Size-dependent thermoelasticity

In this paper a consistent theory is developed for size-dependent thermoelasticity in heterogeneous anisotropic solids. This theory shows that the temperature change can create not only thermal strains, but also thermal mean curvatures in the solids. This formulation is based on the consistent size-dependent continuum mechanics in which the couple-stress tensor is skew-symmetric. Here by including scale-dependent measures in the energy and entropy equations, the general expressions for force-and couple-stresses, as well as entropy density, are obtained. Next, for the linear material the constitutive relations and governing coupled size-dependent thermoelasticity equations are developed. For linear material, one can see that the thermal properties are characterized by the classical symmetric thermal expansion tensor and the new size-dependent skew-symmetric thermal flexion tensor. Thus, for the most general anisotropic case, there are nine independent thermoelastic constants. Interestingly, for isotropic and cubic materials the thermal flexion tensor vanishes, which shows there is no thermal mean curvature.


INTRODUCTION
Within the framework of the consistent continuum mechanics, couple-stresses inevitably appear along with the force-stresses to describe the interaction of constituents of the matter.Accordingly, the force-stress tensor is not symmetric.Neglecting the couple-stresses results in what is called the classical continuum mechanics, which provides a reasonable basis for analyzing the behavior of materials when the size-dependency can be ignored.However, experiments usually do not agree with the classical theory in the regions, where the gradients of strains and rotations are very high.These regions include areas near the boundary surfaces, such as holes, notches and cracks, where the size dependency is observed.This may be especially important in fracture mechanics, where considering the couple stresses not only accounts for the size dependency, it also provides new failure criteria and crack growth law.Therefore, a consistent size-dependent continuum theory is necessary to account for the length scale effect due to the microstructure of materials.The new progress in micromechanics, nanomechanics and nanotechnology requires this advanced size dependent modeling of continuum in different branches of multi-physics disciplines, such as thermoelasticity.
Latin American Journal of Solids and Structures 11 (2014)   Thermoelasticity is a multi-physics discipline, which investigates the interaction of the field of deformation with the field of temperature.It combines, on the basis of the thermodynamics of the irreversible processes, two separately physical phenomena, namely the theory of elasticity and the theory of heat conduction.Classical thermoelasticity describes the relation between temperature change and strain in solids (Biot, 1956;Nowacki, 1962).However, the classical theory cannot address the size dependency, because it does not include any length dependent parameter in its formulation.Therefore, it is necessary to develop a sizedependent thermoelasticity, which accounts for the microstructure of the material by introducing higher gradients of deformation.It is interesting to note that the classical theory of thermoelasticity predicts that for two-dimensional simply connected isotropic regions, a steady state temperature field with zero boundary tractions will not affect the in-plane stress field.This rather surprising conclusion is the result of simplicity of classical thermoelasticity.The new size-dependent thermoelasticity predicts a stress boundary layer for this case, which is more important in small scale bodies.Ahmadi and Firoozbakhsh (1975) have developed a size-dependent thermoelasticity theory by considering the first gradient of the strain tensor.This development is within the framework of the strain gradient theory (Mindlin, 1964;Mindlin and Eshel, 1968).However, the strain gradient tensor is not the fundamental measure of deformation in a consistent sizedependent continuum theory.The micropolar theory has also been extended to include thermal effects by Nowacki (1966a,b,c) and Eringen (1970).Nevertheless, microrotation, which brings extraneous degrees of freedom, is not a proper continuum mechanical concept.
Recently, Hadjesfandiari and Dargush (2011) have developed the consistent continuum mechanics for solids.In this theory, the couple-stress tensor is skew-symmetric and the body couple is not distinguishable from the body force.The work also demonstrates that the stresses are fully determinate, and the measure of deformation energetically-conjugate to couple-stress is the skew-symmetrical mean curvature tensor.This theory involves only true continuum kinematical quantities without recourse to any additional artificial degrees of freedom.An initial incomplete version of this theory was developed by Mindlin andTiersten (1962), andKoiter (1964), which used the displacement field, its corresponding true continuum rotation and the gradient of the rotation as the curvature tensor.However, this curvature tensor is not a proper measure of deformation and creates indeterminacy in the couplestress tensor.
