Fractional Temperature-Dependent BEM for Laser Ultrasonic Thermoelastic Propagation Problems of Smart Nanomaterials

1. Introduction

Many engineering research in recent years have aroused academics' interest in investigating the thermoelastic behavior of materials [1,2,3,4,5,6] due to their potential in geological and engineering applications. Nanotechnology is concerned with developing tools for studying the properties of nanomaterials, whereas nanoscience is concerned with moving and manipulating atoms to achieve the properties required in a particular field of life [7,8]. Nanostructures are one of the most important outcomes of nanotechnology. A structure is classified as a nanostructure if one of its dimensions is 100 nanometers or less. Understanding the mechanical behavior of deformed nanostructures is critical because they are used in a wide range of industries and professions, including engineering, medicine, renewable energy, and military applications. In the industrial sector, certain nanoparticles are used to create filters due to their greater strength as compared to traditional materials [9]. Because of recent advances in nanoscale electronics and photonics [10,11,12], certain nanoparticles can be utilized as drug-carrying materials in the medical profession because they have a unique sensitivity to the place inside the human body to which the drug is supposed to be conveyed. When they reach that location, they accurately release the drug. Encouragement studies have also confirmed the potential for employing nanoparticles as a cancer treatment. Furthermore, gold nanoparticles are employed to detect pregnancies in home pregnancy test kits. Nanowires are being employed in nanoscale biosensors for early illness detection [13,14]. In the field of renewable energy, the panel, which is connected by an electrical circuit and contains hundreds of solar cells, converts solar energy into electrical energy. Military uses for nanomaterials include the creation of nanoscale cylinders with strength and rigidity that have a million times the storage capacity of conventional computers, military clothing that can absorb radar waves for stealth and infiltration, and nanosatellites [15,16,17]. Specific nanomaterials are incorporated into concrete in the building and construction industry to improve its tenacity, rigor, and water resistance. These materials include silica nanoparticles, carbon nanotubes, and titanium dioxide (TiO2). Many nanotechnology applications rely on porothermoelastic interactions that vary with size [18,19,20]. Because size-dependent thermopiezoelectric problems are computationally complex to solve and do not have a general analytical solution, numerical methods for solving them should be developed [21]. The BEM model of Fahmy et al. [22] described the thermopiezoelectricity theory in smart nanomaterials. In the considered BEM model, we introduced a new solution for fractional, temperature-dependent, and wave propagation size-dependent thermopiezoelectricity problems in smart nanomaterials. This paper may be considered as a generalization for [22] with fractional, temperature-dependent, and wave propagation effects which are not considered in [22]. The boundary element method (BEM) is an efficient numerical approach employed to solve partial differential equations [23,24,25,26]. It outperforms the finite element method (FEM) in several ways [27]. Only the problem's boundary needs to be discretized for the BEM. In comparison to FEM, which necessitates discretization of the entire problem domain, it has a substantial advantage. Because it requires less computational work and input data preparation, this feature is critical for solving complex problems in smart nanomaterials. It also improves the feature's usability. Only the BEM formulation procedure can address infinite domain problems with complicated borders and geometrical quirks accurately. The BEM approach is particularly effective for measuring field derivatives such as tractions, heat fluxes, and sensitivities. The BEM solution is provided by the integral representation expression. In the FEM, the solution is only computed at nodal points. As a result, BEM has recently emerged as a reliable, practical, and widely used alternative to FEM for modelling of fractional temperature- and size-dependent thermoelastic problems in smart nanomaterials technology.

In this paper, the temperature-dependent thermoelasticity problems have been solved using the boundary element method (BEM) to understand the mechanical characteristics of deformed smart nanomaterials. The numerical results show the impacts of the fractional parameter, piezoelectric parameter and length scale parameter on the total force-stresses. The numerical results also show temperature-dependent and temperature-independent effects on smart nanomaterials and nonsmart nanomaterials, as well as the viability, effectiveness, and precision of the current BEM methodology.

2. Formulation of the Problem

Consider a cross section of thermoelastic smart nanomaterial in the $x_1{x}_2-\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{e}$, occupies the region $V$ that bounded by $S$ as shown in Figure 1. Assume $n_{\alpha }$ can be written as

$$n_{\alpha }=e_{\alpha \beta }\frac{d{x}_{\beta }}{ds}$$

(1)

where $e_{\alpha \beta }\left(e_{12}=-e_{21}=1,e_{11}=e_{22}=0\right)$.

All quantities in the $x_1{x}_2-\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{e}$ are independent of $x_3$.

