Elastomers small-strain behavior

During the 1940s and early 1950s, Leaderman [L4,L5],Tobolsky [A2,A3, Dll, T5], Ferry [F3], and others made extensive studies of the small-strain behavior of elastomers and related polymers. These studies involved creep (deformation under applied stress), stress relaxation following applied stresses, and imposed oscillatory strains. These and other experimental techniques used have been described in special detail in the monograph of Ferry [F3]. These studies showed that all of these deformations could be represented in terms of the superposition principle of Boltzmann [B26] [see Eq.(43)]. 

So far, we have considered the elasticity of filler networks in elastomers and its reinforcing action at small strain amplitudes, where no fracture of filler-filler bonds appears. With increasing strain, a successive breakdown of the filler network takes place and the elastic modulus decreases rapidly if a critical strain amplitude is exceeded (Fig. 42). For a theoretical description of this behavior, the ultimate properties and fracture mechanics of CCA-filler clusters in elastomers have to be evaluated. This will be a basic tool for a quantitative understanding of stress softening phenomena and the role of fillers in internal friction of reinforced rubbers. 

Brown JD (1997) Nonlinear dynamic behavior of filled elastomers at small strain amplitudes. PhD Thesis, Rensselaer Polytechnic Institute, Troy, New York Chazeau L, Brown JD,Yanyo LC, Sternstein SS (2000) Polym Compos 21 202... 

A rubber-like solid is unique in that its physical properties resemble those of solids, liquids, and gases in various respects. It is solidlike in that it maintains dimensional stability, and its elastic response at small strains (<5%) is essentially Hookean. It behaves like a liquid because its coefficient of thermal expansion and isothermal compressibility are of the same order of magnitude as those of liquids. The implication of this is that the intermolecular forces in an elastomer are similar to those in liquids. It resembles gases in the sense that the stress in a deformed elastomer increases with increasing temperature, much as the pressure in a compressed gas increases with increasing temperature. This gas-like behavior was, in fact, what first provided the hint that rubbery stresses are entropic in origin. 

The equilibrium small-strain elastic behavior of an "incompressible" rubbery network polymer can be specified by a single number—either the shear modulus or the Young s modulus (which for an incompressible elastomer is equal to 3. This modulus being known, the stress-strain behavior in uniaxial tension, biaxial tension, shear, or compression can be calculated in a simple manner. (If compressibility is taken into account, two moduli are required and the bulk modulus. ) The relation between elastic properties and molecular architecture becomes a simple relation between two numbers the shear modulus and the cross-link density (or the... 

The preceding equations provided a reasonable foundation for predicting DE behavior. Indeed the assumption that DEs behave electronically as variable parallel plate capacitors still holds however, the assumptions of small strains and linear elasticity limit the accuracy of this simple model. More advanced non-linear models have since been developed employing hyperelasticity models such as the Ogden model [144—147], Yeoh model [147, 148], Mooney-Rivlin model [145-146, 149, 150] and others (Fig. 1.11) [147, 151, 152]. Models taking into account the time-dependent viscoelastic nature of the elastomer films [148, 150, 151], the leakage current through the film [151], as well as mechanical hysteresis [153] have also been developed. 

Upon stretching along the layer normal, at relatively small strain on a rubbery scale of about 5—7%, smectic elastomers may break up into stripes leading macro-scopically to a cloudy appearance [148]. The striped texture corresponds to a local layer inclination (rotation) relative the average direction of the layer normal. The system prefers the layers to rotate in order to relieve any layer extension deformation in favor of lower-cost rubber distortions at constant layer spacing. This reaction is the rubbery equivalent of the classical instability to avoid layer dilation in low-molecular-mass smectics, described in [149]. However, this type of behavior is not universal because other samples show isotropic rubber behavior [142, 144, 145]. 

The minimum free energy approach to a network has been used to describe the stress-strain behavior of hydrogen-bonded solids at small strains [9]. The present forms of the stress tensor and work function (in two dimensions) have been applied to the fiber network problem [7,8], for which the minimizing constraints were taken to be the stress tensor components. In the case of an elastomer network, the stress constraints must be replaced by deformation constraints if the admissable solutions to the problem are to be restricted to constant volume deformations. 

From the dynamic mechanical investigations we have derived a discontinuous jump of G and G" at the phase transformation isotropic to l.c. Additional information about the mechanical properties of the elastomers can be obtained by measurements of the retractive force of a strained sample. In Fig. 40 the retractive force divided by the cross-sectional area of the unstrained sample at the corresponding temperature, a° is measured at constant length of the sample as function of temperature. In the upper temperature range, T > T0 (Tc is indicated by the dashed line), the typical behavior of rubbers is observed, where the (nominal) stress depends linearly on temperature. Because of the small elongation of the sample, however, a decrease of ct° with increasing temperature is observed for X < 1.1. This indicates that the thermal expansion of the material predominates the retractive force due to entropy elasticity. Fork = 1.1 the nominal stress o° is independent on T, which is the so-called thermoelastic inversion point. In contrast to this normal behavior of the l.c. elastomer... 

The influence of filler is not limited to this enhancement of the non-Newtonian behavior of elastomers. At very small shear rates, filled green compounds also exhibit an additional increase of viscosity that cannot be explained by strain amplification. This effect is usually attributed to the existence of the filler network the direct bonding of reinforcing objects by adsorbed chains implies a increased force to be broken. Obviously this influence can be observed only at very low strain, because a very small increase of interaggregate distances immediately implies a desorption of the bridging elastomeric chains. 

Elastomers are a class of polymers that can be repeatedly strained and then return to the approximate original length on release of the load. Traditional elastomers such as rubber are able to achieve this elastic behavior by having a low glass transition temperature and a small number of chemical crosslinks that form a permanent network... 

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