Second gradient thermoelasticity with microtemperatures

Electronic Research Archive https://www.aimspress.com/journal/era ERA, 33(2): 537–555. DOI: 10.3934/era.2025025 Received: 19 November 2024 Revised: 03 January 2025 Accepted: 08 January 2025 Published: 07 February 2025 Research article Second gradient thermoelasticity with microtemperatures Dorin Ies¸an1and Ram´ on Quintanilla2,* 1Al.I. Cuza University and Octav Mayer Institute of the Mathematics of Romanian Academy, Bd. Carol I, nr. 11, 700506 Iasi, Romania 2Departamento de Matem´ aticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222 Terrassa, Barcelona, Spain *Correspondence: Email: [email protected]; Tel: +34-93-7398162. Abstract: This research was concerned with 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. The master balance laws of micromorphic continua, the theory of the strain gradient of elasticity, and Green-Naghdi thermomechanics were used to derive a second gradient theory. The semigroup theory of linear operators allowed us to prove that the problem of the second gradient thermoelasticity with microtemperatures is well-posed. For the equations of isotropic rigids, we presented a natural extension of the CauchyKovalevski-Somigliana solution of isothermal theory. In the case of stationary vibrations, the fundamental solutions of the basic equations were obtained. Uniqueness and instability of the solutions were obtained in the case of antiplane shear deformations. Keywords: elastic solids with microtemperatures; solids with microstructure; second gradient theory; constitutive equations; well-posed problems 1. Introduction In recent years, Green-Naghdi thermodynamics [1–3] has been used to establish some theories of thermoelasticity that take into account the second-order temperature gradient [4–6]. On the other hand, the balance laws of the continua with microstructure [7–10] 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 [11–13] 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 538 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 CauchyKovalevski-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 t0, and to a fixed system of rectangular Cartesian coordinates Oxj, (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, xjare reference coordinates, yj are spatial coordinates, a superposed dot denotes material time differentiation, and f,jdenotes partial differentiation of fwith respect to xj. We denote by ∂Bthe boundary of B. Following [1, 2], we get the following local balance of entropy: ρ˙η=Si,i+ρ(s+ξ).(2.1) By using the theory of continua with microstructure [9], we can obtain the balance of the first moment of entropy in the form: ρ˙ηj= Λki,k+Si−Hi+ρ(Qi+ξi),(2.2) In the relations (2.1) and (2.2), we have used the following notations: ρis the reference mass density; ηis the entropy per unit mass of the body; Siis the entropy flux vector; sis the external rate of supply of entropy per unit mass; ξis the internal rate of production of entropy per unit mass; ηjis the first entropy moment vector; Λi j is the first entropy flux moment tensor; Hiis the mean entropy flux vector; Qiis the first moment of the external rate of supply of entropy, and ξjis the first moment of the internal rate of production of entropy. The entropy flux Σand the first entropy moment flux vector σjat regular points of ∂Bare given by [1,2,8], Σ = Sjnj, σk= Λjknj,(2.3) where niis the outward unit normal of ∂B. Let θbe the absolute temperature. We denote by xthe center of mass of a generic microelement V. We assume that for x′∈V, we have θ(x′,t)=θ(x,t)+Ti(x,t)(x′ i−xi).