Non-linear behavior of functionally graded elastoplastic beam under torsion

Abstract

The torsional behavior of beams graded in one and two directions under large displacements and angular deformations was analyzed using the power law and sinusoidal functions. Functionally graded material is elastoplastic, consisting of ceramic and metal. A nonlinear finite element method with isoparametric hexahedral elements was used. The finite element formulation was developed by using the updated Lagrangian formulation based on the virtual displacement principle. An iterative solution using Newton-Raphson and updated Newton-Raphson methods was used to solve the nonlinear equation system. The propagation of the plastic region was calculated based on the flow theory of plasticity. Elastoplastic behavior and effective material properties were determined according to the TTO model. Numerical investigations have shown that functionally graded beams behave quite differently from homogeneous beams under torsion. Yielding of the material starts at the outer boundaries of the section of the homogeneous beams, and the plastic region propagates symmetrically. On the other hand, yielding and propagation of plastic regions tend to shift to regions with more ceramic volume with higher effective Young modulus in functionally graded beams. Beams graded in the axial direction have a non-linear variation of rotation angle along the axial direction, unlike beams graded in section and pure metal beams. The amount of non-linearity increases with increasing volume of the ceramic material, which has higher torsional stiffness. Unlike homogeneous beams, the largest shear stresses can occur within the section rather than at the outer boundaries of the section. In beams graded from ceramic to metal using the power law, the section moves along the transverse direction in addition to the rotation. This transverse displacement occurs in the grading direction, and its magnitude is about 3% of the thickness at 12.5° rotation angle. Also, the shear stresses are not zero in the section's midpoint. The effects of material distribution on displacements, stresses, and plastic region propagation were examined, and essential points were reported.

Keywords

• Functionally graded material
• torsion
• hexahedral finite element

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