Bi-modulus materials with different mechanical responses in tension and compression are often found in civil,composite, and biological engineering. Numerical analysis of bimodular materials is strongly nonlinear and convergence is usually a problem for traditional iterative schemes. This paper aims to develop a stabilized computational method for nonlinear analysis of 3D bimodular materials. Based on the parametric variational principle, a unified constitutive equation of 3D bimodular materials is proposed, which allows the eight principal stress states to be indicated by three parametric variables introduced in the principal stress directions.The original problem is transformed into a standard linear complementarity problem（LCP） by the parametric virtual work principle and a quadratic programming algorithm is developed by solving the LCP with the classic Lemke＇s algorithm. Update of elasticity and stiffness matrices is avoided and, thus, the proposed algorithm shows an excellent convergence behavior compared with traditional iterative schemes.Numerical examples show that the proposed method is valid and can accurately analyze mechanical responses of 3D bimodular materials. Also, stability of the algorithm is greatly improved.
A concurrent multiscale method is developed for simulating quasi-static crack propagation in which the failure processes occur in only a small portion of the structure. For this purpose, a multiscale model is adopted and both scales are discretized with finite-element meshes. The extended finite element method is employed to take into account the propagation of discontinuities on the fine-scale subregions. At the same time, for the other subregions, the coarse-scale mesh is employed and is resolved by using the extended multiscale finite element method. Several representative numerical examples are given to verify the validity of the method.
