Printer Friendly

Analysis of high strain hysteresis loss of nonlinear rubbery polymer.


Mathematical constitutive theories for nonlinear large elastic deformation of rubber on the basis of the strain energy density function have been developed (1-7). However, the quantitative evaluation of hysteresis loss, which is defined as the measure of energy dissipated during cyclic deformation of rubber, has not been done. Early attempts of deriving a mathematical expression were not successful, owing to the complexities of the mathematical formulations. Since the hysteresis loss in the rubber compound is helpful in understanding the performance of rubber vulcanizates, an effort is made here to quantify the amount of energy dissipated. Various authors - Ferry (8), Payne et al. (9, 10), Medalia (11), Meinecke (12), and Roland (13) - presented experimental studies regarding hysteresis loss of rubber vulcanizates at low and high strains. It has been reported that at low strain, hysteresis loss per cycle is proportional to the loss modulus with a simplifying assumption of a linear stress-strain relationship (8)

[H.sub.y] = [Pi][(DSA/100).sup.2 G[double prime]([Omega]) (1)

where DSA is the double strain amplitude or peak to peak displacement in percentage and G[double prime]([Omega]) is the loss modulus. Yang and Chen (14) calculated energy loss theoretically for linear viscoelastic material under periodic triangular strain loading by using the Boltzmann superposition principle. They found the results in agreement with Ferry's conclusions that the energy stored during the forward deformation half cycle is completely released during the reverse deformation half cycle. This energy loss in rubber vulcanizates is a complicated phenomenon. In an earlier paper (15), we observed that it is a nonlinear function of strain amplitudes, strain rates, experimental temperatures, and compositional variables such as filler loading, types of filler, resin loading, crosslink density, and the nature of the polymer. In another paper (16), we have quantified low strain hysteresis loss (strain levels less than 100%) over a wide range of frequencies, strain levels, temperatures, and compositional variables. But this model equation is not applicable at strain levels greater than 100%. We have also derived a relationship (17) between heat generation of filled rubber vulcanizates and hysteresis loss. The literature survey indicates that several questions on the hysteresis loss at high strain still remain unanswered.

The objective of this paper is to develop a model equation for quantitative prediction of hysteresis loss (at high strain i.e., up to 400%), over a wide range of frequencies, strain levels, temperatures, and compositional variables. This equation has been tested, using vulcanizates of natural rubber and styrene-butadiene rubber of varying compositions.


Most of the previous attempts to express the viscoelastic behavior of rubber have been based either on models made up of springs and dashpots and the corresponding classical linear equations for vibration (18) or on the strain energy density function (1-7).

Considering the Boltzmann superposition principle, the state of stress in a rubberlike material at any given time depends on the deformation history of the material. If there is a change of strain at time t of amount (de/dt)dt, the stress at some later time [Tau] depends on the product of (de/dt)dt and a function of the elapsed time ([Tau] - t). These products are summed from the beginning of the deformation history of the material to the time at which the stress existing in the material is required. The above can be expressed symbolically by the equation:

[Mathematical Expression Omitted] (2)

or, [Mathematical Expression Omitted] (3)

where A[prime] is a material constant and [Phi]([S.sub.0]) is a stress function.

Figure 1 shows the variation of stress with extension time and duration time for stress relaxation. As usual, the stress increases with an increase in extension time at a constant strain rate, and the stress decays with time at constant strain. It may be inferred, as did earlier authors (18), that f([Tau]- t) may be represented with sufficient accuracy by the simple function [([Tau] t).sup.-m] in the case of stress relaxation and [(([Tau] - t).sup.+m] in the case of extension time. Although the value of the index m is found to depend on the value of ([Tau] - t), it may be considered constant over a wide range of values of ([Tau] t). Figure 2 illustrates that stress is a nonlinear function of strain rates over a temperature range from 25 [degrees] C to 125 [degrees] C. On the basis of Equation 3, ref. 18, and Fig. 2, we can write:

[Mathematical Expression Omitted] (4)

The classical theories of a purely entropic elasticity, represented by Equation 5, show that the stress should be directly proportional to the absolute temperature ([[Theta].sub.0]) at constant extension ratio ([Lambda]). The elastic free energy ([W.sup.*]) in terms of the conformational entropy alone (19) is given by

[Mathematical Expression Omitted] (5)

where [v.sub.1] is the number of network chains per unit volume (the crosslink density), [k.sub.*] is Boltzmann's constant, and [[Lambda].sub.1], [[Lambda].sub.2], [[Lambda].sub.3] the extension ratios in three directions.

