1 Introduction

The classical thermoelasticity theory is generalized and transformed into various thermoelasticity theories, called “generalized thermoelasticity”. One such theory comes from Lord and Shulman [1] and is defined as the generalized thermoelastic theory with relaxation times. Sharma et al. [2] studied the propagation of thermoelastic waves in a uniform isotropic plate in the Lord–Shulman context. Ezzat et al. [3] introduced a state-space formulation of boundary layer magneto hydrodynamic free convection with relaxation time. Abbas and Zenkour [4] presented an article on the LS model of the electromagnetic thermoelastic response of an infinite function gradient cylinder. Sadeghi and Kiani [5] discussed solutions for generalized magneto-thermoelasticity of layers using Green–Lindsay and Lord–Shulman-Green–Lindsay theories. A series of research papers were submitted to Sharif et al. [6,7,8,9,10] as an application of Lord and Shulman’s theory. Diffusion means the transport of material from high concentration to low concentration, which was recently investigated in an interesting study on its applications in industry, geophysics, and semiconductor industry, where the concentration is calculated using Fick’s law. Thermal diffusion in a solid elastic materials results from the coupling of temperature and mass diffusion coefficients, and the pressure field. Nowacki [11,12,13] formulated the theory of thermoelastic diffusion A coupled thermoelastic model with infinite wave propagation velocity is used. Sherief et al. [14] introduced the generalized thermoelastic diffusion theory. Sherief and Saleh [15] studied the half-space problem in the generalized thermoelastic diffusion theory. Abbas and Marin [16] proposed a method to solve the two-dimensional generalized thermoelastic diffusion problem using laser pulses. Afram, and Khader [17] used fractional thermoelastic diffusion theory to solve the problem of two-dimensional half-space. Choudhary et al. [18] studied the interactions of thermo-mechanical in a fractional order generalized thermoelastic solid with the diffusion. Hussein [19] solved the two-dimensional spherical region problem in the generalized thermoelectric theory of diffusion. A number of researchers [20,21,22,23,24,25] have conducted numerous studies on generalized thermoelastic diffusion. The theory of non-local elasticity takes into account the long-distance forces between atoms. Thus, the loads do not only depend on the loads at the individual point considered but on the entire body. Edelen et al. [26,27,28] introduced the theory of non-local elasticity. Khader and Marrouf [29] discussed the influence of non-local thermoelastic interactions on a half-space overlaid via a thick layer. The problem of the non-local theory of thermoelastic materials has been solved by Bachher and Sarkar [30], considering the voids and heat transfer’s fractional derivative. Narendar et al. [31] discussed the prediction of a non-local scale parameter for the dimensions of single-walled chairs and zig-zag, carbon nanotubes via the mechanics of molecular structural, non-local elasticity, and propagation of the wave. Zenkour et al. [32, 33] discussed the non-local thermoelasticity theory of a problem containing a nanobeam vibration. Singh [34] studied the Rayleigh surface waves in a problem of half-space with voids by using the non-local thermoelastic. Several researchers [35,36,37,38,39,40,41,42,43,44] have done much research on the theory of non-local thermoelasticity.

This article demonstrates the diffusion theory of thermoelasticity for a two-dimensional half-space problem. The surface is exposed to thermal shocks and hydrostatic loads. Chemical potential must be a known function of time. The solution is obtained with the direct method using Laplace’s techniques and Fourier’s exponential transform. The local and non-local thermoelastic are discussed, and we study the effect of non-local parameters on the generalized theory of thermoelastic diffusion.

2 Mathematical model of a non-local thermoelasticity with diffusion

The model takes the form of half-space (\(x \ge 0\,,\, - \,\infty < y < \infty\)) under the non-local thermoelastic with diffusion.

In non-local elasticity, the total internal energy is a function of the strain. The internal energy density coincides with those of the lattice dynamics at the discrete points occupied by the atoms [40, 45]. i.e., the stress tensor is dependent on the stress at all points of the body. The function \(\psi \left( {\left| {x - x^{\prime } } \right|} \right)\) is a non-local kernel that describes the effect of distant interactions of material points between x and x’ of the elastic body. The integral of the non-local kernel over the domain of integration is unity.

$$\int\limits_{v} {\psi \left( {\left| {x - x^{\prime } } \right|} \right){\text{d}}v = 1} ,$$

