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Download-D638.Pdf‎>.Access In: 08/14/2014 Materials Research. 2015; 18(5): 918-924 © 2015 DOI: http://dx.doi.org/10.1590/1516-1439.320414 Mechanical Characterization and FE Modelling of a Hyperelastic Material Majid Shahzada*, Ali Kamranb, Muhammad Zeeshan Siddiquia, Muhammad Farhana aAdvance Materials Research Directorate, Space and Upper Atmosphere Research Commission – SUPARCO, Karachi, Sindh, Pakistan bInstitute of Space Technology – IST, Islamabad Highway, Islamabad, Pakistan Received: August 25, 2014; Revised: July 29, 2015 The aim of research work is to characterize hyperelastic material and to determine a suitable strain energy function (SEF) for an indigenously developed rubber to be used in flexible joint use for thrust vectoring of solid rocket motor. In order to evaluate appropriate SEF uniaxial and volumetric tests along with equi-biaxial and planar shear tests were conducted. Digital image correlation (DIC) technique was utilized to have strain measurements for biaxial and planar specimens to input stress-strain data in Abaqus. Yeoh model seems to be right choice, among the available material models, because of its ability to match experimental stress-strain data at small and large strain values. Quadlap specimen test was performed to validate material model fitted from test data. FE simulations were carried out to verify the behavior as predicted by Yeoh model and results are found to be in good agreement with the experimental data. Keywords: hyperelastic material, DIC, Yeoh model, FEA, quadlap shear test 1. Introduction Hyperelastic materials such as rubber are widely used The selection of suitable SEF depends on its application, for diverse structural applications in variety of industries corresponding variables and available data for material ranging from tire to aerospace. The most attractive property parameter identification5. Chagnon et al.6 described four of rubbers is their ability to experience large deformation main qualities of an efficient hyperelastic material model under small loads and to retain initial configuration without as follows: considerable permanent deformation after load is removed1. ▪ It should have the ability to exactly reproduce the entire Their stress-strain behaviour is highly non-linear and a simple ‘S’ shaped response of rubber modulus of elasticity is no longer sufficient. Therefore, characterization of elastic behaviour of highly extensible, ▪ The change of deformation mode should not be nonlinear materials is of great importance2. problematic, i.e. if the model operates satisfactorily in The constitutive behaviour of hyperelastic material is uniaxial tension, it must also be satisfactory in simple shear or in equibiaxial extension derived from SEF ‘W’ based on three strain invariants I1, I2 and I . It is the energy stored in material per unit of reference 3 ▪ The number of fitting material parameters should be volume (volume in the initial configuration) as a function small, in order to decrease the number of experimental 3 of strain at that point in material . tests needed for their determination W= fII( 123,, I) (1) ▪ The mathematical formulation should be simple to render possible the numerical implementation of the where I , I and I are three invariants of Green deformation 1 2 3 model. tensor defined in terms of principal stretch ratios 1λ , λ2 and 4 λ3 given by : For a precise prediction of rubber behaviour, used in an assembly (e.g. flexible joint), by finite element simulation, I =++λλλ222 11 2 3 it should be tested under same loading conditions to which 22 22 22 (2) I2122331=++λλ λλ λλ original assembly will be subjected. The uniaxial tests are 222 easy to perform and are well understood7 but uniaxial data I3= λλλ 123 alone does not produce reliable set of coefficients for material Generally, hyperelastic materials are considered models8, especially if the original assembly experiences incompressible, i.e. I3 = 1; hence only two independent complex stress states. Therefore, biaxial, planar (pure shear) strain measures namely I1 and I2 remain. This implies that and volumetric tests need to be performed along with a 2 ‘W’ is a function of I1 and I2 only ; uniaxial tension test to incorporate the effects of multiaxial stress states in the model. (3) W= W( I12 −− 3I, 3) For specific applications, the tailoring of rubber mechanical *e-mail: [email protected] properties is carried out by the addition of various chemicals. 