Analytical network-averaging of the tube model:

Author’s Accepted Manuscript Analytical network-averaging of the tube model. Rubber elasticity Vu Ngoc Khiêm, Mikhail Itskov

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To appear in: Journal of the Mechanics and Physics of Solids Received date: 23 November 2015 Revised date: 23 May 2016 Accepted date: 24 May 2016 Cite this article as: Vu Ngoc Khiêm and Mikhail Itskov, Analytical networkaveraging of the tube model. Rubber elasticity, Journal of the Mechanics and Physics of Solids, http://dx.doi.org/10.1016/j.jmps.2016.05.030 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.

Analytical network-averaging of the tube model. Rubber elasticity Vu Ngoc Khiˆema,∗, Mikhail Itskova a

Department of Continuum Mechanics, RWTH Aachen University, Kackertstr. 9, 52072 Aachen, Germany

Abstract In this paper, a micromechanical model for rubber elasticity is proposed on the basis of analytical network-averaging of the tube model and by applying a closed-form of the Rayleigh exact distribution function for non-Gaussian chains. This closed-form is derived by considering the polymer chain as a coarse-grained model on the basis of the quantum mechanical solution for finitely extensible dumbbells (Ilg. et al. (2000)). The proposed model includes very few physically motivated material constants and demonstrates good agreement with experimental data on biaxial tension as well as simple shear tests. Keywords: Rubber elasticity, Network-averaging, Micro-mechanics, Rayleigh exact non-Gaussian distribution

1

1. Introduction

2

The fact that many engineering rubber products (such as automotive tires and seals,

3

suspension mounts, building and bridge bearings, elastomeric flex-elements) undergo large

4

strain requires an effective and accurate prediction of the stress-strain relation in complex

5

states of deformation. In the context of constitutive modeling of rubber-like materials, the

6

concept of hyperelasticity is of extreme advantage. Many hyperelastic constitutive models

7

for rubber elasticity have been proposed in the literature. They can mathematically be clas-

8

sified into five main groups: first-invariant (I1 ) based models (Treloar (1943); Yeoh (1990);

9

Gent (1996); Yeoh and Fleming (1997); Elias-Z´ un ˜iga and Beatty (2002); Beatty (2007); ∗

Corresponding author Email address: [email protected] (Vu Ngoc Khiˆem )

Preprint submitted to Journal of the Mechanics and Physics of Solids

May 25, 2016

10

Lopez-Pamies (2010); Khajehsaeid, Arghavani, and Naghdabadi (2013)), two-invariant (I1

11

and I2 ) based models (Mooney (1940); Rivlin (1948); Isihara, Hashitsume, and Tatibana

12

(1951); Rivlin and Saunders (1951); Gent and Thomas (1958); Hart-Smith (1966); Alexan-

13

der (1968); James, Green, and Simpson (1975); Haines and Wilson (1979); Kilian (1981);

14

Diani and Rey (1999); Pucci and Saccomandi (2002); Attard and Hunt (2004); Amin, Wira-

15

guna, Bhuiyan, and Okui (2006); Horgan and Saccomandi (2006); Beda (2007); Carroll

16

(2011)), principal stretches based models (Valanis and Landel (1967); Ogden (1972); Ed-

17

wards and Vilgis (1986); Heinrich and Kaliske (1997); Kaliske and Heinrich (1999); Kl¨ uppel

18

and Schramm (2000); Shariff (2000); Meissner and Matˇejka (2003); Drozdov (2007); Horgan

19

and Murphy (2007); Mansouri and Darijani (2014)), logarithmic strain invariants based mod-

20

els (Hencky (1933); Anand (1986); Lurie (1990); Criscione, Humphrey, Douglas, and Hunter

21

(2000); Xiao (2012); Neff, Ghiba, and Lankeit (2015)), and network models (James and

22

Guth (1943); Flory and Rehner (1943); Flory (1944); Wang and Guth (1952); Treloar (1954);

23

Treloar and Riding (1979); Wu and Van der Giessen (1992); Arruda and Boyce (1993); Wu

24

and Van der Giessen (1993); Beatty (2003); Miehe, G¨oktepe, and Lulei (2004); Diani and

25

Gilormini (2005); Bechir, Chevalier, and Idjeri (2010); Itskov, Ehret, and Dargazany (2010);

26

Kroon (2010); Davidson and Goulbourne (2013)). For further detail on the formulations

27

and comparisons between the mentioned hyperelastic models interested readers are referred

28

to (Boyce and Arruda (2000); Seibert and Sch¨oche (2000); Marckmann and Verron (2006);

29

Steinmann, Hossain, and Possart (2012); Beda (2014)). As reported in Marckmann and Ver-

30

ron (2006), only four models, namely the Ogden model (Ogden (1972)), the extended-tube

31

model (Kaliske and Heinrich (1999)), the Shariff model (Shariff (2000)) and the tube-like

32

microsphere model (Miehe et al. (2004)) are able to capture stress-strain relations for rubber

33

in multiple states of deformation.

34

In comparison to the two models of Valanis-Landel type (Ogden (1972); Shariff (2000)),

35

the tube-like microsphere model (Miehe et al. (2004)) is physically motivated and includes

36

a first order approximation non-Gaussian phantom network (Kuhn and Gr¨ un (1942)) and a

37

topological constraint. Thus, its parameters can be defined by studying the microstructure of

38

the material and could be physically measured. In microsphere type models, polymer chains 2

39

are assumed to be continuously distributed in all spatial directions. The network strain

40

energy function is calculated by integration of the one dimensional free energy of polymer

41

chains over the unit sphere (Wu and Van der Giessen (1992, 1993)). The microsphere concept

42

has extensively been used in various field of research: deformation induced anisotropy in filled

43

elastomers (G¨oktepe and Miehe (2005); Diani, Brieu, and Vacherand (2006); Dargazany and

44

Itskov (2009); Machado, Chagnon, and Favier (2014); Itskov and Knyazeva (2015)), rubber

