Mooney-Rivlin Equation Research Articles

Elastic behavior of chitosan films

The thermoelastic behavior and equilibrium stress–strain properties of chitosan films lightly crosslinked with gluteraldehyde and swollen with water were studied. Precautions were taken to preclude changes in the swelling ratio of swollen sample films during the experiment. The results indicate that at relatively low extensions the elastic behavior of the biopolymer is entropic in origin. The equilibrium stress–strain isotherms of chitosan did not obey Mooney–Rivlin equation because of sharp increases in stress with extension ratio at high extensions. This is attributed mainly to interchain hydrogen-bonded interactions, but a possible contribution due to strain–induced crystallization cannot be ruled out. © 1997 John Wiley & Sons, Inc.

Identification of the Mooney-Rivlin Equation with the Toda Lattice by Synchronization in Chaotic Systems.

In this work, identification of the Mooney-Rivlin equation with the Toda lattice is studied numerically. We choose two materials as a pendulum, whose constitutive equations are the Mooney-Rivlin equation and the Toda lattice equation, and swing them in the two-dimensional vertical plane. In general, the motion of these pendulums becomes chaotic. We examine the chaotic behavior and identify the two nonlinear systems by using ISCS (Index of Synchronization of Chaotic Systems). We find that the dynamic properties of the two nonlinear systems correspond with each other quite well and the Toda lattice is useful for the model of rubber.

Statistical mechanics of fuzzy random polymer networks

A statistical mechanics framework of fuzzy random polymer networks is established based on the theories of fuzzy systems. The entanglement effect is manifested quantitatively by introducing an entanglement tensor and membership function and the amorphous structure is treated as the fuzzy random network made up of macromolecular coils entangled randomly. A random tetrahedral entangled-crosslinked cell is chosen as an average representative unit of the fuzzy random polymer network structure. By making use of the theory of fuzzy probability and statistical mechanics, the expression for the free energy of deformation is given, which fits well with the experimental data on rubber elasticity under various deformation modes. Both classical statistical theory and Mooney-Rivlin equation can be taken as its special cases.

Characterization of reinforcing in rubber.

The author tries to apply a new method to characterize the reinforcing effect of fillers on basis of the Mullins softening and recovery, followed by the evaluation of the stress-elongation curves obtained after repeated deformation cycles and heat treatments of various duration. The apparent activation energy of the recovery process, i.e., the reformation of physical bonds after Mullins softening, was between 30-60kJ/mol. It is predominantly a physical process. By the evaluation of the stress-strain curves according to the Mooney-Rivlin equation, it is established that the constant C1 of the equation depends primarily on the number of the active network chain segments of the stable chemical network formed during vulcanization, which is also the case in filled vulcanizates. On the other hand, constant C2 of the equation depends on the rubber-filler physical interaction, and on the number of effective network chain segments determined by chain entanglements. As these physical network point undergo changes during the deformation, the value of C2 also changes. The constant C2 can be used as a general parameter for the characterization of the activity, the reinforcing effect of fillers.

Characterization of nonideal networks by stress‐strain measurements at large extensions

AbstractIt is shown how a characterization of unfilled, amorphous rubber networks can be evaluated from uniaxial stress‐strain measurement data. Beside the cross‐linking density, the relative scission probability during the curing procedure is evaluated, which determines the amount of dangling free chain ends and the number of trapped entanglements. These values are found from the C2 term of the Mooney‐Rivlin equation by using the predictions of a tube model. A necessary requirement for applying stress‐strain measurements at large extensions is the consideration of the finite extensibility component of the reduced stress. It is taken into account by using a numerical procedure, which derives from a series expansion of the inverse Langevin approximation. The experimental results found at natural rubber networks cross‐linked with thiuram (TMTD) and peroxid (DCP) show that network defects cannot be neglected in the DCP networks. They are assumed to be connected to the worse tensil strength properties compared to the TMTD networks. © 1993 John Wiley & Sons, Inc.

Structure and elasticity of polyurethane networks based on poly(butadiene) diol, 4,4′-diphenylmethane diisocyanate and poly(oxypropylene) triol

Relations between structure and equilibrium mechanical properties of polyurethane networks based on poly(butadiene) diol, 4,4′-diphenylmethane diisocyanate (MDI) and poly(oxypropylene) triol (POPT) prepared at various initial ratios of reactive groups were studied. An inhomogeneous structure of the networks was revealed by small-angle X-ray scattering. The microdomains consist of POPT molecules crosslinked with MDI and they have a diameter of about 4–5 nm. The characteristic distance of microdomains is about 9 nm. The poor mutual miscibility of the reaction components and the existence of hydroxyl group association in the reaction mixture are the probable reasons for microdomain formation. The strain dependence of the equilibrium elastic force obeyed the Mooney-Rivlin equation. The experimental moduli of dry and equilibrium swollen samples were higher than those calculated for the affine limit of the Flory junction fluctuation model on the assumption of a homogeneous network formation.

An analysis of the orientational birefringence of poly(methyl methacrylate) in terms of the Mooney-Rivlin equation

The orientational birefringence of poly(methyl methacrylate) (PMMA) was analysed using the Mooney-Rivlin equation. It is shown that its dependence on strain can be well described by this equation over more than three decades of time (in seconds). More interestingly, the B 1 term (usually found and ascribed to some temporary affine deformation upon stretching in uncrosslinked polymers) is absent for PMMA within the error of experimental measurements. The B 2 term is shown to be governed by the chain relaxation that takes place during stretching, emphasizing its molecular origin associated with the entanglement network. Also, a comparison of B 1 and B 2 for PMMA and polystyrene (PS) indicates clearly a significant difference between the deformation behaviour of the entanglement network for the two polymers. The B 1 term of PS gives an important contribution to the orientational birefringence, and relaxes with time in a similar way to B 2.

Polyurethane elastomers studied by the Mooney–Rivlin equation for rubbers

AbstractA series of poly(ester–urethane) block copolymers were prepared via a two‐step polymerization process. The prepolymer composition was kept constant in all the samples, while the NCO/OH ratio during extension was varied from 0.9 up to 1.2. By this way, chemical and physical cross‐links were constructed into the materials. Equilibrium stress–strain measurements have been carried out for the examination of the elastomeric behavior of the materials tested. Calculation of an extra entropy term resulting from the network deformation is based on a first‐principles derivation of the distribution function, taking into account a new type of network constraints. The process yields the Mooney–Rivlin equation and the constant C2 as a function of physical cross‐links.

Multiaxial deformation of polymer networks

A detailed molecular model for describing the response of a polymer network to multiaxial deformation is developed. The approach allows one to explicitly calculate the strain energy density function W and its dependence on the strain invariants I. The results correctly describe the peculiar behavior of the strain energy derivatives observed experimentally at very low deformation and at high strains. The strong dependence of those derivatives on strain is attributed to slippage of molecular chains through entanglements. Our model predictions also clearly demonstrate the lack of physical meaning associated with the phenomenological Mooney-Rivlin equation

Elasticity theory. 6. Molecular theory of the Mooney-Rivlin equation and beyond

A theory of high elasticity based on the calculation of the distribution function of the gyration tensor for an elastic body is developed. There is but one free parameter in the theory to this order. The potential for bonded interactions is treated in the harmonic approximation, and the nonbonded interactions are handled by expressing the free energy density as a function of the segment density fluctuations

Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.

The contribution of molecular entanglements to the rubber—elastic behaviour of electron-irradiated linear polyethylene

Spun polyethylene monofilaments were electron irradiated at room temperature with doses from 0.7 to 6.0 Mrad, in vacuum. Rubber elasticity experiments were performed on the as spun filaments, and filaments of draw ratio 12:1, at strain rates of 9 · 10 −3 and 9 · 10 −2 s −1. The results were related to the network structure using the Flory simple statistical theory (SST), and the Mooney-Rivlin (MR) equation. The MR plots showed two straight line regions, which were assigned to the affine deformation of chemical crosslinks at low strain, and to the non-affine deformation associated with entanglement slippage at higher strain. The MR equation at low deformation gave acceptable M ̄ c values, in close agreement with those obtained from the SST, provided the gel fraction in each case was assumed to be constant at 1.0, suggesting contributions from non-permanent network chains participating in entanglements. For the spun filaments, a linear relationship between M ̄ c and dose was observed, and extrapolation to zero dose gave a value of 12 000 g mol −1 for the molecular weight between entanglements at 142°C. M ̄ c showed a dependence on the strain rate, providing further evidence for non-permanent entanglement contributions. For the drawn filaments, M ̄ c decreased in a non-linear manner with dose, and was less sensitive to strain rate. This was attributed to the deformation of the network during drawing, whereby some entanglements are pulled out, and some rendered more permanent and effective. At intermediate doses, M ̄ c was seen to decrease up to a draw ratio of 20:1, and then increase at 30:1. This was again attributed to the effect of drawing on the structure of the entanglement network.

Entanglement Tightening and Stress Relaxation

The contribution of entanglements to the modulus of a time independent polymer is satisfactorily accounted for by the C2/λ term in the Mooney-Rivlin equation. On the other hand, the role of entanglements in time dependent polymers is less evident and seems to give rise to a non-linear Mooney plot. A model is introduced which considers an entanglement as a topological knot that tightens with initial deformation and subsequently inhibits chain slippage. The observed maximum in a Mooney plot at small values of λ is predicted, and the experimental data of plasticised PVC polybutadiene and styrenebutadiene copolymer are fitted reasonably well.

Interpretation of the rheological behaviour in elongation of uncrosslinked polystyrene melts in terms of the Mooney-Rivlin equation

The Mooney-Rivlin equation, which has been widely used for crosslinked rubbers, is applied to describe the rheological behaviour in the elongation of uncrosslinked polystyrene melts. Elongational stress as well as flow-birefringence data can be fitted with a Mooney-Rivlin type equation, provided that the temperature is not too high and the value of the strain rate is within a certain range, corresponding to a mainly reversible deformation. Contrary to crosslinked rubbers, the C 1 term of the Mooney-Rivlin equation is much lower than the C 2 term. It has also been shown that C 1 depends more on strain rate than C 2.

A theory of pseudo cross-link

The rubber elasticity is discussed in terms of the pseudo cross-link concept. The classical equation of rubber elasticity deviates from experiments not only at large extension, but also even at small one. The former deviation is to be ascribed in most cases to the limited chain extensibility, because the steep increase of the force at large extension is varnished if the equations represented by the fractional extension derived by Guth, James or the authors is employed. The deviation at small extension is also diminished if the empirical equation of Mooney-Rivlin is used. The Mooney-Rivlin type equation can be derived on the basis of pseudo cross-link theory. The C2-term in the Mooney-Rivlin equation is attributed to the pseudo cross-link which decreases during elongation. When the remaining pseudo cross-link is less than a critical value for viscous flow, the flow resistance partly balanced with the elastic force becomes large. The effect of viscous flow is revealed in the force in retraction as a hysteresis. Filler enhances the hysteresis.

Inhomogeneities and deviations from the Gaussian photoelastic behavior of networks

Abstract The microphase separation of the swelling agent in swollen gels due to a change in temperature leads to an increase in deviations of the photoelastic behavior from the theory of Gaussian networks. These departures are reflected in an increase in the parameter C2 of the Mooney-Rivlin equation, particularly in a pronounced increase in the stress-optical coefficient Ce and in its marked dependence on deformation. While the mechanical behavior is affected in a decisive manner by the swollen gel matrix, optical characteristics are mainly influenced by particles of the excluded liquid phase.

Deformation homogene et plastique du poly(ethylene-terephthalate) amorphe

The drawing behaviour of amorphous poly(ethylene-terephthalate) has been undertaken at temperature (i) T = 80° (homogeneous deformation); (ii) T = 20° (plastic deformation); (iii) and two stage drawing: homogeneous drawn at T = 80° followed by cold drawing at T = 20°. The results (birefringence, draw ratio-shrinkage force) are discussed in term of the gaussian theory of rubber elasticity and the Mooney-Rivlin equation.

Study of the mechanism of high deformations in rubber-like block copolymers

In order to explain the mechanism of elasticity of a number of diene block-copolymers with linear and radial structure of macromolecules by deformation calorimetry, a study was made of the dependence of the heat and energy of uniaxial reversible elongation on deformation up to an elongation of 1000%. Thermodynamic analysis showed that the proportion of intramolecular changes in entropy and energy is independent of deformation. The joint study of results of thermoelasticity and analysis according to the Mooney-Rivlin equation suggests that gradients in the region of high deformations is the consequence of deformation of polystyrene domains and not of limited extensibility of elastomer chains, as is usually assumed for high deformations of elastomers.

Applications of a two-network model for crosslinks and trapped entanglements

The dependence of stress and birefringence on strain in uniaxial extension of crosslinked rubbers can accurately be described by a model in which the properties of a network of crosslinks and a network of trapped entanglements are additive. The crosslink network is neo-Hookean and the entanglement network can conveniently be described by the Mooney-Rivlin equation. When the crosslinks are introduced in a state of strain near the glass transition temperature, the two networks have different reference undeformed states; they can be distinguished by appropriate measurements in the state of ease where their associated stresses are equal and opposite and in the state of deformation where the cross-links were introduced and make no contribution to the stress. When partial relaxation is permitted before crosslinking, trapping probabilities can be calculated for both relaxed and unrelaxed entanglements and compared with the Langley theory. The results are consistent with the terminal mechanism of relaxation in the tube theory of Doi and Edwards.

Study of elastic and thermoelastic properties of ethylene–propylene and ethylene–vinyl acetate copolymers. II. Thermoelastic properties

The thermoelastic properties of ethylene–propylene and ethylene–vinyl acetate copolymers crosslinked to different degrees were studied. An equation was proposed for calculating the relative contribution of the internal energy, fU/f, from the temperature dependence of shear modulus G. Analysis of a relation for calculating fU/f derived on the basis of the Mooney-Rivlin equation was made.