Symmetry, stress tensor

The term in brackets is the linear momentum equation and is equal to zero. The term tijkTjk depends only on the anti-symmetric part of Tjk. Thus, a condition of stress tensor symmetry can be expressed as ... 

In Fig. 20 we show a theoretical dispersion plot using these parameters and a tensile stress = 2.7 x 10 dyn/cm. Due to the symmetry of the modes at X the stress tensor tpy does not affect the surface eigenmodes at this symmetry point. In addition, we have softened the intralayer force constant 4>ii in the first layer by about 10%. With these parameters, we find good agreement between experimental data and theoretical dispersion curves. 

The symmetry of the stress tensor can be established using a relatively straightforward argument. The essence of the argument is that if the stress tensor were not symmetric, then finite shearing stresses would accelerate the angular velocity w of a differential fluid packet without bound—something that obviously cannot happen. 

Fig. 2.14 Top view of the differential element, used to illustrate the symmetry of the stress tensor.

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

For uniform, infinite surfaces the stress tensor is independent of position on the surface and therefore can be taken outside the area integration. For a symmetric double layer system, the end result can be expressed as the osmotic pressure difference due to the double layer that is, the concentration of electrolyte at the plane of symmetry minus the bulk concentration, all multiplied by kT. For symmetric electrolytes this is [1]... 

As discussed in Section 7.7, crystals, particularly organic crystals, usually exist in lower-symmetry nonorthogonal systems for them the off-diagonal terms of P become important. The response of a crystal to the stress tensor P is a series of fractional displacements, small compared to any dimension of the body these fractional displacements are called strains and are denoted by the strain (or dilatation) tensor s ... 

In equations (32-33), 36 denotes the limiting surface of the complementary domain Bg = Sq u Sj u S2 (Fig 12). and o are the total stress tensor and vector, respectively, n is the outward unit vector normal to the surface. Sq and S2 are upstream and downstream surfaces limiting the flow domain imder consideration, perpendicular to the z-axis. In the cases under consideration (ducts involving symmetries), it can be shown that equations (32-33) reduce into one scalar equation [55] ... 

For a steady uniaxial extensional flow, we can assume by symmetry that the stress tensor contains only diagonal components. We can then evaluate the terms in Eq. (A3-13) containing the velocity gradient by using Eq. (A3-1) ... 

A system with an additional axis of symmetry, for example X3, is isotropic. The components of the stress tensor in the system of coordinates x[, x, can be obtained from those in the reference frame of the coordinate system xi, X2, X3 by means of the following cosine directors ... 

To start with, let us determine the stress and the deformation of a hollow sphere (outer radius J 2, inner radius R ) under a sudden increase in internal pressure if the material is elastic in compression but a standard solid (spring in series with a Kelvin-Voigt element) in shear (Fig. 16.1). As a consequence of the radial symmetry of the problem, spherical coordinates with the origin in the center of the sphere will be used. The displacement, obviously radial, is a function of r alone as a consequence of the fact that the components of the strain and stress tensors are also dependent only on r. As a consequence, the Navier equations, Eq. (4.108), predict that rot u = 0. Hence, grad div u = 0. This implies that... 

Taking into account the symmetry of the stress tensor and the expression (13.8) for the deformation tensor, we can write... 

By definition it is a symmetric second-rank tensor. The stress tensor ffy, i,j= 1,3), is also a symmetric second-rank tensor defined as follows (Landau and Lifchitz ) the element Oy is the i component of the force acting on the unit area normal to the axis x. The symmetry of the stress tensor is imposed by the condition of mechanical equilibrium. 

Here Ctju are the stilfness constants and Sijki are the compliance constants. They form two symmetric fourth-rank tensors with 81 elements inverse one to another. For the triclinic symmetry only 21 elements are independent because the strain and stress tensors are symmetric. Consequently the indices i,j and k, I can be permuted and also can be permuted one pair with another. For a crystal symmetry higher than triclinic the number of independent elastic constants is less than 21. 

The following sections develop three subjects the classical approximations for the strain/stress in isotropic polycrystals, isotropic polycrystals under hydrostatic pressure and the spherical harmonic analysis to determine the average strain/stress tensors and the intergranular strain/stress in textured samples of any crystal and sample symmetry. Most of the expressions that are obtained for the peak shift have the potential to be implemented in the Rietveld routine, but only a few have been implemented already. 