45

viscoelasticity (Miehe and G¨oktepe (2005); Linder, Tkachuk, and Miehe (2011)), fibrous

46

biological tissue modeling (Menzel and Waffenschmidt (2009); Alastru´e, S´aez, Mart´ınez,

47

and Doblar´e (2010); Martufi and Gasser (2011); Holzapfel, Unterberger, and Ogden (2014)),

48

strain-induced crystallization in natural rubbers (Dargazany, Khiˆem, and Itskov (2014);

49

Mistry and Govindjee (2014); Guili´e, Thien-Nga, and Le Tallec (2015)). In the following, we are going to discuss the issue of the numerical accuracy in the context of microsphere models. In the tube-like microsphere model, the principal nominal stresses can be obtained as follows √

(1−p)

Pi = µ N Λ i λ

L

−1



λ √ n



u (p−2) u

Λ

Dii

 −

µnU Λ−3 i q



u (q−2) u

Dii

Υ



− ρΛi −1 ,

(1)

where µ, n, U, p, q are material parameters, and L−1 is the inverse of the Langevin function 1 defined by L (β) = coth β − . ρ is a Lagrange multiplier associated to the incompressibility β u u condition. Further, Λi (i = 1, 2, 3) denote the principal stretches, Λ and Υ are the macrou

stretch and macro tube contraction in a direction u, respectively. Dii is the component ii of u

the directional tensor D = u ⊗ u. The micro-stretch λ is evaluated by a p-root operator as (Miehe et al. (2004))   p1 u λ = Λp ,

(2)

3

where h•i denotes an average of • over the unit sphere S 1 h•i = 4π

Z •dS.

(3)

S

51

The integral (3) can be taken analytically only in some special cases. Generally, the m P averaging operator requires a numerical cubature as h•i ≈ wi •i . Therein, m is the

52

number of integration points and wi (i = 1, 2, ..m) are the corresponding weighting factors.

53

In the model by Miehe et al. (2004), 21-point integration scheme by Bazant and Oh (1986)

54

is used. In Ehret, Itskov, and Schmid (2010), it was shown that this integration scheme

55

induces strong anisotropy into initially isotropic constitutive laws. Recently, large errors in

56

the stress-strain relation of microsphere models due to the numerical integration over the

57

unit sphere have been reported by Verron (2015). We also observed that predictions of the

58

tube-like microsphere model (Miehe et al. (2004)) strongly depend on the integration scheme

59

chosen.

50

i=1

60

In the current work, we propose a constitutive model for rubber elasticity based on

61

analytical network-averaging of the tube model and by applying a closed-form of the Rayleigh

62

non-Gaussian distribution. Thus, the model is micromechanically motivated but does not

63

require the numerical integration. The paper is organized as follows: Sect.2 describes the

64

constitutive model and its molecular statistical basis. Predictive capabilities of the model

65

are then demonstrated by comparison with various experimental data and other constitutive

66

models in Sect.3. Finally, advantages of the model and reasons for its accurate results are

67

discussed.

68

2. Constitutive model and its molecular statistical basis

69

2.1. Statistical mechanics of a confined polymer network The rubber network is considered as a composition of several one-dimensional subnetworks dispersed in different directions u in a unit sphere. The molecular conformation in each subnetwork is confined by neighbouring network chains. Following Doi and Edwards 4

(1986); Heinrich, Straube, and Helmis (1988), Kaliske and Heinrich (1999); Miehe et al. (2004), each polymer molecule in a directional subnetwork is considered as a freely jointed chain restricted within a tube-like zone. The conformational probability density function of u

a polymer chain with n segments and normalized end-to-end distance r = λ R lying within a tube of length r and normalized diameter d can be considered as a joint probability of a random walk chain (Treloar (1946, 1975)) Pcexact

 m  n−2 n (n − 1) X (−1)k n−r (n, r) = −k 8πrb2 k! (n − k)! 2 k=0 

(4)

and a topological constraint (Doi and Edwards (1986); Miehe et al. (2004))  π2R Pt (d) = exp −α 2 . 3d 

(5)

71

 n−r , R is the normalized end-to-end distance of polymer chains in Therein m = floor 2 the reference configuration, and α is a normalization factor. Note that end-to-end distances

72

and tube diameters are normalized here by the Kuhn length b. The equivalence of (4) to the

73

Rayleigh exact distribution function for non-Gaussian chains (Lord Rayleigh (1919)) was

74

shown by Volkenstein (1963) and Flory (1969).



70

The joint probability is thus given by the product of (4) and (5) as (cf. Miehe et al. (2004)) n−2   m   n (n − 1) X (−1)k π2R n−r P (r, d) = −k exp −α 2 . 8πrb2 k! (n − k)! 2 3d k=0 

(6)

75

Remark 1. The inverse Langevin function resulting from a first order approximation of

76

the Rayleigh exact distribution function (Kuhn and Gr¨ un (1942)) was implemented in the

77

most network models of rubber elasticity. However, Flory (1969); Jernigan and Flory (1969)

78

pointed out that the approximation by Kuhn and Gr¨ un cannot capture the exact probability

79

density (4).

5

In view of (6), the free energy resulting from (4) is given by ψc (n, r) = −kB T ln Pc ,

80

(7)

where T denotes the absolute temperature and kB is the Boltzmann constant. The additional potential induced by the topological constraint results from (5) as u

ψt = −kB T ln Pt = kB T ω υ ,

(8)

82

 2 π2R D u where ω = α 2 is a geometrical parameter of the tube, υ = is the contraction of 3D d the tube cross-section (Miehe et al. (2004)), and D is the normalized average diameter of

83

the tube in the reference configuration.

84

2.1.1. Closed-form of the non-Gaussian distribution

81

85

The exact non-Gaussian probability distribution function (4) is a discrete function in

86

terms of the end-to-end distance r, and is thus not convenient for mathematical implemen-

87

tations. In order to derive a closed-form of the non-Gaussian distribution, we follow Ilg,

88

Karlin, and Succi (2000) and consider the polymer chain as a coarse-grained model.

