fracture and mechanical behavior of rubber-like polymers under finite deformation in biaxial stress...

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Fracture and mechanicalbehavior of rubber-likepolymers under finitedeformation in biaxial stressfieldSueo Kawabata aa Department of Polymer Chemistry, Faculty ofEngineering , Kyoto University , Kyoto, JapanPublished online: 13 Sep 2006.

To cite this article: Sueo Kawabata (1973) Fracture and mechanical behavior ofrubber-like polymers under finite deformation in biaxial stress field, Journal ofMacromolecular Science, Part B: Physics, 8:3-4, 605-630

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J. MACROMOL. SC1.-PHYS., B8(3-4), 605-630 (1973)

Fracture and Mechanical Behavior of Rubber-like Polymers under Finite Deformation in Biaxial Stress Field

SUE0 KAWABATA Department of Polymer Chemistry Faculty of Engineering Kyoto University Kyoto, Japan

Summary

A study on the mechanical properties of rubbery polymers under finite deformation is presented here using the biaxial stress relaxation method for the experimental study. In the first part the gradient functions of the strain energy density function are determined as functions of I, and I,, which are the invariants of the deformation tensor, by biaxial stress relaxation testing. The following results were found for both vulcanized natural rubber and SBR.

1. Each aW/aI, and aW/aI, is a function of both I, and I,, especially in the small deformation region. However, the functions take on almost constant values and become flat for relatively large deformations.

2. The temperature dependence of these functions are such that aW/aI, is proportional to the absolute temperature in the rubbery region whereas aW/aI, is almost independent of temperature.

3. The relaxation properties of both functions are very close and have almost the same relaxation rate.

In the second part some discussion on the cause of the functional forms of these functions are presented using a structural model composed of 16 stretchable elements. The result is that aW/aI, can be related to the energy required for the elongation of these elements, and aW/aI, corresponds to the energy required for the expansion of surfaces which are formed by the elements. This

605 Copyright Q 1974 by Marcel Lkkker, Inc. All Rights Reherved. Neither this work nor any part may be reproduced or transmitted in any form or by any mrans. electronic or mechanical. including photocopying. microfilming, and recording, or by any infomlation storage and retrieval system, without permission in writing froin the publisher.

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606 S. KAWABATA

implies that aW/aI, corresponds to an intramolecular energy, such as entropy elasticity, and aW/aI, to the energy caused by intermolecular forces in the body. This estimation is also supported by an experiment in which aW/aI, took a negative value in the small deformation region.

using both the biaxial tensile tester and the thin film inflation method. The fracture criterion of constant stretch ratio is proposed. This simple criterion can be deduced from the structural model presented here. It is also shown, using plasticized polyvinyl chloride, that if the material changes its property from rubbery to glassy, this simple relation is no longer applicable.

Finally, the fracture properties of the rubber-like polymers are observed by

INTRODUCTION

A study on the mechanical properties of rubbery polymers with finite deformation as obtained by biaxial s t ress relaxation, as well as their fracture behavior in a biaxial s t ress field, is presented here.

For rubber-like polymers the interrelations between microstructure and mechanical properties have not been fully explained. Ex- perimental observation of their behavior in a multiaxial stress state will yield powerful information about the interrelations between structure and properties for further development of this field.

From continuum mechanics the stress- strain relation for incompressible materials with finite deformation is given by Eqs. (1) and (2) in the orthogonal coordinate system, Xi, which is taken along the principal axes of the deformation:

Where A, (i = 1, 2) is the stretch ratio along the Xi direction, E i are the true stresses acting on the X i surface and given, from engineering stress (3 i, by = Xio (i is not summed) and W is the strain energy density (strain energy per unit volume). I, and I, a re the invariants of the deformation tensor and given as a function of A, by

I , = x;L+x;+< (3)

Here I, (= A;gg) is unity because of the incompressibility of the body.