Similar to the ODF for texture, SODF can be subjected to a Fourier analysis by using generalized spherical harmonics. However, there are three important differences. The first is that in place of one distribution (ODF), six SODFs are analyzed simultaneously. The components of the strain, or the stress tensor can be used for analysis in the sample or in the crystal reference system. The second difference concerns the invariance to the crystal and the sample symmetry operations. The ODF is invariant to both crystal and sample symmetry operations. By contrast, the six SODFs in the sample reference system are invariant to the crystal symmetry operations but they transform similarly to Equation (65) if the sample reference system is replaced by an equivalent one. Inversely, the SODFs in the crystal reference system transform like Equation (65) if an equivalent one replaces this system and remain invariant to any rotation of the sample reference system. Consequently, for the spherical harmonics coefficients of the SODF one expects selection rules different from those of the ODF. As the third difference, the average over the crystallites in reflection (83) is structurally different from Equations (5)+ (11). In Equation (83) the products of the SODFs with the ODF are integrated, which, in comparison with Equation (5), entails a supplementary difficulty. 

Finally, the reader should appreciate a significant difference between the way in which a and r were introduced. In Section 1.2.1 the strain tensor was defined on purely mathematical grounds, whereas the conjugate stress tensor was introduced by purely physical reasoning (i.e., force balance). However, both elastic body arc explicitly excluded. As far as rr is concerned, this is effected by introducing the displacement of mass elements (r) as a vector field and by defining displacement tensor V (r) (see Ekjs. (1.1) and (1-6)] for the stress tensor r, the symmetry property stated in Eki. (1-14) serves to eliminate rotations. 

The coefficent array C,)H relates the stress tensor to the strain tensor and is itself a tensor (called the tensor of elasticity). Fortunately for a homogeneous material the 81 terms in the tensor of elasticity reduce through symmetry considerations to only three,... 

To obtain a simple form of the balance of moment of momentum, we confine its formulation to inertial frame with angular moment (3.88) having point y fixed here (although we use here the inertial frame fixed with distant stars, resulting formulations are valid in any inertial frame as will be shown at the end of this section). Again, the main reason for that is the nonobjectivity of x, y, v in (3.88), cf. (3.25), (3.38) generalization of this balance in the arbitrary frame will be discussed below but we note that the main local result—symmetry of stress tensor (3.93) below—is valid in the arbitrary frame. 

Balance of moment of momentum (3.93) expressed through the symmetry of a stress tensor (at least for mechanically nonpolar materials, cf. Rem. 17) is valid in any frame, even noninertial. Finally we can see that because (3.48) is valid for transformations between any inertial frames, the balances of angular moment related to fixed y (3.89)-(3.91) are valid in any inertial frame and not only in those fixed with distant stars. 

Summary. The first three balance equations are formulated in this section. The balances are necessary conditions to be fulfilled not only in thermodynamics but generally (in continuum mechanics). The balance of mass was formulated locally in several alternatives—(3.62), (3.63), or (3.65). The most important consequence of the balance of momentum is the Cauchy theorem (3.72), which introduces the stress tensor. The local form of this balance is then expressed by (3.76) or (3.77). The most relevant outcome of the balance of moment of momentum is the symmetry of the stress tensor (3.93). Note that in this section also an important class of quantities— the specific quantities—was introduced by (3.66) note particularly their derivative properties (3.67) and (3.68). 

Besides those based on symmetry in Rem. 30, see e.g., [8], another was used by Haupt [72] according to the size of memory for the stress tensor T in an isothermal body materials (mostly solids) are... 

Without going into details we note also that sufficient conditions of minima of a may be discussed and further simpliflcations of these results using objectivity (3.124) and material symmetry may be obtained using Cauchy-Green tensors, Piola-Kirchhoff stress tensors, the known linearized constitutive equations of solids follow, e.g. Hook and Fourier laws with tensor (transport) coefficients which are reduced to scalars in isotropic solids (e.g. Cauchy law of deformation with Lame coefficients) [6, 7, 9, 13, 14]. 

Inserting this result into (4.67) and subtracting (4.56) multiplied by (x—y)A we obtain the local partial moment of momentum balance for constituent a as a symmetry of the partial stress tensor... 

Summary. The balance of momentum postulated for individual constituents leads to the Cauchy s theorem for partial stress tensors (4.53) and the local form of this balance is given by (4.56) or (4.57). The balance of momentum for mixture as a whole is given by (4.63) or (4.64). The balance of moment of momentum postulated for individual constituents gives the symmetry of the partial stress tensor—see (4.70). Analogical balance for mixture as a whole gives symmetry of sum of these tensors, cf. (4.75). Note that in mixture conceptually new quantities entered these balances— especially partial quantities and the interaction forces between constituents. 

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