Establishing Failure Criterion for Elastomers Through Improved Biaxial Testing.
The failure criterion for composites based on finite elasticity developed by Feng et al.  and later extended for polymers and soft biological materials by Feng and Hallquist  depends on testing materials in two deformation modes, in tensile and biaxial condition.
The biaxial extension of elastomers has been accomplished by radial stretching of disks, balloon style extension of thin membranes, and by complex scissor grip systems all with reasonable success. However, observing failure in biaxial extension is problematic because the region where the elastomer is gripped or held will typically experience a higher strain than the biaxial region and fail first. This is certainly true for the radial stretching of disks and for the scissor grip system.
The balloon style biaxial failure has had some success when special molded sheets of elastomer are constructed wherein a thick clamping ring is molded into the elastomer membrane to allow a strain transition at the gripping region. However, this special molding of specimens is often not practical.
The approach described in this paper is to push a hemispherical probe into an elastomer sheet surface stretching it until failure. The data from the tests were used to establish failure criterion in a simplified rubber material model in the finite element code LS-DYNA.
MATERIAL MODEL WITH DAMAGE
Many hyperelastic material models use polynomial fits to derive their constants. The MAT_SIMPLIFIED_RUBBER material model implemented with LS-DYNA is designed to take direct engineering stress strain obtained using standard tensile and compression tests. Strain rate dependency can also be input as part of this material model. To model damage in the polymers unloading data can be provided; this unloading curve is not strain rate dependent. The basis of the MAT_SIMPLIFIED_RUBBER is the Ogden strain energy potential model.
The Ogden strain energy potential shown in equation 1 is based on the principal stretch ratios shown in equation 2. As observed from tests, rubber materials do not exhibit the same path dependent behavior during loading and unloading. This is due to damage in the material. The damage behavior is a function of the elastic deviatoric energy and the maximum deviatoric energy over the deformation path, equation 3. The general implemented Ogden strain energy potential is limited to quasi-static state.
W = [3.summation over (i=1)][n.summation over (j=1)][[[mu].sub.j]/[[alpha].sub.j]]([lambda].sup.*[[alpha].sub.j].sub.i-1)+K(J-1-lnJ) (1)
[[lambda].sup.*.sub.i] = [[lambda].sub.i] [J.sup.[-1/3]] = [[[lambda].sub.i]/[J.sup.[1/3]] J = [[lambda].sub.1][[lambda].sub.2][[lambda].sub.3] = [v/[v.sub.0]](2)
W = (1-d([W.sub.0]/[W.sub.0,max])) [3.summation over (i=1)][n.summation over (j=1)][[[mu].sub.j]/[[alpha].sub.j]]([lambda].sup.*[[alpha].sub.j].sub.i-1)+K(J-1-lnJ)
Where, the damage functional is represented by
[W.sub.0,max]=max([W.sub.0], [W.sub.0,max]) [??] 0 [less than or equal to] [[W.SUB.0]/[W.sub.0,max]][less than or equal to]1[??]0[less than or equal to]D[less than or equal to]1 (4)
Material unloading due to stress relaxation in this material model is represented using Prony Series. Equation 5 below represents the series. Where the Shear Moduli are Gi and the decay constants are Yi.
g(t)= [n.summation over (i=1)][G.sub.i][e.sup.-yit] (5)
Failure in rubber is implemented in terms of strain invariants. The failure surface is defined by Equation 6. Since it is based on energy principle; failure occurs when the strain energy reaches a maximum.
[I.sub.1] ,-[[GAMMA].sub.1]([I.sub.1]-3) (2)+[[GAMMA].sub.2]([I.sub.2]-3)-K=0 (6)
[I.sub.1]=[[lambda].sup.2.sub.1]+[[lambda].sup.2.sub.2]+[[lambda].sup.2.sub.3] and [I.sub.2]=[[lambda].sup.2.sub.1]+[[lambda].sup.2.sub.2]+[[lambda].sup.2.sub.3][[lambda].sup.2.sub.1] (7)
Where, [I.sub.1] and [I.sub.2] are strain invariants and can be written in terms of principle stretches as shown in Equation 7. The Failure surface is defined using the principle stretch ratios as show in Figure 1.
The constants in Equation 6 were calculated using the following Formulae. We have three equations and three unknowns.
Simultaneous equations are used to solve these. For most practical applications [[GAMMA].sub.2] is Zero, leaving two unknowns.
From uniaxial test (compression or tension) principle stretch ratios in axis 2 and 3 are same
[[lambda].sub.2] = [[lambda].sub.3]
For incompressible materials
Substituting these in equation 7 we get Strain invariants in terms of Principle Stretch Ratios for uniaxial test, Equation 9
[I.sub.1] = ([[lambda].sup.2.sub.3]+1)/[[lambda].sub.1] and [l.sub.2]-[[lambda].sub.1] (9)
From Biaxial test data
[[lambda].sub.1] = [[lambda].sub.2] (10)
Substituting equations 8 and 10 into Equation 7 we get strain invariants in terms of principle stretch for biaxial data; see Equation 11
[I.sub.1]=[2[lambda].sup.2.sub.1]+[1/[lambda].sup.4.sub.1] and [I.sub.2]=[[lambda].sup.4.sub.1]+[2/[lambda].sup.2.sub.1] (11)
Deriving Failure Constants - Uniaxial Mode
The uniaxial tension or compression test will provide one data point on the failure envelope. It is a simple and common test to perform. Three uniaxial tensile failure strains of 4.32, 4.4, and 4.53 were obtained from test shown in Figure 2.
