Double curvature bending

Figure 5.23(b) Double curvature bending (typical) F bolt =0.5F k +Q... 

Processing any plastics, reinforced or unreinforced, into curved panels is relatively easy and inexpensive. Panels fit the structural theory that curved shaped can be stiffer to bend than flat shapes of the same weight. However, to withstand external pressure, a square section component will usually be heavier than one that is circular and of the same volume. Both single- and double-curvature designs are widely used to ensure a more effective use of RP materials. 

Textile materials have the special property of achieving complex buckling with smoothly rounded folds of double curvature. In contrast, paper, which bends easily in single curvature, resists double curvature, and, if forced, buckles into sharp point or line discontinuities. 

Recent advances in the mechanics of complex buckling [5] come from a recognition that the formation of spherical surfaces of double curvature, which is the converse of drawing a map of the world on a planar surface, demands inplane (membrane) strain as well as bending. It has also been recognised that the simplest problem in this class is the buckling of a circular specimen pushed inwards at three equally spaced points. The material deforms into a dome of... 

Practically drape determines apparel design and influences fabric quality for specific end uses. When a fabric is draped it can bend in one or more directions. Usually, curtains and drapes bends in one direction, whereas garments and upholstery exhibit a complex three-dimensional form with double curvature. Hence, fabric drape is a complex mathematical problem involving large deformation under low stresses. [4]... 

Nevertheless, the avena coleoptile exhibits a curvature to unilateral UV-illumina-tion with a satisfactory log-linear response/time relationship38) (the bending mode is similar to that observed for the second positive curvature which develops from the coleoptile base cf. 2.2). Fig. 5 338) shows that the double-peaked action spectrum does not match neither flavin (Fig. 5 5,16S)) nor carotenoid absorption (Fig. 5 4,183)), most likely excluding both as photoreceptors. The growth hormone auxin (cf. 2.4 and Scheme 1) has been discussed to be a possible photoreceptor. However, in this case, this is not supported by the action spectrum either. 

Bozic and Svetina [36] analysed a different situation, where addition of membrane constituents happens from the external milieu, and there is no metabolism inside, but there is limited permeability. They supposed that the membrane assumes spontaneous membrane curvature. This is non-zero if the properties of the inside and outside solutions differ, or if the two layers of a bilayer membrane differ in composition, or if some membrane-embedded constituents are asymmetrically shaped. They were able to show that under these assumptions membrane division is possible provided TLkC4 > 1.85, where T is the time taken to double the membrane area, L is the hydraulic permeability of the membrane, k is the bending modulus, and C is the spontaneous membrane curvature. In this model growing vesicles first retain spherical shape, then are distorted to a dumbbell, then to a pair of asymmetric vesicles coupled by a narrow neck, and finally to a pair of spherical vesicles linked by a narrow neck. Separation of the two daughter vesicles occurs as a result of mechanical agitation in the solution. 

Not only do surfactants and cosurfactants lower the interfacial tension, but also their molecular structures affect the curvature of the interface as shown schematically in Fig. 3. The hydrocarbon chains are rather closely packed (about 0.25 nm per chain) they repel one another sideways and as a result have a tendency to bend the interface around the water side. The counterions of the ionic headgroups also repel one another sideways and thus tend to curve the interface around the oil side. The bullQ polar groups of nonionic surfactants have a similar effect. So we understand qualitatively that more cosurfactant promotes W/O rather than O/W microemulsions. More electrolyte compresses the double layer, diminishes the sideways pressure of the double layer, and also promotes W/O microemulsions. The polar groups of PEO nonionics become more compact (less soluble) at higher temperatures, and so with this type of surfactants high temperature leads to W/O microemulsions. 

In this paper, we investigate the domain morphology in the phase separation of copolymer-homopolymer mixtures. Computer simulations are carried out in two dimensions to explore the kinetics of the double-phase separation. An interfadal approach is applied to study the relative stability of the domain patterns and to evaluate the bending and the curvature moduli of a bilayer membrane. 

Fig. 18. Average values of the steady-state radii of curvature for symmetrical double-cantilever beams (1.0. 2.0, 3.0 mm thick) separated by wedges of different sizes being in.sertcd into the interface. A comparison is made between the experimental observations and the predictions of the cohesive-zone model using values of Ti, = 1.4 kJm and d = 100 MPa. The dotted line represents the analytical predictions based on beam-bending theory [58].

---