5. Numerical results and discussion

The BEM that has been used in the current paper can be applicable to a wide variety of FGA structures problems associated with the proposed theory of three temperatures nonlinear uncoupled magneto-thermoelasticity. In order to evaluate the influence of graded parameter on the three temperatures and displacements, the numerical results are carried out and depicted graphically for homogeneous (m ¼ 0) and functionally graded (m ¼ 0:5 and 1:0) structures.

Figure 2. Variation of the electron temperature Te through the thickness coordinate x.

Figures 2–4 show the distributions of the three temperatures Te, Ti and Tp through the thickness coordinate Ox. It was shown from these figures that the three temperatures increase with increasing value of graded parameter m.

Figures 7 and 8 show the distributions of the displacements u<sup>1</sup> and u<sup>2</sup> with the time for boundary element method (BEM), finite difference method (FDM) and finite element method (FEM) to demonstrate the validity and accuracy of our proposed technique. It is noted from numerical results that the BEM obtained results are agree quite well with those obtained using the FDM of Pazera and Jędrysiak [81] and FEM of Xiong and Tian [82] results based on replacing heat

Boundary Element Model for Nonlinear Fractional-Order Heat Transfer in Magneto…

conduction with three-temperature heat conduction.

DOI: http://dx.doi.org/10.5772/intechopen.88255

Variation of the displacement u2 through the thickness coordinate x.

Figure 6.

Figure 7.

Figure 8.

43

Variation of the displacement u1 with time τ.

Variation of the displacement u2 with time τ.

Figures 5 and 6 show the distributions of the displacements u1 and u2 through the thickness coordinate Ox. It was noticed from these figures that the displacement components increase with increasing value of graded parameter m.

Figure 3. Variation of the ion temperature Ti through the thickness coordinate x.

Figure 4. Variation of the photon temperature Tp through the thickness coordinate x.

Figure 5. Variation of the displacement u1 through the thickness coordinate x.

Boundary Element Model for Nonlinear Fractional-Order Heat Transfer in Magneto… DOI: http://dx.doi.org/10.5772/intechopen.88255

Figures 7 and 8 show the distributions of the displacements u<sup>1</sup> and u<sup>2</sup> with the time for boundary element method (BEM), finite difference method (FDM) and finite element method (FEM) to demonstrate the validity and accuracy of our proposed technique. It is noted from numerical results that the BEM obtained results are agree quite well with those obtained using the FDM of Pazera and Jędrysiak [81] and FEM of Xiong and Tian [82] results based on replacing heat conduction with three-temperature heat conduction.

Figure 6. Variation of the displacement u2 through the thickness coordinate x.

Figure 7. Variation of the displacement u1 with time τ.

Figure 8. Variation of the displacement u2 with time τ.

Figures 2–4 show the distributions of the three temperatures Te, Ti and Tp through the thickness coordinate Ox. It was shown from these figures that the three

Figures 5 and 6 show the distributions of the displacements u1 and u2 through the thickness coordinate Ox. It was noticed from these figures that the displacement

temperatures increase with increasing value of graded parameter m.

Mechanics of Functionally Graded Materials and Structures

components increase with increasing value of graded parameter m.

Variation of the ion temperature Ti through the thickness coordinate x.

Variation of the photon temperature Tp through the thickness coordinate x.

Variation of the displacement u1 through the thickness coordinate x.

Figure 3.

Figure 4.

Figure 5.

42
