Couple stress

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... 

One of the apparent results of introducing couple stress is the size-dependent effect. If the problem scale approaches molecular dimension, this effect is obvious and can be characterized by the characteristic length 1. The size effect is a distinctive property while the film thickness of EHL is down to the nanometre scale, where the exponent index of the film thickness to the velocity does not remain constant, i.e., the film thickness, if plotted as a function of velocity in logarithmic scale, will not follow the straight line proposed by Ham-rock and Dowson. This bridges the gap between the lubrication theory and the experimental results. 

The couple stress method can be used for modeling a special case of micro-polar fluids, i.e., the two-phase flow, wherein the constitutive equation is given by [22,34-38]... 

Z = Vp/p is named as the characteristic length which contributes to the couple stress effect. 

Das, N. C., Elastohydrodynamic Lubrication Theory of Line [39] Contacts Couple Stress Fluid Model, Tribal. Trans., Vol. 40, No.2,1997,pp.353-359. 

Lin, J. R., Squeeze Film Characteristics Between a Sphere and [40] a Flat Plate Couple Stress Fluid Model, Camputers Structures, Vol. 75,2000, pp. 73-80. 

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... 

In the case of heat transfer analysis, axial temperature distribution, shown in Figure 3 are specified for the surfaces of both He and sulfuric flow cannels, considering heat transfer coefficients. And outer surface of block is modeled as adiabatic condition. Figures 5 and 6 show the temperature and the stress distributions in the block, respectively. The stress shown in Figure 6 is a coupled stress with thermal stress and static stress caused by the operating pressure difference between He and sulfuric acid. Analytical conditions are as follows ... 

Experimental studies on coupled stress- flow of single rock fractures... 

Hybrid model of Discrete Fracture Network (DFN) and Discrete Element Method (DEM) for coupled stress-flow analysis of rock slope stability (Wang, 2000) ... 

In terms of numerical methods, the dominance by FEM with equivalent continuum approach might not be most suitable for sparsely or moderately fractured hard rocks and more advanced methods and codes using discrete approach are needed. The issue of applicability of the equivalent media approach, the associated scale effects, and uncertainty evaluations need to be fully explored. The processes are dominated by coupled stress-flow problems and effects of thermo-chemical effects need more attention. More works for soils, clays, sands and other similar media, which are equally, if not more, important in the fields of geo-engineering and environments, seem also needed. 

Su B, Zhan M and Wang Y. 1997. Experimental study on the coupled stress and fluid flow in fractured rock masses. Chinese J. of Geotech. Engng., 19(4), pp. 73-77. 

Wu Y. 1996. The coupled stress-ground water flow-model for jointed rock mass. J. of Geological Hazards Environmental... 

Wu Y and Zhang Z. 1996. An application of generalized double porosity media model for coupled stress and groundwater flow analysis in fractured rock mass. J. of Engineering Geology, 1(3), pp. 40-46. 

Oda, M., 1986. An Equivalent Continuum Model for Coupled Stresses and Fluid Flow Analysis in Jointed Rock Masses, Water Resour. Res. 22,... 

Oda, M., 1986. An equivalent continuum model for coupled stress and fluid flow analysis and jointed rock masses. Water Resources Research, V. 22, No. 13, pp. 1845-1856. 

This article will try to clarify relevant physical processes in shale stability in drilling. However, a quantitative framework for coupling stress-strain-strength behavior and transport processes in shale must be based on induced volume changes, and this issue is not explicitly discussed. 

For the basic equations of coupled stress-flow analysis mentioned above, it is very difficult to solve them in closed-form. The transposition method of progression and integration can only be applied for problems of boundary value problems of simple geometry and boundary conditions. Therefore the finite element method (FEM) is used to solve the coupled partial differential equations in this paper. 

The FEM modelling was applied for the simulating the dam behaviour with coupled stress-flow effects under different loading cases according to the water level variations. The tangential displacements of Longyangxia dam s typical dam section on April 16, 1989, April 16, 1990, and April 16, 1996, and December 31, 1999 were calculated and compared with measured data. The results are listed in Table 2 and illustrated in Figure 4. 

The formulation of extended continua can be traced back to the Cosserat brothers [13], who enriched the standard continuum by rotational degrees of freedom. As a consequence, such a continuum does not only transfer stresses but also couples stresses. The Cosserat or micropolar theory is well established now [17, 22, 32]. It is successfully applied if materials with microstructure are modeled on the macroscale, e.g., the equivalence of the Cosserat model is shown for materials with lattice-Hke microstractures [1, 2, 16, 25, 34, 35, 41] and with granular materials [3-5, 8, 15]. Furthermore, the micropolar approach is taken into account if strain localization arises which in the framework of the standard continuum would lead to a mesh dependence in the numerical solution (see, e.g.. Ref [29]). The Cosserat model introduces a so-called internal length A in the theory which allows for the existence of solutions of the bound-... 

Lakes, R. S. (1982), Dynamical study of couple stress effects in human compact bone, J. Biomech. Eng. 1 4(1) 6-11. 

At 21.9 MPa stress load, the mega-coupled composite exhibits a much steeper strain versus time behavior than stampable polypropylene composite and fails after about 46.5 hr. However, this creep behavior is exceptional compared with the corresponding uncoupled GFRP, which fails almost immediately as reported by Silverman (33). This trend indicates that the mega-coupled stress trans-... 

If all gradients Vcr, Vm, and VT are sufficiently small, their products may be discarded. Then, upon dispensing with several intermediate steps one obtains the coupled stress-diffusion expression... 

A somewhat different result, where B = fi/p was obtained elsewhere (Wu 2001). Employing Eshelby s stress (Eshelby 1951) that article presented a scheme that resulted in coupled stress-diffusion formulations as weU. 

Park S K and Gao X-L (2006) Bernoulli-Euler beam model based on a modified couple stress theory, J Micromech Microeng 16 2355-2359. 

To complete our dynamic theory, it is necessary to prescribe forms for the dynamic stress and couple stress. The simplest choice consistent with known effects appears to be that at a material point... 

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