Establishing the skew-symmetric character of the couple-stress tensor resolves the quest for the consistent size-dependent continuum mechanics, which provides a fundamental basis for the development of size-dependent material response.Therefore, this development can be extended quite naturally into many branches of continuum mechanics involving different multi-physics disciplines.Hadjesfandiari (2013) has already developed a size-dependent piezoelectricity for dielectric materials.Here we concentrate to develop the coupled sizedependent thermoelasticity for solids.
Latin American Journal of Solids and Structures 11 (2014) 1679-1708 In the following section, we provide an overview of the thermomechanical equations.This includes the equations for the kinematics, mechanics and thermodynamics of size-dependent small deformation continuum mechanics.Here by including scale-dependent measures in the energy and entropy equations, the general thermoelastic expressions for force-and couplestresses, as well as entropy density, are obtained.In Section 3, the constitutive relations for linear size-dependent heterogeneous anisotropic thermoelastic materials also are derived along with the governing equations.Section 4 provides the general theory of size-dependent thermoelasticity for linear isotropic material.This section includes the size-dependent thermoelastic waves as well.Finally, Section 5 contains a summary and some general conclusions.

GOVERNING SIZE-DEPENDENT THERMOMECHANICAL EQUATIONS
Consider a heterogeneous anisotropic elastic solid material occupying a volume V bounded by a surface A .Assume the body is undeformed and is stress-free at a uniform absolute temperature T 0 when there is no external force.This initial state is referred as the reference or natural state of the body, and the absolute temperature T 0 is called the reference tempera- ture.When the body is subjected to external forces and heat sources, it undergoes a temperature change field ϑ = T − T 0 , and an accompanying deformation specified by the displace- ment field u .These quantities may depend on position and time.Here, we assume infinites- imal deformation and temperature changes, where The parameter l S represents the smallest characteristic length in the body.By these as- sumptions, the mechanical and thermal properties of the material can be taken constants during deformation and temperature change.
The infinitesimal strain and rotation tensors are defined as respectively.Since the true tensor ω ij is skew-symmetrical, one can introduce its corre- sponding dual pseudo rotation vector as where ε ijk is the permutation tensor or Levi-Civita symbol.The consistent infinitesimal pseudo skew-symmetric mean curvature tensor defined by Hadjesfandiari and Dargush (2011) as Since this tensor is also skew-symmetrical, its corresponding dual true mean curvature vector is By some manipulation, one can obtain The external forces, heat sources and heating produce internal stresses and heat fluxes in The force-stress tensor σ ji is generally non-symmetric and can be decomposed as ) where σ ji ( ) and σ ji [ ] are the symmetric and skew-symmetric parts, respectively.Based on the new development in size-dependent continuum mechanics (Hadjesfandiari and Dargush, 2011), the pseudo couple-stress tensor µ ij is skew-symmetrical The components of the force-stress σ ij and couple-stress µ ij tensors in this consistent theory are shown in Fig. 2. Since µ ij is skew-symmetric, the moment-traction m i n ( ) given by ( 9) is tangent to the sur- face.As a result, the couple-stress tensor µ ij creates only bending moment-traction on any arbitrary surface in matter.The force-traction t i n ( ) and the consistent bending moment- traction m i n ( ) acting on an arbitrary surface with unit normal vector n i are shown in Fig. 3.The true couple-stress vector µ i dual to the pseudo-tensor µ ij is defined as Consequently, the surface moment-traction vector tangent to the surface m i n ( ) reduces to To formulate the fundamental equations, we consider an arbitrary part of this heterogeneous anisotropic elastic body occupying a volume V a enclosed by boundary surface A a .In infinitesimal deformation theory, the displacement vector field u is so small that the velocity and acceleration fields can be approximated by !u and !! u , respectively.As a result, the linear and angular equations of motion for this part of the body are written as where f i is the body force per unit mass of the body, and ρ is the mass density.Hadj- esfandiari and Dargush (2011) have shown that the body couple density is not distinguishable from body force in size-dependent couple stress continuum mechanics and its effect is simply equivalent to a system of body force and surface traction.By using Eqs.( 8) and ( 9) for tractions in the equations of motion ( 14) and ( 15), along with the divergence theorem, and noticing the arbitrariness of volume V a , we finally obtain the differential form of the equations of motion as We note that the angular equation of motion ( 17) gives the skew-symmetric part of the force-stress tensor as Latin American Journal of Solids and Structures 11 (2014) 1679-1708 Therefore, the independent linear equation of motion reduces to Now, we consider the first law of thermodynamics for this body.Heat conduction arises from the irreversible transfer of heat energy due to the temperature gradient in the body.
