WITHDRAWN: General constitutive solutions for isotropic hyperelastic materials

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General constitutive solutions for isotropic hyperelastic materials Fuzhang Zhao

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S0022-5096(13)00147-6 http://dx.doi.org/10.1016/j.jmps.2013.08.002 MPS2333

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Received date: 12 June 2013 Accepted date: 7 August 2013 Cite this article as: Fuzhang Zhao, General constitutive solutions for isotropic hyperelastic materials, Journal of the Mechanics and Physics of Solids, http://dx.doi. org/10.1016/j.jmps.2013.08.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

General Constitutive Solutions for Isotropic Hyperelastic Materials Fuzhang Zhao∗ Innova Engineering, Inc. 1 Park Plaza, Suite 980, Irvine, CA 92614, USA

Abstract A general strain energy function in the form of a partial differential equation has been derived based on a balance principle between strain energy and stress work done using a continuum approach. The partial differential equation in terms of three invariants of right Cauchy-Green tensor has then been solved by Lie group methods. With the requirements of translational, rotational, and volumetric deformations, growth conditions, geometric curvatures, and numerical stability, the general solution is boiled down to a particular three-term solution for isotropic hyperelastic materials. The three terms include the I1 c1 term√for monitoring constant slopes of principal translational deformations, the I2 c2 term for capturing curvatures of rotational deformations, and the I14 /I3 c3 term for describing the curvatures of ellipsoidal deformations. The stress tensors and tangent tensors have been worked out for both compressible and incompressible materials. Three coefficients, c1 , c2 , and c3 , have been determined from Treloar’s uniaxial tension data using a linear least square method and applied to predict the pure shear and equibiaxial tension modes. Keywords: Constitutive relationship, Hyperelastic materials, Ellipsoidal deformations, Strain energy function, Stress work done 1. Introduction Many materials, including rubbers, elastomers, and biological tissues, can be characterized as being homogeneous, isotropic, incompressible or nearly ∗

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Preprint submitted to Journal of the Mechanics and Physics of Solids

August 16, 2013

incompressible, finite and nonlinear elastic continuum solids. Constitutive models play an important role in designs and analyses of rubber-like material components. The established models may be classified into two main categories: micro-mechanical and phenomenological. Phenomenological models can be subdivided into two groups: stretch-based and invariant-based. Micro-mechanical models include 3-chain, 8-chain, unit sphere, full-network, Flory-Erman, tube, and extended tube models. Stretch-based phenomenological models cover Ogden, Shariff, and Attard models. Invariant-based phenomenological models include Neo-Hooke, Mooney-Rivlin, Isihara, GentThomas, Swanson, Yeoh, Arruda-Boyce, Gent, Yeoh-Fleming, Carroll, HartSmith, Alexander, van der Waals, Pucci - Saccomandi, and Lopez-Pamies models. To validate constitutive models for rubber-like materials, experimental data from uniaxial tension, biaxial tension, and pure shear tests are usually used. Treloar’s experimental data for vulcanized rubber containing 8% sulfur has been adopted as a benchmark for rubber-like materials (Treloar, 1944). The established models mentioned above have been throughly reviewed and numerically curve-fitted to Treloar’s experimental data by Boyce and Arruda (2000), Steinmann et al. (2012), and Hossain and Steinmann (2013). In addition, Holzapfel, Fung, and other phenomenological models are briefly reviewed and frequently used for anisotropic biological tissues by Holzapfel et al. (2000). For constitutive models, great progress has been made in continuum mechanics. The thermodynamic foundations of constitutive equations at finite strains are archived in the classic treatise on continuum mechanics by Truesdell and Noll (1965). Advanced mathematical foundations of elasticity are presented in the monographs by Marsden and Hughes (1983), Ogden (1984), and Holzapfel (2000). A key assumption regarding rubber-like materials is hyperelasticity such that the deformation is finitely elastic and path independent. For isothermal and adiabatic processes, the strain energy function reduces to a function of only the Green-Lagrange strain tensor, E, as Ψ = Ψ(E).

