Publication Date:
2015-10-22
Description:
Follower forces on a surface are defined as the forces which are keeping their constant directions with regard to the surface. A typical example is pressure, which is acting always in the direction of the normal. The linearization is necessary in order to solve a problem using an implicit scheme in the case of large deformations. Consider, for example, pressure p acting on a surface s , defined by the vector r (ξ 1 , ξ 2 ). In this case, a weak form is written as: (1) Formally, as shown in [1], [2] the following linearization can be formulated as (2) which is recovering an apparent unsymmetrical structure with regard to Δ u and δ u and, therefore, corresponding unsymmetrical tangent matrix. Though, the system is initially conservative an initially looking unsymmetrical structure is derived due to the linearization of the normal in the Cartesian coordinate system. A special complicated transformation for integrals with closed boundaries was necessary to show the symmetry in this case, see in [3], [4]. Nevertheless, as mentioned by Simo in [5], linearization in a covariant form is always leading to the symmetric structure for conservative systems. Application of the covariant derivation has become a standard tool within the geometrically exact theory of contact interaction, see in [6], [7]. The corresponding symmetry of tangent matrices for all conservative cases is then automatically fulfilled, moreover, they are obtained for all contact pairs such as surface-to-surface, curve-to-curve, curve-to-surface in a close covariant form. Using these transformations, one can treat contact tractions not as unknown variables to be computed, but as given external forces which are keeping their direction in the local coordinate system. This case we can call an inverse contact algorithm . The main results of the geometrically exact theory for contact interactions for surface-to-surface contact pairs are outlined first in order to obtain all relationships for the inverse contact algorithm . (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
Electronic ISSN:
1617-7061
Topics:
Mathematics
,
Physics
,
Technology
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