On the Behaviour of Hyperelastic Materials, a Mooney-Riviln approach


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On the Behaviour of Hyperelastic Materials, a Mooney-Riviln approach

aM. Ramamohan Rao

Department of Mechanical Engineering GITAM, Visakhapatnam

bProf. M. R. S. Satayanarayana Department of Mechanical Engineering GITAM, Visakhapatnam

AbstractThis paper emphasizes the superior elastic properties attained, and behavior of the Hyperelastic materials which are basically natural rubber composites with certain polymers. The most elegant aspect is the wide range of properties of these materials finding universal applications in critical and complex environments viz hydrophones and other under water warfare systems. All structural materials used in engineering components have linear elastic properties in the strain range of 10%, in contrast the elastomeric materials possessing high rate of strains even up to 300%. This special category of materials finds their application in shock mounts, Transducer assemblies (Hydrophones), Medical equipment and rubber seals etc. Analysis of these materials need special attention, as mechanical behavior exceeds linear elastic theory. FEA programs involve parameters with Mooney constants in the strain energy potential. In the present investigation two types of rubber material models are studied viz polyurethane (PU) and Neoprene for determining the Mooney constants which are needed for FEA Program.

Keywords Neoprene;Polyurethane(PU); Strain Energy; tensile testing;

  1. INTRODUCTION

    Attention of researchers due to ever increasing demands and for modeling Elastomeric materials behavior under mechanical and geometrical boundary conditions gained popularity in recent times. It is common practice to characterize the mechanical behavior of these materials represent the constitutive equation through a strain energy density function. Hyperelastic materials are often considered for various industrial applications, due to their remarkable properties of flexibility, recovery after load release and resistance to high deformation levels. Many attempts have been made to develop more general hyperelastic models to include different aspects of materials behavior. Henky H [1] derived the elastic behavior of hyperelastic materials, large extensions up to 270 % analytically by a simple function. The deformed and unreformed stresses are the functions of two constants which are bulk modulus and logarithmic extension ratio. The deformed state of stress in tension is high when compared to unreformed state of stress, whereas in case of compression, the deformed state of stress is less when compared with unreformed state of stress. Ellen M. Arruda [2] states that, no existing model which accurately represents the behavior of hyperelastic materials in various deformation states and satisfies the criterion of requiring only a small number of physically based parameters or constants. The main condition is any constitutive model constants should be independent of deformation state to provide predictive capability.

    In Structural Analysis usually elastic material properties are expressed in terms of strain energy density function. This strain energy function into a separable form related to the principal directions is derived by the Mooney [3] and Rivlin [4]. This advance led to the Ogden model [5] which is largely used today. Strain energy density function is expressible in the form of even powered series of the principal stretches. A variety of strain energy density functions have been extracted from Rivlins model. Strain energy density is sum of independent functions of the principal stretches for incompressible materials.

    The strain invariants and coefficients are required for strain energy density function for rubber like materials as determined by D.W. Haines [6]. The strain-energy function is expressed as a power series of invariants. The material models formulated in invariants of the strain tensor are based on a series approach in different powers of the first and second basic invariant. Because of the incompressibility of the material, the third basic invariant is constant and, hence, does not contribute to the stored energy. Formulations of the strain energy function based on Eigen values were presented by Ogden [5]. These models show a good adaptability to the experimental data resulting from the high degree of non- linearity. The examples of these strain energy density functions have been presented in the references [79]. The computations involved in the nonlinear equations of the problem must be set into an appropriate quadratic form for obtaining the optimal efficiency. This can be obtained in general by introducing additional variables and/or differential relations between the variables. In the context of hyperelastic models, strongly nonlinear terms are involved, such as logarithmic or fractional functions.

    Two nonlinear materials viz PU and Neoprene have been modeled by using uni-axial tension/compression test in the present investigation, and the material constants are determined through least-squares-fit procedures. In order to estimate the constants that best fits the curve-fit, a program was written in MATLAB optimization solver which inputs experimental stress-strain data and constraints. To minimize the errors of the program and also to know the closeness of the stress-strain curve with the expected constants an RMS function is created and applied for determining the 2-Mooney, 5-Mooney and 9-Mooney constants.

  2. HYPERELASTIC MATERIALS ANALYSIS

    Strain Energy Density functions requires a number of mathematical constitutive theories of nonlinear, large elastic

    deformations. These theories, coupled with the numerical method, can be used very effectively by the user to analyze the elastomer products operating under highly deformed states. According to this theory in large deformation, rubber is assumed isotropic in elastic behavior and very nearly incompressible. The elastic properties of rubber can be explained in terms of a strain energy function based on the strain invariants I1, I2 and I3. These invariants are as the function of extension ratios 1,2 and 3 have following properties in equations 1,2 and 3.

