Simulation of a Hybrid-Forming Process using Temperature- and Rate-Dependent Viscoplastic Material Models

Hybrid-forming processes for graded structures are quite innovative methods for the production of components with tailored properties, particularly tailored material properties and geometrical shape. In this contribution a hybrid-forming process based on the utilization of locally varying thermo-mechanical effects is investigated [1]. For process optimization and improvement of the resulting work piece the simulation of the entire forming process is necessary in modern engineering. The main topics of this contribution are the simulation of the cyclic thermal loaded forming tool and the simulation of the work piece treated at large deformations with phase transformations. For both materials temperature- and rate-dependent viscoplastic material models are applied and parameter identification using cyclic tensile-compression tests for the forming tool material and phase transformation tests for a low-alloy steel similar to the work piece material is presented. For validation of finite-element-calculations for the forming tool thermal shock experiments are performed with optical deformation measurements. For validation of finite-element-calculations for the work piece numerical results of geometry and structure after heating, forming and cooling are compared to experimental micro sections. Results concerning the forming tool will be used for future lifetime prediction and results concerning the work piece will be used for future specific setting of graded material properties. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electromagnetic flow control in liquid metals using Lorentz force techniques

Flow measurement and flow control in opaque and aggressive liquids such as metal melts are challenging tasks in industrial fluid mechanics. Optical measurement techniques as well as submerging probes or control units cannot be applied in case of highly severe environment; hence, contactless electromagnetic methods are of interest for practical flow control because of relatively high electrical conductivity of investigated materials. In this paper we present working principle and experimental results of two such contactless techniques. First, time-of-flight Lorentz force velocimetry (LFV), which allows to determine the flow rate of liquid metal without information about any fluid properties or magnetic field magnitude by cross-correlation of two Lorentz force signals. Secondly, Lorentz torque velocimetry (LTV) - a technique, which uses an electromagnetic pump with a torque sensor connected to the pump's shaft. In the methods so-called Lorentz force [1] is measured, which has electromagnetic nature and is proportional to velocity and flow rate of conductive liquid. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Suppression of Electromagnetically Induced Free-Surface Instabilities in Liquid Metals

The work presented is devoted to studies of foundations of the suppression techniques for electromagnetically induced instabilities of the free liquid metal surfaces. Such instabilities occur, for instance, in a number of technologies for requested refinement of materials like electron beam evaporation or levitation melting, in which the electromagnetic shaping technique is applied. The instability phenomenon in our case has following magnetohydrodynamic background. High-frequency magnetic field causes Lorentz force density generation within so-called skin-layer in the liquid metal, which works as a surface pressure in turn. With increase of the electromagnetic pressure a destabilizing mechanism progresses within the liquid metal domain, which causes a bifurcation. In this paper we discuss application alternating magnetic fields to increase the bifurcation threshold by passive and active redistribution of the Lorentz force density in the near-surface layers of the liquid metal domain. A novel stabilization system on the basis of a local electromagnetic influence on the liquid metal surface in the high frequency magnetic field is presented. The experimental results are discussed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electromagnetic interaction of moving conductors with magnets: kinematic solutions for infinitely long square and cylindrical geometries

The electromagnetic drag force on a point dipole near a moving conductor caused by the induced electric currents is investigated by numerical and analytical computations. Our focus is on prototypical configurations for Lorentz force velocimetry, i.e. velocity measurement from the electromagnetic drag force on the dipole. We examine the particular cases of conducting infinite bars of square or round cross-section, which are moving with constant velocity in the field of arbitrary oriented magnetic dipole. In addition, we study the laminar liquid-metal flow in a square duct. The motion of the conductor is prescribed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maslov index and Morse theory for the relativistic Lorentz force equation

We study the Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold. The flow of the Jacobi equation along each solution preserves the so-called twisted symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We introduce the notion of F-conjugate plane for each conjugate instant; the restriction of the spacetime metric to the F-conjugate plane is used to compute the Maslov index, which is given by a sort of algebraic count of the conjugate instants. For a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field preserving the flow of the Killing vector field, we introduce a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. We prove that the value of this Morse index equals the Maslov index and we prove the Morse relations for the solutions of the Lorentz force equation in a static spacetime.Mathematics Subject Classification (2002): Primary: 58E10, 83C10; Secondary: 53D12     

A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.     