(1)

The Green-Lagrange strain tensor is defined by E=

1 (C − I) , 2 2

(2)

in which C = FT F is the right Cauchy-Green tensor, F is the deformation gradient tensor, and I is the identity tensor (Malvern, 1969). With Clausius-Duhem inequality, applying (2), the second Piola-Kirchhoff stress tensor, S, takes the form of ∂Ψ(E) ∂Ψ(C) =2 . (3) ∂E ∂C When hyperelastic materials under deformation is isotropic, the strain energy function depends on the invariants of right Cauchy-Green tensor, which was established by the classical work of Rivlin (1948) S=

Ψ = Ψ(I1 , I2 , I3 ),

(4)

in which the three invariants, I1 , I2 , and I3 , of right Cauchy-Green tensor are given by I1 (C) = trC = λ21 + λ22 + λ23 , (5) ] 1[ 2 I2 (C) = I1 − trC2 = I3 trC−1 = λ21 λ22 + λ22 λ23 + λ23 λ21 , (6) 2 I3 (C) = detC = λ21 λ22 λ23 , (7) where λi (i = 1, 2, 3) are principal stretches in mutually orthogonal directions. Substituting (4) into (3)2 and using chain rule of differentiation yields ( ) ∂Ψ ∂I1 ∂Ψ ∂I2 ∂Ψ ∂I3 ∂Ψ(I1 , I2 , I3 ) =2 + + . (8) S=2 ∂C ∂I1 ∂C ∂I2 ∂C ∂I3 ∂C The derivatives of the three invariants with respect to C are given by the following three equations, respectively ∂I1 = I, ∂C

(9)

∂I2 = I1 I − C, (10) ∂C ∂I3 = I3 C−1 . (11) ∂C Substituting the derivatives (9), (10), and (11) into constitutive equation (8) produces the most general form of a stress as a function of the three strain invariants for isotropic hyperelastic materials under finite strains ) ] [( ∂Ψ ∂Ψ ∂Ψ −1 ∂Ψ + I1 I− C + I3 C . (12) S=2 ∂I1 ∂I2 ∂I2 ∂I3 3

For a tangent operator, a fourth-order elasticity tensor, C, is usually introduced to bridge the span between changes in stress dS and changes in right Cauchy-Green tensor dC through 1 dS = C : dC, 2 where the fourth-order elasticity tensor is written as

(13)

∂S ∂ 2Ψ =4 . (14) ∂C ∂C∂C The most general elasticity tensor in terms of the three principal invariants is documented in the book written by Holzapfel (2000) C=2

∂S ∂ 2 Ψ(I1 , I2 , I3 ) =4 (15) ∂C ∂C∂C = δ1 I ⊗ I + δ2 (I ⊗ C + C ⊗ I) + δ3 (I ⊗ C−1 + C−1 ⊗ I) + δ4 C ⊗ C +δ5 (C ⊗ C−1 + C−1 ⊗ C) + δ6 C−1 ⊗ C−1 + δ7 C−1 ⊙ C−1 + δ8 I,

C = 2

where forth-order tensors ∂I/∂C, ∂C/∂C, and ∂C−1 /∂C are defined by ∂I ∂C = O, = I = δik δjl ei ⊗ ej ⊗ ek ⊗ el = ei ⊗ ej ⊗ ei ⊗ ej , (16) ∂C ∂C ∂C−1 1 −1 −1 −1 = −C−1 ⊙ C−1 = − (Cik Cjl + Cil−1 Cjk ), (17) ∂C 2 and the parameters δ1 , δ2 , · · · , δ8 are defined by ( 2 ) 2 ∂ Ψ ∂2Ψ ∂Ψ 2 ∂ Ψ δ1 = 4 + 2I1 + + I1 , (18) ∂I1 ∂I1 ∂I1 ∂I2 ∂I2 ∂I2 ∂I2 ( 2 ) ( ) ∂ Ψ ∂ 2Ψ ∂ 2Ψ ∂2Ψ δ2 = −4 + I1 , δ3 = 4 I 3 + I1 I3 , (19) ∂I1 ∂I2 ∂I2 ∂I2 ∂I1 ∂I3 ∂I2 ∂I3 ( ) 2 ∂ 2Ψ ∂ 2Ψ ∂Ψ 2 ∂ Ψ δ4 = 4 , δ5 = −4I3 , δ6 = 4 I 3 + I3 , (20) ∂I2 ∂I2 ∂I2 ∂I3 ∂I3 ∂I3 ∂I3 ∂Ψ ∂Ψ , δ8 = −4 . (21) δ7 = −4I3 ∂I3 ∂I2 Although the constitutive equation (12) and the fourth-order elasticity tensor (14) are the fundamental constitutive equations in the theory of finite hyperelasticity, the strain energy function, Ψ, is the key to determining all of them. Thus, the main objective is to establish, solve, and apply the strain energy function to construct a general hyperelastic constitutive model. 4