The ψ-function peaks at \(\left| {x - x^{\prime } } \right| = 0\) which generally decreases with increasing size, the influence of the kernel ψ behaves as a Direct-delta function. Eringen [40, 45] proved that the function G satisfies the relation

$$\left( {1 - \varepsilon^{2} \nabla^{2} } \right)\psi \left( {\left| {x - x^{\prime } } \right|} \right) = \delta \left( {\left| {x - x^{\prime } } \right|} \right),$$

where \(\varepsilon = e_{0} a/l\) is an elastic non-local parameter, \(a\) and e0 are internal characteristic length and material constant, respectively, and \(l\) is the external characteristic length. The differential form of the non-local stress tensor τ can be expressed by using Eringen’s assumptions

$$\left( {1 - \varepsilon^{2} \nabla^{2} } \right)\tau = \sigma .$$

The cubical dilatation e is thus given by:

$${e = \frac{{\partial u_{x} }}{\partial x} + \frac{{\partial u_{y} }}{\partial y}}.$$
(1)

Stress components are defined as:

$${\sigma_{xx} = (\lambda + {2}\mu )\frac{{\partial u_{x} }}{\partial x} + \lambda \frac{{\partial u_{y} }}{\partial y} - \gamma (T - T_{{0}} ) - \chi C},$$
(2)
$${\sigma_{yy} = (\lambda + {2}\mu )\frac{{\partial u_{y} }}{\partial y} + \lambda \frac{{\partial u_{x} }}{\partial x} - \gamma (T - T_{{0}} ) - \gamma_{1} C},$$
(3)
$${\sigma_{xy} = \lambda \left( {\frac{{\partial u_{x} }}{\partial y} + \frac{{\partial u_{y} }}{\partial x}} \right)}.$$
(4)

The motion equations are in tensor form \(\sigma_{ij,j} = \rho \ddot{u}_{i}\), and by substituting the stress components \(\sigma_{ij}\), we get [30]

$${\rho \left( {1 - \varepsilon^{2} \nabla^{2} } \right)\frac{{\partial^{{2}} u_{x} }}{{\partial t^{{2}} }} - (\lambda + \mu )\frac{\partial e}{{\partial x}} - \mu \nabla^{{2}} u_{x} + \gamma \frac{\partial T}{{\partial x}} + \gamma_{1} \frac{\partial C}{{\partial x}} = }0,$$
(5)
$${\rho \left( {1 - \varepsilon^{2} \nabla^{2} } \right)\frac{{\partial^{{2}} u_{y} }}{{\partial t^{{2}} }} - (\lambda + \mu )\frac{\partial e}{{\partial y}} - \mu \nabla^{{2}} u_{y} + \gamma \frac{\partial T}{{\partial y}} + \gamma_{1} \frac{\partial C}{{\partial y}} = }0.$$
(6)

The heat conduction equation

$${\frac{\partial }{\partial t}\left[ {1 + \tau_{0} \frac{\partial }{\partial t}} \right](\rho c_{E} T + \gamma T_{{0}} e + aT_{0} C)} - K\nabla^{{2}} T = 0.$$
(7)

The diffusion equation

$${D\gamma_{1} \nabla^{2} e + Da\nabla^{2} T + \left( {1 - \varepsilon^{2} \nabla^{2} } \right)\frac{\partial }{\partial t}\left[ {1 + \tau_{0} \frac{\partial }{\partial t}} \right]C - Db\nabla^{2} C = }0,$$
(8)
$${P = bC - \gamma_{1} e - a(T - T_{{0}} )}.$$
(9)

3 The problem solution

The non-dimensional variables:

$$\begin{gathered} \left\{ {u_{r} ,u_{y} ,x,y} \right\}^{*} = c_{{0}} \eta \left\{ {u_{r} ,u_{y} ,x,y} \right\},\;\sigma_{ij}^{*} = \frac{{\sigma_{ij} }}{\mu },\;\theta^{*} = \frac{{\gamma (T - T_{0} )}}{{\lambda + {2}\mu }}, \hfill \\ C^{*} = \frac{{\gamma_{1} C}}{{\lambda + {2}\mu }},\;(t,\tau_{0} )^{*} = c_{{0}}^{{2}} \eta (t,\tau_{0} ),\,P^{*} = \frac{P}{{\gamma_{1} }}, \hfill \\ \eta = \frac{{\rho c_{E} }}{k}{,}\;c_{{0}}^{{2}} = \frac{{\lambda + {2}\mu }}{\rho }. \hfill \\ \end{gathered}$$

