** Progress in Earth and Planetary Science is the official journal of the Japan Geoscience Union, published in collaboration with its society members.

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Review

Solid earth sciences

202401202401

### Reconsideration of the energy balance in earthquake faulting

#### Mitsuhiro Matsu'uraMitsuhiro Matsuâ€™ura

Theoretical seismology, Background stress field, Earthquake faulting, Moment tensor, Inelastic strain, Elastic potential energy, Radiated seismic energy, Work done for shear faulting, Fault constitutive law, Rupture growth rate

A schematic diagram showing the difference in rupture growth path between the dynamic and quasi-static cases. The thick gray curve τ = τ_{dy} (D) and white curve τ = τ _{qs} (D) represent the dynamic and quasi-static rupture growth paths, respectively. For reference, the slip-weakening curve τ = τ _{fr} (D) is also plotted in black. The symbols τ _{i}, τ _{p}, τ _{r}, and τ _{f} indicate the initial, peak, residual, and final values of the shear stress τ, respectively. The dynamic rupture growth path is characterized by the initial slip-weakening phase, the intermediate high-speed rupture propagation phase, and the final slip deceleration and transient adjustment phase. The area enclosed by the thick gray and white curves, , gives the total seismic energy escaping from the mechanical system concerned.

The occurrence of earthquakes is now understood as brittle shear fracture releasing the elastic potential energy stored in the earth. Since the 1950s, many studies on the energy balance in earthquake faulting have been done, but there seems to be some incoherence among them. The essential reason is because various changes in conceptual framework happened during the last six decades, specifically the introduction of the new paradigm of plate tectonics in the 1960s, the concept of moment tensor as source representation in the 1970s, and the fault constitutive law governing rupture growth in the 1990s. Therefore, it will be worthwhile to reconsider the energetics of earthquake faulting from a current perspective. For this purpose, first of all, we summarize the basic concepts of elastic potential energy and moment tensor and review the general representation of earthquake sources and the origin of background crustal stress to confirm that the effect of earth’s self-gravitation is negligible in the energetics of shear faulting. Next, as a starting point for discussion, we directly derive a basic equation of mechanical energy balance in dynamic shear faulting from the equation of motion for an elastic body subjected to tectonic-origin deviatoric stress. Then, we review the widely accepted formula for indirectly evaluating radiated seismic energy from a simplified energy balance equation and compare with the direct evaluation based on the analytical solution of displacement fields for a point dislocation source in order to call attention to inconsistency between them. The inconsistency comes from the omission of the effects of rupture growth rate in the simplified energy balance equation. So, finally, we review the energy balance at the tips of a propagating shear crack, which naturally leads to the introduction of the slip-weakening fault constitutive law as a fundamental equation governing earthquake rupture. Then, we discuss the whole process of earthquake rupture, consisting of initiation, acceleration, steady propagation, deceleration, and termination from the viewpoint of energy balance.