(2.4) Electronic Research Archive Volume 33, Issue 2, 537–555. 539 We call the functions Timicrotemperatures. As in [1], we consider the thermal displacement αand the thermal microdisplacements βjby ˙α=θ, ˙ βj=Tj.(2.5) We now consider a domain Pat time t, bounded by a surface ∂P, and let Pbe the respective domain in reference configuration, with the boundary ∂P. In view of [1,9, 11,14], we propose an energy balance in the form: ZP ρ(¨ui˙ui+˙e)dv =ZP ρ(fi˙ui+sθ+QiTi)dv (2.6) +Z∂P (ti˙ui+ Σθ+σjTj+Gjθ,j+ ΠjiTi,j)da for every region Pof Band every time. Here uiis the displacement vector, eis the internal energy per unit mass fiis the body force per unit mass, tithe stress vector associated with the surface ∂Pbut measured per unit area of ∂P, and Giand Πi j 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 tj=ti jni(2.7) and tji,j+ρfi=ρ¨ui(2.8) where ti j 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: ZP ρ˙edv =ZP [tji ˙ui,j+ρ˙ηθ +ρ˙ηjTj+Sjθ,j+ Λk jTj,k−(Si−Hi)Ti−ρξθ −ρξjTj]dv (2.9) +Z∂P (Gjθ,j+ ΠjiTi,j)da. With an argument similar to that used to derive the relation (2.7), from (2.9), we obtain (Gj−Gk jnk)θ,j+(Πji −Πk jink)Ti,j=0,(2.10) where Gk j and Πk ji are tensors associated to the surface loads Giand Πji, respectively. With the help of (2.10), we obtain the local expression form of the energy balance ρ˙e=tji ˙ui,j+ρ˙ηθ +ρ˙ηjTj+Fjθ,j+ Γk jTj,k+(Hi−Si)Ti(2.11) +Gk jθ,jk + Πk jiT,i jk −ρξθ −ρξjTj, where the following notation Fj=Sj+Gk j,k,Γk j = Λk j + Πmk j,m,(2.12) is used. Electronic Research Archive Volume 33, Issue 2, 537–555. 540 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 ρ, e,ti j, η, θ, ηj,Tj,Sj,Λk j,Hj,Fj,Γk j, ξ, and ξjdo not change by such motion. The equality (2.11) implies that tji =ti j.(2.13) If we consider the Helmholtz free energy ψby ψ=e−θη −Tjηj,(2.14) and we see that the energy balance may be written in the form: ρ(˙ ψ+θ˙η+Tj˙ηj)=ti j ˙ei j +Fjθ,j(2.15) + Γk jTj,k+(Hi−Si)Ti+Gk jθ,jk + Πki jTi,jk −ρξθ −ρξjTj, where we have introduced the strain tensor 2ei j =ui,j+uj,i.(2.16) 3. Constitutive equations From now on, we define the constitutive equations for ψ, ti j, η, ηj,Sj,Hj,Gk j,Fj,Γk j,Πk j, ξ, and ξj, and we suppose that these are functions of the set V=(ei j, θ, Tj, α,j, βk,j, α,i j, βk,i j). To simplify the writting, we omit the explicit dependence of xkand then the material should be homogeneous and assume that there is no kinematical constraint. In the theory established in [3], the constitutive variables are ei j, θ, and α,j. If we introduce the notation A=ρψ, then Eq (2.15) becomes (∂A ∂ei j −ti j)˙ei j +(∂A ∂θ +ρη)˙ θ+(∂A ∂Ti +ρηi)˙ Ti(3.1) +(∂A ∂α,i −Fi)θ,i+(∂A ∂βj,i −Γi j)Tj,i+(∂A ∂α,i j −Gji)θ,i j +(∂A ∂βi,jk −Πk ji)Ti,jk +ρθξ +(ρξi+Si−Hi)Ti=0. From (3.1), we find that [3] ti j =∂A ∂ei j , ρη =−∂A ∂θ , ρηj=−∂A ∂Tj ,(3.2) Fi=∂A ∂α,i ,Γi j =∂A ∂βj,i ,Gji =∂A ∂α,i j ,Πk ji =∂A ∂βi,jk , and ρξθ +(ρξi+Si−Hi)Ti=0.