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Computational Overlap Coupling Between Micropolar Linear Elastic Continuum Finite Elements and Nonlinear Elastic Spherical Discrete Elements in One Dimension

Abstract : The report presents a one dimensional (1D) problem for overlap coupling between a micropolar linear elastic 1D mixed finite element (FE) model and a 1D string of Hertzian nonlinear elastic discrete element spheres. The 1D micropolar balance equations and linear elastic constitutive equations are derived from the three-dimensional (3D) theory by assuming Timoshenko beam kinematics with axial stretch. They are formulated in weak form, where upon introducing interpolation functions, lead to coupled, linear 1D micropolar FE matrix equations. The nonlinear vector equations for a string of Hertzian nonlinear elastic, glued, discrete element (DE) spheres are derived and solved by the Newton-Raphson method. The 1D micropolar FEs and 1D string of DEs are coupled using a bridging-scale decomposition as a point of departure. 1D numerical examples for full overlap coupling, and partial overlap coupling, for quasi-statics are demonstrated. The formulation is general for quasi-statics and dynamics.