Abstract
Spontaneously propagating cracks in solids emit both pressure and shear waves. When a shear crack propagates faster than the shear wave speed of the material, the coalescence of the shear wavelets emitted by the nearcracktip region forms a shock front that significantly concentrates particle motion. Such a shock front should not be possible for pressure waves, because cracks should not be able to exceed the pressure wave speed in isotropic linearelastic solids. In this study, we present fullfield experimental measurements of dynamic shear cracks in viscoelastic polymers that result in the formation of a pressure shock front, in addition to the shear one. The apparent violation of classic theories is explained by the strainratedependent material behavior of polymers, where the crack speed remains below the highest pressure wave speed prevailing locally around the crack tip. These findings have important implications for the physics and dynamics of shear cracks such as earthquakes.
Introduction
Shock fronts are sharp discontinuities that arise whenever a perturbing feature, such as a crack traveling through a medium, exceeds the characteristic speed of the waves by which the energy is transferred in the medium. In such a situation, the waves coalesce into a sharp shock front, as observed in atmospheric supersonic flight, hypersonic reentry from space, meteoroid transit through the atmosphere, and motion of planets with respect to the solar wind^{1,2,3,4,5}.
In fracture mechanics and geophysics, shear shock fronts have been observed to arise by the coalescence of shear waves emitted by tips of spontaneously propagating shear ruptures exceeding the shear wave speed of the surrounding material^{6,7,8,9}. These ruptures are commonly referred to as intersonic or supershear. The speed of the spontaneously propagating cracks is a fundamental problem that has captivated the interest of the scientific community for several decades due to its implications across multiple scientific and engineering disciplines^{6,9,10,11,12,13,14,15,16,17,18}. In particular, the study of shear cracks propagating along frictional interfaces and the associated shock fronts is relevant to earthquake dynamics^{19,20,21,22}. The formation of the shock fronts is an important problem in its own right, due to implications of this phenomenon for strong ground motion much farther from earthquakeproducing faults than currently accounted for in seismic hazard^{6,7,8,9,19,20,21}.
Spontaneously propagating cracks are driven by elastodynamic waves, where the energy released by the crack motion is transferred through the medium to the crack tip region with the (higher) pressure wave speed and (lower) shear wave speed. It is intuitively evident that a crack cannot exceed the fastest way to transfer energy: the pressure wave speed^{6,10,11,13,14}. Hence, the formation of a shock front may appear impossible for the pressure waves.
In this study, we provide the first experimental evidence of spontaneously propagating shear ruptures forming a pressure shock front and explain the formation by the strainratedependent—and hence spatially variable—stiffening of the material in the vicinity of the rupture tip. The presence of the pressure shock fronts enables us to refer to our cracks as supersonic.
Results
Formation and observation of pressure shock fronts
The presented dynamic shear ruptures are produced in an experimental setup developed to mimic earthquakes in the laboratory^{6,8,9,22} (Fig. 1a; see Methods section). The setup features a quadrilateral specimen made of a polymeric material—either Poly(methyl methacrylate) (PMMA) or Homalite100—with an interface inclined at an angle α (Fig. 1a). The uniform external load P vertically applied to the specimen (yellow arrows) results in a normal and a shear static prestress acting along the interface. The tests exhibited in Figs. 1 and 2 have been conducted under the following experimental conditions: P = 21 MPa and α = 30° for PMMA and P = 25 MPa and α = 29° for Homalite100. To check the repeatability of the experimental outcomes, several tests have been done under nominally the same experimental conditions. The ruptures are triggered by the local brief pressure release due to the rapid sublimation of a NiCr wire filament placed across the specimen’s interface turning into plasma. The rupture initiation is imposed, but the subsequent crack propagation is spontaneous. This laboratory earthquake setup has been successfully employed in the past to study several key rupture phenomena including supershear transition to intersonic speeds^{9}, rupture directionality and limiting speeds due to bimaterial effects^{23}, offfault damage generation^{14}, pulselike to cracklike transitions^{24}, opening of thrust faults^{25}, and friction evolution^{26}.
