Axisymmetric Plane Stress States of an Annulus Subject to Displacive Shear Transformation

[+] Author and Article Information
Yuwei Chi, Thomas J. Pence, Hungyu Tsai

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824

J. Appl. Mech 72(1), 44-53 (Feb 01, 2005) (10 pages) doi:10.1115/1.1828062 History: Received July 08, 2003; Revised August 16, 2004; Online February 01, 2005
Copyright © 2005 by ASME
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Birman,  V., 1999, “Analysis of an Infinite Shape Memory Alloy Plate With a Circular Hole Subjected to Biaxial Tension,” Int. J. Solids Struct., 36(1), pp. 167–178.
Birman,  V., 1997, “Review of Mechanics of Shape Memory Alloy Structures,” Appl. Mech. Rev., 50(11), pp. 629–645.
Bernardini, D., and Pence, T. J., 2002, “Shape-Memory Materials, Modeling,” Encyclopedia of Smart Materials, M. Schwartz, ed., J. Wiley & Sons, New York, pp. 964–979.
Bondaryev,  E. N., and Wayman,  C. M., 1988, “Some Stress-Strain-Temperature Relationships for Shape Memory Alloys,” Metall. Trans. A, 19A(10), pp. 2407–2413.
Gall,  K., Sehitoglu,  H., Maier,  H. J., and Jacobus,  K., 1998, “Stress-Induced Martensitic Phase Transformation in Polycrystalline CuZnAl Shape Memory Alloys Under Different Stress States,” Metall. Mater. Trans. A, 29A(3), pp. 765–773.
Lim,  T. J., and McDowell,  D. L., 1999, “Mechanical Behavior of an Ni-Ti Shape Memory Alloy Under Axial-Torsional Proportional and Nonproportional Loading,” ASME J. Eng. Mater. Technol., 121(1), pp. 9–18.
Giannakopoulos,  A. E., and Olsson,  M., 1993, “Axisymmetric Deformation of Transforming Ceramic Rings,” Mech. Mater., 16(3), pp. 295–316.
Budiansky,  B., Hutchinson,  J. W., and Lambropoulos,  J. C., 1983, “Continuum Theory of Dilitant Transformation Toughening in Ceramics,” Int. J. Solids Struct., 19(4), pp. 337–355.
Nadai, A., 1950, Theory of Flow and Fracture of Solids, McGraw–Hill, New York, pp. 472–481.
Chi,  Y., Pence,  T. J., and Tsai,  H., 2003, “Plane Stress Analysis of a Shape Memory Annular Plate Subject to Edge Pressure,” J. Phys. IV, 112, pp. 245–248.
Briggs,  J. P., and Ostrowski,  J. P., 2002, “Experimental Feedforward and Feedback Control of a One-Dimensional SMA Composite,” Smart Mater. Struct., 11(1), pp. 9–23.
Qidwai,  M. A., and Lagoudas,  D. C., 2000, “On Thermomechanics and Transformation Surfaces of Polycrystalline NiTi Shape Memory Alloy Material,” Int. J. Plast., 16, pp. 1309–1343.
Briggs,  J. P., and Ponte Castaneda,  P., 2002, “Variational Estimates for the Effective Response of Shape Memory Alloy Actuated Fiber Composites,” ASME J. Appl. Mech., 69(4), pp. 470–480.
Delaey,  L., Krishnan,  R. V., Tas,  H., and Warlimont,  H., 1974, “Review: Thermoelasticity, Pseudoelasticity and the Memory Effects Associated With Martensitic Transformations. Part 1: Structural and Microstructural Changes Associated With the Transformations,” J. Mater. Sci., 9, pp. 1521–1535.
Liang,  C., and Rogers,  C. A., 1990, “One-Dimensional Thermomechanical Constitutive Relations for Shape Memory Materials,” J. Intell. Mater. Syst. Struct., 1, pp. 207–234.
Ivshin,  Y., and Pence,  T. J., 1994, “A Constitutive Model for Hysteretic Phase Transition Behavior,” Int. J. Eng. Sci., 32(4), pp. 681–704.
Gillet,  Y., Meunier,  M. A., Brailovski,  V., Trochu,  F., Patoor,  E., and Berveiller,  M., 1995, “Comparison of Thermomechanical Models for Shape Memory Alloy,” J. Phys. IV, 5(C8), pp. 1165–1170.
Budiansky,  B., 1958, “Extension of Michell’s Theorem to Problems of Plasticity and Creep,” Q. Appl. Math., 16, pp. 307–309.
Roberts, S. M., and Shipman, J. S., 1972, Two-Point Boundary Value Problems: Shooting Methods, American Elsevier, New York.