2. General Hyperelastic Constitutive Model For a general hyperelastic constitutive model based on the theory of continuum mechanics, the partial differential equation of a strain energy function must be derived. Then, once the partial differential equation is solved and the function is determined, a general hyperelastic constitutive model can be obtained under commonly accepted conditions. 2.1. Strain Energy Partial Differential Equation Based on the principle of balance, a strain energy at a finite elastic strain must be identical to the stress work done. With the analogy of stress power, ˙ in Holzapfel (2000), the stress work done can which is defined as S : E be formulated as a double-dot product or double contraction of the second Piola-Kirchhoff stress tensor and the Green-Lagrange strain tensor Ψ = S : E.

(22)

Substituting (12) and (2) into (22), we have ( ) ∂Ψ ∂Ψ ∂Ψ ∂Ψ −1 Ψ= + I1 I : (C − I) − C : (C − I) + I3 C : (C − I) . ∂I1 ∂I2 ∂I2 ∂I3 (23) T 2 Expanding (23) with C : C = trC C = trC gives ( ) ) ) ∂Ψ ∂Ψ ∂Ψ ( ∂Ψ ( Ψ= + I1 (I1 − 3) − trC2 − I1 + I3 3 − trC−1 . (24) ∂I1 ∂I2 ∂I2 ∂I3 Eliminating the right Cauchy-Green tensor related terms with (5)1 , (6)1 , (6)2 , (7)1 , noticing trC2 = I12 − 2I2 , and rearranging yields the following partial differential equation Ψ = (I1 − 3)

∂Ψ ∂Ψ ∂Ψ + (2I2 − 2I1 ) + (3I3 − I2 ) . ∂I1 ∂I2 ∂I3

(25)

Now the partial differential equation for the strain energy function (25) is simplified as a function of three scalar invariants only, which satisfies the normalization condition. The strain energy partial differential equation can be generally solved by Lie group methods.

5

2.2. Solution for Strain Energy Function The strain energy partial differential equation can be rearranged as Ψ = I1

∂Ψ ∂Ψ ∂Ψ + 2I2 + 3I3 − Ψ0 , ∂I1 ∂I2 ∂I3

(26)

where Ψ0 is the stress work done corresponding to an initial undeformed configuration, which is given by Ψ0 = 3

∂Ψ ∂Ψ ∂Ψ + 2I1 + I2 . ∂I1 ∂I2 ∂I3

(27)

Although I1 , I2 , and related derivatives appear in Ψ0 , they represent initial stresses corresponding to an initial undeformed configuration. In constitutive modeling, initial stresses are usually not considered, as implied by (13). Thus, in the following integrations, Ψ0 is treated as a constant. Based on Lie group methods by Ibragimov (2009), the characteristic system for the strain energy partial differential equation (26) reads dI1 dI2 dI3 dΨ = = = . I1 2I2 3I3 Ψ + Ψ0 1

(28) 1

Taking its three first integrals ψ1 = I22 /I1 , ψ2 = I33 /I1 , and ψ3 = (Ψ + Ψ0 )/I1 , we obtain the general solution to (26) with the following form [ ] 1 1 Ψ = I1 f (I22 /I1 ) + g(I33 /I1 ) − Ψ0 , (29) where f and g are two arbitrary functions. 2.3. Model for Isotropic Hyperelastic Materials 2.3.1. A Particular Solution For ease of curve fitting, accurate representation of deformation geometry and controllable numerical stability, I select the lowest ordered polynomial, a general linear function for the first arbitrary function, f , which takes the following form of 1 1 f (I22 /I1 ) = c1 + c2 (I22 /I1 ). (30) The second arbitrary function involves I3 , which is closely related to a hydrostatic pressure. Based on the growth conditions, the I3 term should become the denominator of the fraction defined in the function g so that 6