2015; 18(5) Mechanical Characterization and FE Modelling of a Hyperelastic Material 919 Minor changes in chemical composition can alter mechanical 2.4. Reduced Polynomial model properties significantly. Therefore, it is essential to test a This model does not include any dependency on I . particular rubber composition and simulate through FEA to 2 The sensitivity of strain energy function to variation in I is have an apposite SEF. Once determined, this can be used for 2 generally much smaller than the sensitivity to variation in simulating the assembly in which particular rubber has been I . It appears that eliminating the terms containing I from used. FE software packages like Abaqus offer a number 1 2 strain energy function improves the ability of the models to of SEFs to accommodate the nonlinear behavior of rubber predict the behaviour of complex deformation states when and other hyperelastic materials. This study shows that Yeoh limited test data is available12. The Neo-Hookean form is model has an advantage over other available material models first order reduced polynomial model. because of its good match with experimental data over large strain values for given rubber composition. 2.5. Yeoh model In 1993, Yeoh13 proposed a phenomenological model in the 2. Hyperelastic Models in Abaqus form of third-order polynomial based only on first invariant In Abaqus , two types of hyperelastic material models are I1. It can be used for the characterization of carbon-black available and each model defines the strain energy function filled rubber and can capture upturn of stress-strain curve. in a different way9. One is the phenomenological models It has good fit over a large strain range and can simulate which treat the problem from the viewpoint of continuum various modes of deformation with limited data. This leads mechanics and stress-strain behaviour is characterized to reduced requirements for material testing14. The Yeoh without reference to the microscopic structure. Other one model is also called the reduced polynomial model and for is physically motivated models which consider the material compressible rubber can be given as response from the viewpoint of microstructure. A brief i review about the hyperelastic models available in Abaqus 331 2i W=∑∑ CI3i0( 1 −+) ( Jel − 1) (7) exploited during this study is given below. i1= i1= Di 2.1. Mooney-Rivlin model 2.6. Ogden model Two parameters phenomenological model that works Proposed in 1972 by Ogden15,16, this is also a well for moderately large stains in uniaxial elongation and phenomenological model and is based on principal stretches shear deformation10,11. But, it cannot capture the upturn instead of invariants. The model is able to capture upturn (S-curvature) of the force-extension relation in uniaxial test (stiffening) of stress-strain curve and models rubber accurately and the force-shear displacement relation in shear test. For a for large ranges of deformation. Attention should be paid compressible rubber, model has a form not to use this model with limited test (e.g. just uniaxial tension). A good agreement has been observed between 1 2 WC=( I −+ 3) C( I −+ 3) ( J − 1) (4) Ogden model and Treloar’s17 experimental data for unfilled 10 1 01 2 D el 1 rubber for extensions up to 700%. 2.2. Neo-Hookean model NN2µ ααα 1 2i =i λλλiii ++−+ − (8) W∑∑2 ( 1233) ( J1el ) This model is a special case of Mooney-Rivlin form with i1= αi i1= Di C01= 0 and can be used when material data is insufficient. It is simple to use and can make good approximation at λi is the deviatoric principal stretch and μi, αi are relatively small strains. But, it too cannot capture the upturn temperature dependent material properties. Abaqus allows of stress strain curve. up to N = 6 terms in the above form. 1 2 WC=( I −+ 3) ( J − 1) (5) 2.7. Arruda-Boyce model 10 1 D el 1 Based on molecular chain network, it is also called 2.3. Full Polynomial model Arruda-Boyce 8-chain model because it was developed based on representative volume (hexahedron) element For isotropic and compressible rubber, polynomial where 8 chains emanate from the center to the corners of model can be given as the volume18. This is two parameter shear model based only j 2i on I and works well with limited test data. NNi 1 1 W=∑∑ CIij( 1 − 3) ( I 2 −+ 3) ( Jel − 1) (6) 2 ij, = 0 i= 1Di 5 C 1 J1− W=µ i I3ii −+el −ln(J ) (9) ∑ 2i− 2 ( 1 ) el Where: i1= λm D2 Cij= material constants that control the shear behaviour and can be determined from uniaxial, biaxial and planar tests. 3. Experimental Details Di = material constant that control bulk compressibility and set to zero for fully incompressible rubber. It can be 3.1. Material estimated from volumetric test. Natural rubber reinforced by carbon-black was used for Jel = Elastic volume ratio this research work whose chemical composition is given N = Number of terms in strain energy function. Abaqus in Table 1. The nominal rubber mechanical properties as allows up-to 6 terms. determined by various tests are given in Table 2. 920 Shahzad et al. Materials Research Table 1. Chemical composition of rubber. Natural Rubber Carbon-Black Stearic Acid Zinc Oxide MBT Sulphur-80 (gms) (gms) (gms) (gms) (gms) (gms) 100 40 6 3 1 1 Table 2. Mechanical properties of rubber calculated by different tests. Tensile % Shear Shear Strength Elongation Strength Modulus (MPa) (MPa) (MPa) 12 610 4.6 0.36 3.2.
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