Premium
Measuring the modulus of a sphere by squeezing between parallel plates
Author(s) -
Rodriguez Ferdinand,
Patel Suman K.,
Cohen Claude
Publication year - 1990
Publication title -
journal of applied polymer science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.575
H-Index - 166
eISSN - 1097-4628
pISSN - 0021-8995
DOI - 10.1002/app.1990.070400122
Subject(s) - shear modulus , equator , modulus , radius , natural rubber , young's modulus , ideal (ethics) , bulk modulus , elasticity (physics) , physics , spheres , materials science , mathematical analysis , classical mechanics , mathematics , composite material , thermodynamics , law , latitude , computer science , computer security , astronomy , political science
Formulas are derived relating the deformation resulting from the application of a force f to the poles of a sphere of uniform modulus. When the force f is applied, the poles, originally a distance 2 R * apart, now are a distance 2R u apart. Also, the diameter of the sphere at its equator is increased from 2R * to a new value 2R e . For an ideal rubbery material with a shear modulus of g , the force is related to the change in equatorial diameter by\documentclass{article}\pagestyle{empty}\begin{document}$$ \sigma _c = \frac{f}{{\pi R^{*2} }} = G\left\{ {\left( {\frac{R}{{R^* }}} \right)_e^4 - \left( {\frac{{R^* }}{R}} \right)_e^2 } \right\}. $$\end{document} Thus, measurement of the increase in relative radius at the equator as a function of force applied at the poles yields the modulus directly. The change in R v also can be related to G , but a numerical integration is required. Equations can also be derived relating modulus to dimensional changes when Hooke's law is used in place of ideal rubber elasticity.