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7.1.2 Weak Form of Two Body Contact
The balance of linear momentum can be written for each of the two bodies in the reference configuration, where is a weighting function and is the Piola-Kirchhoff stress tensor. The last term corresponds to the virtual work of the contact tractions on body . For notational convenience, the notations and are introduced to denote the collection of the respective mappings and (for 1,2). In other words, The variational principle for the two body system is the sum of (7.1.2-1) for body 1 and 2 and can be expressed as, Or in short, Note that the minus sign is included in the definition of the contact integral . The contact integral can be written as an integration over the contact surface of body 1 by balancing linear momentum across the contact surface: The contact integral can now be rewritten over the contact surface of body 1: In the case of frictionless contact, the contact traction is taken as perpendicular to surface 2 and therefore can be written as, where is the (outward) surface normal and is to be determined from the solution strategy. For example in a Lagrange multiplier method the 's would be the Lagrange multipliers.
By noting that the variation of the gap function is given by equation (7.1.2-6) can be simplified as,