Plastic deformation resolved shear stress

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. 

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... 

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... 

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. 

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. 

The initial dislocation density of the sample was estimated to be less than 10 m ". The orientation of the samples was chosen in order to align the torsion axis as close as possible with the c-axis ( 1°). The maximum resolved shear stress is then applied on the basal planes. The plastic deformation is accommodated by the glide of screw dislocations on the... 

Now that we have the notion of the stress tensor in hand, we seek one additional insight into the nature of forces within solids that will be of particular interest to our discussion of plastic flow in solids. As was mentioned in section 2.2.3, plastic deformation is the result of shearing deformations on special planes. Certain models of such deformation posit the existence of a critical stress on these planes such that once this stress is attained, shearing deformations will commence. To compute the resolved shear stress on a plane with normal n and in a direction s we begin by noting that the traction vector on this plane is given by... 

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. 

Consider a single crystal being subjected to uniaxial tension or compression, as shown in Fig. 6.20. Clearly, the ease with which plastic deformation is activated will depend not only on the ease of dislocation glide for a particular slip system but also the shear stress acting on each system. This is similar to the problem discussed in Section 2.10 (Eq. (2.44)) though one should note the plane normal, the stress direction and the slip direction are not necessarily coplanar, (< +A)5 90°. In other words, slip may not occur in the direction of the maximum shear stress. The resolved shear stress acting on the slip plane in the slip direction is... 

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... 

In (a), the minimum critical resolved shear stress [henceforth CRSS] is 36 MPa. In (b), deformation occurs by a combination of delamination and kink-band formation in individual grains and also by shear-band formation. The multiple modes of deformation allow for plastic behavior in any arbitrary orientation of the compressive load. Notice that the 312- and 211-phases are layered hexagonal carbides and nitrides, having the general formula Mn+iAXn, (MAX), where n = 1 to 3, M is an early transition metal, A is an A-group element (mostly III A and IV A, or groups 13 and 14) and X is either carbon and/or nitrogen. 

Resolved shear stress is a concept related to plastic deformation and associated with shear stress. Similarly, it is reasonable to talk about ceramics exhibiting ductility as a consequence of acting shear stress. To do so, one must consider stress and strain tensors. The stress tensor in Sects. 1.22 and 1.23 (Eqs. 1.13-1.13b) is rewritten here as ... 

Equation (4.6) gives the resolved shear stress . The product in the equation is known as the Schmid factor and determines whether the orientation is favorable for shp. The conditions for slip are given by Schmid s Law and the value of Eq. (4.6), often represented in the literature by Tj, indicating the onset of plastic deformation and called critical resolved shear stress . CRSS is a structure-sensitive property, since it is very dependent on impurities and the way the crystal was grown and handled. 

In contrast with the Takayanagi model, which considers only extensional strains, a major deformation process involves shear in the amorphous regions. Rigid lamellae move relative to each other by a shear process in a deformable matrix. The process is activated by the resolved shear stress a sin y cosy on the lamellar surfaces, where y is the angle between the applied tensile stress o and the lamellar plane normals, which reaches a maximum value for y = 45° (see Chapter 11 for discussion of resolved shear stress in plastic deformation processes). 

The degree of anisotropy of a property may be negligible, but this is not usually the case in indentation hardness measurements on ceramic crystals. Later we will consider the phenomenological aspect of hardness anisotropy to demonstrate that, whatever the ramifications of the theoretical models, the nature of anisotropy is consistent and reproducible for a wide range of ceramics. Then we shall consider the models based on a resolved shear stress analysis and discuss their implications in terms of the role of plastic deformation and indentification of active dislocation slip systems. 

The detailed mechanical behavior of neptunium metal has not been determined. a-Np twins profusely during plastic deformation at room temperature, indicating that few slip systems exist, and that a high resolved shear stress is required for yielding by slip. 

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. 

Many normally brittle solids can undergo considerable plastic deformation when, in addition to an axial stress, they are subjected to a hydrostatic pressure. If the brittle strength of the material is Og and the specimen has a hydrostatic pressure p applied to it, the tensile stress needed to produce brittle fracture is Og + p. This tensile stress gives rise to a shear stress resolved in the slip direction in the slip plane and if this resolved shear stress reaches a value corresponding to the axial yield stress Oy before it reaches Og 4- p the material will deform plastically without brittle fracture taking place. 

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... 

With continued extension of a single crystal, both the number of slip lines and the slip step width increase. For FCC and BCC metals, shp may eventually begin along a second slip system, the system that is next most favorably oriented with the tensile axis. Furthermore, for HCP crystals having few slip systems, if the stress axis for the most favorable slip system is either perpendicular to the slip direction (A = 90°) or parallel to the slip plane = 90°), the critical resolved shear stress is zero. For these extreme orientations, the crystal typically fractmes rather than deforms plastically. 

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