Homoclinic bifurcation of a rate-weakening patch in a viscoelastic medium and effect of strength contrast

Abstract The time-dependent viscoelastic deformation of host rocks is important when considering the dynamics of fault behavior, specifically in brittle–ductile transitional regions or shallow subduction zones, because it relaxes stress heterogeneity and affects loading to the fault. For a rate-and-state fault embedded in a Maxwell viscoelastic medium, a previous study discovered a transition from repeated earthquakes to the permanent stuck of a rate-weakening patch (EQ–ST transition) with decreased viscoelastic relaxation time $${t}_{\text{c}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>t</mml:mi> <mml:mtext>c</mml:mtext> </mml:msub> </mml:math> . This transition differs from the well-known seismic–aseismic transition explained by the Hopf bifurcation at the critical stiffness of an elastic system. To better understand the EQ–ST transition, quantifying the effect of heterogeneous frictional strength is important, because this effect is characteristic to the viscoelastic medium and is absent in the elastic limit. Previous experimental studies suggest a potential contrast in frictional strength $$\Delta {f}_{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mmultiscripts> <mml:mi>f</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:mrow> </mml:math> in such a way that a rate-weakening patch is stronger than a rate-strengthening region containing clay minerals. Here, we conducted two-dimensional, fully dynamic earthquake sequence simulations for a fault in a Maxwell viscoelastic medium; we investigated the EQ–ST transition in the two-dimensional parameter space of $${t}_{\text{c}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>t</mml:mi> <mml:mtext>c</mml:mtext> </mml:msub> </mml:math> and $$\Delta {f}_{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mmultiscripts> <mml:mi>f</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:mrow> </mml:math> . With $$\Delta {f}_{*}=0.3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mmultiscripts> <mml:mi>f</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> <mml:mo>=</mml:mo> <mml:mn>0.3</mml:mn> </mml:mrow> </mml:math> , the EQ–ST transition occurred at about 1 order of magnitude larger $${t}_{\text{c}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>t</mml:mi> <mml:mtext>c</mml:mtext> </mml:msub> </mml:math> than in the case with $$\Delta {f}_{*}=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mmultiscripts> <mml:mi>f</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We constructed a coarse-grained model with only two degrees of freedom based on the spatial average. Consequently, the coarse-grained model behaves remarkably similar to the continuum model, and the EQ–ST transition is associated with a homoclinic bifurcation. The EQ–ST boundary in the parameter space can be quantitatively explained by considering elastic loading due to creep in the rate-strengthening region and unloading by viscoelastic relaxation of the stress heterogeneity comparable to $$\Delta {f}_{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mmultiscripts> <mml:mi>f</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:mrow> </mml:math> . A larger rate-weakening patch is anticipated to become aseismic earlier as $${t}_{\text{c}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>t</mml:mi> <mml:mtext>c</mml:mtext> </mml:msub> </mml:math> decreases, because the elastic loading rate is inversely correlated with the patch size. This may be qualitatively consistent with the change in the size distribution of events around the brittle–ductile transition in observations and laboratory experiments; however, further investigations on, for example, the interactions of events and changes in frictional parameters with depth are required for quantitative discussion. Graphical abstract