Time harmonic analysis of antiplane crack in couple stress elastic materials
Enrico Radi
1, Andrea Nobili
2, Gennady Mishuris
31Dipartimento di Scienze e Metodi dell'Ingegneria - Università di Modena e Reggio Emilia, Reggio Emilia,
Italy
2Dipartimento di Ingegneria "Enzo Ferrari" - Università di Modena e Reggio Emilia, Modena, Italy 3Institute of Mathematical and Physical Sciences, Aberystwyth University, Aberystwyth, UK
Email: enrico.radi@unimore.it
The time harmonic response of a rectilinear and semi-infinite crack in a couple stress (CS) elastic solid under Mode III loading conditions is investigated in the present work. The full-field solution of the dynamic crack problem obtained in [1] through Fourier integral transforms and the Wiener–Hopf technique is generalized here by considering more general loading conditions, consisting in arbitrary reduced stress and couple stress tractions applied at the crack faces.
The solution for quasistatic Mode III crack in indeterminate CS elastic materials was given in [2]. Later, the problem of steady-state Mode III crack propagation was investigated in [3]. In the present work, a travelling wave loading, applied in the form of generalized reduced tractions at the crack faces, is considered as the forcing term. As a result, a complex wave pattern appears, which differs significantly from the Mode III classical elastic solution. The results of the present analysis may be used as a building block to address, by means of superposition, the problem of arbitrary antiplane wave propagation in a cracked CS solid. Resonance is triggered when the applied loading is fed into the crack-tip at Rayleigh speed. Elastodynamic stress intensity factors are given, which generalize the corresponding results presented in [2] for the qusistatic framework. They incorporate the effect of the applied loading frequency and thereby account for the interplay of the diffracted waves.
A remarkable wave pattern appears which consists of entrained waves extending away from the crack, reflected Rayleigh waves moving along the crack surfaces, localized waves irradiating from the crack-tip and body waves scattered around the crack-tip (Fig. 1). Interestingly, the localized wave solution may be greatly advantageous for defect detection through acoustic emission.
Fig. 1. a) displacement wave entrained by the traction applied at the crack faces; b) scattered body waves; c) localized wave traveling along the y direction; d) total wave pattern.
References:
[1] Nobili A., Radi E., Vellender A. (2019) Diffraction of antiplane shear waves and stress concentration in a cracked couple stress elastic material with micro inertia. J. Mech. Phys. Solids, 124, 663-680.
[2] Radi E. (2008) On the effects of characteristic lengths in bending and torsion on mode III crack in couple stress elasticity. Int. J. Solids Struct., 45, 3033-3058
[3] Mishuris G., Piccolroaz A., Radi E. (2012). Steady-state propagation of a mode III crack in couple stress elastic materials. Int. J. Eng. Sci., 61 , 112-128.
