Acta Mech 208, 39–53 (2009) DOI 10.1007/s00707-008-0128-1

Baojiu Lin

A new model for hyperelasticity

Received: 24 September 2008 / Published online: 2 December 2008 © Springer-Verlag 2008

Abstract The essential criterion for a good mathematical model for hyperelasticity is its ability to match the measured strain energy curves under different deformations over a large range. One group of models for hyperelasticity is to express the strain energy as a function of I 1 , I 2 , I 3 , the invariants of the right CauchyGreen deformation tensor. Under the assumption of incompressibility, it can be proved that all valid (I 1 , I 2 ) pairs fall in a region bounded by the I 1 − I 2 locus from deformations under simple extension and equalbiaxial extension (or, equivalently, simple compression). I 1 − I 2 locus from planar extension lies inside the region. Since the strain energy curves from simple deformation modes can be measured from experiments, it is possible to approximately obtain the strain energy under other (I 1 , I 2 ) values by interpolating data from the three measured curves. The proposed model for hyperelasticity is an interpolation algorithm with all mathematical details. The new model is implemented into a user-defined material subroutine in commercial FEA software. It can not only accurately reproduce the measured data from these simple deformation modes but also predict the stress–strain curve under planar extension in a reasonable good accuracy even without using the measured data from planar extension.

1 Introduction The mechanical behavior of rubber-like materials is very complicated. In past decades, great efforts have been focused on the development of mathematical models of rubber-like materials. The classic theory for rubber-like materials is well addressed in [1,2]. Discussion of modeling the mechanical behavior of rubber-like materials can be found in [3]. Broad reviews of mathematical models for the mechanical properties of rubber-like materials for finite element analysis can be found in [4,5]. More literature about the mathematical models and finite element simulations of rubber and rubber-like materials can be found through the bibliographies [6,7]. An introduction on engineering design of rubber parts and mechanical properties testing of rubber can be found in [8,9]. So far, almost all well known mathematical models from these efforts are focused on construction of strain energy functions by best data fitting against the stress–strain data pairs measured under different simple deformation modes. For example, the Neo-Hookean model assumes the strain energy density as a linear function of I 1 and the Mooney-Rivlin model assumes the strain energy density as a linear function of (I 1 , I 2 ). Comparisons among the Mooney-Rivlin model, Ogden model, Neo-Hookean model, Yeoh model, Arruda-Boyce model and the Van der Waals model for rubber-like materials can be found in [10]. A comparison among the Gent model, the Hart-Smith model and Arruda-Boyce model is reported in [11]. Each of these wellknow models has abilities to well match experiment data for certain materials in a certain range of strain. But none of them has the capability to match the testing data from a large range of materials and over a large range of strain. The varieties of the complicated behaviors of rubber-like materials make them very difficult B. Lin (B) ExxonMobil Chemical Company, 388 S Main Street, Akron, OH 44311, USA E-mail: [email protected]

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to be interpreted by a single model. On the other hand, the demand for better designs (high performance, low cost, less materials etc.) in the engineering community has been rapidly pushing the use of finite element simulation techniques into daily design practices. One of the key factors for obtaining correct simulation results by using nonlinear finite element analysis is a right model for mechanical properties of rubber-like materials. Unfortunately, even the very sophisticated commercial nonlinear finite element packages, like ABAQUS, do not provide adequate models to properly describe the mechanical behavior observed in actual materials, such as thermoplastic elastomers. To fill the gap between the lack of the good mathematical models with solid physical meanings and the immediate demand from engineering applications, a new model based on the interpolation of testing data in the (I 1 , I 2 , E) space is proposed in this study. The details of this model are presented in the subsequent sections.

2 Finite deformation theory for incompressible materials In nonlinear continuum mechanics [12], at least two configurations: the current and the initial configurations are required to describe the deformation of a body. Mathematically, a material point in its initial configuration can be identified by its position X in a reference frame. In the current configuration, the same material point can be identified by its current position x in the same reference frame. The motion from X to x can be mathematically expressed as the following function of X and time t: x = x(X, t).

