𝐿,𝐴_𝐾,𝑍_𝐾,𝜃. (5.1) Assuming that this function is analytic in terms of the 𝐄,𝐀,𝐙 values, if this function is expanded in Taylor series around 𝐄=𝟎, 𝐀=𝟎, 𝐙=𝟎, the expression will be obtained for the stress potential:Σ𝐸_𝐾𝐿,𝐴_𝑆,𝑍_𝑌,𝜃=Σ₀(𝜃,𝐗)+Σ_𝐾𝐿(𝜃,𝐗)𝐸_𝐾𝐿+12Σ_{𝐾𝐿𝑀𝑁}(𝜃,𝐗)𝐸_𝐾𝐿𝐸_𝑀𝑁+𝜆_𝑆𝑁(𝜃,𝐗)𝐴_𝑆𝐴_𝑁+Ω_𝑌𝑁(𝜃,𝐗)𝑍_𝑌𝑍_𝑁+𝜁_KLSN(𝜃,𝐗)𝐸_𝐾𝐿𝐴_𝑆𝐴_𝑁+𝜅_{𝐾𝐿𝑌𝑁}(𝜃,𝐗)𝐸_𝐾𝐿𝑍_𝑌𝑍_𝑁+⋯. (5.2) Coefficients in this equation have been defined as follows:Σ₀=Σ0,0,Σ_𝐾𝐿≡𝜕Σ𝜕𝐸_𝐾𝐿0,Σ_{𝐾𝐿𝑀𝑁}≡𝜕²Σ𝜕𝐸_𝐾𝐿𝜕𝐸_𝑀𝑁0,𝜆_𝑆𝑁≡12𝜕²Σ𝜕𝐴_𝑆𝜕𝐴_𝑁0,Ω_𝑌𝑁≡12𝜕²Σ𝜕𝑍_𝑌𝜕𝑍_𝑁0,𝜁_{𝐾𝐿𝑆𝑁}≡16𝜕³Σ𝜕𝐸_𝐾𝐿𝜕𝐴_𝑆𝜕𝐴_𝑁0,𝜅_{𝐾𝐿𝑌𝑁}≡16𝜕³Σ𝜕𝐸_𝐾𝐿𝜕𝑍_𝑌𝜕𝑍_𝑁0. (5.3) Due to the symmetry of the 𝐸_𝐾𝐿 tensor and nondependence on order of the derivatives in the definitions in the expressions (5.3), these coefficients bear the symmetry characteristics indicated below:Σ_𝐾𝐿=Σ_𝐿𝐾,Σ_{𝐾𝐿𝑀𝑁}=Σ_{𝐿𝐾𝑀𝑁}=Σ_{𝐾𝐿𝑁𝑀}=Σ_{𝑀𝑁𝐾𝐿},𝜆_𝑆𝑁=𝜆_𝑁𝑆,Ω_𝑌𝑁=Ω_𝑁𝑌,𝜁_{𝐾𝐿𝑆𝑁}=𝜁_{𝐿𝐾𝑆𝑁}=𝜁_{𝐾𝐿𝑁𝑆},𝜅_{𝐾𝐿𝑌𝑁}=𝜅_{𝐿𝐾𝑌𝑁}=𝜅_{𝐾𝐿𝑁𝑌}. (5.4) The following relations can be written down for the linear theory in continuum mechanics [22]:𝐸_𝐾𝐿≅𝐸_𝐾𝐿≡12𝑈_𝐾,𝐿+𝑈_𝐿,𝐾,𝑒_𝑘𝑙≅𝑒̃_𝑘𝑙=∈_𝑘𝑙≡12𝑢_𝑘,𝑙+𝑢_𝑙,𝑘,∈_𝑘𝑙≅𝜆_𝑘𝐾𝜆_𝑙𝐿𝐸_𝐾𝐿,𝐸_𝐾𝐿≅𝜆_𝑘𝐾𝜆_𝑙𝐿𝑒̃_𝑘𝑙,𝑥_𝑘,𝐾=𝜆_𝑘𝐾+𝑢_𝑘,𝐾,𝑋_𝐾,𝑘=Λ_𝐾𝑘−𝑈_𝐾,𝑘,𝑥_𝑘,𝐾𝑥_𝑙,𝐿=𝜆_𝑘𝐾𝜆_𝑙𝐿,𝑋_𝐾,𝑘𝑋_𝐿,𝑙=𝜆_𝑘𝐾𝜆_𝑙𝐿,𝐸_𝐾𝐿≅𝐸_𝐾𝐿≡𝜆_𝑘𝐾𝜆_𝑙𝐿𝑒̃_𝑘𝑙=12𝜆_𝑘𝐾𝜆_𝑙𝐿𝑢_𝑘,𝑙+𝑢_𝑙,𝑘,𝑥_𝑝,𝑃𝑥_𝑟,𝑅𝐴_𝐾𝐴_𝐿=𝑥_𝑝,𝑃𝑥_𝑟,𝑅𝑋_𝐾,𝑘𝑋_𝐿,𝑙𝑎_𝑘𝑎_𝑙𝜆²_𝑎=𝜆_𝑝𝑃𝜆_𝑟𝑅𝜆_𝑘𝐾𝜆_𝑙𝐿𝑎_𝑘𝑎_𝑙for𝜆_𝑎,𝑥=1_𝑝,𝑃𝑥_𝑟,𝑅𝑍_𝐾𝑍_𝐿=𝑥_𝑝,𝑃𝑥_𝑟,𝑅𝑋_𝐾,𝑘𝑋_𝐿,𝑙𝑧_𝑘𝑧_𝑙𝜆²_𝑧=𝜆_𝑝𝑃𝜆_𝑟𝑅𝜆_𝑘𝐾𝜆_𝑙𝐿𝑧_𝑘𝑧_𝑙for𝜆_𝑧,𝑑=1_𝑝𝑟=𝜕𝐸_𝑃𝑅𝑋𝜕𝑡^𝑃,𝑟𝑋_𝑅,𝑟=𝜕∈_𝑝𝑟𝜕𝑡,𝑑_𝑝𝑟=𝜕𝑢_𝑝,𝑟,̇𝜕𝑡𝜀≈𝜕𝜀,𝐽𝜕𝑡₋₁≅1−𝑢_𝑘,𝑘,𝜌≅𝜌₀1−𝑢_𝑘,𝑘. (5.5) The expression of the spatial form of stress for compressible media with inextensible fiber families can be written down as indicated below: 𝑡_𝑝𝑟=Γ_𝑎𝑎_𝑘𝑎_𝑙+Γ_𝑧𝑧_𝑘𝑧_𝑙+1−𝑢_𝑘,𝑘𝜕Σ𝜕∈_𝑝𝑟. (5.6) In the linear theory, arguments on which Σ depends can be expressed in spatial form as follows:Σ=Σ∈_𝑘𝑙,𝑎_𝑘,𝑧_𝑘,𝜃,𝐗. (5.7) Assuming this function is analytic in terms of ∈_𝑘𝑙,𝑎_𝑘,𝑧_𝑘 and expanding it in the Taylor series around ∈_𝑘𝑙=0,𝑎_𝑠=0,𝑧_𝑦=0 