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J. Japan Inst. Metals, Vol. 38, No. 10 (1974),

pp. 942-949

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A Mathematical Expression of Knoop Hardness Anisotropy of Cubic Crystal and its Application to Sendust Alloy Crystal

**
Kazumi Shimotori**^{1}

^{1}Metals and Ceramics Laboratory, Research and Development Center, Tokyo-Shibaura Electric Co., Ltd., Kawasaki

#### Abstract:

A mathematical expression that describes various Knoop hardness anisotropy curves in a cubic crystal was proposed by phenomenological approach based on the crystallographic symmetry. The general equation is approximately expressed as a power series up to the sixth order, where the variables are directional cosines of the crystallographic plane of indentation and the orientation of the Knoop indenter. In the cubic crystal, all the anisotropic variations can be expressed by 8 constants, but only 3 constants are necessary if the hardness is independent on the plane of indentation. When the equation is solved for the rotating angle of the indenter, ømega , on a given plane, it can be rewritten as _{KHN}=A+B⋅cos 2 ømega+C⋅cos 4 ømega+D⋅cos 6 ømega+E⋅sin 2 ømega+F⋅sin 4 ømega+G⋅sin 6 ømega .

Particular solutions on 001, 011, 111, 012 and 112 were derived from the general equation. Then they were applied to Knoop hardness variations on 001, 011 and 111 of the Sendust alloy crystal, and it was proved that the equation can describe the anisotropy within the error of less than 5% to the total variation in KHN. The difference in the relations between the constants was discussed for *bcc* and *fcc* metals.

(Received 1974/04/11)

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