Elastoplastic Buckling of Functionally Graded Beams using Tamura-Tomota-Ozawa and Ramberg-Osgood Material Models

Abstract

Functionally Graded Materials (FGMs) are an advanced class of composite materials characterized by a gradual, continuous variation of material properties spatially. While buckling of slender Functionally Graded Beams (FGBs) can be analyzed using linear eigenvalue analysis and is well documented, the buckling response of FGBs with low and medium slenderness ratios is under-researched. This behavior is highly complex due to coupled shear effects and material yielding in the elastoplastic region. The goal of this work is to investigate the nonlinear elastoplastic buckling of short and medium metal-ceramic FGBs. The beam kinematics are modeled using the First-Order Shear Deformation Theory (Timoshenko beam theory) within a semi-analytical Ritz framework. To simulate the elastoplastic behavior of FGBs, a modified rule-of-mixtures law, based on the Tamura-Tomota-Ozawa model, is coupled with the Ramberg-Osgood phenomenological constitutive equations. The nonlinear stress-strain behavior of the metal component of the FGB is described using Hencky's total plastic strain model. Finally, an arc-length solver is employed to trace the FGB's nonlinear load-displacement path. Parametric studies are conducted by varying power-law coefficients and thickness ratios, and the results are compared with those from a 3D Finite Element Analysis (FEA) using Abaqus/Standard. The analytical model demonstrates excellent agreement with FEA for beams with a medium length-to-thickness ratio, with a maximum error of just about 4% for thickness ratios > 15. However, some discrepancies are observed when comparing very short FGBs with thickness ratios between 5 and 15. These discrepancies stem from the fundamental divergence between the deformation theory of plasticity employed in this formulation and the flow theory used in FEA models, which highlights the 'plastic buckling paradox', and from differences between 3D FEA and 1D First-Order Shear Deformation Theory, reinforcing the critical need for experimental validation.