Ψ → ∞ as I3 → 0+ . Physically, we need an infinite strain energy to compress a continuum material to contract to a zero volume. Mathematically, in order to accurately describe the curvatures of ellipsoidal deformations, we shall use the third order polynomial. Thus, I choose the second arbitrary function as 1

1

g(I33 /I1 ) = c3 (I33 /I1 )−3 = c3 I13 /I3 .

(31)

Substituting (30) and (31) into (29) gives the particular solution of the strain energy function Ψp = c0 + c1 I1 + c2

√

I2 + c3

I14 , I3

(32)

where c0 = −Ψ0 and the remaining three coefficients, c1 , c2 , and c3 , are material constants to be fixed by experimental data. 2.3.2. Stress Tensors and Elasticity Tensors The stress tensors and tangent operators are indispensable ingredients in any constitutive models for the numerical solution of boundary value problems by iterative approaches such as Newton-Raphson methods. Applying the constitutive equation (12) to the particular solution (32) gives the second Piola-Kirchhoff stress tensor ( ) − 12 −1 3 −1 Sp = 2c1 + 8c3 I1 I3 + c2 I1 I2 I − c2 I2 2 C − 2c3 I14 I3−1 C−1 . (33) The eight parameters for the tangent tensors have been worked out by substituting the particular solution (32) into (18) through (21) − 12

δ1p = 48c3 I12 I3−1 + 2c2 I2 δ3p = −16c3 I13 I3−1 ,

−3

−3

− c2 I12 I2 2 , δ2p = c2 I1 I2 2 , −3

δ4p = −c2 I2 2 ,

δ6p = δ7p = 4c3 I14 I3−1 ,

δ5p = 0, − 21

δ8p = −2c2 I2 .

(34) (35) (36)

Seven out of eight parameters are non-zeros and therefore the particular solution is pretty general. The particular constitutive model for isotropic hyperelastic materials is applicable to both compressible and incompressible conditions. Geometric meanings of the three terms in the particular solution can also be revealed through the second Piola-Kirchhoff stress tensor and the elasticity tensor. The term with the coefficient c1 only appears in the second 7

Piola-Kirchhoff stress tensor equation (33), describing a constant modulus but does not show up in the elasticity tensor. Let us take the non-zero parameters in the elasticity tensor as an example to illustrate the geometric meanings of the three terms in the particular constitutive relationship. The first term with the coefficient c1 does not show up in the parameters for the elasticity tensor, meaning that the curvature of a deformation space is zero due to principal translational stretches. The term related to the coefficient c2 models the curvatures of rotational deformations. The term involving the coefficient c3 captures the curvature of an ellipsoidal deformation, coupling a volumetric or dilatational deformation and an isochoric or distortional deformation. Specific equations for the mean curvature of a general spacial surface and the Gaussian curvature of an ellipsoid can be found in the classic monograph in the field of differential geometry by Carmo (1976). Nevertheless, ellipsoidal deformations preserve numerical stabilities. The particular solution as a constitutive model is equally applicable to incompressible hyperelastic materials. The current model will be used to fit the Treloar’s experimental data of vulcanized rubbers. 3. Model Validation 3.1. Application of the Particular Solution Many experimental test methods can be used to characterize rubber-like materials. Most commonly used methods cover uniaxial tension, uniaxial compression, equibiaxial tension, pure shear, and volumetric compression tests in which detailed reviews are given by Charlton (1994). In order to validate the general constitutive model (32), Treloar’s data will be used, in which a total of 56 pairs of stretch values and nominal stresses from the curves of uniaxial tension, equibiaxial tension, and pure shear tests have been extracted and documented by Steinmann et al. (2012). The nominal stress as a function of a principal stretch for the current model will be derived for uniaxial tension, pure shear, and equibiaxial tension deformation modes, respectively. The three principal nominal stresses in terms of a strain energy function and principal stretches are given by Pa =

∂Ψ ∂I2 ∂Ψ ∂I3 ∂Ψ ∂I1 + + , ∂I1 ∂λa ∂I2 ∂λa ∂I3 ∂λa

8

a = 1, 2, 3.

