A proper treatment of the rubber material service conditions and material degradation phenomena like strain softening is of prime importance in the testing of rubbers specimens for FEA material characterization. The accuracy and reliability of obtained test data depends on how the mechanical conditioning and representational service conditions of the material have been accounted for in the test data. To simulate a component in unused and unaged conditions, the mechanical conditioning requirements are different than the ones for simulating a component that has gone through extensive field service and aging under different environmental conditions. To simulate performance of a material or component by Finite Element Analysis (FEA) it should be tested underthe same deformation modes to which original assembly will be subjected. The uniaxial tension tests are easy to perform and are fairly well understood but if the component assembly experiences complex multiaxial stress states then it becomes imperative to test in other deformation modes. Planar (pure shear), biaxial and volumetric (hydrostatic) tests need to be performed along with uniaxial tension test to incorporate the effects of multiaxial stress states in the FEA model.
Material stiffness degradation phenomena like Mullin’s effect at high strains and Payne’s effect at low strains significantly affect the stiffness properties of rubbers. After the first cycle of applied strain and recovery the material softens, upon subsequent stretching the stiffness is lower for the same applied strain. Despite all the history in testing hyperelastic and viscoelastic materials, there is a lack of a methodical and standard testing protocol for pre-conditioning. Comprehensive studies on the influence of hyperelastic material testing pre-conditioning is not available.
1) Mechanical Testing of Polymers, Metals and Composite Materials 2) Fatigue and Durability Testing 3) Dynamic Mechanical Analysis (DMA) of Materials and Components 4) Hyperelastic, Viscoelastic Material Characterization Testing 5) Data Cards for Input into FEA, CAE softwares 6) FEA Services 7) Custom Test Setups with NI Labview DAQ
An
O-ring or a Seal under energized conditions must maintain good contact force
throughout the functional life of the products. Contact force is generated
between the mating surfaces when one of the mating surfaces deflects and
compresses the seal surface. In order for the sealing to remain effective the
contact surfaces must return to the undeformed original position when the
contacting force is removed. Under these
conditions the deflection of the sealing element must be fully recoverable and
so hyperelastic by nature. If there is
any unrecoverable strain in the material the performance of the seal is diminished
and leak would occur from between the surfaces. The key to designing a good
sealing element is that the good contact force is as high as possible while at
the same time ensuring that the deflection remains hyperelastic in nature.
This requires the use of a material with a good combination of force at a desired deformation characteristic. The relationship between strain and stress is described by the material’s stress-strain curve. Figure 1 shows typical stress-strain curves from a polymer thermoplastic material and thermoset rubber material. Both the materials have plastic strain properties where when the material is stretched beyond the elastic limit there is some permanent deformation and the material does not fully return to its original undeformed condition.
Figure 1: Stress-Strain Curves from Thermplastic and Thermoset Materials
The plastic strain, is the area between the loading and unloading line in both the graphs. In automotive application this permanent plastic strain is observed more easily in under the hood components located near the engine compartments because of the presence of high temperature conditions. If a polymer part such as intake manifold is stressed to a certain and held for a period of time then some of the elastic strain converts to plastic strain resulting in observations of permanent deformation in the component. There are two physical mechanisms by which the amount of plastic strain increases over time, 1) Stress relaxation and 2) Creep. Creep is an increase in plastic strain under constant force, while in the case of Stress relaxation, it is a steady decrease in force under constant applied deformation or strain. Creep is a serious issue in plastic housings or snap fit components, while Stress relaxation is a serious issue in sealing elements. Experimental studies on creep behavior of plastics is carried out using the tensile creep test. The loading is purely under static conditions according to ISO 899-1. The specimens used in the testing are generally as prescribed as 1A and 1B in ISO 527 and ASTM D638. These specimens correspond to the generalized description of specimens according to ISO 3167.
Figure 2: Graphical Representation of Creep and Stress Relaxation
Figure 3 shows the
results from Creep testing of an HDPE material. In Most Finite Element Analysis
software, stress relaxation and creep both can be simulated with the help of experimental
test data.
