1 Introduction

Elastic plates are essential components that have highly important mechanical structures and,as such,have a large range of practical applications,in particular in industrial construction and microchip production. The thin plates,which are basic elements in various mechanical structures,can develop cracks due to wear of the material or by design. For this reason,the mechanical properties of such particular structures have been intensively studied in several papers^{[1, 2, 3, 4, 5]}. For a long time,the effect of transverse shear forces has been ignored in the framework of the classical linear theory under Kirchhoff’s model^[4]. Lately,however,the demand for more stringent accuracy created by the sophistication of industrial design and the advent of very powerful computers has led to a continuous increase in the use of the refined models. The refined models,where the field equations are much more complicated than in the old, classical one,take into account both the deflection of the middle section and the transverse shear forces. To justify their use,a rigorous mathematical analysis is necessary in order to settle the questions of well-posedness,analyticity and stability of the solution before numerical computations can be considered. It is worth recalling that such questions have been studied in several papers^{[6, 7, 8, 9, 10, 11, 12]}.

In this paper,these questions are further investigated,particularly when diffusion effects are considered in the thermoelastic thin plate theory. Thus,many consequent mechanical and mathematical problems need serious analysis.

Maybe we can think that the classical theory of thermoelastic thin plates is a good model to explain the thermal conduction in different structures. However,the research conducted in the development of high technologies after the second world war,confirmed that the field of diffusion in solids cannot be ignored. Thus,the obvious question is,what happens when the diffusion effect is considered with the thermal effect in the theory of thin elastic plate. Diffusion can be defined as the random walk of a set of particles from regions of high concentration to regions of lower concentration. Thermodiffusion in an elastic solid is due to coupling of the fields of strain,temperature,and mass diffusion. The processes of heat and mass diffusion play important roles in many engineering applications,such as satellites problems,returning space vehicles,and aircraft landing on water or land. There is now a great deal of interest in the process of diffusion in the manufacturing of integrated circuits,integrated resistors, semiconductor substrates,and MOS transistors. Oil companies are also interested to this phenomenon to improve the conditions of oil extractions.

The classic thermoelastic diffusion theory was introduced by Nowacki^[13]. Later,Sherief et al.^[14] established the generalized model with relaxation times. Recently,Aouadi et al.^{[15, 16, 17, 18, 19, 20]} have derived the governing equations of some thermoelastic diffusion theories as well as some qualitative properties of the solutions. An overview of different models of this theory was presented in a survey article credited to Aouadi^[21].

This paper aims to provide a consistent theory of homogeneous and isotropic thermoelastic diffusion thin plates. This work,which has not been obtained in any reference yet,represents a first step to understand the fundamental limits of intrinsic thermo-diffusion dissipations arising in elastic structures. However,the arguments of the classic theory can be adapted here to obtain the theory of thermoelastic diffusion thin plates under the Lagnese and Lions model^[4] and according to Norris’ comments^[5]. In Section 2,we develop the main points to obtain the equations of thermoelastic diffusion thin plate under three different models of heat and diffusion transmission. First the classical Fourier’s law is used to describe the thermo-diffusion conduction in the plate. Unfortunately,this law has a serious shortcoming. The corresponding parabolic part in both heat and diffusion equations predicts an unrealistic result: the thermal and diffusion disturbances at one point of the body are instantly felt everywhere in the body, that is they entail infinite propagation speeds. This observation leads one to look for other more general realistic models of transmission. Here,we write the equations of thermoelastic diffusion thin plate according to two generalized models of transmission. We consider the plate with second sound,that is,the distributions of the temperature and diffusion in the plate are described by Cattaneo’s law,instead of classical Fourier’s law of conduction. Then,we consider the case of plate with memory where we use the Gurtin-Pipkin’s law of conduction. Under both generalized models,thermal and diffusion disturbances propagate with finite speed,so that the corresponding equations are of hyperbolic types. In Section 3,using the C0-semigroup theory, we prove the well-posedness of the model derived in the frame of classic laws. 2 Basic equations and priliminaries

Following Nowacki^[13],the basic governing equations of motion,heat and diffusion,in the absence of body forces,external loads,heat and diffusion sources,are given as follows.

