Skip to main content
SpringerLink
Log in

The British School of Elasticity, Plasticity and Defects: Applied Mathematics

  • Chapter
  • First Online:
Continuum Mechanics Through the Twentieth Century

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 196))

  • 2581 Accesses

Abstract

Although pertaining to specific aspects of the development of continuum mechanics in the period of interest, it happens that this coincides with a technical expertise in applied mathematics particularly well cultivated in the United Kingdom, hence, an unavoidable regional bias in spite of the international nature of science. The prevailing influence of some institutions such as the University of Cambridge is obvious, while, unexpectedly, research fostered by technical problems met during the Second World War, also had a strong influence on the selection of projects. A recurring theme is a specific interest in mathematical problems posed by the theory of elasticity, no doubt a consequence of the enduring influence of past “elasticians” of great mathematical dexterity among whom A.E.H. Love must be singled out. A clear-cut emphasis was placed on problems dealing with the existence of field singularities such as happens with cracks, dislocations, and other material defects. Here great names are those of A.A. Griffith, Ian Sneddon, “Jock” Eshelby, and A.N. Stroh. Simultaneously, an “immoderate” but fruitful taste for problems of elastic wave propagation with applications in both mechanics and geophysics was demonstrated and still remains a subject of attraction. Furthermore, a geometrical approach to defect theory was proposed by a group around Bruce Bilby, while Rodney Hill produced among the most powerful results in plasticity theory and homogenisation procedure. Still it is the mathematical dexterity and elegance allied to a deep physical insight that best characterizes most of these works.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Ashby MF, Gibson LJ (1997) Cellular solids structure and properties. Cambridge University Press, Cambridge

    Google Scholar 

  • Ashby MF, Jones DRH (1996, 1998) Engineering materials I, II, Second edn. Butterworth, Oxford

    Google Scholar 

  • Atkinson C, Eshelby JD (1968) Flow of energy into the tip of a moving crack. Int J Fract Mech 4:3–8

    Google Scholar 

  • Ball JM (1977) Convexity conditions and existence theorems in nonlinear elasticity. Arch Rat Mech Anal 65:193–201

    Article  MATH  Google Scholar 

  • Ball JM, James RD (1987) Fine phase mixtures as minimizers of energy. Arch Rat Mech Anal 100:13–52

    Article  MathSciNet  MATH  Google Scholar 

  • Ball JM, Murat F (1984) W1, p quasi-convexity and variational problems for multiple integrals. J Funct Anal 58:225–253

    Article  MathSciNet  MATH  Google Scholar 

  • Batchelor G (1994) The life and legacy of G.I. Taylor. Cambridge University Press, Cambridge

    Google Scholar 

  • Bilby BA, Bullough R, Smith E (1955) Continuous distributions of dislocations: new application of the methods of non-Riemannian geometry. Proc Roy Soc Lond A231:263–273

    MathSciNet  Google Scholar 

  • Bilby BA, Gardner LRT, Stroh AN (1957) Continuous distributions of dislocations and the theory of plasticity, Actes du Xième Congrès International de Mécanique Appliquée (Bruxelles, 1956), Vol VIII. Université de Bruxelles, Belgique, pp 35–44

    Google Scholar 

  • Bilby BA, Swinden KH, Cottrell AH (1964) Plastic yielding from sharp notches. Proc Roy Soc Lond A279:1

    Google Scholar 

  • Chadwick P (1956) Elastic waves propagated in a magnetic field, In: Proceeding Congrès International de Mécanique Appliquée, Brussels, Vol VII. Université de Bruxelles, Belgique

    Google Scholar 

  • Chadwick P (1960) Thermoelasticity. The dynamical theory. In: Sneddon IN, Hill R (eds) Progress in solid mechanics, Vol I. North-Holland, Amsterdam, pp 263–328

    Google Scholar 

  • Chadwick P (1976) Continuum mechanics. George Allen and Unwin, London

    Google Scholar 

  • Chadwick P (1997) The application of the stroh formalism to prestressed elastic media. Math Mech solids 4:379–403

    Article  MathSciNet  Google Scholar 

  • Chadwick P, Smith GD (1977) Foundations of the theory of surface waves in anisotropic elastic materials. Adv Appl Mech 17:303–376

