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Published in: Journal of Engineering Mathematics 1/2017

29-08-2017

On Boussinesq’s problem for a cracked halfspace

Author: A. P. S. Selvadurai

Published in: Journal of Engineering Mathematics | Issue 1/2017

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Abstract

This paper examines the axisymmetric elastostatic problem that deals with the action of a concentrated normal force on the surface of an isotropic elastic halfspace containing a penny-shaped crack. The mathematical formulation of the elasticity problem should take into consideration the sense of action of the concentrated force. The paper presents the development of Fredholm integral equations of the second-kind that are associated with this category of problem and indicates the numerical technique that is adopted for their solution. The numerical results are presented for the stress intensity factors generated at the penny-shaped crack experiencing either opening or closure.

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Literature
1.
go back to reference Boussinesq J (1885) Applications des potentials à l’étude de l’équilibre et du mouvement des solides élastique. Gauthier-Villars, ParisMATH Boussinesq J (1885) Applications des potentials à l’étude de l’équilibre et du mouvement des solides élastique. Gauthier-Villars, ParisMATH
2.
go back to reference Davis RO, Selvadurai APS (1996) Elasticity and geomechanics. Cambridge University Press, Cambridge Davis RO, Selvadurai APS (1996) Elasticity and geomechanics. Cambridge University Press, Cambridge
3.
4.
go back to reference Selvadurai APS (2000) Partial differential equations in mechanics, vol 2. The biharmonic equation, Poisson’s equation. Springer, BerlinCrossRefMATH Selvadurai APS (2000) Partial differential equations in mechanics, vol 2. The biharmonic equation, Poisson’s equation. Springer, BerlinCrossRefMATH
6.
7.
go back to reference Lord Kelvin (1848) Note on the integrations of the equations of equilibrium of an elastic solid. Camb Dublin Math J 3:87–89 Lord Kelvin (1848) Note on the integrations of the equations of equilibrium of an elastic solid. Camb Dublin Math J 3:87–89
8.
go back to reference Love AE (1927) A treatise on the mathematical theory of elasticity. Cambridge University Press, CambridgeMATH Love AE (1927) A treatise on the mathematical theory of elasticity. Cambridge University Press, CambridgeMATH
9.
10.
go back to reference Fröhlich OK (1934) Druckverteilung im baugrunde. Mit besonderer berücksichtigung der plastischen erscheinungen. Julius Springer, WienCrossRef Fröhlich OK (1934) Druckverteilung im baugrunde. Mit besonderer berücksichtigung der plastischen erscheinungen. Julius Springer, WienCrossRef
11.
go back to reference Selvadurai APS (2014) On Fröhlich’s solution for Boussinesq’s problem. Int J Numer Analyt Meth Geomech 38:925–934CrossRef Selvadurai APS (2014) On Fröhlich’s solution for Boussinesq’s problem. Int J Numer Analyt Meth Geomech 38:925–934CrossRef
12.
13.
go back to reference Sneddon IN (1946) The distribution of stress in the neighbourhood of a crack in an elastic solid. Proc R Soc Ser A A228:229–260ADSCrossRefMathSciNet Sneddon IN (1946) The distribution of stress in the neighbourhood of a crack in an elastic solid. Proc R Soc Ser A A228:229–260ADSCrossRefMathSciNet
14.
go back to reference Sack RA (1946) Extension of Griffith’s theory of rupture to three dimensions. Proc Phys Soc 58:729–736ADSCrossRef Sack RA (1946) Extension of Griffith’s theory of rupture to three dimensions. Proc Phys Soc 58:729–736ADSCrossRef
15.
go back to reference Sneddon IN (1972) The use of integral transforms. McGraw Hill, New YorkMATH Sneddon IN (1972) The use of integral transforms. McGraw Hill, New YorkMATH
16.
go back to reference Sneddon IN (ed) (1977) Application of integral transforms in the theory of elasticity. International Centre for Mechanical Sciences, Courses and Lectures No. 220, Springer, Vienna Sneddon IN (ed) (1977) Application of integral transforms in the theory of elasticity. International Centre for Mechanical Sciences, Courses and Lectures No. 220, Springer, Vienna
17.
go back to reference Hunter SC, Gamblen D (1974) The theory of a rigid circular disc ground anchor buried in an elastic soil either with or without adhesion. J Mech Phys Solids 22:371–399ADSCrossRefMATH Hunter SC, Gamblen D (1974) The theory of a rigid circular disc ground anchor buried in an elastic soil either with or without adhesion. J Mech Phys Solids 22:371–399ADSCrossRefMATH
18.
go back to reference Selvadurai APS (1980) Asymmetric displacements of a ground anchor embedded in a transversely isotropic elastic medium of infinite extent. Int J Eng Sci 18:979–986CrossRefMATH Selvadurai APS (1980) Asymmetric displacements of a ground anchor embedded in a transversely isotropic elastic medium of infinite extent. Int J Eng Sci 18:979–986CrossRefMATH
20.
