Second gradient thermoelasticity with microtemperatures

1.   Introduction

In recent years, Green-Naghdi thermodynamics ^[1,2,3] has been used to establish some theories of thermoelasticity that take into account the second-order temperature gradient ^[4,5,6]. On the other hand, the balance laws of the continua with microstructure ^[7,8,9,10] led to a theory of thermodynamics of elastic materials where the inner structure has microelements with microtemperatures. At the begin of this paper, we use the theory of non-simple elastic solids ^[11,12,13] and results from thermodynamics of multipolar continua ^[14] to obtain a second gradient theory of thermoelasticity with microtemperatures. An introduction of the concepts of thermal displacement and thermal microdisplacements as well as the theory of multipolar continua allows us to derive the local form of energy balance and constitutive equations. In ^[3], the authors established a theory of thermoelasticity characterized by constitutive equations that depend on the first gradient of the displacement vector, on temperature, and on the first gradient of thermal displacement. In the present work, we have consider the following new independent constitutive variables: the second gradient of thermal displacement, the second gradient of thermal microdisplacements, as well as microtemperatures. To simplify the writing, we limit our attention only to the introduction of the second-order spatial derivatives of the thermal variables.

We express the field equations of the linear case in terms of components of the displacement vector, thermal displacement, and thermal microdisplacements, and obtain a fourth-order system of equations. The boundary-initial-value problems are also formulated. The semigroup theory of linear operators allows us to prove that the problem of the second gradient thermoelasticity with microtemperatures is well-posed. For the equations of isotropic rigids, we present a natural extension of the Cauchy-Kovalevski-Somigliana solution of the isothermal theory. In the case of stationary vibrations, we establish the fundamental solutions of the basic equations. Anti-plane shear deformations are also considered and uniqueness and instability results are obtained.

The relevance in introducing temperature gradient effects in thermomechanics can be recalled from ^[15].

2.   Balance equations

In this section, we propose a second gradient theory of solids with microtemperatures by using the basic laws of mechanics of materials with microstructure and Green-Naghdi thermomechanics. Throughout this paper, the motion of the body is referred to the reference configuration $B$, occupied by the body at time $t_0$, and to a fixed system of rectangular Cartesian coordinates $Ox_j$, (j = 1, .., 3). Latin subscripts range over the integers (1, 2, 3), and Greek subscripts range over the integers (1, 2). Cartesian tensor notation is used throughout. In what follows, $x_j$ are reference coordinates, $y_j$ are spatial coordinates, a superposed dot denotes material time differentiation, and $f_{, j}$ denotes partial differentiation of $f$ with respect to $x_j$. We denote by $\partial B$ the boundary of $B$. Following ^[1,2], we get the following local balance of entropy:

By using the theory of continua with microstructure ^[9], we can obtain the balance of the first moment of entropy in the form:

In the relations (2.1) and (2.2), we have used the following notations: $\rho$ is the reference mass density; $\eta$ is the entropy per unit mass of the body; $S_i$ is the entropy flux vector; $s$ is the external rate of supply of entropy per unit mass; $\xi$ is the internal rate of production of entropy per unit mass; $\eta_j$ is the first entropy moment vector; $\Lambda_{ij}$ is the first entropy flux moment tensor; $H_i$ is the mean entropy flux vector; $Q_i$ is the first moment of the external rate of supply of entropy, and $\xi_j$ is the first moment of the internal rate of production of entropy. The entropy flux $\Sigma$ and the first entropy moment flux vector $\sigma_j$ at regular points of $\partial B$ are given by ^[1,2,8],

where $n_i$ is the outward unit normal of $\partial B$. Let $\theta$ be the absolute temperature. We denote by ${\bf{x}}$ the center of mass of a generic microelement $V$. We assume that for ${\bf{x}}' \in V$, we have

We call the functions $T_i$ microtemperatures. As in ^[1], we consider the thermal displacement $\alpha$ and the thermal microdisplacements $\beta_j$ by

