Review Article

Celebrating the 100th Anniversary of Inglis Result: From a Single Notch to Random Surface Stress Concentration Solutions

[+] Author and Article Information
Hector E. Medina

Department of Mechanical
and Nuclear Engineering,
Virginia Commonwealth University,
Richmond, VA 23284-3028
e-mail: hedmedina@yahoo.com

Ramana Pidaparti

Department of Mechanical
and Nuclear Engineering,
Virginia Commonwealth University,
Richmond, VA 23284-3028

Brian Hinderliter

Department of Mechanical
and Industrial Engineering,
University of Minnesota-Duluth,
1305 Ordean Court,
Duluth, MN 55812

Or, at least he should be called one of the fathers of the SCF, to give credit to Neuber's account of the facts.

1Corresponding author.

2Neuber explicitly states [1] that Kolosov had developed a similar work and reported it four years earlier [2] in a paper written in Russian.

Manuscript received December 17, 2013; final manuscript received July 17, 2014; published online September 10, 2014. Assoc. Editor: Bart Prorok.

Appl. Mech. Rev 67(1), 010802 (Sep 10, 2014) (9 pages) Paper No: AMR-13-1102; doi: 10.1115/1.4028069 History: Received December 17, 2013; Revised July 17, 2014

We celebrate the first quantitative evidence for the stress concentration effect of flaws analyzed by Inglis. Stress concentration factor (SCF) studies have evolved ever since Inglis' 1913 result related to the problem of the elliptical hole in a plate, which also approximately applies to the half-elliptical notch case. We summarize a hundred years of development of the SCF with the exclusive focus on analytical solutions, with a very specific route: the series of works reviewed and presented herein include a parade of solutions beginning with (and those that followed) Inglis famous result, continue with periodic discrete discontinuities, sinusoidal periodic surfaces, and end with more complex continuous configurations such as random surfaces. Furthermore, we show that the form of Inglis' result is powerful enough to serve as first-order approximation for some cases of multiple discontinuities and even continuous rough topologies. Thus, we proposed the Modified Inglis formula (MIF), to estimate the SCF for a variety of configurations, in honor to Inglis' historical result. The impetus of this review stems from the fact that for many engineering problems involving multiphysical solid–fluid interactions, there is a broad interest to couple stress concentration relationships with thermodynamics, fluid dynamics, or even diffusion equations in order to expand understanding on stress-driven interactions at the solid–fluid interface. Additionally, a handy first-order estimate of the SCF can serve in the initial stage of designs of structures and parts containing discontinuities.

