An Analytical Procedure for the Prediction of the Stress-Strain State in Notches under Multiaxial Fatigue
Keywords:
Finite elements, Plasticity, Notches, Analytical model, Fatigue
Abstract
This work presents an analytical procedure for estimating elastic-plastic stresses and strains in notched shafts subjected to synchronous non-proportional torsional and tensile cyclic loading. The specification of the equivalent stress concentration factor is firstly accomplished. Neuber’s rule in conjunction with the assumed material law provides the relation between the applied loading and the equivalent stress and strain. Principal stresses and strains yield from the corresponding equivalent values incorporating Hencky’s equations. The transformation of the principal stresses and strains to the appropriate coordinate system yields the final result. For the assessment of the analytical procedure, notch stress-strain results from several finite element analyses of an axisymmetric cylindrical shaft with a circumferential groove subjected to multiaxial synchronous fatigue loading are presented. A satisfactory agreement between the analytical and numerical results is observed.
References
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[2]. Barkey M.E., Socie D.F., Hsia K.J., 1993, A Yield Surface Approach to the Estimation of Notch Strains for Proportional and Nonproportional Cyclic Loading, TAM Report No. 709-UILU ENG-93-6007, University of Illinois, Urbana-Champaign, USA.
[3]. Bathe K., 1996, Finite Element Procedures, Prentice Hall, New Jersey.
[4]. Besseling, J.F., 1958, A Theory of Elastic, Plastic, and Creep Deformations of an Initially Isotropic Material Showing Anisotropic Strain-Hardening Creep Recovery and Secondary Creep, J. Appl. Mech., pp. 529-536.
[5]. Dietmann H., 1968, Berechnung der Fliesskurven von Bauelementen bei kleinen Verformungen, readership thesis, University of Stuttgart, Germany.
[6]. Dowling N.E., Brose W.R., Wilson W.K., 1977, Notched Member Fatigue Life Predictions by the Local Strain Approach in Fatigue under Complex Loading, Advances in Engineering, 6, SΑΕ, Warrendale, pp. 55–84.
[7]. Esderts A., Amstutz H., 1994, Betriebsfestigkeit bei mehrachsiger Beanspruchung II, Heft 188, Forschungshefte Forschungskuratorium Maschinenbau e.V., Frankfurt, Germany.
[8]. Filippini M., 2000, Stress Gradient Calculations at Notches, Int. J. Fatigue, 22, pp. 397–409.
[9]. Glinka G., 1985, Energy Density Approach to Calculation of Inelastic Strain–Stress near Notches and Cracks, Eng. Fract. Mech., 22, pp. 485–508.
[10]. Grubisic V., Sonsino C.M., 1982, Influence of Local Strain Distribution on Low-Cycle Fatigue Behavior of Thick-Walled Structures, ASTM-STP 770, pp. 612–629.
[11]. Hardrath H.F., Ohman H., 1951, A Study of Elastic Plastic Stress Concentration Factors due to notches and fillets in plat plates, Technical Note NACA TN 2566.
[12]. Hoffmann M., Seeger T., 1985, A Generalized Method for Estimating Multiaxial Elastic–Plastic Notch Stresses and Strains. Part 1 - Theory, J. Eng. Mater. Technol., 107, pp. 250-254.
[13]. Hoffmann M., Seeger T., 1985, A Generalized Method for Estimating Multiaxial Elastic–Plastic Notch Stresses and Strains. Part 2 - Application and general discussion, J. Eng. Mater. Technol., 107, pp. 255–260.
[14]. Hoffmann M., Seeger T., 1985, Kerbbeanspruchungen I – Ermittlung und Beschreibung mehrachsiger Kerbbeanspruchungen im nichtlinearen Bereich, Heft 115, Forschungshefte Forschungskuratorium Maschinenbau e.V., Frankfurt, Germany.
[15]. Hoffmann M., Seeger T., 1989, Stress–Strain Analysis and Life Predictions of a Notched Shaft under Multiaxial Loading, Multiaxial Fatigue – Analysis and Experiments, SAE AE-14, Warrendale, pp. 81–101.
[16]. Knop M., Jones R., Molent L, Wang C., 1999, On the Glinka and Neuber methods for Calculating Notch Tip Strains under Cyclic Load Spectra, Int. J. Fatigue, 22, pp. 743–755.
