Mixed Formulation
Nearly incompressible materials can cause numerical problems if only displacements are used in the interpolating functions. Small errors in the evaluation of volumetric strain, due to the finite resolution of the discrete model, are exaggerated by the high bulk modulus. This leads to an unstable representation of stresses, and in general to underestimation of the deformation because spurious volumetric stresses might balance also applied shear and bending loads.
When the Nearly incompressible material check box is selected in the Settings window for the material, the negative volumetric stress pw is treated as an additional dependent variable. The resulting mixed formulation is also known as u-p formulation. This formulation removes the effect of volumetric strain from the original stress tensor and replaces it with an interpolated pressure, pw. A separate equation constrains the interpolated pressure to make it equal (in a finite-element sense) to the original pressure calculated from the strains.
Select this setting when the material data is close to incompressibility. For an isotropic material, this happens when Poisson’s ratio approaches 0.5.
The mixed formulation is useful not only for linear elastic materials but also for elastoplastic materials, hyperelastic materials, and viscoelastic materials. The Hyperelastic Material and Plasticity nodes are available with the Nonlinear Structural Materials Module. The Plasticity node is also available with the Geomechanics Module.
The order of the shape function for the auxiliary pressure variable should be one order less than that for the displacements. Thus, it is not recommended to use linear elements for the displacement variables in the domains, where the mixed formulation is turned on. Also note that some iterative solvers do not work well together with mixed formulation because the stiffness matrix becomes indefinite.
For an isotropic linear elastic material, the second Piola-Kirchhoff stress tensor S, computed directly from the strains, is replaced by a modified version:
where I is the unit tensor and the pressure p is calculated from the stress tensor
The auxiliary dependent variable pw is set equal to p using the equation
(2-12)
where κ is the bulk modulus.
The modified stress tensor is used then in calculations of the energy variation.
For orthotropic and anisotropic materials, the auxiliary pressure equation is scaled to make the stiffness matrix symmetric. Note, however, that the stiffness matrix in this formulation is not positive definite and even contains a zero block on the diagonal in the incompressible limit. This limits the possible choices of direct and iterative linear solver.
In case of linear elastic materials without geometric nonlinearity (and also for hyperelastic materials), the stress tensor s in the above equations is replaced by the 2nd Piola-Kirchhoff stress tensor S, and the pressure p with: