Finite Element Modeling of Stress Strain Curve and Micro Stress and Micro Strain Distributions of Titanium Alloys— A Review

Affiliation(s)

Department Mechanical Engineering, National Institute of Technology, Warangal, India.

Department Mechanical Engineering, National Institute of Technology, Warangal, India.

ABSTRACT

Most of the alloys like titanium, steel, brass, copper, etc., are used in engineering applications like automobile, aero- space, marine etc., consist of two or more phases. If a material consists of two or more phases or components it is very difficult to predict the properties like mechanical and other properties based on simple laws such as rule of mixtures. Titanium alloys are capable of producing different microstructures when it subjected to heat treatments, so much of money and time are squandering to study the effect of microstructure on mechanical properties of titanium alloys. This squandering can be reduced with the help of modeling and optimization techniques. There are many modeling tech- niques like Finite element method, Mat lab, Mathematical modeling etc. are available. But Finite element method is widely used for prediction because of capable of producing distributions of stresses and strains at any different loads. From the literature it is observed that there is a good agreement between the calculated and measured stress strain curves. This review paper describes the effect of volume fraction and grain size of alpha phase on the stress strain curve of the titanium alloys. It also can predict the effect of strength ratio on stress strain curve by using FEM. This informa- tion will be of great use in designing and selecting the titanium alloys for various engineering applications.

Most of the alloys like titanium, steel, brass, copper, etc., are used in engineering applications like automobile, aero- space, marine etc., consist of two or more phases. If a material consists of two or more phases or components it is very difficult to predict the properties like mechanical and other properties based on simple laws such as rule of mixtures. Titanium alloys are capable of producing different microstructures when it subjected to heat treatments, so much of money and time are squandering to study the effect of microstructure on mechanical properties of titanium alloys. This squandering can be reduced with the help of modeling and optimization techniques. There are many modeling tech- niques like Finite element method, Mat lab, Mathematical modeling etc. are available. But Finite element method is widely used for prediction because of capable of producing distributions of stresses and strains at any different loads. From the literature it is observed that there is a good agreement between the calculated and measured stress strain curves. This review paper describes the effect of volume fraction and grain size of alpha phase on the stress strain curve of the titanium alloys. It also can predict the effect of strength ratio on stress strain curve by using FEM. This informa- tion will be of great use in designing and selecting the titanium alloys for various engineering applications.

Cite this paper

G. Srinivasu and N. Raja, "Finite Element Modeling of Stress Strain Curve and Micro Stress and Micro Strain Distributions of Titanium Alloys— A Review,"*Journal of Minerals and Materials Characterization and Engineering*, Vol. 11 No. 10, 2012, pp. 953-960. doi: 10.4236/jmmce.2012.1110094.

G. Srinivasu and N. Raja, "Finite Element Modeling of Stress Strain Curve and Micro Stress and Micro Strain Distributions of Titanium Alloys— A Review,"

References

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[2] L. Li, J. K. Sun and X. J. Meng, “The Appliance of Tita- nium Alloy and Its Development,” Titanium Industry. Progres, Vol. 21, No. 19, 2004, pp. 58-65

[3] R. R. Boyer, “An Overview on the Use of Titanium in the Aerospace Industry,” Materials science & Engineering A, Vol. 213, No. 1-2, 2004, pp. 103-114. doi:10.1016/0921-5093(96)10233-1

[4] H. J. Rack and J. I. Qazi, “Titanium Alloys for Biomedi- cal Applications,” Materials Science and Engineering C, Vol. 26, No. 8, 2006, pp. 1269-1277. doi:10.1016/j.msec.2005.08.032

[5] P. J. Bahia, D. Eylon, R. R. Boyer and D. A. Koss, “Beta Titanium Alloys and Their Role in the Titanium Industry —Keynote Lecture, Beta Titalzium Alloys,” Warrendale, PA, TMS, 1993, pp. 3-14.

[6] E. W. Collings, “The Physical Metalh￠rgy of Titalzium Alloys,” ASM, Materials Park, Ohio, 1984, pp. 2-4.

[7] T. W. Duerig and J. C. Williams, R. R. Boyer and H. W. Rosenberg, “Overview: Microstructure and Properties of Beta Titanium Alloys, Beta Titanium Alloys,” Warren- dale, PA, TMS, 1984, pp. 19-67.

[8] D. Eylon, S. Fujishiro, P. J. Postans and F. H. Froes, “High Temperature Alloys—A Review,” Journal of Met- als, Vol. 36, No. 4, 1984, pp. 55-62.

[9] S. Sastry, T. C. Peng, P. J. Mechter and J. E. O’Neal, “The Effect of Microstructure on the Mechanical Properties of Two-Phase Titanium Alloys,” Journal of Metals, Vol. 36, No. 5, 1984, pp. 21-28.

[10] V. J. Colagelo and F. A. Heizer, “Analysis of Metallurgi- cal Failures”, John Wiley, New York, 1987.

[11] J. D. Emburg and F. Zok, “Micromechanisms of Frac- ture”, Proceedings of the 26th Conference of Metallur- gists, Winnipeg, 23-26 August 1987, p. 198

[12] J. Sieniawski, “Scientific Papers of the Rzeszow Univer- sity of Technology—Mechanics No. 10,” Rzeszow Uni- versity, Rzeszow, 1985.

[13] L. J. Hunter, M. Strangwood and P. Bowen, “Effect of Microstructure on the Fracture Behaviour of the α & β Titanium Alloy Ti-4Al-4Mo-2Sn-0.5Si wt.% (IMI 550) ,” In: P. A. Blenkinsop, W. J. Evans and H. M. Flower, Eds., Titanium’95, Science and Technology, The Institute of Materials, London, 1996, p. 925.

[14] J. Sieniawski, “The Effect of Phase Composition on the Fracture Toughness (KIc) of Structural Titanium Alloys,” Proceedings of the ISUMEL-2 Second International Sym- posium of Ukrainian Mechanical Engineers in Lviv, 2-6 September 1995, p. 101.

[15] G. Lutjering, “Influence of Processing on Microstructure and Mechanical Properties of α & β Titanium Alloys,” Materials Science and Engineering: A, Vol. 243, No. 1-2, 1998, pp. 32-45. doi:10.1016/S0921-5093(97)00778-8

[16] W. J. Evans, “Optimising Mechanical Properties in α & β Titanium Alloys,” Materials Science and Engineering: A, Vol. 243, No. 1-2, 1998 pp. 89-96. doi:10.1016/S0921-5093(97)00784-3

[17] C. Cauer and G. Luetjering, “Thermo-Mechanical Proc- essing of High Strength β-Titanium Alloys and Effect on Microstructure and Properties,” Journal of Materials Processing Technology, Vol. 117, No. 3, 2001, pp. 311- 317. doi:10.1016/S0924-0136(01)00788-9

[18] R. Filip, K. Kubiak, W. Ziaja and J. Sieniawski, “The Effect of Microstructure on the Mechanical Properties of Two-Phase Titanium Alloys,” Journal of Materials Proc- essing Technology, Vol. 133, No. 1-2, 2003, pp. 84-89. doi:10.1016/S0924-0136(02)00248-0

[19] S. Malinov, W. Sha and Z. Guo, “Application of Artificial Neural Network for Prediction of Time-Temperature —Transformation Diagrams in Titanium Alloys,” Materials Science and Engineering: A, Vol. 283, No. 1-2, 2000, pp. 1-10. doi:10.1016/S0921-5093(00)00746-2

[20] Y. C. Zhu, W. D. Zeng, Y. Sun, F. Feng and Y. G. Zhou, “Artificial Neural Network Approach to Predict the Flow Stress in the Isothermal Compression of As-Cast TC21 Titanium Alloy”, Computational Materials Science, Vol. 50, No. 5,2011, pp. 1785-1790. doi:10.1016/j.commatsci.2011.01.015

[21] S. Malinov and W. Sha, “Application of Artificial Neural Networks for Modelling Correlations in Titanium Al- loys,” Materials Science and Engineering: A, Vol. 365, No. 1-2, 2004, pp. 202-211. doi:10.1016/j.msea.2003.09.029

[22] J. McBride, S. Malinov and W. Shaa, “Modelling Tensile Properties of Gamma-Based Titanium Aluminides Using Artificial Neural Network,” Materials Science and Engi- neering: A, Vol. 384, No. 1-2, 2004, pp. 129-137. doi:10.1016/j.msea.2004.05.072

[23] S. Malinov, W. Sha and J. J. McKeown, “Modelling the Correlation between Processing Parameters and Proper- ties in Titanium Alloys Using Artificial Neural Network,” Computational Materials Science, Vol. 21, No. 3, 2001, pp. 375-394. doi:10.1016/S0927-0256(01)00160-4

[24] M. Kato, T. Fujii and S. Onaka, “Effects of Shape and Volume Fraction of Second Phase on Stress States in Two-Phase Materials,” Materials Science and Engineer- ing: A, Vol. 285, No. 1-2, 2000, pp. 144-150. doi:10.1016/S0921-5093(00)00639-0

[25] J. R. C. Guimaraes and D. L. Valeriano Alves, “On the Analysis of Stress-Strain Curves by Means of Empirical Equations,” Scripta Metallurgica, Vol. 9, No. 11, 1975, pp. 1147-1148. doi:10.1016/0036-9748(75)90395-6

[26] P. R. Rios, J. R. C. Guimares and K. K. Chawla, “Model- ling the Stress-Strain Curves of Dual Phase Steels,” Scripta Metallurgica, Vol. 15, No. 8, 1981, pp. 899-904. doi:10.1016/0036-9748(81)90274-X

[27] B. K. Kad, M. Dao, J. Robert and Asaro, “Numerical Simulations of Stress-Strain Behavior in Two-Phase α2 + β Lamellar TiAl Alloys,” Materials Science and Engi- neering: A, Vol. 192-193, 1995, pp. 97-103. doi:10.1016/0921-5093(94)03210-6

[28] X. P. Wu, S. R. Kalidindi, C. Necker and A. A. Salem, “Prediction of Crystallographic Texture Evolution and Anisotropic Stress-Strain Curves during Large Plastic Strains in High Purity α-Titanium Using a Taylor-Type Crystal Plasticity Model,” Acta Materialia, Vol. 55, No. 2, 2007, pp. 423-432. doi:10.1016/j.actamat.2006.08.034

[29] S. Malinov, W. Sha and P. Markovsky, “Experimental Study and Computer Modelling of the β ? α + β Phase Transformation in β21s Alloy at Isothermal Conditions,” Journal of Alloys and Compounds, Vol. 348, No. 1-2, 2003, pp. 110-118. doi:10.1016/S0925-8388(02)00804-6

[30] J. D. C. Teixeira, B. Appolaire, E. Aeby-Gautier, S. Denis and L. Hericher, “Modeling of the Phase Transformations in Near-β Titanium Alloys during the Cooling after Forg- ing,” Computational Materials Science, Vol. 42, No. 2, 2008, pp. 266-280. doi:10.1016/j.commatsci.2007.07.056

[31] J. Jinoch, S. Ankem and H. Margolin, “Calculations of Stress-Strain Curve and Stress and Strain Distributions for an α-β Ti-8Mn Alloy,” Materials Science and Engi- neering, Vol. 34, No. 3, 1978, pp. 203-211. doi:10.1016/0025-5416(78)90052-6

[32] S. Neti, M. N. Vijayshankar and S. Ankem, “Finite Ele- ment Method Modeling of Deformation Behavior of Two-Phase Materials Part I: Stress-Strain Relations,” Materials Science and Engineering: A, Vol. 145, No. 1, 1991, pp. 47-54. doi:10.1016/0921-5093(91)90294-W

[33] S. Neti, M. N. Vijayshankar and S. Ankem, “Finite Ele- ment Method Modeling of Deformation Behavior of Two-Phase Materials Part II: Stress and Strain Distribu- tions,” Materials Science and Engineering: A, Vol. 145, No. 1, 1991, pp. 55-64. doi:10.1016/0921-5093(91)90295-X

[34] S. Ankem and H. Margolin, “Finite Element Method (FEM) Calculations of Stress-Strain Behavior of Al- pha-Beta Ti-Mn Alloys: Part I. Stress-Strain Relations,” Metallurgical Transactions A, Vol. 13 No. 4, 1982, pp. 595-601.

[35] N. Ramakrishnan and V. S. Arunachalam, “Finite Ele- ment Methods for Materials Modeling,” Progress in Ma- terials Science, Vol. 42, No. 1-4, 1991, pp. 253-261. doi:10.1016/S0079-6425(97)00031-5

[36] H. Shen and L. C. Brinson, “Finite Element Modeling of Porous Titanium,” International Journal of Solids and Structures, Vol. 44, No. 1, 2007, pp. 320-335. doi:10.1016/j.ijsolstr.2006.04.020

[37] L. Durand, M. Massaoudi, M. Cabie, A. Ponchet, “Me- chanical Behaviour of a Two-Phase Material from the Behaviour of Its Components: Interface Modelling by Fi- nite Element Method,” Materials and Design, Vol. 29, No. 8, 2008, pp. 1609-1615. doi:10.1016/j.matdes.2007.10.002

[38] W. Ziaja, “Finite Element Modelling of the Fracture Be- haviour of Surface Treated Ti-6Al-4V Alloy,” Computa- tional Materials Science and Surface Engineering, Vol. 1, No. 1, 2009, pp. 53-60.

[39] X. Q. Zhao, X. L. Zang, Q. F. Wang, P. Joongkeun and Q. X. Yang, “Numerical Simulation of the Stress-Strain Curve and the Stress and Strain Distributions of the Tita- nium-Duplex Alloy,” Rare Metals, Vol. 27, No. 5, 2008, pp. 463-467. doi:10.1016/S1001-0521(08)60163-1

[40] L. M. Wang, J. J. Xu, L. Yan, Z. D. Liu and G. Yang, “A FEM Study on the Mechanical Responses of Pseudoelas- tic TiNi Alloys to a Particle Normal Loads,” Wear, Vol. 260, No. 6, 2006, pp. 573-579. doi:10.1016/j.wear.2004.12.035

[41] B. Liao, C. L. Zhang, J. Wu, D. Y. Cai, C. M. Zhao, X. J. Ren and Q. X. Yang, “Numerical Simulation of the Stress-Strain Curve of Duplex Weathering Steel,” Mate- rials and Design, Vol. 29, No. 2, 2008, pp. 562-567. doi:10.1016/j.matdes.2006.12.021

[42] H.-F. Dong, J. Li, Y. Zhang, J. Park and Q.-X. Yang, “Numerical Simulation on the Microstress and Mi- crostrain of Low Si-Mn-Nb Dual-Phase Steel,” Interna- tional Journal of Minerals, Metallurgy and Materials, Vol. 17, No. 2, 2010, pp. 173-178. doi:10.1007/s12613-010-0209-8

[43] O. O. Oluwole, P. O. Atanda and B. I. Imasogie, “Finite Element Modeling of Heat Transfer in Salt Bath Fur- naces,” The Journal of Minerals and Materials Charac- terization and Engineering, Vol. 8, No. 3, 2009, pp. 229- 236.

[44] C. C. Ihueze, “The Galerki Approach for Finite Elements of Field Functions: The Case of Buckling in GRP,” The Journal of Minerals and Materials Characterization and Engineering, Vol. 9, No. 4, 2010, pp. 389-409.

[1] W. P. Li, “The Appliance of Titanium Alloy and Its De- velopment,” Light Metals, Vol. 5, No. 53, 2002, pp. 25- 32.

[2] L. Li, J. K. Sun and X. J. Meng, “The Appliance of Tita- nium Alloy and Its Development,” Titanium Industry. Progres, Vol. 21, No. 19, 2004, pp. 58-65

[3] R. R. Boyer, “An Overview on the Use of Titanium in the Aerospace Industry,” Materials science & Engineering A, Vol. 213, No. 1-2, 2004, pp. 103-114. doi:10.1016/0921-5093(96)10233-1

[4] H. J. Rack and J. I. Qazi, “Titanium Alloys for Biomedi- cal Applications,” Materials Science and Engineering C, Vol. 26, No. 8, 2006, pp. 1269-1277. doi:10.1016/j.msec.2005.08.032

[5] P. J. Bahia, D. Eylon, R. R. Boyer and D. A. Koss, “Beta Titanium Alloys and Their Role in the Titanium Industry —Keynote Lecture, Beta Titalzium Alloys,” Warrendale, PA, TMS, 1993, pp. 3-14.

[6] E. W. Collings, “The Physical Metalh￠rgy of Titalzium Alloys,” ASM, Materials Park, Ohio, 1984, pp. 2-4.

[7] T. W. Duerig and J. C. Williams, R. R. Boyer and H. W. Rosenberg, “Overview: Microstructure and Properties of Beta Titanium Alloys, Beta Titanium Alloys,” Warren- dale, PA, TMS, 1984, pp. 19-67.

[8] D. Eylon, S. Fujishiro, P. J. Postans and F. H. Froes, “High Temperature Alloys—A Review,” Journal of Met- als, Vol. 36, No. 4, 1984, pp. 55-62.

[9] S. Sastry, T. C. Peng, P. J. Mechter and J. E. O’Neal, “The Effect of Microstructure on the Mechanical Properties of Two-Phase Titanium Alloys,” Journal of Metals, Vol. 36, No. 5, 1984, pp. 21-28.

[10] V. J. Colagelo and F. A. Heizer, “Analysis of Metallurgi- cal Failures”, John Wiley, New York, 1987.

[11] J. D. Emburg and F. Zok, “Micromechanisms of Frac- ture”, Proceedings of the 26th Conference of Metallur- gists, Winnipeg, 23-26 August 1987, p. 198

[12] J. Sieniawski, “Scientific Papers of the Rzeszow Univer- sity of Technology—Mechanics No. 10,” Rzeszow Uni- versity, Rzeszow, 1985.

[13] L. J. Hunter, M. Strangwood and P. Bowen, “Effect of Microstructure on the Fracture Behaviour of the α & β Titanium Alloy Ti-4Al-4Mo-2Sn-0.5Si wt.% (IMI 550) ,” In: P. A. Blenkinsop, W. J. Evans and H. M. Flower, Eds., Titanium’95, Science and Technology, The Institute of Materials, London, 1996, p. 925.

[14] J. Sieniawski, “The Effect of Phase Composition on the Fracture Toughness (KIc) of Structural Titanium Alloys,” Proceedings of the ISUMEL-2 Second International Sym- posium of Ukrainian Mechanical Engineers in Lviv, 2-6 September 1995, p. 101.

[15] G. Lutjering, “Influence of Processing on Microstructure and Mechanical Properties of α & β Titanium Alloys,” Materials Science and Engineering: A, Vol. 243, No. 1-2, 1998, pp. 32-45. doi:10.1016/S0921-5093(97)00778-8

[16] W. J. Evans, “Optimising Mechanical Properties in α & β Titanium Alloys,” Materials Science and Engineering: A, Vol. 243, No. 1-2, 1998 pp. 89-96. doi:10.1016/S0921-5093(97)00784-3

[17] C. Cauer and G. Luetjering, “Thermo-Mechanical Proc- essing of High Strength β-Titanium Alloys and Effect on Microstructure and Properties,” Journal of Materials Processing Technology, Vol. 117, No. 3, 2001, pp. 311- 317. doi:10.1016/S0924-0136(01)00788-9

[18] R. Filip, K. Kubiak, W. Ziaja and J. Sieniawski, “The Effect of Microstructure on the Mechanical Properties of Two-Phase Titanium Alloys,” Journal of Materials Proc- essing Technology, Vol. 133, No. 1-2, 2003, pp. 84-89. doi:10.1016/S0924-0136(02)00248-0

[19] S. Malinov, W. Sha and Z. Guo, “Application of Artificial Neural Network for Prediction of Time-Temperature —Transformation Diagrams in Titanium Alloys,” Materials Science and Engineering: A, Vol. 283, No. 1-2, 2000, pp. 1-10. doi:10.1016/S0921-5093(00)00746-2

[20] Y. C. Zhu, W. D. Zeng, Y. Sun, F. Feng and Y. G. Zhou, “Artificial Neural Network Approach to Predict the Flow Stress in the Isothermal Compression of As-Cast TC21 Titanium Alloy”, Computational Materials Science, Vol. 50, No. 5,2011, pp. 1785-1790. doi:10.1016/j.commatsci.2011.01.015

[21] S. Malinov and W. Sha, “Application of Artificial Neural Networks for Modelling Correlations in Titanium Al- loys,” Materials Science and Engineering: A, Vol. 365, No. 1-2, 2004, pp. 202-211. doi:10.1016/j.msea.2003.09.029

[22] J. McBride, S. Malinov and W. Shaa, “Modelling Tensile Properties of Gamma-Based Titanium Aluminides Using Artificial Neural Network,” Materials Science and Engi- neering: A, Vol. 384, No. 1-2, 2004, pp. 129-137. doi:10.1016/j.msea.2004.05.072

[23] S. Malinov, W. Sha and J. J. McKeown, “Modelling the Correlation between Processing Parameters and Proper- ties in Titanium Alloys Using Artificial Neural Network,” Computational Materials Science, Vol. 21, No. 3, 2001, pp. 375-394. doi:10.1016/S0927-0256(01)00160-4

[24] M. Kato, T. Fujii and S. Onaka, “Effects of Shape and Volume Fraction of Second Phase on Stress States in Two-Phase Materials,” Materials Science and Engineer- ing: A, Vol. 285, No. 1-2, 2000, pp. 144-150. doi:10.1016/S0921-5093(00)00639-0

[25] J. R. C. Guimaraes and D. L. Valeriano Alves, “On the Analysis of Stress-Strain Curves by Means of Empirical Equations,” Scripta Metallurgica, Vol. 9, No. 11, 1975, pp. 1147-1148. doi:10.1016/0036-9748(75)90395-6

[26] P. R. Rios, J. R. C. Guimares and K. K. Chawla, “Model- ling the Stress-Strain Curves of Dual Phase Steels,” Scripta Metallurgica, Vol. 15, No. 8, 1981, pp. 899-904. doi:10.1016/0036-9748(81)90274-X

[27] B. K. Kad, M. Dao, J. Robert and Asaro, “Numerical Simulations of Stress-Strain Behavior in Two-Phase α2 + β Lamellar TiAl Alloys,” Materials Science and Engi- neering: A, Vol. 192-193, 1995, pp. 97-103. doi:10.1016/0921-5093(94)03210-6

[28] X. P. Wu, S. R. Kalidindi, C. Necker and A. A. Salem, “Prediction of Crystallographic Texture Evolution and Anisotropic Stress-Strain Curves during Large Plastic Strains in High Purity α-Titanium Using a Taylor-Type Crystal Plasticity Model,” Acta Materialia, Vol. 55, No. 2, 2007, pp. 423-432. doi:10.1016/j.actamat.2006.08.034

[29] S. Malinov, W. Sha and P. Markovsky, “Experimental Study and Computer Modelling of the β ? α + β Phase Transformation in β21s Alloy at Isothermal Conditions,” Journal of Alloys and Compounds, Vol. 348, No. 1-2, 2003, pp. 110-118. doi:10.1016/S0925-8388(02)00804-6

[30] J. D. C. Teixeira, B. Appolaire, E. Aeby-Gautier, S. Denis and L. Hericher, “Modeling of the Phase Transformations in Near-β Titanium Alloys during the Cooling after Forg- ing,” Computational Materials Science, Vol. 42, No. 2, 2008, pp. 266-280. doi:10.1016/j.commatsci.2007.07.056

[31] J. Jinoch, S. Ankem and H. Margolin, “Calculations of Stress-Strain Curve and Stress and Strain Distributions for an α-β Ti-8Mn Alloy,” Materials Science and Engi- neering, Vol. 34, No. 3, 1978, pp. 203-211. doi:10.1016/0025-5416(78)90052-6

[32] S. Neti, M. N. Vijayshankar and S. Ankem, “Finite Ele- ment Method Modeling of Deformation Behavior of Two-Phase Materials Part I: Stress-Strain Relations,” Materials Science and Engineering: A, Vol. 145, No. 1, 1991, pp. 47-54. doi:10.1016/0921-5093(91)90294-W

[33] S. Neti, M. N. Vijayshankar and S. Ankem, “Finite Ele- ment Method Modeling of Deformation Behavior of Two-Phase Materials Part II: Stress and Strain Distribu- tions,” Materials Science and Engineering: A, Vol. 145, No. 1, 1991, pp. 55-64. doi:10.1016/0921-5093(91)90295-X

[34] S. Ankem and H. Margolin, “Finite Element Method (FEM) Calculations of Stress-Strain Behavior of Al- pha-Beta Ti-Mn Alloys: Part I. Stress-Strain Relations,” Metallurgical Transactions A, Vol. 13 No. 4, 1982, pp. 595-601.

[35] N. Ramakrishnan and V. S. Arunachalam, “Finite Ele- ment Methods for Materials Modeling,” Progress in Ma- terials Science, Vol. 42, No. 1-4, 1991, pp. 253-261. doi:10.1016/S0079-6425(97)00031-5

[36] H. Shen and L. C. Brinson, “Finite Element Modeling of Porous Titanium,” International Journal of Solids and Structures, Vol. 44, No. 1, 2007, pp. 320-335. doi:10.1016/j.ijsolstr.2006.04.020

[37] L. Durand, M. Massaoudi, M. Cabie, A. Ponchet, “Me- chanical Behaviour of a Two-Phase Material from the Behaviour of Its Components: Interface Modelling by Fi- nite Element Method,” Materials and Design, Vol. 29, No. 8, 2008, pp. 1609-1615. doi:10.1016/j.matdes.2007.10.002

[38] W. Ziaja, “Finite Element Modelling of the Fracture Be- haviour of Surface Treated Ti-6Al-4V Alloy,” Computa- tional Materials Science and Surface Engineering, Vol. 1, No. 1, 2009, pp. 53-60.

[39] X. Q. Zhao, X. L. Zang, Q. F. Wang, P. Joongkeun and Q. X. Yang, “Numerical Simulation of the Stress-Strain Curve and the Stress and Strain Distributions of the Tita- nium-Duplex Alloy,” Rare Metals, Vol. 27, No. 5, 2008, pp. 463-467. doi:10.1016/S1001-0521(08)60163-1

[40] L. M. Wang, J. J. Xu, L. Yan, Z. D. Liu and G. Yang, “A FEM Study on the Mechanical Responses of Pseudoelas- tic TiNi Alloys to a Particle Normal Loads,” Wear, Vol. 260, No. 6, 2006, pp. 573-579. doi:10.1016/j.wear.2004.12.035

[41] B. Liao, C. L. Zhang, J. Wu, D. Y. Cai, C. M. Zhao, X. J. Ren and Q. X. Yang, “Numerical Simulation of the Stress-Strain Curve of Duplex Weathering Steel,” Mate- rials and Design, Vol. 29, No. 2, 2008, pp. 562-567. doi:10.1016/j.matdes.2006.12.021

[42] H.-F. Dong, J. Li, Y. Zhang, J. Park and Q.-X. Yang, “Numerical Simulation on the Microstress and Mi- crostrain of Low Si-Mn-Nb Dual-Phase Steel,” Interna- tional Journal of Minerals, Metallurgy and Materials, Vol. 17, No. 2, 2010, pp. 173-178. doi:10.1007/s12613-010-0209-8

[43] O. O. Oluwole, P. O. Atanda and B. I. Imasogie, “Finite Element Modeling of Heat Transfer in Salt Bath Fur- naces,” The Journal of Minerals and Materials Charac- terization and Engineering, Vol. 8, No. 3, 2009, pp. 229- 236.

[44] C. C. Ihueze, “The Galerki Approach for Finite Elements of Field Functions: The Case of Buckling in GRP,” The Journal of Minerals and Materials Characterization and Engineering, Vol. 9, No. 4, 2010, pp. 389-409.