1. Introduction
The motion of a body can be properly described by the mutual interaction between deformation and temperature distribution. Biot [1] eliminated the classical uncoupled paradox inherent by proposing the coupled thermoelasticity theory with the consideration of the conventional Fourier's formulation. This paradox states that elastic distortion is independent of the change in temperature which is impractical. However, the utmost disadvantage of Biot's [1] proposition is the infinite speed of thermal signals. Afterward, many non-classical heat conduction laws indicating the finite speeds of thermal waves have been demonstrated [2], [3], [4], [5]. Chandrasekharaiah [6] proposed the dual-phase-lag (DPL) thermoelastic model based on the heat conduction law provided by Tzou [7]. A generalization of Tzou's theory, termed the three-phase-lag (TPL) thermoelastic model, was illustrated by Choudhuri [8]. In addition to the phase lags for the heat flux vector and the temperature gradient associated with DPL theory, TPL model introduces another delay parameter which is known as the phase lag for thermal displacement gradient. Unfortunately, both models established by Tzou [7] and Choudhuri [8] proceed towards ill-posed systems in terms of Hadamard. The modified Green-Naghdi theories have been introduced to thermo-diffusion models in several works [9], [10], [11], [12], [13]. The refined models of Green-Naghdi theories are employed to investigate the characteristics of thermomechanical waves in axisymmetric disks [14], [15]. Mashat and Zenkour [16] applied the nonlocal thermoelasticity theory based on Euler-Bernoulli hypothesis under unified DPL Green-Naghdi model for analysing thermoelastic vibration of temperature-dependent nanobeams subjected to a ramp type heating. Abouelregal [17] first detected that the correlation between the modified Fourier law and energy equations in these two theories generates a series of elements in the point spectrum. Hence, the real component becomes infinity which results in the discontinuity of the solution. Quintanilla [18] has derived a novel thermoelastic model subjected to Moore-Gibson-Thompson's (MGT) equation is constituted from a third-order differential equation. This modified theory incorporates a relaxation time parameter into the heat equation of Green-Naghdi theory of type III. This relaxation parameter represents the slackening time due to the fast transient effects of thermal inertia. Recently, implementations of MGT thermoelasticity theory in different material micro-structures have attracted the severe attention of several researchers [19], [20], [21], [22]. Abouelregal et al. [23] have considered a nonlocal isotropic magneto-thermoelastic half-space to conduct a computational study on the basis of MGT heat equation in the presence of a periodically varying heat flow. The characteristics of thermal and mechanical waves in an infinite thermo-viscoelastic orthotropic solid cylinder under MGT theory are analyzed by Abouelregal et al. [24] where the thermal conductivity is assumed to vary with temperature. Thermal conductivity is an important parameter that plays vital roles in different areas, such as mechanical and civil engineering, physics, chemistry, and thermal load analysis in semiconductors. Several analytical and experimental works have implied the dependency of thermal conductivity on temperature change [25], [26], [27]. Guo et al. [28] investigated a modified fractional-order generalized piezothermoelastic model with variable thermal conductivity. Othman et al. [29] elaborated the impact of variable thermal conductivity on a pre-stressed infinite fiber-reinforced plate. In recent years, many eminent researchers have introduced variable thermal conductivity in their studies [30], [31], [32], [33].
Extensive interest in fractional calculus has begun to grow recently as fractional-order differential equations have significant applications in various fields, e.g., continuum mechanics, biophysics, electrical engineering, bioengineering, viscoelasticity, signal and image processing, electrochemistry etc. Caputo [34] fractional derivative has been modified by Diethelm [35] in the following manner:
$D_{a}^{M}f\left( t \right)=\underset{a}{\overset{t}{\mathop \int }}\,K_{M}^{*}\left( t-\varepsilon \right){{f}^{\left( r \right)}}\left( \varepsilon \right)d\varepsilon $
with
$K_{M}^{*}\left( t-\varepsilon \right)=\frac{{{\left( t-\varepsilon \right)}^{r-M-1}}}{ \Gamma \left( r-M \right)}$
where ${{f}^{\left( r \right)}}$ symbolizes the usual $r$ th derivative of the function and $K_{M}^{*}\left( t-\varepsilon \right)$ refers to the fixed kernel function for a given real number M. According to the defined expression, $K_{M}^{*}\left( t-\varepsilon \right)$ has a singularity at t=ε. This formulation of a fractional-order derivative may be appropriately employed in the mechanical problems regarding plasticity, fatigue, and electromagnetic hysteresis [36]. The basic concept of memory dependent derivative (MDD) has been invented by Wang and Li [37]. The description of the first order (M=1) of function f is provided in the following integral form of a common derivative with a kernel function ${{K}^{*}}\left( t-\varepsilon \right)$ (chosen randomly) on a slipping interval $\left[ t-\epsilon,t \right]$:
${{D}_{\epsilon }}f\left( t \right)=\frac{1}{\epsilon }\underset{t-\epsilon }{\overset{t}{\mathop \int }}\,{{K}^{*}}\left( t-\varepsilon \right){f}'\left( \varepsilon \right)d\varepsilon $
where $\epsilon (>0)$ indicates delay time. In general, the memory effect requires weight $0\le {{K}^{*}}\left( t-\varepsilon \right)\le 1$ for $\varepsilon \in \left[ t-\epsilon,t \right]$ in order to achieve a smaller magnitude of ${{D}_{\epsilon }}f\left( t \right)$ as compared to that of the common derivative ${f}'\left( t \right)$. Clearly, the right-hand side of (3) is a weighted mean of ${f}'\left( t \right)$. By assuming our present time as t, it can be stated that $\left[ t-\epsilon,t \right]$ is our past. Thus, it may be concluded that the MDD possesses the feature that the functional value in real time is also dependent on the past time. Therefore, ${{D}_{\epsilon }}$ is known as a non-local operator. The kernel function ${{K}^{*}}\left( t-\epsilon \right)$ can be selected randomly, such as 1, $\left[ 1-\left( t-\varepsilon \right) \right]$, and ${{\left[ 1-\frac{t-\varepsilon }{\epsilon } \right]}^{q}}$, for any $q\left( \in \Re \right)$ which is related to a more realistic phenomenon. These functions are monotonically increasing from 0 to 1 in $\left[ t-\epsilon,t \right]$. An appropriate kernel function can be chosen with respect to the nature of the problem, e.g.
$\begin{align} & {{K}^{*}}\left( t-\varepsilon \right)=1-\frac{2{{a}_{1}}}{\epsilon }\left( t-\varepsilon \right)+\frac{a_{2}^{2}}{{{\epsilon }^{2}}}{{\left( t-\varepsilon \right)}^{2}} \\ & =1\ \ \ \ \ \ \ \text{if}\ \ \ \ {{a}_{2}}=0,{{a}_{1}}=0 \\ & =1-\frac{t-\varepsilon }{\epsilon }\ \ \ \ \ \ \ \ \text{if}\ \ \ \ {{a}_{2}}=0,{{a}_{1}}=0.5 \\ & =1-\left( t-\varepsilon \right)\ \ \ \ \ \ \ \text{if}\ \ \ \ {{a}_{2}}=0,{{a}_{1}}=0.5\epsilon \\ & ={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}\ \ \ \text{if}\ \ \ \ {{a}_{2}}=1,{{a}_{1}}=1. \\ \end{align}$
As a special case, when ${{K}^{*}}\left( t-\varepsilon \right)\equiv 1$ and $\epsilon \to 0$, we get
${{D}_{\epsilon }}f\left( t \right)=\frac{1}{\epsilon }\underset{t-\epsilon }{\overset{t}{\mathop \int }}\,{f}'\left( \varepsilon \right)d\varepsilon =\frac{f\left( t \right)-f\left( t-\epsilon \right)}{\epsilon }\to {f}'\left( t \right).$
The equation written above delineates that $\frac{d}{dt}$ is a limiting case of ${{D}_{\epsilon }}$ as $\epsilon \to 0$.
The heat conduction law in MDD MGT thermoelasticity is written as below following Quintanilla [18]
$\left( 1+\epsilon {{D}_{\epsilon }} \right){{q}_{i}}=-\left[ \kappa \nabla \theta +{{\kappa }^{*}}\nabla \Phi \right].$
The behavior analysis of any particle in the interior of a physical substance is illuminated by the postulates of nonlocality in the light of functional fundamental relations. According to the nonlocal theory, a particular particle characteristic depends on the state of every point in the substance. This aspect makes the nonlocal theory different from the classical theory. The nonlocal atomic interactions have profound usage in solid-state physics for determining material properties. The nonlocal theory deals with the internal characteristic length parameter which precisely estimates the physical features of material at nano-scale. Eringen and Edelen [45] proposed the preliminary idea of nonlocal elasticity. Afterward Eringen's [46], [47], [48] intense efforts in extending this theory opened new paths in different directions. The uniqueness of this theory has been discussed by Altan [49]. Cracium [50] established a nonlocal thermoelastic model. A nonlocal Fourier's law is developed by Challamel et al. [51] subjected to one-dimensional and two-dimensional thermal lattices. Bachher and Sarkar [52] have worked on thermoelastic material with voids under the fractional-order heat equation in the context of Eringen's nonlocal elasticity theory. The propagation of plane waves in a nonlocal thermoelastic half-space with voids under DPL heat conduction law is studied by Mondal et al. [53]. Abouelregal [54] applied the Taylor expansion method for higher-order time derivatives and nonlocality in the analysis of a new nonlocal thermoelasticity theory with dual-phase delays. Enlightened by Eringen's nonlocal theory, Mondal [55] has considered a magneto-thermoelastic rod to study memory response due to moving heat source based on Lord-Shulman theory of thermoelasticity. Recently, the theory of nonlocal elasticity is being embraced in various researches corresponding to different areas [56], [57], [58], [59].
Myriad contributions of porosity in various fields of science and technology, such as chemical engineering, geophysics, biomechanics, bone mechanics, and the petroleum industry [60], [61], [62], [63] have provoked the incorporation of elastic materials with voids into different research topics. By studying the linear theory of elastic materials with voids, Cowin and his co-worker [64], [65] formulated the void volume fraction as a ratio of the bulk density of porous material to that of the complete matrix material. Lesan [66] determined the thermo-mechanical behavior of substances with voids. The theory of double porosity, comprising matrix blocks disconnected by fracture networks, was conceptualized by Barenblatt et al. [67]. The fundamental relations and macroscopic momentum for a dual-porosity/dual-permeability medium were provided by Berryman and Wang [60]. A good quantity of interesting research articles [68], [69], [70], [71] regarding the fluid-saturated double porous material has been published in recent years. Khalili and Selvadurai [62] explored the thermo-hydro-mechanical properties of a completely coupled double porous thermoelastic solid. Straughan [72] explained the conditions of stability and uniqueness of an elastic solid with double porosity. By the virtue of Darcy's law, several articles [72], [73], [74] employed the theory of double porosity which involves the displacement vector and the pressure associated with matrix pores and fractures. The thermoelasticity theory related to double porous solid based on Nunziato-Cowin theory of elastic material with voids has been analyzed by Lesan and Quintanilla [75]. The aforementioned researchers have also exhibited the uniqueness, the criterion of stability, and the condition for the solution to existing. Kumar et al. [76] analyzed Rayleigh wave propagation in a thermoelastic double porous half-space. Recently, many articles [77], [78], [79], [80] have discussed thermo-mechanical interactions in double poro-thermoelastic medium with voids. The micropolar theory of elasticity focuses on the microstructure of the material which is beneficial for modern engineering structures and high-frequency vibrations with short wavelengths. The linear micropolar theory was established by Eringen [81]. In addition to force stresses, this theory supports the couple stresses. Moreover, micropolar solids experience macro-deformations and micro-rotations. Eringen and his co-worker [82], [83] developed the elasticity theory of micropolar solids where particle micromotions within a micro-volume element along its centroid are taken into account. The collaboration between micropolar theory and thermal effects was first presented by Nowacki [84], [85], [86]. Othman and Singh [87] studied the effect of rotation on a micropolar thermoelastic half-space under five generalized thermoelasticity theories. Othman and Lotfy [88] formulated a two-dimensional model consisting of micropolar thermoelastic material with voids for executing the effects of thermal relaxation times on all the physical variables related to deformation. Ezzat and Awad [89] introduced the constitutive relations, uniqueness of solution, and impact of thermal shock in the generalized micropolar thermoelasticity theory with two temperatures. Othman et al. [90] considered a rotating micropolar thermoelastic medium under DPL theory to examine the impact of two temperatures. Hilal et al. [91] studied the effect of the gravitational field on the deformation of a rotating micropolar thermoelastic medium subjected to microtemperatures. Intense use of the theory of micropolar thermoelasticity has begun to grow in the recent decade [92], [93], [94].
Several eminent researchers [91], [92], [95] have explored different thermo-mechanical properties of micropolar thermoelastic medium under distinct thermoelasticity theories. Yadav [94] has analyzed the propation of plane waves in an initially stressed micropolar diffusive porous media in the context of fractional-order thermoelasticity theory. Recently, the thermo-mechanical interaction in a functionally graded double porous thermoelastic model with gravitational effect has been studied by Kalkal et al. [96]. Kumar et al. [97] have considered a micropolar thermoelastic medium with voids subjected to Eringen's nonlocal elasticity theory under DPL and Lord-Shulman (LS) models. Moreover, the applications of MDD in thermoelasticity have gained significant attention from many researchers [40], [41], [42]. MGT thermoelasticity theory subjected to temperature-dependent properties has been considered in several inspiring works [21], [24], [98]. To the best of the authors’ knowledge, no study has been carried out with respect to the collaboration between Eringen's nonlocal theory and micropolar double porous thermoelastic material with voids (MDPTMWV). On the other hand, as far as the authors are concerned, so far MDD MGT heat equation has not been incorporated in any research work. Therefore, the present article intends to determine memory response in a nonlocal MDPTMWV half-space with variable thermal conductivity. In addition to this, the novel concept of MGT equation is applied to observe the analytical and graphical behavior of all the considered physical variables due to deformation. The analytical expressions of the displacements, temperature, void volume fractions, microrotation vector, force stresses, and equilibrated stresses are deduced via the normal mode technique. By utilizing the numerical values of the relevant parameters for magnesium crystal-like material, the effects of kernel functions, thermal conductivity, and nonlocality parameters are determined graphically. Moreover, different generalized thermoelasticity theories with variable conductivity, i.e., Moore-Gibson-Thompson thermoelasticity (MGTTE), Green-Naghdi III thermoelasticity (GN-IIITE), Lord-Shulman thermoelasticity (LSTE), and classical coupled thermoelasticity (CCTE) theories are compared to observe the changes in the distributions of the aforementioned field variables. The linear theory of micropolar elasticity describes the behavior of many new synthetic materials of the elastomer and polymer type as well as it is extensively used for the bone modeling. The theory of micropolar generalized thermoelasticity can be utilized by engineers for designing machine components, such as heat exchangers and boiler tubes. Multiple poroelasticity theory has several benefits in production of gas and energy, nuclear waste treatment, oil acquisition, carbon sequestration, and tissue engineering etc. Furthermore, the nonlocal elasticity theory has widespread applications in mass stream sensors, actuators, frequency synthesis, flexural-mode micromechanical and nanomechanical beam resonators, and ultra sensitive mass detection. In view of these myriad contributions of the nonlocal micropolar double poro-thermoelasticity, authors have gained the motivation to analyze the thermoelastic deformations in the considered model.
2. Basic equations for nonlocal MDPTMWV
2.1. Strain-displacement relation
${{\upsilon }_{ij}}=0.5\left( {{u}_{i,j}}+{{u}_{j,i}} \right).$
2.2. Nonlocal stress-strain-temperature relations
$\begin{align} & \left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{ij}}=\sigma _{ij}^{l}= \Lambda {{\upsilon }_{rr}}{{\delta }_{ij}}+2\mu {{\upsilon }_{ij}}+K\left( {{u}_{j,i}}-{{\epsilon }_{ijr}}{{\varphi }_{r}} \right) \\ & +\beta _{1}^{*}\psi {{\delta }_{ij}}+\beta _{2}^{*}\phi {{\delta }_{ij}}-{{\gamma }_{1}}\theta {{\delta }_{ij}} \\ \end{align}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\mu }_{ij}}=\mu _{ij}^{l}=\alpha {{\varphi }_{r,r}}{{\delta }_{ij}}+\lambda {{\varphi }_{i,j}}+\eta {{\varphi }_{j,i}}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{i}}=\sigma _{i}^{l}={{\vartheta }_{1}}{{\psi }_{,i}}+{{\vartheta }_{2}}{{\phi }_{,i}}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\varsigma }_{i}}=\varsigma _{i}^{l}={{\vartheta }_{2}}{{\psi }_{,i}}+{{\vartheta }_{3}}{{\phi }_{,i}}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{ \Xi }_{1}}= \Xi _{1}^{l}=-\beta _{1}^{*}{{\upsilon }_{jj}}-\zeta _{1}^{*}\psi -\zeta _{2}^{*}\phi +{{\nu }_{1}}\theta $
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{ \Xi }_{2}}= \Xi _{2}^{l}=-\beta _{2}^{*}{{\upsilon }_{jj}}-\zeta _{2}^{*}\psi -\zeta _{3}^{*}\phi +{{\nu }_{2}}\theta $
where $\tau ={{m}_{0}}{{\mu }_{cl}}$, ${{\gamma }_{1}}=\left( 3 \Lambda +2\mu +K \right){{\alpha }_{\theta }}$, and $\theta ={{\theta }_{a}}-{{\theta }_{0}}$ with the assumption $\mid \theta /{{\theta }_{0}}\mid \ll 1$.
2.3. Nonlocal stress equation of motion
$\begin{align} & \left( \Lambda +\mu \right)\nabla \left( \nabla.\mathbf{\vec{u}} \right)+\left( \mu +K \right){{\nabla }^{2}}\mathbf{\vec{u}}+K\nabla \times \mathbf{\vec{\varphi }}+\beta _{1}^{*}\nabla \psi \\ & +\beta _{2}^{*}\nabla \phi -{{\gamma }_{1}}\nabla \theta =\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right)\left( \rho \mathbf{\vec{u}}{{,}_{tt}}-\mathbf{\vec{F}} \right). \\ \end{align}$
2.4. Nonlocal couple stress equation of motion
$\begin{align} & \left( \alpha +\lambda +\eta \right)\nabla \left( \nabla.\mathbf{\vec{\varphi }} \right)-\eta \nabla \times \left( \nabla \times \mathbf{\vec{\varphi }} \right)+K\nabla \times \mathbf{\vec{u}}-2K\mathbf{\vec{\varphi }} \\ & =\rho \left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right)\varpi {{{\mathbf{\vec{\varphi }}}}_{,tt}}. \\ \end{align}$
2.5. Nonlocal void equations of motion
$\sigma _{i,i}^{l}+\text{ } \Xi _{1}^{l}+\rho \left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{l}_{1}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{ \Pi }_{1}}{{\psi }_{,tt}}$
$\varsigma _{i,i}^{l}+ \Xi _{2}^{l}+\rho \left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{l}_{2}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{ \Pi }_{2}}{{\phi }_{,tt}}.$
2.6. Memory-dependent nonlocal MGT heat conduction equation
Following Eq. (5), the generalized memory-dependent nonlocal MGT heat conduction law for MDPTMWV is proposed as
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right)\left( 1+\epsilon {{D}_{\epsilon }} \right){{q}_{i}}=-\left[ \kappa \nabla \theta +{{\kappa }^{*}}\nabla \Phi \right].$
In view of Eringen's nonlocal theory of thermoelasticity [46], the energy equation in the absence of heat source may be written as
$\rho {{\theta }_{0}}{{s}_{,t}}=-{{q}_{i,i}}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right)\rho s={{\left( \rho s \right)}^{l}}={{\gamma }_{1}}{{\upsilon }_{jj}}+{{\nu }_{1}}\psi +{{\nu }_{2}}\phi +a\theta.$
Using Eqs. (17)-(19), the heat conduction equation for MGT thermoelasticity with memory-dependent derivative is given by
${{\left[ \nabla.\left( \kappa \nabla \theta \right)+\nabla.\left( {{\kappa }^{*}}\nabla \Phi \right) \right]}_{,t}}=\left( 1+\epsilon {{D}_{\epsilon }} \right)\left[ {{\left( \rho {{C}_{s}}{{\theta }_{,t}} \right)}_{,t}}+{{\theta }_{0}}{{\left( {{\gamma }_{1}}\upsilon +{{\nu }_{1}}\psi +{{\nu }_{2}}\phi \right)}_{,tt}} \right]$
where ${{ \Phi }_{,t}}=\theta $, $\upsilon ={{\upsilon }_{jj}}$, and $a{{\theta }_{0}}=\rho {{C}_{s}}$. The detailed description of the aforesaid terms along with some other terms mentioned later have been represented in a tabulated form (Table 1).
Table 1. Nomenclature. |
| Symbols | Description | Symbols | Description |
|---|---|---|---|
| $\Lambda $,$\mu $ | Lame's parameters | ${{\epsilon }_{ijk}}$ | Levi-Civita symbol |
| $\alpha $,$\lambda $,$\eta $,$K$ | Micropolar material constants | ${{\upsilon }_{ij}}$ | Strain tensor |
| ${{\sigma }_{ij}}$ | Nonlocal stress tensor | ${{\mu }_{ij}}$ | Nonlocal couple stress tensor |
| ${{\sigma }_{i}}$,${{\varsigma }_{i}}$ | Nonlocal equilibrated stress vectors of matrix and fracture pores, respectively | ${{\Xi }_{1}}$,${{\Xi }_{2}}$ | Nonlocal equilibrated body forces of matrix and fracture pores, respectively |
| $\sigma _{ij}^{l}$ | Local stress tensor | $\mu _{ij}^{l}$ | Local couple stress tensor |
| $\sigma _{j}^{l}$,$\varsigma _{i}^{l}$ | Local equilibrated stress vectors of matrix and fracture pores, respectively | $\Xi _{1}^{l}$,$\Xi _{2}^{l}$ | Local equilibrated body forces of matrix and fracture pores, respectively |
| $\tau $ | Nonlocality parameter | ${{m}_{0}}$ | Material constants |
| ${{\mu }_{cl}}$ | Internal characteristic length | $\upsilon $ | Dilatation term |
| ${{\sigma }_{ij}}$ | Kronecker delta | $\vec{u}$ | Displacement vector |
| ${{u}_{1}}$,${{u}_{3}}$ | Displacement components of solid along x-and z-directions, respectively | $\vec{F}$ | Body force vector |
| $\rho $ | Density of material | $\vec{\varphi }$ | Microrotation vector |
| $\theta $ | Thermodynamic temperature | $\Phi $ | Thermal displacement |
| $\varpi $ | Microinertia | ${{l}_{1}}$,${{l}_{2}}$ | Extrinsic equilibrated body forces |
| ${{q}_{i}}$ | Component of heat flux vector | $\psi $,$\phi $ | Volume fractions of matrix and fracture pores, respectively |
| ${{\theta }_{a}}$ | Absolute temperature | ${{\theta }_{0}}$ | Reference temperature |
| $\kappa $ | Temperature-dependent thermal conductivity | ${{\kappa }^{*}}$ | Temperature-dependent material property corresponding to the rate of conductivity. |
| ${{\kappa }_{0}}$ | Thermal conductivity at reference temperature ${{\theta }_{0}}$ | $\kappa _{0}^{*}$ | Material property corresponding to the rate of conductivity at reference temperature ${{\theta }_{0}}$. |
| $\epsilon $ | Time relaxation parameter | s | Entropy per unit volume |
| Cs | Specific heat at constant strain | ${{\alpha }_{\theta }}$ | coefficient of linear thermal expansion |
| $\beta _{1}^{*}$, $\beta _{2}^{*}$,${{\vartheta }_{1}}$, ${{\vartheta }_{2}}$,${{\vartheta }_{3}}$,$\zeta _{1}^{*}$, $\zeta _{2}^{*}$, $\zeta _{3}^{*}$ | Void parameters | ${{\prod }_{1}}$,${{\prod }_{2}}$ | Equilibrated inertia of matrix and fracture pores, respectively |
| ${{\gamma }_{1}}$,${{v}_{1}}$,${{v}_{2}}$ | Thermal parameters | t | Time |
3. Formulation of the problem
In the present mathematical model, we consider a nonlocal, homogeneous, isotropic semi-infinite two-dimensional medium consisting of MDPTMWV under the purview of memory-dependent MGT thermoelasticity with variable thermal conductivity. It is assumed that the system is in undeformed phase initially at the reference temperature ${{\theta }_{0}}$. The positive z-axis is indicated vertically downward direction and the free surface of the substrate (i.e., z=0) is subjected to both the mechanical load and thermal shock. As the two-dimensional deformation is evaluated in the present article, the displacement vector $\vec{u}=({{u}_{1}}(x,z,t),0,{{u}_{3}}(x,z,t))$ and the microrotation vector $\vec{\varphi }=(0,{{\varphi }_{2}}(x,z,t),0)$. In the present mathematical formulation, some of the thermal properties of the MDPTMWV solid vary with the change in temperature which results in a nonlinear MGT heat conduction equation. Therefore the parameters $\kappa $, ${{\kappa }^{*}}$, and Cs become linearly dependent on temperature. However, the thermal diffusivity constant ${{k}_{d}}\left( =\kappa /\rho {{C}_{s}} \right)$ is assumed to be independent of temperature. Hence, we replace the parameters $\kappa $ and ${{\kappa }^{*}}$ by the relations
$\begin{align} & \kappa ={{\kappa }_{0}}\left( 1+{{\kappa }_{1}}\theta \right) \\ & {{\kappa }^{*}}=\kappa _{0}^{*}\left( 1+{{\kappa }_{2}} \Phi \right) \\ \end{align}$
where ${{\kappa }_{1}}$, ${{\kappa }_{2}}\le 0$ and ${{\kappa }_{0}}$, $\kappa _{0}^{*}$ are thermal parameters at reference temperature ${{\theta }_{0}}$.
