Relationship between Fracture Toughness and Fracture Surface Fractal Dimension in AISI 4340 Steel

Affiliation(s)

NAVAIR Jacksonville, Jacksonville, USA.

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Materials Science & Engineering Department, University of Florida, Gainesville, USA.

NAVAIR Jacksonville, Jacksonville, USA.

ABSTRACT

This study analyzes the relationship between fracture toughness and the fracture surface fractal dimension for a set of twenty-four CT-type AISI 4340 steel specimens heat treated to a variety of tensile strengths. Specimens were tested in accordance with ASTM E 399. Their respective fracture surfaces were plated, polished, photographed under an SEM in BSE mode and measured according to the Richardson method to obtain fractal dimensions. For brittle materials the limited results are consistent with previous literature: increasing fractal dimension with increasing toughness. For partially or fully ductile materials the results indicate a decrease in fractal dimension with an increase in fracture toughness. The data are modeled using a variation of the function applied to ceramics. Fracture in a ductile mode is characterized by the formation of dimples which exhibit fractal characteristics. The results are discussed in terms of the micromechanisms of fracture.

Cite this paper

L. Carney and J. Mecholsky, "Relationship between Fracture Toughness and Fracture Surface Fractal Dimension in AISI 4340 Steel,"*Materials Sciences and Applications*, Vol. 4 No. 4, 2013, pp. 258-267. doi: 10.4236/msa.2013.44032.

L. Carney and J. Mecholsky, "Relationship between Fracture Toughness and Fracture Surface Fractal Dimension in AISI 4340 Steel,"

References

[1] S. R. Lampman, “ASM Handbook Vol. 19: Fatigue and Fracture,” ASM International, Ohio, 1996.

[2] V. J. Colangelo and F. A. Heiser, “Analysis of Metallurgical Failures,” John Wiley & Sons, New York, 1974.

[3] J. J. Mecholsky, “Fractography in Failure Analysis,” American Society for Testing and Materials, Philadelphia, 1978.

[4] J. C. Russ, “Fractal Surfaces,” Plenum Press, New York, 1994.

[5] B. B. Mandelbrot, “The Fractal Geometry of Nature,” WH Freeman Co., New York, 1982.

[6] J. J. Mecholsky Jr., “Fractal Analysis and Fractography: What Can We Learn That’s New?” Key Engineering Materials, Vol. 413, 2009, pp. 145-153.

[7] V. Y. Milman, N. A. Stelmashenko and R. Blumenfeld, “Fracture Surfaces: A Critical Review of Fractal Studies and a Novel Morphological Analysis of Scanning Tunneling Microscopy Measurements,” Progress in Materials Science, 1994, Vol. 38, 1994, pp. 425-474. doi:10.1016/0079-6425(94)90006-X

[8] G. P. Cherepanov, A. S. Balankin and V. S. Ivanova, “Fractal Fracture Mechanics—A Review,” Engineering Fracture Mechanics, Vol. 51, No. 6, 1995, pp. 997-1033. doi:10.1016/0013-7944(94)00323-A

[9] A. M. Gokhale and E. E. Underwood, “A General Method for Estimation of Fracture Surface Roughness: Part I. Theoretical Aspects,” Metallurgical Transactions A, Vol. 21, No. 5, 1990, pp. 1193-1199.

[10] A. M. Gokhale and W. J. Drury, “A General Method for Estimation of Fracture Surface Roughness: Part II. Practical Considerations,” Metallurgical Transactions A, Vol. 21, No. 5, 1990, pp. 1201-1207.

[11] W. Lei and B. Chen, “Fractal Characterization of Some Fracture Phenomena,” Engineering Fracture Mechanics, Vol. 50, No. 2, 1995, pp. 149-155. doi:10.1016/0013-7944(94)00180-P

[12] J. J. Friel and C. S. Pande, “Direct Determination of Fractal Dimension of Fracture Surfaces Using Scanning Electron Microscopy and Stereoscopy,” Journal of Materials Research, Vol. 8, No. 1, 1993, pp. 100-104.

