**1. Introduction**

The classical thermo-elasticity (CTE) theory which was introduced by Duhamel [1] and Neumann [2] characterized the strain-temperature gradients equations in an elastic body, but it has two shortcomings contrary to physical observations: First, the heat conduction equation of this theory does not include any elastic terms. Second, the heat conduction equation is of a parabolic type predicting infinite speeds of thermal waves. Biot [3] developed the classical coupled thermo-elasticity (CCTE) theory to overcome the first shortcoming in CTE. However, both theories

mij,j þ εijkσjk þ ρMi ¼ Jρω€<sup>i</sup> (2)

ωj,i (4)

εiklul,k (5)

� � (6)

� � (8)

(9)

(10)

<sup>p</sup> (11)

2

∇½ �¼ i∇Tið Þ r, τ eið Þ Te � Ti (7)

*<sup>∂</sup>*Tαð Þ r, <sup>τ</sup> ∂τ

� �, <sup>α</sup> <sup>¼</sup> e, <sup>δ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>

<sup>α</sup> , <sup>α</sup> <sup>¼</sup> e, i, <sup>p</sup> <sup>¼</sup> pT<sup>3</sup>þ

ρ T3 p

� � � <sup>β</sup>ijT<sup>α</sup> Cijkl <sup>¼</sup> Cklij <sup>¼</sup> Cjikl*;* <sup>β</sup>ij <sup>¼</sup> <sup>β</sup>ji � � (3)

<sup>2</sup> ui,j <sup>þ</sup> uj,i � �, ri <sup>¼</sup> <sup>1</sup>

<sup>∇</sup> p∇Tpð Þ r, <sup>τ</sup> � � <sup>¼</sup> ep Te � Tp

mij ¼ α ωk,kδij þ αωi,j þ α̿

*Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques…*

The two-dimension three-temperature (2D-3T) radiative heat conduction

**3. A new mathematical modelling of nonlinear generalized micropolar**

With reference to a Cartesian coordinate system ð Þ *x*1, *x*2, *x*<sup>3</sup> , we consider an anisotropic micropolar thermoviscoelastic structure occupies the region R which bounded by a closed surface S, and Sið Þ i ¼ 1, 2, 3, 4 denotes subsets of S such that

The 2D-3T radiative heat conduction Eqs. (6)–(8) can be expressed as [53]

∇½ �þ α∇Tαð Þ r, τ ð Þ¼ r, τ cαρδ<sup>1</sup>

�ρ eið Þ Te � Ti –ρ ep Te � Tp

� �, <sup>α</sup> <sup>¼</sup> p, <sup>δ</sup><sup>1</sup> <sup>¼</sup> <sup>4</sup>

where parameters cα, αð Þ α ¼ e, i, p , , ei, ep are constant inside each subdomain, but they are discontinuous on the interfaces between subdomains.

ρ eið Þ Te � Ti , α ¼ i, δ<sup>1</sup> ¼ 1

ρ ep Te � Tp

ei <sup>¼</sup> <sup>ρ</sup>eiT�2*=*<sup>3</sup> <sup>e</sup> ,ep <sup>¼</sup> <sup>ρ</sup>epT�1*=*<sup>2</sup> <sup>e</sup> , <sup>α</sup> <sup>¼</sup> αT<sup>5</sup>*=*<sup>2</sup>

∇½ �¼� e∇Teð Þ r, τ eið Þ� Te � Ti ep Te � Tp

<sup>ϵ</sup>ij <sup>¼</sup> <sup>ε</sup>ij � <sup>ε</sup>ijkð Þ rk � <sup>ω</sup><sup>k</sup> , <sup>ε</sup>ij <sup>¼</sup> <sup>1</sup>

where

ce

σij ¼ Cijkl ℵ eδij þ αˇ uj,i � εijkω<sup>k</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.90824*

equations can be expressed as [53]

4 ρ cpT<sup>3</sup> p

ci

**thermoviscoelasticity problem**

**3.1 BEM simulation for temperature field**

8 >>><

>>>:

*<sup>∂</sup>*Tið Þ r, <sup>τ</sup> <sup>∂</sup><sup>τ</sup> � <sup>1</sup> ρ

*<sup>∂</sup>*Tpð Þ r, <sup>τ</sup> <sup>∂</sup><sup>τ</sup> � <sup>1</sup> ρ

*<sup>∂</sup>*Teð Þ r, <sup>τ</sup> <sup>∂</sup><sup>τ</sup> � <sup>1</sup> ρ

S1 þ S2 ¼ S3 þ S4 ¼ S*:*

where

and

**189**

ð Þ¼ r, τ

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 of elastic bodies, then Youssef [15] extended this theory to generalized 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 element technique has been formulated in the context of micropolar 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 computational cost.

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 of our proposed model.
