International Journal of Fracture

, Volume 188, Issue 2, pp 119–145 | Cite as

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

Original Paper

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 

References

  1. Achenbach JD (1973) Wave propagation in elastic solids. North-Holland, AmsterdamGoogle Scholar
  2. Antipov YA (2012) Weight functions of a crack in a two-dimensional micropolar solid. Q J Mech Appl Math 65:239–271CrossRefGoogle Scholar
  3. Aravas N, Giannakopoulos AE (2009) Plane asymptotic crack-tip solutions in gradient elasticity. Int J Solids Struct 46:4478–4503CrossRefGoogle Scholar
  4. 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
  5. Atkinson C, Leppington FG (1977) The effect of couple stresses on the tip of a crack. Int J Solids Struct 13:1103–1122CrossRefGoogle Scholar
  6. Bigoni D, Drugan WJ (2007) Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. ASME J Appl Mech 74:741–753CrossRefGoogle Scholar
  7. Cauchy AL (1851) Note sur l’ equilibre et les mouvements vibratoires des corps solides. Comptes-Rendus Acad Paris 32:323–326Google Scholar
  8. 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
  9. Chen JY, Huang Y, Ortiz M (1998) Fracture analysis of cellular materials: a strain gradient model. J Mech Phys Solids 46:789–828CrossRefGoogle Scholar
  10. Cosserat E, Cosserat F (1909) Theorie des Corps Deformables. Hermann et Fils, ParisGoogle Scholar
  11. Dal Corso F, Willis JR (2011) Stability of strain gradient plastic materials. J Mech Phys Solids 59:1251–1267CrossRefGoogle Scholar
  12. Engelbrecht J, Berezovski A, Pastrone F, Braun M (2005) Waves in microstructured materials and dispersion. Philos Mag 85:4127–4141CrossRefGoogle Scholar
  13. Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42:475–487CrossRefGoogle Scholar
  14. 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
  15. Fisher B (1971) The product of distributions. Q J Math 22:291–298CrossRefGoogle Scholar
  16. Freund LB (1972) Energy flux into the tip of an extending crack in an elastic solid. J Elast 2:341–349CrossRefGoogle Scholar
  17. Freund LB (1990) Dynamic fracture mechanics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  18. Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity—I. Theory. J Mech Phys Solids 47:1239–1263CrossRefGoogle Scholar
  19. 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
  20. 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
  21. 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
  22. Gourgiotis PA, Georgiadis HG (2007) Distributed dislocation approach for cracks in couple-stress elasticity: shear modes. Int J Fract 147:83–102CrossRefGoogle Scholar
  23. 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
  24. Gourgiotis PA, Georgiadis HG (2011) The problem of sharp notch in couple-stress elasticity. Int J Solids Struct 48:2630–2641CrossRefGoogle Scholar
  25. 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
  26. 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
  27. 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
  28. 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
  29. Han SY, Narasimhan MNL, Kennedy TC (1990) Dynamic propagation of a finite crack in a micropolar elastic solid. Acta Mech 85:179–191CrossRefGoogle Scholar
  30. 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
  31. 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
  32. 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
  33. Itou S (1972) The effect of couple-stresses dynamic stress concentration around a crack. Int J Eng Sci 10:393–400CrossRefGoogle Scholar
  34. 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
  35. 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
  36. Koiter WT (1964) Couple-stresses in the theory of elasticity. Parts I and II. Proc Ned Akad Wet B67:17–44Google Scholar
  37. Kulakhmetova SA, Saraikin VA, Slepyan LI (1984) Plane problem of a crack in a lattice. Mech Solids 19:101–108Google Scholar
  38. Lakes RS (1983) Size effects and micromechanics of a porous solid. J Mater Sci 18:2572–258CrossRefGoogle Scholar
  39. Lakes RS (1993) Strongly Cosserat elastic lattice and foam materials for enhanced toughness. Cell Polym 12:17–30Google Scholar
  40. Livne A, Bouchbinder E, Svetlizky I, Fineberg J (2010) The near-tip fields of fast cracks. Science 327:1359–1263CrossRefGoogle Scholar
  41. Lubarda VA, Markenskoff X (2000) Conservation integrals in couple stress elasticity. J Mech Phys Solids 48:553–564CrossRefGoogle Scholar
  42. Maranganti R, Sharma P (2007a) Length scales at which classical elasticity breaks down for various materials. Phys Rev Lett 98(195504):1–4Google Scholar
  43. 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
  44. 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
  45. Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448CrossRefGoogle Scholar
  46. 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
  47. Mora R, Waas AM (2000) Mesurement of the Cosserat constant of circular-cell polycarbonate honeycomb. Philos Mag A 80:1699–1713CrossRefGoogle Scholar
  48. Muki R, Sternberg E (1965) The influence of couple-stresses on singular stress concentrations in elastic solids. ZAMP 16:611–618CrossRefGoogle Scholar
  49. 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
  50. Noble B (1958) Methods based on the Wiener-Hopf technique. Pergamon Press, OxfordGoogle Scholar
  51. Nowacki W (1986) Theory of asymmetric elasticity. Pergamon Press, OxfordGoogle Scholar
  52. Ostoja-Starzewski M, Jasiuk I (1995) Stress invariance in planar Cosserat elasticity. Proc R Soc Lond A 451:453–470CrossRefGoogle Scholar
  53. 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
  54. 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
  55. Radi E, Gei M (2004) Mode III crack growth in linear hardening materials with strain gradient effects. Int J Fract 130:765–785CrossRefGoogle Scholar
  56. 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
  57. Ravi-Chandar K (2004) Dynamic fracture. Elsevier, AmsterdamGoogle Scholar
  58. 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
  59. Rice JR (1968b) Mathematical analysis in the mechanics of fracture. In: H Liebowitz (eds) Fracture. Academic Press New York, 2:191–311Google Scholar
  60. Roos BW (1969) Analytic functions and distributions in physics and engineering. Wiley, New YorkGoogle Scholar
  61. Rosakis AJ, Samudrala O, Coker D (1999) Cracks faster than the shear wave speed. Science 284:1337–1340CrossRefGoogle Scholar
  62. 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
  63. 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
  64. 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
  65. 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
  66. Toupin RA (1962) Perfectly elastic materials with couple stresses. Arch Ration Mech Anal 11:385–414CrossRefGoogle Scholar
  67. Vardoulakis I, Georgiadis HG (1997) SH surface waves in a homogeneous gradient elastic half-space with surface energy. J Elast 47:147–165CrossRefGoogle Scholar
  68. Voigt W (1887) Theoretische Studien uber die Elasticitatsverhaltnisse der Krystalle. Abh Ges Wiss Gottingen 34:3–100Google Scholar
  69. Willis JR (1971) Fracture mechanics of interfacial cracks. J Mech Phys Solids 19:353–368CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Civil, Environmental and Mechanical EngineeringUniversity of TrentoTrentoItaly

Personalised recommendations