We now consider a domain $\mathcal {P}$ at time $t$, bounded by a surface $\partial \mathcal {P}$, and let $P$ be the respective domain in reference configuration, with the boundary $\partial P$. In view of ^[1,9,11,14], we propose an energy balance in the form:

for every region $P$ of $B$ and every time. Here $u_i$ is the displacement vector, $e$ is the internal energy per unit mass $f_i$ is the body force per unit mass, $t_i$ the stress vector associated with the surface $\partial \mathcal {P}$ but measured per unit area of $\partial P$, and $G_i$ and $\Pi_{ij}$ are the monopolar and dipolar entropy flux per unit area, respectively. We impose that the dipolar body force and the spin inertia per unit mass are not present (see ^[14]). From (2.6), we can derive the balance of linear momentum so that, by the well-known method, we obtain

and

where $t_{ij}$ is the stress tensor. After the use of the divergence theorem and the equalities (2.1)–(2.3), (2.7), and (2.8), the relation (2.6) can be written in the form:

With an argument similar to that used to derive the relation (2.7), from (2.9), we obtain

where $G_{kj}$ and $\Pi_{kji}$ are tensors associated to the surface loads $G_i$ and $\Pi_{ji}$, respectively. With the help of (2.10), we obtain the local expression form of the energy balance

where the following notation

is used.

Following ^[14], we assume a motion of the body that is different from the given motion by a superposed uniform rigid body angular velocity, and that $\rho, e, t_{ij}, \eta, \theta, \eta_j, T_j, S_j, \Lambda_{kj}, H_j, F_j, \Gamma_{kj}, \xi$, and $\xi_j$ do not change by such motion. The equality (2.11) implies that

If we consider the Helmholtz free energy $\psi$ by

and we see that the energy balance may be written in the form:

where we have introduced the strain tensor

3.   Constitutive equations

From now on, we define the constitutive equations for $\psi, t_{ij}, \eta, \eta_j, S_j, H_j, G_{kj},$ $F_j, \Gamma_{kj}, \Pi_{kj}, \xi$, and $\xi_j$, and we suppose that these are functions of the set $V = (e_{ij}, \theta, T_j, \alpha_{, j}, \beta_{k, j}, \alpha_{, ij}, \beta_{k, ij})$. To simplify the writting, we omit the explicit dependence of $x_k$ and then the material should be homogeneous and assume that there is no kinematical constraint. In the theory established in ^[3], the constitutive variables are $e_{ij}, \theta$, and $\alpha_{, j}$. If we introduce the notation $A = \rho \psi$, then Eq (2.15) becomes

From (3.1), we find that ^[3]

and

We introduce the notations

where $t_0$ is a reference time and $T_0$, $T_j^0, \alpha_0$, and $\beta_j^0$ are given constants. As in ^[3], we consider new thermal variables

From (3.4) and (3.5), we get

From now on, we restrict our attention to the linear theory where the functions $u_i$, $T$, and $\theta_j$ can be written as

where $\epsilon$ is a parameter small enough for squares and higher powers to be neglected, and $u_i'$, $T'$, and $\theta_j'$ are independent of $\epsilon$. As usual, we assume that $A$ is a quadratic form of the variables $e_{ij}, T, \theta_j, \alpha_{, j}, \beta_{k, j}, \alpha_{, ij},$ and $\beta_{k, ij}$ and that $H_i, \xi$, and $\xi_j$ are linear functions of the same variables. We consider the case of a material with a center of symmetry. Thus, we have

The following symmetries are satisfied:

It follows from (3.2), (3.7), and (3.8) that

For isotropic materials, the number of constitutive coefficients is drastically reduced (see ^[12]). From (3.3), we see that the response function $\xi$ vanishes when the microtemperatures $T_j$ vanish. In the linear case, the function $\xi$ that satisfies this requirement must be $\xi = c_jT_j$, where $c_j$ are constants. Since the body has a center of symmetry, we get $\xi = 0$. Thus, from (2.12), (3.3), and (3.9), we find that

If we use these results, then Eqs (2.1) and (2.2) take the form

The equations of the linear theory consist of the equations of motion (2.8), the energy equations (3.11), the constitutive equations (3.9), and the geometrical equations (2.16). In view of (2.12), we can write the Eqs (3.11) in the form

Equations (2.8) and (3.11) can be expressed in terms of the unknowns $u_j$, $\chi$, and $\varphi_i$. Thus, we obtain the equations

where

To the system (3.13), we have to adjoin boundary and initial conditions.