will give us the following expression: Σ∈_𝑘𝑙,𝑎_𝑠,𝑧_𝑦,𝜃,𝐗=Σ₀(𝜃,𝐗)+Σ_𝑘𝑙(𝜃,𝐗)∈_𝑘𝑙+12Σ_{𝑘𝑙𝑚𝑛}(𝜃,𝐗)∈_𝑘𝑙∈_𝑚𝑛+𝜆_𝑠𝑛(𝜃,𝐗)𝑎_𝑠𝑎_𝑛+Ω_𝑦𝑛(𝜃,𝐗)𝑧_𝑦𝑧_𝑛+𝜁_{𝑘𝑙𝑠𝑛}(𝜃,𝐗)∈_𝑘𝑙𝑎_𝑠𝑎_𝑛+𝜅_{𝑘𝑙𝑦𝑛}(𝜃,𝐗)∈_𝑘𝑙𝑧_𝑦𝑧_𝑛+⋯. (5.8) The spatial material tensors Σ_𝑘𝑙, Σ_{𝑘𝑙𝑚𝑛}, 𝜆_𝑠𝑛, Ω_𝑦𝑛, 𝜁_{𝑘𝑙𝑠𝑛}, and 𝜅_{𝑘𝑙𝑦𝑛} in (5.8) bear the same properties as the material tensors of the material Σ_𝐾𝐿,Σ_{𝐾𝐿𝑀𝑁}, 𝜆_𝑆𝑁,Ω_𝑌𝑁, 𝜁_{𝐾𝐿𝑆𝑁}, and 𝜅_{𝐾𝐿𝑌𝑁} and are defined as follows:Σ_𝑘𝑙=𝜆_𝑘𝐾𝜆_𝑙𝐿Σ_𝐾𝐿,Σ_{𝑘𝑙𝑚𝑛}≡𝜆_𝑘𝐾𝜆_𝑙𝐿𝜆_𝑚𝑀𝜆_𝑛𝑁Σ_{𝐾𝐿𝑀𝑁},𝜆_𝑠𝑛=𝜆_𝑠𝑆𝜆_𝑛𝑁𝜆_𝑆𝑁,Ω_𝑦𝑛=𝜆_𝑦𝑌𝜆_𝑛𝑁Ω_𝑌𝑁,𝜁_{𝑘𝑙𝑠𝑛}≡𝜆_𝑘𝐾𝜆_𝑙𝐿𝜆_𝑠𝑆𝜆_𝑛𝑁𝜁_{𝐾𝐿𝑆𝑁},𝜅_{𝑘𝑙𝑦𝑛}≡𝜆_𝑘𝐾𝜆_𝑙𝐿𝜆_𝑦𝑌𝜆_𝑛𝑁𝜅_{𝐾𝐿𝑌𝑁}. (5.9) In order to obtain a correct formulation of the linear theory, expression (5.8) should be quadratic at most in terms of the endlessly small expansion tensor ∈_𝑘𝑙 and temperature change T. For this purpose, the coefficients dependent on 𝜃 in the expression (5.8) have been defined as follows, respectively:Σ₀(𝜃,𝐗)=Σ₀𝑇₀+𝑇,𝐗=𝜌₀(𝐗)𝜓₀𝑇₀,𝐗−𝜌₀(𝐗)𝜂₀𝑇₀1,𝐗𝑇−2𝜌₀(𝐗)1𝑇₀𝐶𝑇₀,𝐗𝑇²Σ+⋯,_𝑘𝑙(𝜃,𝐗)=𝛾_𝑘𝑙𝑇₀,𝐗−𝛽_𝑘𝑙𝑇₀𝜆,𝐗𝑇+⋯,_𝑠𝑛(𝜃,𝐗)=Λ_𝑠𝑛𝑇₀,𝐗−𝜇_𝑠𝑛𝑇₀Ω,𝐗𝑇+⋯,_𝑦𝑛(𝜃,𝐗)=Ω_𝑦𝑛𝑇₀,𝐗−𝜋_𝑦𝑛𝑇₀Σ,𝐗𝑇+⋯,_{𝑘𝑙𝑚𝑛}(𝜃,𝐗)=Σ_{𝑘𝑙𝑚𝑛}𝑇₀+𝑇,𝐗=Σ_{𝑘𝑙𝑚𝑛}𝑇₀,𝜁,𝐗_{𝑘𝑙𝑠𝑛}(𝜃,𝐗)=𝜁_{𝑘𝑙𝑠𝑛}𝑇₀+𝑇,𝐗=𝜁_{𝑘𝑙𝑠𝑛}𝑇₀,𝜅,𝐗_{𝑘𝑙𝑦𝑛}(𝜃,𝐗)=𝜅_{𝑘𝑙𝑦𝑛}𝑇₀+𝑇,𝐗=𝜅_{𝑘𝑙𝑦𝑛}𝑇₀,𝐗. (5.10) Coefficients in this equations have been defined as follows:𝜕𝜓₀(𝑇,𝐗)𝜕𝑇_𝑇=𝑇0≡−𝜂₀𝑇₀,𝜕,𝐗²𝜓₀(𝑇,𝐗)𝜕𝑇²_𝑇=𝑇01≡−𝑇₀𝐶𝑇₀,𝛾,𝐗_𝑘𝑙𝑇₀,𝐗≡Σ_𝑘𝑙𝑇₀,𝐗=𝛾_𝑙𝑘𝑇₀,𝐗,𝛽_𝑘𝑙𝑇₀,𝐗≡−𝜕Σ_𝑘𝑙(𝑇,𝐗)𝜕𝑇_𝑇=𝑇0=𝛽_𝑙𝑘𝑇₀,Λ,𝐗_𝑠𝑛𝑇₀,𝐗≡Λ_𝑛𝑠𝑇₀,𝐗,𝜇_𝑠𝑛𝑇₀,𝐗≡−𝜕𝜆_𝑠𝑛(𝑇,𝐗)𝜕𝑇_𝑇=𝑇0=𝜇_𝑛𝑠𝑇₀,Ω,𝐗_𝑦𝑛𝑇₀,𝐗≡Ω_𝑛𝑦𝑇₀,𝐗,𝜋_𝑦𝑛𝑇₀,𝐗≡−𝜕Ω_𝑦𝑛(𝑇,𝐗)𝜕𝑇_𝑇=𝑇0=𝜋_𝑛𝑦𝑇₀,𝐗. (5.11) In these expressions, 𝜓₀(𝑇₀,𝐗), 𝜂₀(𝑇₀,𝐗), and 𝐶(𝑇₀,𝐗) are scalar; 𝛾_𝑘𝑙(𝑇₀,𝐗), 𝛽_𝑘𝑙(𝑇₀,𝐗), Λ_𝑠𝑛(𝑇₀,𝐗),𝜇_𝑠𝑛(𝑇₀,𝐗), Ω_𝑦𝑛(𝑇₀,𝐗), 𝜋_𝑦𝑛(𝑇₀,𝐗), Σ_{𝑘𝑙𝑚𝑛}(𝑇₀,𝐗), 𝜁_{𝑘𝑙𝑠𝑛}(𝑇₀,𝐗), and 𝜅_{𝑘𝑙𝑦𝑛}(𝑇₀,𝐗) are tensorial material constants, and these coefficients depend on the initial temperature 𝑇₀ of the medium and medium particles in heterogeneous materials. In homogenous materials, the dependence on 𝐗 is eliminated. In