Steady straining

Egolfopoulos, F.N., Zhang, H., and Zhang, Z., Wall effects on the propagation and extinction of steady, strained, laminar premixed flames. Combust. Flame, 109,237,1997. 

A hydrostatic stress can be superposed, but it is caused only by elastic volumetric strain of the composite. The result in Eqn. (39) is, perhaps, not very useful since it is rare that a steady strain rate will be kinematically imposed. When both fiber and matrix creep, the steady solutions for a fixed stress in isothermal states are quite complex but can be computed by numerical inversion of Eqn. (39). The solution can, however, be given for the isothermal case where the fibers do not creep. (For non-fiber composites, this should be... 

Equation (19) implies tirat, for small popidations of broken molecules, the rate of radical formation, dNjdt, ould be constant at constant stress. This was at first tlmu t to be the case from the data obtained in constant loadii -rate tests but it was soon found that stepwise loading tests told a different story. F ie 18 shows data of Becht and Fischer for the stepwise loading of njdon 6. As previously observed by Roylance and De Vries, the rate of radical production under steady strain decreases to zero after a period of minutes (or even seconds), althoi stress relaxation is only of the order of 10%. 

Magnaudet J., Rivero M., Fabre J. (1997) Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow. J. FluidMech. 284, 97-135. 

Comparison of predictions of a flamelet library (based on planar steady strained flames with the same chemical and transport models as for the DNS) and DNS results was also given in Ref. 2. Despite the large scatter, it appears that the agreement is reasonable the evolution of Sc with strain is correctly predicted by the library as shown in Fig. 6 even though the library overestimates the effects of strain on the flame (this effect is probably due to unsteady mechanisms). This result brings more credit to the idea of flamelet libraries than the simple chemistry computations did. 

Uniaxial extensional viscosity and shear viscosity 77+ as functions of time after inception of steady straining for lUPAC A low density polyethylene. The open symbols are elongational viscosities the solid and half-open symbols are shear viscosities. Adapted from Meissner (1972). 

Stressing viscosity (i, for uniaxial, biaxial, and planar extension, stressing viscosity for planar extension, and shear viscosity as functions of time ter inception of steady straining for polyisobutylene. The solid line is the low shear rate limit of. Extension and shear rates are 0.08s except the biaxial which is 0.02s-. From Retting and Lawn, 1991. 

If a creep experiment is carried out at a constant stress that is sufficiently large that the sample is strained into the nonlinear regime before achieving a steady strain rate, the compliance function will be a function of stress as well as time. Data from this type of experiment have been reported by Agarwal and Plazek [ 53 ], but creep has not been widely used for the study of nonlinear viscoelasticity. 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

In the steady-state creep regime of ceramics, almost aU creep mechanisms fit a strain rate dependence of the form (18) ... 

Elasticity is another manifestation of non-Newtonian behavior. Elastic Hquids resist stress and deform reversibly provided that the strain is not too large. The elastic modulus is the ratio of the stress to the strain. Elasticity can be characterized usiag transient measurements such as recoil when a spinning bob stops rotating, or by steady-state measurements such as normal stress ia rotating plates. 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

Steady-propagating plastic waves [20]-[22] also give some useful information on the micromechanics of high-rate plastic deformation. Of particular interest is the universality of the dependence of total strain rate on peak longitudinal stress [21]. This can also be expressed in terms of a relationship between maximum shear stress and average plastic shear strain rate in the plastic wave... 

Although the initial elastic and the primary creep strain cannot be neglected, they occur quickly, and they can be treated in much the way that elastic deflection is allowed for in a structure. But thereafter, the material enters steady-state, or secondary creep, and the strain increases steadily with time. In designing against creep, it is usually this steady accumulation of strain with time that concerns us most. 

Like metals, ceramics creep when they are hot. The creep curve (Fig. 17.4) is just like that for a metal (see Book 1, Chapter 17). During primary creep, the strain-rate decreases with time, tending towards the steady state creep rate... 

Above 0.5 ceramics creep in exactly the same way that metals do. The strain-rate increases as a power of the stress. At steady state (see Chapter 17, eqn. 17.6) this rate is... 

Isometric data from the creep curves may also be superimposed on the creep rupture data in order to give an indication of the magnitudes of the strains involved. Most plastics behave in a ductile manner under the action of a steady load. The most notable exceptions are polystyrene, injection moulding grade acrylic and glass-filled nylon. However, even those materials which are ductile at short times tend to become embrittled at long times. This can cause... 

As reviewed in Chapter 2, loads are often applied abruptly, resulting in significant stress and strain increases. However, the elasticity of most TPs lets recovery usually be complete. Therefore, the steady-state stress and deflection of plastic can be considered identical to that of a product that is loaded gradually. However, when impact becomes severe, failure can result from it. 

When a load is initially applied to a specimen, there is an instantaneous strain or elongation. Subsequent to this, there is the time-dependent part of the strain (creep), which results from the continuation of the constant stress at a constant temperature. In terms of design, creep means changing dimensions and deterioration of product strength when the product is subjected to a steady load over a prolonged period of time. 

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