PSI - Issue 77

Koji Uenishi et al. / Procedia Structural Integrity 77 (2026) 183–189 Uenishi / Structural Integrity Procedia 00 (2026) 000–000

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field and ensuing physical phenomena. In the isotropic, homogeneous linear elastic framework, since there exist two body waves, namely, longitudinal (P) and shear (S) waves, the fundamental mechanical characteristics of the induced phenomena largely depend on two Mach numbers related to the P wave speed c P and S wave speed c S , M P ≡ c / c P and M S ≡ c / c S . However, the wave field itself is symmetric regardless of the Mach numbers if the energy source such as a crack tip moves straightly. As shown analytically below in the next chapter, this symmetry remains unbroken even when the energy source moves along a straight plane of weakness (interface) between identical solids. In order to break the symmetry, normally, the interface is set between dissimilar solids (e.g. Uenishi et al. (1999)), but here, a curved interface is set between identical solids. It is shown that the induced wave field becomes asymmetric with respect to the curved interface, and the wave energy can be greatly confined to some specific regions on the convex side of the interface for a wide range of the Mach numbers. This contribution also plays a role of complementing the recent paper by Uenishi (2025) on waves generated by energy sources moving along curved surfaces, and possible effects of the asymmetric wave confining due to interface curvature are discussed based on actual observations during the Noto Peninsula, Japan, earthquake in 2024. 2. Waves induced by an energy source moving along a straight / curved plane of weakness Assume that an energy source (concentrated Dirac pressure pulse of intensity A ) moves with a constant speed c along a loose interface having finite separation in a two-dimensional, infinitely extending isotropic, homogeneous linear elastic solid. For the problem shown in Fig. 1(a), the steady-state analytical solution (see e.g., Cole and Huth (1958), Fung (1965), Georgiadis and Barber (1993), Rossmanith et al. (1997)) is considered for a straight interface while for the one indicated in Fig. 1(b), the dynamic transient behavior is numerically investigated where an energy source begins to move at time t = 0 from bottom along a circular loose interface. Finite difference techniques with the second order spatiotemporal accuracy are employed, and as mentioned in Uenishi (2025), linear elasticity enables us to set c P as 1 without losing generality. Orthogonal 201 × 201 grid points are employed, with the uniform grid spacing 0.05, time step 0.025 and the energy absorbing outer boundary conditions. The radius r of the circular interface is set at r = 2.5. The analytical and numerical observations are summarized as follows. First, in the subsonic case where the speed of the source is smaller than the P and S wave speeds, i.e. c < c S < c P or M P < M S < 1, the induced steady-state normal ( σ x , σ y ) and shear ( τ xy ) stresses (compression positive) for the right half of Fig. 1(a) are expressed as ⎩⎪⎨ ⎪⎧ = ( ) �� 2 − 2 � 2 − 4 � , = − ( ) �� 2 − 2 2 + 2 �� 2 − 2 � − 4 � , = − 2 ( ) � 2 − 2 �� − � , (1) where ( ) ≡ (2 − 2 ) 2 − 4 β β , θ ≡ + β , θ ≡ + β ( 0 ≤θ ≤π , 0 ≤θ ≤π ), β ≡ � 1 − 2 and β ≡� 1 − 2 . Related isochromatic fringe patterns or contours of the maximum in-plane shear stress τ max shown in Fig. 2(a) left indicate that the induced dynamic stress field looks akin to the static one without movement of the energy source. No Mach-type wave front (singular line) exists and waves are more widely spread, but of course, the wave field is symmetric with respect to the interface. For the curved interface shown in Fig. 2(a) right, as simply imagined, symmetry is broken due to the introduction of the geometrical asymmetry, and the induced waves on the convex side seem to be stronger than those on the concave side. The wave energy is more confined to the interface on the convex side, and the influential depths of the induced waves from the interface look shallower on the convex side than those on the concave side. Note that the relatively slowly moving energy source constantly generates body and Rayleigh (R) waves and some energy concentrated regions due to the faster and widely spread Rayleigh waves (see e.g. Jin et al. (2005), Zhang et al. (2020)) appear ahead of the energy source. Second, if the speed c is still in a subsonic regime but equal to the R wave speed c R , ( )=0 in the above equations (1) and the system becomes resonant. The case c ≈ c R is shown for both straight and curved interfaces in Fig. 2(b) where the wave fields look “sharper” than those in a purely subsonic regime (Fig. 2(a)). Again, compared