# Analytical and numerical treatment of a dynamic crack model

• A. Lalegname
• A. -M. Sändig
• G. Sewell
Original Paper

## Abstract

We discuss the propagation of a running crack in a bounded linear elastic body under shear waves for a simplified 2D-model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked, bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u  =  u(y, t) and the one-dimensional crack tip trajectory h  =  h(t). We assume that the crack grows straight. Based on a paper of Nicaise-Sändig, we derive an improved formula for the ordinary differential equation of motion for the crack tip, where the dynamical stress intensity factor occurs. The numerical simulation is an iterative procedure starting from the wave field at time t  =  t i . The dynamic stress intensity factor will be extracted at t  =  t i . Its knowledge allows us to compute the crack-tip motion h(t i+1) with corresponding nonuniform crack speed assuming (t i+1t i ) is small. Now, we start from the cracked configuration at time t  =  t i+1 and repeat the steps. The wave displacements are computed with the FEM-package PDE2D. Some numerical examples demonstrate the proposed method. The influence of finite length of the crack and finite size of the sample on the dynamic stress intensity factor will be discussed in detail.

## Keywords

Dynamic crack propagation Wave equation Energy balance law Adaptive FEM-Method Computation of dynamic stress intensity factors

## References

1. ASTM E 1823-96 (1996) Standard terminology relating to fatigue ans fracture testing, Annual book of ASTM Standards, vol. 03.01. American Society for Testing and Materials, West ConshohockenGoogle Scholar
2. Atluri SN, Nishioka T (1985) Numerical studies in dynamic fracture mechanics. In: Williams ML, Knauss WG(eds) Dynamic fracture. Martinus Nijhoff Publishers, Dodrecht, pp 119–135Google Scholar
3. Bratov V, Petrov Y (2007) Application of incubation time approach to simulate dynamic crack propagation. Int J Fract 146: 53–60
4. Brenner SC (1999) Multigrid methods for the computation of singular solutions and stress intensity factors I: corner singularities. Math Comput 68(226): 559–583
5. Brenner SC, Scott LR (1994) The mathematical theory of finite element methods. Springer VerlagGoogle Scholar
6. Broberg KB (1999) Cracks and fracture, [u.a.]. Academic Press, San Diego, CaliforniaGoogle Scholar
7. Brokate M, Khludnev A (2004) On crack propagation shapes in elastic bodies. Z Angew Math Phys 55: 318–329
8. Buehler MJ, Gao H, Huang Y (2003) Atomistic and continuum studies of a suddenly stopping supersonic crack. Comput Mater Sci 28: 385–408
9. Ciarlet PG (1976) The finite element method for elliptic problems. North-Holland Publishing CompanyGoogle Scholar
10. Charoenphan S (2002) Computer methods for modeling the progressive damage of composite material plates and tubes. PHD Thesis, University of Wisconsin-MadisonGoogle Scholar
11. Dauge M (1988) Elliptic boundary value problems on corner domains, Lecture notes in mathematics 1341, Springer-Verlag, Berlin-Heidelberg, MR 91a:35078Google Scholar
12. Destuynder P, Jaoua M (1981) Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile. Math Methods Appl Sci 3: 70–87
13. Erdogan F (1968) Crack-propagation theories, Chap. 5. In: (eds) Fracture, vol II. Academic Press, New YorkGoogle Scholar
14. Freund LB (1973) Crack propagation in an elastic solid subjected to general loading. III Stress wave loading. J Mech Phys Solids 21: 47–61
15. Freund LB (1990) Dynamic fracture mechanics. Cambridge University Press, New York
16. Freund LB, Clifton RJ (1974) On the uniqueness of plate elastodynamic solutions for running cracks. J Elast 4(4): 293–299
17. Freund LB, Rosakis AJ (1992) The structure of the near tip field solution during transient elastodynamic crack growth. J Mech Phys Solids 40: 699–719
18. Freund LB, Duffy J and Rosakis AJ (1981) Dynamic fracture initiation in metals and preliminary results on the method of caustics for crack propagation measurements, Cambridge University PressGoogle Scholar
19. Friedman A, Hu B, Velazquez JJL (2000) The evolution of stress intensity factors and the propagation of cracks in elastic media. Arch Ration Mech Anal 152: 103–139
20. Grisvard P (1985) Elliptic problems in nonsmooth domains, Pitman. Boston MR 86m:35044Google Scholar
21. Großmann C, Roos H-G (2005) Numerische Behandlung partieller Differentialgleichungen. B.G. Teubner VerlagGoogle Scholar
22. Gross D (1996) Bruchmechanik. Springer-Verlag, Berlin
23. Kerkhof F (1965) Habilitationsschrift, KarlsruheGoogle Scholar
24. Kobayashi AS, Mall S (1978) Dynamic fracture toughness of Homalite 100. Exp Mech 18: 11–18
