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Research Papers

The Concept of Physical Metric in the Thermomechanical Modeling of Phase Transformations With Emphasis on Shape Memory Alloy Materials

[+] Author and Article Information
Vassilis P. Panoskaltsis, Dimitris Soldatos

Department of Civil Engineering,
Demokritos University of Thrace,
12 Vassilissis Sofias Street,
Xanthi 67100, Greece

Lazaros C. Polymenakos

Autonomic & Grid Computing,
Athens Information Technology,
Peania 19002, Greece

Contributed by the Materials Division of ASME for publication in the Journal of Engineering Materials and Technology. Manuscript received July 14, 2012; final manuscript received February 1, 2013; published online April 2, 2013. Assoc. Editor: Xi Chen.

J. Eng. Mater. Technol 135(2), 021016 (Apr 02, 2013) (17 pages) Paper No: MATS-12-1167; doi: 10.1115/1.4023780 History: Received July 14, 2012; Revised February 11, 2013

In this work we derive a new version of generalized plasticity, suitable to describe phase transformations. In particular, we present a general multi surface formulation of the theory which is capable of describing the multiple and interacting loading mechanisms, which occur during phase transformations. The formulation relies crucially on the consideration of the intrinsic material (“physical”) metric as a primary internal variable and does not invoke any decomposition of the kinematical quantities into elastic and inelastic (transformation induced) parts. The new theory, besides its theoretical interest, is also important for application purposes such as the description and the prediction of the response of shape memory alloy materials. This is shown in the simplest possible setting by the introduction of a material model. The ability of the model in simulating several patterns of the experimentally observed behavior of these materials such as the pseudoelastic phenomenon and the shape memory effect is assessed by representative numerical examples.

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References

Wayman, C. M., 1964, Introduction to the Crystallography of Martensitic Transformation, Macmillan, New York.
Smallman, R. E., and Bishop, R. J., 2000, Modern Physical Metallurgy and Materials Engineering, 6th ed., Butterworth-Heinemann, Stoneham, MA.
Bhattacharya, K., 2003, Microstructure of Martensite. Why It Forms and How It Gives Rise to the Shape—Memory Effect?, Oxford University Press, Oxford.
Raniecki, B., Lexcellent, C., and Tanaka, K., 1992, “Thermodynamic Models of Pseudoelastic Behaviour of Shape Memory Alloys,” Arch. Mech., 44, pp. 261–284.
Abeyaratne, R., and Knowles, J. K., 1993, “A Continuum Model for a Thermoelastic Solid Capable of Undergoing Phase Transitions,” J. Mech. Phys. Solids, 41, pp. 451–571. [CrossRef]
Ivshin, Y., and Pence, T., 1994, “A Thermodynamical Model for a One Variant Shape Memory Material,” J. Intell. Mater. Syst. Struct., 5, pp. 455–473. [CrossRef]
Boyd, J. G., and Lagoudas, D. C., 1994, “Thermomechanical Response of Shape Memory Alloy Composites,” J. Intell. Mater. Syst. Struct., 5, pp. 333–346. [CrossRef]
Leclercq, S., and Lexcellent, C., 1996, “A General Macroscopic Description of the Thermomechanical Behavior of Shape Memory Alloys,” J. Mech. Phys. Solids, 44, pp. 953–980. [CrossRef]
Lubliner, J., and Auricchio, F., 1996, “Generalized Plasticity and Shape Memory Alloys,” Int. J. Solids Struct., 33, pp. 991–1004. [CrossRef]
Panoskaltsis, V. P., Bahuguna, S., and Soldatos, D., 2004, “On the Thermomechanical Modeling of Shape Memory Alloys,” Int. J. Non-Linear Mech., 39, pp. 709–722. [CrossRef]
Saint-Sulpice, L., Arbab Chirani, C., and Calloch, S., 2009, “A 3D Super-Elastic Model for Shape Memory Alloys Taking Into Account Progressive Strain Under Cyclic Loadings,” Mech. Mater., 41, pp. 12–26. [CrossRef]
Malukhin, K., and Ehmann, K., 2012, “A Model of the Kinetics of the Temperature-Induced Phase Induced Phase Transformation in NiTi Alloys and Its Experimental Verification,” J. Intell. Mater. Syst. Struct., 23, pp. 35–44. [CrossRef]
Chan, C. W., Chan, S. H. J., and Man, H. C., 2012, “1-D Constitutive Model for Evolution of Stress Induced R-Phase and Localized Lüders-Like Stress-Induced Martensitic Transformation of Super-Elastic NiTi Wires,” Int. J. Plast., 32–33, pp. 85–105. [CrossRef]
Ball, J. M., and James, R. D., 1987, “Fine Phase Mixtures and Minimizers of Energy,” Arch. Rat. Mech. Anal., 100, pp. 13–52. [CrossRef]
Silhavy, M., 1999, “On the Compatibility of Wells,” J. Elast., 55, pp. 11–17. [CrossRef]
James, R. D., and Hane, K. F., 2000, “Martensitic Transformations and Shape-Memory Materials,” Acta Mater., 48, pp. 197–222. [CrossRef]
Levitas, V. I., and Preston, D. L., 2005, “Thermomechanical Lattice Instability and Phase Field Theory of Martensitic Phase Transformations, Twinning and Dislocations at Large Strains,” Phys. Lett. A, 343, pp. 32–39. [CrossRef]
Arndt, M., Griebel, M., Novác, V., Rubíček, T., and Šittner, P., 2006, “Martensitic Transformation in NiMnGa Single Crystals: Numerical Simulation and Experiments,” Int. J. Plast., 22, pp. 1943–1961. [CrossRef]
Müller, Ch., and Bruhns, O. T., 2006, “A Thermodynamic Finite-Strain Model for Pseudoelastic Shape Memory Alloys,” Int. J. Plast., 22, pp. 1658–1682. [CrossRef]
Christ, D., and Reese, S., 2009, “A Finite Element Model for Shape-Memory Alloys Considering Thermomechanical Couplings at Large Strains,” Int. J. Solids Struct., 46, pp. 3694–3709. [CrossRef]
Levitas, V. I., and Ozsoy, I. B., 2009, “Micromechanical Modeling of Stress-Induced Phase Transformations—Part 1: Thermodynamics and Kinetics of Coupled Interface Propagation and Reorientation,” Int. J. Plast., 25, pp. 239–280. [CrossRef]
Thamburaja, P., and Anand, L., 2000, “Polycrystalline Shape-Memory Materials: Effect of Crystallographic Texture,” J. Mech. Phys. Solids, 49, pp. 709–737. [CrossRef]
Anand, L., and Gurtin, M. E., 2003, “Thermal Effects in the Superelasticity of Crystalline Shape-Memory Materials,” J. Mech. Phys. Solids, 51, pp. 1015–1058. [CrossRef]
Pan, H., Thamburaja, P., and Chau, F. S., 2007, “Multi-Axial Behavior of Shape Memory Alloys Undergoing Martensite Reorientation and Detwinning,” Int. J. Plast., 23, pp. 711–732. [CrossRef]
Manchiraju, S., and Anderson, P. M., 2010, “Coupling Between Phase Transformations and Plasticity: A Microstructure-Based Finite Element Model,” Int. J. Plast., 26, pp. 1508–1526. [CrossRef]
Thamburaja, P., 2010, “A Finite-Deformation-Based Theory for Shape-Memory Alloys,” Int. J. Plast., 26, pp. 1195–1219. [CrossRef]
Panoskaltsis, V. P., Soldatos, D., and Triantafyllou, S. P., 2011, “Generalized Plasticity Theory for Phase Transformations,” Proc. Eng., 10, pp. 3104–3108. [CrossRef]
Panoskaltsis, V. P., Soldatos, D., and Triantafyllou, S. P., 2011, “A New Model for Shape Memory Alloy Materials Under General States of Deformation and Temperature Conditions,” Proceedings of the 7th GRACM International Congress on Computational Mechanics, A. G.Boudouvis and G. E.Stavroulakis, eds., Athens, June 30–July 2.
Lubliner, J., 1974, “A Simple Theory of Plasticity,” Int. J. Solids Struct., 10, pp. 313–319. [CrossRef]
Lubliner, J., 1980, “An Axiomatic Model of Rate-Independent Plasticity,” Int. J. Solids Struct., 16, pp. 709–713. [CrossRef]
Lubliner, J., 1984, “A Maximum-Dissipation Principle in Generalized Plasticity,” Acta Mech., 52, pp. 225–237. [CrossRef]
Lubliner, J., 1987, “Non-Isothermal Generalized Plasticity,” Thermomechanical Couplings in Solids, H. D.Bui and Q. S.Nyugen, eds., IUTAM, North-Holland, Amsterdam, pp. 121–133.
Ramanathan, G., Panoskaltsis, V. P., Mullen, R., and Welsch, G., 2002, “Experimental and Computational Methods for Shape Memory Alloys,” Proceedings of the 15th ASCE Engineering Mechanics Conference, A.Smyth, ed., Columbia University, New York, June 2–5.
Valanis, K. C., 1995, “The Concept of Physical Metric in Thermodynamics,” Acta Mech., 113, pp. 169–184. [CrossRef]
Valanis, K. C., and Panoskaltsis, V. P., 2005, “Material Metric, Connectivity and Dislocations in Continua,” Acta Mech., 175, pp. 77–103. [CrossRef]
Panoskaltsis, V. P., Soldatos, D., and Triantafyllou, S., P., 2011, “The Concept of Physical Metric in Rate-Independent Generalized Plasticity,” Acta Mech., 221, pp. 49–64. [CrossRef]
Marsden, J. E., and Hughes, T. J. R., 1994, Mathematical Foundations of Elasticity, Dover, New York.
Stumpf, H., and Hoppe, U., 1997, “The Application of Tensor Analysis on Manifolds to Nonlinear Continuum Mechanics—Invited Survey Article,” Z. Angew Math. Mech., 77, pp. 327–339. [CrossRef]
Panoskaltsis, V. P., Polymenakos, L. C., and Soldatos, D., 2008, “On Large Deformation Generalized Plasticity,” J. Mech. Mater. Struct., 3, pp. 441–457. [CrossRef]
Panoskaltsis, V. P., Polymenakos, L. C., and Soldatos, D., 2008, “Eulerian Structure of Generalized Plasticity: Theoretical and Computational Aspects,” J. Eng. Mech., 134(5), pp. 354–361. [CrossRef]
Eisenberg, M. A., and Phillips, A., 1971, “A Theory of Plasticity With Non-Coincident Yield and Loading Surfaces,” Acta Mech., 11, pp. 247–260. [CrossRef]
Brezis, H., 1970, “On a Characterization of Flow-Invariant Sets,” Commun. Pure Appl. Math., XXIII, pp. 261–263. [CrossRef]
Simo, J. C., and Hughes, T. J. R., 1998, Computational Inelasticity, Springer, New York.
Simo, J. C., Marsden, J. E., and Krishnaprasad, P. S., 1988, “The Hamiltonian Structure of Elasticity: The Convected Representation of Solids, Rods and Plates,” Arch. Rat. Mech. Anal., 104, pp. 125–183. [CrossRef]
Simo, J. C., and Ortiz, M., 1985, “A Unified Approach to Finite Deformation Plasticity Based on the Use of Hyperelastic Constitutive Equations,” Comput. Meth. Appl. Mech. Eng., 49, pp. 221–245. [CrossRef]
Schutz, B., 1999, Geometrical Methods of Mathematical Physics, Cambridge University Press, Cambridge, UK.
Coleman, B. D., and Gurtin, M., 1967, “Thermodynamics With Internal State Variables” J. Chem. Phys., 47, pp. 597–613. [CrossRef]
Valanis, K. C., 1972, Irreversible Thermodynamics of Continuous Media, Internal Variable Theory, (CISM Courses and Lectures No. 77), International Centre for Mechanical Sciences, Springer, Wien, Germany.
Lexcellent, C., and Laydi, R. M., 2011, “About the Choice of a Plastic-Like Model for Shape Memory Alloys,” Vietnam J. Mech., 33(4), pp. 283–291.
Likhachev, A. A., and Koval, Y. N., 1992, “On the Differential Equation Describing the Hysterisis Behavior of Shape Memory Alloys” Scr. Metal. Mater., 27, pp. 223–227. [CrossRef]
Voyiadjis, G. Z., and Dorgan, R. J., 2007, “Framework Using Functional Forms of Hardening Internal State Variables in Modeling Elasto-Plastic-Damage Behavior,” Int. J. Plast., 23, pp. 1826–1859. [CrossRef]
Khan, A. S., and Huang, S., 1995, Continuum Theory of Plasticity, John Wiley, New York.
Toi, Y., Lee, J. B., and Taya, M., 2004, “Finite Element Analysis of Superelastic Large Deformation Behavior of Shape Memory Alloy Helical Springs,” Comp. Struct., 82, pp. 1685–1693. [CrossRef]
Meyers, A., Xiao, H., and Bruhns, O., 2003, “Elastic Stress Ratcheting and Corotational Stress Rates,” Tech. Mech., 23, pp. 92–102.

Figures

Grahic Jump Location
Fig. 8

Cycle 1: shear stress τ12 versus angle of rotation φ

Grahic Jump Location
Fig. 9

Cycle 1: normal stress τ22 versus angle of rotation φ

Grahic Jump Location
Fig. 1

Finite shear. Shear stress τ12 versus shear strain γ.

Grahic Jump Location
Fig. 2

Finite shear. Normal stress τ11 versus shear strain γ.

Grahic Jump Location
Fig. 3

Plain strain (restrained tension): monotonic loading at various temperatures. Normal stress τ11 versus elongation λ.

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Fig. 4

Plain strain (restrained tension): monotonic loading at various temperatures. Normal stress τ33 versus elongation λ.

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Fig. 5

Plain strain (biaxial extension): monotonic loading at various temperatures. Normal stress τ11 versus elongation λ.

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Fig. 6

Plain strain (biaxial extension): monotonic loading at various temperatures. Normal stress τ33 versus elongation λ.

Grahic Jump Location
Fig. 7

Cycle 1: normal stress τ11 versus angle of rotation φ

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Fig. 10

Cycle 1: martensite fraction ξ versus angle of rotation φ

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Fig. 11

Cycle 2: normal stress τ11 versus angle of rotation φ

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Fig. 12

Cycle 2: shear stress τ12 versus angle of rotation φ

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Fig. 13

Cycle 2: normal stress τ22 versus angle of rotation φ

Grahic Jump Location
Fig. 14

Cycle 2: martensite fraction ξ versus angle of rotation φ

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