Critical resolved shear stress material

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.

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). 

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. 

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. 

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... 

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. 

If we take these effects into account, the Schmid factor has to be replaced in a polycrystalline material by another number, the Taylor factor M. For a face-centred cubic material, M takes a value of 3.1 [34]. The relation between the critical resolved shear stress Tcrit and the yield strength measured in uniaxial tension ap thus is... 

A very simple yield criterion for anisotropic materials is the critical resolved shear stress of Schmid [14]. This is concerned with crystal slip. The law states that yield occurs when the resolved shear stress in the slip direction in the slip plane reaches a critical value. Although this law is extensively used in metal plasticity, it is of restricted application in polymers. 

When the resolved shear stress exceeds the critical resolved shear stress Tcrss (a material property), slip will occur as shown in Figure 8.2. The yield stress can then be written as... 

The strength of a material is related to its defect structure as well as to its bond energy. As stated previously, dislocations can begin to move through the grains when the resolved shear stress reaches or exceeds the critical resolved shear stress. The primary strengthening mechanism then involves inhibiting the motion of dislocations. This can be accomplished in a variety of ways as will be shown. 

Plastic deformation is produced by the shear stresses set up in the material and another basic law of plastic deformation, applicable most simply to single crystals, is that the plastic strain depends only on the shear stress in the slip plane, resolved parallel to the slip direction. Further, appreciable plastic deformation by slip starts when this resolved shear stress reaches a fairly well-defined value called the critical resolved shear stress (c.r.s.s.). This is Schmidt s law and it embodies the result that the yield stress is a characteristic of a given material, other conditions being the same. 

Table 3.2 Critical resolved shear stress and rigidity modulus at room temperature for materials in the form of single crystals...

In response to an apphed tensile or compressive stress, slip in a single crystal commences on the most favorably oriented shp system when the resolved shear stress reaches some critical value, termed the critical resolved shear stress it represents the minimmn shear stress required to initiate slip and is a property of the material that determines when yielding occurs. The single crystal plastically deforms or yields when T (max) = t ss, and the magnitude of the apphed stress required to initiate yielding (i.e., the yield strength o-y) is... 

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... 

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... 

The shear component of the applied stress appears to be the major factor in causing yielding. The uniaxial tensile stress in a conventional stress-strain experiment can be resolved into a shear stress and a dilational (negative compressive) stress normal to the parallel sides of test specimens ofthe type shown in Fig. 11-20. Yielding occurs when the shear strain energy reaches a critical value that depends on the material, according to the von Mises yield criterion, which applies fairly well to polymers. 

It was shown that for most crystalline polymers, including polypropylene and other polyolefins, the tensile drawing proceeds at a much lower stress than kinematically similar channel die compression [10,17]. Lower stress in tension was always associated with cavitation of the material. Usually a cavitating polymer is characterized by larger and more perfect lamellar crystals and cavities are formed in the amorphous phase before plastic yielding of crystals. If the lamellar crystals are thin and defected then the critical shear stress for crystal plastic deformation is resolved at a stress lower than the stress needed for cavitation. Then voiding is not activated. An example of such behavior is low density polyethylene [10]. 

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