The heat conduction energy flow is represented by heat flux or thermal flux vector q i .The amount of heat flux energy flow q n ( ) through the surface element dA with outward directed unit normal vector n i is given by (Carslaw and Jaeger, 1959) The energy equation for the arbitrary volume V a can be written as where U is the internal energy per unit volume and Q is the quantity of heat generated in unit time and unit volume.This equation shows that the rate of change of total energy of the system in the volume V a is equivalent to the power of the external forces, moments and the heat power received by thermal conduction.By substituting from Eqs. ( 8), ( 9) and ( 20), this equation becomes Then by using the divergence theorem and equations of motion ( 16) and ( 17), and noticing the arbitrariness of volume V a , one can obtain the energy balance in differential form The second law of thermodynamics for this volume can be written as where S is the internal entropy per unit volume.By using Eq. ( 20) for the heat flux q n ( ) in Eq. ( 24), along with the divergence theorem, and noticing the arbitrariness of volume V a , we obtain the Clausius-Duhem inequality which can also be written as By combining Eqs. ( 23), and ( 26) to eliminate Q , we have It turns out that for our development we should define the Helmholtz free energy density F by using the Legendre transformation as Then by introducing Eq. ( 29) in Eq. ( 27), we obtain However, for the simple elastic body the free energy density is a function of the independent variables e ij , κ ij and T ; that is If we differentiate F with respect to time, we obtain Latin American Journal of Solids and Structures 11 (2014) 1679-1708 This can be simply written as Then by using Eq. ( 33) in Eq. ( 30), we obtain This inequality should be satisfied for all independent arbitrary rates !e ij , ! κ ij and !T .
Hence, their corresponding coefficients in Eq. ( 34) must vanish.Therefore, we find the following general constitutive relations for the symmetric part of the force-stress tensor σ ji ( ) , couple-stress tensor µ ji and entropy S : As a result, the Clausius-Duhem inequality Eq. ( 34) reduces to which means the transfer of heat in the opposite direction of temperature gradient.This is the simple result of the second law of thermodynamics.
The relations ( 28) and ( 31) show that As a result, we have the rate equation By using Eq. ( 40) in the energy balance Eq. ( 23), we obtain the equation of entropy balance in differential form Let us return to the constitutive equations (35-37).If we further agree to construct the functional F , such that we can write in place of Eqs. ( 35) and ( 36) It should be noticed that if we consider the Helmholtz free energy in the form Latin American Journal of Solids and Structures 11 (2014) 1679-1708 for the couple-stress vector.Then by using this in Eq. ( 18) we obtain the skew-symmetric part of the force-stress tensor σ ji Therefore, for the total force-stresses, we generally have Now, we consider the entropy density function S .The relation (37) shows that By differentiating with respect to time, we have Using Eq. ( 50) in Eq. ( 40) for !U , we obtain This expression shows that the quantity T ∂S ∂T is the measure of internal energy change rate in unit volume of the body during a change in temperature at constant strain and curvature.Consequently, we realize that this quantity is the specific heat at constant strain and curvature and denote it by c v , that is Latin American Journal of Solids and Structures 11 (2014) 1679-1708 Therefore, the relation for energy and entropy rate densities become The relation (54) will be used in our development in the next section.
What has been presented so far is the size-dependent continuum mechanics theory of infinitesimal thermoelastic materials.The fundamental thermoelastic equations in the volume with the constitutive relations We notice that the these governing equations are subject to some prescribed compatible boundary conditions on the boundary A .From a mathematical point of view, we can speci- fy either the displacement vector u i or the force-traction vector t i n ( ) , the tangential compo- nent of the rotation vector ω i or the tangent moment-traction vector m i n ( ) , and the absolute temperature T or the normal heat flux q n ( ) .In a time dependent problem, we are also re- quired to specify the initial conditions at an initial time, say t = 0 , everywhere in the volume V to have a consistent initial boundary value problem.These include initial deformation u i x,0 ( ), velocity !u i x,0 ( ) and absolute temperature T x,0 ( ).