    (1)

    (2)

    (3)

    The incompressible materials require the third invariant which is equal to one, and hence eqs. 1,2 and 3 can further be reduced to eqs. 4 and 5. This is because when the material is compressible the third invariant becomes equal to one, and hence the third stretch ratio can be expressed as a function of the first two in equation 7. Next, when a load is only applied in one principal direction as in the case of uni-axial loading, the second stretch ratio (2) is equal and to the third stretch ratio (3). Thus invariants can be expressed as a function of only two stretch ratios and Equation 8 can be expressed as a direct relationship of the first and second stretch ratio [10].

    (4)

    (5)

    =

    (6)

    Which implies

    (7)

    implies

    (8)

    (18)

    (19)

    (18)

    (19)

    A. Mooney-Rivlin Approach

    It is Shown that in the generalised Mooney's approach [3] most general strain energy function for a homogenous, isotropic, incompressible and elastic material is in eq. 9

    The five constnt mooney-Rivlin function in eq. 11 can obtains if N=2 in Mooney generalized eq. 9

    (

    11)

    To evaluate constants redefining the strain energy density function is needed. The equations 12 to 15 were obtained because , where i is the different axis [4] and of 3 is equal to zero

    (12)

    (13)

    (14)

    (15)

    In case of uni-axial test and to get the stress function a derivation is necessary and a multiplication with is required. The equations 16 and 17 for obtained for 5-Mooney equations by deriving equation 11 w.r.t to I1 and I2.

    (16)

    (17)

    The derivatives of the invariants w.r.t streatch ratios are given in equation number 18 and 19

    Where W is the strain energy density function, I1 and I2 are the measure of distortion in the material, Cij describes the shear behavior of the material, Di introduces the material incompressibility and Jel is the elastic volume strain.

    The two constant mooney-Rivlin function in eq. 10 obtains if N=1 in Mooney generalized eq. 9

  3. EXPERIMENTAL STUDY Experiments were conducted by using Uni-axial

    (9

    )

    (9

    )

    Tensile Test Machine as shown in Fig. 1 having 10KN load cell, it could be used to calculate the Tensile Strength, Shear strength and Adhesive Strength.

    (10)

    Fig 1: Polyurethane Dumbbell type Specimen & Fixing position in M/s INSTRON

    The polyurethane material specimen of 75 mm Gauge Length and Area 6.6 mm2 was considered basing on the guidelines given in ASTM standards (ATM D412). The specimen is fixed between the two grips. During the test, the upper grip moves up at a speed of 500mm/min velocity in the upward direction (axial direction).

    1. Uni-axial Tensile test Specimen dimensions.

      Plain rubber sheet is used for the specimens employing a cutting die, manufactured from the standard dimensions of a stock of length 115 mm width 25 mm and thickness 2.5 mm. The sample specimen is shown below fig 2. For each material namely PU and Neoprene four specimens are prepared as shown in Fig. 3 to Fig. 4 and the tests were carried out and Experimental results are given in table I.

      Fig 2: Dimensions of Dumbbell specimen

      Fig :3 PU Dumbbell specimen

      Fig: 4 NEOPRENE Dumbbell Specimens TABLE 1: EXPERIMENTAL RESULTS

      MODEL

      Width (mm)

      Thk (mm)

      Load Max (N)

      Stress Max (MPa)

      Strain Max

      PU

      5.81

      1.26

      178

      24.39

      4.743

      5.85

      1.27

      176

      23.386

      5.076

      5.80

      1.32

      175

      22.92

      4.99

      Neoprene

      5.68

      2.36

      168.6

      12.57

      4.6

      5.6

      2.53

      163

      11.64

      3.9

      5.6

      2.33

      174.6

      13.2

      4.73

      MODEL

      Width (mm)

      Thk (mm)

      Load Max (N)

      Stress Max (MPa)

      Strain Max

      PU

      5.81

      1.26

      178

      24.39

      4.743

      5.85

      1.27

      176

      23.386

      5.076

      5.80

      1.32

      175

      22.92

      4.99

      Neoprene

      5.68

      2.36

      168.6

      12.57

      4.6

      5.6

      2.53

      163

      11.64

      3.9

      5.6

      2.33

      174.6

      13.2

      4.73

  4. RESULTS AND CONCLUSIONS

    1. Results

      From the practical data of two different test specimens the Mooney constants are extracted as given in table II an table

      III. In order to estimate the best constant that fits the curve- fit, a program was written, to ensure that the results are as good as possible and the program is created as an optimization solver. The estimated Mooney constants are obtained at minimized RMS value from MATLAB OPTIM tool. The corresponding stress strain diagrams and curve fit for different Mooney constants like 2-Mooney, 5-Mooney and 9-Mooney fits are given for all tested specimens in graphs from Fig. 5 to Fig.12.

