Sensitivity Analysis of Hyperelastic Problems Using the Local and Global Complex-Variable Finite Element Methods




Ytuarte, Ernest

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A new sensitivity approach based on the Complex-Variable Finite Element Method (ZFEM) was developed to analyze problems involving hyperelastic materials and large geometric deformations. ZFEM introduces complex-valued algebra into the traditional finite element formulation, yielding highly accurate derivatives of the output variables with respect to parameters of interest, including shape, material, and loading input variables. The sensitivity information can be used to make an appropriate material selection, optimize geometry, or adjust loading conditions of hyperelastic problems. In addition, ZFEM is demonstrated for the computation of the tearing energy, an important parameter for determining crack growth in hyperelastic materials. The method was implemented through User Element (UEL) routines in the commercial software ABAQUS. The Local Complex Finite Element Method (L-ZFEM) was first implemented for nonlinear elastic fracture problems in this work. L-ZFEM is a variation of the traditional ZFEM approach, also denoted as global ZFEM (G-ZFEM), but restricts complex-valued computations to a small area surrounding the crack tip. L-ZFEM is fracture specific but it allows for more efficient computation of the tearing energy. However, it exhibits the typical limitations of local-based approaches like the J-integral, which includes non-convergent tearing energy values for cracks in nonhomogeneous bodies. In contrast, G-ZFEM converges for non-homogeneous bodies at different crack extension directions. Thus, it can be regarded as the reference solution in fracture mechanics applications. The extension of the ZFEM to hyperelasticity opens an area of opportunity for optimizing the design of rubber seals, artificial biological tissues, seismic isolation bearings, and non-linear elastic structural components.


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Complex-Variable, Deformation, Hyperelasticity, Sensitivity, Tearing, ZFEM, Finite element method



Mechanical Engineering