0
TECHNICAL PAPERS

The Mode III Crack Problem in Microstructured Solids Governed by Dipolar Gradient Elasticity: Static and Dynamic Analysis

[+] Author and Article Information
H. G. Georgiadis

Mechanics Division, National Technical University of Athens, 1 Konitsis Street, Zographou GR-15773, Greece e-mail: georgiad@central.ntua.gr

J. Appl. Mech 70(4), 517-530 (Aug 25, 2003) (14 pages) doi:10.1115/1.1574061 History: Received April 28, 2002; Revised December 19, 2002; Online August 25, 2003
Copyright © 2003 by ASME
Your Session has timed out. Please sign back in to continue.

References

Mindlin,  R. D., 1964, “Micro-Structure in Linear Elasticity,” Arch. Ration. Mech. Anal., 16, pp. 51–78.
Green,  A. E., and Rivlin,  R. S., 1964, “Multipolar Continuum Mechanics,” Arch. Ration. Mech. Anal., 17, pp. 113–147.
Green,  A. E., 1965, “Micro-Materials and Multipolar Continuum Mechanics,” Int. J. Eng. Sci., 3, pp. 533–537.
Mindlin,  R. D., and Eshel,  N. N., 1968, “On First Strain-Gradient Theories in Linear Elasticity,” Int. J. Solids Struct., 4, pp. 109–124.
Weitsman,  Y., 1966, “Strain-Gradient Effects Around Cylindrical Inclusions and Cavities in a Field of Cylindrically Symmetric Tension,” ASME J. Appl. Mech., 33, pp. 57–67.
Day,  F. D., and Weitsman,  Y., 1966, “Strain-Gradient Effects in Microlayers,” ASCE J. Eng. Mech., 92, pp. 67–86.
Cook,  T. S., and Weitsman,  Y., 1966, “Strain-Gradient Effects around Spherical Inclusions and Cavities,” Int. J. Solids Struct., 2, pp. 393–406.
Herrmann, G., and Achenbach, J. D., 1968, “Applications of Theories of Generalized Cosserat Continua to the Dynamics of Composite Materials,” Mechanics of Generalized Continua, E. Kroener, ed., Springer, Berlin, pp. 69–79.
Achenbach,  J. D., Sun,  C. T., and Herrmann,  G., 1968, “On the Vibrations of a Laminated Body,” ASME J. Appl. Mech., 35, pp. 689–696.
Vardoulakis, I., and Sulem, J., 1995, Bifurcation Analysis in Geomechanics, Blackie Academic and Professional (Chapman and Hall), London.
Fleck,  N. A., Muller,  G. M., Ashby,  M. F., and Hutchinson,  J. W., 1994, “Strain Gradient Plasticity: Theory and Experiment,” Acta Metall. Mater., 42, pp. 475–487.
Lakes, L., 1995, “Experimental Methods for Study of Cosserat Elastic Solids and Other Generalized Elastic Continua,” Continuum Models for Materials with Microstructure, H.-B. Muhlhaus, ed., John Wiley and Sons, Chichester, pp. 1–25.
Vardoulakis,  I., and Georgiadis,  H. G., 1997, “SH Surface Waves in a Homogeneous Gradient Elastic Half-Space with Surface Energy,” J. Elast., 47, pp. 147–165.
Wei,  Y., and Hutchinson,  J. W., 1997, “Steady-State Crack Growth and Work of Fracture for Solids Characterized by Strain Gradient Plasticity,” J. Mech. Phys. Solids, 45, pp. 1253–1273.
Begley,  M. R., and Hutchinson,  J. W., 1998, “The Mechanics of Size-Dependent Indentation,” J. Mech. Phys. Solids, 46, pp. 2049–2068.
Exadaktylos,  G., and Vardoulakis,  I., 1998, “Surface Instability in Gradient Elasticity With Surface Energy,” Int. J. Solids Struct., 35, pp. 2251–2281.
Huang, Y., Zhang, L., Guo, T. F., and Hwang, K. C., 1997, “Near-Tip Fields for Cracks in Materials With Gradient Effects,” Proceedings of the IUTAM Symposium on Nonlinear Analysis of Fracture, J. R. Willis, ed., Kluwer Academic Publishers, Dordrecht, pp. 231–243.
Zhang,  L., Huang,  Y., Chen,  J. Y., and Hwang,  K. C., 1998, “The Mode-III Full-Field Solution in Elastic Materials With Strain Gradient Effects,” Int. J. Fract., 92, pp. 325–348.
Chen,  J. Y., Huang,  Y., and Ortiz,  M., 1998, “Fracture Analysis of Cellular Materials: A Strain Gradient Model,” J. Mech. Phys. Solids, 46, pp. 789–828.
Georgiadis,  H. G., and Vardoulakis,  I., 1998, “Anti-Plane Shear Lamb’s Problem Treated by Gradient Elasticity With Surface Energy,” Wave Motion, 28, pp. 353–366.
Georgiadis,  H. G., Vardoulakis,  I., and Lykotrafitis,  G., 2000, “Torsional Surface Waves in a Gradient-Elastic Half-Space,” Wave Motion, 31, pp. 333–348.
Georgiadis, H. G., Vardoulakis, I., and Velgaki, E. G., 2002, “Dispersive Rayleigh-Wave Propagation in Microstructured Solids Characterized by Dipolar Gradient Elasticity,” J. Elast., submitted for publication.
