Constitutive modeling of solids at finite deformation using a second-order stress–strain relation

International Journal of Engineering Science 48 (2010) 223–236

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Constitutive modeling of solids at finite deformation using a second-order stress–strain relation H. Darijani a, R. Naghdabadi b,* a b

Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran Mechanical Engineering Department and Institute for Nano Science and Technology, Sharif University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 6 June 2009 Received in revised form 25 August 2009 Accepted 26 August 2009 Available online 14 October 2009 Communicated by M. Kachanov Keywords: Finite deformation Strain measures Hookean-type Constitutive modeling

a b s t r a c t In this paper, a deformation measure is introduced which leads to a class of strain measures in the Lagrangian and Eulerian descriptions. In order to develop a constitutive equation, a second-order constitutive relation based on these strain measures is considered for modeling the mechanical behavior of solids at finite deformation. For this purpose and performance evaluation of the proposed strains, a Hookean-type constitutive equation is considered and the uniaxial loading as well as simple shear and pure shear tests are examined. It is shown that the constitutive modeling based on the proposed strains give results which are in good agreement with the experimental data. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction We recall that the general motion of a continuum is described by x = v(X, t), where x is the spatial position at time t of a material particle with the material coordinate X. A material element dX at the reference configuration is transformed, through the motion, into a material element dx at time t. The relation between dX and dx is given by

dx ¼ FdX

ð1Þ

where F is called deformation gradient tensor. Consider two material elements dX(1) and dX(2) at the reference configuration. Due to the motion characterized by the deformation gradient F, the material elements are transformed into dx(1) and dx(2) at time t:

dxð1Þ ¼ FdX ð1Þ ;

dxð2Þ ¼ FdX ð2Þ

ð2Þ

In order to follow the deformation in the Lagrangian and Eulerian descriptions, tensors of Green strain E and Almansi strain e are defined as follows [1]

dxð1Þ  dxð2Þ  dX ð1Þ  dX ð2Þ ¼ 2dX ð1Þ  EdX ð2Þ dxð1Þ  dxð2Þ  dX ð1Þ  dX ð2Þ ¼ 2dxð1Þ  edxð2Þ

* Corresponding author. Tel.: +98 21 66165546; fax: +98 21 66000021. E-mail address: [email protected] (R. Naghdabadi). 0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.08.006

ð3Þ

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Substituting Eq. (2) into Eq. (3) we obtain

E¼

F TF  I ; 2

e¼

I  F T F 1 2

ð4Þ

where I is the identity tensor. Since det(F) > 0, the polar decomposition theorem states that F is uniquely decomposed as

F ¼ RU ¼ VR

ð5Þ

where U and V are the right and left stretch tensors, respectively. U and V are positive definite symmetric tensors and R is a proper orthogonal rotation tensor, which represents the rotation of the eigenvectors of U, Ni, to the eigenvectors of V, ni.

ni ¼ R N i

ð6Þ

Let ki, (i = 1,2,3) be eigenvalues of the stretch tensors. Indeed, ki are the principal stretches of the deformation. The general class of Lagrangian strain tensor was defined by Hill as [2,3]:

Eðf Þ ¼ FðUÞ ¼

3 X

f ðki ÞN i  N i

ð7Þ

i¼1

Also, the Eulerian description of Eq. (7) is represented as follows

eðf Þ ¼ FðVÞ ¼ REðf Þ RT ¼

3 X

f ðki Þni  ni

ð8Þ

i¼1

where f() is an arbitrary strictly-increasing scalar function satisfying conditions f(1) = 0 and f0 (1) = 1. Also, the symbol  represents the dyadic or tensor product. If in the special case, f() is selected in the form of f ðki Þ ¼ ðkm i  1Þ=m with m as an arbitrary integer, then the Seth–Hill strains are obtained in the form [2,4]

EðmÞ ¼

Um  I m

ð9Þ

The Lagrangian and Eulerian descriptions of the Seth–Hill strain tensors are as follows [5]:

Um  I I  U m ; EðmÞ ¼ m m Vm  I I  V m ðmÞ ¼ ¼ ; e m m

Lagrangian description : EðmÞ ¼ Eulerian description : eðmÞ

ð10Þ

For example, the nominal strain E(1) = U  I, the Green strain E(2) = (U2  I)/2 and the logarithmic strain E(0) = ln(U) are strain measures in the Seth–Hill class that are given by the scalar functions f ðki Þ ¼ ðki  1Þ, f ðki Þ ¼ ðk2i  1Þ=2 and f ðki Þ ¼ lnðki Þ, respectively. In this paper, two deformation measures are introduced considering material elements in the reference and current configurations. According to the properties of these measures, a class of strain measures is proposed. Hookean-type constitutive equations are used for investigating the behavior of the proposed strains. For this purpose, uniaxial loading as well as simple shear and pure shear tests are examined and the results are compared with the corresponding results based on the Seth–Hill strains. 2. Deformation modeling The strain tensors E and e are commonly used to characterize the deformation in the reference and current configurations, respectively. Now, we characterize the deformation using both of reference and current configurations as follows:

dxð1Þ  dxð2Þ  dX ð1Þ  dX ð2Þ ¼ 2dxð1Þ  EdX ð2Þ dxð1Þ  dxð2Þ  dX ð1Þ  dX ð2Þ ¼ 2dX ð1Þ  Edxð2Þ

ð11Þ

Substituting Eq. (2) into Eq. (11) we obtain

E ¼ ðF  F T Þ=2;

e ¼ ET ¼ ðF T  F 1 Þ=2 E

ð12Þ

e are two-point strain measures which are obtained according to Eq. (10). It is noted that strain tensor where tensors E and E e is the same as what was introduced earlier [6]. Based on the polar decomposition theorem and using Eq. (5), Lagrangian E and Eulerian descriptions of these strains (Eq. (12)) become

U  U 1 2 V  V 1 e Eulerian description : D ¼ 2 Lagrangian description : D ¼

ð13Þ

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225

1 According to Eqs. (7) and (8), the scalar function of the strain tensors in Eq. (13) is given by gðki Þ ¼ ðk i ki Þ=2. This function tends to +1 if k goes to +1 and goes to 1 if k approaches zero, whereas the Seth–Hill scalar functions have not these physical properties simultaneously. In the same way, more general measures of strain G(U) or G(V), coaxial with U and V, can be defined so that their characteristic functions are more sensible than those of the corresponding functions of the Seth–Hill measures. Therefore, we define a class of strain measures as follows:

NðgÞ ¼ GðUÞ ¼

3 X

gðki ÞN i  N i

i¼1

‰

ðgÞ

T

ðgÞ

¼ GðVÞ ¼ RN R ¼

3 X

ð14Þ gðki Þni  ni

i¼1

where g(ki) should satisfy the following properties:

gð1Þ ¼ 0;

g 0 ð1Þ ¼ 1;

limk!1 gðkÞ ¼ 1;

g 0 ðkÞ > 0 ðaÞ

limk!0 gðkÞ ¼ 1 ðbÞ

ð15Þ

A family of functions g(ki) subject to the constraints (15)a and (15)b, can be introduced as a linear combination of the terms km, kn(mn > 0 or m, n = 0) in the form of

gðkÞ ¼

1 ðkm  kn Þ mþn

ð16Þ

Now, by substituting Eq. (16) into Eq. (14) we can obtain the Lagrangian and Eulerian descriptions of the proposed strain tensors as follows:

1 ðU m  U n Þ mþn 1 ðV m  V n Þ ¼ mþn

Nðm;nÞ ¼ ‰ðm;nÞ

ð17Þ

for the case n = m, these tensors are simplified as follows

1 ðU m  U m Þ 2m 1 ¼ ðV m  V m Þ 2m

NðmÞ ¼ ‰ðmÞ

ð18Þ

It is noted that these strain tensors (Eq. (18)) for m = 2 become the same as what was introduced earlier [7]. For the special case m = 0, by applying the L’Hopital’s rule we can find the indeterminate forms of these tensors which are equivalent to the logarithmic strain tensors.

lim NðmÞ ¼ lnðUÞ;

m!0

lim ‰ðmÞ ¼ lnðVÞ

m!0

ð19Þ

For m = 1, the two-point description of the strain tensors N(m), ‰ðmÞ become similar to Eq. (12)

E ¼ RNðm¼1Þ ¼ ðF  F T Þ=2 e ¼ RT ‰ðm¼1Þ ¼ ðF T  F 1 Þ=2 E

ð20Þ

We can list the proposed strain tensors in the Eulerian and Lagrangian descriptions as follows

( ðmÞ

N

¼ (

‰

ðmÞ

¼

1 ðU m 2m

 U m Þ m ¼ 1; 2; 3; . . .

lnðUÞ 1 ðV m 2m

lnðVÞ

m¼0 V

m

Þ m ¼ 1; 2; 3; . . .

ð21Þ

m¼0

To put an idea forward for the physical interpretation of the proposed strains we consider the single material element dX(1) = dX(2) = dX = dSN, where N is a unit vector and dS is the length of dX. Suppose that under the deformation, dX and dS transform into dx and ds, respectively, where dx = ds n, then expressions for the longitudinal strain according to different measures are given as follows

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m

ds  dS 1 ¼ ð1  km Þ; mP0 m m mds m m ds  dS 1 N  EðmÞ N ¼ ¼ ðkm  1Þ; mP0 m m mdS 2m 2m ds  dS 1 m N  NðmÞ N ¼ ðk  km Þ; m P 0 or m 6 0 m m ¼ 2m 2mds dS 2m 2m ds  dS 1 m n  NðmÞ n ¼ m P 0 or m 6 0 ðk  km Þ; m m ¼ 2m 2mds dS n  eðmÞ n ¼

ð22Þ

where k = ds/dS denotes the stretch ratio. Fig. 1 illustrates the longitudinal strain based on different strain measures versus stretch k. It can be seen from Fig. 1 that the strain measures on the basis of Eq. (22)3,4 tend to +1 when k goes to +1 and go to 1 when k approaches zero. But, for the Seth–Hill strain measures (Eq. (22)1,2), when k tends to +1 the strain measure (22)1 goes to 1/m, and when k goes to zero the strain measure (22)2 approaches 1/m, which are not physically compatible results. In fact, adding the constraints (14)b in defining the proposed strain measures (17) and (18) yields physically reasonable results.

3. Hookean-type constitutive modeling As cited by Hill [3], a natural generalization of the Hooke’s law leads to the constitutive equation:

T ðmÞ ¼ @w=@EðmÞ ¼ 2lEðmÞ þ ktrðEðmÞ Þ (m)

ð23Þ (m)

where T is conjugate stress of the strain measure E , l = E/2(1 + t), k = Et/(1 + t)(1  2t) and t, E are Poisson’s ratio and Young’s modulus, respectively. Accordingly, the constitutive equation in terms of the proposed strain measure is written as follows

RðmÞ ¼ @w=@NðmÞ ¼ 2lNðmÞ þ ktrðNðmÞ Þ (m)

ð24Þ (m)

where R is the stress tensor conjugate to the strain measure N . It is noted that both of the strain measures E(m) and N(m) as well as the stresses T(m) and R(m) are objective. That means the quantities of strain and stress are not affected by rigid body rotation of the reference frame. In order to use the proposed strain measures N(m) in the constitutive modeling and compare the result with the Seth–Hill strain measures, it is necessary to determine the stress measures conjugate to N(m). For the purpose of comparison these stress measures should be determined in terms of Cauchy stress tensor.

Fig. 1. Comparison of longitudinal strains based on different measures versus stretch.

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Considering the second-order tensors A and B, the double contraction operation is defined as A:B = tr (ATB) = tr(ABT) = Aij Bij, where tr stands for trace. A stress measure K is said to be conjugate to a strain measure H if K : H_ represents power per _ [8]. Therefore, unit reference volume, w

_ ¼ Jr : D ¼ P : F_ ¼ K : H_ w

ð25Þ

where r, P and D are the Cauchy stress, first Piola–Kirchhoff stress and rate of deformation tensors, respectively, and J is the third invariant of the right stretch tensor U. e be conjugate to the strain measure E = (F  FT)/2, then Let stress tensor P

e : ðF_ þ F T F_ T F T Þ e : E_ ¼ P e : 1 ðF  F T Þ ¼ 1 P _ ¼ Jr : D ¼ P : F_ ¼ P w 2 2

ð26Þ

Eq. (26) can consequently be written as follows

trðP F_ T Þ ¼

1 h e _ T e 1 _ 1 i ¼ tr tr P F þ PF FF 2

! # e þ F T P e T F T P F_ T 2

"

ð27Þ

e and P: Since in Eq. (27), tensor F_ T is arbitrary, we can write the following relation between P

e T F T e þ F T P 2P ¼ P

ð28Þ

Similarly, the stress tensor R(1) conjugate to the strain measure N(m=1) = (U  U1)/2 can be determined. Using the relation _ 1 þ U 1 UÞR _ T and Eq. (23), it is concluded that: D ¼ 1=2ðRðUU

  1 _ 1 Jr : D ¼ Jr : RðUU þ U 1 UÞRT ¼ Rð1Þ : 2

_ 1 U_ þ U 1 UU 2

! ð29Þ

We can arrange Eq. (29) as follows

tr

h

 i h  i U 1 RT rR þ RT rRU 1 U_ ¼ tr Rð1Þ þ U 1 Rð1Þ U 1 U_

ð30Þ

The arbitrariness of U_ implies that the stress measure R(1) conjugate to N(1) satisfies the tensor equation

U 1 RT rR þ RT rRU 1 ¼ Rð1Þ þ U 1 Rð1Þ U 1 (2)

2

ð31Þ 2

Likewise, for the strain measure N = (U  U )/4, it is assumed that the stress tensor R Eq. (25) is recast for the two conjugate pairs as follows

Jr : D ¼ R

ð2Þ

:

ðU 2 Þ  ðU 2 Þ 4

! ¼R

ð2Þ

:

F T DF þ F 1 DF T 2

! ¼

FRð2Þ F T þ F 1 Rð2Þ F T 2

(2)

(2)

is conjugate to N . Hence,

! :D

ð32Þ

Consequently, for arbitrary choices of D, we find the expression

2Jr ¼ FRð2Þ F T þ F T Rð2Þ F 1

ð33Þ

Some well-known relations between the Cauchy stress tensor and stress tensors conjugate to the Seth–Hill strain measures are as follows [9]:

Jr ¼ FT ð2Þ F T ;

T ð1Þ ¼ ðT ð2Þ U þ UT ð2Þ Þ=2

Jr ¼ F T T ð2Þ F 1 ;

T ð1Þ ¼ ðT ð2Þ U 1 þ U 1 T ð2Þ Þ=2

ð34Þ

where pairs (T(2), E(2)), (T(2), E(2)), (T(1), E(1)) and (T(1), E(1)) are conjugate stress and strain tensors. 4. Applications In order to investigate the behavior of the proposed strain measures, basic deformations such as the uniaxial loading, simple shear and pure shear tests are examined. The material is considered to be isotropic, homogenous elastic which obeys the Hookean-type constitutive equation. 4.1. Uniaxial loading of a cylinder Consider a circular cylinder under axial load P in the X1 direction which is uniformly distributed at the end cross sections. Because of the symmetry, the description of the material particles deformation is:

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x1 ¼ k1 X 1 ; l k1 ¼ ; L

x2 ¼ k2 X 2 ;

x1 ¼ k3 X 3

r k2 ¼ k3 ¼ R

ð35Þ

where L and l are the reference and current lengths, R and r denote the initial and final radii of the element, respectively. In this case, the deformation gradient tensor is:

F ¼ k1 e1  e1 þ k2 e2  e2 þ k2 e3  e3

ð36Þ

where ei, i = 1,2,3 are the unit vectors in the Cartesian coordinate system. For this deformation R = I, thus, F = U = V. For deformation gradient (36), the constitutive Eq. (24) based on the proposed strains (Eq. (21)1) are written as follows

RðmÞ ¼ 2l

 m  m m m k  km km km  km 1 þ 2k2  2k2 1  k1 2 ðe2  e2 þ e3  e3 Þ þ k 1 e1  e1 þ 2l 2 I 2m 2m 2m

ð37Þ

Using Eq. (37), the nonzero components of R(m) are derived as:

  k  m m m m k  km km þ 1  k1 1 þ 2k2  2k2 2m 1  l  m m  k  m m ðmÞ m ¼ R33 ¼ k  k1 þ 2km k  k2 þ 2  2k2 2m 1 m 2

RðmÞ 11 ¼ RðmÞ 22

l

m

ð38Þ

The stress R(2) and the Cauchy stress are kinematically related through Eq. (33), therefore, the Cauchy stress tensor components for the axially loaded cylinder are given as follows

r11 ¼

1 2k1 k22

r22 ¼ r33 ¼



 ð2Þ k21 þ k2 R11 1 1

2k1 k22



 ð2Þ k22 þ k2 R22 2

ð39Þ

In the case of uniaxial loading, the form of the Cauchy stress tensor r is presented as follows:

r¼

P Ak22

e1  e1

ð40Þ

where A is the initial cross sectional area of the element. From Eq. (40), it is noted that r22 = r33 = 0. Therefore, using Eqs. (38)–(40), it is concluded that:

  m m km ¼ t km 2  k2 1  k1   P k2m  k2m 1 ¼ 1 EA NðmÞ 4mk1

ð41Þ ð42Þ

However, the relation between the force and stretch based on the Seth–Hill strains (Eq. (23)) is as follows [10]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m k2 ¼ 1 þ t  tkm 1

  m ð1  tÞkm P 1 þ 2tk2  t  1 ¼ k1m1 EA EðmÞ mð1  2tÞð1 þ tÞ

ð43Þ

For m = 0, both of Eqs. (42) and (43) yield

P ln ðk1 Þ ¼ EA k1

ð44Þ

Fig. 2 depicts force-stretch curves based on different measures of strain with Poisson’s ratio t = 0.25. As it can be seen from Fig. 2, in finite elastic deformation problems, for m > 1 the Seth–Hill strains, result in softening within the compression zone which is not physically acceptable. Also, for m < 0, these strains yield tensile softening and compressive stiffening. For m = 1, they represent a linear behavior for the material that acts like a string obeying the linear Hooke’s law in every range of deformation. But, based on the proposed strain measures N(m), for m = ±1, ±2 the function ðP=EAÞNðmÞ is monotonically increasing in compression and tension, and no softening effect could be seen. It seems that trend of the results obtained based on the proposed strains are more reasonable than the Seth–Hill strains. Thus, we can characterize the actual behavior of the materials with finite deformation using an appropriate selection of (m, n). For this purpose, it is necessary to use the experimental data of simple traction tests. For example, we refer to uniaxial tension tests carried out by El-Ratal and Mallick [11] on commercial and seat type foams. Based on Eq. (24), a constitutive law in terms of two-parameter proposed strains (Eq. (17)) is considered to establish a correlation between the test data and theory. Using Eq. (24) and considering r2 ¼ r3 ¼ 0 with k2 ¼ k2 , it is concluded that:

H. Darijani, R. Naghdabadi / International Journal of Engineering Science 48 (2010) 223–236

229

Fig. 2. Normalized force–stretch curves of the cylinder in the simple tension for different strain measures.

 m  k k  kn 1 2ðk þ lÞ 1  1 lð2k þ 3lÞ  m ¼ k1  kn 1 m þ n 2ðk þ lÞ

n km 2  k2 ¼ 

Rðm;nÞ 11

ð45Þ

where R(m,n)represents the stress conjugate to the strain measure N(m,n). Based on the Hill’s principal axis method and energy conjugacy notion, an approach was proposed to find the relation between the components of two conjugate stress tensors by Farahani and Naghdabadi [9]. Since the Lagrangian stress tensor R(m,n) is conjugate to N(m,n) and the Lagrangian stress tensor R(2) is conjugate to N(2), referring to [9] it can be concluded that ð2Þ R11 ¼

n 2 mkm 1 þ nk1 Rðm;nÞ 11 m þ n k21 þ k2 1

ð46Þ

Substituting Eq. (45)2 into Eq. (46) and using Eq. (39), the general expression for the Cauchy stress, r1, is derived as follows

r1 ¼

1 J ðm þ nÞ2

lð2k þ 3lÞ 2n ðmk2m  ðm  nÞkmn Þ 1  nk1 1 ðk þ lÞ

ð47Þ

Using the experimental data of El-Ratal and Mallick [11] together with the correlation between the values of the test data and the theory, the material parameters are calculated as follows

l ¼ 5:0367 l ¼ 7:37

For test data of Commercial foam : m ¼ 4; n ¼ 1; k ¼ 2:315; For test data of Seat foam : m ¼ 4; n ¼ 1; k ¼ 3:26;

ð48Þ

Then, Eq. (45)1 is numerically solved for the lateral stretch for a given value of the longitudinal stretch k1 . Fig. 3 depicts a comparison of the theory and the experimental results of uniaxial tension stress versus longitudinal stretch for commercial and seat type foams. The lateral stretch versus longitudinal stretch obtained from the theory and tests are also compared in Fig. 4. As it is observed there is a good agreement between the results. 4.2. The simple shear test This deformation is defined as an isochoric plane deformation for which there is a set of line elements whose orientation is such that they are unchanged in length and direction by the deformation. Let the X1 direction defines this orientation. The kinematics of this test is simply defined in a Lagrangian description as follows [5]

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Fig. 3. Comparison of the theoretical and experimental results of uniaxial tension stress versus longitudinal stretch for commercial and seat foams.

Fig. 4. Comparison of the theoretical and experimental results of uniaxial tension stress versus longitudinal stretch for commercial and seat foams.

x1 ¼ X 1 þ cX 2 x2 ¼ X 2

ð49Þ

x3 ¼ X 3 where c is the shear displacement. The representation of the deformation gradient tensor becomes

F ¼ I þ ce1  e2

ð50Þ

Hence, the right stretch tensor has the following form:

2 2 þ c2 c U ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e1  e1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2  e2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe1  e2 þ e2  e1 Þ þ e3  e3 4 þ c2 4 þ c2 4 þ c2

ð51Þ

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231

Based on the proposed strain measures (Eq. (21)1), it is concluded that tr(N(m)) = 0 for the simple shear test. Thus, for this isochoric deformation, it would be an interesting outcome. On the contrary, for this deformation using the Seth–Hill strains it is obtained that tr(E(m)) – 0. Using the constitutive Eq. (24) with tr(N(m)) = 0, the stress tensor R(m) for the simple shear test becomes as follows

RðmÞ ¼ 2lNðmÞ

ð52Þ

The basis-free Eqs. (31) and (33) are used to determine the Cauchy stress components in terms of the stresses conjugate to the strain measures N(1) and N(2), respectively. For this purpose, using Eqs. (31) and (52), we obtain the nonzero Cauchy stress components for the strain measure N(1) in the form

ðr11 ÞNðm¼1Þ ¼ ðr22 ÞNðm¼1Þ ¼

lc2 2

;

ðr12 ÞNðm¼1Þ ¼ lc

ð53Þ

Following the same approach, using Eqs. (33) and (52) the nonzero Cauchy stress components for the strain measure N(2) are obtained as follows:

ðr11 ÞNðm¼2Þ ¼ ðr22 ÞNðm¼2Þ ¼

lc  4

 2c þ c3 ;

ðr12 ÞNðm¼2Þ ¼

l 2

2c þ c3



ð54Þ

Batra [12,13], Nader [14] and Farahani and Bahai [10] have studied the behavior of different Seth–Hill strain measures using the Hookean-type constitutive equation in the simple shear test. Employing the Hookean-type constitutive equation in terms of the Seth–Hill strain E(2) and its conjugate stress T(2) (second Piola–Kirchhoff stress tensor), the nonzero Cauchy stress components are derived in the form:

 lc2  ð1  tÞlc2 2  3t þ c2 ð1  tÞ ; ðr22 ÞEðm¼2Þ ¼ 1  2t 1  2t  lc  ¼ 1  2t þ c2 ð1  tÞ 1  2t  

ðr11 ÞEðm¼2Þ ¼ ðr12 ÞEðm¼2Þ

ð55Þ

and, for the conjugate pair Eð2Þ ; T ð2Þ , the nonzero Cauchy stress components are obtained as follows:

 ð1  tÞlc2 lc2  ; ðr22 ÞEðm¼2Þ ¼  2  3t þ c2 ð1  tÞ 1  2t 1  2t  lc  ¼ 1  2t þ c2 ð1  tÞ 1  2t

ðr11 ÞEðm¼2Þ ¼  ðr12 ÞEðm¼2Þ

ð56Þ

Based on the proposed strain measures, it is observed that r11 = r22 and r33 = 0. But for the Cauchy stresses based on the Seth–Hill strains r11 + r22 – 0 and r33 – 0. Using Eqs. (23) and (24) for various measures of strain along with different values of m, the Cauchy stress components are calculated and shown in Figs. 5–8 for t = 0.25. In Fig. 5, r12 is an odd function of c for all values of m. It is realized from Fig. 6 that r11 obtained based on the Seth–Hill strains, is bounded to E/(1  2t)(1 + t) for m = 1. Also, when m < 0 then r11 < 0 and when m > 0, r11 > 0. Fig. 7 shows that r22 obtained based on the Seth–Hill strains, is bounded to E/(1  2t)(1 + t) for m = 1 and its behavior is similar to r11. Fig. 8 shows that r33 obtained based on the Seth–Hill strains has the same behavior as r11 and r22(r33 < 0 for m < 0 and r33 > 0 for m > 0). Furthermore, since the considered material is Hookean, the results for c < < 1 should reduce to the results of the infinitesimal strain. As it is observed from Figs. 6–8, the results based on the Seth– Hill strains attain positive values for m > 0 and negative values for m < 0. Therefore, for the Seth–Hill strains the results are inconsistent while the results based on the proposed strains are physically consistent. 4.3. The pure shear test The orientation of the principal axes of strain tensors changes in the isochoric plane deformation of simple shear. If, however, the principal axes are fixed in direction, R = I, the resulting isochoric plane deformation is said to be a pure shear. In practice, the pure shear deformation can be achieved by clamping a rectangular sheet of material between two parallel edges (in X3-direction) and extending the sheet in its plane normal to the clamped edges in one direction. If k1 is the principal stretch in the direction of the extension, then k2 ¼ k1 1 and k3 = 1. For this deformation [5],

x1 ¼ k 1 X 1 ;

x2 ¼ k1 1 X2;

x3 ¼ X 3

ð57Þ

Using Eq. (57) together with the constitutive Eq. (24) for the proposed strains (Eq. (17)1) we obtain

Rðm;nÞ ¼ 2l

 m  m n n k  kn km km  kn1 1 þ k1  k1 1  k1 e1  e1 þ 2l 1 e2  e2 þ k 1 I mþn mþn mþn

ð58Þ

For comparison purposes, we should determine the relation between stress tensor R(m,n) and the Cauchy stress tensor using the notion of conjugacy in the form of

_ ¼ Jr : D ¼ Rðm;nÞ : N_ ðm;nÞ w

ð59Þ

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Fig. 5. Normalized r12 for different strain measures in the simple shear test.

Fig. 6. Normalized r11 for different strain measures in the simple shear test.

Using the same procedure as used in the previous section for obtaining the conjugate stresses and considering that R(m,n) in Eq. (58) is coaxial with U for each (m, n), it is concluded that:

r¼

1 Rðm;nÞ ðmU m þ nU n Þ mþn

ð60Þ

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233

Fig. 7. Normalized r22 for different strain measures in the simple shear test.

Fig. 8. Normalized r33 for different strain measures in the simple shear test.

Applying the condition of r2 = 0 in Eq. (60), the nonzero stress components are obtained as follows

ðr11 ÞNðm;nÞ ¼ 2l ðr33 ÞNðm;nÞ ¼

n m n ðkm 1 þ k1  k1  k1 Þ

ðm þ nÞ2 2l 2

ðm þ nÞ

ðkn1  km 1 Þ

n ðmkm 1 þ nk1 Þ

ð61Þ

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In the same way, the nonzero Cauchy stress components in terms of the Seth–Hill strains (Eq. (23)) are derived as follows:

ðr11 ÞEðmÞ ¼

2l 2m ðk  1Þ; m 1

ðr33 ÞEðmÞ ¼

2l Þ ð1  k2m 1 m

ð62Þ

Figs. 9 and 10 show a comparison between the Cauchy stress components r11 and r33 obtained based on different measures of strain in the pure shear test, respectively. As it is observed in Fig. 9, ðr11 ÞNðm;nÞ (Eq. (61)) is monotonically increasing; when k1 approaches +1 ðr11 ÞNðm;nÞ tends to 1, and when k1 goes to zero ðr11 ÞNðm;nÞ approaches 1. On the other hand, for the Seth–Hill strains with m > 0, ðr11 ÞEðmÞ approaches 1 when k1 goes to 1 and it is asymptotic to the line r11 = 2l/m when k1 goes to zero. For the Seth–Hill strains with m < 0,when k1 goes to zero then ðr11 ÞEðmÞ approaches 1 and it is asymptotic to

Fig. 9. Normalized r11 for different strain measures in the pure shear test.

Fig. 10. Normalized r33 for different strain measures in the pure shear test.

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235

Fig. 11. Comparison of the theoretical and experimental data of pure shear test for the rubber Heuillet and Dugautier [15] and Kawaba.

the line r11 = 2l/m when k1 goes to 1. Furthermore, Fig. 10 shows that r33 obtained based on different measures of strain has the same trend as that of r11. Therefore, the results based on the Seth–Hill strains are not physically acceptable while the trends of the results based on the proposed strains are in agreement with the physical sense. In order to validate the results, it is necessary to use the experimental data of pure shear tests. For this purpose, we refer to the pure shear experiments that have been previously done on a rubber by Heuillet and Dugautier [15] and Kawabata et al. [16]. Based on Eq. (61)1, the Cauchy stress expression including parameters m, n, l is considered to establish a correlation between the stress values obtained from these tests and the theory. Similarly, the unknown parameters are determined as follows:

l ¼ 0:57 l ¼ 0:38

For Heuillet and Dugautier test data ½15 : m ¼ 0:24; n ¼ 1:34; For Kawabata etal: test data ½16 : m ¼ 0:23; n ¼ 1:13;

ð63Þ

Fig. 11 shows a comparison of the predictions for the stress values and those extracted from the experiment results of Heuillet and Dugautier [15] and Kawabata et al. [16]. In both cases there is a good agreement between the results of the experiment and the theory. 5. Conclusions In this paper, we introduced a deformation measure which led to a class of strain measures suitable for constitutive modeling of solids under finite deformation. A Hookean-type constitutive equation was used for investigating the effect of different strain measures. For this purpose, the conjugate stresses of the strain measures in terms of the Cauchy stress components were derived. In order to evaluate the performance of the proposed strains, the uniaxial loading as well as simple shear and pure shear tests were examined and the results were compared with the corresponding ones based on the Seth–Hill strains. It was observed that in uniaxial loading the results based on the Seth–Hill strains show unexpected material hardening or softening in tension and compression. But the results based on the proposed strain measures are monotonically increasing in tension and compression and no softening effect was observed. As, using experimental data of simple traction tests it was validated that a second-order constitutive equation based on proposed strain measures can be advantageous for modeling the mechanical behavior (Figs. 3 and 4). In the simple shear test, the Cauchy stress components based on the Seth–Hill strains attain positive values for m > 0 and negative values for m < 0 (m is the strain index). Since the results for small shear displacement (c < < 1) should reduce to the results of the infinitesimal strain, it is concluded that these results are inconsistent while the results based on the proposed strains are free of such inconsistency. In the pure shear test, both of the Cauchy stress components r11 and r33 obtained based on the proposed strains are monotonically increasing. But for the Seth–Hill strains with m > 0, when k1 tends to 1 these stresses approach 1 and when k1 tends to zero they approach to constant values. Also, for the Seth–Hill strains with m < 0, when k1 goes to zero the values of these stresses approach 1 and when k1 tends to 1 they approach constant values. Therefore, the results based on the Seth–Hill strains are not physically acceptable while the trends of the results based on the proposed strains are in agreement with the physical sense in the pure shear test. In order to validate this claim, pure shear experimental data of Heuillet and Dugautier [15] and Kawabata et al.

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[16] were examined. It was observed that predictions for the stress values from the theory and those extracted from the experiment results are in a good agreement. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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