A C 1 -continuous formulation for 3D finite deformation frictional contact

Abstract

A new 3D smooth triangular frictional node to surface contact element is developed using an abstract symbolical programming approach. The C 1 -continuous smooth contact surface description is based on the six quartic Bézier surfaces. The weak formulation and the penalty method are formulated for the description of large deformation frictional contact problems. The presented approach, based on a non-associated frictional law and elastic-plastic tangential slip decomposition, results into quadratic rate of convergence within the Newton-Raphson iteration loop. The frictional sliding path for the smooth, as well as the simple frictional node to surface contact element presented herein, is defined by the mapping of the current in the last converged configuration. Examples demonstrate the performance of symbolically developed contact elements, as well as the stability and more realistic contact description for the smooth elements in comparison with the simple ones. IntroductionDiscretization of bodies in contact using low order contact elements is significantly influenced by the introduced roughness in the surface geometry caused by the discretization itself. As described in for 2D problems in [5], sliding of a slave node over several master segments (i.e., surfaces in the 3D case), due to the sudden changes in normal vectors, can cause loss of the quadratic convergence rate, numerical instability, rough non-physical behavior or jumps in the velocity field for the dynamic problems. In this paper a way is presented for threedimensional contact problems which helps to avoid these problems without detrimentally influencing rough normal changes between adjacent master contact surfaces. Also the non-uniqueness regarding the definition of the surface normal for regions near the edges of master surfaces is avoided.The importance and advantages of surface smoothing in case of 2D analysis is already described in [1][2][3][4][5]. Following the idea of smoothing an active two-dimensional contact segment by the use of two polynomials defined by midnodes, a C 1 -continuous (smooth) 3D frictional polynomial node to surface contact element is presented herein.Contact between 3D solids is most commonly analyzed using quadrilateral contact elements that are based on 8-node hexahedral (brick) continuum finite elements. However, since the master contact surface is generally curved, the master nodes are not lying on the same plane. Furthermore the normal vectors from one element do not pass on to the adjacent element in a smooth way. To overcome these difficulties two different approaches were developed to approximate curved contact surfaces when using quadrilateral contact elements. In the first approach [8] the curved contact surface is approximated by the flat contact surface, achieved by averaging the normal vectors. In the second approach [9-11] the curved contact surface is approximated by four flat triangles. Hence, both approaches allow a closed form solution for the projection of the slave node when finding the minimum distance.S...

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