Georgiadis,  H. G., and Velgaki,  E. G., 2002, “High-Frequency Rayleigh Waves in Materials With Microstructure and Couple-Stress Effects,” Int. J. Solids Struct., 40, pp. 2501–2520.
Amanatidou, E., and Aravas, N., 2001, “Finite Element Techniques for Gradient Elasticity Problems,” Proceedings of the 6th Greek National Congress of Mechanics, Hellenic Society of Theoretical and Applied Mechanics, 2 , pp. 149–154.
Gazis,  D. C., Herman,  R., and Wallis,  R. F., 1960, “Surface Elastic Waves in Cubic Crystals,” Phys. Rev., 119, pp. 533–544.
Jaunzemis, W., 1967, Continuum Mechanics, MacMillan, New York.
Fung, Y. C., 1965, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, NJ.
Ignaczak, J., 1971, “Tensorial Equations of Motion for Elastic Materials With Microstructure,” Trends in Elasticity and Thermoelasticity (W. Nowacki Anniversary Volume), Wolters-Noordhoff, Groningen, pp. 89–111.
Rice,  J. R., 1968, “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks,” ASME J. Appl. Mech., 35, pp. 379–386.
Rice, J. R., 1968, “Mathematical Analysis in the Mechanics of Fracture,” Fracture, H. Liebowitz, ed., 2 , Academic Press, New York, pp. 191–311.
Atkinson,  C., and Leppington,  F. G., 1974, “Some Calculations of the Energy-Release Rate G for Cracks in Micropolar and Couple-Stress Elastic Media,” Int. J. Fract., 10, pp. 599–602.
Lubarda,  V. A., and Markenscoff,  X., 2000, “Conservation Integrals in Couple Stress Elasticity,” J. Mech. Phys. Solids, 48, pp. 553–564.
Williams,  M. L., 1952, “Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates in Extension,” ASME J. Appl. Mech., 74, pp. 526–528.
Williams,  M. L., 1957, “On the Stress Distribution at the Base of a Stationary Crack,” ASME J. Appl. Mech., 79, pp. 109–114.
Barber, J. R., 1992, Elasticity, Kluwer Academic Publishers, Dordrecht.
van der Pol, B., and Bremmer, H., 1950, Operational Calculus Based on the Two-Sided Laplace Integral, Cambridge University Press, Cambridge, UK.
Carrier, G. A., Krook, M., and Pearson, C. E., 1966, Functions of a Complex Variable, McGraw-Hill, New York.
Roos, B. W., 1969, Analytic Functions and Distributions in Physics and Engineering, John Wiley and Sons, New York.
Mittra, R., and Lee, S. W., 1971, Analytical Techniques in the Theory of Guided Waves, MacMillan, New York.
Gel’fand, I. M., and Shilov, G. E., 1964, Generalized Functions, 1 , Academic Press, New York.
Lauwerier,  H. A., 1963, “The Hilbert Problem for Generalized Functions,” Arch. Ration. Mech. Anal., 13, pp. 157–166.
Knowles,  J. K., and Pucik,  T. A., 1973, “Uniqueness for Plane Crack Problems in Linear Elastostatics,” J. Elast., 3, pp. 155–160.
Van Dyke, M., 1964, Perturbation Methods in Fluid Mechanics, Academic Press, New York.
Rice,  J. R., 1974, “Limitations to the Small Scale Yielding Approximation for Crack Tip Plasticity,” J. Mech. Phys. Solids, 22, pp. 17–26.
Mills,  N. J., 1974, “Dugdale Yielded Zones in Cracked Sheets of Glassy Polymers,” Eng. Fract. Mech., 6, pp. 537–549.
Elssner,  G., Korn,  D., and Ruhle,  M., 1994, “The Influence of Interface Impurities on Fracture Energy of UHV Diffusion Bonded Metal-Ceramic Bicrystals,” Scr. Metall. Mater., 31, pp. 1037–1042.
Bogy,  D. B., and Sternberg,  E., 1967, “The Effect of Couple-Stresses on Singularities due to Discontinuous Loadings,” Int. J. Solids Struct., 3, pp. 757–770.
Prakash,  V., Freund,  L. B., and Clifton,  R. J., 1992, “Stress Wave Radiation From a Crack Tip During Dynamic Initiation,” ASME J. Appl. Mech., 59, pp. 356–365.
Fisher,  B., 1971, “The Product of Distributions,” Quart. J. Math. Oxford, 22, pp. 291–298.
Bueckner,  H. F., 1958, “The Propagation of Cracks and the Energy of Elastic Deformation,” Trans. ASME, 24, pp. 1225–1230.
Barenblatt,  G. I., 1962, “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture,” Adv. Appl. Mech., 7, pp. 55–129.
Nilsson,  F., and Stahle,  P., 1988, “Crack Growth Criteria and Crack Tip Models,” SM Arch., 13(4), pp. 193–238.
Cherepanov, G. P., 1979, Mechanics of Brittle Fracture, McGraw-Hill, New York.
Freund, L. B., 1990, Dynamic Fracture Mechanics, Cambridge University Press, Cambridge, UK.
Vekua, I. N., 1968, New Methods for Solving Elliptic Equations, North-Holland, Amsterdam.
Ablowitz, M. J., and Fokas, A. S., 1997, Complex Variables: Introduction and Applications, Cambridge University Press, Cambridge, UK.

