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

Authors: Melek Usal
Journal: Mathematical Problems in Engineering
 (2010)

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## 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.


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## Body

## 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 (λa2=CKLAKAL=1, λz2=CKLZKZL=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 vectorsA(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] (2.1)akdl=xk,KAKdL,akdldL=xk,KAK.
Rates of extension of fiber family A can be defined as follows:(2.2)λa≡(dldL)A.
If expression (2.2) is substituted into (2.1), the following expression is obtained: (2.3)ak=λa-1xk,KAK.
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(2.4)|A|=|a|=1,akak=1=λa-2xk,Kxk,LAKAL=λa-2CKLAKAL.
Accordingly, the form is found to be
(2.5)λa2=CKLAKAL.
Relationships that are true for the Z-fiber family can be expressed as follows [23, 24]:
(2.6)zk=λz-1xk,KZK,λz≡(dldL)Z,λz2=CKLZKZL,
where dL and dl are, respectively, arc length of fiber before and after deformation, AK and ZK are fiber unit vector components before deformation, ak and zk are fiber unit vector components after deformation, xk,K=∂xk/∂XK is deformation gradient, λa and λz are rates of extension of fiber families, and CKL=xk,Kxk,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:(3.1)ρ̇+ρvk,k=0,ρ(x,t)=ρ0(X)J(x,t)(conservationofmassinmaterialrepresentation),
balance of linear mometum: (3.2)ρv̇p=ρfp+trp,r,
balance of moment of momentum: (3.3)εkrptrp=0,trp=tpr,
conservation of energy:(3.4)ρε̇=tkldkl+qk,k+ρh,
Clausius-Duhem inequality:(3.5)ρθη̇-∇⋅q+1θq⋅∇θ-ρh⩾0.
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[xk,K]=ρ0/ρ(x,t) for jacobian, v̇ for acceleration, tlk for stress tensor, fk for the mechanical volumetric force per unit of mass, ε for internal energy density per unit of mass, qk 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 εijk 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:(4.1)-ρ(ε̇-θη̇)+tkldlk+1θqkθ,k≥0.
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 θ(4.2)ψ≡ε-θη.
As a result, the entropy inequality is transformed as follows, expressed in new terms:
(4.3)-ρ(ψ̇+θ̇η)+tkldlk+1θqkθ,k≥0.
Entropy inequality is obtained as follows in the material form [26]:
(4.4)-(Σ̇+ρ0θ̇η)+12TKLĊKL+1θθ,KQK≥0.
Terms relating to the new values have been provided below:(4.5)Σ≡ρ0ψ,(4.6)ĊKL=2dklxk,Kxl,L⟹dkl=12ĊKLXK,kXL,l,(4.7)TKL≡JXK,kXL,ltkl⟹tkl=J-1xk,Kxl,LTKL,(4.8)QK≡JXK,kqk⟹qk=J-1xk,KQK,(4.9)GK≡θ,K=xk,Kθ,k⟹gk≡θ,k=XK,kθ,K.
Here, Σ stands for thermodynamic stress potential, ψ for generalized free energy density, dkl for deformation (strain) rate tensor, XK,k=∂XK/∂xk for the deformation gradient of the reverse motion, TKL for the stress tensor on material coordinates, QK for the heat flux vector on material coordinates, and GK 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:(4.10)Σ(X,t)=Σ[x(X′,t′),θ(X′,t′),X]X∈V-∞<t′≤t.
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:(4.11)Σ(XK,t)=Σ[CKL(XK,t),GK(XK,t),AK(XK),ZK(XK),θ(XK,t),XK].
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 AK and ZK do not depend on time, the following expression is obtained by taking the material derivative of expression (4.11).(4.12)Σ̇=∂Σ∂CKLĊKL+∂Σ∂θ,KĠ+∂Σ∂θθ̇.
Substituting this expression in (4.4) gives us the following inequality:(4.13)12(TKL-2∂Σ∂CKL)ĊKL-ρ0(η+1ρ0∂Σ∂θ)θ̇-∂Σ∂GKĠK+1θGKQK≥0.
Since we are able to arbitrarily replace the arguments in inequality (4.13) from θ to θ̇, from CKL to ĊKL, and from GK to ĠK, for the inequality (4.13) to be satisfied, the coefficients of θ̇, ĊKL and ĠK will be zero. The coefficient of GK cannot be zero as due to GK’s presence in the arguments of Σ, it cannot be arbitrarily replaced. Equalizing the coefficients of ĊKL, θ̇, and ĠK to zero will give us the following expressions:(4.14)TKL=2∂Σ∂CKL,(4.15)η=-1ρ0∂Σ∂θ,(4.16)∂Σ∂GK=0.
