**2. Formulation of the problem**

The governing equations for three-temperature anisotropic generalized micropolar thermoviscoelasticity problems can be expressed as [58]

$$
\sigma\_{\text{ij,j}} + \mathfrak{p} \mathbf{F}\_{\text{i}} = \mathfrak{p} \ddot{\mathbf{u}}\_{\text{i}} \tag{1}
$$

*Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques… DOI: http://dx.doi.org/10.5772/intechopen.90824*

$$
\rho \mathbf{m}\_{\rm ij,j} + e\_{\rm ijk} \sigma\_{\rm jk} + \rho \mathbf{M}\_{\rm i} = \mathbf{J} \rho \ddot{\mathbf{o}}\_{\rm i} \tag{2}
$$

where

share the second shortcoming. So, several generalized thermoelasticity theories that predict finite speeds of propagation for heat waves have been developed such as extended thermo-elasticity (ETE) theory of Lord and Shulman [4], temperaturerate-dependent thermo-elasticity (TRDTE) theory of Green and Lindsay [5], three linear generalized thermoelasticity theories (type I, II and III) of Green and Naghdi [6, 7], low-temperature thermoelasticity (LTTE) model of Hetnarski and Ignaczak [8], the dual phase-lag (DPL) heat conduction equation of Tzou [9, 10] which has been developed taking into consideration the phonons-electrons interactions to obtain dual phase-lag thermoelasticity (DPLTE) [11, 12], and three-phase-lag thermoelasticity (TPLTE) model of Choudhuri [13] who takes into consideration the phase-lags of heat flux, temperature gradient and thermal displacement gradient. Chen and Gurtin [14], introduced the theory of two-temperatures (conductive temperature and thermodynamic temperature) heat conduction in the context

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

of elastic bodies, then Youssef [15] extended this theory to generalized

element technique has been formulated in the context of micropolar

tional cost.

**188**

of our proposed model.

**2. Formulation of the problem**

thermoelasticity. Fahmy [16] introduced three-temperature nonlinear generalized micropolar-magneto-thermoelasticity theory and developed a new boundary element technique for Modeling and Simulation of complex problems associated with this theory. Theory of micropolar elasticity [17, 18] has been developed for studying the mechanical behavior of polymers and elastomers and applied in many applications [19–24]. Then, Eringen [25] and Nowacki [26] extended it to micropolar thermoelasticity, and then implemented in various applications [27–29]. Because of strong nonlinearity of three-temperatures radiative heat conduction equations, the numerical solution and simulation of such problems are always difficult and require the development of new numerical schemes [30, 31]. In comparison with other numerical methods [32–34], the boundary element method has been successfully applied and performed for solving various applications [35–60]. The boundary

thermoelasticity by Sladek and Sladek [61–63] and Huang and Liang [64]. Through the current paper, the term three-temperatures introduced for the first time in the field of nonlinear generalized micropolar thermoviscoelasticity. Recently, evolutionary algorithms [65, 66] have received much attention of researchers. The genetic algorithm (GA) can deal with the multi-objective and complex shapes problems. Also, it could reach an optimal solution with highly reduced computa-

The main aim of this article is to introduce a new theory called nonlinear generalized micropolar thermoviscoelasticity involving three temperatures. Because of strong nonlinearity, it is very difficult to solve the problems related to this theory analytically. Therefore, we propose a new boundary element model for simulation

and optimization of three temperatures nonlinear generalized micropolar thermoviscoelastic problems associated with this theory. The genetic algorithm (GA) was implemented based on free form deformation (FFD) technique and nonuniform rational B-spline (NURBS) curve as an optimization technique for the considered problems. The numerical results demonstrate the validity and accuracy

