Maximum resolved shear stress

If the maximum resolved shear stress r and the plastic shear strain rate y are defined according to (it is assumed that the Xj and Xj directions are equivalent)... 

As mentioned previonsly, a nnmber of slip systems can operate simnltaneonsly, bnt there will be one system that has the orientation which affords it the largest resolved shear stress of all the slip systems in operation. This system will be the one that has the maximum geometric factor, (cos< f)cosl. )inax. since the applied stress is the same for all slip systems. So, the maximum resolved shear stress, T max. is given by ... 

As the applied stress, ct, increases, the maximum resolved shear stress increases according to Eq. (5.19), finally reaching a critical value, called the critical resolved shear stress, Xcr, at which slip along the preferred plane begins and plastic deformation commences. We refer to the applied stress at which plastic deformation commences as... 

The initial dislocation density of the sample was estimated to be less than 10 m ". The orientation of the samples was chosen in order to align the torsion axis as close as possible with the c-axis ( 1°). The maximum resolved shear stress is then applied on the basal planes. The plastic deformation is accommodated by the glide of screw dislocations on the... 

The ASBs in deformed specimens normally appear as bands with altered microstructure running along directions of maximum resolved shear stress. These bands are seen as distinct, because they are surrounded by larger regions of unaltered nucrostructure. Note that no such bands of altered nucrostructure were observed in the stir zones of the welds examined here. The lack of clear evidence for ASBs in the stir zone during FSW of Ti-6A1-4V may be explained by one or more of the following ... 

Calculate the maximum load a polymer sample in uniaxial tension can sustain before yielding when the maximum resolved shear stress (Xma ) is 10 N/m. The cross-sectional area of the sample is 10 m. ... 

The most compliant directions, and thus those in which tan is a maximum, occur when the maximum resolved shear stress is parallel with the lamellar planes. An inter-lamellar shear mechanism is suggested, the mechanical anisotropy of the j relaxation being determined by the configuration of the lamellae, and not by the molecular orientation within the lamellae, which differs between annealed, a-c and b-c sheets. Differences in the magnitudes of the relaxation peaks for different types of specimen will depend on the spread of deviations from the mean angle of the lamellae, and on the lamellae in the annealed sheet being arranged in cones around the stretch direction rather than in flat layers. 

All the specimens were loaded under compression at a constant stress rate of 20 psi per second in the temperature range of 1000-1600 °C. Loading the specimens at his orientation results in a maximum resolved shear stress on four of the six possible slip systems belonging to the 110 <110 > family. The resolved shear stress vanishes in the other two slip systems, as well as in the 001 <110 > family. The four slip systems consist of two pairs of orthogonal slip systems which intersect at 60°. Figure 5.4 shows the results of compression tests performed with the stress axis in the (111) direction. 

Figure 3.33. Part of a map showing the most 10 highly stressed slip systems beneath and to the sides of a long punch indenter oriented along [110]. The filled faces on the poly-hedra are the most highly stressed slip systems in the three regions I, II, and III separated by the solid lines. The arrows indicate the direction of slip showing convergence in III, rosette in II, and divergence in I. Dotted lines with numbers are contours of maximum resolved shear stress. After. ...

HDPE, where no p relaxation is observed [29], In these sheets, the crystal lamellae make an acute angle of about 40° with the initial draw direction [30], Applying the stress along the initial draw direction then gives the maximum resolved shear stress parallel to lamellar planes. We see from Figure 10.10 that the maximum loss is tan 3o, confirming that the a relaxation in HDPE is primarily an interlamellar shear process from a macroscopic mechanical viewpoint. 

For crystals of reasonably pure, well-annealed metals at a given temperature, slip begins when the resolved shear stress reaches a certain critical value, which is characteristic of each metal. In the case of aluminum, for example, the observed critical shear stress Uco is usually about 4x10 N/m ( 4 bars = 0.4 MPa). Theoretically, for a perfect crystal, the resolved shear stress is expected to vary periodically as the lattice planes slide over each other and to have a maximum value that is simply related to the elastic shear modulus /t. This was first pointed out in 1926 by Frenkel who, on the basis of a simple model, estimated that the critical resolved shear stress was approximately equal to h/Itt (see Kittel 1968). In the case of aluminum (which is approximately elastically isotropic), = C44 = 2.7x10 N/m, so the theoretical critical resolved shear stress is about lO wco for the slip system <100>(100). 

The octahedral shear stress criterion has some appeal for materials that deform by dislocation motion In which the slip planes are randomly oriented. Dislocation motion Is dependent on the resolved shear stress In the plane of the dislocation and In Its direction of motion ( ). The stress required to initiate this motion is called the critical resolved shear stress. The octahedral shear stress might be viewed as the "root mean square" shear stress and hence an "average" of the shear stresses on these randomly oriented planes. It seems reasonable, therefore, to assume that slip would initiate when this stress reaches a critical value at least for polycrystal1ine metals. The role of dislocations on plastic deformation in polymers (even semicrystalline ones) has not been established. Nevertheless, slip is known to occur during polymer yielding and suggests the use of either the maximum shear stress or the octahedral shear stress criterion. The predictions of these two criteria are very close and never differ by more than 15%. The maximum shear stress criterion is always the more conservative of the two. 

