An adaptive FE 2 method for the thermomechanical behaviour of a SMA‐Fiber matrix composite

This contribution deals with the multiscale analysis of a reinforced matrix with shape‐memory‐alloys (SMA). The complicated micro‐structure permits a full discretized macro‐structure, due to that an FE 2 approach is used. Unlike classic materials, the SMA has more complex behavior with a high‐temperature dependency in the loading and unloading case. The micro‐structure of the macroscopic problem consists of a linear‐elastic matrix and a random fiber distribution. The stress response of the composite depends non‐linearly on the deformation, the fiber orientation and the temperature. Due to the non‐linear behavior of the fiber a nested homogenized is employed at each integration point. This results in the so‐called FE 2 method. One disadvantage of the FE 2 method is the high computational effort by solving the boundary value problem (BVP) in every integration point and every iteration step. This motivates the present work to introduce an indicator, which determines whether an accompanying homogenization is needed. Due to the temperature dependency, on the macro‐scale, a coupled thermo‐mechanical problem is solved. In the first homogenization step, the BVP of the RVE is solved with Neumann Boundary conditions. This leads to an overestimation of the strains. The SMA will remain linear elastic until the phase transition condition is reached. The formulation is similar to the classical yield condition of elasto‐plastic materials, however, here a strain instead a stress creteria is used. From the phase transition condition, the temperature dependent indicator is defined and formulated as a limit strain for the linear behavior of the SMA‐fibers. The accompanying homogenization is firstly needed when the limit strain is reached.