TECHNICAL PAPERS
Jul 1, 1984

Rock Fracture Via Strain‐Softening Finite Elements

Publication: Journal of Engineering Mechanics
Volume 110, Issue 7

Abstract

The fracture of rock is assumed to arise from propagation of a blunt crack band with continuously distributed (smeared) microcracks or continuous cracks. This approach, justified by material heterogeneity, is convenient for finite element analysis, and allows analyzing fracture on the basis of triaxial stress‐strain relations which cover the strain‐softening behavior. A simple compliance formulation is derived for this purpose. The practical form of the theory involves two independent material parameters, the fracture energy and the tensile strength. The width of the crack band front is considered as a fixed material property and can be taken as roughly five‐times the grain size of rock. The theory is shown to be capable of satisfactorily representing the test data available in the literature. In particular, good fits are demonstrated for the measured maximum loads, as well as for the measured resistance curves (R‐curves). Statistical analysis of the deviations from the test data is also presented.

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
Barenblatt, G. I., “The Formation of Equilibrium Cracks During Brittle Fracture. General Ideas and Hypothesis. Axially‐Symmetric Cracks,” Prikladnaya Matematika i Mekhanika, Vol. 23, No. 3, 1959, pp. 434–444.
2.
Barenblatt, G. I., “The Mathematical Theory of Equilibrium Crack in the Brittle Fracture,” Advances in Applied Mechanics, Vol. 7, 1962, pp. 55–125.
3.
Baz⁁ant, Z. P., “Instability, Ductility and Size Effect in Strain‐Softening Concrete,” Journal of the Engineering Mechanics Division, ASCE, Vol. 102, No. EM2, Apr., 1976, pp. 331–344.
4.
Baz⁁ant, Z. P., “Crack Band Model for Fracture of Geomaterials,” 4th International Conference on Numerical Methods in Geomechanics, Edmonton, Alberta, June, 1982, (ed. Z. Eisenstein), Proc. Vol. 3 (invited lectures), pp. 1137–1152.
5.
Baz⁁ant, Z. P., “Microplane Model for Fracture Analysis of Concrete Structures,” Proceedings, Symposium on “The Interaction of Non‐nuclear Munitions with Structures,” U.S. Air Force Academy, Colorado Springs, Colo., May, 1983, pp. 49–55.
6.
Baz⁁ant, Z. P., “Mechanics of Fracture and Progressive Cracking in Concrete,” Report No. 83‐2/428m, Center for Concrete and Geomaterials, Northwestern University, Evanston, Ill., Feb., 1983; also Chapt. I in Applications of Fracture Mechanics to Concrete, G. C. Sih, ed., Martinus Nijhoff, The Netherlands, to appear.
7.
Baz⁁ant, Z. P., and Cedolin, L., “Blunt Crack Band Propagation in Finite Element Analysis,” Journal of the Engineering Mechanics Division, ASCE, Vol. 105, No. EM2, Proc. Paper 14529, 1979, pp. 297–315.
8.
Baz⁁ant, Z. P., and Cedolin, L., “Fracture Mechanics of Reinforced Concrete,” Journal of the Engineering Mechanics Division, ASCE, Vol. 106, No. EM6, Proc. Paper 15917, 1980, pp. 1287–1306.
9.
Baz⁁ant, Z. P., and Cedolin, L., “Finite Element Modeling of Crack Band Propagation in Reinforced Concrete,” Report No. 81‐9/640f, Center for Concrete and Geomaterials, Northwestern University, Evanston, Ill., 1981.
10.
Baz⁁ant, Z. P., and Gambarova, P. G., “Rough Cracks in Reinforced Concrete,” Journal of the Structural Division, ASCE, Vol. 106, No. ST4, Proc. Paper No. 15330, 1980, pp. 819–842.
11.
Baz⁁ant, Z. P., and Kim, S. S., “Plastic‐Fracturing Theory for Concrete,” Journal of the Engineering Mechanics Division, ASCE, Vol. 105, No. EM3, Proc. Paper 14653, 1979, pp. 407–428.