Figure 3: Sample Creep Test Results for an HDPE Material
Creep modulus Ec(t) is used to describe the time
dependent material behavior of plastics. It is defined as the ratio of the
applied stress and time-dependent deformation at time (t):
Ec(t) = sigma/epsilom(t) (1)
Creep rate Ec(t)/dt is used to describe the long-term creep
behavior, it is defined from the ratio of deformation or strain increase with
respect to time
dot{Ec(t)} = depsilom/dt (2)
Creep Stages
1)
Primary Creep: The process starts at a rapid rate and slows with time.
Typically it settles down within a few minutes or hours depending upon the
nature of material. Strain rate decreases as strain increases.
2)
Secondary Creep:
At
this state the process has a relatively uniform rate and is known as steady
state creep.
Strain
rate is minimum and constant. Balance between between recovery and strain hardening.
Fracture typically does not occur during this
stage.
3)
Tertiary Creep: This stage shows an accelerated creep rate and terminates with
failure or a fracture. It is associated with both necking and formation of
voids.
An O-ring or a Seal
under energized conditions must maintain good contact force throughout the
functional life of the products. Contact force is generated between the mating
surfaces when one of the mating surfaces deflects and compresses the seal
surface. In order for the sealing to remain effective the contact surfaces must
return to the undeformed original position when the contacting force is removed
or when there are vibratory displacements between the contacting surfaces. Under these conditions the deflection of the
sealing element must be fully recoverable and so hyperelastic by nature. If there is any unrecoverable strain in the
material the performance of the seal is diminished and leak would occur from
between the surfaces. The key to designing a good sealing element is that the
good contact force is as high as possible while at the same time ensuring that
the deflection remains hyperelastic in nature. This requires the use of a
material with a good combination of force at a desired deformation
characteristic. Figure 4 shows the family of curves for a stress relaxation
experiment carried out at multiple strain levels.
Figure 5 shows the
results from a compression stress relaxation test on a rubber material. The
results show the test data over a 3 day period.
Figure 4: Stress Relaxation Curves at Multiple Strain Levels
The
initial rapid relaxation and decrease in force occurs due to chemical process
related degradation of the material, while at longer duration and time frames
the drop in force is due to physical relaxation. Numerous studies have shown
that the relaxation mechanism in polymers and rubbers is dependent on many
factors as the nature and type of polymer, fillers and ingredients used, strain
levels, strain rates and also temperature. The rate of relaxation is generally
found to decrease at lower levels of filler loading and the rate of stress
relaxation increases at higher levels of filler loading. This is attributable
to polymer filler interactions
Figure 5: Sample Continuous Compression Test Results for Nitrile Elastomer Material
The molecular causes of stress relaxation can be classified to be
based on five different processes.
1). Chain Scission: The
decrease in the measured stress over time is shown in Figures 4 and 5
where, 3 chains initially bear the load but subsequently one of
the chains degrade and break down.
2). Bond Interchange: In
this particular type of material degradation process, the chain portions
reorient themselves with respect to their partners causing a decrease in stress.
3). Viscous Flow: This occurs basically due to the slipping of
linear chains one over the other. It is particularly responsible for viscous
flow in pipes and elongation flow under stress.
Figure 6: Chain Scission in an Elastomeric Material
4). Thirion Relaxation: This is a reversible relaxation of the
physical crosslinks or the entanglements in elastomeric networks. Generally an
elastomeric network will instantaneously relax by about 5% through this
mechanism.
5). Molecular Relaxation: Molecular relaxation occurs especially
near Tg (Glass Transition Temperature). The molecular chains
generally tend to relax near the Tg.
References:
1. Sperling, Introduction to Physical Polymer Science, Academic Press, 1994.
2. Ward et al., Introduction to Mechanical Properties of Solid Polymers, Wiley, 1993. 3. Seymour et al. Introduction to Polymers, Wiley, 1971.
3. Ferry, Viscoelastic Properties of Polymers, Wiley, 1980.
4. Goldman, Prediction of Deformation Properties of Polymeric and Composite Materials, ACS, 1994.
5. Menczel and Prime, Thermal Analysis of Polymers, Wiley, 2009.
6. Pete Petroff, Rubber Energy Group Class Notes, 2004.
7. ABAQUS Inc., ABAQUS: Theory and Reference Manuals, ABAQUS Inc., RI, 02.
8. Dowling, N. E., Mechanical Behavior of Materials, Engineering Methods for Deformation, Fracture and Fatigue Prentice-Hall, NJ, 1999.
