Skip to main content
SpringerLink
Log in

A crack tip driving force model for mode I crack propagation along linear strength gradient: comparison with the sharp strength gradient case

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In the present work, based on the Dugdale criterion, a closed-form theoretical crack tip driving force model of simple expressions was proposed for static and fatigue crack propagation along a linear yield strength gradient (YSG). Qualitative and quantitative analyses of the YSG effect on the crack tip driving force are given. The crack tip driving force depends on the yield strength at the crack tip, the YSG within the plastic zone and the applied load. A positive (or negative) YSG has a shielding (or amplifying) effect on the crack tip driving force. A difference of material-toughening mechanisms between smooth and sharp strength gradient was revealed both theoretically and numerically. In the linear YSG case, when a fatigue crack propagates along the YSG at a constant applied cyclic crack tip stress intensity factor, the cyclic crack tip opening displacement decreases (or increases) when the crack propagates towards a higher (or lower)-strength region; however, the cyclic crack tip stress intensity factor increases, no matter when the crack propagates towards a higher- or lower-strength region. This abnormal phenomenon causes theoretical and experimental challenges for characterizing the smooth strength gradient effect on the crack tip driving force.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a :

Distance between crack tip and interface

\(C_{\mathrm{inh}}\) :

Additional material inhomogeneity term

CPS8:

8-Node biquadratic plane stress quadrilateral element

CT:

Compact tension

CTOD:

Crack tip opening displacement

DCB:

Double cantilever beam

E :

Young’s modulus

FCP:

Fatigue crack propagation

FEM:

Finite element method

\(G_{\mathrm{applied}}\) :

Applied energy release rate

J :

J-integral

\(J_{\mathrm{applied}}\) :

Applied J-integral

\(J_{\mathrm{tip}}\) :

Crack tip J-integral

k :

Yield strength gradient

K :

Stress intensity factor

\(K_{\mathrm{applied}}\) :

Applied stress intensity factor

\(K_{\mathrm{cohesive}}\) :

Cohesive stress intensity factor

\(K_{\mathrm{tip}}\) :

Crack tip stress intensity factor

l :

Distance between interface and mathematical crack tip

\(p_{\infty }\) :

Monotonic far field load

\(P_{\mathrm{applied}}\) :

Applied load per thickness

\(r_{\mathrm{hom}}\) :

Crack tip plastic zone size (or Dugdale cohesive zone size) in homogeneous material

r :

Crack tip plastic zone size (or Dugdale cohesive zone size) in YSG case

\(r^{\prime }\) :

Radius of crack tip circular plastic zone

v :

Poisson’s ratio

W :

Specimen width

YSG:

Yield strength gradient

\(\alpha \) :

Ratio of crack length to specimen width

\(\varGamma _{\mathrm{cz}}\) :

Integration contour for J-integral

\(\delta \) :

Crack opening displacement

\(\Delta \) :

Parameter under cyclic or fatigue load

\(\varphi (z)\) :

Complex potential

\({\sigma }^{\mathrm{A}}\) :

Yield stress of material A in plane stress condition

\(\sigma ^{\mathrm{B}}\) :

Yield stress of material B in plane stress condition

\(\sigma _{\mathrm{s}}\) :

Yield stress

\(\sigma _{0}\) :

Cohesive stress at crack tip

\(\upsilon \) :

Crack opening displacement in the y direction from the crack axis in the cohesive zone

References

  1. Zhao, P., Guo, S., Liu, G., Chen, Y., Li, J.: Fast fabrication of W–Cu functionally graded material by high-gravity combustion synthesis and melt-infiltration. J. Nucl. Mater. 445(1), 26–29 (2014)

    Article  Google Scholar 

  2. Liu, T.-J., Wang, Y.-S.: Axisymmetric frictionless contact problem of a functionally graded coating with exponentially varying modulus. Acta Mech. 199(1–4), 151–165 (2008)