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Neuber, H., and Hahn, H., 1966, “Stress Concentration in Scientific Research and Engineering,” ASME Appl. Mech. Rev., 19(3), pp. 187–199.
Kolosov, G. V., 1909, “On an Application of Complex Function Theory to a Plane Problem of the Mathematical Theory of Elasticity (in Russian),” Dissertation, Dorpat University, Yuriev, 1909.
Inglis, C. E., 1913, “Stresses in a Plate Due to the Presence of Cracks and Sharp Corners,” Trans. Inst. Naval Archit., 55, pp. 219–241.
Hopkinson, B., 1921, 1910 Collected Scientific Papers, Cambridge University Press, Cambridge, UK.
Griffith, A., 1921, “The Phenomenon of Rupture and Flow in Solids,” Philos. Trans. R. Soc. London, Ser. A, 221(582–593), pp. 163–198. [CrossRef]
de Saint-Venant, A., 1856, “Memoire sur la torsion des prismes, avec des considerations sur leur flexion, Imprimerie nationale (read June 13, 1853),” [Mem. Divers Savants14, pp. 233–560 (1855)].
Anderson, T. L., 1991, Fracture Mechanics, Fundamentals and Applications, CRC Press, Boca Raton, FL.
Irwin, G., 1957, “Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate,” ASME J. Appl. Mech., 24, pp. 361–364.
Le Tallec, P., and Mouro, J., 2001, “Fluid Structure Interaction With Large Structural Displacements,” Comput. Meth. Appl. Mech. Eng., 190(24–25), pp. 3039–3067. [CrossRef]
Paidoussis, M. P., Price, S. J., and de Langre, E., 1998, Fluid-Structure Interactions: Slender Structures and Axial Flow, Academic Press, London, UK.
Belytschko, T., 1980, “Fluid-Structure Interaction,” Computers and Structures, 12(4), pp. 459–469. [CrossRef]
De Hart, J., Peters, G., Schreurs, P., and Baaijens, F., 2003, “A Three-Dimensional Computational Analysis of Fluid–Structure Interaction in the Aortic Valve,” J. Biomech., 36(1), pp. 103–112. [CrossRef] [PubMed]
Liang, J., and Suo, Z., 2001, “Stress-Assisted Interactions at a Solid-Fluid Interface,” Interface Sci., 9(1–2), pp. 93–104. [CrossRef]
Srolovitz, D. J., 1989, “On the Stability of Surfaces of Stressed Solids,” Acta Metall., 37(2), pp. 621–625. [CrossRef]
Hillig, W. B., and Charles, R. J., 1965, Surface, Stress-Dependent Surface Reaction, and Strength in High Strength Materials, Wiley, New York, pp. 682–703.
Yu, H., 2005, “Crack Nucleation From a Single Notch Caused by Stress-Dependent Surface Reactions,” Int. J. Solids Struct., 42(13), pp. 3852–3866. [CrossRef]
Peterson, R., 1974, Stress Concentration Factors, Wiley, New York.
Pilkey, W., and Pilkey, D., 2008, Peterson's Stress Concentration Factors, 3rd ed., Wiley, New York.
Hardy, S., and Malik, N., 1992, “A Survey of Post-Peterson Concentration Factor Data,” Int. J. Fatigue, 14(3), pp. 147–153. [CrossRef]
Noda, N., and Takase, Y., 2002, “Stress Concentration Formulas Useful for All Notch Shapes in a Flat Test Specimen Under Tension and Bending,” J. Test. Eval., 30(5), pp. 369–381. [CrossRef]
Noda, N., and Takase, Y., 2006, “Stress Concentration Formula Useful for All Notch Shape in a Round Bar (Comparison Between Torsion, Tension and Bending),” Int. J. Fatigue, 28(2), pp. 151–163. [CrossRef]
Yang, Z., Kim, C., Cho, C., and Beom, H. G., 2008, “The Concentration of Stress and Strain in Finite Thickness Elastic Plate Containing a Circular Hole,” Int. J. Solids Struct., 45, pp. 713–731. [CrossRef]
Green, A., 1948, “Three-Dimensional Stress Systems in Isotropic Plates, I,” Philos. Trans. R. Soc. London, Ser. A, 240(825), pp. 561–597. [CrossRef]
