Analysis and numerics for a thermomechanical phase transition model in steel

Abstract

This thesis is concerned with the thermomechanical modeling and numerical treatment of metallurgical phase transitions in steel during quenching. Based on the fundamental principles of thermodynamics, a coupled model consisting of partial and ordinary differential equations is derived. The constitutive equations are adapted to the focus of interest which is deformation induced by phase transitions considered on the macroscale. For the modeling of phase transitions a mixture approach is chosen leading to a system of ordinary differential equations. The deformation caused by phase transitions is included into the model by different expansion coefficients of the respective steel phases and an integral term accumulating deviatoric stresses during transformation, which accounts for transformation induced plasticity (trip). Existence and uniqueness results are obtained utilizing fixed point arguments applied for a series of subproblems until finally the complete original equation system is solved. The fixed point iterations take place in an L-setting, p > 4, to achieve the self-mapping property of the operator by embedding theorems. With respect to the numerical treatment, a scheme for a reduced model, which still captures major effects and includes transformation induced plasticity, is set up. The implementation is done within the finite element framework provided by the toolbox WIAS-pdelib. The single equations are solved sequentially for each time discretization point, the time stepping is carried out by a semi-implicit approach. The resulting code is applied to an experimental setup investigated within the collaborative research center SFB 570 Distortion Engineering in Bremen. The effect of inhomogenous quenching strategies on the so-called out-of-roundness of roller bearing rings is investigated and the outcome of the numerical computations is compared to experimental observations. Motivated by the correspondence of locally varying heat transfer coefficients and shape alteration of the workpiece, a strategy for distortion compensation by means of a gradient method obtained from optimal control theory is introduced.