Symmetries, Conservation Laws, and Cohomology of Maxwell's Equations Using Potentials

New nonlocal symmetries and conservation laws are derived for Maxwell's equations in 3 + 1 dimensional Minkowski space using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. (In contrast the standard potential system using a single vector potential is not duality-invariant.) The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain natural geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling and duality-rotation symmetries, and the well-known Poincaré and dilation symmetries which involve homothetic Killing vectors, the classification yields new geometric symmetries involving Killing–Yano tensors related to rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's equations are constructed from these geometric symmetries by applying a conserved current formula that uses the joint potentials and directly generates conservation laws from any (local or nonlocal) symmetries of Maxwell's equations. This formula is shown to arise through a series of mappings that relate, respectively, symmetries/adjoint-symmetries of the joint potential system and adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived as by-products of the study of cohomology of closed one-forms and two-forms locally constructed from the electromagnetic field and its derivatives to any finite order for all solutions of Maxwell's equations. In particular it is shown that the only nontrivial cohomology consists of the electromagnetic field (two-form) itself as well as its dual (two-form), and that this two-form cohomology is killed by the introduction of corresponding potentials.     

Characterization of short fiber reinforced polymers

This contribution presents ideas, how composite materials can be characterized with respect to experimental testing. The material properties of the investigated short glass fiber reinforced polymer are obtained by providing results from the experiment in order to seperate different material effects, such as elasticity, plasticity, damage, viscoelasticity, compressibility and anisotropy. Therefore, at first, linear uniaxial tensile tests with cyclic loadings have been realized. The application of the material in this work is the machining by a three-dimensional forming process. Hence, multiaxial loadings have to be additionally taken into account matching these conditions. In order to provide more information, biaxial tensile tests have to be realized using a testing device supplying the additional necessary experimental data [1, 2]. A final aim of this work is to develope a verification experiment representing the three-dimensional forming process as realistically as possible, e. g. a Nakajima test [4]. For this case, a three-dimensional optical analysis in order to get the necessary measurement data, is indispensable to realize an inverse method [3] by comparing the information of the complete deformation field as well as the force data of the experiment using a given material model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Lorentz force flow meter for application at blast furnaces: design and calibration

A reference Lorentz force flow meter (LFF) has been developed to measure molten steel mass flow at the end of the runner of an experimental blast furnace. It works according to the principles of Lorentz force velocimetry [1] in which a static magnetic field interacts with a liquid metal stream. The magnetic field lines are generated by an arrangement of permanent magnets and penetrate the entire cross-section of the flow generating eddy currents and a total Lorentz force inside the melt. This force is proportional to the mass flow of the liquid metal and owing Newton's third law, there is a counter force of the same magnitude acting on the magnet system which is connected to a load cell. For accurate flow rate measurements, a “dry and wet calibration” of the LFF needs to be performed [2]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scattering of electromagnetic plane waves by a spheroid uniformly moving in free space

We investigate the scattering process, generated by a plane electromagnetic field that is incident upon a moving perfectly conducting spheroid. An accurate treatment of the electromagnetic waves interaction with scatterers in uniform motion is based on the special relativity principle. In the object's frame the incident wave is assumed to have a wavelength which is much larger than the characteristic dimension of the scatterer and thus the low‐frequency approximation method is applicable to the scattering problem. For the near electromagnetic field we obtain the zeroth‐order low‐frequency coefficients, while in the far field we calculate the leading terms for the scattering amplitude and scattering cross‐section. Finally, using the inverse Lorentz transform, we obtain the same approximations in the observer's frame. Copyright © 2006 John Wiley & Sons, Ltd.     

