Uncertainty Analysis of Dynamic Rupture Measurements Obtained Through Ultrahigh-Speed Digital Image Correlation

Abstract

Background

The full-field behavior of dynamic shear cracks, with their highly transient features, has recently been quantified by employing Digital Image Correlation (DIC) coupled with ultrahigh-speed photography (at 1-2 million frames/sec). The use of ultrahigh-speed DIC has enabled the observation of complex structures associated with the evolution of the dynamic shear fractures under controlled laboratory conditions, providing a detailed description of their distinctive full-field kinematic features. This has allowed to identify, for instance, the spatiotemporal characteristics of sub-Rayleigh and intersonic shear ruptures, and to measure the evolution of dynamic friction during rupture propagation of frictional shear ruptures.

Objective

Capturing such highly transient phenomena represents a challenging metrological process influenced by both ultra-fast imaging procedures and DIC analysis parameters. However, the effect of these parameters on the quantification of the rupture features has not been assessed yet. Here, a simulated experiment framework is presented and employed to evaluate the uncertainties associated with ultrahigh-speed DIC measurements.

Methods

Finite element simulations replicate laboratory experiments of dynamic ruptures spontaneously propagating along frictional interfaces. Experimental images of the specimen acquired with an ultrahigh-speed camera are numerically deformed by the displacement fields obtained from the numerical simulations and are analyzed using the same DIC analysis procedure as in the laboratory experiments.

Results

The displacement, particle velocity, and strain fields obtained from the DIC analysis are compared with the ground-truth fields of the numerical simulations, correlating the measurement resolution with the physical length scale of the propagating Mode II rupture. In addition, the full-field data are employed to estimate the capability of the ultrahigh-speed DIC setup to infer the dynamic friction evolution.

Conclusions

This methodology allows us to quantify the accuracy of the ultrahigh-speed DIC measurements in resolving the complex spatiotemporal structures of dynamic shear ruptures, focusing on the impact of the key correlation parameters.

Appendices

Appendix 1

Friction Description and Rupture Nucleation in the Numerical Simulations

The numerical model adopted the linear slip-weakening formulation for the description of friction evolution [65]. In this formulation, the friction coefficient is given by:

$$\begin{aligned} f= {\left\{ \begin{array}{ll} f_s-(f_s-f_d)\dfrac{\delta }{D_c} \, , &{} \delta \le D_c\\ f_d \, , &{} \delta > D_c \end{array}\right. } \end{aligned}$$

(1)

where $f_s$ and $f_d$ are the static and dynamic friction coefficients and $D_c$ is the critical slip distance over which $f_d$ is reached. We employ $f_s = 0.65$, $f_d = 0.26$, and $D_c = 25$ $\mu$m obtained experimentally [24] under the same loading conditions as used in the simulations.

Dynamic rupture nucleation is obtained by artificially modifying the frictional properties in a small region around the desired nucleation size (Fig. 1(c)). In the nucleation region, the frictional behavior is described by the linear slip-weakening law given by equation (1), with modified parameters ($f_s^{*}=0.22$, $f_d^{*}=0.01$, $D_c^{*}=0.05$ $\mu$m) obtained through an iterative process in order to produce a rupture propagation consistent with the actual experiment. At the beginning of the simulation, the lower frictional strength over the nucleation region compared to the applied pre-stress level result in the initiation of a dynamic rupture. Note that this procedure is similar to reducing the normal stress due to wire explosion with the higher friction coefficients.

The length of the nucleation region $2L^*$ can be obtained from the critical nucleation length $2L_c$, which indicates the minimum size of the initial slip region required to induce a dynamic instability. The half-length $L_c$ is computed as [55, 77]:

$$\begin{aligned} L_c = \dfrac{\mu }{\pi (1-\nu )}\, \dfrac{(f_s-f_d)D_C}{P(\sin \alpha -f_d \cos \alpha )^2}\, , \end{aligned}$$

(2)

where $\mu =E/[2(1+\nu )]$ is the shear modulus and $\nu$ is the Poisson coefficient. The nucleation length was taken 1.2 times $L_c$, so that $2L^*=14.76$ mm.

