OS-V: 0810 Hyperelastic Large Displacement Nonlinear Analysis
with a Pressurized Rubber Disk
In this problem, a rubber disk which is pinned at its circumferential edge is
subjected to pressure load. This causes the disk to bulge into a spherical shape, like a
balloon.
The experimental results were published by Oden (1972) and Hughes & Carnoy
(1981). OptiStruct results are verified with the Oden
and Hughes & Carnoy tests. This example illustrates hyperelastic nonlinear large
displacement solutions with different material models (namely, Mooney and
Ogden).
Benchmark Model
The rubber disk has radius of 7.50 in (190.5 mm) with radially varying element sizes.
Such elements are preferred because the innermost element would be subjected to
maximum extension. Therefore, the innermost elements are shortest in radial length.
Thickness of the disk is 0.5 in (12.7 mm) with 2 elements along the thickness. The
innermost elements are CPENTA and rest of the elements are
CHEXA elements.
The 1, 2, and 3 degrees of freedom of the grids at the circumference are constrained,
and a pressure load of 45 psi is to be applied. The reference results are digitized
from plots in Oden (1972) and Hughes & Carnoy (1981) and used for correlation in
this study. To more accurately correlate the results from the digitized plots, the
total 45 psi pressure load in the OptiStruct run is
divided into multiple continuing nonlinear subcases (using
CNTNLSUB entries to allow subcases to continue solutions from
the end of previous subcases sequentially until the full 45 psi pressure load is
applied). Figure 1. Model with Boundary Conditions
Material
Mooney-Rivlin Model
C10= 80 lb / in2
C01= 20 lb / in2
Ogden Model
C10= 160 lb / in2
C01= 40 lb / in2
Results
It is observed that Mooney material model run with OptiStruct correlates well with the results of Oden (1972).
The Mooney and Ogden material model runs correlate very well in the pressure range
of 0 to 12 psi and closely match with Oden (1972). The Hughes & Carnoy (1981)
results are not a close match in this range of pressures.
Within the pressure range of 12-24 psi there is reasonable correlation among all
results and runs.
From 24 to 31 psi pressures Mooney model is in good agreement with Oden (1972) and
Hughes and Carnoy (1981). The Oden model shows reasonable correlation in this
pressure range. Figure 2. Displacement at Applied Pressure Values for Oden
Test. Hughes & Carnoy and OptiStruct results
using Mooney and Ogden material models Figure 3. Graph of the Oden Test, Hughes and Carnoy Test and
OptiStruct. Mooney and Ogden models
1 Nonlinear finite element shell formulation accounting for large membrane
strains. Thomas J.R. Hughes and Eric Carnoy Division of Applied Mechanics, Durand
Building, Stanford University, Stanford, 1982
2 C. Nyssen, Modeling by finite elements of nonlinear behavior of
aerospatal structures, Thesis, University of Liege, Belgium, 1979
3 J.T. Oden and J.E. Key, Analysis of finite deformations of elastic solids
by the finite element method, Proc. IUTAM Colloquium on High Speed Computing of
Elastic Structures, Liege, Belgium, 1971
4 T.J.R. Hughes and J. Winget, Finite rotation effects in numerical
integration of rate constitutive equations arising in large-deformation analysis,
Internat. J. Numer. Meths. Engrg. 15 (1980) 1862-1867
