Couple stress tensor

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... 

If u denotes the outward unit normal to the surface 5, then the usual tetrahedron argument [161, p.l29] shows that the surface force U and surface moment U are expressible in terms of the stress tensor tij and couple stress tensor lij respectively, through the relations... 

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. 

To proceed further we have to make more specific assumptions about the dynamic contributions Uj and Uj to the stress and couple stress tensors. This means that some relations between these stresses and the motion of the material will have... 

It is instructive at this point to comment upon the constant c. The surface traction exerted by an outer cylinder upon the liquid crystal is given via equations (4.37), (4.128) and (4.129) with 1/ = e,. (notice that tzr = 0 by equations (4.121), (5.245), (5.279), (5.284), (5.285) and (5.286)). Recalling that there are surface, contributions arising from both the stress tensor (4.128) and couple stress tensor (4.129) (cf. the balance law (4.31)), the corresponding moment about the s -axis of such a cylinder of radius r is then, in cylindrical coordinates, with U = UjUj and 1% = hj ji... 

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... 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

The tensor of stress, Oy, has the meaning of the force in direction j on an infinitesimal area with normal in the direction i and is again a symmetric tensor with 6 independent components. In classical elasticity only the force resultant at any point is considered, the couple that must also exist is assumed to be negligible by comparison. However, in polar field theories of elasticity, couple stresses are considered and additional equations of equilibirum required. Classically however only the equation of stress equilibrium... 

Hooke s law relates stress (or strain) at a point to strain (or stress) at the same point and the structure of classical elasticity (see e.g. Love, Sokolnikoff) is built upon this linear relation. There are other relationships possible. One, as outlined above (see e.g. Green and Adkins) involves the large strain tensor Cjj which does not bear a simple relationship to the stress tensor, another involves the newer concepts of micropolar and micromorphic elasticity in which not only the stress but also the couple at a point must be related to the local variations of displacement and rotation. A third, which may prove to be very relevant to polymers, derives from non-local field theories in which not only the strain (or displacement) at a point but also that in the neighbourhood of the point needs to be taken into account. In polymers, where the chain is so much stiffer along its axis than any interchain stiffness (consequent upon the vastly different forces along and between chains) the displacement at any point is quite likely to be influenced by forces on chains some distance away. 

In the most systematic application of this approach, Harlow and co-workers at Los Alamos have derived a transport equation for the full Reynolds stress tensor pu u j. They have coupled this equation with a scalar dissipation transport equation and have utilized with various semi-empirical approximations to evaluate the numerous unknown velocity, velocity-pressure, and velocity-temperature correlations which appear in the formulation. While this treatment is fairly vigorous, extensive compu-... 

This theory has two notable features. The nonlocality of molecular interaction is reflected by the ellipticity of Eq. (19) [cf Eq. (15)]. Thus, the LCP configuration is globally coupled by distortional elasticity. In addition, the elastic stress tensor is asymmetric. The mean-field torque on LCP molecules amounts to a volume torque on the material, which modifies the usual conservation of angular momentum. The antisymmetric part of the stress tensor precisely balances the volume torque computed by averaging the molecular torque. ... 

Problem 2-24. Polar Fluids. There has been a significant amount of theory developed for so-called polar fluids. These are fluids in which it is assumed that the surface couple r and the body couple c are both nonzero. Show that the stress tensor will no longer, in general, be symmetric for such fluids, but will satisfy the relationship. 

First of all, surface rheology is completely described by four rheological parameters elasticity and viscosity of compression/dilatation and of shear. In every case surface flow is coupled with the hydrodynamics of the adherent liquid bulk phase. From interfacial thermodynamics we know that the integration over the deviation of the tangential stress tensor from the bulk pressure represents the interfacial tension y (after Bakker 1928). 

Sepehr et al. [2008] are investigating the viscoelastic Giesekus model [Giesekus, 1982, 1983 Bird et al., 1987] coupled with Eq. (16.41), with Dr described by Doi [1981]. The interactions between polymer and particles were incorporated following suggestions by Fan [1992] and Azaiez [1996]. These authors used Eq. (16.42) with the contribution to stress tensor caused by clay platelets [Eq. (16.43)] and viscoelastic Giesekus matrix expressed as [Fan, 1992]... 

---