Solid/solid phase transformations in polycrystalline metals have a strong influence on the microstructure and the resulting material behavior. For example, microstructural evolution in shape memory alloys is governed by a temperature and/or stress induced transition between austenite and martensite, resulting in the eponymous shape memory effect and superelastic material behavior. The use of variational methods allows the derivation of material models that can be used to predict the complex microstructural evolution. The basic concept of the modeling approach is that each material prefers a state of minimum energy. The formulation of the energetic state of the observed material and the subsequent minimization leads directly to evolution equations describing the state of the material. We have applied the variational concept to capture the behavior of different materials, such as the temperature gradient dependent phase transformations in steel, as well as phase transformations coupled with plastic deformations and inelastic effects in shape memory alloys.
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The general pursuit of technological advancement requires structural components to meet increasingly higher standards. In order to optimize the performance of the materials used, detailed knowledge of the overall as well as microscopic material behavior under specific loading conditions is required. Therefore, we develop a two-scale finite element (FE) and fast Fourier transform (FFT)-based method for the investigation of thermo-mechanically coupled elasto-viscoplastic polycrystalline materials at finite strains. Assuming that the length scale of the microscale is sufficiently smaller compared to the length scale of the macroscale, we consider the macroscopic and microscopic boundary value problems as two coupled subproblems. The macroscopic boundary value problem is solved by the finite element method. At each macroscopic integration point, the microscopic boundary value problem is embedded as a periodic unit cell whose solution fields are computed using fast Fourier transforms and a Newton-Krylov solver. The scale transition is performed by defining the macroscopic quantities by the volume averages of their microscopic counterparts. Finally, we incorporate a solution strategy based on a coarsely discretized microstructure to develop an efficient two-scale simulation scheme.
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Osteoporosis is the most prevalent bone disease worldwide. The disease is characterized by a loss of bone density over time, weakening the bone and increasing the likelihood of fractures. Sonography may be a promising future diagnostics method for the early detection of osteoporosis. We introduced a material model consisting of a macroscopic and microscopic scale to simulate this process. On the microscopic scale, we differentiate between the phases cortical bone and bone marrow. Mechanical, electric and magnetic effects are considered in the model, as the magnetic field strength is the quantity which is measured in experiments and from which conclusions are drawn about the bone health. We applied the finite element square method (FE²) to solve the multiscale coupled partial differential equation (PDE) system. To model the different stages of osteoporosis, we constructed representative volume elements (RVEs) with different volume fraction of cortical bone. We were able to show that the magnetic field strength decreases significantly for later stages of the disease. The solution of the inverse problem – deriving the health condition of bone from magnetic field data – is important for diagnostics. We use artificial neural networks (ANNs) to solve this problem for synthetic data with high accuracy.
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Last update: Dec 17, 2024