 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| A. A. Movchan and L. G. Sil'chenko, "Analysis of buckling induced by the direct thermoelastic transformation under the action of compression stresses," Mech. Solids. 39 (2), 104-114 (2004) |
| Year |
2004 |
Volume |
39 |
Number |
2 |
Pages |
104-114 |
| Title |
Analysis of buckling induced by the direct thermoelastic transformation under the action of compression stresses |
| Author(s) |
A. A. Movchan (Moscow)
L. G. Sil'chenko (Moscow) |
| Abstract |
Various approaches to the stability analysis (buckling analysis) of solids subjected to the direct thermoelastic phase transformation under the action of compression stresses are considered. Two alternative concepts - fixed phase composition and continuing phase transformations - are compared on the basis of the investigation of the nonlinear behavior and buckling of the Shenly column with the supporting beams from a shape memory alloy. The comparison shows that the continuing phase transformation concept leads to more realistic results.
The direct thermoelastic transformation under that action of compression loads is a rather effective method for providing the members from shape memory alloys (SMA) with initial compressive strains. However, it is known [1] that the deformation ability of SMA subjected to direct thermoelastic transformations substantially increases due to both the reduction in the elastic moduli and the development of phase strains. As a result, when subjected to initial force or geometric perturbations, the specimen can deviate from the rectilinear equilibrium shape during the phase transformation, i.e., the buckling induced by direct thermoelastic phase transformations can occur. This type of buckling was discovered experimentally in [2]. It has been established that titanium nickelide specimens that do not buckle either in austenitic or martensitic state in isothermal conditions can buckle under the same loading when being transformed from the austenitic state to the martensitic state. The critical loads for the buckling induced by the direct martensite transformation can be by a factor of 3 to 4 lower than the isothermal critical loads in the martensitic state characterized by the least value of Young's modulus.
There are only a few of publications devoted to the buckling analysis of solids undergoing thermoelastic phase transformations. The buckling of thin-walled SMA elements induced by the compression reaction stresses that appear during the reverse martensite transformation in the prestrained state is studied in [3, 4]. The mechanical behavior of the Shenly column with SMA beams undergoing the direct martensite transformation under the action of a compressive load was investigated in [5] in the uncoupled linearized formulation. The nonlinear behavior of the same column was studied in [6].
The solution of linearized buckling problems for solids
undergoing thermoelastic phase transformations suggests the choice to be made between two alternative hypotheses. In accordance with the first hypothesis, the transition from the unperturbed state to the adjacent equilibrium shape occurs so rapidly that the additional phase transformation does not have enough time to occur. In this case, the phase composition parameter is assumed constant in the linearized equations. We will refer to this hypothesis as the fixed phase composition concept. In [7], within the framework of this concept, the linearized buckling problem is considered for the SMA beam that undergoes the direct martensite transformation under the action of the compression force constant in magnitude and direction. The critical force in this case turns out to be close to Euler's buckling force for a reduced value of Young's modulus, which is typical of the martensitic state.
The alternative hypothesis (the continuing phase transformation concept) implies that the transition to the adjacent equilibrium shape is accompanied by an additional phase transformation.
To investigate the essential features of the buckling induced by thermoelastic phase transformation and to make a choice in favor of one of the alternative hypotheses, we consider in the present paper the nonlinear behavior and buckling of the Shenly column (Fig. 1). The supporting beams of the column are made of a shape memory alloy, have identical cross-sectional areas, and work only in tension/compression. The distance between the beams is b, the length of the rigid body is L, and the magnitude of the force applied to the body at the top point of the symmetry axis is P. This force is directed vertically downward and is constant in magnitude.
We consider both the ideal column, the beams of which have the same length a, and the column with the initial geometric defect due to which the length of beam 1 exceeds that of beam 2 by Δa. In the latter case, a denotes the mean length of the beams. As a result of the initial defect, the symmetry axis of the body is inclined at the angle φ0 =arcsin(Δa/b) to the vertical. We assume that the temperature of both beams is the same at each time instant and decreases synchronously through the temperature interval of the direct martensite transformation, which causes the transformation of the material of the beams from the austenitic state to the martensitic state. |
| References |
| 1. | V. A. Likhachev and V. G. Malinin, Structural-analytical Theory of Strength [in Russian], Nauka, St. Petersburg, 1993. |
| 2. | A. A. Movchan and S. A. Kazarina, "Experimental study of buckling induced by thermoelastic phase transformations under the action of compressive stresses," Probl. Mashinostr. i Nadezhn. Mashin, No. 6, pp. 82-89, 2002. |
| 3. | M. A. Khusainov, "Investigation of the axially symmetric buckling effect for circular plates," Zh. Tekhn. Fiziki, Vol. 67, No. 6, pp. 118-120,
1997. |
| 4. | M. A. Khusainov and O. A. Malukhina, "Buckling analysis of shape memory spherical segments," in Transactions of the Third International Seminar "Modern Issues of Strength" [in Russian], Vol. 2, pp. 185-189, Izd-vo Novgorodsk. Un-ta, Novgorod, 1999. |
| 5. | A. A. Movchan and L. G. Sil'chenko, "The problem of buckling under thermoelastic martensite phase transformations," in Transactions of the Third International Seminar "Modern Issues of Strength" [in Russian], Vol. 1, pp. 307-311, Izd-vo Novgorodsk. Un-ta, Novgorod, 1999. |
| 6. | A. A. Movchan and L. G. Sil'chenko, "The buckling of the Shenly column in the case of creep or direct thermoelastic martensite transformation," Mekhanika Kompozit. Materialov i Konstr., Vol. 6, No. 1, pp. 89-102, 2000. |
| 7. | L. G. Sil'chenko, "Buckling of a compressed beam from a shape-memory alloy induced by the direct thermoelastic phase transformation," in Physics of Deformation and Fracture Processes and Prediction of the Mechanical Behavior of Materials: 36th Seminar "Topical Issues of Strength", Scientific Transactions. Part 1 [in Russian], pp. 359-364, Vitebsk, 2000. |
| 8. | A. A. Movchan, "Micromechanical constitutive equations for shape-memory alloys," Probl. Mashinostr. i Nadezhn. Mashin, No. 6, pp. 47-53, 1994. |
| 9. | A. A. Movchan, "Micromechanical approach to the description of deformation induced by martensite transformations in shape-memory alloys," Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 197-205, 1995. |
| 10. | A. A.Movchan, "The choice of the approximation for the transformation diagrams and models that account for the disappearance of martensite crystals in shape-memory alloys," Zh. Prikl. Mekhaniki i Techn. Fiziki, Vol. 36, No. 2, pp. 79-90, 1995. |
| 11. | A. A. Movchan, "Consideration of the elastic modulus variability and the effect of stresses on the phase composition in shape memory alloys," Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 79-90, 1998. |
| 12. | Ya. G. Panovko and I. I. Gubanova, Stability and Vibration of Elastic Systems [in Russian], Nauka, Moscow, 1987. |
| 13. | C. Liang and C. A. Rogers, "One dimensional thermomechanical constitutive relations for shape memory materials," Appl. Math. Model., Vol. 1, No. 2, pp. 207-235, 1990. |
|
| Received |
10 April 2002 |
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