Gradient damage models applied to dynamic fragmentation of brittle materials

Abstract

This paper is devoted to the use of gradient damage models in a dynamical context. After the setting of the general dynamical problem using a variational approach, one focuses on its application to the fragmentation of a brittle ring under expansion. Although the 1D problem admits a solution where the damage field remains uniform in space, numerical simulations show that the damage field localizes in space at a certain time and then a fragmentation of the ring rapidly occurs. To understand this phenomenon from a theoretical point of view, one develops a stability analysis of the homogeneous response by studying the growth of small perturbations. A dimensional analysis shows that the problem essentially depends on two dimensionless parameters $\tilde{\ell }$ and $\tilde{\dot{\varepsilon }}_0$, $\tilde{\ell }$ being related to the characteristic length present in the damage model and $\tilde{\dot{\varepsilon }}_0$ to the applied expansion rate. Then, since the product $\tilde{\ell }\tilde{\dot{\varepsilon }}_0$ is small in practice, the problem of stability is solved in a closed form by using asymptotic expansions. The comparison with the numerical results allows us to conclude that the time at which the damage localizes and the number of fragments are really governed by the growth of the imperfections. To conclude, a numerical simulation of the fragmentation of a 2D ring is presented.

Asymptotic analysis of the differential equation governing the amplitude of the perturbations

Let us consider the second order linear differential equation

$$\begin{aligned} \eta ^2 \ddot{y}(t)=f(t)y(t) \end{aligned}$$

(70)

where $\eta$ is a small parameter and the dot indicates the derivative with respect to t. In (70) f is a smooth function defined for $t>0$ and such that

$$\begin{aligned} f(t){\left\{ \begin{array}{ll}<0&{}\text{ if } \quad 0<t<t_c\\ =0 &{}\text{ if } \quad t=t_c\\>0&{}\text{ if } \quad t>t_c\end{array}\right. },\qquad \dot{f}(t_c)>0. \end{aligned}$$

In the application we will take

$$\begin{aligned} f(t)=\frac{3t^2-1}{t^4(t^2+1)} \end{aligned}$$

(71)

and in such a case $t_c=1/\sqrt{3}$, $\dot{f}(t_c)=\sqrt{3}/6$. Denoting the general solution by $y_\eta$, one wants to construct an asymptotic approximation of $y_\eta$. For that one must discriminate three different cases according to t is smaller than, close to or greater than the zero of f, $t_c$.

• 1. When$t>t_c$. Then, setting $y(t)=\exp (z(t)/\eta )$, the differential equation (70) becomes

  $$\begin{aligned} \eta \ddot{z}(t)+\dot{z}(t)^2-f(t)=0. \end{aligned}$$

  (72)

  The general solution $z_\eta$ of (72) can be expanded in powers of $\eta$:

  $$\begin{aligned} z_\eta (t)=z_0(t)+\eta z_1(t)+\eta ^2 z_2(t)+\cdots . \end{aligned}$$

  Inserting that expansion into (72) gives for the first two terms:

  $$\begin{aligned} \dot{z}_0(t)^2=f(t),\quad \ddot{z}_0(t)+2\dot{z}_0(t)\dot{z}_1(t)=0. \end{aligned}$$

  One easily deduces that $z_0(t)=\pm F(t)$ where F is defined on $(0,+\infty )$ by

  $$\begin{aligned} F(t)=\left| \int _{t_c}^t\sqrt{|f(s)|}ds\right| \end{aligned}$$

  (73)

  and $z_1(t)=-\frac{1}{4}\ln f(t)+\text{ cst }$. Therefore the general solution $y_\eta$ can be approximated (up to the “second order”) by

  $$\begin{aligned} y_\eta (t)=\frac{a_1 \exp \Big (-\dfrac{F(t)}{\eta }\Big )+b_1 \exp \Big (\dfrac{F(t)}{\eta }\Big )}{f(t)^{1/4}} \end{aligned}$$

  (74)

  where $a_1$ and $b_1$ are two arbitrary constants. (Note that we could refine the approximation by calculating the next terms of the expansions. However, the present approximation up to second order will be sufficient for our purpose.)

• 2. When$0<t<t_c$. Then, setting $y(t)=\exp (\mathsf iz(t)/\eta )$ where $\mathsf i$ is the complex number equal to $\sqrt{-1}$, one obtains by the same procedure that the general solution $y_\eta$ can be approximated (up to the “second” order”) by

  $$\begin{aligned} y_\eta (t)=\frac{a_0 \sin \Big (\dfrac{F(t)}{\eta }+\dfrac{\pi }{4}\Big )+b_0 \cos \Big (\dfrac{F(t)}{\eta }+\dfrac{\pi }{4}\Big )}{|f(t)|^{1/4}} \end{aligned}$$

  (75)

  where $a_0$ and $b_0$ are two arbitrary constants and F(t) is still given by (73). Note that the exponential functions of the previous case are here replaced by sinusoidal functions because of the change of sign of f at $t=t_c$.