Using the above non-dimensional variables to rewrite the above equations

$${\sigma_{xx} = {2}\frac{{\partial u_{x} }}{\partial x} + \beta^{{2}} \left( {e - \theta - C} \right) - {2}e},$$
(10)
$${\sigma_{yy} = {2}\frac{{\partial u_{y} }}{\partial y} - }\beta^{{2}} (e + \theta + C) + {2}e,$$
(11)
$${\sigma_{xy} = \frac{{\partial u_{x} }}{\partial y} + \frac{{\partial u_{y} }}{\partial x}},$$
(12)
$${\beta^{{2}} \left( {1 - \varepsilon^{2} \nabla^{2} } \right)\frac{{\partial^{{2}} u_{x} }}{{\partial t^{{2}} }} = \nabla^{{2}} u_{x} + \frac{\partial }{\partial x}\left[ {\beta^{{2}} \left( {e - \theta - C} \right) - e} \right]},$$
(13)
$${\beta^{{2}} \left( {1 - \varepsilon^{2} \nabla^{2} } \right)\frac{{\partial^{{2}} u_{y} }}{{\partial t^{{2}} }} = \nabla^{{2}} u_{y} + \frac{\partial }{\partial y}\left[ {\beta^{{2}} \left( {e - \theta - C} \right) - e} \right]},$$
(14)
$${\nabla^{{2}} \theta = \frac{\partial }{\partial t}\left[ {1 + \tau_{0} \frac{\partial }{\partial t}} \right](\theta + A_{1} e + A_{2} C)},$$
(15)
$${\nabla^{2} e + A_{3} \nabla^{2} \theta + A_{4} \left( {1 - \varepsilon^{2} \nabla^{2} } \right)\frac{\partial }{\partial t}\left[ {1 + \tau_{0} \frac{\partial }{\partial t}} \right]C = A_{5} \nabla^{2} C,}$$
(16)
$${P = A_{5} C - e - A_{3} \theta },$$
(17)

where \(\begin{gathered} A_{1} = \frac{{\gamma^{2} T_{{0}} }}{{\rho c_{E} (\lambda + {2}\mu )}},\;A_{2} = \frac{{aT_{0} \gamma }}{{\rho c_{E} \gamma_{1} }},\,A_{3} = \frac{{a(\lambda + {2}\mu )}}{{\gamma \gamma_{1} }}, \hfill \\ A_{4} = \frac{{\lambda + {2}\mu }}{{\gamma_{1}^{2} D\eta }},\;A_{5} = \frac{{b(\lambda + {2}\mu )}}{{\gamma_{1}^{2} }},\;\beta^{{2}} = \frac{{\lambda + {2}\mu }}{\mu }. \hfill \\ \end{gathered}\).

We apply the Laplace transform of both sides of the Eqs. (10)–(17), we get

$${\overline{\sigma }_{xx} = {2}\frac{{\partial \overline{u}_{x} }}{\partial x} + \beta^{{2}} \left( {\overline{e} - \overline{\theta } - \overline{C}} \right)2\overline{e}},$$
(18)
$${\overline{\sigma }_{yy} = {2}\frac{{\partial \overline{u}_{y} }}{\partial y} - \beta^{{2}} \left( {\overline{e} + \overline{\theta } + \overline{C}} \right) + 2\overline{e}},$$
(19)
$${\overline{\sigma }_{xy} = \frac{{\partial \overline{u}_{x} }}{\partial y} + \frac{{\partial \overline{u}_{y} }}{\partial x},}$$
(20)
$${\beta^{{2}} s^{2} \left( {1 - \varepsilon^{2} \nabla^{2} } \right)\overline{u}_{x} = \nabla^{{2}} \overline{u}_{x} + \frac{\partial }{\partial x}\left[ {\beta^{{2}} \left( {\overline{e} - \overline{\theta } - \overline{C}} \right) - \overline{e}} \right]},$$
(21)
$${\beta^{{2}} s^{2} \left( {1 - \varepsilon^{2} \nabla^{2} } \right)\overline{u}_{y} = \nabla^{{2}} \overline{u}_{y} + \frac{\partial }{\partial y}\left[ {\beta^{{2}} \left( {\overline{e} - \overline{\theta } - \overline{C}} \right) - \overline{e}} \right]},$$
(22)
$${\nabla^{{2}} \overline{\theta } = s(1 + \tau_{0} s)(\overline{\theta } + A_{1} \overline{e} + A_{2} \overline{C})},$$
(23)
$${\nabla^{2} \overline{e} + A_{3} \nabla^{2} \overline{\theta } + \left( {1 - \varepsilon^{2} \nabla^{2} } \right)A_{4} s(1 + \tau_{0} s)\overline{C} = A_{5} \nabla^{2} \overline{C}},$$
(24)
$${\overline{P} = A_{5} \overline{C} - \overline{e} - A_{3} \overline{\theta }}.$$
(25)