Figliola has made steady flow velocity and shear stress measurements downstream from a 25 mm spherical disc aortic valve (47,89). At a flow rate of 25 1/min he measured a maximum wall shear stress of 722 dynes/cm and an occluder wall shear stress (resolved on the upper side of occluder) of 440 dynes/cm. He also monitored a maximum turbulent shear stress of 545 dynes/ cm2, a 25 mm downstream from the valve. His velocity measurements also showed a large region of stagnation across the outflow face of the disc. Tillman has measured the "wall" (i.e. surface) shear stresses along the orifice ring in the major and minor outflow regions of an aortic valve under pulsatile flow... 

Suppose that yielding occurs in a polyethylene crystal when the critical resolved shear stress, 27.58 X 10 N/m is produced on the 110 type slip plane along a <111> type direction. If the tensile axis coincides with the <110> direction, what maximum axial stress must be applied to cause yielding on a 110 plane in a <111> direction ... 

Consider a single crystal being subjected to uniaxial tension or compression, as shown in Fig. 6.20. Clearly, the ease with which plastic deformation is activated will depend not only on the ease of dislocation glide for a particular slip system but also the shear stress acting on each system. This is similar to the problem discussed in Section 2.10 (Eq. (2.44)) though one should note the plane normal, the stress direction and the slip direction are not necessarily coplanar, (< +A)5 90°. In other words, slip may not occur in the direction of the maximum shear stress. The resolved shear stress acting on the slip plane in the slip direction is... 

One immediately observes, from Eq. (4.6), that the resolved shear stress is zero, either when = 90° or when X = 90°. In the first case, the tensile axis is normal to the slip plane, whereas, in the second case, the tensile axis is parallel to the slip plane. Deformation by slip is not expected when the tensile axis is parallel to the slip direction, because the shear stress is zero. The component of stress, normal to the slip plane, does not influence slip. However, in ceramics, the effect of normal stress on the critical shear stress is not necessarily negligible but, in the following this is not considered. Maximum shear stress is when — X = 45° (Fig. 4.16). 

The authors [6] suggest the term shearability , Sm, for the maximum shear strain that a homogeneous crystal can withstand. It is defined by Sm = argmax o-(s), where a(s), is the resolved shear stress and s is the engineering shear strain in a specified slip system. The relaxed shear stress, (7 in Table 4.2 is normalized by Gr. In this table, experimental and calculated values of the relaxed shear vales of Gr are given. For details on these calculations, refer to the work of Ogata et al. [6]. 

In contrast with the Takayanagi model, which considers only extensional strains, a major deformation process involves shear in the amorphous regions. Rigid lamellae move relative to each other by a shear process in a deformable matrix. The process is activated by the resolved shear stress a sin y cosy on the lamellar surfaces, where y is the angle between the applied tensile stress o and the lamellar plane normals, which reaches a maximum value for y = 45° (see Chapter 11 for discussion of resolved shear stress in plastic deformation processes). 

According to equation (3.10), when the axis of rotation of the slip system is parallel to the axis drawn at 90° to the sliding direction that lies in the plane of sliding, i.e., XY, then sin to = 0. This produces a decreased penetration and therefore an increased hardness. At < = 90° to XY, the slip system rotation axis will be normal to the surface axis XY, the effective resolved shear stress is maximized, leading to a maximum penetration, and this direction will be the softest. 

In an amorphous glassy polymer, work hardening is considered to correspond to stretching of the entanglement network invoked to account for the rubbery plateau above Tg (see Section 14.3.3). This explains the recoverability of the deformation above Tg (in the absence of an applied force, the network retracts to its equilibrium conformation) and it is borne out by the observed correlation between the value of X in the neck and the maximum extensibility of the network, X ax In semicrystalline polymers, the evolution of the crystalline texture may also contribute to work hardening, because the resolved shear stress on activated slip systems tends to decrease as deformation proceeds at constant stress, as will be discussed further (see Section 14.4.3). 

For a single slip the slip direction in the crystal rotates always towards the direction of maximum extension in uniaxial tension it rotates towards the tensile axis, while in uniaxial compression it rotates away from the compression axis. The angle through which the crystal rotates is a simple function of the applied strain [107]. It must be mentioned that the slip takes place when the resolved shear stress on the slip plane reaches a critical value known as critical resolved shear stress. Currently the critical resolved shear stresses for slips are well known for only few polymers. 

The Vickers microhardness of the BP wafers varies from 3000 to 4000 kg/mm depending on orientation (47,51). The periodicity of the hardness curve of the (100) plane shows fourfold symmetry (Pig. 7). Minimum and maximum hardnesses in the (100) plane correspond to the (110) and (001) directions, respectively. By analogy to the anisotropy of hardness in the (001) plane of cubic boron nitride (52) compared with resolved shear stress curves, the primary slip systems of BP are 111 (110). The elastic constants Cu, Cu, and C44 were determined by Brillouin scattering (18) as given in Table 1. 

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