## 2 Monitoring Dynamic Shear Ruptures in the Laboratory
### Post-Processing of the Displacement Fields  
Prior to the computation of strain fields, we filtered high-frequency noise from the displacement fields using a non-local-means (NL-means) filter [23; 30; 31]. This filter enables the displacement fields to be smoothed, while maintaining the original signal pattern. An example of full-field displacement maps of a laboratory rupture obtained using high-speed photography, DIC analysis, and a non-local filter is given in Figure 1. To facilitate comparisons, we report the same experimental rupture discussed in [1; 3], produced with an applied vertical load \(P=23\) MPa and inclination angle \(\alpha=29^{\circ}\), and monitored with a small field of view. This was a super-shear rupture propagating at a speed of 2200 m/s. Super-shear ruptures are interfacial ruptures whose tip propagates at speeds greater than the shear wave speed of the surrounding solid, as first observed in [24; 26; 32; 33]. Note that the full-field maps are cropped and their size is slightly smaller than the reported field of view size. These displacement fields are used throughout the study to test the capability of new algorithms to enforce traction continuity (TC) at the interface. Strains are computed from the filtered displacement fields using finite difference approximation. We employ a frame system \(x_{1}\)-\(x_{2}\) parallel to the interface and use the central difference scheme for pixels away from the boundaries [23]:  
\[\varepsilon_{11}(i,j,k) = \frac{u_{1}(i,j+1,k)-u_{1}(i,j-1,k)}{2\,s} \tag{1}\] \[\varepsilon_{22}(i,j,k) = \frac{u_{2}(i+1,j,k)-u_{2}(i-1,j,k)}{2\,s}\] (2) \[\varepsilon_{12}(i,j,k) = \frac{1}{2}\Bigg{(}\frac{u_{1}(i+1,j,k)-u_{1}(i-1,j,k)}{2\,s}+ \frac{u_{2}(i,j+1,k)-u_{2}(i,j-1,k)}{2\,s}\Bigg{)}, \tag{3}\]  
where \(u_{1}(i,j,k)\) and \(u_{2}(i,j,k)\) are the interface-parallel and interface-normal displacement components, respectively, for pixel \((i,j)\) and frame \(k\), and \(s\) is the step size. Note that the displacements in Equations (1)-(3) are expressed in pixels. Second-order backward and forward difference schemes are used to compute strains at the pixels immediately above and below the interface, respectively [1; 3]. The stress changes with respect to the reference configuration (before rupture) are computed from the strain fields using the standard plane-stress linear elastic constitutive equations:  
\[\left[ \begin{array}{c}\Delta\sigma_{11}(x_{1},x_{2},t)\\ \Delta\sigma_{22}(x_{1},x_{2},t)\\ \Delta\sigma_{12}(x_{1},x_{2},t)\end{array} \right]=\frac{E}{1-\
u^{2}}\left[ \begin{array}{ccc}1&\
u&0\\ \
u&1&0\\ 0&0&1-\
u\end{array} \right]\left[ \begin{array}{c}\varepsilon_{11}(x_{1},x_{ 2},t)\\ \varepsilon_{22}(x_{1},x_{2},t)\\ \varepsilon_{12}(x_{1},x_{2},t)\end{array} \right], \tag{4}\]  
where \(E\) is the [PERSON]'s modulus and \(\
u\) is [PERSON]'s ratio. The actual stresses are given by:  
\[ \begin{array}{c}\sigma_{11}(x_{1},x_{2},t)=\sigma_{11(0)}(x_{1},x_{2})+ \Delta\sigma_{11}(x_{1},x_{2},t)\\ \sigma_{22}(x_{1},x_{2},t)=\sigma_{22(0)}(x_{1},x_{2})+\Delta\sigma_{22}(x_{1 },x_{2},t)\\ \sigma_{12}(x_{1},x_{2},t)=\sigma_{12(0)}(x_{1},x_{2})+\Delta\sigma_{12}(x_{1 },x_{2},t),\end{array}  \tag{5}\]  
where \(\sigma_{11(0)}(x_{1},x_{2})\), \(\sigma_{22(0)}(x_{1},x_{2})=\sigma_{0}\), and \(\sigma_{12(0)}(x_{1},x_{2})=\tau_{0}\) are the pre-stresses at the reference configuration. Since Homalite-100 is a strain-rate sensitive material at the strain rate levels developed during these dynamic ruptures, we use the dynamic values of the elastic constants to compute the stress changes [1; 3]. An example of full-field maps of normal and shear stresses calculated from the displacement fields shown in Figure 1 are given in Figure 2a,b, respectively. The figures clearly show that the experimental technique enables capturing the spatial and temporal features of the ruptures. In particular, the more compressional and dilatational regions surrounding the crack are clearly imaged, as expected for shear ruptures (see [1; 3] for more examples and physical insight).