By using the Eqs. (21) and (22), we deduce:

$${\nabla^{{2}} \overline{\theta } + \nabla^{{2}} \overline{C}} - \left[ {\left( {1 + \varepsilon^{2} s^{2} } \right)\nabla^{{2}} - s^{2} } \right]\overline{e} = 0.$$
(26)

Eliminating \(\overline{\theta }\) and \({\overline{C} }\) from Eqs. (23), (24), and (26), to obtain

$${\left[ {\nabla^{{6}} - a_{{1}} \nabla^{{4}} + a_{{2}} \nabla^{{2}} - a_{{3}} } \right]\overline{e} = {0,}}$$
(27)

where

$$a_{1} = \frac{1}{\Delta }\left\{ \begin{gathered} \left( {1 + \varepsilon^{2} s^{2} } \right)A_{4} s(1 + \tau_{0} s) \hfill \\ + \left[ {s^{2} + s(1 + \tau_{0} s)(1 + \varepsilon^{2} s^{2} + A_{1} )} \right]\left[ {A_{5} + A_{3} + A_{4} \varepsilon^{2} s(1 + \tau_{0} s)} \right] \hfill \\ - A_{3} s^{2} - s(1 + \tau_{0} s)(1 - A_{2} )\left[ {A_{3} \left( {1 + \varepsilon^{2} s^{2} } \right) - 1} \right] \hfill \\ \end{gathered} \right\},$$

\(a_{2} = \frac{1}{\Delta }\left\{ \begin{gathered} s^{3} (1 + \tau_{0} s)\left[ {A_{5} + A_{3} + A_{4} \varepsilon^{2} s(1 + \tau_{0} s)} \right] \hfill \\ + A_{4} s(1 + \tau_{0} s)\left[ {s^{2} + s(1 + \tau_{0} s)(1 + \varepsilon^{2} s^{2} + A_{1} )} \right] \hfill \\ - A_{3} s^{3} (1 + \tau_{0} s)(1 - A_{2} ) \hfill \\ \end{gathered} \right\},\)

$$a_{3} = \frac{{A_{4} s^{4} (1 + \tau_{0} s)}}{\Delta }^{2} ,$$
$$\Delta = \left( {1 + \varepsilon^{2} s^{2} } \right)\left( {\left[ {A_{5} + A_{4} \varepsilon^{2} s(1 + \tau_{0} s)} \right] - 1} \right).$$

\({\overline{\theta } }\) and \({\overline{C} }\) achieved the equations

$${\left[ {\nabla^{{6}} - a_{{1}} \nabla^{{4}} + a_{{2}} \nabla^{{2}} - a_{{3}} } \right]\overline{\theta } = {0}},$$
$${\left[ {\nabla^{{6}} - a_{{1}} \nabla^{{4}} + a_{{2}} \nabla^{{2}} - a_{{3}} } \right]\overline{C} = {0}}.$$

Equation (27) can be written as follows:

$${\left( {\nabla^{{2}} - k_{{1}}^{{2}} } \right)\left( {\nabla^{{2}} - k_{{2}}^{{2}} } \right)\left( {\nabla^{{2}} - k_{{3}}^{{2}} } \right)\overline{e} = {0}},$$
(28)

where \({k_{{1}}^{{2}} ,\;k_{{2}}^{{2}} \;and\;k_{{3}}^{{2}} }\) are the roots of the of Eq.:

$${k^{{6}} - a_{{1}} k^{{4}} + a_{{2}} k^{{2}} - a_{{3}} = {0}}.$$

The solution of Eq. (28) takes the form

$${(\nabla^{{2}} - k_{j}^{{2}} )\overline{e}_{i} = {0}},\;i = 1,\,2,\,3.$$
(29)

To solve Eq. (29), we apply the Fourier transform on Eq. (29), and we obtain

$${(D^{{2}} - \alpha_{i}^{{2}} )\overline{e}_{i}^{*} = {0}},$$
(30)

where \({D = \frac{\partial }{\partial x}}\), and \({\alpha_{i}^{{2}} = q^{{2}} + k_{i}^{{2}} }\).