(1)

The deformation of a body at a typical material point is measured by the geometry change of the point with respect to the points in its neighborhood during a motion. Mathematically, it can be described as dx = FdX

(2)

∂x , ∂X

(3)

with F=

which is known as the deformation gradient tensor. The distance between two material points before and after the deformation is related by dx T dx = dX T F T FdX,

(4)

C = FT F

(5)

where

is called the right Cauchy-Green deformation tensor. It is positive definite if dX
= 0. Thus, there is the principal orientation in which the tensor is diagonal as ⎡ 2 ⎤ λ1 C = ⎣ λ22 ⎦ . (6) λ23 Accordingly, in the same principal orientation, the deformation gradient tensor is written as ⎤ ⎡ λ1 F = ⎣ λ2 ⎦ , λ3

(7)

where λi (i = 1, 2, 3) are called the principal stretch ratios. The change of distance of the material point related to points in its neighborhood is dx T dx − dX T dX = dX T (F T F − I )dX = 2dX T EdX.

(8)

A new model for hyperelasticity

41

Here the Green-Lagrangean strain tensor is defined as E=

1 T (F F − I ). 2

In the principal orientation, the Green-Lagrangean strain tensor is also diagonal, i.e., ⎤ ⎡ ε1 E = ⎣ ε2 ⎦ . ε3

(9)

(10)

The principal strains are related to the principal stretch ratios by 2εi = λi2 − 1.

(11)

Three invariants of the right Cauchy-Green deformation tensor are defined as I1 = λ21 + λ22 + λ23 , I2 = I3 =

λ21 λ22 + λ21 λ23 + λ22 λ23 , λ21 λ22 λ23 = J 2 ,

(12) (13) (14)

where J = det(F) is related to volume deformation in the following way: dv = J dV.

(15)

The geometric meaning of I 1 and I 2 can be explained using Fig. 1, in which a unity cube deforms to a cube with side lengths of λ1 , λ2 , and λ3 . I 1 is the square of the distance from the original point to the corner of the deformed cube. I 2 is the half of the sum of the square of the side surface areas of the deformed cube. It means that under same I 1 , a bigger I 2 value means larger deformation. Similarly, under same I 2 , a bigger I 1 value means larger deformation. For incompressible material, there is an internal constraint of incompressibility J = λ1 λ2 λ3 = 1.

Fig. 1 Geometry of the deformed unity cube

(16)

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Under this constraint, the first and second invariants are rewritten as I1 = λ21 + λ22 + I2 = λ21 λ22 +

1

,

(17)

1 1 + 2. 2 λ1 λ2

(18)

For the simple tension, the deformation gradient tensor is ⎡ λ1 ⎢ √1 F =⎣ λ1

λ21 λ22

⎤ √1 λ1

⎥ ⎦.

(19)

The corresponding I1 and I2 are 2 , λ1 1 I2 = 2λ1 + 2 . λ1 I1 = λ21 +

(20) (21)

For the equal-biaxial extension, the deformation gradient tensor is ⎡ F =⎣

λ1

⎤ λ1

1 λ21

⎦.

(22)

The corresponding I1 and I2 are 1 , λ41 2 I2 = λ41 + 2 . λ1 I1 = 2λ21 +

(23) (24)

The planar extension is another important simple deformation mode. Its deformation gradient tensor is ⎡ ⎤ λ1 1 ⎦. F =⎣ (25) λ1 1 The corresponding I1 and I2 are 1 , λ21 1 I2 = 1 + λ21 + 2 . λ1 I1 = 1 + λ21 +

(26) (27)

Since I 1 = I 2 in the planar extension, it is represented by a straight line laying between the curves from the simple tension and the equal-biaxial extension as shown in Fig. 2. Under the assumption of incompressibility, all the valid point (I 1 , I 2 ) are limited to a region bounded by I 1 and I 2 from simple tension and equal-biaxial extension (or simple compression), as shown in Fig. 2. It can be proved mathematically that for the same I 1 , the simple tension has the minimum I 2 value while the equal-biaxial extension has the maximum I 2 value. Similarly, for the same I 2 , the equal-biaxial extension has the minimum I 1 value while the simple tension has the maximum I 1 value.