25. Kondrat’ev VA (1967) Boundary value problems for elliptic equations in domains with conical or angular points. Trans Moscow Math Soc 16: 227–313
26. Koslov VA, Maz’ya VG, Rossmann J (1997) Elliptic boundary value problems in domains with point singularities, American Mathematical Society, ProvidenceGoogle Scholar
27. Kostrov BV (1966) Unsteady propagation of longitudinal shear cracks. Appl Math Mech 30: 1241–1248
28. Kovtunenko VA (2001) Sensitivity of cracks in 2D-Lam’e problem via material derivatives. Z Angew Math Phys 52: 1071–1087
29. Lee Y, Prakash V (1998) Dynamic fracture toughness versus crack tip speed relationship at lower than room temperature for high strength 4340var structural steel. J Mech Phys Solids 46(10): 1943–1967
30. Maz’ya VG, Plamenevskii BA (1978) On the coefficients in the asymptotics of the solutions of an elliptic boundary value problem in domains with conical points. J Soviet Math 9: 750–764
31. Morozov N, Petrov Y (2000) Dynamics of fracture. Springer-Verlag, Berlin
32. Nazarov SA, Plamenesvkii BA (1994) Elliptic problems in domians with piecewise smooth boundaries, expositions in mathematics, vol 13. de Gruyter, Berlin, MR 95h:35001Google Scholar
33. Nicaise S, Sändig A-M (2007) Dynamical crack propagation in a 2D elastic body The out-of plane state. J Math Anal Appl 329: 1–30
34. Nishioka T, Atluri SN (1986) Computational methods in dynamic fracture. In: Atluri SN (ed) Computational methods in the mechanics of fracture, Chap. 10, Elsevier Science Publishers, pp 335-383Google Scholar
35. Ohyoshi T (1973) Effect of orthotropy on singular stresses produced near a crack tip by incident SH-waves. ZAMM 53: 409–411
36. Owen DM, Zhuang S, Rosakis AJ, Ravichandran G (1998) Experimental determination of dynamic crack initiation and propagation fracture toughness in thin aluminum sheets. Int J of Fract 90: 153–174
37. Ravi-Chandar K, Knauss WG (1982) Dynamic crack-tip stresses under stress wave loading—A comparison of theroy and experiment. Int J Fract 20: 209–222
38. Ravi-Chandar K, Knauss WG (1984) An experimental investigation into the mechanics of dynamic fracture: I. Crack initiation and arrest. Int J Fract 25: 247–262
39. Ravichandran G, Clifton RJ (1989) Dynamic fracture undr plane wave loading. Int J Fract 40: 157–201
40. Rosakis G, Ravichandran G (2000) Dynamic failure mechanics. Int J Solids Struct 37: 331–348
41. Rosakis AJ, Duffy J, Freund LB (1984) The determination od dynamic fracture toughness of AISI 4340 steel by the shadow spot method. J Mech Phys Solids 32: 443–460
42. Rosakis AJ, Liu C, Freund LB (1991) A note on the asymptotic stress field of a non-uniformly propagating dynamic crack. Int J Fract 50: R39–R45
43. Sändig A-M, Nicaise S, Lalegname A (2007) Dynamic crack propagation in a 2D elastic body. The out-of plane case. ICIAM 07. ETH ZürichGoogle Scholar
44. Sewell G PDE2D, University of Texas, El Paso. http://www.pde2d.com
45. Sewell G (2005) The numerical solution of ordinary and partial differential equations, 2nd Edn. WileyGoogle Scholar
46. Schwab C (1998) P- and hp-finite element methods. Oxford University PressGoogle Scholar
47. Schwalbe K-H, Landes JD, Heerens J (2007/14) Classical fracture mechanics methods. Comprehensive structural integrity. Online update, vol 11. GKSS 2007/14Google Scholar
48. Seelig Th (1997) Zur Simulation der dynamischen Rißausbreitung mit einer Zeitbereichs-Randelementmethode. Ph.D. Thesis, TH Darmstadt, GermanyGoogle Scholar
49. Suresh S (1998) Fatigue of materials, 2nd edn. Cambridge University PressGoogle Scholar
50. Takahashi K, Arakawa K (1987) Dependence of crack acceleration on the dynamic stress–intensity factor in polymers. Exp Mech 27: 195–199
51. Tvergaard V, Hutchinson J (1992) The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J Mech Phys Solids 40(6): 1377–1397 http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/exper/gordon/www/fractough.html
52. Williams T et al. (2004) GnuPlot, version 4.0. Technical report, Pixar Corporation, http://www.gnuplot.info/2004
53. Yang B, Ravi-Chandar K (1996) On the role of the process zone in dynamic fracture. J Mech Phys Solids 44(12): 1955–1976
54. Zehnder AT, Rosakis AJ (1990) Dynamic fracture initiation and propagation in 4340 steel under impact loading. Int J Fract 43(4): 271–285
55. Zhang Ch (1993) On wave propagation in cracked solids Habilitationsschrift. TH Darmstadt, GermanyGoogle Scholar
56. Zhang Ch, Gross D (1993) Interaccion of antiplane cracks with elastic waves in transversely isotropic materials. Acta Mechanica 101: 231–247
57. Zhou F, Shioya T (1996) Energy balance analysis on mode-III dynamic crack propagation in fixed sided strip. Int J Fract 80: 33–44