It is obvious that the reference or natural state is different from the initial state.The body in its natural state is without any stresses, strains and curvatures, that is

LINEAR THERMOELASTICITY THEORY
For linear material, we consider the classical Duhamel's generalization of Fourier heat conduction law (Duhamel, 1832) as where k ij is the thermal conductivity tensor.Therefore, the Clausius-Duhem inequality (58) reduces to Based on the non-equilibrium statistical mechanics, Onsager (1931) has shown that the tensor k ij is symmetric Consequently, Eq. ( 64) proves that the non-singular symmetric conductivity tensor k ij is positive definite.By introducing Eq. ( 63) in the entropy balance Eq. ( 57), we obtain for heterogeneous linear materials.
It should be noticed that the classical theory of generalized Fourier heat conduction (63) predicts infinite speed of heat transport, which contradicts the physical facts.As a result, non-classical theories have been developed to incorporate a flux rate term into the Fourier's law of heat conduction and formulated a generalized theory admitting finite speed for thermal signals (e.g., Lord and Shulman, 1967).These theories consider heat propagation as a wave phenomenon rather than a diffusion phenomenon.However, in this paper we have not included this formulation in our development.In a future work, the formulation to incorporate a flux rate term into the Fourier's law of heat conduction in the framework of size-dependent thermoeleasticity will be developed to admit finite speed for thermal signals.Now let us expand the Helmholtz free energy F = F e ij ,κ ij ,T

(
) into the Taylor expansion in the neighborhood of the reference state, where ( e ij = 0 , κ ij = 0 , T = T 0 ), that is ∂e ij e ij + ∂F 0,0,T 0 ( ) The constant F 0,0,T 0 ( ) is the free energy corresponding to the reference state, which can be taken arbitrarily equal to zero.Since there is no initial stresses in the reference state, the quantities ∂F 0,0,T 0 ( ) and ∂F 0,0,T 0 ( ) ∂κ ij also vanish.By assuming the entropy to be zero at the reference state, we take ∂F 0,0,T 0 ( ) ∂T equal zero.Therefore, for linear elastic sizedependent thermoelasicity the free energy F can be considered as the homogeneous quadrat- ic where the coefficients are defined as Latin American Journal of Solids and Structures 11 (2014) 1679-1708 Obviously, these tensors have the symmetry and skew-symmetry relations Since the material is heterogeneous, the mechanical and thermal tensor properties can vary from point to point in space.However, we have assumed that these properties are constants at each point during the deformation and temperature change.
Consequently, by using the Helmholtz free energy density Eq. ( 68) in the general relations (35-37), we obtain the following constitutive relations The tensors a ij and b ij represent the thermoelastic properties of the material.The sym- metric true tensor a ij is the coupling term between the temperature change ϑ and the sym- metric part of force-stress tensor σ ji ( ) in Eq. ( 77), and the skew-symmetric pseudo tensor b ij is the size-dependent coupling term between the temperature and the couple-stress tensor µ ji in Eq. ( 78).We will shortly explore further characters of these thermoelastic tensors and the quantity m .Consequently, for the most general case, the number of distinct components for A ijkl , B ijkl , C ijkl , a ij , b ij and are 21, 6, 18, 6, and 3, respectively.Therefore, the most gen- eral linear thermoelastic anisotropic material, including m , is described by 55 independent constitutive coefficients.
The equations ( 77) and ( 78) can be considered as the generalized Duhamel-Neumann relations for a heterogeneous anisotropic size-dependent body.Interestingly, by using entropy density Eq. ( 79) and free energy Eq. ( 68) in Eq. ( 28), we obtain for density function U .This expression is a positive definite quadratic form without explicit thermomechanical coupling.This shows that the tensors A ijkl , B ijkl and C ijkl containing the elastic constitutive coefficients, and the parameter m are positive definite.The tensor A ijkl is actually equivalent to its corresponding tensor in Cauchy elasticity.
