Variational Formulation for Frictional Interface Dynamics

The dynamic response of component bolted joints often plays a significant role in the overall behavior of a structural system. Accurate finite element simulation of these problems requires proper treatment of the interface conditions. We present a formulation carefully suited to these problems that incorporates discontinuous Galerkin (DG) treatment locally at the interface. The present work is an extension of our previous investigations of friction models within a finite element method for quasi-static problems. The current emphasis is on the treatment of the inertial term and ensuring that artificial resonance is not induced by the discrete interface. The weak imposition of continuity constraints allows the stick-slip behavior at the jointed surface to proceed smoothly, reducing the numerical instability compared to node-to-node contact techniques. As a model problem, we simulate the dynamic response of a lap joint subjected to an impulse axial force assuming Coulomb friction at the interface. INTRODUCTION Mechanical joints are common components of engineering systems to transfer forces between members. The compliance and interfacial friction present in these components often plays a significant role in the overall dynamics of the structure. Often, the primary sources of vibration damping and energy dissipation in an engineered system can be traced back to the local stiffness and damping characteristics of its bolted and riveted joints. Damping can be desirable in order to limit the amplitude of vibrations, while in other cases the associated fretting and wear may be undesirable. The accurate simulation of this dynamic behavior by finite element methods remains a challenging issue because it spans across the scales from nano to macro. While the dynamic response of large contiguous structures can typically be simulated accurately to produce close agreement between experiments and numerical models, the complexities in joints such as local dissipation and compliance that arise at the microscale can lead to sizable discrepancies between simulations and observations. Thus, practical simulation of structures dictates the development of multiscale models that account for the behavior of joints as well as robust numerical platforms in which these models can be embedded. DISCONTINUOUS GALERKIN TREATMENT OF DYNAMIC INTERFACES Toward the development of a viable computational platform, we invoke concepts from the Discontinuous Galerkin method to treat the interface conditions. The key feature of the method is that consistent flux terms are derived to weakly impose displacement continuity and stress equilibrium at the joint interface. Treating these conditions in an integral sense avoids the pitfalls of discrete springs that hinder convergence during stick/slip iterations and also frees the mesh to be nonconforming at the interface. Also, the method does not use Lagrange multipliers and thus avoids the bias that arises from associated master-slave constructs. The derivation of the additional interface terms proceeds in a straightforward manner [1]. Consider the continuum domain partitioned by an interface as shown in Fig. 1. Starting from the balance of momentum written in weighted-residual form, integration by parts is applied to the stress divergence term, taking into account the boundary integrals that emerge along the interface. Then, the conditions of displacement continuity and traction equilibrium are weakly imposed along the boundary through appropriate expressions weighted by the average of the variational displacement field. Combining and simplifying terms leads to a symmetric weak form of elastodynamics. Spatial discretization then proceeds as usual, where nonconforming element faces are permitted along the interface. The integrals from the interface