    Article  MATH  Google Scholar 

  3. Kapuria, S., Kumari, P., Nath, J.: Efficient modeling of smart piezoelectric composite laminates: a review. Acta Mech. 214(1–2), 31–48 (2010)

    Article  MATH  Google Scholar 

  4. Bhattacharyya, A., Subhash, G., Arakere, N.: Evolution of subsurface plastic zone due to rolling contact fatigue of M-50 NiL case hardened bearing steel. Int. J. Fatigue 59, 102–113 (2014)

    Article  Google Scholar 

  5. Trudel, A., Sabourin, M., Lévesque, M., Brochu, M.: Fatigue crack growth in the heat affected zone of a hydraulic turbine runner weld. Int. J. Fatigue 66, 39–46 (2014)

    Article  Google Scholar 

  6. Suresh, S., Sugimura, Y., Tschegg, E.: The growth of a fatigue crack approaching a perpendicularly-oriented, bimaterial interface. Scr. Metall. Mater. 27(9), 1189–1194 (1992)

    Article  Google Scholar 

  7. Sugimura, Y., Grondin, L., Suresh, S.: Fatigue crack growth at arbitrary angles to bimaterial interfaces. Scr. Metall. Mater. 33(12), 2007–2012 (1995)

    Article  Google Scholar 

  8. Sugimura, Y., Lim, P., Shih, C., Suresh, S.: Fracture normal to a bimaterial interface: effects of plasticity on crack-tip shielding and amplification. Acta Metall. Mater. 43(3), 1157–1169 (1995)

    Article  Google Scholar 

  9. Pippan, R., Flechsig, K., Riemelmoser, F.: Fatigue crack propagation behavior in the vicinity of an interface between materials with different yield stresses. Mater. Sci. Eng. A 283(1), 225–233 (2000)

    Article  Google Scholar 

  10. Jiang, F., Deng, Z., Zhao, K., Sun, J.: Fatigue crack propagation normal to a plasticity mismatched bimaterial interface. Mater. Sci. Eng. A 356(1), 258–266 (2003)

    Article  Google Scholar 

  11. He, J.-W., Jia, M., Wen, S.-P., Li, N.: Cracking in the soft interface layer of an Al-alloy laminate. Int. J. Fatigue 25(5), 421–426 (2003)

    Article  Google Scholar 

  12. Ukadgaonker, V.G., Bhat, S., Jha, M., Desai, P.: Fatigue crack growth towards the weld interface of alloy and maraging steels. Int. J. Fatigue 30(4), 689–705 (2008)

    Article  MATH  Google Scholar 

  13. Velu, M., Bhat, S.: Experimental investigations of fracture and fatigue crack growth of copper-steel joints arc welded using nickel-base filler. Mater. Des. 67, 244–260 (2015)

    Article  Google Scholar 

  14. Kim, A., Suresh, S., Shih, C.: Plasticity effects on fracture normal to interfaces with homogeneous and graded compositions. Int. J. Solids Struct. 34(26), 3415–3432 (1997)

    Article  MATH  Google Scholar 

  15. Kim, A., Besson, J., Pineau, A.: Global and local approaches to fracture normal to interfaces. Int. J. Solids Struct. 36(12), 1845–1864 (1999)

    Article  MATH  Google Scholar 

  16. Kolednik, O., Predan, J., Fischer, F.D.: Reprint of “Cracks in inhomogeneous materials Comprehensive assessment using the configurational forces concept”. Eng. Fract. Mech. 77(18), 3611–3624 (2010)

    Article  Google Scholar 

  17. Pippan, R., Riemelmoser, F.: Fatigue of bimaterials. Investigation of the plastic mismatch in case of cracks perpendicular to the interface. Comput. Mater. Sci. 13(1), 108–116 (1998)

    Article  Google Scholar 

  18. Wäppling, D., Gunnars, J., Stahle, P.: Crack growth across a strength mismatched bimaterial interface. Int. J. Fract. 89(3), 223–243 (1998)

