Variational Methods in Contact Mechanics

Contact mechanics examines the deformation and interaction of bodies at interfaces where separation, adhesion and friction may occur. Variational methods reformulate contact problems as inequalities derived from energy principles, allowing the incorporation of unilateral constraints and non-smooth interface laws within a unified mathematical framework. In classical variational inequalities, convex energy functionals and linear operators describe idealised frictionless contact or perfectly adhered interfaces. Extension to hemivariational inequalities introduces nonconvex, locally Lipschitz potentials that capture complex frictional behaviour, slip thresholds and adhesion hysteresis. Existence, uniqueness and stability of solutions hinge on monotonicity properties, compactness arguments and fixed-point theorems. Numerical realisation often relies on time-discretisation schemes, such as the Rothe method, and penalty or duality approaches to enforce contact constraints, yielding convergent algorithms with a posteriori error estimates. Practical implementations address elastic, viscoelastic and fluid-structure interactions in engineering, geomechanics and biomechanics, bridging rigorous analysis with computational innovation to predict wear, optimise designs and control interface performance under varying load and environmental conditions.

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Variational Methods in Contact Mechanics publication trend

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Technical terms

Variational inequality: A mathematical statement expressing equilibrium as the stationarity of an energy functional subject to inequality constraints.

Hemivariational inequality: A generalisation involving nonconvex, nondifferentiable energy potentials, typically handled via Clarke’s generalized gradient.

Unilateral contact: A constraint allowing bodies to separate but preventing interpenetration, modelled by inequality conditions on normal displacement.

Normal compliance: A contact law that permits small interpenetration penalised by a potential, representing elastic deformation of thin interface layers.

Rothe method: A time-discretisation technique that transforms an evolutionary variational problem into a sequence of elliptic problems for numerical or theoretical analysis.

References

1. Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics. Numerical Algorithms (2019).
2. Boundary optimal control of a nonsmooth frictionless contact problem. Computers & Mathematics with Applications (2019).
3. On Flows of Bingham‐Type Fluids with Threshold Slippage. Advances in Mathematical Physics (2017).