It is quite challenging to capture the highly dynamic evolution of these shear ruptures in the laboratory, as they take mere tens of microseconds to span the experimental samples. The fullfield visualization of supersonic cracks employed here is enabled by our recently developed dynamic imaging technique^{26} based on a combination of highspeed photography and digital image correlation (DIC; see Methods section). A sequence of 128 highspeed digital images—with temporal sampling of up to two million frames per second—is converted into a temporal series of displacement fields by the DIC method, with algorithms tailored to treat displacement discontinuities along an interface^{26}. The particle velocity and strain fields are computed by temporal and spatial differentiation of the displacement fields, respectively. The strainrate maps are obtained by time differentiation of the strain fields. In a set of repeated experiments, an array of four straingage stations have been placed along the interface to capture the rupture’s arrival time and to confirm the supersonic nature of the rupture (Supplementary Fig. 1).
The fullfield images of the particle velocity, strain, and strainrate fields during dynamic ruptures in our experiments exhibit two pairs of sharp fronts diverging from the rupture tip, associated with the formation of the pressure and shear shock fronts (Fig. 1a, inset; Fig. 2 and Supplementary Fig. 2; and Supplementary Movie 1). The shear shock fronts, occurring when the rupture exceeds the shear wave speed, have been observed using photoelasticity^{6,8,9,14,22,25}, a technique sensitive to shear deformations. Our newly developed highspeed DIC technique reveals the additional formation of a pressure shock front. The pressure shock fronts are most visible in the distribution of the volumetric strain rate, \({\mathrm{tr}}\left( {{\dot{\boldsymbol \varepsilon }}} \right)\) (Fig. 2c, d), whereas the shear shock fronts are most noticeable in the distribution of the shear strain rate, \(\dot \varepsilon _{12}\) (Fig. 2e, f), consistent with the properties of the corresponding waves.
Confirming the shock nature of the pressure features
How can we confirm that these features are indeed pressure shock fronts and not some other expression of a pressure wave field that would be present around any crack tip^{8}? As the shock front is the envelope of coalescing waves, the defining feature of a shock front is the kinematic relationship that holds among the inclination angle β of the shock front, the wave speed (either c_{s} or c_{p} depending on the front), and rupture speed V_{r}^{1,5,7,8,9}:
The inclination angle of the shock front is the angle that the front forms with the path of the propagating feature, in our case the specimen's interface.
To verify this relation for the pressure shock fronts, we need to find the rupture speed, inclination angle, and wave speed of the material. We compute the rupture speed V_{r} from the temporal series of velocity maps by tracking the rupture tip location along the interface at each frame (Fig. 1c and Methods section). This leads to the steady rupture speed of V_{r }= 2.57 km s^{−1} within the window of observation for the experiment with PMMA shown in Figs. 1, 2 (left column), 3 (top inset), Supplementary Figure 2, and Supplementary Movie 1. The inclination angle varies along the pressure shock front (Fig. 2); for the steady rupture speed, the kinematic relation (Eq. 1) would imply that the wave speeds are decreasing in the interfacenormal direction. Such an observation is consistent with the viscoelastic response to spatially variable strain rates, with the higher strain rates closer to the crack tip leading to more viscoelastic stiffening and hence higher wave speeds. Indeed, several experimental studies in polymers, including PMMA^{18,27,28,29,30,31} and Homalite100^{7,8,27}, have accounted for their viscoelastic nature by considering the specimens as still uniformly linear elastic but with uniformly altered (stiffer) values of elastic constants during their dynamic response. Some of those studies^{29,30,31} considered two sets of uniform material properties: unstiffened (lowfrequency) ones ahead of the rupture arrival and stiffened (highfrequency) ones for the spatial locations along the interface behind the rupture tip. Several studies^{6,7,8,18,27,28,31} observed crack tip speeds similar to the ones reported in this work but did not recognize their significance, comparing the crack tip speeds to the uniformly higher dynamic wave speeds and concluding that the cracks are intersonic, a wellknown phenomenon^{7,8,18,28,31}, where pressure shock fronts cannot exist. Our findings emphasize the qualitative importance of the viscoelastic effects in creating the spatially heterogeneous stiffening—due to spatially inhomogeneous strain rates—that has not yet been considered, which is key to the formation of the pressure shock fronts.
