Dietmar Hömberg

A mathematical model for induction hardening including mechanical effects

Abstract We investigate a mathematical model for induction hardening of steel. It accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We take care of effects like transformation strain and transformation plasticity induced by the phase transitions and allow for physical parameters depending on the respective phase volume fractions. The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L1 data. We prove existence of a weak solution to the complete system using a truncation argument.

PID control of laser surface hardening of steel

We discuss control strategies for the laser surface hardening of steel. The goal is to achieve a prescribed hardening depth avoiding surface melting. Our mathematical model consists of a system of ordinary differential equations (ODEs) for the phase volume fractions coupled with the heat equation. The system is solved semi-implicitely using the finite element method. To obtain a uniform hardening depth the first attempt is to use proportional integral differential (PID) control to achieve a constant temperature in the hot spot of the laser beam on the surface. However, the numerical results prove that this is not sufficient. We show that the best strategy is to control the temperature close to the lower boundary of the hardening zone. Then one can compute the optimal temperature in the hot spot of the beam and use it as the setpoint for the pyrometer control of the real process

Irreversible Phase Transitions in Steel

We present a mathematical model for the austenite–pearlite and austenite–martensite phase transitions in eutectoid carbon steel. The austenite–pearlite phase change is described by the Additivity Rule. For the austenite–martensite phase change we propose a new rate law, which takes into account its irreversibility. We investigate questions of existence and uniqueness for the three-dimensional model and finally present numerical calculations of a continuous cooling transformation diagram for the eutectoid carbon steel C1080.

Optimal Shape Design of Inductor Coils for Surface Hardening

We study a mathematical model for induction hardening of steel. It consists of a vector potential formulation of Maxwells equations coupled with a heat equation and an evolution equation for the volume fraction of the high temperature phase in steel called austenite. An important task for practical applications of induction hardening is to find the optimal coupling distance between inductor and workpiece. To this end we control the volume fraction of austenite with respect to perturbations of the coupling distance. The coil is modeled as a tube and is defined by a regular curve. We formulate the shape optimization problem over the set of admissible curves and prove the existence of an optimal curve. We apply the material derivative method for the shape sensitivity analysis of the state system. Finally, the shape gradient is specified for an optimal curve and the first order necessary optimality conditions are established.

On safe crack shapes in elastic bodies

According to the Griffith criterion, a crack propagation occurs, provided that the derivative of the energy functional with respect to the crack length reaches some critical value. We consider a generalization of this criterion to the case of nonlinear cracks satisfying a nonpenetration condition and investigate the dependence of the shape derivative of the energy functional on the crack shape. In the paper, we find the crack shape which gives the maximal deviation of the energy functional derivative from a given critical value and, in particular, prove that this optimality problem admits a solution.

On the evaluation of dilatometer experiments

The goal of this article is a mathematical investigation of dilatometer experiments. These are used to detect the kinetics of solid–solid phase transitions in steel upon cooling from the high-temperature phase. Usually, the data are only used for measuring the start and end temperatures of the phase transition. In the case of several coexisting product phases, expensive microscopic investigations have to be performed to obtain the resulting fractions of the different phases. In contrast, it is shown in this article that in the case of at most two product phases the complete phase transition kinetics including the final phase fractions are uniquely determined by the dilatometer data. Numerical results confirm the theoretical result.

On an inverse problem related to laser material treatments

Motivated by the growing number of industrially important laser material treatments, we investigate an inverse problem related to the controllability on a curve for a semilinear parabolic equation. We prove the local in time well posedness and a global stability result for the inverse problem in the three-dimensional case. As an application we consider the control of laser surface hardening. We show that our theory justifies the application of PID-control to this situation and present numerical simulations for a PID control of laser hardening. Moreover, the result of an industrial case study is presented confirming the practical applicability of our approach.

Quasistationary problem for a cracked body with electrothermoconductivity

In resistance spot welding two workpieces are pressed together by electrodes. Owing to the Joule effect and the high resistivity in the contact area between the workpieces, the welding current leads to an increase in temperature, until finally a weld nugget is formed (see Fig. 1). For a complete description of the process, one has to take into account mechanical, thermal and electrical effects, as well as the free boundary between liquid metal and solid. To the knowledge of the authors, mathematical models up to now have only considered the thermal and electrical effects, neglecting mechanics (see, for example, [15]). A mathematical model for the special case of impulse resistance welding has been developed in [7]. The basic equations to obtain the displacement u = (u1, u2), the temperature θ and the electric potential φ are the quasistatic balance law of momentum, the balance of internal energy as well as the quasistatic balance law of electrical charge. In the framework of isotropic linearized thermoelasticity, we can formulate the balance laws in the undeformed domain.

Parameter identification in non-isothermal nucleation and growth processes

We study non-isothermal nucleation and growth phase transformations, which are described by a generalized Avrami model for the phase transition coupled with an energy balance to account for recalescence effects. The main novelty of our work is the identification of temperature dependent nucleation rates. We prove that such rates can be uniquely identified from measurements in a subdomain and apply an optimal control approach to develop a numerical strategy for its computation.

Path-Planning with Collision Avoidance in Automotive Industry

An optimal control problem to find the fastest collision-free trajectory of a robot is presented. The dynamics of the robot is governed by ordinary differential equations. The collision avoidance criterion is a consequence of Farkas’s lemma and is included in the model as state constraints. Finally an active set strategy based on backface culling is added to the sequential quadratic programming which solves the optimal control problem.