Load-strain tangent

Neutral Loading. The strain lies on the elastic limit surface = 0, but the tangent to the strain history points along the tangent to the elastic limit surface = 0. It is assumed that A = 0. The material is said to be undergoing neutral loading and the elastic limit surface is stationary. 

This modulus value is often arbitrarily chosen, although several methods have been suggested for arriving at a suitable value. One is to plot a secant modulus based on 1% strain or that is 0.85% of the initial tangent modulus (Chapter 2, SHORT-TERM LOAD BEHAVIOR). However, for many plastics, particularly the crystalline TPs, this method is too restrictive, so in most practical situations the limiting strain is decided in consultation... 

The flexural modulus is the ratio, within the elastic limit, of stress to corresponding strain. It is calculated by drawing a tangent to the steepest initial straight-line portion of the load-deflection curve and using an appropriate formula. 

Equation (14.7) corresponds to the slope of the tangent to the curve cr, vs. e drawn from the point e = -1 or X, = 0. Figure 14.7 shows the true stress versus nominal strain curves for polymer samples A and B. Curves B] and B2 are compatible with curve B of Figure 14.6. The so-called Considere construction, Eq. (14.7), is satisfied with the tangent to the curves drawn from E = — 1. The tangential point corresponds to the maximum observed in the curve vs. and therefore with the maximum load that the specimen can support. In practice, the Considere construction is used as a criterion to decide when a polymer will form an unstable neck or form a neck accompanied by cold drawing. 

A relation has been presented by Martin et al. (1975) that relates the increase in residual pore water (u) for each load cycle to fhe rebound tangent modulus (E ), porosity (n), bulk modulus of an air/water mixture and the volumetric strain per cycle (As d), which is given by Equation 9.17. 

Fig. L(a) Three common working definitions of the yield point for metals. (1) Load maximum (2) tangent method, (3) firoofistress" or "strain-offset method. (The proof-strain is commonly taken to be 0 /%, but is quite arbitrary.) (b) Load elongation curves for polymers. (I) Brittle, (2) strain softening, (3) cold-drawing, (4) strain-hardening, (5) rubbery. Typical definitions of the yield point are marked by arrows on curves (2), (3) and (4). Any one polymer can show behaviour raiding from (1) to (5) depending on test conditions, e.g. temperature, strain-rate, tension...

A typical example of recent studies of time-temperature-modulus relationships may be found in papers by Moehlenpah et al. (1970, 1971), who examined crosslinked epoxy resins filled with glass beads, fibers, or air bubbles. The initial tangent modulus in compression was seen to increase with a decrease in strain rate flexural and tensile moduli were reported to behave in a similar fashion. The WLF shift factor was essentially independent of the type of filler used and of the mode of loading. Kerner s equation was found to hold for the particulate composites in the glassy range. 

Articular joints are exposed to compressive forces that are applied very quickly, as well as to very large shear forces. Stammen et al. [53] recognized PVA-C as a viable option for total joint replacement but only if the load-bearing properties could be matched with those of natural tissue. Studies of the compressive tangent modulus and shear tangent modulus were undertaken for the PVA-C product, and a limited strain-rate dependence under unconfined compression was displayed. 

These dynamic moduli correspond to the initial tangent moduli of the stress-strain curve for an instantaneously applied load and are usually higher than those obtained in static tests. The frequency and nature of discontinuities within a rock mass affect its deformability. In other words, a highly discontinuous rock mass exhibits a iower compressional wave velocity than a massive rock mass of the same type. The influence of discontinuities on the deformability of a rock mass can be estimated from a comparison of its in situ compressional velocity, /pf, and the laboratory sonic velocity, /p, determined from an intact specimen taken from the rock mass. The velocity ratio, /pf/t/pi, reflects the deformability and so can be used as a quality index. A comparison of the velocity ratio with other rock quality indices is given in Table 2.7. 

Values and units Flexural modulus (MPA) Flexural strength, at rupture (MPa) Flexural strength, at maximum strain (MPa) At conventional deflection which is 1.5 X height therefore 4 mm specimens would have a maximum strain at 3.5%. Tangent modulus (MPa) Secant modulus (MPa) Flexural strength, (at rupture ) (MPa) Flexural yield strength (MPa) Maximum allowable strain in the outer fibers is 0.05 mm/mm. The point where the load does not increase with increased deflection, provided it occurs before the maximum strain rate. 

In some cases there is no observed load drop and another definition of yield stress is required. One approach is to determine the stress where the two tangents to the initial and final parts of the load elongation curve intersect (Figure 11.8(b)). An alternative is to attempt to define an initial linear slope on the stress-strain curve and then to draw a line parallel to this that is offset by a specified straia, say 2 per cent. The interception of this line with the stress strain curve then... 

Many unreinforced and reinforced plastics have a definite tensile modulus of elasticity where deformation is directly proportional to then-loads below the proportional limits. Since stress is proportional to load and strain to deformation, stress is proportional to strain. Fig. 2.4 shows this relationship. The top curve is where the S-S straight line identifies a modulus and a secant modulus based at a specific strain rate at point C that could be the usual 1% strain. Bottom curve secant moduli of different plastics are based on a 85% of the initial tangent modulus. 

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