89

Accordingly, the force applied on a coarse-grained segment of the polymer chain can be

90

calculated based on a dumbbell model consisting of two particles joined by a finitely exten-

91

sible spring. The spring can be considered as a matter wave, whose motion is described by

92

93

94

the Schr¨odinger equation for a particle trapped in an infinite spherical well. The probability hn  πr i2 density of the ground state of the Schr¨odinger equation is given by Peq = sin πr n (Ilg et al. (2000)). κ The polymer chain is considered as a coarse-grained model with n identical coarse2 9 grained segments. The value κ = 2 arises from the spring constant of the linear coarseπ grained model. Thus, the probability distribution of the end-to-end distance of the polymer

6

chain with

κ n coarse-grained segments can be given by 2 κ

Pc (r) =

n 2 Y

Peq = A

i=1

 πr iκn hn sin , πr n

(9)

95

where the normalization A = Pcexact (n, 10−2 ) guarantees identical mode likelihoods (the

96

greatest probability of occurring) between (4) and (9). It can be seen that the proposed

97

probability distribution function (9) shows good agreement with the exact non-Gaussian

98

distribution (see Fig.1). Substituting the closed-form (9) into (4), the non-Gaussian free energy can be expressed by ψc (n, r) = −kB T ln [Pc (r)] = nkB T κ ln

99

πr  πr  + C, n sin n

(10)

where C is a constant resulting from the condition of the energy-free reference configuration. Thus, the chain force can be obtained as the first derivative of the free energy with respect to r as (cf. Ilg et al. (2000)) fc (n, r) = kB T κ

hn r

− π cot

 πr i n

.

(11)

102

κπ 2 kB T Note that the chain stiffness can be obtained from the first derivative of (11) as = 3n 3kB T , which is identical to the spring constant of the linear coarse-grained model (see e.g. n de Gennes (1979)). The chain force (11) arises from the closed-form of the Rayleigh exact

103

distribution function, and thus is not a representation of the inverse Langevin function.

100

101

104

Remark 2. In contrast to other works on molecular-statistical basis of constitutive

105

models (see e.g. Boyce (1996); Horgan and Saccomandi (2002); Ehret (2015)) applying

106

approximations of the inverse Langevin function by either generalized power series or Pad´e

107

approximants, our modeling target is to derive a closed-form of the exact non-Gaussian

108

probability distribution (4) (see Fig.1). The trigonometric basis of the probability distribution 7

−3

5 exact n=8 model

c

0.012

Probability distribution P (r)

Probability distribution Pc(r)

0.014

0.01 0.008 0.006 0.004

x 10

exact n=16 model

4

3

2

1

0.002 0 0

2

4 6 End−to−end distance r

0 0

8

5

(a) −3

2

x 10

exact n=32 model

exact n=24 model

2.5

c

Probability distribution P (r)

Probability distribution Pc(r)

x 10

2 1.5 1 0.5 0 0

20

(b)

−3

3

10 15 End−to−end distance r

5

10 15 20 End−to−end distance r

25

(c)

1.5

1

0.5

0 0

10 20 30 End−to−end distance r

(d)

9 and b = 1nm for different chain lengths: (a) nonπ2 Gaussian distribution with n = 8, (b) n = 16, (c) n = 24, and (d) n = 32. The limit of chain extensibility can be seen: the likelihood of a polymer chain reaching its contour length (r = n) is zero. Figure 1: Validation of the closed-form (9) with κ =

109

function applied here can be derived from a radial form of the Schr¨ odinger equation, and thus

110

can be physically motivated. By this means, the model also captures the chain extensibility

111

limit.

8

40

112

2.1.2. Stretch and tube contraction averaging In this section, we extend the full-network average-stretch approach by Beatty (2003) to a generalized network-averaging concept for the tube model. Accordingly, in contrast to microsphere models, each directional chain in the network is represented by a mean ¯ and an average-tube chain confined in a mean tube undergoing an average-tube stretch λ contraction υ¯. Thus, the strain energy of the whole network can be expressed by   ¯ υ¯ , Ψ = Nc ψmean λ,

(12)

113

where Nc is the number of chains per unit referential volume, and ψmean is the free energy

114

of the mean chain. The average macro-stretch is evaluated as the root mean square of the macroscopic stretch over the unit sphere S. The square of the macroscopic stretch in a direction u = sin θ cos ϕN 1 + sin θ sin ϕN 2 + cos θN 3 specified by the spherical coordinates θ and ϕ is given by u

Λ2 = Λ21 sin2 θcos2 ϕ + Λ22 sin2 θsin2 ϕ + Λ23 cos2 θ,

(13)

where N i (i = 1, 2, 3) are principal directions. By means of the surface integral, the average macro-stretch is derived as (Kearsley (1989)) v s  u Z2π Zπ r u u u 1 I1 u ¯= Λ Λ2 = t , dϕ Λ2 sin θdθ = 4π 3 0

(14)

0

115

where I1 denotes the first principal invariant of the right Cauchy-Green tensor C. Note that

116

the 4-chain model (Wang and Guth (1952)) and 8-chain model (Arruda and Boyce (1993))

117

yield identical macro-stretch as the full-network average macro-stretch (see e.g. Yeoh and

118

Fleming (1997); Beatty (2003)). For a directional subnetwork containing Mc polymer chains in series, the normalized 9

u u



Reference state

Current state u

n1

n2

nM c

R1

R2

RM c

1

u

u

2

M

c

u

l L

Mc

L   Ri i 1

Cross-link

Polymer chains u

Figure 2: Deformation in an inhomogeneous subnetwork: Λ is the macro-stretch applied on the whole u subnetwork, and λ is the microstretch exerted on a polymer chain.

referential length of the subnetwork is L =

Mc P

Ri , where Ri denotes the normalized end-to-

i=1

end distance of the ith polymer chain (i.e. a strand between two cross-links). As the real rubber network is inhomogeneous, polymer chains have different number of chain segments ni (see Fig.2). The macro-stretch at failure of the whole subnetwork can be expressed by Mc P u

Λ f ailure =

i=1 M Pc

ni .