(3.3) We introduce the notations θ(x,t0)=T0,Tj(x,t0)=T0 j, α(x,t0)=α0, βj(x,t0)=β0 j,(3.4) Electronic Research Archive Volume 33, Issue 2, 537–555. 541 where t0is a reference time and T0,T0 j, α0, and β0 jare given constants. As in [3], we consider new thermal variables T=θ−T0, θi=Ti−T0 i, χ =Zt 0 Tds, φi=Zt 0 θids.(3.5) From (3.4) and (3.5), we get α=χ+T0(t−t0)+α0, βj=φj+T0 j(t−t0)+β0 j, α,i=χ,i, βi,j=φi,j,(3.6) ˙χ=T,˙φi=θi. From now on, we restrict our attention to the linear theory where the functions ui,T, and θjcan be written as ui=ϵu′ i,T=ϵT′, θj=ϵθ′ j where ϵis a parameter small enough for squares and higher powers to be neglected, and u′ i,T′, and θ′ jare independent of ϵ. As usual, we assume that Ais a quadratic form of the variables ei j,T, θj, α,j, βk,j, α,i j,and βk,i j and that Hi, ξ, and ξjare linear functions of the same variables. We consider the case of a material with a center of symmetry. Thus, we have 2A=Ai jrsei jers −2bi jei jT+2Ci jrsei jφr,s+2Di jrsei jχ,rs −aT2−2Li jTφi,j(3.7) −2Ni jTχ,i j −Bi jθiθj−2Ci jθiχ,j−2dipqrφp,qrθi+Ki jχ,iχ,j+2Mipqrχ,iφp,qr +Ei jrsφi,jφr,s+2Hi jrsφi,jχ,rs +Ui jkpqrφi,jkφp,qr +Qi jrsχ,i jχ,rs. The following symmetries are satisfied: Ai jrs =Ajirs =Arsi j,bi j =bji,Ci jrs =Cjirs,Di jrs =Djirs =Djisr,(3.8) Bi j =Bji,Ni j =Nji,dipqr =diprq,Ki j =Kji,Ei jrs =Ersi j,Mipqr =Miprq, Hi jrs =Hi jsr,Ui jkpqr =Upqri jk =Uik jpqr,Qi jrs =Qrsi j =Qjirs. It follows from (3.2), (3.7), and (3.8) that ti j =Ai jrsers −bi jT+Ci jrsφr,s+Di jrsχ,rs, ρη =bi jei j +aT +Li jφi,j+Ni jχ,i j, ρηi=Bi jθj+Ci jχ,j+dipqrφp,qr,(3.9) Fj=−Ci jθi+Ki jχ,i+Mjpqrφp,qr, Γi j =Crs jiers −LjiT+Ejirsφr,s+Hjirsχ,rs, Gji =Drsi jers −Ni jT+Qi jrsχ,rs +Hrsi jφr,s, Πk ji =−dri jkθr+Msi jkχ,s+Ui jkpqrφp,qr. For isotropic materials, the number of constitutive coefficients is drastically reduced (see [12]). From (3.3), we see that the response function ξvanishes when the microtemperatures Tjvanish. In the linear Electronic Research Archive Volume 33, Issue 2, 537–555. 542 case, the function ξthat satisfies this requirement must be ξ=cjTj, where cjare constants. Since the body has a center of symmetry, we get ξ=0. Thus, from (2.12), (3.3), and (3.9), we find that ρξi=Hi−Si.(3.10) If we use these results, then Eqs (2.1) and (2.2) take the form ρη =Sk,k+ρs, ρηi= Λki,k+ρQi.(3.11) 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 Fk,k−Gk j,k j −ρ˙η=−ρs,Γi j,i−Πrk j,rk −ρ˙ηj=−ρQj.(3.12) Equations (2.8) and (3.11) can be expressed in terms of the unknowns uj,χ, and φi. Thus, we obtain the equations Ai jrsur,s j −bi j ˙χ,j+Djirkχ,rk j +Ci jrsφr,s j +ρfi=ρ¨ui,(3.13) −Drpqkur,pqk −bi j ˙ui,j+Ki jχ,i j −Qi jrsχ,i jrs −a¨χ−Rpq jrφp,qr j −pi j ˙φi,j=−ρs, Crs jkur,sk −pjk ˙χ,k+ζp jqr ˙φp,qr +Rjkrsχ,rsk +Ejkrsφr,sk −Ujrspqmφp,qmsr −Bjk ¨φk=−ρQj, where Rjkrs =Hjkrs −Mr jks, ζp jqr =dp jqr −djpqr,pjk =Ljk +Cjk.