The rotation in terms of deformation displacement vector $\left(u_1,u_2\right)$ and electric field in terms of electric potential $\phi$ can be expressed as

$$\begin{gathered} \mathsf{\Omega }={\mathsf{\Omega }}_3=\frac{1}{2}\left(u_{2,1}-u_{1,2}\right) \\ =\frac{1}{2}{e}_{\alpha \beta }{u}_{\beta ,\alpha } \end{gathered}$$

(2)

$$E_{\alpha }=-{\phi }_{,\alpha }$$

(3)

The strain tensor, mean curvature vector and true couple-stress can be written as follows

$$\begin{gathered} {\epsilon }_{\alpha \beta } \\ =\frac{1}{2}\left(u_{\alpha ,\beta }+u_{\beta ,\alpha }\right) \end{gathered}$$

(4)

$$\begin{gathered} k_{\alpha }=e_{\alpha \beta }{k}_{3\beta } \\ =\frac{1}{2}{e}_{\alpha \beta }{\mathsf{\Omega }}_{,\beta } \end{gathered}$$

(5)

$$M_i=\frac{1}{2}{e}_{ijk}{M}_{kj}$$

(6)

where $k_1=k_{32}=\frac{1}{2}{\mathsf{\Omega }}_{,2},k_2=-k_{31}=-\frac{1}{2}{\mathsf{\Omega }}_{,1}$ and $k_{3\alpha }=-k_{\alpha 3}=\frac{1}{2}{\mathsf{\Omega }}_{,\alpha }$.

and $M_{\alpha }={\epsilon }_{\alpha \beta }{M}_{3\beta },M_{ij}=-M_{ji},M_1=-M_{23},M_2=M_{13},\mathrm{a}\mathrm{n}\mathrm{d}{M}_3=M_{21}=0$

The force-stress tensor can be divided into two sections

$$\begin{gathered} {\sigma }_{\alpha \beta }={\sigma }_{\left(\alpha \beta \right)}+{\sigma }_{\left[\alpha \beta \right]}, \\ {\sigma }_{3\alpha }={\sigma }_{\alpha 3} \\ =0 \end{gathered}$$

(7)

The electric displacement $D_{\alpha }$ is given as

$$D_{\alpha }=e_0{E}_{\alpha }+P_{\alpha }$$

(8)

The governing equations for entropy balance, force equilibrium, moment equilibrium equations and Gauss’s law for electric field of considered smart nanomaterial can be expressed as

$$-q_{\alpha ,\alpha }+Q=0$$

(9)

$$\begin{gathered} {\sigma }_{\beta \alpha ,\beta }+F_{\alpha } \\ =0 \end{gathered}$$

(10)

$$\begin{gathered} {\sigma }_{\left[\beta \alpha \right]}=-M_{\left[\alpha ,\beta \right]},{\sigma }_{\left[21\right]}=-{\sigma }_{\left[12\right]} \\ =-M_{\left[1,2\right]} \end{gathered}$$

(11)

$$D_{\alpha ,\alpha }={\rho }_E$$

(12)

$$\begin{gathered} {\left[{\sigma }_{\left(\beta \alpha \right)}-M_{\left[\alpha ,\beta \right]}\right]}_{,\beta }+F_{\alpha } \\ =0 \end{gathered}$$

(13)

Now, we present the following constitutive equations of considered smart nanomaterial

The heat flux

$$q_{\alpha }=-k{\mathsf{\Theta }}_{,\alpha }$$

(14)

The force-stress, couple-stress and electric displacement are

$$\begin{gathered} {\sigma }_{\left(\alpha \beta \right)}=\lambda {\epsilon }_{\gamma \gamma }{\delta }_{\alpha \beta }+2\mu {\epsilon }_{\alpha \beta } \\ -\left(3\lambda +2\mu \right)\overline{\alpha }\mathsf{\Theta }{\delta }_{\alpha \beta } \end{gathered}$$

(15)

$$\begin{gathered} M_{\alpha }=-8\mu l^2{k}_{\alpha }+2f{E}_{\alpha }, \\ {l}^2=\frac{\eta }{\mu } \end{gathered}$$

(16)

$$\begin{gathered} D_{\alpha } \\ =e{E}_{\alpha } \\ +4f{k}_{\alpha } \end{gathered}$$

(17)

The force-traction, couple-traction and normal electric displacement are

$$t_{\alpha }={\sigma }_{\beta \alpha }{n}_{\beta }$$

(18)

$$\begin{gathered} m=e_{\beta \alpha }{M}_{\alpha }{n}_{\beta } \\ =M_2{n}_1-M_1{n}_2 \end{gathered}$$

