Stress tensor principal directions

At each point on the vessel at a given time, there are three principal stresses, ui, ct2, and three stress differences Si 2/ S2, and 83,1. Hie principal stresses are calculated from the six components of the stress tensor. The directions of the principal stresses may change during the cycle... 

For any state of deformation at a point, we can find three planes on which there are only normal deformations (tensile or compressive). As with the stress tensor, the directions of these three planes are called principal directions and the deformations are called principal deformations /, or principal extensions. Determining of the principal extensions is an eigenvalue problem comparable to determining the principal stresses in the preceding section. All the same equations hold. Thus from eq. 1.3.5 principal extensions are the three roots or eigenvalues of... 

The principal strain rates are eigenvalues of the strain-rate tensor (matrix). As described more fully in Section A.21, the direction cosines that describe the orientation of the principal strain rates are the eigenvectors associated with the eigenvalues. In solving practical fluids problems, there is rarely a need to determine the principal strain rates or their orientations. Rather, these notions are used theoretically with the Stokes postulates to form general relationships between the strain-rate and stress tensors. It is perhaps worth noting that in solid mechanics, the principal stresses and strains have practical utility in understanding the behavior of materials and structures. 

Perhaps surprisingly, it turns out that the complex series of operations represented by Eq. 2.151 leads to a relatively simple result that is independent of the particular principal-coordinate directions. The stress tensor in a given coordinate system is related to the strain-rate tensor in the same coordinate system as... 

Determine the principal axes for the stress tensor. Why are the principal directions the same for the full stress tensor and the deviatoric stress tensor How does this result relate to the Stokes postulates that are used in the derivation of the Navier-Stokes equations ... 

Given a state at a point (e.g., stress or strain rate) that can be described by a symmetric tensor in some orthogonal coordinate system, it is always possible to represent that particular state in a rotated coordinate system for which the tensor has purely diagonal components. The axes for such a rotated coordinate system are called the principal axes, and the diagonal components are called the principal components. Finding the principal states and the principal directions is an eigenvalue problem. 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

The molecular deformation ratio K in the directions I and II can be estimated in the following way the difference Aa between the principal stresses in the x-y plane can be readily calculated from the birefringence A (measured parallel and perpendicular to the direction of extinction) and the stress-optical coefficient C for molten polystyrene (C= 4.8xlO Pa , see Chapter III.l). According to the classical network theory, the stress tensor is proportional to the Cauchy deformation tensor which means that the network deformation along the principal directions of the stress tensor are X and 1/ where ... 

It is also noted that the principal stresses for extensional flow are in the same direction as those for the strain rate. Thus, the stress tensor also has only diagonal components... 

Momentum flux n. In hydrodynamics, an interpretation of shear stress t in which each of the six shear components of the stress tensor is viewed as the rate of flow, per unit of shear area, and perpendicular to that area, of momentum directed along a principal axis in the surface of shear. In laminar flow through a circular orifice, with radial coordinate r and axial coordinate z, the only non-zero shear component is trz> the flux of z-directed momentum in the redirection. Newton s law of viscosity for this situation becomes... 

When a stress perpendicular to a surface acts at a point on that surface in the absence of shear stresses, that stress is called principal stress . In other words, it is possible to find an orientation such that, for any point on a surface represented by the elementary cube, the shear stresses on the cube s surfaces vanish. Thus, fxy = %z = fzx = 0 and only three normal stresses (of the nine) remain. The three perpendicular planes and the three coordinates are the principal planes and the principal axes, respectively. Directions along the principal axes are known as principal directions . The stress components vary as the orientation of the orthogonal, coordinate axes changes. Figure 1.14 illustrates the principal stresses and their directions for a point inside a body, compared to the initial system of coordinates and the stress tensor after rotation. 

In the common case of a symmetric tensor, fulfilling Ay = Aji, the eigenvalues are real numbers and the eigenvectors are perpendicular to each other. This, for example, is the case in stress or strain states in a classical continuum. The eigenvalues in this case are called principal stresses and principal strains, respectively the eigenvectors are the principal directions or axes. 

Let the object for uniaxial extension be a cylindrical rod with a cross-sectional area S to which an extension force F is applied. The ratio F/S = Pis the extensional stress. Alternatively, this is a component of a stress tensor in the selected coordinate system, Oyy = 032 = 2 = F. The action of the force resulted in a stressed state that is uniform in all directions (we are neglecting nonuniformities and local concentrations of stresses in certain special areas, such as the mounting points of the load or of the rod itself). The selected coordinate system in this case is the principal one. From symmetry considerations at F = Oyy, this tensor is as follows ... 

