Stress tensor eigenvalue

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. 

The derivatives with respect to x sample the off-diagonal behaviour of F > and generate terms related to the current density j and the quantum stress tensor er. The first-order term is proportional to the current density, and this vector field is the x complement of the gradient vector field Vp. The second-order term is proportional to the stress tensor. Considered as a real symmetric matrix, its eigenvalues and eigenvectors will characterize the critical points in the vector field J and its trace determines the kinetic energy densities jK(r) and G(r). The cross-term in the expansion is a dyadic whose trace is the divergence of the current density. 

To transform between different axis sets, we use the same method as for deformation gradients as given by Equation (3.21). Note that, unlike the deformation gradient, the stress tensor is always symmetric. This is the consequence of the equilibrium of torques applied to a material element, as pointed out in Chapter 2. As a second-order tensor, the stress is subject to the same axis transformation operations as the deformation gradient and Cauchy-Green measure (Equations (3.21) and (3.22)). The principal stresses are the eigenvalues... 

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. 

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... 

Table 1 The eigenvalues of the electronic stress tensor (A1, A2, -I3) and the tensile to compressive ratio defined in (21), computed at the bcp of the indicated bond...

Fig. 3 Stress tensor analysis along the reaction path from the enol to the aldehyde form of acetaldehyde. The eigenvectors of the stress tensor at the critical points of the electron density are shown, labeled by their eigenvalues in increasing order, X < X2 < /I3. The stress tensor was computed at the B3LYP/6-311G(d,p) level. Further details can be found in the paper of Guevara-Garcia et al. [32]...

Numerical tests performed with sets of 25-250 focal mechanisms are presented. The stress tensor is fixed for all datasets. The focal mechanisms are selected to satisfy the Mohr-Coulomb failure criterion (see Fig. 12a, b) and subsequently they are used for the calculation of moment tensors. The moment tensors were contaminated by uniform noise ranging from 0 to 50 % of the norm of the moment tensor (calculated as the maximum of absolute values of the moment tensor eigenvalues). The noisy moment tensors were decomposed back into strikes, dips and rakes of noisy focal mechanisms inverted for stress. The deviation between the true and noisy fault normals and slips attained values from 0° to 25°... 

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. 

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. 

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. 

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