9. Srinivas, K., and Dharaiya, D., Material And Rheological Characterization For Rapid Prototyping Of Elastomers Components, American Chemical Society, Rubber Division, 170th Technical Meeting, Cincinnati, 2006.
Our automotive rubber components design, development and testing services provides you with the technical insight you need to ensure your components provide the necessary stiffness, vibration isolation and stability to the vehicle.
The application of computational mechanics analysis
techniques to elastomers presents unique challenges in modeling the following
characteristics:
– The load-deflection behaviour of an elastomer is markedly
non-linear.
– The recoverable strains can be as high 400 % making it
imperative to use the large
deflection theory.
– The stress-strain characteristics are highly dependent on
temperature and rate effects are pronounced.
– Elastomers are nearly incompressible.
– Viscoelastic effects are significant.
The ability to model the special elastomer characteristics
requires the use of sophisticated material models and non-linear Finite element
analysis tools that are different in scope and theory than those used for metal
analysis. Elastomers also call for superior analysis methodologies as
elastomers are generally located in a system comprising of metal-elastomer parts
giving rise to contact-impact and complex boundary conditions. The presence of
these conditions require a judicious use of the available element technology
and solution techniques.
FEA Support Testing
Most commercial FEA software packages use a curve-fitting
procedure to generate the material constants for the selected material model.
The input to the curve-fitting procedure is the stress-strain or stress-stretch
data from the following physical tests:
1 Uniaxial
tension test
2 Uniaxial
compression test OR Equibiaxial tension test
3 Planar
shear test
4 Volumetric
compression test
A minimum of one test data is necessary, however greater
the amount of test data, better the quality of the material constants and the
resulting simulation. Testing should be carried out for the deformation modes
the elastomer part may experience during its service life.
Curve-Fitting
The stress-strain data from the FEA support tests is used
in generating the material constants using a curve-fitting procedure. The
constants are obtained by comparing the stress-strain results obtained from the
material model to the stress-strain data from experimental tests. Iterative
procedure using least-squares fit method is used to obtain the constants, which
reduces the relative error between the predicted and experimental values. The
linear least squares fit method is used for material models that are linear in
their coefficients e.g Neo-Hookean, Mooney-Rivlin, Yeoh etc. For material
models that are nonlinear in the coefficient relations e.g. Ogden etc, a
nonlinear least squares method is used.
Verification and Validation
In the FEA of elastomeric components it is
necessary to carry out checks and verification steps through out the analysis.
The verification of the material model and geometry can be carried out in three
steps,
_ Initially a single element
test can be carried out to study the suitability of the chosen material model.
_ FE analysis of a tension
or compression support test can be carried out to study the material
characteristics.
_ Based upon the feedback
from the first two steps, a verification of the FEA model
can be carried out by applying the main
deformation mode on the actual component
on any suitable testing machine and verifying the results computationally.
Figure 1: Single Element Test
Figure(1) shows the single element
test for an elastomeric element, a displacement
boundary condition is applied on a face, while constraining the movement of the opposite face. Plots A and B show the deformed and undeformed plots for the single element. The load vs. displacement values are then compared to the data obtained from the experimental tests to judge the accuracy of the hyperelastic material model used.
Figure 2: Verification using an FEA Support Test
Figure (2) shows the verification
procedure carrying out using an FEA support test.
Figure shows an axisymmetric model of the
compression button. Similar to the single
element test, the load-displacement values from
the Finite element analysis are compared to the experimental results to check
for validity and accuracy. It is possible that the results may match up very
well for the single element test but may be off for the FEA support test verification
by a margin. Plot C shows the specimen in a testing jig. Plot D and E show the undeformed
and deformed shape of the specimen.