[13] M. Coster and J. L. Chermant, “Recent Developments in Quantitative Fractography,” International Metals Review, Vol. 28, No. 4, 1983, pp. 228-250.

[14] Z. P. Bazant, “Scaling Laws in Mechanics of Failure,” Journal of Engineering Materials and Technology, Vol. 119, No. 9, 1993, pp. 1828-1844.

[15] D. E. Roach and A. D. Fowler, “Dimensionality Analysis of Patterns: Fractal Measurements,” Computer and Geosciences, Vol. 19, No. 6, 1993, pp. 849-869. doi:10.1016/0098-3004(93)90055-A

[16] L. L. Mishnaevsky, “Methods of the Theory of Complex Systems in Modelling of Fracture: A Brief Review,” Engineering Fracture Mechanics, Vol. 56, No. 1, 1997, pp. 47-56. doi:10.1016/S0013-7944(96)00110-5

[17] L. V. Meisel, “Perimeter-Area Analysis, the Slit-Island Method and the Fractal Characterization of Metallic Fracture Surfaces,” Journal of Physics D: Applied Physics, Vol. 24, No. 6, 1991, pp. 942-952. doi:10.1088/0022-3727/24/6/020

[18] A. Carpinteri, “Fractal Nature of Material Microstructure and Size Effects on Apparent Mechanical Properties,” Mechanics of Materials, Vol. 18, No. 2, 1994, pp. 89-101. doi:10.1016/0167-6636(94)00008-5

[19] A. Carpinteri, G. Ferrara and L. Imperato, “Scaling Laws for Strength and Toughness of Disordered Materials: A Unified Theory Based on Fractal Geometry,” Engineering Fracture Mechanics, Vol. 48, No. 5, 1994, pp. 673-689. doi:10.1016/0013-7944(94)90175-9

[20] D. E. Passoja, “Fundamental Relationships between Energy and Geometry in Fracture,” In: J. Varner and V. D. Frechette, Eds., Fractography of Glasses and Ceramics, American Ceramic Society, Westerville, 1988, pp. 101-126.

[21] J. J. Mecholsky, T. J. Mackin and D. E. Passoja, “Self-Similar Crack Propagation in Brittle Materials,” In: J. Varner and V. D. Frechette, Eds., Fractography of Glasses and Ceramics, American Ceramic Society, Westerville, 1988, pp. 127-134.

[22] J. J. Mecholsky, D. E. Passoja and K. S. Feinberg-Ringel, “Quantitative Analysis of Brittle Fracture Surfaces Using Fractal Geometry,” Journal of the American Ceramic Society, Vol. 72, No. 1, 1989, pp. 60-65. doi:10.1111/j.1151-2916.1989.tb05954.x

[23] J. J. Mecholsky and S. W. Freiman, “Relationship between Fractal Geometry and Fractography,” Journal of the American Ceramic Society, Vol. 74, No. 12, 1991, pp. 3136-3138. doi:10.1111/j.1151-2916.1991.tb04313.x

[24] J. J. Mecholsky, “Quantitative Fractography: An Assessment,” In: J. R. Varner and V. D. Frechette, Eds., Ceramics Transaction: Fractography of Glasses and Ceramics II, American Ceramic Society, Westerville, 1991, pp. 413-451.

[25] J. J. Mecholsky, Fractography, Fracture Mechanics and Fractal Geometry: An Integration,” In: J. R. Varner, V. D. Frechette and G. D. Quinn, Eds., Ceramic Transactions: Fractography of Glasses and Ceramics III, American Ceramic Society, Westerville, 1996, pp. 385-393.

