Elasticity

Solid Mechanics and its Applications

Volumn 172, Issue , 2010, Pages 1-553

  Barber, J R ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

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EID: 80755167223     PISSN: 09250042     EISSN: None     Source Type: Book Series    
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References (167)

1. J.R.Barber, Intermediate Mechanics of Materials, McGraw-Hill, New York, 2000, §2.2.
2. This problem was investigated by E.Reissner, Note on the theorem of the symmetry of the stress tensor, Journal of Mathematics and Physics Vol. 23 (1944), pp.192-194.
3. See also D.B.Bogy and E.Sternberg, The effect of couple-stresses on the corner singularity due to an asymmetric shear loading, International Journal of Solids and Structures, Vol. 4 (1968), pp.159-174.
4. V.V.Meleshko and A.M.Gomilko, Infinite systems for a biharmonic problem in a rectangle, Proceeedings of the Royal Society of London, Vol. A453 (1997), pp.2139-2160. (Pubitemid 127759064)
5. M.L.Williams, Stress singularities resulting from various boundary conditions in angular corners of plates in extension, ASME Journal of Applied Mechanics, Vol. 19 (1952), pp.526-528.
6. D.B.Bogy, Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions, ASME Journal of Applied Mechanics, Vol. 38 (1971), pp.377-386.
7. A problem of this kind was solved by G.G.Adams, A semi-infinite elastic strip bonded to an infinite strip, ASME Journal of Applied Mechanics, Vol. 47 (1980), pp.789-794. (Pubitemid 11415181)
8. J.Dundurs and M.S.Lee, Stress concentration at a sharp edge in contact problems, Journal of Elasticity, Vol. 2 (1972), pp.109-112
9. M.Comninou, Stress singularities at a sharp edge in contact problems with friction, Zeitschrift für angewandte Mathematik und Physik, Vol. 27 (1976), pp.493-499.
10. I.N.Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951.
11. The basic theory of the Mellin transform and its relation to the Fourier transform is explained by I.N.Sneddon, loc. cit. For applications to elasticity problems for the wedge, see E.Sternberg and W.T.Koiter, The wedge under a concentrated couple: A paradox in the two-dimensional theory of elasticity, ASME Journal of Applied Mechanics, Vol. 25 (1958), pp.575-581
12. W.J.Harrington and T.W.Ting, Stress boundary-value problems for infinite wedges, Journal of Elasticity, Vol. 1 (1971), pp.65-81.
13. A problem of this kind was considered by G.Tsamasphyros and P.S.Theocaris, On the solution of the sector problem, Journal of Elasticity, Vol. 9 (1979), pp.271- 281.
14. This requires that the eigenfunction series is complete for this problem. The proof is given by R.D.Gregory, Green's functions, bi-linear forms and completeness of the eigenfunctions for the elastostatic strip and wedge, Journal of Elasticity, Vol. 9 (1979), pp.283-309. (Pubitemid 10424061)
15. For a more detailed discussion of elastic contact problems, see K.L.Johnson, Contact Mechanics, Cambridge University Press, 1985
16. and G.M.L.Gladwell, Contact Problems in the Classical Theory of Elasticity, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980.
17. N.I.Muskhelishvili, Singular Integral Equations, (English translation by J.R.M.Radok, Noordhoff, Groningen, 1953).
18. For applications to elasticity problems, including the above contact problem, see N.I.Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, (English translation by J.R.M.Radok, Noordhoff, Groningen, 1953). For a simpler discussion of the solution of contact problems involving singular integral equations, see K.L.Johnson, loc. cit, §2.7.
19. C.Cattaneo, Sul contatto di corpi elastici, Accademia dei Lincei, Rendiconti, Ser 6, Vol. 27 (1938), pp.342-348, 433-436, 474-478.