    Article  Google Scholar 

  19. Riemelmoser, F.O., Pippan, R.: The J-integral at Dugdale cracks perpendicular to interfaces of materials with dissimilar yield stresses. Int. J. Fract. 103(4), 397–418 (2000)

    Article  Google Scholar 

  20. Bhat, S., Ukadgaonker, V.G.: Dugdale cohesive zone modeling to evaluate J integral at the interface of strength mismatched steels: a simplified numerical approach. Finite Elem. Anal. Des. 46(7), 601–610 (2010)

    Article  Google Scholar 

  21. Bhat, S., Narayanan, S.: A computational model and experimental validation of shielding and amplifying effects at a crack tip near perpendicular strength-mismatched interfaces. Acta Mech. 216(1–4), 259–279 (2011)

    Article  MATH  Google Scholar 

  22. Bhat, S., Ukadgaonker, V.G.: Solution of singular integrals in mathematical model of mode I crack near strength mismatched interface. Am. J. Comput. Math. 2(02), 156 (2012)

    Article  Google Scholar 

  23. Bhat, S., Ukadgaonker, V.G.: Finite element modeling of crack tip blunting for estimation of energy release rate component of mode I crack near a strength mismatched interface. Appl. Math. 4(05), 1 (2013)

    Article  Google Scholar 

  24. Bhat, S.: Tip parameter approximation and fatigue growth of crack towards inclined weld interface between strength mismatched steels. Int. J. Damage Mech 20, 752–782 (2010)

    Article  Google Scholar 

  25. Bhat, S., Ukadgaonker, V.G.: A refined theoretical model for mode I crack near the interface of strength and plasticity mismatched materials in K dominant regime. Int. J. Damage Mech 24(8), 1194–1213 (2015)

    Article  Google Scholar 

  26. Simha, N., Fischer, F., Kolednik, O., Chen, C.: Inhomogeneity effects on the crack driving force in elastic and elastic–plastic materials. J. Mech. Phys. Solids 51(1), 209–240 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Simha, N., Fischer, F., Kolednik, O., Predan, J., Shan, G.: Crack tip shielding or anti-shielding due to smooth and discontinuous material inhomogeneities. Int. J. Fract. 135(1–4), 73–93 (2005)

    Article  MATH  Google Scholar 

  28. Kolednik, O., Predan, J., Shan, G., Simha, N., Fischer, F.: On the fracture behavior of inhomogeneous materials—a case study for elastically inhomogeneous bimaterials. Int. J. Solids Struct. 42(2), 605–620 (2005)

    Article  MATH  Google Scholar 

  29. Kahl, M., Reifsnider, K.: Influence of local variations of yield strength on plastic zones at crack tips. Eng. Fract. Mech. 4(4), 653–664 (1972)

    Article  Google Scholar 

  30. Reifsnider, K., Kahl, M.: Effect of local yield strength gradients on fatigue crack propagation. Int. J. Mech. Sci. 16(2), 105–119 (1974)

    Article  Google Scholar 

  31. Kermanidis, A., Pantelakis, S.: Fatigue crack growth analysis of 2024 T3 aluminium specimens under aircraft service spectra. Fatigue Fract. Eng. Mater. Struct. 24(10), 699–710 (2001)

    Article  Google Scholar 

  32. Kermanidis, A., Pantelakis, S.G.: Prediction of crack growth following a single overload in aluminum alloy with sheet and plate microstructure. Eng. Fract. Mech. 78(11), 2325–2337 (2011)

    Article  Google Scholar 

  33. Predan, J., Gubeljak, N., Kolednik, O.: On the local variation of the crack driving force in a double mismatched weld. Eng. Fract. Mech. 74(11), 1739–1757 (2007)

  34. Kolednik, O., Predan, J., Gubeljak, N., Fischer, D.F.: Modeling fatigue crack growth in a bimaterial specimen with the configurational forces concept. Mater. Sci. Eng. A 519(1), 172–183 (2009)

