Modeling of hyperelastic incompressible behavior of functionally graded material under bending load

In this paper, the behavior of inhomogeneous functionally graded rubber with large deformations and under bending loading is modeled by assuming an incompressible hyper-elastic material. In hyper-elastic inhomogeneous functionally graded materials, mechanical properties continuously changes from one point to another in the specified direction. In the other words, they gradually become material from material to another. For modeling the nonlinear behavior of material, hyperelasticity theory and strain energy density functions, which are a function of the invariants of the left deformation Cauchy-Green tensor, are used. In order to be able to apply the existing energy functions to inhomogeneous functionally graded materials, they must be changed, therefore, the changes in the constant of the energy functions are assumed power shape and in the direction of the curvature radius after bending, due to the inhomogeneous functionally graded of the material. Since many materials are inhomogeneous, using the assumption of inhomogeneous functionally graded of the material is one of the most practical methods. In this paper, the modeling of the hyperelastic behavior of inhomogeneous functionally graded rubber is done under bending loading and extraction of the Cauchy stress relations governing the cross-section caused by this loading. For modeling, the generalized Mooney-Rivlin energy function is used and the properties change in radial direction are considered and heterogeneity variations are also investigated.