Couple stress vector

Prom this relationship it is seen that the generalised body and surface forces are related to the body moment K and couple stress vector 1 by, respectively. 

Prom equations (2.142) and (2.146) it follows that at equilibrium the couple stress tensor lij and couple stress vector 1 are given by... 

As in the nematic liquid crystal case, it is clear that (6,78) represents a balance of forces at equilibrium. This can be seen by applying arguments that parallel those in Remark (i) on page 40. Similarly, it can be shown that the equilibrium equations (6.79) and (6.80) actually arise from a balance of moments, as has been demonstrated by Stewart and McKay [267], by means of an appropriate extension to the Ericksen identity (B.6) (cf. [181]). For the present, however, we restrict our attention to the derivation of the couple stress tensor Uj and its associated couple stress vector Analogous to the discussion for the nematic case in Remark (i) on page 40 when the balance of moments was discussed, consider an arbitrary, infinitesimal, rigid body rotation a for which... 

By other hand, the spatial form of the stress resultant h and the stress couple in vectors can be estimated from the stress vector Pi according to... 

We recall here the physical interpretation of the stress tensor in Cartesian coordinates [161, pp. 131-132]. Let tj be the stress vector (surface force) representing the force per unit area exerted by the material outside the coordinate surface upon the material inside (where the unit outward normal to this surface is in the direction e ). The component Uj then represents the component of this stress vector at a point on the coordinate surface. For example, if the x coordinate surface has unit outward normal i/ = (1,0,0) then the stress vector at a point on this coordinate surface is simply ti = BiUjUj = eita = (tii, 21, 3i)- A similar interpretation arises for the couple stress tensor. The components tn, 22 and 33 are called the normal stresses or direct stresses and the components ti2, t2i> i3, 3i, 23, 32 are called the shear stresses. 

Following usual conventions, repeated indices indicate summation and fy denotes df/dXj. The permutation S5mibol is used to present the vector cross product in indicial notation. Due to the anisotropic nature, traction and body couples can exist, and thus the angular momentum equation must be considered. For purely viscous fluids this equation says simply that the deviatoric stresses are symmetric. 

The first point reflects the fact that the dynamics is independent of the orientation in space. Points 2 and 3 manifest that both the total angular momentum and the parity are conserved. The Coriolis coupling arises from the continuous rotation of the body-fixed system with the scattering vector. Finally we stress that for J = Q, = 0 Equation (11.7) goes over into (3.20). 

The vector coupling coefficients of Eq. (1) are not symmetric in the -numbers. Therefore, in order to stress the inherent symmetry, the highly symmetric function, V [here denoted V(j)], was defined by Fano and Racah. It is related to the vector coupling coefficient by the expression... 

The sources of tectonic earthquakes can be studied very closely in some cases where their results become visible on faults at the earth s surface (see Fig. 11.1). Exact geodetic mapping on faults allows the slow movement of one side against the other to be followed. Due to this movement, high stresses can build up at the fault slowly over a long time until the friction at one point is exceeded. When this occurs, one side of the fault suddenly slips with respect to the other side by a displacement vector up to some metres at very large earthquakes and part of die stored elastic energy is set free as earthquake tremors. Those observations lead Reid [1910] to his so-called shear stress hypothesis as schematically shown in Fig. 11.2. This mechanism is nowadays called a double-couple mechanism. 

By virtue of the assumptions of unidirectional electrostatic fields and planar mechanical stress, the electromechanically coupled constitutive relations have been modified significantly. In example, the formulation on the left-hand side of Eq. (4.10a) reduces to Eq. (4.28) or (4.30). A transformation of coordinates on the considered plane may be performed as a rotation around the axis normal to this plane. In the case of Eq. (4.30), the base vector C2 represents the axis of rotation and thus this planar rotation may be formulated as follows ... 

Both effects are manifestations of the same fimdamental property of the substance linear coupling between a second-rank tensor (like strain or stress) and a first-rank tensor (such as electric field vector). This is emphasized by the fact that the same coefficients d j, enter both in Eq. (8.45) and (8.46). 

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