Hence, [W.sup.*] [varies] [[Theta].sub.0] (6)

As the stress depends on the temperature, it should be proportional to the temperature rise during deformation of rubber vulcanizates. The above can be expressed as

[Phi]([S.sub.0]) [varies] d[Theta] (7)

Combining Equations 4 and 7 on the principle of joint variance yields

[Mathematical Expression Omitted] (8)

where A[double prime] is a material constant (negative).

According to the thermodynamics and the calorimetric principles, under quasi-adiabatic conditions the temperature rise (d[Theta]) in an extension from the unstrained length ([l.sub.0]) to the final strain [Mathematical Expression Omitted] is given by

[Mathematical Expression Omitted] (9)

where [C.sub.L] is the specific heat, m[double prime] is the mass of the vulcanizate, and [(ds[prime]/dl).sub.[[Theta].sub.0,p[prime]]] is the change of entropy on extension dl at constant temperature [[Theta].sub.0] and pressure p[prime].

From thermodynamics, we can write

[(ds[prime]/dl).sub.[[Theta].sub.0[center dot]p[prime]]] = - [(ds/d[Theta]).sub.l.p[prime]] (10)

where [(ds/d[Theta]).sub.l,p[prime]] is the change of stress (s) with the change of temperature (d[Theta]) at constant length l and pressure p[prime].

Substituting Equation 10 into Equation 9, the temperature rise may be represented as

[Mathematical Expression Omitted] (11)

By substituting Equation 11 into Equation 8, we obtain the stress response function as follows:

[Mathematical Expression Omitted] (12)

In order to derive the dependence of stress on temperature ([[Theta].sub.0]) under any specified condition (i.e., constant volume or constant pressure), Equation 13, derived by Treloar (19), may be introduced into Equation 12:

[Mathematical Expression Omitted] (13)

In this expression, [Mathematical Expression Omitted] is a function of strain and time, and [Mathematical Expression Omitted] defined as mean square end-to-end distance, is a function of temperature, but is independent of the volume of the specimen. On the other hand, [Mathematical Expression Omitted] is a function of the volume of the specimen. [Beta] is the volume expansion coefficient.

The mean square chain length, [Mathematical Expression Omitted], in the unstrained state is the same as the mean square vector length, [Mathematical Expression Omitted], of a corresponding set of free chains. From statistical theory of rubber elasticity (19), [Mathematical Expression Omitted] may also be expressed as

[Mathematical Expression Omitted] (14)

where r is the distance between the ends of a chain.

The radial distribution function [Omega](r) in the nonGaussian region was first derived by Kuhn and Grun (19) and given by

[Mathematical Expression Omitted] (15)

where [C.sup.*] is a normalization constant, [L.sup.-1] is the inverse Langevin function and n[prime] is the number of monomer units in a polymer chain, each of which has a length l[prime].

From Equations 14 and 15, we can write

[Mathematical Expression Omitted] (16)

Taking logarithms and differentiation of Equation 16 with respect to temperature with a simplifying assumption that l[prime] is independent of temperature ([[Theta].sub.0]) because of the very low thermal expansion coefficient of the molecules in the crosslinked state, we obtain

[Mathematical Expression Omitted] (17)

Introducing Equation 17 into Equation 13, we therefore write

[Mathematical Expression Omitted] (18)

Now insertion of Equation 18 into Equation 12, we obtain

[Mathematical Expression Omitted] (19)

If the strain is a simple harmonic function of time such that

[Mathematical Expression Omitted] (20)

then stress becomes

[Mathematical Expression Omitted] (21)

where [Omega] is the frequency and [Delta] is the phase angle.