Discussion
We find that the nonuniform stiffening due to viscoelastic effects, and hence spatially variable wave speeds, can indeed explain our experimental observations, including the inclination angles of the pressure shock front observed in our experiments. We use the published data on how strain rates affect the Young’s and shear moduli^{32,33,34,35,36,37} (Fig. 3), focusing on the PMMA due to the abundance of available data. We employ the approximation of quasielastic solid^{38,39} (see Methods section), in which the functional form for the material properties is that of a linearelastic solid but each effective material constant is assumed to depend on the local, instantaneous level of the strain rate. As a consequence of this approximation, the effective wave speeds of the polymers investigated here are functions of the strain rate^{32,33,34,35,36,37} (Fig. 3). For the areas not yet reached by the crack or waves, the strain rate is near zero; we select the lowstrainrate (LSR) value of \(\left {{\dot{\boldsymbol \varepsilon }}} \right = 10^{  4}{\kern 2pt} {\mathrm{s}}^{  1}\) to represent that regime, and the corresponding pressure and shear wave speeds are \(c_{\mathrm{p}}^{{\mathrm{LSR}}} = 1.79{\kern 2pt} {\mathrm{km}}{\kern 1pt} {\mathrm{s}}^{  1}\) and \(c_{\mathrm{s}}^{{\mathrm{LSR}}} = 0.86{\kern 2pt} {\mathrm{km}}{\kern 1pt} {\mathrm{s}}^{  1}\), respectively (Fig. 3 and Fig. 1c). As \(V_{\mathrm{r}} \, > \, c_{\mathrm{p}}^{{\mathrm{LSR}}}\) (Fig. 1c), the rupture propagates supersonically with respect to the effective pressure wave speed of the far field, which experiences the LSR conditions. At the crack tip, much higher strain rates—of the order of \(\left {{\dot{\mathrm \varepsilon }}} \right = 4 \times 10^3{\kern 1pt} {\mathrm{s}}^{  1}\)—develop (Fig. 3, bottom inset), constituting the highstrainrate (HSR) regime. The corresponding effective pressure and shear wave speeds are \(c_{\mathrm{p}}^{{\mathrm{HSR}}} = 2.85{\kern 2pt} {\mathrm{km}}{\kern 1pt} {\mathrm{s}}^{  1}\) and \(c_{\mathrm{s}}^{{\mathrm{HSR}}} = 1.37{\kern 2pt} {\mathrm{km}}{\kern 1pt} {\mathrm{s}}^{  1}\), respectively (Fig. 3 and Fig. 1c). Therefore, the rupture propagates intersonically with respect to the HSR wave speeds (\(c_{\mathrm{s}}^{{\mathrm{HSR}}} < V_{\mathrm{r}} = 2.57{\kern 2pt} {\mathrm{km}}{\kern 1pt} {\mathrm{s}}^{  1} < c_{\mathrm{p}}^{{\mathrm{HSR}}}\)), in local agreement with basic physics and energyreleaserate analytical models^{6,10,11,13,14} of rupture growth in linearelastic solids. At several locations along the pressure shock front (Fig. 3, upper inset, cyan star symbols), the local inclination angle β_{p} is measured to range between 71° and 83° and, based on the relation (Eq. 1), the corresponding values of the local pressure wave speed, \(c_{\mathrm{p}}^{{\mathrm{ISR}}} = V_{\mathrm{r}}{\kern 1pt} {\mathrm{sin}}( {\beta _{\mathrm{p}}} )\), can be obtained and range from 2.43 to 2.54 km s^{−1}. When we plot these pressure wave speed values against the strainrate magnitudes \(\left {{\dot{\boldsymbol \varepsilon }}} \right^{c_{\mathrm{p}}}\) (from 1.7 × 10^{2} to 4.9 × 10^{2} s^{−}^{1}) measured at the corresponding locations, we find that they are in excellent correspondence with the viscoelastic response (Fig. 3). These intermediate pressure wave speeds of 2.43 to 2.54 km s^{−1} are also below the rupture speed of V_{r} = 2.57 km s^{−1}, confirming that the rupture tip travels faster than not only the LSR pressure wave speed but also the pressure wave speeds at the examined locations of the pressure shock front.
