Incompressible and nearly incompressible behavior is often encountered in solid mechanics. Natural rubber is almost incompressible and the ratio of its bulk modulus to the shear modulus is in range of thousands.

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Standard displacement formulation of elastic problems tends to produce inaccurate results as Poisson’s ratio reaches 0.5 or when the material becomes completely incompressible. The displacement response of the model is poor and elements lock due to volume preserving constraints imposed by incompressible materials. This phenomenon is known as volumetric locking.

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Consider an elastic solid body with almost incompressible material. The stress strain relationship for such materials is given as

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the pressure strain relationship is given as

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where p is the hydrostatic pressure, k is the bulk modulus and     is volumetric strain. 

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If we increase the bulk modulus to infinity, the volumetric strain reduces to zero and pressure cannot be calculated from the above equation. Hence displacement-based finite element method cannot predict the stresses within an incompressible material.

 

Advanced element formulations like B-Bar, Enhanced Assumed Strain and Mixed u/p formulations are used to address the problem of incompressible materials and associated volumetric locking response.

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As an example, we consider the linear analysis of the Cook’s Membrane as shown in Fig 1. The Cook’s membrane problem is a tapered panel clamped at one end and subjected to an in-plane shear loading on the free end. The material properties are taken to be E = 250 and nu = 0:4999 such that a nearly incompressible response is obtained.