Substituting Eq. (21) into Eq. (20) we obtain the nonlinear MDD MGT heat conduction equation in the following form
$\begin{align} & {{\kappa }_{0}}{{\left[ \nabla.\left( \left( 1+{{\kappa }_{1}}\theta \right)\nabla \theta \right) \right]}_{,t}}+\kappa _{0}^{*}{{\left[ \nabla.\left( \left( 1+{{\kappa }_{2}} \Phi \right)\nabla \Phi \right) \right]}_{,t}} \\ & =\left( 1+\epsilon {{D}_{\epsilon }} \right)\left[ {{\left( \rho {{C}_{s}}{{\theta }_{,t}} \right)}_{,t}}+{{\theta }_{0}}{{\left( {{\gamma }_{1}}\upsilon +{{\nu }_{1}}\psi +{{\nu }_{2}}\phi \right)}_{,tt}} \right] \\ \end{align}$
By following Youssef [99], let us introduce new functions which are as follows
${{T}_{1}}=\underset{0}{\overset{\theta }{\mathop \int }}\,\left( 1+{{\kappa }_{1}}\vartheta \right)d\vartheta \ {{T}_{2}}=\underset{0}{\overset{ \Phi }{\mathop \int }}\,\left( 1+{{\kappa }_{2}}\vartheta \right)d\vartheta.$
Integrating Eq. (23) we get
${{T}_{1}}=\theta \left( 1+0.5{{\kappa }_{1}}\theta \right)\ {{T}_{2}}= \Phi \left( 1+0.5{{\kappa }_{2}} \Phi \right).$
Differentiating (23) with respect to $i\left( =x,z \right)$ we obtain
$\nabla {{T}_{1}}=\frac{\kappa }{{{\kappa }_{0}}}\nabla \theta \ \nabla {{T}_{2}}=\frac{{{\kappa }^{*}}}{\kappa _{0}^{*}}\nabla \Phi .$
Again differentiating (25) with respect to $i\left( =x,z \right)$ we get
${{\nabla }^{2}}{{T}_{1}}=\nabla.\left( \frac{\kappa }{{{\kappa }_{0}}}\nabla \theta \right)\ {{\nabla }^{2}}{{T}_{2}}=\nabla.\left( \frac{{{\kappa }^{*}}}{\kappa _{0}^{*}}\nabla \Phi \right).$
Differentiating (23) with respect to t we yield
${{T}_{1,t}}=\frac{\kappa }{{{\kappa }_{0}}}{{\theta }_{,t}}\ {{T}_{2,t}}=\frac{{{\kappa }^{*}}}{\kappa _{0}^{*}}{{ \Phi }_{,t}}.$
Using the relation $\theta ={{ \Phi }_{,t}}$, from Eq. (27) we get
${{T}_{2,t}}=\frac{{{\kappa }^{*}}}{\kappa _{0}^{*}}\theta.$
With the help of Eqs. (26)–(28) and assuming $\mid \theta /{{\theta }_{0}}\mid \ll 1$ we get the following result
${{\left( {{\nabla }^{2}}{{T}_{2}} \right)}_{,t}}\approx {{\nabla }^{2}}{{T}_{1}}.$
In view of Eq. (29), Eq. (22) is transformed as follows
${{\left( {{\nabla }^{2}}{{T}_{1}} \right)}_{,t}}+\frac{\kappa _{0}^{*}}{{{\kappa }_{0}}}{{\nabla }^{2}}{{T}_{1}}=\left( 1+\epsilon {{D}_{\epsilon }} \right)\left[ \frac{1}{{{k}_{d}}}{{T}_{1,tt}}+\frac{{{\theta }_{0}}}{{{\kappa }_{0}}}{{\left( {{\gamma }_{1}}\upsilon +{{\nu }_{1}}\psi +{{\nu }_{2}}\phi \right)}_{,tt}} \right].$
Neglecting body forces and in view of Eq. (24), the constitutive relations (7)-(12) and equations of motion (13)-(16) for the considered two-dimensional model become
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{xx}}=\sigma _{xx}^{l}=\left( \Lambda +2\mu +K \right){{u}_{1,x}}+ \Lambda {{u}_{3,z}}+\beta _{1}^{*}\psi +\beta _{2}^{*}\phi -{{\gamma }_{1}}\left( \frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}}}{{{\kappa }_{1}}} \right),$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{zz}}=\sigma _{zz}^{l}= \Lambda {{u}_{1,x}}+\left( \Lambda +2\mu +K \right){{u}_{3,z}}+\beta _{1}^{*}\psi +\beta _{2}^{*}\phi -{{\gamma }_{1}}\left( \frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}}}{{{\kappa }_{1}}} \right),$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{zx}}=\sigma _{zx}^{l}=\left( \mu +K \right){{u}_{1,z}}+\mu {{u}_{3,x}}-K{{\varphi }_{2}},$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\mu }_{zy}}=\mu _{zy}^{l}=\eta {{\varphi }_{2,z}}\ \left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\mu }_{xy}}=\mu _{xy}^{l}=\eta {{\varphi }_{2,x}}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{x}}=\sigma _{x}^{l}={{\vartheta }_{1}}{{\psi }_{,x}}+{{\vartheta }_{2}}{{\phi }_{,x}}\ \left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{z}}=\sigma _{z}^{l}={{\vartheta }_{1}}{{\psi }_{,z}}+{{\vartheta }_{2}}{{\phi }_{,z}},$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\varsigma }_{x}}=\varsigma _{x}^{l}={{\vartheta }_{2}}{{\psi }_{,x}}+{{\vartheta }_{3}}{{\phi }_{,x}}\ \left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\varsigma }_{z}}=\varsigma _{z}^{l}={{\vartheta }_{2}}{{\psi }_{,z}}+{{\vartheta }_{3}}{{\phi }_{,z}},$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{ \Xi }_{1}}= \Xi _{1}^{l}=-\beta _{1}^{*}\left( {{u}_{1,x}}+{{u}_{3,z}} \right)-\zeta _{1}^{*}\psi -\zeta _{2}^{*}\phi +{{\nu }_{1}}\left( \frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}}}{{{\kappa }_{1}}} \right)$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{ \Xi }_{2}}= \Xi _{2}^{l}=-\beta _{2}^{*}\left( {{u}_{1,x}}+{{u}_{3,z}} \right)-\zeta _{2}^{*}\psi -\zeta _{3}^{*}\phi +{{\nu }_{2}}\left( \frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}}}{{{\kappa }_{1}}} \right)$
$\begin{align} & \left( \Lambda +2\mu +K \right){{u}_{1,xx}}+\left( \Lambda +\mu \right){{u}_{3,xz}}+\left( \mu +K \right){{u}_{1,zz}}-K{{\varphi }_{2,z}}+\beta _{1}^{*}{{\psi }_{,x}} \\ & +\beta _{2}^{*}{{\phi }_{,x}}-{{\gamma }_{1}}{{\left( \frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}}}{{{\kappa }_{1}}} \right)}_{,x}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right)\rho {{u}_{1,tt}}, \\ \end{align}$
$\begin{align} & \left( \mu +K \right){{u}_{3,xx}}+\left( \Lambda +\mu \right){{u}_{1,zx}}+\left( \Lambda +2\mu +K \right){{u}_{3,zz}}+K{{\varphi }_{2,x}}+\beta _{1}^{*}{{\psi }_{,z}} \\ & +\beta _{2}^{*}{{\phi }_{,z}}-{{\gamma }_{1}}{{\left( \frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}}}{{{\kappa }_{1}}} \right)}_{,z}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right)\rho {{u}_{3,tt}}, \\ \end{align}$
$\eta {{\nabla }^{2}}{{\varphi }_{2}}+K\left( {{u}_{1,z}}-{{u}_{3,x}} \right)-2K{{\varphi }_{2}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right)\rho \varpi {{\varphi }_{2,tt}},$
$\begin{align} & {{\vartheta }_{1}}{{\nabla }^{2}}\psi +{{\vartheta }_{2}}{{\nabla }^{2}}\phi -\beta _{1}^{*}\left( {{u}_{1,x}}+{{u}_{3,z}} \right)-\zeta _{1}^{*}\psi -\zeta _{2}^{*}\phi \\ & +{{\nu }_{1}}\left( \frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}}}{{{\kappa }_{1}}} \right)=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{ \Pi }_{1}}{{\psi }_{,tt}}, \\ \end{align}$
$\begin{align} & {{\vartheta }_{2}}{{\nabla }^{2}}\psi +{{\vartheta }_{3}}{{\nabla }^{2}}\phi -\beta _{2}^{*}\left( {{u}_{1,x}}+{{u}_{3,z}} \right)-\zeta _{2}^{*}\psi -\zeta _{3}^{*}\phi \\ & +{{\nu }_{2}}\left( \frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}}}{{{\kappa }_{1}}} \right)=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{ \Pi }_{2}}{{\phi }_{,tt}}. \\ \end{align}$
Let us introduce the non-dimensional variables in the following form
$\begin{align} & (x',z',\tau ')=\frac{{{c}_{1}}}{{{k}_{d}}}(x,z,\tau )\text{ }(u_{1}^{'},u_{3}^{'})=\frac{{{c}_{1}}}{{{k}_{d}}}({{u}_{1}},{{u}_{3}})\text{ }(t',\epsilon ')=\frac{c_{1}^{2}}{{{k}_{d}}}(t,\epsilon ) \\ & \theta '=\frac{\theta }{{{\theta }_{0}}}\text{ }\ \kappa_{1}^{'},\kappa_{2}^{'}={{\theta}_{0}}({{\kappa}_{1}},{{\kappa}_{2}})\text{}(T_{1}^{'},T_{2}^{'})=\frac{1}{{{\theta}_{0}}}({{T}_{1}},{{T}_{2}})\text{}\varphi_{2}^{'}\\&={{\varphi}^{2}}(\psi',\phi')=\frac{{{\prod}_{1}}c_{1}^{4}}{\beta_{1}^{*}k_{d}^{2}}(\psi,\phi)\\&\sigma_{ij}^{'}=\frac{{{\sigma}_{ij}}}{\Lambda+2\mu+K}\text{}\mu_{ij}^{'}=\frac{{{c}_{1}}}{{{\gamma}_{1}}{{\theta}_{0}}{{k}_{d}}}{{\mu}_{ij}}(\sigma_{i}^{'},\varsigma_{i}^{'})\\&=\frac{{{k}_{d}}}{{{\vartheta}_{1}}{{c}_{1}}}({{\sigma}_{i}},{{\varsigma}_{i}})\text{}{{c}_{1}}=\sqrt{\frac{\Lambda+2\mu+K}{\rho}}.\\\end{align}$
With the help of these non-dimensional variables mentioned in Eq. (44), Eqs. (31)–(43) and (30) for linearity are transformed as follows (omitting primes)
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{xx}}=\sigma _{xx}^{l}={{u}_{1,x}}+{{r}_{1}}{{u}_{3,z}}+{{r}_{2}}\psi +{{r}_{3}}\phi -{{r}_{4}}{{T}_{1}}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{zz}}=\sigma _{zz}^{l}={{r}_{1}}{{u}_{1,x}}+{{u}_{3,z}}+{{r}_{2}}\psi +{{r}_{3}}\phi -{{r}_{4}}{{T}_{1}}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{zx}}=\sigma _{zx}^{l}={{r}_{5}}{{u}_{1,z}}+{{r}_{6}}{{u}_{3,x}}-{{r}_{7}}{{\varphi }_{2}},$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\mu }_{zy}}=\mu _{zy}^{l}={{r}_{8}}{{\varphi }_{2,z}}\text{ }\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\mu }_{xy}}=\mu _{xy}^{l}={{r}_{8}}{{\varphi }_{2,x}}$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{x}}=\sigma _{x}^{l}={{r}_{9}}{{\psi }_{,x}}+{{r}_{10}}{{\phi }_{,x}}\text{ }\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\sigma }_{z}}=\sigma _{z}^{l}={{r}_{9}}{{\psi }_{,z}}+{{r}_{10}}{{\phi }_{,z}},$
$\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\varsigma }_{x}}=\varsigma _{x}^{l}={{r}_{10}}{{\psi }_{,x}}+{{r}_{11}}{{\phi }_{,x}}\text{ }\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\varsigma }_{z}}=\varsigma _{z}^{l}={{r}_{10}}{{\psi }_{,z}}+{{r}_{11}}{{\phi }_{,z}},$
${{u}_{1,xx}}+\left( {{r}_{1}}+{{r}_{6}} \right){{u}_{3,xz}}+{{r}_{5}}{{u}_{1,zz}}-{{r}_{7}}{{\varphi }_{2,z}}+{{r}_{2}}{{\psi }_{,x}}+{{r}_{3}}{{\phi }_{,x}}-{{r}_{4}}{{T}_{1,x}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{u}_{1,tt}},$
${{r}_{5}}{{u}_{3,xx}}+\left( {{r}_{1}}+{{r}_{6}} \right){{u}_{1,zx}}+{{u}_{3,zz}}+{{r}_{7}}{{\varphi }_{2,x}}+{{r}_{2}}{{\psi }_{,z}}+{{r}_{3}}{{\phi }_{,z}}-{{r}_{4}}{{T}_{1,z}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{u}_{3,tt}},$
${{r}_{12}}{{\nabla }^{2}}{{\varphi }_{2}}+{{r}_{13}}\left( {{u}_{1,z}}-{{u}_{3,x}} \right)-{{r}_{14}}{{\varphi }_{2}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\varphi }_{2,tt}},$
${{r}_{15}}{{\nabla }^{2}}\psi +{{r}_{16}}{{\nabla }^{2}}\phi -\left( {{u}_{1,x}}+{{u}_{3,z}} \right)-{{r}_{17}}\psi -{{r}_{18}}\phi +{{r}_{19}}{{T}_{1}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\psi }_{,tt}},$
${{r}_{20}}{{\nabla }^{2}}\psi +{{r}_{21}}{{\nabla }^{2}}\phi -{{r}_{22}}\left( {{u}_{1,x}}+{{u}_{3,z}} \right)-{{r}_{23}}\psi -{{r}_{24}}\phi +{{r}_{25}}{{T}_{1}}=\left( 1-{{\tau }^{2}}{{\nabla }^{2}} \right){{\phi }_{,tt}},$
${{\nabla }^{2}}{{T}_{1,t}}+{{r}_{26}}{{\nabla }^{2}}{{T}_{1}}=\left( 1+\epsilon {{D}_{\epsilon }} \right){{\left[ {{T}_{1}}+{{r}_{27}}\left( {{u}_{1,x}}+{{u}_{3,z}} \right)+{{r}_{28}}\psi +{{r}_{29}}\phi \right]}_{,tt}},$
where ${{r}_{i}}$'s, i=1(1)29 are delineated in Appendix A1.
4. Solution of the problem: normal mode technique
Normal mode technique has been employed in this study to derive the expressions of the displacement components, microrotation vector, thermodynamic temperature, volume fractions of matrix and fracture pores, normal, shear, and couple stress tensors, and equilibrated stress vectors. The primary aim of this method is to decompose the aforesaid variables and to provide accurate solutions without any presumed limitations on the physical variables that are present in the field equations. With regard to the normal mode, we assume the solution of the field variables in the following form:
$\begin{align} & \left[ {{u}_{1}},{{u}_{3}},{{\varphi }_{2}},{{T}_{1}},\psi,\phi,{{\sigma }_{ij}},{{\mu }_{ij}},{{\sigma }_{i}},{{\varsigma }_{i}} \right]\left( x,z,t \right) \\ & =\left[ {{{\bar{u}}}_{1}},{{{\bar{u}}}_{3}},{{{\bar{\varphi }}}_{2}},{{{\bar{T}}}_{1}},\bar{\psi },\bar{\phi },{{{\bar{\sigma }}}_{ij}},{{{\bar{\mu }}}_{ij}},{{{\bar{\sigma }}}_{i}},{{{\bar{\varsigma }}}_{i}} \right]\left( z \right){{e}^{\omega t+\iota nx}} \\ \end{align}$
where $\omega \left( ={{\omega }_{R}}+\iota {{\omega }_{I}} \right)$ is a complex wave frequency, n is the wave number towards x-direction, and ${{\bar{u}}_{1}}\left( z \right)$, ${{\bar{u}}_{3}}\left( z \right)$, ${{\bar{\varphi }}_{2}}\left( z \right)$, ${{\bar{T}}_{1}}\left( z \right)$, $\bar{\psi }\left( z \right)$, $\bar{\phi }\left( z \right)$, ${{\bar{\sigma }}_{ij}}\left( z \right)$, ${{\bar{\mu }}_{ij}}\left( z \right)$, ${{\bar{\sigma }}_{i}}\left( z \right)$, ${{\bar{\varsigma }}_{i}}\left( z \right)$ are the amplitudes of the aforementioned physical variables.
