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Theory Manual Version 3.6
 Subsection 7.1.8: Facet-To-Facet Sliding Up Section 7.1: Sliding Interfaces Subsubsection Slip Kinematics 

7.1.9 Sliding-Elastic

This algorithm was presented in [114] and differs considerably from the previous two. Consider a domain consisting of two bodies and with respective boundaries and . The two bodies are in contact over portions of\strikeout off\uuline off\uwave off \uuline default\uwave default and , respectively denoted and . The contribution of contact to the external virtual work may be written as where is a virtual velocity, represents the traction on , and is an elemental area of . In contact analyses, the tractions on are equal and opposite, , and the contact surfaces are shared, hence we may select one surface to perform the integration over. The virtual work arising from contact may be written as an integral over the primary surface only, yielding eq.(7.1.9-1) is commonly referred to as the contact integral.
To evaluate the directional derivatives of along increments in the displacements of , as required for an iterative technique such as Newton's method, it is necessary to formulate the integration over an invariant domain so that the directional derivative may be brought inside the integral without concern for variations in the domain of integration. In our approach the integral is formulated over the invariant parametric space of [12, 28].
Each surface is expressed in parametric form using coordinates , and material points are identified through their parametric coordinates. On each surface , covariant basis vectors are given by Here, is the spatial representation of surface as it deforms over time , in terms of contravariant surface parametric coordinates . These covariant basis vectors are tangent to , and it follows that the unit outward normal to is Furthermore, the elemental area on is evaluated as Therefore the contact integral of eq.(7.1.9-1) may be placed into matrix form and rewritten as where represents the invariant parametric space of surface and integration is performed over points with prescribed parametric coordinates . Since represents a material frame, it is possible to linearize by applying the directional derivative operator directly to the integrand, where it is understood that for any function , To proceed with this linearization, it is necessary to formulate the kinematics of points on and provide expressions for the contact traction that can differentiate between frictional stick and slip.
 Subsection 7.1.8: Facet-To-Facet Sliding Up Section 7.1: Sliding Interfaces Subsubsection Slip Kinematics 

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