It is understood from expression (4.16) that the stress potential does not depend on GK. Therefore, arguments on which the stress potential depends are expressed as follows:(4.17)Σ=Σ(CKL,AK,ZK,θ).
Thus, inequality (4.13) is reduced to the following form:(4.18)1θGKQK≥0.
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 depends(4.19)QK=QK(CKL,AK,ZK,GK,θ).
In consideration of expression (4.19), inequality (4.18) is written down as follows:(4.20)GKQK(CKL,AK,ZK,GK,θ)≥0orQ(C,A,Z,G,θ,X)⋅G≥0.
In inequality (4.20), when GK=0, QK must also be equal to zero. Accordingly, maintaining the order of independent constitutive variables in expression (4.20), the following expression should be written:(4.21)QK(CKL,AK,ZK,0,θ)=0.
On the other hand, internal energy density ε can be written as follows from the expressions (4.2), (4.5), and (4.15):(4.22)ε=1ρ0(Σ-∂Σ∂θθ).
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 QK, 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). LetS̲̲=[SKL] 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 (4.23)XK′=SKLXL,XL=SLKTXK′,S̲̲-1=S̲̲T
established using the orthogonal matrix S̲̲ should remain form invariant. Mathematically, this indicates the validity of the transformations(4.24)Σ(S̲̲C̲̲S̲̲T,S̲̲A̲,S̲̲Z̲,θ)=Σ(C̲̲,A̲,Z̲,θ),(4.25)Q̲(S̲̲C̲̲S̲̲T,S̲̲A̲,S̲̲Z̲,S̲̲G̲,θ)=Q̲(C̲̲,A̲,Z̲,G̲,θ).
The following conditions should be satisfied since the fiber families are assumed to be inextensible [22, 27]:(4.26)λa2=CKLAKAL=1,λz2=CKLZKZL=1.
Thus, the constitutive equation for the stress is obtained as follows in spatial and material coordinates:(4.27)tkl=Γaakal+Γzzkzl+2xk,Kxl.L∂Σ∂CKL,TKL=ΓaAKAL+ΓzZKZL+2∂Σ∂CKL.
In these expressions, Γa andΓz 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 temperatureT0 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 relationCKL=δKL+2EKL exists between the Green deformation tensor and strain tensor, EKL can be expressed as EKL≅ẼKL≡(1/2)(UK,L+UL,K)in a linear theory, and the arguments of the stress potential given by expression (4.17) can be written down as follows:(5.1)Σ=Σ(ẼKL,AK,ZK,θ).
Assuming that this function is analytic in terms of the Ẽ,A,Z values, if this function is expanded in Taylor series around Ẽ=0, A=0, Z=0, the expression will be obtained for the stress potential:(5.2)Σ(ẼKL,AS,ZY,θ)=Σ0(θ,X)+ΣKL(θ,X)ẼKL+12ΣKLMN(θ,X)ẼKLẼMN+λSN(θ,X)ASAN+ΩYN(θ,X)ZYZN+ζKLSN(θ,X)EKLASAN+κKLYN(θ,X)EKLZYZN+⋯.
Coefficients in this equation have been defined as follows:(5.3)Σ0=Σ(0̲̲,0̲),ΣKL≡∂Σ∂ẼKL|0,ΣKLMN≡∂2Σ∂ẼKL∂ẼMN|0,λSN≡12∂2Σ∂AS∂AN|0,ΩYN≡12∂2Σ∂ZY∂ZN|0,ζKLSN≡16∂3Σ∂EKL∂AS∂AN|0,κKLYN≡16∂3Σ∂EKL∂ZY∂ZN|0.
Due to the symmetry of the ẼKL tensor and nondependence on order of the derivatives in the definitions in the expressions (5.3), these coefficients bear the symmetry characteristics indicated below:(5.4)Σ=ΣLKKL,ΣKLMN=ΣLKMN=ΣKLNM=ΣMNKL,λSN=λNS,ΩYN=ΩNY,ζKLSN=ζLKSN=ζKLNS,κKLYN=κLKYN=κKLNY.
