Post-Buckling Analysis With Frictional Contacts Combining Complementarity Relations and an Arc-Length Method

Abstract

A linear complementarity problem formulation combined with an arc-length method is presented for post-buckling analysis of geometrically non-linear structures with frictional contact constraints. The arc-length method with updated normal plane constraint is used to trace the equilibrium paths of the structures after limit points. Under the proportional loading assumption, the unknown load scale parameter used in the arc-length method is expressed in terms of contact forces, and eliminated to formulate as a linear complementarity problem. The unknown contact variables such as contact status and contact forces can be directly solved in this formulation without any ad hoc technique. Complicated non-linear buckling behaviours, such as snap-buckling, can be efficiently solved by the developed method, as shown by several buckling and post-buckling problems with frictional contact constraints. KEY WORDS post-buckling frictional contact; complementarity; arc-length methodRecently, Kwak et al." -I 3 have derived a linear complementarity problem which is mathematically complete and needs no contact iteration even with friction. This is in contrast with other approaches in the literature such as Kikuchi and Oden,14 well studied in the frame of a variational formulation for friction problems.

Nonlinear analysis of structural elements under unilateral contact constraints by a Ritz type approach

“…Usually these methods are based on the use of the special finite elements derived to simulate the impenetrability condition between two surfaces (Wriggers and Imhof, 1993;Wriggers, 2002). On the other hand, the unilateral constraints can be maintained in the formulation, retaining the original philosophy of the problem, and an alternative linear complementary problems (LCPs) can be obtained and solved by, for example, Lemke's or Dantig's algorithms (Lemke, 1968;Ascione and Grimaldi, 1984;Joo and Kwak, 1986;Barbosa, 1986;Silveira, 1995;Koo and Kwak, 1996;Silva, 1998;Silva et al, 2001;Wriggers, 2002;Pereira, 2003;Holanda and Gonc ßalves, 2003).…”

“…For example, finite element applications including contact constraints in the stability of rods, limit-point behaviour of thick rubber spherical shells and large deformation post-buckling behavior of structures are presented, respectively, by Wriggers (1984), Endo et al (1984) and Simo et al (1986). Algorithms specifically designed to trace complex nonlinear equilibrium paths, such as arc-length procedures (Crisfield, 1991), have been used by Wriggers et al (1987), Stein et al (1990), Bjö rkman (1992), Koo and Kwak (1996) and Silveira and Gonalves (2001), among others, to solve stability problems of structures with unilateral contact constraints. Tschö pe et al (2003a,b) extended methods for the detection of critical points to problems with inequality constraints.…”

The unilateral contact buckling problem of geometrically perfect and imperfect beams

“…Besides these techniques, there exist methodologies where the unilateral constraints are maintained in the formulation of the problem. These techniques lead to quadratic programming or linear complementarity problems which can be solved by a variety of mathematical methods, for example, Lemke's or Dantig's algorithms (Lemke, 1968;Ascione and Grimaldi, 1984;Joo and Kwak, 1986;Barbosa, 1986;Silveira, 1995;Koo and Kwak, 1996;Hexiang et al, 1999;Silva et al, 2001;Silveira and Concalves, 2001;Wriggers, 2002;Pereira, 2003;Hollanda and Concalves, 2003). Mathematical programming resembles the force method of structural analysis, thus the local contact loads are set as unknowns and the solution is obtained by minimizing a given function (Chand et al, 1976;Fisher and Melosh, 1987).…”

“…Thus, special techniques have to be considered in order to calculate the post-critical equilibrium paths. Such algorithms, which have the potential of tracing the complex nonlinear equilibrium paths, have been developed by Wriggers (Wriggers et al, 1987), Stein (Stein et al, 1990), Bjorkman (Bjorkman, 1992), Koo and Kwak (Koo and Kwak, 1996), Silveira and Concalves (Silveira and Concalves, 2001), Tscope et al (Tscope et al, 2003a,b).…”

“…Even in that case, the classical variational techniques confront many difficulties which arise from the unilateral constraints. To this end, evolutionary mathematical theories and algorithms have been proposed by some researchers, leading however to a much more complex formulation (Wriggers et al, 1987;Stein et al, 1990;Bjorkman, 1992;Koo and Kwak, 1996;Silveira and Concalves, 2001;Tscope et al, 2003a,b). The proposed methodology avoids the numerical treatment of the certain constrained BVP by considering the basic differential equations of a beam in bending and adapting them appropriately in order to take into account the unilateral contact conditions.…”