It is interesting to note that the Helmholtz free energy density F can also be written in terms of the mean curvature vector as where and the symmetry relations hold.It is seen that the tensor B ij is positive definite.Note that the number of distinct components for true tensors B ij and C ijk , and true vector b i are 6, 18 and 3, respectively.By using the Helmholtz free energy density Eq. ( 81) in the general relations ( 35), ( 46) and (37), we obtain the following constitutive relations The skew-symmetric part of the force-stress tensor for a homogeneous material is found as We notice the appearance of the third order gradient of deformation κ i, j in Eq. ( 90) for the skew-symmetric part of force-stress tensor σ ji is the result of angular equation of motion (18), which is not a constitutive equation.Consequently, the constitutive relation for the total force-stress tensor in a homogeneous material is The physical character of the symmetric tensor a ij and skew-symmetric tensor b ij can be investigated by writing Eqs. ( 77) and ( 78) as where Latin American Journal of Solids and Structures 11 (2014) 1679-1708 The true tensor ij α and pseudo tensor ij β are the new thermoelastic tensors with the ob- vious symmetry and skew-symmetry relations It is seen that the quantities α ij ϑ and β ij ϑ in Eqs. ( 92) and ( 93) are the strain and mean curvature tensors created by the temperature change ϑ = T − T 0 , respectively.Hence, they represent the thermal strain and thermal mean curvature tensors in an unrestrained body While the symmetric tensor α ij is the classical coefficient of thermal expansion tensor, the new size-dependent skew-symmetric tensor β ij may be called the coefficient of thermal flex- ion tensor.Although, there is generally no restriction on the tensors α ij and β ij , in most cases α ij is positive definite.It is obvious that for the most general anisotropic case, the number of distinct components for α ij and β ij are 6, and 3, respectively.Consequently, the constitutive relations ( 92) and ( 93) can be written as The tensors e ij − e ij Th and κ ij −κ ij Th represent the mechanical strain and mean curvature tensors, respectively.Similarly, Eqs. ( 87) and ( 88) can also be written as Latin American Journal of Solids and Structures 11 (2014) 1679-1708 where These set of 9 independent equations ( 104) and ( 105) give 9 independent coefficients α ij and β i .The dual relation between thermal flexion true vector β i and thermal flexion pseudo tensor β ij is given by Hence, the thermal mean curvature vector in the unrestrained body is defined as Therefore It should be noticed that the effect of thermal flexion moduli always appears along with couple-stresses.This means that if we neglect the couple stresses ( B ij = 0 , C ijk = 0 ), all other size-dependent effects disappear as well ( β ij = 0 ).Now, we try to define the positive quantity m in the Helmholtz free energy Eq. ( 81).For this purpose, let us return to the Taylor expansion Eq. ( 67) for Helmholtz free energy and neglect all powers higher than second in e ij and κ ij , but keep all terms depending on tem- perature and collect them in one function C = C ϑ ( ).Consequently, for Helmholtz free en- ergy we have where C = C ϑ ( ) is a function depending on temperature term only with C 0 ( ) = 0 .By using this form of F in the entropy density function Eq. ( 37), we obtain Consequently, Eq. ( 52) gives which can be written as By integrating this relation, and noticing that at the natural state, F = 0 and S = 0 , we obtain ∂C ϑ ( ) For the linear theory c v is constant.Thus, this integral becomes Latin American Journal of Solids and Structures 11 (2014) 1679-1708 By using this in Eq. ( 111) for the entropy density, we obtain However, by the restriction << , for small temperature change, this rela- tion can be written as By comparing Eqs. ( 117) and ( 89), we find Therefore, when and When the constitutive relations force-stress tensor (91), entropy density (117) and thermal flux vector (63) are written in terms of displacements and temperature for homogeneous material, we obtain Latin American Journal of Solids and Structures 11 (2014) 1679-1708 By carrying these forms into the linear motion Eq. ( 16) and the entropy balance Eq. ( 41), one obtains the governing equations for size-dependent homogeneous thermoelasticity as follows: These are the coupled size-dependent thermoelasticity equations.It should be noticed that the expression a ij ! e ij + b i !κ i ( ) T 0 is the coupling term in the entropy balance Eq. ( 125), which involves the thermal and mechanical terms.If this term can be neglected, we obtain the uncoupled heat conduction equation which is no longer size-dependent.