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RUBBER-LIKE POLYMERS 607

(5)

In the case of uniaxial extension along the X, axis, we obtain the relations

- u

u, = u3 = 0

and A, = 1/dh,, or using engineering stress u,,

From the classical kinetic theory of rubber elasticity, the stored- energy function W (same as strain energy density) is given by [l]

W = +NkT(XH + 2 + < - 3) (6)

or

W = +NkT(I, - 3) (7)

where k, T , and N a re Boltzman's constant, absolute temperature and number of chains per unit volume, respectively.

Putting

G = NkT (8)

and from Eqs. (3) and (7), the energy gradient functions used in Eqs. (1) and (2) can be reduced to

aW/aI, = G/2, aW/aI, = 0 (9)

Equation (9) implies that the aW/aI,, has a constant value for constant N and T and is independent of the deformation, and also that ~ W / J I , is zero.

ible materials, From Mooney's expression for W for rubber-like and incompress-

w = Cl(I1 - 3) + c&, - 3) (10)

where C, and C, a re constants and a re material parameters. This form for W is the simplest approximation of the form [2]

W =C Cmn(Il - 3)"(I, - 3)" m, "

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608 S. KAWABATA

From Eq. (10) we obtain

aw/a1, = c, aw/a1, = c, (1 2)

The kinetic theory, given by Eq. (8), shows C, = G/2 and C, = 0 in Mooney's expression. The stress-strain relation for uniaxial deformation for Mooney's expression [3], obtained'by substituting Eq. (12) in Eq. (5), is

The Mooney-Rivlin plot, that is, the plot of al/2(X, - l / X ? ) against l / h , gives the values of C, and C, or aW/aI, and aW/aI, under the assumption that both functions must be constant for any set of I, and I,.

Although this method is very useful because of i ts simple proce- dure, it is not able to give the true values of aW/aI, when those values a re functions of either I, or I, or of both of them. To obtain the functional forms of W for isotropic and incompressible materials, the biaxial stretching method can be applied. From Eqs. (1) and (2) we obtain [2]

and

For relatively small deformation such a s I,, I, < 3.5, a very precise technique for measuring the Xi values is required to calculate accu- rate values of aW/aI, and aW/aI, from the above equations.

materials by Treloar [l] reduced the stress-strain relation for incompressible

where Ei is true s t ress and can be calculated from X,a, (i is not summed).

the above by assuming a separable symmetric function for W as Recently, Valanis and Landel [4] introduced equations similar to

w = W(XJ + w(x,) + W(X,) (1 7)

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and found a simple method for determining the form of W. This assumption is based on a simple model which is the same as the model used for the entropy calculation in the classical theory.

strain relations for incompressible and isotropic materials a r e ob- t aine d :

By using Eq. (17), the following representations for the stress-

where ~ ’ ( x ) = aw/ax. From Eq. (6),

By substituting Eq. (19) in Eq. (6) we obtain the same representation as Eq. (18). Both forms of the above equations and Treloar’s Eq. (16) a r e the same except for the difference in whether w’(X)/X is constant or not. The reason is that both deductions a re based on the simple model of noninterrelated chains.

experiment data, and have shown that Valanis and Landel calculated the form of w’(X) from some biaxial

w’(A) = 2K In A (20)

for the range of 0.6 < A < 2.5.

W from experiment with no difficulties such as arose in the use of Eqs. (14) and (15). But the important thing is whether the assumption on which the theory for Eq. (17) is based is acceptable or not. When we consider the complicated situation with respect to the molecular-assembly system of elastomers, this acceptance is still doubtful. For this reason, aW/aI, are used for material parameters in this article and calculated by the use of Eqs. (14) and (15) for I,,

Equation (18) gives a very simple method for finding the form of

I, a 3.5.

EXPERIMENTAL SURVEY OF THE ENERGY DENSITY FUNCTIONS

Functional Forms of aW/aI, and aW/aI,

Biaxial tensile equipment, designed by the author, was used for this experiment. The main part of this equipment is shown in Fig. 1.