It is noteworthy that the effectively supersonic rupture propagation observed in our experiments, in the sense of forming the pressure shock front, cannot be explained by a hyperelastic behavior at the crack tip. Hyperelasticity, in which the stiffening occurs with larger strains (in contrast to the strainrate effects of viscoelasticity) has been suggested by several numerical studies as a potential mechanism for supersonic crack propagation^{12,15,16}. However, the constitutive response of PMMA does not manifest hyperelastic stiffening in tension^{32,35}, compression^{34,35,36,37} (Supplementary Fig. 3a and b), or shear^{40} experiments for the levels of strains produced by the dynamic cracks in our experiments, which are smaller than 3.2 × 10^{−3} (Supplementary Fig. 3c). In addition, theoretical studies of crack propagation in lattice models^{17,41} suggested that supersonic solutions may exist in the absence of any stiffening, hyperelastic or viscoelastic, near the crack tip. However, our study cannot examine the relevance of those solutions, as the viscoelastic polymers we study do exhibit documented significant viscoelastic stiffening^{32,34,35,36,37,40}, which is fully consistent with our experimental findings and hence dominates the experimental response.
To summarize, our experimental results capture pressure shock fronts forming in viscoelastic polymers by spontaneously propagating ultrafast inplane shear ruptures, and demonstrate the importance of taking into account the nonuniform viscoelastic stiffening in the vicinity of the rupture front to explain the existence and angle of these features. The presented experimentally obtained ruptures are a striking example of how spatially nonuniform local material stiffening and the associated change in energy transfer can completely modify the largerscale processes, leading to the formation of pressure shock fronts and hence apparent spontaneous supersonic crack propagation. Effectively, the dynamics of the process induces a transient heterogeneity in the elastic properties. The nonuniform strainrate fields associated with the rupture tip and the resulting nonuniform viscoelastic stiffening are essential for the formation of a pressure shock front, in addition to the shear one. These findings are important for a number of engineering and geological applications, as they demonstrate how high and nonuniform strain rates at the crack tip can induce a nonuniform viscoelastic response in the materials that may be treated as uniformly linear elastic under many other conditions. It is noteworthy that most materials, including rocks^{42,43}, exhibit viscoelasticity at the highstrainrate regimes characteristic of rapidly propagating dynamic cracks. In studies of dynamic earthquake ruptures, the main emphasis so far has been on how high stresses at the rupture tip can induce damage and hence decrease the effective elastic properties and wave speeds^{6,44,45,46}. Our study illustrates the potential of a significant counteracting phenomenon in which the local elastic properties are transiently increased due to viscoelastic effects, promoting faster rupture propagation, potentially all the way to apparently supersonic ruptures with respect to the wave speeds in most of the bulk.