order to simplify notation, we will not indicate the arguments (𝑇₀,𝐗) of such coefficients. Substituting the expressions (5.11) and (5.10) in (5.8) gives us the following expression:Σ∈_𝑘𝑙,𝑎_𝑠,𝑧_𝑦,𝑇₀+𝑇,𝐗=𝜌₀𝜓₀−𝜌₀𝜂₀𝜌𝑇−⁰𝑐2𝑇₀𝑇²+𝛾_𝑘𝑙∈_𝑘𝑙−𝛽_𝑘𝑙𝑇∈_𝑘𝑙+Λ_𝑠𝑛𝑎_𝑠𝑎_𝑛−𝜇_𝑠𝑛𝑇𝑎_𝑠𝑎_𝑛+Ω_𝑦𝑛𝑧_𝑦𝑧_𝑛−𝜋_𝑦𝑛𝑇𝑧_𝑦𝑧_𝑛+12Σ_{𝑘𝑙𝑚𝑛}∈_𝑘𝑙∈_𝑚𝑛+𝜁_{𝑘𝑙𝑠𝑛}∈_𝑘𝑙𝑎_𝑠𝑎_𝑛+𝜅_{𝑘𝑙𝑦𝑛}∈_𝑘𝑙𝑧_𝑦𝑧_𝑛+⋯. (5.12) If derivative in (5.6) is taken from (5.12) and used in substitution, the following expression is obtained:𝑡_𝑝𝑟=Γ_𝑎𝑎_𝑝𝑎_𝑟+Γ_𝑧𝑧_𝑝𝑧_𝑟+1−𝑢_𝑘,𝑘−𝛽_𝑝𝑟𝑇+Σ_{𝑝𝑟𝑚𝑛}∈_𝑚𝑛+𝜁_{𝑝𝑟𝑠𝑛}𝑎_𝑠𝑎_𝑛+𝜅_{𝑝𝑟𝑦𝑛}𝑧_𝑦𝑧_𝑛. (5.13) Due to the Σ_{𝑝𝑟𝑚𝑛}=Σ_{𝑝𝑟𝑛𝑚} symmetry property of the coefficient Σ_{𝑝𝑟𝑚𝑛} in this expression, the constitutive equation given by expression (5.13) can be converted into the following form in terms of linear constituents of the displacement gradient:𝑡_𝑝𝑟=Γ_𝑎𝑎_𝑝𝑎_𝑟+Γ_𝑧𝑧_𝑝𝑧_𝑟−𝛽_𝑝𝑟𝑇+Σ_{𝑝𝑟𝑚𝑛}𝑢_𝑚,𝑛+𝜁_{𝑝𝑟𝑠𝑛}𝑎_𝑠𝑎_𝑛+𝜅_{𝑝𝑟𝑦𝑛}𝑧_𝑦𝑧_𝑛−𝜁_{𝑝𝑟𝑠𝑛}𝑎_𝑠𝑎_𝑛𝑢_𝑘,𝑘−𝜅_pryn𝑧_𝑦𝑧_𝑛𝑢_𝑘,𝑘. (5.14) In a composite material reinforced by two arbitrary independent and inextensible fiber families, the medium is assumed to be anisotropic, compressible, homogeneous, dependent on temperature gradient and showing linear elastic behavior. Equation (5.14) is the linear constitutive equation of stress. First and second terms on the right part of (5.14) are caused by the inextensibility of the fibers. Γ_𝑎 and Γ_𝑧-fiber stretch, both are determined through field equations and boundary conditions. These two terms are reaction stresses and cannot be expressed by any constitutive equation. The third term expresses the temperature effect, and the fourth term expresses the contribution of the elastic deformation to the stress. Regarding the fifth and the sixth term, two interpretations are possible. The first one states that if the medium is not loaded in any way (𝑇=constant,𝐸_𝐾𝐿=0), it will not switch to stress and therefore physically it should be 𝜁_{𝑝𝑟𝑠𝑛}=0 