The solution of Eq. (30) is bounded at x > 0, and takes the form

$${\overline{e}^{ * } (x,\;q,\;s) = \sum\limits_{{i = {1}}}^{{3}} {A_{i} \left( {q,s} \right)} \;e^{{ - \alpha_{i} x}} }.$$
(31)

Similarly, by using the Eqs. (23), (24), and (26) we can get the solutions of \(\overline{\theta }^{*}\) and \({\overline{C} }\)

$${\overline{\theta }^{ * } \left( {x,\;q,\;s} \right) = \sum\limits_{{i = {1}}}^{{3}} {\frac{\left( {s\left[ {1 + \tau_{0} s} \right]\left( {k_{i}^{{2}} \left[ {A_{2} (1 + \varepsilon^{2} s^{2} ) + A_{1} } \right] - A_{2} s^{2} } \right)} \right)}{{k_{i}^{{2}} \left[ {k_{i}^{{2}} - s\left[ {1 + \tau_{0} s} \right](1 - A_{2} )} \right]}}A_{i} e^{{ - \alpha_{i} x}} } },$$
(32)
$${\overline{C}^{ * } \left( {x,\;q,\;s} \right) = \sum\limits_{{i = {1}}}^{{3}} \begin{gathered} \frac{(1 + \varepsilon^{2} s^{2} )k_{i}^{{4}} - k_{i}^{{2}} \left[ {s^{2} + s(\varepsilon^{2} s^{2} + 1 + A_{1} )\left[ {1 + \tau_{0} s} \right]} \right] + \left. {s^{3} \left[ {1 + \tau_{0} s} \right]} \right)}{{k_{i}^{{2}} \left[ {k_{i}^{{2}} - s\left[ {1 + \tau_{0} s} \right](1 - A_{2} )} \right]}} \hfill {A_{i} \;e^{{ - \alpha_{i} x}} } . \hfill \\ \end{gathered} }$$
(33)

To find the displacement \(u_{x}\), we apply the Fourier transform to Eq. (21) to get

$${\left[ {D^{{2}} - n^{{2}} } \right]\overline{u}_{x}^{*} = \frac{ - 1}{{\left( {1 + \beta^{2} s^{2} \varepsilon^{2} } \right)}}\sum\limits_{{i = {1}}}^{{3}} {} q_{i} \left[ {{1} - \frac{{\beta^{{2}} s^{{2}} }}{{k_{i}^{{2}} }}} \right]A_{i} e^{{ - \alpha_{i} x}} },$$
(34)

where \({n^{{2}} = \frac{{q^{{2}} + \beta^{{2}} s^{{2}} \left( {1 + q^{2} \varepsilon^{2} } \right)}}{{\left( {1 + \beta^{2} s^{2} \varepsilon^{2} } \right)}}}.\)

By using Eqs. (31–33) into Eq. (34), we get

$${\left[ {D^{{2}} - n^{{2}} } \right]\;\overline{u}_{x}^{*} = \frac{ - 1}{{\left( {1 + \beta^{2} s^{2} \varepsilon^{2} } \right)}}\sum\limits_{{i = {1}}}^{{3}} {} q_{i} \left[ {{1} - \frac{{\beta^{{2}} s^{{2}} }}{{k_{i}^{{2}} }}} \right]\;A_{i} e^{{ - \alpha_{i} x}} }.$$
(35)

The solution of Eq. (35) is bounded as \(x \to \infty\), and given by

$${\overline{u}_{x}^{*} = Bq^{{2}} e^{ - nx} - \sum\limits_{{i = {1}}}^{{3}} {\frac{{\alpha_{i} A_{i} }}{{k_{i}^{{2}} }}e^{{ - \alpha_{i} x}} ,} }$$
(36)

where \(B = B(q,s)\) is a constant depending on s and q.

In order to find \(u_{y}\), we use the Eqs. (1), (31) and (36)

$${\overline{u}_{y}^{*} = iq\left[ {nBe^{ - nx} - \sum\limits_{{i = {1}}}^{{3}} {\frac{{A_{i} }}{{k_{i}^{{2}} }}e^{{ - \alpha_{i} x}} } } \right]}.$$
(37)

By substituting Eqs. (31), (32), (33), (36), and (37) into equations into Eqs. (18), (20), and (25) we get the transformed terms stress and chemical potential.

