Deprecated: Caller from MathObject::readFromCache ignored an error originally raised from MathObject::readFromCache: [1054] Unknown column 'math_mathml' in 'field list' in /var/www/html/includes/debug/MWDebug.php on line 385
Deprecated: Caller from LCStoreDB::get ignored an error originally raised from MathObject::readFromCache: [1054] Unknown column 'math_mathml' in 'field list' in /var/www/html/includes/debug/MWDebug.php on line 385 Top-hat transform - formulasearchengine
Top-hat transform
From formulasearchengine
Revision as of 13:45, 4 November 2013 by en>Tevye guy(Added the alternate name "bottom hat transform" for black top-hat transform. Citation I include should be improved (i.e. more reliable source))
The original model proposed by Yeoh had a cubic form with only dependence and is applicable to purely incompressible materials. The strain energy density for this model is written as
where are material constants. The quantity can be interpreted as the initial shear modulus.
Today a slightly more generalized version of the Yeoh model is used.[3] This model includes terms and is written as
When the Yeoh model reduces to the neo-Hookean model for incompressible materials.
For consistency with linear elasticity the Yeoh model has to satisfy the condition
where is the shear modulus of the material.
Now, at ,
Therefore, the consistency condition for the Yeoh model is
Stress-deformation relations
The Cauchy stress for the incompressible Yeoh model is given by
Uniaxial extension
For uniaxial extension in the -direction, the principal stretches are . From incompressibility . Hence .
Therefore,
Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the directions with the direction constrained, the principal stretches are . From incompressibility . Hence .
Therefore,
A version of the Yeoh model that includes dependence is used for compressible rubbers. The strain energy density function for this model is written as
where , and are material constants. The quantity is interpreted as half the initial shear modulus, while is interpreted as half the initial bulk modulus.
When the compressible Yeoh model reduces to the neo-Hookean model for incompressible materials.
References
↑Yeoh, O. H., 1993, "Some forms of the strain energy function for rubber," Rubber Chemistry and technology, Volume 66, Issue 5, November 1993, Pages 754-771.
↑Rivlin, R. S., 1948, "Some applications of elasticity theory to rubber engineering", in Collected Papers of R. S. Rivlin vol. 1 and 2, Springer, 1997.
↑Selvadurai, A. P. S., 2006, "Deflections of a rubber membrane", Journal of the Mechanics and Physics of Solids, vol. 54, no. 6, pp. 1093-1119.