The generalized stokes integral theorem for a covariant pseudotensor field

Ориентируемые континуумы играют важную роль в микрополярной теории упругости, все реализации которой возможны только в рамках псевдотензорного формализма и представления об ориентируемом многообразии. Особенно это касается теории микрополярных гемитропных упругих сред. В настоящей работе рассматриваются различные формулировки интегральной теоремы Стокса для асимметричного ковариантного пседотензорного поля, заданного веса. Тем самым достигается распространение известной интегральной формулы Стокса на случай псевдотензоров. Последнее обстоятельство позволяет использовать, указанное обобщение для микрополярных континуумов. Исследование существенно опирается на класс специальных координатных систем. Oriented continua play an important role in the micropolar theory of elasticity, all realizations of which are possible only within the framework of the pseudotensor formalism and the orientable manifold concept. This especially concerns the theory of micropolar hemitropic elastic media. In this paper, we consider various formulations of the Stokes integral theorem for an asymmetric covariant pseudotensor field of a given weight. This extends the well-known Stokes integral formula to the case of pseudotensors. The latter circumstance makes it possible to use the manifistated generalization for micropolar continua. The study relies heavily on the class of special coordinate systems.

On a pseudotensor generalization of the Hugoniot-Hadamard linking boundary conditions

В представляемой работе исследуются особенности связывающих двусторонних граничных условий на поверхностях разрывов, распространяющихся в сплошных средах (в частности, в микрополярных континуумах). Теория Югонио-Адамара, существенно развитая Г.И. Быковцевым, распространения поверхностей разрывов физических полей обобщена на случай псевдотензорного полевого описания. Вводятся понятия фундаментального ориентирующего псевдоскаляра и псевдоскалярного времени. Исследуется геометрия поверхностей уровня псевдоскалярного поля, представляющих интерес для механики наращиваемых тел. Вводится понятие псевдонормали к поверхности. Обсуждаются вопросы дифференцирования по псевдоскалярному времени и его преобразования при зеркальных отражениях и инверсиях пространства. Получены геометрические и кинематические условия совместности первого порядка в терминах псевдотензоров. Выведены условия совместности для слабых разрывов перемещений и микровращений в микрополярном континууме. The present work deals with the linking boundary conditions formulated on the both sides of a propagating wave surface (in particular, in micropolar continua). The Hugoniot-Hadamard theory of physical fields wave surfaces propagation, essentially developed by G.I. Bykovtsev, is generalized to the case of a pseudotensor field description. The concepts of fundamental orienting pseudoscalar and pseudoscalar time are introduced and discussed. The geometry of level surfaces of a given pseudoscalar field is studied. The concept of a pseudovector normal to a surface is introduced. The pseudoscalar time derivative is proposed and discussed. Geometric and kinematic first order compatibility conditions are obtained in terms of pseudotensors. The compatibility conditions are derived for weak discontinuities of displacements and microrotations due to defromations of the micropolar solid.

Applications of micropolar SPH in geomechanics

A smoothed particle hydrodynamics code based on micropolar continua for geomaterials is developed for problems involving large deformation and shear strain localization. Two typical geotechnical problems, i.e., biaxial compression test and sand column collapse, are simulated using classical and micropolar model to demonstrate the performance of the newly proposed method. A parameter study is given on the scale effect in the micropolar continua.

Comparative 3D DEM simulations of sand–structure interfaces with similarly shaped clumps versus spheres with contact moments

Three-dimensional simulations of a monotonic quasi-static interface behaviour between initially dense cohesionless sand and a rigid wall of different roughness during tests in a parallelly guided direct shear test under constant normal stress are presented. Numerical modelling was carried out by the discrete element method (DEM) using clumps in the form of convex non-symmetric irregularly shaped grains. The clumps had an aspect ratio of 1.5. A regular grid of triangular grooves (asperities) along the wall with a different height at the same distance was assumed. The numerical results with clumps were directly compared under the same conditions with our earlier DEM simulations using pure spheres with contact moments with respect to the peak and residual interface friction angle, width of the interface shear zone, ratio between grain slips and grain rotations, distribution of contact forces and stresses. The difference between the behaviour of clumps and pure spheres with contact moments proved to be noticeable in the post-peak regime due to a different particle shape. The rolling resistance model with pure spheres was proved to be limited for capturing particle shape effects. Three different boundary conditions along the interface were proposed for micropolar continua, considering grain rotations and grain slips, wall grain moments and wall grain forces, and normalized interface roughness. The numerical results in this paper offer a better understanding of the interface behaviour of granular bodies in DEM and FEM simulations.

PSEUDOTENSOR FORMULATION OF THE MECHANICS OF HEMITROPIC MICROPOLAR MEDIA

The possibility of applications of relative tensors concepts to the mechanics of micropolar continuum and, in particular, for the hemitropic micropolar continua is considered. The fundamental tensors and orienting relative scalars in three-dimensional space are introduced. Permutation symbols and absolute Levi-Civita tensors are investigated in further details. Algebraic and differential properties of the relative tensors are discussed. The weights of the fundamental kinematic tensors are determined. The wryness tensor and the asymmetric strain tensor are constructed in terms of the vectors of micro-rotation and displacements. Notions of force and couple traction vectors, associated force and associated couple stress vector, force and couple stresses tensors are discussed in the frameworks of relative tensors algebra. The weights of the basic micropolar elasticity tensors are determined and discussed. The constitutive form of the micropolar elastic potential is introduced as an absolute scalar in order to obtain micropolar constitutive equations. In the linear case, the elastic potential is a quadratic form whose coefficients are pseudoscalars. The weights of the constitutive pseudoscalars are calculated. The dimensionless constitutive micropolar constants and constitutive constants with physical dimensions are discriminated. Statics and dynamics of micropolar elastic continua are developed in terms of relative tensors. Dynamic equations involving displacements and microrotations in the case of semi-isotropic (hemitropic) symmetry are derived and represented by the pseudotensor technique. The paper can be considered as a script of fundamental formulas and concepts related to the algebra and differentiation of relative tensors of arbitrary rank.