Keywords: multi-body contact, pseudo-rigid bodies, Coulomb friction, contact dynamics, semismooth Newton step.

Micro-modelling of masonry as a composition of elastic units and nonlinear joints is computationally expensive. This fact is often stated in literature and as so provides motivation for development of suitable macroscopic models. A majority of those focus on a two-dimensional case. This is partly due to availability of experimental data rendering mostly in-plane behaviour. It is also down to the effort necessary to develop a constitutive framework, which in consequence makes other aspects of modelling less relevant.

It is of interest here, whether a computationally inexpensive three-dimensional micro-modelling framework can be designed. The employed numerical devices tend to minimise computational effort. Nevertheless, they maintain some desirable properties. In particular, finite kinematics is used in order to assure consistent representation of large rotations. The price of non-linearity is alleviated by the assumption of spatially homogeneous deformations. Frictional contact behaviour of joints is governed by the Signorini-Coulomb law. The resulting nonlinear complementary problem is solved with a semismtooh Newton method [5] combined with an iterative Gauss-Seidel scheme [3]. A sequence of incremental steady state solutions is obtained by means of dynamic relaxation.

The current development can be well summarised as a combination of the Contact Dynamics method by Jean and Moreau [1,2] and the theory of pseudo-rigid bodies by Cohen and Muncaster [4]. To some extent the novelty of the proposed approach lays in adopting the semismooth Newton step by Hüeber et al. [5] in this context. Some aspects of dynamic relaxation are also discussed, where a special care is necessary in order to deal with actively operating rigid modes. Force and displacement controlled simulations are presented and compared with available experimental findings. Even though an acceptable agreement of the results is achieved, the numerical formulation does not avoid some amount of dependence on to the rate of the applied control loading.

1

M. Jean, "The non-smooth contact dynamics method", Computer Methods in Applied Mechanics and Engineering, 177, pp. 235-257, 1999. doi:10.1016/S0045-7825(98)00383-1

2

J. Moreau, "Numerical aspects of the sweeping process", Computer Methods in Applied Mechanics and Engineering, 177, pp. 329-349, 1999. doi:10.1016/S0045-7825(98)00387-9

3

F. Jourdan, P. Alart, and M. Jean, "A gauss-seidel like algorithm to solve frictional contact problems", Computer Methods in Applied Mechanics and Engineering, 155, pp. 31-47, 1998. doi:10.1016/S0045-7825(97)00137-0

4

H. Cohen and R. G. Muncaster, "The Theory of Pseudo-rigid Bodies", Springer, New York, 1988.

5

S. Hueber, G. Stadler, and B. I. Wohlmuth, "A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction", To appear, 2007. doi:10.1137/060671061

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