Logarithmic linear strain

A plot of the logarithm of strain rate decreases as a function of the logarithm of time and is linear. The same behavior has been observed for undisturbed and remolded wet or dry clay, NC and OC clay, and sand (Singh and Mitchell, 1968). In general, different soil types exhibit varying amounts of time-dependent deformations and stress variations with time. These variations are also exhibited by their secondary compression and creep characteristics. 

Such total finite strains are known as natural or logarithmic strains, the use of which was first suggested by Ludwig (1909). The engineering strain is frequently used as a measure of finite linear strain (e.g., the percentage elongation in a simple tensile test is usually quoted). The relationship between these two measures can be derived as follows ... 

To account for rate effects after yielding in soUds Ludwik, (1909) and Prandtl, (1928) observed that for some materials the yield stress in uniaxial tension was linearly related to the logarithm of strain rate and suggested use of the equation,... 

A still simpler and more accessible approach to predict the creep strain of engineering components is based on a logarithmicly linear approximation of the creep modulus, E(t) ... 

For low-cycle fatigue of un-cracked components where (imax or iCT inl am above o-y, Basquin s Law no longer holds, as Fig. 15.2 shows. But a linear plot is obtained if the plastic strain range defined in Fig. 15.3, is plotted, on logarithmic scales, against the cycles to failure, Nf (Fig. 15.4). TTiis result is known as the Coffin-Manson Law ... 

Tests have shown that when total strain is plotted against the logarithm of the total creep time (ie NT or total experimental time minus the recovery time) there is a linear relationship. This straight line includes the strain at the end of the first creep period and thus one calculation, for say the 10th cycle allows the line to be drawn. The total creep strain under intermittent loading can then be estimated for any combinations of loading/unloading times. 

Diffusion systems are based upon the ability of the antibiotic to diffuse through agar and cause the inhibition of the sensitive assay strains. Since the substrate to be assayed is applied in a "point source," diffusion occurs radially. A circular zone of inhibition forms and the size of the zone is a function of the concentration. This function is expressed as a linear relationship between the size of the zone of inhibition and the logarithm of the concentration. 

The signals produced by the strain gauge can be amplified linearly or logarithmically. In the linear mode the displacement of the X-V recorder along the X or pressure axis is inversely proportional to the pore radius as given by equation (10.24)... 

Ductile-brittle transitions are more accurately located by variable temperature tests than by altering impact speed in an experiment at a fixed temperature. This is because a linear fall in temperature is equivalent to a logarithmic increase in straining rate. The ductile-brittle transition concept can be clarified by sketches such as that in Fig. 11-25. In the brittle region, the impact resistance of a material is related to its LEFM properties, as described above. In the mixed mode failure... 

Tanaka [155] has attempted to make a quantitative estimate of the contribution of ring strain and basicity to reactivity of cyclic ethers in cationic copolymerization. Free energy of polymerization was used as a measure of ring strain. The relationship he derived related the logarithm of relative reactivity, l/r , of m-membered ring ethers with i substituents to n-membered ring compounds with j substituents to a linear combination of the differences in basicity, A(pXb)m,i- ,/ and in free energy, viz. 

As discussed in Sect. 4, in the fluid, MCT-ITT flnds a linear or Newtonian regime in the limit y 0, where it recovers the standard MCT approximation for Newtonian viscosity rio of a viscoelastic fluid [2, 38]. Hence a yrio holds for Pe 1, as shown in Fig. 13, where Pe calculated with the structural relaxation time T is included. As discussed, the growth of T (asymptotically) dominates all transport coefficients of the colloidal suspension and causes a proportional increase in the viscosity j]. For Pe > 1, the non-linear viscosity shear thins, and a increases sublin-early with y. The stress vs strain rate plot in Fig. 13 clearly exhibits a broad crossover between the linear Newtonian and a much weaker (asymptotically) y-independent variation of the stress. In the fluid, the flow curve takes a S-shape in double logarithmic representation, while in the glass it is bent upward only. 

The Standard recommends plotting of the deformation of the specimen (as thickness of the specimen, as strain or as void ratio) as ordinate on a linear scale and the corresponding applied pressure in kN/m2 as abscissa on a logarithmic scale. From this, the coefficient of volume compressibility or the coefficient of consolidation may be evaluated as specified in the Standard. 

ISO 8013 is confined to static strain conditions and can seriously underestimate the creep that occurs under dynamic loading. The creep rate in cycled rubber is higher than that predicted by a simple Boltzmann superposition, but linearity is still observed between creep and logarithmic time or the logarithm of the number of cycles, as long as a physical mechanism applies [39,40]. The increase is most striking with strain-crystallizing elastomers such as natural rubber. 

Normally a logarithmic time scale is used to plot the creep curve, as shown in Figure 3.9b, so that the time dependence of strain after long periods can be included. If a material is linearly viscoelastic (Equation 3.17), then at any selected time each line in a family of creep curves (with equally spaced stress levels) should be offset along the strain axis by the same amount. Although this type of behavior maybe observed for plastics at low strains and short times, in most cases the behavior is nonlinear, as indicated in Figure 3.9c. 

Figure 7 shows a plot of the logfnj vs log(r values at 3.5% strain and 30 °C and 110 °C [note that the data are plotted as referenced to f, = 1800 sj. The data show good linearity, the gradient of which is the double logarithmic tiiift rate. 

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