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61 0 S. UAWABATA

x 2 SYSTEM

- SERVO CONTROLLED MOVING BAR

FIG. 1. Biaxial tensile tester, its appearance and its schematic diagram.

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RUBBER-LIKE POLYMERS 61 1

Measurements were carried out mainly by biaxial s t ress relaxation testing. The portion shown in Fig. 1 is covered by a temperature box in which the temperature is controlled up to 150°C, and all this equipment is put in a small room in which the temperature is controlled from -20°C to room temperature. U s e of both temperature systems enables us to control the temperature in the range from -20 to 150°C. The size of samples is 10 x 10 x 0.1 cm.

Vulcanized rubbers such as natural rubber (NR), styrene-butadiene rubber (SBR), and acrylonitrite-butadiene rubber (NBR) have been measured to date, and it has been confirmed that all of them have similar functional forms for aW/aI, in their rubbery state.

A s an example, Fig. 2 shows 1 min isochronal data for aW/aI, and aW/aI,, respectively, for NR (Sample 1) at room temperature, where the composition is shown in Table 1. Figure 3 show similar data for a SBR vulcanizate (Sample 2).

TABLE 1

Samplesa

Sample NR-RSS-1 ZnO Stearic acid Dicup MgO

NR 1 100 3 1 4 1

Sample JSR-SBR. -1500 S MBT Stearic acid ZnO

SBR 2 100 1 1.5 1 5

SBR 3 100 1.5 1.5 1 5

a1450C, 30 rnin cure.

Generally speaking, changes in value of aW/aI,, and also aW/aI,, with changes in I, and I, are small except in the small deformation region. Another feature of the functional form is that the change of aW/aI, on a constant I, line is complementary to the change of aW/aI, on the same I, line.

Figure 4 shows aW/aI, and aW/aI, on the I, = I, line. (This deformation is obtained by the stretching holding X, = 1 or X, = 1 and called strip biaxial extension.) As seen in Fig. 5, this deformation goes through the center of the domain of possible I, and I,, and the values of aW/aIi on this line can be considered to be approximately the mean value for uniaxial and homogeneous biaxial deformation.

Thus the general trend of aW/aI, functions can be estimated from this curve. A s seen in Fig. 4, the change of the value of aW/aI, with

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61 2 S. KAWABATA

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"E -

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614 S. KAWABATA

1

4 5 6 7 8 I1 (= 12)

FIG. 4. aW/aI, and aW/aI, on I, = I, line for NR and SBR.

increasing I, is relatively small, but aW/aI, is evidently a decreasing function of I, (= IJ.

Temoerature Deoendence

A s expected from Eq. ( B ) , the value of aW/aI, should be proportional to temperature ("K) i f the classical theory is accepted. Fig- ure 6 shows the temperature dependence of both aW/aI, and aW/aI, for NR, Fig. 7 for SBR from 1 min isochronal plots from the s t ress relaxation measurements. The temperature dependence of aW/aT, is clearly shown and looks as i f it is proportional to absolute temperature except for the low temperature region where the effect of the glassy state seems to appear.

Time Dependence

Six deformations were selected so as to cover a wide region of deformations. The s t ress relaxation behavior of the aW/aI, and aW/aI, for each of these points is shown in Fig. 8 for SBR (Sample 3). The same result was also obtained for NR.

relaxation in Fig. 8 is dependent on the material but is independent of deformation.

aW/aI, and aW/aI, have the same relaxation rate: the slope of the

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616 S. KAWABATA

1 -5

% 5 1.0- Y

; ;0- / 0 0 0 , .’ - 0 .’ /

- o 0.1 min - o 1.0 min

10. min . c 0.5

Temp. FIG. 6. Temperature dependence of aW/aI, and aW/aI, for NR (Sample 1).

The following separable property can be obtained from the above experimental results:

The relation shown in Eq. (21) is only valid for the rubbery state and for unfilled rubber.

AN EXPLANATION OF aW/aI, AND aW/aI,

Consider a cubic model which consists of 16-stretchable elements, PIP,, P,P,, . . ., in an undeformed body a s shown in Fig. 9, and assume an affine deformation for this body.

