Disclinations and GND tensor effects on the multislip flow rule in crystal plasticity

Abstract

<jats:p> This paper deals with new elastoplastic models for crystalline materials with microstructural defects, such as dislocations and disclinations, which are consistent with the multislip plastic flow rule, and compatible with the free energy imbalance principle. The defect free energy function is a function of the disclination tensor and its gradient, and of the geometrically necessary dislocation (GND) tensor, via the Cartan torsion. By applying the free energy imbalance, the appropriate viscoplastic (diffusion-like) evolution equations are derived for shear plastic rates (in slip systems) and for the disclination tensor. The two sets of differential (or partial differential, i.e., non-local) equations describe the rate form of the adopted disclination–dislocation model. The first set is typical for finite deformation formalism, while the second set refers to the evolution equations with respect to the reference configuration. The dislocation appears to be a source for producing disclination defects. A pure dislocation elastoplastic model is also proposed. Multislip models with disclination within the small deformation approach are derived from the finite deformation models. The initial and boundary value problems are formulated and the incremental (rate) equilibrium equation leads to a variational equality for the velocity field, at any time, which is coupled with the rate type models for the set of variables. First, the elastic problem is solved for a certain time interval by assuming that the existing defects inside the body remain inactive. Subsequently, the variational equality is solved for the velocity field, at any time, if the slip systems are activated. Consequently, the state of the body with defects is defined by the solution of the differential-type equations, when the velocity field is known for a certain time interval. Appropriate initial conditions are necessary, including those associated with defects which became active. Finally, an update algorithm must be provided in order to compute the fields at the current moment. </jats:p>

Applications of vector bundle to finite elasto-plasticity

“…We mention now that only the geometrical background is discussed here. When an axiomatic characterization of crystalline materials is done, the definition of the plastic and elastic distortions has to be introduced simultaneously with constitutive and evolution equations (see, for instance, Cleja-Ţigoiu [10, 24]).…”

“…The concepts of Riemann–Cartan geometry, such as they are described by Le and Stumpf [8], Clayton [19], Yavari and Goriely [20], [21], Steinmann [22], led to different scalar and tensor measures of defects. The non-vanishing torsion of certain connection (which is associated with plastic distortion) is related to the dislocation density, the so-called geometrically necessary dislocations, i.e., GND tensor, while the non-vanishing curvature tensor is related to the disclinations (see De Wit [18], Clayton et al [14], Yavari and Goriely [20, 21], Cleja-Ţigoiu et al [23], and Cleja-Ţigoiu [24]).…”

Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity

“…A is related to the second-order tensor α = (F p ) −1 curl(F p ) (which is defined in [34], the geometrically necessary dislocation, GND-tensor) by…”

“…following the procedure proposed in [34]. Consequently, the second-order torsion tensor, N , associated with Cartan torsion, written in (88), via Formula (79) 1 , reads N = (F p ) −1 curlF p + (C p ) −1 (tr Λ)I − (Λ)…”

“…The measures of the micro-structural defects are defined as third-order tensor fields, following [34], in terms of the elements that characterize the plastic connection, emphasized through the decomposition theorem. To justify the given definitions, we return to the classical framework of continuum mechanics.…”