(37)

In a uniaxial tension test, we have the relations of λ1 = λ and λ22 = λ23 = λ based on the incompressible condition of I3 = λ21 λ22 λ23 = 1. Thus, the first two invariants for the uniaxial tension deformation mode are 1 2 I1u = λ2 + , I2u = 2λ + 2 . (38) λ λ −1

Substituting (38) into (32) and then into (37) gives 1 − λ−3 c2 + 8(λ2 + 2λ−1 )3 (λ − λ−2 )c3 . Pu = 2(λ − λ−2 )c1 + √ 2λ + λ−2

(39)

In a pure shear test, we have the relations of λ1 = λ, λ2 = 1, and λ3 = λ−1 based on the incompressible condition of I3 = λ21 λ22 λ23 = 1. Thus, the first two invariants for pure shear deformation mode are the same I1s = I2s = λ2 + 1 +

1 . λ2

(40)

Substituting (40) into (32) and then into (37) gives (λ − λ−3 ) Ps = 2(λ − λ−3 )c1 + √ c2 + 8(λ2 + λ−2 + 1)3 (λ − λ−3 )c3 . (41) 2 −2 λ +λ +1 In an equibiaxial tension test, we have the relations of λ1 = λ2 = λ and λ3 = λ−2 based on the incompressible condition of I3 = λ21 λ22 λ23 = 1. Thus, the first two invariants for equibiaxial tension deformation mode are I1b = 2λ2 +

1 , λ4

I2b = λ4 +

2 . λ2

(42)

Substituting (42) into (32) and then into (37) gives (λ3 − λ−3 ) Pb = 2(λ − λ−5 )c1 + √ c2 + 8(2λ2 + λ−4 )3 (λ − λ−5 )c3 . λ4 + 2λ−2

(43)

The second Piola-Kirchhoff stress tensor with the incompressibility condition of I3 = 1 simplifies as ) ( −1 − 12 3 I − c2 I2 2 C. Spi = 2c1 + 8c3 I1 + c2 I1 I2 (44) The eight parameters for the tangent tensors for incompressible materials have been worked out as Cpi = δ1pi I ⊗ I + δ2pi (I ⊗ C + C ⊗ I) + δ4pi C ⊗ C + δ8pi I, 9

(45)

where

−3

− 12

−3

− c2 I12 I2 2 , δ2pi = c2 I1 I2 2 ,

δ1pi = 48c3 I12 + 2c2 I2

− 23

− 21

δ4pi = −c2 I2 , δ8pi = −2c2 I2 .

(46) (47)

3.2. Fitting Model to Treloar’s Experimental Data Treloar’s uniaxial tension experimental data has been chosen to fit the current uniaxial tension model (39) by a linear least square method. The total accumulated error between the model and experiment is given by n ∑ ]2 [ ϵ= Pu (λj ) − PuTreloar (λj ) ,

(48)

j=1

where PuTreloar is Treloar’s data in uniaxial tension test and n is the total number of pairs of experimental data. The error, ϵ, can be minimized with the following conditions ∂ϵ = 0, ∂ck

k = 1, 2, 3.

(49)

Substituting (39) into (48) and applying (49) yields ∑ ∑ ∑ ∑ A j A j c1 + Bj Aj c2 + Cj Aj c3 = PuTreloar (λj )Aj , ∑ ∑

Aj Bj c1 + A j C j c1 +

∑ ∑

Bj Bj c2 + Bj Cj c2 +

∑ ∑

Cj Bj c3 = Cj Cj c3 =

∑ ∑

(50)

PuTreloar (λj )Bj ,

(51)

