Rubber near-incompressibility

When a material is stretched there is also contraction in the direction perpendicular to the direction of stretching. The ratio of the lateral contraction to the longitudinal extension is Poisson s ratio. For incompressible materials, Poisson s ratio is 0.5 and as rubbers are very nearly incompressible they have values close to this. 

An example of the success of the temporary network model for a practical application is shown in Fig. 3-11. Here, the predictions of Eq. (3-24) are compared to experimental force-deflection data for impact tests in which a heavy flat-bottomed object is dropped onto a flat circular pad of dissipative Sorbothane rubber at various velocities and two different temperatures. Since the material is nearly incompressible under these conditions, the impact... 

Rubbers are very nearly incompressible Le. G < jT (where K is the bulk modulus). 

Some other complications in the course of numerical simulation of rubber and tires are rubber near-incompressibUity and reinforcement in tires. The rubber nearincompressibility makes the system of finite element equations ill conditioned since in this case the volumetric stiffness greatly exceeds the shear stiffness. The nearincompressibility and incompressibility conditions are essentially constraints imposed on the solution, and depending on the ratio of the number of discrete equations and discrete number of constraints solution may or may not exist. Therefore, the design of specific finite elements to satisfy these conditions becomes very important. 

Ordinary liquids and liquid crystals are nearly incompressible. In ordinary fluid dynamics the incompressibility approximation under the constraint div v = 0 has frequently been utilized. In a soft elastomer such as vulcanized rubber, where shear modulus is very small as compared with bulk modulus, the incompressibility approximation has also been usefully employed. The constraint of the incompressibility approximation, div v = 0 for ordinary fluids or divergence of displacement vector for elastic (isotropic) materials, does not modify any other terms of the equations of motion div v = 0, or divergence of displacement vector, is a solutirai of the equations of motion, provided that pressure p is chosen as an appropriate harmoiuc function (V p = 0). However, for anisotropic matters, such as liquid crystals or anisotropic solids (crystals), since the div v = 0 or its elastic version cannot be a special solution of equations of motion, the incompressibility approximation requires a careful consideration [12, 18]. 

For crosslinked incompressible rubbers, Poisson s ratio is nearly equal to 0.5, so that (20)... 

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