Plastic behaviour, ideal

The transition from ideal elastic to plastic behaviour is described by the change in relaxation time as shown by the stress relaxation in Fig. 66. The immediate or plastic decrease of the stress after an initial stress cr0 is described by a relaxation time equal to zero, whereas a pure elastic response corresponds with an infinite relaxation time. The relaxation time becomes suddenly very short as the shear stress increases to a value equal to ry. Thus, in an experiment at a constant stress rate, all transitions occur almost immediately at the shear yield stress. This critical behaviour closely resembles the ideal plastic behaviour. This can be expected for a polymer well below the glass transition temperature where the mobility of the chains is low. At a high temperature the transition is a... 

In the present paper the behaviour of cracks in tubes made of a fiber-reinforced composite is studied. It is assumed that the fibers are oriented circumferentially and that the material behaviour follows the concept of an ideal plastic body. 

In terms of rheology ceramic bodies hold a special position between ideal elastic and ideal plastic bodies, as they exhibit Bingham behaviour. Plotted on a shear stress/shearing speed graph, ceramic plastic bodies start to deform only after having reached a certain shear stress tq, the so-called yield point. 

In order that the bipartite beam should attain the higher bending strength as compared to a glulam beam of the same size and wood quality, the theoretical models indicate that the adhesive layers must exhibit a very special load-bearing behaviour. The adhesive manufacturer was able to produce some adhesive layers (like 027-2 in Fig. 7) which attained the requisite load-bearing behaviour for the ideal elastic adhesive layer. However, he was not quite able to fully attain the behaviour required for the ideal plastic adhesive layer. We decided to perform further tests with two selected adhesive layers (009-05 and 13-1 in Fig. 7), which came close to the desired performance needed for the ideal-plastic adhesive layer. There was a need to estimate the performance of bipartite beams with these adhesive connections. A programme based on the Excel solver function was developed to calculate the beam behaviour for these and other adhesive layers as follows. 

Molten polymers are viscoelastic materials, and so study of their behaviour can be complex. Polymers are also non-ideal in behaviour, i.e. they do not follow the Newtonian liquid relationship of simple liquids like water, where shear-stress is proportional to shear strain rate. Unlike Newtonian liquids, polymers show viscosity changes with shear rate, mainly in a pseudoplastic manner. As shear rate increases there is a reduction in melt viscosity. This is true of both heat-softened plastics and rubbers. Other time-dependent effects will also arise with polymer compounds to complicate the rheological process behaviour. These may be viscosity reductions due to molecular-mass breakdown or physical effects due to thixotropic behaviour, or viscosity increases due to crosslinking/branching reactions or degradation. Generally these effects will be studied in rotational-type rheometers and the extrusion-type capillary rheometer. 

Although most types of fibre do not exhibit fully plastic behaviour, the stress/strain characteristic of carbon and organic fibres is not linear to failure. The modulus of carbon fibres may increase with increasing extension and hence ideally one should distinguish between tangent and secant modulus (section 1.4). 

The next step is to cope with the unrealistic high pressure values above material strength. Due to these high pressure values, the material will deform plastically. For this reason a linear-elastic, ideal-plastic material behaviour was introduced. This is done by the implementation of a new variable into the elasticity equation (4). The value of ftpiast has to be calculated iteratively ... 

Fig. 6. Film thickness and pressure distribution with linear-elastic, ideal-plastic material behaviour...

Since the pressure drops down at die fictitious solid contact spots, there will be no plastic deformation at these spots themselves. High pressure values can only be found directly upstream of the bump. Hence, the implementation of the linear-elastic, ideal-plastic material behaviour without Paum leads to an increasing plastic deformation upstream of the fictitious solid contact spot during the iterations and not at the contact itself. So no useful solution can be obtained. 

The present review shows how the microhardness technique can be used to elucidate the dependence of a variety of local deformational processes upon polymer texture and morphology. Microhardness is a rather elusive quantity, that is really a combination of other mechanical properties. It is most suitably defined in terms of the pyramid indentation test. Hardness is primarily taken as a measure of the irreversible deformation mechanisms which characterize a polymeric material, though it also involves elastic and time dependent effects which depend on microstructural details. In isotropic lamellar polymers a hardness depression from ideal values, due to the finite crystal thickness, occurs. The interlamellar non-crystalline layer introduces an additional weak component which contributes further to a lowering of the hardness value. Annealing effects and chemical etching are shown to produce, on the contrary, a significant hardening of the material. The prevalent mechanisms for plastic deformation are proposed. Anisotropy behaviour for several oriented materials is critically discussed. 

A very simple explanation of the effect of notching has been given by Orowan [95], For a deep, symmetrical tensile notch, the distribution of stress is identical to that for a flat frictionless punch indenting a plate under conditions of plane strain [102] (Figure 12.31). The compressive stress on the punch required to produce plastic deformation can be shown to be (2 + 7t)K, where K is the shear yield stress. For the Tresca yield criterion the value is l.Sloy and for the von Mises yield criterion the value is 2.82oy, where 0 is the tensile yield stress. Hence for an ideally deep and sharp notch in an infinite solid the plastic constraint raises the yield stress to a value of approximately 2>Oy which leads to the following classification for brittle-ductile behaviour first proposed by Orowan [95] ... 

Examples of load-deflection curves in Figure 10.30 show the possible deficiencies of this type. Curve a describes a small strain hardening while curve a represents a rapid decrease of load after cracking and both are characterized by the same toughness index I5. Similarly, for curves b and b index I o is the same and only index I5 shows small difference, while the material s behaviour is completely different. Also, for curves c and c only I5 and I o are different while the values of I20 are the same. These examples prove that the determination of the toughness index is only a useful method of comparison for the effective material s behaviour with that of an idealized linear elastic-plastic material. However, it cannot be considered as a perfect material characteristic. 

Figure 4.42 Behaviour of ideally elastic and ideally elastic-plastic materials in flexure, showing the stress and strain distribution in flexure at three different stages (I, II, III) and the resulting load-deflection curves.

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