[26] B. B. Mandelbrot, D. E. Passoja and A. J. Paullay, “Fractal Character of Fracture Surfaces of Metals,” Nature, Vol. 308, 1984, pp. 721-722. doi:10.1038/308721a0

[27] C. S. Pande, L. R. Richards and S. Smith, “Fractal Characteristics of Fractured Surfaces,” Journal of Materials Science Letters, Vol. 6, No. 3, 1987, pp. 295-297. doi:10.1007/BF01729330

[28] C. S. Pande, L. E. Richards, L. E. Louat, B. D. Dempsey and A. J. Schwoeble, “Fractal Characterization of Fractured Surfaces,” Acta Metallurgica, Vol. 35, No. 7, 1987, pp. 1633-1637. doi:10.1016/0001-6160(87)90110-6

[29] G. Bo and Z. H. Lai, “Fractal Characteristics of J-R Resistance Curves of Ti-6A1-4V Alloys,” Engineering Fracture Mechanics, Vol. 44, No. 6, 1993, pp. 991-995. doi:10.1016/0013-7944(93)90119-D

[30] W. A. Spitzig, G. E. Pellisier, C. D. Beachem, A. J. Brothers, M. Hill and W. R. Warke, “Electron Fractography,” American Society for Testing and Materials, Philadelphia, 1968.

[31] G. R. Yoder and V. Weiss, “Superplasticity in Eutectoid Steel,” Metallurgical Transactions, Vol. 3, No. 3, 1972, pp. 675-681. doi:10.1007/BF02642750

[32] K. K. Ray and G. Mandal, “Study of Correlation between Fractal Dimension and Impact Energy in a High Strength Low Alloy Steel,” Acta Metallurgica et Materialia, Vol. 40, No. 3, 1992, pp. 463-469. doi:10.1016/0956-7151(92)90394-T

[33] O. A. Hilders and D. Pilo, “On the Development of a Relation between Fractal Dimension and Impact Toughness,” Materials Characterization, Vol. 38, No. 3, 1997, pp. 121-127. doi:10.1016/S1044-5803(96)00148-9

[34] T. J. Hill, A. Della Bona and J. J. Mecholsky, “Establishing a Protocol for Measurements of Fractal Dimensions in Brittle Materials,” Journal of Materials Science, Vol. 36, No. 11, 2001, pp. 2651-2657. doi:10.1023/A:1017900526824

[35] T. J. Mackin, J. J. Mecholsky and D. E. Passoja, “Crack Propagation in Brittle Materials as a Fractal Process,” Materials Research Society, Warrendale, 1987.

[36] E. E. Underwood and K. Banerji, “Fractals in Fractography” Materials Science and Engineering, Vol. 80, 1986, pp. 1-14. doi:10.1016/0025-5416(86)90297-1

[37] K. Wiencek, A. Czarski and T. Skowronek, “Fractal Characterization of Fractured Surfaces of a Steel Containing Dispersed Fe3C Carbide Phase,” Materials Characterization, Vol. 46, No. 2-3, 2001, pp. 235-238. doi:10.1016/S1044-5803(01)00129-2

[38] E. Bouchaud, G. Lapasset and J. Planes, “Fractal Dimension of Fractured Surfaces: A Universal Value?” Europhysics Letters, Vol. 13, No. 1, 1990, pp. 73-79.

[39] E. Bouchaud, G. Lapasset and J. Planes, “Statistics of Branched Fracture Surfaces,” Physical Review B, 1993, Vol. 48, No. 5, 1993, pp. 2917-2928. doi:10.1103/PhysRevB.48.2917

[40] E. Bouchaud, “Scaling Properties of Cracks,” Journal of Physics: Condensed Matter, Vol. 9, No. 21, 1997, pp. 4319-4344. doi:10.1088/0953-8984/9/21/002

[41] Z. Q. Mu and C. W. Lung, “Studies on the Fractal Dimension and Fracture Toughness of Steel,” Journal of Physics D: Applied Physics, Vol. 21, No. 5, 1988, pp. 848-850. doi:10.1088/0022-3727/21/5/031

[42] R. E. Williford, “Scaling Similarities between Fracture Surfaces, Energies, and a Structure Parameter,” Scripta Metallurgica, Vol. 22, No. 2, 1988, pp. 1749-1754. doi:10.1016/S0036-9748(88)80333-8

[43] American Society for Testing and Materials, “Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials,” 1990.

