Heterogeneous materials are widely used in engineering applications. Materials with different properties can be combined to obtain composites with enhanced capabilities. The application of these materials can be observed across many different industries, with much interest given to designing and predicting their response to applied loads. Utilizing finite elements such materials are able to be simulated. If such materials are modeled using a direct numerical model (DNM), the finest length scale is represented by the mesh discretization. However, for large, industrial problems this is done at great computational cost. Multi-scale formulations have been used to decrease the computational cost, making use of a periodic representative unit cell (RUC). Such cells are embedded into the coarse scale domain’s integration points (FE2), which can also be computationally expensive. Methods have been produced to further reduce the cost of multi-scale formulations by embedding the macro-scale solution with the relevant micro-scale features. Such methods are referred to as Transformation Field Analysis (TFA).
This work presents a new approach to the traditional TFA developed by Dvorak, expanding on the multi-scale enrichment techniques developed by Fish. TFA relies upon the development of influence functions and concentration tensors to upscale information on the fine scale to the coarse scale, with information being passed between a fine scale mesh and a single coarse scale element (Local TFA). A new model is proposed, one that allows communication of eigenstrain effects between all fine scale meshes and all coarse scale elements (Nonlocal TFA).
A 2D plane-strain mechanical model is developed to solve 2-scale problems. An isotropic linear hardening model is used with J2 plasticity to represent eigenstrains as plastic strains. A model reduction is also used to demonstrate how Nonlocal TFA performs in comparison to Local TFA with piecewise continuous strains. The models are discretized and implemented into a numerical solver. A direct numerical model is also provided for verification purposes, with a method of manufactured solutions used to verify the DNM. Afterwards, the results are discussed.
This work may be expanded to include cohesive interfaces between the matrix and particles, as well as the inclusion of fracture mechanics, allowing the model to capture separation of particles from the matrix, as well as other micro mechanical effects.