(i) The equation of motion

(ii) The stress-strain-temperature-diffusion relation

(iii) The displacement-strain relation

(iv) The entropy-strain-temperature-diffusion relation

(v) The energy equation

(vi) The equation of conservation of mass

(vii) The chemical-strain-temperature-diffusion relation

where λ and μ are Lamé’s constants. β₁ = (3λ + 2μ)α_t and β₂ = (3λ + 2μ)α_c,where αt and α_c are,respectively,the coefficients of linear thermal and diffusion expansion. θ = T − T₀ is small temperature increment,where T is the absolute temperature of the medium,and T0 is the reference temperature. ρ is the mass density per unit volume,and cE is the specific heat at a constant strain. u_i are the components of the displacement vector u,t_ij are the components of the symmetric stress tensor T,and e_ij are the components of the strain tensor. P is the chemical potential per unit mass,S is the entropy per unit mass,C is the concentration of the diffusive material in the elastic body, is a measure of diffusive effect,and w is a measure of thermodiffusion effect.

The expressions of heat and diffusion flux vectors q and p will be given later under three different models of transmission.

In the following sections,we need a different alternative form,where the chemical potential is used as a state variable instead of the concentration. From (7),we obtain

Substituting (8) into (1)-(6),we obtain the desired alternative form as follows: where are physical positive constants. It is easy to obtain from (10) that which implies that

Both last inequalities are needed to stabilize thermoelastic diffusion systems^{[15, 16, 17, 18, 19, 20]}. 3 Motion thin plate equation

We now consider a thermoelastic diffusion plate occupying the domain −∞ < x,y < ∞, and . We suppose that the thickness of the plate h is smaller in one order of magnitude than its span or diameter so that it can be termed as a thin plate. Suppose also that the plate is traction free on the surfaces: t₁₃ = t₂₃ = t₃₃ = 0 on z = ±h/2.

To describe what thin plates models has in common,we derive the motion equations. The first equation of (9) can be written as

The repeated Greeks indicate summation over 1 and 2. The development so far is independent of the models of heat and diffusion transmission. Integrating the first term of (12) with respect to z and using the fact that the shear and the normal stresses vanish on the surfaces yield^[5] where w is the averaged through-thickness displacement, The in-plane equilibrium equation (the second term of (12)) then implies where the first moment of the stresses (or simply the moments) is given by According to the Kirchhoff hypothesis for thin plates,we consider the following assumptions:

(i) The mid plane remains unstrained subsequent to bending.

(ii) Plane sections initially normal to the mid surface remain plane and normal to that after the bending. This means that we can take e₁₃ = e₂₃ = 0. The deflection of the plate is thus associated principally with bending strains. Consequently the normal strain e33,resulting from transverse loading,may be neglected.

(iii) The normal stress component t₃₃ is much smaller than the other stress components and may also be neglected.

Thus,using the above assumptions (ii) and (iii),the symmetric stress tensor tαβ satisfies the following equations:

where δαβ is a metric tensor on the surface of the plate. From the second term of (17),we can obtain and from the third term of (17),we can obtain

According to the assumption (ii),the strain-displacement relations become

Now,u_3,3 = 0 implies that This means that the lateral deflection does not vary over the plate thickness. Integrating the expressions of e₁₃ and e₂₃ in (20) gives us From the assumption (i),one can deduce that u₁₀ = u₂₀ = 0,and consequently (22) becomes which confirms (18). Substituting (23) into (20),we get The divergence of ut appearing in (9) is given by

By using (19),we get

From (23),we get which provides the strains at any point in the plate. Δ in (25) means the in-plane Laplacian.

Furthermore,(15) becomes

where I = h³/12.

