Diffraction of antiplane shear waves and stress concentration in a cracked couple stress elastic material with micro inertia

Abstract

We investigate diffraction of reduced traction shear waves applied at the faces of a stationary crack in an elastic solid with microstructure, under antiplane deformation. The material behaviour is described by the indeterminate theory of couple stress elasticity and the crack is rectilinear and semi-infinite. The full-field solution of the crack problem is obtained through integral transforms and the Wiener-Hopf technique. A remarkable wave pattern appears which consists of entrained waves extending away from the crack, reflected Rayleigh waves moving along the crack, localized waves irradiating from the cracktip with, possibly, super-Rayleigh speed and body waves scattered around the crack-tip. Interestingly, the localized wave solution may be greatly advantageous for defect detection through acoustic emission. Dynamic stress intensity factors are presented, which generalize to Elastodynamics the corresponding results already obtained in the static framework. The correction brings out the important role of wave diffraction on stress concentration.

A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materials

“…Upon approaching the grazing condition, this displacement field may be expanded in terms of the emergence angle to yield precisely the Rayleigh-like wave expressed by the purely imaginary zero of the Rayleigh function. It is exactly this wave that is found in [6] radiating from the tip of a semi-infinite crack under dynamic loadings. It is pointed out that no Rayleigh-like wave is supported in CE, for no evanescent bulk mode exists.…”

“…having let w(ξ 2 ) = e 1 exp (λ 1 ξ 2 ) + e 2 exp (λ 2 ξ 2 ) , (3.33) provided that branch cuts in the square root are taken as to give positive real part on the real axis, see [6]. Plugging this form into the boundary conditions (3.13) and demanding for non-trivial solutions to exist, yields the Rayleigh function…”

“…We also point out that, at variance with [34], for a half-plane we are not free to choose the sign in front of square roots λ 1,2 , that is univocally determined by the choice of the branch cuts. Such choice is determined by Sommerfeld's condition and by the boundedness requirement at infinity, as detailed in [36] and in [6].…”

On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory

“…An excellent historical overview of GCM, together with extensive bibliographic details, may be found in [17]. Among GCM theories, we mention the theory of micro-polar elasticity [2,3,25], the couple-stress and strain-gradient elasticity theories [35,23] and the nonlocal theory of elasticity [7]. In particular, following [7], "linear theory of nonlocal elasticity, which has been proposed independently by various authors [...], incorporates important features of lattice dynamics and yet it contains classical elasticity in the long wave length limit".…”

Diffraction and Reflection of Antiplane Shear Waves in a Cracked Couple Stress Elastic Material

“…The solution differs significantly from the classical result given in [2] for isotropic elastic materials. Indeed, unlike classical elasticity, antiplane shear Rayleigh waves are supported in couple stress materials [4]. A complicated wave pattern appears which consists of entrained waves extending away from the crack, reflected Rayleigh waves, localized waves irradiating from and body waves scattered by the crack-tip.…”