• 3. In the neighborhood of$t_c$. When t tends to $t_c$ from below or from above, the approximations (75) and (74) diverge because $f(t_c)=0$. One must construct another approximation when t is close to $t_c$ and match it to the two previous ones. Note that, when t is close to $t_c$, $f(t)\approx \dot{f}(t_c)(t-t_c)$ and $F(t)\sim \frac{2}{3}\sqrt{\dot{f}(t_c)}|t-t_c|^{3/2}$. So, in order that F(t) be of the order of $\eta$, one makes the following rescaling of t:

  $$\begin{aligned} t=t_c+\eta ^{2/3} T, \end{aligned}$$

  T being the rescaled time. Then, using the approximation $f(t)\approx \dot{f}(t_c)(t-t_c)$, the differential equation (70) becomes

  $$\begin{aligned} \frac{d^2 y}{d T^2}=\dot{f}(t_c)T y. \end{aligned}$$

  Its general solution involves the two Airy functions $\mathsf {A_i}$ and $\mathsf {B_i}$ defined by the following indefinite integrals:

  $$\begin{aligned} \mathsf {A_i}(t)= & {} \frac{1}{\pi }\int _{0}^{+\infty }\cos \left( \frac{s^3}{3}+ts\right) ds,\\ \mathsf {B_i}(t)= & {} \frac{1}{\pi }\int _{0}^{+\infty }\left( \exp \left( -\frac{s^3}{3}+ts\right) \right. \\&\left. +\sin \left( \frac{s^3}{3}+ts\right) \right) ds. \end{aligned}$$

  Specifically, the general solution $y_\eta$ of (70) can be approximated by

  $$\begin{aligned} y_\eta (t_c+\eta ^{2/3} T)= & {} a_*\mathsf {A_i}\Big (\dot{f}(t_c)^{1/3}T\Big )\nonumber \\&+b_*\mathsf {B_i}\Big (\dot{f}(t_c)^{1/3}T\Big ), \end{aligned}$$

  (76)

  where $a_*$ and $b_*$ are two arbitrary constants.

In order that the three asymptotic approximations correspond to the same function $y_\eta$ on the half real line, one must adjust the six constants. For that, one must write the matching conditions when $|t-t_c|$ is small but |T| is large. On one hand, one uses the fact that

$$\begin{aligned}&F(t_c+\eta ^{2/3} T)\approx \frac{2}{3}\eta \sqrt{\dot{f}(t_c)}\,T^{3/2},\\&f(t_c+\eta ^{2/3} T)\approx \eta ^{2/3}\dot{f}(t_c)\,T, \end{aligned}$$

and, in the other hand, one uses the following behaviors of the Airy functions near $\pm \infty$:

$$\begin{aligned}&\mathsf {A_i}(t)\approx \displaystyle {\left\{ \begin{array}{ll}\dfrac{\exp \left( -\frac{2}{3}t^{3/2}\right) }{2\sqrt{\pi }t^{1/4}}&{}\text{ when }\quad t\rightarrow +\infty \\ \dfrac{\sin \left( \frac{2}{3}|t|^{3/2}+\frac{\pi }{4}\right) }{\sqrt{\pi }|t|^{1/4}}&{}\text{ when }\quad t\rightarrow -\infty \end{array}\right. },\\&\mathsf {B_i}(t)\approx \displaystyle {\left\{ \begin{array}{ll}\dfrac{\exp \left( \frac{2}{3}t^{3/2}\right) }{\sqrt{\pi }t^{1/4}}&{}\text{ when }\quad t\rightarrow +\infty \\ \dfrac{\cos \left( \frac{2}{3}|t|^{3/2}+\frac{\pi }{4}\right) }{\sqrt{\pi }|t|^{1/4}}&{}\text{ when }\quad t\rightarrow -\infty \end{array}\right. }. \end{aligned}$$

Comparing (74) and (75) with (76), one gets

$$\begin{aligned}&a_0=\left( {\eta }{\dot{f}(t_c)}\right) ^{1/6}\frac{a_*}{\sqrt{\pi }}=2a_1\\&b_0=\left( {\eta }{\dot{f}(t_c)}\right) ^{1/6}\frac{b_*}{\sqrt{\pi }}=b_1. \end{aligned}$$

Therefore, the general solution of (70) can be approximated by

$$\begin{aligned} y_\eta (t)={\left\{ \begin{array}{ll}\left( 2 a_1 \sin \Big (\dfrac{F(t)}{\eta }+\dfrac{\pi }{4}\Big )+b_1 \cos \Big (\dfrac{F(t)}{\eta }+\dfrac{\pi }{4}\Big )\right) |f(t)|^{-1/4}&{}\text{ when }\quad 0<t<t_c-O(\eta ^{2/3})\\ \dfrac{2a_1\sqrt{\pi }}{\eta ^{1/6}\dot{f}(t_c)^{1/6}}\mathsf {A_i}\Big (\dot{f}(t_c)^{1/3}\dfrac{t-t_c}{\eta ^{2/3}}\Big )+\dfrac{b_1\sqrt{\pi }}{\eta ^{1/6}\dot{f}(t_c)^{1/6}}\mathsf {B_i}\Big (\dot{f}(t_c)^{1/3}\dfrac{t-t_c}{\eta ^{2/3}}\Big )&{}\text{ when }\quad |t-t_c|\le O(\eta ^{2/3})\\ \left( a_1 \exp \Big (-\dfrac{F(t)}{\eta }\Big )+b_1 \exp \Big (\dfrac{F(t)}{\eta }\Big )\right) {f(t)^{-1/4}}&{}\text{ when }\quad t>t_c+O(\eta ^{2/3}) \end{array}\right. }, \end{aligned}$$

(77)

the two constants $a_1$ and $b_1$ being fixed by initial conditions.