Sternberg, E., and Sadowsky, M., 1949, “Three-Dimensional Solution for the Stress Concentration Around a Circular Hole in a Plate of Arbitrary Thickness,” ASME J. Appl. Mech., 16, pp. 27–38.
Youngdahl, C. K., and Sternberg, E., 1966, “Three-Dimensional Stress Concentration Around a Cylindrical Hole in a Semi-Infinite Elastic Body,” ASME J. Appl. Mech., 33(4), pp. 855–865. [CrossRef]
Folias, E. S., and Wang, J., 1990, “On the Three-Dimensional Stress Field Around a Circular Hole in a Plate of Arbitrary Thickness,” Comput. Mech., 6(5–6), pp. 379–391. [CrossRef]
Li, X., Kasai, T., Nakao, S., Tanaka, H., Ando, T., Shikida, M., and Sato, K., 2005, “Measurement for Fracture Toughness of Single Crystal Silicon Film With Tensile Test,” Sens. Actuators, A, 119(1), pp. 229–235. [CrossRef]
Timoshenko, S., and Goodier, J. N., 1970, Theory of Elasticity, 3rd ed., Mcgraw-Hill, Maidenhead.
Dundurs, J., 1967, “Dependence of Stress on Poisson's Ratio in Plane Elasticity,” Int. J. Solids Struct., 3(6), pp. 1013–1021. [CrossRef]
Grant, R., Lorenzo, M., and Smart, J., 2007, “The Effect of Poisson's Ratio on Stress Concentrations,” J. Strain Anal. Eng. Des., 42(2), pp. 95–104. [CrossRef]
Young, W. C., and Budynas, R. G., 2001, Roark's Formulas for Stress and Strain, McGraw-Hill, Maidenhead.
Morris, I., O’ Donnell, P., Delassus, P., and McGloughlin, T., 2004, “Experimental Assessment of Stress Patterns in Abdominal Aortic Aneurysms Using the Photoelastic Method,” Strain, 40(4), pp. 165–172. [CrossRef]
ESDU Data Sheet 89048, 1989, Elastic Stress Concentration Factors, Geometric Discontinuities in Rods and Tubes of Isotropic Materials, Engineering Science Data Unit, London.
Atzori, B., Lazzarin, P., and Meneghetti, G., 2003, “Fracture Mechanics and Notch Sensitivity,” Fatigue Fract. Eng. Mater. Struct., 26(3), pp. 257–267. [CrossRef]
Neuber, H., 1958, Kerbspannunglehre, 2nd ed., Springer-Verlag, Berlin, Germany.
Cotrell, A. H., 1963, “Mechanics of Fracture,” Tewksbury Symposium of Fracture, University of Melbourne, Australia, pp. 1–27.
Hutchinson, J. W., 1968, “Singular Behaviour at the End of a Tensile Crack in a Hardening Material,” J. Mech. Phys. Solids, 16(1), pp. 13–31. [CrossRef]
Ashby, M. F., 1966, “Work Hardening of Dispersion-Hardened Crystals,” Philos. Mag., 14(132), pp. 1157–1178. [CrossRef]
Xia, L., and Shih, C. F., 1995, “Ductile Crack Growth-I. A Numerical Study Using Computational Cells With Microstructurally-Based Length Scales,” J. Mech. Phys. Solids, 43(2), pp. 233–259. [CrossRef]
Sieradzki, K., and Newman, R. C., 1985, “Brittle Behavior of Ductile Metals During Stress-Corrosion Cracking,” Philos. Mag. A, 51(1), pp. 95–132. [CrossRef]
Suo, Z., and Gong, X., 1993, “Notch Ductile-to-Brittle Transition Due to Localized Inelastic Band,” ASME J. Eng. Mater. Technol., 115(3), pp. 319–326. [CrossRef]
Balankin, A., Susarrey, O., Mora, C., Patiño, J., Yoguez, A., and Garcia, E., 2011, “Stress Concentration and Size Effect in Fracture of Notched Heterogeneous Material,” Phys. Rev. E, 83(01), p. 015101 (R). [CrossRef]
Fichter, W. B., 1970, Stress Concentration in Filament-Stiffened Sheets of Finite Length, NASA TN D-5947.
Franklin, H., 1970, “Hole Stress Concentration in Filamentary Structures,” Fibre Sci. Technol., 2(3), pp. 241–249. [CrossRef]
Pindera, J. T., 1999, “Actual Three-Dimensional Stresses in Notches, Crack Tips and Lamination Planes,” Composites Part B, 30, pp. 189–203. [CrossRef]
