Fundamental Study on Nonlinear Finite Element Method to Achieve High Accuracy

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The finite element method is widely used in the structural analysis of engineering products for the reasonable design and developments. The present study is concerned with the constitutive equation and variational principles and their effects on numerical analysis in order to achieve high accuracy in the non-linear finite element analysis during three years from 1990 to 1992. Hypoelasticity, defined as the rate type constitutive equation, is used to describe plastic behaviors of metal alloys, on the other hand, hyperelasticity, defined as the potential type constitutive equation, is used for rubber materials. Hypoelasticity widely used with Jaumann stress derivative gives oscillation in stress-deformation curve if linear kinematic hardening law is adopted in simple shear problem. Hyperelasticity, based on Green-Lagrange strain measured from the undeformed configuration, cannot predict accurately the experimental fact that shear stress respond behind shear deformation in simple shear problem. For the analysis of hyperelasticity, the employed mixed variational principle does not include the initial stress term, which significantly affects bifurcation analysis. In order to overcome such shortcomings in constitutive equation and mixed variational principles, the following points are studied consecutively.(1) By using rotationless strain regarded as the generalized logarithmic strain, hyperelastic constitutive equation is proposed.(2) Theoretical aspects of hypoelasticity is studied from the viewpoint of corotational equation and non-coaxiality relating to intermediate configurations.(3) Constitutive equation for large plastic strain is formulated based on internal time theory and compared to the experiments.(4) Mixed variational principles taking account of the initial stress term and stability of solutions are studied theoretically.(5) Numerical analysis of hypo and hyperelasticity is carried out by using rotationless strain.