12.
Baz⁁ant, Z. P., and Oh, B. H., “Crack Band Theory for Fracture of Concrete,” Materials and Structures(RILEM, Paris), Vol. 16, 1983, pp. 155–177.
13.
Baz⁁ant, Z. P., and Oh, B. H., “Concrete Fracture via Stress‐Strain Relations,” Report No. 81‐10/665c, Center for Concrete and Geomaterials, Northwestern University, Evanston, Ill., 1981.
14.
Baz⁁ant, Z. P., and Panula, L., “Statistical Stability Effects in Concrete Failure,” Journal of the Engineering Mechanics Division, ASCE, Vol. 104, 1978, No. EM5, pp. 1195–1212.
15.
Baz⁁ant, Z. P., and Tsubaki, T., “Slip‐Dilatancy Model for Cracked Reinforced Concrete,” Journal of the Structural Division, ASCE, Vol. 106, No. ST9, Proc. Paper 15704, 1980, pp. 1947–1966.
16.
Carpinteri, A., “Static and Energetic Fracture Parameters for Rocks and Concretes,” Report, Istituto di Scienza delle Costruzioni‐Ingegneria, University of Bologna, Italy, 1980.
17.
Carpinteri, A., “Experimental Determination of Fracture Toughness Parameters KIC and JIC for Aggregate Materials,” “Advances in Fracture Research,” Proc. Paper, 5th International Conference on Fracture, Cannes, France, D. Frangois, ed., Vol. 4, 1981, pp. 1491–1498.
18.
Cedolin, L., and Baz⁁ant, Z. P., “Effect of Finite Element Choice in Blunt Crack Band Analysis,” Computer Methods in Applied Mechanics and Engineering, Vol. 24, No. 3, 1980, pp. 305–316.
19.
Dugdale, D. S., “Yielding of Steel Sheets Containing Slits,” Journal of Mechanics and Physics of Solids, Vol. 8, 1960, pp. 100–104.
20.
Evans, R. H., and Marathe, M. S., “Microcracking and Stress‐Strain Curves for Concrete in Tension,” Matériaux et Constructions, Vol. 1, No. 1, 1968, pp. 61–64.
21.
Heilmann, H. G., Hilsdorf, H. H., and Finsterwalder, K., “Festigkeit und Verformung von Beton unter Zugspannungen,” Deutscher Ausschuss für Stahlbeton, Heft 203, W. Ernst & Sohn, West Berlin, 1969.
22.
Hillerborg, A., Modéer, M., and Petersson, P. E., “Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements,” Cement and Concrete Research, Vol. 6, 1976, pp. 773–782.
23.
Hoagland, R. G., Hahn, G. T., and Rosenfield, A. R., “Influence of Microstructure on Fracture Propagation in Rock,” Rock Mechanics, Vol. 5, 1973, pp. 77–106.
24.
Hughes, B. P., and Chapman, G. P., “The Complete Stress‐Strain Curve for Concrete in Direct Tension,” Bulletin RILEM, No. 30, 1966, pp. 95–97.
25.
Ingraffea, A. R., “Numerical Modeling of Fracture Propagation,” Report, Cornell University, 1983 (also, to appear in Rock Fracture Mechanics, H. P. Rossmanith, ed., The International Center for Mechanical Sciences, Udine, Italy, 1983).
26.
Kfouri, A. P., and Miller, K. J., “Stress Displacement Line Integral and Closure Energy Determinations of Crack Tip Stress Intensity Factors,” Int. Journal of Pres. Ves. and Piping, Vol. 1, No. 3, 1974, pp. 179–191.
27.
Kfouri, A. P., and Rice, J. R., “Elastic/Plastic Separation Energy Rate for Crack Advance in Finite Growth Steps,” in “Fracture 1977” (Proc. Paper of the 4th Intern. Conf. on Fracture, held in Waterloo, Ontario), D. M. R. Taplin, ed., University of Waterloo Press, Vol. 1, 1977, pp. 43–59.
28.
Knauss, W. G., “On the Steady Propagation of a Crack in a Viscoelastic Sheet; Experiments and Analysis,” Reprinted from the Deformation in Fracture of High Polymers, H. H. Kausch, ed., Plenum Press, 1974, pp. 501–541.
29.
Kupfer, H. B., and Gerstle, K. H., “Behavior of Concrete under Biaxial Stress,” Journal of the Engineering Mechanics Division, ASCE, Vol. 99, No. EM4, Proc. Paper 9917, 1973, pp. 853–866.
