Advanced constitutive laws for computational magnetics: The case of magneto-mechanical behaviour

Ferromagnetic materials in electromagnetic devices are subjected to multiaxial stress states. Stress significantly affects the material behaviour, and appropriate modelling tools are required to describe this coupling effect. Multiscale approaches for magneto-elastic behaviour can provide a useful description of the complex interaction between magneto-mechanical loading and material response. However, these models are not easily implementable into numerical simulations. This paper gives a brief overview of different strategies to extract information from multiscale approaches so as to use them in standard numerical tools for the design of electromagnetic devices. It is known for long that mechanical stress state significantly influences the magnetic and magneto-mechanical behaviour of ferromagnetic materials [1]. The different stages required in the manufacturing of electrical machines and electromagnetic devices, as well as the operational conditions, generate a variety of mechanical stress states in the ferromagnetic materials used in their fabrication. These mechanical loadings usually have a detrimental effect on the material behaviour and hence on the device performance [2 – 4]. It is then highly desirable that the designer of electromagnetic devices can have access to constitutive equations representative for the magneto-mechanical behaviour of ferromagnetic materials. The stress being described by a second order tensor (six independent components), and the magnetic field by a vector (three independent components), it is easy to understand that the possible combinations for the magneto-mechanical loading are huge. Many experimental and numerical studies are restricted to 1D configurations considering uniaxial stress (usually tensile) applied in the direction parallel to the magnetic field. Although these studies are very useful to explore the mechanisms of magneto-mechanical coupling, they obviously need to be generalised in order to cover the practical configurations encountered in most real devices. The problem seems too complex to be described by macroscopic phenomenological approaches, as often used in the modelling of – uncoupled – magnetic behaviour. A variety of approaches have been developed to describe and understand magneto-mechanical coupling effects, based on different levels of approximations. Among them, multiscale approaches (MSA [5 – 11]) appear particularly promising to capture the complexity of magneto-elastic behaviour. These approaches are based on an energetic description of the magneto-elastic equilibrium at the magnetic domain scale combined to a statistic description of the average magnetic domain distribution. A main strength of MSA is that they can naturally deal with heterogeneous and anisotropic materials. They also naturally introduce the multiaxiality of magneto-mechanical loadings (3D stress state and any relative orientation between stress and magnetic field). Some of the MSA can also consider hysteresis effects. However MSA remain usually too complex to be easily implemented into numerical analysis tools for the design of electromagnetic systems. Hence modelling strategies are required to take benefit from the output of MSA while keeping the numerical burden acceptable. A first idea is to use MSA as a “numerical testing machine” and consider it as a tool to identify the material parameters of macroscopic models. Indeed, macroscopic constitutive models, notably those based on thermodynamic approaches [12 – 15], are capable of tackling the multiaxility of magneto-mechanical effects. However they are more reliable if the modelling parameters are obtained from more complex configurations than simple uniaxial magneto-elastic tests. An option is to develop multiaxial experimental setups [16 – 19], but the use of MSA is an alternative path. The modelling parameters of MSA can usually be identified on simple experimental tests due to their univocal physical meaning. MSA can then be used to perform “numerical tests” in any loading condition suitable for the identification of macroscopic model parameters. Another strategy is to derive simplified models from full MSA, usually neglecting the effects of material heterogeneity [20 – 22]. The multiaxial feature of MSA is preserved, and the constitutive equations can incorporate anisotropy [22], [23] or hysteresis effects [21], [24]. Such approaches have been shown to be efficient to describe the behaviour of simple test structures [25], transformers [22], [26] or electrical machines under magneto-mechanical loading [23], [24]. The simplest – but also less accurate - strategy is probably to use the concept of equivalent stress [27], [28] or equivalent strain [29]. An equivalent stress is a uniaxial stress that – applied in the direction parallel to the magnetic field – would have the same effect as the real multiaxial stress existing in the structure. Heterogeneity and initial anisotropy effects are neglected, but it is a simplified way to take into account the multiaxiality of stress in electromagnetic structures without developing fully multiaxial magneto-mechanical numerical tools. This approach has been applied successfully to evaluate the potentiality of smart materials to perform flux weakening in electrical machines [30] or to analyse the effect of stress on the losses in electrical machines [31].