[44] Society of Automotive Engineers, “Heat Treatment of Steel Parts: Aerospace Materials Specification 2759C,” 2000.

[45] American Society for Testing and Materials, “Test Method for Microhardness of Materials,” Philadelphia, 1999.

[46] Louis Raymond, “Metal Progress Datasheet: Nomogram for Determining ASTM Grain Size,” Metals Progress, Vol. 126, No. 6, 1984, p.34.

[47] Media Cybernetics, “Materials-Pro Analyzer User Guide,” Silver Springs, 1996.

[48] J. K. West and L. L. Hench, “The Effect of Environment on Silica Fracture: Vacuum, Carbon Monoxide, Water and Nitrogen,” Philosophical Magazine A, Vol. 77, No. 1, 1998, pp. 85-113. doi:10.1080/13642819808206385

[49] K. Ishikawa, “Fractals in Dimple Patterns of Ductile Fracture,” Journal of Materials Science Letters, Vol. 9, No. 4, 1990, pp. 400-402. doi:10.1007/BF00721011

[50] W. M. Garrison Jr., “A Micromechanistic Interpretation of the Influence of Undissolved Carbides on the Fracture Toughness of a Low Alloy Steel,” Scripta Metallurgica, Vol. 20, No. 5, 1986, pp. 633-636. doi:10.1016/0036-9748(86)90480-1

[51] W. M. Garrison Jr. and N. R. Moody, “The Influence of Inclusion Spacing and Microstructure on the Fracture Toughness of Steel,” Metallurgical Transactions A, Vol. 18, No. 7, 1987, pp. 1257-1263.

[52] W. M. Garrison and N. R. Moody, “Ductile Fracture,” Journal of Physics and Chemistry of Solids, Vol. 48, No. 11, 1987, pp. 1035-1074. doi:10.1016/0022-3697(87)90118-1

[53] W. M. Garrison Jr., “A Microstructural Interpretation of the Fracture Strain and Characteristics Fracture Distance,” Scripta Metallurgica, Vol. 18, No. 6, 1984, pp. 583-586. doi:10.1016/0036-9748(84)90345-4

[54] W. M. Garrison Jr., A. L. Wojcieszynski and L. E. Iorio, Conference Proceedings of 1997 TMS Annual Meeting, Orlando, 1997, pp. 361-372.

[55] R. O. Ritchie, W. L. Server and R. A. Wullaert, “Critical Fracture Stress and Fracture Strain Models for the Prediction of Lower and Upper Shelf Toughness in Nuclear Pressure Vessel Steels,” Metallurgical Transactions A, Vol. 10, No. 10, 1979, pp. 1557-1570. doi:10.1007/BF02812022

[1] S. R. Lampman, “ASM Handbook Vol. 19: Fatigue and Fracture,” ASM International, Ohio, 1996.

[2] V. J. Colangelo and F. A. Heiser, “Analysis of Metallurgical Failures,” John Wiley & Sons, New York, 1974.

[3] J. J. Mecholsky, “Fractography in Failure Analysis,” American Society for Testing and Materials, Philadelphia, 1978.

[4] J. C. Russ, “Fractal Surfaces,” Plenum Press, New York, 1994.

[5] B. B. Mandelbrot, “The Fractal Geometry of Nature,” WH Freeman Co., New York, 1982.

[6] J. J. Mecholsky Jr., “Fractal Analysis and Fractography: What Can We Learn That’s New?” Key Engineering Materials, Vol. 413, 2009, pp. 145-153.

