Absolute curvatures in integral geometry

1980 ◽  
Vol 88 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Adrian J. Baddeley
Keyword(s):  
Cross Section ◽  
Integral Geometry ◽  
Convex Bodies ◽  
Practical Applications ◽  
Geometrical Probability ◽  
Lower Dimensional ◽  
Canonical Status ◽  
Surface Integrals

Surface integrals of curvature arise naturally in integral geometry and geometrical probability, most often in connection with the Quermassintegrale or cross-section integrals of convex bodies. They enjoy many desirable properties, such as the ability to be determined by summing or averaging over lower-dimensional sections or projections. In fact the Quermassintegrale are the only functionals of convex bodies to meet certain, quite reasonable, requirements. The conclusion has often been drawn, especially in practical applications, that the Quermassintegrale and their associated curvature integrals have a canonical status to the exclusion of all other quantities.

scholarly journals Inequalities for the Surface Area of Projections of Convex Bodies

2018 ◽  
Vol 70 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Alexander Koldobsky ◽  
Petros Valettas
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Maximal Surface ◽  
Projection Body ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

A third note on recent research in geometrical probability

1974 ◽  
Vol 6 (01) ◽  
pp. 103-130 ◽  
Author(s):  
D. V. Little
Keyword(s):  
Pattern Recognition ◽  
Integral Geometry ◽  
Random Search ◽  
Euclidean Spaces ◽  
Geometrical Probability

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, flats, and networks in Euclidean spaces, pattern recognition, random coverage and packing, random search, stereology, and probabilistic aspects of integral geometry.

Flexoelectric effect induced in an anisotropic bar with cubic symmetry under torsion

2019 ◽  
Vol 25 (3) ◽  
pp. 820-837
Author(s):  
AR El Dhaba ◽  
ME Gabr
Keyword(s):  
Cross Section ◽  
Gradient Elasticity ◽  
Strain Gradient Elasticity ◽  
Cubic Symmetry ◽  
Rectangular Shape ◽  
Practical Applications ◽  
Flexoelectric Effect ◽  
The Cross ◽  
Gradient Elasticity Theory ◽  
Physical Quantities

In this article, we study the flexoelectricity induced in a prismatic anisotropic bar due to torsion. The simplified strain gradient elasticity theory is considered in this study. The bar is uniform, that is, any cross-section of the bar has a rectangular shape with cubic internal structure symmetry. The traction and higher traction forces effect on the deflection and spontaneous polarization of the bar with different boundary conditions are also discussed. The induced wedge forces are also considered during this study. The magnesium oxide (MgO) physical quantities values are chosen to present a numerical example as one of the practical applications of the problem. The results are discussed and introduced graphically. The most interesting finding in this study is the wedge force directions. When the displacements inside the cross-section of the bar are uniformly distributed, the resultant wedge forces have the same inclination with the cross-section boundary. Meanwhile, if the displacement is not uniformly distributed, the wedge force inclinations with the cross-section boundary are not equal.

scholarly journals Estimates for measures of lower dimensional sections of convex bodies

2017 ◽  
Vol 306 ◽  
pp. 880-904 ◽  
Author(s):  
Giorgos Chasapis ◽  
Apostolos Giannopoulos ◽  
Dimitris-Marios Liakopoulos
Keyword(s):  
Convex Bodies ◽  
Lower Dimensional

scholarly journals Centrally symmetric convex bodies and sections having maximal quermassintegrals

2012 ◽  
Vol 49 (2) ◽  
pp. 189-199
Author(s):  
E. Makai ◽  
H. Martini
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
The Body ◽  
Symmetric Convex ◽  
Local Theorem ◽  
Centrally Symmetric ◽  
Euclidean Unit Ball ◽  
Analogous Theorem ◽  
Lower Dimensional ◽  
Dimensional Surface

Let d ≧ 2, and let K ⊂ ℝd be a convex body containing the origin 0 in its interior. In a previous paper we have proved the following. The body K is 0-symmetric if and only if the following holds. For each ω ∈ Sd−1, we have that the (d − 1)-volume of the intersection of K and an arbitrary hyperplane, with normal ω, attains its maximum if the hyperplane contains 0. An analogous theorem, for 1-dimensional sections and 1-volumes, has been proved long ago by Hammer (see [2]). In this paper we deal with the ((d − 2)-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small C2-perturbations, or C3-perturbations of the Euclidean unit ball, respectively.