Analysis of the dynamic response by using sine functions is not a problem, but the use of complex functions gives easier mathematical formulation and better understanding of the process of deformation. On the basis of complex variables, the input and the output function can be written as

[Mathematical Expression Omitted] (22)

Differentiating Equation 22 with respect to time (t), and then substituting into Equation 19 and rearranging, we obtain

[Mathematical Expression Omitted] (23)

where [Mathematical Expression Omitted] is a function of strain. To find out the stress at any point, we have to replace [Mathematical Expression Omitted] in terms of strain. For this purpose we apply the continuum theory of finite deformation.

Now considering the continuum theory of finite deformation, the stress-strain equation of homogeneous isotropic and elastic materials such as vulcanized rubber can be derived from the strain energy density function, [W.sub.*], where [W.sub.*] is the elastic energy stored in a deformed body According to Ogden (19), the strain energy function can be written as

[W.sub.*] = W([I.sub.1]) + W([I.sub.2]) + W([I.sub.I3])(24)

where [I.sub.1], [I.sub.2], and [I.sub.3] are the strain invariants.

[Mathematical Expression Omitted] (25)

and [[Lambda].sub.1], [[Lambda].sub.2], and [[Lambda].sub.3] are the principal extension ratios.

For incompressible material [I.sub.3] = 1. Hence Equation 24 reduces to

[Mathematical Expression Omitted] (26)

where [C.sub.j] and [b.sub.j] are material coefficients.

In the ease of homogeneous uniaxial deformation, the stress [Mathematical Expression Omitted] can be derived from W.

[Mathematical Expression Omitted] (27)

where [Lambda] = extension ratio

= 1 + dl/[l.sub.0]

= 1 + e (28)

By inserting Equations 27 and 28 into Equation 23, we have

[Mathematical Expression Omitted] (29)


[Mathematical Expression Omitted]

Now, we assume that stress ([S.sub.0]) is a nonlinear function of [Phi]([S.sub.0]). Equation 29 can be expressed as

[Mathematical Expression Omitted] (30)

We design the experiment in the following manner. We elongate the test piece at a certain strain rate up to [e.sub.k+1]. Then, the test piece is retracted at the same strain rate. The stress-strain curve obtained is shown in Fig. 3.

The complete cycle consists of two segments: a deformation segment from time (v - 1)[Tau][prime] + [t.sub.k-1] to (v - 1)[Tau][prime] + [t.sub.k] and a recovery segment from (v - 1)[Tau][prime] + [t.sub.k-1] to (v - 1)[Tau][prime] + [t.sub.K], where v represents the number of cycles, which varies from 1 to N[prime] + 1; k is an integer that varies from 1 to k and k[double prime] from k + 1 to k + [k.sub.2].

It is observed that the value of the integer m varies with time in the following manner. Between (v - 1)[Tau][prime] + [t.sub.k-1] and (v - 1)[Tau][prime] + [t.sub.k], let m have the value [Mathematical Expression Omitted] and between (v - 1)[Tau][prime] + [t.sub.k-1], and (v - 1)[Tau][prime] + [t.sub.k], let m have the value [Mathematical Expression Omitted]. In this time interval the strain value varies in the following manner: [Mathematical Expression Omitted] to [Mathematical Expression Omitted] in the time interval (v - 1)[Tau][prime] + [t.sub.k-1] to (v - 1)[Tau][prime] + [t.sub.k] and [Mathematical Expression Omitted] to [Mathematical Expression Omitted] in the time interval (v - 1)[Tau][prime] + [t.sub.k[double prime]-1] + [t.sub.k[double prime]-1] to (v - 1)[Tau][prime] + [t.sub.k].

The stress at any cycle and at any point, according to Equation 30, may be represented as

[Mathematical Expression Omitted] (31)

The first term on the right-hand side in Equation 31 represents the stress during the deformation part of cycle and the second term portrays the recovery part of the cycle.