(19)

$$d=D_{\alpha }{n}_{\alpha }$$

(20)

Thus, the total force-stress tensor is

$$\begin{gathered} {\sigma }_{\beta \alpha }=\lambda {\epsilon }_{\gamma \gamma }{\delta }_{\alpha \beta }+2\mu {\epsilon }_{\alpha \beta }+2\mu l^2{e}_{\alpha \beta }{\nabla }^2\mathsf{\Omega } \\ -\frac{E}{1-2\nu }\overline{\alpha }\mathsf{\Theta }{\delta }_{\alpha \beta } \end{gathered}$$

(21)

where $E=2\mu \left(1+v\right),\lambda =2\mu \frac{v}{1-2v}$

The fractional-order temperature-dependent heat equation is

$$\begin{gathered} D_{\tau }^a\mathsf{\Theta }\left(\mathbf{x},\tau \right)=\xi \nabla \left[\lambda \left(\mathsf{\Theta }\right)\nabla \mathsf{\Theta }\left(\mathbf{x},\tau \right)\right]+\xi Q(\mathbf{x},\mathsf{\Theta },\tau ),\xi \\ =\frac{1}{\rho \left(\mathsf{\Theta }\right)c\left(\mathsf{\Theta }\right)} \end{gathered}$$

(22)

in which

$$\begin{gathered} Q\left(\mathbf{x},\mathsf{\Theta },\tau \right)=\overline{Q}\left(\mathbf{x},\mathsf{\Theta },\tau \right)+\frac{1-R}{{\mathbf{x}}_0}{e}^{\left(-\frac{{\mathbf{x}}_a}{{\mathbf{x}}_0}\right)J\left(\tau \right)},J\left(t\right) \\ =\frac{J_0\tau }{{\tau }_1^2}{e}^{-\frac{\tau }{{\tau }_1}},a=1,2,3 \end{gathered}$$

As a result, equations (9), (10) and (12) may be expressed as

$$\begin{gathered} k{\nabla }^2\mathsf{\Theta }+Q \\ =0 \end{gathered}$$

(23)

$$\begin{gathered} \lambda u_{\beta ,\beta \alpha }+\mu (\left(1+l^2{\nabla }^2\right)u_{\beta ,\beta \alpha }+\left(1-l^2{\nabla }^2\right){\nabla }^2{u}_{\alpha }) \\ -\frac{E}{1-2\nu }\overline{\alpha }{\mathsf{\Theta }}_{,\alpha }+F_{\alpha }=0 \end{gathered}$$

(24)

$$\begin{gathered} e{\nabla }^2\phi +{\rho }_E=0, \\ e=e_r{e}_0 \end{gathered}$$

(25)

Now, we can introduce the following definitions for $q$, $t_{\alpha }$, $m$, and $d$ as follows

The normal heat flux

$$\begin{gathered} q=q_{\alpha }{n}_{\alpha } \\ =-k\frac{\partial \mathsf{\Theta }}{\partial n} \end{gathered}$$

(26)

The force-traction vector

$$\begin{gathered} t_{\alpha }={\sigma }_{\beta \alpha }{n}_{\beta }=\left(\lambda {\epsilon }_{\gamma \gamma }{\delta }_{\alpha \beta }+2\mu {\epsilon }_{\alpha \beta }+2\mu l^2{e}_{\alpha \beta }{\nabla }^2\mathsf{\Omega } \\ -\frac{E}{1-2\nu }\overline{\alpha }\mathsf{\Theta }{\delta }_{\alpha \beta }\right)n_{\beta } \end{gathered}$$

(27)

The couple-traction

$$\begin{gathered} m=e_{\beta \alpha }{\mu }_{\alpha }{n}_{\beta } \\ =4\mu l^2\frac{\partial \mathsf{\Omega }}{\partial n}-2f\frac{\partial \phi }{\partial s} \end{gathered}$$

(28)

The normal electric displacement

$$\begin{gathered} d=D_{\alpha }{n}_{\alpha }=-e\frac{\partial \phi }{\partial n}+ \\ 2f\frac{\partial \mathsf{\Omega }}{\partial s} \end{gathered}$$

(29)