The following relations can be written down for the linear theory in continuum mechanics [22]:(5.5)EKL≅ẼKL≡12(UK,L+UL,K),ekl≅ẽkl=∈kl≡12(uk,l+ul,k),∈kl≅λkKλlLẼKL,ẼKL≅λkKλlLẽkl,xk,K=λkK+uk,K,XK,k=ΛKk-UK,k,xk,Kxl,L=λkKλlL,XK,kXL,l=λkKλlL,EKL≅ẼKL≡λkKλlLẽkl=12λkKλlL(uk,l+ul,k),xp,Pxr,RAKAL=xp,Pxr,RXK,kXL,lakalλa2=λpPλrRλkKλlLakal(forλa=1),xp,Pxr,RZKZL=xp,Pxr,RXK,kXL,lzkzlλz2=λpPλrRλkKλlLzkzl(forλz=1),dpr=∂EPR∂tXP,rXR,r=∂∈pr∂t,dpr=∂(up,r)∂t,ε̇≈∂ε∂t,J-1≅1-uk,k,ρ≅ρ0(1-uk,k).
The expression of the spatial form of stress for compressible media with inextensible fiber families can be written down as indicated below: (5.6)tpr=Γaakal+Γzzkzl+(1-uk,k)∂Σ∂∈pr.
In the linear theory, arguments on which Σ depends can be expressed in spatial form as follows:(5.7)Σ=Σ(∈kl,ak,zk,θ,X).
Assuming this function is analytic in terms of ∈kl,ak,zk and expanding it in the Taylor series around ∈kl=0,as=0,zy=0 will give us the following expression: (5.8)Σ(∈kl,as,zy,θ,X)=Σ0(θ,X)+Σkl(θ,X)∈kl+12Σklmn(θ,X)∈kl∈mn+λsn(θ,X)asan+Ωyn(θ,X)zyzn+ζklsn(θ,X)∈klasan+κklyn(θ,X)∈klzyzn+⋯.
The spatial material tensors Σkl, Σklmn, λsn, Ωyn, ζklsn, and κklyn in (5.8) bear the same properties as the material tensors of the material ΣKL,ΣKLMN, λSN,ΩYN, ζKLSN, and κKLYN and are defined as follows:(5.9)Σkl=λkKλlLΣKL,Σklmn≡λkKλlLλmMλnNΣKLMN,λsn=λsSλnNλSN,Ωyn=λyYλnNΩYN,ζklsn≡λkKλlLλsSλnNζKLSN,κklyn≡λkKλlLλyYλnNκKLYN.
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:(5.10)Σ0(θ,X)=Σ0(T0+T,X)=ρ0(X)ψ0(T0,X)-ρ0(X)η0(T0,X)T-12ρ0(X)1T0C(T0,X)T2+⋯,Σkl(θ,X)=γkl(T0,X)-βkl(T0,X)T+⋯,λsn(θ,X)=Λsn(T0,X)-μsn(T0,X)T+⋯,Ωyn(θ,X)=Ωyn(T0,X)-πyn(T0,X)T+⋯,Σklmn(θ,X)=Σklmn(T0+T,X)=Σklmn(T0,X),ζklsn(θ,X)=ζklsn(T0+T,X)=ζklsn(T0,X),κklyn(θ,X)=κklyn(T0+T,X)=κklyn(T0,X).
Coefficients in this equations have been defined as follows:(5.11)∂ψ0(T,X)∂T|T=T0≡-η0(T0,X),∂2ψ0(T,X)∂T2|T=T0≡-1T0C(T0,X),γkl(T0,X)≡Σkl(T0,X)=γlk(T0,X),βkl(T0,X)≡-∂Σkl(T,X)∂T|T=T0=βlk(T0,X),Λsn(T0,X)≡Λns(T0,X),μsn(T0,X)≡-∂λsn(T,X)∂T|T=T0=μns(T0,X),Ωyn(T0,X)≡Ωny(T0,X),πyn(T0,X)≡-∂Ωyn(T,X)∂T|T=T0=πny(T0,X).
In these expressions, ψ0(T0,X), η0(T0,X), and C(T0,X) are scalar; γkl(T0,X), βkl(T0,X),  Λsn(T0,X),μsn(T0,X), Ωyn(T0,X), πyn(T0,X), Σklmn(T0,X), ζklsn(T0,X), and κklyn(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:(5.12)Σ(∈kl,as,zy,T0+T,X)=ρ0ψ0-ρ0η0T-ρ0c2T0T2+γkl∈kl-βklT∈kl+Λsnasan-μsnTasan+Ωynzyzn-πynTzyzn+12Σklmn∈kl∈mn+ζklsn∈klasan+κklyn∈klzyzn+⋯.