(15)

Ri

i=1

The micro-stretch at failure of the weakest polymer chain is 

u

λ f ailure = min

ni Ri

 .

(16)

In the following, we postulate the macro-micro transition proposed by Heinrich et al. (1988) as u

u

q λ =Λ .

(17)

10

The stretch amplification exponent q appearing there can be obtained for the state at failure by assuming that the subnetwork fails when the weakest polymer chain reaches its contour length. Thus, inserting (16) into (17) we obtain 



 ni ln min Ri q= M . Mc c P P ln ni − ln Ri i=1

(18)

i=1

119

In the special case when all polymer chains have the same number of chain segments and

120

the same referential end-to-end distance, q = 1 in view of (18), which results in an affine

121

deformation. According to the generalized network-averaging concept, only the mean chain is considered. Substitution of the average-stretches into (17) yields thus ¯=Λ ¯ q. λ

(19)

122

As seen from (18), the stretch amplification exponent q is related to the degree of inho-

123

mogeneity of the rubber network. We assume that the network density only influences the

124

number of subnetworks, but does not effect the degree of inhomogeneity in each subnetwork

125

and consequently q. Replacing in (10) the normalized end-to-end distance by the root mean square end-to√ end distance of the freely jointed chain R = n according to (12), the strain energy of the ideal network can be written as π ¯ π ¯ √ λ sin √ · λ n n    + C = µc κn ln , Ψc = µc κn ln π ¯ π ¯ sin √ λ sin √ λ n n

126

(20)

where µc = Nc kB T . In the next step, the average macro tube contraction is evaluated as the root mean square of the macroscopic tube contraction over the unit sphere S. The square of the macroscopic 11

contraction of an infinitesimal area with the unit normal u is given by u
Υ2 = Λ21 Λ22 cos2 θ + Λ23 sin2 θ Λ21 cos2 ϕ + Λ22 sin2 ϕ .

(21)

In the same manner as (14), the average macro tube contraction is derived as (Kearsley (1989))

¯ = Υ

s

u

Υ2



v u Z2π Zπ r u u I2 u1 dϕ Υ2 sin θdθ = =t , 4π 3 0

127

(22)

0

where I2 denotes the second principal invariant of the right Cauchy-Green tensor C. In contrast to Miehe et al. (2004), we assume that the average micro tube contraction ¯ In view of (8), the network strain and the average macro tube contraction coincide υ¯ = Υ. energy due to the topological constraint can be expressed as  Ψt = µ t

I2 3

 12 ,

(23)

where µt = Nc kB T ω = Nc kB T α

128

π2R . 3D2

(24)

2.2. Constitutive model As a result of (6),(20) and (23), the strain energy function can be given by  sin Ψ (I1 , I2 ) = Ψc (I1 ) + Ψt (I2 ) = µc κn ln sin

q I1 2 "  # 1/2 I2 3 −1 ,   q ! + µt 3 π I1 2 √ n 3

π √ n



(25)

129

where µc κ is the effective shear modulus of the ideal network, and µt is the topological shear

130

modulus. Note that (25) automatically fulfills the condition of the energy-free reference 12

131

configuration. For incompressible materials characterized by the constraint detF = 1, the second PiolaKirchhoff stress tensor takes the form ∂Ψ µc κ S=2 −ρC−1 = ∂C 3

"


q #   q −1  − 12 √ π I31 2 n I1 2 µt I2 q I+ K−ρC−1 , (26)
2q − π n cot √n I 3 3 3 1

3

where K = I1 I − C and ρ denotes a Lagrange multiplier. Thus, the principal nominal stresses can be expressed as

Pi =

132

133

µc κ 3

135

136

√

n
2q − π n cot

I1 3

π


2q #   q −1  − 12  ρ  I1 2 µt I2 √ q Λi + Λi Λ2j + Λ2k − , (27) 3 3 3 Λi n I1 3

where i 6= j 6= k 6= i = 1, 2, 3. The fourth order tangent modulus is calculated as C=2

134

"

∂S = (4Ψ00 c + 4Ψ0 t ) I I + 4Ψ00 t K K − 4Ψ0 t (I ⊗ I)s ∂C 
s  −2ρ C−1 C−1 − C−1 ⊗ C−1 ,

(28)

∂ 2 Ψc ∂ 2 Ψt 00 ,Ψ t= , and the fourth order tensor products and ⊗ are dewhere Ψ c = ∂I1 ∂I1 ∂I2 ∂I2 fined by (A B) : C = A (B : C), and (A ⊗ B) : C = ACB (Itskov (2015)). Furthermore,
the symmetrization operator in (28) is defined by As : X = A : 12 X + XT (Itskov (2015)). 00

137

3. Results and discussion

138

3.1. Comparisons with experimental data

139

In this section, we validate the performance of the proposed model by comparing it to

140

various sets of experimental data from Treloar (1944), Kawabata, Matsuda, Tei, and Hawai

141

(1981), and Yeoh and Fleming (1997).

142

The model is first fitted to experimental data by Treloar on uniaxial tension, pure shear

143

and equibiaxial tension of vulcanized rubbers containing 8phr sulfur (Treloar (1944)). A

144

simultaneous fitting to the three set of experimental data is necessary due to the strong cor13

3 Treloar [1944]

Treloar [1944] Model uniaxial

Model pure shear Model equibiaxial Nominal stress P11 ( MPa )

Nominal stress P11 ( MPa )

6

4

2

0 1

2

3

4 5 Stretch Λ1

6

7

8

2

1

0 1

2

3 Stretch Λ1

4

Figure 3: Prediction of the proposed model (25) with µc κ = 0.2975[M P a], n = 19.0438, q = 0.9192, µt = 0.1835[M P a] in comparison with the experimental data by Treloar (1944).

