Symplectic solution for size-dependent contact analysis

A novel symplectic framework suitable for size-dependent contact analysis has been established. The governing equations in symplectic form are derived for the contact between a no-slip rigid punch and a finite-sized plane with horizontal exponential material gradient. The dual Hamiltonian transformation of a quasi-Hamiltonian operator with asymmetric lower-left block is constructed. Analytical eigensolutions are obtained that lead to derivation of the coefficients in symplectic expansion via Hamiltonian mixed energy variational principle. The unique formulation of symplectic expansion in principle stems from the specific distribution of eigenvalues. Local phase transition is observed and analysis shows that size effects originate from the existence of real eigenvalues. A typical numerical example is presented to illustrate the efficiency of this symplectic approach. This new approach to contact analysis lays a solid theoretical foundation for material characterisation that utilises high-throughput testing methodologies and functionally graded specimens.