  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
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A. A. Movchan and L. G. Sil’chenko, "Buckling of a circular plate made of a shape memory alloy due to a reverse thermoelastic martensite transformation," Mech. Solids. 43 (1), 100111 (2008) 
Year 
2008 
Volume 
43 
Number 
1 
Pages 
100111 
DOI 
10.3103/S0025654408010093 
Title 
Buckling of a circular plate made of a shape memory alloy due to a reverse thermoelastic martensite transformation 
Author(s) 
A. A. Movchan (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya 4, Moscow, 125047, Russia, movchan47@mail.ru)
L. G. Sil’chenko (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya 4, Moscow, 125047, Russia, seliger@mail.ru) 
Abstract 
We solve the axisymmetric buckling problem for a circular plate made of a shape memory alloy undergoing reverse martensite transformation under the action of a compressing load, which occurs after the direct martensite transformation under the action of a generally different (extending or compressing) load. The problem was solved without any simplifying assumptions concerning the transverse dimension of the supplementary phase transition region related to buckling. The mathematical problem was reduced to a nonlinear eigenvalue problem. An algorithm for solving this problem was proposed. It was shown that the critical buckling load under the reverse transition, which is obtained by taking into account the evolution of the phase strains, can be many times lower than the same quantity obtained under the assumption that the material behavior is elastic even for the least (martensite) values of the elastic moduli. The critical buckling force decreases with increasing modulus of the load applied at the preliminary stage of direct transition and weakly depends on whether this load was extending or compressing.
In shape memory alloys (SMA), mutually related processes of strain and direct (from the austenitic into the martensite phase) or reverse thermoelastic phase transitions may occur. The direct transition occurs under cooling and (or) an increase in stresses and is accompanied by a significant decrease (nearly by a factor of three in titan nickelide) of the Young modulus. If the direct transition occurs under the action of stresses with nonzero deviator, then it is accompanied by accumulation of macroscopic phase strains, whose intensity may reach 8%. Under the reverse transition, which occurs under heating and (or) unloading, the moduli increase and the accumulated strain is removed. For plates compressed in their plane, in the case of uniform temperature distribution over the thickness, one can separate trivial processes under which the strained plate remains plane and the phase ratio has a uniform distribution over the thickness. For sufficiently high compressing loads, the trivial process of uniform compression may become unstable in the sense that, for small perturbations of the plate deflection, temperature, the phase ratio, or the load, the difference between the corresponding perturbed process and the unperturbed process may be significant.
The results of several experiments concerning the buckling of SMA elements are given in [1, 2], and the statement and solution of the corresponding boundary value problems can be found in [311]. The experimental studies [2] and several analytic solutions obtained for the Shanley column [3, 4], rods [57], rectangular plates under direct [8] and reverse [9] transitions showed that the processes of thermoelastic phase transitions can significantly (by several times) decrease the critical buckling loads compared with their elastic values calculated for the less rigid martensite state of the material. Moreover, buckling does not occur in the onephase martensite state in which the elastic moduli are minimal but in the twophase state in which the values of the volume fractions of the austenitic and martensite phase are approximately equal to each other. This fact is most astonishing for buckling, studied in the present paper, under the reverse transition in which the Young modulus increases approximately half as much from the beginning of the phase transition to the moment of buckling. In [39] and in the present paper, the static buckling criterion is used. Following this criterion, the critical load is defined to be the load such that a nontrivial solution of the corresponding quasistatic problem is possible under the action of this load. If, in the problems of stability of rods and SMA plates, small perturbations of the external load are added to small perturbations of the deflection (the critical force is independent of the amplitude of the latter), then the critical forces vary depending on the value of perturbations of the external load [5, 8, 9]. Thus, in the case of small perturbations of the load, the problem of stability of SMA elements becomes indeterminate. The solution of the stability problem for SMA elements also depends on whether the small perturbations of the phase ratio and the phase strain tensor are taken into account.
