Accurate prediction of stress-strain behavior of metals as a function of arbitrary temperature and strain rate paths has remained a challenge. The Mechanical Threshold Stress constitutive model is one formalism that has emerged following several decades of research. Vast experience has accumulated with the application of the Mechanical Threshold Stress model over a wide variety of pure metals and alloys. Out of this has arisen common trends across metal systems. The magnitude of activation energies presents one example of this, where these variables consistently increase in magnitude as the obstacle to dislocation motion transitions from short range to long range. Trends in strain hardening are also observed. In Face-Centered Cubic metals the magnitude of strain hardening scales with the stacking fault energy; trends in Body-Centered Cubic metals are less clear. Model parameters derived for over twenty metals and alloys are tabulated. Common trends should guide future application of the MTS model and further model development.
KeywordsMechanical Threshold StressConstitutive BehaviorDeformation KineticsStrain HardeningInternal State Variable
Follansbee, P.S. and Kocks, U.F. (1988) A Constitutive Description of the Deformation of Copper Based on the Use of the Mechanical Threshold Stress as an Internal State Variable. Acta Metallurgica, 36, 81-93. https://doi.org/10.1016/0001-6160(88)90030-2
Follansbee, P.S. (2022) Fundamentals of Strength: Principles, Experiment, and Applications of an Internal State Variable Constitutive Formulation. 2nd Edition, Springer Nature, Cham. https://doi.org/10.1007/978-3-031-04556-1
Follansbee, P.S. (2014) On the Definition of State Variables for an Internal State Variable Constitutive Model Describing Metal Deformation. Materials Sciences and Applications, 5. https://doi.org/10.4236/msa.2014.58062
Kocks, U.F., Argon, A.S. and Ashby, M.F. (1975) Thermodynamics and Kinetics of Slip. In: Chalmers, B., Christian, J.W. and Massalski, T.B., Eds., Progress in Materials Science, Pergamon Press, Oxford, 4.
Follansbee, P.S. (2022) MTS Model Application to Materials Not Starting in the Annealed Condition. Materials, 15, Article 7874. https://doi.org/10.3390/ma15227874
Kocks, U.F. (1976) Laws for Work Hardening and Low Temperature Creep. Journal of Engineering Materials Technology, 98, 76-85. https://doi.org/10.1115/1.3443340
Follansbee, P.S., Huang, J.C. and Gray, G.T. (1990) Low-Temperature and High-Strain-Rate Deformation of Nickel and Nickel-Carbon Alloys and Analysis of the Constitutive Behavior According to an Internal State Variable Model. Acta Metallurgica et Materialia, 38, 1241-1254. https://doi.org/10.1016/0956-7151(90)90195-M
Follansbee, P.S. (2010) Analysis of Deformation Kinetics in Seven Body-Centered Cubic Pure Metals Using a Two-Obstacle Model. Metallurgical and Materials Transactions A, 41, 3080-3090. https://doi.org/10.1007/s11661-010-0372-6
Briggs, T.L. and Campbell, J.D. (1972) The Effect of Strain Rate and Temperature on the Yield and Flow of Polycrystalline Niobium and Molybdenum. Acta Metallurgica, 20, 711-724. https://doi.org/10.1016/0001-6160(72)90100-9
Risebrough, N.R. (1965) The Deformation Characteristic of Zinc and Cadmium. Ph.D. Thesis, University of British Columbia, Vancouver.
Liu, H., Huang, C.X., Wu, S.D. and Zhang, Z.F. (2008) Tensile Deformation and Fracture Behaviors of High Purity Polycrystalline Zinc. Materials Science and Engineering: A, 490, 117-125. https://doi.org/10.1016/j.msea.2008.01.004
Dalla Torre, F.H., Pereloma, E.V. and Davies, C.H.J. (2006) Strain Hardening Behaviour and Deformation Kinetics of Cu Deformed by Equal Channel Angular Extrusion from 1 to 16 Passes. Acta Materialia, 54, 1135-1146. https://doi.org/10.1016/j.actamat.2005.10.041
Mannan, S.L. and Rodriguez, P. (1975) Grain Size Dependence of the Deformation Behaviour of Cadmium. Acta Metallurgica, 23, 221-228. https://doi.org/10.1016/0001-6160(75)90187-X
Follansbee, P.S. (2022) Application of the Mechanical Threshold Stress Model to Large Strain Processing. Materials Sciences and Applications, 13, 300-316. https://doi.org/10.4236/msa.2022.135016
Carreker Jr., R.P. (1957) Tensile Deformation in Silver as a Function of Temperature, Strain Rate, and Grain Size. JOM, 9, 112-115. https://doi.org/10.1007/BF03398466
Malone, S. (2019) Deformation Twinning in Silver. Senior Capstone Project, Saint Vincent College, Latrobe, Pennsylvania, USA.
Gray III, G.T., Chen, S.R. and Vecchio, K.S. (1999) Influence of Grain Size on the Constitutive Response and Substructure Evolution of Monel 400. Metallurgical and Materials Transactions A, 30, 1235-1247. https://doi.org/10.1007/s11661-999-0273-8
Paton, N.E., Williams, J.C. and Rauscher, G.P. (1973) The Deformation of α-Phase Titanium. In: Jaffee, R.I. and Burte, H.M., Eds., Titanium Science and Technology, Plenum Press, New York, 1049-1069.
Chen, S.R. and Gray, G.T. (1997) Influence of Twinning on the Constitutive Response of Zr: Experiments and Modeling. 5th International Conference on Mechanical and Physical Behaviour of Materials under Dynamic Loading, Toledo, 22-26 September 1997, C3:741-C3:746. https://doi.org/10.1051/jp4:19973126
Suzuki, H., Hashizume, S., Yabuki, Y., Ichihara, Y., Nakajima, S. and Kenimochi, K. (1968) Studies on the Flow Stress of Metals and Alloys. Report of the Institute of Industrial Science, the University of Tokyo.
Takuda, H., Morishita, T., Kinoshita, T. and Shirakawa, N. (2005) Modelling of Formula for Flow Stress of a Magnesium Alloy AZ31 Sheet at Elevated Temperatures. Journal of Materials Processing Technology, 164-165, 1258-1262.
Yin, C.A., Doner, M. and Conrad, H. (1983) Deformation Kinetics of Commercial Ti-50A (0.5 at. Pct Oeq) at Low Temperatures (T https://doi.org/10.1007/BF02668896
Doner, M. and Conrad, H. (1973) Deformation Mechanisms in Commercial Ti (0.5 at. Pct Oineq) at Intermediate and High Temperatures (0.3-0.6 tinm). Metallurgical Transactions, 4, 2809-2817. https://doi.org/10.1007/BF02644581
Follansbee, P.S. and Gray, G.T. (1989) An Analysis of the Low Temperature and Low and High Strain Rate Deformation of Ti-6Al-4V. Metallurgical Transactions A, 20, 863-874. https://doi.org/10.1007/BF02651653
Conway, J.B., Stentz, R.H. and Berling, J.T. (1974) Fatigue, Tensile and Relaxation Behavior of Stainless Steels. Report Commissioned by the U.S. Atomic Energy Commission Division of Reactor Research and Development. https://doi.org/10.2172/4239699
Follansbee, P.S. (2012) An Internal State Variable Constitutive Model for Deformation of Austenitic Stainless Steel. Journal of Engineering Materials and Technology, 134, 041007. https://doi.org/10.1115/1.4006822
Nojima, T. (1988) Constitutive Equation of Some Kinds of Steels at High Rates of Strain. In: Chiem, C.Y., Kunze, H.D. and Meyer, L.W., Eds., Impact Loading and Dynamic Behaviour of Materials, DGM Informationsgesellschaft Verlag, Bremen, 357-364.
Nemat-Nasser, S. and Guo, W.G. (2000) High Strain-Rate Response of Commercially Pure Vanadium. Mechanics of Materials, 32, 243-260. https://doi.org/10.1016/S0167-6636(99)00056-3
Hoge, K.G, and Mukherjee, A.K. (1977) The Temperature Dependence and Strain Rate Dependence of the Flow Stress of Tantalum. Journal of Materials Science, 12, 1666-1672. https://doi.org/10.1007/BF00542818
Groger, R. and Vitek, V. (2008) Multiscale Modeling of Plastic Deformation of Molybdenum and Tungsten. III. Effects of Temperature and Plastic Strain Rat. Acta Materialia, 56, 5426-5439. https://doi.org/10.1016/j.actamat.2008.07.027
Brunner, D. and Glebovsky, V. (2000) Analysis of Flow-Stress Measurements of High-Purity Tungsten Single Crystals. Materials Letters, 44, 144-152. https://doi.org/10.1016/S0167-577X(00)00017-3
Gray, G.T. and Chen, S.R. (2007) MST-8 Constitutive Properties & Constitutive Modeling. Los Alamos National Laboratory, LA-CP-07-1590.
Norstrom, L.A. (1977) The Influence of Nitrogen and Grain Size on Yield Strength in Type AISI 316L Austenitic Stainless Steel. Metal Science, 11, 208-212. https://doi.org/10.1179/msc.1977.11.6.208
Fleischer, R.L. (1963) Substitutional Solution Hardening. Durcissement de solution par substitution Verfestigung in substitutionsmischkkistallen. Acta Metallurgica, 11, 203-209. https://doi.org/10.1016/0001-6160(63)90213-X
Follansbee, P.S. (2015) Structure Evolution in Austenitic Stainless Steels—A State Variable Model Assessment. Materials Sciences and Applications, 6, 457-463. https://doi.org/10.4236/msa.2015.66049
Follansbee, P.S. (2016) Analysis of Deformation in Inconel 718 When the Stress Anomaly and Dynamic Strain Aging Coexist. Metallurgical and Materials Transactions A, 47, 4455-4466. https://doi.org/10.1007/s11661-016-3609-1
Nalawade, S.A., Sundararaman, M., Kishore, R. and Shah, J.G. (2008) The Influence of Aging on the Serrated Yielding Phenomena in a Nickel-Base Superalloy. Scripta Materialia, 59, 991-994. https://doi.org/10.1016/j.scriptamat.2008.07.004
Kocks, U.F. and Mecking, H. (2003) Physics and Phenomenology of Strain Hardening: The FCC Case. Progress in Materials Science, 48, 171-273. https://doi.org/10.1016/S0079-6425(02)00003-8
Lee, J.Y., Yang, M.K., Song, L., Vitos, L. and Kwon, S.K. (2017) The Behaviour of Stacking Fault Energy upon Interstitial Alloying. Scientific Reports, 7, Article No. 11074. http://www.nature.com/scientificreports https://doi.org/10.1038/s41598-017-11328-4
Gallagher, P.C.J. (1970) The Influence of Alloying, Temperature, and Related Effects on the Stacking Fault Energy. Metallurgical Transactions, 1, 2429-2461. https://doi.org/10.1007/BF03038370
Chen, S.R. and Gray, G.T. (1996) Constitutive Behavior of Tantalum and Tantalum-Tungsten Alloys. Metallurgical and Materials Transactions A, 27, 2994-3006. https://doi.org/10.1007/BF02663849
Johnson, G.R. and Cook, W.H. (1983) A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates, and High Temperatures. Proceedings 7th International Symposium on Ballistics, The Hague, 19-21 April 1983, 541-547.
Zerilli, F.J. and Armstrong, R.W. (1987) Dislocation-Mechanics-Based Constitutive Relations for Material Dynamics Calculations. Journal of Applied Physics, 61, 1816-1825. https://doi.org/10.1063/1.338024
Voce, E. (1948) The Relationship between Stress and Strain for Homogeneous Deformation. Journal of the Institute of Metals, 74, 537-562.
Estrin, Y. (1996) Dislocation-Density-Related Constitutive Modeling. In: Krausz, A.S. and Krausz, K., Eds., Unified Constitutive Laws for Plastic Deformation, Academic Press Inc., San Diego, 69-106. https://doi.org/10.1016/B978-012425970-6/50003-5
Follansbee, P.S. (2020) Further Analysis of Dynamic Strain Aging in Context of an Internal State Variable Constitutive Formalism. Metallurgical and Materials Transactions A, 51, 1275-1285. https://doi.org/10.1007/s11661-019-05591-5