12.4 Analytical Model The deformed specimen in Fig. 12.4 suggests an analytical model that can be useful for explaining and predicting the material behavior around a drilled hole. Figure 12.5 schematically illustrates a cross-section of the model, which is axisymmetric with center axis at the left. The model consists of a circular disk of thickness W-h on the left, joined to an outer annular ring of thickness W, inner radius a, and “infinite” outer radius on the right. The two parts share a common lower surface. The drilled hole of depth h is located above the disk. The residual stresses relieved by hole cutting act on the hole surface as shown in Fig. 12.1, creating a radial force per unit angle S ¼σ h a. An internal force F and moment M are created between the inner disk and outer ring. They combine to create a net radial force C ¼S F and a net moment per unit angle Q¼S (W h)/2 + F h/2 M on the outer ring. From consideration of equilibrium of the forces and moments between the inner disk and outer ring, and of compatibility of displacement and rotation of the boundary between them, it may be shown [15] that: Q ¼ S 2hk1K2 þ2K2 k1 þk2 ð Þ W h ð Þ h2k1k2 þ4 k1 þk2 ð Þ K1 þK2 ð Þ ð12:4Þ C ¼ S 4k2 K1 þK2 ð Þþhk1k2W h2k1k2 þ4 k1 þk2 ð Þ K1 þK2 ð Þ ð12:5Þ where the in-plane stiffnesses k1 and k2 and the bending stiffnesses K1 andK2 are for the axisymmetric “A” case: k1 ¼ F Vr að Þ ¼ W h ð Þ E 1 ν k2 ¼ C Vr að Þ ¼ WE 1þν ð 12:6Þ K1 ¼ M θ að Þ ¼ W h ð Þ3 E 12 1 ν ð Þ K2 ¼ Q θ að Þ ¼ W3 E 12 1þν ð Þ ð 12:7Þ where the subscripts 1 and 2 respectively refer to the inner disk and outer ring. The displacements Ur(r) at radius r on the surface of the outer ring combine contributions from the centerline radial displacement Vr(r) and the centerline rotationθ(r). Following the method of Vinson [16] the combination of the two contributions is: Ur rð Þ ¼ Ca EWr 1þν ð Þ þ W 2 Qa212 1þν ð Þ EW2r ð 12:8Þ Similarly, for the anticlastic “B” case: k1 ¼ F Ur að Þ ¼ W h ð Þ E 1þν k2 ¼ C Ur að Þ ¼ WE 3 ν ð 12:9Þ Fig. 12.5 Analytical model of a hole in a finite thickness specimen 12 Residual Stress Measurements in Finite-Thickness Materials by Hole-Drilling 93
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