Shear stress resolved

When a tensile stress is applied to a crystal, it may be resolved into two perpendicular shear components, one making angle p with the normal to the slip plane and the other making angle A with the slip direction. The resolved shear stress is then 

When the resolved shear stress from the applied tensile stress exceeds the crss, the material yields by shearing along a slip direction in a slip plane. 

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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)... 

Figure 4,2, Medal struck in Austria to commemorate the 50th anniversary of the discovery of the critical shear stress law by Erich Schmid. The image represents a stereographic triangle with "isobars showing crystal orientations of constant resolved shear stress (courtesy H.P. Stiiwe).

Mark, Polanyi and Schmid, of the constant resolved shear-stress law, which specifies that a crystal begins to deform plastically when the shear stress on the most favoured potential slip plane reaches a critical value. 

DEFORMATION MODE AND CRITICAL RESOLVED SHEAR STRESS... 

For the [l52] orientation, where the resolved shear stress for <110] ordinary slip is 1.3 times that for <101] superlattice slip, <101] superlattice slip is observed up to the peak temperature. Above the peak, <110] ordinary slip occurs mostly on 111. Near the peak temperature, <101] dislocations are mostly in their screw orientations with many... 

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. 

The continuous chain model includes a description of the yielding phenomenon that occurs in the tensile curve of polymer fibres between a strain of 0.005 and 0.025 [ 1 ]. Up to the yield point the fibre extension is practically elastic. For larger strains, the extension is composed of an elastic, viscoelastic and plastic contribution. The yield of the tensile curve is explained by a simple yield mechanism based on Schmid s law for shear deformation of the domains. This law states that, for an anisotropic material, plastic deformation starts at a critical value of the resolved shear stress, ry =/g, along a slip plane. It has been... 

In view of the development of the continuous chain model for the tensile deformation of polymer fibres, we consider the assumptions on which the Coleman model is based as too simple. For example, we have shown that the resolved shear stress governs the tensile deformation of the fibre, and that the initial orientation distribution of the chains is the most important structural characteristic determining the tensile extension below the glass transition temperature. These elements have to be incorporated in a new model. 

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... 

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.

Equations (5.20) and (5.21) are valid for an applied tensile or compressive stress and can be used in the case of twinning as well. However, Ter for twinning is usually greater than Xcr for shear. Some values of the critical resolved shear stress for slip in some common metals, and their temperature dependence, are shown in Figure 5.13. Note that the critical resolved shear stress for HCP and FCC (close-packed) structures rises only modestly at low temperatures, whereas that for the BCC and rock salt structures increases significantly as temperature decreases. 

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

Figure 5.13 Temperature variation of critical resolved shear stress for single-crystal metals of different crystal structures. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission John Wiley Sons, Inc.

Experimentally, it has been observed for single crystals of a number of metals that the critical resolved shear stress is a function of the dislocation density, Pd -... 

Figure 11.7 Velocity vs. resolved shear stress for dislocation motion in LiF single...

The connection between processing conditions and crystalline perfection is incomplete, because the link is missing between microscopic variations in the structure of the crystal and macroscopic processing variables. For example, studies that attempt to link the temperature field with dislocation generation in the crystal assume that defects are created when the stresses due to linear thermoelastic expansion exceed the critically resolved shear stress for a perfect crystal. The status of these analyses and the unanswered questions that must be resolved for the precise coupling of processing and crystal properties are described in a later subsection on the connection between transport processes and defect formation in the crystal.. 

Figure 2. Thermal conductivity of several common semiconductor materials plotted against the best estimates for the critical resolved shear stress (CRSS) for the crystal. As explained in the text, materials with low thermal conductivity and low CRSS are hardest to grow.

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