Elasticity, Fracture and Flow

IN this monograph I have attempted to set out, in as elemen­ tary a form as possible, the basic mathematics of the theories of elasticity, plasticity, viscosity, and rheology, together with a discussion of the properties of the materials involved and the way in which they are idealized to form a basis for the mathe­ matical theory. There are many mathematical text-books on these subjects, but they are largely devoted to methods for the solution of special problems, and, while the present book may be regarded as an introduction to these, it is also in­ tended for the large class of readers such as engineers and geologists who are more interested in the detailed analysis of stress and strain, the properties of some of the materials they use, criteria for flow and fracture, and so on, and whose interest in the theory is rather in the assumptions involved in it and the way in which they affect the solutions than in the study of special problems. The first chapter develops the analysis of stress and strain rather fully, giving, in particular, an account of Mohr's repre­ sentations of stress and of finite homogeneous strain in three dimensions. In the second chapter, on the behaviour of materials, the stress-strain relations for elasticity (both for isotropic and simple anisotropic substances), viscosity, plas­ ticity and some of the simpler rheological models are described.

Contenu

I Stress and Strain.- 1. Introductory.- 2. Stress. Definitions and notation.- 3. Stresses in two dimensions.- 4. Stresses in three dimensions.- 5. Mohr's representation of stress in three dimensions.- 6. Displacement and strain. Introduction.- 7. The geometry of finite homogeneous strain in two dimensions.- 8. Finite homogeneous strain in three dimensions.- 9. Mohr's representation of finite homogeneous strain without rotation.- 10. Infinitesimal strain in two dimensions.- 11. Infinitesimal strain in three dimensions.- II Behaviour of Actual Materials.- 12. Introductory.- 13. The stress-strain relations for a perfectly elastic isotropic solid.- 14. Special cases: biaxial stress and strain.- 15. Strain-energy.- 16. Anisotropic substances.- 17. Finite hydrostatic strain.- 18. Natural strain.- 19. The equations of viscosity.- 20. Fracture and yield.- 21. The maximum shear stress theory of fracture and its generalizations.- 22. Mohr's theory of fracture.- 23. Earth pressure.- 24. The Griffith theory of brittle strength.- 25. Strain theories of failure.- 26. The tensile test on ductile materials.- 27. Yield criteria.- 28. The yield surface.- 29. The equations of plasticity.- 30. Substances with composite properties.- III Equations of Motion and Equilibrium.- 31. Introductory.- 32. Simple problems illustrating the behaviour of elastic, viscous, plastic and Bingham substances.- 33. The elastic equations of motion.- 34. The elastic equations of equilibrium.- 35. Special cases of the equations of elasticity.- 36. Special problems in elasticity.- 37. Wave propagation.- 38. Elastic waves.- 39. The equations of motion of a viscous fluid.- 40. Special problems in viscosity.- 41. Plastic flow in two dimensions.- IV Applications.- 42. Introductory.- 43. Experimental results on the mechanical properties of rocks.- 44. Systems having one or more planes of weakness.- 45. Porous media.- 46. Further discussion of criteria for failure.- 47. Stresses and faulting in the crust.- 48. The Coulomb-Navier theory in terms of invariants.- 49. The representation of two-dimensional stress fields.- 50. Stresses around openings.- 51. The use of the complex variable.- 52. Displacements.- 53. Underground measurements and their results.- 54. Measurement of rock properties.- 55. Effects of flaws, size and stress gradient.- 56. The complete stress-strain curve.- V Applications to Structural Geology.- 57. Introductory.- 58. Combination of strains.- 59. Determination of finite strain from deformed objects.- 60. Progressive deformation.- 61. Analysis of strain in folding.- 62. Instability theory: folding and kinking.- 63. Development of preferred orientations of ellipsoidal particles.- Notation.- Author Index.