As we mentioned, the prescribed boundary conditions on the surface of the body can be any compatible combination of u i , ω i , T and t i n ( ) , m i n ( ) , q n ( )

ISOTROPIC LINEAR THERMOELASTIC MATERIAL
For a heterogeneous isotropic material, the symmetry relations require where the quantities can vary in the domain from point to point.The moduli λ and µ have the same meaning as the Lamé constants for an isotropic material in Cauchy elasticity.As we know, these two constants are related by Latin American Journal of Solids and Structures 11 (2014) 1679-1708 where ν is the Poisson ratio, and E is the modulus of elasticity or Young's modulus.It is seen that where K is the bulk modulus of elasticity K = E 3 1− 2ν ( ) . We also notice that the material constant η accounts for the couple-stresses in the isotropic material.
It is seen that for the isotropic material the coefficient of thermal flexion tensor β ij van- ishes and there is only one coefficient of the thermal expansion α .Therefore, the relations (98) and ( 99) show This means that no thermal shear strain and thermal mean curvature are created by the temperature change ϑ .As a result, for the tensor a ij , we obtain As a result, the Helmholtz free energy and internal energy densities become The following restrictions are necessary for positive definite energy density U The ratio specifies the characteristic material length l , which accounts for size-dependency in the small deformation couple stress elasticity theory under consideration here.Consequently, the constitutive relations for the symmetric part of the force-stress tensor, the couple-stress vector, and the heat flux vector can be written as Interestingly, Eq. ( 149) can be written as or Latin American Journal of Solids and Structures 11 (2014) 1679-1708 For the skew-symmetric part of the force-stress tensor in a homogeneous material, we have which can also be written as Therefore, for the total force-stress tensor, we have For the governing equations, we have These are the coupled size-dependent thermoelasticity equations for a homogeneous isotropic material.As we noticed, the expression 3λ + 2µ ( ) αT 0 !e kk is the coupling term in the energy Eq. ( 158), which involves the thermal and mechanical terms.If this term can be neglected, we obtain the uncoupled conduction equation The governing equations ( 157) and ( 158) can also be written in vectorial form as Based on the Helmholtz decomposition theorem, we can introduce the scalar potential φ and vector potential A such that where the first and second terms in the right hand side of Eq. ( 162) represent the dilatational (irrotational) and equivoluminal (shear) components of deformation, respectively.Because of gauge freedom, we can arbitrarily take the gauge condition As a result, the equation of motion (160) can be decomposed in two equations for the potentials φ and A as where we have assumed the decomposition When the heat sources and body forces are absent, we obtain the governing equations where c 1 and c 2 are the velocities of dilatational and shear waves in classical elastic waves, respectively, defined as It is seen that the dilatational (irrotational) wave equation ( 167) is coupled to the temperature equation ( 169).This means that the propagation of dilatational wave is accompanied by a generation of heat.Therefore, the dilatational wave is damped and its energy is partly converted to heat.However, the shear wave equation ( 168) is uncoupled from the tempera-Latin American Journal of Solids and Structures 11 (2014) 1679-1708 ture.This means that this wave is not damped, but undergoes dispersion, as a result of sizedependency.Furthermore, the shear wave does not generate any temperature change.

CONCLUSIONS
The consistent size-dependent continuum mechanics is a practical theory, which enables us to develop many different multi-physics formulations that may govern the behavior of solid continua at the smallest scales.The size-dependent formulations have the priority because of their importance in nanomechanics and nanotechnology.Here, we have developed the sizedependent thermoelascticity, which shows the possible thermal mean curvature in anisotropic material.The most general anisotropic linear elastic material is described by 55 independent constitutive coefficients.This includes three thermal flexion coefficients relating thermal mean curvatures to temperature change.
The new size dependent thermoelascticity clearly shows that the thermal mean curvature disappears in isotropic couple stress materials, where the two Lamé parameters, one length scale, and one thermal expansion completely characterize the behavior.
The present theory shows that couple-stresses are necessary for the development of any thermomechanical size-dependent effect.Additional aspects of linear thermoelasticity, including reciprocal theorem, fundamental solutions and computational mechanics formulations, will be addressed in forthcoming work.Beyond this, the present theory should be useful for the development of other coupled size-dependent multi-physics formulations, such as piezothermo-elasticity, which are also important for analysis at small scales.

Figure 1
Figure 1 Components of force-and couple-stress tensors in the original inconsistent theory.

Figure 2
Figure 2 Components of force-and couple-stress tensors in the present consistent theory.

Figure
Figure 3 Force-traction t n ( ) and the consistent bending moment-traction m n ( ) .