A new model for the volume fraction of martensitic transformations
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I.
j 5 1 2 exp [w VDSA→M (MS 2 T )]
INTRODUCTION
FOR
over half a century, martensitic transformations have been the subject of continuing studies which have intensified in recent years. This is mostly due to the interest in applications involving the unique mechanical behavior of shape-memory alloys. To understand the kinetics of these transformations, it is necessary to know quantitatively the volume fraction of martensite (j) formed upon cooling or under loading. For example, an internal variables model has been proposed by Tanaka et al.[1] and recently modified by Liang and Roger[2,3] and Brinson[4] for modeling the mechanical behavior of shape-memory alloys. It is based on Tanaka and Nagaki’s model[5] for the second-order phase transformation, which uses j as an internal variable in addition to temperature and applied stress. Many attempts have been made to study the relationship between the volume fraction j and the quenching temperature T, particularly in dynamically stabilized steels. Two empirical relationships have been proposed for steels. The first one, which will be called the ‘‘power relationship,’’ has been given by Harris and Cohen[6] as 5.32
j 5 1 2 6.95 3 10215 [455 2 (Ms 2 T )]
[1]
where Ms is the temperature at which, in the absence of external stress, martensite first forms on cooling from the austenite region and T is the temperature in degrees Kelvin. The second empirical relationship, which will be called the ‘‘exponential relationship,’’ has been proposed by Koistinen and Marburger[7] as
j 5 1 2 exp [21.10 3 1022 (Ms 2 T )]
[2]
A theoretical exponential model has been derived by Magee,[8] based on the assumption that the number of new plates formed per unit volume is linearly proportional to the chemical driving force. The proportionality constant is w and the result is similar to Eq. [2]: H.Y. YU is Materials Research Engineer with the Mechanics and Materials Branch, Materials Science and Technology Division, Naval Research Laboratory, Washington, DC 20375-5343. Manuscript submitted November 22, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS A
[3]
where V is the average volume of the newly formed plates and DSA→M is the entropy change at the transition. Another theoretical model has been proposed by Fisher et al.,[9] which will be referred to as the FHT model and is as follows:
j 5 1 2 [ q n0 1 1]2m
[4]
where n0 is the number of martensite nuclei, q is the number of austenite grains per mole of lattice sites, and m is a parameter that is a function of the shape of the martensite. The agreement between this theory and experiment is satisfactory, but it can be contended that the significance of the agreement is diminished by the arbitrary choice of values for the two adjustible parameters in the expressions for q and m. The exponential relationships (Eqs. [2] and [3]) have been used extensively to describe the volume fraction of martensitic transformations.[10–14] For example, in the internal variables model for shape-memory alloys, Tanaka et al.[10] adapted the exponential for
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