Figure 1: Arbitrary Finite Element structure with a joint For the characterization of the contact inside the joint, the involved surfaces and corresponding DOF are subdivided into a master and a slave surface. ª º ¬ ¼ T T T IJ IJ,M IJ,S x x x (3) where the vector of master DOF IJ,M x is of the dimension (m x 1) and the one of the slave DOF IJ,S x is of the dimension (s x 1). An additional (g x 1) vector g is introduced holding the relative normal displacement between the master and the slave surface at the master DOF. The quantity g is equal to the number of master surface FE nodes. A negative value at the i-th entry of g means, that the i-th node of the master surface penetrates the slave surface while a positive entry indicates gaping. For the sake of simplicity it is assumed that s=m and that the master and slave nodes do coincide at the unstressed reference position. The latter assumptions lead to a simple computation of g in the form of ª º ¬ ¼ IJ,M IJ,S g Q x x (4) where the invariant (g x m) matrix Q projects the physical displacements into a displacement along the corresponding normal vector of each master node. Finally a (g x 1) vector a is introduced. The component ai (i = 1..g) of the latter vector holds the joint area which is associated with master FE node number i. The sum of all entries of a gives the entire joint area. The literature offers several approaches for doing computational contact mechanics. For this contribution the self contact problem under consideration is treated as a quadratic programming problem. In the latter context the static problem is completely described by a minimization problem in the form of ª¬ º¼ T T B 1 min( ) 2 x x Kx f x (5) so that t ig 0 for i = 1..g, (6) see [9] and [10]. The (n x n) matrix Kdenotes the stiffness matrix of full rank. In case of FE structures the condition i > n is always fulfilled because the vector of joint DOF is a subset of the entire DOF vector x. When the minimization problem (5) is not written in x but the conditions (6) are, the problem over constraining can occur. This is the case when a transformation in the form of x ĭq (7) is applied where the (n x r) matrix ĭ holds in its columns r Ritz type trial vectors. The (r x 1) vector q is a generalized coordinate. The minimization problem takes on the form ª¬ º¼ T T red red 1 min( ) 2 x q K q f q (8) so that 31
RkJQdWJsaXNoZXIy MTMzNzEzMQ==