The critical resolved shear stress

It was stated in section 6.2.3 that a dislocation will start to move if a sufficiently large shear stress acts on the slip system. This stress value is called critical resolved shear stress Tcrit- It is not equal to the yield strength rp of an isotropic material under shear loading because in the latter case different slip systems have to be activated that are usually not parallel to the shear stress. For a single crystal, the yield criterion (cf. section 3.3.1) is 

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 

If we now relate both forces to the area they are acting upon and use = 

Putting this into equation (6.6) results in the resolved shear stress (or Schmid stress) 

This equation determines the shear stress in a slip system resulting from the external stress and the orientation of the system. The factor cos A cos 0 is known as the Schmid factor. If the resolved shear stress reaches the critical value Tcrit, the material yields. The yield criterion for uniaxial loading is thus 

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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. 

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). 

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

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. 

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 ... 

Figure 6.27 Effect of various divalent impurities on the critical resolved shear stress in various alkali halides. (After Cook and Pharr, 1994.)...

FIGURE 17.6 Geometry used to determine the critical resolved shear stress. 

The critical resolved shear stress, Terss, is the minimum shear stress required to initiate slip for a particular slip system defined when a = 0/. 

High-temperature reinforcement by in situ precipitation of TiC and TiB2 from supersaturated solid solutions has already been used with interesting results. In the TiC-TiB2 system, either TiC or TiB2 can be the host crystal for the corresponding minority phase or the precipitate [278-280]. The addition of a small fraction of boron to TiC can increase the critical resolved shear stress at 1600°C by a factor of six if TiB2 precipitates are formed at the (111) slip plane of TiC. 

Fig. 9.15 The normal-stress dependence of the critical resolved shear stress on the (100) plane in the [001] direction of HDPE (from Bartczak et al. (1992a) courtesy of the ACS).

Thus, Q/b should be proportional, if not equal, to the critical resolved shear stress (CRSS) of an 1KB dislocation loop. In the present authors studies to date, this has been shown repeatedly to be the case [137-139]. 

Most ceramics are brittle at low and medium temperatures, and can be deformed plastically above the brittle-to-ductile transition temperature. The critical resolved shear stress (CRSS) then decreases rapidly with increasing temperature. In many cases there is a linear relationship between 1( (CRSS) and temperature, as first shown by Castaing for semiconductor crystals [17]. Examples are shown in Figures 9.1-9.4. For MgO in Figure 9.1 [5], the relationship is well obeyed for both easy slip on the... 

Kink nucleation on partial dislocations has also been considered [27] because of the experimental observation that the critical resolved shear stress for 110 (111) slip in molybdenum disilicide decreases when substitutional alloying elements are added that decrease the stacking fault energy, and increases when substitutional elements are added that increase the stacking fault energy. This modification may apply not only to other intermetallics but also to ceramics such as spinel, where increasing... 

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