Yield stress, 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... 

Results for polystyrene (1 ) indicate that the bands start to occur at a tensile yield stress defined (1 ) as the point at which stress is no longer proportional to strain. For our spec imen, tl onset of oscillation occurs at a stress of 3.9 x 10 dynes/cm, and it tensile yield stress as defined above is... 

Yield stress defined by the conditions y = 0 Critical stress defined by the conditions y(t) = 0... 

The yield stress (defining the onset of plastic deformation) = force at yield point/original cross-sectional area. 

We will discuss here the anisotropic yield behaviour of oriented polymers but there is a need for a few preliminary remarks regarding the topic of yield in general. In describing the deformation of many crystalline materials, especially metals and ceramics, it is often convenient to introduce the idealisation of an elastic-plastic transition . The term elastic is used to describe the components of the strain which are proportional to the applied stresses, and which are completely recovered on removal of the stresses. Plastic strains are observed only for stresses greater than or equal to the yield stress and are not recovered on removal of the stress. The yield stress defines the elastic-plastic transition. 

The denominator <5 (Tg — T) accounts actually for the interchain effects (friction between chains) on the yield stress. Thus, the reduced yield stress defined above by Eq. 11.5 should be only a function of an intrachain property, characterized by the chain stiffness (Wu 1990, 1992) ... 

Fig. 31 Relative yield stresses (defined as the ratio of the yield stress of the gel with PE-b-PEP to that without the additive) of the 4wt% wax (C28 through C36) gels as a function of PE-b-PEP 1.5K/5K concentration at 0°C. Mass fractions are reported on a total solution concentration basis. The letters correspond to the micrographs shown in Fig. 32. The yield stresses of the unmodified and 1 wt% PE-b-PEP 1.5K/5K modified C32 wax are measured at -5°C. Open triangles C24, closed triangles C28) open circles C32, closed circles C36...

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

Viscosity has been replaced by a generahzed form of plastic deformation controlled by a yield stress which may be determined by compression e)meriments. Compare with Eq. (20-48). The critical shear rate describing complete granule rupture defines St , whereas the onset of deformation and the beginning of granule breakdown defines an additional critical value SVh... 

It is apparent therefore that a materials resistance to crack growth is defined not just by its inherent toughness but by its ratio of toughness to yield stress. Some typical values of K d(Ty are given in Table 2.2. 

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. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

Microindentation hardness normally is measured by static penetration of the specimen with a standard indenter at a known force. After loading with a sharp indenter a residual surface impression is left on the flat test specimen. An adequate measure of the material hardness may be computed by dividing the peak contact load, P, by the projected area of impression1. The hardness, so defined, may be considered as an indicator of the irreversible deformation processes which characterize the material. The strain boundaries for plastic deformation, below the indenter are sensibly dependent, as we shall show below, on microstructural factors (crystal size and perfection, degree of crystallinity, etc). Indentation during a hardness test deforms only a small volumen element of the specimen (V 1011 nm3) (non destructive test). The rest acts as a constraint. Thus the contact stress between the indenter and the specimen is much greater than the compressive yield stress of the specimen (a factor of 3 higher). 

The mechanical concepts of stress are outlined in Fig. 1, with the axes reversed from that employed by mechanical engineers. The three salient features of a stress-strain response curve are shown in Fig. la. Initial increases in stress cause small strains but beyond a threshold, the yield stress, increasing stress causes ever increasing strains until the ultimate stress, at which point fracture occurs. The concept of the yield stress is more clearly realised when material is subjected to a stress and then relaxed to zero stress (Fig. Ih). In this case a strain is developed but is reversed perfectly - elastically - to zero strain at zero stress. In contrast, when the applied stress exceeds the yield stress (Fig. Ic) and the stress relaxes to zero, the strain does not return to zero. The material has irreversibly -plastically - extended. The extent of this plastic strain defines the residual strain. 

Consistency, working time, setting time and hardening of an AB cement can be assessed only imperfectly in the laboratory. These properties are important to the clinician but are very difficult to define in terms of laboratory tests. The consistency or workability of a cement paste relates to internal forces of cohesion, represented by the yield stress, rather than to viscosity, since cements behave as plastic bodies and not as Newtonian liquids. The optimum stiffness or consistency required of a cement paste depends upon its application. 

In an attempt to relate the grain size in a metal to its mechanical properties quantitatively, Fetch and Hall (9 and references therein) proposed an expression relating grain size d with hardness H in a metal. Hardness is defined in this case as the yield stress, the stress at which value the material experiences the onset of permanent deformation. Thus,... 

It is convenient to introduce the concepts of material flow function, FF, and flow factor, ff. The material flow function, FF, relates the unconfined yield stress, To, to the corresponding major consolidating stress, cri, and is determined experimentally from the yield locus of the material, as shown in Fig. 8.9. The material flow function is presented as a plot of To versus flow factor, ff, is defined by... 

Considering the equilibrium of a stable arch, show that the minimum diameter of the orifice in a conical hopper, d,ma, which is defined to account for the arching, can be related to the unconfined yield stress of the bulk material by the relation... 

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