438 M. Breitfuss et al. illustrated in Fig. 41.1b, the virtual work of the zero thickness element is given by ıWe DıWe; CıWe;CD Z e; jc;h ıue; t e; jc d Z e;C jc;h ıue;C t e;C jc d D Z e; jc;h ıue; te jc d Z e;C jc;h ıue;C te jc d : (41.9) The contact stresses, e.g. for normal direction, depend on the gap function g D ue;C ue; T ew DhNeue;C h Neue; h i T ew D ue;C h ue; h T Ne Ne T ew (41.10) via the “constitutive” relation te jc D tnew with tn D( 0 if g 0; cng if g<0 (41.11) with the penalty parameter cn. For evaluation of the integral in Eq. (41.9) the triangular shaped interface e jc;h depicted in Fig. 41.2b is projected onto a uniform triangle as depicted in Fig. 41.2c. The JacobianJ of this transformation is given by 2 66 4 @ @ @ @ 3 77 5 D 2 66 4 @u @ @v @ @u @ @v @ 3 77 5 2 64 @ @u @ @v 3 75DJ 2 64 @ @u @ @v 3 75: (41.12) With this transformation the virtual work of the zero thickness element is given by ıWe D 1 Z 0 1 Z 0 ıue;C ıue; te jc detJ d d D 1 Z 0 1 Z 0 ıue;C h ıue; h T Ne Ne T te jc detJ d d (41.13) where the stress equivalent nodal force vector of the zero thickness element is found fe contact D 1 Z 0 1 Z 0 Ne Ne T te jc detJ d d : (41.14) Finally Eq. (41.4) is extended with the nodal contact force vectors obtained from the zero thickness elements within the joint interface. The discretization of the contact problem Eq. (41.1) is given by MRuh CKuh Dfext Cfcontact : (41.15) 41.4 Model Order Reduction An approximation of the nodal displacements uh in Eq. (41.15) is given by uh ˆq (41.16) with the matrix ˆholding the displacement trial functions or modes 'i and the according generalized coordinates q. As described in [3] a mode base ˆclassic DŒ‰CM ˆNM (41.17)
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