Energy balance criteria for viscoelastic fracture
Authors:
J. M. Golden and G. A. C. Graham
Journal:
Quart. Appl. Math. 48 (1990), 401-413
MSC:
Primary 73M25; Secondary 73B30, 73F15
DOI:
https://doi.org/10.1090/qam/1074956
MathSciNet review:
MR1074956
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Abstract: An energy balance criterion of the Griffith type has been used to derive conditions that are valid, in the isothermal noninertial approximation, for the growth of cracks in viscoelastic bodies. These bodies are acted upon by general position and time-dependent load. The conditions have the same form as the instability conditions obtained for the corresponding problems in elasticity theory and, in particular, are independent of crack velocity. The analysis relies upon an exact calculation of the displacement and stress fields that is derived in the Appendix with the aid of extensions to viscoelasticity of the Kolosov-Muskhelishvili equations of elasticity theory.
A. A. Griffith, The phenomenon of rupture and flow in solids, Phil. Trans. Roy. Soc. A 221, 163–198 (1921)
A. A. Griffith, The theory of rupture, Proc. 1st Intern. Congr. Appl. Mech., Delft, 1924. p. 55–63
P. I. Vincent and K. V. Gotham, Effect of crack propagation velocity on the fracture energy of poly(methylmethacrylate), Nature 210, 1254 (1966)
E. C. Francis, C. H. Carlton, and G. H. Lindsey, Viscoelastic fracture of solid propellants in pressurization loading conditions, J. Spacecraft 11, 691–696 (1974)
W. G. Knauss, Fracture of solids possessing deformation rate sensitive material properties, The Mechanics of Fracture, AMD-Vol. 19, F. Erdogan (Editor), The American Society of Mechanical Engineers, New York, 1976, p. 69–103
M. E. Gurtin, Thermodynamics and the cohesive zone in fracture, J. Appl. Math. Phys. (ZAMP) 30, 991–1003 (1979)
J. R. Willis, Crack propagation is viscoelastic media, J. Mech. Phys. Solids 15, 229–240 (1967)
B. V. Kostrov and L. V. Nikitin, Some general problems of mechanics of brittle fracture, Arch. Mech. Stosow. 6, 749–776 (1970)
W. S. Blackburn, Steady crack growth in a linear viscoelastic material, Int. J. Fract. Mech. 7, 354–356 (1971)
G. A. C. Graham, Two extending crack problems in linear viscoelasticity, Quart. Appl. Math. 27, 497–507 (1970)
G. A. C. Graham, Quasi-static crack growth in linear viscoelastic bodies that are acted upon by alternating tensile and compressive loads, Proc. Roy. Irish Acad. 75A, 263–269 (1975)
G. P. Cherepanov, Crack propagation in continuous media, J. Appl. Math. Mech. 31, 503–512 (1967)
R. J. Nuismer, On the governing equation for quasi-static crack growth in linearly viscoelastic materials, J. Appl. Mech. 41, 631–634 (1974); 42, 521 (1975)
R. M. Christensen and L. N. McCartney, Viscoelastic crack growth, Int. J. Fract. 23, R11–R13 (1983)
W. G. Knauss, The mechanics of polymer fracture, Appl. Mech. Rev. 26, 1–17 (1973)
L. N. McCartney, Crack growth laws for a variety of viscoelastic solids using energy and COD fracture criteria, Int. J. Fract. 15, 31–40 (1979); 16, R27–R30, R109–R110 (1980)
R. A. Schapery, On the analysis of crack initiation and growth in nonhomogeneous viscoelastic media, Fracture Mechanics (R. Burridge, Ed.), American Mathematical Society, Providence. 1979, pp. 137–152
R. A. Schapery, Time-dependent fracture: continuum aspects of crack growth, Encyclopedia of Materials Science and Engineering, Vol. 7, M. B. Bever (Editor-in-Chief), Pergamon Press, Oxford, 1986, pp. 5043–5053
A. A. Kaminskii, Investigations in the field of the mechanics of the fracture of viscoelastic bodies, Soviet Applied Mechanics 16, 741–759 (1980); Mechanics of fracture of viscoelastic bodies (in Russian), Naukova Dumka, Kiev, 1980