[17]. Kühnapfel K.F., 1976, Kerbdehnungen und Kerbspannungen bei elasto-plastischer Beanspruchung; rechnerische Ermittlung, Vergleich mit Versuchsergebnissen, phDthesis, TH Aachen, Germany.
[18]. Lemaitre J., Chaboche J.L., 1990, Mechanics of Solid Materials, Cambridge University Press.
[19]. von Mises, R,. 1913, Mechanik der festen Körper im plastisch-deformablen Zustand, Nachr. Konigl. Ges. Wiss. Göttingen, math. phys., pp. 582-593.
[20]. Mowbray D.F., McConnelee J.E., 1073, Applications of Finite Element Stress Analysis and Stress–Strain Properties in Determining Notch Fatigue Specimen Deformation and Life, ASTM STP 519, pp. 151–169.
[21]. Mroz Z., 1967, On the Description of Anisotropic Work Hardening, J. Mech. Phys. Solids., 17, pp. 163-175.
[22]. Neuber H., 2001, Kerbspannungslehre, 4th edition, Springer-Verlag.
[23]. Neuber H., 1961, Theory of Stress Concentration for Shear strained Prismatical Bodies with Arbitrary Nonlinear Stress–Strain Law, J. Appl. Mech., 28, pp. 544–550.
[24]. Noda N.A., Takase Y., 1999, Stress Concentration Formulae useful for any Shape of Notch in a Round Test Specimen under Tension and under Bending, Fatigue Fract. Engng Mater. Struct., 22, pp.1071–1082.
[25]. Peterson R.E., 1974, The Stress Concentration Factors Handbook, John Wiley and Sons, New York.
[26]. Prager W., 1955, The Theory of Plasticity: Α Survey of Recent Achievements. Proc. Inst. Mech. Engrs., 169, pp. 41-57.
[27]. Saal H., 1975, Näherungsformeln für die Dehnformzahl, Materialprüfung, 17, pp. 395- 398.
[28]. Savaidis A., 1994, Finite-Element Untersuchungen und Entwicklung eines N€aherungsverfahrens zur Beschreibung mehrachsiger elastisch-plastischer Kerbbeanspruchungen bei synchroner nichtproportionaler zyklischer Belastung, 52, Institut für Stahlbau und Werkstoffmechanik, TU Darmstadt, Germany.
[29]. Savaidis A., Savaidis G., Zhang Ch., 2001, FE Fatigue Analysis of Notched Elastic-Plastic Shaft Under Multiaxial Loading Consisting of Constant and Cyclic Components, Int. J. Fatigue, 23, pp. 303-315.
[30]. Savaidis A., Savaidis G., Zhang Ch., 2001, Elastic-Plastic FE Analysis of a Notched Shaft Under Multiaxial Nonproportional Cyclic Loading, Theor. Appl. Fract. Mech., 32, pp. 87-97.
[31]. Savaidis A., Savaidis G., Zhang Ch., 2002, Numerical Analysis of Notched Elastic-Plastic Structures Under Multiaxial Variable Amplitude Loading, Computers and Structures, 80, pp. 1907-1918.
[32]. Savaidis A., Savaidis G., Zhang Ch., 2003, Approximate Elastic-Plastic Analysis of a Notched Bar Under Tensile and Torsional Fatigue, Theor. Appl. Fract. Mech., 40, pp. 85-96.
[33]. Seeger T., Beste A., 1977, Zur Weiterentwicklung von Näherungsformeln für die Berechnung von Kerbbeanspruchungen im elastisch-plastischen Bereich, VDI Verlag, Reihe 18, Nο 2, pp. 1–56.
[34]. Seeger T., Heuler P., 1980, Generalized Application of Neuber’s Rule, J. Test. Eval., 8, pp. 199–204.
[35]. Sih G.C. (Ed.), 1978, Stress Analysis of Notch Problems, Mechanics of Fracture, 5, Noordhoff International Publishing, Leyden.
[36]. Stowell E.Z., 1950, Stress and Strain Concentration at a Circular Hole in an Infinite Plate, Technical Note NACA TN 2073.
[37]. Topper T.H., Wetzel R.M., Morrow J.D., 1969, Neuber’s Rule Applied to Fatigue of Notched Specimens, J. Mater., 1, pp. 200–209.
[38]. Walker E.K., 1977, Multiaxial Stress–Strain Approximations for Notch Fatigue, J. Test. Eval., 5, pp. 106–113.
[39]. Ziegler Η., 1959, Α Modification of Prager's Hardening Rule. Quart. Αppl. Math., 17, pp. 55-60.
