Part VIII - Quantitative Metallography of Surface Rumpling

The distribution of displacements of originally plane surfaces by an invariant-plane strain has been analyzed statistically. The displacement ratio (the magnitude of the displacement of a point originally a unit distance from the invariant plane) can be determined from an analysis either of tilts of random surfaces or of displacements of random scratches scribed on a randowz surface. In either case the mean of the tangent of the observed displacement angle is related to the displacement ratio, and the number of observations used in determining the mean is related to the confidence limits placed on the estimate of the displacement ratio. In estimating the effect of applied stress on a phase transformation whose displacement effects can be approximated as an invariant-plane strain,* the dis- placement ratio of the invariant-plane strain is of primary importance. Two procedures for determining such displacement ratios are considered here. In the tilt method the measured quantity is taken to be the angle through which a random polish plane is rotated by the invariant-plane strain. In the scratch method the measured quantity is taken to be the angle between the original direction of a random scratch in a random polish plane and the projection of its final direction in the polish plane. In order to obtain precise results from either of the methods, it is necessary to measure the relative volume change associated with the strain in a separate experiment. TILT METHOD A locally homogeneous invariant-plane strain, P, with displacement ratio, S, may be written&apos; where ; and < are unit vectors indicating the displacement direction and invariant-plane normal, respectively, as pictured in Fig. 1. Eq. [I] describes a uniaxial strain if the direction of displacement is parallel to the invariant-plane normal and a pure shear strain if the displacement direction is perpendicular to the invariant-plane normal. The original orientation of the random polish plane relative to the invariant plane and to the displacement direction is shown in Fig. 2. The angle 03 between < and u is related to the relative volume change, ?V/V, and to the displacement ratio by As a result of the invariant-plane strain, the polish plane is rotated about its intersection with the invariant plane by the angle y, which is always taken to be positive. The tangent of this angle in terms of the above parameters is