Skip to main content
Log in

Effect of notch depth and radius on the critical fracture load of bainitic functionally graded steels under mixed mode I + II loading

  • Published:
Physical Mesomechanics Aims and scope Submit manuscript

Abstract

Functionally graded steels are produced from austenitic stainless steel and carbon steel by controlling the chemical distribution of chromium, nickel and carbon atoms at the remelting stage through electroslag remelting process. In the present paper, the strain-energy density criterion is employed to assess the critical load of rounded V-notched components made of functionally graded bainitic steel. A crack arrester configuration under mixed mode loading is considered. The flow (yield/ultimate) strength and fracture toughness are assumed to vary exponentially along the notch depth direction while the Young’s modulus and the Poisson’s ratio are assumed to be constant. The control volume, which is a reminiscent of Neuber’s elementary structural volume, depends on the ultimate tensile strength σut and the fracture toughness K IC in the case of brittle or quasi-brittle materials subjected to static loadings. Since, σut and K IC are not constant along the notch depth, the control volume which can be obtained numerically as a function of the variation of these material properties through the specimen width. Different values of the notch root radius (from 0.2 to 2.0 mm) and notch depth (from 5 to 7 mm) are considered. The assessed critical fracture loads are in sound agreement with the experimental results.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Neuber, H., Theory of Notch Stresses: Principles for Exact Calculation of Strength with Reference to Structural Form and Material, Berlin: Springer Verlag, 1958.

    Google Scholar 

  2. Peterson, R.E., Notch Sensitivity, Metal Fatigue, Sines, G. and Waisman, J.L., Eds., New York: McGraw Hill, 1959, pp. 293–306.

    Google Scholar 

  3. Tanaka, K., Engineering Formulae for Fatigue Strength Reduction due to Crack Like Notches, Int. J. Fract., 1983, vol. 22, pp. R39–R46.

    Article  Google Scholar 

  4. Lazzarin, P., Tovo, R., and Meneghetti, G., Fatigue Crack Initiation and Propagation Phases near Notches in Metals with Low Notch Sensitivity, Int. J. Fat., 1997, vol. 19, no. 8–9, pp. 647–657.

    Article  Google Scholar 

  5. Taylor, D., Geometrical Effects in Fatigue: A Unifying Theoretical Model, Int. J. Fat., 1999, vol. 21, no. 5, pp. 413–420.

    Article  ADS  Google Scholar 

  6. Taylor, D., The Theory of Critical Distances, Eng. Fract. Mech., 2008, vol. 75, no. 7, pp. 1696–1705.

    Article  Google Scholar 

  7. Susmel, L. and Taylor, D., The Theory of Critical Distances to Predict Static Strength of Notched Brittle Components Subjected to Mixed-Mode Loading, Eng. Fract. Mech., 2008, vol. 75, no. 3–4, pp. 534–550.

    Article  Google Scholar 

  8. Susmel, L. and Taylor, D., The Theory of Critical Distances to Estimate the Static Strength of Notched Samples of Al6082 Loaded in Combined Tension and Torsion. Part II: Multiaxial Static Assessment, Eng. Fract. Mech., 2010, vol. 77, no. 3, pp. 470–478.

    Article  Google Scholar 

  9. Radaj, D., Näherungsweise Berechnung der Formzahl von Schweißnähten, Schw Schn., 1969, vol. 21, pp. 97–105, 151–158.

    Google Scholar 

  10. Berto, F., Lazzarin, P. and Radaj, D., Fictitious Notch Rounding Concept Applied to Sharp V-Notches: Evaluation of the Microstructural Support Factor for Different Failure Hypotheses. Part I: Basic Stress Equations, Eng. Fract. Mech., 2008, vol. 75, no. 10, pp. 3060–3072.

    Article  Google Scholar 

  11. Berto, F., Lazzarin, P., and Radaj, D., Fictitious Notch Rounding Concept Applied to Sharp V-Notches: Evaluation of the Microstructural Support Factor for Different Failure Hypotheses. Part II: Microstructural Support Analysis, Eng. Fract. Mech., 2009, vol. 76, no. 9, pp. 11511175.

    Article  Google Scholar 

  12. Radaj, D., Lazzarin, P., and Berto, F., Fatigue Assessment of Welded Joints under Slit-Parallel Loading Based on Strain Energy Density or Notch Rounding, Int. J. Fat., 2009, vol. 31, no. 10, pp. 1490–1504.

    Article  Google Scholar 

  13. Berto, F. and Lazzarin, P., Fictitious Notch Rounding Approach of Pointed V-Notch under In-plane Shear, Theor. Appl. Fract. Mec., 2010, vol. 53, pp. 127–135.

    Article  Google Scholar 

  14. Berto, F., Lazzarin, P., and Radaj, D., Fictitious Notch Rounding Concept Applied to V-Notches with Root Holes Subjected to In-plane Shear Loading, Eng. Fract. Mech, 2012, vol. 79, pp. 281–294.

    Article  Google Scholar 

  15. Berto, F., Fictitious Notch Rounding Concept Applied to V-Notches with End Holes under Mode 3 Loading, Int. J. Fatigue., 2012, vol. 38, pp. 188–193.

    Article  Google Scholar 

  16. Radaj, D., Lazzarin, P., and Berto, F., Generalised Neuber Concept of Fictitious Notch Rounding. Int. J. Fatigue., 2013, vol. 51, pp. 105–115.

    Article  Google Scholar 

  17. Gomez, F.J., Elices, M., and Valiente, A., Cracking in PMMA Containing U-Shaped Notches, Fatigue Fract. Eng. Mater. Struct., 2000, vol. 23, no. 9, pp. 795–803.

    Article  Google Scholar 

  18. Elices, M., Guinea, G.V., Gómez, J., and Planas, J., The Cohesive Zone Model: Advantages, Limitations and Challenges, Eng. Fract. Mech., 2002, vol. 69, no. 2, pp. 137163.

    Google Scholar 

  19. Planas, J., Elices, M., Guinea, G.V., Gómez, F.J., Cendón, D.A., and Arbilla, I., Generalizations and Specializations of Cohesive Crack Models, Eng. Fract. Mech., 2003, vol. 70, no. 14, pp. 1759–1776.

    Article  Google Scholar 

  20. Gomez, F.J. and Elices, M., Fracture of Components with V-Shaped Notches, Eng. Fract. Mech., 2003, vol. 70, no. 14, pp. 1913–1927.

    Article  Google Scholar 

  21. Gomez, F.J., Elices, M., and Planas, J., The Cohesive Crack Concept: Application to PMMA at −60°C, Eng. Fract. Mech., 2005, vol. 72, no. 8, pp. 1268–1285.

    Article  Google Scholar 

  22. Gomez, F.J. and Elices, M., Fracture Loads for Ceramic Samples with Rounded Notches, Eng. Fract. Mech., 2006, vol. 73, no. 7, pp. 880–894.

    Article  Google Scholar 

  23. Lazzarin, P. and Zambardi, R., A Finite-Volume-Energy Based Approach to Predict the Static and Fatigue Behaviour of Components with Sharp V-Shaped Notches, Int. J. Fract., 2001, vol. 112, no. 3, pp. 275–298.

    Article  Google Scholar 

  24. Lazzarin, P. and Berto, F., Some Expressions for the Strain Energy in a Finite Volume Surrounding the Root of Blunt V-Notches, Int. J. Fract., 2005, vol. 135, no. 1–4, pp. 161–185.

    Article  MATH  Google Scholar 

  25. Berto, F., Lazzarin, P., Gomez, F.J., and Elices, M., Fracture Assessment of U-Notches under Mixed Mode Loading: Two Procedures Based on the Equivalent Local Mode I’ Concept, Int. J. Fract, 2007, vol. 148, no. 4, pp. 415–433.