and 𝜅_{𝑝𝑟𝑦𝑛}=0, because being loaded with fibers is not sufficient for a medium to automatically get stressed. Another interpretation can be as follows. No parameters have been used related with cross-section thickness of fibers neither in this study nor in other studies examining macroscopic behavior of fiber-reinforced media. In other words, distribution of fibers is present in the medium only as a topologic object that just causes anisotropy, which means it is completely geometric. In this regard, there is no constraint that would prevent us from reinforcing fiber on the molecular scale. Therefore, if a distribution can be practically placed into the medium in the form of a molecular chain, it is possible to suggest that this will alter the present ionic distribution and stress the medium with no other effect. In this case, coefficients 𝜁_{𝑝𝑟𝑠𝑛} and 𝜅_{𝑝𝑟𝑦𝑛} in fifth and sixth terms will be different from zero and will thus gain a physical meaning. These terms can be interpreted as internal stress contribution stimulated by dislocation. The seventh and eighth terms show the stress formed by interaction of the deformation field with contributions of fiber fields.

If it is assumed that the medium is without fibers, expression (5.14) will be reduced to retain the third and fourth terms indicating the contribution to the stress of the temperature and strain tensor. Accordingly, in this study, terms of (5.14) have been obtained under the mentioned assumptions and are reduced to generally known classical expressions in special cases. This supports the opinion proving the reliability of the model we have created. These new terms are the expressions of constitutive equation on spatial coordinates for stress on a mathematical model created for fabricated composites, specifically for materials involving a distribution of two absolutely arbitrary fibers.

