Impact-induced transition from damage to perforation

We investigate the impact-induced damage and fracture of a bar-shaped specimen of heterogeneous materials focusing on how the system approaches perforation as the impact energy is gradually increased. A simple model is constructed which represents the bar as two rigid blocks coupled by a breakable interface with disordered local strength. The bar is clamped at the two ends, and the fracture process is initiated by an impactor hitting the bar in the middle. Our calculations revealed that depending on the imparted energy, the system has two phases: at low impact energies the bar suffers damage but keeps its integrity, while at sufficiently high energies, complete perforation occurs. We demonstrate that the transition from damage to perforation occurs analogous to continuous phase transitions. Approaching the critical point from below, the intact fraction of the interface goes to zero, while the deformation rate of the bar diverges according to power laws as function of the distance from the critical energy. As the degree of disorder increases, farther from the transition point the critical exponents agree with their zero disorder counterparts; however, close to the critical point a crossover occurs to a higher exponent.

Figure 1

Model of the specimen under three-point bending. The specimen is composed of two rigid blocks of extensions $a$ and $b$ coupled by an elastic interface. The interface is discretized in terms of breakable fibers with stochastic strength. The deformation of the bar due to impact is characterized by the deflection $\delta$ of its middle point. The impact loading results in a linear deformation profile along the interface, where the top and bottom fibers have the smallest $l_1$ and largest $l_N$ length, respectively.

Figure 2

Probability distribution $p\left({\varepsilon }_c\right)$ of the breaking thresholds for three different values of the width $W/{\varepsilon }_c^0=0.25,0.5,0.75$. The average strength of fibers ${\varepsilon }_c^0$ is fixed; however, the degree of disorder can be tuned by varying the width $W$ of the distribution. The value of $W$ increases from top to bottom.

Figure 3

The evolution of the energies during an impact process at the critical impact energy $E_0=E_c$ as the function of deflection $\delta$ (black lines). In the final state $E_c$ is completely dissipated by fiber breaking, hence, $E_{el}\left({\delta }_c\right)=0$ and $E_{\mathrm{dis}}\left({\delta }_c\right)=E_c$. Simulations were performed for an interface of $N={10}^6$ fibers with $W={\varepsilon }_c^0$ for the threshold distribution (7). The red symbols highlight the case of a subcritical impact at $E_0\approx E_c/2$.

Figure 4

The remaining velocity $v_r$ of the impactor as a function of the imparted energy $E_0$. In the damage phase the impactor stops $v_r=0$ at a partial failure of the specimen, while beyond the perforation limit $E_0>E_c$ it keeps moving after the specimen broke into two pieces. The impactor is assumed to have a unit mass.

Figure 5

The critical energy $E_c$ of perforation as a function of the amount of disorder $W/{\varepsilon }_0^c$. The continuous line represents the analytical solution Eq. (12), while the symbols stand for numerical measurements. $E_c$ is divided by the critical energy of the zero disorder case.

Figure 6

The effect of the number of fibers $N$ and the degree of disorder $W$ on the damage threshold ${\delta }_d$ and on the critical deflection ${\delta }_c$ of the system. The damage threshold ${\delta }_d(N,W)$ (a) and the critical deflection ${\delta }_c(N,W)$ (b) are presented as function of $N$ for several values of $W$. Both quantities converge to asymptotic values ${\delta }_d(\infty ,W)$ and ${\delta }_c(\infty ,W)$ in the limit $N\to \infty$. The deviation from the asymptotic value has a universal power-law dependence on the number of fibers in (c) and (d). The bold straight lines represent power laws of exponent $-1/2$. Rescaling the deviations by an appropriate power of $W$, the curves obtained at different disorders can be collapsed on top of each other. This scaling is demonstrated for ${\delta }_c(N,W)$ in (e). The asymptotic values of the damage threshold and critical deflection as a function of $W$ are shown in (f).

Figure 7

Maximal deflection ${\delta }_m$ of the bar reached as a function of the impact energy $E_0$ for several disorder strengths $W$. The value of ${\delta }_m$ is nondimensionalized by division with the critical deflection ${\delta }_c^0$ of the zero disorder case. The value of $W$ increases from bottom to top.

Figure 8

Fraction of intact fibers $n=N_i/N$ as a function of the impact energy $E_0$ for several values of the degree of disorder $W$. The value of $W$ increases from right to left.

Figure 9

Derivative of the maximal deflection ${\delta }_m$ with respect to the impact energy $E_0$ as a function of the relative distance from the critical point $(E_c-E_0)/E_c$ for several values of the degree of disorder $W$. The dashed lines represent power laws of exponent $1/2$ and $2/3$. As the disorder $W$ increases the crossover point between the two power-law regimes shifts to the right. The value of $W$ increases from left to right.

Figure 10

The fraction of intact fibers as a function of the relative distance from the critical point $\mathrm{\Delta }=(E_c-E_0)/E_c$ for several values of the degree of disorder $W$. The dashed lines represent power laws of exponent $1/2$ and $2/3$. The value of $W$ increases from left to right.

Figure 11

The value of the crossover point ${\mathrm{\Delta }}^{*}$ as a function of the degree of disorder $W/{\varepsilon }_0^c$. The straight line represents a power law of exponent 2.0.