Mooney-Rivlin Equation Research Articles

Volume Change Accompanying Rubber Extension

Abstract : A new expression for the volume dilation accompanying rubber extension is used to describe published data. This new equation is shown to describe the data better than either that derived from simple kinetic theory or that derived from the Mooney-Rivlin Equation. (Author)

On the Second Order Approximation in the Phenomenological

From the phenomenological theory of finite deformation, a stress-strain relation was obtained with a second order approximation to the stored energy function w with higher degree accuracy than that shown by L.R.G. Treloar or R.S. Rivlin.In the phenomenological theory of large elastic deformation, the stored energy function w can be written in terms of three strain invariantes I1, I2 and I3, and one way of expressing this is in terms of power series:(1)where I1=α12+α22+α32, I2=α12α22+α22α32+α12α32, I3=α12α22α32 and αi, (i=1, 2, 3) are the principal extension ratios. If the principal extension ratios α1, α2 and α3 are small and assumed to be nearly in unity, the second order approximation to the expression (1) for w will be given as follows, in the case of incompressible materials(2)because the squared term (I>1-3)2 is equivalent to the term (I2-3) through the principal extension ratios, from the view-point of the second order approximation. From this type of the stored energy function, the following stress-strain relation has been obtained(3)for the case of simple extension or uniaxial compression defind by the extension ratio or the compression ratio α, where A≡2(c100-6c200), B≡2(4c200+c010), C≡4c200. Upon comparing the resulting stress-strain relation with the experimental stress-strain behavior, it seems that this relation is valid for rubber vulcanizates, and even for larger deformations to which the assumption that deformations are small (α«2) is inapplicable, and that this relation almost coincides with the observed stress-strain relation in the extension range wider than that predicted by Mooney-Rivlin equation.

Large Elastic Behavior of SBR and BR Vulcanizates

In the preceding paper1), a report was made of the study enactted by one of the authors of the stress-strain relation that was obtained from the phenomenological theory of finite deformation. So far as the incompressible materials were concerned, the relation was written as follows:(1)where f denoted the tension at extension ratio α, A, B, and C were constants.In this paper, the applicability of the Eq. (1) to the stress-strain behavior of styrene-butadiene rubber (SBR) and butadiene rubber (BR) was tested, and the temperature dependences of the values of A, B, C were examined. This is, however, only a preliminary examination, because the data discussed in this paper were determined while making the study of fracture phenomena, and little attention was paid to attaining equilibrium state in obtaining the data.SBR and BR were both cured with tetramethyl thiuram disulfide and contained 5 parts of zinc oxide and 2 parts of stearic acid per 100 parts of rubber. The stress-strain data were determined by means of an Instron type tester at various temperatures between-40 and 120°C at a crosshead speed 50mm/min.It was found that at temperature over 30°C, fracture occurred at low extensions (α>3) and Mooney-Rivlin equation (Same tipe equation as Eq. (1) where C=0) was in agreement with the data over 30°C, and that at temperatures below 30°C, the experimental stress-strsin relations almost coincided with the relation of Eq. (1) in the extension range α<4.5, while Mooney-Rivlin equation coincided with the data only in the extension range α<3. It was found also that the values of A for SBR and BR showed almost linear increase with temperature above 30°C, the values of B and C decreased with increase of temperature and C were almost 0 for the data above 30°C.

Rubber Elasticity

Rubber elasticity is treated in terms of a general random network bead-spring model which expresses chain connectivity through Dirac delta functions. In contrast to the classical theory, the present starting point is the segmental distribution function, from which, besides elasticity, most polymer properties are deducible, e.g., relaxation spectra, intrinsic viscosities, etc. The method is that developed by Fixman and by Imai. The salient results are: (a) macroscopic observables correspond to mean-square vector quantities; (b) rubber elasticity is a function only of the strain invarient I1 (not of I2); in particular: (c) the C2 term of the Mooney–Rivlin equation reflects effects of network connectivity (e.g., the functionality of cross links), not of I2; (d) The new one-parameter stress–strain equation here derived fits classical data on one- and two-dimensional strain. The quantity corresponding to the “front factor” in the classical equation is ∼ 1.8, in reasonable agreement with measurements by Ferry et al.

The effects of strain on the equilibrium properties of rubber networks

The coefficients of isothermal compressibility, cubical expansion and thermal pressure, have been measured for lightly crosslinked butyl rubber over a range of temperatures and at several elongations. The “network contribution” to these coefficients is found to be negligibly small as predicted by analyses based on either the Mooney-Rivlin or Gaussian equations of state.

Correlation by physical and chemical methods of the transverse crosslinking density in unsaturated polyester copolymers

The following relationship correlated equilibrium stress with deformation according to the statistical theory of high elasticity [1, 2]: σ ∞ = 2 C ( λ −1/ λ 2) in which σ ∞- equilibrium stress, kg/cm 2, λ-deformation; λ = l/ l 0, C-elasticity constant, kg/cm 2, C- NkT/2 ( N-number of chains per unit volume, k-Boltzmann constant, T-abs. temperature, °K). The statistical theory equation can be regarded as a special case of the phenomenological Mooney-Rivlin equation, which establishes the relation between equilibrium stress and deformation in the range of small and medium deformation due to uni-axial elongation [3–5]: σ ∞ = 2C 1 (λ−1/λ 2) + 2C 2· 1 λ (λ− 1 λ 2 ) in which C 1, C 2 are constants, and C 1 has the same meaning as the C of eqn. (1).

