Stress plane strain flow

The corresponding mean stress on the punch for plane strain uniaxial compression is 2k. Hence the constraint factor C = (1 + irl2)2kf2k = 2.57. The relation between the plane strain flow stress 2k and the axisymmetric (3-D) flow stress a-y is 2k = 2o-yf 3). Therefore the 3-D constraint factor will be... 

Another interesting result is obtained from the comparison of the scattering of the sheared samples in the z direction to that of an isotropic sample. Fig. 30 shows the Zimm plot of sample C in the directions x and z for a specimen cut in the x-z plane. The corresponding curve for an isotropic sample, which has also been plotted in the same figure, is found to be identical to within experimental error to the curve in the z direction. This result indicates that the position correlations within the chain in the neutral direction of the shear flow are not affected by the flow, at least up to the values of stress and strain used in the present study. In particular. Table 7 shows that the mean square chain dimension in that direction, Rg 2> which has been determined either in the x-z or in the y-z planes for the various samples, is round to be equal to the radius of gyration of an isotropic sample to within experimental error (Rg 2=82 lA). The same result has been found by Lindner in dilute solutions [31]. 

A powder l not necessarily maintain a shear stress-imposed strain rate gradient in the fluid sense. Due to force instabilities, it will search for a characteristic shp plane, with one mass of powder flowing against the next, an example being rat-hole discharge from a silo. 

The pop-in concept was first developed by Boyle, Sulhvan, and Krafft [3] and forms the basis of the current Kic test method that is embodied in ASTM Test Method E-399 [2]. The basic concept is based on having material of sufficient thickness so that the developing plane stress plastic zone at the surface would not reheve the plane strain constraint in the midthickness region of the crack front at the onset of crack growth (see Fig. 4.2). It flowed logically from the case of the penny-shaped crack, as shown in Fig. 4.4 in the previous subsection. 

The 2D TM modelling was carried out with the finite element numerical code JobFem (Stille, 1982). The aim of the modelling was to predict the thermo-induced stresses and the resulting total stresses in the pillar under the period of heating. The in-situ stresses were extracted from Examine3D for two horizontal sections in the pillar located 0.5 and 1.5 m below the tunnel floor. Heal flow and temperature induced stresses have been calculated under plane strain conditions. 

The idealization of the two coupled crystalline and amorphous components of HDPE as joined sandwich elements and their interactive plastic deformation by crystal plasticity and amorphous flow comes close to the assumptions of the Sachs model of interaction. Thus, the composite model employing a Sachs-type interaction law does indeed result in quite satisfactory predictions both for the stress-strain curve and for the texture development in plane-strain compression flow and even in other modes of deformation (Lee et al. 1993b). In the following sections we discuss the application of the composite model to plane-strain compression flow and compare the findings of the model with results from corresponding experiments. 

Fig. 9.28 Model results for the normalized equivalent macroscopic flow stress 5e/ro as a function of the equivalent macroscopic strain Se for several deformation histories of tension, plane-strain compression, uniaxial compression, and simple shear compared with experimental results of plane-strain compression (o) and uniaxial compression ( ), where To = 7.8 MPa is the plastic-shear resistance of HDPE (from Lee et al. (1993a) courtesy of Elsevier).

Figure 9.28 shows the stress train eurves of a whole eomplement of deformation modes with all flow stresses normalized with tq, giving the dependenees of e/ro, the normalized global equivalent deviatorie shear resistances, on Se, the global equivalent plastic strain. The predieted stress strain eurve for plane-strain eompression agrees well with the data points of the Gal ski et al. experiments. We note that the predicted response for uniaxial tension is also elose to the predietion for plane-strain compression and that these two, as examples of irrotational flow, differ markedly from the simple shear results and also from the experimental results and the predictions for uniaxial compression, in comparison with the experimental results of Bartczak et al. (1992b). 

O Bradaigh and Pipes originally studied the plane stress flows of ideal fiber reinforced fluids (IFRF) and have used a penalty method to impose the fiber inextensibility constraint with biquadratic velocity/bilinear discontinuous tension elements. A linear and quasistatic scheme has been used to calculate instantaneous velocities which are multiplied by the time step to displace the mesh. Such an approach involves the buildup of considerable error unless time steps are extraordinarily small. Recently, this model has been improved by developing a mesh updating scheme that incorporates finite incompressibility and inextensibility constraints [8]. The new model also uses large displacement contact/friction elements to model tool contact and interply slip between layers of IFRF, in plane strain. 

The best designs of rheometers use geometries so that the forces/ deformation can be reduced by subsequent calculation to stresses and strains, and so produce material parameters. It is very important that the principle of material independence is observed when parameters are measured on the rheometers. The flow within the rheometers should be such that the kinematic variables and the constitutive equations describing the flow must be unaffected by any rigid rotation of both body and coordinate system - in other words, the response of the material must not be dependent upon the position of the observer. When designing rheometers, care is taken to see that the rate of deformation satisfies this principle for simple shear flow or viscometric flow. The flow analyzed can be considered as viscometric (simple shear) flow if sets of plane surfaces (known as shear planes) are seen to exist and each is moving past the other as a solid plane, i.e. the distance between every two material points in the plane remains constant. 

In plane strain compression, the plastic flow ends shortly and strain hardening induced by straightening of the entanglement network sets in. Since no or little chain disentanglement occurs, the stress reaches very high values at a relatively low true strain. 

Go is the flow stress (roughly the average of the yield and ultimate stresses) a is a constant that varies with the stress state and strain hardening coefficient and is about 0.3 for carbon steel under plane strain [117]... 

Unless otherwise stated it is now assumed that, in addition to the condition of axisymmetry, planes normal to the axial direction will remain plane (plane strain) and all the properties are isotropic. Thus there will be only three components of stress, tr, (Tj, and cr, which will be referred to generally as a. Both the fuel and the cladding material are subjected to a number of complex interacting physical phenomena thermal gradients, elastic strains, plastic flow, creep deformation, and, most important of all, the volume changes induced by radiation. 

Extensional flows occur when fluid deformation is the result of a stretching motion. Extensional viscosity is related to the stress required for the stretching. This stress is necessary to increase the normalized distance between two material entities in the same plane when the separation is s and the relative velocity is ds/dt. The deformation rate is the extensional strain rate, which is given by equation 13 (108) ... 

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

The word viscosity comes from the Latin word for mistletoe, viscum. Anyone familiar with this plant is aware that it exudes a viscous sticky sap when harvested. Viscosity is defined after Isaac Newton in his Principia as the ratio of stress to shear rate and is given the symbol T. Stress (a) in a fluid is simply force/area, like pressure, and has the units of pascals (Pa S.I. units) or dynes/cm2 (c.g.s.). Shear rate or strain rate (y or dyldt) is the differential of strain (y) with respect to time. Strain is simply the change in shape of a volume of fluid as a result of an applied stress and has no units. The shear rate is in fact a velocity gradient, not a flow rate. It has the bizarre units of 1/time (sec-1) and is the velocity at a given point in the fluid divided by the distance of that point from the stationary plane. 

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