Continuum elements with incompatible modes

The lower-order quadrilateral continuum elements in Abaqus of type CPS4I, CPE4I, CAX4I, CPEG4I, and C3D8I, as well as the related hybrid elements, are enhanced by incompatible modes to improve the bending behavior.

The following topics are discussed:

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Solid (continuum) elements


In addition to the displacement degrees of freedom, incompatible deformation modes are added internal to the elements. The primary effect of these degrees of freedom is to eliminate the so-called parasitic shear stresses that are observed in regular displacement elements if they are loaded in bending.

In addition, these degrees of freedom eliminate artificial stiffening due to Poisson's effect in bending. In regular displacement elements the linear variation of the axial stress due to bending is accompanied by a linear variation of the stress perpendicular to the bending direction, which leads to incorrect stresses and an overestimation of the stiffness. The incompatible modes prevent such a stress from occurring.

In the nonhybrid elements (except CPS4I) additional incompatible modes are added to prevent locking of the elements for approximately incompressible material behavior. For fully incompressible material behavior, hybrid elements must be used. In these elements pressure degrees of freedom are added to enforce a linear pressure variation inside the element. In the hybrid elements the additional incompatible modes used to prevent locking are not included.

The incompatible mode elements perform almost as well as second-order elements in many situations if the elements have an approximately rectangular shape. The performance is considerably less if the elements have a parallelogram shape. For trapezoidal element shapes the performance is not much better than the performance of regular displacement elements.

Because of the internal degrees of freedom (4 for CPS4I; 5 for CPE4I, CAX4I, and CPEG4I; and 13 for C3D8I) the elements are somewhat more expensive than regular displacement elements. However, the additional degrees of freedom do not substantially increase the wavefront size since they can be eliminated immediately. In addition, it is not necessary to use selectively reduced integration, which partially offsets the cost of the additional degrees of freedom.

The geometrically linear incompatible mode formulation used in Abaqus is related to the work presented by Simo and Rifai (1990). Simo's formulation is very similar to much earlier work done by Wilson et al. (1973) and Taylor et al. (1976). The nonlinear formulation is based on work by Simo and Armero (1992).