Shakedown analysis method for composites based on homogenization theory

Direct methods of plastic analysis are widely used in composites analysis to determine material strength for safety assessment or lightweight optimization design. Multi-scale processing of periodic heterogeneous composite material is needed due to its existing of microstructure. The standard method is to determine the macroscopic properties from the calculation results of microcosmic representative volume elements (RVEs) by using the homogenization theory. However, in current practice, there are some disadvantages of transforming the micro strain domain to the macro stress shakedown domain when considering multiple external loads. The domain cannot fully demonstrate the shakedown condition, and it is impossible to evaluate a known loading combination only from the knowledge of whether the load leads to the shakedown state. To overcome this disadvantage, a new comprehensive approach was proposed to enhance endurance limit strength of composites under variable loads for long term. Considering the example of in-plane strength analysis, for microcosmic RVEs, a new set of boundary condition was defined in the form of uniform strain. The boundary condition was derived from the elastic response under unit loads by using Hook’s law and stiffness matrix. The resulting elastic stress field was used later for plastic shakedown analysis. Based on the lower bound theorem of plastic mechanics, optimization programming for load factor was performed, and after proper mathematical reformulation, the conic quadratic optimization problem could be solved efficiently. Macro-stress shakedown domain can be obtained after scale-transformation of the RVE results. The bases of this stress domain are unidirectional stress in geometry space. The stress amplitude of a structure can be evaluated by this domain for determining the shakedown state in a simple and practical manner. Further, changes in the boundary condition of RVE do not affect the limit and elastic analysis. Finally, few numerical examples were presented for verification and illustration. This approach can be expanded to three dimensions and employed for more complex structures.