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Theorems of Pappus and Guldinus: Problem Solving01:12

Theorems of Pappus and Guldinus: Problem Solving

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Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
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Line, Surface, and Volume Integrals01:15

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A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a prescribed path. If the prescribed path is closed, the integrals reduce to a closed-line integral. The closed-contour integral of the vector field is referred to in terms of the circulation of the vector field around the closed path. A vector with zero circulation around every closed path is called a conservative field, while one with non-zero...
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Electric Field of a Non Uniformly Charged Sphere01:22

Electric Field of a Non Uniformly Charged Sphere

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Gauss's law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for determining the expressions for the electric field for a particular charge distribution if the electric flux is known.
Consider a non-uniformly charged sphere, for which the density of charge depends only on the distance from a point in space and not on the direction. Such a sphere has a spherically symmetrical charge distribution. Here, the electric...
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Theorems of Pappus and Guldinus01:10

Theorems of Pappus and Guldinus

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The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
For finding the surface area, consider a differential line element that generates a ring with surface area dA when revolved.
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Gravitational Potential Energy for Extended Objects01:07

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Consider a system comprising several point masses. The coordinates of the center of mass for this system can be expressed as the summation of the product of each mass and its position vector divided by the total mass:
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Gravitation Between Spherically Symmetric Masses01:14

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The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, assuming that the center of mass is concentrated at the respective centers of the bodies.
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Related Experiment Video

Updated: Oct 12, 2025

A 3D Spheroid Model as a More Physiological System for Cancer-Associated Fibroblasts Differentiation and Invasion In Vitro Studies
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Volume Integral Equation Method Solution for Spheroidal Inclusion Problem.

Jungki Lee1, Mingu Han1

  • 1Department of Mechanical and Design Engineering, Hongik University, Sejong City 30016, Korea.

Materials (Basel, Switzerland)
|November 27, 2021
PubMed
Summary
This summary is machine-generated.

The volume integral equation method (VIEM) provides accurate numerical analysis for solids with inclusions. This study establishes VIEM results as reference values for elastostatic inclusion problems under various loadings.

Keywords:
boundary element method (BEM)isotropic/anisotropic inclusion problemsstandard finite element method (FEM)volume integral equation method (VIEM)

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Area of Science:

  • Solid mechanics
  • Computational mechanics
  • Materials science

Background:

  • Analyzing inclusions in solid materials is crucial for predicting material behavior.
  • Traditional methods face challenges with complex geometries and anisotropic properties.
  • Numerical methods are essential for solving intricate elastostatic problems.

Purpose of the Study:

  • Introduce the volume integral equation method (VIEM) for analyzing inclusions in infinite solids.
  • Demonstrate VIEM's versatility for three-dimensional elastostatic inclusion problems.
  • Provide benchmark results for validating other analytical and numerical techniques.

Main Methods:

  • The study employs the volume integral equation method (VIEM).
  • Numerical simulations are performed for infinite isotropic solids with single spheroidal inclusions.
  • Analysis covers both isotropic and anisotropic inclusions (spherical, prolate, oblate).

Main Results:

  • VIEM results are presented for inclusions under uniform remote tensile loading.
  • VIEM results are also presented for inclusions under remote shear loading.
  • The study validates VIEM for various inclusion types and loading conditions.

Conclusions:

  • The volume integral equation method (VIEM) is a versatile tool for elastostatic inclusion analysis.
  • The generated results serve as reliable reference values for future research.
  • VIEM offers a robust approach for complex material modeling involving inclusions.