Invariant plane strain

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. 

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. 

Figure 2. Optical micrograph showing an Fe-24Pt alloy partially transformed to martensite. The surface upheavals caused by martensite formation are analyzed to be an invariant plane strain. Both phases are ordered...

The value of E depends upon the values of the elonents in the stress and strain tensors. Under plane stress conditions, one of die principal stresses is zero and E is equal to Young s modulus, E. However, under plane strain conditions, the strain in one of the principal axes is zero and E = E/(l — v ) where v is Poisson s ratio. For most polymers 0.3 < v < 0.5 and the values of both Gic and Kic invariably are much greater when measured in plane stress. For the purposes both of toughness comparisons and component design, die plane strain values of Gic and Kic are preferred because th are the minimum values fm any given material. In order to achieve plane strain conditions, the following criteria need to be satisfied ... 

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. 

In two dimensional analyses (plane strain), which are frequently used in glacier studies, the third invariants are zero. Failure of the second assumption would imply that a sample in simple shear would expand or contract in the direction perpendicular to the shear plane and, because ice is incompressible, this would be accompanied by contraction or expansion in the shear plane. A plane-strain model could not then be used. 

By plotting Hugoniot curves in the pressure-particle velocity plane (P-u diagrams), a number of interactions between surfaces, shocks, and rarefactions were solved graphically. Also, the equation for entropy on the Hugoniot was expanded in terms of specific volume to show that the Hugoniot and isentrope for a material is the same in the limit of small strains. Finally, the Riemann function was derived and used to define the Riemann Invarient. 

For uniaxial (hexagonal) symmetry the 6 strain components are subdivided in two (invariant) one-dimensional subsets (indicated by the superscript a, and subscripts 1 and 2 for the volume dilatation and the axial deformation, respectively), and two different two-dimensional subsets, indicated by y for deformations in the (hexagonal) plane, and by e for skew deformations. These modes are also depicted in fig. 3. In this case, the magnetostriction can be expressed as... 

Provided that the out-of-plane positions are occupied this is invariably true as the in-plane chelate rings are more strained than the out-of-plane rings (65). 

Here u is the position of a layer plane and z is the position coordinate locally parallel to the director n, where n is parallel to the average molecular axis, which is assumed to remain normal to the layer plane, du/dz = e is the compressional (or dilational) strain. Thus, layer bending and layer compression are characterized by a splay (or layer-bend) modulus K and a compression modulus B. Other kinds of distortion present in nematics, such as bend or twisting of the director n, are not compatible with layers that remain nearly parallel, and hence are forbidden. Equation (10-36) is not invariant to rotations of frame, and its validity is limited to weak distortions a rotationally invariant expression has been given by -Grinstein and Pelcovits (1981).---------------------------------------------------------... 

The first invariant represents rate of change of volume, which is zero for incompressible fluids. The third invariant IIIo is zero for plane flows. The second invariant IIo represents a mean rate of deformation including all shearing and extensional components. It is convenient to define, for all flows, a generalized strain rate as... 

In this section, the effect on substrate curvature of the variation of mismatch strain and material properties through the thickness of layered films is analyzed. The derivation of the Stoney formula (2.7) in Section 2.1 refers only to the resultant membrane force in the film any through-the-thickness variation of mismatch strain in the film is considered only peripherally. Film thickness was taken into account explicitly in Section 2.2, but it was assumed there that mismatch strain and elastic properties of the material were uniform throughout the film. However, there are situations of practical significance for which this is not the case. Two of the most common cases are compositionally graded films in which the mismatch strain and the elastic properties vary continuously through the thickness of the film, and multi-layered films for which the mismatch strain and the elastic properties are discontinuous, but piecewise constant, from layer to layer throughout the thickness of the film. In both cases, the mismatch strain and the material properties are assumed to be uniform in the plane of the interface. With reference to the cylindrical r, 6,. z—coordinate system introduced in Section 2.1, the mismatch strain and film properties are now assumed to vary with z for fixed r and 6, but both are invariant with respect to r and 9, for fixed z. 

The strain components for any deformation are subject to the conditions of strain compatibility which assure that the strain field can be derived from a physically realizable displacement field. In the present case, the only compatibility condition which is not satisfied identically is eaif(2) = 0. This condition follows directly from the symmetry and translational invariance characteristics of the system and is independent of material response. If the notation introduced above is preserved here, whereby k is the spherical curvature of the substrate midplane and 2np is the location of the plane for... 

The necessary condition established in Sections 6.2 and 6.3 for dislocation formation in a uniformly strained film can readily be extended to cases where the mismatch strain varies in an arbitrary way through the thickness of the film, provided that the material is still elastically homogeneous. With reference to Figure 6.11, the mismatch strain is a function of the y—coordinate and, in this case, it is written as em(y) The mismatch is independent of the X and z coordinates, thus preserving the important property of invariance under translation in the plane. The fact that the stress components (Txx and (Tzz in the background mismatch field vary with y does not disturb the equilibrium state of the background field, even if that variation is discontinuous in the y—direction. 

The crystal fraction of all specimens was calculated, as shown in Fig. 3.a), based on the equator diffraction peaks (200 and 120 planes) [7]. From these quantitative values, we can corroborate that the crystalline phase increases with strain during stretching. However, the change in this property is almost invariable up to 300% strain and then a significant increase is achieved, in accordance with tensile results. The molecular chains, which are initially entangled, are rearranged along the draw axes, and hence, the polymer shows anisotropic physical properties. We believe that the above observations are evidence of an amorphous/semi-crystalline transition near 300% strain. 

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