Science China Physics, Mechanics and Astronomy

, Volume 54, Issue 7, pp 1309–1318

# Transient near tip fields in crack dynamics

• Yuri Petrov
• Alexander Utkin
Research Paper

## Abstract

Transient effects of stress-strain fields in the vicinity of a stationary crack tip under high rate loads are discussed. Exact analytical solutions to near tip stresses are compared to fields prescribed by leading terms (one or several) of Williams asymptotic expansion. Influence of load application mode, time (or, which is the same, distance from a crack tip) and Poisson’s ratio on this discrepancy is extensively examined. Some effects connected with crack tip propagation speed are also discussed. Significant inconsistencies between real (or received in numerical solutions of state equations - e.g. finite element computations) crack tip fields and stress intensity factor (SIF) singular field observed by numerous researchers are explained. The scope of problems where SIF field can be used for correct prediction of dynamic stress-strain fields in the crack tip region is established. Possibility to correctly approximate fields that are not SIF dominated, accounting additional terms of Williams expansion, is studied.

## Keywords

transient crack tip fields dynamic fracture high-rate loads asymptotic expansions

## List of main symbols

σij

components of the Cauchy stress tensor

t

time

r,θ

polar coordinates with origin at the crack tip

KI

the first term of the asymptotic expansion of stresses surrounding the tip of the mode I loaded crack (the mode I stress intensity factor)

Rn

the second and the following terms of the asymptotic expansion of stresses surrounding the crack tip

ϕij

angular functions

x(x1, x2)

Cartesian coordinate

W=W(t,x)

displacement field

u,v

components of displacement

µ

shear modulus

ν

Poisson’s ratio

E

Young’s modulus

$$c_1 = \sqrt {\frac{{1 - v}} {{\left( {1 + v} \right)\left( {1 - 2v} \right)}}\frac{E} {\rho }} = \sqrt {\frac{{2\left( {1 - v} \right)}} {{1 - 2v}}\frac{\mu } {\rho }}$$

longitudinal wave speed

$$c_2 = \sqrt {\frac{\mu } {\rho }}$$

transversal wave speed

cR

Rayleigh wave speed

f(t)

P

H

Heaviside step function equal to 0 if the argument is negative and equal to 1 otherwise

φ,ψ

longitudinal and transversal wave potentials

ρ

mass density

$$\gamma = {{c_2 } \mathord{\left/ {\vphantom {{c_2 } {c_1 }}} \right. \kern-\nulldelimiterspace} {c_1 }} = \sqrt {\frac{{1 - 2v}} {{2\left( {1 - v} \right)}}}$$

ratio of longitudinal and transverse wave speeds

γR=cR/c1

ratio of Rayleigh and longitudinal wave speeds

Si

the sum of the first i terms of the asymptotic expansion

σc

the critical value for tensile stress (ultimate strength)

KIC

the critical value for stress intensity factor

τ

the microstructural time of a brittle fracture process (or fracture incubation time) — a parameter characterizing the response of the studied material to applied dynamic loads. τ is constant for a given material and does not depend on problem geometry, the way a load is applied, the shape of a load pulse and its amplitude

d

characteristic size of fracture process zone

## References

1. 1.
Yoffe E H. The moving Griffith crack. Philos Mag, 1951, 42: 739–750
2. 2.
Ang W T. Transient response of a crack in an anisotropic strip. Acta Mech, 1987, 70: 97–109
3. 3.
Ang W T. A crack in an anisotropic layered material under the action of impact loading. J Appl Mech, 1988, 55: 120–125
4. 4.
Freund L B. The mechanics of dynamic shear crack propagation. J Geophys Res, 1979, 84: 2199–2209
5. 5.
Achenbach J D. Extension of a crack by a shear wave. Z Angew Math Phys, 1970, 21: 887–900
6. 6.
Achenbach J D. Crack propagation generated by a horizontally polarized shear wave. J Mech Phys Solids, 1970, 18: 245–259
7. 7.
Achenbach J D. Dynamic effects in brittle fracture. Mechanics Today. Nemat-Nasser S, ed. New York: Pergamon, Elmsford, 1974. 1–57Google Scholar
8. 8.
Eshelby J D. The elastic field of a crack extending nonuniformly under general anti-plane loading. J Mech Phys Solids, 1969, 17: 177–199
9. 9.
Broberg K B. The propagation of a brittle crack. Arch Fysik, 1960, 18: 159–192
10. 10.
11. 11.
Kostrov B V. Unsteady propagation of longitudinal shear cracks. Appl Math Mech, 1966, 30: 1241–1248
12. 12.
Kostrov B V, Nikitin L V. Some general problems of mechanics of brittle fracture. Arch Mchaniki Stosowanej, 1970, 22: 749–775
13. 13.
Kostrov B V. On the crack propagation with variable velocity. Int J Fract, 1975, 11: 47–56
14. 14.
Williams M L. On the stress distribution at the base of a stationary crack. J Appl Mech, 1957, 24: 109–114
15. 15.
Irwin G R. Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech, 1957, 24: 361–364Google Scholar
16. 16.
Ma C C, Freund L B. The extent of the stress intensity factor during crack growth under dynamic loading conditions. J Appl Mech, 1986, 53: 303–310
17. 17.
Filippov A V. Some problems of diffraction of plane elastic waves (in Russian). Appl Math Mech, 1965, 20: 688–703Google Scholar
18. 18.
Petrov Y V, Utkin A A. Asymptotic representation of stresses surrounding the crack tip in dynamic problems of elasticity. IPME RAS, St.-Petersburg, 2001Google Scholar
19. 19.
Petrov Y V, Morozov N F. On the modeling of fracture of brittle solids. J Appl Mech, 1994, 61: 710–712
20. 20.
Petrov Y V, Morozov N F, Smirnov V I. Structural macromechanics approach in dynamics of fracture. Fatigue Fract Eng Mater Struct, 2003, 26: 363–372
21. 21.
Morozov N, Petrov Y. Dynamics of fracture. Berlin: Springer-Verlag, 2000
22. 22.
Freund L B. Dynamic Fracture Mechanics. Cambridge: Cambridge University Press, 1990
23. 23.
Broberg K B. The near-tip field at high crack velocities. Int J Fract, 1989, 39: 1–13
24. 24.
Broberg K B. Intersonic bilateral slip. Geophys J Int, 1994, 119: 706–714
25. 25.
Broberg K B. Intersonic mode II crack expansion. Arch Mech, 1995, 47: 859–871
26. 26.
Rosakis A J, Samudrala O, Coker D. Cracks faster than the shear wave speed. Science, 1999, 284: 1337–1340

© Science China Press and Springer-Verlag Berlin Heidelberg 2011

## Authors and Affiliations

• 1
• 2
• Yuri Petrov
• 1
• 2
Email author
• Alexander Utkin
• 1
1. 1.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSt.-PetersburgRussia
2. 2.St.-Petersburg State UniversitySt.-PetersburgRussia