Rubber elastic shear modulus

Ordinary liquids and liquid crystals are nearly incompressible. In ordinary fluid dynamics the incompressibility approximation under the constraint div v = 0 has frequently been utilized. In a soft elastomer such as vulcanized rubber, where shear modulus is very small as compared with bulk modulus, the incompressibility approximation has also been usefully employed. The constraint of the incompressibility approximation, div v = 0 for ordinary fluids or divergence of displacement vector for elastic (isotropic) materials, does not modify any other terms of the equations of motion div v = 0, or divergence of displacement vector, is a solutirai of the equations of motion, provided that pressure p is chosen as an appropriate harmoiuc function (V p = 0). However, for anisotropic matters, such as liquid crystals or anisotropic solids (crystals), since the div v = 0 or its elastic version cannot be a special solution of equations of motion, the incompressibility approximation requires a careful consideration [12, 18]. 

NRB (natural rubber bearing) Elastic shear modulus Upper +10 +6 +10... 

An increase in the swelling degree usually results in lowering elastic modulus. According to the rubber elasticity theory [116-118] the shear modulus of the gel G can be expressed as ... 

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

Equation (52) allows us to estimate the impact of viscoelastic braking on the capillary flow rate. As an example, we will consider that the liquid is tricresyl phosphate (TCP, 7 = 50 mN-m t = 0.07 Pa-s). The viscoelastic material is assumed to have elastic and viscoelastic properties similar to RTV 615 (General Electric, silicone rubber), i.e., a shear modulus of 0.7 MPa (E = 2.1 MPa), a cutoff length of 20 nm, and a characteristic speed, Uo, of 0.8 mm-s [30]. TCP has a contact angle at equilibrium of 47° on this rubber. 

According to the rubber elasticity theory ( 1, 2), the equilibrium shear modulus, Ge, is proportional to the concentration of EANC s and an additional contribution due to trapped entanglements may also be considered ... 

Figure 5.11 Dependence of the reduced equilibrium shear modulus, Ge/wg// 7" on the molar ratio of [OH]/[NCO] groups, ah, for poly(oxypropylene)triol (Niax LG 56)-4,4 -diisocyanatodiphenylmethane system (—-) limits of the Flory-Erman junction fluctuation rubber elasticity theory. The dependence has been reconstructed from data of ref. [78]...

Here, v is Poisson s ratio which is equal to 0.5 for elastic materials such as hydrogels. Rubber elasticity theory describes the shear modulus in terms of structural parameters such as the molecular weight between crosslinks. In the rubber elasticity theory, the crosslink junctions are considered fixed in space [19]. Also, the network is considered ideal in that it contained no structural defects. Known as the affine network theory, it describes the shear modulus as... 

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

From the theory of rubber elasticity it is quite clear that the shear modulus at time zero must read ... 

Stress/strain relationships are commonly studied in tension, compression, shear or indentation. Because in theory all stress/strain relationships except those at breaking point are a function of elastic modulus, it can be questioned as to why so many modes of test are required. The answer is partly because some tests have persisted by tradition, partly because certain tests are very convenient for particular geometry of specimens and partly because at high strains the physics of rubber elasticity is even now not fully understood so that exact relationships between the various moduli are not known. A practical extension of the third reason is that it is logical to test using the mode of deformation to be found in practice. 

Chain entanglements are the cause of rubber-elastic properties in the liquid. Below the "critical" molecular mass (Mc) there are no indications of a rubbery plateau. The length of the latter is strongly dependent on the length of the molecular chains, i.e. on the molar mass of the polymer. From the shear modulus of the pseudo rubber plateau the molecular weight between entanglements may be calculated ... 

Using shear modulus obtained at different temperature to conclude that model of rubber cannot be applied to dough elasticity. 

Dependence of the Shear Modulus on the Concentration. The experimental results of Figure 3a show that the plateau values increase with the detergent concentration. Unfortunately, we were not able to reach the rubber plateau for all concentrations for lack of the frequency range. From the theory of networks it is possible to calculate the number of elastically effective chains between the crosslinks from the shear modulus Gq of the rubber plateau (12). If the network... 

According to the theory of rubber elasticity [29], the "equilibrium" shear modulus Ge°(T), above Tg, of a polymer crosslinked beyond its gel point, is determined by Mc. Equation 11.22... 

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

Modulus of Elasticity. It is assumed that these gels are deformed according to the theory of rubber elasticity, which tells us that the shear modulus is given (5, Chap. 11) by... 

The simplest model is the statistical theory of rubber-like elasticity, also called the affine model or neo-Hookean in the solids mechanics community. It predicts the nonlinear behavior at high strains of a rubber in uniaxial extension with Fq. (1), where ctn is the nominal stress defined as F/Aq, with F the tensile force and Aq the initial cross-section of the adhesive layer, A is the extension ratio, and G is the shear modulus. 

The deformation ability of networks strongly swollen with benzene and those slightly swollen in cyclohexane was unexpectedly found to be the same. What is surprising here is the absence of any correlation between the volume increase of model networks on swelling and their deformation under compression or elongation [130], as it would have to foUow from the classic theory of rubber elasticity. This theory does not predict any difference between the extensional modulus and the shear modulus that controls the swelling. Nevertheless, the experimental ratio of Ce(CH)/ Ce(BZ) = 6 is twice as large as the ratio of E(CH)/E(BZ) = 3 (irrespective ofp) [123]. 

Neither the uniform strain model nor the uniform stress model is appropriate for this microstructure. Consequently, the elastic moduli of polyurethanes lie between the limits set by Eqs (4.11) and (4.12). For a network chain of Me = 6000, the rubber elasticity theory of Eq. (3.20) predicts a shear modulus of about 0.4 MPa. The hard blocks will have the typical 3GPa Young s modulus of glassy polymers. Increases in the hard block content cause the Young s modulus to increase from 30 to 500 MPa (Fig. 7.13). For automobile panel applications it is usual to have a high per cent of hard blocks so that the room temperature flexural modulus is 500 MPa. 

Under small deformations rubbers are linearly elastic solids. Because of high modulus of bulk compression (about 2000 MN/m ) compared with the shear modulus G (about 0.2-5 MN/m ), they may be regarded as relatively incompressible. The elastic behavior under small strains can thus be described by a single elastic constant G. Poisson s ratio is effectively 1/2, and Young s modulus E is given by 3G, to good approximation. 

It is customary to characterize the modulus, stiffness, or hardness of rubbers by measuring their elastic indentation by a rigid die of prescribed size and shape under specified loading conditions. Various nonlinear scales are employed to derive a value of hardness from such measurements (Soden, 1952). Corresponding values of shear modulus G for two common hardness scales are given in Figure 1.18. 

The shear modulus of a rubber is inversely proportional to the average network chain length according to the Gaussian theory of rubber elasticity [19]. Therefore, examination of the storage modulus above the Tg provides the evidence for network hydrolytic resistance. Figure 12.13 shows that the storage modulus (Eg)... 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

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