Milan Jirasek and Sonia Marfia: IJNME-2005

International Journal for Numerical Methods in Engineering, 63 (2005), 77-102.

NONLOCAL DAMAGE MODEL WITH DISPLACEMENT AVERAGING

Milan Jirásek
Czech Technical University in Prague , Czech Republic
and
Sonia Marfia
University of Cassino, Italy

Abstract

Continuum damage models describe the changes of material stiffness and strength, caused by the evolution of defects, in the framework of continuum mechanics. In many materials, a fast evolution of defects leads to stress-strain laws with softening, which creates serious mathematical and numerical problems. To regularize the model behavior, various generalized continuum theories have been proposed. Integral-type nonlocal damage models are often based on weighted spatial averaging of a strain-like quantity. This paper explores an alternative formulation with averaging of the displacement field. Damage is assumed to be driven by the symmetric gradient of the nonlocal displacements. It is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium. Around physical boundaries of the analyzed body, both formulations differ and the nonlocal displacement model generates spurious damage in the boundary layers. The paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the nonlocal weight function. Alternatively, an implicit gradient formulation could be used. Issues of algorithmic implementation, computational efficiency and smoothness of the resolved stress fields are discussed.

Conclusions

The main results of the present study can be summarized as follows:

Efficiency of the numerical averaging scheme for damage models with nonlocal strain can be increased by a modification that collects information from nodes rather than from Gauss points.
A nonlocal damage formulation in which damage is driven by the symmetric gradient of the nonlocal displacement field can act as a localization limiter, but the weight function must be constructed such that the averaging operator preserves not only a constant field but also a linear field.
With an appropriate choice of model parameters, the nonlocal damage model based on nonlocal displacements gives similar localized damage patterns as the model based on nonlocal strain, and the global characteristics of the failure process, such as the load-displacement diagram, are also very similar.
The model based on nonlocal strains leads to stress oscillations in certain regions of the process zone. The oscillations are caused by the unbalanced quality of approximation of the local and nonlocal strain.
If the nonlocal displacements are computed at nodes and then interpolated using the same shape functions as for the local displacements, the model based on nonlocal displacements substantially reduces stress oscillations.
The tangent stiffness matrix of the nonlocal displacement model can be assembled from local element contributions on which a linear transformation of columns is applied. This stiffness matrix is always nonsymmetric and has a high bandwidth.
The model with nonlocal displacements could be reformulated in the implicit gradient format, where the nonlocal field is defined as the solution of a Helmholtz-type differential equation. With a suitable choice of boundary conditions, linear fields are preserved and spurious damage does not appear.

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CVUT / 16 May 2005 / milan.jirasek@fsv.cvut.cz