Young shape

The Young shapes form what are sometimes referred to as standard irreps of S N), since they are adapted to the subgroup chain... 

U(l), which involve the transformations of progressively fewer and fewer orbitals—only the first m — 1, then the first m — 2, and so on. For Sat the irreps were labelled by Young shapes and their basis vectors by the corresponding standard Young tableaux for U(m) they are again labelled by Young shapes (which indicate the index symmetry of the tensor bases), while their basis vectors refer to standard Weyl tableaux. 

In the approach due to Gel fand and Tsetelin, the symmetry-adapted CFs of the standard irreps of U(m) are identical with those based on (10.3.4), but, instead of using a Young shape to indicate a particular irrep, a GeVfand tableau is employed. Such a tableau is a triangular array of integers (there is one array for each basis function) of the form... 

Returning to equilibrium shapes, these have been determined both experimentally and by solution of the Young-Laplace equation for a variety of situations. Examples... 

An approximate treatment of the phenomenon of capillary rise is easily made in terms of the Young-Laplace equation. If the liquid completely wets the wall of the capillary, the liquids surface is thereby constrained to lie parallel to the wall at the region of contact and the surface must be concave in shape. The... 

Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. 

The axisymmetric drop shape analysis (see Section II-7B) developed by Neumann and co-workers has been applied to the evaluation of sessile drops or bubbles to determine contact angles between 50° and 180° [98]. In two such studies, Li, Neumann, and co-workers [99, 100] deduced the line tension from the drop size dependence of the contact angle and a modified Young equation... 

The corroded tubercle floor is almost always a dish-shaped depression, much wider than it is deep (Fig. 3.23). Undercutting is very rare. The metal-loss width almost exactly matches the tubercular mound width. Corrosion rates exceeding 50 mil per year are rare, except when tubercles are young. Average local corrosion rates are usually 20 mil per year or less. 

Use a mechanics of materials approach to determine the apparent Young s modulus for a composite material with an inclusion of arbitrary shape in a cubic element of equal unit-length sides as In the representative volume element (RVE) of Figure 3-17. Fill in the details to show that the modulus is... 

The PGS obtained by Wang and coworkers was a kind of thermoset elastomer with the Young s modulus of 0.282 0.025 MPa, a tensile strain of at least 267 zE 59.4%, and a tensUe strength was at least 0.5 MPa. The mechanical properties of PGS were well consisted with that of some common soft tissues. Although PGS is a thermoset polymer, its prepolymer can be processed into various shapes by solving it in common organic solvents such as 1,3-dioxolane, tetrahydrofuran, isopropanol, ethanol, and iV,M-dimethylformamide. Porous scaffolds can be fabricated by salt leaching. 

For item 3, if an assumption is made that the seal completely fills its housing, the bulk modulus is the operative one. If the seal does not completely fill the housing, the shape factor effects apply. This is a very tricky area as Young s modulus and bulk modulus differ by at least two orders of magnitude. 

It is well known that when liquid droplets form on a flat substrate they adopt spherical cap shapes (neglecting gravity effects) with a contact angle 6. This angle depends solely on the interfacial energies as described by the Young s equation ... 

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