Effect of Stress Singularity Magnitude on Scaling of Strength of Quasi-Brittle Structures
Publication: Journal of Engineering Mechanics
Volume 140, Issue 5
Abstract
Engineering structures are often designed to have complex geometries, which could introduce stress singularities that are weaker than the conventional crack-tip singularity. Extrapolating the results of small-scale laboratory tests to predict the response of a full-scale structure comprised of quasi-brittle materials requires an understanding of how the weak stress singularities modify the classical energetic and statistical scaling theories of quasi-brittle fracture. Through a theoretical and numerical study, a new scaling law for quasi-brittle fracture is derived, which explicitly relates the nominal structural strength to the structure size and the magnitude of the stress singularity. The theoretical analysis is based on a generalized weakest-link model that combines the energetic scaling of fracture with the finite weakest-link model. The model captures the transition from the energetic scaling to statistical scaling as the strength of the stress singularity diminishes. The new scaling law is in close agreement, for the entire range of stress singularities, with the size effect curves predicted through finite-element simulations of concrete beams containing an arbitrary-angle V-notch under Mode-I fracture.
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References
Bažant, Z. P. (1984). “Size effect in blunt fracture: Concrete, rock, metal.” J. Eng. Mech., 518–535.
Bažant, Z. P. (2004). “Scaling theory for quasibrittle structural failure.” Proc. Natl. Acad. Sci. USA, 101(37), 13400–13407.
Bažant, Z. P. (2005). Scaling of structural strength, 2nd Ed., Elsevier, London.
Bažant, Z. P., Caner, F. C., Carol, I., Adley, M. D., and Akers, S. A. (2000). “Microplane model M4 for concrete. I: Formulation with work-conjugate deviatoric stress.” J. Eng. Mech., 944–953.
Bažant, Z. P., and Kazemi, M. T. (1990). “Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete.” Int. J. Fract., 44(2), 111–131.
Bažant, Z. P., Le, J.-L., and Bazant, M. Z. (2009). “Scaling of strength and lifetime distributions of quasibrittle structures based on atomistic fracture mechanics.” Proc. Natl. Acad. Sci. USA, 106(28), 11484–11489.
Bažant, Z. P., Le, J.-L., and Hoover, C. G. (2010). “Nonlocal boundary layer (NBL) model: Overcoming boundary condition problems in strength statistics and fracture analysis of quasibrittle materials.” Proc., 7th Int. Conf. on Fracture Mechanics of Concrete (CD-ROM), Jeju, Korea, 1–8.
Bažant, Z. P., and Novák, D. (2000). “Energetic-statistical size effect in quasibrittle failure at crack initiation.” ACI Mater. J., 97(3), 381–392.
Bažant, Z. P., and Pang, S.-D. (2006). “Mechanics based statistics of failure risk of quasibrittle structures and size effect on safety factors.” Proc. Natl. Acad. Sci. USA, 103(25), 9434–9439.
Bažant, Z. P., and Pang, S.-D. (2007). “Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture.” J. Mech. Phys. Solids, 55(1), 91–134.
Bažant, Z. P., Pang, S.-D., Vořechovský, M., and Novák, D. (2007). “Energetic-statistical size effect simulated by SFEM with stratified sampling and crack band model.” Int. J. Numer. Methods Eng., 71(11), 1297–1320.
Bažant, Z. P., and Planas, J. (1998). Fracture and size effect in concrete and other quasibrittle materials, CRC Press, Boca Raton, FL.
Bažant, Z. P., and Xi, Y. (1991). “Statistical size effect in quasi-brittle structures. II: Nonlocal theory.” J. Eng. Mech., 2623–2640.
Bažant, Z. P., and Yu, Q. (2006). “Size effect on strength of quasibrittle structures with reentrant corners symmetrically loaded in tension.” J. Eng. Mech., 1168–1176.
Bažant, Z. P., and Yu, Q. (2009). “Universal size effect law and effect of crack depth on quasi-brittle structure strength.” J. Eng. Mech., 78–84.
Cannone Falchetto, A., Le, J.-L., Turos, M. I., and Marasteanu, M. O. (2013). “Indirect determination of size effect on strength of asphalt mixture at low temperatures.” Mater. Struct., in press.
Carpenter, W. C. (1984). “Mode I and mode II stress intensities for plates with cracks of finite opening.” Int. J. Fract., 26(3), 201–214.
Carpinteri, A. (1987). “Stress-singularity and generalized fracture toughness at the vertex of reentrant corners.” Eng. Fract. Mech., 26(1), 143–155.
Dassault Systèmes Simulia. (2011). ABAQUS 6.11 documentation, Providence, RI.
da Vinci, L. (1945). The notebooks of Leonardo da Vinci, Vol. 3, E. McCurdy, ed., Cape, London.