(3.14) 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 Bconsists of the union of a finite number of smooth surfaces, smooth curves (edges), and points (corners). Let Cbe 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 Z∂P (ti˙ui+ Σθ+σjTj+Gjθ,j+ ΠjiTi,j)da (4.1) =Z∂P [tki ˙ui+(Fk−Grk,r)θ+(Γk j −Πmk j,m)Tj+Gk jθ,j+ Πk jiTi,j]nkda. We will use the notations D f =f,jnj,Di=(δi j −ninj)∂ ∂xj ,(4.2) where δi j is the Kronecker delta. Then we obtain Gk jθ,jnk=Gk j nknlDθ−θ(Gk jnk)+Dj(Gk jnkθ),(4.3) Electronic Research Archive Volume 33, Issue 2, 537–555. 543 Πk jiTi,jnk= Πk jinknjDTi−TiDj(Πk jink)+Dj(Πk jinkTi). As in [11,12], from (4.1) and (4.3), we get Z∂P (ti˙ui+ Σθ+σjTj+Gjθ,j+ ΠjiTi,j)da (4.4) =Z∂P (ti˙ui+ Φ1θ+ Φ2Dθ+ ΨjTj+WiDTi)da +ZC (Yθ+ ΩiTi)ds where Φ1=(Fk−Grk,r)nk−Dj(nsGs j)+(Djnj)nsnpGsp,Φ2=Grsnrns,(4.5) Ψi=(Γki −Πrki,r)nk−Dj(nsΠs ji)+(Djnj)nsnpΠspi,Wi= Πrsinrns, Y=<Grsnrys>, Ωi=<Πrsinrys>, yi=ϵirk srnk. Here, skare the components of the unit vector tangent to C,<f>denotes the difference of the limits of ffrom both sides of C, and ϵjrk is the alternating symbol. The first boundary-initial-value problem is characterized by the boundary conditions ui=u∗ i, χ =χ∗, φi=φ∗ i,Dχ=ζ∗,Dφi=γ∗ ion ∂B×I,(4.6) where u∗ i, χ∗, φ∗ i, ζ∗, and γ∗ iare prescribed functions. For the second boundary-initial-value problem, the boundary conditions are [11] ti=t∗ i,Φ1= Φ∗ 1,Φ2= Φ∗ 2,Ψi= Ψ∗ i,Wi=W∗ ion ∂B×I,(4.7) Y=Y∗,Ωi= Ω∗ ion C×I, where t∗ i,Φ∗ 1,Φ∗ 2,Ψ∗ i,W∗ i,Y∗, and Ω∗ iare given. The initial conditions are ui(x,0) =u0 i(x),˙ui(x,0) =v0 i(x), χ(x,0) =χ0(x),˙χ(x,0) =χ1(x),(4.8) φ(x,0) =φ0 i(x),˙φi(x,0) =ν0 i(x),x∈B, where u0 i,v0 i, χ0, χ1, φ0 i, and ν0 iare 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 ρand the thermal capacity aare strictly positive. (ii) The matrix Bi j is positive definite. (iii) The quadratic form W(ei j, κi jk, χ,i, χ,ji, φ,i, φ,i j)= Electronic Research Archive Volume 33, Issue 2, 537–555. 544 Ai jrsei jers +2Ci jrsei jφr,s+2Di jrsei jχ,rs +Ki jχ,iχ,j+2Mipqrχ,iφp,qr +Ei jrsφi,jφr,s+2Hi jrsφi,jχ,rs +Ui jkpqrφi,jkφp,qr +Qi jrsχ,i jχ,rs, is positive definite, i.e., there exists a positive constant Csuch that: W≥C(ei jei j +χ,rχ,r+χ,rsχ,rs +φr,sφr,s+φp,qrφp,qr). Let us to propose the problem as an abstract problem in a suitable Hilbert space. We will work on the space H=W1,2 0(B)×L2(B)×W2,2 0(B)×L2(B)×W2,2 0(B)×L2(B), where W1,2 0,W2,2 0, and L2are the usual Sobolev spaces, W2,2 0=[W2,2 0]3and L2=[L2]3. The elements in this space can be denoted by U=(u,v, χ, θ, φ,ϕ). We consider the scalar product associated to the norm ||(u,v, χ, θ, φ,ϕ)||2(5.1) =ZB [ρvivi+aθ2+Bi jϕiϕj+W(ei j, χ,i, χ,ji, φ,i, φ,i j)]dv. Now, we want to see our problem as a Cauchy problem in H. We define the operator A                           u v χ θ φ ϕ                           =                           v M θ υ ϕ Ω                           (5.2) where M=(Mi),Ω=(ωi), and Mi=ρ−1(Ai jrsur,s j −bi jθ,j+Ci jrsφr,s j +Djirkχ,rk j), ωi=Fi j(Crs jkur,sk −pjkθ,k+ζp jqrϕp,qr +Rjkrsχ,rsk +Ejkrsφr,sk −Ujrspqmφp,qmsr), υ=a−1(−Drpqkur,pqk −bi jvi,j+Ki jχ,i j −Qi jrsχ,i jrs −Rpq jrφp,qr j −pi jϕi,j), where Fi jBjk =δik. We note that our problem can be written as dU dt =AU +F(t),U(0) =(u0,v0, χ0, χ1,φ0,ν0),(5.3) where F(t)=(0,f(t),0, ρa−1s,0, ρFi jQj). Electronic Research Archive Volume 33, Issue 2, 537–555. 