go back to reference Selvadurai APS (1993) The axial loading of a rigid circular anchor plate embedded in an elastic halfspace. Int J Numer Anal Meth Geomech 17:43–353CrossRefMATH Selvadurai APS (1993) The axial loading of a rigid circular anchor plate embedded in an elastic halfspace. Int J Numer Anal Meth Geomech 17:43–353CrossRefMATH
21.
go back to reference Selvadurai APS (1994) On the problem of a detached anchor plate embedded in a crack. Int J Solids Struct 31:1279–1290CrossRefMATH Selvadurai APS (1994) On the problem of a detached anchor plate embedded in a crack. Int J Solids Struct 31:1279–1290CrossRefMATH
22.
go back to reference Selvadurai APS (2002) Mechanics of a rigid circular disc bonded to a cracked elastic halfspace. Int J Solids Struct 39:6035–6053CrossRefMATH Selvadurai APS (2002) Mechanics of a rigid circular disc bonded to a cracked elastic halfspace. Int J Solids Struct 39:6035–6053CrossRefMATH
23.
go back to reference Smith FW, Alavi MJ (1971) Stress intensity factors for a penny-shaped crack in a halfspace. Eng Fract Mech 3:241–254CrossRef Smith FW, Alavi MJ (1971) Stress intensity factors for a penny-shaped crack in a halfspace. Eng Fract Mech 3:241–254CrossRef
24.
go back to reference Srivastava KN, Singh K (1969) The effect of a penny-shaped crack on the distribution of stress in a semi-infinite solid. Int J Eng Sci 7:469–490CrossRefMATH Srivastava KN, Singh K (1969) The effect of a penny-shaped crack on the distribution of stress in a semi-infinite solid. Int J Eng Sci 7:469–490CrossRefMATH
25.
go back to reference Erdogan F, Arin K (1971) Penny-shaped crack in an elastic layer bonded to dissimilar halfspaces. Int J Eng Sci 9:213–232CrossRefMATH Erdogan F, Arin K (1971) Penny-shaped crack in an elastic layer bonded to dissimilar halfspaces. Int J Eng Sci 9:213–232CrossRefMATH
26.
go back to reference Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. Wiley, New YorkMATH Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. Wiley, New YorkMATH
27.
go back to reference Kassir MK, Sih GC (1975) Mechanics of fracture three-dimensional crack problems, vol 2. Noordhoff International, LeydenMATH Kassir MK, Sih GC (1975) Mechanics of fracture three-dimensional crack problems, vol 2. Noordhoff International, LeydenMATH
28.
go back to reference Cherepanov GP (1979) Mechanics of brittle fracture (Translation Editors R. de Witt and W.C. Cooley). McGraw-Hill, New York Cherepanov GP (1979) Mechanics of brittle fracture (Translation Editors R. de Witt and W.C. Cooley). McGraw-Hill, New York
29.
go back to reference Broberg KB (1999) Cracks and fracture. Academic Press, San Diego Broberg KB (1999) Cracks and fracture. Academic Press, San Diego
30.
go back to reference Green AE, Zerna W (1968) Theoretical elasticity. Oxford University Press, OxfordMATH Green AE, Zerna W (1968) Theoretical elasticity. Oxford University Press, OxfordMATH
31.
go back to reference Little RW (1973) Elasticity. Prentice-Hall, Upper Saddle RiverMATH Little RW (1973) Elasticity. Prentice-Hall, Upper Saddle RiverMATH
32.
go back to reference Barber JR (2002) Elasticity. Kluwer Academic Publishers, DordrechtMATH Barber JR (2002) Elasticity. Kluwer Academic Publishers, DordrechtMATH
33.
go back to reference Podio-Guidugli P (2014) Elasticity for geotechnicians: a modern exposition of Kelvin, Boussinesq, Flamant, Cerruti, Melan and Mindlin problems. Solid mechanics and its applications, vol 204. Springer, SwitzerlandMATH Podio-Guidugli P (2014) Elasticity for geotechnicians: a modern exposition of Kelvin, Boussinesq, Flamant, Cerruti, Melan and Mindlin problems. Solid mechanics and its applications, vol 204. Springer, SwitzerlandMATH
34.
go back to reference Baker CTH (1977) The numerical treatment of integral equations. Clarendon Press, OxfordMATH Baker CTH (1977) The numerical treatment of integral equations. Clarendon Press, OxfordMATH
35.
go back to reference Delves LM, Mohamed JL (1985) Computational methods for integral equations. Cambridge University Press, CambridgeCrossRefMATH Delves LM, Mohamed JL (1985) Computational methods for integral equations. Cambridge University Press, CambridgeCrossRefMATH
36.
go back to reference Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge University Press, CambridgeCrossRefMATH Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge University Press, CambridgeCrossRefMATH
40.
41.
go back to reference Selvadurai APS (2000) Fracture evolution during indentation of a brittle elastic solid. Mech Cohesive-Frict Mater 5:325–339CrossRef Selvadurai APS (2000) Fracture evolution during indentation of a brittle elastic solid. Mech Cohesive-Frict Mater 5:325–339CrossRef
Metadata
Title
On Boussinesq’s problem for a cracked halfspace
Author
A. P. S. Selvadurai
Publication date
29-08-2017
Publisher
Springer Netherlands
Published in
Journal of Engineering Mathematics / Issue 1/2017
Print ISSN: 0022-0833
Electronic ISSN: 1573-2703
DOI
https://doi.org/10.1007/s10665-017-9934-6

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