1 Introduction

A substance (such as an atom, ion, or molecule) always moves from a high concentration region to a low concentration region. This phenomenon can be described by the classical Fick's law. Nowadays, the diffusion principle has been increasingly administered in various fields, especially in geophysics and industry. For instance, it can be used to measure the diffusion coefficients of various cations in minerals and improve the surface attributes of metals, such as wear, corrosion resistance, and hardness. Recently, in order to obtain more efficient extraction of oil from oil deposits, the process of thermoelastic diffusion has become a focus in oil industry. In addition, the thermodiffusion process also contributes to studies in fields associated with the emergence of semiconductor devices and the advancement of microelectronics.

In the preceding cases, temperature plays an important role.Thermodiffusion in solids is one of the transport processes.Therefore, it is imperative to investigate the coupling effects among the temperature field, the diffusion field, and the strain field. Customarily, the diffusion process is formulated by using the Fick's law. However, it does not consider the mutual exchange between the introduced substance and the substance or the influence of temperature on the interplay. The earliest critical assessment was reported in the work of Oriani^[1]. Nowacki^[2]predicted an infinite speed of propagation of the thermoelastic wave by formulating the coupled thermoelastic model in the theory of thermoelastic diffusion. Kumar et al.^[3] studied the reflection and refraction phenomenon in an elastic half-space medium in the context of thermoelastic diffusion theory, and further derived the amplitude ratios of reflected and refracted waves. Subsequently, the generalized thermoelastic diffusion theories were extended, which can predict a finite speed of propagation for the thermoelastic wave and diffusive wave. The most celebrated theory is the generalized thermoelastic diffusion theory, which has been established^[4]by incorporating the Lord-Shulman (L-S) model^[5]. Following this theory, Sherief and El-Maghraby^[6] investigated a thick plate subjected to a time-dependent thermal shock and a time-dependent chemical potential shock. Sherief and Hussein^[7]obtained an analytical solution of the two-dimensional generalized thermoelastic diffusion problem for a half-space by using the double transform techniques. With the finite element method in the time domain directly, Xia et al.^[8] solved the dynamic response problem of an infinite body with a cylindrical cavity in order to avoid losing precision in the application of the integrated transformation method. By using the normal mode analysis, Othman et al.^[9] worked on the exploration of disruptions in a homogeneous, isotropic elastic medium in the physical domain. By using the Fourier transform, Ram et al.^[10] obtained a general solution of generalized thermodiffusion in an elastic solid. Kumar and Kansal^[11] investigated the Rayleigh wave propagation in a homogeneous, transversely isotropic, thermoelastic diffusive half-space. The thermodiffison process, which is important in the field of oil extraction, was considered by Deswal and Choudhary^[12]. Aouadi^[13] proposed a novel theory of thermoelastic diffusion for a thin plate, and derived the governing equations under three different thermoelastic diffusion theories. He et al.^[14] investigated a two-dimensional generalized thermoelastic diffusion problem. Li et al.^[15-16]studied the plane waves of generalized electromagnetothermoelastic with diffusion for a half-space and evaluated the transient responses for a medium with variable thermal conductivity and diffusivity under thermal and chemical shock.

Lately, fractional calculus and fractional differential equations have attracted much attention because of their wide applications in many fields of applied science such as mechanics of solids, heat conduction, viscoelasticity, electrochemistry, and signal processing. The fractional calculus can also be used to modify many existing models. Due to its potential applicability in a variety of problems, it has been applied in areas of mathematical physics and other engineering applications, including theoretical studies and numerical methods, such as the existence and uniqueness of solutions of fractional differential equations (see, e.g., Refs. [17]-[19])and corresponding algorithms vital for dealing with engineering troubles^[20]. Recent studies have demonstrated that the employment of fractional-order derivatives and integrals to deal with the formulation of some specific problems is more economical and practical than the classical method. Inspired by the fractional calculus, especially in anomalous diffusion, Chen^[21] and Chen et al.^[22-23] introduced the concept of fractional anomalous diffusion in detail and further developed a fractal derivative model of anomalous diffusion. Suzuki et al.^[24] put forward a fractional heat transfer model and applied it to analyze an anomalous thermal diffusion problem, which showed that the analytical solutions match with the numerical results. Sweilam et al.^[25] presented a novel numerical approach to deal with the space fractional-order diffusion equation, and the comparison of the results with other numerical methods showed that this method is simple and effective. Kumar and Gupta^[26] studied different characteristics of the plane wave propagation in a thermoelastic diffusive medium by applying the methodology of fractional calculus.Liu et al.^[27] developed a new integral transform method which can be extended to study other practical problems related to fractional-order differential equations. El-Karamany and Ezzat^[28-29] introduced two different models of fractional heat conduction law for a non-homogeneous anisotropic elastic solid. In a recent work, Li et al.^[30] proposed a size-dependent generalized diffusion-thermoelasticity theory by introducing fractional calculus.

