Pressure shock fronts formed by ultra-fast shear cracks in viscoelastic materials

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 near-crack-tip 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 linear-elastic solids. In this study, we present full-field 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 strain-rate-dependent 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.

Recent Milestones in Unraveling the Full-Field Structure of Dynamic Shear Cracks and Fault Ruptures in Real-Time: From Photoelasticity to Ultrahigh-Speed Digital Image Correlation

“…In a recent study [76], we have observed the formation of pressure shock fronts associated with the spontaneous propagation of shear ruptures in viscoelastic polymers, apparently contradicting classic linear elasticity theory. This contradiction is explained with the high-strain-rate dependence of the polymers inducing local stiffening of the material and locally increasing the effective wave speeds as the crack tip is approached, resulting in the rupture tip propagating slightly less than, or equal to, the local, near-tip pressure wave speed but above the wave speed prevailing in far-field, thus giving rise to the formation of a pressure shock front, in addition to a trailing shear Mach Cone [76]. In other words, the wave speeds, instead of having a single value as in linear elasticity, depending on the local value of the strain rate field at each point, and as such, they exhibit a field variation themselves.…”

“…In order to confirmed that the shock fronts visualized are indeed Mach features, rather than some other manifestation of the corresponding waves, we have verified that the characteristic geometric feature of such fronts (i.e., inclination angle β with respect to the interface) satisfies the kinematic relation characteristic of Mach waves [76]: β = sin −1 (c/V r ), where V r is the measured rupture speed and c is either the pressure or shear wave speed (depending on the front considered). We have explained this behavior by the viscoelastic effects of the polymers resulting in a significant increase in the elastic moduli and effective wave speed with strain rates [76]. The high degree of variability of elastic moduli with strain rates is extensively documented in the literature [145][146][147][148][149].…”

“…3, [77]), the pressure shock fronts become clearly visible when plotting the volumetric strain rate tr(ε) ( Fig. 11(c)), while the shear shock fronts are prominent in the shear strain rateε 12 [76]. At the crack tip where |ε| is the highest, the effective wave speeds are higher compared with the undeformed solid in the bulk, and the rupture speed can approach this higher limit.…”

“…In the new phase of evolution of our "Laboratory Earthquake" setup, we have recently been able to quantify the full-filed behavior of dynamic ruptures using digital image correlation combined with ultrahigh-speed photography [75][76][77][78]. The setup (Fig.…”

“…12(a) Formation of pressure shock fronts in Homalite-100, a high-strain rate sensitive material: (a) particle velocity in the fault-parallel directionu 1 , (b) strain rate magnitude |ε|, and (c) volumetric strain rate tr(ε) (figure produced using the data from Ref. [76]). The sequence shows that the pressure shock front fully developing as the strain rate magnitude increases.…”