OS-V: 0810 Hyperelastic Large Displacement Nonlinear Analysis with a Pressurized Rubber Disk

In this problem, a rubber disk 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).

Model Files

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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).
1. Model with Boundary Conditions


Material

Mooney-Rivlin Model
C10= 80 lb / in2
C01= 20 lb / in2
Ogden Model
μ 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH8oqBdaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIYaaaaa@3D40@
C10= 160 lb / in2
α 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHXoqydaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIYaaaaa@3D29@
μ 2 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH8oqBdaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIYaaaaa@3D40@
C01= 40 lb / in2
α 2 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaHXoqydaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIYaaaaa@3D29@

Results

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.
2. Displacement at Applied Pressure Values for Oden Test. Hughes & Carnoy and OptiStruct results using Mooney and Ogden material models


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