6. Determination of Heat Flux Vector Constitutive Equation in Linear Thermoelasticity

Here, the approach assumed for the stress potential has been assumed for the heat flux vector. Accordingly, the heat flux vector can be found through a power series expansion in terms of components of the arguments on which it depends, around a reference location selected as the natural condition. Considering that 𝐄 can be substituted by 𝐄 in the linear theory, arguments on which the heat flux vector depends, entropy inequality, and constraint caused by this inequality have been found as follows:𝑄_𝑅=𝑄_𝑅𝐸,𝐴,𝑍,𝐺,𝜃,𝑋𝐐𝑄,𝐄,𝐀,𝐙,𝐆,𝜃,𝐗⋅𝐆≥0,_𝑅=𝑄_𝑅𝐄,𝐀,𝐙,,0,𝜃,𝐗=0. (6.1)(6.2)(6.3) Expanding the function (6.1) in a Taylor series around 𝐄=𝟎,𝐀=𝟎, 𝐙=𝟎, 𝐆=𝟎 will give us the following expression: 𝑄_𝑅𝐄,𝐀,𝐙,𝐆,𝜃,𝐗=𝑄_𝑅(𝐆,𝜃,𝐗)+𝜕𝑄_𝑅𝐄,𝐆,𝜃,𝐗𝜕𝐸_{𝐿𝑀𝐄=0}𝐸_𝐿𝑀+𝜕𝑄_𝑅(𝐀,𝐆,𝜃,𝐗)𝜕𝐴_𝐿𝐴_{𝑀𝐀=𝟎}𝐴_𝐿𝐴_𝑀+𝜕𝑄_𝑅(𝐙,𝐆,𝜃,𝐗)𝜕𝑍_𝐿𝑍_{𝑀𝐙=𝟎}𝑍_𝐿𝑍_𝑀. (6.4) Here, the definitions as in the following are used𝑄_𝑅(𝐆,𝜃,𝐗)≡𝑄_𝑅(𝜃,𝐗)+𝜕𝑄_𝑅(𝐆,𝜃,𝐗)𝜕𝐺_{𝐿𝐆=𝟎}𝐺_𝐿=𝐵_𝑅(𝜃,𝐗)+𝐵_𝑅𝐿(𝜃,𝐗)𝐺_𝐿𝐻+⋯,_𝑅𝐿𝑀(𝐆,𝜃,𝐗)≡𝜕𝑄_𝑅𝐄,𝐆,𝜃,𝐗𝜕𝐸_{𝐿𝑀𝐄=𝟎}=𝐵_𝑅𝐿𝑀()𝑌𝜃,𝐗+⋯,_𝑅𝐿𝑀(𝐆,𝜃,𝐗)≡𝜕𝑄_𝑅(𝐀,𝐆,𝜃,𝐗)𝜕𝐴_𝐿𝐴_{𝑀𝐀=𝟎}=𝐷_𝑅𝐿𝑀()𝑁𝜃,𝐗+⋯,_𝑅𝐿𝑀(𝐆,𝜃,𝐗)≡𝜕𝑄_𝑅(𝐙,𝐆,𝜃,𝐗)𝜕𝑍_𝐿𝑍_{𝑀𝐙=𝟎}=𝐹_𝑅𝐿𝑀(𝜃,𝐗)+⋯. (6.5) Using the above-mentioned definitions in the series expansion given by expression (6.4) can give us the following expression:𝑄_𝑅𝐄,𝐀,𝐙,𝐆,𝜃,𝐗=𝐵_𝑅(𝜃,𝐗)+𝐵_𝑅𝐿(𝜃,𝐗)𝐺_𝐿+𝐵_𝑅𝐿𝑀(𝜃,𝐗)𝐸_𝐿𝑀+𝐷_𝑅𝐿𝑀(𝜃,𝐗)𝐴_𝐿𝐴_𝑀+𝐹_𝑅𝐿𝑀𝑍_𝐿𝑍_𝑀+⋯. (6.6) Due to the symmetry of the tensor 𝐄 and independence of derivatives in the definitions in expressions (6.5) from the order, these coefficients bear the symmetry characteristics given below:𝐵_𝑅𝐿𝑀=𝐵_𝑅𝑀𝐿,𝐷_𝑅𝐿𝑀=𝐷_𝑅𝑀𝐿,𝐹_𝑅𝐿𝑀=𝐹_𝑅𝑀𝐿. (6.7) Since 𝐆=𝟎⇒𝐐=𝟎 due to the constraint in (6.3), the following expression can be written down from the relation (6.6):0=𝐵_𝑅(𝜃,𝐗)+𝐵_𝑅𝐿𝑀(𝜃,𝐗)𝐸_𝐿𝑀+𝐷_𝑅𝐿𝑀(𝜃,𝐗)𝐴_𝐿𝐴_𝑀+𝐹_𝑅𝐿𝑀(𝜃,𝐗)𝑍_𝐿𝑍_𝑀+⋯. (6.8) Since expression (6.8) is zero for any arbitrary deformation measure, coefficients in this equation should be zero. Therefore,𝐵_𝑅(𝜃,𝐗)=𝐵_𝑅𝐿𝑀(𝜃,𝐗)=𝐷_𝑅𝐿𝑀(𝜃,𝐗)=𝐹_𝑅𝐿𝑀(𝜃,𝐗)=0. (6.9) Accordingly, (6.6) is reduced to the following form:𝑄_𝑅=𝑄_𝑅𝐄,𝐀,𝐙,𝐆,𝜃,𝐗=𝐵_𝑅𝐿(𝜃,𝐗)𝐺_𝐿=𝐵_𝑅𝐿(𝜃,𝐗)𝜃_,𝐿. (6.10) Substituting expression (6.10) in the inequality (6.2) gives us the following expression:𝐵_𝑅𝐿(𝜃,𝐗)𝜃_,𝐿𝜃_,𝑅≥0or𝐵_𝑅𝐿(𝜃,𝐗)𝐺_𝐿𝐺_𝑅≥0. (6.11) Therefore, the tensor 𝐵_𝑅𝐿(𝜃,𝐗) should satisfy the following condition for any temperature gradient:𝐵_𝑅𝐿𝜃_,𝑅𝜃_,𝐿≥0or𝐵_(𝑅𝐿)𝜃_,𝑅𝜃_,𝐿≥0. (6.12) The 𝐵_𝑅𝐿 tensor is named conductivity coefficient tensor. Inequality (6.12) tells us that the symmetric part of this tensor is positive definite. For the linear theory, the coefficient 𝐵_𝑅𝐿 is expressed as follows in similarity to the coefficient Σ_𝑃𝑅:𝐵_𝑅𝐿(𝜃,𝐗)=𝐵_𝑅𝐿𝑇₀+𝑇,𝐗=𝐵_𝑅𝐿𝑇₀,𝐗+𝜕𝐵_𝑅𝐿(𝑇,𝐗)𝜕𝑇₀𝑇+⋯. (6.13) Moreover, the coefficient 𝜃_,𝐿 can be written down as follows:𝜃_,𝐿=𝑇₀+𝑇_,𝐿=𝑇_,𝐿. (6.14) Substituting expression (6.14) in (6.10) and omitting the nonlinear term (𝑇)(𝑇_,𝐿), the