Zweben, C., 1974, “An Approximate Method of Analysis for Notched Unidirectional Composites,” Eng. Fract. Mech., 6(1), pp. 1–10. [CrossRef]
Van Dyke, P., and Hedgepeth, J. M., 1969, “Stress Concentrations From Single-Filament Failures in Composite Materials,” Text. Res. J., 39(7), pp. 618–626.
Chang, F. K., and Chang, K. Y., 1987, “A Progressive Damage Model for Laminated Composites Containing Stress Concentrations,” J. Compos. Mater., 21(9), pp. 834–855. [CrossRef]
Fukuda, H., and Kawata, K., 1976, “On the Stress Concentration Factor in Fibrous Composites,” Fibre Sci. Technol., 9(3), pp. 189–203. [CrossRef]
Hedgepeth, J. M., 1961, Stress Concentrations in Filamentary Structures, NASA TN D-882.
Hedgepeth, J. M., and Van Dyke, P., 1967, “Local Stress Concentrations in Imperfect Filamentary Composite Materials,” J. Compos. Mater., 1, pp. 294–309.
Sherman, D., 1961, “Weighty Medium Weakened by Periodically Located Circular and Noncircular Holes,” Inzh. Zh., 1(1), pp. 92–103.
Nisitani, H., 1968, “Method of Approximate Calculation for Interference of Notch Effects and Its Application,” Bull. JSME, 11(47), pp. 725–738. [CrossRef]
Savin, G., 1968, Distribution of Stresses Around Holes [in Russian], Naukova Dumka, Kiev.
Mironenko, N., 1988, “Periodic and Doubly Periodic Plane Problems of the Theory of Elasticity for Domains With Curvilinear Holes,” Prikl. Mekh., 24(6), pp. 91–97.
Heywood, R., 1952, Designing by Photoelasticity, 1st ed., Chapman and Hall, London.
Castagnetti, D., and Dragoni, E., 2013, “Stress Concentration in Periodic Notches: A Critical Investigation of Neuber's Method,” Materialwiss. Werkstofftech, 44(5), pp. 364–371. [CrossRef]
Savruk, M. P., and Kazberuk, A., 2009, “Stresses in an Elastic Plane With Periodic System of Closely Located Holes,” Mater. Sci., 45(6), pp. 831–844. [CrossRef]
Belotserkovskii, C., and Lifanov, I., 1985, Numerical Methods in Singular Integral Equations [in Russian], Nauka, Moscow.
Gao, H., 1991, “A Boundary Perturbation Analysis for Elastic Inclusions and Interfaces,” Int. J. Solids Struct., 28(6), pp. 703–725. [CrossRef]
Gao, H., 1991, “Stress Concentration at Slightly Undulating Surfaces,” J. Mech. Phys. Solids, 39(4), pp. 443–458. [CrossRef]
Green, A. E., and Zerna, W., 1968, Theoretical Elasticity, 2nd ed., Oxford University, London.
Medina, H., and Hinderliter, B., 2013, “Method for Generating and Realising Replicates of Randomly Roughened Surfaces, Tested on Poly Methyl Methacrylate,” Experimental Techniques, (published online). [CrossRef]
Hinderliter, B., and Croll, S., 2008, “Predicting Coating Failure Using the Central Limit Theorem and Physical Modeling,” J. Mater. Sci., pp. 6630–6641. [CrossRef]
Medina, H., and Hinderliter, B., “Stress Concentration at Slightly Roughened Random Surfaces: Analytical Solution,” Int. J. Solids Struct.51(10), pp. 2012–2018. [CrossRef]
Hahn, S. L., 1996, Hilbert Transforms, in the Transforms and Applications Handbook, A.Poularakis, ed., CRC Press, Boca Raton, FL, Chap. 7.
Titchmarsh, E. C., “Conjugate Trigonometric Integrals,” 1924, Proc. London Math. Soc., 24, pp. 109–130.
Medina, H., and Hinderliter, B., 2012, “Use of Poly (Methyl Methacrylate) in the Study of Randomly Damaged Surfaces: I. Experimental Approach,” Polymer, 53, pp. 4525–4532. [CrossRef]