[7] V. Y. Milman, N. A. Stelmashenko and R. Blumenfeld, “Fracture Surfaces: A Critical Review of Fractal Studies and a Novel Morphological Analysis of Scanning Tunneling Microscopy Measurements,” Progress in Materials Science, 1994, Vol. 38, 1994, pp. 425-474. doi:10.1016/0079-6425(94)90006-X

[8] G. P. Cherepanov, A. S. Balankin and V. S. Ivanova, “Fractal Fracture Mechanics—A Review,” Engineering Fracture Mechanics, Vol. 51, No. 6, 1995, pp. 997-1033. doi:10.1016/0013-7944(94)00323-A

[9] A. M. Gokhale and E. E. Underwood, “A General Method for Estimation of Fracture Surface Roughness: Part I. Theoretical Aspects,” Metallurgical Transactions A, Vol. 21, No. 5, 1990, pp. 1193-1199.

[10] A. M. Gokhale and W. J. Drury, “A General Method for Estimation of Fracture Surface Roughness: Part II. Practical Considerations,” Metallurgical Transactions A, Vol. 21, No. 5, 1990, pp. 1201-1207.

[11] W. Lei and B. Chen, “Fractal Characterization of Some Fracture Phenomena,” Engineering Fracture Mechanics, Vol. 50, No. 2, 1995, pp. 149-155. doi:10.1016/0013-7944(94)00180-P

[12] J. J. Friel and C. S. Pande, “Direct Determination of Fractal Dimension of Fracture Surfaces Using Scanning Electron Microscopy and Stereoscopy,” Journal of Materials Research, Vol. 8, No. 1, 1993, pp. 100-104.

[13] M. Coster and J. L. Chermant, “Recent Developments in Quantitative Fractography,” International Metals Review, Vol. 28, No. 4, 1983, pp. 228-250.

[14] Z. P. Bazant, “Scaling Laws in Mechanics of Failure,” Journal of Engineering Materials and Technology, Vol. 119, No. 9, 1993, pp. 1828-1844.

[15] D. E. Roach and A. D. Fowler, “Dimensionality Analysis of Patterns: Fractal Measurements,” Computer and Geosciences, Vol. 19, No. 6, 1993, pp. 849-869. doi:10.1016/0098-3004(93)90055-A

[16] L. L. Mishnaevsky, “Methods of the Theory of Complex Systems in Modelling of Fracture: A Brief Review,” Engineering Fracture Mechanics, Vol. 56, No. 1, 1997, pp. 47-56. doi:10.1016/S0013-7944(96)00110-5

[17] L. V. Meisel, “Perimeter-Area Analysis, the Slit-Island Method and the Fractal Characterization of Metallic Fracture Surfaces,” Journal of Physics D: Applied Physics, Vol. 24, No. 6, 1991, pp. 942-952. doi:10.1088/0022-3727/24/6/020

[18] A. Carpinteri, “Fractal Nature of Material Microstructure and Size Effects on Apparent Mechanical Properties,” Mechanics of Materials, Vol. 18, No. 2, 1994, pp. 89-101. doi:10.1016/0167-6636(94)00008-5

[19] A. Carpinteri, G. Ferrara and L. Imperato, “Scaling Laws for Strength and Toughness of Disordered Materials: A Unified Theory Based on Fractal Geometry,” Engineering Fracture Mechanics, Vol. 48, No. 5, 1994, pp. 673-689. doi:10.1016/0013-7944(94)90175-9

[20] D. E. Passoja, “Fundamental Relationships between Energy and Geometry in Fracture,” In: J. Varner and V. D. Frechette, Eds., Fractography of Glasses and Ceramics, American Ceramic Society, Westerville, 1988, pp. 101-126.

[21] J. J. Mecholsky, T. J. Mackin and D. E. Passoja, “Self-Similar Crack Propagation in Brittle Materials,” In: J. Varner and V. D. Frechette, Eds., Fractography of Glasses and Ceramics, American Ceramic Society, Westerville, 1988, pp. 127-134.

[22] J. J. Mecholsky, D. E. Passoja and K. S. Feinberg-Ringel, “Quantitative Analysis of Brittle Fracture Surfaces Using Fractal Geometry,” Journal of the American Ceramic Society, Vol. 72, No. 1, 1989, pp. 60-65. doi:10.1111/j.1151-2916.1989.tb05954.x

[23] J. J. Mecholsky and S. W. Freiman, “Relationship between Fractal Geometry and Fractography,” Journal of the American Ceramic Society, Vol. 74, No. 12, 1991, pp. 3136-3138. doi:10.1111/j.1151-2916.1991.tb04313.x

[24] J. J. Mecholsky, “Quantitative Fractography: An Assessment,” In: J. R. Varner and V. D. Frechette, Eds., Ceramics Transaction: Fractography of Glasses and Ceramics II, American Ceramic Society, Westerville, 1991, pp. 413-451.

[25] J. J. Mecholsky, Fractography, Fracture Mechanics and Fractal Geometry: An Integration,” In: J. R. Varner, V. D. Frechette and G. D. Quinn, Eds., Ceramic Transactions: Fractography of Glasses and Ceramics III, American Ceramic Society, Westerville, 1996, pp. 385-393.

[26] B. B. Mandelbrot, D. E. Passoja and A. J. Paullay, “Fractal Character of Fracture Surfaces of Metals,” Nature, Vol. 308, 1984, pp. 721-722. doi:10.1038/308721a0

[27] C. S. Pande, L. R. Richards and S. Smith, “Fractal Characteristics of Fractured Surfaces,” Journal of Materials Science Letters, Vol. 6, No. 3, 1987, pp. 295-297. doi:10.1007/BF01729330

[28] C. S. Pande, L. E. Richards, L. E. Louat, B. D. Dempsey and A. J. Schwoeble, “Fractal Characterization of Fractured Surfaces,” Acta Metallurgica, Vol. 35, No. 7, 1987, pp. 1633-1637. doi:10.1016/0001-6160(87)90110-6

[29] G. Bo and Z. H. Lai, “Fractal Characteristics of J-R Resistance Curves of Ti-6A1-4V Alloys,” Engineering Fracture Mechanics, Vol. 44, No. 6, 1993, pp. 991-995. doi:10.1016/0013-7944(93)90119-D

[30] W. A. Spitzig, G. E. Pellisier, C. D. Beachem, A. J. Brothers, M. Hill and W. R. Warke, “Electron Fractography,” American Society for Testing and Materials, Philadelphia, 1968.

[31] G. R. Yoder and V. Weiss, “Superplasticity in Eutectoid Steel,” Metallurgical Transactions, Vol. 3, No. 3, 1972, pp. 675-681. doi:10.1007/BF02642750

[32] K. K. Ray and G. Mandal, “Study of Correlation between Fractal Dimension and Impact Energy in a High Strength Low Alloy Steel,” Acta Metallurgica et Materialia, Vol. 40, No. 3, 1992, pp. 463-469. doi:10.1016/0956-7151(92)90394-T

[33] O. A. Hilders and D. Pilo, “On the Development of a Relation between Fractal Dimension and Impact Toughness,” Materials Characterization, Vol. 38, No. 3, 1997, pp. 121-127. doi:10.1016/S1044-5803(96)00148-9

[34] T. J. Hill, A. Della Bona and J. J. Mecholsky, “Establishing a Protocol for Measurements of Fractal Dimensions in Brittle Materials,” Journal of Materials Science, Vol. 36, No. 11, 2001, pp. 2651-2657. doi:10.1023/A:1017900526824

[35] T. J. Mackin, J. J. Mecholsky and D. E. Passoja, “Crack Propagation in Brittle Materials as a Fractal Process,” Materials Research Society, Warrendale, 1987.

[36] E. E. Underwood and K. Banerji, “Fractals in Fractography” Materials Science and Engineering, Vol. 80, 1986, pp. 1-14. doi:10.1016/0025-5416(86)90297-1

[37] K. Wiencek, A. Czarski and T. Skowronek, “Fractal Characterization of Fractured Surfaces of a Steel Containing Dispersed Fe3C Carbide Phase,” Materials Characterization, Vol. 46, No. 2-3, 2001, pp. 235-238. doi:10.1016/S1044-5803(01)00129-2

[38] E. Bouchaud, G. Lapasset and J. Planes, “Fractal Dimension of Fractured Surfaces: A Universal Value?” Europhysics Letters, Vol. 13, No. 1, 1990, pp. 73-79.