3. Boundary Conditions

The temperature and displacement boundary conditions under consideration are

$$\mathsf{\Theta }=\overline{\mathsf{\Theta }} on S_{\mathrm{T}}$$

(30)

$$q=\overline{q} on S_q,S_T\cup S_q=S,S_T\cap S_q=\mathrm{\varnothing }$$

(31)

$$u_{\alpha }={\overline{u}}_{\alpha } on S_u$$

(32)

$$t_{\alpha }={\overline{t}}_{\alpha } on S_t,S_u\cup S_t=S,S_u\cap S_t=\mathrm{\varnothing }$$

(33)

where

$$\mathsf{\Omega }=\overline{\mathsf{\Omega }} on S_{\omega }$$

(34)

$$m=\overline{m} on S_m,S_{\omega }\cup S_m=S,S_{\omega }\cap S_m=\mathrm{\varnothing }$$

(35)

and

$$\phi =\overline{\phi } on S_{\phi }$$

(36)

$$d=\overline{d} on S_d,S_{\phi }\cup S_d=S,S_{\phi }\cap S_d=\mathrm{\varnothing }$$

(37)

where $S_{\mathrm{T}}$, $S_q$, $S_u$, $S_t$, $S_{\mathsf{\Omega }}$, $S_m$, $S_{\phi }$ and $S_d$ are the sprcified boundary values for $\mathrm{T}$, $q$, $u_{\alpha }$, $t_{\alpha }$, $\mathsf{\Omega }$, $m$, $\phi$ and $d$, respectively.

4. Boundary Element Implementation

By using Caputo's formula and Eq. (22), we can write [28,29]

$$D_{\tau }^a{\mathsf{\Theta }}^{f+1}+D_{\tau }^a{\mathsf{\Theta }}^f\approx \sum _{J=0}^k{W}_{a,J}\left({\mathsf{\Theta }}^{f+1-J}\left(\mathbf{x}\right)-{\mathsf{\Theta }}^{f-J}\left(\mathbf{x}\right)\right)$$

(38)

where

$$W_{a,0}=\frac{{\left(∆\tau \right)}^{-a}}{\Gamma \left(2-a\right)}\mathrm{a}\mathrm{n}\mathrm{d}{W}_{a,J}=W_{a,0}\left({\left(J+1\right)}^{1-a}-{\left(J-1\right)}^{1-a}\right)$$

(39)

By using Eq. (38), Eq. (22) may be written as

$$\begin{gathered} W_{a,0}{\mathsf{\Theta }}^{f+1}\left(\mathbf{x}\right)-\lambda \left(\mathbf{x},\mathsf{\Theta }\right){\mathsf{\Theta }}_{,ii}^{f+1}\left(\mathbf{x}\right)-{\lambda }_{,i}\left(\mathbf{x},\mathsf{\Theta }\right){\mathsf{\Theta }}_{,i}^{f+1}\left(\mathbf{x}\right)=W_{a,0}{\mathsf{\Theta }}^f\left(\mathbf{x}\right)-\lambda \left(\mathbf{x}\right){\mathsf{\Theta }}_{,ii}^f\left(\mathbf{x}\right) \\ -{\lambda }_{,i}\left(\mathbf{x},\mathsf{\Theta }\right){\mathsf{\Theta }}_{,j}^f\left(\mathbf{x}\right)-\sum _{J=1}^f{W}_{a,J}\left({\mathsf{\Theta }}^{f+1-J}\left(\mathbf{x}\right)-{\mathsf{\Theta }}^{f-J}\left(\mathbf{x}\right)\right)+h_m^{f+1}\left(\mathbf{x},\mathsf{\Theta },\tau \right)+h_m^f\left(\mathbf{x},\mathsf{\Theta },\tau \right) \end{gathered}$$

(40)

By using Kirchhoff transformation $T={\int }_{T_0}^T\frac{\lambda \left(\mathsf{\Theta }\right)}{{\lambda }_0}d\mathsf{\Theta }$ [30], Eq. (22) may be written as follows [31]

$${\nabla }^{2}T\left(\mathbf{x},\mathsf{\tau }\right)+\frac{1}{{\lambda }_0}h\left(\mathbf{x},T,\mathsf{\tau }\right)=\frac{{\rho }_0{c}_0}{{\lambda }_0}\frac{\partial T(\mathbf{x},\mathsf{\tau })}{\partial \mathsf{\tau }}+Nl\left(\mathbf{x},T,\dot{T}\right)$$

(41)

which can be expressed as [31]

$${\nabla }^{2}T\left(\mathbf{x},\mathsf{\tau }\right)+\frac{1}{{\lambda }_0}{h}_{Nl}\left(\mathbf{x},T,\dot{T},\mathsf{\tau }\right)=\frac{{\rho }_0{c}_0}{{\lambda }_0}\frac{\partial T(\mathbf{x},\mathsf{\tau })}{\partial \mathsf{\tau }}$$