Another relevant quantity that characterizes the spatial resolution of the dynamic crack is the cohesive zone length $\varLambda _0$, which indicates the portion of the interface, behind the fracture tip, where the static shear stress transitions to its dynamic value [55, 65, 78]. For a Mode II crack, the cohesive zone length of a crack advancing with a near-zero rupture speed can be estimated as follows:

$$\begin{aligned} \varLambda ^0 = \dfrac{9\pi }{32(1-\nu )}\dfrac{\mu D_c}{(f_s-f_d)P\cos ^2\alpha }\, . \end{aligned}$$

(3)

Note that this formula describes the cohesive zone associated with a quasi-static rupture, while the cohesive length for dynamic rupture is generally smaller [1, 66]. According to equation (3), the estimated cohesive zone length is $\varLambda ^0=9.72$ mm.

Appendix 2

Computation of the Velocity and Strain Fields

The particle-velocity maps are obtained from the displacement fields using a central difference scheme for the time differentiation, i.e:

$$\begin{aligned}&\dot{u}_{1}(x_1, x_2,t) = \dfrac{u_1(x_1, x_2,t+1)-u1(x_1, x_2,t-1)}{\Delta t}\, , \end{aligned}$$

(4)

$$\begin{aligned}&\dot{u}_{2}(x_1, x_2,t) = \dfrac{u_2(x_1, x_2,t+1)-u2(x_1, x_2,t-1)}{\Delta t} \, , \end{aligned}$$

(5)

Similarly, the strain fields are obtained at each timestep t, for data points away from the boundaries, using a central difference algorithm:

$$\begin{aligned}&\varepsilon _{11}(x_1, x_2) = \dfrac{u_1(x_1, x_2+p)-u_1(x_1, x_2-p)}{2 p}\, , \end{aligned}$$

(6)

$$\begin{aligned}&\varepsilon _{22}(x_1, x_2) = -\dfrac{u_2(x_1+p, x_2)-u_2(x_1-p, x_2)}{2 p} \, , \end{aligned}$$

(7)

$$\begin{aligned} \begin{aligned} \varepsilon _{12}(x_1, x_2) =&\ \frac{1}{2} \biggl [ -\dfrac{u_1(x_1+p, x_2)-u_1(x_1-p, x_2)}{2 p} \\&+\frac{u_2(x_1, x_2+p)-u_2(x_1, x_2-p)}{2 p} \biggr ] \, , \end{aligned} \end{aligned}$$

(8)

where p indicates the pitch distance between two measurement points. Here we choose p = 1 according to the stepsize of 1 pixel imposed in the DIC analysis. Close to the contact interface, the strain fields are derived using the backward and forward difference scheme for data points above and below the interface, respectively. For instance, according to the forward finite difference scheme, the strain components are computed through three points as:

$$\begin{aligned}&\varepsilon _{11}(x_1, x_2) = \dfrac{-u_1(x_1,x_2+2p)+4u_1(x_1,x_2+p)-3u_1(x_1,x_2)}{2p}\, ,\end{aligned}$$

(9)

$$\begin{aligned}&\varepsilon _{22}(x_1, x_2) = \dfrac{-u_2(x_1+2p,x_2)+4u_2(x_1+p,x_2)-3u_2(x_1,x_2)}{2p}\, , \end{aligned}$$

(10)

$$\begin{aligned} \begin{aligned} \varepsilon _{12}(x_1, x_2) =&\ \frac{1}{2} \biggl [ -\frac{-u_1(x_1+2p,x_2)+4u_1(x_1+p,x_2)-3u_1(x_1,x_2)}{2p} \\&+\frac{-u_2(x_1,x_2+2p)+4u_2(x_1,x_2+p)-3u_2(x_1,x_2)}{2p} \biggr ] \, . \end{aligned} \end{aligned}$$

(11)