The application of computational mechanics analysis techniques to elastomers presents
unique challenges in modeling the following characteristics:
1) The load-deflection behavior of an elastomer is markedly non-linear.
2) The recoverable strains can be as high 400 % making it imperative to use the large
deflection theory.
3) The stress-strain characteristics are highly dependent on temperature and rate effects
are pronounced.
4) Elastomers are nearly incompressible.
5) Viscoelastic effects are significant.
The inability to apply a failure theory as applicable to metals increases the complexities regarding the failure and life prediction of an elastomer part. The advanced material models available today define the material as
hyperelastic and fully isotropic. The strain energy density (W) function is used to describe the material behavior.
To help you better understand, we broke down everything you need to know about materials, testing, FEA verifications and validations etc.
2. Computational Mechanics in the design and development of polymeric components
3. Why there are recommended testing protocols
4. Curve-fitting the Material Constants.
5. Verifications and Validations of FEA Solutions
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Abaqus – Tips and Tricks: When to use what Elements?
For a 3D stress analysis, ABAQUS offers different typess of linear and quadratic hexahedral elements, a brief description is as below;
Linear Hexahedral: C3D8 further subdivided as C3D8R, C3D8I, and C3D8H
Quadratic Hexahedral: C3D20 further subdivided as C3D20R, C3D20I, and C3D20H, C3D20RH
Linear Tetrahedral: C3D4 further subdivided as C3D4R, and C3D4H
Quadratic Tetrahedral: C3D10 further subdivided as C3D10M, C3D10I and C3D10MH
Prisms: C3D6 further subdivided as C3D6R, and C3D6H
In three-dimensional (3D) finite element analysis, two types of element shapes are commonly utilized for mesh generation: tetrahedral and hexahedral. While tetrahedral meshing is highly automated, and relatively does a good job at predicting stresses with sufficient mesh refinement, hexahedral meshing commonly requires user intervention and is effort intensive in terms of partitioning. Hexahedral elements are generally preferred over tetrahedral elements because of their superior performance in terms of convergence rate and accuracy of the solution.
The preference for hexahedral elements(linear and uadratic) can be attributed to the fact that linear tetrahedrals originating from triangular elements have stiff formulations and exhibit the phenomena of volumetric and shear locking. Hexahedral elements on the other hand have consistently predicted reasonable foce vs loading (stiffness) conditions, material incompressibility in friction and frictionless contacts. This has led to modeling situations where tetrahedrals and prisms are recommended when there are frictionless contact conditions and when the material incompressibility conditiona can be relaxed to a reasonable degree of assumption.
A general rule of thumb is if the model is relatively simple and you want the most accurate solution in the minimum amount of time then the linear hexahedrals will never disappoint.
Modified second-order tetrahedral elements (C3D10, C3D10M, C3D10MH) all mitigate the problems associated with linear tetrahedral elements. These element offer good convergence rate with a minimum of shear or volumetric locking. Generally, observing the deformed shape will show of shear or volumetric locking and mesh can be modified or refined to remove these effects.
C3D10MH can also be used to model incompressible rubber materials in the hybrid formulation. These variety of elements offer better distribution of surface stresses and the deformed shape and pattern is much better. These elements are robust during finite deformation and uniform contact pressure formulation allows these elements to model contact accurately.
The following are the recommendations from the house of Abaqus(1);
Minimize mesh distortion as much as possible.
A minimum of four quadratic elements per 90o should be used around a circular hole.
A minimum of four elements should be used through the thickness of a structure if first-order, reduced-integration solid elements are used to model bending.
Abaqus Theory and Reference Manuals, Dassault Systemes, RI, USA
Finite element analysis is widely used in the design and analysis of elastomer components in the automotive and aerospace industry. Numerous publications [7-11] addressed the applications of FEA and have established the method as a reliable tool to predict stress analysis parameters under different loading conditions. In this report we have studied the different material models available to simulate elastomer behavior. Nonlinear Finite element code Abaqus® was used to develop the 2D Axisymmetric and 3D models. Generalized continuum axisymmetric and hexahedral elements were used to model the structures in two and three dimensions using these hyperelastic material models. Testing of materials to characterize the rubber compounds has been carried out in-house and the material constants have been developed using the regression analysis based curve-fitting procedure.
MATERIAL TESTING AND CHARACTERIZATION PROCEDURE:
The application of computational mechanics analysis techniques to elastomers presents unique challenges in modeling the following characteristics:
The load-deflection behavior 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.
Elastomers are nearly incompressible.
Viscoelastic effects are significant.
Finite element codes like Abaqus® Ansys®, LS-Dyna® and MSC-Marc® generally require the input of test data covering the maximum range, which the elastomer product experiences in service conditions. Test data from the main principal deformation modes are generally used as shown in Figure 1. When designing a product from scratch all the four tests can be used to generate the constants for the design but for failure analysis one may not have enough material to carry out all the tests.
Figure 1: AdvanSES Material Characterization Tests
STRAIN ENERGY FUNCTIONS:
In the FE analysis of elastomeric materials, the material is characterized by using different forms of the strain energy density function. The strain energy density of a solid can be defined as the work done per unit volume to deform a material from a stress free reference or original state to a final state. The strain energy density functions have been derived using a 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 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 ai are material parameters and for an incompressible material Di=0.
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 lb2, 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.
Figure 2 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 material testing system (MTS) and results verified computationally. Figure 2 shows a bushing on a testing jig, and the plot 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 2: Product Testing and FEA Model Verification Results
REFERENCES:
1) Gent, N. A., Engineering with Rubber: How to Design Rubber Components Hanser Publishers, NY, 92
2) Srinivas, K., Material Characterization and Finite Element Analysis (FEA) of High Performance Tires, Internation Rubber Conference at the India Rubber Expo, 2005.
3) ABAQUS Inc., ABAQUS: Theory and Reference Manuals HKS Inc., RI, 02
4) Bathe, K. J., Finite Element Procedures Prentice-Hall, NJ, 96
5) Dowling, N. E., Mechanical Behavior of Materials, Engineering Methods for
Deformation, Fracture and Fatigue Prentice-Hall, NJ, 99
6) Morman, K., and Pan, T. Y. Application of Finite Element Analysis in theDesign
of Automotive Elastomeric Components, Rubber Chemistry and Technology, 1988
7) Gall, R. et al. Some Notes on the Finite Element Analysis of Tires, Tire Science and Technology, Vol. 23 No. 3, 1995
8) Surendranath, H. and Kuessner, M. Assessment using Finite Element Analysis, Tire Technology International, 2003
9) Zhang, X., Rakheja, S., and Ganesan, R. Stress Analysis of the Multi-Layered System of a Truck Tire, Tire Science and Technology, Vol. 30 No. 4, 2002
10) 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.