PuTreloar (λj )Cj ,

(52)

where Aj =

2(λj − λ−2 j ),

1 − λ−3 j −2 3 Bj = √ , Cj = 8(λ2j + 2λ−1 j ) (λj − λj ). (53) −2 2λj + λj

Numerically solving (50), (51), and (52), along with (53), simultaneously obtains the three coefficients as c1 = 0.1409441, c2 = 0.1425925 and c3 = 3.1970322 × 10−7 . The comparison between the current model and Treloar’s uniaxial tension test data is shown in Fig. 1. The current model is excellently suited for fitting Treloar’s uniaxial tension test data. To further demonstrate the predictive capacity of the model, the three coefficients obtained from 10

8

Current Hyperelastic Model Treloar’s Uniaxial Tension

Nominal Stress, P (MPa)

7 6 5 4 3 2 1 0 1

2

3

4

5

6

7

8

Principal Stretch, λ

Figure 1: Comparison between Model and Treloar’s Uniaxial Tension Data.

fitting Treloar’s uniaxial tension test data have been used in the pure shear equation (41) and the equibiaxial tension equation (43), respectively. The predictions of Treloar’s pure shear and equibiaxial tension test data are shown in Fig. 2 and Fig. 3, respectively. The pure shear prediction results are quite accurate. For Treloar’s equibiaxial tension test data, however, the model’s prediction is slightly higher than experimental data at larger stretches. 4. Discussions on Models In the general solution (29) of strain energy function, Ψ, there are two functions, f and g, that look pretty arbitrary. It is worth emphasizing that all terms should have the same dimension as that of I1 or λ2 even though they are already dimensionless. The general constitutive model (32) has been derived from the general solution of the strain energy partial differential equation, which is suitable for both incompressible and compressible isotropic hyperelastic materials. For incompressible materials, I3 = 1, the current model reduces to Carroll’s isochoric strain energy model (Carroll, 2011). Carroll’s model is an accurate fit for Treloar’s uniaxial tension data for incompressible rubbers. The current uniaxial tension equation (39) is composed of three terms, in which the contribution of each term is depicted in Fig. 4. The first term 11

Nominal Stress, P (MPa)

2.0

Current Model Prediction Treloar’s Pure Shear Data

1.5

1.0

0.5

0.0 1

2

3

4

5

Principal Stretch, λ

Figure 2: Current Model Prediction for Treloar’s Pure Shear Data.

(I1 c1 ), also known as neo-Hooke term, is the first invariant term, which√models principal stretches and possesses positive slopes. The second term ( I2 c2 ) describes the rotational deformations including twisting and bending. As we know, the energy needed for rotational deformations is much smaller than that of translational deformations, demonstrated by Fig. 4. Although the square root of the I2 term contributes insignificant stress magnitude, it contributes a substantial amount of nonlinearity as depicted in the stand alone c2 term plot in Fig. 5. Its slopes change from positive √ 1/3to zero and down to negative. At the stationary point of λ0 = (4 + 3 2) ≈ 2.02, a maximum stress is reached. At the two extremes, the stress contribution goes to negative infinity as the stretch approaches zero and the stress contribution reduces to zero as the stretch approaches infinity. Thus, rotational deformations exist through the whole extension process and the compression process. Models based on I1 terms only will not be able to correctly describe simple torsion experiments as observed by Horgan and Saccomandi (1999). Since the first two terms mentioned above are not enough to capture responses at relatively large deformations shown in Fig. 4, the third term (I14 /I3 c3 ) is crucial. For incompressible materials, I3 = 1, I14 c3 models finite deformations due to increasing crystallinity and decreasing amorphous phase, in which the deformations are purely shape changes. For compressible materials, however, I14 /I3 c3 accurately captures the curvatures of ellipsoidal deformations, 12

Nominal Stress, P (MPa)

3.0

Current Model Prediction Treloar’s Equibiaxial Tension

2.5

2.0

1.5

1.0

0.5

0.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Principal Stretch, λ

Figure 3: Current Model Prediction for Treloar’s Equibiaxial Tension Data.

