Resolved Shear Stress Models

Fig. 4-5. The measured anisotropy in Knoop hardness of p-BN as compared with predictions of the resolved shear stress model [20].

From the above example it can be seen that a complex system needing careful analysis is present in each case, but the underlying fact is that the type of dislocation and their interactions are intimately concerned with the stress-strain field imposed by the geometry of the indenter. The implication of this is that hardness anisotropy is an obvious manifestation of dislocation interactions and indenter facet geometry. Simplified interpretations of this have been sought, of which the Brookes Resolved Shear Stress model, given in Section 3.6.1, is an important development. 

The effect this has on developing predictive equations is discussed in Section 3.6.1 where resolved shear stress models are considered. 

These methods are now strongly suggesting a work-hardening function to account for anisotropy peaks as dislocations lock this feature becomes the dominant one in the most recent developments of hardness anisotropy theory that finally move away from the resolved shear stress models. 

The benefits of simplicity inherent in the resolved shear stress models were lost when Hirsch and co-workers, examining the plastic zone beneath (111) and (111) faces of GaAs indented by a Vickers diamond, carefully sectioned the crystals and found that hemispherical symmetry of the plastic zone is not evident. Hagan has shown the hemispherical symmetry but only for nonciystalline solids. This demonstration pointed to the fact that the plastic zone is anisotropic and the realization that a new model I must be developed based on the stresses caused by a straight punch. From... 

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. 

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). 

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... 

These tests were performed using an Instron Model 8561 (single screw) machine in air and the furnace was adapted to perform four-point bend tests. The rates indicated in Fig. 2.3 relate to crosshead displacement. Figure 2.4 shows the resolved shear stress at yield for the specimens tested ate = 4.2 x 10 s above Tc at the indicated orientations. The mechanism for slip is dislocation glide, which explains the orientation dependence of yield, as seen in Fig. 2.4. Thus, the BDT temperature, Tc, of the sapphire (AI2O3) varies not only with the strain rate, but also with the crystallographic orientation of the fracture plane. 

AHq is the energy for cross-slip, cr is the critical resolved shear stress, a is the applied stress and C and c are constants. A model of creep controlled by cross-slip from the 111 to the 100 plane in the temperature range of 530-680 °C over the stress range of 360-600 MN was found to be in good agreement with the experimental results. The energy to form a restriction between the partials, namely to recombine the Shockley partials, was evaluated on the basis of Dorn s expression [8]. (See also Hemker et al. [51] for the creep mechanism at intermediate temperatures in Ni3Al). 

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 most consistently successful models for explaining and predicting the nature of anisotropy in indentation hardness have been those based on adaptations of the Schmid-Boas resolved shear stress criteria. Thus slip is initiated when the critical resolved shear stress (t ) is reached for the most favorably oriented slip system. Then ... 

In an isotropic polycrystalline polymer whose microstructure consists of stacked lamellae arranged in the form of spherolites, the slip systems activated depend on the local orientation of the lamellae with respect to the applied stress and, as deformation proceeds, these orientations are modified. To calculate the evolution of the crystalline texture, one can consider the polymer to behave as a crystalline aggregate. Although the entropic contribution of chain orientation in the amorphous regions may also need to be considered, the major contribution to work hardening in tension is rotation of the slip planes toward the tensile axis, so that the resolved shear stress in the slip direction diminishes. This results in a fiber texture in the limit of large deformations, such that the crystallites are oriented with their c axis (the chain axis) parallel to the stretch direction. Despite the relative success of such models, they do not explicitly address the micro-mechanisms involved in the transformation of the spherulitic texture into a fiber texture. One possibility is that the... 

Fig. 3.8 Measured dislocation cell size d vs. calculated normalized resolved shear stress xjCb along the [110] direction of a 6-inch VCz GaAs wafer (empty squares) [77[ and estimated from von Mises stress modeling in VCz GaAs crystals (gray squares) [75] in...

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

Real-time wall shear stress is difficult to monitor precisely because it varies in space and time. MEMS sensors provide high spatial resolution to resolve variations in shear stress in a 3D bifurcation model for small-scale hemodynamics. The application of MEMS sensors with backside wire bonding G ig. fib) captured the spatial variations in shear stress in a 3D bifurcation model (Fig. 8). The measured skin friction coefficients at various positions correlated well with values derived from the exact Navier-Stokes solution of the flow within the bifurcation [13]. Therefore, the development of MEMS sensors has enabled the precise measurements of spatial variations in shear stress for small-scale hemodynamics otherwise difficult with conventional technologies such as computed tomography (CT scan), magnetic resonance imaging (MRI), ultrasound, and laser Doppler velocimetry. 

Rouhanizadeh M et al (2006) MEMS sensors to resolve spatial variations in shear stress in a 3D blood vessel bifurcation model. IEEE Sens J 6(l) 78-88... 

The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. 

---