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11th World Congress on Computational Mechanics (WCCM XI)5th European Conference on Computational Mechanics (ECCM V)

6th European Conference on Computational Fluid Dynamics (ECFD VI)

July 20–25, 2014, Barcelona, Spain

USE OF UZAWA ALGORITHM FOR SIMULATINGFRICTIONAL CONTACT BETWEEN CRACK FACES IN ABODY CONTAINING RANDOMLY ORIENTED CRACKS

Morteza Nejati∗, Adriana Paluszny and Robert W. Zimmerman

Department of Earth Science and Engineering, Imperial College,London, United Kingdom, m.nejati11@imperial.ac.uk

Key words: 3D frictional contact, crack, Augmented Lagrangian method, Uzawa algo-rithm.

Micro-structured materials, including rock and concrete, contain micro-cracks that signif-icantly influence their deformation behaviour. As they are often subjected to compressiveloads, the numerical simulation of their deformation requires the treatment of frictionalcontact over the crack faces. Due to the need for high contact precision, a sophisticatednumerical approach is required to simulate the behaviour of these materials. This paperfocusses on simulating the frictional contact of crack faces in a body containing randomlyoriented cracks, using the finite element method. The main novelty of this work is thenumerical simulation of the deformation behaviour of solids containing dense internalfrictional contacts. Based on the concept of infinitesimal strain theory, displacements aresmall enough to assume a geometrically linear problem, and therefore the contact sur-faces can be discretised using an isoparametric interpolation of crack face elements. Thisrequires matching meshes at the crack faces. As an unstructured mesh is used, due tothe complexity of the geometry, the crack faces are discretised using quadratic triangularelements. Linear elastic behaviour is also assumed to model the intact material. Thepenalty method was first used to apply the contact constraint on the crack faces. How-ever, in the cases of high contact precision, large penalties cause ill-conditioning, and sospecial augmented Lagrange techniques are required to solve the problem [1]. The Uzawaalgorithm [2] was implemented to apply the constraint using penalty parameters as lowas the Young’s modulus of the material. In the Uzawa algorithm, penalty contributionsare used together with constant Lagrange multipliers in an inner loop. The Lagrangemultipliers are then updated within an outer loop to converge to their exact values. Theinterfacial behaviour was modelled using τf = µσn + τ0, in which τf is the tangentialtraction, µ is the friction coefficient and τ0 is the cohesion stress. The backward Eulerintegration scheme together with the return mapping strategy was employed to apply thetangential constraints on the crack faces. Figure 1 shows the boundary conditions appliedfor analysing a singly-cracked body under uniform compressive stress. The distribution of

Morteza Nejati, Adriana Paluszny and Robert W. Zimmerman

contact tractions on the crack faces is shown in Figure 2. Figures 3 and 4 show the finiteelement mesh and contact traction distribution for a multiply-cracked body containingtwenty-seven non-intersecting cracks. Very smooth contact tractions are obtained, evenwhen very low penalty parameters are used. The results show that the Uzawa algorithmcan be effectively applied to very high contact precision problems for which the penaltymethod does not perform well due to ill-conditioning issues.

Figure 1: Applied boundary conditions.Figure 2: Contact tractions for a

singly-cracked body (µ=0.6, τ0 = 0).

Figure 3: Finite element mesh for amultiply-cracked body.

Figure 4: Contact tractions for amultiply-cracked body (µ=0.6, τ0 = 0).

REFERENCES

[1] P. Wriggers. Computational Contact Mechanics , Second Edition, Berlin, Springer-Verlag, 2006.

[2] J.C. Simo and T.A. Laursen. An augmented lagrangian treatment of contact problemsinvolving friction. Computers and Structures, Vol. 42, 97–116, 1992.

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