Stress ellipsoid

Internal-pressure design rules and formulas are given for cylindrical and spherical shells and for ellipsoidal, torispherical (often called ASME heads), hemispherical, and conical heads. The formulas given assume membrane-stress failure, although the rules for heads include consideration for buckling failure in the transition area from cylinder to head (knuckle area). 

External-pressure failure of shells can result from overstress at one extreme or n om elastic instability at the other or at some intermediate loading. The code provides the solution for most shells by using a number of charts. One chart is used for cylinders where the shell diameter-to-thickness ratio and the length-to-diameter ratio are the variables. The rest of the charts depic t curves relating the geometry of cyhnders and spheres to allowable stress by cui ves which are determined from the modulus of elasticity, tangent modulus, and yield strength at temperatures for various materials or classes of materials. The text of this subsection explains how the allowable stress is determined from the charts for cylinders, spheres, and hemispherical, ellipsoidal, torispherical, and conical heads. 

The terms that are linear in the stresses are useful in representing different strengths in tension and compression. The terms that are quadratic in the stresses are the more or less usual terms to represent an ellipsoid in stress space. However, the independent parameter F,2 is new and quite unlike the dependent coefficient 2H = 1/X in the Tsai-Hill failure criterion on the term involving interaction between normal stresses in the 1- and 2-directions. 

The tensile stresses acting in the direction of converging stream lines can ellipsoidally deform the big particles, but not so much as to form fine fibrils from small particles (region B). The matrix are also elongated in the converging section. As they pass the die exit (region C), recoil of the matrix occurs to release the stored energy... 

Based on a lot of experimental observations, criteria for the drop stability can be defined as below the U curve, namely We < We.cn, the interfacial stress can equilibrate the shear stress, and the drop will only deform into a stable prolate ellipsoid. Above this curve, the viscous shear stress becomes larger than the interfacial stress. The drop is at first extended and finally breaks up into smaller droplets. 

In passing from spherical particles to particles of an anisodiametrical shape (ellipsoids or fibers) the stress resisted by the filler is the higher the more pronounced the anisodiametricity of the particles [142]. 

Einstein coefficient b, in (5) for viscosity 2.5 by a value dependent on the ratio between the lengths of the axes of ellipsoids. However, for the flows of different geometry (for example, uniaxial extension) the situation is rather complicated. Due to different orientation of ellipsoids upon shear and other geometrical schemes of flow, the correspondence between the viscosity changed at shear and behavior of dispersions at stressed states of other types is completely lost. Indeed, due to anisotropy of dispersion properties of anisodiametrical particles, the viscosity ceases to be a scalar property of the material and must be treated as a tensor quantity. 

When the probe makes contact with the film, it generates a radial stress field around the point of contact. If the film is isotropic, it deforms in a uniform ring around the probe, as shown in Fig. 8.11 a). If the film is oriented, it deforms in a non-uniform manner. When the film is mildly oriented, the deformation area becomes ellipsoidal, as we see in Fig. 8.11 b), with its long axis... 

It is simple to understand the connection between the shear modulus and a. A sphere can be deformed into a prolate ellipsoid either by mechanical stress, or by an electric field. The input work required is measured by G = shear modulus in the first case and by a in the second case. Equating the input work needed in each case and solving for G, yields ... 

On the one hand in a flow field shear stresses t are exerted on a droplet which cause a deformation into an ellipsoid on the other hand the surface area of the droplet is increased by this deformation, so that the interfacial energy first effect becomes smaller in comparison to the second one, which results in an eqnilibrium at which no droplets are broken-np (expressed in the capillary number Ca = tI(cj/R), (see MT 9.1.5). 

Equation (3-33) shows how the inertia term changes the pressure distribution at the surface of a rigid particle. The same general conclusion applies for fluid spheres, so that the normal stress boundary condition, Eq. (3-6), is no longer satisfied. As a result, increasing Re causes a fluid particle to distort towards an oblate ellipsoidal shape (Tl). The onset of deformation of fluid particles is discussed in Chapter 7. 

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... 

Fig. 1.1. Laboratory coordinate system x direction of flow (also 1-direction), y direction of velocity gradient (also 2-direction), I, II principal directions of stress, y orientation angle if stress-ellipsoid, vx velocity (in -direction), q velocity gradient...

This equation relates sM to the orientation of the stress-ellipsoid [cf. eq. (1.3)]. This result is first quoted by Lodge. It differs by a factor of one half from that for the (completely recoverable) simple shear s of a perfectly elastic isotropic solid (50) ... 

The elastic energy of inhomogeneous, anisotropic, ellipsoidal inclusions can be studied using Eshelby s equivalent-inclusion method. Chang and Allen studied coherent ellipsoidal inclusions in cubic crystals and determined energyminimizing shapes under a variety of conditions, including the presence of applied uniaxial stresses [11]. 

Any exact theory, unless the geometry is simple, involves hopelessly complicated calculations of stress distributions even if the elements are large enough for these to be valid (which is not the case for small assemblies of polymer chains). In principle (see e.g. Chen and Young91 ) any geometry may be treated, but ellipsoids and parallelepipeds are the most usual. 

The strain component S12 is usually the deformation of the body along axis 1, due to a force along axis 2 the strain tensor s is usually symmetrical, = s and thus, of the nine terms of s, at most six are unique. Both P and s can be represented as ellipsoids of stress and strain, respectively, and can be reduced to a diagonal form (e.g., P j along some preferred orthogonal system of axes, oblique to the laboratory frame or to the frame of the crystal, but characteristic for the solid the transformation to this diagonal form is a... 

However, for nonspherical particles, rotational Brownian motion effects already arise at 0(0). In the case of ellipsoidal particles, such calculations have a long history, dating back to early polymer-solution rheologists such as Simha and Kirkwood. Some of the history of early incorrect attempts to include such rotary Brownian effects is documented by Haber and Brenner (1984) in a paper addressed to calculating the 0(0) coefficient and normal stress coefficients for general triaxiai ellipsoidal particles in the case where the rotary Brownian motion is dominant over the shear (small rotary Peclet numbers)—a problem first resolved by Rallison (1978). 

II. In a secondary step, once the local tensile stress reaches a critical value (equal to the stress at which macroscopic yielding can take place), individual blocks, about 10 to 30 nm in size are pulled out of the crystal ribbons. Due to this local yielding process, submicroscopic defects having an ellipsoidal shape are created between the lamellae Typical dimensions are 10 to 50 nm for the large and 2 to 6 nm for the short half axis thus, these microvoids are still, by an order of magnitude, smaller than microcopically visible crazes However,... 

Fig. 5. Distribution of craze related tractions on and around the ellipsoidal craze cavity under stress k(x) is the premor-dial craze cavity, u(x) the additional displacements of the craze border upon the insertion of the expanded craze lentil and the application of the external stress resulting in the final shape of the craze border f(x)...

One fundamental question concerning the new approach seems to have passed unnoticed despite the many arguments that have been raised aginst it. In the derivation of the and 5-functions, Scheraga and Mandelkern have assumed that the three hydrodynamic properties can be represented by an identical equivalent ellipsoid. If either F, or p of this fictitious ellipsoid is arbitrarily fixed the other is automatically defined. But there is no a priori reason to assume that the particles under shearing stress and sedimentation should fit the same hydrodynamic model. If Ve is kept identical the appropriate p value for describing one property might differ from the... 

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