**6. Numerical results and discussion**

In order to illustrate the numerical results of this study, we consider a monoclinic graphite-epoxy as an anisotropic smart material which has the following constants [35].

The elasticity tensor is expressed as

$$\mathbf{C}\_{pjkl} = \begin{bmatrix} 430.1 & 130.4 & 18.2 & 0 & 0 & 201.3 \\ 130.4 & 116.7 & 21.0 & 0 & 0 & 70.1 \\ 18.2 & 21.0 & 73.6 & 0 & 0 & 2.4 \\ 0 & 0 & 0 & 19.8 & -8.0 & 0 \\ 0 & 0 & 0 & -8.0 & 29.1 & 0 \\ 201.3 & 70.1 & 2.4 & 0 & 0 & 147.3 \end{bmatrix} \text{GPa} \tag{83}$$

The mechanical temperature coefficient is

$$
\beta\_{pj} = \begin{bmatrix}
\mathbf{1.01} & \mathbf{2.00} & \mathbf{0} \\
\mathbf{2.00} & \mathbf{1.48} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{7.52}
\end{bmatrix} \cdot \mathbf{10}^6 \text{ N/km}^2 \tag{84}
$$

The thermal conductivity tensor is

$$k\_{pj} = \begin{bmatrix} 5.2 & 0 & 0 \\ 0 & 7.6 & 0 \\ 0 & 0 & 38.3 \end{bmatrix} \text{W/km} \tag{85}$$

Mass density *<sup>ρ</sup>* <sup>¼</sup> 7820 kg*=*m<sup>2</sup> and heat capacity *<sup>c</sup>* <sup>¼</sup> 461 J*=*kg k.

The technique that has been proposed in the current chapter can be applicable to a wide range of three-temperature nonlinear thermal stress problems of FGA structures. The main aim of this chapter is to assess the impact of MDD and anisotropy on the three-temperature nonlinear thermal stress distributions.

*The CPU time and the number of iterations for some communication–avoiding Krylov subspace solvers.*

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart… DOI: http://dx.doi.org/10.5772/intechopen.92829*

The proposed technique that has been implemented in the current study can be applicable to a wide variety of FGA smart structure problems involving three temperatures. All the physical parameters satisfy the initial and boundary conditions. The efficiency of our BEM modeling technique has been improved using an explicit staggered algorithm based on communication-avoiding Arnoldi procedure to decrease the computation time.

**Figure 5** shows the variations of the three temperatures Te, Ti and Tp with the time τ in the presence of MDD. **Figure 6** shows the variations of the three temperatures Te, Ti and Tp with the time τ in the presence of MDD. It can be seen from **Figures 5** and **6** that the MDD has a significant effect on the temperature distributions.

**Figure 5.** *Variation of the three-temperature (with memory) with time* τ*.*

**Figure 6.** *Variation of the three-temperature (without memory) with time* τ*.*

**6. Numerical results and discussion**

**Solvers DOF**

The elasticity tensor is expressed as

The mechanical temperature coefficient is

*βp j* ¼

The thermal conductivity tensor is

2 6 4

*kp j* ¼

2 6 4

Mass density *<sup>ρ</sup>* <sup>¼</sup> 7820 kg*=*m<sup>2</sup> and heat capacity *<sup>c</sup>* <sup>¼</sup> 461 J*=*kg k.

on the three-temperature nonlinear thermal stress distributions.

constants [35].

**108**

**Table 2.**

*Cp jkl* ¼

*Advanced Functional Materials*

In order to illustrate the numerical results of this study, we consider a monoclinic graphite-epoxy as an anisotropic smart material which has the following

CA–Arnoldi Residual 6.81E–12 5.38E–12 4.13E–11 4.17E–11 7.57E–11

CA–GMRES Residual 2.98E–12 1.90E–12 1.28E–11 1.36E–11 1.22E–11

CA– Lanczos Residual 7.20E–11 3.35E–11 2.72E–11 3.97E–11 8.33E–11

*The CPU time and the number of iterations for some communication–avoiding Krylov subspace solvers.*

CPU time (s) 4.96 10.78 99.24 134.26 293.29 Iterations 25 25 25 25 25

CPU time (s) 5.06 11.49 126.38 164.09 445.51 Iterations 50 50 50 50 50

CPU time (s) 5.05 11.47 139.07 180.49 514.72 Iterations 22 26 28 30 32

*:*1 130*:*4 18*:*2 0 0 201*:*3 *:*4 116*:*7 21*:*0 0 0 70*:*1 *:*2 21*:*0 73*:*60 0 2*:*4 0 0 0 19*:*8 �8*:*0 0 0 00 �8*:*0 29*:*1 0 *:*3 70*:*1 2*:*4 0 0 147*:*3

> 1*:*01 2*:*00 0 2*:*00 1*:*48 0 0 07*:*52

> > 5*:*20 0 0 7*:*6 0 0 0 38*:*3

The technique that has been proposed in the current chapter can be applicable to a wide range of three-temperature nonlinear thermal stress problems of FGA structures. The main aim of this chapter is to assess the impact of MDD and anisotropy

3 7

> 3 7 5

**965 1505 3380 3964 6005**

<sup>5</sup> � <sup>10</sup><sup>6</sup> <sup>N</sup>*=*km<sup>2</sup> (84)

W*=*km (85)

GPa (83)

In order to study the anisotropy and MDD effects on the nonlinear thermal stresses, we assume the following four cases: A, B, C, and D, where case A denotes the nonlinear thermal stress distribution in the isotropic material without MDD effect, case B denotes the nonlinear thermal stress distribution in isotropic material with MDD effect, case C denotes the nonlinear thermal stress distribution in anisotropic material without MDD effect, and case D denotes nonlinear thermal stress distribution in anisotropic material with MDD effect.

**Figures 7**–**9** show the variation of the nonlinear thermal stresses σ11, σ<sup>12</sup> and σ<sup>22</sup> with the time τ. It is clear from these figures that both anisotropy and MDD have a significant influence on the nonlinear thermal stress distributions.

Since there are no available results for the considered problem in the literature. Therefore, we only considered the one-dimensional special case for the variations of the nonlinear thermal stress σ<sup>11</sup> with the time τ as shown in **Figure 10**. The validity and accuracy of our proposed technique was confirmed by comparing graphically

**Figure 7.** *Variation of the nonlinear thermal stress* σ*<sup>11</sup> with time* τ*.*

our BEM results with those obtained using the FDM of Pazera and Jędrysiak [95] and FEM of Xiong and Tian [96] results based on replacing one-temperature heat conduction with the total three-temperature *T* ð Þ T ¼ Te þ Ti þ Tr heat conduction. It can be noticed that the BEM results are found to agree very well with the FDM

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart…*

The main aim of this chapter is to introduce a new MDD model based on BEM for obtaining the transient three-temperature nonlinear thermal stresses in FGA smart structures. The governing equations of this model are very hard to solve analytically because of nonlinearity and anisotropy. To overcome this, we propose a

and FEM results.

*Variation of the nonlinear thermal stress* σ*<sup>11</sup> with time* τ*.*

*Variation of the nonlinear thermal stress* σ*<sup>22</sup> with time* τ*.*

*DOI: http://dx.doi.org/10.5772/intechopen.92829*

**Figure 10.**

**Figure 9.**

**7. Conclusion**

**111**

**Figure 8.** *Variation of the nonlinear thermal stress* σ*<sup>12</sup> with time* τ*.*

*A Novel MDD-Based BEM Model for Transient 3T Nonlinear Thermal Stresses in FGA Smart… DOI: http://dx.doi.org/10.5772/intechopen.92829*
