Constitutive Equations For Isotropic Materials Research Articles

A coupled electromagnetic–thermomechanical approach for the modeling of electric motors

Future developments of lighter, more compact and powerful motors – driven by environmental and sustainability considerations in the transportation industry – involve higher stresses, currents and electromagnetic fields. Strong couplings between mechanical, thermal and electromagnetic effects will consequently arise and a consistent multiphysics modeling approach is required for the motors’ design. Typical simulations – the bulk of which are presented in the electrical engineering literature – involve a stepwise process, where the resolution of Maxwell’s equations provides the Lorentz and magnetic forces which are subsequently used as the external body forces for the resolution of Newton’s equations of motion.The work presented here proposes a multiphysics setting for the equations governing electric motors. Using the direct approach of continuum mechanics, a general framework that couples the electromagnetic, thermal and mechanical fields is derived using the basic principles of thermodynamics. Particular attention is paid to the derivation of the coupled constitutive equations for isotropic materials under small strain but arbitrary magnetization.Due to the complex geometry of a typical electric motor, numerical solutions of the governing equations are in order. To gain insight, the theory is hereby applied to obtain the analytical solution of an idealized asynchronous motor for which we calculate the electric current, magnetic, stress and temperature fields as a function of the applied current and slip parameter. The different components of the stress tensor and body force vector are compared to their purely mechanical counterparts due to inertia, quantifying the significant influence of electromagnetic phenomena.

On elastoplastic deformation of grey cast iron

Abstract Many engineering materials appear an independent from the kind of loading elastoplastic material behavior. With respect to the sign convention we get, for instance, the same stress-strain diagrams for tension and compression. For materials which contain voids, pores, etc., different elastoplastic stress–strain curves can be obtained in tension and compression tests. In addition to these experimental observations, such materials often show inelastic volumetric deformations. Grey cast iron is a typical example of this kind of material. Constitutive equations for this special elastoplastic behavior are derived as a particular case of generalized constitutive equations for isotropic materials based on a potential. The yield condition is established on the second invariant of the stress deviator and, additionally, on the first invariant of the stress tensor and on a hardening parameter. The plastic potential depends on the same arguments, but a nonassociated flow rule is assumed. For the hardening function, a modification of the plastic work hardening model is used. All parameters of the model are identified by tests. The verification is realized by different independent tests.

Finite strain generalization of smallstrain constitutive relations for any finite strain tensor and additive volumetric-deviatoric split

The paper deals with finite strain generalization of small-strain constitutive equations for isotropic materials for which the strain is split into a volumetric part and a deviatoric part (the latter characterizing the isochoric strain, i.e. a strain at constant volume). The volumetric-deviatoric split has so far been handled by a multiplicative decomposition of the transformation tensor; but the existing sophisticated complex constitutive models for small strains of cohesive pressure-sensitive dilatant materials, such as concrete and geomaterials, involve an additive decomposition and would be difficult to convert a multiplicative decomposition. It is shown that an additive decomposition of any finite strain tensor, and of the Green-Lagrange strain tensor in particular, is possible, provided that the higher-order terms of the deviatoric strain tensor are allowed to depend on the volumetric strain. This dependence is negligible for concrete and geomaterials because the volumetric strains are normally small, whether or not the deviatoric strains are large. Furthermore, the related question of the choice of the finite-strain measure to be used for the finite-strain generalization is analysed. A transformation of the Green-Lagrange finite strain tensor whose parameters approximately reflect the degrees of freedom equivalent to replacing the small strain tensor by any other possible finite strain measure is proposed. Finally a method by which the stress tensor that is workconjugate to any finite strain tensor can be converted to the Green-Lagrange strain tensor is presented.