$${\overline{\sigma }_{xx}^{*} = \sum\limits_{{i = {1}}}^{{3}} {\frac{{n^{2} }}{{k_{i}^{{2}} }}A_{i} e^{{ - \alpha_{i} x}} - {2}Bq^{2} ne^{ - nx} } },$$
(38)
$${\overline{\sigma }_{xy}^{*} = iq\left[ {{2}\sum\limits_{{i = {1}}}^{{3}} {\frac{{A_{i} }}{{k_{i}^{{2}} }}\alpha_{i} e^{{ - \alpha_{i} x}} - \left( {n^{{2}} + q^{{2}} } \right)Be^{ - nx} } } \right]},$$
(39)
$${\overline{P}^{*} = \sum\limits_{{i = {1}}}^{{3}} {\left\{ \begin{gathered} - 1 + A_{5}\frac{(1 + \varepsilon^{2} s^{2} )k_{i}^{{4}} - k_{j}^{{2}} \left[ {s^{2} + s(\varepsilon^{2} s^{2} + 1 + A_{1} )\left[ {1 + \tau_{0} s} \right]} \right] + s^{3} \left[ {1 + \tau_{0} s} \right]}{{k_{i}^{{2}} \left[ {k_{i}^{{2}} - s\left[ {1 + \tau_{0} s} \right](1 - A_{2} )} \right]}} \hfill \\ \begin{gathered} - A_{3} \frac{s\left[ {1 + \tau_{0} s} \right] {\left( k_{i}^{{2}} \left[ {A_{2} (1 + \varepsilon^{2} s^{2} ) + A_{1} } \right] - A_{2} s^{2}\right)}}{{k_{i}^{{2}} \left[ {k_{i}^{{2}} - s\left[ {1 + \tau_{0} s} \right](1 - A_{2} )} \right]}} \hfill \\ \end{gathered} \hfill \\ \end{gathered} \right\}} A{}_{i}e^{{ - \alpha_{i} x}} }.$$
(40)

Boundary conditions (B.C.) are given by:

$$\begin{gathered} \theta ({0,}\;y,\;t) = f_{{1}} (t),\;\sigma_{xx} ({0,}\;y,\;t) = f_{{2}} (t), \hfill \\ \sigma_{xy} ({0,}\;y,\;t) = {0,}\;P({0,}\;y,\;t) = f_{{3}} (t) \hfill \\ \end{gathered} .$$
(41)

Applying the Fourier and Laplace transforms of Eq. (41), we get

$$\begin{gathered} \overline{\theta }^{*} ({0,}\;q,\;s) = \overline{f}_{{1}}^{*} (s,\;q),\;\overline{\sigma }_{xx}^{*} ({0,}\;q,\;s) = \overline{f}_{{2}}^{*} \left( {s,q} \right), \hfill \\ \overline{\sigma }_{xy}^{*} ({0},\;q,\;s) = {0,}\;\overline{P}^{*} \left( {0,q,s} \right) = \overline{f}_{{3}}^{*} \left( {s,q} \right) \hfill \\ \end{gathered}$$
(42)

By substituting from Eqs. (32), (38), (39), and (40) into Eq. (42), we get the systems of equations of the parameters \(A_{1} ,\,A_{2} ,\,A_{3} ,\,{\text{and}}\,B\)

$$\begin{gathered} \sum\limits_{{i = {1}}}^{{3}} {\frac{ \left\{ {s\left[ {1 + \tau_{0} s} \right]\left( {k_{i}^{{2}} \left[ {A_{2} (1 + \varepsilon^{2} s^{2} ) + A_{1} } \right] - A_{2} s^{2} } \right)} \right\}}{{k_{i}^{{2}} \left[ {k_{i}^{{2}} - s\left[ {1 + \tau_{0} s} \right](1 - A_{2} )} \right]}}} \hfill A_{i} \; = \overline{f}_{{1}}^{*} (s,\;q) \hfill \\ \end{gathered} ,$$
(43)
$$\sum\limits_{{i = {1}}}^{{3}} {\frac{{n^{2} }}{{k_{i}^{{2}} }}A_{i} } - {2}Bq^{2} n = \overline{f}_{{2}}^{*} (s,\;q),$$
(44)
$${2}\sum\limits_{{i = {1}}}^{{3}} {\frac{{A_{i} }}{{k_{i}^{{2}} }}\alpha_{i} - \left( {n^{{2}} + q^{{2}} } \right) - B = 0} ,$$
(45)
$${ \sum\limits_{{i = {1}}}^{{3}} {\left\{ \begin{gathered} - 1 + A_{5}\frac{(1 + \varepsilon^{2} s^{2} )k_{i}^{{4}} - k_{j}^{{2}} \left[ {s^{2} + s(\varepsilon^{2} s^{2} + 1 + A_{1} )\left[ {1 + \tau_{0} s} \right]} \right] + s^{3} \left[ {1 + \tau_{0} s} \right]}{{k_{i}^{{2}} \left[ {k_{i}^{{2}} - s\left[ {1 + \tau_{0} s} \right](1 - A_{2} )} \right]}} \hfill \\ \begin{gathered} - A_{3} \frac{s\left[ {1 + \tau_{0} s} \right] {\left( k_{i}^{{2}} \left[ {A_{2} (1 + \varepsilon^{2} s^{2} ) + A_{1} } \right] - A_{2} s^{2}\right)}}{{k_{i}^{{2}} \left[ {k_{i}^{{2}} - s\left[ {1 + \tau_{0} s} \right](1 - A_{2} )} \right]}} \hfill \\ \end{gathered} \hfill \\ \end{gathered} \right\}} A{}_{i}=\overline{f}_{{3}}^{*} \left( {s,q} \right) }.$$
(46)