545 The domain of the operator Ais the subspace of elements of our Hilbert space such that v∈W1,2 0,ϕ∈W2,2 0, θ ∈W2,2, Ai jrsur,s j +Djirkχ,rk j −bi jθ,j∈L2, −Drpqkur,pqk −Qi jrsχ,i jrs −Rpq jrφp,qr j ∈L2, and Crs jkur,sk +Rjkrsχ,rsk −Ujrspqmφp,qmsr ∈L2. This domain is a dense subset of our space. After an easy but laborious calculation, we can see that <AU,U>=0, for every element Uat the domain of the operator. The next step in our approach is to show that zero belongs to the resolvent of the operator. Let U∗ be in H. We must prove that the equation AU =U∗ admits a solution. That is v=u∗, θ =χ∗,ϕ=φ∗, M=v∗, υ =θ∗,Ω=ϕ∗. We substitute the first three equations into the others to find that Ai jrsur,s j +Ci jrsφr,s j +Djirkχ,rk j =ρv∗ i+bi jχ∗ ,j, −Drpqkur,pqk +Ki jχ,i j −Qi jrsχ,i jrs −Rpq jrφp,qr j =aθ∗+bi ju∗ i,j+pi jφ∗ i,j, Crs jkur,sk +Rjkrsχ,rsk +Ejkrsφr,sk −Ujrspqmφp,qmsr =Bjkϕ∗ k+pjkχ∗ ,k−ζp jqrφ∗ p,qr. If we denote αi1=ρv∗ i+bi jχ∗ ,j, α2=aθ∗+bi ju∗ i,j+pi jφ∗ i,j, α3j=Bjkϕ∗ k+pjkχ∗ ,k−ζp jqrφ∗ p,qr,A1i=Ai jrsur,s j +Ci jrsφr,s j +Djirkχ,rk j, A2=−Drpqkur,pqk +Ki jχ,i j −Qi jrsχ,i jrs −Rpq jrφp,qr j, A3j=Crs jkur,sk +Rjkrsχ,rsk +Ejkrsφr,sk −Ujrspqmφp,qmsr, then our system can be written as Ai1=αi1,A2=α2,Ai3=αi3. Electronic Research Archive Volume 33, Issue 2, 537–555. 552 We do not assume any condition on the coefficients µ, C2,E2, and b1, but we are going to obtain a couple of qualitative results for our system. Nevertheless we will need to suppose that ρand bare 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: H(t)=ZD (ρu2+bφ2)dx+ω∗(t+t0)2,(9.2) where ω∗and t0are two non-negative real numbers to be selected. We have ˙ H(t)=2ZD (ρu˙u+bφ˙φ)dx+2ω∗(t+t0), and ¨ H(t)=2ZD (ρu¨u+bφ¨φ)dx+2ZD (ρ|˙u|2+b|˙φ|2)dx+2ω∗. We can notice that ZD (ρu¨u+bφ¨φ)dx=−ZD (µ|∇u|2+2C2∇φ∇φ+E2|∇φ|2+b1|∆φ|2)dx ZD (ρ|˙u|2+b|˙φ|2)dx−2E(0). Therefore ¨ H(t)=4ZD (ρ|˙u|2+b|˙φ|2)dx+2(ω∗−2E(0)). A simple use of Holder’s inequality allows us to conclude H(t)¨ H(t)−(˙ H(t))2≥ −2(ω∗+2E(0))H(t). If we put homogeneous initial conditions, we have that E(0) =0 and if we take ω∗=0, we can conclude that H(t)¨ H(t)−(˙ H)2≥0.(9.3) This inequality allows us to establish (see [19], p. 19) that H(t)≤ H(0)1−t/T∗H(T∗)t/T∗ for all tbetween 0 and T∗. Thus, in the case where we impose null initial data, we obtain that H(t)=0 for all tin 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 ω∗=−2E(0) and again conclude the previous inequality. We can also get (see [19], p. 20) H(t)≥ H(0) exp t˙ H(0) H(0) .(9.4) We note that we can always select t0large enough to guarantee that ˙ H(0) >0. When E(0) =0 and ˙ H(0) >0, we also conclude the growth estimator. Electronic Research Archive Volume 33, Issue 2, 537–555. 553 Theorem 4. Let us to suppose that ρand bare 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,˙ 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 ais 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 Ram´ on 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. 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