20. R.D.Mindlin, Compliance of elastic bodies in contact, ASME Journal of Applied Mechanics, Vol. 17 (1949), pp.259-268.
21. Notice that the assumption is that all points in -c < x < c are in a state of stick throughout the loading process. It is therefore possible to integrate (12.77) and write the boundary condition in terms of h(x) instead of ?h(x). In general, this is possible as long as no point passes from a state of slip to one of stick during the loading. In particular, the stick zone must not advance into the slip zone during loading. For an exhaustive study of the effect of loading history in frictional contact problems, see J.Dundurs and M.Comninou, An educational elasticity problem with friction: Part 1, Loading and unloading paths for weak friction, ASME Journal of Applied Mechanics, Vol. 48 (1981), pp.841-845;
22. Part 2: Unloading for strong friction and reloading, ibid., Vol. 49 (1982), pp.47-51;
23. Part 3: General load paths, ibid, Vol. 50 (1983), pp.77-84.
24. The effect of non-monotonic loading in the related problem of two contacting spheres was considered by R.D.Mindlin and H.Deresiewicz, Elastic spheres in contact under varying oblique forces, ASME Journal of Applied Mechanics, Vol. 21 (1953), pp.327-344.
25. The history-dependence of the friction law leads to quite complex arrangements of slip and stick zones and consequent variation in the loadcompliance relation. These results also find application in the analysis of oblique impact, where neither normal nor tangential loading is monotonic (see N.Maw, J.R.Barber and J.N.Fawcett, The oblique impact of elastic spheres, Wear, Vol. 38 (1976), pp.101-114).
26. K.L.Johnson, Energy dissipation at spherical surfaces in contact transmitting oscillating forces, Journal of Mechanical Engineering Science, Vol. 3 (1961), pp.362- 368.
27. J.Jäger, Half-planes without coupling under contact loading. Archive of Applied Mechanics Vol. 67 (1997), pp.247-259. (Pubitemid 127608068)
28. M.Ciavarella, The generalized Cattaneo partial slip plane contact problem. I Theory, II-Examples. International Journal of Solids and Structures. Vol. 35 (1998), pp.2349-2378.
29. D.A.Hills and D.Nowell, D. Mechanics of Fretting Fatigue. Kluwer, Dordrecht, 1994
30. M.P.Szolwinski and T.N.Farris, Mechanics of fretting fatigue crack formation. Wear Vol. 198 (1996), pp.93-107.
31. M.Ciavarella and D.A.Hills, Some observations on the oscillating tangential forces and wear in general plane contacts, European Journal of Mechanics A-Solids. Vol. 18 (1999), pp.491-497.
32. Note that this applies to the uncoupled problem (β = 0) only. With dissimilar materials, there is generally a leading slip zone and there can also be an additional slip zone contained within the stick zone. This problem is treated by R.H.Bentall and K.L.Johnson, Slip in the rolling contact of dissimilar rollers, International Journal of Mechanical Sciences, Vol. 9 (1967), pp.389-404.
33. F.W.Carter, On the action of a locomotive driving wheel, Proceedings of the Royal Society of London, Vol. A112 (1926), pp.151-157.
34. For a more extensive discussion of frictional problems of this type, see K.L.Johnson, Proceedings of the Royal Society of London, loc. cit., Chapters 5,7,8.
35. L.Bragg and J.F.Nye, A dynamical model of a crystal structure, Proceedings of the Royal Society of London, Vol. A190 (1947), pp.474-481.
36. For more details of the extensive development of the field of Fracture Mechanics, the reader is referred to the many excellent texts on the subject, such as M.F.Kanninen and C.H.Popelar, Advanced Fracture Mechanics, Clarendon Press, Oxford, 1985
37. H.Leibowitz, ed., Fracture, An Advanced Treatise, 7 Vols., Academic Press, New York, 1971.
38. Stress intensity factors for a wide range of geometries are tabulated by G.C.Sih, Handbook of Stress Intensity Factors, Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, PA, 1973.
39. I.S.Gradshteyn and I.M.Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980.
40. See for example J.Dundurs, Concentrated force in an elastically embedded disk, ASME Journal of Applied Mechanics, Vol.30 (1963), pp.568-570
41. J. Dundurs and G. P. Sendeckyj, Edge dislocation inside a circular inclusion, Journal of the Mechanics and Physics of Solids, Vol.13 (1965), pp.141-147.