4 Inversion of the double Laplace–Fourier transform

A numerical technique was employed to calculate the values of the functions by reversing the double transforms mentioned earlier. Initially, a Fourier expansion-based numerical approach was utilized to invert the Laplace transforms [46]. Subsequently, the integrals present in the formula for the inverse of the Fourier transform were assessed numerically using a subroutine, which was a modified version of “qint” from the book [46]. The FORTRAN programming language was exclusively employed on a personal computer for this purpose.

5 Numerical results and discussion

For numerical evaluations, the problem constants have been given as [15]

$${\begin{gathered} D = {0}{\text{.85}} \times {10}^{{ - {8}}} ,T_{{0}} = {239}\;,\;\tau = {0}{\text{.2}},\;c_{E} = {383}{\text{.1}},\;\tau_{{0}} = {0}{\text{.02, }} \hfill \\ \alpha_{t} = {1}{\text{.78}} \times {10}^{{ - {5}}} ,\;k = {386}\;,\,\,\sigma = {8954}\;,\mu = {3}{\text{.86}} \times {10}^{{{10}}} ,\; \hfill \\ \alpha_{c} = {1}{\text{.98}} \times {10}^{{ - {4}}} \;,\,\;\lambda = {7}{\text{.76}} \times {10}^{{{10}}} ,\;\;a = {1}{\text{.2}} \times {10}^{{4}} ,\; \hfill \\ b = {0}{\text{.9}} \times {10}^{{6}} ,\,\eta = {8886}{\text{.73}}{.} \hfill \\ \end{gathered} }$$

\(f(y,t)\) are taken the form,

$${f_{{1}} (y,t) = \theta_{{0}} H\left( {h - \left| y \right|} \right)},$$
(47)
$${f_{{2}} (y,t) = f_{{0}} H\left( {h - \left| y \right|} \right),}$$
(48)
$${f_{{3}} (y,t) = P_{{0}} H\left( {h - \left| y \right|} \right).}$$
(49)

That is, by applying heat to the surface of the half-space in a narrow band 2 h wide around the y-axis to keep it at θ0, while the rest of the surface is at zero temperature. Equations (47–49) after applying Laplace and the Fourier transforms give the formula.

$${\overline{f}_{{1}}^{*} \left( {q,s} \right) = \theta_{{0}} \sqrt {\frac{{2}}{\pi }} \left( {\frac{{{\text{sin}}\left( {qh} \right)[1 - i\pi q\partial (q)]}}{qs}} \right)},$$
(50)
$${\overline{f}_{{2}}^{*} \left( {q,s} \right) = f_{{0}} \sqrt {\frac{{2}}{\pi }} \left( {\frac{{{\text{sin}}\left( {qh} \right)[1 - i\pi q\partial (q)]}}{qs}} \right)},$$
(51)
$${\overline{f}_{{3}}^{*} \left( {q,s} \right) = P_{{0}} \sqrt {\frac{{2}}{\pi }} \left( {\frac{{{\text{sin}}\left( {qh} \right)[1 - i\pi q\partial (q)]}}{qs}} \right)}.$$
(52)

Figures 1, 2, 3, 4 and 5, represent the temperature distributions, displacement, concentration, stress, and chemical potential, for different time values \(t = 0.05,\;t = 0.07,\;t = 0.1\).And we have taken \(\varepsilon = 0\) (local thermoelastic diffusion) [14, 15], were estimated on the x-axis \(\left( {y = 0} \right)\). Solid lines characterize the case when \(t = 0.05\), also the dash lines characterize the case when \(t = 0.07\), and the dot lines characterize the case when \(t = 0.1\).

