Stress tensor and averaging in mechanics of continuous media: PMM vol. 39, n≗2, 1975, pp. 374–379

Abstract

The problem of determining the mean stress and the other macrovariables originates upon passing from the equations of motion which are valid in the microscale, to the macroscopic equations which describe the motion of continuous media (such as a turbulized fluid, an elastic medium with microdefects, the suspension of gas bubbles or solid particles in a fluid, etc.). The mean value of the stress tensor over a volume was introduced in the monograph [1], and precisely this quantity was used in the governing relations to compute the Einstein viscosity of suspensions. Moreover, some effective representation in terms of integrals over surfaces [1] was used in specific calculations of these means with respect to the volume. Later, Batchelor [2], and some other authors after him [3], used precisely these means with respect to the volume as the stresses in the macroequations of motion by assuming the equivalence between the average with respect to a volume and with respect to a surface. Hence, in particular, the absolute symmetry of the macrostress tensor follows in the above-mentioned cases. In this paper it is shown that the average of the microstress tensor and the microflux of the momenta with respect to the volume according to the rule in [1] determines only some symmetric part of the complete macrostress tensor. For the simple case of viscous fluid moving inhomogenously over a microlevel, this mean of the tensor with respect to the volume is related linearly to the mean strain rates. Moreover, the presentation used in [1] permits clarification of the essential difference between the mean stresses with respect to the volume and with respect to the surfaces, in the general case. The method of integrating the microequations with respect to the vioume [4–6] naturally results in the appearance of stresses in the macroequations, which are the means with respect to the differential macroareas. It is essential that the macrostress tensory is hence generally nonsymmetric although the equations of motion in the microscale correspond to symmetric continuum mechanics. It is this consideration which permitted the development of the continuum equations of motion of a suspension, which reflects the effect of nonequllibrium intrinsic rotation of the suspended particles [7], and the case of a turbulized fluid with anisotropies of eddy character is set in conformity to the nonzero antisymmetric part of the Reynolds stresses [8].