The present work aims to establish the theory of fractional-order generalized thermoelastic diffusion for anisotropic and linearly thermoelastic diffusive media. The corresponding generalized variational principle is also presented using the semi-inverse method. Ultimately, the dynamic response of a semi-infinite medium with one end subjected to a thermal shock and a chemical potential shock is investigated by using the Laplace transform. The effects of fractional order parameters on the variations of different field quantities are graphically analyzed.

2 Governing equations and general theory

To obtain a unified theory of fractional-order generalized thermoelastic diffusion, fractional calculus is proposed. The equation of motion has the following form:

(1)

where σ_ij is the stress tensor, ρ is the mass density, u_i is the elastic displacement vector, and f_i is the body force.

The generalized form of stain-displacement relations is given by

(2)

where ε_ij is the strain vector.

With the introduction of fractional calculus, the generalized heat conduction law (i.e., the L-S model^[5], the Green-Lindsay (G-L)model^[31], and the Green-Naghdi (G-N)model^[32-33]) and the mass diffusion law^[4] may be expressed as

(3)

(4)

where q_i, η_i, θ, P, κ, D, and κ^* are the heat flux vector, the diffusing mass vector, the conductive temperature, the chemical potential, the coefficient of thermal conductivity, the coefficient of diffusion, and the material constant characteristic in the G-N model^[32-33], respectively with ℓ_i(i =1, 2, ..., 6) and ω _i(i = 1, 2, 3) put forward here to simplify the model introduced in this research. In addition, Ipoints to an integral operator expounded as^[34]

(5)

where Γ(α) is the gamma function, 0≤α ≤2, and I⁰f(t)=f(t).

The definition of fractional derivatives is commonly based on the Riemann-Liouville integral. In an integral form, the most widely adopted ones are the Riemann-Liouville and Caputo fractional derivatives. In this paper, the Riemann-Liouville type of fractional-order derivative is adopted^[34],

(6)

As a matter of fact, the fractional calculus has explicit physical meanings. Youssef^[35] pointed out that the physical significance of fractional order 0 < α < 1 represents weak conductivity, α=1 represents normal conductivity, and 1 < α < 2 represents strong conductivity. Based on the investigational results of heat conduction applied to processed meat by Mitra et al.^[36], Ghazizadeh et al.^[37] assessed the fractional order and discovered that 0 < α < 1 for meat may be applicable for porous materials and synthetic materials containing microcracks.

In the absence of a heat source, the equations for energy conservation and mass conversation are

(7a)

(7b)

where S and C are the entropy density and the concentration of diffusive material, respectively.

For linear and anisotropic thermoelastic diffusive media, the coupled constitutive relations are given by^[38]

(8)

(9)

(10)

where c, χ ^σ, and γ ^σ are material constants. In addition, τ, τ ₀, τ ₁, and τ₂ are the relaxation times, and ℓ₀ and ω ₀ serve the same purpose as ℓ_i(ω _i).

Equations (1) -(10) are the basic governing equations of the force field, the temperature field, and the diffusion field. Substituting Eq.(8) into Eq.(1), and then combining Eq.(2), we obtain

(11)

The governing equations for the diffusion field and the temperature field are obtained by combining Eqs.(3), (6), and (7a) (or Eqs.(4), (6), and (7b)) and then introducing Eq.(9) (or Eq.(10)) as

(12)

(13)

where

The initial conditions of the problem need to be introduced by the following boundary conditions. On the surfaces s₁ and s₂, the displacement and traction are specified as

(14)

On the surfaces s₃ and s₄, the temperature and heat flux satisfy

(15)

On the surfaces s₅ and s₆, the chemical potential and diffusion flux satisfy

(16)

In the above boundary conditions, s₁ + s₂ = s₃+ s₄ = s₅ + s₆ = s covers the total boundary surface.

Now, the initial boundary value problem in the context of fractional-order generalized thermoelastic diffusion theory has been proposed.

