Resolved shear stress deformation

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... 

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 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. 

The ASBs in deformed specimens normally appear as bands with altered microstructure running along directions of maximum resolved shear stress. These bands are seen as distinct, because they are surrounded by larger regions of unaltered nucrostructure. Note that no such bands of altered nucrostructure were observed in the stir zones of the welds examined here. The lack of clear evidence for ASBs in the stir zone during FSW of Ti-6A1-4V may be explained by one or more of the following ... 

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... 

In this section we will deal with attempts to describe the yield criterion for anisotropic polymers in terms of a critical resolved shear stress by analogy with the deformation of metal single crystals. Further discussion of the single crystal approach follows in section 11.3.2 where we discuss the structural reorganisation occurring within deformation bands. 

IDD is synonymous with the preferred orientation of the crystal c-axes suggesting an interpretation of the deformation in terms of the extended molecular chains sliding past each other parallel to their main axis which is called the c-slip process . Keller and Rider supported this proposal by observing that the yield stress varied with specimen orientation in a way which could, for an appreciable range of orientations, be interpreted by a modified critical resolved shear stress criterion. Their proposal was that yield occurs when the applied shear stress parallel to the preferred c-axis direction reaches a critical value, which depends on the normal stress on that slip plane. For a specimen cut with the tensile axis at an angle 6 to the IDD this criterion can be written... 

Because of chain inextensibility, the shear rate of any slip system is not dependent on the normal-stress component in the chain direction (Parks and Ahzi 1990). This renders the crystalline lamellae rigid in the chain direction. To cope with this problem operationally, and to prevent global locking-up of deformation, a special modification is introduced to truncate the stress tensor in the chain direction c. Thus, we denote by S° this modification of the deviatoric Cauchy stress tensor S in the crystalline lamella to have a zero normal component in the chain direction, i.e., by requiring that 5 c,c = 0, where c,- and c,- are components of the c vector (Lee et al. 1993a). The resolved shear stress in the slip system a can then be expressed as r = where R is the symmetrical traceless Schmid tensor of stress resolution associated with the slip system a. The components of the symmetrical part of the Schmid tensor / , can be defined as = Ksfw" + fs ), where if and nj are the unit-vector components of the slip direction and the slip-plane normal of the given slip system a, respectively. 

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... 

Figure 9.25 Critical resolved shear stress versus mol% of oxide solute (for cations in the form of Cr +, Ti andTi ) in sapphire deformed at 1500°C. The curves through the data correspond toa c dependence. Data from Ref [203].

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 ... 

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