Convex bodies with non‐convex cross‐section bodies

Mathematika ◽  
1999 ◽  
Vol 46 (1) ◽  
pp. 127-129 ◽  
Author(s):  
Ulrich Brehm
Keyword(s):  
Cross Section ◽  
Convex Bodies

Out-of-the-Lifting-Plane Stability Analysis of Crane Telescopic Boom with Cable Exerted at the Top

2013 ◽  
Vol 446-447 ◽  
pp. 474-478
Author(s):  
Nian Li Lu ◽  
Liang Du ◽  
Shi Ming Liu ◽  
Yuan Xue
Keyword(s):  
Cross Section ◽  
Carrying Capacity ◽  
Stability Problem ◽  
Recurrence Formula ◽  
Variable Cross Section ◽  
Critical Force ◽  
Variable Cross ◽  
Usual Practice ◽  
Characteristic Equations ◽  
Practical Applications

To enhance the carrying capacity of the crane variable cross-section telescopic boom, the usual practice is using the cable at its top end, it makes the out-of-the-lifting plane stability problem of crane telescopic boom become solving the Euler critical force with follower force. This paper established the deflection differential equations of crane telescopic boom model which under actions of cable, with proper boundary conditions, the recurrence formula of buckling characteristic equations were presented, and some practical applications were given. The influence on buckling critical force of crane boom due to the ratio of the length of cable and crane boom was discussed. Took certain four-sectioned telescopic boom as example, the destabilizing critical force was calculated, the result showed that in comparison with the ANSYS method, the buckling characteristic equations in this paper is completely correct.

An Inverse Design Method of Buckling-Guided Assembly for Ribbon-Type 3D Structures

2019 ◽  
Vol 87 (3) ◽  
Author(s):  
Zheng Xu ◽  
Zhichao Fan ◽  
Yanyang Zi ◽  
Yihui Zhang ◽  
Yonggang Huang
Keyword(s):  
Cross Section ◽  
3D Structure ◽  
Experimental Studies ◽  
Functional Materials ◽  
Optimization Method ◽  
Inverse Design ◽  
Torsional Angle ◽  
3D Structures ◽  
Practical Applications ◽  
The Cross

Abstract Mechanically guided three-dimensional (3D) assembly based on the controlled buckling of pre-designed 2D thin-film precursors provides deterministic routes to complex 3D mesostructures in diverse functional materials, with access to a broad range of material types and length scales. Existing mechanics studies on this topic mainly focus on the forward problem that aims at predicting the configurations of assembled 3D structures, especially ribbon-shaped structures, given the configuration of initial 2D precursor and loading magnitude. The inverse design problem that maps the target 3D structure onto an unknown 2D precursor in the context of a prescribed loading method is essential for practical applications, but remains a challenge. This paper proposes a systematic optimization method to solve the inverse design of ribbon-type 3D geometries assembled through the buckling-guided approach. In addition to the torsional angle of the cross section, this method introduces the non-uniform width distribution of the initial ribbon structure and the loading mode as additional design variables, which can significantly enhance the optimization accuracy for reproducing the desired 3D centroid line of the target ribbon. Extension of this method allows the inverse design of entire 3D ribbon configurations with specific geometries, taking into account both the centroid line and the torsion for the cross section. Computational and experimental studies over a variety of elaborate examples, encompassing both the single-ribbon and ribbon-framework structures, demonstrate the effectiveness and applicability of the developed method.

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