Grahic Jump Location
(a) Typical one-dimensional stress-strain curve for T>Af. (b) Temperature dependence of threshold stresses σsFTfFTsRT and σfRT.
Grahic Jump Location
Generic phase distribution within the shape memory alloy annulus. The inner ring M (ri≤r≤rM) consists of martensite. The outer ring A (rA≤r≤ro) consists of austenite. The intermediate ring C (rM<r<rA) consists of an austenite/martensite mixture.
Grahic Jump Location
Martensite phase fraction as a function of effective stress for T>Af
Grahic Jump Location
Typical stress distributions (normalized by σs) for a plate with M phase distribution [σ(r)≥σf]. Here E=50 GPa,σf=4σs=300 MPa,ro/ri=2 and (pi,po)=(1,5)σs. Solid lines represent the case of α=0.05. For comparison, stresses for an elastic solution (α=0) under the same boundary conditions are plotted as dashed lines.
Grahic Jump Location
The FML (fully martensitic loop) and FAL (fully austenitic loop) for E=50 GPa, α=0.05, σf=4σs=300 MPa and ro/ri=2. Thrusts inside the smaller ellipse (FAL) produce a fully austenitic plate, while thrusts outside the bigger loop (FML) produce a fully martensitic plate.
Grahic Jump Location
FMLs for various σf values. Here E=50 GPa, α=0.05, σs=75 MPa and ro/ri=2. The smallest loop is the common FAL, and the other curves are FMLs, each marked with corresponding value of σfs.
Grahic Jump Location
Structure map (type I) for ro/ri=2. Here E=50 GPa, α=0.05 and σf=4σs=300 MPa. Note that there is no ACM region. Thrust pair (pi,po)=(5,0.5)σs, marked as a dot, produces a CM phase distribution. The resulting stresses are shown in Fig. 10.
Grahic Jump Location
Structure map (type II) for ro/ri=5. Here E=50 GPa, α=0.05 and σf=4σs=300 MPa. All six types of phase distributions are present. Thrust pair (pi,po)=(5,0.5)σs, marked as a dot, produces a ACM phase distribution. The resulting stresses are shown in Fig. 11.
Grahic Jump Location
Structure map for the special value ro/ri=3.58. Lower values produce type I structure maps, while higher values produces type II. Here E=50 GPa, α=0.05 and σf=4σs=300 MPa. There is no ACM region, and ADL contacts MEL at two points. One of the two points (shown as a dot) represents (pi,po)=(3.565,0.995)σs, producing a C phase distribution with σ(ri)=σf and σ(ro)=σs. The resulting stresses are shown in Fig. 12.
Grahic Jump Location
Stresses (normalized by σs) for a CM phase distribution. Here E=50 GPa, α=0.05, σf=4σs=300 MPa,ro/ri=2 and (pi,po)=(5,0.5)σs. An elastic solution (α=0) is shown for comparison (dashed lines).
Grahic Jump Location
Stresses (normalized by σs) for a ACM phase distribution. Here E=50 GPa, α=0.05, σf=4σs=300 MPa,ro/ri=5 and (pi,po)=(5,0.5)σs. An elastic solution (α=0) is shown for comparison (dashed lines).
Grahic Jump Location
Stresses (normalized by σs) for a special C phase distribution with rM=ri and rA=ro, for E=50 GPa, α=0.05, σf=4σs=300 MPa,ro/ri=3.58 and (pi,po)=(3.565,0.995)σs. An elastic solution (α=0) is shown for comparison (dashed lines).



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