Figure(3) shows the verification
procedure that can be carried out to verify the FEA
Model as well as the used material model. The procedure also validates the boundary conditions if the main deformation mode is simulated on an testing machine and results verified computationally. Plot F shows a bushing on a testing jig, plots G and H show the FEA model and load vs. displacement results compared to the experimental results. It is generally observed that verification procedures work very well for plane strain and axisymmetric cases and the use of 3-D modeling in the present procedure provides a more rigorous verification methodology.
Figure 3: FEA Model Verification using an Actual Part
AdvanSES provides Hyperelastic, Viscoelastic Material Characterization Testing for CAE & FEA softwares.
Unaged and Aged Properties and FEA Material Constants for all types of Polymers and Composites. Mooney-Rivlin, Ogden, Arruda-Boyce, Blatz-ko, Yeoh, Polynomials etc.
Non-linear Viscoelastic Dynamic Properties of Polymer, Rubber and Elastomer Materials
Static testing of materials as per ASTM D412, ASTM D638, ASTM D624 etc can be categorized as slow speed tests or static tests. The difference between a static test and dynamic test is not only simply based on the speed of the test but also on other test variables em- ployed like forcing functions, displacement amplitudes, and strain cycles. The difference is also in the nature of the information we back out from the tests. When related to poly- mers and elastomers, the information from a conventional test is usually related to quality control aspect of the material or the product, while from dynamic tests we back out data regarding the functional performance of the material and the product.
Tires are subjected to high cyclical deformations when vehicles are running on the road. When exposed to harsh road conditions, the service lifetime of the tires is jeopardized by many factors, such as the wear of the tread, the heat generated by friction, rubber aging, and others. As a result, tires usually have composite layer structures made of carbon-filled rubber, nylon cords, and steel wires, etc. In particular, the composition of rubber at different layers of the tire architecture is optimized to provide different functional properties. The desired functionality of the different tire layers is achieved by the strategical design of specific viscoelastic properties in the different layers. Zones of high loss modulus material will absorb energy differently than zones of low loss modulus. The development of tires utilizing dynamic characterization allows one to develop tires for smoother and safer rides in different weather conditions.
Figure Locations of Different Materials in a Tire Design
The dynamic properties are also related to tire performance like rolling resistance, wet traction, dry traction, winter performance and wear. Evaluation of viscoelastic properties of different layers of the tire by DMA tests is necessary and essential to predict the dynamic performance. The complex modulus and mechanical behavior of the tire are mapped across the cross section of the tire comprising of the different materials. A DMA frequency sweep
test is performed on the tire sample to investigate the effect of the cyclic stress/strain fre- quency on the complex modulus and dynamic modulus of the tire, which represents the viscoelastic properties of the tire rotating at different speeds. Significant work on effects of dynamic properties on tire performance has been carried out by Ed Terrill et al. at Akron Rubber Development Laboratory, Inc.
Non-linear Viscoelastic Tire Simulation Using FEA
Non-linear Viscoelastic tire simulation is carried out using Abaqus to predict the hysteresis losses, temperature distribution and rolling resistance of a tire. The simulation includes several steps like (a) FE tire model generation, (b) Material parameter identification, (c) Material modeling and (d) Tire Rolling Simulation. The energy dissipation and rolling re- sistance are evaluated by using dynamic mechanical properties like storage and loss modu- lus, tan delta etc. The heat dissipation energy is calculated by taking the product of elastic strain energy and the loss tangent of materials. Computation of tire rolling is further carried out. The total energy loss per one tire revolution is calculated by;
Ψdiss = ∑ i2πΨiTanδi, (.27)
i=1
where Ψ is the elastic strain energy,
Ψdiss is the dissipated energy in one full rotation of the tire, and
Tanδi, is the damping coefficient.
The temperature prediction in a rolling tire shown in Fig (2) is calculated from the loss modulus and the strain in the element at that location. With the change in the deformation pattern, the strains are also modified in the algorithm to predict change in the temperature distribution in the different tire regions.
Rubber components under multiaxial cyclic loading conditions are often considered to have failed or degraded when there is a change is the stiffness of the component and it is no longer able to provide the performance it is designed for. Elastomeric polymer components are widely used in many industries like automotive, aerospace and biomedical applications due to their good vibration isolation and energy absorption characteristics. The type of loading normally encountered by these components in service is mutiaxial in nature. Fatigue failure is thus a major consideration in their design and availability of testing techniques to predict fatigue life under these complex conditions is a necessity.