With the help of Eq. (57), the non-dimensional coupled governing Eqs. (51)-(56) can be rewritten as follows
$\left( {{S}_{1}}{{D}^{2}}-{{S}_{2}} \right){{\bar{u}}_{1}}+{{S}_{3}}D{{\bar{u}}_{3}}-{{r}_{7}}D{{\bar{\varphi }}_{2}}-{{S}_{4}}{{\bar{T}}_{1}}+{{S}_{5}}\bar{\psi }+{{S}_{6}}\bar{\phi }=0$
${{S}_{3}}D{{\bar{u}}_{1}}+\left( {{S}_{7}}{{D}^{2}}-{{S}_{8}} \right){{\bar{u}}_{3}}+{{S}_{9}}{{\bar{\varphi }}_{2}}-{{r}_{4}}D{{\bar{T}}_{1}}+{{r}_{2}}D\bar{\psi }+{{r}_{3}}D\bar{\phi }=0$
${{r}_{13}}D{{\bar{u}}_{1}}-{{S}_{10}}{{\bar{u}}_{3}}+\left( {{S}_{11}}{{D}^{2}}-{{S}_{12}} \right){{\bar{\varphi }}_{2}}=0$
${{S}_{13}}{{\bar{u}}_{1}}-D{{\bar{u}}_{3}}+{{r}_{19}}{{\bar{T}}_{1}}+\left( {{S}_{14}}{{D}^{2}}-{{S}_{15}} \right)\bar{\psi }+\left( {{r}_{16}}{{D}^{2}}-{{S}_{16}} \right)\bar{\phi }=0$
${{S}_{17}}{{\bar{u}}_{1}}-{{r}_{22}}D{{\bar{u}}_{3}}+{{r}_{25}}{{\bar{T}}_{1}}+\left( {{r}_{20}}{{D}^{2}}-{{S}_{18}} \right)\bar{\psi }+\left( {{S}_{19}}{{D}^{2}}-{{S}_{20}} \right)\bar{\phi }=0$
${{S}_{21}}{{\bar{u}}_{1}}+{{S}_{22}}D{{\bar{u}}_{3}}+\left( {{S}_{23}}{{D}^{2}}+{{S}_{24}} \right){{\bar{T}}_{1}}+{{S}_{25}}\bar{\psi }+{{S}_{26}}\bar{\phi }=0$
where $D=\frac{\partial }{\partial z}$ and ${{S}_{i}}$'s, i=1(1)26 are defined in Appendix A2.
By simplifying Eqs. (58)-(63), we obtain the conditions for the existence of a non-trivial solution leading to the characteristic equation in the following form
$\underset{i=1}{\overset{7}{\mathop \sum }}\,{{R}_{i}}{{D}^{14-2i}}\left( {{{\bar{u}}}_{1}},{{{\bar{u}}}_{3}},{{{\bar{\varphi }}}_{2}},{{{\bar{T}}}_{1}},\bar{\psi },\bar{\phi } \right)\left( z \right)=0$
where ${{R}_{i}}$'s are obtained by collecting the coefficients of ${{D}^{14-2i}}$ s, i=1(1)7.
As $z\to \infty $, the solution of Eq. (64) can be expressed in the following form
$\left[ {{{\bar{u}}}_{1}},{{{\bar{u}}}_{3}},{{{\bar{\varphi }}}_{2}},{{{\bar{T}}}_{1}},\bar{\psi },\bar{\phi } \right]\left( z \right)=\underset{i=1}{\overset{6}{\mathop \sum }}\,\left[ 1,H_{i}^{\left( 1 \right)},H_{i}^{\left( 2 \right)},H_{i}^{\left( 3 \right)},H_{i}^{\left( 4 \right)},H_{i}^{\left( 5 \right)} \right]{{M}_{i}}{{e}^{-{{\delta }_{i}}z}}$
where ${{\delta }_{i}}$, i=1(1)6 be the roots of Eq. (64) with positive real parts and the functions ${{M}_{i}}$, i=1(1)6 require to be determined.
The terms $H_{i}^{\left( 1 \right)}$, $H_{i}^{\left( 2 \right)}$, $H_{i}^{\left( 3 \right)}$, $H_{i}^{\left( 4 \right)}$, $H_{i}^{\left( 5 \right)}$, i=1(1)6 in Eq. (65) can be expressed in the following form
$H_{i}^{\left( 1 \right)}=-\frac{\mathop{\sum }_{j=0}^{9}\delta _{i}^{19-2j}{{G}_{122+j}}}{\mathop{\sum }_{j=0}^{9}\delta _{i}^{18-2j}{{G}_{112+j}}}$
$H_{i}^{\left( 2 \right)}=\frac{{{r}_{13}}{{\delta }_{i}}+{{S}_{10}}H_{i}^{\left( 1 \right)}}{{{S}_{11}}\delta _{i}^{2}-{{S}_{12}}}$
$H_{i}^{\left( 3 \right)}=-\frac{H_{i}^{\left( 1 \right)}\mathop{\sum }_{j=0}^{3}\delta _{i}^{7-2j}{{G}_{99+j}}+\mathop{\sum }_{j=0}^{3}\delta _{i}^{6-2j}{{G}_{108+j}}}{\mathop{\sum }_{j=0}^{4}\delta _{i}^{8-2j}{{G}_{103+j}}}$
$H_{i}^{\left( 4 \right)}=-\frac{H_{i}^{\left( 1 \right)}\left( \delta _{i}^{3}{{G}_{45}}+{{\delta }_{i}}{{G}_{46}} \right)+H_{i}^{\left( 3 \right)}\left( \delta _{i}^{2}{{G}_{47}}+{{G}_{48}} \right)+\left( \delta _{i}^{2}{{G}_{52}}+{{G}_{53}} \right)}{\delta _{i}^{4}{{G}_{49}}+\delta _{i}^{2}{{G}_{50}}+{{G}_{51}}}$
$H_{i}^{\left( 5 \right)}=-\frac{{{r}_{22}}{{\delta }_{i}}H_{i}^{\left( 1 \right)}+{{r}_{25}}H_{i}^{\left( 3 \right)}+\left( {{r}_{20}}\delta _{i}^{2}-{{S}_{18}} \right)H_{i}^{\left( 4 \right)}+{{S}_{17}}}{{{S}_{19}}\delta _{i}^{2}-{{S}_{20}}}$
where ${{G}_{j}}$, j=1(1)131 are defined in Appendix A3.
In view of Eqs. (57) and (65) we have
$\left[ {{u}_{1}},{{u}_{3}},{{\varphi }_{2}},{{T}_{1}},\psi,\phi \right]\left( x,z,t \right)=\underset{i=1}{\overset{6}{\mathop \sum }}\,\left[ 1,H_{i}^{\left( 1 \right)},H_{i}^{\left( 2 \right)},H_{i}^{\left( 3 \right)},H_{i}^{\left( 4 \right)},H_{i}^{\left( 5 \right)} \right]{{M}_{i}}{{e}^{-{{\delta }_{i}}z}}{{e}^{\omega t+\iota nx}}.$
By substituting Eq. (71) in Eqs. (45)-(50), the nonlocal stress components can be acquired as
$\begin{align} & \left[ {{\sigma }_{xx}},{{\sigma }_{zz}},{{\sigma }_{zx}},{{\mu }_{zy}},{{\mu }_{xy}},{{\sigma }_{x}},{{\sigma }_{z}},{{\varsigma }_{x}},{{\varsigma }_{z}} \right]\left( x,z,t \right) \\ & =\underset{i=1}{\overset{6}{\mathop \sum }}\,\left[ H_{i}^{\left( 6 \right)},H_{i}^{\left( 7 \right)},H_{i}^{\left( 8 \right)},H_{i}^{\left( 9 \right)},H_{i}^{\left( 10 \right)},H_{i}^{\left( 11 \right)},H_{i}^{\left( 12 \right)},H_{i}^{\left( 13 \right)},H_{i}^{\left( 14 \right)} \right]{{M}_{i}}{{e}^{-{{\delta }_{i}}z}}{{e}^{\omega t+\iota nx}} \\ \end{align}$
where the terms $H_{i}^{\left( 6 \right)}$, $H_{i}^{\left( 7 \right)}$, $H_{i}^{\left( 8 \right)}$, $H_{i}^{\left( 9 \right)}$, $H_{i}^{\left( 10 \right)}$, $H_{i}^{\left( 11 \right)}$, $H_{i}^{\left( 12 \right)}$, $H_{i}^{\left( 13 \right)}$, $H_{i}^{\left( 14 \right)}$, i=1(1)6 in Eq. (72) can be expressed in the following form
$\left. \begin{align} & H_{i}^{\left( 6 \right)}=\frac{\iota n-{{r}_{1}}{{\delta }_{i}}H_{i}^{\left( 1 \right)}-{{r}_{4}}H_{i}^{\left( 3 \right)}+{{r}_{2}}H_{i}^{\left( 4 \right)}+{{r}_{3}}H_{i}^{\left( 5 \right)}}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})} \\ & H_{i}^{\left( 7 \right)}=\frac{\iota n{{r}_{1}}-{{\delta }_{i}}H_{i}^{\left( 1 \right)}-{{r}_{4}}H_{i}^{\left( 3 \right)}+{{r}_{2}}H_{i}^{\left( 4 \right)}+{{r}_{3}}H_{i}^{\left( 5 \right)}}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})} \\ & H_{i}^{\left( 8 \right)}=\frac{-{{r}_{5}}{{\delta }_{i}}+\iota n{{r}_{6}}H_{i}^{\left( 1 \right)}-{{r}_{7}}H_{i}^{\left( 2 \right)}}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})}\text{ }H_{i}^{\left( 9 \right)}=\frac{-{{r}_{8}}{{\delta }_{i}}H_{i}^{\left( 2 \right)}}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})}\text{ }H_{i}^{\left( 10 \right)}=\frac{\iota n{{r}_{8}}H_{i}^{\left( 2 \right)}}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})} \\ & H_{i}^{\left( 11 \right)}=\frac{\iota n\left( {{r}_{9}}H_{i}^{\left( 4 \right)}+{{r}_{10}}H_{i}^{\left( 5 \right)} \right)}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})}H_{i}^{\left( 12 \right)}=\frac{-{{\delta }_{i}}\left( {{r}_{9}}H_{i}^{\left( 4 \right)}+{{r}_{10}}H_{i}^{\left( 5 \right)} \right)}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})} \\ & H_{i}^{\left( 13 \right)}=\frac{\iota n\left( {{r}_{10}}H_{i}^{\left( 4 \right)}+{{r}_{11}}H_{i}^{\left( 5 \right)} \right)}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})}H_{i}^{\left( 14 \right)}=\frac{-{{\delta }_{i}}\left( {{r}_{10}}H_{i}^{\left( 4 \right)}+{{r}_{11}}H_{i}^{\left( 5 \right)} \right)}{1-{{\tau }^{2}}(\delta _{i}^{2}-{{n}^{2}})} \\ \end{align} \right\}$
5. Boundary conditions
In the present work, a nonlocal, homogeneous, isotropic micropolar double porous thermoelastic material with voids half-space is considered which occupies the region $z\ge 0$. The free surface (i.e., z=0) of the half-space is subjected to both mechanical and thermal load (${{L}_{M}}$ and ${{L}_{T}}$, respectively), as displayed in Fig. 1. For the purpose of determining the unknowns ${{M}_{i}}$, i=1(1)6, the appropriate boundary conditions at z=0 for the considered problem are given by
Fig. 1. Geometry of the problem. |
5.1. Thermal boundary condition
$\theta \left( x,0,t \right)={{L}_{T}}\left( x,t \right)$
where ${{L}_{T}}\left( x,t \right)={{L}_{T}}{{e}^{\omega t+\iota nx}}$ and ${{L}_{T}}$ is the magnitude of thermal load.
In view of Eqs. (24) and (74) we get
${{T}_{1}}\left( x,0,t \right)=L_{T}^{*}\left( x,t \right)$
where $L_{T}^{*}\left( x,t \right)=L_{T}^{*}{{e}^{\omega t+\iota nx}}$ and $L_{T}^{*}={{L}_{T}}\left( 1+0.5{{\kappa }_{1}}{{L}_{T}}{{e}^{\omega t+\iota nx}} \right)$.
5.2. Normal and tangential stresses boundary conditions
${{\sigma }_{zz}}\left( x,0,t \right)=-{{L}_{M}}\left( x,t \right)$
${{\sigma }_{zx}}\left( x,0,t \right)=0$
where ${{L}_{M}}\left( x,t \right)={{L}_{M}}{{e}^{\omega t+\iota nx}}$ and ${{L}_{M}}$ is the magnitude of normal mechanical load.
