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A mixed mode local symmetric fracture criterion for geo-materials
Published in W.A. Hustrulid, G.A. Johnson, Rock Mechanics Contributions and Challenges: Proceedings of the 31st U.S. Symposium, 2020
W.W. El-Tahan, G.H. Staab, S.H. Advani, J.K. Lee
One can say that in K1 - K2 space, KICC=σCπa should be the limit for K1 compression. Therefore, we have a straight line and a point (K1 = KICC) on the envelope. The simplest convex curve passing through this point and intersecting the line appears to be a circular sector. We choose this circle to be tangential to the linear envelope and preserve continuity of the function and its derivative, as depicted in Figure 4. The equation of the linear segment is K2/KIC+μ(K1/KIC)=K2C/KIC
Integral Calculus
Published in Richard C. Dorf, Ronald J. Tallarida, Pocket Book of Electrical Engineering Formulas, 2018
Richard C. Dorf, Ronald J. Tallarida
For the circular sector of angle 2α and radius R, the area A is αR2; the integral needed for x′, expressed in polar coordinates is ∬xdA=∫−aα∫0R(rcosθ)rdrdθ[R33sinθ]−α+α=23R3sinα
Shape of epithelia
Published in A. Šiber, P. Ziherl, Cellular Patterns, 2018
4.11 Imagine a disk of tissue where only cells in a given narrow sector divide, producing new cells in the circumferential direction (panel c in Figure 1.4). The new cells induce stress in the tissue if the disk is restricted to a plane, and the elastic energy can be decreased considerably if the disk is allowed to buckle out of the plane. This growth mode can be represented as an insertion of a circular sector (new tissue) into the disk (existing tissue). A similar shape may be expected under more general circumstances where growth is strongly anisotropic and more pronounced in the circumferential direction than in the radial direction. The physics sketched may be associated with the shape of sympetalous flowers—their petals are fused into a single sheet of plant tissue (these include the well‐known Brugmansia and sweet potato flowers, but also hedge binweed). Discuss the general features of the shape of the buckled disk and estimate its bending energy.
The intersection of two petals: a computer-assisted extension of another old geometric problem
Published in International Journal of Mathematical Education in Science and Technology, 2022
An elementary solution to this problem is explained in, e.g. Simonson (2011, Section 12.3). For a summary, see Figure 4 (left). First, the area of the horizontally-shaded region is the area of the -circular-sector OYZ subtracted by that of the equilateral triangle OYZ: . This, subtracted from the area of the -circular-sector ZYX, is the area of the vertically-shaded region: . Finally, notice that the required area is the area of the square subtracted by four times the area of the vertically-shaded region [cf. Figure 4 (right)]: .
A hybrid dislocation
Published in Philosophical Magazine, 2021
To solve it exactly, it is possible to start from two old works [10,11] that involve a heterogeneous elastic bicrystal formed of two semi-infinite crystals connected by a flat interface. In these two references, the two bodies are ‘glued together’, i. e., traction and displacement both are continuous across the interface. One of these works leads to an analytical formulation of the elastic field produced by a point force applied in one of the crystals, while the other gives the field produced by a dislocation loop in the bicrystal if the Green tensor of each crystal is known, a question which has not yet received a complete answer to the knowledge of the authors. The guiding idea is to calculate the elastic field of a HD as the limit of a flat dislocation loop whose perimeter tends towards infinity by retaining a straight part at the interface. To be more precise, we consider a bicrystal deformed by a closed dislocation loop whose perimeter (Figure 1) is that of a circular sector OHK with a very large interfacial radius OH = OK. In other words, the further away the H point is from origin O, the better the evaluation of the elastic field at a point close to O. The same remark is used in Hirth and Lothe [12] to calculate the local field of a straight dislocation from a large closed dislocation loop.
Novel, compact, circular-sectored antenna for Ultra-Wideband (UWB) communications
Published in Electromagnetics, 2020
Arnab De, Bappadittya Roy, Anup Kumar Bhattacharjee
The Geometry of the proposed antenna configuration is demonstrated in Figure 1. To design the proposed one at first, a rectangular shape reference antenna is considered and denoted as Antenna 1 having a dimension of 40 × 40 mm2 as shown in Figure 1(a). In the second step, the patch structure is changed and made as a circular sector shape of radius (a) shown in Figure 1(b), where the ground plane structure is same as Antenna 1. In the next step, the ground structure is modified and another circular sector part of radius (b) is introduced in the ground plane which is shown in Figure 1(c). Now the structure consists of two circular sectored parts, in which one element on the top is the patch, shown in Figure 1(b) and the other sectored part is the ground plane on the back surface, shown in Figure 1(c). In the last step, the final geometry is achieved by embedding two symmetrical circular slots in the patch of radius (r1 = r2) 5.0 mm. The distance (d) is the center-to-center distance of the two circles. The radius of the circles plays an important part in enlarged bandwidth and compactness. A Rogers RT/Duroid 5880 substrate having thickness of 0.787 mm, dielectric constant (Ɛr) of 2.2 and loss tangent (tan δ) = 0.002 is used for the structure. The overall dimension of the antenna as well as width and length of the substrate are 40 mm and 40 mm, respectively. The designed prototype is fed by a co-axial probe of 50 Ω impedance with the help of an SMA connector of radius 0.5 mm. The feed point is optimized to match the input impedance over the entire UWB range. Table 1 shows all the dimensions of the parameters starting from Antenna 1–4.