Abstract
Cosserat type continuum theories have been employed by many authors to study cracks in elastic solids with microstructures. Depending on which theory was used, different crack tip stress singularities have been obtained. In this paper, a microstructure continuum theory is used to model a layered elastic medium containing a crack parallel to the layers. The crack problem is solved by means of the Fourier transform. The resulting integrodifferential equations are discretized using the Chebyshev polynomial expansion method for numerical solutions. By using the present theory, the explicit internal length effects upon the crack opening displacement and stress field can be observed. It is found that the stress field near the crack tip is not singular according to the microstructure continuum solution although the level of the opening stress shows an increasing trend until it gets very close to the crack tip. The rising portion of the near tip opening stress is used to project the stress intensity factor which agrees fairly well with that obtained using the FEM to perform stress analyses of the cracked layered medium with the exact geometry. The numerical solutions also indicate that treating the layered medium as an equivalent homogeneous classical elastic solid is not adequate if cracks are present and accurate stress intensity factors in the original layered medium is desired.
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References
Achenbach JD, Sun CT, Herrmann G (1968) On the vibrations of a laminated body. J Appl Mech 35: 689–696
Aifantis EC (1999) Gradient deformation models at nano, micro, and macro scales. ASME J Eng Mater Tech 121: 189–202. doi:10.1115/1.2812366
Atkinson C, Leppington FG (1977) The effect of couple stress on the tip of a crack. Int J Solids Struct 13: 1103–1122. doi:10.1016/0020-7683(77)90080-4
Cosserat E, Cosserat F (1909) Theorie des Corps Deformables. A. Hermann & Fils, Paris
Eringen AC, Speziale CG, Kim BS (1977) Crack tip problem in non-local theory. J Mech Phys Solids 25: 339–355. doi:10.1016/0022-5096(77)90002-3
Eringen AC, Suhubi ES (1964) Nonlinear theory of micro-elastic solids. Int J Eng Sci 2: 189–203. doi:10.1016/0020-7225(64)90004-7
Fannjiang AC, Chan YS, Paulino GH (2002) Strain gradient elasticity for antiplane shear cracks: a hypersingular integrodifferential equation approach. SIAM J Appl Math 62: 1066–1091. doi:10.1137/S0036139900380487
Fleck N, Hutchinson J (1993) A phenomenological theory for strain gradient effects in plasticity. J Mech Phys Solids 41: 1825–1857. doi:10.1016/0022-5096(93)90072-N
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: 439–465. doi:10.1016/S0022-5096(96)00089-0
Kennedy TC, Kim JB (1993) Dynamic analysis of cracks in micropolar elastic materials. Eng Fract Mech 44: 207–216. doi:10.1016/0013-7944(93)90045-T
Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16: 51–78. doi:10.1007/BF00248490
Paul HS, Sridharan K (1981) The problem of a griffith crack in micropolar elasticity. Int J Eng Sci 19: 563–579. doi:10.1016/0020-7225(81)90090-2
Sih GC, Liebowitz H (1968) . In: Liebowitz H (ed) Fracture: an advanced treatise, vol 2. Academic Press, New York, pp 67–190
Sternberg E, Muki R (1967) The effect of couple-stress on the stress concentration around a crack. Int J Solids Struct 3: 69–95. doi:10.1016/0020-7683(67)90045-5
Sun CT, Achenbach JD, Herrmann G (1968) Continuum theory for a laminated medium. J Appl Mech 35: 467–475
Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11: 385–414. doi:10.1007/BF00253945
Vardoulakis I, Exadaktylos G, Aifantis E (1996) Gradient elasticity with surface energy: mode-III crack problem. Int J Solids Struct 33: 4531–4559. doi:10.1016/0020-7683(95)00277-4
Zhang L, Huang Y, Chen JY, Hwang KC (1998) The mode III full-field in elastic materials with strain gradient effects. Int J Fract 92: 325–348. doi:10.1023/A:1007552621307
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Huang, G.L., Sun, C.T. Multiscale continuum modeling of a crack in elastic media with microstructures. Int J Fract 160, 109–118 (2009). https://doi.org/10.1007/s10704-009-9399-3
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DOI: https://doi.org/10.1007/s10704-009-9399-3