(26) is the standard complete plate equation for elastic,thermoelastic,or thermoelastic diffusion plates.

Thus,substituting (19) and (23) into the constitutive relation (17),we obtain the extended traction equation as follows:

On the other hand,the expression of M_αβ given by (16) can be written in terms of w,Θ, and Φ as follows:

where are the first moments of temperature and chemical potential,respectively.

Furthermore,(26) can be reduced to

where are provided by (28). Substituting (31) into (30),we get where

(32) is the common equation of motion of a thermoelastic diffusion thin plate under any law of heat or diffusion transmission. When the diffusion effect is ignored (γ² = 0),(32) reduces to the standard thermoelastic thin plate equation^{[2, 3, 4, 5]}.

In the following subsections,we complete (32) by the equations of heat and diffusion,under three different laws of heat and diffusion transmission,according to the corresponding Lagnese and Lions model^[4] and the corrections and criticism made by Norris^[5]. It is worth noting that the equation of motion (32) remains the same for the three following systems derived from the three different laws of transmission. 4 Thermo-diffusion thin plate equations

To complete (32) by the equations of heat and diffusion,we substitute (25) into (9),

where ν₁ and ν₂ are given by (33),and

It is easy to prove that the positive constants c^*,k^*,and r^* verify (11). 4.1 Classic laws

Now,we write the heat and diffusion equations according to classic Fourier’s law for heat and Fick’s law for diffusion transmission,respectively,

where k is the coefficient of thermal conductivity,and is the coefficient of diffusion conductivity.

Substituting the divergence of (36) into (34),we conclude

Multiplying (37) by z and then integrating,we get

where Θ and Φ are defined by (29),and ξ₁(x,y,t) and ξ₂(x,y,t) are the first moments of θ,_zz and P,_zz,respectively,i.e., Applying the assumed zero flux conditions on the plate surfaces, yields In order to relate ξ₁ to Θ,Lagnese and Lions^[4] assumed that θ(x,y,z,t) is linear in z,i.e., Substituting (42) into the first equation of (41),we get Similarly,using the same arguments,we can obtain

Substituting (43) and (44) into (38),we get

Adding (32) to (45),we obtain the classical equations of thermoelastic diffusion thin plate in the frame of the Lagnese and Lions model,i.e.,

where

It is worth noting that the positive coefficients verify (11). 4.2 Cattaneo’s laws

In this case,a modified law of heat conduction,the Cattaneo’s law^[22],i.e.,

replaces the classic Fourier law (the first term of (36)). The positive parameter τ₀ is the relaxation time describing the time lag in the response of the heat flux to a gradient in the temperature. Similarly,the second term of (36) is replaced by^[14] where τ is the diffusion relaxation time,which will ensure that the equation satisfied by the concentration will also predict finite speeds of propagation of matter from one medium to the other.

Substituting the divergence of (48) and (49) into (34),we conclude

Multiplying (50) by z and then integrating,we get where Θ and Φ are defined by (29),and ξ₁ and ξ₂ are given by (39)-(44). Substituting (43) and (44) into (51),we get Adding (32) to (52),we obtain the equations of thermoelastic diffusion thin plate under Cattaneo’s laws in the frame of Lagnese and Lions model where the coefficients are given by (47). By using Cattaneo’s laws for heat and diffusion transmissions,the last two terms of (53) become hyperbolic,and hence,automatically eliminate the paradox of infinite speed. It is easy to see that (53) can be reduced to the classical thin plate equation (46) by taking τ₀ = τ = 0. 4.3 Gurtin-Pipkin’s law

According to Gurtin-Pipkin’s theory^[23],the linearized constitutive equations of q and p are given by^{[20, 24]}

where k(s) and (s) are,respectively,heat and diffusion conductivity relaxation kernels. Substituting the divergence of (54) into (34),we get

Multiplying (55) by z and then integrating,we get

Substituting (43) and (44) into (56),we get

Adding (32) to (57),we obtain the equations of thermoelastic diffusion thin plate with memory in the frame of Lagnese and Lions model

where the coefficients are given by (47). The convolution terms,appearing in (58), introduced by Gurtin-Pipkin’s law entail the finite propagation speed for both thermal and diffusion disturbances. (54) can be reduced to the classical Fourier’s law by taking k(s) = (s) = δ(s) (the Dirac distribution at zero). Besides,if we take

and differentiate the first and second terms of (54) with respect to t,one can easily arrive (formally) at (48) and (49),respectively.