Methods
Laboratory setup
The laboratory setup employed in this study is the described in details in previous works^{6,7,9,22,24,25,26}. Our specimen configuration features either a 200 × 250 × 12.5 mm^{3} PMMA or a 200 × 200 × 10 mm^{3} Homalite100 plate. The sample is separated into two identical halves by an oblique cut at an angle α (Fig. 1a and Supplementary Fig. 1a). The juxtaposition of these two halves creates an interface (greenshaded area). In order to obtain repeatable and desired tribological conditions, these surfaces are polished to near opticalgrade finish and beadblasted by employing glass particles in the range of 104–211 mm diameter^{7,26}. A uniform load P is vertically applied to the specimen, resulting in a resolved normal (σ_{0 }= P cos^{2}α) and shear (τ_{0} = P sin α cos α) stress on the interface. Rupture nucleation is obtained by means of the rapid discharge of an electric potential through a 0.08 mm NiCr wire filament, placed across the interface (Fig. 1a and Supplementary Fig. 1a). Before initiation, electrical charges are accumulated in a capacitor bank in order to achieve a potential of 1.5 (for tests with Homalite100) to 2 kV (for tests with PMMA). The wire’s rapid sublimation produces a short pressure pulse, inducing the rupture initiation by locally frictionally weakening the interface. In this study, we present three tests conducted on PMMA and one test on Homalite100. All experiments performed with PMMA have an applied farfield load of P = 21 MPa; one configuration features an inclination angle of α = 30° (Figs. 1 and 2; Fig. 3, top inset; Supplementary Figs. 2 and 3c; and Supplementary Movie 1) and the other two are at α = 29° (Fig. 3, bottom inset; and Supplementary Fig. 1). The two tests with α = 29° are used to verify the rupture propagation speed (Supplementary Fig. 1), of whom one configuration employs a smaller imaging window in order to achieve higher accuracy with the fullfield technique (Fig. 3, bottom inset; and Supplementary Fig. 1a and c); and the other one features an array of three strain gauges (Supplementary Fig. 1). The experiment with Homalite100 is characterized by a farfield load of P = 25 MPa and an inclination angle of α = 29° (Fig. 2, right column). The fullfield images of velocity, strain, and strain rates are obtained by the employment of our dynamic imaging technique based on the combination of ultrahighspeed photography, DIC algorithms^{47} and postprocessing analysis^{26}. The strainrate magnitude field is computed from the strainrate components as the Frobenius norm of the tensor: \(\left {{\dot{\boldsymbol \varepsilon }}} \right = {\dot{\boldsymbol \varepsilon }}_{\mathrm{F}} = \sqrt {{\dot{\boldsymbol \varepsilon }} \\ gt: {\dot{\boldsymbol \varepsilon }}} = \sqrt {\dot \varepsilon _{ij}\dot \varepsilon _{ij}}\), assuming planestress conditions.
Wavespeed computation
Effective wave speeds due to viscoelastic stiffening are assumed to be a function of the strain rate, by adopting linearelastic relations with the values of elastic moduli dependent on the local level of strain rate. The elastic modulus of PMMA is tracked as a function of the strain rate using measurements derived from the literature^{32,34,35,36,37,48,49,50,51,52,53,54,55} (Fig. 3), ranging from quasistatic compression tests (10^{−5} s^{−1}) to highly dynamic conditions (10^{4} s^{−1}). As these measurements are from uniaxial tests and they need to be related to the threedimensional strainrate fields of our tests, we compute the strainrate magnitude from the corresponding tensor as \(\left {{\dot{\boldsymbol \varepsilon }}} \right = \sqrt {\dot \varepsilon _{ij}\dot \varepsilon _{ij}}\). The pressure and shear wave speeds are then calculated as a function of the strainrate magnitude using the linearelastic relations for plane strain with the elastic moduli depending on the specific level of strain rate (Fig. 3): \(c_{\mathrm{p}} = \sqrt {E\left( {1  \nu } \right)/\left[ {\rho \left( {1 + \nu } \right)\left( {1  2\nu } \right)} \right]}\) and \(c_{\mathrm{s}} = \sqrt {E/\left[ {2\rho \left( {1 + \nu } \right)} \right]}\), assuming density ρ = 1180 kg m^{−3} (measured) and a constant Poisson’s ratio of ν = 0.35^{35,37}. Leastsquare fits of the wave speeds versus strainrate magnitude (Fig. 3, red line) are then used to determine the LSR and HSR wavespeed values discussed in the main text and presented in Fig. 1c and Supplementary Figure 1c. The functional form of the fitted curve is \(a + b{\kern 1pt} {\mathrm{log}}\left( {\left {{\dot{\boldsymbol \varepsilon }}} \right} \right)\), where a = 2.24 and b = 5.9 × 10^{−2} for the pressure wave speed.
