Finite-size effects in a model for plasticity of amorphous composites

We discuss the plastic behavior of an amorphous matrix reinforced by hard particles. A mesoscopic depinning-like model accounting for Eshelby elastic interactions is implemented. Only the effect of a plastic disorder is considered. Numerical results show a complex size dependence of the effective flow stress of the amorphous composite. In particular, the departure from the mixing law shows opposite trends associated to the competing effects of the matrix and the reinforcing particles, respectively. The reinforcing mechanisms and their effects on localization are discussed. Plastic strain is shown to gradually concentrate on the weakest band of the system. This correlation of the plastic behavior with the material structure is used to design a simple analytical model. The latter nicely captures reinforcement size effects in ${(logN/N)}^{1/2}$, where $N$ is the linear size of the system, observed numerically. Predictions of the effective flow stress accounting for further logarithmic corrections show a very good agreement with numerical results.

Figure 1

Effect of a plastic inclusion in an elastic matrix submitted to a pure shear biaxial loading $\mathrm{\Sigma }={\mathrm{\Sigma }}_{yy}=-{\mathrm{\Sigma }}_{xx}$. Only the spherical inclusion experiences plasticity. The first row represents the plastic strain ${\varepsilon }^{\mathrm{pl}}$, the second row the elastic strain ${\varepsilon }^{\mathrm{el}}$, and the third row the total strain $\varepsilon ={\varepsilon }^{\mathrm{el}}+{\varepsilon }^{\mathrm{pl}}$. The strain fields are represented by a color scale on the reference mesh on the left column and on a deformed mesh on the right column. One recognizes the traditional quadrupolar symmetry associated to the Eshelby inclusion.

Figure 2

Sketch of a simple plastic behavior. Plasticity sets in at yield stress ${\mathrm{\Sigma }}^{\mathrm{Y}}$, a hardening stage follows until a stress plateau is reached. The latter stress value defines the flow stress ${\mathrm{\Sigma }}^{\mathrm{F}}$. The plastic strain ${\varepsilon }^{\mathrm{pl}}$ is defined as the total strain $\varepsilon$ minus the elastic strain ${\varepsilon }^{\mathrm{el}}$.

Figure 3

Variation of the ultimate yield strength ${\mathrm{\Sigma }}^{\mathrm{F}}$ with the system size $N$ for a mere amorphous matrix ($\varphi =0$) with a yield stress ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$. The line corresponds to the power-law expression ${\mathrm{\Sigma }}^{\mathrm{F}}={\mathrm{\Sigma }}^{*}+\frac{A}{N}$. As shown in the inset this evolution is consistent with the numerical data.

Figure 4

Variation of the standard deviation $\delta {\mathrm{\Sigma }}^{\mathrm{F}}$ of the ultimate yield strength ${\mathrm{\Sigma }}^{\mathrm{F}}$ with ${\mathrm{\Sigma }}^{\mathrm{F}}$ for the amorphous matrix with a yield stress ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$ for system sizes $N=16,32,64,128,256$. The standard deviation is obtained for 40 realizations. As expected for a critical transition, a linear behavior is obtained. An extrapolation at zero standard deviation gives an estimate of the critical threshold, ${\mathrm{\Sigma }}^{*}$ at infinite size.

Figure 5

Variation of the ultimate yield strength ${\mathrm{\Sigma }}^{\mathrm{F}}$ with the system size $N$ for the amorphous matrix ($\varphi =0$) and amorphous composites with various fractions of hard inclusions. The yield stress of the amorphous matrix is ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$, and the yield stress of the inclusions is ${\mathrm{\Sigma }}^{\mathrm{H}}=10$. Depending on $\varphi$ the yield strength shows either a decreasing or an increasing trend with increasing system size.

Figure 6

Variation of the ultimate yield strength ${\mathrm{\Sigma }}^{\mathrm{F}}$ with the fraction $\varphi$ of hard inclusions for a yield stress ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$ of the matrix and a yield stress ${\mathrm{\Sigma }}^{\mathrm{H}}=10$ of the hard inclusions and for five different system sizes $N=16, N=32, N=64, N=128$, and $N=256$. The same data are shown in the inset in semilogarithmic scale.

