Physics-based numerical implementation framework towards multi-scale contact problem

This paper establishes a general computational framework to solve the muti-scale contact problem by integrating the statistical contact model with the finite element format. Compared to existing models, the proposed method is applicable to most geometric configurations and can effectively evaluate the pressure distribution. In this work, a modified Karush-Kuhn-Tucker (KKT) condition is proposed by the assumption that asperity height obeys the Gaussian distribution. Therefore, in the variational formula, the contact contribution is decomposed into body contribution and asperity contribution, corresponding to the nominal smooth surface and roughness, respectively. Then the linearization and constraint enforcement of these two components are derived, followed by a nonlinear Newton-Raphson-based iterative algorithm. The contact patch test and Hertz contact test are conducted, and the predicted results are consistent with the theoretical and experimental values, confirming the effectiveness and accuracy of the proposed approach. It is worth noting that in the Hertz contact test, the contact pressure distribution varies progressively with the roughness level and external force, tending to the Hertz limit or Gaussian limit. This means that the proposed method can be applied to any roughness and load. Finally, the contact behaviors of the transmission interface of a piezoelectric actuator, i.e., a typical multi-scale contact problem, are studied as an engineering application case.