heat flux vector is written down as follows: 𝑄_𝑅=𝐵_𝑅𝐿𝑇₀,𝐗𝑇_,𝐿. (6.15) The expression of the spatial form of heat flux vector for compressible media can be written down as indicated below: 𝑞_𝑟=1−𝑢_𝑘,𝑘𝑄_𝑅𝑥_𝑟,𝑅. (6.16) If (6.15) is substituted into expression (6.16), using expressions (5.5) and omitting the nonlinear term (𝑢_𝑘,𝑘𝑇_,𝑙), the spatial form of the heat flux vector follows as𝑞_𝑟=𝐵_𝑟𝑙𝑇₀,𝐗𝑇_,𝑙. (6.17) The spatial tensor 𝐵_𝑟𝑙 of the material in (6.17) has the same symmetry characteristics as the tensor 𝐵_𝑅𝐿 and is defined as follows:𝐵_𝑟𝑙≡𝜆_𝑟𝑅𝜆_𝑙𝐿𝐵_𝑅𝐿. (6.18) Equation (6.17) is the Fourier heat transfer law, which defines linear heat transfer, and it can be written down as follows in the vectorial form:𝐪=𝐁∇𝑇. (6.19)

7. Determination of Field Equations

Before proceeding to obtain the field equations, let us discuss the meaning of the tensor 𝛽_𝑝𝑟 in (5.14). Firstly, let us define the tensor Σ⁻¹_{𝑝𝑟𝑚𝑛}, which is the reversed tensor Σ_{𝑝𝑟𝑚𝑛} and has the same symmetry properties as this tensor, as follows:Σ_{𝑝𝑟𝑚𝑛}Σ⁻¹_{𝑚𝑛𝑘𝑙}≡12𝛿_𝑝𝑘𝛿_𝑟𝑙+𝛿_𝑝𝑙𝛿_𝑟𝑘,Σ⁻¹_{𝑝𝑟𝑚𝑛}=Σ⁻¹_{𝑟𝑝𝑚𝑛}=Σ⁻¹_{𝑚𝑛𝑝𝑟}=Σ⁻¹_{𝑝𝑟𝑛𝑚}. (7.1) The tensor 𝛼_𝑝𝑟 comprised of thermal expansion coefficients that can be easily measured physically can be defined as follows:𝛼_𝑝𝑟≡Σ⁻¹_{𝑝𝑟𝑚𝑛}𝛽_𝑚𝑛=𝛼_𝑟𝑝. (7.2) To find the reverse of expression (7.2), let us multiply both sides of the equation by the tensor Σ_{𝑘𝑙𝑝𝑟}. Then, using a suitable index replacement, the following can be written down: 𝛽_𝑝𝑟=Σ_{𝑝𝑟𝑚𝑛}𝛼_𝑚𝑛. (7.3) Substituting expression (7.3) in (5.14) will give us the following:𝑡_𝑝𝑟=Γ_𝑎𝑎_𝑝𝑎_𝑟+Γ_𝑧𝑧_𝑝𝑧_𝑟+𝜁_{𝑝𝑟𝑠𝑛}𝑎_𝑠𝑎_𝑛−𝑢_𝑘,𝑘𝑎_𝑠𝑎_𝑛+𝜅_{𝑝𝑟𝑦𝑛}𝑧_𝑦𝑧_𝑛−𝑢_𝑘,𝑘𝑧_𝑦𝑧_𝑛+Σ_{𝑝𝑟𝑚𝑛}𝑢_𝑚,𝑛−𝛼_𝑚𝑛𝑇. (7.4) The following expression can be written down in regard to a linear theory:𝜌𝑣̇_𝑘≅𝜌₀1−𝑢_𝑘,𝑘𝜕𝑣_𝑘𝜕𝑡≅𝜌₀𝜕𝜕𝑡𝜕𝑢_𝑘𝜕𝑡≅𝜌₀𝜕²𝑢_𝑘𝜕𝑡². (7.5) In a linear theory, the Cauchy equations of motion can be written as follows substituting the expressions (7.5) and (5.5) in (3.2):𝑡_{𝑘𝑙,𝑙}+𝜌₀1−𝑢_𝑙,𝑙f_k=𝜌₀𝜕²𝑢_𝑘𝜕𝑡². (7.6) Considering that the medium is homogenous and omitting the term (𝜌₀𝑢_𝑙,𝑙f_k), let us calculate the divergence of the stress given by (7.4) and substitute it in (7.6) to obtain the following field equation under the above-mentioned assumptions:𝜌₀𝜕²𝑢_𝑝𝜕𝑡²=Σ_{𝑝𝑟𝑚𝑛}𝑢_{𝑚,𝑛𝑟}−𝛼_𝑚𝑛𝑇_,𝑟+𝜌₀f_p+Γ_𝑎,𝑟𝑎_𝑝𝑎_𝑟+Γ_𝑎𝑎_𝑝,𝑟𝑎_𝑟+𝑎_𝑝𝑎_𝑟,𝑟+Γ_𝑧,𝑟𝑧_𝑝𝑧_𝑟+Γ_𝑧𝑧_𝑝,𝑟𝑧_𝑟+𝑧_𝑝𝑧_𝑟,𝑟+𝜁_{𝑝𝑟𝑠𝑛}𝑎_𝑠,𝑟𝑎_𝑛+𝑎_𝑠𝑎_𝑛,𝑟−𝜁_{𝑝𝑟𝑠𝑛}𝑎_𝑠,𝑟𝑎_𝑛+𝑎_𝑠𝑎_𝑛,𝑟𝑢_𝑘,𝑘−𝜁_{𝑝𝑟𝑠𝑛}𝑢_{𝑘,𝑘𝑟}𝑎_𝑠𝑎_𝑛+𝜅_{𝑝𝑟𝑦𝑛}𝑧_𝑦,𝑟𝑧_𝑛+𝑧_𝑦𝑧_𝑛,𝑟−𝜅_{𝑝𝑟𝑦𝑛}𝑧_𝑦,𝑟𝑧_𝑛+𝑧_𝑦𝑧_𝑛,𝑟𝑢_𝑘,𝑘−𝜅_{𝑝𝑟𝑦𝑛}𝑢_{𝑘,𝑘𝑟}𝑧_𝑦𝑧_𝑛. (7.7) Expression (7.7) gives us a field equation with the unknowns 𝑢_𝑘,Γ_𝑎,Γ_𝑧. The solution of this field equation under initial and boundary conditions forms the mathematical structure of a boundary value problem to consider.