Hysteresis Loss or Energy Dissipation

Since the energy loss per unit time t per unit volume of a material is the product of the instantaneous stress and the rate of strain, i.e.,

[Mathematical Expression Omitted] (32)

the total hysteresis loss per unit volume in a deformation cycle is given by

[Mathematical Expression Omitted] (33)

The hysteresis loss in the steady state, i.e., in the Nth cycle, is

[Mathematical Expression Omitted] (34)

For complex mathematical formulations, it is very difficult to determine So from Equation 31. As an approximation, we therefore consider the integral above (Equation 33) to be written as follows:

[Mathematical Expression Omitted] (35)

Substituting Equation 31 into Equation 35 with a simplifying assumption (m = 1, n = 1) of nonlinear function of stress-strain rate and extension or retraction time, we find that the hysteresis loss for a complete cycle is

[Mathematical Expression Omitted] (36)


[M.sup.*] = 0.013A[double prime][[Theta].sub.0][l.sub.0][c.sub.j]/3[C.sub.L]

Here [M.sup.*], [K.sup.*], A[double prime], and [C.sub.j] are constants; [Delta], [Omega], [[Theta].sub.0], [Beta], and [C.sub.L] are phase angle, frequency, temperature, volume expansion coefficient, and sp. heat respectively; [e.sub.2] is 100% constant swain and [e.sub.3] is the variable strain level (magnitude of [e.sub.2] and [e.sub.3] are taken in strain, not in percentage); n[double prime] is an integer; [l.sub.0] is the unstrained length; and [t.sub.2] is the time to complete a cycle at 100% strain level.

The energy stored per unit volume in a cycle is

[Mathematical Expression Omitted] (37)


The ratio of dissipated to stored energy per cycle is

[[[E.sub.Energy dissipated]/Energy stored].sup.[K.sup.*]] = 2[Pi] tan [Delta] (38)

Rearranging Equation 38, we obtain

[E.sub.Energy dissipated]/[E.sub.Energy stored] = [[2[Pi] tan [Delta]].sup.1/[K.sup.*]] (39)

which agrees with Ferry's conclusions (8), when [K.sup.*] = 1.


The formulations of the various mixes are given in Table 1. The nature of polymers (mixes [A.sub.i] to [L.sub.i]), the loading of filler (mixes [A.sub.i] and [B.sub.i]), the nature of filler (mixes [I.sub.i], [J.sub.i], [K.sub.i] and LI), the loading of resin (mixes CI and DI) and the crosslink density (mixes EI, FI, GI and HI) were varied.

Natural rubber (RMA-4), SBR-1502, stearic acid, Si-69 (Degussa, AG), silica and aromatic oil were supplied by Birla Tyres Ltd., Balasore. ISAF (N220) was supplied by Phillips Carbon Black Ltd., Durgapur. Zinc oxide and clay were obtained from the local market. High styrene resin and sulfur were supplied by Bengal Waterproof Ltd., Panihati. Polymerized 1,2-dihydro 2,2,4-trimethyl quinoline (TMQ), N-(1,3-dimethyl butyl)-N[prime]-Phenyl-p-phenylene-diamine (6-PPD), N-cyclohexyl thiophthalimide (PVI) and 2-(4-morpholinyl mercapto)-benzthiazole sulphenamide (BSM) were supplied by ICI Ltd., Rishra.


The ingredients were mixed with rubber on a two-roll mill (0.15 m x 0.33 m, Schwabenthan, Germany) at a temperature of 50 [degrees] C and a friction ratio 1:1.1.


The curing characteristics of the mixes were evaluated from a Rheometer R-100 according to ASTM D-2084-81. The molding of the tensile sheets was carried out at a temperature of 150 [degrees] C, 4 MPa pressure, and optimum cure time (too in minutes) using a David Bridge Press, Castleton, England.



High strain hysteresis loss is defined as the energy dissipated in stretching, when the specimens were deformed to strain level greater than 100%. Hysteresis loss was determined on the tensile dumbbells (ASTM D-412-80, type 2 die) over a range of temperatures (25 [degrees] C to 100 [degrees] C) and strain rates (1.9 x [10.sup.-3] [sec.sup.-1] to 9.5 x [10.sup.-2] [sec.sup.-1]) and extensions 1% to 400% using a Zwick Universal Testing machine 1445 equipped with an environmental chamber. The samples were mounted in mechanical clamps 44 mm apart and the cross-head was adjusted to give zero tension. The temperature was controlled to [+ or -]0.5 [degrees] C. The samples were preconditioned in the Zwick heating chamber for 10 min before testing. For the purpose of obtaining hysteresis energy loss continuously, the stress-strain curves were recorded on tape and fed into the attached computer. The measurements were continued up to eight cycles.

