Morphological study of elastic-plastic-brittle transitions in disordered media

We use a spring lattice model with springs following a bilinear elastoplastic-brittle constitutive behavior with spatial disorder in the yield and failure thresholds to study patterns of plasticity and damage evolution. The elastic–perfectly plastic transition is observed to follow percolation scaling with the correlation length critical exponent $\nu \approx 1.59$, implying the universality class corresponding to the long-range correlated percolation. A quantitative analysis of the plastic strain accumulation reveals a dipolar anisotropy (for antiplane loading) which vanishes with increasing hardening modulus. A parametric study with hardening modulus and ductility controlled through the spring level constitutive response demonstrates a wide spectrum of behaviors with varying degree of coupling between plasticity and damage evolution.

Figure 1

(a) The bilinear constitutive force-displacement $\left(F\text{−}\delta \right)$ law implemented at the spring level showing the elastic-plastic transition at ${\delta }_y$, elastic unloading at ${\delta }_u$, and failure at ${\delta }_f$. $\delta$ is the absolute change in length of the spring. The elastic spring stiffness is denoted by $k$, while the postyielding spring stiffness is $k_T$. (b) The diamond spring lattice of size $L=8$ (solid lines) along with the square grid (dotted lines) that is used to generate a pixelated image by assigning a pixel to each spring. The boundary conditions used are such that the bottom edge nodes are fixed and the upper edge nodes are displaced in small increments in the out-of-plane direction, while periodic boundary conditions are used on the side edges.

Figure 2

[(a)–(d)] Set of yielded springs (black pixels) evolving with the increasing external load applied with the corresponding normalized average strain value $\varepsilon$ (see Fig. 3 for the corresponding stress and $v_y$ values). White pixels represent the grains in an elastic state. The simulation is performed with $E_T/E=0.2$ on lattice size $L=256$.

Figure 3

(a) The normalized average stress-strain curve and (b) volume fractions $v_y$ and $v_y^{\text{all}}$ for $E_T/E=0.2$ on lattice size $L=256$ shown in Fig. 2. Since the elastic unloading activity is negligible in this case, both curves cannot be differentiated. The elastic-plastic transition in terms of the order parameters $e_T,v$, and $v_y^{\text{all}}$ [Eq. (2)] is given in (c).

Figure 4

Effect of hardening on the elastic-plastic transition in $v\text{−}{e}_T$ space for $E_T/E=\{0,0.05,0.2,0.5\}$ on $L=256$ is shown in (a)–(d), respectively. The solid lines are for $v$, while the dotted lines represent $v_{\text{all}}$. The dash-dot lines represent $v_{\text{rp}}$ curves obtained using random bond percolation on the same spring lattice. The difference between $v_{\text{all}}$ and $v$ is the measure of the amount of unloading activity which is significant only for the perfectly plastic case.

Figure 5

[(a)–(f)] Evolving set of plastic grains for the perfectly plastic case $\left(E_T/E=0\right)$ with increasing loading. Black pixels represent springs that are currently in the plastic state, while white pixels represent elastic or elastically unloading springs. As $e_T\to 0$, the elastic unloading is observed in some specific zones due to formation of shear bands throughout the domain. At $e_T=0$ the yield line formed is pointed out with an arrow in (f).

Figure 6

[(a)–(e)] Number of springs yielded along a given $y$ ordinate $p\left(y\right)$ plotted against $y$ for $E_T/E=\{0,0.05,0.2,0.4,0.8\}$ for a single realization. $p\left(y\right)$ is obtained by dividing the loading into six stages.

Figure 7

[(a)–(d)] Evolving set of plastic grains for $E_T/E=100$ with increasing loading. Black pixels represent springs that are currently in the plastic state, while white pixels represent elastic springs. Panel (a) shows that even with sufficient plastic volume fraction the lattice exhibits an insignificant change in $e_T$ from 1. When observed closely, vertical streaks of increasing strength can be spotted in (b)–(d).

Figure 8

Effect of hardening on the elastic-plastic transition in $v\text{−}{e}_T$ space for $E_T/E=\{2,4,10,100\}$ on $L=256$, shown with solid lines in (a)–(d), respectively. The $v\text{−}{e}_T$ curves for the random bond percolation simulations at the same $E_T/E$ values are shown with the dash-dot lines. As $E_T/E\to 1$ the $v\text{−}{e}_T$ curves approach the line with unit slope $\left(v=e_T\right)$.

Figure 9

The crossing probability $R(L,p)$ with systems of increasing sizes for the elastic–perfectly plastic transition problem obtained using Eq. (3). The elastic–perfectly plastic transition can be observed to behave similarly to a percolation problem where ${lim}_{L\to \infty }R(L,p)$ is a step function at the critical threshold $p_c$.

