Elastic bending

K. S. Pister and S. B. Dong, Elastic Bending of Layered Plates, Journal of the Engineering Mechanics Division, ASCE, October 1959, pp. 1-10. 

Flexural modulus is the force required to deform a material in the elastic bending region. It is essentially a way to characterize stiffness. Urethane elastomers and rigid foams are usually tested in flexural mode via three-point bending and tite flexural (or flex ) modulus is obtained from the initial, linear portion of the resultant stress-strain curve. 

The conformational analysis of PIB is performed, with explicit allowance for elastic bending of the chain C-C-C bond angles. 

Dobbie, I., Linari, M., Piazzesi, G., Reconditi, M., Koubassova, N., Ferenczi, M. A., Lombardi, V., and Irving, M. (1998). Elastic bending and active tilting of myosin heads during muscle contraction. Nature 396, 383-387. 

The elastic bending modulus Kc for lipid bilayers was found to be of the order of 10 x 10-20 J.6 For the lyotropic lamellar liquid crystals, the additional presence of a cosurfactant at the interface shouldleadto smaller values. Various experimental measurements21 provided values ranging between 0.08 and 5 x 10-20 J. 

In what follows, the two fluctuating interfaces will be replaced by many small, independent surfaces of area S, separated by a distance z (see Figure 4). The (metastable) distribution of the distances between the surfaces, in an ensemble subjected to a constant pressure, will be assumed Boltzmannian. It will be also assumed that the fluctuating interfaces have constant total areas and an elastic bending modulus Kq. Let us first consider that the interfaces interact per unit area through a harmonic potential... 

Figure 5.10d illustrates a case of contact mode. The contact position (z) is on the repulsive side of the short-range force curve. The arrow pointed to the left on Figure 5.10d represents the phenomenon of tip jump-to-contact with the sample caused by cantilever elastic bending when... 

The second step in an iteration, while simulating the evolution of the ultrathin trilayer solid-fluid-solid films, is to calculate the pressures acting on the fluid. The van der Waals pressure is obtained locally from the height of the film using Eq. (4). The elastic bending pressure is also obtained from the height of the film, but this time a finite-difference approximation is required to obtain the derivatives in discrete form. In particular,... 

Consequently, the elastic bending stresses reach a maximum value at the inner surface of... 

Sueh experiments have been useful in determining the interfaee fracture energies of laminates. " Often it is more convenient is such tests to apply the forees in elastic bending, as shown in Fig. 14.19(a). 

Figure 14.19. (a) Elastic bending causing detachment of fitm. (b) Plastic bending. 

Ez,b,k=modulus of elasticity, bending about minor axis Gxy=shear modulus of flange material... 

For = 0> the free energy density in the bulk includes only the flexoelectric and the elastic (bend) terms ... 

Our tentative interpretation is that the thin PTFE layers are patchy. Then, for liquid crystal layers thinner than the crossover thickness, the molecules on top of the patches will be perpendicular to the surface, while those between patches will be parallel. (We disregard here the case of very thin layers discussed above, when all molecules are parallel.) With increasing liquid crystal thickness, however, the elastic bend energy in the transition between the two orientations becomes gradually more important, and at some transition thickness a single orientation, parallel to the surface, may become energetically favored. Conceivably, a compromise inclined orientation could also be the results. 

Fig. 4.5 SiCNW morphologies of the elastic bend (a, c) and the final plastic bend of (b) and (d). The location indicated by the white arrow in (c) is the triggering point of plastic defcmnation. e is an enlarged TEM toight field image. The black arrows in (e) show the top and bottom deformation-induced amcnphous zones. Re ninted with permission from Han et al. [5]. With kind permission of the American Chemical Society...

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