Shear equilibrium elastic

Rheologists have long believed that all fluids are viscoelastic in behavior. As a result, the deformation of any fluid from the imposition of a stress is the sum of an elastic deformation, which is recoverable, and viscous flow, which is not recoverable. For fluids of low viscosity at moderate rates of shear, the elastic recovery is extremely rapid and the relaxation time is extremely short. As a result, the elastic portion of the deformation is too small to measure, and the fluid is considered simply viscous. When viscoelastic fluids are stressed, some of the energy involved is stored elastically, various parts of the system being deformed into new nonequilibrium positions relative to one another. The remainder is dissipated as heat, various parts of the system flowing into new equilibrium positions relative to one another. 

Els, E2s and Ess are modules of elasticity, respectively, of fast, slow and equilibrium elastic deformations (mN/m) X is elasticity of adsorption layers (%) r s and q are shear surface viscosities (mN-s/m), with t represents Bingam viscosity of partially destroyed structure Pki and Pk2 are Shvedov and Bingam critical shear stresses for flow of adsorption layers, respectively x is the period of relaxation. Experiments have been performed at pH 7.8. 

Deflection temperature Environmental stress cracking Fatigue strength Equilibrium elastic line Shear bands Failure envelope Crazing... 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

Fig. 2.10. Certain high strength solids with low thermal conductivity show a loss or reduction of shear strength when loaded above the Hugoniot elastic limit. The idealized behavior of such solids upon loading is shown here. The complex, heterogeneous nature of such yield phenomena probably results in processes that are far from thermodynamic equilibrium.

Fig. 2.15. Release wavespeeds at very high pressure can be determined by experiments in which the sample thickness is varied for fixed thickness of a high velocity impactor. Data on aluminum alloy 2024 are shown. As indicated in the figure, shear velocity (C ) and Poisson s ratio (cr) at pressure can be calculated from the elastic and bulk speeds if thermodynamic equilibrium is assumed (after McQueen et al. [84M02]).

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 ... 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

The dynamic surface tension of a monolayer may be defined as the response of a film in an initial state of static quasi-equilibrium to a sudden change in surface area. If the area of the film-covered interface is altered at a rapid rate, the monolayer may not readjust to its original conformation quickly enough to maintain the quasi-equilibrium surface pressure. It is for this reason that properly reported II/A isotherms for most monolayers are repeated at several compression/expansion rates. The reasons for this lag in equilibration time are complex combinations of shear and dilational viscosities, elasticity, and isothermal compressibility (Manheimer and Schechter, 1970 Margoni, 1871 Lucassen-Reynders et al., 1974). Furthermore, consideration of dynamic surface tension in insoluble monolayers assumes that the monolayer is indeed insoluble and stable throughout the perturbation if not, a myriad of contributions from monolayer collapse to monomer dissolution may complicate the situation further. Although theoretical models of dynamic surface tension effects have been presented, there have been very few attempts at experimental investigation of these time-dependent phenomena in spread monolayer films. 

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]...

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. 

A typical evolution of equilibrium mechanical properties during reaction is shown in Fig. 6.1. The initial reactive system has a steady shear viscosity that grows with reaction time as the mass-average molar mass, Mw, increases and it reaches to infinity at the gel point. Elastic properties, characterized by nonzero values of the equilibrium modulus, appear beyond the gel point. These quantities describe only either the liquid (pregel) or the solid (postgel) state of the material. Determination of the gel point requires extrapolation of viscosity to infinity or of the equilibrium modulus to zero. 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

A viscoelastic fluid has the appearance of a solid body it deforms and wholly recovers below t0, and only partly recovers above t0. From 0 to t0, the fluid undergoes an elastic conformational transition above t0, the fluid undergoes an irreversible transition, whence the mass begins to flow toward a new equilibrium position. Carrageenan-water-polyol systems have been suggested to be industrially useful in consideration of their significant t0 followed by shear-thinning (Tye, 1988). 

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