--

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RUBBER- LIKE POLYMERS 61 7

2.0 -

1.5-

% 0

x 1.0-

.- z 2 - 0.5

c I 1 1 I 1 I 1

260 300 340 380 I 1 I I I I 1

-20 0 20 40 60 80 1OO'C Temp.

FIG. 7. Temperature dependence of aW/aI, and aW/aI, for SBR (Sample 2).

The edges of the cube are taken along the principal axes of the deformation of this body. When the model is stretched by principal stretch ratios Xi (i = 1, 2, 3), an edge element PIP, is stretched by XI, P,P, by h, and so on.

Now, i f the strain energy of the edge elements caused by the stretching by X is assumed to be proportional to X2 following entropy elasticity, the total energy stored in all edge elements by their stretching becomes the sum of each X2 of all the elements.

For quantitative analysis the sum of the energy of six edge elements should be taken instead of the 12 edge elements (the other six edge elements should not be considered because of continuity of structure; that is, they belong to the next cubic structures).

- -

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S , KAWAB AT A 618

0.1 1 10 t ,min

FIG. 8. Time dependence of aW/aI , and aW/aI, for SBR (Sample 2), P,, P,, P,, S,, S4, and S, correspond to the deformation given by stretch ratios hl = 1.629,1.592, 1.452,1.465,1.762, and 2.306, and X2 = 0.908,0.998,1.222,0.999,1,000, and 1.012, respectively .

Undeformed

0 Xl

FIG. 9. A model for structural analysis.

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Then if we assume the entropy elasticity,

We, = %K(X: +

= K I ,

+ h;)

_ _ - where K is constant and equal to G in Eq. (8). On the other hand, each of the diagonal elements P,P,, P,P,, . . .,

in Fig. 9 is elongated at the same time and their stretch ratios a re given by (X: + % + X:)1/2 for all diagonal elements, and the square of it becomes I, also.

Quantitatively, as the length of the each diagonal elements is equal to v/3 times the length of an edge element, the energy stored in a diagonal element is d 3 times the energy stored in an edge element.

Assuming also entropy elasticity, we obtain for the diagonal elements the energy function

4 63 Wa = --I, 2

and the total energy stored in all elements becomes

W, = We, + W,, = K ( l + 2d3)I, (24)

If we assume that the energy required for stretching those elements is caused only by entropy elasticity of independent molecular chains, W takes the same form as Eq. (6) which is introduced in the classical kinetic theory. That is, in this case the energy change caused by stretching of the body by X is equal to h2 - 1 and we also obtain the energy change

W = K(l + 2d3)(1, - 3) (25)

We obtain the same result from Eqs. (24) and (25), i.e.,

aw a11 - = constant

and

On the other hand, let us consider the expansion of area S, (sur-

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620 S. KAWABATA

face perpendicular to X, axis) shown in Fig. 10. The ratio is given by X2h. For S, it is h,h and for S, it is XI&.

FIG. 10. Surfaces formed by the elements.

Now, assume that the energy required for the surface expansion is proportional to A,, the square of the expansion ratio A. Then we obtain the energy required for the expansion of the side planes of this cube as

where K,, is constant, Here we have to consider only the surfaces S,, S,, and S, because of the continuity of the structure model.

Next, let us consider the planes formed by the diagonal elements. For example, surface P,P,P,. There a re a total of 1 2 such surfaces in this cube. For surface P,PGP,, the expansion ratio becomes h,(Xg + < ) l f 2 under the assumption of affine transformation of material points in the body. Because its square is A:g + X;g, the total A2 for these planes is given by

Following the same assumption, we have for the diagonal elements the energy Wds:

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Thus the total energy becomes

From Eq. (30) we have

3 = 0 , -- aw - K,, + 8Kd, = constant a11 a 1 2

From the above rough sketch of the deformation mechanism it is estimated that aW/aI, should correspond to elongations of the edge elements, o r of molecular chains, and aW/aI, to expansion of the surfaces formed by three diagonal elements; in other words, expansion of the distances between the edge elements, o r between molecular chains, against their intermolecular forces.