145

relation between the principal invariants I1 and I2 in these test protocols (see also Urayama

146

(2006)). The resulting material constants are listed in Table 1. The same set of material

147

constants is also used to predict the stress-strain relation in a series of biaxial tensions by

148

Kawabata et al. (1981). The rubber sample in this experiment was reported to have similar

149

chemical composition and mechanical response as the vulcanized rubber used in Treloar’s

150

experiment (Marckmann and Verron (2006)). The biaxial experiments were carried out with

151

different fixed values of stretch Λ1 , while varying the stretch Λ2 (in the orthogonal direc-

152

tion) from the uniaxial tension to the equibiaxial tension. This benchmark is very tough

153

because there are only three models (Kaliske and Heinrich (1999); Shariff (2000); Miehe

154

et al. (2004)) known in literature able to capture both Kawabata et al. (1981) and Treloar

155

(1944) data using the same set of material constants (Marckmann and Verron (2006)). As

Table 1: Material constants of the proposed model (25) for the experimental data by Treloar (1944). Note that due to slippage of polymer chains through entanglements, the micro-stretch can be less than the macrostretch (Ball, Doi, Edwards, and Warner (1981);Edwards and Vilgis (1988);Kroon (2010)), thus q can be less than 1.

µc κ[M P a] n 0.2975 19.0438 14

q 0.9192

µt [M P a] 0.1835

5

0.4

Λ1 = 1.24

0.25

Λ1 = 1.20

0.2

Λ1 = 1.14

Λ1 = 1.16 Λ1 = 1.12

0.15

Λ1 = 1.10 Λ1 = 1.08

0.1

Λ1 = 1.06 Λ1 = 1.04

0.05

22

0.3

( MPa )

1.2

Nominal stress P

Nominal stress P22 ( MPa )

0.35

Λ1 = 3.7

1

Λ1 = 3.4 Λ1 = 3.1

0.8

Λ1 = 2.8 Λ1 = 2.5

0.6

Λ1 = 2.2 0.4

Λ1 = 1.9 Λ1 = 1.6

0.2

Λ1 = 1.3

Model 0 0.9

1

1.1 Stretch Λ2

1.2

Model 0 0.5

1.3

(a)

1

1.5

2 Stretch Λ2

2.5

3

(b)

Figure 4: Prediction of the proposed model (25) against a series of biaxial tensions by Kawabata et al. (1981): (a) at small values of Λ1 , and (b) at large values of Λ1 .

156

seen in Figs.3 and 4, the proposed model demonstrates very good agreement with both sets

157

of experimental data as well.

158

In Yeoh and Fleming (1997), a set of tension, compression and simple shear tests for

159

vulcanized natural rubbers with four different amounts of curatives (sulfur and accelerator):

160

0.5 (A), 1.0 (B), 1.5 (C) and 2.0 phr (D) were reported. As the samples differ from each

161

other only in the amount of curatives, the material parameters depending on the network

162

density (µc , n and µt ) should be proportional to the amount of sulfur and accelerator. To verify this concept, we first evaluated the four material parameters of the model by fitting it to the experimental data from the sample D (see Fig.5). Note that under simple shear, the nominal shear stress P12 (Fig.5, right) is given by  P12 =

163

∂Ψ ∂Ψ + ∂I1 ∂I2

 2γ,

(29)

where γ is the amount of shear. To simulate the behavior of samples A, B, and C, the material parameters are further

15

3.5

4

Yeoh & Fleming [1997] (D) Model simple shear (D)

Yeoh & Fleming [1997] (D) Model uniaxial (D)

2

Shear stress P12 ( MPa )

Nominal stress P11 ( MPa )

1.2

0

−2

−4

1 0.8 0.6 0.4 0.2

−6 0

1

2

Stretch Λ

3

4

0 0

5

0.5

1 1.5 Amount of shear γ

1

2

Figure 5: Prediction of the proposed model versus stress-strain responses of the sample (D) from the experiments by Yeoh and Fleming (1997). Fitted parameters: µc κ = 0.2886[M P a], n = 120.9714, q = 1.3187, µt = 0.0873[M P a].

assumed to depend linearly on the amount of curatives as follows µc(i) = Nc(i) kB T = µcD ϕ(i) ,

n(i) =

nD , ϕ(i)

(30)

166

Nc(i) denotes the ratio between network densities, and i = A, B, C. As a result NcD √ π2 n of (24) and (30) the topological shear moduli can be determined as µt(i) = Nc(i) kB T α = 3D2 √ µtD ϕ(i) . As mentioned in Section 2.1.2, the stretch amplification is independent of the

167

network density, so that the material parameter q takes the same value for the four samples.

168

The so obtained values of material constants are given in Table 2. Note that the macro-

169

stretches at rupture Λrupture (which can be obtained from I1 = 3n q ) predicted by the model

170

(Table 2) are comparable to the extension ratios at break determined with the ASTM D412

171

standard for the four samples (Yeoh and Fleming (1997)). Despite the low number of fitting

172

parameters, good agreement with experimental results is observed (see Fig.6). By means of

173

(30), the model can simulate rubbers of the same compound with various network densities

174

based on only one set of fitting parameters.

164

165

where ϕ(i) =

1

16

3

4

2.5 ( MPa )

2

12

0

Shear stress P

Nominal stress P11 ( MPa )

2

Yeoh & Fleming [1997] (C) (B) (A) Model simple shear (C) (B) (A)

−2 Yeoh & Fleming [1997] (C) (B) (A) Model uniaxial (C) (B) (A)

−4

−6 0

1

2

Stretch Λ1

3

4

1.5

1

0.5

0 0

5

(a)

1

2 3 Amount of shear γ

4

(b)

Figure 6: Predictions (without fitting) of the proposed model versus stress-strain responses of the samples (A), (B) and (C) from the experiments by Yeoh and Fleming (1997): (a) uniaxial tension, and (b) simple shear. As seen in Figs.5 and 6 there is a shift in the simple shear responses from convex shapes (samples (C) and (D)) to concave shapes (samples (A) and (B)). It might be due to nonlinear correlations between the material parameters and the network density. Table 2: Material constants of the proposed model (25) used for the comparison to the experimental data by Yeoh and Fleming (1997). Fitted parameters are shaded.