Large Deformation Tensile Properties of Elastomers. I. Temperature Dependence of C1 and C2 in the Mooney-Rivlin Equation

Abstract Tensile stress strain data were determined at various extension rates and temperatures on unfilled butyl and silicone vulcanizates and on six hydrofluorocarbon (Viton A-HV) vulcanizates that contain between 0.90 and 12.2×10−5 mole of effective network chains per unit volume of gel rubber. From one min isochronal data, values of C1 and C2 in the Mooney-Rivlin equation were derived. For the butyl and silicone vulcanizates from, respectively, −20 to 150° C and —45 to 200° C, C1 increases with temperature at a rate which is in reasonable agreement with published data from force-temperature measurements. Because C2 is sensibly temperature independent over these extended temperature ranges, it is concluded that C2 is a finite quantity under equilibrium conditions. Time dependent behavior was observed on the Viton A-HV vulcanizates between —5 and 230° C. For each vulcanizate between about 25 and 230° C, C1273/T is temperature independent, but it increases with decreasing temperature below about 25° C. Except at the lowest temperatures, C2273/T decreases with increasing-temperature, the rate of decrease becoming progressively less with an increase in crosslink density. Above 25° C the ratio C2/C1, which ranges from about 90 to 0.6, decreases with an increase in either temperature or crosslink density. The time-dependent behavior of C2273/T appears to arise primarily from molecular processes whose relaxation times are considerably longer than the terminal relaxation time of individual chains.

Thermodynamic relations for rubberlike elasticity on simple elongation

Thermodynamic relations pertaining to rubberlike elasticity on simple elongation are derived systematically in terms of the four alternate sets of independent variables (T, V, L), (T, p, L), (T,V, α) and (T, p, λ), where λ is the elongation ratio and α is the same corrected for volume dilation. Formulae, which facilitate changes of independent variables from one set to another, are given. The volume dilation coefficient and the correction terms, required to evaluate the energetic contribution to the elastic force from measurements performed under constant pressure, are all derived as a function of the thermoelastic equation of state. Approximate expressions for estimating these quantities, which were proposed previously on the basis of either the infinitesimal theory of elasticity or the statistical theory of gaussian networks, are examined critically in light of the exact relations derived here; also the nature of the approximations involved is pointed out. Upon analysis of volume dilation data available in the literature, new approximate expressions for these quantities, which apply when the stress-strain relation obeys the Mooney-Rivlin equation, are proposed.

Network characterization of silicone rubber vulcanizates. Part I. solvent, temperature, and specimen shape effects on effective crosslink density determinations

AbstractThe effective crosslink densities were determined for four silica‐filled silicone rubber vulcanizates which represent different degrees of crosslinking. The determinations were made from compression‐deflection measurements of rubber specimens at an equilibrium swelling condition. Nine solvents representing a wide range of dilatant strengths were investigated at four equilibrium temperatures (8, 23, 41, and 60°C.) for their effect on the crosslinking determinations. Specimen shape effects were also investigated. Explicit equations of state were derived empirically and were found to be similar to the well‐known Mooney‐Rivlin equation in which two independent parameters are required for the definition of a dilated rubber network.

Correlation of vulcanizate properties with polymer–black interaction

The specific surface activity of carbon black, as indicated by amount of rubber bound to the black during the mixing process, has been found to be the major factor in determining the modulus of the vulcanizate at high extensions. However, modulus at low extension is independent of specific surface activity and mostly influenced by other black characteristics, such as “structure” and surface area. Further, a relationship of the Guth-Gold type based upon elastic theory was applied for predicting modulus of filled vulcanizates at high extension from that of an identically cured gum vulcanizate. The relationship is where Φ is the reinforcement volume or volume fraction of filler plus volume fraction of bound polymer. This relationship was also used to calculate the actual elongation in a filled vulcanizate to compensate for that portion that does not elongate at all, and this actual extension ratio (α') was utilized in the Mooney-Rivlin equation. The Mooney-Rivlin equation is the theoretical expression for the modulus of a vulcanizate as it changes with elongation and cure level (or number of effective chains per unit volume). The equation has been successfully used previously for gum vulcanizates, but not for filled vulcanizates. With the use of the corrected elongation (α'), experiment and theory are in much better agreement than previously, and a general concept of the influence of black upon vulcanizate properties can be postulated.

Stress-Strain Relation of Pure-Gum Rubber Vulcanizates in Compression and Tension

Abstract The empirical function of Martin, Roth, and Stiehler represented by Eq. (1) where A has the value of 0.38 may be regarded as an adequate representation of the available experimental data covering both the compression and tension of pure-gum vulcanizates. The stress and strain are to be measured after a constant time of creep. The approximate validity extends over the range 0.5&amp;lt;L&amp;lt;3.5. The success of the single empirical function in representing data obtained in both compression and tension over a range as great as this is regarded as very significant. In the range of values of L from 0.5 to 1.0 (compression) the empirical function gives results in agreement with the predictions of the statistical theory of rubber elasticity. After representing the stress and strain in a transition region extending from L=1.0 to 1.5, the empirical function gives approximately constant coefficients C1and C2 in the Mooney equation over the range from 1.5 to about 3.5. The behavior of the empirical function in the transition region from L=1.0 to L=1.5 shows that the statistical theory of elasticity fails to represent the experimental data even at the lowest elongations while proving satisfactory in the compression region. The Mooney-Rivlin equation also fails to furnish adequate representation of the experimental data at low elongations. The most satisfactory method of determining Young's modulus from experimental observations of stress and strain in pure-gum vulcanizates involves a plot of log F/(L−1−L−2) against (L−L−1). For observations within the range of L=0.75 to 2.0 the simpler plot of F/(L−1−L−2) against (L - 1) is thoroughly satisfactory. In both cases M is obtained from the intercept, and the constant A in Eq. (1) is determined from the slope.