Because 𝜃=𝑇₀+𝑇 and 𝜕𝑇/𝜕𝜃=1, the entropy and the internal energy density given in expressions (4.15) and (4.22) can be written down as follows:1𝜂=−𝜌₀𝜕Σ𝜕𝑇𝜕𝑇1𝜕𝜃=−𝜌₀𝜕Σ1𝜕𝑇,𝜀=𝜌₀Σ−𝑇₀+𝑇𝜕Σ𝜕𝑇. (7.8)(7.9) Substituting (7.8) in expression (7.9) will give us the following expression:Σ𝜀=𝜌₀+𝑇₀+𝑇𝜂. (7.10) Taking the derivative of Σ given by expression (5.12) according to 𝑇 and substituting it in (7.8) after related operations will allow us to express entropy in terms of the displacement gradient component as follows:𝜂=𝜂₀+𝑐𝑇𝑇₀+𝛽_𝑘𝑙𝜌₀𝑢_𝑘,𝑙+𝜇_𝑠𝑛𝜌₀𝑎_𝑠𝑎_𝑛+𝜋_𝑦𝑛𝜌₀𝑧_𝑦𝑧_𝑛. (7.11) Let us now substitute expressions (5.12) and (7.11) in (7.10) and make necessary arrangements to obtain the internal energy density as follows:𝜀=𝜀₀𝑇+𝑐𝑇+²2𝑇₀+𝑇₀𝛽_𝑘𝑙𝜌₀𝑢_𝑘,𝑙+12𝜌₀Σ_{𝑘𝑙𝑚𝑛}𝑢_𝑘,𝑙𝑢_𝑚,𝑛+1𝜌₀Λ_𝑠𝑛+𝜁_{𝑘𝑙𝑠𝑛}𝑢_𝑘,𝑙𝑎_𝑠𝑎_𝑛+Ω_𝑦𝑛+𝜅_{𝑘𝑙𝑦𝑛}𝑢_𝑘,𝑙𝑧_𝑦𝑧_𝑛. (7.12)𝜀₀ coefficient in this equation has been defined as 𝜀₀=𝜓₀+𝑇₀𝜂₀, where 𝜀₀, 𝜓₀, and 𝜂₀ are, respectively, internal energy density, free energy density, and entropy density in natural condition. Taking a material derivative of expression (7.12) and considering that 𝜌=𝜌₀(1−𝑢_𝑘,𝑘) give us the following expression: 𝜌𝜀̇=𝜌₀1−𝑢_𝑚,𝑚𝜀̇=𝜌₀𝑐𝑇1+𝑇₀𝜕𝑇𝜕𝑡+𝑇₀𝛽_𝑘𝑙𝜕𝑢_𝑘,𝑙𝜕𝑡+Σ_{𝑘𝑙𝑚𝑛}𝜕𝑢_𝑘,𝑙𝑢𝜕𝑡^𝑚,𝑛+𝜁_{𝑘𝑙𝑠𝑛}𝜕𝑢_𝑘,𝑙𝑎𝜕𝑡^𝑠𝑎_𝑛+𝜅_{𝑘𝑙𝑦𝑛}𝜕𝑢_𝑘,𝑙𝑧𝜕𝑡^𝑦𝑧_𝑛−𝜁_{𝑘𝑙𝑠𝑛}𝜕𝑢_𝑘,𝑙𝑢𝜕𝑡^𝑚,𝑚𝑎_𝑠𝑎_𝑛−𝜅_{𝑘𝑙𝑦𝑛}𝜕𝑢_𝑘,𝑙𝑢𝜕𝑡^𝑚,𝑚𝑧_𝑦𝑧_𝑛 (7.13) The term 𝑞_𝑟,𝑟 is obtained as follows from (6.17):𝑞_𝑟,𝑟=𝐵_{𝑟𝑙,𝑟}𝑇_𝑙+𝐵_(𝑟𝑙)𝑇_,𝑙𝑟=𝐵_(𝑟𝑙)𝑇_,𝑙𝑟. (7.14) Let us now substitute the expressions (7.13), (7.14), (7.4), and (5.5) in the equation conservation of energy given by expression (3.4) and make necessary arrangements to make the following field equations linear in terms of 𝑢_𝑘,𝑙 and 𝑇:𝜌₀𝑐𝜕𝑇+𝑇𝜕𝑡⁰𝛽_𝑘𝑙−Γ_𝑎𝑎_𝑘𝑎_𝑙−Γ_𝑧𝑧_𝑘𝑧_𝑙𝜕𝑢_𝑘,𝑙𝜕𝑡=𝛽_(𝑘𝑙)𝑇_,𝑙𝑘+𝜌₀1−𝑢_𝑙,𝑙ℎ. (7.15) In a composite material reinforced by two arbitrary independent and inextensible fiber families, where the medium is assumed to be anisotropic, compressible, homogeneous, dependent on temperature gradient, and showing linear elastic behavior, (7.15) is a heat transfer equation.

8. Conclusions

As an approach in this study, the stress potential and heat flux vector functions have been assumed to be analytic and expanded in Taylor series in terms of their arguments on which they depend. The type and the number of terms taken in the series expansion have been determined based on the assumption that mechanical interactions and temperature changes are linear. On the other hand, since the matrix material has to remain insensitive to directional changes along fibers, even-numbered exterior products of vector fields representing fiber distributions have been considered. The reference position of the medium has been selected at the uniform temperature 𝑇₀ and stress-free natural condition, from which position the medium has been assumed to move away by small displacements, and small temperature changes. Accordingly, the forms in spatial coordinates of the constitutive equations of the stress and the heat flux vector have been presented by (5.14) and (6.17). The constitutive equation of