Dunn, M. L., Suwito, W., and Cunningham, S. J. (1996). “Stress intensities at notch singularities.” Eng. Fract. Mech., 34(29), 3873–3883.
Dunn, M. L., Suwito, W., and Cunningham, S. J. (1997). “Fracture initiation at sharp notches: Correlation using critical stress intensities.” Int. J. Solids Struct., 34(29), 3873–3883.
Fisher, R. A., and Tippet, L. H. C. (1928). “Limiting forms of the frequency distribution of the largest and smallest member of a sample.” Proc. Camb. Philos. Soc., 24(2), 180–190.
Gomez, F. J., and Elices, M. (2003). “A fracture criterion for sharp V-notched samples.” Int. J. Fract., 123(3–4), 163–175.
Grassl, P., and Bažant, Z. P. (2009). “Random lattice-particle simulation of statistical size effect in quasi- brittle structures failing at crack initiation.” J. Eng. Mech., 85–92.
Hoover, C. G., Bažant, Z. P., Vorel, J., Wendner, R., and Hubler, M. H. (2013). “Comprehensive concrete fracture tests: Description and results.” Eng. Frac. Mech., 114, 92–103.
Jirásek, M., and Bauer, M. (2012). “Numerical aspects of the crack band approach.” Comp. Struct., 110–111, 60–78.
Jirásek, M., and Zimmermann, T. (1998). “Rotating crack model with transition to scalar damage.” J. Eng. Mech., 277–284.
Le, J.-L. (2011). “General size effect on strength of bi-material quasibrittle structures.” Int. J. Fract., 172(2), 151–160.
Le, J.-L., Bažant, Z. P., and Bazant, M. Z. (2011). “Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures. I: Strength, crack growth, lifetime and scaling.” J. Mech. Phys. Solids, 59(7), 1291–1321.
Le, J.-L., Bažant, Z. P., and Yu, Q. (2010) “Scaling of strength of metal-composite joints. II: Interface fracture analysis.” J. Appl. Mech., 77(1), 011012.
Le, J.-L., Eliáš, J., and Bažant, Z. P. (2012). “Computation of probability distribution of strength of quasibrittle structures failing at macro-crack initiation.” J. Eng. Mech., 888–899.
Le, J.-L., and Xue, B. (2013). “Energetic-statistical size effect in fracture of bimaterial hybrid structures.” Eng. Frac. Mech., 111, 106–115.
Lee, J., and Fenves, G. L. (1998). “Plastic-damage model for cyclic loading of concrete structures.” J. Eng. Mech., 892–900.
Leguillon, D. (2002). “Strength or toughness? A criterion for crack onset at a notch.” Eur. J. Mech. A, Solids, 21(1), 61–72.
Liu, X. H., Suo, Z., and Ma, Q. (1998). “Split singularities: Stress field near the edge of silicon die on polymer substrate.” Acta Mater., 47(1), 67–76.
Lubliner, J., Oliver, J., Oller, S., and Onate, E. (1989). “A plastic-damage model for concrete.” Int. J. Solids Struct., 25(3), 299–326.
Mariotte, E. (1686). Traité du mouvement des eaux, M. de la Haire, ed., Michallet, Paris.
Mazars, J. (1986). “A model for a unilateral elastic damageable material and its application to concrete.” Fracture toughness and fracture energy of concrete, Elsevier, Amsterdam, Netherlands, 61–71.
Ritchie, R. O., Knott, J. F., and Rice, J. R. (1973). “On the relation between critical tensile stress and fracture toughness in mild steel.” J. Mech. Phys. Solids, 21(6), 395–410.
Seweryn, A. (1994). “Brittle fracture criterion for structures with sharp notches.” Eng. Fract. Mech., 47(5), 673–681.
Sinclair, G. B., Okajima, M., and Griffin, J. H. (1984). “Path independent integrals for computing stress intensity factors at sharp notches in elastic plates.” Int. J. Numer. Methods Eng., 20(6), 999–1008.
Weibull, W. (1939). The phenomenon of rupture in solids, Generalstabens Litografiska Anstalts Förlag, Stockholm, Sweden.
Williams, M. L. (1952). “Stress singularities resulting from various boundary conditions in angular corners of plates in extension.” J. Appl. Mech., 19(4), 526–528.
Zegeye, E., Le, J.-L., Turos, M., and Marasteanu, M. O. (2012). “Investigation of size effect in asphalt mixture fracture testing at low temperature.” Road Mater. Pavement Des., 13(S1), 88–101.
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© 2014 American Society of Civil Engineers.
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Received: Sep 12, 2012
Accepted: Jul 8, 2013
Published online: Jul 10, 2013
Published in print: May 1, 2014
Discussion open until: Jun 8, 2014
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