Figure 10

Convergence of the estimator $p_{\text{avg}}\left(L\right)$ to $p_c$ according to Eq. (9) with $p_c=0.2844$. The fit is obtained by excluding the three leftmost points corresponding to $L=8,$ 16, and 24 that show deviation from the straight line due to higher-order corrections.

Figure 11

Convergence of the standard deviation $\Delta \left(L\right)$ according to Eq. (10). The fit is obtained by excluding the three leftmost points corresponding to $L=8,16$, and 24 that show deviation from the straight line due to higher-order corrections.

Figure 12

Convergence of $R^{\prime }(L,p_c)$ according to Eq. (11). The fit is obtained by excluding the three leftmost points corresponding to $L=8,16$, and 24 that show deviation from the straight line due to higher-order corrections.

Figure 13

The least-squares fit obtained for $R(L,p_c)$ according to Eq. (12).

Figure 14

The collapsed cumulative yield probability distribution $R(L,p)$ for different $L$ plotted against the reduced variable $[p_{\text{avg}}\left(L\right)-p]/\Delta \left(L\right)$.

Figure 15

Normal distribution fits of $R(L,p)$ with a black dotted line of slope 1 for comparison. Here $\Phi (.)$ is the standard normal distribution and $[p_{\text{avg}}\left(L\right)-p]/\Delta \left(L\right)$ is the standard normal variable. The collapse of data for different $L$ indicates that the normal distribution adequately describes $R(L,p)$.

Figure 16

[(a)–(d)] Distribution of the accumulated plastic strain ${\gamma }_s$ after the completion of the elastic-plastic transition for $E_T/E=\{0,0.05,0.2,0.5\}$. The ${\gamma }_s$ values are normalized with the maximum ${\gamma }_s^{\text{max}}$ to visualize the distribution from 0 (white) to 1 (black). The plastic strain distribution can be observed to be more homogenized (reducing anisotropy) as $E_T/E$ increases.

Figure 17

[(a)–(d)] The power spectra of the plastic strain maps at $L=256$ and $E_T/E=\{0,0.05,0.2,0.5\}$ averaged over 64, 10, 10, and 10 realizations, respectively. The strength of the dipolar symmetry diminishes with increasing $E_T/E$, implying uniform plastic strain accumulation.

Figure 18

The power spectral density $S(k_x,k_y)$ (averaged over 64 realizations each) of the plastic strain maps along $k_x=0$ (red triangles) and $k_y=0$ (blue circles) for $E_T/E=\{0,0.05,0.2,0.5\}$ are shown in (a)–(d), respectively. For $E_T/E=0$, the power spectra follows a power-law scaling $S\left(k\right)\sim k^{\alpha }$ with exponents $\alpha =-1.48\pm 0.02$ for $k_x=0$ and $\alpha =-0.43\pm 0.03$ for $k_y=0$. For $E_T/E>0$ we can observe that $\alpha \approx 0$ and the anisotropy diminishes with increasing $E_T/E$ as the two curves get closer.

Figure 19

Angular dependence of the scaling coefficient $\alpha \left(\theta \right)$ of the averaged power spectral density $S\left(k\right)$ $(\sim k^{\alpha \left(\theta \right)})$ of the plastic strain maps for $E_T/E=0$.

Figure 20

Normalized average stress-strain plots are shown for $E_T/E=\{0,0.2,0.5,0.8\}$ in (a)–(d), respectively, at $N=\{2,5,10\}$. The lines with increasing $N$ can be identified as we move from left to right along the arrow shown. The limiting failure stress can be observed to increase with increasing $N$ as well as $E_T/E$.

Figure 21

The volume fraction of the yielded springs $v_y$ (dotted line) and failed springs $v_f$ (solid line) are shown for $E_T/E=\{0,0.2,0.5,0.8\}$ in (a)–(d), respectively, at $N=\{2,5,10\}$. The $v_y$ and $v_f$ lines for increasing $N$ can be identified as one moves from left to right along the arrow.

Figure 22

(a) The volume fraction of the yielded springs $v_y$ (dotted line) and failed springs $v_f$ (solid line) are shown for $E_T/E=0$ at $N=100$. (b) The yielded springs (black pixels) and failed springs [red (gray) pixels] for $E_T/E=0$ at $N=100$ at increasing load steps. Localization of damage at the yield surface can be observed from (iv) to (vi).

Figure 23

Damage evolution images for $E_T/E=\{0,0.8\}$ at $N=\{2,5,10\}$ with black pixels for yielded springs and red (gray) pixels for failed springs on $L=256$. For each $\{E_T/E,N\}$, four images characterizing the transition are shown. The final fracture surface formed is shown in (iv) for each case.