3 Generalized variational principle

In order to establish a reliable numerical method for the fractional generalized diffusion theory, a generalized variational principle is needed. In this section, to derive such a principle, the semi-inverse method is used^[39]. The dual-convolutional variational principles of elastic dynamics were established^[40]. The time-derivative term in the fundamental equations can be given by^[41]

(17)

where φ indicates an arbitrary function, and ∆t = t -t_{n -1} is the equal step length. In view of Eq.(17), Eqs.(7) and (8) can be written sequentially as

(18)

(19)

(20)

where

When 0 < β ≤1, the Caputo definition of the time fractional derivative provides

In Eq.(21), a summation is used to replace the integral term and estimate the first-order time derivative by the first-order backward contrast, i.e.,

(22)

According to Eq.(22), Eq.(3) (0 ≤α ≤1) has the following form:

(23)

where

According to Eq.(17), Eq.(23) satisfies the equation

(24)

where

When 1 ≤α ≤2, Eq.(3) may be written as

(25)

In view of Eq.(22), one obtains

(26)

where

According to Eq.(17), Eq.(26) takes the form

where

For convenience, the comprehensive form of Eqs.(24) and (27) is

(28)

From Eqs.(4), (17), and (22), with a similar method, the general relation of η _i and P has the form of

(29)

which can be discussed in the following two cases.

When 0 < α ≤1, one obtains

(30)

where

When 1 ≤α ≤2, one obtains

(31)

where

An energy-like trial function with independent elements (σ_ij, ε_ij, u_i, θ, q_i, P, η_i) may be established as

(32)

where

(33)

in which F and G_i(i = 1, 2, ..., 12) are unidentified functions to be determined below.

The stationary condition concerning σ_ij has the structure

(34)

To fulfill Eq.(2), one obtains

(35)

which designates that F is not related to σ_ij and its derivatives.

The stationary condition for ε_ij revealed in Eq.(32) can be expressed as

(36)

In consideration of Eq.(20), one acquires

(37)

Introducing Eq.(37) into Eq.(33) leads to

(38)

The trial Euler equation for u_i in Eq.(32) can be written as

(39)

Set

(40)

One discovers that Eq.(39) attains Eq.(1). Substituting Eq.(40) into Eq.(38) yields

(41)

The stationary condition with respect to θ in Eq.(32) is

(42)

Considering Eq.(18), one obtains

(43)

Introducing Eq.(43) into Eq.(41), one obtains

(44)

The trail Euler equation for q_i in Eq.(32) has the form

(45)

In view of Eq.(28), one gets

(46)

Substituting Eq.(46) into Eq.(44) yields

(47)

The stationary condition with respect to P in Eq.(32) is

(48)

Considering Eq.(19), one obtains

(49)

Introducing Eq.(49) into Eq.(47), one gets

(50)

The trial Euler equation for η _i in Eq.(32) has the form

(51)

In view of Eq.(29), one has

(52)

Substituting Eq.(52) into Eq.(50) yields

(53)

from which, one gets

(54)

Implementing Green's theory on the boundary, one acquires

(55)

Taking into consideration the boundary equations on s_k(k =1, 2, ..., 6), one gains from Eq.(55) that

(56)

Finally, substituting Eqs.(54) and (56) into Eq.(32) yields the generalized variational principle for fractional-order generalized thermoelastic diffusion. Note that the numerous special theorems can be procured by establishing acceptable constraints.

4 Solution of problem in Laplace transform domain 4.1 Problem formulation

In this section, we consider a semi-infinite thermoelastic diffusive material, which is assumed to be unstrained and unstressed. The problem may be simplified as a one-dimensional (1D) problem, of which the boundary at one end is traction free and subjected to a thermal shock θ ₁H(t) and a chemical potential shock P₁H(t). The initial boundary conditions can be expressed as follows:

(57)

The equations for the 1D case are

(58)

where

In the above equations, λ and µ are the Lame's constants, and α_t, α_c, a, b, and c_E are the coefficient of linear thermal expansion, the coefficient of linear diffusion expansion, the measurement of thermo-diffusion effect, the measurement of diffusive effect, and the specific heat at constant strain, respectively. Consider the following dimensionless quantities:

(59)

Then, the governing equations may be written as (dropping the asterisks for convenience)

(60)

(61)

where

4.2 Laplace transform

Substituting the Laplace transform defined by the formula

(62)

into Eqs.(60) and (61) and then utilizing the homogeneous initial circumstances, we obtain

(63)

(64)

where

Eliminating θ and P in Eq.(63), the following sixth-order partial differential equation is obtained:

(65)

where

In a similar manner, we can show that θ and P satisfy the equations

(66)

(67)

Equation (65) can be factorized as

(68)

where k₁, k₂, and k₃ are the roots with positive real portions of the distinctive equation,

(69)

The solution of Eq.(68) for x ≥0 has

(70)

where A_i = A_i(s) are parameters only relying on s.