If derivative in (5.6) is taken from (5.12) and used in substitution, the following expression is obtained:(5.13)tpr=Γaapar+Γzzpzr+(1-uk,k)(-βprT+Σprmn∈mn+ζprsnasan+κprynzyzn).
Due to the Σprmn=Σprnm symmetry property of the coefficient Σprmn 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:(5.14)tpr=Γaapar+Γzzpzr-βprT+Σprmnum,n+ζprsnasan+κprynzyzn-ζprsnasanuk,k-κprynzyznuk,k.
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. Γa and Γz-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,EKL=0), it will not switch to stress and therefore physically it should be ζprsn=0 and κpryn=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 ζprsn and κpryn 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 thatE can be substituted by Ẽ 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:(6.1)QR=QR(Ẽ̲̲,A̲,Z̲,G̲,θ,X̲),(6.2)Q(Ẽ,A,Z,G,θ,X)⋅G≥0,(6.3)QR=QR(Ẽ,A,Z,,0,θ,X)=0.
Expanding the function (6.1) in a Taylor series around Ẽ=0,A=0, Z=0, G=0 will give us the following expression: (6.4)QR(Ẽ,A,Z,G,θ,X)=QR(G,θ,X)+∂QR(Ẽ,G,θ,X)∂ELM|E=0ẼLM+∂QR(A,G,θ,X)∂ALAM|A=0ALAM+∂QR(Z,G,θ,X)∂ZLZM|Z=0ZLZM.
Here, the definitions as in the following are used(6.5)QR(G,θ,X)≡QR(θ,X)+∂QR(G,θ,X)∂GL|G=0GL=BR(θ,X)+BRL(θ,X)GL+⋯,HRLM(G,θ,X)≡∂QR(Ẽ,G,θ,X)∂ELM|E=0=BRLM(θ,X)+⋯,YRLM(G,θ,X)≡∂QR(A,G,θ,X)∂ALAM|A=0=DRLM(θ,X)+⋯,NRLM(G,θ,X)≡∂QR(Z,G,θ,X)∂ZLZM|Z=0=FRLM(θ,X)+⋯.
Using the above-mentioned definitions in the series expansion given by expression (6.4) can give us the following expression:(6.6)QR(Ẽ,A,Z,G,θ,X)=BR(θ,X)+BRL(θ,X)GL+BRLM(θ,X)ẼLM+DRLM(θ,X)ALAM+FRLMZLZM+⋯.
Due to the symmetry of the tensor Ẽ and independence of derivatives in the definitions in expressions (6.5) from the order, these coefficients bear the symmetry characteristics given below:(6.7)BRLM=BRML,DRLM=DRML,FRLM=FRML.
Since G=0⇒Q=0 due to the constraint in (6.3), the following expression can be written down from the relation (6.6):(6.8)0̲=BR(θ,X)+BRLM(θ,X)ẼLM+DRLM(θ,X)ALAM+FRLM(θ,X)ZLZM+⋯.
Since expression (6.8) is zero for any arbitrary deformation measure, coefficients in this equation should be zero. Therefore,(6.9)BR(θ,X)=BRLM(θ,X)=DRLM(θ,X)=FRLM(θ,X)=0.
Accordingly, (6.6) is reduced to the following form:(6.10)QR=QR(Ẽ,A,Z,G,θ,X)=BRL(θ,X)GL=BRL(θ,X)θ,L.
Substituting expression (6.10) in the inequality (6.2) gives us the following expression:(6.11)BRL(θ,X)θ,Lθ,R≥0orBRL(θ,X)GLGR≥0.
Therefore, the tensor BRL(θ,X) should satisfy the following condition for any temperature gradient:(6.12)BRLθ,Rθ,L≥0orB(RL)θ,Rθ,L≥0.
The BRL 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 BRL is expressed as follows in similarity to the coefficient ΣPR:(6.13)BRL(θ,X)=BRL(T0+T,X)=BRL(T0,X)+∂BRL(T,X)∂T|0T+⋯.
Moreover, the coefficient θ,L can be written down as follows:(6.14)θ,L=(T0+T),L=T,L.
Substituting expression (6.14) in (6.10) and omitting the nonlinear term (T)(T,L), the heat flux vector is written down as follows: (6.15)QR=BRL(T0,X)T,L.
The expression of the spatial form of heat flux vector for compressible media can be written down as indicated below: (6.16)qr=(1-uk,k)QRxr,R.
