Locally Embedded Strong Discontinuities at finite strains – Numerical Implementation

Abstract

In this paper, a geometrically nonlinear finite element approximation for highly localized deformation in structures undergoing material failure in the form of strain softening, is developed. The basis for its numerical implementation in this class of problems is defined through the elaboration of Strong Discontinuity Approach-fundamentals. Proposed numerical model uses an Enhanced Assumed Strain Concept for the additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, this part of the deformation mapping is condensed out at the material level, without the application of static condensation technique. The resulting set of constitutive equations is formally identical to that of standard plasticity and therefore, can be solved using the return-mapping algorithm. No assumptions regarding the interface law connecting the displacement discontinuity with the conjugate traction vector are made. As a result, the proposed numerical solution can be applied to a broad range of different mechanical problems including mode-I fracture in brittle materials or the analysis of shear bands. Kinematics induced by strong discontinuitiesIn what follows, a body Ω ⊂ R 3 is separated by a hyperplane ∂ s Ω ⊂ Ω of class C 1 (piecewise). Based on this partition, an approximation of the displacement field of the typeis adopted. In Equation (1), H s denotes the indicating function of the subset Ω + , [ [u] ] the displacement discontinuity and ϕ a smooth ramp function (see [1]). By applying the generalized derivative D(•) to the deformation mapping defined by Equation (1), the deformation gradient is assumed to be of the typewith δ s representing the DIRAC-delta distribution and N the normal vector of ∂ s Ω. Since the enhanced strains are modeled in an incompatible fashion, it is admissible to neglect the gradient of the displacement jump (GRAD [[u] ] = 0), cf. [1,2]. In the proposed EAS concept, only the fieldū is approximated globally conforming, i.e. by using standard shape functions. Constitutive equations: Cohesive lawsAccording to Equation (2), the strains in Ω ± are regularly distributed. Hence, classical constitutive equations can be applied. For the sake of simplicity, we assume a hyperelastic material response characterized by the energy functionalHere, C is the right CAUCHY-GREEN tensor and λ, µ are the LAMÉ constants. The inelastic deformations are taken into account by means of a cohesive law of the type T | ∂sΩ =:in terms of the traction vector T and the displacement jump. The coupling between the constitutive equations in Ω ± and those in ∂ s Ω is provided by the condition of traction continuity. With the positive definiteness of a norm || • || and applying a pull back operation, this continuity condition readsSince the identityT = C · S · N holds (C · S are the MANDEL stresses), the con...