A new model for hyperelasticity

43

Fig. 2 The region for valid (I1 , I2 ) pairs

3 The new model for hyperelasticity The objective for modeling the hyperelasticity is to find the best estimation of the strain energy density at any point inside the region of valid (I 1 , I 2 ) pairs. The strain energy density curves under simple tension and equalbiaxial extension can be obtained experimentally. In other words, the strain energy curves along the boundaries of the valid region for (I 1 , I 2 ) pairs are known experimentally. In addition, the strain energy density curve from a planar extension can also be obtained without a big effort. The locus from the planar extension sits inside the region. Taking the SantopreneTM thermoplastic rubber grade 121-73W175 as an example, the stress–strain curves from the three simple deformation modes are plotted in Fig. 3 and the corresponding curves of strain energy density as functions of I1 are plotted in Fig. 4, and the same strain energy density curves are also plotted in the 3D space of (I 1 , I 2 , E) in Fig. 5. Alternatively to fit any pre-defined mathematical model, the attempt in this study is to construct a strain energy density function by interpolating the strain energy density curves obtained experimentally. The proposed new model actually is a numerical interpolation algorithm for the construction of the strain energy density function. As mentioned in the preceding section, I 2 represents the deformation of the surface area of a unity

Fig. 3 Stress–strain curves under simple deformation modes

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Fig. 4 Curves of strain energy density vs I1 under simple deformation modes

Fig. 5 Curves of strain energy density of SantopreneTM thermoplastic rubber grade 121-73W175 in (I1 , I2 , E) space under simple deformation modes at 23◦ C

cube while I 1 represents the deformation of the distance of the diagonal corners. For a given I 1 , the simple tension deformation corresponds to the lowest deformation in the surface area of the unity cube, while the equalbiaxial extension corresponds to the largest deformation of the surface area of the unity cube. Consequently, we can believe that the strain energy is largest under equal-biaxial extension, and smallest under a simple tension. Therefore, it is reasonable to assume that the strain energy at a given I 1 can be described by the following function: ˆ 1 , t), E(I1 , I2 ) = E(I

(28)

where the parameter t is defined as t=

I2 − I2eb (I1 ) st (I ) − I eb (I ) I21 1 1 2

.

(29)

A new model for hyperelasticity

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I2st (I1 ) and I2eb (I1 ) are the I 2 values on the locus from the simple tension and the equal-biaxial extension corresponding to I 1 . When t = 0, the strain energy can be determined by the strain energy curve from the simple tension testing, i.e., ˆ 1 , 0) = E st (I1 ). E(I

(30)

When t = 1, it equals the strain energy curve from the equal-biaxial extension testing, i.e., ˆ 1 , 1) = E eb (I1 ). E(I

(31)

When a planar extension testing data is available, an additional known strain energy value is ˆ 1 , t pt ) = E pt (I1 ) E(I

(32)

at the point t pt =

I1 − I2eb (I1 ) . st I2 (I1 ) − I2eb (I1 )

(33)

The above three known strain energy points are sufficient to construct a strain energy density function as ˆ 1 , t) = co + c1 t + c2 t 2 E(I1 , I2 ) = E(I

(34)

c0 = E st (I1 ), 2 − (E pt (I ) − E st (I )) (E eb (I1 ) − E st (I1 ))t pt 1 1 , c1 = 2 t pt − t pt

(35)

with the coefficients

c2 =

−(E eb (I1 ) − E st (I1 ))t pt + (E pt (I1 ) − E st (I1 )) . 2 −t t pt pt

(36) (37)

Taking SantopreneTM thermoplastic rubber grade 121-73W175 as an example, the strain energy density function constructed according to Eq. (34) is plotted in Fig. 6.

4 The derivation of the stress–strain relationship According to the principle of virtual work, the second Piola–Kirchhoff stress tensor is related to the right Cauchy-Green deformation tensor by ∂ E(I1 , I2 ) δεi j ∂ci j   ∂ E(I1 , I2 ) ∂ I1 ∂ E(I1 , I2 ) ∂ I2 =2 + δεi j , ∂ I1 ∂ci j ∂ I2 ∂ci j

σi j δεi j = 2

(38)

where ci j are the components of the right Cauchy-Green deformation tensor. The derivatives of I 1 and I 2 with respect to ci j can be found in [13,14], as ∂ I1 = δi j , ∂ci j ∂ I2 = I1 δi j − ci j . ∂ci j

(39) (40)

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Fig. 6 The strain energy density surface in the (I1 , I2 , E) space

Here δi j is the Kronecker tensor. The derivatives of E(I 1 , I 2 ) with respect to I 1 and I 2 can be obtained from ∂ E(I1 , I2 ) ∂ E(I1 , I2 ) d I1 + d I2 ∂ I1 ∂ I2   ˆ ˆ ˆ 1 , t) = ∂ E(I1 , t) ∂t dI1 + ∂t d I2 + ∂ E(I1 , t) dI1 . = d E(I ∂t ∂ I1 ∂ I2 ∂ I1

dE(I1 , I2 ) =

(41)

From Eq. (41), the partial derivatives of the strain energy density function with respect to the invariants can be found as

ˆ 1 , t) ˆ 1 , t) ∂t ∂ E(I1 , I2 ) ∂ E(I ∂ E(I d I1 , = + (42) ∂ I1 ∂t ∂ I1 ∂ I1 ˆ 1 , t) ∂t ∂ E(I1 , I2 ) ∂ E(I = d I2 , ∂ I2 ∂t ∂ I2

(43)

where d I st (I ) d I2eb (I1 ) d I2st (I1 ) − d2 I 1 ∂t d I1 − d I1 1 = st − t , ∂ I1 I2 (I1 ) − I2eb (I1 ) I2st (I1 ) − I2eb (I1 ) 1 ∂t = st , ∂ I2 I2 (I1 ) − I2eb (I1 ) ˆ 1 , t) ∂ E(I = c1 + 2c2 t, ∂t ˆ 1 , t) ∂ E(I dc0 dc1 dc2 2 = + t+ t , ∂ I1 dI1 dI1 dI1

(44) (45) (46) (47)

and the details of the derivation of the partial derivatives of I 2 and ci with respect to I 1 are presented in the Appendix.

A new model for hyperelasticity

47

5 The derivation of the tangent modulus In the implementation of nonlinear finite element analysis algorithm, the tangent modulus tensor is required for improving convergence in numerically solving the nonlinear algebraic equations and for analysis of stability. The tangent modulus tensor is defined as the derivatives of the Piola-Kirchhoff stress tensor with respect to the right Cauchy-Green deformation tensor. The relation for the tangent modulus can be derived from Eq. (38) as ∂σi j δεi j δεkl ∂ckl

∂ 2 E(I1 , I2 ) ∂ I1 ∂ 2 E(I1 , I2 ) ∂ I2 ∂ I1 ∂ E(I1 , I2 ) ∂ 2 I1 =4 + + ∂ckl ∂ I1 ∂ I2 ∂ckl ∂ci j ∂ I1 ∂ci j ∂ckl ∂ I12



∂ 2 E(I1 , I2 ) ∂ I2 ∂ I2 ∂ E(I1 , I2 ) ∂ 2 I2 ∂ 2 E(I1 , I2 ) ∂ I1 δεi j δεkl . + + + ∂ I1 ∂ I2 ∂ckl ∂ckl ∂ci j ∂ I2 ∂ci j ∂ckl ∂ I22

Di jkl δεi j δεkl =

(48)

The second order partial derivatives of I 1 and I 2 with respect to the right Cauchy-Green deformation tensor from Eqs. (42) and (47) can be found in [13–15] as ∂ 2 I1 = 0, ∂ci j ∂ckl

(49)

∂ci j ∂ 2 I2 = δkl − . ∂ci j ∂ckl ∂ckl

(50)

The second order derivatives of the strain energy density function with respect to I 1 and I 2 can be derived from Eqs. (41) to (45) through the following relation: ˆ 1 , t) d2 E(I1 , I2 ) = d2 E(I 2 ˆ 1 , t)  ∂t ∂t ∂ 2 E(I dI + dI = 1 2 ∂ 2t ∂ I1 ∂ I2   ˆ 1 , t) ∂ 2 t ˆ 1 , t) ∂ E(I ∂ 2t ∂ 2 E(I + dI dI + 2 dI dI dI1 dI1 , + 1 1 2 1 2 ∂t ∂ I1 ∂ I1 ∂ I2 ∂ 2 I1 where

 2  d2 I2st (I1 ) d I2eb (I1 ) d2 I2st (I1 ) d I2eb (I1 ) d I2st (I1 ) − dI 2 − t d I12 − d I12 ∂ 2t ∂t d I1 − d I1 1 = −2 − , ∂ I1 I2st (I1 ) − I2eb (I1 ) ∂ I12 I2st (I1 ) − I2eb (I1 ) ∂ 2t = 0, ∂ I22 ˆ 1 , t) ∂ 2 E(I = 2c2 , ∂t 2 ˆ 1 , t) ∂ 2 E(I d2 c0 d2 c1 d2 c2 2 = + t + t . ∂ I12 dI12 dI12 dI12

(51)

(52) (53) (54) (55)

The details of the derivation of the second order partial derivatives of I2 and ci with respect to I1 are presented in the Appendix. 6 Validation of the new model An ABAQUS user-defined subroutine for hyperelastic materials based on this new model is developed. The testing data from the simple tension, equal-biaxial extension, and planar extension of three different grades:

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Santoprene™ Theromplatic Rubber Grade 121-73W175 Stress-Strain Curves under Simple Tension at 23°C 2.5

Stress (MPa)

2.0

1.5

1.0

Experimental Data New Model without PT Data Abaqus Marlow Model

0.5

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

New Model with PT Data Abaqus Mooney-Rivlin Model

0.35

0.40

0.45

0.50

0.45

0.50

Strain

Fig. 7 The stress–strain curves from simple tension of 121-73W175 Santoprene™ Theromplatic Rubber Grade 121-73W175 Stress-Strain Curves under Equal-biaxial Extension at 23°C 3.5 3.0

Stress (MPa)

2.5 2.0 1.5 1.0 0.5 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Strain Experimental Data Abaqus Mooney-Rivlin Model

New Model with PT Data Aabqus Marlow Model

New Model without PT Data

Fig. 8 The stress–strain curves from equal-biaxial extension of 121-73W175 Santoprene™ Theromplatic Rubber Grade 121-73W175 Stress-Strain Curves under Planar Extension at 23°C 3.0

Stress (MPa)

2.5 2.0 1.5 1.0 0.5 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Strain Experimental Data Abaqus Mooney-Rivlin Model

New Model with PT Data Abaqus Marlow Model

Fig. 9 The stress–strain curves from planar extension of 121-73W175

New Model without PT Data

0.50

A new model for hyperelasticity

49

Santoprene™ Theromplatic Rubber Grade 121-67W175 Stress-Strain Curves under Simple Tension at 23°C 2.5

Stress (MPa)

2.0

1.5

1.0

0.5

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Strain Experimental Data Abaqus Mooney-Rivlin Model

New Model without PT Data

New Model with PT Data Abaqus Marlow Model

Fig. 10 The stress–strain curves from simple tension of 121-67W175 Santoprene™ Theromplatic Rubber Grade 121-67W175 Stress-Strain Curves under Equal-biaxial Extension at 23°C 2.5

Stress (MPa)

2.0

1.5

1.0

0.5

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Strain Experimental Data Abaqus Mooney-Rivlin Model

New Model with PT Data Abaqus Marlow Model

New Model without PT Data

Fig. 11 The stress–strain curves from equal-biaxial extension of 121-67W175 Santoprene™ Theromplatic Rubber Grade 121-67W175 Stress-Strain Curves under Planar Extension at 23°C 3.0

Stress (MPa)

2.5

2.0

1.5

1.0

0.5

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Strain Experimental Data New Model without PT Data Abaqus Marlow Model

Fig. 12 The stress–strain curves from planar extension of 121-67W175

New Model with PT Data Abaqus Mooney-Rivlin Model

0.50

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B. Lin

Santoprene™ Theromplatic Rubber Grade 121-58W175 Stress-Strain Curves under Simple Tension at 23°C 1.4 1.2

Stress (MPa)

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.45

0.50

Strain Experimental Data New Model without PT Data

New Model with PT Data Abaqus Mooney-Rivlin Model

Abaqus Marlow Model

Fig. 13 The stress–strain curves from simple tension of 121-58W175

Santoprene™ Theromplatic Rubber Grade 121-58W175 Stress-Strain Curves under Equal-biaxial Extension at 23°C

1.8 1.6

Stress (MPa)

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Strain Experimental Data Abaqus Mooney-Rivlin Model

New Model with PT Data Abaqus Marlow Model

New Model without PT Data

Fig. 14 The stress–strain curves from equal-biaxial extension of 121-58W175

121-73W175, 121-58W175, and 121-67W175 (with hardness of 58, 67, and 73 shore A) of SantopreneTM thermoplastic rubber are used in validation. The simulation results from ABAQUS using this new user-defined subroutine for hyperelastic materials are plotted in Figs. 7–15. For comparison, the predictions from material models built in ABAQUS are also plotted in Figs. 7–15. It clearly shows that the new model can exactly reproduce the testing data obtained in all three different deformation modes, while the Marlow method from ABAQUS can match the testing data from one deformation mode only. In Figs. 7–15, only the simple tension testing data is used in the ABAQUS Marlow model (see [16]). Using a simplified version of the current new model, in which only the simple tension and equal-biaxial extension are used and the linear interpolation method is used to construct the strain energy density function in the (I 1 , I 2 , E) space, the simulation can still predict the planar extension testing in a reasonable good accuracy as shown in Figs. 9, 12, and 15.

