Stress-strain relations temperature dependence

Figure 18.17 shows that the characteristics of the stress-strain curve depend mainly on the value of n the smaller the n value, the more rapid the upturn. Anyway, this non-Gaussian treatment indicates that if the rubber has the idealized molecular network strucmre in the system, the stress-strain relation will show the inverse S shape. However, the real mbber vulcanizate (SBR) that does not crystallize under extension at room temperature and other mbbers (NR, IR, and BR at high temperature) do not show the stress upturn at all, and as a result, their tensile strength and strain at break are all 2-3 MPa and 400%-500%. It means that the stress-strain relation of the real (noncrystallizing) rubber vulcanizate obeys the Gaussian rather than the non-Gaussian theory. 

FIGURE 18.18 Temperature dependence of the stress-strain relation in carbon black-filled natural mbber (NR). (From Fukahori, Y. and Seki, W Polymer, 33, 1058, 1992.)... 

Mechanical Damping. The elastic energy behavior of resins at low temperatures was determined from the stress-strain relation. Small inelastic contributions, not detectable with that method can be determined by damping measurements using a free torsion pendulum. The mechanical loss, tan 8m, is plotted vs. temperature in Fig. 10. Low-temperature values were similar for the resins considered. No major structural dependence is indicated. (It is not within the scope of this paper to consider the relaxation maxima of resins.)... 

Elastic properties serve an obvious utility in mechanics of materials, e.g., stress-strain relations and dislocation characteristics (Fisher and Dever, 1967 Fisher and Alfred, 1968). Moreover, elastic properties and their temperature dependencies provide important information and understanding of such physical characteristics as magnetic behavior, polymorphic transformations, and other fundamental lattice phenomena. In this section the elastic properties and their temperature dependencies are presented for all the rare earth metals except promethium, for which there is no data. To the writer s knowledge this is the first one-source compilation of the temperature dependencies of the elastic properties of the rare earth metals. 

The strain measures for dry (unswollen) vulcanizates of a large number of natural rubbers, butadiene-styrene and butadiene-acrylonitrile copolymers, polydimethylsiloxanes, polymethylmethacrylates, polyethylacrylates and polybutadienes with different degrees of crosslinking and measured at various temperatures re confined within the shaded area in Fig. 1. These measures were determined from the stress as a function of extension at (or near) equilibrium, i.e. by applying Eq. (7). Therefore they only reproduce the equilibrium stress-strain relation for the crossllnked rubbers. In all cases the strain dependence of the tensile force (and hence of the tensile stress) was expressed in terms of the well-known Mooney-Rivlin equation, equating the equilibrium tensile stress to ... 

In the strain-hardening region The true stress linearly increased with the Gaussian strain modified by the transient Poisson s ratio data and the slope of the line, corresponding to the strain-hardening coefficient, showed a positive dependence on temperature. In addition, the true stress-strain relation curves could be described by a linear relationship with the Gaussian strain modified by transient Poisson s ratio data. 

The theory relating stress, strain, time and temperature of viscoelastic materials is complex. For many practical purposes it is often better to use an ad hoc system known as the pseudo-elastic design approach. This approach uses classical elastic analysis but employs time- and temperature-dependent data obtained from creep curves and their derivatives. In outline the procedure consists of the following steps ... 

Viscoelasticity A combination of viscous and elastic properties in a plastic with the relative contribution of each being dependent on time, temperature, stress, and strain rate. It relates to the mechanical behavior of plastics in which there is a time and temperature dependent relationship between stress and strain. A material having this property is considered to combine the features of a perfectly elastic solid and a perfect fluid. 

Polymers are viscoelastic materials meaning they can act as liquids, the visco portion, and as solids, the elastic portion. Descriptions of the viscoelastic properties of materials generally falls within the area called rheology. Determination of the viscoelastic behavior of materials generally occurs through stress-strain and related measurements. Whether a material behaves as a viscous or elastic material depends on temperature, the particular polymer and its prior treatment, polymer structure, and the particular measurement or conditions applied to the material. The particular property demonstrated by a material under given conditions allows polymers to act as solid or viscous liquids, as plastics, elastomers, or fibers, etc. This chapter deals with the viscoelastic properties of polymers. 

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

In all expressions the Einstein repeated index summation convention is used. Xi, x2 and x3 will be taken to be synonymous with x, y and z so that o-n = axx etc. The parameter B will be temperature-dependent through an activation energy expression and can be related to microstructural parameters such as grain size, diffusion coefficients, etc., on a case-by-case basis depending on the mechanism of creep involved.1 In addition, the index will depend on the mechanism which is active. In the linear case, n = 1 and B is equal to 1/3t/ where 17 is the linear shear viscosity of the material. Stresses, strains, and material parameters for the fibers will be denoted with a subscript or superscript/, and those for the matrix with a subscript or superscript m. 

Stress is related to strain through constitutive equations. Metals and ceramics typically possess a direct relationship between stress and strain the elastic modulus (2) Polymers, however, may exhibit complex viscoelastic behavior, possessing characteristics of both liquids and solids (4.). Their stress-strain behavior depends on temperature, degree of cure, and thermal history the behavior is made even more complicated in curing systems since material properties change from a low molecular weight liquid to a highly crosslinked solid polymer (2). ... 

To obtain information on the actual sizes Of of the relaxation clusters in glassy polymers it is necessary to associate the simulation results with experimental results on shear activation volumes Av that relate directly to the product Qfy. This, together with the simulation results for the transformation shear strain (= 7 ) j offsr a means of estimating the actual values of Of of relaxing clusters and the number of molecular segments n in the clusters that take part in the shear relaxations. Before considering such an evaluation, we first present experimental results on the temperature dependences of the yield stresses of several flexible-chain and stiff-chain glassy polymers. 

We develop first the considerations related to shear response in a ID context of plastic-shear flow to state the basic kinetic response of the solid, where s stands for an applied shear stress, t stands for a threshold plastic-shear resistance, and y is taken to be the plastic-shear strain yP. As a useful simplification, we first consider the material to be rigid on the basis that the plastic-shear increments are large, in comparison with the elastic-strain increments. At temperatures T > OK, for which the elastic moduli of the solid are significantly lower than at 0 K, we expect that the rate-independent plastic-shear resistance z temperature dependence as the elastic-shear modulus (Chapter 4). Then, where the plastic response in a rate-independent manner is initiated when s = z(T), under conditions of s < z T), a plastic response is still possible by thermal assistanee and occurs at a (plastic) shear rate of (Argon 1973)... 

Correspondingly, eq. (13.2) representing the strain-rate dependence of the plastic resistance is taken to be given by a standard uniaxial reference experiment at a reference strain rate e gf, typically of magnitude 10 s that evokes a reference tensile uniaxial plastic resistance o-j-gf, which in this case would be the tensile yield stress o-q. The form of the idealized power law relating eg to Ug is given by the exponent m of the equivalent stress, which must be temperature-dependent in a form given in Chapter 8 as... 

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