Examines the hyperelastic behavior of a hexahedral element under enforced displacement using different material models
such as Arruda Boyce, reduced polynomial, Yeoh and Ogden model.
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.
Examines the hyperelastic behavior of a hexahedral element under enforced displacement using different material models
such as Arruda Boyce, reduced polynomial, Yeoh and Ogden model.
Examines the hyperelastic behavior of a hexahedral element under enforced
displacement using different material models such as Arruda Boyce, reduced polynomial, Yeoh
and Ogden model.
In 1944, L.G.R. Treloar 1 performed experiments on 8% rubber to obtain the
uniaxial stress-strain curve which has been digitized and utilized for this
simulation.
Benchmark Model
A single hexahedral element with 8-node CHEXA is used to perform
the hyperelastic simulation. The element is of size 10 x 10 x 10 millimeters. The
nodes 1, 2, 7, and 8 are constrained through zero-length CLEAS
elements in 5 degrees of freedom (2,3,4,5,6) and a small value of spring stiffness
is assigned to restrain the element (this does not affect the results of the
simulation). Enforced displacement of 70 mm in dof 1 is applied on nodes 3, 4, 5,
and 6 using SPCD entry. The material stress-strain curve for 8%
sulphur rubber has been digitized from the Treloar 1 paper.
Units: mm, s, Mg, N, MPa
Material
The MATHE Bulk Data Entry is used to input hyperelastic material
data for the model.
Field
Model
Identifies material model
NU (Poisson's Ratio)
Both 0.4997 and 0.495 values can be used
0.4997 demonstrates a better fit for the incompressible rubber
material
TAB1
Defines the Uniaxial tension-compression data
TAB2
Defines Equi-biaxial data
TAB4
Defines Shear data
Required engineering stress versus strain material data is generated from test data
as curve input for different material laws. The hyperelastic data gathered by
Treloar for 8% sulfur rubber test data is used. 図 2 and 図 3 show the Treloar test data for the 3 strain states
most important in characterizing a hyperelastic material, uniaxial tension, equal
biaxial extension and pure shear. 2
For RPOLY material model, 3 test data Uniaxial, Biaxial, and Planar Stress-Strain
data were used, as they provided a good fit in OptiStruct. For Yeoh, Ogden, and ABOYCE material models, only Uniaxial test stress-strain
data is used in the MATHE entry for curve fitting in OptiStruct (since only using uniaxial test data provided the
best fit for these three material models). For all material models, the curve fit
was better when the Poisson’s ratio of 0.4997 is used, instead of 0.495.
Results
OptiStruct outputs true stress and strains, which cannot
be converted to engineering (nominal) stress-strain using the typical conversion
equations since the equations are not valid at higher strains (above 200%).
Instead, the engineering stress-strain values were calculated by:(1)
(2)
Where,
True stress
Engineering stress
True strain
Engineering strain
For engineering stress, combined SPC forces at nodes 3, 4, 5, and 6 was calculated at
each increment and divided by the original area of the element face 3, 4, 5, 6 (100
sq. mm) to get the engineering stress. For engineering strain, the change in the
length of the element along X-direction was calculated at each increment and divided
by original length (10 mm) to get the engineering strain.
The results were plotted for stress-strain curve fit considered by OptiStruct, digitized stress-strain test data and engineering
stress-strain calculated from OptiStruct results.
Reduced Polynomial (RPOLY) Model
The test data, OptiStruct curve fit
and OptiStruct output data is:
The RPOLY model correlates well with the test data, and
also the fit for the material model from test data is good.
Ogden Model
The test data, OptiStruct curve
fit and OptiStruct results are plotted.
OptiStruct results and fit
correlate well with test data until 300% strain, and also continues to
be reasonably close beyond 300%.
Arruda-Boyce (ABOYCE) Model
The test data, OptiStruct curve fit and OptiStruct results are plotted.
The ABOYCE model correlates very well with test results
until 100% strain and continues to be a reasonably close match beyond
100%.
Yeoh Model
The test data, OptiStruct curve fit
and OptiStruct results are plotted.
Additionally, Radioss results are also
illustrated for Yeoh material model.
OptiStruct results correlate
well with the test data and fit until about 525% strain and continues to
be a reasonably good match beyond 525%. The results obtained by OptiStruct are in good agreement with both test
result and Radioss output.
1 Treloar, L. R. G. "Stress-strain data for vulcanised rubber under
various types of deformation" Transactions of the Faraday Society 40
(1944): 59-70
2 Miller, Kurt. "Testing Elastomers for Hyperelastic Material Models
in Finite Element Analysis" Axel Products, Inc., Ann Arbor, MI (2017).
Last modified April 5, 2017