Yield surface defined

The distance from the crack tip, along the X-axis, at which the von Mises equivalent stress falls below the yield stress, defines the size of the plastic zone, r. For the plane stress case of unconstrained yielding, which corresponds to the free surface of the specimen in Figure 4, this gives... 

The physical removal of surface material is called sputtering, where energetic, chemically inert ions such as Ar" or Xe" are accelerated toward the wafer and physically eject material from the surface. The yield is defined as the ratio of the number of ejected surface atoms to the number of incoming ions per given ion energy. 

For a monolayer film, the stress-strain curve from Eqs. (103) and (106) is plotted in Fig. 15. For small shear strains (or stress) the stress-strain curve is linear (Hookean limit). At larger strains the stress-strain curve is increasingly nonlinear, eventually reaching a maximum stress at the yield point defined by = dT Id oLx x) = 0 or equivalently by c (q x4) = 0- The stress = where is the (experimentally accessible) static friction force [138]. By plotting T /Tlx versus o-x/o x shear-stress curves for various loads T x can be mapped onto a universal master curve irrespective of the number of strata [148]. Thus, for stresses (or strains) lower than those at the yield point the substrate sticks to the confined film while it can slip across the surface of the film otherwise so that the yield point separates the sticking from the slipping regime. By comparison with Eq. (106) it is also clear that at the yield point oo. 

The identification of geraniol can be confirmed by its conversion into citral, Cj Hj O, its aldehyde, which has a very characteristic odour and yields well-defined crystalline derivatives. Five parts of the alcohol fraction are shaken with 2-5 parts of chromic acid and four parts of concentrated sulphuric acid dissolved in 100 parts of water. The mixture is warmed in the water-bath for a few minutes, when crude citral separates on the surface of the liquid. This is purified by steam distillation and conversion into its sulphonic acid compound in the... 

The potential of block copolymers in forming supramolecular structures on surfaces has been studied in great detail by Stupp and co-workers [36]. Their studies have concentrated on the supramolecular properties of diblock and triblock rod-coil polymers which consist of one or two components that are conformationally flexible and one component with a more rod-like character at one of the ends of the polymer chain ( c in Figure 4.25). This latter component can be crystallized and is expected to yield well-defined structural arrangements. The Stupp-type copolymers are based on diblock styrene-b-isoprene coils combined with various rod-like components. The synthesis of such materials is carried out in two stages. In the first step, the diblock coil part is produced, as shown in Figure 4.26. The... 

The trial functions representing variations in are given by eqn (5.69) and substitution of (r) for into n] yields fi]- At the point of variation, = and fJ] equals fi]. The variations 5ij/ and dij/ are not given prescribed values on any of the boundaries, including the boundary of the subsystem. Instead only the natural boundary condition, that V,t/ nj and Vji/ n, together with ij/ and, vanish on all infinite boundaries, will be invoked. The functional [(, fi] is to be varied not only with respect to however, but also with respect to the surface defining the subsystem fJ. Only by having the surface itself considered to be a function of

[Pg.155]

Definitions.- Deactivation of catalysts is typically considered a necessary evil. Yet the unsteady-state changes in properties of the catalyst surface that almost invariably lead to a decrease in the rate of loss of the reactant generally result in a different slate of products being formed, and sometimes this difference is for the better. In this review, we define the activity of the catalyst as the rate of reactant loss, typically per unit mass of catalyst. The selectivity of the catalyst towards a particular product species is defined here as the amount of the species formed per amount of the reactant lost, whereas the yield is defined as the rate of formation of the species per unit mass of catalyst, i.e., the product of activity and selectivity. (Obviously, other definitions of these terms exist. In particular, Wheeler and others consider selectivity to be the ratio of the desired product formation rate to that of undesired product(s). Qualitatively equivalent results are obtained with either definition.)... 

The XSW field established inside the crystal and above the crystal surface induces photoelectron emission from atoms within the field. The excited atoms (ions), in turn, emit characteristic fluorescence X-rays and Auger electrons. In the dipole approximation, the photoelectric effect cross section is proportional to the E-field intensity at the center of the atom. (It is necessary to consider higher-order multi-pole terms in the photoelectric cross-section under special conditions, as discussed by Fischer et al. (1998) and Schreiber et al. (2001). For this review, we will assume the dipole approximation.) Therefore, with the XSW intensity from Equation (6), the normalized X-ray fluorescence yield is defined as... 

Abstract The paper presents results of an experimental study of thermal effects on the mechanical behaviour of a saturated clay, with emphasis on the determination of the onset of yielding. The study was performed on CM clay (Kaolin) using a temperature-controlled u-iaxial apparatus. Applied temperatures were between 22 °C and 90 °C. Various methods are used to identify the yield points (pseudo-elastic limit) and to define the shape of the yield surface in the invariant stress space p (effective mean pressure)- q (deviatoric stress). Yield surface obtained at 90 °C is compared with results at ambient temperature. Based on this comparison, thermo-mechanical yielding is discussed and yield limit evolution with temperature is presented. 

The Modified Cam-Clay yield surface is defined by the following expression ... 

Where c is the cohesion, is the friction angle in compression path, po is the pre-consolidation pressure, which defines the size of the yield surface, and ni is a coefficient introduced to take into account the effect of the third stress invariant. 

Experiment results of triaxial tests on saturated samples allow us to define the yield surface of oillike and water-like plugs. The transition between these two cases is characterized by results of suction controlled oedometer tests. 

Photolysis rates must be proportional to the total number of photons absorbed per unit time since only one molecule would be activated per photon absorbed according to the Stark-Einstein law. The reaction quantum yield would define the extent to which the activated molecules are transformed. The near-surface (<50 cm) specific rate of iight absorption, in water at a given wavelength is expressed in terms of the solar irradiance, Z(A) (photons cm s or meinsteins cm s ) and the molar extinction coefficient, s(A) ... 

To describe this kind of anisotropic material behavior precisely, the yield surface is needed. For quite simple but well-known and established yield surfaces, the so-called r-values are used to describe this behavior. The r-value (normal anisotropy) is defined by the ratio of width to thickness strain (measured as ttue strains). The index indicates the angle between drawing and rolling direction of the material. Usually, the r-values are determined in 0°, 45°, and 90°. 

The computed trial elastic stress is relaxed by the plastic-corrector method, as described in Table 5.1. In Table 5.1, Qg is the hardening tensor, 2 is the inelastic multiplier, which shows the magnitude of inelastic deformation, and fi is the shear modulus. The second-order tensor hy defines the direction of the inelastic flow for example, in the case of the associated formulation it is hy = dif//dffy, where the yield surface is = 0. The deviatoric stress and back stress tensors are, respectively, identified by Sy = [Pg.195]

Eq. (5) is specialized for the internal force components defining the yield surfaces... 

The first of the concepts of broad applicability is the existence of a yield surface in stress space this same quantity is known by a variety of names, including yield locus, the elastic limit surface and yield function. The function defining the surface is denoted by and its role is to identify the boundary of the elastic range in a space spanned by the stress components (7jj. In the most general circumstances to be considered here, this function takes the form... 

Immediate consequences of the postulate are that the yield function defines a convex surface in stress spacef and that the plastic strain rate e . is normal to the yield surface at any point at which the surface is smooth. Generalization of the feature of normality to surfaces with an edge or an apex is straightforward. The feature of normality implies that... 

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