# Steady-state propagation of a mode II crack in couple stress elasticity

- 330 Downloads
- 19 Citations

## Abstract

The present work deals with the problem of a semi-infinite crack steadily propagating in an elastic body subject to plane-strain shear loading. It is assumed that the mechanical response of the body is governed by the theory of couple-stress elasticity including also micro-rotational inertial effects. This theory introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying microstructure and has proved to be very effective for modeling complex microstructured materials. It is assumed that the crack propagates at a constant sub-Rayleigh speed. An exact full field solution is then obtained based on integral transforms and the Wiener–Hopf technique. Numerical results are presented illustrating the dependence of the stress intensity factor and the energy release rate upon the propagation velocity and the characteristic material lengths in couple-stress elasticity. The present analysis confirms and extends previous results within the context of couple-stress elasticity concerning stationary cracks by including inertial and micro-inertial effects.

## Keywords

Dynamic fracture Couple-stress elasticity Microstructure Mode-II crack Micro-rotational inertia Complex materials## Notes

### Acknowledgments

Panos A. Gourgiotis gratefully acknowledges support from the European Union FP7 project “Modelling and optimal design of ceramic structures with defects and imperfect interfaces” under contract number PIAP-GA-2011-286110-INTERCER2. Andrea Piccolroaz would like to acknowledge the Italian Ministry of Education, University and Research (MIUR) for the Grant FIRB 2010 Future in Research “Structural mechanics models for renewable energy applications” (RBFR107AKG).

## References

- Achenbach JD (1973) Wave propagation in elastic solids. North-Holland, AmsterdamGoogle Scholar
- Antipov YA (2012) Weight functions of a crack in a two-dimensional micropolar solid. Q J Mech Appl Math 65:239–271CrossRefGoogle Scholar
- Aravas N, Giannakopoulos AE (2009) Plane asymptotic crack-tip solutions in gradient elasticity. Int J Solids Struct 46:4478–4503CrossRefGoogle Scholar
- Atkinson C, Leppington FG (1974) Some calculations of the energy-release rate G for cracks in micropolar and couple-stress elastic media. Int J Frac 10:599–602CrossRefGoogle Scholar
- Atkinson C, Leppington FG (1977) The effect of couple stresses on the tip of a crack. Int J Solids Struct 13:1103–1122CrossRefGoogle Scholar
- Bigoni D, Drugan WJ (2007) Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. ASME J Appl Mech 74:741–753CrossRefGoogle Scholar
- Cauchy AL (1851) Note sur l’ equilibre et les mouvements vibratoires des corps solides. Comptes-Rendus Acad Paris 32:323–326Google Scholar
- Chang CS, Shi Q, Liao CL (2003) Elastic constants for granular materials modeled as first-order strain-gradient continua. Int J Solids Struct 40:5565–5582CrossRefGoogle Scholar
- Chen JY, Huang Y, Ortiz M (1998) Fracture analysis of cellular materials: a strain gradient model. J Mech Phys Solids 46:789–828CrossRefGoogle Scholar
- Cosserat E, Cosserat F (1909) Theorie des Corps Deformables. Hermann et Fils, ParisGoogle Scholar
- Dal Corso F, Willis JR (2011) Stability of strain gradient plastic materials. J Mech Phys Solids 59:1251–1267CrossRefGoogle Scholar
- Engelbrecht J, Berezovski A, Pastrone F, Braun M (2005) Waves in microstructured materials and dispersion. Philos Mag 85:4127–4141CrossRefGoogle Scholar
- Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42:475–487CrossRefGoogle Scholar
- Fleck NA, Hutchinson JW (1997) Strain gradient plasticity. In: Hutchinson JW, Wu TY (eds) Advances in applied mechanics, vol 33. Academic Press, New York, pp 295–361Google Scholar
- Fisher B (1971) The product of distributions. Q J Math 22:291–298CrossRefGoogle Scholar
- Freund LB (1972) Energy flux into the tip of an extending crack in an elastic solid. J Elast 2:341–349CrossRefGoogle Scholar
- Freund LB (1990) Dynamic fracture mechanics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity—I. Theory. J Mech Phys Solids 47:1239–1263CrossRefGoogle Scholar
- Georgiadis HG (2003) The mode-III crack problem in microstructured solids governed by dipolar gradient elasticity: static and dynamic analysis. ASME J Appl Mech 70:517– 530Google Scholar
- Georgiadis HG, Velgaki EG (2003) High-frequency Rayleigh waves in materials with microstructure and couple-stress effects. Int J Solids Struct 40:2501–2520CrossRefGoogle Scholar
- Georgiadis HG, Vardoulakis I, Velgaki EG (2004) Dispersive Rayleigh-wave propagation in microstructured solids characterized by dipolar gradient elasticity. J Elast 74:17–45CrossRefGoogle Scholar
- Gourgiotis PA, Georgiadis HG (2007) Distributed dislocation approach for cracks in couple-stress elasticity: shear modes. Int J Fract 147:83–102CrossRefGoogle Scholar
- Gourgiotis PA, Sifnaiou MD, Georgiadis HG (2010) The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity. Int J Fract 166:179–201CrossRefGoogle Scholar
- Gourgiotis PA, Georgiadis HG (2011) The problem of sharp notch in couple-stress elasticity. Int J Solids Struct 48:2630–2641CrossRefGoogle Scholar
- Gourgiotis PA, Georgiadis HG, Sifnaiou MD (2012) Couple-stress effects for the problem of a crack under concentrated shear loading. Math Mech Solids 17:433–459CrossRefGoogle Scholar
- Gourgiotis PA, Georgiadis HG, Neocleous I (2013) On the reflection of waves in half-spaces of microstructured materials governed by dipolar gradient elasticity. Wave Motion 50:437– 455Google Scholar
- Graff KF, Pao YH (1967) The effects of couple-stresses on the propagation and reflection of plane waves in an elastic half-space. J Sound Vib 6:217–229CrossRefGoogle Scholar
- Grentzelou CG, Georgiadis HG (2008) Balance laws and energy release rates for cracks in dipolar gradient elasticity. Int J Solids Struct 45:551–567CrossRefGoogle Scholar
- Han SY, Narasimhan MNL, Kennedy TC (1990) Dynamic propagation of a finite crack in a micropolar elastic solid. Acta Mech 85:179–191CrossRefGoogle Scholar
- Huang Y, Zhang L, Guo TF, Hwang KC (1997) Mixed mode near tip fields for cracks in materials with strain-gradient effects. J Mech Phys Solids 45:439–465CrossRefGoogle Scholar
- Huang Y, Chen JY, Guo TF, Zhang L, Hwang KC (1999) Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects. Int J Fract 100:1–27CrossRefGoogle Scholar
- Hwang KC, Jiang H, Huang Y, Gao H, Hu N (2002) A finite deformation theory of strain gradient plasticity. J Mech Phys Solids 50:81–99CrossRefGoogle Scholar
- Itou S (1972) The effect of couple-stresses dynamic stress concentration around a crack. Int J Eng Sci 10:393–400CrossRefGoogle Scholar
- Itou S (1981) The effect of couple-stresses on the stress concentration around a moving crack. Int J Math Math Sci 4:165–180CrossRefGoogle Scholar
- Itou S (2013) Effect of couple-stresses on the mode I dynamic stress intensity factors for two equal collinear cracks in an infinite elastic medium during passage of time-harmonic stress waves. Int J Solids Struct 50:1597–1604CrossRefGoogle Scholar
- Koiter WT (1964) Couple-stresses in the theory of elasticity. Parts I and II. Proc Ned Akad Wet B67:17–44Google Scholar
- Kulakhmetova SA, Saraikin VA, Slepyan LI (1984) Plane problem of a crack in a lattice. Mech Solids 19:101–108Google Scholar
- Lakes RS (1983) Size effects and micromechanics of a porous solid. J Mater Sci 18:2572–258CrossRefGoogle Scholar
- Lakes RS (1993) Strongly Cosserat elastic lattice and foam materials for enhanced toughness. Cell Polym 12:17–30Google Scholar
- Livne A, Bouchbinder E, Svetlizky I, Fineberg J (2010) The near-tip fields of fast cracks. Science 327:1359–1263CrossRefGoogle Scholar
- Lubarda VA, Markenskoff X (2000) Conservation integrals in couple stress elasticity. J Mech Phys Solids 48:553–564CrossRefGoogle Scholar
- Maranganti R, Sharma P (2007a) Length scales at which classical elasticity breaks down for various materials. Phys Rev Lett 98(195504):1–4Google Scholar
- Maranganti R, Sharma P (2007b) A novel atomistic approach to determine strain-gradient elasticity constants: tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies. J Mech Phys Solids 55:1823–1852CrossRefGoogle Scholar
- Maugin GA (2010) Mechanics of generalized continua: What do we mean by that?. One hundred years after the Cosserats. In: Maugin GA, Metrikine AV (eds) Mechanics of generalized continua. Springer, New York, pp 3–13CrossRefGoogle Scholar
- Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448CrossRefGoogle Scholar
- Mishuris G, Piccolroaz A, Radi E (2012) Steady-state propagation of a mode III crack in couple stress elastic materials. Int J Eng Sci 61:112–128CrossRefGoogle Scholar
- Mora R, Waas AM (2000) Mesurement of the Cosserat constant of circular-cell polycarbonate honeycomb. Philos Mag A 80:1699–1713CrossRefGoogle Scholar
- Muki R, Sternberg E (1965) The influence of couple-stresses on singular stress concentrations in elastic solids. ZAMP 16:611–618CrossRefGoogle Scholar
- Nieves MJ, Movchan AB, Jones IS, Mishuris GS (2013) Propagation of Slepyan’s crack in a non-uniform elastic lattice. J Mech Phys Solids. doi: 10.1016/j.jmps.2012.12.006
- Noble B (1958) Methods based on the Wiener-Hopf technique. Pergamon Press, OxfordGoogle Scholar
- Nowacki W (1986) Theory of asymmetric elasticity. Pergamon Press, OxfordGoogle Scholar
- Ostoja-Starzewski M, Jasiuk I (1995) Stress invariance in planar Cosserat elasticity. Proc R Soc Lond A 451:453–470CrossRefGoogle Scholar
- Piccolroaz A, Mishuris G, Radi E (2012) Mode III interfacial crack in the presence of couple stress elastic materials. Eng Fract Mech 80:60–71CrossRefGoogle Scholar
- Polyzos D, Fotiadis DI (2012) Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models. Int J Solids Struct 49:470–480CrossRefGoogle Scholar
- Radi E, Gei M (2004) Mode III crack growth in linear hardening materials with strain gradient effects. Int J Fract 130:765–785CrossRefGoogle Scholar
- 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 45:3033–3058CrossRefGoogle Scholar
- Ravi-Chandar K (2004) Dynamic fracture. Elsevier, AmsterdamGoogle Scholar
- Rice JR (1968a) A path independent integral and the approximate analysis of strain concentration by notches and cracks. ASME J Appl Mech 35:379–386CrossRefGoogle Scholar
- Rice JR (1968b) Mathematical analysis in the mechanics of fracture. In: H Liebowitz (eds) Fracture. Academic Press New York, 2:191–311Google Scholar
- Roos BW (1969) Analytic functions and distributions in physics and engineering. Wiley, New YorkGoogle Scholar
- Rosakis AJ, Samudrala O, Coker D (1999) Cracks faster than the shear wave speed. Science 284:1337–1340CrossRefGoogle Scholar
- Sciarra G, Vidoli S (2012a) The role of edge forces in conservation laws and energy release rates of strain-gradient solids. Math Mech Solids 17:266–278CrossRefGoogle Scholar
- Sciarra G, Vidoli S (2012b) Asymptotic fracture modes in strain-gradient elasticity: size effects and characteristic lengths for isotropic materials. J Elast doi: 10.1007/s10659-012-9409-y
- Sternberg E (1960) On the integration of the equations of motion in the classical theory of elasticity. Arch Ration Mech Anal 6:34–50CrossRefGoogle Scholar
- Sternberg E, Muki R (1967) The effect of couple-stresses on the stress concentration around a crack. Int J Solids Struct 3:69–95CrossRefGoogle Scholar
- Toupin RA (1962) Perfectly elastic materials with couple stresses. Arch Ration Mech Anal 11:385–414CrossRefGoogle Scholar
- Vardoulakis I, Georgiadis HG (1997) SH surface waves in a homogeneous gradient elastic half-space with surface energy. J Elast 47:147–165CrossRefGoogle Scholar
- Voigt W (1887) Theoretische Studien uber die Elasticitatsverhaltnisse der Krystalle. Abh Ges Wiss Gottingen 34:3–100Google Scholar
- Willis JR (1971) Fracture mechanics of interfacial cracks. J Mech Phys Solids 19:353–368CrossRefGoogle Scholar