Laminar magnetohydrodynamic duct flow in the presence of a magnetic dipole

We study magnetohydrodynamic flow of a liquid metal in a straight duct. The magnetic field is produced by an exterior magnetic dipole. This basic configuration is of fundamental interest for Lorentz force velocimetry (LFV), where the Lorentz force opposing the relative motion of conducting medium and magnetic field is measured to determine the flow velocity. The Lorentz force acts in equal strength but opposite direction on the flow as well as on the dipole. We are interested in the dependence of the velocity on the flow rate and on strength of the magnetic field as well as on geometric parameters such as distance and position of the dipole relative to the duct. To this end, we perform numerical simulations with an accurate finite-difference method in the limit of small magnetic Reynolds number, whereby the induced magnetic field is assumed to be small compared with the external applied field. The hydrodynamic Reynolds number is also assumed to be small so that the flow remains laminar. The simulations allow us to quantify the magnetic obstacle effect as a potential complication for local flow measurement with LFV. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bending of sinusoidal functionally graded piezoelectric plate under an in-plane magnetic field

This work investigates the bending of a simply supported functionally graded piezoelectric plate under an in-plane magnetic field. The extended sinusoidal plate theory for piezoelectric plate is adopted. The governing equations are derived by the principle of the virtual work considering the Lorentz magnetic force obtained from the Maxwell's relation. The effect of magnetic field, electric loading and gradient index on the displacement, electric potential, stress and electric displacement are numerically presented and discussed in detail. These conclusions will be of particular interest to the future analysis of piezoelectric plate in magnetic field.     

Flow rate measurements in turbulent liquid metal channel flow using Time-of-Flight Lorentz force velocimetry

Non-contact flow control and flow measurements in hot and aggressive metal melts are big challenges in metallurgical applications. Time-of-Flight Lorentz force velocimetry (ToF LFV) is an electromagnetic measurement technique to meet these challenges. Our experimental results demonstrate that this method is well suited to measure flow rate in turbulent liquid metal channel flow without knowledge of both melt and magnetic field properties. Moreover, the measured flow profiles are in very good agreement with predictions of numerical simulations using the commercial program Package FLUENT MHD. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gravitation and Electromagnetism as Geometrical Objects of a Riemann-Cartan Spacetime Structure

In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature R of a particular kind of Riemann-Cartan spacetime structure. In our model the gravitational and electromagnetic fields are identified as geometrical objects of the structure.We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of our theory encodes the same information as the one contained in Chern-Simons term ${{\bf A} \wedge {\it d}{\bf A}}$ that is proportional to the spin density of the electromagnetic field. Next we show that by adding to the geometrical Lagrangian a term describing the interaction of a electromagnetic current with a general electromagnetic field plus the gravitational field, together with a term describing the matter carrier of the current we get Maxwell equations with source term and Einstein equation having as source term the sum of the energy-momentum tensors of the electromagnetic and matter terms. Finally modeling by dust charged matter the carrier of the electromagnetic current we get the Lorentz force equation. Moreover, we prove that our theory is gauge invariant. We also briefly discuss our reasons for the present enterprise.     

Formalism of two potentials for the numerical solution of Maxwell’s equations

A new formulation of Maxwell’s equations based on the introduction of two vector and two scalar potentials is proposed. As a result, the electromagnetic field equations are written as a hyperbolic system that contains, in contrast to the original Maxwell system, only evolution equations and does not involve equations in the form of differential constraints. This makes the new equations especially convenient for the numerical simulation of electromagnetic processes. Specifically, they can be solved by applying powerful modern shock-capturing methods based on the approximation of spatial derivatives by upwind differences. The cases of an electromagnetic field in a vacuum and an inhomogeneous material are considered. Examples are given in which electromagnetic wave propagation is simulated by solving the formulated system of equations with the help of modern high-order accurate schemes.     

Simulation of direct chill casting under the influence of a low-frequency electromagnetic field

A comprehensive, multiphysics, meshless, numerical model is developed for the simulation of direct chill casting under the influence of a low-frequency electromagnetic field. The model uses mixture-continuum-mass, momentum and energy-conservation equations to simulate the solidification of axisymmetric aluminium-alloy billets. The electromagnetic-induction equation is coupled with the fluid flow and used to calculate the Lorentz force. The involved partial-differential equations are solved with the meshless-diffuse-approximate method by employing second-order polynomial shape functions and a 13-noded local support. An explicit time-stepping scheme is used. The boundary conditions for the heat transfer involve the effects of hot-top, mould chill and direct chill. The use of a meshless method and the automatic node-arrangement generation made it possible to investigate the complicated flow structures in geometrically complex inflow conditions, including sharp and curved edges, in a straightforward way. A time-dependent adaptive computational node arrangement is used to decrease the calculation time. The model is demonstrated by casting an Al-5.25wt%Cu aluminium alloy billet with a radius of 120 mm. Results on simplified and realistic inflow geometry are considered and compared. The effect of the low-frequency electromagnetic force on the temperature, liquid fraction and fluid flow are investigated under different current densities and frequencies.     