Similarly, the solutions of Eqs.(66) and (67) can be written as

(71)

(72)

where A'_i and A"_i are parameters depending on s only.

Substituting Eqs.(70) -(72) into Eq.(63), we get

(73)

(74)

Thus, we have

(75)

(76)

Substituting Eqs.(70), (75), and (76) into Eq.(64), we get

(77)

By using the Laplace transform of the boundary circumstances (57) together with Eqs.(70), (75), (76), and (77), we obtain the following series of linear formulas. Then, we can obtain the unknown parameters A₁, A₂, and A₃.

(78)

(79)

(80)

4.3 Numerical inversion of transforms

It is necessary to perform the Laplace inversion for the considered variables obtained in the Laplace transform domain.

The inversion of the Laplace transform is defined as

(81)

where s is the Laplace transform parameter.

Taking s = d + iy, and using the Fourier sequences in the span of [0, 2L], we obtain the estimated formula from Honig and Hirdes^[42],

(82)

where

(83)

The Korrecktur and the ε-algorithm methods are used to reduce the discretization error and the truncation error, therefore accelerating convergence. The details of these methods were mentioned in Ref. [42].

5 Numerical results and discussion

The copper material is chosen for numerical evaluation, and the constants are given by

Because we study the problem of diffusion in solids, which belongs to weak diffusion, the value of fractional-order parameter is in the range of 0 < α ≤1. The non-dimensional temperature, chemical potential, and displacement are illustrated in Figs. 1-5.In the calculation, we specify τ ₀=τ ₁=0.02 and τ= τ ₂=0.2. The numerical results are presented to evaluate the influence of fractional-order parameter on the response.

In the calculation, θ ₁ and P₁ are selected to be constants, i.e., θ ₁ = P₁ = 1.

Figures 1 and 2 show the distributions of temperature and chemical potential along x at the instant t = 0.05 under differentα. As shown in Fig. 1, when α = 1, it triggers a jump at see Eq.(60)), which is referred to the wave front of the thermal wave.These results agree well with the predicted thermal wave front of Sherief^[42] at x = 0.352. It is also found that the smaller the fractional order α is, the farther the thermal signals arrive. The diffusive wave front see Eq.(60)) presented in Fig. 2 approximates the indication of Sherief^[43] (x = 0.923). As seen from Fig. 2, the smaller the fractional order α is, the smoother the distribution of the chemical potential is, i.e., the diffusive wave travels faster when the fractional order α is smaller.Furthermore, variations of the temperature and chemical potential for a small α are smoother than those for a larger one.

Figures 3 and 4 present the time history of the temperature and chemical potential at the point x = 0.1. Compared with the results depicted in Figs. 3 and 4, it can be concluded that the more reduced the fractional order, the larger the velocity of the thermal wave and diffusive wave. In other words, when the fractional order is small, the chemical potential (or temperature) of the point rises earlier, as shown in Figs. 3 and 4. Moreover, it can be observed that the curves of the temperature and chemical potential are smoothed by a small fractional-order parameter (the jump is shown when α = 1 is eliminated), and the distributions of temperature and chemical potential coincide under different fractional orders for a longer time.

Figure 5 displays the distributions of displacement along x at t=0.08 under varying α. Figure 5 shows that the displacement has significant alterations at the positions A andB. The point A is the location the elastic wave arrives at x=0.0483 (, see Eq.(60)), while the second one B represents the wave front of a thermal wave. More importantly, the fractional order can smooth the larger variation atB, but to a lesser effect on the first one (at A). This is logical because the fractional calculus is presented in the governing equations of the heat conduction and mass diffusion (see Eq.(60)). In addition, we ascertain that the fractional order may slightly diminish the displacement.

Figures 1-5 show the variations of thermoelastic wave and diffusive wave. It can be clearly observed that the non-zero values of all considered variables are only in a bounded region and identically vanish outside this region, which is entirely dominated by the nature of the finite speed of the thermoelastic wave and diffusive wave. Importantly, by incorporating the fractional-order parameter, it can preferably reflect the real wave propagation of thermoelastic wave, as well as diffusive wave.

6 Concluding remarks

The fractional-order generalized thermoelastic diffusion theory is established for anisotropic and linearly elastic diffusive media. A generalized variational theorem for the novel theory is developed by using the semi-inverse method. A numerical example is presented to demonstrate that the fractional order has a significant impact on the response. Notably, it can smooth the response when the material is subjected to a sudden shock, which commonly exists in practical engineering problems.