It has been emphasised that the Ivantsov and Trivedi solutions are shape-preserving; this is illustrated in Figure 7.21, with a series of identical parabolae which show the sequence of growth. The parabolae are equidistant along X to illustrate a constant rate of lengthening. An observer located at the origin would also notice that the plate thickens along Y but at an ever decreasing rate. It follows necessarily from the parabolic shape that the thickness must increase with /2. The position of the interface with respect to the moving coordinates х,У (Figure 7.16) is given by setting Xn = 1:
Making the substitutions X = 0 and Y = Y, gives
where Y, is the plate half-thickness along Y at X = 0. The plate thickness will therefore vary parabol- ically with time at a rate that is smaller than V(.
Figure 7.21 The translation of the same parabola along X in the sequence 1-6 illustrates the nature of the shapepreserving solution. The distance between adjacent parabolae is identical along X to represent the constant growth rate in that direction.
Figure 7.22a shows how the plate aspect ratio и = Y,/vtt decreases as the plate lengthens. The formulation of the theory requires that the plate exists at time zero, with length X = 0.5r and Y, = r because the focus is located at X = 0; the initial aspect ratio is therefore 2, which is an artefact of the model, though the decrease in aspect ratio as the plate lengthens is observed experimentally. The Widmanstatten ferrite plates start off rather stubby and become slender as they grow (Figure 7.1). Consistent with many observations, the theory also predicts smaller aspect ratios at lower transformation temperatures.
The rate at which the plates thicken should compare against that at which allotriomorphs of ferrite thicken during diffusion-controlled growth. The thickness Z of the allotriomorph is given by (Chapter 4)
where am is the parabolic-thickening rate constant for one-dimensional growth; using this relation.
Figure 7.22 (a) The calculated aspect ratios of Widmanstatten ferrite plates as a function of the plate length and transformation temperature, (b) A comparison of the parabolic thickening of Widmanstatten ferrite versus that of allotriomorphic ferrite under identical conditions. The growth of Widmanstatten ferrite is modelled as in , but for comparison with allotriomorphic ferrite, 50 J mol-1 of stored energy associated with Widmanstatten ferrite is neglected; the thickness of the allotriomorphic ferrite is calculated at the time corresponding to a particular length of the aw- The neglect of stored energy does not affect the relative positions of any of the curves. The plate lengthening rates are 0.76 x 10-6 and 0.098 x 10~6m s_l for 672 and 721°C respectively.
the ratio of plate to allotriomorph thickening rates is given by
This ratio is found to be less than unity (Figure 7.22b) because of the dependence of the plate thickening rate on the tip radius. The parabolic shape is geometrically constrained by the radius at the tip; a finer radius leads to a smaller thickening rate.
The model described above can be compared against measurements  that confirm that the thickening rate always decreases with time Figure 7.23. In some cases the plates thickened smoothly (curve B) whereas in others it increased abruptly between sequences of smooth thickening (curves D and E). The discontinuous thickening was attributed to irregularities in the shape of the interface. Consistent with theory, the plate thickness was in many cases smaller than expected for an allotriomorph (curves B-C), but the opposite result was found in other instances (curve E), with very large variations even within the same austenite grain. In some cases (curve C) the plates simply stopped thickening. These detailed variations cannot be explained but might have something to do with the fact that the growth theory neglects the effect of elastic and plastic strains caused by the displacive mechanism of transformation, on the shape of the transformation product.
Figure 7.23 Schematic illustration of the different Widmanstatten ferrite plate thickening behaviours reported by Kinsman, Eichen and Aaronson . Curve A represents the onedimensional carbon diffusion-controlled thickening of allotriomorphic ferrite.