Figure 7

Stress-strain curves for a yield stress of the matrix ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$, a system size of $N=64$, volume fractions of hard sites $\varphi =\{0,\cdots ,0.25\}$, and for a yield stress of the hard inclusions of ${\mathrm{\Sigma }}^{\mathrm{H}}=4$ (a), ${\mathrm{\Sigma }}^{\mathrm{H}}=10$ (b), ${\mathrm{\Sigma }}^{\mathrm{H}}=40$ (c), and ${\mathrm{\Sigma }}^{\mathrm{H}}={10}^8$ (d). The volume fractions which are not in the caption correspond to volume fractions below $0.06$ and for which at this size the effect of adding hard inclusions is not very obvious.

Figure 8

Evolution of the distribution of local plastic thresholds $P\left({\sigma }^{\mathrm{c}}\right)$ of the pure amorphous matrix upon plastic deformation. The system studied here has size $N=64$ and the initial yield stress of the matrix is ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$. A gradual exhaustion effect is observed until a stationary distribution is reached.

Figure 9

Evolution of the distribution of effective plastic thresholds $P\left({\sigma }_c^{\mathrm{eff}}\right)$ of the pure amorphous matrix, where ${\sigma }_c^{\mathrm{eff}}={\sigma }^{\mathrm{c}}-{\sigma }^{\mathrm{el}}$ upon plastic deformation. The system studied here has size $N=64$ and the initial yield stress of the matrix is ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$. The sharp lower front is associated to the emergence of a global yield stress. The latter gradually increases upon plastic deformation (hardening) until it reaches its final value (the stress plateau of the stress-strain curve).

Figure 10

Top: Stress-strain curve for amorphous composites of system size $N=64$, with a fraction $\varphi =0.12$ of hard inclusions of yield stress ${\mathrm{\Sigma }}^{\mathrm{H}}=4,10,40,{10}^8$. The harder the inclusions, the longer the hardening regime. Bottom: Stress-strain curve for amorphous composites with a growing fraction of extremely hard particles. The higher the fraction, the larger the hardening slope.

Figure 11

Evolution of the distribution of effective plastic thresholds $P\left({\sigma }_c^{\mathrm{eff}}\right)$, where ${\sigma }_c^{\mathrm{eff}}={\sigma }^{\mathrm{c}}-{\sigma }^{\mathrm{el}}$ upon plastic deformation. The system studied here has size $N=64$ and corresponds to a volume fraction $\varphi =0.16$ of hard sites of yield stress ${\mathrm{\Sigma }}^{\mathrm{H}}=10$ and an initial yield stress of the matrix ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$. The building of internal stresses gradually smears out the peak associated with hard particles. Conversely, the sharp front corresponding to the global yield stress increases upon plastic deformation longer than in the case of the sole amorphous matrix.

Figure 12

[(a)–(c)] Maps of the relative plastic strain ${\varepsilon }_{i,j}^{\mathrm{pl}}/\langle {\varepsilon }^{\mathrm{pl}}\rangle$ for a system size $N=64$, a yield stress of hard sites ${\mathrm{\Sigma }}^{\mathrm{H}}=10$, and volume fractions of hard inclusions $\varphi ={10}^{-3}, {10}^{-2}, {10}^{-1}$, respectively. The initial yield stress of the matrix is ${\sigma }^{\mathrm{c}}\in [0.5;1.5]$. [(d)–(f)] Maps of the associated final configurations of plastic thresholds ${\sigma }^{\mathrm{c}}$. The positions of hard sites are visible in dark red.

Figure 13

Maps of incremental plastic strain $\mathrm{\Delta }{\varepsilon }^{\mathrm{pl}}$ for a volume fraction of hard inclusions $\varphi ={10}^{-1}$, a system size $N=64$, a yield stress of hard sites ${\mathrm{\Sigma }}^{\mathrm{H}}=10$, and different values of the average plastic strain $\langle {\varepsilon }^{\mathrm{pl}}\rangle$.