For the sake of brevity,we prove in the following section the well-posedness of the classical model only. 5 Well-posedness of classical model

In this section,we consider an open bounded domain ,where the dimension d ≥ 2 may be arbitrary,but d = 2 corresponds to models derived in the previous section.

We study the well-posedness of system (46) subject to initial conditions

We consider clamped boundary conditions for w and Dirichlet boundary condition for Θ and Φ,i.e., where v = (v2,v2,· · · ,vn) stands for the unit outer normal to ∂Ω,and .

We now transform the initial-boundary-value problem given by (46),(59),and (60) to an abstract problem on a suitable Hilbert space.

Put wt = v in (46),and (46) is reduced to the system of equations

where I is the identity operator. Set

Then,(61) can be written into the Sobolev equation,

where

Let

where are the Hilbert spaces.

We define the inner products of as follows:

The following theorem is proved by using the same argument used in the proof of Theorem 11.11^[25].

Theorem 5.1 The problem (63) is well-posed and is governed by a (C₀) contraction semigroup on K.

Proof From Goldstein^[25],it is sufficient to prove that is a positive and self-adjoint operator on ,that 0 belongs to the resolvent set of ,i.e.,0 ∈ ρ(),that , and that in a maximal dissipative operator on . Since it is easy and classic to prove that 0 ∈ ρ(),and that ,we shall only prove that is a positive and self-adjoint operator on . For all ,we have

Since the positive coefficients verify the condition (11),we deduce that . Let,then,ζ be a number chosen in such a way that . Thus,the Cauchy-Schwartz and Young’s inequalities lead to Thus,(64) becomes

Thus, is a positive operator on . Since

the operator is self-adjoint on .

To prove that is a maximal dissipative operator on ,we have for all

from which it follows that

which proves that is a dissipative operator on . To show R(I −B) = ,we shall prove the existence of a vector of functions satisfying which in terms of the components gives

Inserting the first equation into the others,we have

To solve (68),we consider the following bilinear form defined on by

We show that the following variational equation has a unique solution H₀¹ (Ω). The bilinear form F is coercive,because

According to the Lax-Milgram theorem,there exists a unique solution of (69) satisfying

If we put φ = Φ = 0 in (69),

which implies that Φ is a weak solution of the equation

Therefore,Φ ∈ H²(Ω) × H1 0 (Ω). If we put φ = ψ = 0 in (69),

which implies that Φ is a weak solution of the equation

Therefore,Θ ∈ H²(Ω) × H1 0 (Ω). Finally,if we put φ = ψ = 0 in (69),

which implies that w is a weak solution of the equation

Therefore,w ∈ H⁴(Ω) × H₀² (Ω). Combining these facts implies that R(I −B) = H. 6 Conclusions

We summarize the obtained results as follows.

(i) We have derived the basic equations of thermoelastic diffusion thin plate under three different laws of heat and diffusion transmission according to the Lagnese and Lions model^[4] and the corrections and criticism made by Norris^[5].

(ii) We have proved the mathematical well-posedness for the proposed mechanical model of thermoelastic diffusion thin plates.

(iii) By comparison with other models proposed for thin plates,the model proposed in this paper is more reasonable in predicting the propagation of thermal,diffusion,and elastic waves in thin plates. We believe that the mechanical and mathematical results presented can reveal the usefulness of this model in many engineering applications,which is exactly what this paper aims at.