Deriving Failure Constants - Biaxial Mode
Given the large strain to failure observed in the uniaxial tensile test described earlier, along with limitation in preparing specially molded test samples, we felt the membrane inflation test described by Feng  was limiting to derive the failure constants in biaxial mode.
The balloon style biaxial failure has been successful when special molded sheets of elastomer are constructed wherein a thick clamping ring is molded into the elastomer membrane to allow a strain transition at the gripping region. However, this special molding of specimens is often not practical.
The approach described here is to push a hemispherical probe into an elastomer sheet surface stretching it until failure. The sheet is pneumatically clamped between rings to minimize slipping of elastomer relative to the gripping ring as shown in Figure 4.
The force required to push the probe into the sheet and the probe displacement are measured. The probe displaces the sheet until failure. Because the strain field is not even throughout the sheet and is somewhat localized near the probe tip, the elastomer is likely to fail near the clamping region as seen in Figure 5.
Recognizing that there is some friction at the probe contact and that the strain may localize, the solution is not perfect but it is quite practical.
The failure force of the samples are a result of this test as shown in Figure 6. The principal stretch ratios and failure strains are derived through an inverse relation through a finite element model. The test data are then used to develop the failure constants K and [GAMMA].
Figure 7 shows the finite element model of the hemispherical probe test. The maximum principal strains appear at the pole position of the sample.
Some friction calibration tests were done to obtain the correct peak strains by correlating the force versus displacement of the physical test presented in Figure 6. Figure 8 shows the FEA predicted force curve that matches with test.
Material Model Characterization
The failure parameters derived in the earlier sections are part of the requirement for the material model to failure. To fully utilize the versatility of this model additional hyperelastic parameters are incorporated, especially material behavior in compression and strain rates which will be discussed here.
Material data was obtained from tensile, compressive, and biaxial test where the samples were not subjected to failure but were cycled at different strain levels. Figures 9 and 10 show the uniaxial tension and compression tests conducted at strain rates of [0.1s.sup.-1], [1s.sup.-1], [10s.sup.-1], and [100s.sup.-1]. As expected the material is more sensitive to strain rates especially under compression load.
The Bulk Modulus of the material was calculated from confined compressive test. Figure 11 shows the estimated modulus.
The short term stress relaxation tests carried out were used to obtain Shear Modulus Gi and the decay constants are Yi. Figure 12 shows the data that were used.
Data from tests were filtered to remove noise in the data and improve material model stability. The tensile and compressive tests were combined to get the entire data range in a single curve. The tensile test and compressive test data were characterized and validated using single element models. This helped in faster turnaround and calibration of the data. Figure 13 shows the comparison of the test data with model fitting.
We ensure material unloading behavior is used for both tension and compression. A LOG_LOG interpolation is used to ensure smooth strain rate behavior during transitioning from one strain rate data to another.
Stress relaxation data was used to get Prony series constants. The fitted response was compared against test data and found to be in good agreement. Figure 14 shows the comparison of the test.
The material model was further tested for stability under different strain rates and then held under constant state of strain to evaluate stress redistribution over time using simple tension tests and compression tests as shown in Figure 15. The analysis response of the material and stability over time, which were very good, is shown in Figure 16. It is important to note that the tuned friction co-efficient derived for hemispherical CAE tests do not affect the compression tests.
The elastomer material that exhibits sensitivity to strain rates, load path, and damage was characterized using MAT_SIMPLIFIED_ RUBBER material model. The damage limits were established using two experiments--a tensile to failure test and a biaxial test. The biaxial test pushes a hemispherical probe into an elastomer sheet, stretching it to failure. The biaxial testing method developed was simple, repeatable, and can handle large strains to failure. The biaxial testing method also does not require specially molded elastomer sheets.
At present, a finite-element analysis is used to calculate peak strains. At the risk of adding complexity to a simple experiment, a better test method will be to use digital image correlation strain measurement on the surface of the elastomer opposite the probe contact point to directly define the strain field.
(1.) A simplified approach to the simulation of rubber-like materials under dynamic loading., Du Bois Paul A., 4th European LS-Dyna Users Conference
(2.) On Constitutive Equation for Elastomers and Elastomeric Foams, Feng William W., Halquist John O., 4th European LS-Dyna Users Conference
(3.) A Failure Criterion for Polymers and Soft Biological Materials, Feng William W., Halquist John O, 5th European LS-Dyna Users Conference
(4.) A Simplified Approach for Strain-Rate Dependent Hyperelastic Materials with Damage, Benson D.J, Kolling S., Du Bois P. A., 9th International LS-Dyna Users Conference.
(5.) LS-Dyna User Manual and Theoretical Manual, Livermore Software Technology Corporation.
The lead author Raju Gandikota can be contacted at
The authors would like to thank Eric DeHoff and Kishore Pydimarry at Honda R&D Americas, Inc. for their support and permission to publish this work.
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|Publication:||SAE International Journal of Materials and Manufacturing|
|Date:||Jul 1, 2017|
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