In this work, a novel approach for the simulation of phase-transformations is investigated. The main idea of the model presented is to make use of actual Gibbs energy barriers, which are then used for the computation of transformation probabilities. The model is embedded into a non-affine micro-sphere formulation with volumetric-deviatoric split – inducing that the underlying Helmholtz free energy depends on volumetric and deviatoric strain measures as independent variables. The Helmholtz free energy of each phase takes the form of an elliptic paraboloid in volumetric-deviatoric strain space. After carrying out a Legendre-transformation of the free energy, the resulting Gibbs energy of each phase also takes an elliptic-paraboloidal form. The actual Gibbs energy barriers between different phases are computed by the numerical minimization of non-linear parametric Gibbs energy intersection curves. Based on the Gibbs energy barriers, an approach from statistical physics is used to obtain an evolution law for the volume fractions. Moreover, the model is coupled to von-Mises-type plasticity, where we consider linear proportional hardening for simplicity.
The overall energy of the phase mixture is obtained from the contributions of the individual constituents, resulting – for a three-phase material – in an energy landscape with three local minima. Depending on the external loads applied, the energy landscape of the material changes as illustrated in Fig. 1. The figure shows the time- and load-dependent overall Gibbs energy landscape of a three-phase material mixture with A denoting the Gibbs energy minimum of the austenitic parent phase, M+ denoting the Gibbs energy minimum of the martensitic tensile phase, and M- denoting the local energy minimum of the martensitic compression phase. In Fig. 2, the according cyclic deformation of a one-dimensional bar as well as its evolution of volume fractions is depicted. Here, black denotes the austenitic parent phase, blue indicates the martensitic tensile phase, and red represents the volume fraction of the martensitic compression phase.
The scalar-valued micro-model is embedded into a non-affine micro-sphere formulation in order to extend the model to three dimensions. Physically speaking, the micro-sphere response is related to the response of a polycrystal with a statistically uniform spatial distribution of underlying single-crystals. In consequence, the micro-sphere represents the mesoscopic level of our multiscale-model. The macroscopic strain tensor is split into volumetric and deviatoric part, where both parts are then projected onto each integration direction of the micro-sphere. The obtained scalar-valued volumetric and deviatoric strain measures in each integration direction are then used as input variables on the micro-level, where the Gibbs energy landscape is computed based on volumetric and deviatoric micro-strains.
The stress-strain response of the micro-sphere subjected to a homogeneous mixed volumetric-deviatoric load is presented in Fig. 3. An illustration of the evolution of volume fractions for each individual integration direction of the micro-sphere is given in Fig. 4. Note that the example presented is based on 21 integration directions per hemi-sphere, i.e. 42 integration directions for the whole micro-sphere. Finally, the obtained spatial distribution of plastic strains are provided in Fig. 5.
Fig. 3: Stress-strain response | Fig. 4: Spatial distribution of volume fractions | Fig. 5: Spatial distribution of plastic strains |
The model presented is capable of capturing the temperature-dependent pseudo-elastic/pseudo-plastic behavior of shape memory alloys. For lower temperatures, shape memory alloys behave pseudo-plastic, i.e. after one initial tensile load cycle, non-zero strains remain after the tensile load vanishes and a stress-free configuration is reached. Figs. 6 and 7 show that for lower temperatures at around 0°C, the model predicts a pseudo-plastic response for the given shape memory alloy material parameters. As the temperature increases, the stress-strain-response clearly tends towards a pseudo-elastic behavior.
Note that the stress-strain responses provided in Fig. 6 are obtained with non-active plasticity. In contrast to that, Fig. 7 shows temperature-dependent stress-strain cycles with active plasticity. The results show that the consideration of plasticity within the material leads to a smoother stress-strain-response, while at the same time plasticity prevents the material from reaching high stress tips at maximum tensile strain.
Fig. 6: Temperature-dependent stress-strain response; plasticity not active | Fig. 7: Temperature-dependent stress-strain response with active plasticity |
Dr.-Ing. Richard Ostwald, Dr.-Ing. Thorsten Bartel, Prof. Dr.-Ing. Andreas Menzel