the stress expressed by (5.14) in terms of the tensor 𝛼_𝑝𝑟 comprised of thermal expansion coefficients has been expressed by (7.4). To obtain field equations, constitutive equation of the stress given by (7.4) has been substituted into the Cauchy equation of motion, yielding field equation (7.7). Values in the equation of conservation of energy given by expression (3.4) have been substituted into (3.4), yielding field equation (7.15). Solution of the field equations along with initial and boundary conditions in conformity with the structure of the problem to be used in practice will constitute the structure of a boundary value problem to consider. Unknowns in the field equations (7.7) and (7.15), 𝐮(𝐱,𝑡), Γ_𝑎, and Γ_𝑧. Γ_𝑎 and Γ_𝑧, which are Lagrange coefficients, can be calculated using the field equations and boundary conditions. After 𝐮 has been designated, the stress distribution is obtained from (7.4). After the stress distribution is found as a tensor field, the stress vector at a desired cross-section can be easily calculated from the expression 𝐭_(𝐧)𝑟=𝑛_𝑝𝑡_𝑝𝑟. Here, it needs to be considered that the fiber distributions 𝑎_𝑘(𝐱,𝑡) and 𝑧_𝑘(𝐱,𝑡) after deformation for inextensible fibers in terms of fiber distributions before deformation are 𝑎_𝑘=𝑥_𝑘,𝐾𝐴_𝐾(𝐗) and 𝑧_𝑘=𝑥_𝑘,𝐾𝑍_𝐾(𝐗).

Besides, considering the equations of motion (7.7), we can see the internal thermomechanical forces affecting the medium. Type of the terms on the right is in the dimension of force per unit of volume. The first term on the right represents force created by the elastic deformation, the second term is force created by the temperature gradient, the third term is mechanical body force, and the fourth and the sixth terms are similar to molecular considerations in (5.14). The fifth, seventh, eighth, and eleventh terms are forces caused by curvature of the fibers. The ninth and twelfth terms are forces caused by interaction of fibers and their curvature with the deformation field. The tenth and thirteenth terms represent forces caused by interaction of the deformation field with distributions of fiber fields. In other words, by drawing a free body diagram of a material element in the medium, it is possible to see all such force contributions acting on the element. As stated before, our field equations where we can apply initial and boundary conditions for the aforementioned media are (7.7) and (7.15). A more detailed discussion of these equations will be contained in future works.

Acknowledgments

The author would like to thank the anonymous referees who made much useful and detailed suggestions that helped her to correct some errors and improve the quality of this paper. The author is also grateful to professor Horst Ecker for his editorial helping and valuable comments which have greatly aided in the improvement of this paper.