Effect of Phase Transformations on Hardness of Semiconductors
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Ppt - pressure of a reversible phase transformation (PPt = const); T - temperature; Tm - melting temperature; cc - material's constant, introduced in [1], and accounting for dissipation of work of deformation to heat in the core (based on the experimental data, it was recommended to take (x = 1 for metals and polymers, and a = 1/3 for ceramics [1]); c - radius of a hemisphere that forms the boundary between the plastic core and plastic zone (Fig. 1); b - radius of the hemisphere that forms the boundary between the plastic and elastic zones, b _ c (Fig. 1); b* - radius of the hemisphere that limits the zone of reversible phase transformations (Fig. 1); v, = 2re3/3 - volume of the plastic core; v2 - volume of the plastic zone; Vlpt- volume of the zone of reversible phase transformations in the core; v 2,, - volume of the zone (c _•r _ Ppt (Fig. 1). Eqs. (1)-(3) and (5)-(7) lead to the following dependence: 3 8v/vl = [3 Y(1-v)/E + A exp(-3ppt/(2Y))](b/c)3, which relates the penetration volume 5v with the radius of the plastic zone, b. From eq. (8), we can conclude that the size of the plastic zone depends on the phase transformation, i.e., on A and Ppt. Since the radial pressure pl on the core surface is [ 1,21]:
(6) (7)
(8)
p, = 2/3 11[1+ ln(b/c)3], using (8), we can obtain:
2~ r~ l v [3 Y(1 1
-
8 vEY)](9 v) +EAexp (- 3 ppt / (2Y))]
(9)
Here and below if x in the expression ln(x) is less than 1, then x is set equal to 1. For a cone, eq. (8) can be written as: 1
3
2 coty = [3 Y(l-v)/E + A exp(-3pp,/(2Y1))](b/c) 2 If we assume that the surface of the core 7tc is equal to the surface of the impression of a Vickers 2 pyramid d /2 (d is the diagonal of the impression) with the angle between the faces 2y (for the Vickers pyramid 2y = 1360), eq. (8) can be rewritten as:
4-1 --n cot y = [3Y(1 - v)/ E+ Aexp(-3ppt / (2Y))](b/ c)3
(10)
and for A = 0 it is the same as eq. (6) in Ref. 1. Using the same approach, equations for other indenters (Knoop, Berkovich, Rockwell, etc.) can be obtained. If phase transformations occur exclusively in the core (b* = c), eq (8) can be simplified: v/v 1 =(3 Y(I-v)/E)(b/c)3 + A, (II) because b/c = exp(3ppt/(2 1)) (see eq. (4)). Using (8) and (9) at 8v/vl = 4l '-7/ coty, and taking into account that HV/siny =p =pl + aY (see eq. (8) in Ref. 1), we can obtain an expression for the widely used Vickers hardness HV: HY =2y~l+3oa+ln -JtEcot•y (12)
sin y
3
2
12 Y(1 - v) + 4AE exp(-3 Ppt / (2 Y))
Eq. (12) transforms to Tanaka's equation [1] for A = 0.
Phase Transformations Within the Core (pl
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