5.3. Couple stress boundary condition
${{\mu }_{zy}}\left( x,0,t \right)=0.$
5.4. Equilibrated stresses boundary conditions
${{\sigma }_{z}}\left( x,0,t \right)=0$
${{\varsigma }_{z}}\left( x,0,t \right)=0.$
With the help of the above boundary conditions (74)-(80)and using the expressions (71) and (72) we get the system of six nonhomogeneous equations represented in the following matrix form
$\left( \begin{matrix} H_{1}^{(3)} & H_{2}^{(3)} & H_{3}^{(3)} & H_{4}^{(3)} & H_{5}^{(3)} & H_{6}^{(3)} \\ H_{1}^{(7)} & H_{2}^{(7)} & H_{3}^{(7)} & H_{4}^{(7)} & H_{5}^{(7)} & H_{6}^{(7)} \\ H_{1}^{(8)} & H_{2}^{(8)} & H_{3}^{(8)} & H_{4}^{(8)} & H_{5}^{(8)} & H_{6}^{(8)} \\ H_{1}^{(9)} & H_{2}^{(9)} & H_{3}^{(9)} & H_{4}^{(9)} & H_{5}^{(9)} & H_{6}^{(9)} \\ H_{1}^{(12)} & H_{2}^{(12)} & H_{3}^{(12)} & H_{4}^{(12)} & H_{5}^{(12)} & H_{6}^{(12)} \\ H_{1}^{(14)} & H_{2}^{(14)} & H_{3}^{(14)} & H_{4}^{(14)} & H_{5}^{(14)} & H_{6}^{(14)} \\ \end{matrix} \right)\left( \begin{matrix} {{M}_{1}} \\ {{M}_{2}} \\ {{M}_{3}} \\ {{M}_{4}} \\ {{M}_{5}} \\ {{M}_{6}} \\ \end{matrix} \right)=\left( \begin{matrix} L_{T}^{*} \\ -{{L}_{M}} \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{matrix} \right).$
The solution of the above system of Eq. (81) yields the expressions of ${{M}_{i}}$'s, i=1(1)6 in the following form
${{M}_{1}}=\frac{{{ \Omega }_{1}}}{ \Omega }{{M}_{2}}=\frac{{{ \Omega }_{2}}}{ \Omega }{{M}_{3}}=\frac{{{ \Omega }_{3}}}{ \Omega }{{M}_{4}}=\frac{{{ \Omega }_{4}}}{ \Omega }{{M}_{5}}=\frac{{{ \Omega }_{5}}}{ \Omega }{{M}_{6}}=\frac{{{ \Omega }_{6}}}{ \Omega }$
where Ω represents the 6×6 matrix present in Eq. (81) and ${{ \Omega }_{i}}$, i=1(1)6 are obtained by substituting the ith column by ${{\left( \begin{matrix} L_{T}^{*} & -{{L}_{M}} & 0 & 0 & 0 & 0 \\ \end{matrix} \right)}^{T}}$.
Substituting (82) into Eqs. (71) and (72), we determine the following analytical expressions of the displacements, temperature, void volume fractions, microrotation vector, and stress fields for a MDPTMWV substrate in the context of MDD MGT thermoelasticity with variable conductivity
$\begin{align} & \left[ {{u}_{1}},{{u}_{3}},{{\varphi }_{2}},{{T}_{1}},\psi,\phi \right]\left( x,z,t \right) \\ & =\frac{1}{ \Omega }\underset{i=1}{\overset{6}{\mathop \sum }}\,\left[ 1,H_{i}^{\left( 1 \right)},H_{i}^{\left( 2 \right)},H_{i}^{\left( 3 \right)},H_{i}^{\left( 4 \right)},H_{i}^{\left( 5 \right)} \right]{{ \Omega }_{i}}{{e}^{-{{\delta }_{i}}z}}{{e}^{\omega t+\iota nx}} \\ \end{align}$
$\begin{align} & \left[ {{\sigma }_{xx}},{{\sigma }_{zz}},{{\sigma }_{zx}},{{\mu }_{zy}},{{\mu }_{xy}},{{\sigma }_{x}},{{\sigma }_{z}},{{\varsigma }_{x}},{{\varsigma }_{z}} \right]\left( x,z,t \right) \\ & =\frac{1}{ \Omega }\underset{i=1}{\overset{6}{\mathop \sum }}\,\left[ H_{i}^{\left( 6 \right)},H_{i}^{\left( 7 \right)},H_{i}^{\left( 8 \right)},H_{i}^{\left( 9 \right)},H_{i}^{\left( 10 \right)},H_{i}^{\left( 11 \right)},H_{i}^{\left( 12 \right)},H_{i}^{\left( 13 \right)},H_{i}^{\left( 14 \right)} \right] \\ & {{ \Omega }_{i}}{{e}^{-{{\delta }_{i}}z}}{{e}^{\omega t+\iota nx}}. \\ \end{align}$
$\theta \left( x,z,t \right)=\frac{-1+\sqrt{1+2{{\kappa }_{1}}{{T}_{1}}\left( x,z,t \right)}}{{{\kappa }_{1}}}.$
6. Particular cases
Some particular cases have been considered in this section. Case-I delineates the absence of memory dependent derivative and variable thermal conductivity parameters. In addition to the conditions described in case-I, the nonlocality parameter and micropolar effect are neglected in case-II. Under case-I, equations of motion and heat conduction equation for MGTTE model, GN-IIITE model, GN-IITE model, LSTE model, and CCTE model are summarized and documented as follows:
6.1. Case I: the absence of memory-dependent derivative and variable thermal conductivity parameters
In the absence of memory-dependent derivative (i.e., ${{D}_{\epsilon }}f\left( t \right)\to {f}'\left( t \right)$) and variable thermal conductivity parameters (i.e., ${{\kappa }_{1}}={{\kappa }_{2}}=0$), the heat conduction equations of the following models can be acquired
(i) From Eq. (22), the MGT heat conduction equation in the absence of MDD and variable thermal conductivity is given by
${{\kappa }_{0}}{{\nabla }^{2}}{{\theta }_{,t}}+\kappa _{0}^{*}{{\nabla }^{2}}\theta =\left( 1+\epsilon \frac{\partial }{\partial t} \right)\left[ \rho {{C}_{s}}{{\theta }_{,tt}}+{{\theta }_{0}}{{\left( {{\gamma }_{1}}\upsilon +{{\nu }_{1}}\psi +{{\nu }_{2}}\phi \right)}_{,tt}} \right]$
which is in well agreement with the heat conduction equation of Quintanilla [18].
(ii) Substituting $\epsilon =0$ in Eq. (86) we get the heat conduction equation for GN-IIITE model as follows
${{\kappa }_{0}}{{\nabla }^{2}}{{\theta }_{,t}}+\kappa _{0}^{*}{{\nabla }^{2}}\theta =\rho {{C}_{s}}{{\theta }_{,tt}}+{{\theta }_{0}}{{\left( {{\gamma }_{1}}\upsilon +{{\nu }_{1}}\psi +{{\nu }_{2}}\phi \right)}_{,tt}}.$
(iii) Ignoring the first term in the left-hand side of Eq. (87), the heat conduction equation for GN-IITE model is obtained as
$\kappa _{0}^{*}{{\nabla }^{2}}\theta =\rho {{C}_{s}}{{\theta }_{,tt}}+{{\theta }_{0}}{{\left( {{\gamma }_{1}}\upsilon +{{\nu }_{1}}\psi +{{\nu }_{2}}\phi \right)}_{,tt}}.$
(iv) Replacing $\kappa _{0}^{*}$ by 0 in Eq. (86) we achieve the following heat conduction equation for LSTE model
${{\kappa }_{0}}{{\nabla }^{2}}\theta =\left( 1+\epsilon \frac{\partial }{\partial t} \right)\left[ \rho {{C}_{s}}{{\theta }_{,t}}+{{\theta }_{0}}{{\left( {{\gamma }_{1}}\upsilon +{{\nu }_{1}}\psi +{{\nu }_{2}}\phi \right)}_{,t}} \right].$
(v) Setting $\epsilon =0$ in Eq. (89), the heat conduction equation for CCTE model is expressed as
${{\kappa }_{0}}{{\nabla }^{2}}\theta =\rho {{C}_{s}}{{\theta }_{,t}}+{{\theta }_{0}}{{\left( {{\gamma }_{1}}\upsilon +{{\nu }_{1}}\psi +{{\nu }_{2}}\phi \right)}_{,t}}.$
6.2. Case II: the absence of nonlocal and micropolar effect
In addition to the case I, when the nonlocal and micropolar effect is also absent in the model, viz., $\tau =\alpha =\lambda =\eta =K=\varpi =0$, the stress-strain-temperature relations and equations of motion for a local double porous thermoelastic material with voids can be obtained as follows:
$\left. \begin{align} & {{\sigma }_{ij}}=\sigma _{ij}^{l}=\Lambda {{\upsilon }_{rr}}{{\delta }_{ij}}+2\mu {{\upsilon }_{ij}}+\beta _{1}^{*}\psi {{\delta }_{ij}}+\beta _{2}^{*}\phi {{\delta }_{ij}}-{{\gamma }_{1}}\theta {{\delta }_{ij}} \\ & {{\sigma }_{i}}=\sigma _{i}^{l}={{\vartheta }_{1}}{{\psi }_{,i}}+{{\vartheta }_{2}}{{\phi }_{,i}} \\ & {{\varsigma }_{i}}=\varsigma _{i}^{l}={{\vartheta }_{2}}{{\psi }_{,i}}+{{\vartheta }_{3}}{{\phi }_{,i}} \\ & {{\Xi }_{1}}=\Xi _{1}^{l}=-\beta _{1}^{*}{{\upsilon }_{jj}}-\zeta _{1}^{*}\psi -\zeta _{2}^{*}\phi +{{v}_{1}}\theta \\ & {{\Xi }_{2}}=\Xi _{2}^{l}=-\beta _{2}^{*}{{\upsilon }_{jj}}-\zeta _{2}^{*}\psi -\zeta _{3}^{*}\phi +{{v}_{2}}\theta \\ & (\Lambda +\mu )\nabla (\nabla.\vec{u})+\mu {{\nabla }^{2}}\vec{u}+\beta _{1}^{*}\nabla \psi +\beta _{2}^{*}\nabla \phi -{{\gamma }_{1}}\nabla \theta =\rho {{{\vec{u}}}_{,tt}}-\vec{F} \\ & \sigma _{i,i}^{l}+\Xi _{1}^{l}+\rho {{l}_{1}}={{\prod }_{1}}{{\psi }_{,tt}} \\ & \varsigma _{i,i}^{l}+\Xi _{2}^{l}+\rho {{l}_{2}}={{\prod }_{2}}{{\phi }_{,tt}} \\ \end{align} \right\}$
Eq. (91) is in good agreement with the Eqs. (2)-(9) of Gupta et al. [80] when the functionally graded parameter, gravity, and rotation are absent and the vertical depth is taken along z-direction.
7. Validation of the present model with pre-established models
Some notable cases are discussed in a tabulated format (Table 2), which validate the considered model.
8. Numerical data and graphical observations
The variations in the displacements, microrotation vector, temperature, void volume fractions, stress tensors, and equilibrated stress vectors are graphically illustrated with respect to the numerical values of different physical quantities related to magnesium crystal-like material in order to observe the graphical behavior of the aforementioned field attributes. By following Khalili [102] and Deswal and Kalkal [103], the numerical values corresponding to the relevant parameters are presented in Table 3. The numerical values of other parameters used for graphical implementations are ${{\omega }_{R}}=2$, ${{\omega }_{I}}=1$, $n=1.2$, ${{L}_{T}}=1$, ${{L}_{M}}=1$, and ${{\kappa }_{1}}={{\kappa }_{2}}=-10$. In Figs. 2–21, variation in the real component of the dimensionless field variables are plotted against the vertical depth z on the plane x=0.5 and at time t=0.5s. Figs. 2–11 portray the effect of the variation in kernel function $\left( {{K}^{*}} \right)$ on the considered non-dimensional field variables $\left( {{u}_{1}},{{u}_{3}},{{\varphi }_{2}},\theta,\psi,\phi,{{\sigma }_{zz}},{{\sigma }_{zx}},{{\sigma }_{z}},{{\varsigma }_{z}} \right)$ for both variable and constant thermal conductivity. In Figs. 12–21, all field variables are compared separately on the basis of the generalized theories of MGTTE, GN-IIITE, LSTE, and CCTE in the presence and absence of nonlocality parameter.
