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Thursday, May 20, 2010

A Kinetic-Based Constitutive Law for Martensitic Transformations

P. Adler, Medtronic Cardiovascular, Santa Rosa, CA; S. Kari, FEA Solutions, Windsor, CA; J. allen, Medtronic Vascular, Santa Rosa, CA

Examination of a stress-strain curve characteristic of a Nitinol sample processed to exhibit superelasticity at body temperature reveals two major factors controlling flow behavior during stress-induced martensitic transformation. At low strains the operation of the transformation acts as a dynamic softening mechanism. At higher strains the matrix becomes saturated with transformation product and deformation by more traditional slip, mechanical twinning, or variant redistribution (in the case of Nitinol) mechanisms ensue with a concomittant increase in hardening rate. The former effect is amenable to quantitative treatement in the case of a stress-induced martensitic transformation at low temperatures (and stresses) where material flow is entirely controlled by transformation kinetics i.e., at Ms< T <Msσ (termed Md in Nitinol literature). Within this range of temperatures the volume fraction of martensite, f, scales linearly with strain i.e., f = kε, and the initial rate of transformation, ∂f/∂t, can be expressed in terms of the nucleation activation energy, Q, as ∂f/∂t = nsVνexp –Q/RT where ns, V, ν are the density of nucleation sites, instantaneous mean martensite plate volume, and nucleation attempt frequency, respectively. The observed linear dependence of Q on the transformation molar free-energy change, Q = A + BΔG, allows derivation of the critical driving force required to achieve a given ∂f/∂t as ΔGcrit(∂f/∂t) = -1/B(A+RT ln (∂f/∂t)/nsVν) which shows a linear dependence on temperature. These equations are sufficient to predict a strain-rate sensitivity of the flow stress of the form: ∂σ/∂(∂ε/∂t) = -RT/[B(∂ΔG/∂σ)] with the resulting constitutive relation: σ(ε, ∂ε/∂t, T) = -[B(∂ΔG/∂σ)]-1 [A+BΔGcrit + RT ln (∂f/∂t)/nsVν]. This equation describes the macroscopic flow behavior of alloys exhibiting a stress-induced martensitic phase transformation based on the dynamic evolution of martensitic nucleation kinetics. Comparison with non-thermoelastic alloys is quite good. Modifications necessary to account for thermoelastic behavior are discussed.

Summary: Building on our presentation describing the defect-dissociation mechanics and related thermodynamic driving forces leading to barrierless nucleation, quantitative models describing the kinetics of stress-assisted martensitic transformations have allowed prediction of the constitutive flow relations in non-thermoelastic alloys. These models are described along with experiments designed to incorporate the effects of interfacial reversibility as occur in thermoelastic alloys such as Nitinol.