indicating that isochoric deformations and volumetric deformations are inseparable. When I3 decreases, the hydrostatic pressure will increase. Greater stresses are generally necessary to overcome the hydrostatic pressure part of volume changes for certain shape changes. The pressure effect is particularly important for biological tissues with a rich equivalent liquid phase, which are usually modeled as biphasic materials, for example, by Holmes and Mow (1990). 5. Concluding Remarks The partial differential equation for isotropic hyperelastic constitutive models has been formulated based on the balance principle between strain energy and stress work done. The strain energy function as a general function of the first three invariants has then been solved using Lie group methods. The general solution can be applied to constitutively model isotropic hyperelastic materials. Treloar’s three experimental test data for the rubber with 8% sulfur, a generally accepted benchmark, has been adopted to integrate with the general solution. According to the features of Treloar’s uniaxial tension test curve, geometric constraints, physical constraints, and numerical stability concerns, a three-term particular solution (32) for strain energy function has been extracted from the general solution (29). The 13

6

1st Term 2nd Term 3rd Term

Nominal Stress, P (MPa)

5

4

3

2

1

0 1

2

3

4

5

6

7

8

Principal Stretch, λ

Figure 4: Individual Terms of Current Uniaxial Tension Model.

three-term model includes the I1 c1 term for monitoring a constant modulus √ due to principal translational deformations, the I2 c2 term for capturing curvatures of rotational deformations, and the I14 /I3 c3 term for describing curvatures of ellipsoidal deformations, indicting that isochoric deformations and volumetric deformations are inseparable for general isotropic hyperelastic materials. The stress tensors and tangent tensors have been worked out, in which the particular constitutive solution works for both compressible and incompressible materials. The three coefficients, c1 , c2 , and c3 , have been determined by a linear least square method using Treloar’s uniaxial tension data and applied to predict Treloar’s pure shear and equibiaxial tension data. Excellent agreement between the theory and the experiments has been achieved. Boyce, M.C., Arruda, E.M., 2000. Constitutive models of rubber elasticity: a review. Rubber Chem. Technol. 73, 504–523. Carroll, M.M., 2011. A strain energy function for vulcanized rubbers. J. Elast. 103, 173–187. Charlton, D.J., Yang, J., Teh, K.K., 1994. A review of methods to characterize rubber elastic behavior for use in finite element analysis. Rubber Chem. Technol. 67, 481–503. 14

0.08

Nominal Stress, P (MPa)

0.06 0.04 0.02 0.00 Second Term

-0.02 -0.04 -0.06 -0.08 0

1

2

3

4

5

6

7

8

Principal Stretch, λ

Figure 5: Second Term Details of the Current Uniaxial Tension Model.

Carmo, M.P., 1976. Differential Geometry of Curves and Surfaces. PrenticeHall, Englewood Cliffs, New Jersey. Holmes, M.H., Mow, V.C., 1990. The nonlinear characteristics of soft gels and hydrated connective tissues in ultrafiltration. J. Biomech. 23, 1145–1156. Holzapfel, G.A., 2000. Nonlinear Solid Mechanics. John Wiley & Sons, Chichester. Holzapfel, G.A., Gasser, T.C., Ogden, R.W., 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elast. 61, 1–48. Horgan, C.O., Saccomandi, G., 1999. Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility. J. Elast. 59, 159–170. Hossain, M., Steinmann, P., 2013. More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study. J. Mech. Behav. Mater. 22, 27–50. Ibragimov, N.H., 2009. A Practical Course in Differential Equations and Mathematical Modelling. Higher Education Press, Beijing. 15

Malvern, L.E., 1969. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, New Jersey. Marsden, J.E., Hughes, T.J.R., 1983. Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, New Jersey. Ogden, R.W., 1984. Non-linear Elastic Deformations, Ellis Horwood, Chichester. Rivlin, R.S., 1948. Large elastic deformations of isotropic materials IV. Further developments of the general theory. Philos. TR. R. Soc. S.-A. 241, 379–397. Steinmann, P., Hossain, M., Possart, G., 2012. Hyperelastic models for rubber-like materials: consistent tangent operators and suitability for Treloar’s data. Arch. Appl. Mech. 82, 1183–1217. Treloar, L.R.G., 1994. Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70. Truesdell, C., Noll, W., 1965. The non-linear field theories of mechanics. In: Fl¨ ugge, S. (Eds.) Encyclopedia of Physics, III/3. Springer-Verlag, Berlin.

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