Table 2. Some notable cases and the corresponding models. |
| Case | Condition | Model |
|---|---|---|
| I | ${{D}_{\epsilon }}f\left( t \right)\to f\prime \left( t \right)$, ${{\kappa }_{1}}={{\kappa }_{2}}=\tau =\alpha =\lambda =\eta =K=\varpi =\epsilon =0$ | Case I represents the local double porous thermoelastic medium with voids under GN-IIITE theory, which is in good agreement with Kalkal et al. [96] in the absence of functionally graded parameter, gravity, and thermal relaxation times |
| II | ${{D}_{\epsilon }}f\left( t \right)\to f\prime \left( t \right)$, $\begin{align} & {{\kappa }_{1}}={{\kappa }_{2}}=\tau =\alpha =\lambda =\eta =K=\varpi =\kappa _{0}^{*}= \\ & \beta _{2}^{*}={{\vartheta }_{2}}={{\vartheta }_{3}}=\zeta _{2}^{*}=\zeta _{3}^{*}={{\nu }_{2}}=0 \\ \end{align}$ | Case II represents the local porous thermoelastic medium with one type of voids under LSTE theory, which is in good agreement with Gunghas et al. [100] in the absence of functionally graded parameter and gravity |
| III | ${{D}_{\epsilon }}f\left( t \right)\to f\prime \left( t \right)$, $\begin{align} & {{\kappa }_{1}}={{\kappa }_{2}}=\tau =\alpha =\lambda =\eta =K \\ & =\varpi =\epsilon =\beta _{1}^{*}=\beta _{2}^{*}={{\vartheta }_{1}}={{\vartheta }_{2}}={{\vartheta }_{3}}=\zeta _{1}^{*}=\zeta _{2}^{*} \\ & =\zeta _{3}^{*}={{\nu }_{1}}={{\nu }_{2}}=0 \\ \end{align}$ | Case III represents the local thermoelastic medium under GN-IIITE theory, which is in good agreement with Gunghas et al. [101] in the absence of functionally graded parameter, rotation, and magnetic field |
| IV | $\begin{align} & \tau =\epsilon =\kappa _{0}^{*}=\beta _{1}^{*}=\beta _{2}^{*}={{\vartheta }_{1}}={{\vartheta }_{2}}={{\vartheta }_{3}}=\zeta _{1}^{*}=\zeta _{2}^{*} \\ & =\zeta _{3}^{*}={{\nu }_{1}}={{\nu }_{2}}=0 \\ \end{align}$ | Case IV represents the local micropolar thermoelastic medium with variable conductivity under memory-dependent CCTE theory, which is in good agreement with Said [30] in the absence of rotation, thermal relaxation times, and magnetic field |
| V | $\begin{align} & {{\kappa }_{1}}={{\kappa }_{2}}=\tau =\alpha =\lambda =\eta =K=\varpi =\kappa _{0}^{*}=\beta _{1}^{*}= \\ & \beta _{2}^{*}={{\vartheta }_{1}}={{\vartheta }_{2}}={{\vartheta }_{3}}=\zeta _{1}^{*}=\zeta _{2}^{*}=\zeta _{3}^{*}={{\nu }_{1}}={{\nu }_{2}}=0 \\ \end{align}$ | Case V represents the local thermoelastic medium under memory-dependent LSTE theory, which is in good agreement with Othman and Mondal [43] in the absence of rotation |
Table 3. Numerical values of various constants. |
| Symbols | Values | Symbols | Values |
|---|---|---|---|
| Λ | 9.4×1010 N m−2 | μ | 4×1010 N m−2 |
| K | 1010 N m−2 | η | 0.779×10−9 N |
| ϖ | 0.2×10−19 m2 | ρ | 1740 kg m−3 |
| ${{\vartheta }_{1}}$ | 1.3×10−5 N | ${{\vartheta }_{2}}$ | 0.12×10−5 N |
| ${{\vartheta }_{3}}$ | 1.1×10−5 N | $\zeta _{1}^{*}$ | 1.2×1010 N m−2 |
| $\zeta _{2}^{*}$ | 1.23×106 N m−2 | $\zeta _{3}^{*}$ | 2.21×1010 N m−2 |
| $\beta _{1}^{*}$ | 0.9×1010 N m−2 | $\beta _{2}^{*}$ | 0.1×1010 N m−2 |
| ${{\prod }_{1}}$ | 0.1456×10−12 N m−2 s2 | ${{\prod }_{2}}$ | 0.1546×10−12 N m−2s2 |
| ${{v}_{1}}$ | 0.16×105 N m−2 K−1 | ${{v}_{2}}$ | 0.219×105 N m−2 K−1 |
| ${{\vartheta }_{t}}$ | 1.78×10−5 K−1 | ${{\theta }_{0}}$ | 293 K |
| Cs | 383.1 J kg−1 K−1 | ${{\kappa }_{0}}$ | 3860 W m−1 K−1 |
| $\tau $ | 7×10−6 m | $\epsilon $ | 0.02 s |
Fig. 2. Distribution of ${{u}_{1}}$ for different kernel functions and variable thermal conductivity parameters. |
8.1. Effect of kernel function $\left( {{K}^{*}} \right)$ and thermal conductivity parameters $\left( {{\kappa }_{1}},{{\kappa }_{2}} \right)$
In Figs. 2–11, horizontal displacement component (${{u}_{1}}$), vertical displacement component (${{u}_{3}}$), microrotation component (${{\varphi }_{2}}$), temperature (θ), volume fraction of matrix pores (ψ), volume fraction of fractures (ϕ), normal stress (${{\sigma }_{zz}}$), tangential stress (${{\sigma }_{zx}}$), equilibrated stress of matrix pores (${{\sigma }_{z}}$), and equilibrated stress of fractures (${{\varsigma }_{z}}$) are plotted against z for different kernels ${{K}^{*}}$ in the presence and absence of thermal conductivity parameters $\left( {{\kappa }_{1}},{{\kappa }_{2}} \right)$, respectively. For the purpose of graphical implementations, the kernel functions are taken to be 1 (when ${{a}_{1}}={{a}_{2}}=0$), $1-\frac{t-\varepsilon }{\epsilon }$ (when ${{a}_{2}}=0$, ${{a}_{1}}=0.5$), $1-\left( t-\varepsilon \right)$ (when ${{a}_{2}}=0$, ${{a}_{1}}=0.5\epsilon $), and ${{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$ (when ${{a}_{2}}=1$, ${{a}_{1}}=1$) as mentioned in Eq. (4). In Figs. 2–11, curves 1 and 5, curves 2 and 6, curves 3 and 7, and curves 4 and 8 are drawn for the kernel functions ${{K}^{*}}=1$, ${{K}^{*}}=1-\frac{t-\varepsilon }{\epsilon }$, ${{K}^{*}}=1-\left( t-\varepsilon \right)$, and${{K}^{*}}={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$, respectively. Solid curves (i.e. curves 1,2,3 and 4) correspond to the variable thermal conductivity, whereas dashed curves (i.e. curves 5,6,7 and 8) are drawn for implementing the constant thermal conductivity.
It is noticed from Fig. 2 that ${{u}_{1}}$ begins to increase up to a certain value of z and approaches 0 with respect to ascending z values. Moreover, in absence of the variable thermal conductivity, curves attain much higher values of ${{u}_{1}}$ than the curves with variable thermal conductivity. Also, ${{u}_{1}}$ attains maximum magnitude for ${{K}^{*}}={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$ followed by the values corresponding to ${{K}^{*}}=1-\frac{t-\varepsilon }{\epsilon }$, 1, and $1-\left( t-\varepsilon \right)$ up to a certain z value, regardless of the fact that the thermal conductivity parameter is present or absent. According to Fig. 3, u3 starts to decrease up to a certain z value and approaches 0 as z assumes ascending values. It is also revealed that ${{u}_{3}}$ attains maximum value for ${{K}^{*}}=1$ followed by the values associated with ${{K}^{*}}=1-\frac{t-\varepsilon }{\epsilon }$, ${{K}^{*}}={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$, and ${{K}^{*}}=1-\left( t-\varepsilon \right)$ within a particular range of z, in the case of both the variable and constant thermal conductivity. When ${{\kappa }_{1}}={{\kappa }_{2}}=-10$, ${{u}_{3}}$ achieves greater values than in case of ${{\kappa }_{1}}={{\kappa }_{2}}=0$. However, comparison between Figs. 2 and 3 reveals that thermal conductivity parameters has more favorable impact on ${{u}_{1}}$ than ${{u}_{3}}$.
Fig. 3. Distribution of ${{u}_{2}}$ for different kernel functions and variable thermal conductivity parameters. |
Fig. 4 exhibits the effect of variation in ${{\kappa }_{1}}$, ${{\kappa }_{2}}$, and ${{K}^{*}}$ on ${{\varphi }_{2}}$. According to Fig. 4, curves plotted in case of ${{K}^{*}}=1$, ${{K}^{*}}=1-\frac{t-\varepsilon }{\epsilon }$, and ${{K}^{*}}={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$ start to decrease up to a certain z and after a sharp decrement in the ${{\varphi }_{2}}$, curves slightly ascend and approach 0 simultaneously. Curves for ${{K}^{*}}=1-\left( t-\varepsilon \right)$ show ascending nature up to an approximated z value of 0.7 beyond which they approach 0. Furthermore, when $z\in \left[ 0,0.4 \right]$, ${{\varphi }_{2}}$ assumes lowest values for ${{K}^{*}}=1-\left( t-\varepsilon \right)$ and highest values for ${{K}^{*}}={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$, regardless of the fact that the thermal conductivity parameter is present or absent. Furthermore, for the individual kernel, the curve with variable thermal conductivity shows a higher ${{\varphi }_{2}}$ value than the curves with constant thermal conductivity within a particular region of z. Fig. 5 displays the impact of the variation in ${{K}^{*}}$ on $\theta $ in the presence and absence of thermal conductivity parameters. All the curves corresponding to ${{\kappa }_{1}}={{\kappa }_{2}}=0$ have a common initial point of magnitude $\theta =1.2$, whereas θ value is −1 for all curves associated with ${{\kappa }_{1}}={{\kappa }_{2}}=-10$, in the vicinity of source which satisfies the boundary condition (74). According to Fig. 5, as z increases, θ in the absence of thermal conductivity parameters decreases, whereas in the presence of thermal conductivity parameters, θ shows ascending nature with increasing z values, irrespective of all ${{K}^{*}}$. Beyond z=3, each curve eventually approaches 0. For ${{K}^{*}}=1$, θ attains individual maximum value in each case (${{\kappa }_{1}}={{\kappa }_{2}}=0$ and ${{\kappa }_{1}}={{\kappa }_{2}}=-10$). When the variation in ${{\kappa }_{1}}$, ${{\kappa }_{2}}$ is considered, very minute differences among the curves can be visualized for variable thermal conductivity, whereas a considerable difference is observed between curves for constant thermal conductivity in an entire range of z.
Fig. 4. Distribution of ${{\varphi }_{2}}$ for different kernel functions and variable thermal conductivity parameters. |
Fig. 5. Distribution of θ for different kernel functions and variable thermal conductivity parameters. |
Figs. 6 and 7 display the effects of the kernel function and thermal conductivity parameters on ψ and ϕ, respectively. Both figures reveal that ψ and ϕ display decreasing nature with ascending z values. Beyond z=2, all curves approach 0. However, for each ${{K}^{*}}$, the magnitude of ψ is higher than the magnitude of ϕ at the free surface (i.e., z=0). From Figs. 6 to 7 it is also observed that both ψ and ϕ acquire individual maximum magnitudes for K*=1, followed by the values for ${{K}^{*}}=1-\left( t-\varepsilon \right)$, ${{K}^{*}}=1-\frac{t-\varepsilon }{\epsilon }$, and ${{K}^{*}}={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$, regardless of the ${{\kappa }_{1}}$, ${{\kappa }_{2}}$ values, within the z range of [0.5,1.5] approximately. Furthermore, ψ and ϕ are inversely proportional to the thermal conductivity parameters up to a certain z value.
Fig. 6. Distribution of ψ for different kernel functions and variable thermal conductivity parameters. |
Fig. 7. Distribution of ϕ for different kernel functions and variable thermal conductivity parameters. |
Fig. 8 reveals variation in ${{\sigma }_{zz}}$ with respect to ${{K}^{*}}$, ${{\kappa }_{1}}$, and ${{\kappa }_{2}}$. According to the observations, ${{\sigma }_{zz}}$ begins from the value −1.2 at z=0, which justifies the boundary condition (76). From Fig. 8, it is monitored that each curve decreases initially for the z values close to 0, whereas after reaching a certain depth, ${{\sigma }_{zz}}$ assumes ascending values as z proceeds towards the value of 1. Each curve eventually approaches 0. For each case ( ${{\kappa }_{1}}={{\kappa }_{2}}=0$ and ${{\kappa }_{1}}={{\kappa }_{2}}=-10$ ), ${{\sigma }_{zz}}$ attains distinct highest and lowest magnitude for ${{K}^{*}}={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$ and ${{K}^{*}}=1$, respectively, within a particular domain of z. However, when $z\in \left[ 0.1,2 \right]$, ${{\sigma }_{zz}}$ displays higher magnitude in case of ${{\kappa }_{1}}={{\kappa }_{2}}=0$ than in case of ${{\kappa }_{1}}={{\kappa }_{2}}=-10$. The impact of ${{K}^{*}}$, ${{\kappa }_{1}}$, and ${{\kappa }_{2}}$ on ${{\sigma }_{zx}}$ is exhibited in Fig. 9. From the figure it is observed that at z=0, all the curves initially start from ${{\sigma }_{zx}}$ value of 0 which is a fair match to the boundary condition (77). ${{\sigma }_{zx}}$ is observed to increase up to a certain value of z. After reaching the peak, it diminishes and approaches 0 with respect to ascending z values. This fact is well agreed by all the curves. At z=0.3, ${{\sigma }_{zx}}$ acquires the highest and lowest magnitudes for ${{K}^{*}}=1-\left( t-\varepsilon \right)$ and ${{K}^{*}}={{\left( 1-\frac{t-\varepsilon }{\epsilon } \right)}^{2}}$, respectively, regardless of the fact that the thermal conductivity parameter is present or absent. Moreover, an inverse proportionality relationship exists between ${{\sigma }_{zx}}$ and thermal conductivity parameters within an approximate z range of [0.1,2].
Fig. 8. Distribution of ${{\sigma }_{zz}}$ for different kernel functions and variable thermal conductivity parameters. |
Fig. 9. Distribution of ${{\sigma }_{zx}}$ for different kernel functions and variable thermal conductivity parameters. |
Figs. 10 and 11 illustrate the influence of ${{K}^{*}}$, ${{\kappa }_{1}}$, and ${{\kappa }_{2}}$ on ${{\sigma }_{z}}$ and ${{\varsigma }_{z}}$, respectively. All the curves in each figure have coincident initial coordinate (0,0). These facts agree well with the boundary conditions (79) and (80). Fig. 10 executes that all curves corresponding to ${{\sigma }_{z}}$ move upwards and form sharp peaks with unequal heights at different distances from the origin. Afterward, sudden drop followed by slight increment in ${{\sigma }_{z}}$ values are observed until all curves approach 0. The pattern of the curves in Fig. 11 is quite similar to that of in Fig. 10. According to the comparative study between both figures, for each ${{K}^{*}}$, ${{\varsigma }_{z}}$ is lesser than ${{\sigma }_{z}}$ at z values near to 0. From Figs. 10 and 11 it is also observed that ψ and ϕ become maximum when ${{K}^{*}}=1-\left( t-\varepsilon \right)$, within the z range of (0,0.1] approximately. As z proceeds towards 0.5, both ${{\sigma }_{z}}$ and ${{\varsigma }_{z}}$ acquire comparatively larger values in case of ${{K}^{*}}=1$ than the other three kernels. The relationship between equilibrated stresses and kernel alters beyond z=0.5. Furthermore, ${{\sigma }_{z}}$ and ${{\varsigma }_{z}}$ are inversely proportional to the thermal conductivity parameters up to an approximate z value of 0.5.
Fig. 10. Distribution of ${{\sigma }_{z}}$ for different kernel functions and variable thermal conductivity parameters. |
Fig. 11. Distribution of ${{\varsigma }_{z}}$ for different kernel functions and variable thermal conductivity parameters. |
Fig. 12. Distribution of ${{u}_{1}}$ for different nonlocal and local thermoelasticity theories. |
8.2. Comparative analysis of different nonlocal and local thermoelasticity theories
The behaviors of ${{u}_{1}}$, ${{u}_{3}}$, ${{\varphi }_{2}}$, $\theta $, $\psi $, $\phi $, ${{\sigma }_{zz}}$, ${{\sigma }_{zx}}$, ${{\sigma }_{z}}$, ${{\varsigma }_{z}}$ under different thermoelastic models with variable thermal conductivity, viz. MGTTE, GN-IIITE, LSTE, and CCTE in the presence and absence of nonlocality parameter $\left( \tau \right)$ have been analyzed in Figs. 12–21, respectively. Curves 1 and 5, curves 2 and 6, curves 3 and 7, and curves 4 and 8 are drawn for MGTTE, GN-IIITE, LSTE, and CCTE theories, respectively. Solid curves (i.e. curves 1,2,3 and 4) correspond to the nonlocal model, whereas dashed curves (i.e. curves 5,6,7 and 8) represent the local model.