When the coordinate axes are chosen to align with the principal directions of the stress, the stress tensor in uniaxial elongation is... 

All parameters depend on the time and the shear rate. Steady-state conditions are obtained for t — CO. Variable (°, ) denotes the steady state values of the shear stress. The anisotropic character of the flowing solutions give rise to additional stress components, which are different in all three principal directions. This phenomenon is called the Weissenberg effect, or the normal stress phenomenon. From a physical point of view, it means that all diagonal elements of the stress tensor deviate from zero. It is convenient to express the mechanical anisotropy of the flowing solutions by the first and second normal stress difference ... 

However, in practice it is often difficult to figure out the rotations of the coordinate system at every point in the material, so as to line it up with the principal directions. Furthermore, it is usually more convenient to leave the coordinates in the laboratory frame. Thus, we normally do not measure the principal stresses (except for purely extension deformations) but rather calculate them from the measured stress tensor. We show this next. 

Thus, a is the magnitude of the principal stress and n its direction. As we saw earlier (eq. 1.2.10), the stress tensor is the machine that gives us the traction vector on any plane through the dot operation. Thus,... 

Equation 1.3.5 is a cubic and will have three roots, the eigenvalues a, G2, and 03. If the tensor is symmetric all these roots will be real. The roots are then the principal values of T,y and n,-, the principal directions. Mth them T can be transformed to a new tensor such that it will have only three diagonal components, the principal stress tensor, eq. 1.3.1. 

Determine the invariants and the magnitudes and directions of the principal stresses for the stress tensor given in Example 1.2.2. Check the values for the invariants using the principal stress magnitudes. 

After passing through a glass specimen, the light polarization is modified because glass becomes anisotropic when submitted to stresses (residual or transient Appendix G). The refractive index is not imique (Chapter 4, Appendix A) but depends on the stress tensor. The indices along principal directions are... 

Earthquake Mechanisms and Stress Field, Fig. 1 Definition of the stress tensor in original Cartesian coordinate system (a) and in the rotated system of principal stress directions (b)... 

If the above-mentioned assumptions are reasonably satisfied, the stress inversion methods are capable to determine four parameters of the stress tenson three angles defining the directions of the principal stress directions, Ci, <72 nd <73, and shape ratio R. The methods are unable to recover the remaining two parameters of the stress tensor. Therefore, the stress tensor is usually searched with the normalized maximum compressive stress... 

For instance, if T is the stress tensor, the three principal values (A.j, X2, and X3) are called principal stresses, and the directions in which they are acting are called principal directions. ... 

In an incompressible medium, the rheological state at a point is, as far as the stress is concerned, completely described by the shear stresses (three in a general flow) and in the differences of the normal or direct stresses. We shall denote components of the stress tensor by (TijiiJ= 1,2, 3) and suppose the stress tensor is symmetric, so that Gij=Oji. (If Cij is a shear stress if i=j, Gij is a direct or normal stress.) We can change axes by rotation to reduce Gij to principal form—in these (mutually orthogonal) principal axes the principal stresses are g, G2 and 0-3 and all shear stresses vanish. The simplest stress-optical relation is to suppose that the dielectric tensor (Kij) and the stress tensor are coaxial, i.e. have the same principal axes, and that the differences in principal stresses are proportional to the corresponding differences in (principal) refractive indices. Hence if 2( 3) is a principal axis, and g and G2 lie in the xy plane, we have the simplest stress-optic relation as... 

This alternative dramatically affects the results of quantitative calculations because, although the main axes of deformation and stress tensor coincide, the ratio of the principal values is different in general. In the case of linear elongation in Z-direction, the deformation in the X- and Y-direction is decreased whereas the X- and Y-components of the stress tensor (defined positive in traction) increase as a result of the decreasing elongation of the chains in these directions. 

Note that the x, y, and z directions are arbitrary. However, it is beneficial and customary to choose them along the principal axes of the component to be analyzed. Failure theories are based on the principal stresses. Consider a general 3D state of stress on an infinitesimal cube, where all components of the stress tensor are nonzero. F rincipal stresses are the eigenvalues of the stress tensor. Mathematically, the eigenvalues are a stress tensor where the off-diagonal terms, that is, shear stresses are zero. 

Determination of the fourth-rank tensor term F. 2 remains. Basically, F.,2 cannot be found from any uniaxial test in the principal material directions. Instead, a biaxial test must be used. This fact should not be surprising because F-,2 is the coefficient of the product of a. and 02 in the failure criterion. Equation (2.140). Thus, for example, we can impose a state of biaxial tension described by a, = C2 = c and all other stresses are zero. Accordingly, from Equation (2.140),... 

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