If (6.15) is substituted into expression (6.16), using expressions (5.5) and omitting the nonlinear term (uk,kT,l), the spatial form of the heat flux vector follows as(6.17)qr=Brl(T0,X)T,l.
The spatial tensor Brl of the material in (6.17) has the same symmetry characteristics as the tensor BRL and is defined as follows:(6.18)Brl≡λrRλlLBRL.
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:(6.19)q=B∇T.


## 7. Determination of Field Equations

Before proceeding to obtain the field equations, let us discuss the meaning of the tensorβpr in (5.14). Firstly, let us define the tensor Σprmn-1, which is the reversed tensor Σprmn and has the same symmetry properties as this tensor, as follows:(7.1)ΣprmnΣmnkl-1≡12(δpkδrl+δplδrk),Σprmn-1=Σrpmn-1=Σmnpr-1=Σprnm-1.
The tensor αpr comprised of thermal expansion coefficients that can be easily measured physically can be defined as follows:(7.2)αpr≡Σprmn-1βmn=αrp.
To find the reverse of expression (7.2), let us multiply both sides of the equation by the tensor Σklpr. Then, using a suitable index replacement, the following can be written down: (7.3)βpr=Σprmnαmn.
Substituting expression (7.3) in (5.14) will give us the following:(7.4)tpr=Γaapar+Γzzpzr+ζprsn(asan-uk,kasan)+κpryn(zyzn-uk,kzyzn)+Σprmn(um,n-αmnT).
The following expression can be written down in regard to a linear theory:(7.5)ρv̇k≅ρ0(1-uk,k)∂vk∂t≅ρ0∂∂t∂uk∂t≅ρ0∂2uk∂t2.
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):(7.6)tkl,l+ρ0(1-ul,l)fk=ρ0∂2uk∂t2.
Considering that the medium is homogenous and omitting the term (ρ0ul,lfk), 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:(7.7)ρ0∂2up∂t2=Σprmn(um,nr-αmnT,r)+ρ0fp+(Γa),rapar+Γa(ap,rar+apar,r)+(Γz),rzpzr+Γz(zp,rzr+zpzr,r)+ζprsn(as,ran+asan,r)-ζprsn(as,ran+asan,r)uk,k-ζprsnuk,krasan+κpryn(zy,rzn+zyzn,r)-κpryn(zy,rzn+zyzn,r)uk,k-κprynuk,krzyzn.
Expression (7.7) gives us a field equation with the unknowns uk,Γa,Γz. The solution of this field equation under initial and boundary conditions forms the mathematical structure of a boundary value problem to consider.Becauseθ=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:(7.8)η=-1ρ0∂Σ∂T∂T∂θ=-1ρ0∂Σ∂T,(7.9)ε=1ρ0[Σ-(T0+T)∂Σ∂T].
Substituting (7.8) in expression (7.9) will give us the following expression:(7.10)ε=Σρ0+(T0+T)η.
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:(7.11)η=η0+cTT0+βklρ0uk,l+μsnρ0asan+πynρ0zyzn.
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:(7.12)ε=ε0+c(T+T22T0)+T0βklρ0uk,l+12ρ0Σklmnuk,lum,n+1ρ0[(Λsn+ζklsnuk,l)asan+(Ωyn+κklynuk,l)zyzn].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-uk,k) give us the following expression: (7.13)ρε̇=ρ0(1-um,m)ε̇=ρ0c(1+TT0)∂T∂t+T0βkl∂uk,l∂t+Σklmn∂uk,l∂tum,n+ζklsn∂uk,l∂tasan+κklyn∂uk,l∂tzyzn-ζklsn∂uk,l∂tum,masan-κklyn∂uk,l∂tum,mzyzn
The term qr,r is obtained as follows from (6.17):(7.14)qr,r=Brl,rTl+B(rl)T,lr=B(rl)T,lr.
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 uk,l and T:(7.15)ρ0c∂T∂t+(T0βkl-Γaakal-Γzzkzl)∂uk,l∂t=β(kl)T,lk+ρ0(1-ul,l)h.
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 temperatureT0 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 αpr 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), Γa, and Γz. Γa and Γz, 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=nptpr. Here, it needs to be considered that the fiber distributions ak(x,t) and zk(x,t) after deformation for inextensible fibers in terms of fiber distributions before deformation are ak=xk,KAK(X) and zk=xk,KZK(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.



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*Source: 404398-2010-12-09.xml*