Plane-stress flow

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 state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

The plane stress yield zones that spread from the root of the notch envelope, distort, and neutralize the craze. At temperatures just below the ductile-brittle transition temperature the shear flow begins at the notch, but the craze breaks before the shear zones have grown enough to encompass the craze. 

For plane laminar flows, such as Couette or Poiseuille flows (see Figure 3), the velocity and the stress depend only on the first coordinate a I, where I is the interval (0,1) for... 

Problem 3-17. Plane Couette Flow Driven by Oscillating Shear Stress. Consider an initially motionless incompressible Newtonian fluid confined between two infinite plane boundaries separated by a gap width d as depicted in the figure. Suppose the plane y = 0 is oscillated back and forth with an oscillatory shear stress r = to sin cot. 

Problem 3-19. Start-Up of Plane Couette Flow Driven by a Constant Shear Stress. 

The shear stress acting on the drop is given by r c times the velocity gradient u (d)/d. The flow type is probably close to plane hyperbolic flow, and Figure 11.8 shows that in this case Wecr will not strongly depend on the viscosity... 

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 flow (hagram of a typical calculation is shown in Fig. 7. Note that both short and continuous fiber are handled in the same manner. These calculations, while tedious, are analytically simple. The "plane stress", the Qji t rms, are employed because lamination neglects the mechanical propeffies through the ply thictoess. These stiffnesses are sometimes regrouped into new constants called "invariants", the U, terms, for analytical semplicity. 

CLASSICAL TWO-BLADE IMPELLER, 471 Flow in Horizontal Planes, 471 Flow in Vertical Planes, 471 Stresses, 471... 

In the GHS model, the no-slip condition is assumed between the mold surface and the charge surface layer. Hence, the transverse shear stress across the narrow gap between the mold surfaces is the dominant factor that drives material flow, and the in-plane stresses are negligible. The material flow configuration is illustrated in Fig. 3.28. 

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... 

In the case for the plane Poiseuille flow, the stress exerted by the wall on the fluid is linked to the pressure gradient ... 

We assume that the friction stress r exerted by the flow is uniform on the side wall. This assumption is questionable in the case of a complex cross-sectional geometry, but is wholly valid in the case of the plane Poiseuille flow or for the Poiseuille flow in a drcnlar pipe. Therefore, it is written as ... 

Stress intensity factor is given as Ka=a- naf alw), whereas alw- Oy f alW) -> 1.0. In the above expression /(a/w) is the correction factor which depends on the width (w) and flow size (a). The critical value of stress intensity, Kic. is commonly known as plane stress fracture toughness. It is expressed in units of MPa. S-N diagram A plot of stress amplitude against the number of cycles to failure. 

Even if Re, is small, the stress field around a sphere can interact with the wall, causing it to migrate inward. Ho and Leal (1974) have analyzed this problem extensively for both drag (Couette) and pressure-driven (plane Poiseuille) flow. They show that if the spheres are small enough and the flow rate is low. Brownian motion can keep the particles uniformly distributed. Their criteria for neglecting migration are... 

The last component is the tangential stress in the case of a plane parallel flow. 

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