(42)

in which

$$Nl\left(\mathbf{x},T,\dot{T}\right)=\left[\frac{\rho \left(T\right)c\left(T\right)}{\lambda \left(T\right)}-\frac{{\rho }_0{c}_0}{{\lambda }_0}\right]\dot{T}$$

(43)

$$h_{Nl}\left(\mathbf{x},T,\dot{T},\mathsf{\tau }\right)=h\left(\mathbf{x},T,\mathsf{\tau }\right)+\left[{\rho }_0{c}_0-\frac{{\lambda }_0}{\lambda \left(T\right)}\rho \left(T\right)c\left(T\right)\right]\dot{T}$$

(44)

The fundamental solution of (40) can be used to define the integral equation corresponding to (42) as [32].

$$\begin{gathered} C\left(P\right)T\left(P,{\overline{\tau }}_{n+1}\right)+a_0{\int }_{\mathsf{\Gamma }}^{}{\int }_{{\overline{\tau }}_n}^{{\overline{\tau }}_{n+1}}T\left(\mathrm{Q},\mathsf{\tau }\right)q^{*}(P,{\overline{\tau }}_{n+1};Q,\mathsf{\tau })d\mathsf{\tau }\mathrm{d}\mathsf{\Gamma } \\ =a_0{\int }_{\mathsf{\Gamma }}^{}{\int }_{{\overline{\tau }}_n}^{{\overline{\tau }}_{n+1}}q\left(\mathrm{Q},\mathsf{\tau }\right)T^{*}(P,{\overline{\tau }}_{n+1};Q,\mathsf{\tau })d\mathsf{\tau }\mathrm{d}\mathsf{\Gamma } \\ +\frac{a_0}{{\lambda }_0}{\int }_{\mathsf{\Omega }}^{}{\int }_{{\overline{\tau }}_n}^{{\overline{\tau }}_{n+1}}{h}_{Nl}\left(\mathrm{Q},T,\dot{T},\mathsf{\tau }\right)T^{*}(P,{\overline{\tau }}_{n+1};Q,\mathsf{\tau })d\mathsf{\tau }\mathrm{d}\mathsf{\Omega } \\ +{\int }_{\mathsf{\Omega }}^{}T(\mathrm{Q},{\overline{\tau }}_n)T^{*}(P,{\overline{\tau }}_{n+1};Q,\mathsf{\tau })\mathrm{d}\mathsf{\Omega },a_0=\frac{{\lambda }_0}{{\rho }_0{c}_0} \end{gathered}$$

(45)

By using the same technique of Fahmy [31], where the radial point interpolation method (RPIM) and Cartesian transformation method (CTM) [33,34,35,36] have been used to treat the domain integrals in Eq. (45) which is resulted from fractional-order temperature-dependent heat conduction equation (22).

The boundary integral equations can now be expressed as follows [37,38,39]

$$\begin{gathered} c^{Q^{*}}\left(\xi \right)T\left(\xi \right)-{\int }_S^{}{q}^{Q^{*}}\left(\mathrm{x},\xi \right)T\left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ =-{\int }_S^{}{T}^{Q^{*}}\left(\mathrm{x},\xi \right)q\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)+{\int }_V^{}{T}^{Q^{*}}\left(\mathrm{x},\xi \right)Q\left(\mathrm{x}\right)dV\left(\mathrm{x}\right) \end{gathered}$$

(46)

$$\begin{gathered} c_{\alpha \beta }\left(\xi \right)u_{\alpha }\left(\xi \right)+{\oint }_S^{}{t}_{\alpha \beta }^{F^{*}}\left(\mathrm{x},\xi \right)u_{\alpha }\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)+{\int }_S^{}{m}_{\beta }^{F^{*}}\left(\mathrm{x},\xi \right)\mathsf{\Omega }\left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ +{\int }_S^{}{h}_{\beta }^{F^{*}}\left(\mathrm{x},\xi \right)T\left(x\right)dS\left(\mathrm{x}\right)+{\int }_S^{}{d}_{\beta }^{F^{*}}\left(\mathrm{x},\xi \right)\phi \left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ ={\int }_S^{}{u}_{\alpha \beta }^{F^{*}}\left(\mathrm{x},\xi \right)t_{\alpha }\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)+{\int }_S^{}{\mathsf{\Omega }}_{\beta }^{F^{*}}\left(\mathrm{x},\xi \right)m\left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ +{\int }_V^{}{u}_{\alpha \beta }^{F^{*}}\left(\mathrm{x},\xi \right)F_{\alpha }\left(\mathrm{x}\right)dV+{\int }_S^{}{f}_{\beta }^{F^{*}}\left(\mathrm{x},\xi \right)q\left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ -{\int }_V^{}{f}_{\beta }^{F^{*}}\left(\mathrm{x},\xi \right)Q\left(\mathrm{x}\right)dV \end{gathered}$$