Medina, H., and Hinderliter, B., 2013, “Stress, Strain, and Energy at Fracture of Degraded Surfaces: Study of Replicates of Rough Surfaces,” ASME J. Eng. Gas Turbines Power, 136(3), p. 032502. [CrossRef]
Medina, H., and Hinderliter, B., 2013, “Where Do Random Rough Surfaces Fail? Part I: Fracture Loci Safety Envelopes at Early Stages of Degradation,” J. Energy Power Eng., 7, pp. 907–916. Available at: http://davidpublishing.org/show.html?12885
Chen, X., and Gibson, J. M., 1998, “Experimental Evidence of a Gaussian Roughness at Si(111)/SiO2 Interfaces,” Phys. Rev. Lett., 81(22), pp. 4919–4922. [CrossRef]
Bennet, H. E., and Porteus, J. O., 1961, “Relation Between Surface Roughness and Specular Reflectance at Normal Incidence,” J. Opt. Soc. Am. A, 51(2), pp. 123–129. [CrossRef]
Majumdar, A., and Tien, C. L., 1990, “Fractal Characterization and Simulation of Rough Surfaces,” Wear, 136(2), pp. 313–327. [CrossRef]
Voss, R. F., 1988, Fractals in Nature: From Characterization to Simulation, H.-O. Peitgen, and D. Saupe, eds, Springer, New York, pp. 21–70.
Adler, R. J., and Firman, D., 1981, “A Non-Gaussian Model for Random Surfaces,” Philos. Trans. R. Soc. London, Ser. A, 303(1479), pp. 433–462. [CrossRef]
Cerit, M., Genel, K., and Eksi, S., 2009, “Numerical Investigation on Stress Concentration of Corrosion Pit,” Eng. Fail. Anal., 16(7), pp. 2467–2472. [CrossRef]
Sun, K., Samuel, G., and Guo, B., 2005, “Effect of Stress Concentration Factors Due to Corrosion on Production String Design,” Old Prod. Facil., 20(4), pp. 334–339. [CrossRef]
Roberge, Pierre, R., 2008, Corrosion Engineering Principles and Practice, McGraw-Hill, New York.
Sasaki, K., and Burstein, G. T., 1996, “The Generation of Surface Roughness During Slurry Erosion-Corrosion and Its Effect on the Pitting Potential,” Corros. Sci., 38(12), pp. 2111–2120. [CrossRef]
Pidaparti, R. M., Koombua, K., and Appajoysula, S. R., 2009, “Corrosion Pit Induced Stresses Prediction From SEM and Finite Element Analysis,” Int. J. Comput. Methods Eng. Sci. Mech.10(2), pp. 117–123. [CrossRef]
Turnbull, A., Wright, L., and Crocker, L., 2010, “New insight into the pit-to-crack transition from finite element analysis of the stress and strain distribution around a corrosion pit,” Corrosion Science, 52(4), pp. 1492–1498. [CrossRef]
Aoki, S., and Kishimoto, K., 1990, “Application of BEM to Galvanic Corrosion and Cathodic Protection,” Topics in Boundary Element Research, Electrical Engineering Applications, Vol. 7, pp. 65–86. [CrossRef]
Brebbia, C. A., Telles, J. C. F., and Wrobel, L. C., 2012, Boundary Element Techniques: Theory and Applications in Engineering, Springer, London.
Ernst, P., and Newman, R. C., 2001, “Pit Growth Studies in Stainless Steel Foils. I. Introduction and Pit Growth Kinetics,” Corros. Sci., 44(5), pp. 927–941. [CrossRef]
Benjamin, A. C., Cunha, D. J. S., Silva, R. C. C., Guerreiro, J. N. C., Campello, G. C., and Roveri, F. E., 2007, “Stress Concentration Factors for a Drilling Riser Containing Corrosion Pits,” ASME Paper No. OMAE2007-29281. [CrossRef]
Huang, X.-G., and Xu, J.-Q., 2013, “3D Analysis for Pit Evolution and Pit-to-Crack Transition During Corrosion Fatigue,” J. Zhejiang Univ. Sci. A, 14(4), pp. 292–299. [CrossRef]
Xiao-Guang, H., and Jin-Quan, X., 2012, “Pit Morphology Characterization and Corrosion Fatigue Crack Nucleation Analysis Based on Energy Principle,” Fatigue Fract. Eng. Mater. Struct., 35(7), pp. 606–613. [CrossRef]