Here, the energy required for the surface expansion has been assumed to be proportional to the square of its expansion ratio, A'. But the situation is not so simple as in the case of the stretching of edge elements. However, if the surface tension caused by inter- elements o r intermolecular forces is proportional to its area, the assumption is not unreasonable, and the results deduced from this assumption will not be far from the actual behavior of materials.

For further discussion, let us consider a simple system composed of the four elements shown in Fig. 11 and the forces acting in this system. If each of the elements is stretchable and the force required for its stretching is proportional to its length, the attractive force acting on the Element CD toward Element AB, which is caused by the Elements AC and BD, becomes a linear relation with A, which is the stretch ratio of AC and BD as shown as Line 1, Fig. 12 .

FIG. 11. Four elements model having inter- and intraelement force for considering surface energy.

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622 S. KAWABATA

f h

.g u) I \surface expansion

FIG. 12. Forces acting on Element CD.

If the system is in its equilibrium state, there should be a repulsive force between Elements CD and AB. But i f we take a larger value of X, the repulsive force will be gradually changed to an attractive force and then the attractive force will increase with increasing X. After passing the maximum point of force, the attractive force will decrease with increasing X. This process is shown in Fig. 1 2 as Line 2. The dotted line marked 3 is the resultant of both forces. The point on which this resultant line crosses the zero level gives the equilibrium position of Element CD and is shown by X = 1.

Now let us see that the interelement force will be repulsive in the small deformation region of X, near the equilibrium state (between P and Q on Line 2), and that it then changes to an attractive force. This implies that aW/aI, will have a negative value in the early stage of deformation, that is, in the small deformation region, and then it changes to a positive value with increasing X, after which it again decreases with increasing A.

deformation experiment; however, there was no information avail- able on the functional form of aW/aI, in the small deformation region. An experiment especially designed for the small deformation region was carried out using a larger sample of NR to increase the accuracy in measurement of stretch ratios; the results obtained by this experiment a r e shown in Fig. 13. It is clearly seen that aW/aI, takes negative values in the small deformation region. The same relation has been observed for SBR and plasticized PVC. The data support the idea that aW/aI, is due to intermolecular forces.

Figure 4 show the decreasing portion of aW/aI, in a strip biaxial

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1.01

NR 22°C

3.15

FIG. 13. Small deformation region for Sample 1 (NR). aW/aI, takes negative value in the small deformation region at room temperature.

FRACTURE OF THE RUBBER-LIKE MATERIALS IN A BIAXIAL STRESS FIELD

The fracture phenomena seems to be helpful for analyzing the relation between the microstructure and the mechanical behavior of polymers because it depends strongly on the microstructure. Constant rate biaxial stretching was done by two methods; one used the biaxial tensile tester, which was used for measurement of the mechanical properties already described, while the other used the method of inflation of a thin sheet [ti]. Some modifications were made in these techniques [6]. A s shown in Fig. 14, the tes t sheet for the biaxial tensile tester has a thin, round part, and it is expected that fracture will s tar t in this thin part where the stretch ratio is larger than in the surrounding thick part. This modification was made for the purpose of avoiding fracture initiation in the clamping zone at the r im of the sample.

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624 S. KAWABATA

FIG. 14. Biaxial strength testing using the biaxial tensile tester; a sample having a thin region in its central part is used for this experiment.

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In the second method, as shown in Fig. 15, equal biaxial extension (that is, X, = h) can be obtained by inflation with air using a circular hole. On the other hand, an elliptic hole can be used to obtain unequal biaxial extension (that is, X, # 1,) as seen in Fig. 15. Figure 15 shows a block diagram of the apparatus.

critical stretch ratio X, such that fracture occurs when X, or x, reaches this critical ratio. That is, the fracture criterion is

The experimental results have a very simple relation; there is a

Figure 16 shows the breaking stretch ratios for SBR (Sample 3) and Fig. 17 the breaking s t resses themselves, as calculated from the stress-strain relation of this material. Figure 18 shows the breaking stretch ratio for NR (Sample 1). The same relation as in Fig. 16 is obtained.