Dynamic Mechanical Properties

The elastic modulus (E[prime]), loss modulus (E[prime]), and loss tangent (tan[Delta]) were calculated from hysteresis loop by [TABULAR DATA FOR TABLE 2 OMITTED] methods described elsewhere (20). The loops were obtained from the stress-strain measurements (forward and retraction curves). At high deformation, the hysteresis loops were not ellipsoidal, and instead appeared as banana-shaped loops. The skewed loop was transformed into a nearly elliptical loop by plotting true stress [(F[prime]/[A.sub.0])(1 + e)] in place of engineering stress [(F[prime]/[A.sub.0])] as a function of strain, where F[prime] is the force, [A.sub.0] is the original cross-sectional area, and e is the strain.

Young's Modulus

The Young's modulus was measured from the initial slope (below 50% elongation) of the stress-strain curves in a Zwick UTM model 1445 according to ASTM D 412-80 over a range of temperatures from 25 [degrees] C to 100 [degrees] C and strain rates 1.9 x [10.sup.-3] [sec.sup.-1] to 9.5 x [10.sup.-2] [sec.sup.-1].

Volume Expansion Coefficient

The values of the volume expansion coefficient of rubber were taken from the literature (21).


The calculation of the hysteresis loss using Equation 36 would involve the measurement of [K.sup.*], [M.sup.*], phase angle ([Delta]), and volume expansion coefficient ([Beta]) at a frequency ([Omega]) and an operating temperature ([[Theta].sub.0]).

In order to obtain the material constant [K.sup.*] and [M.sup.*], hysteresis loss was performed at different strain levels. The constants were obtained by curve fitting using known values of hysteresis loss, frequency, phase angle, volume expansion coefficient, and temperature. [TABULAR DATA FOR TABLE 3 OMITTED] For convenience, we define another parameter [M.sup.**], so that

[M.sup.**] = [[[M.sup.*]].sup.1/[K.sup.*]] (40)

(It may be recalled that Equation 36 involves the quantity [[Hy].sup.[K.sup.*]]. But in actual practice, the quantity [Hy] is required.) Tables 2 and 3 give the values of [K.sup.*] and [M.sup.**] for the vulcanizates [A.sub.i] to [L.sub.i].

From Tables 2 and 3, it is clear that the material constant [M.sup.**] is a function of the number of cycles, temperatures, strain rates, and material compositions. The material constant ([M.sup.**]) decreases with an increase of temperature and number of cycles, but increases with an increase of strain rate. Another material constant, [Mathematical Expression Omitted], may be defined as the product of Young's modulus and square root of the strain rate. The values of material constant, [M.sup.**], are plotted against the material constant [Mathematical Expression Omitted] in Fig. 4 for both NR and SBR vulcanizates over a range of strain rates, number of cycles, temperatures, and material compositions. The variation in compositions include nature and level of filler, resin, and vulcanization system. The following equation is obtained from Fig. 4:

[Mathematical Expression Omitted] (41)

The value of [A.sup.*] has been found to be 0.5. An excellent correlation is observed between the material constant [M.sup.**] and [Mathematical Expression Omitted].

On the other hand, the other material constant, [K.sup.*], is independent of temperature, strain rate, number of cycles, and material composition. The unique attribute of this correspondence is that the material constant ([K.sup.*] and [M.sup.**]), which are obtained from Equation 36, are exactly the same as the material constants [K.sup.*] and [M.sup.**] obtained in our earlier work (16).

Figure 5 illustrates the effect of DSA (double strain amplitude) on the loss tangent (tan [Delta]) at different strain rates. At small strain, tan [Delta] is small. As the strain amplitude is increased, tan [Delta] passes through a maximum value. The loss tangent increases with the strain rates at any double strain amplitude. At low DSA, the agglomerate structure of the carbon black does not break down. The high energy dissipation in the region of the maxima is the result of frictional force due to agglomeration and deagglomeration of the filler. This mechanism decreases at high strain amplitudes where the agglomerated structures break down. Figure 6 shows the plot of the loss tangent versus DSA at first cycle and the fourth cycle on a semilogarithmic plot. The nature of this curve is almost same as that shown in Fig. 5. It may be noted that tan [[Delta].sub.max] value appears at 0.2 DSA. These results are in agreement with the literature values (22). Thus we can infer that a standard analysis (20) of the distorted hysteresis loops obtained at large DSA is sufficiently accurate for our analysis.

The hysteresis loss at different frequencies, strain levels, and temperatures has been computed at any given cycle. The results of hysteresis loss are plotted against DSA for both NR and SBR vulcanizates in Figs. 7 and 8, respectively. The experimental values are also shown in the same curve. The hysteresis loss increases with an increase in strain levels because of more breakdown of rubber-filler agglomerate structures and the higher coefficient of friction between molecules. At 2 DSA, there is a sharp increase because of the breakdown of rubber-filler agglomerate structures. The theoretical values, calculated by using Equation 36, are in good accord with the experimental values (within [+ or -] 1%). It also predicts that the hysteresis loss decreases with an increase of temperature, owing to the decrease in coefficient of friction between the molecules, melting of the immobilized rubber shell around the filler surface, and the decrease of rubber-filler interaction. The hysteresis loss has also been calculated using Ferry's equation (Equation 1) and is shown in Fig. 7. At low DSA, there is some agreement, but divergence is observed at higher DSAs, because hysteresis loss is a complicated function of strain level. Our calculation shows better correspondence at higher DSA.


Hysteresis loss of nonlinear viscoelastic natural rubber and styrene-butadiene rubber vulcanizates has been quantified in terms of strain levels, frequency, volume expansion coefficient, phase angle, operating temperature, and some material constants ([K.sup.*] and [M.sup.*]) by using the Boltzmann superposition principle, a statistical theory of rubber elasticity, and phenomenological theory. The model equation has been tested by variations of strain levels (100% to 400%), compounding variables (loading of carbon black, silica clay resin, and curatives), temperatures (25 [degrees] C to 100 [degrees] C) and strain rates (0.19 x [10.sup.-2] [sec.sup.-1] to 9.5 x [10.sup.-2] [sec.sup.-1]). The experimental data are in good accord with the theoretical prediction, when hysteresis loss is plotted against double strain amplitude.


Equation 31 given in the text can be solved as follows:

Performing the integration with respect to the corresponding variables, then summing a geometric series and rearranging Equation 31, the stress at any point in the first cycle is

[Mathematical Expression Omitted] (A1)


A = [Mathematical Expression Omitted]

[Omega] = 2[Pi]f[prime][f[prime] = number of cycles per unit time]

F = n[Omega]

H = Sin (n[Tau]/2) Cos [Delta]

I = Cos(n[Tau]/2) Sin [Delta]

K= Sin (n[Tau]/2) Sin [Delta]

L = Cos (n[Tau]/2) Cos [Delta]

[(N).sub.a] = H Sin([Ft.sub.a]) + I Sin([Ft.sub.a]) + K Cos([Ft.sub.a])

- L Cos([Ft.sub.a])

[(d).sub.a] = L S([Ft.sub.a]) + H Cos([Ft.sub.a]) + I Cos([Ft.sub.a])

- K Sin([Ft.sub.a])

[b[prime].sub.j] = [b.sub.j] - 1, [b[double prime].sub.j] = (1 + 0.5[b.sub.j])

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[f.sub.k[double prime]] = Imaginary part of [a.sub.k[double prime]] [exp.sup.i[Delta]]/[e.sub.k[prime]]

[h.sub.k[double prime]] = Imaginary part of [b.sub.k[double prime]] [exp.sup.i[Delta]]/[e.sub.k[prime]]

[P.sub.k], [Q.sub.k], [V.sub.k], [T.sub.k], and [U.sub.k] are constants.


