Efficient and robust numerical treatment of a gradient-enhanced damage model at large deformations

Abstract

The modeling of damage processes in materials constitutes an ill-posed mathematical problem which manifests in mesh-dependent finite element results. The loss of ellipticity of the discrete system of equations is counteracted by regularization schemes of which the gradient enhancement of the strain energy density is often used. In this contribution, we present an extension of the efficient numerical treatment, which has been proposed in [1], to materials that are subjected to large deformations. Along with the model derivation, we present a technique for element erosion in the case of severely damaged materials. Efficiency and robustness of our approach is demonstrated by two numerical examples.

Efficient numerical treatment of a gradient damage model for materials undergoing large deformations

“…Here, (2) denotes the balance of linear momentum and (3) the damage evolution inequality, where r is the dissipation parameter (cf [1]). For the discretization of ( 2) standard H 1 -conforming finite elements with Hexahedral Q 1 -Lagrange interpolation functions are used.…”

“…Applying the extended Hamilton principle to the corresponding total potential including rate independent dissipation due to damage (cf. [5]) and writing the obtained dissipation inequality in strong form yields the following problem (for a more detailed discussion see also [1]): Find u and f , such that…”

“…For the discretization of ( 2) standard H 1 -conforming finite elements with Hexahedral Q 1 -Lagrange interpolation functions are used. Meanwhile, (3) is discretized with a finite difference scheme for unstructured meshes (for details, see [1]). Solution procedure: In an incremental load step solution procedure, equations ( 2) and (3) are solved in a staggered manner: For each load step n ← n + 1, after the finite element solution u n ← u n+1 (obtained by the Newton Raphson procedure of the linearized system (2)), the damage field for each element e is updated as follows:…”

“…

In this contribution we present a gradient damage formulation (cf. [1]) for hyperelastic materials undergoing large deformations, which is based on the small strain formulation of [2]. Through discretizing the gradient extended damage field with the neighbored element method the damage update can be performed at the element level.

…”

Bayesian inversion for unified ductile phase-field fracture

“…Recently, in the setting of continuum mechanics, a new perspective was proposed for embedding microscopic mechanisms into the macromechanical continuum formulation, based on a multi-field incremental variational framework for gradient-extended standard dissipative solids [1,2]. Typical examples are theories of gradient-enhanced damage [3,4,5,6], phase-field models [7,8,9], and strain gradient plasticity [10,11,12]. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure size with respect to the macro-structure size.…”