A Phase Field Approach for Dynamic Fracture

In phase field fracture models cracks are indicated by the value of a scalar field variable which interpolates smoothly between broken and undamaged material. The evolution equation for this crack field is coupled to the mechanical field equations in order to model the mutual interaction between the crack evolution and mechanical quantities. In finite element simulations of crack growth at comparatively slow loading velocities, a quasi-static phase field model yields reasonable results. However, the simulation of fast loading or the nucleation of new cracks challenges the limits of such a formulation. Here, the quasi-static phase field model predicts brutal crack extension with an artificially high crack speed. In this work, we analyze to which extend a dynamic formulation of the mechanical part of the phase field model can overcome this paradox created by the quasi-static formulation. In finite element simulations, the impact of the dynamic effects is studied, and differences between the crack propagation behavior of the quasi-static model and the dynamic formulation are highlighted. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The numerical modelling of DC electromagnetic pump and brake flow

The interaction of an externally imposed magnetic and electric field on the laminar flow of a conducting fluid in a channel is studied using computational techniques. The Navier-Stokes equations and the equations describing the electromagnetic field are solved simultaneously in a single control volume-type computational fluid dynamic code, in a moderate Hartmann number and interaction parameter regime. The flow considered is two-dimensional, with an imposed magnetic field acting in the third dimension over the central region of the channel and decaying exponentially in the remainder. A pair of electrodes placed at right angles to the magnetic field exercises control over the resultant Lorentz force and hence the velocity profile shape. This configuration has application in direct-current electromagnetic pumps or, conversely, electromagnetic brakes. The initial parabolic flow profile acquires an M-shape / W-shape mode in the magnetic field fringe regions, corresponding to a pump / brake. A novel coupled procedure is described to model magnetohydrodynamic phenomena and is used to explore the effects of the Reynolds number, interaction parameter, and applied voltage on the pump / brake configuration.     

Material-based process-chain optimization in metal forming

The characteristics of a manufacturing product are influenced by a variety of different factors, such as the material properties of the base product. The prediction of properties that give optimal results in metal forming applications is a complex task but of high interest for the manufacturer. To realize such a prediction scheme, the process chain is split up into individual process steps and for each of them an inverse modeling is required. The specific aim of this work is to present an approach for the inverse problem formulation of a process step and to solve it using methods of machine learning. Moreover, the challenges that often arise due to the ill-posed nature of inverse problems will be discussed. The main focus is on the crystallographic texture of metals, which strongly affects the deformation behavior during a process step and highly influences the characteristics of the final product. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of a viscoplastic material model on a combined quasi-static-electromagnetic forming process

The prediction and simulation of material behavior by finite element methods has become indispensable. Furthermore, various phenomena in forming processes lead to highly differing results. In this work, we have investigated the process chain on a cross-shaped cup in cooperation between the Institute of Applied Mechanics (IFAM) of the RWTH Aachen and the Institute of Forming Technology and Lightweight Construction (IUL) of the TU Dortmund. A viscoplastic material model based on the multiplicative decomposition of the deformation gradient in the context of hyperelasticity has been used [1,2]. The finite strain constitutive model combines nonlinear kinematic and isotropic hardening and is derived in a thermodynamically consistent setting. This anisotropic viscoplastic model is based on the multiplicative decomposition of the deformation gradient in the context of hyperelasticity. The kinematic hardening component represents a continuum extension of the classical rheological model of Armstrong-Frederick kinematic hardening. The constitutive equations of the material model are integrated in an explicit manner and implemented as a user material subroutine in the commercial finite element package LS-DYNA with the electromagnetical module. The aim of the work is to show the increasing formability of the sheet by combining quasi-static deep drawing processes with high speed electromagnetic forming. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)