Acuna, N., Gonzalez-Sanchez, J., Ku-Basulto, G., and Dominguez, L., 2006, “Analysis of the Stress Intensity Factor Around Corrosion Pits Developed on Structures Subjected to Mixed Loading,” Scr. Mater., 55(4), pp. 363–366. [CrossRef]
Sharma, M. M., and Ziemian, C. W., 2008, “Pitting and Stress Corrosion Cracking Susceptibility of Nanostructured Al-Mg Alloys in Natural and Artificial Environments,” J. Mater. Eng. Perform., 17(6), pp. 870–878. [CrossRef]
Turnbull, A., Wright, L., and Crocker, L., 2010, “New Insight Into the Pit-to-Crack Transition From Finite Element Analysis of the Stress and Strain Distribution Around a Corrosion Pit,” Corros. Sci., 52, pp. 1492–1498. [CrossRef]
Turnbull, A., Horner, D. A., and Connolly, B. J., 2009, “Challenges in Modelling the Evolution of Stress Corrosion Cracks From Pits,” Eng. Fract. Mech., 76(5), pp. 633–640. [CrossRef]
Horner, D. A., Connolly, B. J., Zhou, S., Crocker, L., and Turnbull, A., 2011, “Novel Images of the Evolution of Stress Corrosion Cracks From Corrosion Pits,” Corros. Sci., 53(11), pp. 3466–3485. [CrossRef]
Pidaparti, R. M., and Patel, R. K., 2010, “Investigation of a Single Pit/Defect Evolution During the Corrosion Process,” Corros. Sci., 52(9), pp. 3150–3153. [CrossRef]
Pidaparti, R. M., and Appajoysula, S. R., 2008, “Analysis of Pits Induced Stresses Due to Metal Corrosion,” Corros. Sci., 50(7), pp. 1932–1938. [CrossRef]
Pidaparti, R., and Patel, R., 2008, “Correlation Between Corrosion Pits and Stresses in Al Alloys,” J. Mater. Lett., 62(30), pp. 4497–4499. [CrossRef]
Pidaparti, R., and Patel, R., 2011, “Modeling the Evolution of Stresses Induced by Corrosion Damage in Metals,” J. Mater. Eng. Perform., 20(7), pp. 1114–1120. [CrossRef]
Pidaparti, R. M., and Johnson, A. C., 2013, “Evaluation of Stress Environment Around Pits in Nickel Aluminum Bronze Metal Under Corrosion and Cyclic Stresses,” Struct. Durability Health Monit., 9(1), pp. 87–98. [CrossRef]
Rhinoceros 4.0, 2012, Rhino 4.0 Tutorial, McNeel North America WA.
SolidWorks 2012, 2012, SolidWorks Tutorial, SolidWorks Corporation, Concord, MA.
ansys 14.0, 2012, ansys Tutorial, ANSYS, Inc., Canonsburg, P.


Grahic Jump Location
Fig. 1

(a) Elliptical hole in a infinite elastic thin plate. (b) Semi-elliptical notch in a semi-infinite elastic thin plate. In both cases, remote load is applied perpendicular to major axis and at location very far from the discontinuity.

Grahic Jump Location
Fig. 3

(a) 3D specimen with hole through. (b) 3D notched specimen.

Grahic Jump Location
Fig. 4

(a) Single-notch stress lines. (b) Periodic-discrete-notch configuration. Stress is relaxed due to neighboring notches.

Grahic Jump Location
Fig. 5

A sinusoidal surface. The maximum SCF is found at the troughs of the waves [61].

Grahic Jump Location
Fig. 6

Random rough surface with a prescribed finite autocorrelation length and whose heights are normally distributed. An analytical solution for the root-mean-square SCF of the 2D version was derived and reported in Ref. [65].

Grahic Jump Location
Fig. 7

An example case to illustrate the random surfaces resulting from corrosion, the stress distributions, and the SCF (K in the figures) for: (a) and (b) an early stage corroded surface, and (c) and (d) the same surface at a later stage of corrosion. (a) and (c) are atomic force scans and (b) and (d) are finite element models.



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