    Article  MATH  Google Scholar 

  • Chadwick P, Sneddon IN (1958) Plane waves in an elastic solid conducting heat. J Mech Phys Solids 6:223–230

    Article  MathSciNet  MATH  Google Scholar 

  • Cherepanov GP (ed) (1998) Fracture: a topical encyclopaedia of current knowledge. Krieger, Malabar

    Google Scholar 

  • Collins IF, Houlsby GT (1997) Application of thermomechanical principles to the modelling of geotechnical materials. Proc Roy Soc Lond A453:1975–2001

    Google Scholar 

  • Cottrell AH, Bilby BA (1949) Dislocation theory of yielding and strain ageing of iron. Proc Phys Soc Lond A62(349):49–62

    Google Scholar 

  • Eshelby JD (1951) Force on an elastic singularity. Phil Tran Roy Soc Lond A244:87–112

    Article  MathSciNet  Google Scholar 

  • Eshelby JD (1956) Continuum Theory of lattice Defects. In: Seitz F, Turnbull D (ed) Progress in solid state physics, Vol 3. Academic, New York, p 79

    Google Scholar 

  • Eshelby JD (1957) The determination of the elastic field of inclusion and related problems. Proc Roy Soc Lond A241:376–396

    MathSciNet  Google Scholar 

  • Eshelby JD (1961) Elastic inclusions and inhomogeneities. In: Sneddon IN, Hill RM (eds) Progress in solid mechanics, Vol 2. North-Holland, Amsterdam, pp 89–140

    Google Scholar 

  • Eshelby JD (1970) Energy relations and the energy-momentum tensor in continuum mechanics. In: Kanninen MF, Adler WF, Rosenfeld AR, Jaffe RI (eds) Inelastic behavior of solids, McGraw Hill, New York, pp 77–114

    Google Scholar 

  • Eshelby JD (1971) Fracture mechanics. Sci Prog 59:161–179

    Google Scholar 

  • Eshelby JD (1975a) Elastic energy-momentum tensor. J Elast 5:321–335

    Article  MathSciNet  MATH  Google Scholar 

  • Eshelby JD (1975b) Calculation of energy-release rates. In: Sih GC (ed) Prospects of fracture mechanics, Nordhoff, Leyden

    Google Scholar 

  • Eshelby JD (1982) Aspects of dislocation theory. In: Hopkins HG, Sewell MJ (eds) Mechanics of solids (The Rodney Hill 60th Anniversary Volume), Pergamon Press, Oxford, pp 185–255

    Google Scholar 

  • Eshelby JD, Bilby BA (1968) Dislocations and the theory of fracture. In: Liebowitz H (ed) Fracture, an advanced treatise, Vol 1. Academic, New York, pp 9–182

    Google Scholar 

  • Eshelby JD, Frank FC, Nabarro FRN (1951) The equilibrium of linear arrays of dislocations. Phil Mag 42:351–364

    MathSciNet  MATH  Google Scholar 

  • Fleck NA, Willis JR (2009) A mathematical basis for strain gradient plasticity, I, II. J Mech Phys Solids 57:181–197 and 57:1045–1057

    Google Scholar 

  • Fu YB (2007) Hamiltonian interpretation of the Stroh formalism in anisotropic elasticity. Proc Roy Soc Lond A463:3073–3087

    Google Scholar 

  • Gilman JJ (1998) Alan Arnold Griffith, An appreciation. In: Cherepanov GP (ed) Fracture: a topical encyclopaedia of current knowledge, Krieger, Malabar, pp xix–xxii

    Google Scholar 

  • Griffith AA (1920) The phenomena of rupture and flow in solids. Phil Trans Roy Soc Lond 221:16–98

    Google Scholar 

  • Griffith AA, Taylor GI (1917) The use of soap films in solving torsion problems. Proc Inst Mech Eng 93(1):755–809

    Google Scholar 

  • Hill R (1948) A variational principle of maximum plastic work in classical plasticity. Q J Mech Appl Math 1:18–28

    Article  MATH  Google Scholar 

  • Hill R (1950) The mathematical theory of plasticity. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids 12:213–222

