Computers and Structures, 80 (2002), 1279-1293.


Milan Jirásek and Borek Patzák
Swiss Federal Institute of Technology
LSC - ENAC, EPFL, 1015 Lausanne, Switzerland


This paper deals with the computational analysis of strain localization problems using nonlocal continuum damage models of the integral type. The general framework for a consistent derivation of  the ``nonlocal'' tangent stiffness is presented. The properties  of the tangent stiffness matrix are discussed and the corresponding assembly procedure is described. The quadratic rate of convergence of the Newton-Raphson iteration procedure is demonstrated and the efficiency of the proposed technique is compared to the standard approach based on the secant or elastic stiffness matrices. Performance of iterative solvers for the linearized equilibrium equations for large problems is examined.

Summary and Conclusions

The exact, fully consistent tangent stiffness matrix has been derived for a class of nonlocal isotropic damage models. Due to the nonlocal interactions, the stiffness matrix is  nonsymmetric and its bandwidth increases compared to the ``local'' stiffness matrices. The nonlocal character of the constitutive law makes the element stiffness matrix depend not only on the degrees of freedom associated with the given element but also on the  degrees of freedom associated with the nodes of all the elements whose integration points are within the interaction radius. This leads to nonstandard contributions that must be taken into account in the assembly procedure. From the computational point of view it is therefore essential to use an averaging function with a bounded support, in order to keep the global stiffness  matrix sparse.

The general methodology has been applied both to the classical nonlocal damage model proposed by Pijaudier-Cabot and Bazant (1987) and  to a modified version of that model suitable for quasibrittle materials, in which the propagation of damage is driven mainly by the tensile strains and stresses. It has been shown that, despite the seemingly complicated character of the consistent nonlocal stiffness, the corresponding numerical algorithms can be implemented into an existing nonlocal finite element code in a straightforward manner.

The proposed methodology has been tested on three examples, including a three-dimensional one. It has been demonstrated that the procedure is robust and the equilibrium iterations converge at a quadratic rate, which confirms that the linearization is indeed consistent. In terms of efficiency, the proposed technique is quite competitive, especially when  higher accuracy is required. Another possible area of application is the sensitivity analysis based on the exact linearization, which is an important ingredient of various optimization techniques.

For large-scale problems, especially in three dimensions, the performance of the incremental-iterative solution strategy is substantially increased by using an indirect, iterative solver, instead of a direct solver based on the factorization of the stiffness matrix. In the present study, the Generalized Minimum Residual Method with preconditioning by incomplete LU decomposition has been successfully adopted.

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EPFL / 2 October 2002 /