    Article  MATH  Google Scholar 

  26. Ayatollahi, M.R., Berto, F., and Lazzarin, P., Mixed Mode Brittle Fracture of Sharp and Blunt V-Notches in Polycrystalline Graphite, Carbon, 2011, vol. 49, no. 7, pp. 24652474.

    Article  Google Scholar 

  27. Lazzarin, P., Berto, F., and Ayatollahi, M.R., Brittle Failure of Inclined Key-hole Notches in Isostatic Graphite under In-plane Mixed Mode Loading, Fatigue Fract. Eng. Mater. Struct., 2013, vol. 36, no. 9, pp. 942–955.

    Article  Google Scholar 

  28. Berto, F., Lazzarin, P., and Ayatollahi, M.R., Brittle Fracture of Sharp and Blunt V-Notches in Isostatic Graphite under Torsion Loading, Carbon, 2012, vol. 50, no. 5, pp. 1942–1952.

    Article  Google Scholar 

  29. Berto, F., Lazzarin, P., and Matvienko, Y.G., J-Integral Evaluation for U- and V-Blunt Notches under Mode I Loading and Materials Obeying a Power Hardening Law, Int. J. Fat., 2007, vol. 146, pp. 33–51.

    MATH  Google Scholar 

  30. Lazzarin, P., Campagnolo, A., and Berto, F., A Comparison among Some Recent Energy- and Stress-Based Criteria for the Fracture Assessment of Sharp V-Notched Components under Mode I Loading, Theor. Appl. Fract. Mec., DOI: 10.1016/j.tafmec.2014.03.001.

  31. Berto, F., Elices, M., Lazzarin, P., and Zappalorto, M., Fracture Behaviour of Notched Round Bars Made of PMMA Subjected to Torsion at Room Temperature, Eng. Fract. Mech., 2012, vol. 90, pp. 143–160.

    Article  Google Scholar 

  32. Berto, F., Cendon, D.A., Lazzarin, P., and Elices, M., Fracture Behaviour of Notched Round Bars Made of PMMA Subjected to Torsion at −60°C, Eng. Fract. Mech., 2013, vol. 102, pp. 271–287.

    Article  Google Scholar 

  33. Berto, F., Lazzarin, P., and Marangon, C., Brittle Fracture of U-Notched Graphite Plates under Mixed Mode Loading, Mater. Design., 2012, vol. 41, pp. 421–432.

    Article  Google Scholar 

  34. Berto, F. and Barati, E., Fracture Assessment of U-Notches under Three Point Bending by Means of Local Energy Density, Mater. Design., 2011, vol. 32, pp. 822–830.

    Article  Google Scholar 

  35. Berto, F. and Ayatollahi, M.R., Fracture Assessment of Brazilian Disc Specimens Weakened by Blunt V-Notches under Mixed Mode Loading by Means of Local Energy, Mater. Design., 2011, vol. 32, pp. 2858–2869.

    Article  Google Scholar 

  36. Berto, F., Campagnolo, A., Elices, M., and Lazzarin, P., A Synthesis of Polymethylmethacrylate Data from U-Notched Specimens and V-Notches with End Holes by Means of Local Energy, Mater. Design., 2013, vol. 49, pp. 826833.

    Article  Google Scholar 

  37. Lazzarin, P., Lassen, T., and Livieri, P., A Notch Stress Intensity Approach Applied to Fatigue Life Predictions of Welded Joints with Different Local Toe Geometry, Fatigue Fract. Eng. Mater. Struct., 2003, vol. 26, no. 1, pp. 49–58.

    Article  Google Scholar 

  38. Livieri, P. and Lazzarin, P., Fatigue Strength of Steel and Aluminium Welded Joints Based on Generalised Stress Intensity Factors and Local Strain Energy Values, Int. J. Fract., 2005, vol. 133, no. 3, pp. 247–276.

    Article  Google Scholar 

  39. Lazzarin, P., Berto, F., and Radaj, D., Fatigue-Relevant Stress Field Parameters of Welded Lap Joints: Pointed Slit Tip Versus Keyhole Notch, Fatigue Fract. Eng. Mater. Struct., 2009, vol. 32, no. 9, pp. 713–735.

    Article  Google Scholar 

  40. Radaj, D., Berto, F., and Lazzarin, P., Local Fatigue Strength Parameters for Welded Joints Based on Strain Energy Density with Inclusion of Small-Size Notches, Eng. Fract. Mech., 2009, vol. 76, no. 8, pp. 1109–1130.

    Article  Google Scholar 

  41. Radaj, D., Lazzarin, P., and Berto, F., Fatigue Assessment of Welded Joints under Slit-Parallel Loading Based on Strain Energy Density or Notch Rounding, Int. J. Fatigue, 2009, vol. 31, pp. 1490–1504.

    Article  Google Scholar 

  42. Berto, F., Croccolo, D., and Cuppini, R., Fatigue Strength of a Fork-Pin Equivalent Coupling in Terms of the Local Strain Energy Density, Mater. Design., 2008, vol. 29, pp. 1780–1792.

    Article  Google Scholar 

  43. Berto, F., Lazzarin, P., and Yates, J., Multiaxial Fatigue Behaviour of Quasi-Sharp V-Notched Specimens Made of 39NiCrMo3 Steel: A Non-Conventional Application of the Local Energy, Fatigue Fract. Eng. Mater. Struct., 2011, vol. 34, no. 11, pp. 921–943.

    Article  Google Scholar 

  44. Berto, F. and Lazzarin, P., Fatigue Strength of Structural Components under Multi-axial Loading in Terms of Local Energy Density Averaged on a Control Volume, Int. J. Fatigue, 2011, vol. 33, no. 8, pp. 1055–1065.

    Article  Google Scholar 

  45. Berto, F., Lazzarin, P., and Marangon, C., Fatigue Strength of Notched Specimens Made of 40CrMoV13.9 under Multiaxial Loading, Mater. Design, 2014, vol. 54, pp. 5766.

    Article  Google Scholar 

  46. Berto, F., Lazzarin, P., and Tovo, R., Multiaxial Fatigue Strength of Severely Notched Cast Iron Specimens, Int. J. Fatigue, DOI: http://dx.doi.org/10.1016/j.ijfatigue.2014.01.013.

  47. Ferro, P., Lazzarin, P., and Berto, F., Fatigue Properties of Ductile Cast Iron Containing Chunky Graphite, Mater. Sci. Eng. A, 2012, vol. 554, pp. 122–128.

    Article  Google Scholar 

  48. Berto, F. and Lazzarin, P., Fatigue Strength of Al7075 Notched Plates Based on the Local SED Averaged Over a Control Volume Science China Physics, Mech. Astron., 2014, vol. 57, pp. 30–38.

    Article  Google Scholar 

  49. Berto, F., A Review on Coupled Modes in V-Notched Plates of Finite Thickness: a Generalised Approach to the Problem, Phys. Mesomech., 2013, vol. 16, no. 4, pp. 378390.

    Article  Google Scholar 

  50. Harding, S., Kotousov, A., Lazzarin, P., and Berto, F., Transverse Singular Effects in V-shaped Notches Stressed in Mode II, Int. J. Fat., 2010, vol. 164, pp. 1–14.

    MATH  Google Scholar 

  51. Berto, F., Lazzarin, P., Kotousov, A., and Harding, S., Out-of-Plane Singular Stress Fields in V-Notched Plates and Welded Lap Joints Induced by In-plane Shear Load Conditions, Fatigue. Fract. Eng. M., 2010, vol. 34, pp. 291–304.

    Article  Google Scholar 

  52. Kotousov, A., Lazzarin, P., Berto, F., and Harding, S., Effect of the Thickness on Elastic Deformation and Quasi-Brittle Fracture of Plate Components, Eng. Fract. Mech., 2010, vol. 77, pp. 1665–1681.