    Article  Google Scholar 

  • Hill R (1987) The essential structure of constitutive laws for metal composites and polycrystals. J Mech Phys Solids 15:79–95

    Article  Google Scholar 

  • Hopkins HG, Sewell MJ (eds) (1982) Mechanics of solids: The Rodney Hill 60th anniversary volume. Pergamon Press, Oxford

    MATH  Google Scholar 

  • Houlsby GT, Puzin AM (2002) Rate-dependent plasticity models derived from potential functions. J Rheol 46(1):113–126

    Article  Google Scholar 

  • Houlsby GT, Puzin AM (2006) Principles of hyperplasticity: an approach to plasticity theory based on thermodynamic principles. Springer, London

    Google Scholar 

  • Jeffreys H, Jeffreys BW (1946) Methods of mathematical physics. Cambridge University Press, UK

    MATH  Google Scholar 

  • Knops R (1958) On the variation of Poisson’s ratio in the solution of elastic problems. Q J Mech Appl Math 11:326–350

    Article  MathSciNet  MATH  Google Scholar 

  • Knops R (1966) Two-dimensional electrostriction. Q J Mech Appl Math 16:377–388

    Article  MathSciNet  Google Scholar 

  • Knops R (2001) Elements of elastic stability theory. In: Hayes M, Saccomandi G (eds) Topics in finite elasticity (CISM Lecture notes, Udine, Italy), Springer, Vienna, pp 169–230

    Google Scholar 

  • Knops R, Payne LE (1971) Uniqueness theorems in linear elasticity. Springer, New York

    Book  MATH  Google Scholar 

  • Knops R, Wilkes EW (1973) Theory of elastic stability. In: Truesdell CA, Flügge S (eds) Handuch der Physik, Bd. VI/3, Springer, Berlin

    Google Scholar 

  • Kröner E (1972) Statistical continuum mechanics, (CISM Lecture Notes Udine Italy). Springer, Vienna

    Google Scholar 

  • Lekhnitskii SG (1963) Theory of elasticity of an anisotropic body (translation from the Russian). Holden-Day, San Francisco

    Google Scholar 

  • Love AEH (1892) A Treatise on the mathematical theory of elasticity, (1944, 4th edn. Dover reprint, New York)

    Google Scholar 

  • Love AEH (1911) Some problems of geodynamics, Cambridge University Press, Cambridge, (where “Love waves” are introduced)

    Google Scholar 

  • Mandel J (1971) Plasticité classique et viscoplasticité (CISM Lecture Notes Udine Italy). Springer, Vienna

    Google Scholar 

  • Maugin GA (1985) Nonlinear electromechanical effects and applications. World Scientific, Singapore

    Google Scholar 

  • Maugin GA (1992) The thermomechanics of plasticity and fracture. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Maugin GA (1993) Material inhomogeneities in elasticity. Chapman & Hall, London

    MATH  Google Scholar 

  • Maugin GA (2011) Configurational forces. CRC/Chapman & Hall/Taylor and Francis, Boca Raton

    MATH  Google Scholar 

  • Morrey CB (1952) Quasi-convexity and the lower semi-continuity of multiple integrals. Pacific J Math 2:25–53

    Article  MathSciNet  MATH  Google Scholar 

  • Mott NF, Sneddon IN (1948) Wave mechanics and its applications. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Rayleigh JWS (1887) On waves propagated along the plane surface of an elastic solid, Proc. Lond Math Soc 17:4–11

    Google Scholar 

  • Sneddon IN (1946) The distribution of stress in the neighbourhood of a crack in an elastic solid. Proc Roy Soc Lond A187:229–260

    MathSciNet  Google Scholar 

  • Sneddon IN (1951) Fourier transforms. McGraw Hill, New York

    Google Scholar 

  • Sneddon IN, Berry DS, (1958) The classical theory of elasticity. In: Flügge S (ed) Handbuch der Physik, Vol VI. Springer, Berlin, pp 1–126

    Google Scholar 

  • Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. John Wiley, New York

    MATH  Google Scholar 

  • Stoneley R (1924) Elastic waves at the surface of separation of two solids. Proc Roy Soc Lond A106:416–428

    Google Scholar 

  • Straughan B, (1991) The energy method, stability, and nonlinear convection, Springer, Berlin (2nd edn 2004)

