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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**A. A. Griffith,*The phenomenon of rupture and flow in solids*, Phil. Trans. Roy. Soc. A**221**, 163-198 (1921)**[2]**A. A. Griffith,*The theory of rupture*, Proc. 1st Intern. Congr. Appl. Mech., Delft, 1924. p. 55-63**[3]**P. I. Vincent and K. V. Gotham,*Effect of crack propagation velocity on the fracture energy of poly(methylmethacrylate)*, Nature**210**, 1254 (1966)**[4]**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)**[5]**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**[6]**M. E. Gurtin,*Thermodynamics and the cohesive zone in fracture*, J. Appl. Math. Phys. (ZAMP)**30**, 991-1003 (1979)**[7]**J. R. Willis,*Crack propagation is viscoelastic media*, J. Mech. Phys. Solids**15**, 229-240 (1967)**[8]**B. V. Kostrov and L. V. Nikitin,*Some general problems of mechanics of brittle fracture*, Arch. Mech. Stosow.**6**, 749-776 (1970)**[9]**W. S. Blackburn,*Steady crack growth in a linear viscoelastic material*, Int. J. Fract. Mech.**7**, 354-356 (1971)**[10]**G. A. C. Graham,*Two extending crack problems in linear viscoelasticity*, Quart. Appl. Math.**27**, 497-507 (1970)**[11]**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)**[12]**G. P. Cherepanov,*Crack propagation in continuous media*, J. Appl. Math. Mech.**31**, 503-512 (1967)**[13]**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)**[14]**R. M. Christensen and L. N. McCartney,*Viscoelastic crack growth*, Int. J. Fract.**23**, R11-R13 (1983)**[15]**W. G. Knauss,*The mechanics of polymer fracture*, Appl. Mech. Rev.**26**, 1-17 (1973)**[16]**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)**[17]**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**[18]**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**[19]**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**[20]**A. A. Kaminskii and V. M. Pestrikov,*Kinetics of crack growth in aging viscoelastic bodies*, Soviet Applied Mechanics**22**, 790-797 (1986)**[21]**Olivier Coussy,*Un modèle de viscoélasticité confinée en mécanique de la rupture. Propagation subcritique*, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre**302**(1986), no. 2, 53–56 (French, with English summary). MR**977370****[22]**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)**[23]**G. Goleniewski,*Dynamic crack growth in a viscoelastic material*, Int. J. Fract.**37**, R39-R44 (1988)**[24]**J. M. Golden and G. A. C. Graham,*Fatigue crack growth in viscoelastic materials*, Arabian J. Sci. Engrg.**9**(1984), no. 2, 77–85 (English, with Arabic summary). MR**765094****[25]**G. I. Barenblatt,*The mathematical theory of equilibrium cracks in brittle fracture.*, Advances in Applied Mechanics, Vol. 7, Academic Press, New York, 1962, pp. 55–129. MR**0149728****[26]**J. M. Golden and G. A. C. Graham,*Boundary value problems in linear viscoelasticity*, Springer-Verlag, Berlin, 1988. MR**958684****[27]**F. Erdogan,*Crack-propagation theories*, Fracture, An Advanced Treatise, Vol. II, Mathematical Fundamentals, H. Liebowitz (Editor), Academic Press, New York, 1968, pp. 497-590**[28]**L. N. McCartney,*Mechanics of matrix-cracking in brittle-matrix fibre-reinforced composites*, Proc. Roy. Soc. Lond.**A409**, 329-350 (1987)**[29]**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**[30]**R. M. Christensen,*Theory of Viscoelasticity: An Introduction*, Academic Press, New York, 1982**[31]**I. S. Sokolnikoff,*Mathematical theory of elasticity*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. 2d ed. MR**0075755****[32]**M. E. Gurtin and Eli Sternberg,*On the linear theory of viscoelasticity*, Arch. Rational Mech. Anal.**11**(1962), 291–356. MR**0147047**, https://doi.org/10.1007/BF00253942**[33]**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**(1973), no. 1, 123–140 (English, with French, German, Italian and Russian summaries). MR**0455723**, https://doi.org/10.1016/0020-7225(73)90074-8**[34]**A. E. Green and W. Zerna,*Theoretical elasticity*, Second edition, Clarendon Press, Oxford, 1968. MR**0245245****[35]**Lars V. Ahlfors,*Complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR**510197****[36]**Ian N. Sneddon,*Fourier Transforms*, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1951. MR**0041963****[37]**N. I. Muskhelishvili,*Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending*, Translated from the Russian by J. R. M. Radok, P. Noordhoff, Ltd., Groningen, 1963. MR**0176648****[38]**A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,*Tables of integral transforms. Vol. II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR**0065685****[39]**J. H. Michell,*On the Direct Determination of Stress in an Elastic Solid, with application to the Theory of Plates*, Proc. London Math. Soc.**S1-31**, no. 1, 100. MR**1576699**, https://doi.org/10.1112/plms/s1-31.1.100**[40]**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*, Internat. J. Fracture**14**(1978), no. 6, 639–649 (English, with French summary). MR**600045**, https://doi.org/10.1007/BF00116002**[41]**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)**[42]**L. M. Kačanov,*On the kinetics of crack growth*, J. Appl. Math. Mech.**25**(1961), 739–745. MR**0136119**, https://doi.org/10.1016/0021-8928(61)90043-0**[43]**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)**[44]**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**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73M25,
73B30,
73F15

Retrieve articles in all journals with MSC: 73M25, 73B30, 73F15

Additional Information

DOI:
https://doi.org/10.1090/qam/1074956

Article copyright:
© Copyright 1990
American Mathematical Society