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Plane asymptotic crack-tip solutions in gradient elasticity. (English) Zbl 1176.74155

Summary: An asymptotic crack-tip solution under conditions of plane strain is developed for a material that obeys a special form of linear isotropic strain gradient elasticity. In particular, an elastic constitutive equation of the form \(\tau =\tau ^{(0)} - \ell ^{2}\nabla ^{2}\tau ^{(0)}\) is considered, where \((\tau ,\mathbf {\epsilon })\) are the stress and strain tensors, \(\tau ^{(0)}=\lambda \epsilon _{kk}\delta +2\mu \epsilon, (\lambda ,\mu )\) are the Lamé constants, and \(\ell \) is a material length. Both symmetric (mode-I) and antisymmetric (mode-II) solutions are developed. The asymptotic solution predicts finite strains at the crack-tip. The mode-I crack-tip displacement field \(\mathbf u\) is of the form
\[ \begin{aligned} u_1 = Ax_1 & + \ell\left(\frac{r}{\ell}\right)^{3/2}[A_1 \tilde u_{11} (\theta, \nu ) + A_2 \tilde u_{12} (\theta, \nu )] +O(r^2),\\ u_2 = Bx_2 & + \ell\left(\frac{r}{\ell}\right)^{3/2}[A_1 \tilde u_{21} (\theta , \nu ) + A_2 \tilde u_{22} (\theta, \nu )] +O(r^2), \end{aligned} \] where \((x_1,x_2)\) and \((r,\theta )\) are crack-tip Cartesian and polar coordinates, respectively, \(\nu \) is Poisson’s ratio, and \((A,B,A_1,A_2)\) are dimensionless constants determined by the complete solution of a boundary value problem. The \(A\)- and \(B\)- terms above correspond to uniform normal strains parallel \((\epsilon _{11})\) and normal \((\epsilon _{22})\) to the crack line, which do not contribute to the crack-tip “energy release rate” (\(J\)-integral). Detailed finite element calculations are carried out for an edge-cracked-panel (ECP) loaded by point forces and the asymptotic solution is verified. The region of dominance of the asymptotic solution for the ECP geometry analyzed is found to be order \(\ell /10\). The “energy release rate” is found to decrease with increasing \(\ell \).

MSC:

74R10 Brittle fracture
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics

Software:

Mathematica
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[1] Aifantis, E. C.: On the role of gradients in the localization of deformation and fracture, Int. J. Eng. sci. 30, 1279-1299 (1992) · Zbl 0769.73058
[2] Aifantis, E. C.: Exploring the applicability of gradient elasticity to certain micro/nano reliability problems, Microsyst. technol. 15, 109-115 (2009)
[3] Altan, S. B.; Aifantis, E. C.: On the structure of mode III crack-tip in gradient elasticity, Scripta metall. Mater. 26, 319-324 (1992)
[4] Amanatidou, E.; Aravas, N.: Mixed finite element formulations of strain-gradient elasticity problems, Comput. methods appl. Mech. eng. 191, 1723-1751 (2002) · Zbl 1098.74678
[5] Barenblatt, G. I.: The mathematical theory of equilibrium cracks in brittle fracture, Adv. appl. Mech. 7, 55-129 (1962)
[6] Bergez, D.: Determination of stress intensity factors by use of path-independent integrals, Mech. res. Commun. 1, 179-180 (1974) · Zbl 0344.73075
[7] Bleustein, J. L.: A note on the boundary conditions of toupin’s strain-gradient theory, Int. J. Solids struct. 3, 1053-1057 (1967)
[8] Boyce, W. E.; Diprima, R. C.: Elementary differential equations and boundary value problems, (2001) · Zbl 0178.09001
[9] Budiansky, B.; Rice, J. R.: Conservation laws and energy-release rates, J. appl. Mech. 40, 201-203 (1973) · Zbl 0261.73059
[10] Bui, H. D.: Associated path independent J-integrals for separating mixed modes, J. mech. Phys. solids 31, 439-448 (1983) · Zbl 0523.73074
[11] Burridge, R.: An influence function for the intensity factor in tensile fracture, Int. J. Eng. sci. 14, 725-734 (1976) · Zbl 0346.73059
[12] Chan, Y. -S.; Paulino, G. H.; Fannjiangand, A. C.: Gradient elasticity theory for mode III fracture in functionally graded materials – part II: Crack parallel to the material gradation, J. appl. Mech. 75, 061015-1-061015-11 (2008)
[13] Chang, J. H.; Pu, L. P.: Finite element calculation of energy release rate prior to crack kinking in 2-D solids, Int. J. Numer. methods eng. 39, 3033-3046 (1996) · Zbl 0882.73063
[14] Chang, J. H.; Pu, L. P.: A finite element approach for J2 calculation in anisotropic materials, Comput. struct. 62, 635-641 (1997) · Zbl 0903.73069
[15] Chang, J. H.; Yeh, J. B.: Calculation of J2 integral for 2-D cracks in rubbery materials, Int. J. Fracture 59, 683-695 (1998)
[16] Chen, J. Y.; Huang, Y.; Ortiz, M.: Fracture of cellular materials, a strain gradient model, J. mech. Phys. solids 46, 789-828 (1998) · Zbl 1056.74508
[17] Chen, J. Y.; Wei, Y.; Huang, Y.; Hutchinson, J. W.; Hwang, K. C.: The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses, Eng. fract. Mech. 64, 625-648 (1999)
[18] Eischen, J. W.: An improved method for computing the J2 integral, Eng. fract. Mech. 26, 691-700 (1987)
[19] Exadaktylos, G.: Gradient elasticity with surface energy: mode-I crack problem, Int. J. Solids struct. 35, 421-426 (1998) · Zbl 0930.74050
[20] Exadaktylos, G.; Vardoulakis, I.; Aifantis, E. C.: Cracks in gradient elastic bodies with surface energy, Int. J. Fracture 79, 107-119 (1996) · Zbl 0919.73237
[21] Fisher, B.: The product of distributions, Q. J. Math. 22, 291-298 (1971) · Zbl 0213.13104
[22] Fleck, N. A.; Hutchinson, J. W.: Strain gradient plasticity, Adv. appl. Mech. 33, 95-361 (1997) · Zbl 0894.73031
[23] Freund, L. B.: Energy flux into the tip of an extending crack in an elastic solid, J. elasticity 2, 341-349 (1972)
[24] Georgiadis, H. G.: The mode III crack problem in micro-structured solids governed by dipolar gradient elasticity: static and dynamic analysis, J. appl. Mech. 70, 517-530 (2003) · Zbl 1110.74453
[25] Georgiadis, H. G.; Grentzelou, C. G.: Energy theorems and the J-integral in dipolar gradient elasticity, Int. J. Solids struct. 43, 5690-5712 (2006) · Zbl 1120.74341
[26] Germain, P.: Sur lapplication de la méthode des puissances virtuelles en mécanique des milieux continus, C. R. Acad. sci. Paris 274, 1051-1055 (1972) · Zbl 0242.73005
[27] Germain, P.: The method of virtual power in continuum mechanics. Part 2: microstructure, SIAM J. Appl. math. 25, 556-575 (1973) · Zbl 0273.73061
[28] Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus, J. mécanique 12, 235-274 (1973) · Zbl 0261.73003
[29] Gilman, J. J.: Direct measurements of the surface energies of crystals, J. appl. Phys. 31, 2208-2218 (1960)
[30] Herrmann, A. Golebiewska; Herrmann, G.: On energy-release rates for a plane crack, J. appl. Mech. 48, 525-528 (1981) · Zbl 0463.73098
[31] Grentzelou, C. G.; Georgiadis, H. G.: Balance laws and energy release rates for cracks in dipolar gradient elasticity, Int. J. Solids struct. 45, 551-567 (2008) · Zbl 1167.74325
[32] Griffith, A. A.: The phenomena of rupture and flow in solids, Philos. trans. Roy. soc. Lond. ser. A 221, 163-198 (1920)
[33] Hellen, T. K.; Blackburn, W. S.: The calculation of stress intensity factors for combined tensile and shear loading, Int. J. Fracture 11, 605-617 (1975)
[34] Hibbitt, H. D.: ABAQUS/EPGEN – a general purpose finite element code with emphasis in nonlinear applications, Nucl. eng. Des. 7, 271-297 (1977)
[35] Huang, Y.; Zhang, L.; Guo, T. F.; Hwang, K. C.: Mixed mode near-tip fields for cracks in materials with strain gradient effects, J. mech. Phys. solids 45, 439-465 (1997) · Zbl 1049.74521
[36] Ishikawa, H.: A finite element analysis of stress intensity factors for combined tensile and shear loading by only a virtual crack extension, Int. J. Fracture 16, R243-R246 (1980)
[37] Ishikawa, H., Kitagawa, H., Okamura, H., 1979. J integral of a mixed mode crack and its application. In: Proceedings of the Third International Conference on Mechanical Behaviour of Materials (ICM3), vol. 3, Cambridge, England, August 1979, pp. 447 – 455.
[38] Karlis, G. F.; Tsinopoulos, S. V.; Polyzos, D.; Beskos, D. E.: Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity, Comput. methods appl. Mech. eng. 196, 5092-5103 (2007) · Zbl 1173.74458
[39] Karlis, G. F.; Tsinopoulos, S. V.; Polyzos, D.; Beskos, D. E.: 2D and 3D boundary element analysis of mode-I cracks in gradient elasticity, Comput. model. Eng. sci. 28, 189-207 (2008) · Zbl 1232.74119
[40] Ang, J. -H. Kim; Paulino, G. H.: Finite element evaluation of mixed mode stress intensity factors in functionally graded materials, Int. J. Numer. methods eng. 53, 1903-1935 (2002) · Zbl 1169.74612
[41] Koiter, W. T.: Couple-stresses in the theory of elasticity. I, Proc. K. Ned. akad. Wet. (B) 67, 17-29 (1964) · Zbl 0119.39504
[42] Koiter, W. T.: Couple-stresses in the theory of elasticity. II, Proc. K. Ned. akad. Wet. (B) 67, 30-44 (1964) · Zbl 0124.17405
[43] Laird, C., 1967. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In: Fatigue Crack Propagation, ASTM STP, vol. 415, pp. 131 – 168.
[44] Lawn, B.: Fracture of brittle solids, (1993)
[45] McClintock, F.A., 1962. On the plasticity of the growth of fatigue cracks. In: Drucker, D.C., Gilman, J.J., (Eds.), Fracture of Solids, vol. 20, 1962, pp. 65 – 102.
[46] Mindlin, R. D.: Microstructure in linear elasticity, Arch. ration. Mech. anal. 16, 51-78 (1964) · Zbl 0119.40302
[47] Mindlin, R. D.; Eshel, N. N.: On first strain-gradient theories in linear elasticity, Int. J. Solids struct. 4, 109-124 (1968) · Zbl 0166.20601
[48] Nikishkov, G. P.; Vainshtok, V. A.: Method of virtual crack growth for determining stress intensity factors kiandkii, Strength mater. 13, 696-701 (1981)
[49] Orowan, E.: Fracture and strength of solids, Rep. progr. Phys. 12, 185-232 (1949)
[50] Parks, D. M.: A stiffness derivative finite element technique for determination of crack tip stress intensity factors, Int. J. Fracture 10, 487-502 (1974)
[51] Parks, D. M.: The virtual crack extension method for nonlinear material behavior, Comput. methods appl. Mech. eng. 12, 353-364 (1977) · Zbl 0368.73085
[52] Paulino, G. H.; Fannjiangand, A. C.; Chan, Y. -S.: Gradient elasticity theory for mode III fracture in functionally graded materials – part I: Crack perpendicular to the material gradation, J. appl. Mech. 70, 531-542 (2003) · Zbl 1110.74622
[53] Radi, E.: 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 (2008) · Zbl 1169.74548
[54] Raju, I. S.; Shivakumar, K. N.: An equivalent domain integral method in the two-dimensional analysis of mixed mode crack problems, Eng. fract. Mech. 37, 707-725 (1990)
[55] Rice, J. R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. appl. Mech. 35, 379-386 (1968)
[56] Rice, J. R.; Drucker, D. C.: Energy changes in stressed bodies due to void and crack growth, Int. J. Fracture 3, 19-27 (1967)
[57] Ru, C. Q.; Aifantis, E. C.: A simple approach to solve boundary-value problems in gradient elasticity, Acta mech. 101, 59-68 (1993) · Zbl 0783.73015
[58] Sha, G. T.: On the virtual crack extension technique for stress intensity factors and energy release rate calculations for mixed fracture mode, Int. J. Fracture 25, R33-R42 (1984)
[59] Shi, M. X.; Huang, Y.; Hwang, K. C.: Fracture in a higher-order elastic continuum, J. mech. Phys. solids 48, 2513-2538 (2000) · Zbl 1010.74057
[60] Sladek, J.; Sladek, V.; Zhang, Ch.: Evaluation of the stress intensity factors for cracks in continuously nonhomogeneous solids. Part I: Interaction integral, Mech. adv. Mater. struct. 15, 438-443 (2008)
[61] Unger, D. J.; Aifantis, E. C.: The asymptotic solution of gradient elasticity for mode III, Int. J. Fracture 71, R27-R32 (1995)
[62] Unger, D. J.; Aifantis, E. C.: Strain gradient elasticity theory for anti-plane shear cracks. Part I: Oscillatory displacements, Theor. appl. Fract. mech. 34, 243-252 (2000)
[63] Unger, D. J.; Aifantis, E. C.: Strain gradient elasticity theory for anti-plane shear cracks. Part II: Monotonic displacements, Theor. appl. Fract. mech. 34, 253-265 (2000)
[64] Vardoulakis, I.; Exadaktylos, G.; Aifantis, E. C.: Gradient elasticity with surface energy: mode-III crack problem, Int. J. Solids struct. 33, 4531-4559 (1996) · Zbl 0919.73237
[65] Wolfram, S.: The Mathematica book, (2003)
[66] Zhao, J.; Pedroso, D.: Strain gradient theories in orthogonal curvilinear coordinates, Int. J. Solids struct. 45, 3507-3520 (2008) · Zbl 1169.74334
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