42. K.Aderogba, An image treatment pf elastostatic transmission from an interface layer, Journal of the Mechanics and Physics of Solids, Vol. 51 (2003), pp.267-279. (Pubitemid 35412414)
43. nd.ed., Clarendon Press, Oxford, 1959.
44. See for example J.Dundurs, Distortion of a body caused by free thermal expansion, Mechanics Research Communications, Vol. 1 (1974), pp.121-124.
45. For more details, see J.R.Barber, Some implications of Dundurs' Theorem for thermoelastic contact and crack problems, Journal of Strain Analysis, Vol. 22 (1980), pp.229-232. (Pubitemid 12441866)
46. J.R.Barber, Intermediate Mechanics of Materials, McGraw-Hill, Burr Ridge, 2000, Chapter 6
47. th edn., 1993, Chapter 6
48. W.B.Bickford, Advanced Mechanics of Materials, Addison Wesley, Menlo Park, 1998, Chapter 3.
49. See for example E.T.Copson, An Introduction to the Theory of Functions of a Complex Variable, Clarendon Press, Oxford, 1935, §4.21
50. G.F.Carrier, M.Krook and C.E.Pearson, Functions of a Complex Variable, McGraw-Hill, New York, 1966, §2.2.
51. Explicit forms of the integral (2.11) in terms of the strain components were developed by E.Cesaro and are known as Cesaro integrals. See, for example, A.E.H.Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn., Dover, 1944, §156A.
52. For a more rigorous discussion of this question, see A.E.H.Love, loc.cit. or B.A.Boley and J.H.Weiner, Theory of Thermal Stresses, John Wiley, New York, 1960, §§3.6-3.8.
53. H.M.Westergaard, Theory of Elasticity and Plasticity, Dover, New York, 1964, §66.
54. th.edn., Dover, New York, 1944, §188.
55. See for example, H.M.Westergaard, loc. cit. §69 and R.D.Mindlin, Note on the Galerkin and Papcovich stress functions, Bulletin of the American Mathematical Society, Vol. 42 (1936), pp.373-376.
56. For further discussion of the question, see R.A.Eubanks and E.Sternberg, On the completeness of the Boussinesq-Papcovich stress functions, Journal of Rational Mechanics and Analysis, Vol. 5 (1956), pp.735-746.
57. P.M.Naghdi and C.S.Hsu, On a representation of displacements in linear elasticity in terms of three stress functions, Journal of Mathematics and Mechanics, Vol. 10 (1961), pp.233-246
58. T.Tran Cong and G.P.Steven, On the respresentation of elastic displacement fields in terms of three harmonic functions, Journal of Elasticity, Vol. 9 (1979), pp.325- 333.
59. J.Boussinesq, Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques, Gauthier-Villars, Paris, 1885.
60. nd.edn., Clarendon Press, Oxford, 1968, pp.165-176.
61. Other relations between the Boussinesq potentials are demonstrated by J.P.Bentham, Note on the Boussinesq-Papkovich stress functions, J.Elasticity, Vol. 9 (1979), pp.201-206.
62. See for example A.S.Saada, Elasticity, Pergamon Press, New York, 1973, §6.7.
63. A.E.Green and W.Zerna, Elasticity, loc. cit. §§5.8-5.10.
64. A.E.Green, The elastic equilibrium of isotropic plates and cylinders, Proceedings of the Royal Society of London, Vol. A195 (1949), pp.533-552.
65. W. E. Williams, A solution of the steady-state thermoelastic equations, Zeitschrift für angewandte Mathematik und Physik, Vol. 12 (1961), pp.452-455.
