Keywords: nonlocal damage models, nonlocal displacements, gradient models, consistent tangent matrix, quadratic convergence.

Nonlocal damage models are used to model failure of quasi-brittle materials [1]. Nonlocality -needed to correct the pathological mesh-dependence exhibited by local models- can be incorporated into the model in two different ways. In integral-type models [2,3,4], a nonlocal state variable is computed as the weighted average of the local state variable in a neighbourhood of the point under consideration. In gradient-type models [5], on the other hand, higher-order derivatives (typically second-order) are added to the partial differential equation that describes the evolution of the nonlocal variable. Both approaches yield similar results and are in some cases equivalent [6].

Apart from the state variable, other variables can be selected to incorporate nonlocality. In fact, a number of proposals can be found in the literature. Either scalar (for instance: damage) or vectorial (for instance: strain) Gauss-point quantities may be transformed into the corresponding nonlocal quantities. The existing approaches for integral-type models are compared in [7] by means of a simple 1D numerical test (bar under uniaxial tension).

A new proposal is made here: to use nonlocal displacements to regularize the problem. The two versions are proposed, discussed and compared in the paper: integral-type (nonlocal displacements obtained as the weighted average of standard, local displacements), see [8], and gradient-type (nonlocal displacements obtained as the solution of a second-order partial differential equation).

As discussed and illustrated by means of numerical examples, the regularization capabilities of this new model are very similar to that of the standard model. Moreover, nonlocal displacements lead to mechanically sound and computationally efficient models. For the integral-type regularization, the consistent tangent matrix is much simpler to compute than for the standard approach (nonlocal state variable). This is due to the fact that nonlocality is incorporated into the model completely "upstream" of the constitutive model (i.e. at the level of displacements, the primal unknown in the finite element analysis). Nonlocality is represented in the consistent tangent matrix by a constant matrix of geometrical nonlocal connectivity, which needs to be computed only once, at the beginning of the analysis. The need for cumbersome double loops in Gauss points of the standard approach [9] is suppressed.

In the gradient approach, the boundary conditions on the regularization partial differential equation have a clear physical interpretation: nonlocal displacements must coincide with local (i.e. standard) displacements in the boundary. This contrasts favourably with the controversial issue of prescribing boundary conditions for the nonlocal state variable in standard gradient models. The expression of the consistent tangent matrix is also simpler, thanks to the linear relation between local and nonlocal displacements. This gradient-enhancement of the displacement field is a very simple way to incorporate nonlocality into a finite element code equipped with standard (i.e. local models) nonlinear capabilities.

1

J. Lemaitre and J.-L. Chaboche. "Mechanics of solid materials", Cambridge University Press, Cambridge, 1990.

2

G. Pijaudier-Cabot and Z.P. Bazant. "Nonlocal damage theory", J. Eng. Mech.-ASCE, 118(10):1512-1533, 1987. doi:10.1061/(ASCE)0733-9399(1987)113:10(1512)

3

Z.P. Bazant and G. Pijaudier-Cabot. "Nonlocal continuum damage, localization instability and convergence", J. Appl. Mech.-Trans. ASME, 55(2):287-293, 1988.

4

J. Mazars and G. Pijaudier-Cabot. "Continuum damage theory - application to concrete", J. Eng. Mech.-ASCE, 115(2):345-365, 1989. doi:10.1061/(ASCE)0733-9399(1989)115:2(345)

5

R. de Borst, J. Pamin, R.H.J. Peerlings, and L.J. Sluys. "On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials", Comput. Mech., 17(1-2):130-141, 1995. doi:10.1007/BF00356485

6

A. Huerta and G. Pijaudier-Cabot. "Discretization influence on regularization by two localization limiters", J. Eng. Mech.-ASCE, 120(6):1198-1218, 1994. doi:10.1061/(ASCE)0733-9399(1994)120:6(1198)

7

M. Jirásek. "Nonlocal models for damage and fracture: comparison of approaches", Int. J. Solids Struct., 35(31-32):4133-4145, 1998. doi:10.1016/S0020-7683(97)00306-5

8

A. Rodríguez-Ferran, I. Morata, and A. Huerta. "Efficient and reliable nonlocal damage models", Comput. Methods Appl. Mech. Eng., 2004. to appear. doi:10.1016/j.cma.2003.11.015

9

M. Jirásek and B. Patzák. "Consistent tangent stiffness for nonlocal damage models", Comput. Struct., 80(14-15):1279-1293, 2002. doi:10.1016/S0045-7949(02)00078-0

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