Tensile strength equations

Both tensile strength equations are iRustiated in Figure 12 using data typical of graphite fibers in epoxy resin. 

This relation is similar to the relation between the tensile strength-particle size relation for the filter cakes given by Schubert (4). Kaolin has the second highest shear strength values at two pressures opposite to its highest particle size, 6 microns. This is due to the sheet structure of kaolin and this structure makes kaolin stronger than the other cakes and this effect appears as f (shape factor) as in the case of tensile strength equations defined by Schubert (3,4 ) for the calcite cakes. 

The most direct test of the tensile strength hypothesis would be to compare the value of Tq calculated from the closure point of the isotherm by Equation (3.61) with the tensile strength of the bulk liquid determined directly. Unfortunately, experimental measurement of the tensile strength is extremely difficult because of the part played by adventitious factors such as the presence of solid particles and dissolved gases, so that the values in the literature vary widely (between 9 and 270 bar for water at 298 K, for example). 

It is, however, possible to calculate the tensile strength of a liquid by extrapolation of an equation of state for the fluid into the metastable region of negative pressure. Burgess and Everett in their comprehensive test of the tensile strength hypothesis, plot the theoretical curves of T /T against zjp, calculated from the equations of state of van der Waals, Guggenheim, and Berthelot (Fig. 3.24) (7], and are the critical temperature and critical... 

Fig. 3.24 Test of the tensile strength hysteresis of hysteresis (Everett and Burgess ). TjT, is plotted against — Tq/Po where is the critical temperature and p.. the critical pressure, of the bulk adsorptive Tq is the tensile strength calculated from the lower closure point of the hysteresis loop. C), benzene O. xenon , 2-2 dimethyl benzene . nitrogen , 2,2,4-trimethylpentane , carbon dioxide 4 n-hexane. The lowest line was calculated from the van der Waals equation, the middle line from the van der Waals equation as modified by Guggenheim, and the upper line from the Berthelot equation. (Courtesy Everett.)...

Theoretical Strength of Agglomerates. Based on statistical-geometrical considerations, Rumpf developed the following equation for the mean tensile strength of an agglomerate in which bonds ate localized at the points of particle contact (9) ... 

When the void space in an agglomerate is completely filled with a Hquid (Fig. Ic), the capillary state of wetting is reached, and the tensile strength of the wet particle matrix arises from the pressure deficiency in the Hquid network owing to the concave Hquid interfaces at the agglomerate surface. This pressure deficiency can be calculated from the Laplace equation for chcular capillaries to yield, for Hquids which completely wet the particles ... 

It is evident, by comparing equations 2 and 3, that tensile strength in the pendular state is about one-tbhd that in the capillary state. Intermediate Hquid contents in the funicular state (Fig. lb) yield intermediate values that can be approximated as foUows ... 

By definition, a brittle material does not fail in shear failure oeeurs when the largest prineipal stress reaehes the ultimate tensile strength, Su. Where the ultimate eompressive strength, Su, and Su of brittle material are approximately the same, the Maximum Normal Stress Theory applies (Edwards and MeKee, 1991 Norton, 1996). The probabilistie failure eriterion is essentially the same as equation 4.55. 

From equations 4.12 and 4.13, the mean and standard deviation for the ultimate tensile strength, Su, for steel ean be derived ... 

The latter equation contains constants with well-known values and can therefore be used to predict the fracture stress of most polymers. For example, the bond dissociation energy Do, is about 80 kcal/mol for a C-C bond. For polystyrene, the modulus E 2 GPa, A. 4, p = 1.2 g/cm, = 18,000, and we obtain the fracture stress, o A1 MPa, which compares well with reported values. Polycarbonate, with similar modulus but a lower M. = 2,400 is expected to have a fracture stress of about 100 MPa. In general, letting E 1 GPa, p = 1.0 g/cm, and Do — 335 kJ/mol, the tensile strength is well approximated by... 

Only a small amount of work has been done up to now concerning the prediction of bond strengths and other properties based on the results of the analysis of the resin. Ferg et al. [59] worked out correlation equations evaluating the chemical structures in various UF-resins with different F/U molar ratios and different types of preparation on the one hand and the achievable internal bond as well as the subsequent formaldehyde emission on the other hand. These equations are valid only for well defined series of resins. The basic aim of such experiments is the prediction of the properties of the wood-based panels based on the composition and the properties of the resins used. For this purpose various structural components are determined by means of - C NMR and their ratios related to board results. Various papers in the chemical literature describe examples of such correlations, in particular for UF, MF, MUF and PF resins [59-62]. For example one type of equation correlating the dry internal bond (IB) strength (tensile strength perpendicular to the plane of the panel) of a particleboard bonded with PF adhesive resins is as follows [17]... 

Designers of most structures specify material stresses and strains well within the pro-portional/elastic limit. Where required (with no or limited experience on a particular type product materialwise and/or process-wise) this practice builds in a margin of safety to accommodate the effects of improper material processing conditions and/or unforeseen loads and environmental factors. This practice also allows the designer to use design equations based on the assumptions of small deformation and purely elastic material behavior. Other properties derived from stress-strain data that are used include modulus of elasticity and tensile strength. 

A shaft subject to torque is generally considered to have failed when the strength of the material in shear is exceeded. For a torsional load the shear strength used in design should be the published value or one half the tensile strength, whichever is less. The maximum shear stress on a shaft in torsion is given by the following equation ... 

Because many plastics are relatively flexible, analysis should consider how much deflection might result from the loadings and elevated temperatures the products might see in service. The equations for predicting such deflections should use the modulus of the material its tensile strength is not pertinent. Usually, the most effective way to reduce de-... 

When the butyl rubber was compounded with up to 30 percent of polyisobutylene, which, lacking the unsaturated isoprene units, did not enter into the cross-linking reaction, the tensile strengths were, of course, considerably reduced. They were found nevertheless to be accurately represented by the same equation, (53), provided merely that Sa is taken as the fraction of the composite specimen consisting of network chains subject to orientation. Thus, in this case... 

From a test of the tablet s flexure, tensile strength (oy) is calculated from the following equation ... 

Rumpf (R4) has derived an explicit relationship for the tensile strength as a function of porosity, coordination number, particle size, and bonding forces between the individual particles. The model is based on the following assumptions (1) particles are monosize spheres (2) fracture occurs through the particle-particle bonds only and their number in the cross section under stress is high (3) bonds are statistically distributed across the cross section and over all directions in space (4) particles are statistically distributed in the ensemble and hence in the cross section and (5) bond strength between the individual particles is normally distributed and a mean value can be used to represent each one. Rumpf s basic equation for the tensile strength is... 

Table 7 Estimates of the ultimate tensile strength for perfectly parallel orientation of the chains in the polymer fibre calculated with the equation aL=2.3 f5 V(gec) and / =0.1...

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