A cohesive zone model for dynamic shear faulting based on experimentally inferred constitutive relation and strong motion source parameters

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<jats:p>To understand constitutive behavior near the rupture front during an earthquake source shear failure along a preexisting fault in terms of physics, local breakdown processes near the propagating tip of the slipping zone under mode II crack growth condition have been investigated experimentally and theoretically. A physically reasonable constitutive relation between cohesive stress τ and slip displacement <jats:italic>D</jats:italic>, τ = (τ<jats:sub><jats:italic>i</jats:italic></jats:sub> − τ<jats:sub><jats:italic>d</jats:italic></jats:sub>)[1 + α log (1 + β<jats:italic>D</jats:italic>)] exp (−<jats:sub>η</jats:sub><jats:italic>D</jats:italic>) + τ<jats:sub><jats:italic>d</jats:italic></jats:sub>, is put forward to describe dynamic breakdown processes during earthquake source failure in quantitative terms. In the above equation, τ<jats:sub><jats:italic>i</jats:italic></jats:sub> is the initial shear stress on the verge of slip, τ<jats:italic><jats:sub>d</jats:sub></jats:italic> is the dynamic friction stress, and α, β, and η are constants. This relation is based on the constitutive features during slip failure instabilities revealed in the careful laboratory experiments. These experiments show that the shear stress first increases with ongoing slip during the dynamic breakdown process, the peak stress is attained at a very (usually negligibly) small but nonzero value of the slip displacement, and then the slip‐weakening instability proceeds. The model leads to bounded slip acceleration and stresses at and near the dynamically propagating tip of the slipping zone along the fault in an elastic continuum. The dynamic behavior near the propagating tip of the slipping zone calculated from the theoretical model agrees with those observed during slip failure along the preexisting fault much larger than the cohesive zone. The model predicts that the maximum slip acceleration <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jgrb6952-math-0001.gif" xlink:title="equation image" /> be related to the maximum slip velocity <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jgrb6952-math-0002.gif" xlink:title="equation image" /> and the critical displacement <jats:italic>D<jats:sub>c</jats:sub></jats:italic> by <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jgrb6952-math-0003.gif" xlink:title="equation image" /> , where <jats:italic>k</jats:italic> is a numerical parameter, taking a value ranging from 4.9 to 7.2 according to a value of τ<jats:sub><jats:italic>i</jats:italic></jats:sub>/τ<jats:sub><jats:italic>p</jats:italic></jats:sub> (τ<jats:sub><jats:italic>p</jats:italic></jats:sub> being the peak shear stress) in the present model. The model further predicts that <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jgrb6952-math-0004.gif" xlink:title="equation image" /> be expressed in terms of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jgrb6952-math-0005.gif" xlink:title="equation image" /> and the cut off frequency <jats:italic>f</jats:italic><jats:sub>max</jats:sub><jats:sup><jats:italic>s</jats:italic></jats:sup> of the power spectral density of the slip acceleration on the fault plane as <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jgrb6952-math-0006.gif" xlink:title="equation image" /> and that <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jgrb6952-math-0007.gif" xlink:title="equation image" /> in terms of <jats:italic>D<jats:sub>c</jats:sub></jats:italic> and <jats:italic>f</jats:italic><jats:sub>max</jats:sub><jats:sup><jats:italic>s</jats:italic></jats:sup> as <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/jgrb6952-math-0008.gif" xlink:title="equation image" /> . These theoretical relations agree well with the experimental observations and can explain interrelations between strong motion source parameters for earthquakes. The pulse width of slip acceleration on the fault plane is directly proportional to the time <jats:italic>T<jats:sub>c</jats:sub></jats:italic> required for the crack tip to break down, and <jats:italic>f</jats:italic><jats:sub>max</jats:sub><jats:sup><jats:italic>s</jats:italic></jats:sup> is inversely proportional to <jats:italic>T<jats:sub>c</jats:sub></jats:italic>.</jats:p>

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