A new model for hyperelasticity

51

Santoprene™ Theromplatic Rubber Grade 121-58W175 Stress-Strain Curves under Planar Extension at 23°C 1.6 1.4

Stress (MPa)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Strain Experimental Data Abaqus Mooney-Rivlin Model

New Model with PT Data Abaqus Marlow Model

New Model without PT Data

Fig. 15 The stress–strain curves for planar extension of 121-58W175

7 Conclusions The new model presented in this paper is able to precisely follow stress–strain curves measured from all three deformation models over entire range of tested strain. It can predict planar extension well even only with testing data from simple tension and equal-biaxial extension (or simple compression). This model is also stable in the whole tested strain range. Acknowledgment The author is grateful to his colleague Dr. Maria D. Ellul for review and discussion of this paper.

Appendix: Derivation of partial derivatives of I2 with respect to I1 (1) For simple tension dI1 2 = 2λ1 − 2 dλ1 λ2 2 dI2 =2− 3 dλ1 λ1 λ21 dλ1 = dI1 2(λ31 − 1)

(56) (57) (58)

λ3 + 3λ1 d2 λ1 = − 13 dI1 dλ1 2(λ1 − 1)2 dI2 dλ1 1 dI2 = = dI1 dλ1 dI1 λ1 d2 I 2 d2 I2 dλ1 1 = =− 2 3 dI dλ dI dI1 2(λ1 − 1) 1 1 1

(60)

4 dI1 = 4λ1 − 5 dλ1 λ1

(62)

λ51 dλ1 = dI1 4(λ61 − 1)

(63)

(59)

(61)

(2) For equal-biaxial extension

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d2 λ1 dI1 dλ1 dI2 dλ1 dI2 dI1 d2 I 2 dI1 dλ1 d2 I2 dI12

=−

4 λ10 1 + 5λ1

4(λ61 − 1)2 4 = 4λ31 − 3 λ1 dI2 dλ1 = = λ21 dλ1 dI1 = 2λ1 =

(64) (65) (66) (67)

λ61 2(λ61 − 1)

(68)

(3) The derivatives of ci with respect to I 1 are dc0 dE st (I1 ) = (69) dI1 d I1 d2 c0 d2 E st (I1 ) = (70) dI12 dI12  eb  d E (I1 ) − d E st (I1 ) t − (E eb (I ) − E st (I )) dt pt + d E pt (I1 ) − d E st (I1 ) − pt 1 1 2t pt − 1 dt pt dc2 d I1 d I1 d I1 d I1 d I1 = − c2 2 dI1 t 2pt − t pt t pt − t pt dI1

(71) d2 c2 dI12 =

     eb 2 eb 2 st st d2 t dt d2 E pt (I1 ) − d2 E st (I1 ) − d E 2(I1 ) − d E 2(I1 ) t pt −2 d Ed I(I1 )− d Ed I(I1 ) d Ipt −(E eb (I1 )− E st (I1 )) pt + d I1 d I1 d I12 d I12 d I12 1 1 1 t 2pt − t pt

−2

2t pt − 1 d2 t pt dc2 2t pt − 1 dt pt 2C2 dt pt dt pt − − c 2 dI1 t 2pt − t pt dI1 t 2pt − t pt dI1 dI1 t 2pt − t pt dI12

d I eb (I ) dII2st (I1 ) d I2eb (I1 ) − dI 1 − 2d I 1 dt pt dI 1 1 = st − t pt st 1 eb dI1 I2 (I1 ) − I2 (I1 ) I2 (I1 ) − I2eb (I1 ) d2 t pt dI12

−

d2 I2eb (I1 ) d I12

= st − t pt I2 (I1 ) − I2eb (I1 )

d2 II2st (I1 ) d2 I2eb (I1 ) − d I12 d I12 I2st (I1 ) − I2eb (I1 )

dc1 dE eb (I1 ) dE st (I1 ) dc2 = − − dI1 dI1 dI1 dI1 d2 c1 d2 E eb (I1 ) d2 E st (I1 ) d2 c2 = − − dI12 dI12 dI12 dI12

(72)

(73) d II2st (I1 ) d I2eb (I1 ) dt pt d I1 − d I1 −2 dI1 I st (I1 ) − I eb (I1 ) 2 2

(74) (75) (76)

References 1. 2. 3. 4.

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