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L. Schovanec and J. R. Walton, The energy release rate for a quasi-static mode I crack in a nonhomogeneous linearly viscoelastic body, Engineering Fracture Mechanics 28, 445–454 (1987)
G. Goleniewski, Dynamic crack growth in a viscoelastic material, Int. J. Fract. 37, R39–R44 (1988)
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L. N. McCartney, Mechanics of matrix-cracking in brittle-matrix fibre-reinforced composites, Proc. Roy. Soc. Lond. A409, 329–350 (1987)
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G. A. C. Graham and G. C. W. Sabin, Steady state solutions for a cracked standard linear viscoelastic body, Mech. Res. Commun. 8, 361–368 (1981)
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A. A. Kaminskii and Ya. Ya. Rushchitskii, Applicability of the Volterra principle in the analysis of crack propagation in hereditary elastic media, Soviet Applied Mechanics 5, 102–108 (1969)
- I. N. Sneddon and M. Lowengrub, Crack problems in the classical theory of elasticity, John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0258339
A. A. Griffith, The phenomenon of rupture and flow in solids, Phil. Trans. Roy. Soc. A 221, 163–198 (1921)
A. A. Griffith, The theory of rupture, Proc. 1st Intern. Congr. Appl. Mech., Delft, 1924. p. 55–63
P. I. Vincent and K. V. Gotham, Effect of crack propagation velocity on the fracture energy of poly(methylmethacrylate), Nature 210, 1254 (1966)
E. C. Francis, C. H. Carlton, and G. H. Lindsey, Viscoelastic fracture of solid propellants in pressurization loading conditions, J. Spacecraft 11, 691–696 (1974)
W. G. Knauss, Fracture of solids possessing deformation rate sensitive material properties, The Mechanics of Fracture, AMD-Vol. 19, F. Erdogan (Editor), The American Society of Mechanical Engineers, New York, 1976, p. 69–103
M. E. Gurtin, Thermodynamics and the cohesive zone in fracture, J. Appl. Math. Phys. (ZAMP) 30, 991–1003 (1979)
J. R. Willis, Crack propagation is viscoelastic media, J. Mech. Phys. Solids 15, 229–240 (1967)
B. V. Kostrov and L. V. Nikitin, Some general problems of mechanics of brittle fracture, Arch. Mech. Stosow. 6, 749–776 (1970)
W. S. Blackburn, Steady crack growth in a linear viscoelastic material, Int. J. Fract. Mech. 7, 354–356 (1971)
G. A. C. Graham, Two extending crack problems in linear viscoelasticity, Quart. Appl. Math. 27, 497–507 (1970)
G. A. C. Graham, Quasi-static crack growth in linear viscoelastic bodies that are acted upon by alternating tensile and compressive loads, Proc. Roy. Irish Acad. 75A, 263–269 (1975)
G. P. Cherepanov, Crack propagation in continuous media, J. Appl. Math. Mech. 31, 503–512 (1967)
R. J. Nuismer, On the governing equation for quasi-static crack growth in linearly viscoelastic materials, J. Appl. Mech. 41, 631–634 (1974); 42, 521 (1975)
R. M. Christensen and L. N. McCartney, Viscoelastic crack growth, Int. J. Fract. 23, R11–R13 (1983)
W. G. Knauss, The mechanics of polymer fracture, Appl. Mech. Rev. 26, 1–17 (1973)
L. N. McCartney, Crack growth laws for a variety of viscoelastic solids using energy and COD fracture criteria, Int. J. Fract. 15, 31–40 (1979); 16, R27–R30, R109–R110 (1980)
R. A. Schapery, On the analysis of crack initiation and growth in nonhomogeneous viscoelastic media, Fracture Mechanics (R. Burridge, Ed.), American Mathematical Society, Providence. 1979, pp. 137–152
R. A. Schapery, Time-dependent fracture: continuum aspects of crack growth, Encyclopedia of Materials Science and Engineering, Vol. 7, M. B. Bever (Editor-in-Chief), Pergamon Press, Oxford, 1986, pp. 5043–5053