According to this, the problem of stability of SMA elements can be solved in the framework of several statements (concepts, hypotheses) which differ in the set of quantities whose perturbations are admissible (taken into account) in the process of solving the problem. The variety of these statements applied to the problem of buckling of SMA elements under direct martensite transformation is briefly described in [4, 5]. But, in the problem of buckling under the reverse transformation, some of these statements must be changed.
The main question which we should answer when solving the problem of stability of SMA elements is whether small perturbations of the phase ratio (the volume fraction of the martensite phase q) are taken into account, because an appropriate choice significantly varies the results of solving the stability problem. If, under the transition to the adjacent form of equilibrium, the phase ratio of all points of the body is assumed to remain the same, then we deal with the "fixed phase ratio" concept.
The opposite approach can be classified as the "supplementary phase transition" concept (which occurs under the transition to the adjacent form of equilibrium). It should be noted that, since SMA have temperature hysteresis, the phase ratio in SMA can endure only onesided small variations. But if we deal with buckling under the inverse transformation, then the variation in the volume fraction of the martensite phase cannot be positive.
The phase ratio is not an independent variable, like loads or temperature, but, due to the constitutive relations, its variations occur together with the temperature variations and, in the framework of connected models for a majority of SMA, together with variations in the actual stresses. Therefore, the presence or absence of variations in q is determined by the presence or absence of variations in the temperature, deflection, and load, as well as by the system of constitutive relations used in this particular problem. In the framework of unconnected models which do not take the influence of actual stresses on the phase ratio into account, the "fixed phase ratio" concept corresponds to the case of absence of temperature variations. The variations in the phase ratio may also be absent in connected models in the case of specially chosen values of variations in the temperature and (or) in the external load, as well as in the case of SMA of CuMn type, for which the influence of the actual stresses on the phase compound is absent or negligible.
In the framework of the "fixed phase ratio" hypothesis, the stability problem for SMA elements has a solution coinciding in form with the solution of the corresponding elastic problem, with the elastic moduli replaced by the corresponding functions of the phase ratio.
In the framework of the supplementary phase transition” concept, the result of solving the stability problem essentially depends on whether the small perturbations of the external loads are taken into account in the process of solving the problem. The point is that, when solving the problem in the connected setting, the supplementary phase transition region occupies, in general, not the entire crosssection of the plate but only part of it, and the location of the boundary of this region depends on the existence and the value of these small perturbations. More precisely, the existence of arbitrarily small perturbations of the actual load can result in finite changes of the configuration of the supplementary phase transition region and hence in finite change of the critical values of the load.
Here we must distinguish the "fixed load" hypothesis where no perturbations of the external loads are admitted and the "variable load" hypothesis in the opposite case. The conditions that there no variations in the external loads imply additional equations for determining the boundary of the supplementary phase transition region. If the "supplementary phase transition" concept and the "fixed load" concept are used together, then the solution of the stability problem of SMA is uniquely determined in the same sense as the solution of the elastic stability problem under the static approach.
In the framework of the "variable load" concept, the result of solving the stability problem for SMA ceases to be unique. But one can find the upper and lower bounds for the critical forces which correspond to the cases of total absence of the supplementary phase transition: the upper bound corresponds to the critical load coinciding with that determined in the framework of the "fixed phase ratio" concept, and the lower bound corresponds to the case where the entire crosssection of the plate experiences the supplementary phase transition. The first version does not need any additional name, and the second version can be called as the "allround supplementary phase transition" hypothesis.
In the present paper, the above concepts are illustrated by examples of solving problems about axisymmetric buckling of a circular freely supported or rigidly fixed plate experiencing reverse martensite transformation under the action of an external force uniformly distributed over the contour. We find analytic solutions in the framework of all the abovelisted statements except for the case of free support in the "fixed load" concept, for which we obtain a numerical solution. 
References 
1.  M. A. Khusainov, "Investigation of the Axisymmetric Buckling of Round Plates," Zh. Tekh. Fiz. 67(6), 118120 (1997) [Tech. Phys. (Engl. Transl.) 42 (6), 692694 (1997)]. 
2.  A. A. Movchan and S. A. Kazarina, "Experimental Investigation of the Effect of Buckling Caused by Thermoelastic Phase Transformations under the Action of Compression Stresses," Probl. Mashinostr. Nadezhn. Mashin, No. 6, 8289 (2002). 
3.  A. A. Movchan and L. G. Sil’chenko, "The Stability of a “Shenley Column” during Creep or during Direct Thermoelastic Martensite Transformation," Mekh. Kompos. Mater. Konstr. 6(1), 89102 (2000). 
4.  A. A. Movchan and L. G. Sil’chenko, "Analysis of Buckling Induced by the Direct Thermoelastic Transformation under the Action of Compression Stresses," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 132144 (2004) [Mech. Solids (Engl. Transl.) 39 (2), 104114 (2004)]. 
5.  A. A. Movchan and L. G. Sil’chenko, "Buckling of a Rod Undergoing Direct or Reverse Martensite Transformation under Compressive Stresses," Zh. Prikl. Mekh. Tekh. Fiz. 44(3), 169178 (2003) [J. Appl. Mech. Tech. Phys. (Engl. Transl.) 44 (3), 442449 (2003)]. 
6.  L. G. Sil’chenko, "On the Buckling of a SMA Rod Caused by Reactive Compressing Stresses under a Reverse Martensite Transformation," Mekh. Komp. Mater. Konstr. 10(3), 393406 (2004). 
7.  L. G. Sil’chenko, "Stability of a Constrained SMA Rod under a Reverse Martensite Transformation," Mekh. Komp. Mater. Konstr. 10(4), 566576 (2004). 
8.  A. A. Movchan and L. G. Sil’chenko, "The Stability of a Plate of ShapeMemory Alloy in a Direct Thermoelastic Phase Transition," Prikl. Mat. Mekh., 68(1), 6072 (2004) [J. Appl. Math. Mech. (Engl. Transl.) 68 (1), 5364 (2004)]. 
9.  A. A. Movchan and L. G. Sil’chenko, "Analytical Solution of the Coupled Buckling PROBLEM for a Plate from a Shape Memory Alloy Subjected to Inverse Martensite Transformation," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 164178 (2004) [Mech. Solids (Engl. Transl.) 39 (5), 134145 (2004)]. 
10.  G. A. Malygin, "Euler Instability of Bidirectional Shape Memory Effect in a Titanium Nickelide Strip," Fiz. Tverd. Tela 45(12), 2233 (2003) [Phys. Solid State (Engl. Transl.) 45 (12), 23422347 (2003)]. 
11.  G. A. Malygin and M. A. Khusainov, "Stability of the Mechanical Behavior of an Arched TiNi Strip under the Conditions of the Constrained Shape Memory Effect," Zh. Tekh. Fiz. 74(10), 5763 (2004) [Tech. Phys. (Engl. Transl.) 49 (10), 13011307 (2004)]. 
12.  A. A. Movchan, "Micromechanical Constitutive Equations for Shape Memory Allows," Probl. Mashinostr. Nadezhn. Mashin, No. 6, 4753 (1994). 
13.  A. A. Movchan, "Micromechanical Description of the Deformation due to Martensite Transformations in ShapeMemory Alloys," Izv. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 197205 (1995) [Mech. Solids (Engl. Transl.) 30 (1), 186192 (1995)]. 
14.  A. A. Movchan, "Selecting a PhaseDiagram Approximation and a Model of the Disappearance of Martensite Crystals for Shape Memory Alloys," J. Prikl. Mekh. Tekh. Fiz. 36(2), 173181 (1995) [J. Appl. Mech. Texh. Phys. (Engl. Transl.) 36 (2), 300307 (1995)]. 
15.  A. A. Movchan, So Newnt, and S. A. Kazarina, "Thermodynamical Description of the Mechanical Behavior of Shape Memory Alloys," Deformatsiya i Razrushenie Materialov, No. 8, 1419 (2005). 
16.  A. S. Vol’mir, Stability of Deformable Systems (Nauka, Moscow, 1967) [in Russian]. 

Received 
21 November 2005 
Link to Fulltext 
http://www.springerlink.com/content/66881375317u71m3 
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