**Figure 9.**

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

**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

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

distribution in anisotropic material with MDD effect.

*Advanced Functional Materials*

**Figure 8.**

**110**

**Figure 7.**

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

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

significant influence on the nonlinear thermal stress distributions.

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

**Figure 10.** *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 and FEM results.

#### **7. Conclusion**

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 new boundary element formulation for solving such equations. Since the CA kernels of the s-step Krylov methods are faster than the kernels of standard Krylov methods. Therefore, we used an explicit staggered algorithm based on CA-Arnoldi procedure to solve the resulted linear equations. The computational performance of the proposed technique has been performed using communication-avoiding Arnoldi procedure. The numerical results are presented highlighting the effects of MDD on the three-temperature distributions and the influence of MDD and anisotropy on the nonlinear thermal stresses of FGA smart structures. The numerical results also demonstrate the validity and accuracy of the proposed technique. It can be concluded from numerical results of our current general problem that all generalized and nonlinear generalized thermoelasticity theories can be combined with the three-temperature radiative heat conduction to describe the deformation of FGA smart structures in the context of memory-dependent derivatives. From the research that has been performed, it is possible to conclude that the proposed BEM technique is effective and stable for transient three-temperature thermal stress problems in FGA smart structures.

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The numerical results for our complex and general problem can provide data references for computer scientists and engineers, geotechnical and geothermal engineers, designers of new materials, and researchers in material science as well as for those working on the development of anisotropic smart structures. In the application of three-temperature theories in advanced manufacturing technologies, with the development of soft machines and robotics in biomedical engineering and advanced manufacturing, transient thermal stresses will be encountered more often where three-temperature radiative heat conduction will turn out to be the best choice for thermomechanical analysis in the design and analysis of advanced smart materials and structures.
