Quasi-Static and Implicit-Dynamic Finite Element Solution of Large Deformation Elastic Adhesive Contacts Using a Volumetric Interaction Scheme

Abstract

Adhesive forces, mediated by van der Waals’ and other interactions, dominate the contact response in the micron and sub-micron regimes. Understanding adhesion is especially important in biological systems (interaction of cells with pathogens, bio-locomotion, and drug delivery), mechanical systems (nano-indentation), and Micro-Electro-Mechanical Systems (MEMS), among many others. Classical adhesive contact models like the JKR, DMT, and Maugis’ models apply in the small-deformation regime for regular bodies. Despite attempts by Shull, Lin, and others, enabling large deformation and arbitrary shapes is infeasible in such semi-analytical schemes, necessitating the use of finite element analysis (FEA).

Existing FE models use volume-to-volume (V2V), surface-to-surface (S2S), point to volume (P2V) or point to surface (P2S) interactions. S2S (e.g. Fan et al.) are computationally efficient but are not accurate enough to simulate strong adhesion in soft bodies due to inherent approximations. In these paradigms, a well-known FE scheme is the Coarse-Grained-Contact-Model (CGCM) developed by Sauer and co-workers. While CGCM is quite general, it uses a modification of the classical continuum, which is complicated to implement. More importantly, adhesion involves inherent ‘jump-to’ and ‘jump-off’ instabilities, which have not received adequate attention in the existing simulation literature. Moreover, these instabilities are more pronounced in soft materials, and necessitate new supporting algorithms and computational approaches for successful simulation. Lastly, for applications, it is important for solvers to demonstrate the ability to simulate adhesive systems with realistic material and interaction parameters.

In the present work, a V2V, interaction-based, continuum FE model is developed for large deformation plane strain adhesive contacts, with all interacting bodies considered to be elastic. A tree-based, ultra-fine, structured mesh generator is developed to accurately model interactions while reducing the associated computational expense. A k-d tree based algorithm is implemented to compute the interactions, reducing the computational cost. Both quasi-static and implicit dynamic solvers are developed. The quasi-static solver uses a custom path-following algorithm which can tackle ‘jump-to’ and ‘jump-off’ instabilities for a wide range of problems. The dynamic solver provides an alternative solution strategy to resolve only the stable branches of the solution curve and is especially useful for soft materials with strong adhesion.

The solutions obtained by the quasi-static solver and the dynamic solver in the low-velocity limit show good agreement, except, obviously, in the snap-back zone. In the past, dynamics solvers for adhesive problems (Johnson et al.; Jayadeep et al.) have typically focused on the impact ('unforced') regime rather than on the constant-velocity ('forced') regime, which is often more important in applications. Some studies were carried out to validate various aspects of these solvers, including checks on the accuracy of interaction force calculations, mesh convergence behavior, and various limiting cases. Several model applications were considered to study and test these solvers, including cylinders and elliptical cylinders interacting with half-spaces, and a multi-body problem involving two cylinders and a half-space. Apart from the load-displacement and load-gap curves, a complete set of sub-surface strain fields and transmitted contact tractions is presented. The temporal evolution of the pressure peaks near the edges of contact is clearly revealed, flipping from tensile to compressive as the bodies approach each other very closely. The simulations show that tensile peaks always occur near the 'edge of contact' even in a highly repulsion-dominated regime. The solvers developed in the present work are expected to be useful to explore a spectrum of adhesive contact problems that arise in applications.