    Google Scholar 

  • Straughan B (1998) Explosive instabilities in mechanics. Springer, Berlin

    Book  MATH  Google Scholar 

  • Stroh AN (1954) The formation of cracks as a result of plastic flow. Proc Roy Soc Lond. A223:404–414

    MathSciNet  Google Scholar 

  • Stroh AN (1955a) The formation of cracks in plastic flow. II. Proc Roy Soc Lond A232:548–560

    Google Scholar 

  • Stroh AN (1955b) Brittle fracture and yielding. Phil Mag 46:968–972

    Google Scholar 

  • Stroh AN (1957) A theory of the fracture of metals. Adv Phys 6:418–465

    Article  Google Scholar 

  • Stroh AN (1958) Dislocations and cracks in anisotropic elasticity. Phil Mag 3:597–606

    Article  MathSciNet  Google Scholar 

  • Stroh AN (1960) A simple model of a propagating crack. J Mech Phys Sol 8:119–122

    Article  Google Scholar 

  • Stroh AN (1962a) Steady state problems in anisotropic elasticity. J Math Phys 41:77–103

    MathSciNet  MATH  Google Scholar 

  • Stroh AN (1962b) The force on a moving dislocation. Phys Rev 128:55–61

    Article  Google Scholar 

  • Talbot DRS, Willis JR (1982) Variational estimates for dispersion and attenuation of waves in random composites, I, II. Int J Solids Struct 18:673–683 and 18:685–698

    Google Scholar 

  • Taylor GI (1934) The mechanism of plastic deformation of crystals Part I-Theoretical. Proc Roy Soc Lond A145:362–404

    Google Scholar 

  • Taylor GI, Quinney H (1931) The plastic distorsion of metals. Phil Trans Roy Soc Lond 230:323–362

    MATH  Google Scholar 

  • Ting TCT (1996) Anisotropic elasticity: theory and applications. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Westergaard HM (1939) Bearing pressure and cracks. J Appl Mech 6A:49–53

    Google Scholar 

  • Willis JR (1964) Anisotropic elastic inclusion problems. Q J Mech Appl Math 17:157–174

    Article  MathSciNet  MATH  Google Scholar 

  • Willis JR (1965) Dislocations and inclusions. J Mech Phys Solids 13:377–395

    Article  Google Scholar 

  • Willis JR (1967) Second-order effects of dislocations in anisotropic crystals. Int J Eng Sci 5:171–180

    Article  MATH  Google Scholar 

  • Willis JR (1971) Fracture mechanics of interfacial cracks. J Mech Phys Solids 19:353–368

    Article  MATH  Google Scholar 

  • Willis JR (1972) The penny-shaped crack on an interface. Q J Mech Appl Math 25:367–385

    Article  MATH  Google Scholar 

  • Willis JR (1977) Bounds and self-consistent estimates fore the overall properties of anisotropic composites. J Mech Phsy Solids 25:185–202

    Article  MATH  Google Scholar 

  • Willis JR (1980) A polarization approach to the scattering of elastic waves, I, II. J Mech Phys Solids 28:287–305, 307–327

    Google Scholar 

  • Willis JR (1981a) Variational principles for dynamic problems for inhomogeneous elastic media. Wave Potion 3:1–11

    Article  MathSciNet  MATH  Google Scholar 

  • Willis JR (1981b) Variational and related methods for the overall properties of composites. In: Yih CS (ed) Advances in applied mechanics, vol 21. Academic Press, New York, pp 1–78

    Google Scholar 

  • Willis JR (1991) On methods for bounding the overall properties of nonlinear composites. J Mech Phys Solids 39:73–86

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, Place Jussieu 4, Paris Cedex 05, 75252, France

    Gérard A. Maugin

Authors
  1. Gérard A. Maugin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gérard A. Maugin .

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Maugin, G.A. (2013). The British School of Elasticity, Plasticity and Defects: Applied Mathematics. In: Continuum Mechanics Through the Twentieth Century. Solid Mechanics and Its Applications, vol 196. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6353-1_6

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-6353-1_6

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-6352-4

  • Online ISBN: 978-94-007-6353-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation