Stress theories comparison

Three points are obvious from a comparison of the maximum principal stress theory and the maximum shear stress theory ... 

The K-BKZ Theory Comparison with Experiment. The first data required to test the K-BKZ model is single-step stress relaxation data to determine the material parameters of interest. This is best seen from the following example for a simple shearing history. From equation 49, the shear stress for a simple shear deformation can be expressed as (see Ref 72)... 

Within the context of pressure vessel design codes, the comparison of the allowable strength of the material is always done with respect to the stress intensities. This puts the comparison in terms of the appropriate failure theory either the maximum shear stress theory (Tresca criterion) or the maximum distortion energy theory (von Mises criterion). These failure theories have been discussed in some detail in Chapter 3. 

Figure 9.17 graphically shows a comparison of the four methods of analysis for a two-dimensional stress system (Xr = 0). The upper right quadrant represents tension for both fjc and fy. The upper left quadrant represents fy in tension and/r in compression, and the lower right quadrant represents in tension and fy in compression. The wlnl lines in the figure represent the locus of the conditions at which yield is assumed to begin according to the four theories. The square a-b-c-d represents the maximum stress theory. Point a represents equal tension in both the X and y perpendicular directions, both of which are considered to be equal to the yield-point stress obtained from a simple tensile lest. 

Graphical Comparisons of He Variour Mi es. A graphical comparison of the membrane theory, the ASME modified membrane theory, the principal-stress theory, the... 

Comparison Between Theory and Experiment Comparisons between theory and experiment have been made for many materials. Shown in Fig. 2.20 are the graphs in stress space for the equations for the three theories given above. Also shown is experimental data on five different metals as well as four different polymers. It will be noted that cast iron, a very brittle material agrees well with the maximum normal stress theory while the ductile materials of steel and aluminum tend to agree best with the von Mises criteria. Polymers tend to be better represented by von Mises than the other theories. 

The purpose of these comparisons is simply to point out how complete the parallel is between the Rouse molecular model and the mechanical models we discussed earlier. While the summations in the stress relaxation and creep expressions were included to give better agreement with experiment, the summations in the Rouse theory arise naturally from a consideration of different modes of vibration. It should be noted that all of these modes are overtones of the same fundamental and do not arise from considering different relaxation processes. As we have noted before, different types of encumbrance have different effects on the displacement of the molecules. The mechanical models correct for this in a way the simple Rouse model does not. Allowing for more than one value of f, along the lines of Example 3.7, is one of the ways the Rouse theory has been modified to generate two sets of Tp values. The results of this development are comparable to summing multiple effects in the mechanical models. In all cases the more elaborate expressions describe experimental results better. 

Perhaps the most significant complication in the interpretation of nanoscale adhesion and mechanical properties measurements is the fact that the contact sizes are below the optical limit ( 1 t,im). Macroscopic adhesion studies and mechanical property measurements often rely on optical observations of the contact, and many of the contact mechanics models are formulated around direct measurement of the contact area or radius as a function of experimentally controlled parameters, such as load or displacement. In studies of colloids, scanning electron microscopy (SEM) has been used to view particle/surface contact sizes from the side to measure contact radius [3]. However, such a configuration is not easily employed in AFM and nanoindentation studies, and undesirable surface interactions from charging or contamination may arise. For adhesion studies (e.g. Johnson-Kendall-Roberts (JKR) [4] and probe-tack tests [5,6]), the probe/sample contact area is monitored as a function of load or displacement. This allows evaluation of load/area or even stress/strain response [7] as well as comparison to and development of contact mechanics theories. Area measurements are also important in traditional indentation experiments, where hardness is determined by measuring the residual contact area of the deformation optically [8J. For micro- and nanoscale studies, the dimensions of both the contact and residual deformation (if any) are below the optical limit. 

Obviously, the assumptions involved in the foregoing derivation are not entirely consistent. A transverse strain mismatch exists at the boundary between the fiber and the matrix by virtue of Equation (3.8). Moreover, the transverse stresses in the fiber and in the matrix are not likely to be the same because v, is not equal to Instead, a complete match of displacements across the boundary between the fiber and the matrix would constitute a rigorous solution for the apparent transverse Young s modulus. Such a solution can be found only by use of the theory of elasticity. The seriousness of such inconsistencies can be determined only by comparison with experimental results. 

Comparison with Statistical Theory at Moderate Strains. So far we have shown, that a transition between the two limiting classical theories, i.e. affine theory and phantom theory, is possible by a suitable choice of the network microstructure. This argument goes beyond the revised theory by Ronca and Allegra and by Flory, which predicts such a transition as a result of increasing strain, thus explaining the experimentally observed strain dependence of the reduced stress. 

Throughout the chapter, the importance of network formation theories in understanding and predicting structural development is stressed. Therefore, a short expose on network formation theories is given in this chapter. Although the use of theoretical modeling of network build-up and comparison with experiments play a central role in this chapter, most mathematical relations and their derivation are avoided and only basic postulates of the theories are stated. The reader can always find references to literature sources where such mathematical relations are derived. 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

Kim, J.K., Baillic, C. and Mai, Y.W. (1992). Fnterfacial debonding and fiber pull-out stresses, part F. A critical comparison of existing theories with experiments. J. Mater. Sci. 27, 3143-3154. 

Fig, 4,9, Comparisons of mean fiber fragment length, 2L, as a function of applied stress, (t, between experiments and theory for carbon fiber-epoxy matrix composites with (a) XAl fiber and (b) XAIOO fiber ( ) experiment (---------) prediction (..) debond length. After Zhou ct al. (1995a, b). 

Fig. 4.26. Comparisons between experiments and theory of (a) maximum debond stress, trj, and (b) initial frictional pull-out stress for carbon fiber-epoxy matrix composites. After Kim et al. (1992).

Comparison of Experiments and The Argon Theory. The measured compression modulus E and yield stress O were first converted into JA, and T. by using the... 

The value of Ci is obtained from the plot of o/2(X - A ) vs. 1A and extrapolating to 1A = 0. By comparison with the theory of elasticity, it has been proposed that Cl = 1/2 NRT, where N is cross-link density, R the gas constant, and T the absolute temperature (of the measurement). To assure near-equilibrium response, stress-strain measurements are carried out at low strain rate, elevated temperature, and sometimes in the swollen state. °... 

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

Lodge,A.S., Meissner,J. Comparison of network theory predictions with stress/time data in shear and elongation for a low density polyethylene melt. Rheol. Acta 12, 41-47 (1973). 

Volume-fraction effects on particle coarsening rates have been observed experimentally. For comparisons between theory and experiment, data from liquid+solid systems are far superior to those from solid+solid systems, as the latter are potentially strongly influenced by coherency stresses. Hardy and Voorhees studied Sn-rich and Pb-rich solid phases in Pb-Sn eutectic liquid over the range

presented data in support of the volume-fraction effect, as shown in Fig. 15.9 [7],... 

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