Shear yield

When a solid undergoes shear yielding, the local packing of its constituent units—atoms, molecules, or ions—changes to a new configuration that is stable in the absence of stresses. In glassy and semicrystalline polymers the plastic deformation takes place by means of local shear strains, without any appreciable changes in volume or density. 

Semicrystalline polymers must be considered two-phase mixtures of amorphous regions between lamellar crystals. It has been demonstrated that the yield stress increases with increasing crystallinity when the deformation process occurs at temperatures above the glass transition temperature of the amorphous phase and below, but close to, the melting 

Although the uniaxial tension test is the one most widely used, it has two drawbacks when it is used to provide information on the yielding of polymers. First, the tensile stress applied can lead to brittle fracture before yield takes place, and second, yield occurs in an inhomogeneous way due to the formation of a neck accompanying the tensile test. In any case, given that the section of sample decreases as the stress increases, cj cy . 

If the strain takes place at constant volume, then 

Bearing in mind the definitions of true stress, and nominal stress, a , given in the previous section and Eqs. (14.3) and (14.4), the following relation is easily deduced  

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If the sum of the mechanical allowances, c, is neglected, then it may be shown from equation 15 that the pressure given by equation 33 is half the coUapse pressure of a cylinder made of an elastic ideal plastic material which yields in accordance with the shear stress energy criterion at a constant value of shear yield stress = y -... 

A proposed mechanism for toughening of mbber-modifted epoxies based on the microstmcture and fracture characteristics (310—312) involves mbber cavitation and matrix shear-yielding. A quantitative expression describes the fracture toughness values over a wide range of temperatures and rates. 

Example Approximate calculation of the hardness of solids. This concept of shear yielding - where we ignore the details of the grains in our polycrystal and treat the material as a continuum - is useful in many respects. For example, we can use it to calculate the loads that would make our material yield for all sorts of quite complicated geometries. 

Let US now look at how this contact geometry influences friction. If you attempt to slide one of the surfaces over the other, a shear stress fj/a appears at the asperities. The shear stress is greatest where the cross-sectional area of asperities is least, that is, at or very near the contact plane. Now, the intense plastic deformation in the regions of contact presses the asperity tips together so well that there is atom-to-atom contact across the junction. The junction, therefore, can withstand a shear stress as large as k approximately, where k is the shear-yield strength of the material (Chapter 11). 

Normal - Toleranees, ultimate tensile strength, uniaxial yield strength and shear yield strength of some metallie alloys... 

The largest design dependent strength variable is material strength, either ultimate tensile strength (Su), uniaxial yield strength (Sy), shear yield strength (Ty) or some... 

For ductile materials subjected to pure shear, the reliability is the probabilistic requirement to avoid shear yielding ... 

Sy = yield strength Su = ultimate tensile strength L = loading stress Ty = shear yield strength. 

Ty = shear yield strength J = polar seeond moment of area r = radius of shaft. 

The shear yield strength for duetile metals is a linear funetion of the uniaxial yield strength. Therefore, for pure torsion from equation 4.56 ... 

The eoeffieient of variation of the yield strength, and lienee the shear yield strength for steels, is typieally = 0.08. This means that the eoeffieient of variation of the... 

The torque that can be transmitted by a shaft without yielding was given in equation 4.88. Rearranging for the shear yield strength and the variables in this example gives ... 

Solving equation 4.94 using Monte Carlo simulation for the variables involved, the shear yield strength required for the pump shaft material is found to have a Normal distribution with parameters ... 

Ultimate shear strength Shear yield strength... 

Because strain measurements are difficult if not impossible to measure, few values of yield strength can be determined by testing. It is interesting to note that tests of bolts and rivets have shown that their strength in double shear can at times be as much as 20% below that for single shear. The values for the shear yield point (kPa or psi) are generally not available however, the values that are listed are usually obtained by the torsional testing of round test specimens. 

Indeed, it has been observed that the onset of yielding of isotropic polymers is approximately constant, 0.02< [<0.025, which implies that 0.04shear yield strain, the plastic shear deformation of the domain satisfies a plastic shear law. For temperatures below the glass transition temperature, the continuous chain model enables the calculation of the tensile curve of a polymer fibre up to about 10% strain [6]. Figure 7 shows the observed stress-strain curves of PpPTA fibres with different moduli compared to the calculated curves. 

As for the derivation of Eqs. 122,123 and 124 only the transitions 1—>2 have been counted, these equations do not describe recovery processes, where the transitions 2 —>1 are important as well. These approximations have been made for convenience s sake, but neither imply a limitation for the model, nor are they essential to the results of the calculations. Equation 124 is the well-known formula for the relaxation time of an Eyring process. In Fig. 65 the relaxation time for this plastic shear transition has been plotted versus the stress for two temperature values. It can be observed from this figure that in the limit of low temperatures, the relaxation time changes very abruptly at the shear yield stress Ty = U0/Q.. Below this stress the relaxation time is very long, which corresponds with an approximation of elastic behaviour. 

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... 

The tensile curve of a polymer fibre is characterised by the yield strain and by the strain at fracture. Both correspond with particular values of the domain shear strain, viz. the shear yield strain j =fl2 with 0.04rotation angle of -0y=fl2 and the critical shear strain 0-0b=/iwith /f=0.1. For a more fundamental understanding of the tensile deformation of polymer fibres it will be highly interesting to learn more about the molecular phenomena associated with these shear strain values. 

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