Resolved shear stress calculation

By developing the constraint term, with particular respect to the displacement of material in the slip direction, Brookes, O Neill, and Redfem improved the correlation between hardness values and the resolved shear stress calculation. Moreover, they extended the applicability of this approach to the complete range of crystalline solids. They proposed that... 

Excellent agreement between experiment and onr calculations is obtained when considering the low temperature deformation in the hard orientation. Not only are the Peierls stresses almost exactly as large as the experimental critical resolved shear stresses at low temperatures, but the limiting role of the screw character can also be explained. Furthermore the transition from (111) to (110) slip at higher temperatures can be understood when combining the present results with a simple line tension model. 

Figure 5.12 Schematic illustration of the relationship between tensile axis, slip plane, and slip direction used in calculating the resolved shear stress for a single crystal. Reprinted, by permission, from W. Callister, Materials Science and Engineering An Introduction, 5th ed., p. 160. Copyright 2000 by John Wiley Sons, Inc.

Combine your information to calculate the resolved shear stress, r, using Eq. (5.18). 

An explanation of the tendency for crystalline solids to deform plastically at stresses that are so much smaller than the calculated critical resolved shear stress was first given in 1934 independently by Taylor, Oro-wan, and Polanyi. They introduced the concept of the dislocation into physics and showed that the motion of dislocations is responsible for the deformation of metals and other crystalline solids. At low temperatures, where atomic diffusion is low, dislocations move almost exclusively by slip. 

As will become evident shortly, calculation of the resolved shear stress will serve as a cornerstone of our analysis of plasticity in single crystals. 

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 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]. 

Only in special cases will the external shear stress be exactly parallel to the slip system. Usually, it is thus necessary to calculate the resolved shear stress i. e., the stress component acting as shear stress on the considered slip system in the slip direction. If we restrict ourselves to the case of uniaxial loading as in a tensile test, the calculation of this component is not too difficult. We... 

The resolved shear stress can also be calculated for arbitrary stress states a. To calculate, we first calculate the traction vector t on the slip plane with normal vector n... 

The resolved shear stress in the shp direction m can be calculated from this by projecting t onto m ... 

Since crystals have several different slip systems, as explained in section 6.2.4, the resolved shear stress on all possible slip planes has to be calculated to determine whether the material will yield. The slip system that is oriented most... 

In the previous section, we saw how the resolved shear stress required to activate a slip system in a single crystal can be calculated from the external stress. If Tcrit is the critical stress needed to activate slip, the external stress is connected to Tcrit by the Schmid factor cos A cos 0 for a single crystal. 

Most engineering alloys are poly crystalline. To calculate the yield strength from the critical resolved shear stress in an isotropic, polycrystalline material, we have to take into account that the grains are oriented in an arbitrary manner. We thus have to take the average of all possible crystal orientations. 

Figure 3.23. Reciprocal mean effective resolved shear stress curves for Knoop indentation calculated for (111) planes of a fluorite cubic crystal for the (IIOKIIO) slip system.

Figure 3.26. Reciprocal mean effective resolved shear stress curves for Knoop indentation calculated for (a) 0001 (1120), (b) 1100 (U20>, (c) 3301 (1120), (d) U01 (1120), (e) 1210 <10l0) slip systems on (0001) planes of hexagonal crystals.

By using this method, the stress due to the distribution of hydrogen around the dislocation can be calculated. Figure 5-41 shows the shear component induced in the slip plane. Hydrogen induces a resolved shear stress opposite to the dislocation s one. In other words, segregating hydrogen decreases the resolved shear stress of the dislocation. A parallel can be determined between the effect of emitted dislocations on a crack... 

In an isotropic polycrystalline polymer whose microstructure consists of stacked lamellae arranged in the form of spherolites, the slip systems activated depend on the local orientation of the lamellae with respect to the applied stress and, as deformation proceeds, these orientations are modified. To calculate the evolution of the crystalline texture, one can consider the polymer to behave as a crystalline aggregate. Although the entropic contribution of chain orientation in the amorphous regions may also need to be considered, the major contribution to work hardening in tension is rotation of the slip planes toward the tensile axis, so that the resolved shear stress in the slip direction diminishes. This results in a fiber texture in the limit of large deformations, such that the crystallites are oriented with their c axis (the chain axis) parallel to the stretch direction. Despite the relative success of such models, they do not explicitly address the micro-mechanisms involved in the transformation of the spherulitic texture into a fiber texture. One possibility is that the... 

Fig. 18. Orientation dependence of the critical resolved shear stress (CRSS) in Ta and Mo at ambient pressure, as calculated with the present MGPT potentials and compared with experimental estimates based on the observed yield stress [70-72].

Fig. 3.8 Measured dislocation cell size d vs. calculated normalized resolved shear stress xjCb along the [110] direction of a 6-inch VCz GaAs wafer (empty squares) [77[ and estimated from von Mises stress modeling in VCz GaAs crystals (gray squares) [75] in...

Figure 7.7 Geometric relationships between the tensile axis, slip plane, and slip direction used in calculating the resolved shear stress for a single crystal.

Consider a cylindrical single crystal with cross-sectional area A. It yields at a tensile load P. The angles between the tensile axis and the normal to the slip plane and the slip direction are (p and X, respectively, as illustrated in Figure 3.8. The area of the slip plane inclined at the angle (p is y4/cos< ), and the stress component in the slip direction is PcosX. The critical resolved shear stress, can therefore be calculated by Equation 3.12. The single crystal starts to yield when the shear stress reaches the critical resolved shear stress (Figure 3.8) ... 

But we want the tensile yield strength, A tensile stress a creates a shear stress in the material that has a maximum value of t = a/2. (We show this in Chapter 11 where we resolve the tensile stress onto planes within the material.) To calculate cr from t,, we combine the Taylor factor with the resolution factor to give... 

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