66. Further consequences of this result are discussed in J.R.Barber, Some implications of Dundurs' theorem for thermoelastic contact and crack problems, Journal of Strain Analysis, Vol. 22 (1980), pp.229-232. (Pubitemid 12441866)
67. S.P.Timoshenko and J.N.Goodier, Journal of Strain Analysis, loc. cit., §136.
68. K.Aderogba, On eigenstresses in dissimilar media, Philosophical Magazine, Vol. 35 (1977), pp.281-292.
69. This problem was first solved by R.D.Mindlin, Force at a point in the interior of a semi-infinite solid, Physics, Vol.7 (1936), pp.195-202.
70. A more compendious list of solutions for various singularities in the interior of the half space was given by R.D.Mindlin and D.H.Cheng, Nuclei of strain in the semi-infinite solid, Journal of Applied Physics, Vol.21 (1950), pp.926-930.
71. This solution is known as the centre of rotation. 2 See for example I.S.Gradshteyn and I.M.Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980, §8.70.
72. See I.S.Gradshteyn and I.M.Ryzhik, loc. cit. §§8.81, 8.82, 8.91 for this and more results concerning Legendre functions and polynomials.
73. nd edn., 1959, §16.
74. See for example, A.I.Lur'e, Three-Dimensional Problems of the Theory of Elasticity, Interscience, New York, 1964, Chapter 8.
75. L.M.Milne-Thomson, Antiplane Elastic Systems, Springer, Berlin, 1962.
76. D.Iȩsan, On Saint-Venant's Problem, Archive for Rational Mechanics and Analysis, Vol. 91 (1986), pp.363-373.
77. J.J.Kalker, Variational principles of contact elastostatics, Journal of the Institute for Mathematics and its Applications, Vol. 20 (1977), pp.199-221.
78. N.Kikuchi and J.T.Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988.
79. For more details of this argument see J.R.Barber, Determining the contact area in elastic contact problems, Journal of Strain Analysis, Vol. 9 (1974), pp.230-232.
80. See also R.T.Shield, Load-displacement relations for elastic bodies, Zeitschrift für angewandte Mathematik und Physik, Vol. 18 (1967), pp.682-693.
81. K.L.Johnson, K.Kendall and A.D.Roberts, Surface energy and the contact of elastic solids, Proceedings of the Royal Society of London, Vol. A324 (1971), pp.301- 313.
82. J.A.Greenwood, Adhesion of elastic spheres, Proceedings of the Royal Society of London, Vol. A453 (1997), pp.1277-1297. (Pubitemid 127758619)
83. D.A.Spence, An eigenvalue problem for elastic contact with finite friction, Proceedings of the Cambridge Philosophical Society, Vol. 73 (1973), pp.249-268, has shown that this argument also extends to problems with Coulomb friction at the interface, in which case, the zones of stick and slip also remain self-similar with monotonically increasing indentation force.
84. See I.S.Gradshteyn and I.M.Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980, §1.622.1.
85. rd. edn., 1970, §98.
86. I.N.Sneddon, Note on a boundary-value problem of Reissner and Sagoci, Journal of Applied Physics, Vol. 18 (1947), pp.130-132.
87. G.M.L.Gladwell, Contact Problems in the Classical Theory of Elasticity, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980.
88. E.C.Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1937.
89. I.W.Busbridge, Dual integral equations, Proceedings of the London Mathematical Society, Ser.2 Vol. 44 (1938), pp.115-129
90. I.N.Sneddon, The elementary solution of dual integral equations, Proceedings of the Glasgow Mathematical Association, Vol. 4 (1960), pp.108-110.
91. G.M.L.Gladwell, Proceedings of the Glasgow Mathematical Association, loc. cit., Chapters 5,10.
92. A.E.Green and W.Zerna, Proceedings of the Glasgow Mathematical Association, loc. cit.
93. W.D.Collins, On the solution of some axisymmetric boundary-value problems by means of integral equations, II: Further problems for a circular disc and a spherical cap, Mathematika, Vol. 6 (1959), pp.120-133.
94. A.E.H.Love, Boussinesq's problem for a rigid cone, Quarterly Journal of Mathematics, Vol. 10 (1939), pp.161-175.
95. I.N.Sneddon, Boussinesq's problem for a flat-ended cylinder, Proceedings of the Cambridge Philosophical Society, Vol. 42 (1946), pp.29-39.