The authors acknowledge the financial support provided by DRDO, New Delhi, for carrying out this research work.


1. H. G. Kilian, M. Strauss, and W. Hamm, Rubber Chem. Technol., 67, 1 (1994).

2. D. J. Charlton, J. Yang, and K. K. Teh, Rubber Chem. Technol., 67, 481 (1994).

3. A. R. Johnson, C. J. Quigley, and J. L. Mead, Rubber Chem. Technol. 67, 904 (1994).

4. J. Padovan, H. Parris, and J. Ma, Rubber Chem. Technol., 68, 77 (1995).

5. C. J. Quigley, J. Mead, and A. H. Johnson, Rubber Chem. Technol., 68, 230 (1995).

6. S. Kawabata, Y. Yamashita, H. Ooyama, and S. Yoshida, Rubber Chem. Technol., 68, 311 (1995).

7. A. Sarkar, A. K. Bhowmick, and S. Majumdar, Rubber Chem. Technol., 64, 696 (1991).

8. J. D. Ferry, Viscoelastic Properties of Polymers, Third Edition, John Wiley and Sons, Inc. (1980).

9. J. A. C. Harwood, A. R. Payne, and R. E. Whittaker. Rubber Chem. Technol., 44, 690 (1971).

10. J. A. C. Harwood and A. R. Payne, J. Appl. Polymer Science, 10, 1203 (1966).

11. A. I. Medalia, Rubber Chem. Technol., 51, 436 (1978).

12. E. A. Meinecke and M. I. Taftaf, Rubber Chem. Technol., 61, 534 (1988).

13. C. M. Roland, Rubber Chem. Techno., 62, 880 (1989).

14. T. Yang and Y. Chen, J. Polym. Sci., Part B, Polym. Phys., 20, 1437 (1982).

15. K. K. Kar and A. K. Bhowmick, J. Appl. Polym. Sci., 64, (1997).

16. K. K. Kar and A. K. Bhowmick, Polymer, (submitted).

17. K. K. Kar and A. K. Bhowmick, J. Appl. Polym. Sci., 64, 1541 (1997).

18. H. McCallion and D. M. Davies, Rubber Chem. Technol., 30, 1045 (1957).

19. L. R. G. Treloar, The Physics of Rubber Elasticity, Third Edition, Clarendon Press, Oxford, England (1975).

20. J. M. Caruthers, R. E. Cohen, and A. I. Medalia, Rubber Chem. Technol., 49, 1076 (1976).

21. J. Brandrup and E. H. Immergut, Polymer Handbook, Third Edition, John Wiley and Sons Inc. (1989).

22. J. E. Mark, B. Erman, and F. R. Eirich, in Science and Technology of Rubber, Second Edition, Academic Press, New York (1994).
COPYRIGHT 1998 Society of Plastics Engineers, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 1998 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Kar, Kamal K.; Bhowmick, Anil K.
Publication:Polymer Engineering and Science
Date:Jan 1, 1998
Previous Article:Residual stresses, shrinkage, and warpage of complex injection molded products: numerical simulation and experimental validation.
Next Article:An integrated analysis for a coextrusion process.

Related Articles
Viscoelastic-plastic deformation behavior of polypropylene after cyclic preloadings.
Stress-strain behavior and dynamic mechanical properties of poly(1,1-dimethyl-1-sila-cis-pent-3-ene), poly(1-methyl-1-phenyl-1-sila-cis-pent-3-ene)...
A state variable model for high strength polymers.
Time dependent response of polycarbonate and microcellular polycarbonate.
Fracture toughness of acrylonitrile-butadiene-styrene by J-integral methods.
Biaxial Stress Relaxation of High Impact Polystyrene (HIPS) Above the Glass Transition Temperature.
Nonlinear Viscoelastic and Viscoplastic Response of Glassy Polymers.
Cyclic deformation behavior of an epoxy polymer. Part II: predictions of viscoelastic constitutive models.
Predictions of two nonlinear viscoelastic constitutive relations for polymers under multiaxial loadings.
Rheological study of aqueous dispersions of in situ gelable thermosensitive polymer nanogels.

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters