Initiated in 1991 the cooperative effort (between IPPT, Tokyo Metropolitan Institute of Technology, University of Tsukuba - Japan and Besancon University - France, cf. [1-17]) to model and test NiTi SMA behavior at complex stress state has brought a number of important results that enable reliable design of structure elements at the temperature range of pseudoelasticity. Some of these results will be discussed in this lecture.
a) 3D phenomenological theory of pseudoelasticity has been developed. The adopted form of the macroscopic Gibbs thermodynamic potential for two-phase multi-variant medium was derived using the concepts of mesomechanics. It depends merely on single internal variable z (volume fraction of martensite) since the dependence of the energy of eigenstrain incompatibilities on the other possible displacement - type internal variables are eliminated by assumption that macro-particle is in unconstrained thermodynamic equilibrium (use is made of the concept of " optimal internal rearrangement"). States of unstable thermodynamic phase equilibrium (with respect to state variable z) are identified as critical states at which the forward and reverse martensitic transformations occur (generalization of I. Müller concept to account for a complex stress state). Thus, the initial pseudoelastic flow and strain recovery conditions follow directly from the adopted form of the thermodynamic potential. The analytical expressions are counterparts of Clausius-Clapeyron relation of the classical two-phase fluid thermodynamics. The general formal kinetic law for time variation of z (in the course of forward and reverse phase transformations) was postulated in the form that includes particular laws presented in earlier metallurgical literature. It satisfies the entropy inequality (2D law of thermodynamics). All equations were simplified by an extra requirement of the solid to be isotropic.
b) To verify the reliability of predictions of the theory a special experimental program was developed at IPPT. The two-step deformation behavior of thin-walled tubular specimen (outer diameter-19mm, thickness 1.5mm, length 122mm) made of Ti-51.0at%Ni alloy was investigated under loading-unloading cycle (stress rate controlled, 1 MPa/s) along 5 proportional paths of torsion-tension (compression) type, at ambient temperatures 260, 280, 300, 310, 315, 322.5K (Ms=253K, As=268K, Af=306K, Rs=306, RAs=305K). The alloy and specimen were manufactured and prepared for testing in the Institute of Material Science at Tsukuba University (Japan) under supervision of professor S. Miyazaki (details can be found in [11]). All mechanical tests were performed in the Laboratory of Strength of Materials (IPPT) under assistance of Professor dr L. Dietrich and Dr. G. Socha. The straining effects (low stress level - first step) associated with B2 ("austenite", high temperature phase) →R (rhombohedral) phase transition (p.t.) and pre-existing R-phase reorientation (referred to as R-plasticity) were investigated separately from those coupled with R→M (monoclinic B19' martensite) and B2→M p.t. (high stress level - second step). The advanced data processing enabled to exhibit the experimental trends important in thermodynamics of B2↔M pseudoelasticity (T=310, 315 and 322.5K). In particular, all constants and functions occurring in the developed theory of pseudoelasticity were identified, namely
i) thermostatic constants (occurring in the thermodynamic potential): two elastic constants (assumed isotropy), internal energy and entropy of formation of martensite at stress free state, amplitude of overall eigenstrains, energy of coherence, two constants occurring in the shape function of the surface of critical states (in stress space) at which pseudoelastic flow (associated with forward B2→M stress induced p.t.) and strain recovery (associated with reverse M→B2 p.t.) are initiated.
ii) thermodynamic constants: two constants occurring in formal equations of transformation kinetics.
The specific form of Clausius-Clapeyron equations are proposed. The theoretical and experimental pencils of straight lines on effective stress-temperature plane for different paths show good agreement. The observed dissymmetry of the initial pseudoelastic yield and strain recovery surfaces is described by employing the third invariant of stress deviator into theoretical framework. The specific mathematical form of those surfaces is proposed. They are illustrated on shear stress-axial stress plane for different temperatures.
On the basis of analysis of obtained loading and unloading branches of internal hysteretic loops the new kinetic law for martensitic forward and reverse p.t was proposed for investigated alloy.
Partial verification of the theory was made by comparison of : a) the variations in time of the measured and predicted strain components, b) experimental and theoretical hysteresis loops in the plane of effective stress-equivalent overall eigenstrain for different paths. The observed discrepancies are not essential from the practical view point.
References
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