Invariant plane

An invariant-plane strain consists of a simple shear on a plane, plus a normal strain perpendicular to the plane of shear (see Section 24.1 and Fig. 24.1). This is a combination of Cases 2 and 3. The expression for Ags then follows directly from Eqs. 19.26 and 19.27, with the result that Age is proportional to c/a. Age is therefore minimized for a disc-shaped inclusion lying in the plane of shear. 

The term invariant-plane strain comes from the fact that the plane of shear in an invariant plane strain is both undistorted and unrotated. Hence the plane of shear is a plane of exact matching of the coherent inclusion and the matrix. In martensitic transformations, this matching is met closely on a macroscopic but not a microscopic scale (see Section 24.3). 

Martensitic transformations involve a shape deformation that is an invariant-plane strain (simple shear plus a strain normal to the plane of shear). The elastic coherency-strain energy associated with the shape change is often minimized if the martensite forms as thin plates lying in the plane of shear. Such a morphology can be approximated by an oblate spheroid with semiaxes (r, r, c), with r c. The volume V and surface area S for an oblate spheroid are given by the relations... 

However, important differences exist. Martensite and its parent phase are different phases possessing different crystal structures and densities, whereas a twin and its parent are of the same phase and differ only in their crystal orientation. The macroscopic shape changes induced by a martensitic transformation and twinning differ as shown in Fig. 24.1. In twinning, there is no volume change and the shape change (or deformation) consists of a shear parallel to the twin plane. This deformation is classified as an invariant plane strain since the twin plane is neither distorted nor rotated and is therefore an invariant plane of the deformation. 

These operations do not occur separately and in any particular sequence but are simply a convenient way to conceptualize the transformation as a series of operations, each of which can be analyzed separately, but which working together produce a martensitic structure containing an invariant plane. As such, they can be imagined to occur in any sequence. For purposes of analysis, it is convenient to imagine that the lattice-invariant deformation occurs first, followed by the lattice deformation, followed finally by the rigid-body rotation. We now show that a lattice-invariant shear by slip followed by the lattice deformation analyzed above can produce an undistorted plane. 

The geometrical features of shear deformation are shown in Fig. 24.5. Here, the shear is on the K plane in the direction of d. The initial unit sphere is deformed into an ellipsoid and the Ki plane is an invariant plane. The K2 plane is rotated by the shear into the K 2 position and remains undistorted. A reasonable slip system to assume for the lattice-invariant shear deformation is slip in a (111) direction on a 112 plane in the b.c.t. lattice, which corresponds to slip in a (110) direction on a 110 plane in the f.c.c. lattice. 

Invariant Plane by Addition of Rigid-Body Rotation... 

The plane containing a and c in Fig. 24.9 is the plane in the f.c.c. phase that initially contained the vectors a" and c". If the b.c.t. phase is now given a rigid-body rotation so that a" —> a and c" —> c, the undistorted plane in the b.c.t. phase will be returned to its original inclination in the f.c.c.-axis system and will therefore be an invariant plane of the overall deformation. In the present case, this can be achieved by a rotation around the axis indicated by u in Fig. 24.9 (see Exercise 24.3). The solution of the problem is now complete. The invariant plane is known, and the orientation relationship between the two phases and total shape change can be determined from the combined effects of the known lattice-invariant deformation, lattice deformation, and rigid-body rotation. 

If S is the lattice-invariant deformation tensor and R the rigid-body rotation tensor, the total shape deformation tensor, E, producing the invariant plane can be expressed as... 

Because of the four-fold symmetry of the [001] pole figures in Figs. 24.6-24.9, additional symmetry-related invariant planes can be produced. Also, further work shows that additional invariant planes can be obtained if a lattice-invariant shear corresponding to a = 7.3° rather than a = 11.6° (see Fig. 24.8) is employed [5]. Multiple habit planes are a common feature of martensitic transformations. 

The crystallographic model for martensite described above is primarily due to Wechsler et al. [1], A similar model, employing a different formalism but leading to essentially equivalent results, has also been published by Bowles and MacKen-zie [2-4]. In both models, a search is made for an invariant (or near-invariant) plane which is then proposed as the habit plane, since the selection of this plane... 

Let us establish conditions for the existence of additional linear laws of conservation. Consider one invariant plane P E,z = const. > 0. Let there... 

For Nitinol - at the transition Ms, atoms begins to shear uniformly throughout the crystal. As the temperature is lowered the atomic shear continues to increase. At temperature, Mf, the atoms shear to their maximum point and assume a new structure. Thus, between Ms and Mr temperature interval the crystal structure of Nitinol is undefined and belongs neither to austenite nor to martensite . Therefore, thermodynamically, it should be classified as the second-order transformation. This is illustrated in Fig. 3. Conventionally - above Ms temperature, the whole crystal assume a crystal structure identified as austenite . At Ms temperature, a new crystal structure of martensite begins to form through two-dimensional (planar) atomic shear. The two crystal structures of austenite and martensite therefore share an identical plane known as Invariant Plane. As the temperature is lowered, the two dimensional shear (or more correctly, shift ) continue to take place one plane at a time such that the Invariant Plane moves in the direction as to increase the volume of martensite at the expense of austenite . Ultimately, at Mt temperature the whole crystal becomes martensite . Since between Ms and Mf any given micro-volume of the crystal must belong to either the austenite or the martensite , the transformation is of the first-order thermodynamically. This case is pictorially illustrated in Fig. 4. 

Two phases share a common plane (invariant plane) which undergo shifting one at a time as a function of temperature... 

At Ms temperature TiNi initiates a uniform (inhomogeneous) distortion of its lattice — through a collective atomic shear movement. The lower the temperature, the greater the magnitude of shear movements. As a result, between Ms and Mr temperature the crystal structure is not definable. In sharp contrast, other known martensitic transformations initiate a nonuniform (heterogeneous) nucleation at Ms and thereafter the growth of martensite is achieved by shifting of a two dimensional plane known as invariant plane [28] at a time. Thus, between Ms and Mr temperature the crystal structure is that of austenite and/or martensite . 

Semimajor axis, in AU, where 1 AU = Earth-Sun mean distance — 1.496 X 10 cm. Inclination of orbit relative to an invariable plane. 

What we have learned from this analysis is that the invariant plane allowing for... 

For instance let us consider two lengths r and R satisfying the necessary conditions (32) and (34). What is then the range of possible latitudes p of the vector R with respect to the invariable plane, the plane normal to the angular momentum c That range is of course symmetrical with respect to p= 0 the absolute value of the possible latitudes has a maximum pm-, that can be obtained with (31) and is given by ... 

By varying the initial conditions we can alter the dimensions of the parallelepiped and so displace the invariable planes. It follows that in this case (i.e. no identical commensurabilities) the directions in the/-dimensional j-space which are the axes of the co-ordinates in which the Hamilton-Jsicobi equation is separable have an absolute significance, and that only the scale of each individual variable can be altered. 

If the p jlar axis of the co-ordinate system be taken in the direction of the resultant angular momentum P=, /2ir, the angular separation of the line of nodes from a fixed line in the invariable plane is a cyclic variable conjugate to P. For the other co-ordinates let us take the radius vector r of the outer electron and the conjugate momentum p together with the angular separation ifi of the outer electron from the line of nodes and the conjugate momentum... 

It has been known for some time that the formation of martensite causes roughening (in Bain s words) of a prepolished surface, as shown in Figme 2. Careful analysis of the tilted martensite platelets shows, however, that an invariant plane strain (IPS) has occurred (Bowles, 1951). Such a distortion resembles a shear but also includes a volume-change component. In an IPS the displacement of any point is in a common direction and is proportional to its distance from the undistorted and unrotated invariant plane. 

We now face a dilemma in the case of iron alloys or steels (or in general), the measured IPS is inconsistent with the correct structural change as given by the Bain strain on the other hand, the upsetting produced by the Bain strain is not an IPS. Incidentally, the invariant plane is the habit plane of the martensite plates, as shown in Figure 2. Modem crystallographic theories of martensite formation such as those of Bowles and Mackenzie (1954) (BM) and Wechsler et al. (1953) (WLR) rectify these apparent inconsistencies, so we will proceed to discuss these BM and WLR theories, which are fundamentally identical but differ in mathematical order. 

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