Advertisement

Stability Analysis of Incipient Folding and Faulting of an Elasto-Plastic Layer on a Viscous Substratum

Chapter

Abstract

The initiation of two modes of instability, viz. folding and faulting, is investigated theoretically for an elasto-plastic, frictional- cohesive layer which is underlain by a viscous substratum. The destabilizing factors that are allowed to come into play are the tectonic stress, buoyancy forces resulting from a gravitationally unstable density stratification, and the redistribution of material at the top surface by erosion and deposition processes. The bending stiffness of the overburden, a non-linear function of stress, has a stabilizing influence. A variational formulation of the stability problem allows one to detect the onset of global modes of instability, such as folds and surface modes for compression as well as neck-type modes for extension. Predictions of these global modes remain valid as long as the local condition of strong ellipticity is satisfied. Failure of this condition marks the onset of discontinuities in the velocity field, which are characteristic of localized faulting. The sensitivity of these predictions to the assumed behaviour of the overburden and substratum materials is explored for a prototype representative of a dip section through the Campos salt basin on the Brazilian continental margin. These results illustrate the importance of a proper selection of analogue materials for the design of physical laboratory models. This point is underscored by employing a deformation theory of plasticity which could be seen to reproduce in a simple manner the accommodation of bulk deformation by slip along a population of pervasive small faults in sedimentary rocks as well as in analogue materials, such as sands, used in the laboratory. A historical account of the use of deformation theories of plasticity in stability analyses and a derivation of the relevant incremental moduli are also given in this paper.

