A thermomechanical formulation of finite element schemes for micropolar continua

Abstract

Recent studies highlighted the advantages of the thermomechanical approach for developing models of material behaviour. This approach ensures compliance with thermodynamical laws since constitutive relations are derived from a consideration of thermodynamical potentials. Interestingly, the same thermomechanical techniques can also be used to formulate the finite element models used to implement these constitutive relations. Thus the key advantage of this type of finite element formulation is that a direct link to the underlying physics of the problem is extended through to the model's implementation. Here, we show how the thermomechanical approach can be applied to finite element schemes for models based on micropolar continuum theory. Material points that make up a micropolar continuum possess rotational degrees of freedom, in addition to the conventional translational degrees of freedom. Hence, the equations governing boundary value problems for this class of materials differ from those of their classical counterparts---both from the viewpoint of the constitutive law and the governing conservation laws. We outline the development of finite element schemes for elastic and plastic micropolar materials using the thermomechanical approach. The analysis indicates that while the traditional Galerkin method admits a range of weighting functions, the second law of thermodynamics provides an additional constraint that narrows the choice of admissible functions.