Shear critical resolved

DEFORMATION MODE AND CRITICAL RESOLVED SHEAR STRESS... 

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. 

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... 

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. 

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 -... 

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.

In nearly all metal-forming operations, slip is the dominant method of deformation, although twinning can be significant in some materials. Slip occurs when the shear stress is high enough to cause layers of atoms to move relative to one another. The critical resolved shear stress is lowered when the crystalline lattice is not perfect but contains linear defects called dislocations. Slip-induced plasticity was covered in Chapter 9 of the companion to this text (Lalena and Cleary, 2005) and is reviewed here only briefly. The interested reader is advised to consult Lalena and Cleary (2005), Honeycombe (1984), or Dieter (1976). 

Figure 10.6. Application of a tensile force to a cylindrical single crystal causes a shear stress on some crystal planes. When the shear stress is equal to the critical-resolved shear stress (the yield stress), glide proceeds along the slip direction of the planes.

According to Schmid s law (Schmid and Boas, 1935), plastic flow in a pure and perfect single crystal occurs when the shear stress, acting along (parallel to) the shp direction on the slip plane, reaches some critical value known as the critical resolved shear stress, Tc. From Figure 10.6, it can be seen that the component of the force acting in the slip-direction is Fcos A and it acts over the plane of area A/cos (where A is the cross-sectional area). Thus the resolved shear stress is as = (F/A)cos cos A. Schmid s law states that slip occurs at some critical value of cTj, denoted as... 

Slip relies on chemical bond breaking and bond reformation as two planes of atoms pull apart. It is observed that the critical resolved shear stress required to cause plastic deformation in real materials is much lower (by several orders of magnitude) than the shear stress required in deforming perfect defect-free crystals, the so-called ideal shear stress. The latter is equivalent to the stress required for the simultaneous ghding motion (bond breaking and reformation) of aU the atoms in one plane, over another plane. 

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). 

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. 

TABLE 1.11. Effect of Orientation and Temperature on the Critical Resolved Shear Stresses for Slip of Tungsten Single Crystals in Tension [1.50]... 

Temperature (K) Orientation of tensile axis Initial yield stress (kgmm- ) Critical resolved shear stress on observed slip system (kg mm ) Observed slip systems Systems with the highest resolved shear stress... 

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. 

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