4.   Boundary-initial-value problems

Now, we study the boundary conditions and formulate the basic boundary-initial-value problems. We assume that the boundary of $B$ consists of the union of a finite number of smooth surfaces, smooth curves (edges), and points (corners). Let $C$ be the union of the edges. As in ^[11,12], to obtain the form of the boundary conditions, we must study the surface integral in (2.5). By using (2.3), (2.7), and (2.10), we find that

We will use the notations

where $\delta_{ij}$ is the Kronecker delta. Then we obtain

As in ^[11,12], from (4.1) and (4.3), we get

where

Here, $s_k$ are the components of the unit vector tangent to $C$, $< f >$ denotes the difference of the limits of $f$ from both sides of $C$, and $\epsilon_{jrk}$ is the alternating symbol. The first boundary-initial-value problem is characterized by the boundary conditions

where $u_i^*, \chi^*, \varphi_i^*, \zeta^*$, and $\gamma^*_i$ are prescribed functions.

For the second boundary-initial-value problem, the boundary conditions are ^[11]

where $t_i^*, \Phi_1^*, \Phi_2^*, \Psi_i^*, W_i^*, Y^*$, and $\Omega_i^*$ are given. The initial conditions are

where $u_i^0, v_i^0, \chi^0, \chi^1, \varphi_i^0$, and $\nu_i^0$ are given.

5.   An existence result

Now, we provide an existence and uniqueness result for the problem determined by the system of Eqs (3.13), with the initial condition (4.8) and the homogeneous version of the boundary conditions (4.6). We will use the theory of contractive linear semigroups ^[16].

We assume once and for all that:

(ⅰ) The mass density $\rho$ and the thermal capacity $a$ are strictly positive.

(ⅱ) The matrix $B_{ij}$ is positive definite.

(ⅲ) The quadratic form

is positive definite, i.e., there exists a positive constant $C$ such that:

Let us to propose the problem as an abstract problem in a suitable Hilbert space. We will work on the space

where $W_0^{1, 2}, W_0^{2, 2}$, and $L^2$ are the usual Sobolev spaces, ${\bf{W}}_0^{2, 2} = [W_0^{2, 2}]^3$ and ${\bf{L}}^2 = [L^2]^3$. The elements in this space can be denoted by $\mathcal {U} = ({\bf{u}}, {\bf{v}}, \chi, \theta, {{ \mathit{\boldsymbol{\varphi}}}}, {{ \mathit{\boldsymbol{\phi}}}})$.

We consider the scalar product associated to the norm

Now, we want to see our problem as a Cauchy problem in ${\mathcal {H}}$. We define the operator

where

and

where $F_{ij}B_{jk} = \delta_{ik}$.

We note that our problem can be written as

where

The domain of the operator $\mathcal {A}$ is the subspace of elements of our Hilbert space such that

and

This domain is a dense subset of our space.

After an easy but laborious calculation, we can see that

for every element ${\mathcal{U}}$ at the domain of the operator.

The next step in our approach is to show that zero belongs to the resolvent of the operator. Let $\mathcal {U}^*$ be in $\mathcal {H}$. We must prove that the equation

admits a solution. That is

We substitute the first three equations into the others to find that

If we denote

then our system can be written as

To solve this last system, we note that $(\alpha_{i1}, \alpha_2, \alpha_{i3})\in [W^{-1, 2}]^3\times[W^{-2, 2}]^4$ and that the form

defines a form in $[W^{1, 2}_0]^3\times[W^{2, 2}_0]^4$ which is bounded and coercive. In view of the Lax-Migram lemma ^[17], we can guarantee the existence of solutions to our system as well as the existence of a positive constant $K$ (independent of the point) such that

In view of the previous arguments, we can conclude that ${\mathcal {A}}$ generates a contractive semigroup. Therefore, we obtained the following result:

Theorem 1. Let us assume that $f_i, s$, and $Q_i$ are functions continuous at the domain of the operator ${\mathcal {A}}$ and of class $C^{1}$ in $L^2$. Then, there exists a unique solution to the problem (5.3) that is continuous in the domain of the operator and is of class $C^{1}$ in the Hilbert space ${\mathcal {H}}$.

