Shear plane, definition

There have been a number of past attempts to unify hardness measurements but they have not succeeded. In several cases, hardness numbers have been compared with scalar properties that is, with cohesive energies (Plendl and Gielisse, 1962) or bulk moduli (Cohen, 1988). But hardness is not based on scalar behavior, since it involves a change of shape and is anisotropic. Shape changes (shears) are vector quantities requiring a shear plane, and a shear direction for their definition. In this book, the fact that plastic... 

It is sometimes important to specify a vector with a definite length perhaps to indicate the displacement of one part of a crystal with respect to another part, as in an antiphase boimdary or crystallographic shear plane. In such a case, the direction of the vector is written as above, and a prefix is added to give the length. The prefix is usually expressed in terms of the unit cell dimensions. For example, in a cubic crystal, a displacement of two unit cell lengths parallel to the b axis would be written 2ao[0 1 0]-... 

The zeta potential is the potential at the surface between a stationary solution and a moving charged colloid particle. This surface defines the plane of shear. Its definition is somewhat imprecise because the moving charged particle will have a certain number of counterions attached to it (for example ions in the Stern layer, plus some bound solvent molecules), the combined flowing object being termed the electrokinetic unit. The stability of colloidal suspensions is often interpreted in terms of the zeta potential, because, as we shall see, it is more readily accessible than the surface potential (Eq. 3.7), which describes the repulsive interaction between electric double layers. 

The electric potential of a particle varies gradually from the value of the surface potential (Vo) to zero. For the majority of particles without a complex surface structure, the variation is monotonic. Thus, electric potential at the shear plane is in between vj/o and zero (Figure 6.2). This potential is called the zeta potential ( potential). Zeta potential is different from surface potential /o but is a measurable amount. Since the shear plane is located at the frontier of the particle surface, any interfacial interaction of particle with other species (ions, neighboring particles, etc.) in medium will be encountered at the shear plane. Zeta potential actually has more direct influence compared with surface potential. Zeta potential is determined by many factors, mainly 1) surface potential, 2) potential curve, which is determined by the concentrations and valences of the co- and counter ions in the system and the dielectric constant of the continuous phase, and 3) location of the shear plane. There is no definite relationship between surface potential and zeta potential. For example, in different environments and particle surface conditions, the same zeta potential may correspond to different surface potentials and the same surface potential may result in different zeta potentials. Figure 6.2 describes the relation between... 

The second equality in (4.161) was obtained using the Parodi relation (4.96). Similar equivalent descriptions of tj 6, (j>) can be found in the literature see, for example, Moscicki [205], who has an equally valid description using exactly these viscosities but with different definitions of 0 and . Further examples include those mentioned by G willer [99, Eqn.(7)j and Schneider and Kneppe [247, p.456] (note that in both References [99, 247] the Authors interchange the definitions of the above r/i and r/2). The three cases in Fig. 4.1 then correspond to the three types of experiment which allow the measurements of (a) 771 when 0 = 0 and 0 = 0, (b) 772 when 0 =, and (c) 773 when 0 = 0 and 0 = f. The contribution of 7712 to the apparent viscosity 77 is clearly maximised when the director is immobilised in the shear plane at an angle of to v, in which event 0 = f and 0 = 0, so that... 

Figure 2.4. Definition of a displacement (Burgers or shear) vector b (a) a Burgers vector around a dislocation (defect) A in a perfect crystal there is a closure failure unless completed by b (b) a schematic diagram of a screw dislocation—segments of crystals displace or shear relative to each other (c) a three-dimensional view of edge dislocation DC formed by inserting an extra half-plane of atoms in ABCD (d) a schematic diagram of a stacking fault. (Cottrell 1971 reproduced by the courtesy of Arnold Publishers.)...

The shear-strain rate r = e r is given by the average rate at which the vertex angle decreases, which is the same definition as in the r-z plane ... 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

Instead of Je Lodge (46) derived another quantity called constrained shear recovery sIn this case a shear recovery is considered, where the liquid is constrained by boundary planes which are rigid and do not change their mutual distance during recovery of the liquid. Subscript oo means that the recovery is measured after an infinite time, reckoned from the moment that the shear stress is made zero. According to Lodge, a quite different type of recovery occurs, when the mentioned restrictions are released. This fact has already been noted in the first paragraph of this section. In the definition of Je, however, the mentioned restrictions are tacitly made. 