Sample D 2.0phr C 1.5 B 1.0 A 0.5

175

µc κ[M P a] 0.4073 0.3055 0.2037 0.1018

n 13.5034 18.0045 27.0068 54.0136

q µt [M P a] 1.024 0.2200 1.024 0.1905 1.024 0.1556 1.024 0.1100

Λrupture 6.147 7.084 8.646 12.141

M ΛAST rupture 6.2 6.8 7.9 8.1

3.2. Numerical results for inhomogeneous deformations

176

Our model was implemented into a commercial finite element code via a user defined

177

material subroutine. In this section, it is illustrated by an example with large inhomogeneous

178

deformations. The objective of this numerical study is to assess the accuracy of the proposed

179

constitutive model in predicting complex strain fields. An unfilled silicone rubber sheet with

180

the dimensions 62x82.5x1.75mm and five 20mm diameter holes C1 , C2 , C3 , C4 and C5 is

181

subjected to a tensile force of 20N (Meunier, Chagnon, Favier, Org´eas, and Vacher (2008)).

182

A cut was further made along the direction connecting the centers of the holes C3 and C5 .

183

The detailed locations of the holes on the sheet are given in Fig.7. Stretches Λ2 (in the 17

61.5

X  mm  Y  mm 

82.5

82.5

C4

C5

C3

O

0.0

0.0

C1

47.5

21.2

C2

14.0

23.0

C3

31.5

40.5

C4

47.5

59.0

C5

14.5

58.0

Measured path C2

O

C1

62

Figure 7: Geometry of the rubber sheet specimen used in the tension test (Meunier et al. (2008)).

184

loading direction) at different points along a predefined path were measured by a digital

185

image correlation system (Meunier et al. (2008)). The measured path was selected so that

186

it goes through the high deformation regions around and between C4 and C5 .

187

First, the model is fitted to experimental data of tensile and compressive tests, con-

188

strained tension and compression (pure shear), as well as equibiaxial tension by Meunier

189

et al. (2008) (see Fig.8). The values of the resulting material constants are given in Table 3. The rubber sheet discretized by 2538 reduced integration linear brick elements with hybrid Table 3: Material parameters of the proposed model (25) for the experimental data by Meunier et al. (2008).

µc κ[M P a] n q µt [M P a] 0.2559 3.2935 0.8662 0.0507 190 191

formulation was fixed at the bottom and subjected to the vertical force. The deformed shape

192

of the sheet resulting from the simulation is shown in Fig.9. Stretch values (Λ2 ) experimen-

193

tally measured in some specified points are compared with the numerical values in Fig.10.

194

Good agreement between the simulation results and the experimental data is observed. 18

1 Meunier et al [2008] Model uniaxial

Meunier et al [2008] Model equibiaxial

( MPa )

1.2

11

0 −0.5

Nominal stress P

Nominal stress P11 ( MPa )

0.5

−1 −1.5

0.8

0.4

−2 −2.5

0.5

1

Stretch Λ1

1.5

0 1

2

1.2

1.4

(a)

1.6 Stretch Λ1

1.8

2

2.2

(b) 0 Meunier et al [2008] Model constrained compression

Meunier et al [2008] Model constrained tension

−0.25

Nominal stress P

11

( MPa )

Nominal stress P11 ( MPa )

0.8

0.6

0.4

0.2

0 1

−0.5

−0.75

−1

−1.25

1.2

1.4

1.6 Stretch Λ

1.8

2

2.2

1

(c)

−1.5

0.6

0.7

0.8 Stretch Λ1

0.9

(d)

Figure 8: Prediction of the proposed model (25) with µc κ = 0.2559[M P a], n = 3.2935, q = 0.8662, µt = 0.0507[M P a] in comparison to the experimental data by Meunier et al. (2008): (a) uniaxial tension and compression, (b) equibiaxial tension, (c) constrained tension, (d) constrained compression.

195

3.3. Comparison to other rubber elasticity models

196

For further validation of the model, it is juxtaposed to four other constitutive models,

197

namely the extended-tube model (Kaliske and Heinrich (1999)), the tube-like microsphere

198

model (Miehe et al. (2004)), the Shariff model (Shariff (2000)), and the Kroon model (Kroon

199

(2010)). The series of biaxial tension tests by Kawabata et al. (1981) again serves as a

200

benchmark example. 19

1

presented in Fig. 13. The finite element model geometry was designed using the dimensions given in the table of Fig. 3. A 2D plane stress model was designed since (i) the hickness is fairly homogeneous (1.75 mm  0.02 mm) (ii) he lateral surfaces of the plates are free of load. The lower side of the model is fixed and the upper face is subjected to

a path was studied to compare the local simulated elongation to the experimental elongation. The chosen path passes through the upper holes, as shown in Fig. 13, in order to study the most stretched parts of the plate. The curves describing the experimental and simulated major elongation along the path are presented

LE, Max. Principal (Avg: 75%) +9.478e−01 +8.689e−01 +7.901e−01 +7.112e−01 +6.323e−01 +5.534e−01 +4.745e−01 +3.956e−01 +3.168e−01 +2.379e−01 +1.590e−01 +8.011e−02 +1.224e−03

(a)

(b)

Fig. 12. Photograph of the deformed plate for an applied force of 20 N: (a) deformation of the pattern, (b) colour map: major principal elongation (result of 7D).

Figure 9: Deformed configuration of the rubber sheet: (a) in the experiment (Meunier et al. (2008)), and (b) in the numerical simulation by the proposed model. Meunier et al [2008] FEM result 2.5

Stretch Λ2

2.25 2 1.75 1.5 1.25 1 0

10

20

30 40 X coordinate [mm]

50

60

Figure 10: Comparison between numerical and experimental values (Meunier et al. (2008)) of stretch Λ2 under the tension of the rubber sheet. Material parameters of the proposed model (25) used in the simulation are µc κ = 0.2559[M P a], n = 3.2935, q = 0.8662, µt = 0.0507[M P a].

