Research Papers

On Averaging Interface Response During Dynamic Rupture and Energy Partitioning Diagrams for Earthquakes

[+] Author and Article Information
Hiroyuki Noda1

Institute for Research on Earth Evolution,Japan Agency for Marine-Earth Science and Technology, Yokohama, 236-0001, Japanhnoda@jamstec.go.jp

Nadia Lapusta

Professor of Mechanical Engineering and Geophysics, Division of Engineering and Applied Science, Division of Geological and Planetary Sciences,  California Institute of Technology, Pasadena, CA, 91125lapusta@its.caltech.edu

Described in detail at http://scecdata.usc.edu/cvws/.


Corresponding author.

J. Appl. Mech 79(3), 031026 (Apr 12, 2012) (12 pages) doi:10.1115/1.4005964 History: Received December 23, 2011; Revised January 22, 2012; Posted February 13, 2012; Published April 12, 2012; Online April 12, 2012

Earthquakes occur as dynamic shear cracks and convert part of the elastic strain energy into radiated and dissipated energy. Local evolution of shear strength that governs this process, which is variable in space and time, can be studied from laboratory experiments and rupture models. At the same time, increasingly accurate measurements of radiated energy and other quantities characterize earthquakes in a rupture-averaged way. Here, we present and study two approaches to averaging frictional dissipation during dynamic rupture. The first one is based on the actual progression of dissipation, but the associated averaged shear stress does not reflect the local friction behavior. The second one is constructed to preserve prevailing features of local stress-slip response and performs well in the examples studied. The developed approach should be useful for visualizing energy partitioning in dynamic models and linking them to observations using diagrams that reflect dominant features of local stress evolution.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

(a) Schematic conceptual representation of energy partitioning per unit area during dynamic rupture, in which the change in strain energy ΔW goes into radiated energy ER and frictional dissipation D. The issue is how to construct such a diagram that, at the same time, retains some information about the assumed local frictional behavior. (b) Schematics of a single-degree-of-freedom spring-slider system for which such diagram can be trivially constructed. In the system, the mass is ignored; the radiation is represented by a term proportional to slip rate of the block and illustrated on the diagram by a dashpot.

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Figure 2

An example of dynamic rupture on a planar interface embedded into a 3D elastic space. Top left: The rectangular rupture domain is prestressed to slip mostly in the x1 direction, with an overstressed patch to initiate rupture. Top right: The assumed linear slip-weakening behavior on the interface that acts as a boundary condition to the elastodynamic equations in the bulk. Bottom: Slip accumulation every 0.5 s during dynamic rupture for two cross-sections of the rupture domain.

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Figure 3

Schematics of the initial and final states of a dynamic rupture process on a planar interface embedded into a 3D linear elastic space

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Figure 4

Illustration of the virtual work rate and the strain energy change for the case of Fig. 2 The exact virtual work rate calculated using Eq. 8 is plotted as the black solid line. The approximate rate from Eq. 12, which assumes the same direction of final slip for all ruptured points (gray dashed line), is indistinguishable from the exact one on this scale, for reasons discussed in the text.

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Figure 5

(a)-(b) Average dissipative shear stress τDt on the fault from Eq. 17 for the rupture case of Fig. 2, plotted as a function of time in (a) and average slip in (b). Note that τDt is quite different from the linear slip-weakening friction behavior experienced by each point on the fault (dashed line in (b)). (c)–(d) Rupture area-averaged shear stress τ¯1 plotted against time in (c) and slip in (d). This relatively simple area-averaging of shear stress is different from both the dissipative stress τDt and the assigned local friction behavior. τ¯1 does not integrate to the total dissipation, and hence it cannot be used in the energy partitioning diagrams.

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Figure 6

Illustration of steps in constructing average dissipation stress based on local friction behavior, using the case of Fig. 2 as an example. (Top row) Local scalar dependence of dissipative stress on slip. (Bottom row) The local dependence is stretched/compressed along the slip axis to adjust the local final slip values to the overall fault-averaged final slip δ¯fin, and then compressed/stretched along the stress axis to keep the same area (A1 to A4 ) under the curves and hence the same dissipation density.

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Figure 7

Average dissipation stress based on local friction behavior, τDδ(δ'1). This function of average stress versus slip can be used in the energy partitioning diagram of the type Fig. 1.

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Figure 8

Energy partitioning diagrams for the dynamic rupture of (Fig. 2) with (a) τDt and (b) τDδ . ΔW/A and D/A are given by the striped and gray areas, respectively. ER /A is the difference between them.

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Figure 9

(a) A dynamic process in which a singular Mode III crack propagates with a constant rupture speed VR and then abruptly arrests. (b) Energy partitioning diagram for this example.

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Figure 10

(a) Spatio-temporal distribution of slip in a pulse-like rupture scenario. The rupture occurs in a rectangular domain as illustrated in Fig. 2. The left part of the panel shows the rupture propagation in the Mode III direction and the right part of the panel shows the rupture propagation in the Mode II direction. (b) Local evolution of shear stress with slip at several points on the fault. (c)–(d) Fault-averaged rupture behavior using τDt in (c) and τ in (d). Note that the dissipation rate curve τDδ reproduces the main features of the local behavior.

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Figure 11

Example of dynamic rupture obtained in a long-term simulation of fault slip. (a) Fault geometry: A seismogenic region is embedded into a stable domain, with far-field slip with 10−9 m/s. (b) Accumulation of slip along the mid-depth of the fault (axis x3 ). Slow slip is shown by dashed lines plotted every 10 years and occasional fast dynamic rupture events are illustrated by solid lines plotted every 1 s for periods of high enough slip rate. (c) Shear stress versus slip behavior of three points along the mid-depth of the fault. The middle panel gives the behavior typical of most rupture points. (d)–(e) Rupture-averaged behavior for the event indicated by filling in (b).



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