      TABLE II: EXTRACTED MOONEY CONSTANTS OF PU FROM MATLAB PROGRAM

      Model

      PU

      constants

      2-Mooney

      5-Mooney

      9-Mooney

      C10(MPa)

      1.636

      -0.452

      -0.452

      C01(MPa)

      0.728

      4.853

      4.853

      C20(MPa)

      0

      0

      C11(MPa)

      0.109

      0.119

      C02(MPa)

      0

      0

      C30(MPa)

      0

      C21(MPa)

      -0.003

      C12(MPa)

      0.003

      C03(MPa)

      0

      Bulk Modulus

      23640

      44010

      44010

      Error %

      104.4

      29.15

      29.15

      TABLE III: EXTRACTED MOONEY CONSTANTS OF NEOPRENE FROM MATLAB PROGRAM

      Model

      Neoprene

      constants

      2-Mooney

      5-Mooney

      9-Mooney

      C10(MPa)

      1.141

      -0.141

      1.073

      C01(MPa)

      -1.141

      0.968

      -0.494

      C20(MPa)

      0.037

      0.008

      C11(MPa)

      -0.03

      0.68

      C02(MPa)

      -0.007

      -0.536

      C30(MPa)

      0

      C21(MPa)

      -0.194

      C12(MPa)

      0.198

      C03(MPa)

      -0.002

      Bulk Modulus

      0

      8270

      5790

      Error %

      122.9566

      15.04

      14.4

    2. PU Curve Fit

      Fig 5: Stress vs. Strain behavior of PU

      Fig 6: Curve fit for PU Material with 2-Mooney Constants

      Fig 7: Curve fit for PU Material with 5-Mooney Constants

      Fig 8: Curve fit for PU Material with 9-Mooney Constants

      Neoprene Curve Fit

Fig 9: Stress vs. Strain behavior of Neoprene

Fig 10: Curve fit for Neoprene Material with 2-Mooney Constants

Fig 11: Curve fit for Neoprene Material with 5-Mooney Constants

Fig 12: Curve fit for Neoprene Material with 9-Mooney Constants

V. CONCLUSIONS

  • PU material showed that they have high elongations up to 500% and has a 104% error for 2-Mooney fit.

  • The observed error in PU material reduced to 29% for 5- Mooney and 9-Mooney curve fit.

  • Further it is seen that Neoprene material has high elongations up to 470% and has 123% error for 2- Mooney fit.

  • Reduction in error for Neoprene is limited only to 15% for 5-Moomey and 9-Mooney curve fit.

REFERENCES

  1. Hencky, H., the Elastic Behavior of Vulcanized Rubber, Rubber Chemistry and Technology, Volume 6, Pages 217-224 (1932)

  2. Ellen, M., Arruda and Mary, C., Boyce,. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, Journal of the Mechanics and Physics of Solids, Volume 41, Issue 2, Pages 38941 (1993)

  3. Mooney, M., A Theory of Large Elastic Deformation, Journal of Applied Physics, Volume 11, Pages 582-592 (1940)

  4. Rivlin, R.S., Large elastic deformations of Isotropic materials. II Some Uniqueness Theorems for Pure, Homogeneous Deformation, Mathematical and Physical Sciences, Volume 240 A, Pages 491-508 (1948)

  5. Ogden R.W., Large deformation isotropic elasticity on the correlation of theory and experiment for incompressible rubber like solids, Volume 326, pp 565-584 (1971)

  6. Haines, D.W. and Wilson W.D., Strain-energy density function for rubber like materials, Journal of the Mechanics and Physics of Solids, Volume 27, Issue 4, Pages 345360. (1979)

  7. Sasso M et al Characterization of hyperelastic rubber-like materials by biaxial and uniaxial stretching tests based on optical methods, Polymer Testing, Elsevier, Volume 27, Pages 9951004 (2008)

  8. Lectez, A.S., Verron, E and Huneau, B., How to identify a hyperelastic constitutive equation for rubber-like materials with multi-axial tension torsion experiments, International Journal of Non-Linear Mechanics, Volume 65, Pages 260270 (2014)

  9. Julie Lambert-Diani and Christian Rey, New phenomenological behavior laws for rubbers and thermoplastic elastomers, European Journal of Mechanics – A/Solids, Volume 18, Issue 6, Pages 1027 1043 (1999)

  10. Oscar Lopez-Pamies., A new I1-based hyperelastic model for rubber elastic materials. ComptesRendusMecanique , Volume 338, Issue 1, Pages 311 (2010)

  11. Leonardo Hoss and Rogerio J. Marczak, A new constitutive models for rubber-like materials, Mechanica Computational, Volume XXIX, pp 2759-2773 (2010)

  12. Xiao-Ming Wang, Hao Li, Zheng-Nan Yin And Heng Xiao., Multiaxial strain energy functions of rubber like Materials: An Explicit approach Based On Polynomial Interpolation, Rubber Chemistry and Technology, Volume 87, Issues 1 Pages 168-183 (2014)

  13. Nah C., Lee G.B., Lim J.Y., Kim Y.H., Sen Gupta R. and Gent A.N., Problems in determining the elastic strain energy function for rubber, International Journal of Non-Linear Mechanics, Volume 45, pp 232 235 (2010)

  14. Numerical and experimental investigation of hyperelastic materials for under water applications By M Ramamohan Rao A Doctrorial Thesis sumitted to GITAM University (2016).

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