It is well-known that we also obtain

which establishes the continuous dependence of solutions on initial and supply terms. Thus, we can say that under the conditions (ⅰ), (ⅱ), and (ⅲ), the problem of the second gradient thermoelasticity with microtemperatures is well-posed.

6.   Basic equations for isotropic solids

Now, we state the constitutive equations for isotropic materials and express the basic equations in terms of the unknown functions $u_i$, $\chi$, and $\varphi_i$. For centrosymmetric and isotropic materials, the number of independent parameters is greatly reduced. Using the general forms of isotropic tensors ^[12] and taking into account the symmetries (3.8), we obtain

where $\lambda, \mu, \beta, C_{\alpha}, d_{\alpha}, a, \gamma, \nu, b, \zeta, k, \mu_{\alpha}, E_j, \gamma_r$, and $\kappa_{\alpha}, (\alpha = 1, 2; r = 1, 2, .., 5)$, are material coefficients. By using (3.6) and the relations

we obtain the following constitutive equations

For isotropic solids, the equations for the unknown functions $u_j$, $\chi$, and $\varphi_i$ are established by substituting the functions in (6.2) into Eqs (2.8) and (3.12). Thus we obtain the equations:

where

The problem of heat flow in rigid materials with microtemperatures is characterized by the following equations:

7.   A representation theorem

In this section, we establish a representation theorem for solutions of the system (6.5). We will use the notations

Equations (6.5) can be written in the form

A counterpart of the Cauchy-Kovalevski-Somigliana solution of the classical elastodynamics is given by the following theorem.

Theorem 2. Let

where the functions $\Phi$ and $V_j$ satisfy

Then $\chi$ and $\varphi_j$ satisfy the Eqs (7.2).

Proof. We note that

After substitution of $\chi$ and $\varphi_j$ given by (7.3) in (7.2), then in view of (7.5) and (7.4), we see

With the help of (7.4), we obtain the desired result.

In continuum mechanics, such solution representations have been used to establish the fundamental solutions of the field equations. In classical thermoelasticity, these solutions led to the introduction of the single layer potential and the double layer potential. The method of potentials has been used to reduce boundary value problems of steady vibration theory to singular integral equations and to prove existence theorems (see, e.g., ^[18]).

8.   Fundamental solutions

In this section, we use the representation (7.1) to establish the fundamental solutions in the case of steady vibrations. We assume that

where $\omega$ is a given frequency, ${\bf{x}} = (x_1, x_2, x_3), i = (-1)^{1/2}$, and ${\mbox {Re}}[f]$ is the real part of the function $f$. Let us introduce the notations

We obtain a differential system for the amplitudes $\chi'$ and $\varphi_j'$. To simplify the notation, we omit the primes so that these equations can be written as

The next theorem is a consequence of previous results.

Theorem 3. Let

where the functions $F$ and $U_j$ satisfy

Then $\chi$ and $\varphi_j$ satisfy the Eqs (8.3).

If we introduce the notations

then from (8.2), we get

It is easy to see that if $k_j^2 (j = 1, ..., 4)$ are the roots of the equation

then we have

We denote by $k_5^2$ and $k_6^2$ the roots of the following equation:

From (8.6) and (8.10), we obtain

Similarly we find

where $k_7^2$ and $k_8^2$ are the roots of the equation

Let us assume that

where $f$ is a given function. From (8.5), we see that in this case we can take $F = e$ and $U_j = 0$, where $e$ satisfies the equation

We denote by $k_j \, \, (j = 1, 2, .., 8)$ the roots with positive real parts and assume that they are different. If the functions $g_j$ satisfy the equations:

then the function $e$ can expressed as

where

We now assume that $s = \delta({\bf{x}}- {\bf{y}})$, where $\delta(.)$ is the Dirac delta and ${\bf{y}}$ is a fixed point. In this case, the solution of the Eq (8.15) is given by

where $r = \vert {\bf{x}}- {\bf{y}}\vert$. If we take in (8.4), $U_j = 0$, and $F = e_0$, then we obtain the solution

Let us assume that

where $j$ is fixed. We see that in this case, Eqs (8.5) are satisfied if we take $F = 0$ and $U_i = e_0 \delta_{ij}$. From (8.4), we find the following solution:

The functions $\chi^{(k)}$ and $\varphi^{(k)}_i \, \, (k = 1, ..., 4)$, given by (8.20) and (8.22), represent the fundamental solutions of the system of equations describing steady vibrations.

9.   Anti-plane shear deformations

In this last section, we use the previous results to study the problem of anti-plane shear deformations. We assume that the domain $B$ from here on refers to a right cylinder of length $h$ with the cross-section ${\mathcal {D}}$. We select the coordinate frame in such a way that the $x_3$-axis is parallel to the generator of the cylinder. Body loads are assumed to be of the form

where $I = (0, T^*)$. We look for a solution in the form

We can observe that these functions satisfy our system in the case that the following system

holds. Throughout this section, the Laplacian operator is considered in the two-dimensional case. We can see that the first two equations are coupled to each other, but both are decoupled from the temperature equation. First of all, we will concentrate our attention on the first two equations. We impose the boundary conditions

as well as the initial conditions

For this problem, the energy satisfies:

whenever we assume $f = Q = 0$, because

where the last equality follows after the use of the divergence theorem and boundary conditions.

We do not assume any condition on the coefficients $\mu, C_2, E_2$, and $b_1$, but we are going to obtain a couple of qualitative results for our system. Nevertheless we will need to suppose that $\rho$ and $b$ are two positive real numbers. To do this, it will be convenient to work with a function that will allow us to conclude the desired results. We define this function in the form:

where $\omega^*$ and $t_0$ are two non-negative real numbers to be selected. We have

and

We can notice that

Therefore

A simple use of Holder's inequality allows us to conclude

If we put homogeneous initial conditions, we have that $E(0) = 0$ and if we take $\omega^* = 0$, we can conclude that

This inequality allows us to establish (see ^[19], p. 19) that

for all $t$ between $0$ and $T^*$. Thus, in the case where we impose null initial data, we obtain that ${\mathcal {H}}(t) = 0$ for all $t$ in the interval and, consequently we obtain the null solution. This allows us to conclude the uniqueness of the solutions.

If we now go back to the general case and suppose that the initial energy is negative, we can take $\omega^* = -2E(0)$ and again conclude the previous inequality. We can also get (see ^[19], p. 20)

We note that we can always select $t_0$ large enough to guarantee that $\dot {{\mathcal {H}}}(0) > 0$. When $E(0) = 0$ and $\dot {{\mathcal {H}}}(0) > 0$, we also conclude the growth estimator.

Theorem 4. Let us to suppose that $\rho$ and $b$ are positive. Then:

(ⅰ) The initial-boundary-value problem for the anti-plane shear deformations has a unique solution.

(ⅱ) When $E(0) < 0$ or ($E(0) = 0, \, \, \dot {{\mathcal {H}}}(0) > 0$), then the solution is exponentially unstable.

A similar argument could prove the uniqueness and instability of the solutions for the temperature equation in the case that we only assume that $a$ is strictly positive.

10.   Conclusions

The results obtained in this paper can be summarized as follows:

(a) We present a linear theory of thermoelasticity with microtemperatures where the second thermal displacement gradient and the second gradient of microtemperatures are included in the classical set of independent constitutive variables.

(b) We express the field equations of the linear theory in terms of components of the displacement vector, thermal displacement, and thermal microdisplacement, s and obtain a fourth-order system of equations. The boundary-initial-value problems are also formulated.

(c) The semigroup theory of linear operators is used to prove that the problem of the second gradient thermoelasticity with microtemperatures is well-posed.

(d) We establish a counterpart of the Cauchy-Kovalevski-Somigliana solution of the isothermal theory.

(e) In the case of stationary vibrations, we establish the fundamental solutions of the field equations.

(f) Uniqueness and instability of the solutions are obtained in the case of anti-plane shear deformations.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors thank the referees for their helpful suggestions. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflict of interest

Ramón Quintanilla is an editorial board member for Electronic Research Archive and was not involved in the editorial review or the decision to publish this article. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.