Formulation and analysis of a three-dimensional finite element implementation for adhesive contact at the nanoscale

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Abstract

A three-dimensional finite element model for nanoscale contact problems with strong adhesion is presented. The contact description is based on the Lennard–Jones potential, which is suitable to describe van der Waals attraction between interacting bodies. The potential is incorporated into the framework of nonlinear continuum mechanics, and two different formulations, a body force (BF) and a surface force (SF) formulation, are derived. It is demonstrated that the model is highly accurate for contact surfaces where the minimum local curvature radius of the surface roughness is as low as 8 nm. The finite element implementation of the two formulations is provided and the overall contact algorithm is discussed. The numerical accuracy of the finite element discretization is analyzed in detail. It is shown that the SF formulation is more efficient than the BF formulation but loses accuracy as the strength of adhesion increases. The model has applications in computational biomechanics as is demonstrated by the computation of the adhesion of a gecko spatula.

Introduction

With the advent of nanotechnology, engineering applications at small scales are becoming increasingly important. At the nanoscale the interactions between two bodies are affected strongly by long ranging intermolecular forces [5]. This work presents a computational contact formulation that captures intermolecular forces such as van der Waals adhesion. van der Waals forces have been identified as the main forces governing gecko adhesion [1]. The gecko shows that adhesion mechanisms tend to be very flexible, so that they can adapt to rough surfaces and thus maximize the contact area for adhesion. The high flexibility leads to an important modeling requirement that is often disregarded by contemporary contact and adhesion models: the nonlinear kinematics of large motions and deformations. Therefore, the need exists to develop contact models for adhesion, that accurately capture the nonlinearities of large deformations. A suitable framework for this is given by the equations of nonlinear continuum mechanics [3], which also allows for the description of complex 3D geometries and nonlinear material behavior. Nonlinear models are usually solved by numerical approaches like the finite element method [27]. A requirement in the development of finite element methods lies in the numerical efficiency which is especially challenging for 3D contact problems [26].

The developments presented here are based on the nanoscale contact model of [23], [21]. There, the authors incorporate the Lennard–Jones potential, which is suitable to model van der Waals adhesion, into a nonlinear continuum mechanical framework, and show that this leads to a frictionless contact formulation based on body forces. The authors further show that a straight-forward finite element implementation of the formulation (denoted by ‘method 1’) is highly inefficient. The authors then proceed to develop an efficient contact formulation that is based on surface tractions and is denoted by ‘method 3’. It is shown that a finite element formulation of this method is much more efficient than the straight-forward finite element implementation. In [23], [21] 2D finite element formulations are presented.

The present paper picks up the surface traction formulation and complements it by a new formulation, denoted as the ‘body force formulation’. For both formulations, we develop a 3D finite element implementation and its corresponding contact algorithm. We also derive the algorithmically consistent tangent matrix for the special case of the Signorini contact problem. We then subject both formulations to a rigorous analysis of their accuracy: first, we examine and analyze all the assumptions that are introduced in the derivation of the two formulations. In the second analysis we assess the discretization error introduced by the finite element formulation. Third, we investigate the model error introduced in the surface formulation due to large deformations and strong adhesion. We also examine the physical behavior of nanoscale contact and discuss the numerical difficulties that can appear in computations.

The presented approach bears some similarities to cohesive zone models that have been developed for fracture [29], [14] and delamination [12]. On the other hand, the approach is clearly distinct from computational contact models that have been formulated for macroscale adhesion [17], [25].

Apart from gecko adhesion [1], the presented contact formulation has a long list of further applications. Examples include rubber adhesion [6], adhesion in MEMS [32], adhesion in nanoindentation [31], cell adhesion [11], [13], self cleaning surfaces [2], particle adhesion to water droplets [9], thin film peeling [7] and delamination [10], interactions of nanoparticles [8] and nanostructures [18], rough surface adhesion [15] and synthetic adhesion mechanisms [16].

The remainder of this paper is structured as follows: Section 2 presents the derivation of the two contact formulations. In the derivation the curvature of the neighboring body is neglected. The influence of this assumption is assessed in Section 3. In Section 4 the finite element equations of the two formulations are derived and the contact solution algorithm is presented. The accuracy of the finite element discretization is then examined in Section 5, while Section 6 provides a detailed analysis of the finite element model and its potential difficulties. In Section 7 an application of the model to biomechanical adhesion is considered. The paper concludes with Section 8.

Section snippets

The coarse-grained contact model

This section provides an overview of the coarse-grained contact model of [23], extends the model to 3D, introduces a new contact formulation based on body forces, and discusses all assumptions used in the derivation. The coarse-grained contact model combines a molecular interaction potential with a continuum mechanical contact formulation in order to describe nanoscale contact phenomena like van der Waals adhesion. The model formulation is general, allowing for large deformations and arbitrary

Accuracy of the half-space assumption

In this section we examine and evaluate the error introduced by the half-space approximation which is used to derive Eq. (7) from Eq. (4). The derivation itself is also reported here. In accordance with standard contact notation, we also denote the body Bk, on which the contact forces bk act, as the slave body, and denote the neighboring body B, which causes the contact force bk, as the master body. The contact force acting at xkBk is found by first projecting the slave point perpendicularly

3D finite element formulation

We now present the finite element formulation of the body force formulation, given by Eqs. (1), (2), (8), and the surface force formulation, given by Eqs. (10), (16), (19). We derive the finite element arrays for the two formulations and discuss the overall contact algorithm.

The finite element method offers a systematic solution strategy to approximately solve the weak form governing the problem. To facilitate the numerical integration of Eqs. (1), (10) the integration domains Bk and Bk are

Accuracy of the FE discretization

This section serves to assess the discretization error introduced by the finite element approximation of the body force formulation (35), and the surface force formulation (36). We thereby verify the correctness of the FE expressions and compare the efficiency of the two methods. As a test case we consider the interaction between a quarter torus segment (body B1) and a sphere (body B2) as is illustrated in Fig. 4. The radius of the sphere is chosen as 25r0, the radii of the torus are 5r0 and 50r

Analysis of the FE formulation

The purpose of this section is to demonstrate the convergence and mesh independence of the proposed 3D finite element formulation and to illustrate its behavior for strong adhesion and large deformations. As a simple, straight-forward model problem we analyze the nanoindentation of an elastic block by a rigid spherical indenter, as is shown in Fig. 6. The size of the block is chosen as 2R0×2R0×R0, the radius of the sphere is denoted by R; for the examples in this section we have used R=0.9R0.

Applications

Initially the coarse-grained contact model (CGCM) has been implemented in 2D and used to study several important applications. In [23] the deformation of carbon nano-tube cross-sections is studied and validated against existing numerical results. Ref. [21] show that the CGCM agrees well with the analytical adhesion model of [6]. The CGCM also agrees well with lattice statics computations as is shown in [24]. In [20] the CGCM is combined with a 3D nonlinear beam formulation to model the adhesion

Conclusion

This paper presents two three-dimensional formulations for adhesive contact, denoted as the body force (BF) and surface force (SF) formulation. Both contact formulations are based on an molecular interaction potential that is incorporated into the framework of nonlinear continuum mechanics, as is shown in Section 2. The approach is for example suitable to describe van der Waals interaction between deformable solids, as is considered in Section 7. In the derivation of both contact formulations,

Acknowledgment

The authors thank the student Matthias Holl for his contribution to the spatula modeling.

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