96. A.E.Green and W.Zerna, Proceedings of the Glasgow Mathematical Association, loc.cit., §§5.8-5.10.
97. C.M.Segedin, The relation between load and penetration for a spherical punch, Mathematika, Vol. 4 (1957), pp.156-161.
98. N.Maw, J.R.Barber and J.N.Fawcett, The oblique impact of elastic spheres, Wear, Vol. 38 (1976), pp.101-114.
99. W.D.Collins, On the solution of some axisymmetric boundary-value problems by means of integral equations, IV: Potential problems for a circular annulus, Proceedings of the Edinburgh Mathematical Society, Vol. 13 (1963), pp.235-246.
100. W.D.Collins, On the solution of some axisymmetric boundary-value problems by means of integral equations, II: Further problems for a circular disc and a spherical cap, Mathematika, Vol. 6 (1959), pp.120-133.
101. C.M.Segedin, The relation between load and penetration for a spherical punch, Mathematika, Vol. 4 (1957), pp.156-161.
102. E.T.Copson, On the problem of the electrified disc, Proceedings of the Edinburgh Mathematical Society, Vol. 8 (1947), pp.14-19.
103. A.E.Green, On Boussinesq's problem and penny-shaped cracks, Proceedings of the Cambridge Philosophical Society, Vol. 45 (1949), pp.251-257.
104. This solution was first given by K.L.Johnson, K.Kendall and A.D.Roberts, Surface energy and the contact of elastic solids, Proceedings of the Royal Society of London, Vol.A324 (1971), pp.301-313.
105. I.N.Sneddon, The distribution of stress in the neighbourhood of a crack in an elastic solid, Proceedings of the Royal Society of London, Vol. A187 (1946), pp.226- 260.
106. This problem was first solved by A.L.Florence and J.N.Goodier, The linear thermoelastic problem of uniform heat flow disturbed by a penny-shaped insulated crack, International Journal of Engineering Science, Vol. 1 (1963), pp.533-540.
107. D.B.Bogy, Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading, ASME Journal of Applied Mechanics, Vol. 35 (1968), pp.460-466.
108. D.B.Bogy, The plane solution for joined dissimilar elastic semistrips under tension, ASME Journal of Applied Mechanics., Vol. 42 (1975), pp.93-98.
109. See for example J.R.Barber, The penny-shaped crack in shear and related contact problems, International Journal of Engineering Science, Vol. 13, (1975), pp.815- 832.
110. L.Rubenfeld, Non-axisymmetric thermoelastic stress distribution in a solid containing an external crack, International Journal of Engineering Science, Vol. 8 (1970), pp.499-509
111. J.R.Barber, Steady-state thermal stresses in an elastic solid containing an insulated penny-shaped crack, Journal of Strain Analysis, Vol. 10 (1975), pp.19-24.
112. This problem was first considered by V.I.Mossakovskii and M.T.Rybka, Generalization of the Griffith-Sneddon criterion for the case of a non-homogeneous body, Journal of Applied Mathematics and Mechanics, Vol. 28 (1964), 1277-1286
113. and by F.Erdogan, Stress distribution on bonded dissimilar materials containing circu lar or ring-shaped cavities, ASME Journal of Applied Mechanics, Vol. 32 (1965), pp.829-836.
114. A formulation for the penny-shaped crack with more general loading was given by J.R.Willis, The penny-shaped crack on an interface,Quarterly Journal of Mechanics and Applied Mathematics, Vol. 25 (1972), pp.367-382.
115. F.G.Tricomi, Integral Equations, Interscience, New York, 1957, Chapter 2.
116. N.I.Muskhelishvili, Singular Integral Equations, (English translation by J.R.M.Radok, Noordhoff, Groningen, 1953)
117. M.K.Kassir and A.M.Bregman, The stress-intensity factor for a penny-shaped crack between two dissimilar materials, ASME Journal of Applied Mechanics, Vol. 39 (1972), pp.308-310.