Rupture speed computation
The rupture speed is computed by tracking the rupture tip along the interface using the temporal sequence of the fullfield images. In analogy with numerical simulations of shear ruptures^{11,56}, we identify the rupture tip as the location where the slip velocity exceeds a preset threshold \(\dot \delta _{{\mathrm{th}}}\). The slip velocity \(\dot \delta\) is obtained from the difference of the \(\dot u_1\) particle velocity component parallel to the interface, immediately above and below it. In our calculations, we use \(\dot \delta _{{\mathrm{th}}} = 2.5{\kern 1pt} {\mathrm{m}}{\kern 2pt} {\mathrm{s}}^{  1}\) as a threshold for the slip velocity, as it is sufficiently above the noise level to avoid spurious oscillations and still well below the peaks of the slip velocity, which are in the range of 10 to 20 m s^{−1}. Changing the threshold within ± 1 m s^{−1} does not produce a substantial difference in the arrival times. Adopting this procedure, the rupture arrival time is identified at each location along the interface and the rupture speed is computed with a secondorderaccurate centraldifference scheme, using the sequence of locations and rupture arrival times (Fig. 1c and Supplementary Fig. 1c). To validate the rupture speed obtained from this procedure, we also compute it from the arrival times at a set of three straingage measurement locations, each measuring the direct strain in the direction parallel to the interface, ε_{11} (Supplementary Fig. 1). In this calculation, we select the threshold of the strain signal to be \(\left {\varepsilon _{11,{\mathrm{th}}}} \right = 10^{  3}\) (Supplementary Fig. 1b, horizontal dashed line). Two nominally identical experiments have been conducted on PMMA, under a farfield load of P = 21 MPa and an inclination angle of α = 29°. The rupture speed has been measured using either the DIC technique (employing a small field of view) or the strain gages (Supplementary Fig. 1). (Strain measurements by DIC and strain gages cannot be performed simultaneously in our experiments, as the highpower flash illumination, required for the highspeed image acquisition^{26,57}, releases a strong electromagnetic pulse that interferes with the strain gages, compromising their ability to measure physical strains.) The electric discharge, delivered to the NiCr wire for triggering, also induces an electromagnetic pulse that last several tens of microseconds. This disturbance produces spurious oscillations that overlap with the strain signals, in particular in the proximity of the wire notch (Supplementary Fig. 1a and b). However, the main features associated with rupture propagation are still clearly identifiable and allow precise rupture arrival time calculations, the results of which are in excellent agreement with the DIC ones (Supplementary Fig. 1c).
Data availability
Data supporting the findings of this study are available from the corresponding author upon request.
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Acknowledgements
This study was supported by the US National Science Foundation (NSF) (grant EAR 1321655 and EAR1651235), the US Geological Survey (USGS) (grant G16AP00106), and the Southern California Earthquake Center (SCEC), contribution number 6276. SCEC is funded by NSF Cooperative Agreement EAR1033462 and USGS Cooperative Agreement G12AC20038. We thank Drs Ravichandran and Knauss for helpful discussions.
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M.G., V.R., A.J.R., and N.L. contributed to developing the main ideas, interpreting the results, and producing the manuscript. M.G. and V.R. performed the measurements on PMMA and Homalite100, respectively. V.R. contributed in overseeing the experimental work.
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Correspondence to V. Rubino.
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