In Fig. 12, ${{u}_{1}}$ in case of all theories starts from different negative values. ${{u}_{1}}$ assumes ascending values as z increases. After a certain z value, each curve eventually approaches 0. ${{u}_{1}}$ attains considerably greater values in case of nonlocal LSTE theory than in case of the other three nonlocal thermoelasticity theories within the z range of [0,1.2]. ${{u}_{1}}$ under nonlocal GN-IIITE theory assumes the lowest magnitude until the curves intersect at z=0.5. Beyond z=0.5, nonlocal MGTTE theory inherits lowest ${{u}_{1}}$ value until all curves converge to 0. Moreover, up to z=1.2 approximately, curves 5 and 6 possess the minimum and maximum ${{u}_{1}}$ values, respectively, in the absence of nonlocality parameter. Fig. 13 shows that for each thermoelasticity theory, ${{u}_{3}}$ begins with distinct positive values. Beyond z=0, each curve decreases for initial z values and eventually approaches 0. ${{u}_{3}}$ achieves the maximum value in case of nonlocal MGTTE theory, whereas under nonlocal CCTE theory, ${{u}_{3}}$ assumes the lowest value until the curves coincide with each other. On the other hand, in the absence of nonlocality parameter, LSTE theory and MGTTE theory correspond to maximum and minimum ${{u}_{3}}$ values, respectively, within an approximate z range of [0,2]. Furthermore, it is also observed that ${{u}_{3}}$ is inversely proportional to τ up to a certain z value.
Fig. 13. Distribution of ${{u}_{3}}$ for different nonlocal and local thermoelasticity theories. |
Fig. 14 reveals variation in ${{\varphi }_{2}}$ with respect to local and nonlocal thermoelasticity theories. According to the observation, in the absence of nonlocality parameter, LSTE theory and MGTTE theory inherit maximum and minimum ${{\varphi }_{2}}$ values at the free surface. However, minimal variation is observed between the ${{\varphi }_{2}}$ values for GN-IIITE and LSTE theories. As z increases, all curves converge to 0. It is observed from Fig. 15 that curves corresponding to local and nonlocal thermoelasticity theories with variable conductivity have a common initial point of magnitude θ=−1, in the vicinity of source which satisfies the boundary condition (74). Beyond z=0, θ for every thermoelasticity theory attains ascending values until it approaches to 0. Among all nonlocal thermoelasticity theories, θ achieves highest and lowest values for MGTTE and CCTE theory, respectively, whereas in case of local thermoelasticity theories, θ is maximum for LSTE theory and minimum for MGTTE theory.
Fig. 14. Distribution of ${{\varphi }_{2}}$ for different nonlocal and local thermoelasticity theories. |
Fig. 15. Distribution of θ for different nonlocal and local thermoelasticity theories. |
Figs. 16 and 17 display the impacts of different local and nonlocal thermoelasticity theories on ψ and ϕ, respectively. Both figures reveal that ψ and ϕ display decreasing nature with ascending z values. Beyond z=2.5, all curves approach 0. However, for each thermoelasticity theory, the magnitude of ψ is higher than the magnitude of ϕ at z=0. ψ assumes maximum value in case of nonlocal CCTE theory and minimum value for nonlocal MGTTE theory at the free surface. As z increases, curves show contrary behavior and tend to 0 gradually. However, when τ=0, ψ assumes highest and lowest magnitude for MGTTE and LSTE theory, respectively, at the free surface. Curves in Fig. 17 behave similar to the curves in Fig. 16.
Fig. 16. Distribution of ψ for different nonlocal and local thermoelasticity theories. |
Fig. 17. Distribution of ϕ for different nonlocal and local thermoelasticity theories. |
Fig. 18 reveals variation in ${{\sigma }_{zz}}$ with respect to different local and nonlocal thermoelasticity theories with variable conductivity. According to the observation, ${{\sigma }_{zz}}$ begins from the value −1.2 at z=0, which justifies the boundary condition (76). ${{\sigma }_{zz}}$ augments within the initial diminutive range of z under every theory. Afterward, curves descend up to certain depths at different z values and sudden increments in the σzz values are noticed until all curves approach 0. When, τ=7×10−6, ${{\sigma }_{zz}}$ becomes maximum and minimum in case of LSTE and MGTTE theories within the z range of [0.3,0.7], respectively. Beyond z=0.7, ${{\sigma }_{zz}}$ acquires comparatively higher values in the context of nonlocal CCTE theory than the other three theories. On the other hand, when $z\in \left( 0,0.7 \right]$, ${{\sigma }_{zz}}$ assumes maximum value in case of local GN-IIITE theory and minimum value for local MGTTE theory. The impact of different thermoelasticity theories on ${{\sigma }_{zx}}$ is exhibited in Fig. 19. From the figure it is observed that at z=0, all the curves initially start from ${{\sigma }_{zx}}$ value of 0 which is an exact match to the boundary condition (77). ${{\sigma }_{zx}}$ is observed to increase up to a certain value of z. After reaching the peak, it diminishes and approaches 0 with respect to ascending z values. This fact is well agreed by all the curves except the curves for local LSTE (curve 7) and GN-IIITE (curve 6) theory. Both curves 6 and 7 descend up to certain depths at initial z values and then increases until they converge to 0. In case of local model, ${{\sigma }_{zx}}$ assumes highest and lowest magnitudes for MGTTE and LSTE theories, respectively, when $z\in \left( 0,0.5 \right]$.
Fig. 18. Distribution of ${{\sigma }_{zz}}$ for different nonlocal and local thermoelasticity theories. |
Fig. 19. Distribution of ${{\sigma }_{zx}}$ for different nonlocal and local thermoelasticity theories. |
Figs. 20 and 21 demonstrate the influence of different thermoelasticity theories on ${{\sigma }_{z}}$ and ${{\varsigma }_{z}}$, respectively. All the curves in each figure have coincident initial coordinate (0,0). These facts agree well with the boundary conditions (79) and (80). Fig. 20 executes that all curves corresponding to ${{\sigma }_{z}}$ move upwards and form sharp peaks with unequal heights at different distances from the origin. Afterward, sudden drop followed by slight increment in ${{\sigma }_{z}}$ values are observed until all curves approach 0. The pattern of the curves in Fig. 21 is quite similar to that of in Fig. 20. According to the comparative study between both figures, ${{\varsigma }_{z}}$ is lesser than ${{\sigma }_{z}}$ under every thermoelasticity theory at z values adjacent to 0. Different thermoelasticity theories have oscillatory impact on ${{\sigma }_{z}}$ and ${{\varsigma }_{z}}$ as monitored on the basis of magnitudes.
Fig. 20. Distribution of ${{\sigma }_{z}}$ for different nonlocal and local thermoelasticity theories. |
Fig. 21. Distribution of ${{\varsigma }_{z}}$ for different nonlocal and local thermoelasticity theories. |
9. Conclusions
Some concluding remarks of the present study are as follows:
• The normal mode technique is adopted in the area of thermoelasticity and applied to those particular problems in which the coupled relationships between stress, strain, and temperature exist. This method provides accurate solutions without any presumed limitations on the physical variables that are present in the field equations. The normal mode technique is beneficial to solve problems in different fields, such as hydrodynamics and thermoelasticity.
• Major changes have been visualised between the plotted curves related to all field variables due to the presence and absence of nonlocality parameter in a wide range of the distance parameter which reflects that nonlocality parameter dominates every physical quantity within a very diminutive range of distance.
• The classical field theory, applicable in the length and time domain, provides sufficiently accurate outcomes when external characteristic length becomes greater than internal characteristic length. By contrast, when both characteristic lengths become equal, the nonlocal theory ensures the precision of the obtained solutions rather than the local models.
• The CCTE, LSTE, GN-IITE, and GN-IIITE theories can be obtained as particular cases of MGTTE theory.
• The presence and absence of variable thermal conductivity parameters has major effect on the distributions of field variables of a MDPTMWV substrate.
• kernel functions have a favorable impact on the distributions of all physical field variables.
• All the field variables of the MDPTMWV substrate approach zero with the ascending depth which agrees on the concept of generalized thermoelasticity.
• The usefulness of the current model in the real-world can not be ignored because of it's several applications. Results accomplished in this analytical study can be employed in different practical areas, such as earthquake engineering, material science, carbon sequestration, and seismology.
Declaration of Competing Interest
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.
Acknowledgments
Authors acknowledge the Council of Scientific and Industrial Research(CSIR) by the project (Grant Number 25(0296)/19/EMR-II) entitled Mathematical Modeling of Elastic Waves in Fractional-Order Thermoelastic Solids with Micro-configurations and Initially Stressed Media for providing financial support for this research work.
Appendix A1
${{r}_{1}}=\frac{\Lambda }{\Lambda +2\mu +K}\text{ }{{r}_{2}}=\frac{\beta _{1}^{{{*}^{2}}}k_{d}^{2}}{{{\prod }_{1}}c_{1}^{4}(\Lambda +2\mu +K)}$
${{r}_{3}}=\frac{\beta _{1}^{*}\beta _{2}^{*}k_{d}^{2}}{{{\prod }_{1}}c_{1}^{4}(\Lambda +2\mu +K)}\text{ }{{r}_{4}}=\frac{{{\gamma }_{1}}{{\theta }_{0}}}{\Lambda +2\mu +K}$
