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| Titel |
Transitions to Chaos in Dieterich-Ruina Friction |
| VerfasserIn |
B. Erickson, B. Birnir, D. Lavallée |
| Konferenz |
EGU General Assembly 2009
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 11 (2009) |
| Datensatznummer |
250019726
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| Zusammenfassung |
| We began investigations into the Dieterich-Ruina (D-R) friction law in previous work by
studying the behavior of a single slider-block under this law. We found transitions
to chaos in the numerical solution to this system when a specific parameter was
increased. This parameter, ε = is the ratio of the stress parameters (B - A)
and A in the D-R friction law. The parameter A = ∂τ∕∂log(v), where τ is the
frictional stress and v is the velocity of the slider, is a measure of the direct velocity
dependence (sometimes called the “direct effect") while (A - B) = ∂τss∕∂log(vss), is
a measure of the steady-state velocity (vss) dependence. When compared to the
slip weakening friction law, the parameter (B - A) plays a role of a stress drop
while A corresponds to the strength excess. We found that transitions to chaos for a
single block occur for ε ≈ 9. We also studied the behavior of a system of three
blocks under the D-R friction law, finding that transitions to chaos occurred for
smaller values of this parameter ε. Taking this study a step further, we derive the
elastic wave equation in 1-d under the Dieterich-Ruina friction law to obtain the
position u(x,t), the slip relative to the driver plate. We solve the resulting PDE
by first discretizing in space using the method of lines, and solving the system of
ODEs using an explicit 4th order Runge-Kutta numerical scheme. During the spatial
discretization, the PDE is reverted to a system of ODEs corresponding to a chain of spring
blocks where the spatial mesh determines the number of resulting blocks. Using
uniformly distributed initial conditions and periodic boundary conditions, we find
that numerical solutions to this PDE bifurcate under critical values of the same
parameter ε. We are interested in values of ε for which chaotic regimes result, as a
function of the spatial mesh, or number of blocks. For 100 blocks, we find that
transitions to chaotic solutions occur for ε ≈ 4.7; a smaller value than the value
observed for the chaotic regime for 1 block. Furthermore, our numerical solution
of the PDE suggests that for this range of parameter values, the chaotic behavior
occurs only in time; the spatial structure of the slip is preserved. We compute the
structure function, S1(x,t), to explore statistically stationary states of the solution. We
also compute the average of the spectra computed for many runs with different
initial conditions in order to study the distribution of frequency. Results from these
studies may have implications in the re-normalization of the Dieterich-Ruina friction
law. |
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