The Finite Element Method for the Analysis of Non-Linear and ...

Contact Analysis-Solution Approach
The Constraint Function Method
Let w be a function of λ and g such that the solutions of w(g, λ) = 0
satisfy the Normal conditions
Let v be a function of τ and ˙u such that the solutions v( ˙u, τ) = 0 satisfy
the Tangential Conditions. Then, the contact conditions are given by:
w(g, λ) = 0 v( ˙u, τ) = 0
These can now be imposed on the principle of virtual work using either a
penalty approach or a Lagrange Multiplier method. Variables λ and τ can
be considered Lagrange multipliers, and so we consider their variations δλ,
δτ. By multiplying by the variations and integrating in the domain we
obtain the constraint equation:
∫
S IJ [δλ w(g, λ) + δτ u( ˙u, τ)]dS IJ = 0
The governing equations of motion in this case are now both the principle
of virtual work and the constraint equation
Institute of Structural Engineering Method of Finite Elements II 42