Plane-strain flow

This flow has the same extension component as the simple shear flow but has no rotational component. 3°. Plane straining flow ... 

We start with the fully plastic, non-hardening plane-strain flow field of a deeply double-edge-notched plate of a rigid plastic material where the two symmetrically placed notches simulate two opposing sharp cracks shown in Fig. 12.12. The material is considered to have a tensile yield strength (To (= T) or a yield strength in shear of k = (To/ /3-... 

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... 

Using this concept, Erwin [9] demonstrated that the upper bound for the ideal mixer is found in a mixer that applies a plane strain extensional flow or pure shear flow to the fluid and where the surfaces are maintained ideally oriented during the whole process this occurs when N = 00 and each time an infinitesimal amount of shear is applied. In such a system the growth of the interfacial areas follows the relation given by... 

The first case considered is solute desorption during unconfined compression. We consider a two dimensional plane strain problem, see Fig. 1. A sinusoidal strain between 0 and 15 % is applied at 0.001 Hz, 0.01 Hz, 0.1 Hz and 1 Hz. To account for microscopic solute spreading due to fluid flow a dispersion parameter is introduced. Against the background of the release of newly synthesized matrix molecules the diffusion parameter is set to the value for chondroitin sulfate in dilute solution Dcs = 4 x 10 7 cm2 s-1 [4] The dispersion parameter Dd is varied in the range from 0 mm to 1 x 10 1 mm. The fluid volume fraction is set to v = 0.9, the bulk modulus k = 8.1 kPa, the shear modulus G = 8.9 kPa and the permeability K = lx 10-13m4 N-1 s-1 [14], The initial concentration is normalized to 1 and the evolution of the concentration is followed for a total time period of 4000 s. for the displacement and linear discontinuous. For displacement and fluid velocity a 9 noded quadrilateral is used, the pressure is taken linear discontinuous. 

The constitutive equations are the Oldroyd-B model and a modified Oldroyd-B model in which the viscosity depends on the rate of strain. In [79], Laure et al. study the spectral stability of the plane Poiseuille flow of two viscoelastic fluids obeying an Oldroyd-B law in two configurations the first one is the two layer Poiseuille flow in the second case the same fluid occupies the symmetric upper and lower layers, surrounding the central fluid. (See Figure 9.)... 

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. 

Figure 9.2. Energy consumption as a function of change of the interface area in extensional (uniaxial, biaxial and plane strain) and simple shear flows. The most efficient is the biaxial stretch, the worst (by a factor of 500,000 ) is shear (after Erwin, 1991).

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. 

In Section 9.3 we present experimental studies of plastic deformation in two semi-crystalline polymers, HDPE and Nylon-6, both in tension and in plane-strain compression flow, from initial spherulitic morphologies to large plastic strains. In these studies, the evolving morphological alterations were monitored closely by a complementary array of techniques involving light microscopy and X-ray diffraction and scattering both in the crystalline and in the amorphous components. 

Large-strain plastic flow in HOPE in plane-strain compression... 

In the plastic response of semi-crystalline polymers the starting material has an initial spherulitic morphology and, in the process of simple extensional flow, either in tension or in plane-strain compression, ends up with a highly perfect... 

Rg. 9.8 A polarized-light micrograph of highly textured fIDPE after plane-strain compression to a CR of 12.1 e — 2.49), showing nearly perfectly aligned fibrils viewed from the constraint direction. The long direction of the micrograph is parallel to the flow direction (from Gal ski et al. (1992) courtesy of the ACS). 

Rg. 9.11 The evolution of the crystallographic texture, followed by wide-angle X-ray scattering (WAXS), viewing the plane-strain-compressed material from the flow direction (FD), showing the systematic aggregation of the normals of the principal slip planes (200), (020), and (002) after strains of (a)... 

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). 

Fig. 9.29 Predicted (WAXS) crystallographic textures of the (002) and (200) plane normals of HDPE in plane-strain compression, viewed from the flow direction (FD), after various equivalent strains of (a) = 0.8 and (b) = 1.3 (from Lee et al. (1993a) courtesy of...

Thus, on the whole, the computer simulation of large-strain (up to e 1.3) deformation of HDPE in plane-strain compression flow presented here is able to develop good predictions for texture development in ranges where morphological catastrophes are absent. This is in spite of the many simplifying assumptions introduced into the model to make it tractable. 

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