The onset of the frictional motion of dissimilar materials

Hadar Shlomai; David S. Kammer; Mokhtar Adda-Bedia; Jay Fineberg

doi:10.1073/pnas.1916869117

New Research In

Edited by Alain Karma, Northeastern University, Boston, MA, and accepted by Editorial Board Member Herbert Levine March 27, 2020 (received for review September 27, 2019)

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Fig. 1.
Slow ruptures are bimaterial cracks. (A, Upper) Experimental system with PMMA (pink) and PC (blue) blocks. Strain gauges on block faces (z = 0 and z = 6 mm) are mounted at y locations ∼3.5 and ∼7 mm above and below the interfaces (in green); on one face at 12 locations. An additional three gauges are mounted at y = ∼7-mm height on opposing faces at the same x locations in order to verify that strain measurements are z independent (see SI Appendix). F_N and F_S are normal and shear loads applied via load cells. (A, Lower) Instantaneous real contact area, A(x,z,t), measurements are performed along the entire interface by total internal reflection of an incident sheet of light everywhere except at contacting points (3). The transmitted light is roughly proportional to A(x,z,t). (B) Measured strain field compared to theory and simulation. The spatial structure of Δεijx−xtip at height h = 7 mm inside the stiff (Left) and soft (Right) materials; x_tip is the rupture tip location. Strain variations, Δεij, are obtained by subtracting initial/residual values. Δεijx−xtip for both materials are normalized as Δε∼ij=Δεijx−xtip/Δεxx0,stiff where Δεxx0,stiff is the peak of the stiffer material ahead of x_tip (gray arrow). Colors: measurements for ruptures in the range C_f = 70 to 600 ms⁻¹ = 0.08 to 0.6CSsoft. Corresponding Δε∼ij(x−xtip) predicted by bimaterial LEFM (black) and numerical simulations (dashed orange) for frictionless crack-like ruptures under plane stress conditions at C_f = 0.38CSsoft compare well with measurements. (B, Right Insets) Angular function measurements Eij(Cf,θ)=Δε∼ij(r,θ)⋅r, for heights h = 3.5 and h = 7 mm within the soft material at C_f = 160 ms⁻¹ = 0.18CSsoft collapse to the predicted Eij(Cf,θ) for bimaterial LEFM cracks (black line). (C) Space–time evolution A(x,z,t) of slow ruptures propagating at Cf=0.6 CSsoft along the positive direction. (Left) A(x,t) = < A(x,z,t)>. (Right) Corresponding A(x,z,t) at times, t, denoted by the colored bars at the left. A(x,z,t) was normalized at t = 0.
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Fig. 2.
Slip-pulse evolution. (A) A(x,z, t) of a typical slip pulse propagating at Cf=1.024 CSsoft in the positive direction along a bimaterial interface. (Left) A(x,t) = <A(x,z,t)>. (Right) Corresponding A(x,z,t) to times, ti, denoted by the colored bars at left. A(x,z, t) was normalized at t = 0. The blue arrow denotes the large drop in A(x,t) at the slip-pulse tip that is characteristic of a slip pulse. (B) Slip pulses are transonic (>CSsoft) and typically evolve with the propagation distance. Δεxx of the slip pulse in A at increasing spatial positions. Note that, while the slip-pulse amplitude increases, its width remains fixed to approximately the width of the A(x,t) reduction in A. Maximal values, Δεxxmax and (Inset) Δεyymax, are defined by red arrows. (C) Strain amplitudes for slip pulses (purple) are up to an order of magnitude larger than those of (blue) bimaterial cracks (<CSsoft) despite the narrow range CSsoft<Cf<1.04 CSsoft of slip-pulse velocities compared with the large range for cracks speeds 0<Cf<CSsoft. CSsoft is denoted by the dashed red line. (Inset) Maximal strain amplitudes Δεyymax as a function of Δεxxmax. All strain units are milli-strain.
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Fig. 3.
Slip pulses are bimaterial transonic cracks localized by friction. (A) Measured Δσij(x−xtip) at height h = 7 mm in stiff (Left) and soft (Right) blocks for a slip pulse at Cf=1.035CSsoft under applied normal loading conditions of σyy=3.1 MPa. Stress fields (green) beyond the leading edge of transonic slip pulses compare well with transonic crack solutions described by (black line) the analytic solution (see SI Appendix for details) with q=0.17. Signals are normalized as in Fig. 1C. (B) The exponent q of Eq. 1 for CSsoft<Cf<1.04 CSsoft. Material properties correspond to experiments. (Inset) q over the range 1<Cf/CSsoft<γ. (C) Including frictional coupling causes slip-pulse localization. (Green) Measurements of Δσxxsoft(x−xtip) are compared with frictionless transonic crack solutions (dotted blue line) and a numerical solution at Cf/CSsoft=1.006 that incorporates Coulomb friction (red line) with a kinetic friction coefficient of 0.64. Signals, σ∼xxsoft, are scaled to have the same amplitude. (D) Incorporating friction leads to interface separation at the interface. Numerical frictionless cracks at Cf/CSsoft=1.015 (blue line) are compared with the numerical solution incorporating friction (red line). The solutions were not rescaled. (Upper) σyy at the interface (y=0) for both solutions. Including friction induces σyy(x,0)=0, indicating separation. The interface separation coincides with slip-pulse amplitudes significantly larger than those of the transonic cracks (Lower) providing an effective localization mechanism.
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Fig. 4.
The limiting velocity Clim for transonic rupture is determined by the velocity where σyy at the interface behind xtip becomes compressive. (A) σyy(x,0) profiles of the analytical solution at different transonic velocities. For all velocities, σyy(x>xtip,0) is compressive (<0). Behind xtip, σyy(x<xtip,0) changes its sign from reduction of its background value to increased compression (σyy<0). (B) A-Cf (Eq. 1) describes the magnitude of σyy behind xtip. A−Cf changes its sign at a velocity of Cf=1.041CSsoft. (C) Probability distribution of all measured ruptures in the positive direction below CSstiff. Clim, which is denoted by the purple arrow, corresponds to the highest measured rupture velocity in the positive direction. The red line denotes CSsoft. (Inset) An expanded view of the measured distribution of Cf in the transonic region.