A. A. Kaminskii, Investigations in the field of the mechanics of the fracture of viscoelastic bodies, Soviet Applied Mechanics 16, 741–759 (1980); Mechanics of fracture of viscoelastic bodies (in Russian), Naukova Dumka, Kiev, 1980
A. A. Kaminskii and V. M. Pestrikov, Kinetics of crack growth in aging viscoelastic bodies, Soviet Applied Mechanics 22, 790–797 (1986)
O. Coussy, Un modèle de viscoélasticité confinée en mécanique de la rupture. Propagation subcritique, C. R. Acad. Sci. Paris Série II 302, 53–56 (1986)
L. Schovanec and J. R. Walton, The energy release rate for a quasi-static mode I crack in a nonhomogeneous linearly viscoelastic body, Engineering Fracture Mechanics 28, 445–454 (1987)
G. Goleniewski, Dynamic crack growth in a viscoelastic material, Int. J. Fract. 37, R39–R44 (1988)
J. M. Golden and G. A. C. Graham, Fatigue crack growth in viscoelastic materials, Arabian J. Sci. Engrg. 9, 77–85 (1984)
G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Advances in Applied Mechanics, Vol. 7, H. L. Dryden and Th. von Karman (Editors), Academic Press, New York and London, 1962
J. M. Golden and G. A. C. Graham. Boundary volume problems in linear viscoelasticity, Springer-Verlag, Berlin, Heidelberg, 1988
F. Erdogan, Crack-propagation theories, Fracture, An Advanced Treatise, Vol. II, Mathematical Fundamentals, H. Liebowitz (Editor), Academic Press, New York, 1968, pp. 497–590
L. N. McCartney, Mechanics of matrix-cracking in brittle-matrix fibre-reinforced composites, Proc. Roy. Soc. Lond. A409, 329–350 (1987)
J. N. Goodier, Mathematical theory of equilibrium cracks, Fracture, An Advanced Treatise, Vol II, Mathematical Fundamentals, H. Liebowitz (Editor), Academic Press, New York, 1968, pp. 1–66
R. M. Christensen, Theory of Viscoelasticity: An Introduction, Academic Press, New York, 1982
I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956
M. E. Gurtin and E. Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 11, 291–356 (1962)
G. A. C. Graham and G. C. W. Sabin, The correspondence principle of linear viscoelasticity for problems that involve time-dependent regions, Internat. J. Engrg. Sci. 11, 123–140 (1973)
A. E. Green and W. Zerna, Theoretical Elasticity, Clarendon Press, Oxford, 1968
L. V. Ahlfors, Complex Analysis, McGraw Hill, New York, 1966
I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951
N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, 1963
A. Erdelyi, Tables of Integral Transforms, Vol. II, McGraw Hill, New York, 1954
J. H. Michell, On the direct determination of stress in an elastic solid, with application to the theory of plates, Proc. Lond. Math. Soc. 31, 100–124 (1899)
G. A. C. Graham and G. C. W. Sabin, The opening and closing of a growing crack in a linear viscoelastic body that is subject to alternating tensile and compressive loads, Int. J. Fract. 14, 639–649 (1978)
G. A. C. Graham and G. C. W. Sabin, Steady state solutions for a cracked standard linear viscoelastic body, Mech. Res. Commun. 8, 361–368 (1981)
L. M. Kachanov, On the kinetics of crack growth, J. Appl. Math. Mech. 25, 739–745 (1961)
A. A. Kaminskii and Ya. Ya. Rushchitskii, Applicability of the Volterra principle in the analysis of crack propagation in hereditary elastic media, Soviet Applied Mechanics 5, 102–108 (1969)
I. N. Sneddon and M. Lowengrub, Crack Problems in the Classical Theory of Elasticity, Wiley, New York, 1969
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© Copyright 1990
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