Figures

Grahic Jump Location
Monopolar (external) and dipolar (internal) forces acting on an ensemble of subparticles in a material with microstructure
Grahic Jump Location
A crack under a remotely applied antiplane shear loading. The contour Γ surrounding the crack tip serves for the definition of the J-integral.
Grahic Jump Location
William’s method: the near-tip fields of (i) a finite length crack, (ii) an edge crack, and (iii) a cracked strip correspond to the field generated in a body with a semi-infinite crack
Grahic Jump Location
Branch cuts for the functions (β,γ)
Grahic Jump Location
Contour integrations for the factorization of the kernel function in Eq. (42)
Grahic Jump Location
Rectangular-shaped contour surrounding the crack tip for the evaluations of the J-integral and the energy release rate
Grahic Jump Location
Contour integration for the evaluation of the complex integral in Eq. (66)
Grahic Jump Location
Graphs of the exact gradient (total monopolar stress), asymptotic gradient (total monopolar stress), and classical KIII field solutions in normalized forms
Grahic Jump Location
Variation of the exact total monopolar stress (according to the gradient theory) with (x/h) for the cases c=h2 and c=(0.01h)2. The graphs depict that the cohesive zone is small as compared to the intrinsic material length h and that the stress ahead of the cohesive zone exhibits a bounded maximum.
Grahic Jump Location
Branch cuts for the functions (β̄,γ̄)
Grahic Jump Location
Contour integrations for the factorization of the kernel function defined in Eq. (88)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In