(47)

$$\begin{gathered} c^{\mathsf{\Omega }}\left(\xi \right)\mathsf{\Omega }\left(\xi \right)+{\int }_S^{}{t}_{\alpha }^{C^{*}}\left(\mathrm{x},\xi \right)u_{\alpha }\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)+{\oint }_S^{}{m}^{C^{*}}\left(\mathrm{x},\xi \right)\mathsf{\Omega }\left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ +{\oint }_S^{}{d}^{C^{*}}\left(\mathrm{x},\xi \right)\phi \left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ ={\int }_S^{}{u}_{\alpha }^{C^{*}}\left(\mathrm{x},\xi \right)t_{\alpha }\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)+{\int }_S^{}{\mathsf{\Omega }}^{C^{*}}\left(\mathrm{x},\xi \right)m\left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ +{\int }_V^{}{u}_{\alpha }^{C^{*}}\left(\mathrm{x},\xi \right)F_{\alpha }\left(\mathrm{x}\right)dV \end{gathered}$$

(48)

$$\begin{gathered} c^{\phi }\left(\xi \right)\phi \left(\xi \right)+{\oint }_S^{}{m}^{R^{*}}\left(\mathrm{x},\xi \right)\mathsf{\Omega }\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)+{\oint }_S^{}{d}^{R^{*}}\left(\mathrm{x},\xi \right)\phi \left(\mathrm{x}\right)dS\left(\mathrm{x}\right) \\ ={\int }_S^{}{\phi }^{R^{*}}\left(\mathrm{x},\xi \right)d\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)-{\int }_V^{}{\phi }^{R^{*}}\left(\mathrm{x},\xi \right){\rho }_E\left(\mathrm{x}\right)dV \end{gathered}$$

(49)

The integral equations (46) - (49) in absence of body forces and volume charge density can be written in matrix form as follows

$$\begin{gathered} \left[\begin{array}{c}\begin{array}{c}{c}^{{\mathcal{Q}}^{\mathrm{*}}}\left(\xi \right)T\left(\xi \right)\\ c_{\alpha \beta }\left(\xi \right)u_{\alpha }\left(\xi \right)\end{array}\\ \begin{array}{c}{c}^{\omega }\left(\xi \right)\mathsf{\Omega }\left(\xi \right)\\ c^{\varphi }\left(\xi \right)\phi \left(\xi \right)\end{array}\end{array}\right]+{\oint }_S^{}\left[\begin{array}{cccc}{-q}^{{\mathcal{Q}}^{\mathrm{*}}}& 0& 0& 0\\ h_{\beta }^{F^{\mathrm{*}}}& t_{\alpha \beta }^{F^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& m_{\beta }^{F^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& d_{\beta }^{F^{\mathrm{*}}}\left(\mathrm{x},\xi \right)\\ 0& t_{\alpha }^{C^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& m^{C^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& d^{C^{\mathrm{*}}}\left(\mathrm{x},\xi \right)\\ 0& 0& m^{R^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& d^{R^{\mathrm{*}}}\left(\mathrm{x},\xi \right)\end{array}\right]\left[\begin{array}{c}\begin{array}{c}T\left(\mathrm{x}\right)\\ u_{\alpha }\left(\mathrm{x}\right)\end{array}\\ \begin{array}{c}\mathsf{\Omega }\left(\mathrm{x}\right)\\ \phi \left(\mathrm{x}\right)\end{array}\end{array}\right]dS\left(\mathrm{x}\right) \\ ={\int }_s^{}\left[\begin{array}{cccc}{-\vartheta }^{{\mathcal{Q}}^{\mathrm{*}}}& 0& 0& 0\\ f_{\beta }^{F^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& u_{\alpha \beta }^{F^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& {\mathsf{\Omega }}_{\beta }^{F^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& 0\\ 0& u_{\alpha }^{C^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& {\mathsf{\Omega }}^{C^{\mathrm{*}}}\left(\mathrm{x},\xi \right)& 0\\ 0& 0& 0& {\phi }^{R^{\mathrm{*}}}\left(\mathrm{x},\xi \right)\end{array}\right]\left[\begin{array}{c}\begin{array}{c}q\left(\mathrm{x}\right)\\ t_{\alpha }\left(\mathrm{x}\right)\end{array}\\ \begin{array}{c}m\left(\mathrm{x}\right)\\ d\left(\mathrm{x}\right)\end{array}\end{array}\right]dS\left(\mathrm{x}\right) \end{gathered}$$

