Abstract
A mechanical model based upon elasticity theory for the deformation around a pressurized elliptical hole in an homogeneous, isotropic solid has found application in many areas of engineering, rock mechanics, and structural geology. The explicit equations for stress and displacement around such a hole are given. An apl computer program for calculating these stresses and displacements also is presented. These equations and this program should ease future usage of this model by engineers and geologists.
Similar content being viewed by others
References
Anderson, E. M., 1938, The dynamics of sheet intrusion: Proc. Royal Soc. Edinburgh, v. 58, pt. 3, p. 242–251.
Anderson, E. M., 1951, The dynamics of faulting and dyke formation with applications to Britain: Oliver and Boyd, Edinburgh and London, 206 p.
Brace, W. F., 1960, An extension of Griffith theory of fracture to rocks: Jour. Geophysical Res. v. 65, no. 10, p. 3477–3480.
Brace, W. F., 1964, Brittle fracture of rocks,in State of stress in the Earth's crust: Am. Elsevier Publ. Co., Inc., New York, p. 111–178.
Chinnery, M. A., 1961, The deformation of the ground surface around surface faults: Bull. Seismological Soc. America, v. 51, no. 3, p. 355–372.
Gramberg, J., 1970, The “ellipse-with-notch” theory to explain axial cleavage fracturing of rocks (a natural extension of the first Griffith theory): Intern. Jour. Rock Mechanics and Mining Sci. v. 7, no. 5, p. 537–559.
Griffith, A. A., 1921, The phenomena of rupture and flow in Solids: Philosophical Trans., Royal Soc. London, v. 221, ser. A, p. 163–198.
Griffith, A. A., 1924, Theory of rupture: Proc. First Intern. Cong. Applied Mechanics: Techniscke Backhandel Wattman, Delft., p. 55–63.
Hoek, E., and Bieniawski, Z. T., 1965, Brittle fracture propagation in rock under compression: Intern. Jour. of Fracture Mechanics, v. 1, no. 3, p. 137–155.
Inglis, C. E., 1913, Stresses in a plate due to the presence of cracks and sharp corners: Royal Inst. of Naval Architects Trans., v. 55 pt. 1, p. 219–230.
Irwin, G. R., 1957, Analysis of stresses and strains near the end of a crack traversing a plate: Jour. Applied Mechanics, v. 24, p. 361–364.
Irwin, G. R., 1960, Fracture mechanics,in Structural mechanics—proceedings of first symposium of naval structural mechanics: Pergamon Press, New York, p. 557–592.
Jaeger, J. C., and Cook, N. G. W., 1969, Fundamentals of rock mechanics: Methuen and Co. Ltd., London, 513 p.
Kolosoff, G., 1914, Uber einige Eigenschaften des ebenen Problems der Elastizitatstheorie: Zeitschrift fur Math. und Physik, v. 62, p. 384–409.
Lajtai, E. Z., 1971, A theoretical and experimental evaluation of the Griffith theory of brittle fracture: Tectonophysics, v. 11, no. 2, p. 129–156.
McClain, W. C., 1968, Rock mechanics in the disposal of radioactive wastes by hydraulic fracturing: Felsmechanick und Ingenieurgeol., v. 6, p. 131–161.
McClintock, F. A., and Walsh, J. B., 1962, Friction of Griffith cracks under pressure: Proc. Fourth U.S. Cong. on Applied Mechanics, Am. Soc. Mechanical Eng., New York, p. 1015–1021.
Muskhelishvili, N. I., 1963, Some basic problems of the mathematical theory of elasticity (4th ed.): P. Noordhoff, Ltd., Groningen, 718 p.
Neuber, H., and Hahn, H. G., 1966, Stress concentration in scientific research and engineering: App. Mechanics Rev., v. 19, no. 3, p. 187–199.
Obert, L., Duvall, W. I., and Merrill, R., 1960, Design of underground openings in competent rocks: U.S. Bur. Mines, Bull. 587, 36 p.
Ode, H., 1956, A note concerning the mechanism of artificial and natural hydraulic fracturing systems: Colorado Sch. Mines Quart. v. 51, no. 1, p. 19–29.
Pollard, D. D., 1972, Derivation and evaluation of a model for sheet intrusions: Tectonophysics, in press.
Pollard, D. D., and Johnson, A. M., 1972, Mechanics of growth of laccolithic intrusions in the Henry Mountains, Utah: Tectonophysics, in press.
Roberts, J. L., 1970, The intrusion of magma into brittle rocks,in Mechanism of igneous intrusion: Gallery Press, Liverpool, p. 287–338.
Savin, G. N., 1961, Stress concentration around holes: Pergamon Press, New York, 430 p.
Secor, D. T., 1965, Role of fluid pressure in jointing: Am. Jour. Sci., v. 263, no. 8, p. 633–646.
Secor, D. T., 1969, Mechanics of natural extension fracturing at depth in the Earth's crust,in Proceedings, conference on research in tectonics (kink bands and brittle deformation): Geol. Survey of Canada, GSC Paper 68-52, p. 3–48.
Sneddon, I. N., 1946, The distribution of stress in the neighborhood of a crack in an elastic solid: Proc. Royal Soc. London, v. 187, ser. A, p. 229–260.
Sneddon, I. N., and Lowengrub, M., 1969, Crack problems in the classical theory of elasticity: John Wiley & Sons, Inc., New York, 221 p.
Stevenson, A. C., 1945, Complex potentials in two-dimensional elasticity: Proc. Royal Soc. London, v. 184, ser. A, p. 129–179.
Sun, R. J., 1969, Theoretical size of hydraulically induced horizontal fractures and corresponding surface uplift in an idealized medium: Jour. Geophysical Res., v. 74, no. 25, p. 5995–6011.
Thomas, G. B., 1961, Calculus and analytic geometry: Addison-Wesley Publ. Co., Inc., Mass., 1010 p.
Timoshenko, S., and Goodier, J. N., 1951, Theory of elasticity: McGraw-Hill Book Co., Inc., New York, 506 p.
Walsh, J. B., 1965a, The effect of cracks on the compressibility of rock. Jour. Geophysical Res., v. 70, no. 2, p. 381–389.
Walsh, J. B., 1965b, The effect of cracks on the uniaxial elastic compression of rocks: Jour. Geophysical Res., v. 70, no. 2, p. 399–411.
Walsh, J. B., 1965c, The effect of cracks in rocks in Poisson's ratio: Jour. Geophysical Res., v. 70, no. 20, p. 5249–5257.
Weertman, J., 1971a, Theory of water-filled crevasses in glaciers applied to vertical magma transport beneath oceanic ridges: Jour. Geophysical Res., v. 76, no. 5, p. 1171–1183.
Weertman, J., 1971b, Velocity at which liquid-filled cracks move in the Earth's crust or in glaciers: Jour. Geophysical Res. v. 76, no. 35, p. 8544–8553.
Westergaard, H. M., 1939, Bearing pressures and cracks: Am. Soc. Mechanical Eng., Trans., v. 61, p. A49-A53.
Williams, M. L., 1957, On the stress distribution at the base of a stationary crack: Jour. Appl. Mechanics, v. 24, no. 1, p. 109–114.
Williams, M. L., 1959, The stresses around a fault or crack in dissimilar media: Bull. Seismological Soc. America, v. 49, no. 2, p. 199–204.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pollard, D.D. Equations for stress and displacement fields around pressurized elliptical holes in elastic solids. Mathematical Geology 5, 11–25 (1973). https://doi.org/10.1007/BF02114084
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02114084