PSI - Issue 77

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

185

3

a

b

y

r

x

Speed c

Energy source A

Speed c

P wave speed c P S wave speed c S

Energy source A

Fig. 1. An energy source (concentrated Dirac pressure pulse of intensity A ) moving along a (a) straight or (b) curved circular (radius r = 2.5) loose plane of weakness (interface with separation). Each energy source travels at a constant speed c in a two-dimensional linear elastic solid (longitudinal (P) wave speed c P and shear (S) wave speed c S ). with the concave side, the induced (this time, R-type) waves on the convex side of the curved interface are much stronger but the influential depths are shallower. Third, in the transonic (supershear) case where the speed c is smaller than the P wave speed but larger than the S wave speed, i.e. c S < c < c P or M P < 1 < M S , the induced steady-state stresses for the right half of Fig. 1(a) are given by

⎩⎪⎪ ⎨ ⎪ ⎪⎧ = ( ) �� 2 − 2 � 2 4 + � 2 − 2 � 2 2 + 2 2 +4 ( , , ) � , = − ( ) �� 2 − 2 2 + 2 �� 2 − 2 � 4 + � 2 − 2 � 2 = 2 ( ) � 2 − 2 �� � 2 − 2 � 2 − 4 2 2 + 2 2 + ( , , ) � ,

2 + 2 2 +4 ( , , ) � ,

(2)

where ( ) ≡ (2 − 2 ) 4 +16 β 2 2 , ( , , ) ≡ 4 πβ δ ( + ) − (2 − 2 ) 2 /( + ) , ≡ � 2 − 1 and δ ( ) is Dirac delta function. The Dirac delta function ( + ) included in ( , , ) in (2) indicates that Mach-type S wave fronts (S Mach fronts) exist but Mach-type P wave fronts do not in this case, as shown in Fig. 2(c). If the interface is straight, the Mach fronts are also straight (Fig. 2(c) left), and if curved, the Mach fronts are also curved (Fig. 2(c) right). The general tendency observed in the subsonic cases, more strongly confined wave energy on the convex side of the curved interface, are also observable here. Last, in the supersonic case where the speed c is larger than both P and S wave speeds, i.e. c S < c P < c or 1< M P < M S , the induced steady-state stresses 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 � [ ( + ) − ( + )] , (3) where ( ) ≡ (2 − 2 ) 2 +4 and ≡ � 2 − 1 . The Dirac delta function ( + ) as well as ( + ) in (3) indicates that, as depicted in Fig. 2(d), there exist both Mach-type P and S wave fronts (P and S Mach fronts) in the wave field. The P curved Mach front on the convex side looks stronger than its counterpart on the concave side of the curved interface.