${{r}_{5}}=\frac{\mu +K}{\Lambda +2\mu +K}\text{ }{{r}_{6}}=\frac{\mu }{\Lambda +2\mu +K}\text{ }{{r}_{7}}=\frac{K}{\Lambda +2\mu +K}$
${{r}_{8}}=\frac{\eta }{\rho k_{d}^{2}}\text{ }{{r}_{9}}=\frac{\beta _{1}^{*}k_{d}^{2}}{{{\prod }_{1}}c_{1}^{4}}\text{ }{{r}_{10}}=\frac{{{\vartheta }_{2}}\beta _{1}^{*}k_{d}^{2}}{{{\vartheta }_{1}}{{\prod }_{1}}c_{1}^{4}}$
${{r}_{11}}=\frac{{{\vartheta }_{3}}\beta _{1}^{*}k_{d}^{2}}{{{\vartheta }_{1}}{{\prod }_{1}}c_{1}^{4}}\text{ }{{r}_{12}}=\frac{\mu }{\rho \varpi c_{1}^{2}}\text{ }{{r}_{13}}=\frac{k_{d}^{2}}{\rho \varpi c_{1}^{4}}\text{ }{{r}_{14}}=\frac{2Kk_{d}^{2}}{\rho \varpi c_{1}^{4}}$
${{r}_{15}}=\frac{{{\vartheta }_{1}}}{{{\prod }_{1}}c_{1}^{2}}\text{ }{{r}_{16}}=\frac{{{\vartheta }_{2}}}{{{\prod }_{1}}c_{1}^{2}}\text{ }{{r}_{17}}=\frac{\zeta _{1}^{*}k_{d}^{2}}{{{\prod }_{1}}c_{1}^{4}}$
${{r}_{18}}=\frac{\zeta _{2}^{*}k_{d}^{2}}{{{\prod }_{1}}c_{1}^{4}}\text{ }{{r}_{19}}=\frac{{{v}_{1}}{{\theta }_{0}}}{\beta _{1}^{*}}\text{ }{{r}_{20}}=\frac{{{\vartheta }_{2}}}{{{\prod }_{2}}c_{1}^{2}}\text{ }{{r}_{21}}=\frac{{{\vartheta }_{3}}}{{{\prod }_{2}}c_{1}^{2}}\text{ }{{r}_{22}}=\frac{\beta _{2}^{*}{{\prod }_{1}}}{\beta _{1}^{*}{{\prod }_{2}}}$
${{r}_{23}}=\frac{\zeta _{2}^{*}k_{d}^{2}}{{{\prod }_{2}}c_{1}^{4}}\text{ }{{r}_{24}}=\frac{\zeta _{3}^{*}k_{d}^{2}}{{{\prod }_{2}}c_{1}^{4}}\text{ }{{r}_{25}}=\frac{{{v}_{2}}{{\prod }_{1}}{{\theta }_{0}}}{{{\prod }_{2}}\beta _{1}^{*}}\text{ }{{r}_{26}}=\frac{\kappa _{0}^{*}{{k}_{d}}}{{{\kappa }_{0}}c_{1}^{2}}\text{ }{{r}_{27}}=\frac{{{\gamma }_{1}}{{k}_{d}}}{{{\kappa }_{0}}}$
${{r}_{28}}=\frac{{{v}_{1}}\beta _{1}^{*}k_{d}^{3}}{{{\kappa }_{0}}{{\prod }_{1}}c_{1}^{4}}\text{ }{{r}_{29}}=\frac{{{v}_{2}}\beta _{1}^{*}k_{d}^{3}}{{{\kappa }_{0}}{{\prod }_{1}}c_{1}^{4}}.$
Appendix A2
${{S}_{1}}={{r}_{5}}+{{\tau }^{2}}{{\omega }^{2}}\text{ }{{S}_{2}}={{n}^{2}}+{{\omega}^{2}}+{{n}^{2}}{{\omega }^{2}}{{\tau }^{2}}\text{ }{{S}_{3}}=\iota n({{r}_{1}}+{{r}_{6}})$
${{S}_{4}}=\iota n{{r}_{4}}\text{ }{{S}_{5}}=\iota n{{r}_{2}}\text{ }{{S}_{6}}=\iota n{{r}_{3}}\text{ }{{S}_{7}}=1+{{\tau }^{2}}{{\omega}^{2}}$
${{S}_{8}}={{n}^{2}}{{r}_{5}}+{{\omega}^{2}}+{{n}^{2}}{{\tau }^{2}}{{\omega}^{2}}\text{ }{{S}_{9}}=\iota n{{r}_{7}}\text{ }{{S}_{10}}=\iota n{{r}_{13}}\text{ }{{S}_{11}}={{r}_{12}}+{{\tau }^{2}}{{\omega}^{2}}$
${{S}_{12}}={{n}^{2}}{{r}_{12}}+{{r}_{14}}+{{\omega}^{2}}\text{+}{{n}^{2}}{{\tau }^{2}}{{\omega}^{2}}\text{ }{{S}_{13}}=-\iota n\text{ }{{S}_{14}}={{r}_{15}}\text{+}{{\tau }^{2}}{{\omega}^{2}}$
${{S}_{15}}={{n}^{2}}{{r}_{15}}+{{r}_{17}}+{{\omega}^{2}}\text{+}{{n}^{2}}{{\tau }^{2}}{{\omega}^{2}}\text{ }{{S}_{16}}={{n}^{2}}{{r}_{16}}+{{r}_{18}}\text{ }{{S}_{17}}=-\iota n{{r}_{22}}$
${{S}_{18}}={{n}^{2}}{{r}_{20}}+{{r}_{23}}\text{ }{{S}_{19}}={{r}_{21}}+{{\omega}^{2}}{{\tau }^{2}}\text{ }{{S}_{20}}={{n}^{2}}{{r}_{21}}+{{r}_{24}}+{{\omega}^{2}}\text{+}{{n}^{2}}{{\tau }^{2}}{{\omega}^{2}}$
${{S}_{21}}=\iota n{{\omega}^{2}}{{r}_{27}}\text{(1+}G\text{(}\epsilon,\omega\text{)) }{{S}_{22}}={{\omega}^{2}}{{r}_{27}}\text{(1+}G\text{(}\epsilon,\omega\text{))}$
${{S}_{23}}=-{{r}_{26}}-{{\omega}^{2}}\text{ }{{S}_{24}}={{\omega}^{2}}\text{(1+}G\text{(}\epsilon,\omega\text{))}+{{n}^{2}}({{\omega}^{2}}+{{r}_{26}})$
$\begin{align} & {{S}_{25}}={{\omega}^{2}}{{r}_{28}}\text{(1+}G\text{(}\epsilon,\omega\text{)) }{{S}_{26}}={{\omega}^{2}}{{r}_{29}}\text{(1+}G\text{(}\epsilon,\omega\text{))}G\text{(}\epsilon,\omega\text{)} \\ & =\left( 1-{{e}^{-\omega\epsilon }} \right)-\frac{2{{a}_{1}}}{\epsilon \omega}(1-{{e}^{-\omega\epsilon }}(1+\epsilon \omega)) \\ & +\frac{a_{2}^{2}}{{{\epsilon }^{2}}{{\omega}^{2}}}(2-{{e}^{-\omega\epsilon }}(2+2\omega\epsilon +{{\omega}^{2}}{{\epsilon }^{2}})) \\ \end{align}$
Appendix A3
$G_1=S_3 S_{11} G_2=-r_7 S_{10}-S_3 S_{12} G_3=S_4 S_{11} G_4=-S_4 S_{12}$
$G_5=-S_5 S_{11} G_6=S_5 S_{12} G_7=-S_6 S_{11} G_8=S_6 S_{12} G_9=-S_1 S_{11}$
$G_{10}=-r_7 r_{13}+S_1 S_{12}+S_2 S_{11} G_{11}=-S_2 S_{12} G_{12}=S_7 S_{11} $
$G_{13}=-S_7 S_{12}-S_8 S_{11} G_{14}=S_8 S_{12}+S_9 S_{10} G_{15}=r_4 S_{11}$
$G_{16}=-r_4 S_{12} G_{17}=-r_2 S_{11} G_{18}=r_2 S_{12} G_{19}=-r_3 S_{11} G_{20}=r_3 S_{12}$
$G_{21}=-S_3 S_{11} G_{22}=S_3 S_{12}+r_{13} S_9 G_{23}=S_{19} G_{12}$
$G_{24}=-S_{20} G_{12}+S_{19} G_{13}-r_{22} G_{19} G_{25}=-S_{20} G_{13}+S_{19} G_{14}-r_{22} G_{20}$
$G_{26}=-S_{20} G_{14} G_{27}=S_{19} G_{15} G_{28}=-S_{20} G_{15}+S_{19} G_{16}-G_{19} r_{25}$
$G_{29}=-S_{20} G_{16}-G_{20} r_{25} G_{30}=S_{19} G_{17}-G_{19} r_{20}$
$G_{31}=-S_{20} G_{17}+S_{19} G_{18}+S_{18} G_{19}-G_{20} r_{20} G_{32}=-S_{20} G_{18}+S_{18} G_{20}$
$G_{33}=S_{19} G_{21} G_{34}=-S_{20} G_{21}+S_{19} G_{22}-S_{17} G_{19}$
$G_{35}=-S_{20} G_{22}-S_{17} G_{20} G_{36}=S_{19} S_{22} G_{37}=S_{26} r_{22}-S_{20} S_{22}$
$G_{38}=-S_{19} S_{23} G_{39}=-S_{19} S_{24}+S_{20} S_{23} \quad G_{40}=S_{26} r_{25}+S_{20} S_{24}$
$G_{41}=S_{26} r_{20}-S_{19} S_{25} G_{42}=-S_{18} S_{26}+S_{20} S_{25} G_{43}=-S_{19} S_{21}$
$G_{44}=S_{17} S_{26}+S_{20} S_{21} G_{45}=S_{19}-r_{16} r_{22} G_{46}=-S_{20}+S_{16} r_{22} $
$G_{47}=S_{19} r_{19}-r_{16} r_{25} G_{48}=-r_{19} S_{20}+S_{16} r_{25} G_{49}=S_{14} S_{19}-r_{16} r_{20}$
$G_{50}=-S_{14} S_{20}-S_{15} S_{19}+S_{18} r_{16}+S_{16} r_{20} G_{51}=S_{15} S_{20}-S_{16} S_{18}$
$G_{52}=S_{13} S_{19}-S_{17} r_{16} G_{53}=-S_{13} S_{20}+S_{16} S_{17} G_{54}=S_{19} G_1$
$G_{55}=-S_{20} G_1+S_{19} G_2-r_{22} G_7 G_{56}=-S_{20} G_2-r_{22} G_8$
$G_{56}=S_{19} G_3 G_{57}=-S_{20} G_3+S_{19} G_4-r_{25} G_7 G_{59}=-S_{20} G_4-r_{25} G_8$
$G_{60}=S_{19} G_5-r_{20} G_7 G_{61}=-S_{20} G_5+S_{19} G_6+S_{18} G_7-r_{20} G_8$
$G_{62}=-S_{20} G_6+S_{18} G_8 \quad G_{63}=S_{19} G_9 \quad G_{64}=-S_{20} G_9+S_{19} G_{10}$
$G_{65}=-S_{20} G_{10}+S_{19} G_{11}-S_{17} G_7 G_{66}=-S_{20} G_{11}-S_{17} G_8$
$G_{67}=G_{49} G_{54} G_{68}=-G_{45} G_{60}+G_{49} G_{55}+G_{50} G_{54}$
$G_{69}=-G_{45} G_{61}-G_{46} G_{60}+G_{49} G_{56}+G_{50} G_{55}+G_{51} G_{54}$
$\begin{aligned}G_{70}& =-G_{45} G_{62}-G_{46} G_{61}+G_{50} G_{56}+G_{51} G_{55} G_{71}=-G_{46} G_{62} \\ & +G_{51} G_{56} G_{72}=G_{49} G_{57} G_{73}=-G_{47} G_{60}+G_{49} G_{58}+G_{50} G_{57}\end{aligned}$
$G_{74}=-G_{47} G_{61}-G_{48} G_{60}+G_{49} G_{59}+G_{50} G_{58}+G_{51} G_{57}$
$\begin{aligned}G_{75}& =-G_{47} G_{62}-G_{48} G_{61}+G_{50} G_{59}+G_{51} G_{58} \quad G_{76}=-G_{48} G_{62} \\ & +G_{51} G_{59} \quad G_{77}=G_{49} G_{63} \quad G_{78}=G_{49} G_{64}+G_{50} G_{63}\end{aligned}$
$\begin{aligned}G_{79}& =G_{49} G_{65}+G_{50} G_{64}+G_{51} G_{63}-G_{52} G_{60} G_{80}=G_{49} G_{66}+G_{50} G_{65} \\ & +G_{51} G_{64}-G_{52} G_{61}-G_{53} G_{60} G_{81}=G_{50} G_{66}+G_{51} G_{65} \\ & -G_{52} G_{62}-G_{53} G_{61} \quad G_{82}=G_{51} G_{66}-G_{53} G_{62} \quad G_{83}=G_{23} G_{49}\end{aligned}$
$\begin{aligned}G_{84}& =G_{23} G_{50}+G_{24} G_{49}-G_{30} G_{45} G_{85}=G_{23} G_{51}+G_{24} G_{50} \\ & +G_{25} G_{49}-G_{30} G_{46}-G_{31} G_{45} G_{86}=G_{24} G_{51}+G_{25} G_{50} \\ & +G_{26} G_{49}-G_{31} G_{46}-G_{32} G_{45} G_{87}=G_{25} G_{51}+G_{26} G_{50} \\ & -G_{32} G_{46} \quad G_{88}=G_{26} G_{51} \quad G_{89}=G_{27} G_{49} \quad G_{90}=G_{27} G_{50} \\ & +G_{28} G_{49}-G_{30} G_{47} G_{91}=G_{27} G_{51}+G_{28} G_{50}+G_{29} G_{49} \\ & -G_{30} G_{48}-G_{31} G_{47} G_{92}=G_{28} G_{51}+G_{29} G_{50}-G_{31} G_{48}-G_{32} G_{47}\end{aligned}$
$\begin{aligned}G_{93}& =G_{29} G_{51}-G_{32} G_{48} \quad G_{94}=G_{33} G_{49} \quad G_{95}=G_{34} G_{49}+G_{33} G_{50} \\ & -G_{30} G_{52} \quad G_{96}=G_{35} G_{49}+G_{34} G_{50}+G_{33} G_{51}-G_{31} G_{52} \\ & -G_{30} G_{53} \quad G_{97}=G_{35} G_{50}+G_{34} G_{51}-G_{32} G_{52}-G_{31} G_{53}\end{aligned}$
$\begin{aligned}G_{98}& =G_{35} G_{51}-G_{32} G_{53} \quad G_{99}=G_{36} G_{49} \quad G_{100}=G_{36} G_{50}+G_{37} G_{49} \\ & -G_{41} G_{45} \quad G_{101}=G_{36} G_{51}+G_{37} G_{50}-G_{41} G_{46}-G_{42} G_{45}\end{aligned}$
$G_{102}=G_{37} G_{51}-G_{42} G_{46} \quad G_{103}=G_{38} G_{49}$
$\begin{aligned}G_{104}& = G_{38} G_{50}+G_{39} G_{49} G_{105}=G_{38} G_{51}+G_{39} G_{50}+G_{40} G_{49} \\ & -G_{41} G_{47} G_{106}=G_{39} G_{51}+G_{40} G_{50}-G_{41} G_{48}-G_{42} G_{47}\end{aligned}$
$\begin{aligned}G_{107}& = G_{40} G_{51}-G_{42} G_{48} G_{108}=G_{43} G_{49} G_{109}=G_{44} G_{49}+G_{43} G_{50} \\ & -G_{41} G_{52} G_{110}=G_{44} G_{50}+G_{43} G_{51}-G_{42} G_{52}-G_{41} G_{53}\end{aligned}$
$G_{111}= G_{44} G_{51}-G_{42} G_{53} G_{112}=-G_{72} G_{83}+G_{67} G_{89}$
$G_{113}= -G_{73} G_{83}-G_{72} G_{84}+G_{68} G_{89}+G_{67} G_{90}$
$G_{114}= \sum_{j=0}^2\left(G_{69-j} G_{89+j}-G_{74-j} G_{83+j}\right) $
$G_{115}= \sum_{j=0}^3\left(G_{70-j} G_{89+j}-G_{75-j} G_{83+j}\right)$
$G_{116}= \sum_{j=0}^4\left(G_{71-j} G_{89+j}-G_{76-j} G_{83+j}\right)$
$G_{117}= -\sum_{j=0}^4 G_{76-j} G_{84+j}+\sum_{j=0}^3 G_{71-j} G_{90+j}$
$G_{118}= -\sum_{j=0}^3 G_{76-j} G_{85+j}+\sum_{j=0}^2 G_{71-j} G_{91+j}$
$G_{119}= -\sum_{j=0}^2 G_{76-j} G_{86+j}+\sum_{j=0}^1 G_{71-j} G_{92+j}$
$G_{120}= -G_{76} G_{87}-G_{75} G_{88}+G_{71} G_{93} G_{121}=-G_{76} G_{88}$
$G_{122}= G_{77} G_{89} G_{123}=G_{78} G_{89}+G_{77} G_{90}-G_{72} G_{94}$
$G_{124}= \sum_{j=0}^2 G_{79-j} G_{89+j}-\sum_{j=0}^1 G_{73-j} G_{94+j}$
$G_{125} =\sum_{j=0}^3 G_{80-j} G_{89+j}-\sum_{j=0}^2 G_{74-j} G_{94+j}$
$G_{126} =\sum_{j=0}^4 G_{81-j} G_{89+j}-\sum_{j=0}^3 G_{75-j} G_{94+j}$
$G_{127} =\sum_0^4\left(G_{82-j} G_{89+j}-G_{76-j} G_{94+j}\right) $
$G_{128} =\sum_0^3\left(G_{82-j} G_{90+j}-G_{76-j} G_{95+j}\right)$
$G_{129} =\sum_0^2\left(G_{82-j} G_{91+j}-G_{76-j} G_{96+j}\right) $
$G_{130} =G_{82} G_{92}+G_{81} G_{93}-G_{76} G_{97}-G_{75} G_{98} $
$G_{131} =G_{82} G_{93}-G_{76} G_{98}$