    Article  Google Scholar 

  35. Rakin, M., Kolednik, O., Medjo, B., Simha, N., Fischer, F.: A case study on the effect of thermal residual stresses on the crack-driving force in linear-elastic bimaterials. Int. J. Mech. Sci. 51(7), 531–540 (2009)

    Article  Google Scholar 

  36. Sistaninia, M., Kolednik, O.: Effect of a single soft interlayer on the crack driving force. Eng. Fract. Mech. 130, 21–41 (2014)

    Article  Google Scholar 

  37. Kolednik, O., Zechner, J., Predan, J.: Improvement of fatigue life by compliant and soft interlayers. Scr. Mater. 113, 1–5 (2016)

    Article  Google Scholar 

  38. Kolednik, O., Suresh, S.: The influence of the yield strength gradient on the fracture resistance in FGMs. In: Materials science forum (1999), pp. 963–970. Trans Tech Publ

  39. Kolednik, O.: The yield stress gradient effect in inhomogeneous materials. Int. J. Solids Struct. 37(5), 781–808 (2000)

    Article  MATH  Google Scholar 

  40. Nazari, A., Mohandesi, J.A., Riahi, S.: Modeling fracture toughness of functionally graded steels in crack arrester configuration. Comput. Mater. Sci. 50(4), 1578–1586 (2011)

    Article  Google Scholar 

  41. Nazari, A., Mohandesi, J.A., Riahi, S.: Fracture toughness of functionally graded steels. J. Mater. Eng. Perform. 21(4), 558–563 (2012)

    Article  Google Scholar 

  42. Yang, B., Zhang, K., Chen, G., Luo, G., Xiao, J.: A quantitative analysis of the effect of laser transformation hardening on crack driving force in steels. Surf. Coat. Technol. 201(6), 2208–2216 (2006)

    Article  Google Scholar 

  43. Yang, B.-Q., Zhang, K., Chen, G.-N., Luo, G.-X., Xiao, J.-H.: Effect of a laser pre-quenched steel substrate surface on the crack driving force in a coating-steel substrate system. Acta Mater. 55(13), 4349–4358 (2007)

    Article  Google Scholar 

  44. Dugdale, D.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8(2), 100–104 (1960)

    Article  Google Scholar 

  45. Tada, H., Paris, P.C., Irwin, G.R.: The stress analysis of cracks handbook. Del Research Corporation, St. Louis (1985)

    Google Scholar 

  46. Rice, J.R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35(2), 379–386 (1968)

    Article  Google Scholar 

  47. Rice, J.R.: Mechanics of crack tip deformation and extension by fatigue. ASTM STP 415, 247–309 (1967)

    Google Scholar 

  48. Suresh, S., Sugimura, Y., Tschegg, E.K.: The growth of a fatigue crack approaching a perpendicularly-oriented, bimaterial interface. Scr. Metall. Mater. 27(9), 1189–1194 (1992)

    Article  Google Scholar 

  49. Siegmund, T., Bo, W.: Simulation of fatigue crack growth at plastically mismatched bi-material interfaces. Int. J. Plast. 22(9), 1586–1609 (2006)

    Article  MATH  Google Scholar 

  50. ASTM Standard: E647-13a. Standard test method for measurement of fatigue crack growth rates (2013)

Download references

Author information

Authors and Affiliations

  1. AML, School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China

    Yi-Bo Shang & Hui-Ji Shi

  2. CPI Nuclear Power Institute, 18 Xizhimen St., Beijing, 100044, China

    Zhao-Xi Wang

  3. Suzhou Nuclear Power Research Institute, Suzhou, 215004, China

    Guo-Dong Zhang

Authors
  1. Yi-Bo Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui-Ji Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhao-Xi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guo-Dong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui-Ji Shi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shang, YB., Shi, HJ., Wang, ZX. et al. A crack tip driving force model for mode I crack propagation along linear strength gradient: comparison with the sharp strength gradient case. Acta Mech 227, 2683–2702 (2016). https://doi.org/10.1007/s00707-016-1628-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-016-1628-z

Keywords

Search

Navigation