Curvature measurement

Bates DM, Watts DG. Relative curvature measures of nonlinearity. / R Stat Soc B 1980 42 1-25. 

Figure 9.4 The examples of curved surfaces schematic diagrams showing the principle of surface curvature measurement for sphere (a) and cylinder (b) (c) is the diametric section of meniscus around a zone of contact of two particles (d)-(f) three basic types of surfaces (d) is ellipsoidal, (e) hyperbolic (f) parabolic (g —(g3) are the examples of the surfaces of constant mean surface curvature (minimal surface).

One can say that the mirror folds the length axis at the mirror, so that emergent rays to a real image at the left represent a positive value of s. We are forced also to declare that the mirror also flips the sign of the surface radius. For reflective surfaces, the radius of curvature is defined to be the directed distance from a surface to its center of curvature, measured with respect to the axis used for the emergent light. With this qualification the convention for the signs of s and R is the same for mirrors as for refractive surfaces. 

To predict the residual stress distributions in sprayed FGM, in-situ curvature measurement is necessary because the values of quenching stress are difficult to... 

Now, if we vary the direction of the plane through the NN" axis, this curvature will of course change, and as K varies, it achieves a minimum and a maximum (which are perpendicular to each other) known as the principal curvatures, K, and k2, and the corresponding directions are called principal directions. The principal curvatures measure the maximum and minimum bending of a surface at each point. However, it has been shown mathematically that the sum of the curvatures (k, + K, = 1 /Rt + /R2) is independent of how the first plane is oriented when the second plane is always at right angles to it, and consequently the direction of the planes is a matter of choice. 

Stress Determination by Curvature Measurement (Almen-Type Test)... 

To circumvent the drawbacks of the sin2 P-technique including the problem of a non-linearity of the modulus of elasticity, attempts have been made to apply other tests to obtain estimates of the residual stresses. The curvature measurement is probably one of the most widely used method for determining residual stress and involves measuring the bending of the coated sample in response to both quenching and thermal stresses. From the measured radius of curvature, the stress can be calculated according to the Stoney equation (Stoney, 1909) as... 

Several laboratories are experimenting with pulsed magnetic fields of sufficient strength (10 —10 gauss) to provide useful track curvature measurements in emulsions. 

Fig. 4. Measuring matrix a) scheme, b) general view 1) open gauges for circumference force measuring, 2) closed gauges for curvature measuring, 3) additional curvature gauges...

Fig. 2. Correlation between rate of curvature triangles) and magnitude of downward auxin transport polarity across the root cap circles). Rate of curvature was estimated from plots of curvature (measured from projected negatives) vs time...

The curvature of the substrate can be determined by a variety of techniques. For all of these techniques, to increase the sensitivity of the curvature measurement, a thin substrate, or the order of 100 [tm thick, may be used, keeping in mind that... 

Chronis et al. reported a similar lens array with a thinner PDMS membrane of 40 pm and smaller lens diameter of 200 pm [4]. They also used finite element analysis software (ANSYS) to simulate the mechanical deformation of the membrane under uniform pressure. Figure 6.4a shows a cross sectional view of a single deformed membrane simulated by ANSYS. An important finding was the difference between the maximum deflection of the outer (top) surface of the membrane and the inner (bottom) one. The inner surface of the membrane deflects more than the outer one, resulting in a smaller radius of curvature. Figure 6.4b illustrates the outer and inner radius of curvature measured and calculated from the interferometric lens profile and ANSYS simulations, respectively [4]. 

The change in substrate curvature induced during film deposition or temperature excursion provides valuable insight into the evolution of mismatch stress in the thin film. As noted earlier, a particularly appealing feature of curvature measurement is that extraction of the membrane force / from substrate curvature by recourse to the Stoney formula (2.7) does not involve the material properties of the film, provided that the film is sufficiently thin compared to the substrate. Substrate curvature measurements also provide a means to assess the functional properties of thin films in photonic and microelectronic applications. For example, strain in films can modify the electronic transport characteristics of layered semiconductor systems through modification of the bandstructure of the material (Singh 1993). 

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