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.

Martensitic phase transformations are characterised by spontaneous shape changes of the underlying crystalline lattice, for example from body-centered-cubic (bcc) austenite to body-centered-tetragonal (bct) martensite where effects from diffusion are negligible. Such deformations cause inelastic strains which are referred to as Bain strains. The crystal structures of the underlying material's phases furthermore determine the number of possible martensite variants. In the case of bcc to bct transformations, the number of martensite variants is 3. The cornerstone of the constitutive model is laid by phase energy densities for each martensite variant - the energy wells - from which a representative averaged energy density for the phase mixture is assembled. In this context the assumption of homogeneously distributed strain for the phases leads to a highly non-quasiconvex energy landscape.

The goal of this contribution is to enhance a statitics-based phase-transformation model introduced by Govindjee and Hall (2000) in order to take into account plasticity as well as the interaction between phase-transformation and plasticity effects by introducing a so-called plastic inheritance law. Therefore we extended the Helmholtz free energy functions for each material phase, taking into account plastic strains as new variables for each individual phase. Based on the extended multi-well energy potentials, the probabilistic phase transformation model is then derived. In addition, the differential equations driving the evolution of plasticity as well as the potential-based derivation of the individual plastic driving forces are provided. The coupling of phase-transformation and plasticity effects is incorporated by means of a staggered algorithm.

In this work, an efficient model for the simulation of polycrystalline materials undergoing solid to solid phase transformations is presented. As a basis, a one-dimensional, thermodynamically consistent phase-transformation model is embedded into a micro-sphere formulation, facilitating to simulate three-dimensional boundary value problems. To solve the underlying evolution equations, an explicit integration algorithm that could be proved to be unconditionally A-stable was developed. In addition to the investigation of homogeneous deformation states, representative finite element examples are discussed.

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