[40]. Zheng M., Niemi E., 1997, Analysis of the Stress Concentration Factor for a Shallow Notch by the Slip-Line Field Method, Int. J. Fatigue, 19, p. 191-194.
[2]. Barkey M.E., Socie D.F., Hsia K.J., 1993, A Yield Surface Approach to the Estimation of Notch Strains for Proportional and Nonproportional Cyclic Loading, TAM Report No. 709-UILU ENG-93-6007, University of Illinois, Urbana-Champaign, USA.
[3]. Bathe K., 1996, Finite Element Procedures, Prentice Hall, New Jersey.
[4]. Besseling, J.F., 1958, A Theory of Elastic, Plastic, and Creep Deformations of an Initially Isotropic Material Showing Anisotropic Strain-Hardening Creep Recovery and Secondary Creep, J. Appl. Mech., pp. 529-536.
[5]. Dietmann H., 1968, Berechnung der Fliesskurven von Bauelementen bei kleinen Verformungen, readership thesis, University of Stuttgart, Germany.
[6]. Dowling N.E., Brose W.R., Wilson W.K., 1977, Notched Member Fatigue Life Predictions by the Local Strain Approach in Fatigue under Complex Loading, Advances in Engineering, 6, SΑΕ, Warrendale, pp. 55–84.
[7]. Esderts A., Amstutz H., 1994, Betriebsfestigkeit bei mehrachsiger Beanspruchung II, Heft 188, Forschungshefte Forschungskuratorium Maschinenbau e.V., Frankfurt, Germany.
[8]. Filippini M., 2000, Stress Gradient Calculations at Notches, Int. J. Fatigue, 22, pp. 397–409.
[9]. Glinka G., 1985, Energy Density Approach to Calculation of Inelastic Strain–Stress near Notches and Cracks, Eng. Fract. Mech., 22, pp. 485–508.
[10]. Grubisic V., Sonsino C.M., 1982, Influence of Local Strain Distribution on Low-Cycle Fatigue Behavior of Thick-Walled Structures, ASTM-STP 770, pp. 612–629.
[11]. Hardrath H.F., Ohman H., 1951, A Study of Elastic Plastic Stress Concentration Factors due to notches and fillets in plat plates, Technical Note NACA TN 2566.
[12]. Hoffmann M., Seeger T., 1985, A Generalized Method for Estimating Multiaxial Elastic–Plastic Notch Stresses and Strains. Part 1 - Theory, J. Eng. Mater. Technol., 107, pp. 250-254.
[13]. Hoffmann M., Seeger T., 1985, A Generalized Method for Estimating Multiaxial Elastic–Plastic Notch Stresses and Strains. Part 2 - Application and general discussion, J. Eng. Mater. Technol., 107, pp. 255–260.
[14]. Hoffmann M., Seeger T., 1985, Kerbbeanspruchungen I – Ermittlung und Beschreibung mehrachsiger Kerbbeanspruchungen im nichtlinearen Bereich, Heft 115, Forschungshefte Forschungskuratorium Maschinenbau e.V., Frankfurt, Germany.
[15]. Hoffmann M., Seeger T., 1989, Stress–Strain Analysis and Life Predictions of a Notched Shaft under Multiaxial Loading, Multiaxial Fatigue – Analysis and Experiments, SAE AE-14, Warrendale, pp. 81–101.
[16]. Knop M., Jones R., Molent L, Wang C., 1999, On the Glinka and Neuber methods for Calculating Notch Tip Strains under Cyclic Load Spectra, Int. J. Fatigue, 22, pp. 743–755.
[17]. Kühnapfel K.F., 1976, Kerbdehnungen und Kerbspannungen bei elasto-plastischer Beanspruchung; rechnerische Ermittlung, Vergleich mit Versuchsergebnissen, phDthesis, TH Aachen, Germany.
[18]. Lemaitre J., Chaboche J.L., 1990, Mechanics of Solid Materials, Cambridge University Press.
[19]. von Mises, R,. 1913, Mechanik der festen Körper im plastisch-deformablen Zustand, Nachr. Konigl. Ges. Wiss. Göttingen, math. phys., pp. 582-593.
[20]. Mowbray D.F., McConnelee J.E., 1073, Applications of Finite Element Stress Analysis and Stress–Strain Properties in Determining Notch Fatigue Specimen Deformation and Life, ASTM STP 519, pp. 151–169.