30.
Liu, T. C. Y., Nilson, A. H., and Slate, F. O., “Biaxial Stress‐Strain Relations for Concrete,” Journal of the Structural Division, ASCE, Vol. 98, No. ST5, Proc. Paper 8905, 1972, pp. 1025–1034.
31.
Løland, K. Z., “Continuous Damage Model for Load‐Response Estimation of Concrete,” Cement and Concrete Research, Vol. 10, 1980, pp. 395–402.
32.
Lorain, M., “On the Application of the Damage Theory to Fracture Mechanics of Concrete,” A State‐of‐the‐Art Report, Civil Engineering Department, I.N.S.A. 31077 Toulouse, Cedex, France, 1981.
33.
Mazars, J., “Mechanical Damage and Fracture of Concrete Structures,” 5th International Conference on Fracture, D. Frančois, ed., Cannes, France, Vol. 4, 1981, pp. 1499–1506.
34.
Ngo, D., and Scordelis, A. C., “Finite Element Analysis of Reinforced Concrete Beams,” Journal of the American Concrete Institute, Vol. 64, No.3, Mar., 1967, pp. 152–163.
35.
Parker, A. P., The Mechanics of Fracture and Fatigue, E. F. N. Spon, Ltd., Methuen, London, 1981.
36.
Petersson, P. E., “Fracture Energy of Concrete: Method of Determination,” Cement and Concrete Research, Vol. 10, pp. 78–89,
and “Fracture Energy of Concrete: Practical Performance and Experimental Results,” Cement and Concrete Research, Vol. 10, 1980, pp. 91–101.
37.
Rashid, Y. R., “Analysis of Prestressed Concrete Pressure Vessels,” Nuclear Engineering and Design, Vol. 7, No. 4, Apr., 1968, pp. 334–344.
38.
Rüsch, H., and Hilsdorf, H., “Deformation Characteristics of Concrete Under Axial Tension,” Voruntersuchungen, Bericht Nr. 44, Munich, May, 1963.
39.
Schmidt, R. A., “Fracture‐Toughness Testing of Limestone,” Experimental Mechanics, Vol. 16, No. 5, 1976, pp. 161–167.
40.
Schmidt, R. A., “Fracture Mechanics of Oil Shale—Unconfined Fracture Toughness, Stress Corrosion Cracking, and Tension Test Results,” Proceedings, 18th U.S. Symposium Rock Mechs., Paper 2A2, Colorado School of Mines, Golden, Colo., 1977.
41.
Schmidt, R. A., and Lutz, T. J., “KIC and JIC of Westerly Granite—Effect of Thickness and In‐Plane Dimensions,” ASTM STP 678, S. W. Freiman, ed., American Society for Testing and Materials, Philadelphia, Pa., 1979, pp. 166–182.
42.
Suidan, M., and Schnobrich, W. G., “Finite Element Analysis of Reinforced Concrete,” Journal of the Structural Division, ASCE, Vol. 99, No. ST10, Oct., 1973, pp. 2109–2122.
43.
Wnuk, M. P., “Quasi‐Static Extension of a Tensile Crack Contained in Viscoelastic Plastic Solid,” Journal of Applied Mechanics, ASME, Vol. 41, No. 1,

Information & Authors

Information

Published In

Go to Journal of Engineering Mechanics
Journal of Engineering Mechanics
Volume 110Issue 7July 1984
Pages: 1015 - 1035

History

Published online: Jul 1, 1984
Published in print: Jul 1984

Permissions

Request permissions for this article.

Authors

Affiliations

Zdene⁁k P. Baz⁁ant, F. ASCE
Prof. of Civ. Engrg. and Dir., Center of Concrete and Geomaterials, Technological Inst., Northwestern Univ., Evanston, Ill. 60201
Byung H. Oh, A. M. ASCE
Visiting Scholar, Dept. of Civ. Engrg., Northwestern Univ., Evanston Ill. 60201; Visiting Research Engr., Portland Cement Assoc., Skokie, Ill. 60077; presently, Asst. Prof. of Civ. Engrg., Seoul National Univ., Seoul, Korea

Metrics & Citations

Metrics

Citations

Download citation

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited by

View Options

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share