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Dynamic Properties of Polymer, Rubber and Elastomer Materials

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 cate- gorized 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.

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Limitations of Hyperelastic Material Models

Limitations of Hyperelastic Material Models

Introduction:

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 Jel is 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

W(I1,I2) = C10 (I1-3) + C01 (I2–3)…………………………………………………………(4)

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 µi and 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:

  1. ABAQUS Inc., ABAQUS: Theory and Reference Manuals, ABAQUS Inc., RI, 02
  2. Attard, M.M., Finite Strain: Isotropic Hyperelasticity, International Journal of Solids and Structures, 2003
  1. Bathe, K. J., Finite Element Procedures Prentice-Hall, NJ, 96
  2. Bergstrom, J. S., and Boyce, M. C., Mechanical Behavior of Particle Filled Elastomers,Rubber Chemistry and Technology, Vol. 72, 2000
  3. Beatty, M.F., Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers and Biological Tissues with Examples, Applied Mechanics Review, Vol. 40, No. 12, 1987
  4. 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
  5. Blatz, P. J., Application of Finite Elasticity Theory to the Behavior of Rubber like Materials, Transactions of the Society of Rheology, Vol. 6, 196
  6. 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.
  7. Boyce, M. C., and Arruda, E. M., Constitutive Models of Rubber Elasticity: A Review, Rubber Chemistry and Technology, Vol. 73, 2000.
  8. 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.
  9. Srinivas, K., Material Characterization and Finite Element Analysis (FEA) of High Performance Tires, Internation Rubber Conference at the India Rubber Expo, 2005.
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Fatigue Design Optmization of Torque Arm Bush Mount for Heavy Truck Applications

Design Development and Finite Element Analysis (FEA) of Torque Arm Bush Mount for Heavy Truck Applications

Abstract:

A Torque Arm Bush is a metal-elastomer bonded component that forms an integral part of a heavy truck bogie or suspension system. Many different designs exist in the market today and each one with its own unique geometry, material and load application conditions. This analysis demonstrates the hyperelastic material characterization testing, material constant generation and FEA on the component to predict the service performance.

Methodology:

The physics involved in the simulation are complex and can be summarized as follows:

  1. Elastomer performance is markedly non-linear.
  2. Loading conditions like axial, radial, conical, torsional must be defined in multiple steps as per the service conditions and loading cycles.
  3. Large strain deformation with contacts

Figure 1: Hyperelastic Material Characterization Testing               Figure 2: FEA Model of the Torque Arm Bush Mount Assembly

Approach:

  • Material Study and Characterization to understand static and dynamic material properties.
  • Develop material constants and design concepts based on load-deflection and performance characteristics.
  • Use Finite Element Analysis (FEA) to optimize the design and understand FMEA.
  • Provide assembly modeling & drawings for prototype manufacturing.

Figure 3: Shows the Comparison between FEA and Experimental Testing Results

Figure 4: Deformed Shape and Stress-Strain Distribution in the Torque Arm Bush Mount

Results and Discussion:

The principal deformation modes of a heavy duty suspension component were modeled in Abaqus using hyperelastic analysis. High stresses were noted along the curvature locations in the design under conical deformations and confirmed by fatigue testing. This  locations were identified as ‘hot-spots’ and are fatigue-critical locations. The geometrical and material parameters were optimized to better mitigating the stresses and reduce the fatigue failures.

References:

  1. Dassault Systemes, Abaqus theory and reference manuals
  2. Yunhi, Yu, Nagi G Naganathan, Rao V Dukkipati, A literature review of automotive vehicle engine mounting systems, Mechanism and Machine Theory Volume 36, Issue 1, January 2001.
  3. Srinivas, K., Material Characterization And CAE For Non-Metallic Materials & Manufacturing Processes, SAE Symposium on CAE Applications for Automotive Structures, Detroit, November 2005.

Technicals:

  • Advanced Softwares like Abaqus, Static testing machines are available in-house and design iterations can be carried out on the fly.
  • Full material characterization capabilities of polymeric materials for FEA
  • Capabilities for fatigue durability testing In-house.
  • Advanced material testing facilities like DMA, DSC, TGA and TMA also available.

 

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