[21]. Mroz Z., 1967, On the Description of Anisotropic Work Hardening, J. Mech. Phys. Solids., 17, pp. 163-175.
[22]. Neuber H., 2001, Kerbspannungslehre, 4th edition, Springer-Verlag.
[23]. Neuber H., 1961, Theory of Stress Concentration for Shear strained Prismatical Bodies with Arbitrary Nonlinear Stress–Strain Law, J. Appl. Mech., 28, pp. 544–550.
[24]. Noda N.A., Takase Y., 1999, Stress Concentration Formulae useful for any Shape of Notch in a Round Test Specimen under Tension and under Bending, Fatigue Fract. Engng Mater. Struct., 22, pp.1071–1082.
[25]. Peterson R.E., 1974, The Stress Concentration Factors Handbook, John Wiley and Sons, New York.
[26]. Prager W., 1955, The Theory of Plasticity: Α Survey of Recent Achievements. Proc. Inst. Mech. Engrs., 169, pp. 41-57.
[27]. Saal H., 1975, Näherungsformeln für die Dehnformzahl, Materialprüfung, 17, pp. 395- 398.
[28]. Savaidis A., 1994, Finite-Element Untersuchungen und Entwicklung eines N€aherungsverfahrens zur Beschreibung mehrachsiger elastisch-plastischer Kerbbeanspruchungen bei synchroner nichtproportionaler zyklischer Belastung, 52, Institut für Stahlbau und Werkstoffmechanik, TU Darmstadt, Germany.
[29]. Savaidis A., Savaidis G., Zhang Ch., 2001, FE Fatigue Analysis of Notched Elastic-Plastic Shaft Under Multiaxial Loading Consisting of Constant and Cyclic Components, Int. J. Fatigue, 23, pp. 303-315.
[30]. Savaidis A., Savaidis G., Zhang Ch., 2001, Elastic-Plastic FE Analysis of a Notched Shaft Under Multiaxial Nonproportional Cyclic Loading, Theor. Appl. Fract. Mech., 32, pp. 87-97.
[31]. Savaidis A., Savaidis G., Zhang Ch., 2002, Numerical Analysis of Notched Elastic-Plastic Structures Under Multiaxial Variable Amplitude Loading, Computers and Structures, 80, pp. 1907-1918.
[32]. Savaidis A., Savaidis G., Zhang Ch., 2003, Approximate Elastic-Plastic Analysis of a Notched Bar Under Tensile and Torsional Fatigue, Theor. Appl. Fract. Mech., 40, pp. 85-96.
[33]. Seeger T., Beste A., 1977, Zur Weiterentwicklung von Näherungsformeln für die Berechnung von Kerbbeanspruchungen im elastisch-plastischen Bereich, VDI Verlag, Reihe 18, Nο 2, pp. 1–56.
[34]. Seeger T., Heuler P., 1980, Generalized Application of Neuber’s Rule, J. Test. Eval., 8, pp. 199–204.
[35]. Sih G.C. (Ed.), 1978, Stress Analysis of Notch Problems, Mechanics of Fracture, 5, Noordhoff International Publishing, Leyden.
[36]. Stowell E.Z., 1950, Stress and Strain Concentration at a Circular Hole in an Infinite Plate, Technical Note NACA TN 2073.
[37]. Topper T.H., Wetzel R.M., Morrow J.D., 1969, Neuber’s Rule Applied to Fatigue of Notched Specimens, J. Mater., 1, pp. 200–209.
[38]. Walker E.K., 1977, Multiaxial Stress–Strain Approximations for Notch Fatigue, J. Test. Eval., 5, pp. 106–113.
[39]. Ziegler Η., 1959, Α Modification of Prager's Hardening Rule. Quart. Αppl. Math., 17, pp. 55-60.
[40]. Zheng M., Niemi E., 1997, Analysis of the Stress Concentration Factor for a Shallow Notch by the Slip-Line Field Method, Int. J. Fatigue, 19, p. 191-194.
Published
2008-11-15
How to Cite
1.
PITATZIS N, SAVAIDIS A, SAVAIDIS G. An Analytical Procedure for the Prediction of the Stress-Strain State in Notches under Multiaxial Fatigue. The Annals of “Dunarea de Jos” University of Galati. Fascicle IX, Metallurgy and Materials Science [Internet]. 15Nov.2008 [cited 5May2024];31(2):5-0. Available from: https://www.gup.ugal.ro/ugaljournals/index.php/mms/article/view/3093
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