An asymptotic numerical method to solve compliant Lennard-Jones-based contact problems involving adhesive instabilities

Abstract

For compliant adhesive contact problems based on the Lennard-Jones potential, the non-convexity of the latter leads to jump-in and jump-off instabilities which can hardly be traced by using classical algorithms. In this work, we combine an adapted Asymptotic Numerical Method (ANM) and the multiscale Arlequin method to trace efficiently these instabilities. The ANM is used to trace the entire unstable solution path in a branch-by-branch manner. The Arlequin method is used to achieve a refined resolution in the vicinity of the contact surface and to reduce possible spurious numerical oscillations due to coarse surface discretizations. Numerical results, validated by comparison with available ones, reveal the accuracy, efficiency and robustness of the proposed global methodology.

Appendices

A Elementary matrices and vectors for a linear element

Here the elementary matrices and vectors in Eq. (23) are derived for a two-node linear contact element (see Fig. 19). Let $N_{pe}$ denote the number of collocation points in the considered element. Let us adopt the following notations:

$$\begin{aligned} \varvec{u}_{1}&=\left[ u_{1}^x,u_{1}^y \right] ,\quad \varvec{u}_{2}=\left[ u_{2}^x,u_{2}^y \right] \nonumber \\ \varvec{n}_{j}&=\left[ n_{j}^x,n_{j}^y \right] ,\quad \varvec{\theta }_{j}=\left[ s_j,x_j,y_j,z_j \right] ,1\leqslant j\leqslant N_{pe} \end{aligned}$$

(43)

Fig. 19

A two-node linear element

Let the unknowns associated with this element be arranged as follows:

$$\begin{aligned} \{{\bar{\varvec{U}}}\}^{e}=\left[ \varvec{u}_{1},\varvec{u}_{2},\varvec{\theta }_{1},\cdots , \varvec{\theta }_{N_{pe}}\right] ^{T} \end{aligned}$$

(44)

Then the elementary matrices and vectors are calculated as follows:

$$\begin{aligned}&\left[ \varvec{K}_{U\varTheta } \right] ^{e}\nonumber \\&\quad =\sum \limits _{j=1}^{N_{pe}}w_j \begin{bmatrix} N_{1j}n_j^x ((z_{j})_0-(x_{j})_0)&-N_{1j}n_j^x (s_{j})_0&0&N_{1j}n_j^x (s_{j})_0\\ N_{1j}n_j^y ((z_{j})_0-(x_{j})_0)&-N_{1j}n_j^y (s_{j})_0&0&N_{1j}n_j^y (s_{j})_0\\ N_{2j}n_j^x ((z_{j})_0-(x_{j})_0)&-N_{2j}n_j^x (s_{j})_0&0&N_{2j}n_j^x (s_{j})_0\\ N_{2j}n_j^y ((z_{j})_0-(x_{j})_0)&-N_{2j}n_j^y (s_{j})_0&0&N_{2j}n_j^y (s_{j})_0\\ \end{bmatrix} \end{aligned}$$

(45)

$$\begin{aligned}&\left[ \varvec{K}_{\varTheta U} \right] ^{e}=\sum \limits _{j=1}^{N_{pe}}w_j\frac{(s_{j})_0}{(d_{j})_0} \begin{bmatrix} N_{1j}n_j^x&N_{1j}n_j^y&N_{2j}n_j^x&N_{2j}n_j^y\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{bmatrix}\nonumber \\ \end{aligned}$$

(46)

$$\begin{aligned}&\left[ \varvec{K}_{\varTheta \varTheta } \right] ^{e}=\sum \limits _{j=1}^{N_{pe}}w_j \begin{bmatrix} 1&0&0&0\\ -\,2(s_{j})_0&1&0&0\\ 0&-\,2(x_{j})_0&1&0\\ 0&0&-\,2(y_{j})_0&1 \end{bmatrix} \end{aligned}$$

(47)

$$\begin{aligned}&\{ \varvec{F}_w \}^{e}=\sum \limits _{j=1}^{N_{pe}} w_j \begin{Bmatrix} (s_{j})_0/(d_{j})_0 \varvec{w}\cdot \varvec{n}_j\\ 0\\ 0\\ 0 \end{Bmatrix} \end{aligned}$$

(48)

$$\begin{aligned}&\{ \varvec{F}_u^n \}^{e}=\sum \limits _{j=1}^{N_{pe}} \sum \limits _{i=1}^{n-1} w_j \begin{Bmatrix} ((z_{j})_{n-i}-(s_{j})_i(x_{j})_{n-i}) N_{1j}n_j^x\\ ((z_{j})_{n-i}-(s_{j})_i(x_{j})_{n-i}) N_{1j}n_j^y\\ ((z_{j})_{n-i}-(s_{j})_i(x_{j})_{n-i}) N_{2j}n_j^x\\ ((z_{j})_{n-i}-(s_{j})_i(x_{j})_{n-i}) N_{2j}n_j^y \end{Bmatrix}\nonumber \\ \end{aligned}$$

(49)

$$\begin{aligned}&\{ \varvec{F}_{\theta }^n \}^{e}=\sum \limits _{j=1}^{N_{pe}} \sum \limits _{i=1}^{n-1} w_j \begin{Bmatrix} -(s_{j})_i(d_{j})_{n-i}/(d_{j})_0\\ (s_{j})_i(s_{j})_{n-i}\\ (x_{j})_i(x_{j})_{n-i}\\ (y_{j})_i(y_{j})_{n-i} \end{Bmatrix} \end{aligned}$$

(50)

where $w_j$ is a weight associated with the collocation point $p_j$, and $(d_{j})_i$ is the signed-distance at point $p_j$ at order i. $N_1$ and $N_2$ are the standard shape functions associated with the two nodes. In all expressions, for $1\leqslant j\leqslant N_{pe}$ and $1\leqslant i\leqslant n$, the subscript j indicates a quantity evaluated at the collocation point $p_j$ and the subscript i outside a parenthesis indicates a quantity evaluated at order i.

B A local ANM branch of the normalized LJ interaction

The ANM should be applied in a branch-by-branch manner because the solution of each local branch has a finite range of validity. To illustrate this issue, let us apply the ANM to trace the equilibrium path of the following single algebraic equation:

$$\begin{aligned} \bar{p}=\left( \frac{1}{\bar{d}}\right) ^9-\left( \frac{1}{\bar{d}}\right) ^3 \end{aligned}$$

(51)

Equation (51) is a normalized form of the LJ interaction Eq. (2), where the signed-distance and the contact pressure are normalized as $\bar{d}=\frac{d}{d_e}$ and $\bar{p}=p\frac{3 d_e}{8\varDelta \gamma }$, respectively. Two different initializations corresponding to $\bar{d}_0=4$ and $\bar{d}_0=1.3$ are tested and different truncation orders $N=3,5,10,15~\text {and}~20$ are tested. Numerical experiments are conducted separately using Method-1 and Method-2 depending whether auxiliary fields are treated as independent unknowns or not.

Fig. 20

A single ANM branch to trace the normalized LJ interaction Eq. (51). a$\bar{d}_0=4$, Method-1, b$\bar{d}_0=1.3$, Method-1, c$\bar{d}_0=4$, Method-2, d$\bar{d}_0=1.3$, Method-2

Figure 20 shows the solution of one ANM branch, where in Figs. 20a and c, each curve is plotted for a varies from 0 to 3 and in Fig. 20b, a varies from 0 to 1.25 for all curves. However, in Fig. 20d, the values of a at the ending point for each curve is not the same (which varies from 0.2 to 0.3) due to a too rapid deviation from the reference curve. It can be seen that in all cases, the solution has a finite range of validity since the approximation is only acceptable up to a finite value of a. Beyond a certain step length a, the solution deviates rapidly from the real solution path. All the factors considered (i.e. the truncation order, starting point, whether Method-1 or Method-2 is used, etc.) have significant influence on the range of validity. Indeed, the starting point indicates the local nonlinearity of the equilibrium path. At the starting point $\bar{d}_0=4$, the local nonlinearity is moderate since the contribution of the term $\left( \frac{1}{\bar{d}}\right) ^9$ can be neglected, thus a large step size is adequate for an accurate approximation. Moreover, in this case with a moderate local nonlinearity, using Method-1 or Method-2 makes no large difference on the solution. However, at the starting point $\bar{d}_0=1.3$, a very strong local nonlinearity is encountered and the range of validity is drastically reduced. In such a case with a strong local nonlinearity, Method-1 is preferred to give a relatively large range of validity. In all cases considered, augmenting the truncation order can increase the range of validity. However, beyond order 15, continuing augmenting the truncation order does not make much improvement of the solution since very high-order corrections are too small in magnitude.

Fig. 21

Comparison of the ANM results with the Guduru theory: a$B/R=0.1$, $A/B=0.01$, b$B/R=0.1$, $A/B=0.05$

C Comparison of the ANM results with the Guduru theory

Figure 21 shows the comparison of our numerical results, obtained with the asymptotic numerical and the Arlequin methods, with the results predicted within the Guduru theory for the weakly and the strongly wavy surfaces, considered in Sect. 5.3. One can see the significant difference between the results obtained by the two approaches, especially for strongly wavy surface. This stems clearly from the fact that the model assumptions on which rely the two approaches are different. Actually, Guduru’s theory assumes, in particular, a continuous contact zone. The latter hypothesis is not accurate, especially for the strongly wavy surface. Moreover, since the Guduru theory is derived from the JKR one, it does not include adhesive interaction for large separations, as can be seen in Fig. 21.