[39] E. Bouchaud, G. Lapasset and J. Planes, “Statistics of Branched Fracture Surfaces,” Physical Review B, 1993, Vol. 48, No. 5, 1993, pp. 2917-2928. doi:10.1103/PhysRevB.48.2917

[40] E. Bouchaud, “Scaling Properties of Cracks,” Journal of Physics: Condensed Matter, Vol. 9, No. 21, 1997, pp. 4319-4344. doi:10.1088/0953-8984/9/21/002

[41] Z. Q. Mu and C. W. Lung, “Studies on the Fractal Dimension and Fracture Toughness of Steel,” Journal of Physics D: Applied Physics, Vol. 21, No. 5, 1988, pp. 848-850. doi:10.1088/0022-3727/21/5/031

[42] R. E. Williford, “Scaling Similarities between Fracture Surfaces, Energies, and a Structure Parameter,” Scripta Metallurgica, Vol. 22, No. 2, 1988, pp. 1749-1754. doi:10.1016/S0036-9748(88)80333-8

[43] American Society for Testing and Materials, “Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials,” 1990.

[44] Society of Automotive Engineers, “Heat Treatment of Steel Parts: Aerospace Materials Specification 2759C,” 2000.

[45] American Society for Testing and Materials, “Test Method for Microhardness of Materials,” Philadelphia, 1999.

[46] Louis Raymond, “Metal Progress Datasheet: Nomogram for Determining ASTM Grain Size,” Metals Progress, Vol. 126, No. 6, 1984, p.34.

[47] Media Cybernetics, “Materials-Pro Analyzer User Guide,” Silver Springs, 1996.

[48] J. K. West and L. L. Hench, “The Effect of Environment on Silica Fracture: Vacuum, Carbon Monoxide, Water and Nitrogen,” Philosophical Magazine A, Vol. 77, No. 1, 1998, pp. 85-113. doi:10.1080/13642819808206385

[49] K. Ishikawa, “Fractals in Dimple Patterns of Ductile Fracture,” Journal of Materials Science Letters, Vol. 9, No. 4, 1990, pp. 400-402. doi:10.1007/BF00721011

[50] W. M. Garrison Jr., “A Micromechanistic Interpretation of the Influence of Undissolved Carbides on the Fracture Toughness of a Low Alloy Steel,” Scripta Metallurgica, Vol. 20, No. 5, 1986, pp. 633-636. doi:10.1016/0036-9748(86)90480-1

[51] W. M. Garrison Jr. and N. R. Moody, “The Influence of Inclusion Spacing and Microstructure on the Fracture Toughness of Steel,” Metallurgical Transactions A, Vol. 18, No. 7, 1987, pp. 1257-1263.

[52] W. M. Garrison and N. R. Moody, “Ductile Fracture,” Journal of Physics and Chemistry of Solids, Vol. 48, No. 11, 1987, pp. 1035-1074. doi:10.1016/0022-3697(87)90118-1

[53] W. M. Garrison Jr., “A Microstructural Interpretation of the Fracture Strain and Characteristics Fracture Distance,” Scripta Metallurgica, Vol. 18, No. 6, 1984, pp. 583-586. doi:10.1016/0036-9748(84)90345-4

[54] W. M. Garrison Jr., A. L. Wojcieszynski and L. E. Iorio, Conference Proceedings of 1997 TMS Annual Meeting, Orlando, 1997, pp. 361-372.

[55] R. O. Ritchie, W. L. Server and R. A. Wullaert, “Critical Fracture Stress and Fracture Strain Models for the Prediction of Lower and Upper Shelf Toughness in Nuclear Pressure Vessel Steels,” Metallurgical Transactions A, Vol. 10, No. 10, 1979, pp. 1557-1570. doi:10.1007/BF02812022