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Plate-Bending Theory
Published in Ansel C. Ugural, Plates and Shells, 2017
An examination of Equations 3.3 shows that a circle of curvature can be constructed similarly to Mohr's circle of strain. The curvatures therefore transform in the same manner as the strains. Figure 3.3 shows a plate element and a circle of curvature in which n and t represent perpendicular directions at a point on the midsurface. The principal or maximum and minimum curvatures are indicated by κ1 and κ2. The planes associated with these curvatures are called the principal planes of curvature. The curvature and the twist of a surface vary with the angle θ, measured in the clockwise direction from the set of axes xy to the x′y′ set. The product of the principal curvatures defines a so-called Gaussian curvature.
Problems with Explicit Solutions
Published in Vasilios Alexiades, Alan D. Solomon, Mathematical Modeling of Melting and Freezing Processes, 2018
Vasilios Alexiades, Alan D. Solomon
where Γ > 0 depends on the ratio of surface-tension coefficient to latent heat, see §2.4.F, also [PORTER-EASTERLING], [KURZ-FISHER], [TRIVEDI]. The mean curvature of a surface is the average of its two principal curvatures: 12κ=12(1R1+1R2), with R1, R2 the principal radii of curvature (in our notation κ denotes the sum of the principal curvatures). For a planar (flat) interface, we have κ = 0, so Tf = Tm, and the supercooled liquid must be warmed up to Tm in order to freeze. On the other hand, at a protrusion of curvature κ > 0, the freezing point is lowered by Γκ, so supercooled liquid of temperature Tm−Γκ will freeze on it making it grow. In fact, the more pointed the tip the more it is favored for growth! This is the mechanism by which columnar and dendritic growth takes place, creating a fascinating variety of morphologies, (Figure 1.1.2. Note that the Gibbs-Thomson effect is strictly multidimensional since in one-dimension the interface is just a point, having no curvature, so the effect cannot be felt there.
Basic Ray Optics
Published in Daniel Malacara-Hernández, Brian J. Thompson, Fundamentals and Basic Optical Instruments, 2017
Orestes Stavroudis, Maximino Avendaño-Alejo
We think the most useful definition is that the caustic is the locus of principal centers of curvature of a wavefront. In general, every surface has two principal curvatures at each of its points. This definition then shows clearly that the caustic is a two-sheeted surface. On the other hand, a system of rays originating from a common object is called a normal congruence.
Modelling the role of membrane mechanics in cell adhesion on titanium oxide nanotubes
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Matej Daniel, Kristina Eleršič Filipič, Eva Filová, Tobias Judl, Jaroslav Fojt
The formation of a protrusion requires deformation of the membrane from the mostly planar shape into the shape of a thin cylinder. The bending energy of the membrane is commonly described by Helfrich energy (Helfrich 1973). The elastic strain energy proposed by Helfrich depends on the mean (H) and Gaussian curvature. As we do not expect the change in cell topology by protrusion formation, the Gauss term could be neglected because of the Gauss-Bonnet theorem (Bassereau et al. 2018). where kb is the bending modulus of the cell membrane and C0 is the spontaneous curvature. Spontaneous curvature, or more precisely the spontaneous mean curvature, presents a penalty for the mean curvature asymmetry (Chabanon and Rangamani 2018). The mean curvature can be expressed as an average of principal curvature values C1 and C2 defined as the inverse values of corresponding radii of curvatures R1 and R2, respectively (A).
Local contact characteristics of threaded surfaces in a planetary roller screw mechanism
Published in Mechanics Based Design of Structures and Machines, 2020
Shangjun Ma, Linping Wu, Geng Liu, Xiaojun Fu
According to the differential geometry theory (Ciarlet and Li 2007), the principal curvature of a surface at a point is the normal curvature of the surface along the principal direction at that point. Therefore, the principal curvature at a point on the surface is represented as follows: In the expression, when the surface bends at the contact point along the forward direction of is set to “” Otherwise, is set to “” As shown in Figs. 1 and 2, for surface the principal curvature is set to “” For surfaces and the principal curvature is set to “”
Classification and selection of sheet forming processes with machine learning
Published in International Journal of Computer Integrated Manufacturing, 2018
Elia Hamouche, Evripides G. Loukaides
The principal curvatures for a given surface describe how the surface bends by different amounts in different directions at each point (Guggenheimer 1977). By calculating the principal curvatures at every point of a 3D surface, a 2D representation of the geometry can be generated in order to describe fully the curvature to the classifier. For a given 3D differentiable surface in Euclidean space, a unit normal vector can be drawn at each point on the surface. The normal plane will contain a normal vector and a tangent vector to the surface which cuts the surface in a plane curve. This curve will have different curvatures for different normal planes at each point. The principal curvatures for each point are denoted as and and refer to the maximum and minimum values of this curvature.