    Article  Google Scholar 

  53. Kotousov, A., Lazzarin, P., Berto, F., and Pook, L.P., Three-Dimensional Stress States at Crack Tip Induced by Shear and Anti-plane Loading, Eng. Fract. Mech., 2013, vol. 108, pp. 65–74.

    Article  Google Scholar 

  54. Berto, F., Lazzarin, P., and Kotousov, A., On Higher Order Terms and Out-of-plane Singular Mode, Mech. Mater., 2011, vol. 43, pp. 332–341.

    Article  Google Scholar 

  55. Berto, F., Lazzarin, P., Kotousov, A., and Pook, L., Induced Out-of-Plane Mode at the Tip of Blunt Lateral Notches and Holes under In-plane Shear Loading, Fatig. Fract. Eng. M, 2012, vol. 35, pp. 538–555.

    Article  Google Scholar 

  56. Berto, F., Lazzarin, P., and Marangon, Ch., The Effect of the Boundary Conditions on In-plane and Out-of-plane Stress Field in Three Dimensional Plates Weakened by Free-clamped V-notches, Phys. Mesomech., 2012, vol. 15, no. 1–2, pp. 26–36.

    Article  Google Scholar 

  57. Berto, F. and Lazzarin, P., On Higher Order Terms in the Crack Tips Field, Int. J. Fract., 2010, vol. 161, no. 2, pp. 221–226.

    Article  MATH  Google Scholar 

  58. Berto, F. and Lazzarin, P., Multiparametric Full-Field Representations of the In-plane Stress Fields Ahead of Cracked Components under Mixed Mode Loading, Int. J. Fatigue, 2013, vol. 46, pp. 16–26.

    Article  Google Scholar 

  59. Lazzarin, P., Berto, F., Gomez, F.J., and Zappalorto, M., Some Advantages Derived from the Use of the Strain Energy Density over a Control Volume in Fatigue Strength Assessments of Welded Joints, Int. J. Fatigue, 2008, vol.30, no. 8, p. 1345.

    Article  Google Scholar 

  60. Lazzarin, P., Berto, F., and Zappalorto, M., Rapid Calculations of Notch Stress Intensity Factors Based on Averaged Strain Energy Density from Coarse Meshes: Theoretical Bases and Applications, Int. J. Fatigue, 2010, vol. 32, no. 10, p. 1559.

    Article  Google Scholar 

  61. Lazzarin, P., Afshar, R., and Berto, F., Notch Stress Intensity Factors of Flat Plates with Periodic Sharp Notches by Using the Strain Energy Density, Theor. Appl. Fract. Mech., 2012, vol. 60, pp. 38–50.

    Article  Google Scholar 

  62. Berto, F., Lazzarin, P., and Afshar, R., Simple New Expressions for the Notch Stress Intensity Factors in an Array of Narrow V-Notches under Tension, Int. J. Fat., 2012, vol. 176, pp. 237–244.

    Google Scholar 

  63. Berto, F. and Lazzarin, P., A Review of the Volume-Based Strain Energy Density Approach Applied to V-Notches and Welded Structures, Theor. Appl. Fract. Mech., 2009, vol. 52, no. 3, pp. 183–194.

    Article  Google Scholar 

  64. Berto, F. and Lazzarin, P., Recent Developments in Brittle and Quasi-Brittle Failure Assessment of Engineering Materials by Means of Local Approaches, Mat. Sci. Eng., 2014, vol. 75, pp. 1–48.

    Article  Google Scholar 

  65. Berto, F., A Brief Review of Some Local Approaches for the Failure Assessment of Brittle and Quasi-Brittle Materials, Adv. Mater. Sci. Eng., 2014, pp. 1–10.

    Google Scholar 

  66. Jha, D.K., Kant, T., and Singh, R.K., A Critical Review of Recent Research on Functionally Graded Plates, Comp. Struct., 2013, vol. 96, pp. 833–849.

    Article  Google Scholar 

  67. Aghazadeh Mohandesi, J. and Shahosseinie, M.H., Transformation Characteristics of Functionally Graded Steels Produced by Electroslag Remelting, Metall. Mater. Trans. A, 2005, vol. 36, pp. 3471–3476.

