*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.

If you wish to receive the complete paper, just send me an e-mail.

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