Fig. 1
figure 1

Temperature distribution (Different Time)

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Fig. 2
figure 2

Displacement distribution  (Different Time)

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Fig. 3
figure 3

Concentration distribution  (Different Time)

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Fig. 4
figure 4

Stress distribution  (Different Time)

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Fig. 5
figure 5

Chemical potential distribution  (Different Time)

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In each figure, we can see that all the considered functions have a non-zero value only in a certain region of space, and outside of this region, they vanish in the same way through passage of time, this area distends. i.e., the function value increases over time. The boundary of this region is defined by the wavefront propagating at a finite speed. This does not apply to uncoupled and coupled theories of thermo elasticity, there’s inherently infinite propagation velocity, and all functions examined and have non-zero values everywhere in between, although these values can be very small. Even though these answers were derived mathematically with five-digit precision, the presence of a wavefront at a given role affects all considered functions due to the interaction between the equations that control it. The first waves and the second wave were found to be mainly thermo-mechanical in nature, although the third affected on the scattering of the function values before and after the arrival of the wavefront, similar to the propagation of waves in generalized thermoelasticity [14].

Figures 6, 7, 8, 9 and 10, represent the temperature distributions, displacement, concentration, stress, and chemical potential for three values of time \(\varepsilon = 0,\;\varepsilon = 0.03,\;\varepsilon = 0.05\). And we have taken \(t = 0.1\) (non-local thermoelastic diffusion), was estimated on the x-axis \(\left( {y = 0} \right)\). Solid lines represent the case when \(\varepsilon = 0\), the dash lines represent the case when \(\varepsilon = 0.03\), and the dot lines represent the case when \(\varepsilon = 0.05\).

Fig. 6
figure 6

Temperature distribution (Effect Non-local Parameter)

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Fig. 7
figure 7

Displacement distribution  (Effect Non-local Parameter)

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Fig. 8
figure 8

Concentration distribution (Effect Non-local Parameter)

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Fig. 9
figure 9

Stress distribution (Effect Non-local Parameter)

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Fig. 10
figure 10

Chemical potential distribution (Effect Non-local Parameter)

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Figure 6 represents the temperature distribution. We note from the graph that the behavior of the temperature does not change except at the front of the wave (slight change), and this is due to the effect of non-local elasticity. Figure 7 represents the displacement distribution. We note from the graph that the displacement increases from the negative value at \(x = 0\) to a positive value until the peak value is swallowed at \(x = 0.12\,,(\varepsilon = 0)\), \(x = 0.18\,,(\varepsilon = 0.03)\), \(x = 0.26\,,\,(\varepsilon = 0.05)\) then the position of the wavefront changes suddenly (\(\varepsilon = 0\)) [19] until it fades. In the case of non-local elasticity, we notice that the curve gradually increases until it reaches its maximum and value, then it gradually decreases. Figure 8 represents the concentration distribution; we notice that at \(\varepsilon = 0\), there are two discontinuities \(x = 0.1\,,\,x = 0.69\) [15]. In the case of non-local elasticity, we notice that the curve gradually decreases (has one discontinuity at \(x = 0.7\)), because the effect is not only at the individual point under consideration but at all points of the body [26, 27]. Figure 9 represents the stress distribution. We notice that at \(\varepsilon = 0\), the carve starts from the boundary condition and gradually increases until at \(x = 0.1\) (discontinuity) [15], Then it changes its direction and decreases gradually. In the case of non-local elasticity, the curve starts from the boundary condition and increases gradually in the negative direction until it reaches a maximum value at \(x = 0.1\) (continuity). Then it gradually decreases, i.e., the theory of non-local elasticity takes into account the action forces in the distance between atoms. Consequently, the loads do not only depend on the strains in a single point under consideration, but at all points of the body [26, 27]. Figure 10 represents the chemical potential distribution. We notice that at \(\varepsilon = 0\), the chemical potential at first decreases suddenly \(x = 0.1\), and then gradually decrease. In the case of non-local elasticity, the chemical potential at the beginning increases gradually until it reaches the maximum value at \(x = 0.08\), then it gradually decreases, and then it oscillates.

6 Conclusions

This paper demonstrates the theory of diffusion thermoelasticity for a two-dimensional half-space problem; the surface is subjected to thermal shock and hydrostatic loading. Local and non-local thermoelasticity are discussed, and the influence of non-local parameters on the general theory of thermoelastic diffusion is investigated. Based on the above analysis, we can draw the following conclusions: The conclusion of generalized thermoelastic diffusion is confirmed based on existing literature. Numerical results for non-local and local thermoelasticity are presented graphically. Non-local effects significantly affect the thermal transition behavior compared to non-local elastic materials. The results discussed in this article will be useful to researchers working on the advancement of thermoelasticity and novel materials and provide a theoretical basis for the design of nanoscale structures.