Keywords

Yield Surface Localize Fault Stress Rate Deformation Theory Plasticity Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arthur, J.R.F., Dunstan, T., Al-Ani, Q.A.J, and Assadi, A. (1977). Plastic deformation and failure in granular media. Géotechnique, 27, 53–74CrossRefGoogle Scholar
  2. Batdorf, S.B. and Budiansky, B. (1949). A mathematical theory of plasticity based on the concept of slip. U.S. N.A.C.A. Technical Note 1871.Google Scholar
  3. Biot, M.A. (1961). Theory of folding of stratified visco-elastic media and its implication in tectonics and orogenesis. Geol. Soc. Am. Bull., 72, 1595–1620.CrossRefGoogle Scholar
  4. Biot, M.A. and Odé, H. (1965). Theory of gravity instability with variable overburden and compaction. Geophysics, 30, 213–227.CrossRefGoogle Scholar
  5. Breckels, I.M. and van Eekelen, H.A.M. (1982). Relationship between horizontal stress and depth in sedimentary basins, J. Pet. Tech., 2191–2199Google Scholar
  6. Budiansky, B. (1959). A reassessment of deformation theories of plasticity J. of Applied Mechanics, 259–264Google Scholar
  7. Cobbold, P.R. and Szatmari, P. (1991). Radial gravitational gliding on passive margins. Tectonophysics, 188, 249–289.CrossRefGoogle Scholar
  8. Costin, L.S. (1985). Damage mechanics in the post-failure regime, Mechanics of Materials, 4, 149–160.CrossRefGoogle Scholar
  9. Demercian, S., Szatmari, P. and Cobbold, P.R. (1993). Style and pattern of salt diapirs due to thin-skinned gravitational gliding, Campos and Santos basins, offshore Brazil. Tectonophysics, 228, 393–433.CrossRefGoogle Scholar
  10. Drucker, D.C. and Prager, W. (1952). Soil Mechanics and plastic analysis or limit design. Q. J. Appl. Math., 10, 157–165.Google Scholar
  11. Hill, R. (1950). The Mathematical Theory of Plasticity, Clarendon Press, Oxford.Google Scholar
  12. Hill, R. (1958). A general theory of uniqueness and stability in elasic-plastic solids. J. Mech. Phys. Solids, 6, 236–249.CrossRefGoogle Scholar
  13. Hill, R. (1967). The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids, 15, 79–95.CrossRefGoogle Scholar
  14. Horii, H. and Nemat-Nasser, S. (1983). Estimate of stress intensity factors for interact-ing cracks. In: Advances in Aerospace Structures, Material and Dynamics, 111–117, edited by U. Yuceoglu, R.L. Sierakowski & D.A. Glasgow, ASME, New-York. Google Scholar
  15. Hutchinson, J.W. (1970). Elastic-plastic behaviour of polycrystalline metals and com-posites. Proc. Roy. Soc. Lond., A-319, 247–272.CrossRefGoogle Scholar
  16. Hutchinson, J.W. (1974). Plastic Buckling. Adv. Appl. Mech., 14, pp. 67–144.CrossRefGoogle Scholar
  17. Koiter, W.T. (1953). Stress-strain relations, uniqueness and variational theorem for elastic-plastic materials with a singular yield surface. Q. Appl. Math., 11, 350–354.Google Scholar
  18. Kötter, F. (1903). Die Bestimmung des Druckes an Gekrümmten Gleitflächen, Ber. Akad. der Wiss., Berlin.Google Scholar
  19. Lehner, F.K. & Kachanov, M.L. (1995). On the stress-strain relations for cracked elastic materials in compression. In: Mechanics of Jointed & Faulted Rock, 49–61 edited by H.P. Rossmanith, Balkema, Rotterdam. Google Scholar
  20. Leroy, Y.M. & Triantafyllidis, N. (1996). Stability of a frictional, cohesive layer on a substratum: variational formulation and asymptotic solution. J. Geophys. Res., 101, B8, 17795–17811.CrossRefGoogle Scholar
  21. Leroy, Y.M. & Triantafyllidis, N. (1998). Stability of layered geological structures: an asymptotic solution. In: Material Instabilities in Solids, 15–25, edited by E. van der Giessen & R. de Borst, J. Wiley and Sons, New York. Google Scholar
  22. Leroy, Y.M. and Sassi, W. (1999). A plasticity model for discontinua. This Volume. Google Scholar
  23. Mandel, J. (1962). Essais sur modèles réduits en mécanique des terrains. Etude des conditions de similitude. Rev. de l’Industrie Minérale, 44, 611–620. (English trans-lation in Int. J. Rock Mech. Mining Sci., 1, 31–42, 1963.).Google Scholar
  24. Mandl, G. (1988). Mechanics of tectonic faulting. Elsevier, Amsterdam.Google Scholar
  25. Mandl, G. and Fernandez Luque, R. (1970). Fully developed plastic shear flow of granular materials. Géotechnique, 20, 277–307.CrossRefGoogle Scholar
  26. Massin, P., Triantafyllidis, N. & Leroy, Y.M. (1996). Stability of density-stratified two-layer system. C. R. Acad. Sei., Sér. IIa, Tectonique, 322, 407–413.Google Scholar
  27. McGarr, A. and Gay, N.C. (1978). State of stress in the Earth’s crust. Ann. Rev. Earth Planet. Sci., 6, 405–436.CrossRefGoogle Scholar
  28. Neale, K.W. (1981). Phenomenological constitutive laws in finite plasticity. Solid Mechanics Archive, 6, 79–128.Google Scholar
  29. Nettleton, L.L. and Elkins, A. (1947). Geologic models made from granular materials. Trans. Am. Geophys. Union, 28, 451–466.CrossRefGoogle Scholar
  30. Nieuwland, D.A. (1994). Personal communication.Google Scholar
  31. Odé, H. (1960). Faulting as a velocity discontinuity in plastic deformation. In: Rock Deformation, edited by D. Griggs & J. Handin, Geol. Soc. Am. Memoir 79, pp. 293–321. CrossRefGoogle Scholar
