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Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
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A spherical capacitor consists of two concentric conducting spherical shells of radii R1 (inner shell) and R2 (outer shell). The shells have  equal and opposite charges of +Q and −Q, respectively. For an isolated conducting spherical capacitor, the radius of the outer shell can be considered to be infinite.
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Newton's law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, the Moon, and other planets, the law holds with an assumption that masses of spherical objects are concentrated at their respective centers.
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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a...
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Updated: Jan 20, 2026

Spherical Coordinates
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Collapse of Orthotropic Spherical Shells.

Gautam Munglani1,2, Falk K Wittel1, Roman Vetter1

  • 1Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zürich, Stefano-Franscini-Platz 3, CH-8093 Zürich, Switzerland.

Physical Review Letters
|September 7, 2019
PubMed
Summary
This summary is machine-generated.

Orthotropic elastic spherical shells exhibit complex buckling behaviors and collapse patterns, differing significantly from isotropic shells. Material orthotropy dictates shell shape and energy evolution, revealing natural structures. Keywords: orthotropic shells, buckling, collapse, material orthotropy.

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Area of Science:

  • Solid Mechanics
  • Materials Science
  • Structural Engineering

Background:

  • Isotropic spherical shells have well-understood buckling behaviors.
  • The influence of material orthotropy on shell buckling is less explored.
  • Understanding shell collapse is crucial for engineering and biological applications.

Purpose of the Study:

  • To investigate the buckling and collapse of orthotropic elastic spherical shells.
  • To explore the morphological phase space influenced by shell slenderness and orthotropy.
  • To compare numerical findings with experimental results.

Main Methods:

  • Extensive numerical simulations of orthotropic elastic spherical shells.
  • Experimental validation using fabricated polymer shells.
  • Analysis of buckling pathways and strain energy evolution.

Main Results:

  • A rich morphological phase space with three distinct regimes was identified.
  • Buckling pathways and strain energy evolution strongly depend on material orthotropy.
  • Numerical simulations agreed well with experimental observations.

Conclusions:

  • Material orthotropy significantly alters the buckling and collapse behavior of spherical shells.
  • Robust orthotropic structures, resembling natural forms like stomatocytes and pollen grains, were discovered.
  • The study suggests material orthotropy as a key factor in understanding the shape of natural collapsed shells.