    Article  Google Scholar 

  68. Aghazadeh Mohandesi, J., Shahossinie, M.H., and Paras-tar Namin, R., Tensile Behavior of Functionally Graded Steels Produced by Electroslag Remelting, Metall. Mater. Trans. A, 2006, vol. 37, pp. 2125–2132.

    Article  Google Scholar 

  69. Nazari, A., Aghazadeh Mohandesi, J., and Tavareh, S., Microhardness Profile Prediction of a Graded Steel by Strain Gradient Plasticity Theory, Comput. Mater. Sci., 2011, vol. 50, pp. 1781–1784.

    Article  Google Scholar 

  70. Nazari, A., Aghazadeh Mohandesi, J., and Tavareh, S., Modeling Tensile Strength of Austenitic Graded Steel Based on the Strain Gradient Plasticity Theory, Comput. Mater. Sci., 2011, vol. 50, pp. 1791–1794.

    Article  Google Scholar 

  71. Abolghasemzadeh, M., Samareh Salavati Pour, H., Berto, F., and Alizadeh, Y., Modeling of Flow Stress of Bainitic and Martensitic Functionally Graded Steels under Hot Compression, Mater. Sci. Eng. A, 2012, vol. 534, pp. 329338.

    Article  Google Scholar 

  72. Nazari, A. and Aghazadeh Mohandesi, J., Impact Energy of Functionally Graded Steels with Crack Divider Configuration, J. Mater. Sci. Technol., 2009, vol. 25, pp. 847852.

    Google Scholar 

  73. Nazari, A. and Aghazadeh Mohandesi, J., Modeling Impact Energy of Functionally Graded Steels in Crack Divider Configuration, Mater. Sci. Technol., 2010, vol. 26(11), pp. 1377–1383.

    Article  Google Scholar 

  74. Nazari, A., Aghazadeh Mohandesi, J., and Riahi, S., Modeling Impact Energy of Functionally Graded Steels in Crack Divider Configuration Using Modified Stress-Strain Curve Data, Int. J. Damage Mech., 2012, vol. 21, pp. 27–50.

    Article  Google Scholar 

  75. Samareh Salavati Pour, H., Berto, F., and Alizadeh, Y., A New Analytical Expression for the Relationship Between the Charpy Impact Energy and Notch Tip Position for Functionally Graded Steels, Acta Metallurg. Sinica (Eng. Lett.), 2013, vol. 10(3), pp. 232–240.

    Article  Google Scholar 

  76. Salavati, H. and Berto, F., Prediction the Charpy Impact Energy of Functionally Graded Steels, Eng. Solid Mech., 2013, vol. 2, pp. 21–28.

    Google Scholar 

  77. Barati, E., Alizadeh, Y., and Aghazadeh Mohandesi, J., J-Integral Evaluation of Austenitic-Martensitic Functionally Graded Steel in Plates Weakened by U-Notches, Eng. Fract. Mech., 2010, vol. 77, pp. 3341–3358.

    Article  Google Scholar 

  78. ASTM E1820. Standard Test Method for Measurement of Fracture Toughness, Annual Book of ASTM Standards, 2001, vol. 03.01.

    Google Scholar 

  79. Yosibash, Z., Bussiba, A., and Gilad, I., Failure Criteria for Brittle Elastic Materials, Int. J. Fract., 2004, vol. 125, pp. 307–333.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to F. Berto.

Additional information

Original Text © H. Salavati, Y. Alizadeh, F. Berto, 2014, published in Fizicheskaya Mezomekhanika, 2014, Vol. 17, No. 4, pp. 29–39.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Salavati, H., Alizadeh, Y. & Berto, F. Effect of notch depth and radius on the critical fracture load of bainitic functionally graded steels under mixed mode I + II loading. Phys Mesomech 17, 178–189 (2014). https://doi.org/10.1134/S1029959914030023

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1029959914030023

Keywords

Navigation