INDEX stress tensor

In the above mentioned field equations the number of unknown quantities does not correspond to the number of equations, thus we have to conclude the problem with the constitutive equations for the partial stress tensors T , the interaction forces p", the partial internal energies ea and the partial heat flows q . From the evaluation of the entropy inequality of the saturated porous body, see de Boer [4], we obtain for the solid phase and the mobile phases with Index j3 = L, G the constitutive relations for T and p ... 

The viscoelastic stress-strain equation, Equation (4) can be expressed in finite element formulation which relates the stress tensor a.. at time index n and cell centre (ij) to the corresponding strain tensor arising from the movement of the adjoining cell corners. Using backward differences for the time step, at time index n. 

In solid photoelasticimetry, birefringence is related to local stresses through the stress optical law, which expresses that the principal axes of stress and refractive index tensors are parallel and that the deviatoric parts of the refractive index and stress tensors are proportional ... 

First level tensors are vectors. As explained above, they are indicated by one index. Second level tensors are characterised by two indices. The stress tensor is such a quantity. It has nine components rn, t12, r13, r21. .. r33. The abbreviation normally written is r, where j and i each assume the values 1, 2, 3. In calculations with tensors, the following rule is used. If an index only appears... 

In Ty, the first index designates the axis to which the considered area element is perpendicular, and the second index designates a projection of the stress on the appropriate axis. Thus, Ty comprises nine components depending on x and t. These are components of the tensor of second rank, T, called the stress tensor. In abbreviated form, the correlations of Eq. (4.10) can be written as ... 

Here Oy are the components of stress tensor, the upper index S in Eq. (3.1b) shows, that the coefficients describe the surface energy. 

Generally, any vector of a surface force acting on a body can be decomposed into tangential (fx,fy, shear) and normal (Z, pressure) components (index j = x, y, z) as shown in Fig. 8.2a. In its turn, the element of surface A is a vector characterized by its area A and outward-directed unit vector s that has also three projections in the Cartesian laboratory frame (index /). Therefore, the second rank stress tensor [1] is defined as... 

Here p is the density, t the time, Xi the three Cartesian coordinates, and o,- the components of velocity in the respective directions of these coordinates. In equation 2, the index j may assume successively the values 1, 2, 3 gj is the component of gravitational acceleration in the j direction, and atj the appropriate component of the stress tensor (see below). (A third equation, describing the law of conservation of energy, can be omitted for a process at constant temperature the discussion in this chapter is limited to isothermal conditions.) Now, many experiments are purposely designed so that both sides of equation 1 are zero, and so that in equation 2 the inertial and gravitational forces represented by the first and last terms are negligible. In this case, the internal states of stress and strain can be calculated from observable quantities by the constitutive equation alone. For infinitesimal deformations, the appropriate relations for viscoelastic materials involve the same geometrical form factors as in the classical theory of equilibrium elasticity they are described in connection with experimental methods in Chapters 5-8 and are summarized in Appendix C. 

Many students with engineering or physics backgrounds are already familiar with the stress tensor. They may skip ahead to the next section. The key concepts in this section are understanding (1) that tensors can operate on vectors (eq. 1.2.10), (2) standard index notation (eq. 1.2.21), (3) symmetry of the stress tensor (eq. 1.2.37), (4) the concept of pressure (eq. 1.2.44), and(5)normalstressdifferences(eq. 1.2.45). 

This numbering of components leads to a convenient index notation. As indicated the nine scalar components of the stress tensor can be represented by 7,7, where i and j can take the values from 1 to 3 and the unit vectors Xi, X2. Xa become x,. Thus, we can write the stress tensor with its unit dyads as... 

The stress-optical relation (SOR) lies at the very heart of the use of flow birefringence in rheology (Janeschitz-Kriegl, 1969, 1983 A.S. Lodge, 1955 Tsvetkov, 1964 Fuller, 1990). Given a polymer liquid undergoing flow, both a stress tensor r and an index of refraction tensor n can be defined, The SOR comprises two statements about these tensors ... 

The principal axes of the stress tensor and the refi tive index tensor are collinear. 

The differences in principal values of the stress tensor and the refractive index tensor are proportional the constant of proportionality is called the stress-optic coefficient, C. 

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... 

Here, represents the Cauchy stress tensor, p is the mass density, and ft and m, are the body forces and displacements in the i direction within a bounded domain Q. The two dots over the displacements indicate second derivative in time. The indices i and j in the subscripts represent the Cartesian coordinates x, y, and z. When a subscript follows a comma, this indicates a partial derivative in space with respect to the corresponding index. For the special case of elastic isotropic solids, the stress tensor can be expressed in terms of strains following Hooke s law of elasticity, and the strains, in turn, can be expressed in terms of displacements. The resulting expression for the stress tensor is... 

For readers not familiar with this notation, a few words of explanation may be useful. The indices on the typical component of the stress tensor have the following meaning. The second index indicates that this component of the stress acts in the Xj direction, while the first index indicates that it acts on a plane normal to the X axis. A component is positive when it acts on a fluid element in the plus Xj direction on the face of that element having the larger value of Xj. Thus, a tensile stress has a positive value, while a compressive stress is negative. Note that this sign convention is not used universally. 

The divergence is again taken with respect to the second index and y is an arbitrary scalar which absorbs all contributions parallel to the director n. Noting the first of EQNS (35) this is effectively the Euler-Lagrange equation of static theory with a dynamic term, g, added. We can also rewrite the first of EQNS (S) representing balance of linear momentum by substituting expression (17) for the stress tensor, and adding an inner product of EQN (38) with Vn to obtain... 

Generally the linear approximation suffices, but, because the refractive index can be measured with considerable precision, the change in the impermeability tensor due to stress and electric field should be written as... 

Bulk material properties can be determined quite simply using this model. For example, consider the calculation of the second-moment tensor, Q = (u u ), which is required for the stress and refractive index tensors. Using the independent alignment approximation, we have... 

Finally, it should be stressed that the position of an index in a sequence is significant, since all operators (and coefficients) will eventually be written in antisymmetrized form. We can shed some light on the sign change for the transformation operator for covariant and contravariant tensors by examining the following equations ... 

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