Plastic deformation of a film-substrate with inhomogeneous inclusions under contact loading

Abstract

In this paper, a semi-analytic solution is developed to investigate the plastic deformation of a film-substrate with inhomogeneous inclusions subjected to contact loading. In this solution, the surface pressure distribution and contact area can be determined by solving a set of governing equations via a modified conjugate gradient method. The inhomogeneous inclusions and the coating material are modeled as homogeneous inclusions with known initial eigenstrains plus unknown equivalent eigenstrains, according to the Eshelby’s equivalent inclusion method. A plasticity loop and an incremental loading process are used to obtain the accumulative plastic strain iteratively. This model considers not only the interactions among the contact loading body, embedded inhomogeneous inclusions and film materials, but also the plastic deformation of the film-substrate system. This solution is of great significance to understand the plastic deformation mechanism of a film-substrate with inhomogeneous inclusions under contact loading.

Appendices

Appendix A

Based on the previous study of Zhou et al. [5] regarding the elastic behaviors of a half-space with arbitrary inhomogeneous inclusions under contact loading, the stress field when considering the plastic deformation can be obtained using the EIM and stress superposition:

$$\begin{aligned} \sigma _{\alpha ,\beta ,\gamma }= & {} \sum \limits _{\zeta =0}^{N_{y}-1} \sum \limits _{\varphi =0}^{N_{x}-1} {M_{\alpha -\xi ,\beta -\zeta ,\gamma }^{n}p_{\xi ,\zeta }} +\sum \limits _{\zeta =0}^{N_{y}-1} \sum \limits _{\varphi =0}^{N_{x}-1} {M_{\alpha -\xi ,\beta -\zeta ,\gamma }^{f}f_{\xi ,\zeta }} \nonumber \\&+\sum \limits _{\varphi =0}^{N_{z}-1} \sum \limits _{\zeta =0}^{N_{y}-1} \sum \limits _{\xi =0}^{N_{x}-1} {A_{\alpha -\xi ,\beta -\zeta ,\gamma -\varphi }\varepsilon _{\xi ,\zeta ,\varphi }^{p}} \nonumber \\&+\sum \limits _{\varphi =0}^{N_{z}-1} \sum \limits _{\zeta =0}^{N_{y}-1} \sum \limits _{\xi =0}^{N_{x}-1} {A_{\alpha -\xi ,\beta -\zeta ,\gamma -\varphi }{(\varepsilon }_{\xi ,\zeta ,\varphi }^{*}} +\varepsilon _{\xi ,\zeta ,\varphi }^{**}) \nonumber \\&\quad (0\le \alpha \le N_{x} -1,0\le \beta \le N_{y}-1, 0\le \gamma \le N_{z}-1), \end{aligned}$$

(A1)

where $p_{\xi ,\zeta }$ represents the pressure in a discretized surface area centered at ($x_{\xi },y_{\zeta },0)$, respectively;$M_{\alpha -\xi ,\beta -\zeta ,\gamma }^{n}$ and $M_{\alpha -\xi ,\beta -\zeta ,\gamma }^{f}$ are $6\times 1$ matrix forms of the influence coefficients induced by $p_{\xi ,\zeta }$ and $f_{\xi ,\zeta }$ on a surface element, respectively; $A_{\alpha -\xi ,\beta -\zeta ,\gamma -\varphi }$ is a $6\times 6$ matrix form of the influence coefficients which relate the eigenstresses $\sigma _{\alpha ,\beta ,\gamma }^{p}$ and $\sigma _{\alpha ,\beta ,\gamma }^{*}$ at the observation point ($x_{a},y_{a},z_{a})$ in the cuboid [$\alpha ,\beta ,\gamma$] to the initial eigenstrains $\varepsilon _{\xi ,\zeta ,\varphi }^{p}$ and equivalent eigenstrains $\varepsilon _{\xi ,\zeta ,\varphi }^{*}$ plus plastic strain $\varepsilon _{\xi ,\zeta ,\varphi }^{**}$ in the cuboid [$\xi ,\zeta ,\varphi ,$], respectively, and the detailed expressions of $M_{\alpha -\xi ,\beta -\zeta ,\gamma }^{n}$, $M_{\alpha -\xi ,\beta -\zeta ,\gamma }^{f}$ and $A_{\alpha -\xi ,\beta -\zeta ,\gamma -\varphi }$ can be found in the work by Zhou et al. [5]; the value of $\varepsilon _{\xi ,\zeta ,\varphi }^{**}$ equals zero when the elastic deformation has just occurred and it would be nonzero when plastic deformation happens.

The surface u(x, y) can be decomposed into two parts: (1) the elastic displacement $u^{0}(x,y)$ due to contact loading and (2) the eigendisplacement $u^{*}(x,y)$ due to the subsurface eigenstrains (the initial eigenstrain $\varepsilon _{\xi ,\zeta }^{p}$, the equivalent eigenstrains $\varepsilon _{\xi ,\zeta ,\varphi }^{*}$ and the accumulated plastic strain $\varepsilon _{\xi ,\zeta ,\varphi }^{**})$. It can be written as

$$\begin{aligned} u\left( x,y \right)= & {} \sum \limits _{\zeta =0}^{N_{y}-1} \sum \limits _{\varphi =0}^{N_{x}-1} {Q_{\alpha -\xi ,\beta -\zeta ,\gamma }^{n}p_{\xi ,\zeta }} +\sum \limits _{\zeta =0}^{N_{y}-1} \sum \limits _{\varphi =0}^{N_{x}-1} {Q_{\alpha -\xi ,\beta -\zeta ,\gamma }^{f}f_{\xi ,\zeta }} \nonumber \\&+\sum \limits _{\varphi =0}^{N_{z}-1} \sum \limits _{\zeta =0}^{N_{y}-1} \sum \limits _{\xi =0}^{N_{x}-1} {{S_{\xi -\alpha ,\zeta -\beta ,\varphi }(\varepsilon _{\xi ,\zeta }^{p}+\varepsilon }_{\xi ,\zeta ,\varphi }^{*}+\varepsilon _{\xi ,\zeta ,\varphi }^{**})}, \end{aligned}$$

(A2)

where $Q_{\alpha -\xi ,\beta -\zeta ,\gamma }^{n}$ and $Q_{\alpha -\xi ,\beta -\zeta ,\gamma }^{f}$ are the influence coefficients induced by $p_{\xi ,\zeta }$ and $f_{\xi ,\zeta }$ on a surface element, respectively; $S_{\xi -\alpha ,\zeta -\beta ,\varphi }$ are the coefficients relating to the surface displacement to the initial eigenstrain $\varepsilon _{\xi ,\zeta }^{p}$, equivalent eigenstrain $\varepsilon _{\xi ,\zeta }^{*}$ and effective equivalent plastic strain $\varepsilon _{\xi ,\zeta }^{**}$. The detailed expressions of $Q_{\alpha -\xi ,\beta -\zeta ,\gamma }^{n}$, $Q_{\alpha -\xi ,\beta -\zeta ,\gamma }^{f}$ and $S_{\xi -\alpha ,\zeta -\beta ,\varphi }$ can be found in the work by Zhou et al. [5].

Appendix B

The linear isotropic hardening law describing the size of the yield surface as a function of the accumulated plastic strain p is given as

$$\begin{aligned} \sigma =\sigma _{\mathrm{Y}}+\frac{E_{\mathrm{t}}}{1-\frac{E_{\mathrm{t}}}{E_{S}}}p, \end{aligned}$$

(B1)

where $\sigma _{\mathrm{Y}}$ is the initial yield stress, $E_{S}$ is the Young’s modulus, and $E_{\mathrm{t}}$ is the plastic tangential modulus.

The current study follows the notion postulated by Nelias et al. [43] to calculate the increment in plastic strain. Yielding occurs when the condition $f\left( p+{\varDelta } p \right) =0$ is satisfied in the plastic zone. The actual increment in the accumulated plastic strain $\Delta p$ can be obtained through the Newton–Raphson iteration scheme. The yield function can be expanded and approximated as

$$\begin{aligned} f^{(n+1)}=f^{(n)}+{{\varDelta } p}^{(n)}f_{,p}^{(n)}=0. \end{aligned}$$

(B2)

Between two consecutive iterative steps, the correction of the accumulated plastic strain ${{\varDelta } p}^{(n)}$ can be expressed as

$$\begin{aligned} {{\varDelta } p}^{(n)}=-\frac{f^{( n )}}{f_{,p}^{( n )}}=\frac{f^{( n )}}{g_{,p}^{( n )}-\sigma _{VM,p}^{( n )}}. \end{aligned}$$

(B3)

All of the related variables are updated as follows:

$$\begin{aligned} \sigma _{VM}^{\left( n+1 \right) }= & {} \sigma _{VM}^{\left( n \right) }+\sigma _{VM,p}^{\left( n \right) }{{\varDelta } p}^{(n)},\nonumber \\ p^{(n+1)}= & {} p^{(n)}+{{\varDelta } p}^{(n)} ,\nonumber \\ g^{(n+1)}= & {} g(p^{\left( n+1 \right) }). \end{aligned}$$

(B4)

Here, $p^{(1)}$, $\sigma _{VM}^{\left( 1 \right) }$ and $\sigma _{ij}^{\left( 1 \right) }$ are the initial effective plastic strain, the equivalent von Mises stress and the Cauchy stress components, respectively. The calculation ends if the convergence condition is satisfied:

$$\begin{aligned} \left| \frac{f^{\left( n+1 \right) }}{g^{\left( n+1 \right) }} \right| =\left| \frac{\sigma _{VM}^{\left( n+1 \right) }-g^{\left( n+1 \right) }}{g^{\left( n+1 \right) }} \right| < \hbox {tolerance}. \end{aligned}$$

(B5)

The steps indicated in Eqs. (B2)–(B4) are repeated until the iteration converges. According to the plastic flow rule, the estimation of the plastic strain increment is determined:

$$\begin{aligned} \Delta \varepsilon _{ij}^{p}=\left[ p^{\left( n+1 \right) }-p^{(1)}\right] \frac{3S_{ij}^{\left( n+1 \right) }}{2\sigma _{VM}^{\left( n+1 \right) }}. \end{aligned}$$

(B6)