(50)

Now, it is convenient to rewrite Eq. (50) in compact index-notation form as

$$c_{IJ}\left(\xi \right)u_I\left(\xi \right)+{\oint }_S^{}{t}_{IJ}^{\mathrm{*}}\left(\mathrm{x},\xi \right)u_I\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)={\int }_s^{}{u}_{IJ}^{\mathrm{*}}\left(\mathrm{x},\xi \right)t_I\left(\mathrm{x}\right)dS\left(\mathrm{x}\right)$$

(51)

This leads to the following linear algebraic equations system

$$\overline{T}\overline{u}=\overline{U}\overline{t}$$

(52)

that can also be expressed as

$$AX=B$$

(53)

5. Numerical Results and Discussion

To demonstrate the numerical computations calculated using the proposed methodology, we consider the temperature-dependent thermoelastic smart nanomaterial [40,41] using the boundary conditions depicted in Figure 2, to exemplify the numerical computations computed by the suggested methodology. Under thermal and piezoelectric loadings, the considered thermoelastic smart nanomaterial deforms and becomes electrically polarized. As illustrated in Figure 3, the BEM discretization used 42 border elements and 68 internal points.

Figure 2. Geometry of the considered thermoelastic smart nanomaterial.

Figure 2. Geometry of the considered thermoelastic smart nanomaterial.

Figure 3. BEM model of the current problem.

Figure 3. BEM model of the current problem.

Table 1. Considered properties of pure copper (Cu) nanoparticles [42].

T(oC)	0	500	900
C(J/kg) oK	385	433	480
$\mathit{\rho }(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3)$	8930	8686	8458

Table 1. Considered properties of pure copper (Cu) nanoparticles [42].

T(oC)	0	500	900
C(J/kg) oK	385	433	480
$\mathit{\rho }(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3)$	8930	8686	8458

The thermal conductivity pure copper (Cu) nanoparticles is

$$\mathit{\lambda }=400\left(1-\frac{\mathit{T}}{6000}\right)$$

The solid line indicates Case A, which stands for temperature-dependent smart nanomaterials $\left(f=-1\right).$ Case B is shown by the dashed line, which represents for temperature-dependent nonsmart nanomaterials $\left(f=0\right)$. The dotted line indicates Case C, which stands for temperature-independent smart nanomaterials $\left(f=-1\right)$. Case D is shown by the dash-dot line, which represents for temperature-independent nonsmart nanomaterials $\left(f=0\right)$.

In the present paper, to solve linear systems generated by BEM discretization efficiently, we used stable communication avoiding S-step – generalized minimal residual method (SCAS-GMRES) of Zan et al. [43] to reduce the number of iterations and computation time. The SCAS-GMRES) [43], fast modified fast modified diagonal and toeplitz splitting (FMDTS) of Xin and Chong [44], and unconditionally convergent - respectively scaled circulant and skew-circulant splitting (UC-RSCSCS) of Zi et al. [45] were compared during our solution of the current problem in Table 2. This table shows the number of iterations (Iter.), processor time (CPU time), relative residual (Rr), and error (Err.) calculated for different length scale values. According to Tab. 2, the SCAS-GMRES iterative method requires the least amount of IT and CPU time, implying that it outperforms the FMDTS and UC-RSCSCS iterative methods.

Table 3 explains the numerical solutions obtained for total force-stress ${\mathit{\sigma }}_{11}$ at points $\overline{\mathit{A}}$ and $\overline{\mathit{B}}$ for various length scale values $(l=0.01,0.1\mathrm{a}\mathrm{n}\mathrm{d}1.0)$. Table 3 additionally provides the finite element method (FEM) data of Sladek et al. [46], and the analytical data of Yu et al. [47] for our investigated problem. As demonstrated in Table 3, the BEM data are very consistent with the FEM and analytical data. As a result, the proposed BEM's validity and precision have been demonstrated.

From Figure 4, it is obvious that the total force-stress ${\sigma }_{11}$ increases, decreases then increases tends to zero as $x_1$ tends to infinity for different theories.

From Figure 5, it is obvious that the total force-stress ${\sigma }_{12}$ decreases, increases, decreases then increases tends to zero as $x_1$ tends to infinity for different theories.

From Figure 6, it is obvious that the total force-stress ${\sigma }_{11}$ increase, decrease then increases tends to zero as $x_1$ tends to infinity also it is shown that the total force-stress ${\sigma }_{22}$ increases with the small values of $x_1$ and then decrease and increase with the large values of the $x_1$

From Figure 7, it is obvious that the total force-stress ${\sigma }_{11}$, decreases with an increasing of $x_1$ but it increases with an increasing of fractional order parameter $a$.

From Figure 8, it is clear that the total force-stress ${\sigma }_{12}$, increases and decreases with an increasing of $x_1$, and tends to zero as $x_1$ tends to infinity, it is display also that the values of total force-stress ${\sigma }_{12}$ almost coincide at the different values of fractional order parameter $a$, except for the interval $1.25<x_1<2.20$ where we find that the total force-stress ${\sigma }_{12}$ decreases with an increasing of fractional order parameter $a$

From Figure 9, it is obvious that the total force-stress ${\sigma }_{22}$, increase, decrease and tend to zero as $x_1$ tends to infinity also it is clear that the total force-stress ${\sigma }_{22}$ decreases with the increasing of fractional order parameter $a$.

From Figure 10, it is obvious that the total force-stress ${\sigma }_{11}$ increase, decrease and tend to zero as $x_1$ tends to infinity also it is clear that the total force-stress ${\sigma }_{11}$ decreases with the increasing of piezoelectric parameter f.

From Figure 11, it is seen that big values of of piezoelectric parameter f are very origin comparing with small values of piezoelectric parameter f.

From Figure 12, it is clear that the total force-stress ${\sigma }_{22}$, increase, decrease and tend to zero as $x_1$ tends to infinity, it is display also that the values of total force-stress ${\sigma }_{22}$ almost coincide at the different values of piezoelectric parameter f, except for the interval $1.25<x_1<2.20$ where we find that the total force-stress ${\sigma }_{12}$ decreases with an increasing of piezoelectric parameter f.

From Figure 13, it is obvious that the total force-stress ${\sigma }_{22}$, increase, decrease and tends to zero as $x_1$ tends to infinity also it is clear that the total force-stress ${\sigma }_{11}$ decreases with the small values of length scale parameter $l$ and then increase and inclined with the large values of length scale parameter $l$.

From Figure 14, it is obvious that the total force-stress ${\sigma }_{12}$, increase, decrease and tends to zero as $x_1$ tends to infinity also it is clear that the total force-stress ${\sigma }_{12}$ decreases with the small values of length scale parameter $l$ and then increase and inclined with the large values of length scale parameter $l$

From Figure 15, it is concluded that the total force-stress ${\sigma }_{22}$ along $x_1$-axis increases for the small values of $x_1$ with an increasing of length scale parameter $l$, also it is clear that the total force-stress ${\sigma }_{22}$ decreases and increasing with the large values of $x_1$ and tends to zero as $x_1$ tends to infinity.

6. Conclusion

A new BEM model for temperature- and size-dependent fractional thermoelastic problems in smart nanomaterials is introduced.

A new efficient BEM methodology is developed for treating temperature-dependent and size-dependent thermoelastic problems in smart nanomaterials.

The BEM efficiency has been shown by the usage of the SCAS-GMRES, which minimizes memory needs and processing time.

The suggested model includes thermoelastic and piezoelectric impacts, which allows us to explain the differences between temperature--dependent smart nanomaterials, temperature--independent smart nanomaterials, temperature--dependent nonsmart nanomaterials, and temperature--independent nonsmart nanomaterials.

The numerical data are plotted to show the impacts of the fractional order parameter, temperature, and size on the total force-stresses.

The computational effectiveness of the suggested methodology has been established.

The proposed BEM approach has been shown to be valid and accurate.

We can conclude from current study that our proposed BEM technique is practicable, feasible, effective, and superior to FDM or FEM.

The proposed methodology can be utilized to examine a wide range of thermoelastic problems in smart nanomaterials that are temperature and size dependent.

It can be argued that our research has a wide range of applications, including shape memory alloys, environmental sensors, photovoltaic cells, nanoceramics, sunscreens, air purifiers, food packaging, flame retardants, antibacterial cleansers, filters, smart coatings, and thin films.

Recent numerical calculations for issues with smart nanomaterials may be of interest to nanophysicists, nanochemists, nanobiologists, in addition to mathematicians with expertise in nanotechnology, quantum computing, artificial intelligence and optogenetics.