20

0.4

0.3

( MPa )

1.2 Λ = 1.24 1

0.25

Λ = 1.20

0.2

Λ = 1.14

22

1

Λ1 = 1.16

Nominal stress P

Nominal stress P22 ( MPa )

0.35

1

Λ = 1.12 1

0.15

Λ1 = 1.10 Λ1 = 1.08

0.1

Λ1 = 1.06

Λ1 = 3.4 Λ1 = 3.1

0.8

Λ1 = 2.8 Λ1 = 2.5

0.6

Λ1 = 2.2 0.4

Λ1 = 1.9 Λ1 = 1.6

0.2

Λ1 = 1.04

0.05

Λ1 = 3.7

1

Λ1 = 1.3 Extended tube

Extended tube 0 0.9

1

1.1 Stretch Λ2

1.2

0 0.5

1.3

1

1.5

2 Stretch Λ

2.5

3

2

(a)

(b)

Figure 11: Prediction of the extended-tube model by Kaliske and Heinrich (1999) versus a series of biaxial tensions by Kawabata et al. (1981): (a) at small values of Λ1 , and (b) at large values of Λ1 . Material parameters Gc = 0.202[M P a], Ge = 0.158, β = 0.178, δ = 0.0856 (Marckmann and Verron (2006)) were obtained by fitting to experimental data by Treloar (1944).

The extended-tube model by Kaliske and Heinrich (1999) has only four material parameters and according to Marckmann and Verron (2006) agrees well with the experimental data of Treloar (1944) and Kawabata et al. (1981). The strain energy function of the model is given by   3 
Gc (1 − δ 2 ) (I1 − 3) 2Ge X  −β 2 Ψ= + ln 1 − δ (I − 3) + Λ − 1 , 1 i 2 1 − δ 2 (I1 − 3) β 2 i=1

(31)

201

where Gc , δ, Ge , and β are material constants, while δ is related to the limit of chain

202

extensibility. The first term in (31) is responsible for the phantom network, while the

203

second term represents the topological constraint. However, a connection between the first

204

part of (31) and the non-Gaussian theory of rubber elasticity is not explicitly visible (for

205

comparison, see also the original phantom network strain energy function in Edwards and

206

Vilgis (1986)). We further reproduce predictions of the extended-tube model in the series

207

of biaxial tensions by Kawabata et al. (1981). The material constants of the extended-tube

208

model were taken from Marckmann and Verron (2006). As seen in Fig.11, the extended-tube

209

model slightly underestimates the stress in the small range of deformations. 21

3.5

0.4 1.2

0.3

Nominal stress P22 ( MPa )

Nominal stress P

22

( MPa )

0.35 Λ1 = 1.24

0.25

Λ1 = 1.20

0.2

Λ1 = 1.14

Λ1 = 1.16 Λ1 = 1.12

0.15

Λ1 = 1.10 Λ1 = 1.08

0.1

Λ1 = 1.06

Λ1 = 3.4 Λ1 = 3.1

0.8

Λ1 = 2.8 Λ1 = 2.5

0.6

Λ1 = 2.2 0.4

Λ1 = 1.9 Λ1 = 1.6

0.2

Λ = 1.04

0.05

Λ1 = 3.7

1

Λ1 = 1.3

1

Microsphere 0 0.9

1

1.1 Stretch Λ2

1.2

Microsphere 0 0.5

1.3

(a)

1

1.5

2 Stretch Λ2

2.5

3

(b)

Figure 12: Prediction of the tube-like microsphere model (Miehe et al. (2004)) against a series of biaxial tensions by Kawabata et al. (1981): (a) at small values of Λ1 , and (b) at large values of Λ1 . Material parameters µ = 0.292[M P a], n = 22.01, U = 0.744, p = 1.472, q = 0.1086 (Miehe et al. (2004)) were obtained by fitting to experimental data by Treloar (1944).

210

Stress-stretch diagrams resulting from the tube-like microsphere model (Miehe et al.

211

(2004)) based on the 21-point integration scheme by Bazant and Oh (1986) are plotted in

212

Fig.12 versus the above mentioned experimental data by Kawabata et al. (1981). The ma-

213

terial constants of the tube-like microsphere model were taken from Miehe et al. (2004).

214

Comparing Fig.4 and Fig.12, one can observe that the proposed model demonstrates bet-

215

ter agreement with the experimental data than the tube-like microsphere model at large

216

stretches although the latter model appears to be better in the small deformation regime.

217

Note, however, that our model does not require numerical integration over the unit sphere

218

and thus is apparently advantageous from the numerical point of view. Indeed, for a nu-

219

merical integration scheme with n points, every calculation should be repeated n-times,

220

which renders a microsphere-type model approximately n-times slower in comparison to the

221

corresponding analytical model. Marckmann and Verron (2006) also reported that the four-term version of the Shariff model (Shariff (2000)) is among the three most accurate hyperelastic models for rubber-like materials. This model is based on the hypothesis of Valanis and Landel (1967). Accordingly, 22

3.5

0.4

1.2

0.3

Nominal stress P22 ( MPa )

Nominal stress P22 ( MPa )

0.35 Λ = 1.24 1

0.25

Λ1 = 1.20

0.2

Λ1 = 1.14

Λ1 = 1.16 Λ1 = 1.12

0.15

Λ1 = 1.10 Λ1 = 1.08

0.1

1

Λ = 3.4 1

Λ = 3.1

0.8

1

Λ = 2.8 1

Λ = 2.5

0.6

1

Λ = 2.2 1

0.4

Λ = 1.9 1

Λ = 1.6

Λ1 = 1.06

1

0.2

Λ1 = 1.04

0.05

Λ = 3.7

1

Λ1 = 1.3 Shariff

Shariff 0 0.9

1

1.1 Stretch Λ2

1.2

0 0.5

1.3

1

(a)

1.5

2 Stretch Λ2

2.5

3

(b)

Figure 13: Prediction of the Shariff model (33) against a series of biaxial tensions by Kawabata et al. (1981): (a) at small values of Λ1 , and (b) at large values of Λ1 . Material parameters E = 1.072[M P a], α1 = 0.896, α2 = 0.0398, α3 = 0.88 · 10−5 , α4 = 0.0273 (Marckmann and Verron (2006)) were obtained by fitting to experimental data by Treloar (1944).

the strain energy function of a hyperelastic material can be represented as Ψ = W (Λ1 ) + W (Λ2 ) + W (Λ3 ) .

222

(32)

The four-term version of the Shariff model predicts the following principal Cauchy stresses σi = E

4 X

αk Φk (Λi )

k=0

2 ln Λi + α1 e1−Λi + Λi − 2 + α2 eΛi −1 − Λi = E 3 # 3 (Λi − 1) + α4 (Λi − 1)3 , i = 1, 2, 3, +α3 3.6 Λi

(33)

223

where E is the Young modulus, αk are conjugative parameters of the regular functions

224

Φk (Λi ), and α0 = 1.

225

Predictions of this model (Shariff (2000)) are demonstrated in Fig.13 against stress-

226

stretch diagrams from the biaxial tension tests by Kawabata et al. (1981). The five values of

227

material constants used in this simulation were obtained by Marckmann and Verron (2006). 23

3.5

228

As seen in Fig.13, the Shariff model demonstrates very good agreement with experimental

229

data by Kawabata et al. (1981) in the wide range of Λ1 . Nevertheless, in the range of small

230

strains, the proposed model is more accurate than that one of Shariff. Additionally, this

231

phenomenological model includes more material parameters, most of which do not have any

232

physical interpretation. Kroon recently proposed an 8-chain model based on the tube-like concept (Miehe et al. (2004)) and including five material parameters. The strain energy function of the model is given by (Kroon (2010))  λc Ψ = cc n  √ L−1 n



λ √c n

L−1

 + ln



λc √ n

sinh L−1





λc √ n


α   + cnc λ2nc − 1 + ccon nυ,

(34)

235

where cc , n, cnc , α and ccon are material constants. λcrand λnc are non-affine micro-stretches I1 related to the first principal invariant by λc λnc = . This model demonstrates good 3 agreement with experimental data from uniaxial tension, pure shear and equibiaxial tension.

236

However, the non-affine micro-stretches appearing in (34) must be obtained by solving a

237

nonlinear scale transition equation, which can be numerically expensive. Note also that

238

in order to capture experimental data by Treloar (1944) and Kawabata et al. (1981), two

239

separate sets of material constants are needed for this model (Kroon (2010)).

240

3.4. Discussion

233

234

241

What are the reasons for accurate results provided by certain constitutive models of

242

rubber elasticity in general, and by the proposed model in particular? From the physical

243

point of view this is due to the non-Gaussian statistical foundation, the topological constraint

244

(as can be seen in (1), (25) and (31)) and the non-affine amplification, which all render

245

the model closer to the real polymer network. For phenomenological models, of special

246

importance from the mathematical point of view is the contribution of the second principal

247

invariant (see also Fried (2002); Wineman (2005); Horgan and Smayda (2012)). The tube

248

constraint in the extended-tube model (31) and the efficiency of the Shariff phenomenological

249

model (Shariff (2000)) may also be considered as an implicit consequence of the second 24

250

principal invariant I2 as both these models are formulated in terms of the principal stretches.

251

On the other hand, good performance of the proposed model suggests, however, that the

252

tube-contraction amplification applied for example in Miehe et al. (2004) is not crucial.

253

The extended-tube model, the Shariff model and the proposed model can provide very

254

good agreement with experimental data in multiple states of deformation. Thus, the micro-

255

sphere modeling concept appears to be dispensable for isotropic material modeling. So when

256

may the numerical integration over the unit sphere be advantageous? This is for example the

257

case for anisotropic materials, such as soft fibrous tissues (due to spatial dispersion of col-

258

lagen fibers) or rubbers subjected to stress softening or strain-induced crystallization which

259

takes place only in stretching directions (Gehman and Field (1939); Murakami, Senoo, Toki,

260

and Kohjiya (2002); Toki, Sics, Ran, Liu, Hsiao, Murakami, Senoo, and Kohjiya (2002),

261

Trabelsi et al. (2003)). For constitutive modeling of isotropic rubber-like materials, models

262

including the first two principal invariants appear to be sufficient.

263

4. Conclusions

264

In this paper, we presented a physically based constitutive model for rubber elasticity. In

265

contrast to microsphere-based models (see e.g. Miehe et al. (2004); Dal and Kaliske (2009);

266

Mistry and Govindjee (2014)), the numerical integration over the unit sphere is avoided

267

by using the analytical network-averaging concept. Furthermore, our model is based on a

268

closed-form of the Rayleigh exact non-Gaussian distribution function, which can capture

269

the limit of chain extensibility. The model is described by only four material constants

270

with the clear physical meaning (see Table 4) and demonstrates very good agreement with

271

various experimental data. The proposed model is designed for large strain rubber elasticity.

272

However, in case of moderate strains, it can be simplified to the strain energy of the ideal

273

network with only two parameters µc κ and n (i.e. µt = 0 and q = 1), since the effects of

274

the topological constraint and stretch amplification are negligible there. Therefore, it can

275

enrich understanding in physics of rubber elasticity, and could also be a suitable option for

276

industrial applications. 25

Table 4: Physical meaning of the material constants.

277

Parameters

Physical meaning

µc κ n q µt

Ideal network effective shear modulus [MPa] Average number of chain segments Stretch amplification exponent Topological shear modulus [MPa]

Acknowledgements

278

The authors would like to thank Professor Gr´egory Chignon (University of Grenoble,

279

Laboratory TIMC-IMAG, France) for providing experimental data of the unfilled silicone

280

rubber and geometrical details of the rubber sheet.

281

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