In real world applications all materials and products are subjected to a wide variety of vibrating or oscillating forces. Fatigue testing consists of applying a cyclic load to a test specimen or the component to determine in-service performance during situations similar to real world working conditions.
Advanced Scientific and Engineering Services (AdvanSES) specializes in Fatigue Testing of Automotive Components including hoses, engine mounts, vibration isolators, silent bushes etc. Fatigue testing can be carried out in stress & force control, strain control or displacement control. The deformation modes under which fatigue tests are generally carried out are tension – tension, compression – compression, tension – compression and compression – tension.
ASTM D 5992 Test Standard applies to Dynamic Properties of Rubber Vibration Products such as springs, dampers, and flexible load-carrying devices, flexible power transmission couplings, vibration isolation components and mechanical rubber goods. The standard applies to to the measurement of stiffness, damping, and measurement of dynamic modulus.
Dynamic testing is performed on a variety of rubber parts and components like engine mounts, hoses, conveyor belts, vibration isolators, laminated and non-laminated bearing pads, silent bushes etc. to determine their response to dynamic loads and cyclic loading.
Personalized consultation from AdvanSES engineers can streamline testing and provide the necessary tools and techniques to accurately evaluate material performance under field service conditions.
The quantities of interest for measurements are tan delta, loss modulus, storage modulus, phase etc. All of these properties are viscoleastic properties and require instruments, techniques and measurement practices of the highest quality.
ASTM D5992 covers the methods and process available for determining the dynamic prop- erties of vulcanized natural rubber and synthetic rubber compounds and components. The standard covers the sample shape and size requirements, the test methods, and the pro- cedures to generate the test results data and carry out further subsequent analysis. The methods described are primarily useful over the range of temperatures from cryogenic to 200◦C and for frequencies from 0.01 to 100 Hz, as not all instruments and methods will accommodate the entire ranges possible for material behavior.
Figures(.43and.44) show the results from a frequency sweep test on five (5) different elastomer compounds. Results of Storage modulus and Tan delta are plotted.
Figure .43: Plot of Storage Modulus Vs Frequency from a Frequency Sweep Test
The frequency sweep tests have been carried out by applying a pre-compression of 10 % and subsequently a displacement amplitude of 1 % has been applied in the positive and negative directions. Apart from tests on cylindrical and square block samples ASTM D5992 recommends the dual lap shear test specimen in rectangular, square and cylindri- cal shape specimens. Figure (.45) shows the double lap shear shapes recommended in the standard.
Figure .44: Plot of Tan delta Vs Frequency from a Frequency Sweep Test
Polymeric rubber components are widely used in automotive, aerospace and biomedical systems in the form of vibration isolators, suspension components, seals, o-rings, gaskets etc. Finite element analysis (FEA) is a common tool used in the design and development of these components and hyperelastic material models are used to describe these polymer materials in the FEA methodology. The quality of the CAE carried out is directly related to the input material property and simulation technology. Nonlinear materials like polymers present a challenge to successfully obtain the required input data and generate the material models for FEA. In this brief article we review the limitations of the hyperleastic material models used in the analysis of polymeric materials.
Theory:
A material model describing the polymer as isotropic and hyperelastic is generally used and a strain energy density function (W) is used to describe the material behavior. The strain energy density functions are mainly derived using statistical mechanics, and continuum mechanics involving invariant and stretch based approaches.
Statistical Mechanics Approach
The statistical mechanics approach is based on the assumption that the elastomeric material is made up of randomly oriented molecular chains. The total end to end length of a chain (r) is given by
Where µ and lm are material constants obtained from the curve-fitting procedure and Jelis the elastic volume ratio.
Invariant Based Continuum Mechanics Approach
The Invariant based continuum mechanics approach is based on the assumption that for a isotropic, hyperelastic material the strain energy density function can be defined in terms of the Invariants. The three different strain invariants can be defined as
I1 = l12+l22+l32
I2 = l12l22+l22l32+l12l32
I3 = l12l22l32
With the assumption of material incompressibility, I3=1, the strain energy function is dependent on I1 and I2 only. The Mooney-Rivlin form can be derived from Equation 3 above as
With C01 = 0 the above equation reduces to the Neo-Hookean form.
