Elastic Equation Of State Research Articles

Nonlinear elastic equation of state of solids subjected to uniaxial homogeneous loading

Applying the finite deformation theory to a solid, which possesses either cubic or isotropic symmetry at stress-free natural state and is subsequently loaded homogeneously in uniaxial direction, one obtains a stress (or strain) dependence of the Young's modulus, Poisson's ratio, and a volume (or density) change, together with a nonlinear elastic relation between stress and strain. These are all expressed in terms of the second and third order elastic constants of the solid material. These expressions are illustrated with examples of cubic silicon crystal, isotropic carbon steel, Pyrex glass, and polystyrene at the relaxed state.

Mutually consistent estimates of upper mantle composition from seismic velocity contrasts at 400 km depth

Seismologically determined properties of the 400 km discontinuity may be compared to experimentally determined properties of the associated phase transformation in order to place constraints upon upper mantle bulk composition. Disagreement among previous studies is commonly ascribed to differences in elastic equations of state (especially to assumptions about pressure and temperature derivatives) between studies. However, much of the disparity between studies is actually due to the selection of different seismic data functionals (P-wave velocity,S-wave velocity, etc.) for comparison to minnral clasticity calculations, rather than to the differences in elasticity data sets and equations of state. Within any given study, bulk sound velocity comparisons generally yield more olivine-rich compositional estimates than doP-wave velocity comparisons, which in turn indicate more olivine thanS-wave velocities. Indeed, such variation in compositional estimates within a given study (arising from choice of data functional) exceeds the variation between studies (arising from elastic equation of state approx mations). it can be argued that bulk sound velocities are better constrained seismologically than densities and, being independent of assumptions about shear moduli, should provide more reliable compositional estimates thanP-orS-wave velocities.

Theoretical interpretation of the swelling of elastomers

The swelling dilation moduli reported in the preceding paper are here analyzed with use of a recently proposed theory of rubber elasticity. The theory predicts that the elastic equation of state is dependent on solution thermodynamic parameters, in agreement with the observations. It is successful in accounting for the solvent effects, which is a step forward; it also succeeds in predicting that the Mooney-Rivlin C 2 term decreases with increasing extent of swelling. Unfortunately, agreement with experiment cannot be achieved with a single consistent set of parameters

Elastic properties of swollen real polymer networks

In this paper a new set of stress-strain relations (uniaxial, equi-biaxial, unequibiaxial extensions and pure shear) for swollen networks were presented. Which are derived from the molecular theory of rubber elasticity with constraints of junctions and trapped entanglements and with crosslinks, trapped entanglements and carbon black-polymer interactions. They successed in relating the elastic equation of state to the volume fraction of polymer in swollen networks by three molecular parameters C′1, C′2 and C′3. The relation of stress-strain for uniaxial extension was verified by experiments. It is shown that this relation can successfully predict the dependence of V2 on the C′1, C′2 and C′3, and the contribution of modulus for swollen networks from the trapped entanglement which it shows that role of entanglements can only approach to a limited value, never to zero.

Calorimetric study of rubber elasticity

The thermodynamics of simple elongation of a large number of rubberlike materials has been studied by deformation calorimetry with particular reference to the energy contributions to rubber elasticity. Expressions for mechanical work, heat and change of internal energy on deformation have been derived from the elastic equation of state. New equations for determination of the relative intramolecular energy contribution based on the analysis of elastic inversions of heat and internal energy are proposed. For the majority of polymers studied the free energy of deformation contains a significant intramolecular energy contribution. The temperature coefficients dln〈r 2〉 o dT of the unperturbed dimensions of chains obtained by calorimetry are compared with literature data. Calorimetric results for four networks in the region of 1.02 < λ < 2 have shown that the relative intramolecular energy contribution is independent of deformation and that strain-induced volume dilation and intermolecular energy and entropy changes resulting from this dilation follow the statistical theory only up to λ < 1.3 and at larger deformations the statistical theory underestimates the intermolecular changes. The data obtained for styrene-butadiene thermoplastic elastomers up to λ = 10 demonstrate that the intramolecular energy contribution is independent of deformation in the whole range of deformations. It is concluded that this is a consequence of the validity of the Gaussian statistics at large deformations.

Evaluation of the Theories of Rubberlike Elasticity Using Swollen Networks Crosslinked at Different Degrees of Dilution

Abstract A recently reported study described the preparation of elastomeric networks by crosslinking poly(dimethylsiloxane) chains in both the undiluted state and in solution. Measurements of the moduli of these networks in the unswollen state gave results supporting the theory of rubberlike elasticity developed by Flory and coworkers, rather than the corresponding theory due to James and Guth. These results were not entirely unambiguous, however, because of complications due to departures (“2C2 corrections”) from the expected elastic equation of state. Much of this uncertainty has now been removed by measurements of the moduli of such networks in the swollen state, a condition under which the effects of the 2C2 contributions are known to be minimized. These new results strongly support the earlier conclusion that the Flory theory is the more nearly correct. Nonequilibrium effects, as well as the 2C2 corrections, diminish upon incorporation of a diluent into the crosslinked network; similar decreases occur if diluent had been present during the crosslinking process, even if such diluent is removed prior to the stress-strain measurements. Analysis of these additional results suggests that the 2C2 correction cannot be entirely due to nonequilibrium effects.

The elastic free energy and the elastic equation of state: Elongation and swelling of polydimethylsiloxane networks

AbstractThe elastic free energy for a tetrafunctional network is here expressed as where λ1, etc., are the principal extension ratios relative to the state of reference (wherein 〈r2〉 = 〈r2〉0), and C1, C2, and m are arbitrary parameters. The free energy of a swollen system is taken to be the sum G = Gmix + Gel of Gel and the free energy of mixing \documentclass{article}\pagestyle{empty}\begin{document}$ G_{{\rm mix}} = RT\left[ {n_1 \ln v_1 + n_2 \ln v_2 + \bar \chi n_1 v_2 } \right] $\end{document}. Stress‐strain relations and the chemical potential μ1 are derived from G as functions of the elasticity parameters C1, C2, and m, and of the thermodynamic interaction parameter χ or of \documentclass{article}\pagestyle{empty}\begin{document}$\chi = \bar \chi + n_1 v_2^{ - 1} \partial \bar \chi /\partial n_1 $\end{document}. The dilation of the semi‐open system subject to deformation when exposed to diluent at fixed activity is derived as the sum of the dilation at fixed composition and the dilation due to absorption of diluent. Experiments are reported on the dependence of the equilibrium retractive force on elongation for cross‐linked polydimethylsiloxanes (PDMS) exposed to benzene or hexamethyldisiloxane vapor at regulated activities. Volume fractions of samples covered the range v2 = 1.00 to v2 ≈ 0.30. With the choice of m = ½ the elastic behavior of a given polymer is well represented by one combination of values for C1 and C2 at all dilutions by either diluent. The dependence of the Mooney‐Rivlin (C2) term on volume is thus established, at least for PDMS, and the scope of the semi‐empirical free energy expression and its consequents is greatly enlarged. Values of χ deduced from the equilibrium swelling of the unstrained networks exposed to benzene at various activities are in excellent agreement with those obtained previously from vapor pressures and osmotic pressures on linear PDMS. The results of Allen, Kirkham, Padget, and Price on the elastic behavior of natural rubber are discussed, with particular reference to the coefficients of dilation with elongation which they determined at fixed composition. The present results lend strong support to the principle of additivity of the free energies attributable to the network and to the bulk liquid system, respectively. This principle is fundamental to the analysis of rubber elasticity.

Volume dependence of the C2 correction in the Mooney‐Rivlin elastic equation of state

AbstractStress‐strain isotherms at 25°C are reported for a series of unswollen networks of cis‐1, 4‐polybutadiene which differ both in degree of crosslinking and in the volume at which the crosslinks were introduced. The former quantity was varied by variation of the amount of high energy radiation used to bring about the crosslinking, the latter, by crosslinking the polymer in solution as well as in the bulk, undiluted state. The results are interpreted in terms of the equation f*/(α−α−2) = 2C1 + 2C2α−1 where f* is the force per unit area of the undistorted, unswollen sample, α is the relative length of the sample, and C1 and C2 are empirical constants, the latter serving as a measure of the departure of the dependence of stress on strain from that indicated by the statistical theory of rubber elasticity. At equal values of the “modulus” f*/(α−α−2), the correction C2 is found to be very small for networks prepared in solution and to increase markedly with increase in concentration of polymer in the system undergoing crosslinking. In addition, the networks prepared in solution exhibit considerably smaller nonequilibrium, relaxation effects than those crosslinked in the undiluted state.

STUDY ON THE FORCE-TEMPERATURE BEHAVIOR OF POLYETHYLENE TEREPHTHALATE FIBER

The thermodynamic equation of state for elasticity may be expressed:The force-temperature behavior of the Polyethylene Terephthalate fibers, the drawn spun fibers with the various degrees of orientation and with the different degrees of polymerisation, were observed at the constant elongation (1%).The results observed are:-1) The retractive forces increase with the initial birefringenes of the undrawn fibers and the draw ratio, 2) The tangents of the force-temperature curves are negative at the lower draw ratio, but change to positive at the higher draw ratio, i.e., the thermoelastic inversion occurs at a_??_certain draw ratio, 3) The draw ratio at which the thermoelastic inversion occurs decreases with increased the birefringence of the undrawn fibers and with decrease in degree of polymerisation