# Algorithms for Nonlinear Contact Constraints with Application to Stability Problems of Rods and Shells

• P. Wriggers
• W. Wagner
• E. Stein
Conference paper

## Extended Abstract

In engineering applications often stability or postbuckling behaviour of structural members like thin shells and beams have to be investigated. This can only be done by considering the geometrically nonlinear behaviour. Finite deflections and rotations have to be included for the post-buckling range. In some engineering applications the structures will undergo contact. This adds another type of nonlinearity to the formulation due to the inequality constraints. Here, different contact algorithms will be investigated and compared. Furthermore, the coupling of these types of nonlinearities and the related algorithms are the main aspects of the paper. The contents of the paper may be outlined as follows.

In section 1 we give a summary of the contact formulation. The constraint conditions for the frictionless contact between deformable bodies are introduced via a perturbed Lagrangian procedure for the case of finite deformations. The formulation leads to different contact algorithms based on the Lagrangian multiplier method and the penalty approach. To overcome numerical problems associated with the Lagrangian multiplier approach a partitioning method will be used to solve the nonlinear set of equations. Finally, another method the augmented Lagrangian technique is introduced which is widely used for the solution of constraint problems.

Sections 2 and 3 are concerned with the algorithmic part. Here, methods which are used for the uncoupled problems are combined. The arc-length method for tracing limit points is employed together with active set strategies for the determination of contact regions in the context of an overall Newton iteration. A main object is the developement of a consistent Hessian matrix for the global Newton iteration which up to now is only possible for a fixed set of active constraints.

In section 4, we develop an axisymmetric shell element as a special case of a shell theory including large deflections and moderate rotations. A Mindlin type of shell theory is employed which is derived using direct tensor notation. The resulting element formulation is based on linear shape functions for the three independent variables using only one point numerical integration for all terms. To include finite rotations, an updated Lagrangian procedure is used which then restricts the application of the element only to small strains. This is valid for most of the engineering shell problems. Although there exist theories for elastic axisymmetric shells including finite rotations and strains the foregoing formulation has been used because of its simplicity. In addition, incremental algorithms have to be employed for tracing the postbuckling behaviour of structures. These algorithms allow in most applications only small load steps near limit points. Therefore, for this class of problems the fully nonlinear theory exhibits no advantage over a simple updating scheme.

In section 5, the performance of the algorithms and the element formulation is demonstrated for thin and moderate thick beam and shell applications by numerical examples.

## Keywords

Load Step Lagrangian Multiplier Method Load Deflection Curve Contact Algorithm Penalty Approach
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