Cylinder under internal pressure

As the strains are so large, the results should compare very closely with the exact, rigid-plastic solution of Prager and Hodge (1951). That exact solution gives the stresses as follows:

where k is the yield stress in pure shear ($1/\sqrt{3}$ times the yield stress in pure tension); a₀ is the initial inside radius; b₀ is the initial outside radius; r₀ is the radius, in the initial configuration, of the material point at which the stresses are being calculated; and $a$ is the current value of the inside radius. The form of the solution shows that we need only compare the radial stress, since the other stresses are obtained directly from that component.

The results for the axisymmetric element models are summarized in Figure 2 and Figure 3, while Figure 4 and Figure 5 show the results for the plane strain models. Figure 2 compares the pressure versus inside radius given by the CAX8H and CAX8 finite element models to that given by the exact, rigid-plastic solution. All of the axisymmetric models agree very closely with the exact solution with the exception of that using the fully integrated 8-node (CAX8) element. Recall that the solution is obtained by prescribing the motion of the inside surface of the cylinder, so the pressure is calculated for the finite element models from the reaction forces conjugate to these prescribed displacements.

Figure 3 compares the stress calculated by Abaqus at the point initially halfway through the cylinder wall ($r_0/a_0=$ 1.5) to the exact, rigid-plastic solution. Again, with the exception of the CAX8 element model, there is excellent agreement with the analytical solution.

Figure 4 and Figure 5 show similar results to Figure 2 and Figure 3 for the plane strain models. Again, with the exception of the fully integrated 8-node (CPE8) element model, all of the plane strain models show excellent agreement with the exact, rigid-plastic solution.

The pure displacement 8-node elements (CAX8 and CPE8) give poor results because the strains are calculated directly from the interpolation functions at each integration point, and the incompressibility requirement causes a severe oscillation in the mean pressure stress throughout each element. However, in the hybrid, 8-node elements the mean pressure stress is interpolated independently, so an accurate value is obtained for this variable. In addition, the 4-node elements in Abaqus are constant strain/stress elements for this case (because these elements are coded with a constant hoop strain value and use “selective reduced integration,” in which the volume strain is computed at the centroid only), and so also provide accurate pressure stress values.

Results for models using the fully integrated versions of axisymmetric and plane strain elements are shown here to caution the user. With rare exceptions the fully integrated 8-node quadrilaterals are not as effective as the reduced integration versions of the same elements; the reduced integration 8-node quadrilaterals are, hence, almost always recommended over their fully integrated counterparts. This particular problem gives a dramatic illustration of a difficulty encountered with full integration in a problem in which the bulk behavior of the material is very much stiffer than the shear behavior, a type of behavior commonly encountered.