The governing equations for three-temperature anisotropic generalized

σij,j þ ρFi ¼ ρu€<sup>i</sup> (1)

micropolar thermoviscoelasticity problems can be expressed as [58]

$$\boldsymbol{\sigma}\_{\text{ij}} = \mathbf{C}\_{\text{ijkl}} \,\mathsf{N} \,\mathsf{e} \,\mathsf{S}\_{\text{ij}} + \mathsf{d} \left(\mathbf{u}\_{\text{j},\text{i}} - \boldsymbol{\varepsilon}\_{\text{ij}k} \boldsymbol{\alpha}\_{\text{k}}\right) - \beta\_{\text{ij}} \mathbf{T}\_{\text{a}} \left(\mathbf{C}\_{\text{ijkl}} = \mathbf{C}\_{\text{klij}} = \mathbf{C}\_{\text{jkl}}, \beta\_{\text{ij}} = \beta\_{\text{ji}}\right) \tag{3}$$

$$\mathbf{m}\_{\mathbf{i}\mathbf{j}} = \mathbf{a}\,\mathbf{a}\_{\mathbf{k},\mathbf{k}}\mathbf{\delta}\_{\mathbf{i}\mathbf{j}} + \overline{\mathbf{a}}\mathbf{a}\mathbf{o}\_{\mathbf{i},\mathbf{j}} + \overline{\mathbf{a}}\mathbf{a}\mathbf{o}\_{\mathbf{j},\mathbf{i}} \tag{4}$$

$$
\varepsilon\_{\rm ij} = \varepsilon\_{\rm ij} - \varepsilon\_{\rm ijk} (\mathbf{r}\_{\rm k} - \alpha\_{\rm k}), \varepsilon\_{\rm ij} = \frac{1}{2} \left( \mathbf{u}\_{\rm i,j} + \mathbf{u}\_{\rm j,i} \right), \mathbf{r}\_{\rm i} = \frac{1}{2} \varepsilon\_{\rm ikl} \mathbf{u}\_{\rm l,k} \tag{5}
$$

The two-dimension three-temperature (2D-3T) radiative heat conduction equations can be expressed as [53]

$$\mathbf{c}\_{\mathbf{e}} \frac{\partial \mathbf{T}\_{\mathbf{e}}(\mathbf{r}, \mathbf{r})}{\partial \mathbf{r}} - \frac{\mathbf{1}}{\rho} \nabla [\mathbb{K}\_{\mathbf{e}} \nabla \mathbf{T}\_{\mathbf{e}}(\mathbf{r}, \mathbf{r})] = -\mathcal{W}\_{\mathbf{e}i}(\mathbf{T}\_{\mathbf{e}} - \mathbf{T}\_{i}) - \mathcal{W}\_{\mathbf{e}p}(\mathbf{T}\_{\mathbf{e}} - \mathbf{T}\_{\mathbf{p}}) \tag{6}$$

$$\mathbf{c}\_{\mathbf{i}} \frac{\partial \mathbf{T}\_{\mathbf{i}}(\mathbf{r}, \mathbf{r})}{\partial \mathbf{r}} - \frac{\mathbf{1}}{\rho} \nabla [\mathbb{K}\_{\mathbf{i}} \nabla \mathbf{T}\_{\mathbf{i}}(\mathbf{r}, \mathbf{r})] = \mathbb{W}\_{\mathbf{e}\mathbf{i}}(\mathbf{T}\_{\mathbf{e}} - \mathbf{T}\_{\mathbf{i}}) \tag{7}$$

$$\frac{4}{\rho} \mathbf{c}\_{\mathrm{p}} \mathbf{T}\_{\mathrm{p}}^{3} \frac{\partial \mathbf{T}\_{\mathrm{p}}(\mathbf{r}, \boldsymbol{\pi})}{\partial \boldsymbol{\pi}} - \frac{\mathbf{1}}{\rho} \nabla \left[ \mathbb{K}\_{\mathrm{p}} \nabla \mathbf{T}\_{\mathrm{p}}(\mathbf{r}, \boldsymbol{\pi}) \right] = \mathcal{W}\_{\mathrm{ep}} \left( \mathbf{T}\_{\mathrm{e}} - \mathbf{T}\_{\mathrm{p}} \right) \tag{8}$$
