Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature

Abstract

The paper deals with elasto-plastic materials, whose structural damage is produced by lattice defects, within the constitutive framework of finite elasto-plasticity with second order deformations. For second order deformations, which are represented by distortions and connections, we adopt the decomposition rule into second order elastic and plastic components. The split of the plastic connection into independent components, which involve Bilby's connection type and a second order tensor, called disclination, plays a fundamental role in defining kinematical variables. The lattice defects are modeled by dislocations and disclinations, which are introduced within the finite deformation formalism and based on the differential geometry arguments. The constitutive framework is compatible with the free energy imbalance, postulated in the configuration with torsion, in such a form to include the presence of the disclinations. The model includes macro forces, which are power conjugated with the appropriate rate of the second order elastic deformations, and micro forces which are related with the plastic behaviour, as well as with the disclinations. The micro balance equations which are associated with the presence of disclinations are derived from a virtual power principle relative to the disclination mechanism. As a consequence of the principle of the free energy

Elasto-plastic model with second order defect density tensor

“…The evolution equation which describes the defects, namely (27) 2 , contains the micro force and micro momentum, which are related through the micro balance equation for micro forces. The micro balance equation is postulated in a form, similar to that proposed in Cleja-Ţigoiu [5], with respect to the anholonomic configuration K , but for a different type of defects…”

“…When we pass to the reference configuration, the divergence operator is transformed following the procedure developed in [5]. We arrive at…”

“…In the model we considered that the defect tensor h d produces the variation of the internal power, as well as its gradient, which is power conjugated with micro (defect) momentum. The free energy imbalance is used to find the thermodynamic restrictions, the non-local evolution equations for defect density tensor, following the procedure proposed in [4], [5] and the ideas developed by Gurtin [10]. A description of the model can be realized for a simple expression of the free energy density…”

“…The tensor F d enters the multiplicative decomposition of the deformation gradient into its elastic, damaged and plastic components F = F e F d F p . This point of view has been adopted by Cleja-Ţigoiu in [5], following the papers by Brünig [1], but introducing simplifications in the succession of the damaged and the undamaged configurations made by [1], [2].…”

“…with Λ a second order tensor (which has been viewed to be a measure of disclination in [5]) the following representation for the plastic connection can be derived…”