118. M.Comninou, The interface crack, ASME Journal of Applied Mechanics, Vol. 44 (1977), pp.631-636.
119. M.Comninou and D.Schmueser, The interface crack in a combined tension-compression and shear field, ASME Journal of Applied Mechanics, Vol. 46 (1979), pp.345-358.
120. A.K.Gautesen and J.Dundurs, The interface crack in a tension field, ASME Journal of Applied Mechanics, Vol. 54 (1987), pp.93-98;
121. A.K.Gautesen and J.Dundurs, The interface crack under combined loading, ASME Journal of Applied Mechanics, ibid., Vol. 55 (1988), pp.580-586.
122. L.M.Keer, S.H.Chen and M.Comninou, The interface penny-shaped crack reconsidered, International Journal of Engineering Science, Vol. 16 (1978), pp.765-772.
123. D.A.Hills and J.R.Barber, Interface cracks, International Journal of Mechanical Sciences, Vol. 35 (1993), pp.27-37. (Pubitemid 23612199)
124. See for example, H.A.Richard, Examination of brittle fracture criteria for overlapping mode I and mode II loading applied to cracks, in G.C.Sih, E.Sommer and W.Dahl, eds., Application of Fracture Mechanics to Structures, Martinus Nijhoff, The Hague, 1984, pp.309-316. (Pubitemid 15503650)
125. J.R.Rice, Elastic fracture mechanics concepts for interfacial cracks, ASME Journal of Applied Mechanics, Vol. 55 (1988), pp.98-103. (Pubitemid 18590050)
126. For a more comprehensive review of variational methods in elasticity, the reader is referred to S.G.Mikhlin, Variational Methods in Mathematical Physics, Pergamon, New York, 1964 and S.P.Timoshenko and J.N.Goodier, loc. cit., Chapter 8.
127. For more detailed discussion of the finite element method, see O.C.Zienkiewicz, The Finite Element Method, McGraw-Hill, New York, 1977
128. K.J.Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982
129. T.J.R.Hughes, The Finite Element Method, Prentice Hall, Englewood Cliffs, NJ, 1987.
130. nd.ed. 1956, §27.
131. See for example, J.E.Marsden and T.J.R.Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, 1983, Chapter 6.
132. E.Sternberg and R.A.Eubanks, On the concept of concentrated loads and an extension of the uniqueness theorem in the linear theory of elasticity, Journal of Rational Mechanics and Analysis, Vol. 4 (1955), pp.135-168.
133. E.Sternberg and V.Koiter, The wedge under a concentrated couple: A paradox in the two-dimensional theory of elasticity, ASME Journal of Applied Mechanics, Vol.25 (1958), pp.575-581
134. X. Markenscoff, Some remarks on the wedge paradox and Saint Venant's principle, ASME Journal of Applied Mechanics, Vol.61 (1994), pp.519-523.
135. J.D.Renton, Generalized beam theory applied to shear stiffness, International Journal of Solids and Structures, Vol. 27 (1991), pp.1955-1967
136. R.T.Shield, Load-displacement relations for elastic bodies, Zeitschrift für angewandte Mathematik und Physik, Vol. 18 (1967), pp.682-693.
137. An approximate solution to this problem for a punch of fairly general plan-form is given by V.I.Fabrikant, Flat punch of arbitrary shape on an elastic half-space, International Journal of Engineering Science, Vol. 24 (1986), pp.1731-1740.
138. Fabrikant's solution and the above reciprocal theorem argument are used as the bases of an approximate solution of the general smooth punch problem by J.R.Barber and D.A.Billings, An approximate solution for the contact area and elastic compliance of a smooth punch of arbitrary shape, International Journal of Mechanical Sciences, Vol. 32 (1990), pp.991-997.
139. This proof was first given by J.N.Goodier, Proceedings of the 3rd U.S. National Congress on Applied Mechanics, (1958), pp.343-345.
140. For a good historical survey of the development of potential function methods in Elasticity, see H.M.Westergaard, Theory of Elasticity and Plasticity, Dover, New York, 1964, Chapter 2.
