Elastoplastic nonlinear analysis of functionally graded beams utilizing the symplectic method

Abstract

Elastoplastic nonlinear bending behaviors of ceramic–metal composite functionally graded material (FGM) beams are investigated using the symplectic method in Hamilton system with the classical beam theory. The mechanical properties of two-phase materials are assumed to be power function of the thickness, and the elastoplastic material properties of the FGM beams are given by the linear rule of mixtures. The bilinear hardening elastoplastic constitutive equations are established based on the Tamura–Tomota–Ozawa hybrid reinforcement model. And the von-Mises yield condition is introduced to judge whether the beams have entered the plastic deformation. The canonical equations are solved by the symplectic method, and the loads and deflections are transformed into the symplectic eigenvalues and symplectic eigensolutions in symplectic space and obtained by analytical solutions. Examples of bending behaviors varying with load, deflection, slenderness ratio and power law index are presented. The obtained results reveal that the zones of the elastic bending and the elastoplastic bending as well as the peak values of the deflections vary with material and geometric properties.

Appendices

Appendix A

$$k_{1} = \cos \sqrt \beta \cosh \sqrt \beta ,$$

$$k_{2} = \sin \sqrt \beta \sinh \sqrt \beta ,$$

$$k_{3} = \cos \sqrt \beta \sinh \sqrt \beta ,$$

$$k_{4} = \sin \sqrt \beta \cosh \sqrt \beta .$$

$$c_{1}^{{\text{C - S}}} = - 1,$$

$$c_{2}^{{\text{C - S}}} = \frac{{k_{1} \left( {k_{3} + k_{4} } \right) - k_{2} \left( {k_{3} - k_{4} } \right) - 1}}{{k_{2} \left( {k_{3} + k_{4} } \right) + k_{1} \left( {k_{3} - k_{4} } \right)}},$$

$$c_{3}^{{\text{C - S}}} = \frac{{k_{1}^{2} + k_{2}^{2} - k_{1} }}{{k_{1} \left( {k_{3} - k_{4} } \right) + k_{2} \left( {k_{3} + k_{4} } \right)}},$$

$$c_{4}^{{\text{C - S}}} = - \frac{{k_{1}^{2} + k_{2}^{2} - k_{1} }}{{k_{1} \left( {k_{3} - k_{4} } \right) + k_{2} \left( {k_{3} + k_{4} } \right)}};$$

$$c_{1}^{{\text{S - S}}} = - 1,$$

$$c_{2}^{{\text{S - S}}} = 0,$$

$$c_{3}^{{\text{S - S}}} = \frac{{k_{1} k_{3} + k_{2} k_{4} - k_{3} }}{{k_{3}^{2} + k_{4}^{2} }},$$

$$c_{4}^{{\text{S - S}}} = \frac{{k_{1} k_{4} - k_{2} k_{3} - k_{4} }}{{k_{3}^{2} + k_{4}^{2} }}.$$

$${\text{C}}_{{{1}i}}^{{\text{C - S}}} = - 1,$$

$${\text{C}}_{2i}^{{\text{C - S}}} = \frac{{\sinh \sqrt {\beta_{i} } \cosh \sqrt {\beta_{i} } + \sin \sqrt {\beta_{i} } \cos \sqrt {\beta_{i} } - 1}}{{\sinh \sqrt {\beta_{i} } \cosh \sqrt {\beta_{i} } - \sin \sqrt {\beta_{i} } \cos \sqrt {\beta_{i} } }},$$

$${\text{C}}_{3i}^{{\text{C - S}}} = \frac{{\cos^{2} \sqrt {\beta_{i} } + \sinh^{2} \sqrt {\beta_{i} } - \cos \sqrt {\beta_{i} } \cosh \sqrt {\beta_{i} } }}{{\sinh \sqrt {\beta_{i} } \cosh \sqrt {\beta_{i} } - \sin \sqrt {\beta_{i} } \cos \sqrt {\beta_{i} } }},$$

$${\text{C}}_{4i}^{{\text{C - S}}} = - \frac{{\cos^{2} \sqrt {\beta_{i} } + \sinh^{2} \sqrt {\beta_{i} } - \cos \sqrt {\beta_{i} } \cosh \sqrt {\beta_{i} } }}{{\sinh \sqrt {\beta_{i} } \cosh \sqrt {\beta_{i} } - \sin \sqrt {\beta_{i} } \cos \sqrt {\beta_{i} } }};$$

$${\text{C}}_{1i}^{{\text{S - S}}} = - 1,$$

$${\text{C}}_{2i}^{{\text{S - S}}} = 0,$$

$${\text{C}}_{3i}^{{\text{S - S}}} = \frac{{\sinh \sqrt {\beta_{i} } \cosh \sqrt {\beta_{i} } - \cos \sqrt {\beta_{i} } \sinh \sqrt {\beta_{i} } }}{{\cos^{2} \sqrt {\beta_{i} } \sinh^{2} \sqrt {\beta_{i} } + \sin^{2} \sqrt {\beta_{i} } \cosh^{2} \sqrt {\beta_{i} } }},$$

$${\text{C}}_{4i}^{{\text{S - S}}} = \frac{{\cos \sqrt {\beta_{i} } \sin \sqrt {\beta_{i} } - \sin \sqrt {\beta_{i} } \cosh \sqrt {\beta_{i} } }}{{\cos^{2} \sqrt {\beta_{i} } \sinh^{2} \sqrt {\beta_{i} } + \sin^{2} \sqrt {\beta_{i} } \cosh^{2} \sqrt {\beta_{i} } }}.$$

Appendix B

See Fig. 7.

Fig. 7

Method flowchart