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radius r/o
Central thick. to :
Young mod. Uniform load Flexural sag Max. stress
E = 2.05 106 daN.cm2 q = 5.280 daN.cm2 2o = — 0.300 mm ar = ± 50.02 daN.mm2
radius r/o
Central thick. to :
Young mod. Uniform load Flexural sag Max. stress
E = 2.05 106 daN.cm2 q = 5.280 daN.cm2 2o = — 0.300 mm ar = ± 50.02 daN.mm2
T20{pi = 1} from one of those equations and inject a value (dU/ dp) p1 = e0 unknown, where pi is small. Equations (2.32b) and (2.32a) provide dT20/dp and d2U/dp2, respectively, the latter gives Ui+2. Thus, all elements are known to increment for the next step p2 = p1 + Sp with a very small Sp. We can continue the integration in the radial direction by successive increases pi+1 = pi + Sp. Thus, by changing the starting values of the radial elongation e0, the numerical process is repeated up to satisfy at edge T20{1} = 0.
The maximum radial stresses orr on each surface of the substrate is the sum of two components
Figures 2.4 and 2.5 display the reduced thickness T20 = t/to, the radial deformation u and the maximum stress orr resulting from the integration for a cycloidlike form (VTD Type 1) and a tuliplike form (VTD Type 2). The air pressure load generates convex flexures all over the zoom range. The basic sag used for the integration is
Fig. 2.5 Type 2: Axial force at center and edge reaction Tuliplike VCM. Results of integrations from large deformation theory. Zoom: [f/oof/2.5]. The substrate material is Fe87 Cr13 stainless steel alloy in a quenched state (after Ferrari [18])
Fig. 2.5 Type 2: Axial force at center and edge reaction Tuliplike VCM. Results of integrations from large deformation theory. Zoom: [f/oof/2.5]. The substrate material is Fe87 Cr13 stainless steel alloy in a quenched state (after Ferrari [18])
Poisson's ratio v =
Diameter 2 a =
Central thick. to =
Young mod. Uniform load Flexural sag Max. stress
E = 2.05 106 daN.cm2 F = 9.700 daN z0 =  0.400 mm ur = ± 116.4 daN.mm"'
Poisson's ratio v =
Diameter 2 a =
Central thick. to =
Young mod. Uniform load Flexural sag Max. stress
E = 2.05 106 daN.cm2 F = 9.700 daN z0 =  0.400 mm ur = ± 116.4 daN.mm"'
and negative in both cases (R < 0, q > 0 and F > 0). In the integrations for VTDs Type 1 and 2, the thickness distributions are determined for the basic fratio value f/3.33 considered as the mean value of the zoom range, thus determining z0. At the limit of the zoom range, the maximum deformationratio in Type 1 reaches z0/t0 =  1.33 at f/2.5, i.e. a flexure sag larger than the thickness.
The forces Nr and Nt are both positive at the central zone of the substrate. At the outer part, the force Nr decreases to zero at the perimeter but the force Nt becomes negative. For much larger deformations, this could entail an elastic instability traduced by multiple wavelets along the edge similarly as analytically developed by Casal [7] in his theory of membranes.
The relations between the load q  or central force F  and the deformationratio z/t0 over the zoom ranges [ f/^ f/2.5 ] have been determined by Ferrari [20]. These curves show the important nonlinearity as well for a VTD Type 1 as for a VTD Type 2 (Fig. 2.6).
From these results, the load q or F can be represented in an odd power series of the deformation ratio z/t0 by
Axial Force  
• o2 F 

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