141. J.Dundurs, Discussion on 'Edge bonded dissimilar orthogonal elastic wedges under normal and shear loading', ASME Journal of Applied Mechanics, Vol. 36 (1969), pp.650-652.
142. For an alternative proof of this result see C.R.Calladine and J.A.Greenwood, Line and point loads on a non-homogeneous incompressible elastic half-space, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 31 (1978), pp.507- 529. (Pubitemid 9422252)
143. I.N.Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951.
144. A.Erdelyi, ed., Tables of Integral Transforms, Bateman Manuscript Project, California Institute of Technology, Vol.1, McGraw-Hill, New York, 1954.
145. M.I.G.Bloor and M.J.Wilson, An approximate solution method for the biharmonic problem, Proceedings of the Royal Society of London, Vol. A462, pp.1107- 1121.
146. R.D.Gregory, Green's functions, bi-linear forms and completeness of the eigenfunctions for the elastostatic strip and wedge, J.Elasticity, Vol. 9 (1979), pp.283- 309; (Pubitemid 10424061)
147. yy (0, y) are prescribed, J.Elasticity, Vol. 10 (1980), pp.57-80;
148. R.D.Gregory, The traction boundary value problem for the elastostatic semi-infinite strip; existence of solution and completeness of the Papkovich-Fadle eigenfunctions, J.Elasticity, Vol. 10 (1980), pp.295- 327. These papers also include extensive references to earlier investigations of the problem.
149. V.V.Meleshko and A.M.Gomilko, Infinite systems for a biharmonic problem in a rectangle, Proceeedings of the Royal Society of London, Vol. A453 (1997), pp.2139- 2160. (Pubitemid 127759064)
150. The reader should be warned that there are several typographical errors in this paper, some of which are corrected in V.V.Meleshko and A.M.Gomilko, Infinite systems for a biharmonic problem in a rectangle: further discussion, Proceeedings of the Royal Society of London, Vol. A460 (2004), pp.807-819. (Pubitemid 38333261)
151. For more information about elastodynamic problems, the reader is referred to the classical texts of J.D.Achenbach, Wave Propagation in Elastic Solids, North Holland, Amsterdam, 1973
152. and A.C.Eringen and E.S.Şuhubi, Elastodynamics, Academic Press, New York, 1975.
153. For a more detailed discussion of the impact of elastic bodies, see K.L.Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985, §11.4
154. W.Goldsmith, Impact, Arnold, London, 1960.
155. A.S.Saada, Elasticity, Pergamon Press, New York, 1973, §6.9.
156. G.N.Savin, Stress Concentration around Holes, Pergamon Press, Oxford, 1961.
157. R.E.Peterson, Stress Concentration Design Factors, John Wiley, New York, 1974.
158. J.H.Michell, On the direct determination of stress in an elastic solid, with application to the theory of plates, Proceedings of the London Mathematical Society, Vol. 31 (1899), pp.100-124.
159. See J.R.Barber, Intermediate Mechanics of Materials, McGraw-Hill, Burr Ridge, 2000, Chapter 5.
160. See for example J.R.Barber, Intermediate Mechanics of Materials, McGraw-Hill, Burr Ridge, 2000, Chapter 4.
161. N.I.Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, P.Noordhoff, Groningen, 1963
162. A.C.Stevenson, Some boundary problems of two-dimensional elasticity, Philosophical Magazine, Vol. 34 (1943), pp.766-793
163. A.C.Stevenson, Complex potentials in two-dimensional elasticity, Proceedings of the Royal Society of London, Vol. A184 (1945), pp.129-179
164. A.E.Green and W.Zerna, Theoretical Elasticity, Clarendon Press, Oxford, 1954
165. I.S.Sokolnikoff, Mathematical Theory of Elasticity, 2nd.ed., McGraw-Hill, New York, 1956.
166. L.M.Milne-Thomson, Antiplane Elastic Systems, Springer, Berlin, 1962
167. L.M.Milne-Thomson, Plane Elastic Systems, 2nd edn, Springer, Berlin, 1968.