  32. Ogden, R.W. (1984). Non-Linear Elastic Deformations, Ellis Horwood, Chichester, England.Google Scholar
  33. Olsen, W.A. (1992). The formation of yield-surface vertex in rock. In: Proc. 33rd U.S. Symposium Rock Mech., pp. 701–705, Balkema, Rotterdam. Google Scholar
  34. Poirier, J.P. (1980). Shear localization and shear instability in materials in the ductile field. J. Structural Geology, 2, 135–142.CrossRefGoogle Scholar
  35. Poliakov, A.N.B., Podladchikov, Y. and Talbot, C. (1993). Initiation of salt diapirs with frictional overburdens: numerical experiments. Tectonophysics, 228, 199–210.CrossRefGoogle Scholar
  36. Ramberg, H.R. and Stephansson, O. (1964). Compression of floating elastic and viscous plates affected by gravity, a basis for discussing crustal buckling. Tectonophysics, 1, 101–120.CrossRefGoogle Scholar
  37. Rice, J.R. (1976). The localization of plastic deformation. In: Theoretical and Applied Mechanics, Proc. of the 14th IUTAM Conference, edited by W. Koiter, pp. 207–220, North Holland, Amsterdam. Google Scholar
  38. Rudnicki, J.W. (1984). A class of elastic-plastic constitutive laws for brittle rock. J. of Rheology, 28, 759–778.CrossRefGoogle Scholar
  39. Rudnicki, J.W. and Rice, J.R. (1975). Conditions for the localization of the deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids, 23, 371–394.Google Scholar
  40. Rudnicki, J.W. and Chau, K.T. (1996). Multiaxial response of microcrack constitutive model for brittle rock. In: NARMS’96, 2, edited by M, Aubertin, F. Hassani and H. Mitri, pp. 1707–1714, Balkema, Rotterdam. Google Scholar
  41. Sanders, J.L. (1954). Plastic stress-strain relations based on linear loading functions. In: Proc. Second US Nat. Cong. Appl. Mech., edited by P.M. Naghdi, ASME, 455–460.Google Scholar
  42. Sassi, W. and Faure, J.L. (1996). Role of faults and layer interfaces on the spatial variation of stress regime in basins: Inference from numerical modelling. Tectonophysics, in press.Google Scholar
  43. Shanley, F.R. (1947). Inelastic column theory. J. of the Aeronautical Sciences, 14, 261–267.CrossRefGoogle Scholar
  44. Shi, Z.H. and Horii, H. (1989). Microslip model of strain localization in sand deformation. Mechanics of Materials, 8, 89–102.CrossRefGoogle Scholar
  45. Smoluchowski, M. (1909). Über ein gewisses Stabilitätsproblem der Elastizitätslehre und dessen Beziehung zur Entstehung von Faltengebirgen. Abhandl. Akad. Wiss Krakau, Math. Kl, 3–20. Google Scholar
  46. Sokolovski, V.V. (1960). Statics of soil media. Butterworth, London.Google Scholar
  47. Spencer, A.J.M. (1964). A theory of the kinematics of ideal soils under plane strain conditions. J. Mech. Phys. Solids, 12, 337–351.CrossRefGoogle Scholar
  48. Spencer, A.J.M. (1971). Discussion of “Fully developed shar flow of granular materials, by G. Mandl and R. Fernandez-Luque”. Géotechnique, 21, 190–192.CrossRefGoogle Scholar
  49. Spencer, A.J.M. (1982). Deformation of ideal granular materials. In: H.G. Hopkins & M.J. Sewell (editors), aMechanics of Solids, Pergamon.Google Scholar
  50. Stören, S. and Rice, J.R. (1975). Localized necking in thin sheets. J. Mech. Phys. Solids, 23, 421–441.CrossRefGoogle Scholar
  51. Triantafyllidis, N., (1985). Puckering instability phenomena in the hemispherical cup test. J. Mech. Phys. Solids, 33, 117–139.CrossRefGoogle Scholar
  52. Triantafyllidis, N., Needleman, A. and Tvergaard, V. (1982). On the development of shear bands in pure banding. Int. J. Solids and Structures, 18, 121–138.CrossRefGoogle Scholar
  53. Triantafyllidis, N. and Lehner, F. K. (1993). Interfacial instability of density-stratified two-layer systems under initial stress. J. Mech. Phys. Solids, 41, 117–142.CrossRefGoogle Scholar
  54. Triantafyllidis, N. and Leroy, Y.M. (1994). Stability of a frictional material layer resting on a viscous half-space. J. Mech. Phys. Solids, 42, 51–110.CrossRefGoogle Scholar
  55. Triantafyllidis, N. & Leroy, Y.M. (1997). Stability of a frictional, cohesive layer on a substratum: validity of asymptotic solution and influence of material properties, J. Geophys. Res., 102, B9, 20551–20570.CrossRefGoogle Scholar
  56. Tvergaard, V., Needleman, A. and Lo, K.K. (1981). Flow localization in the plane strain tensile test. J. Mech. Phys. Solids, 2, 115–142.CrossRefGoogle Scholar
  57. Vendeville, B. & Cobbold, P.R. (1987). Glissements gravitaire synsédimentaires et failles normales listriques: modéles expérimentaux. C. R. Acad. Sci. Paris, 305, Série II, 1313–1319.Google Scholar
  58. Vendeville, B.C. and Jackson, M.P.A. (1992). The rise of diapirs during thin-skinned extension. Marine and Petroleum Geology, 9, 331–353.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  1. 1.Laboratoire de Mécanique des Solides Ecole PolytechniqueU.M.R. C.N.R.S. no. 7649Palaiseau CedexFrance
  2. 2.Department of Aerospace EngineeringThe University of MichiganAnn ArborUSA

Personalised recommendations