Stress symmetry

Bazua,E.R., Williams, M.C. A molecular formulation of the internal viscosity in polymer dynamics, and stress symmetry. J. Chem. Phys. 59,2858-2868 (1973). 

Note that the condition of stress symmetry is not valid if there is a significant body couple per unit mass c in the field. In this case, we can easily show, following the same steps that we used in going from (2-35) to (2-40), that... 

During machining the stress symmetry is disturbed, and it must subsequently be re-established by fine annealing lasting about a inonth. 

The principle of the conservation of angular momentum is often used to argue for the symmetry of the extra stress that is, Xxy = Tyx, and so forth. In that case there are six independent components, not nine. The angular momentum argument requires an explicit but often unstated assumption that there is no structure in the fluid that is capable of generating local torques, which seems generally to be the case for polymers (except perhaps for liquid crystalline polymers), and, except for a brief introduction to liquid crystals in Chapter 13, we will assume stress symmetry throughout. 

It should be stressed that although these symmetry considerations may allow one to anticipate barriers on reaction potential energy surfaces, they have nothing to do with the thermodynamic energy differences of such reactions. Symmetry says whether there will be symmetry-imposed barriers above and beyond any thermodynamic energy differences. The enthalpies of formation of reactants and products contain the information about the reaction s overall energy balance. 

Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations S of the point group. In the case of NH3, the characters of the resultant 12x12 representation matrices form a reducible representation... 

Section 28.7). One more advantage of a tubular section is that it exerts equal forces at all points of the eticlosure and relieves it and the conductor from any undue stresses. Octagonal and hexagonal sections are also used as they also have near-symmetry. 

Boundary conditions are special treatments used for internal and external boundaries. For example, the center line in cylindrical geometry is an internal boundary that is modeled as a plane of symmetry. External boundaries model the world outside the mesh. The outermost row of elements is often used to implement the boundary condition as shown in Fig. 9.13. The mass, stress, velocity, etc., of the boundary elements are defined by the boundary conditions rather than the governing equations. External boundary conditions are typically prescribed through user input. 

There are few problems of praetleal interest that ean be adequately approximated by one-dimensional simulations. As an example of sueh, eertain explosive blast problems are eoneerned with shoek attenuation and residual material stresses in nominally homogeneous media, and these ean be modeled as one-dimensional spherieally symmetrie problems. Simulations of planar impaet experiments, designed to produee uniaxial strain loading eonditions on a material sample, are also appropriately modeled with one-dimensional analysis teehniques. In faet, the prineipal use of one-dimensional eodes for the eomputational analyst is in the simulation of planar Impaet experiments for... 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

If there is one plane of material property symmetry, the stress-strain relations reduce to... 

If there are two orthogonal planes of material property symmetry for a material, symmetry will exist relative to a third mutually orthogonal plane. The stress-strain relations in coordinates aligned with principal material directions are... 

The strain-stress relations for the five most common material property symmetry cases are shown in Equations (2.18) to (2.22) ... 

For plane stress in the 1-2 plane of a unidirectional lamina with fibers in the 1-direction, < = T. 3 = r23 = 0- However, from the cross section of such a lamina in Figure 2-39, Y = Z from the obvious geometrical symmetry of the material construction. Thus, Equation (2.126) leads to... 

The particular cross-ply laminate to be examined [4-10] has three layers, so is symmetric about its middle surface. Thus, no coupling exists between bending and extension. Under the condition N = N and all other loads and moments are zero, the stresses in the (symmetric) outer layers are identical. One outer layer is called the 1-layer and has fibers in the x-direction (see Figure 4-39). The inner layer is called the 2-layer and has fibers in the y-direction. The other outer layer is the 3-layer, but because of symmetry there is no need to refer to it. The cross-ply ratio, M, is, 2, so the thickness of the inner layer is ten times that of each of the outer layers (actually, the inner layer" is ten like-oriented lamina Each lamina is. 005 in (.1270 mm) thick, so the total laminate thickness is. 060 in (1.524 mm). 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

An important issue in the thermodynamics of confined fluids concerns their symmetry which is lower than that of a corresponding homogeneous bulk phase because of the presence of the substrate and its inherent atomic structure [52]. The substrate may also be nonplanar (see Sec. IV C) or may consist of more than one chemical species so that it is heterogeneous on a nanoscopic length scale (see Sec. VB 3). The reduced symmetry of the confined phase led us to replace the usual compressional-work term —Pbuik F in the bulk analogue of Eq. (2) by individual stresses and strains. The appearance of shear contributions also reflects the reduced symmetry of confined phases. 

While the smooth substrate considered in the preceding section is sufficiently reahstic for many applications, the crystallographic structure of the substrate needs to be taken into account for more realistic models. The essential complications due to lack of transverse symmetry can be dehneated by the following two-dimensional structured-wall model an ideal gas confined in a periodic square-well potential field (see Fig. 3). The two-dimensional lamella remains rectangular with variable dimensions Sy. and Sy and is therefore not subject to shear stresses. The boundaries of the lamella coinciding with the x and y axes are anchored. From Eqs. (2) and (10) one has... 

Figure 13.17 Molecular structure of some sulfides of arsenic, stressing the relationship to the AS4 tetrahedron (point group symmetry in parentheses).

In a general concept of a symmetry-restricted anharmonic theory Krumhansl relates the phonon anomalies to the electron band topology. The latter is directly determined by the competition of nearest neighbour interactions which in turn can be a function of stress, composition and temperature Nagasawa, Yoshida Makita simulated the <110> ... 

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