Yield strain static

The macroscopic stress cr is this force times the number of interparticle bonds that cross a unit area of the sample this latter factor should scale as 0 /a (Russel et al. 1989). As long as the local applied force increases with increased strain, cr increases with increasing strain, and the gel maintains its mechanical stability. But once the strain reaches the point -that the slope W of the potential is a maximum (see Fig. 7-23), any further strain produces a decreasing force, and the interparticle structure breaks apart. This corresponds to the point of yield. Thus, the yield strain yy is given by the condition that the second derivative W of W D) is zero that is, W" Dy) = 0, where Dy = lyyU + (yy + 1)Dq is the value of D for which W" = 0. Very roughly, we might expect that W is a maximum (W — 0) when separation D = Dy % on the order of twice Dq, the value of D at static equilibrium. This would imply that the yield strain yy is roughly hence, for particles 100 nm in... 

The study of mechanical yielding of gels, glasses, and pastes represents a very artive field of current research. A number of different experimental means have been irsed to determine the yielding conditions, static and dynamic alike. Both strain- and stress-controlled rheometry have been employed. The interested reader is referred to indicative references, " some of which describe glasses from star polymers or star-like micelles. Below, we address some aspects of the so-called flow curve, that is, shear stress versus shear rate. 

What is remarkable is the reduction of tolerable maximum strain after only 105 cycles for the glass fiber-reinforced laminates to only approx. 25% of yield strain in the tensile test. The carbon reinforced laminates still tolerate 75 to 80% of their maximum static strain after 107 cycles, demonstrating their superior cyclic behavior. The aramid fiber-reinforced laminates tolerate only 30% of their maximum static strain after 107 cycles however, they exhibit a striking reduction in maximum strain only at higher cycles of approx. 105. 

To illustrate the effect of radial release interactions on the structure/ property relationships in shock-loaded materials, experiments were conducted on copper shock loaded using several shock-recovery designs that yielded differences in es but all having been subjected to a 10 GPa, 1 fis pulse duration, shock process [13]. Compression specimens were sectioned from these soft recovery samples to measure the reload yield behavior, and examined in the transmission electron microscope (TEM) to study the substructure evolution. The substructure and yield strength of the bulk shock-loaded copper samples were found to depend on the amount of e, in the shock-recovered sample at a constant peak pressure and pulse duration. In Fig. 6.8 the quasi-static reload yield strength of the 10 GPa shock-loaded copper is observed to increase with increasing residual sample strain. 

Figure 6.8. Plot of the quasi-static reloaded yield stress of shock-loaded copper versus the natural logarithm of residual strain for a 10 GPa symmetric shock with 1 /is pulse duration.

Gray and Follansbee [44] quasi-statically tested OFE copper samples that had been shock loaded to 10 GPa and pulse durations of 0.1 fis, 1 /rs, and 2 fus. The quasi-static stress-strain curves are shown in Fig. 7.10 with the response of annealed starting copper included for comparison. The yield strength of shock-loaded copper is observed to increase with pulse duration, as the work-hardening rate is seen to systematically decrease. 

For a monolayer film, the stress-strain curve from Eqs. (103) and (106) is plotted in Fig. 15. For small shear strains (or stress) the stress-strain curve is linear (Hookean limit). At larger strains the stress-strain curve is increasingly nonlinear, eventually reaching a maximum stress at the yield point defined by = dT Id oLx x) = 0 or equivalently by c (q x4) = 0- The stress = where is the (experimentally accessible) static friction force [138]. By plotting T /Tlx versus o-x/o x shear-stress curves for various loads T x can be mapped onto a universal master curve irrespective of the number of strata [148]. Thus, for stresses (or strains) lower than those at the yield point the substrate sticks to the confined film while it can slip across the surface of the film otherwise so that the yield point separates the sticking from the slipping regime. By comparison with Eq. (106) it is also clear that at the yield point oo. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

Microindentation hardness normally is measured by static penetration of the specimen with a standard indenter at a known force. After loading with a sharp indenter a residual surface impression is left on the flat test specimen. An adequate measure of the material hardness may be computed by dividing the peak contact load, P, by the projected area of impression1. The hardness, so defined, may be considered as an indicator of the irreversible deformation processes which characterize the material. The strain boundaries for plastic deformation, below the indenter are sensibly dependent, as we shall show below, on microstructural factors (crystal size and perfection, degree of crystallinity, etc). Indentation during a hardness test deforms only a small volumen element of the specimen (V 1011 nm3) (non destructive test). The rest acts as a constraint. Thus the contact stress between the indenter and the specimen is much greater than the compressive yield stress of the specimen (a factor of 3 higher). 

The allowable strength of materials is higher under dynamic loads, which produce high strain rates, than under static loads. This results in higher resistance to dynamic loads. The most important increases are in the compression strength of concrete and the yield strength of the steel reinforcement. 

For steel, the modulus of elasticity is the same in the elastic region and yield plateau for static and dynamic response. In the strain hardening region the slope of the stress-strain curve is different for static and dynamic response, although this difference is not important for most structural design applications. 

A strength increase is also produced at ultimate strength (F ) for steels however, the ratio f dynamic to static strength is less than at yield. A typical stress-strain curve describing dynamic and static response of steel is shown in Figure 5.5. Elongation at failure is relatively unaffected by the dynamic response of the material. 

In addition, other measurement techniques in the linear viscoelastic range, such as stress relaxation, as well as static tests that determine the modulus are also useful to characterize gels. For food applications, tests that deal with failure, such as the dynamic stress/strain sweep to detect the critical properties at structure failure, the torsional gelometer, and the vane yield stress test that encompasses both small and large strains are very useful. 

Nevertheless, if one assumes a static, rather than a thermodynamic, equilibrium, one can attempt to estimate the dependence of the yield stress Oy and the modulus G on the shape and depth of the interparticle potential. Imagine that a gel is subjected to a shear strain Y that homogeneously displaces particles from their positions of static equilibrium. Pairs of particles are pulled apart by this strain, and separations between particle centers of mass should increase roughly by an amount yrQ, where ro = 2a + Dq is the separation between centers of mass in the absence of strain. Hence, the imposition of a strain y increases the gap between particle surfaces from Dq to... 

Quasi-static Young s modulus measured by Hertzian indentation (b) Data taken from ref [5] (c) Measured by Dynamic Mechanical Thermal Analysis (D.M.T.A) at 1 Hz (T is taken as the temperature of the maximum in tan 5) (d) (7y and Oy are the yield stress under uniaxial and plane strain compression, respectively, for an equivalent strain rate of 5x10" s" (see ref... 

Deformability and Wet Mass Rheology The static yield stress of wet compacts has previously been reported in Fig. 21-113. However, the dependence of interparticle forces on shear rate clearly impacts wet mass rheology and therefore deformabihty. Figure 21-117 illustrates the dynamic stress-strain response of compacts, demonstrating that the peak flow or yield stress increases proportionally with compression velocity [Iveson et al., Powder Technol., 127, 149 (2002)]. Peak flow stress of wet unsaturated compacts (initially pendular state) can be seen to also increase with Ca as follows (Fig. 21-118) ... 

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