Figure 19 shows uniaxial stress-strain relations for vulcanized NR and SBR. So-called crystallization can be obtained in the former material but is not observed in SBR. It is interesting that the rule shown in Formula (32) is common for these two materials even though the two samples differ from each other in their behavior in the ultimate stretch region, as seen in Fig. 19.

Let us again consider the unit model shown in Fig. 9. If we assume that the breaking of any one of the elements in the model leads to fracture of the whole unit, the result indicated by Eq. (32) is very likely.

On the other hand, if the material changes from the rubbery to glassy state, this simple relation will no longer be applicable. A s shown in Fig. 20, plasticized PVC changes its fracture behavior from rubber-like behavior to a different type as the content of plasticizer decreases. .The same relation is also obtained from experiments at different temperatures for the PVC specimen.

CONCLUSION

Although a theoretical explanation of the mechanical behavior of the rubber-like materials is not yet completed, important information has been obtained by these biaxial experiments on rubber-like polymers. In addition, a rough sketch of a microdeformation mechanism for this material has been attempted for the purpose of ex- plaining the form of the strain energy density function by using a simple structural model. Further development of kinetic and struc-

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626 S. KAWABATA

FIG. 15. Equipment for inflation of thin film for equal biaxial (a) and unequal biaxial (b) deformation, and the schematic diagram (c).

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TEMPERATURE CONTROL CABINET

, I A

C LIGHT SOURCE D LIGHT ACCEPTER E MULTIPLE STROBO FLASH F TEMPERATURE CONDITION1 WG

COIL G VALVE H ELECTROMAGNETIC VALVE I PRESSURE TRANSDUCER

\

FIG. 15 (continued).

SBR 30’C

A (Y ; .-

1 0

i, FIG. 16. Breaking stretch ratios for SBR (Sample 3), 30°C.

tural analysis is expected as more information about the mechanical properties of rubber-like polymers during finite deformation is obtained.

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628 S. KAWABATA

SBR 30'C 2 u, (kglcmz)

FIG. 17. Breaking stresses (engineering stress) for SBR (Sample 3).

N R 30°C

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30'C Tensile rate lO%/sec x ; Break point

h

0, x - 4 0 - b

I

1 2 3 4 5 6 7 8 9

FIG. 19. Uniaxial tensile properties for NR (Sample 1) and SBR (Sample 2). A

- ,

-... ",.."I

FIG. 20. x-2; Breaking stretch ratios for plasticized A l e PVC samples 370,410, and, 440

indicate that the content of plasticizer are 37,41, and 44 pph PVC in weight, respectively.

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630 S. KAWABATA

Acknowledgments

The author thanks Messrs. K. Fukuma, T. Akagi, and S. Imai for their assistance in these experiments, and he also thanks the Japan Synthetic Rubber Co. and Mitsubishi-Monsanto Chemicals Co. for their assistance in the preparation of the samples used in this experiment.

REFERENCES

L. R. G. Treloar, The Physics of Rubber Elasticity, Oxford Univ. Press, London, 1958. A. C. Eringen, Nonlinear Theory of Continuous Media, McGraw-Hill, New York, 1962. M. Mooney,J. Appl. Phys., 2,582 (1940). K. G. Valanis and R. F. Landel, Bid., 38,2997 (1967). R. A. Dickie and T. L. Smith, J. Polym. Sci., A-2, 7,687 (1969). S . Kawabata, Preprint of the 18th Rheology Conference at Odawara, Japan, 1970.

Received by Editor November 22, 1972 Accepted b y Editor January 18, 1973

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fracture and mechanical behavior of rubber-like polymers under finite deformation in biaxial stress...

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