POISE (P). A unit of dynamic viscosity. The unit is expressed in dyne second per square centimeter The centipoise (cP) is more commonly used The formal definition of viscosity arises from the concept put forward by Newton that under conditions of parallel flow, the shearing stress is proportional to the velocity giadieut. If lire force acting on each of two planes of aiea A parallel to each oilier, moving parallel lo each other with a relative velocity V, and separated by a perpendicular distance X, be denoted by F. the shearing stress is F/A and the velocity gradient, which will be linear for a true liquid, is V/X. Thus, Ft A = q V/X, where the constant if is the viscosity coefficient or dynamic viscosity of the liquid. The poise is the CGS unit of dynamic viscosity. 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

Viscosity and Plasticity—Viscosity and plasticity are closely related. Viscosity may be defined as the force required to move a unit-area of plane surface with unit-speed relative to another parallel plane surface, from which it is separated by a layer of the liquid of unit-thickness. Other definitions have been applied to viscosity, an equivalent one being the ratio of shearing stress to rate of shear. When a mud or slurry is moved in a pipe in more or less plastic condition the viscosity is not the same for all rates of shear, as in the case of ordinary fluids. A material may be called plastic if the apparent viscosity varies with the rate of shear. The physical behavior of muds and slurries is markedly affected by viscosity. However, consistency of muds and slurries is not necessarily the same as viscosity but is dependent upon a number of factors, many of which are not yet clearly understood. The viscosity of a plastic material cannot be measured in the manner used for liquids. The usual instrument consists of a cup in which the plastic material is placed and rotated at constant speed, causing the deflection of a torsional pendulum whose bob is immersed in the liquid. The Stormer viscosimeter, for example, consists of a fixed outer cylinder and an inner cylinder which is revolved by means of a weight or weights. 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

By Newton s definition the viscosity or, more appropriately, the viscosity coefficient, jj, of a fluid in a laminar steady-state flow is expressed as the tangential force, F, per unit area. A, required to maintain a unit rate of shear (or velocity gradient), G, in the liquid. If the liquid fills the space between two parallel planes of area. A, one of which moves at a constant distance, r, from the other with a relative velocity, u, then we have... 

In PEI the DCG process, as in any polymer, is active. The epsilon CTPZ, however, was not observed. No plane strain shear bands have yet been observed. Some form of localized crack tip shear process can be activated, however, as evidenced by the inversion transition that occured at higher stresses (at the higher temperatures). The fracture surface did not show fracture to occur on a slanted 45 degree plane. The fracture plane was still normal to the leading direction. The fracture surface, however, was not smooth, as seen with craze fracture, but has a definite roughened texture which is associated with active localized shearing. This texture is often described as honeycomb or tufted. 

This is a structure-building component in which two constituent parts of the structure are twin-related across the interface. The twin plane changes the composition of the host crystal by a definite amount (which may be zero). Ordered, closely spaced arrays of twin planes will lead to homologous series of phases. Disordered twin planes will lead to non-stoichiometric phases in which the twin planes serve as the defects. There is a close parallel between chemical twinning and crystallographic shear (see Section IR-11.6.3). 

In HCP metals, the twin plane is normally (10T2). The twinning shear is not well understood in a gross sense, it takes place in the direction [2Tl] for metals with cja ratios less than -s/s (Be, Ti, Mg) and in the reverse direction [2lT] for metals with cja larger than yjh (Zn, Cd), but the direction of motion of individual atoms during shear is not definitely known. Figure 2-23(c) illustrates the usual form of a twin band in HCP metals, and it will be noted that the composition plane, although probably parallel or nearly parallel to the twin plane, is not quite flat but often exhibits appreciable curvature. 

Clearly, the solution breaks down in the limit r —> 0. In fact, according to (7-80), an infinite force is necessary to maintain the plane = 0 in motion at a finite velocity U, and this prediction is clearly unrealistic. Presumably, one of the assumptions of the theory breaks down, although a definitive resolution of the difficulty does not exist at the present time. The most plausible explanation is that the no-slip boundary condition is inadequate in regions of extremely high shear stress. However, as discussed in Chapter 2, this issue is still subject to debate. 

In deriving the equations for the flow of a simple viscous fluid the theoretical physicist uses a definition of viscosity based on a mathematical statement rather than a physical model. Let us define viscosity from a physical point of view. Consider two planes in the body of a fluid a distance dy apart, as shown in Fig. 4-1. If we apply tangential stress along one of these planes and observe a rate of shear yi then... 

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