Figure 14

Fractions $f_{\min }$ and $f_{\max }$ of incremental plastic strain $\mathrm{\Delta }{\varepsilon }^{\mathrm{pl}}$ borne by the weakest and the strongest slip systems $B_{\min }$ and $B_{\max }$ containing the smallest and the largest amount of particles, respectively. The system studied here has size $N=64$ and corresponds to a yield stress of hard sites of ${\mathrm{\Sigma }}^{\mathrm{H}}=10$.

Figure 15

Variation of the probability $P\left(B\right)$ of having at least one hard inclusion on each diagonal, defined in Eq. (9), with the volume fraction $\varphi$ of hard inclusions for different system sizes $N=16, N=32, N=64, N=128, N=1024$. Inset: Variation of the critical fraction ${\varphi }_c$ defined in Eq. (10) with the system size $N$.

Figure 16

Variation of the rescaled yield strength $({\mathrm{\Sigma }}^{\mathrm{F}}(\varphi ,N)-{\mathrm{\Sigma }}^{\mathrm{A}})/({\mathrm{\Sigma }}^{\mathrm{H}}-{\mathrm{\Sigma }}^{\mathrm{A}})/{\varphi }_c\left(N\right)$ with the rescaled volume fraction $\varphi /{\varphi }_c\left(N\right)$ for a yield stress ${\mathrm{\Sigma }}^{\mathrm{H}}=10$ of hard inclusions and different system sizes $N=16$, 32, 64, 128, 256. Inset: Variation of $[{\mathrm{\Sigma }}^{\mathrm{F}}(\varphi ,N)-{\mathrm{\Sigma }}^{\mathrm{A}}]/({\mathrm{\Sigma }}^{\mathrm{H}}-{\mathrm{\Sigma }}^{\mathrm{A}})$ with $\varphi$ for the same systems.

Figure 17

Size scaling of the rescaled flow stress ${\sigma }_R(\varphi ,N)$, defined in Eq. (21), of amorphous composites with concentration of hard particles ranging from $\varphi =0.04$ to $\varphi =0.5$. The yield stress of hard sites is ${\mathrm{\Sigma }}^{\mathrm{H}}=10$. Filled symbols on the vertical axis correspond to the infinite size limit, i.e., the result of the mixing law. The lines indicate the expected scaling behavior in ${(logN/N)}^{1/2}$.

Figure 18

Effect of the concentration of hard particles on the rescaled flow stress of amorphous composites for different system sizes $N=16$, 32, 64, 128, 256. The yield stress of hard sites is ${\mathrm{\Sigma }}^{\mathrm{H}}=10$. The straight line corresponds to the mixing law expected at infinite size. The dashed lines are the analytical predictions of Eq. (21) accounting for logarithmic corrections.

Figure 19

Hardening modulus versus fraction of hard sites for system sizes $N=64$, 128, 256 (symbols). The continuous line shows the mean-field expression $M\left(\varphi \right)=\varphi /(1-\varphi )$ and the discontinuous lines the size-dependent prediction $M(\varphi ,N)=f(\varphi ,N)/[1-f(\varphi ,N\left)\right]$ where $f$ is the fraction of hard sites in the weakest band.

Figure 20

Variation of the function $g\left(N\right)=(\varphi -f)/(1+r\left(N\right))/\sqrt{2h_N}$ with $N$ for different values of $\varphi$. The opened symbols correspond to a numerical evaluation of $g\left(N\right)$ using $10000$ iterations of $N$ drawings from a binomial distribution with parameters $N$ and $\varphi$. The solid and opened symbols of the same color and shape correspond to results for $\varphi$ and $(1-\varphi )$, respectively. The solid lines correspond to the analytical estimation given in Eq. (A19) in the limit of large $N$.

Figure 21

Variation of the function $g\left(N\right)=(\varphi -{\sigma }^R)/[1+r\left(N\right)]/\sqrt{2h_N}$ with $N$ for different values of $\varphi$. The solid symbols and the symbol A correspond to simulation results, with a yield stress of hard sites of ${\mathrm{\Sigma }}^{\mathrm{H}}=10$. The opened symbols of the same color and same shape correspond to a numerical evaluation of $g\left(N\right)$ using $10000$ iterations of $N$ drawings from a binomial distribution with parameters $N$ and $\varphi$. The solid lines correspond to the analytical estimation given in Eq. (A19) in the limit of large $N$.