Stretch Based Continuum Mechanics Approach
The Stretch based continuum mechanics approach is based on the assumption that the strain energy potential can be expressed as a function of the principal stretches rather than the invariants. The Stretch based Ogden form of the strain energy function is defined as
where µiand αi are material parameters and for an incompressible material Di=0.
Neo-Hookean and Mooney-Rivlin models described above are hyperelastic material models where, the strain energy density function is calculated from the invariants of the left Cauchy-Green deformation tensor, while in the Ogden material model the strain energy density function is calculated from the principal deformation stretch ratios.
The Neo-Hookean model, one of the earliest material model is based on the statistical thermodynamics approach of cross-linked polymer chains and as can be studied is a first order material model. The first order nature of the material model makes it a lower order predictor of high strain values. It is thus generally accepted that Neo-Hookean material model is not able to accurately predict the deformation characteristics at large strains.
The material constants of Mooney-Rivlin material model are directly related to the shear modulus ‘G’ of a polymer and can be expressed as follows:
G = 2(C10+ C01) …………………………….…(6)
Mooney-Rivlin model defined in equation (4) is a 2nd order material model, that makes it a better deformation predictor that the Neo-Hookean material model. The limitations of the Mooney-Rivlin material model makes it usable upto strain levels of about 100-150%.
Ogden model with N=1,2, and 3 constants is the most widely used model for the analysis of suspension components, engine mounts and even in some tire applications. Being of a different formulation that the Neo-Hookean and Mooney-Rivlin models, the Ogden model is also a higher level material models and makes it suitable for strains of upto 400 %. With the third order constants the use of Ogden model make it highly usable for curve-fitting with the full range of the tensile curve with the typical ‘S’ upturn.
Discussion and Conclusions:
The choice of the material model depends heavily on the material and the stretch ratios (strains) to which it will be subjected during its service life. As a rule-of-thumb for small strains of approximately 100 % or l=2.0, simple models such as Mooney-Rivlin are sufficient but for higher strains a higher order material model as the Ogden model may be required to successfully simulate the ”upturn” or strengthening that can occur in some materials at higher strains.
REFERENCES:
ABAQUS Inc., ABAQUS: Theory and Reference Manuals, ABAQUS Inc., RI, 02
Attard, M.M., Finite Strain: Isotropic Hyperelasticity, International Journal of Solids and Structures, 2003
Bathe, K. J., Finite Element Procedures Prentice-Hall, NJ, 96
Bergstrom, J. S., and Boyce, M. C., Mechanical Behavior of Particle Filled Elastomers,Rubber Chemistry and Technology, Vol. 72, 2000
Beatty, M.F., Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers and Biological Tissues with Examples, Applied Mechanics Review, Vol. 40, No. 12, 1987
Bischoff, J. E., Arruda, E. M., and Grosh, K., A New Constitutive Model for the Compressibility of Elastomers at Finite Deformations, Rubber Chemistry and Technology,Vol. 74, 2001
Blatz, P. J., Application of Finite Elasticity Theory to the Behavior of Rubber like Materials, Transactions of the Society of Rheology, Vol. 6, 196
Kim, B., et al., A Comparison Among Neo-Hookean Model, Mooney-Rivlin Model, and Ogden Model for Chloroprene Rubber, International Journal of Precision Engineering & Manufacturing, Vol. 13.
Boyce, M. C., and Arruda, E. M., Constitutive Models of Rubber Elasticity: A Review, Rubber Chemistry and Technology, Vol. 73, 2000.
Srinivas, K., Material Characterization and FEA of a Novel Compression Stress Relaxation Method to Evaluate Materials for Sealing Applications, 28th Annual Dayton-Cincinnati Aerospace Science Symposium, March 2003.
Srinivas, K., Material Characterization and Finite Element Analysis (FEA) of High Performance Tires, Internation Rubber Conference at the India Rubber Expo, 2005.
