Dynamic instabilities of frictional sliding at a bimaterial interface
E. A. Brener, M. Weikamp, R. Spatschek, Y. Bar-Sinai, E. Bouchbinder,
Volume: 89. Pages: 149--173
DOI: 10.1016/j.jmps.2016.01.009
Published: 2016
Abstract
Abstract Understanding the dynamic stability of bodies in frictional
contact steadily sliding one over the other is of basic interest
in various disciplines such as physics, solid mechanics, materials
science and geophysics. Here we report on a two-dimensional linear
stability analysis of a deformable solid of a finite height H, steadily
sliding on top of a rigid solid within a generic rate-and-state friction
type constitutive framework, fully accounting for elastodynamic effects.
We derive the linear stability spectrum, quantifying the interplay
between stabilization related to the frictional constitutive law
and destabilization related both to the elastodynamic bi-material
coupling between normal stress variations and interfacial slip, and
to finite size effects. The stabilizing effects related to the frictional
constitutive law include velocity-strengthening friction (i.e. an
increase in frictional resistance with increasing slip velocity,
both instantaneous and under steady-state conditions) and a regularized
response to normal stress variations. We first consider the small
wave-number k limit and demonstrate that homogeneous sliding in this
case is universally unstable, independent of the details of the friction
law. This universal instability is mediated by propagating waveguide-like
modes, whose fastest growing mode is characterized by a wave-number
satisfying kH ∼ O ( 1 ) and by a growth rate that scales with H−1.
We then consider the limit kH → ∞ and derive the stability phase
diagram in this case. We show that the dominant instability mode
travels at nearly the dilatational wave-speed in the opposite direction
to the sliding direction. In a certain parameter range this instability
is manifested through unstable modes at all wave-numbers, yet the
frictional response is shown to be mathematically well-posed. Instability
modes which travel at nearly the shear wave-speed in the sliding
direction also exist in some range of physical parameters. Previous
results obtained in the quasi-static regime appear relevant only
within a narrow region of the parameter space. Finally, we show that
a finite-time regularized response to normal stress variations, within
the framework of generalized rate-and-state friction models, tends
to promote stability. The relevance of our results to the rupture
of bi-material interfaces is briefly discussed.