Mixed Formulation
Nearly incompressible materials can cause numerical problems if only displacements are used in the interpolating functions. Small errors in the evaluation of the volumetric strain, due to the finite resolution of the discrete model, are exaggerated by the high bulk modulus (or low bulk modulus to shear modulus ratios). This leads to an unstable representation of stresses, and in general, to an underestimation of the deformation, as spurious volumetric stresses might balance applied shear and bending loads.
When the Pressure formulation is selected in the Use mixed formulation list, the volumetric stress pw is used as an additional dependent variable. The resulting mixed formulation is also known as a u-p formulation. This formulation removes the effect of the volumetric strain from the original stress tensor, and replaces it with an interpolated pressure, pw. A separate equation constrains the auxiliary pressure variable to make it equal (in an average sense) to the original pressure which is calculated from the strains and material model.
When the Strain formulation is selected in the Use mixed formulation list, the volumetric strain εw is instead used as the additional dependent variable.
The mixed formulation is beneficial when the material data is such that the deformation is close to being incompressible. For an isotropic elastic material, this happens when the Poisson’s ratio approaches 0.5.
The mixed formulation is useful not only for linear elastic materials but also for nonlinear elastic materials, elastoplastic materials, hyperelastic materials, and viscoelastic materials.
The order of the shape function for the auxiliary variable (pw or εw) should be of lower order than that of the displacement field. Note that some iterative solvers do not work well together with mixed formulation because the stiffness matrix becomes indefinite.
When the Pressure formulation is selected for isotropic linear elastic materials, the stress tensor s, computed directly from the strains, is replaced by a modified version:
where I is the unit tensor. The pressure p is calculated from the stress tensor as
This is equivalent to define
The auxiliary dependent variable pw is set equal to p using the equation
(3-26)
where K is the bulk modulus. Scaling by the bulk modulus is necessary, since typical values for the auxiliary pressure pw are in the order of 106 to 109 Pa, while typical values for the displacement degrees of freedom are orders of magnitude smaller.
The modified stress tensor is then used then in calculations of the energy variation.
When the Strain formulation is selected for isotropic linear elastic materials, the auxiliary volumetric strain εw is used instead of the auxiliary pressure pw, and it is the set equal to the volumetric strain εvol using the equation
(3-27)
The modified stress tensor then reads
The advantage of using the Strain formulation is that the values for the auxiliary strain εw often are of a similar order of magnitude as the displacement degree of freedom.
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 with geometric nonlinearity (and also for hyperelastic materials), the stress tensor s in the above equations is replaced by the second Piola–Kirchhoff stress tensor S, see Nearly Incompressible Hyperelastic Materials.
For Plane Stress problems (in 2D solid mechanics, shell, or membrane), it is also possible to use an implicit incompressibility formulation. Based on the plane stress equation (the normal stress is equal to zero), the pressure is analytically found and the incompressibility constraint is enforced using the auxiliary transverse strain degree of freedom.