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.

Understanding Micropolar Theory in the Earth Sciences I: The Eigenfrequency $$\omega _r$$

Even though micropolar theories are widely applied for engineering applications such as the design of metamaterials, applications in the study of the Earth’s interior still remain limited and in particular in seismology. This is due to the lack of understanding of the required elastic material parameters present in the theory as well as the eigenfrequency $$\omega _r$$ ω r which is not observed in seismic data. By showing that the general dynamic equations of the Timoshenko’s beam is a particular case of the micropolar theory we are able to connect micropolar elastic parameters to physically measurable quantities. We then present an alternative micropolar model that, based on the same physical basis as the original model, circumvents the problem of the original eigenfrequency $$\omega _r$$ ω r laking in seismological data. We finally validate our model with a seismic experiment and show it is relevant to explain observed seismic dispersion curves.

Understanding Micropolar Theory in the Earth Sciences II: The Seismic Moment Tensor

Seismic events produced by block rotations about vertical axis occur in many geodynamic contexts. In this study, we show that these rotations can be accounted for using the proper theory, namely micropolar theory, and a new asymmetric moment tensor can be derived. We then apply this new theory to the Kaikōura earthquake (2016/11/14), Mw 7.8, one of the most complex earthquakes ever recorded with modern instrumental techniques. Using advanced numerical techniques, we compute synthetic seismograms including a full asymmetric moment tensor and we show that it induces measurable differences in the waveforms proving that seismic data can record the effects of the block rotations observed in the field. Therefore, the theory developed in this work provides a full framework for future dynamic source inversions of asymmetric moment tensors.

Finite element implementation and application of a sand model in micropolar theory

In traditional finite element failure analyses of geotechnical structures, the micro grain rotations cannot be modelled and numerical solutions are mesh dependent. In this study, a user element including rotational degree of freedom has been developed based on micropolar theory (Cosserat theory), then an enhanced non-associated sand model is calibrated with laboratory data and used to model the plane strain tests. The simulated results demonstrate the polarized model is able to model reasonably the sand behavior as well as the grain rotations in the localized region. What’s more, with this enhanced model, the mesh independent numerical solutions in terms of mechanical responses, shear bands thickness and orientations have been obtained. Article highlights In failure analysis of geostructures, significant rotations of soil grains have been observed to occur in the strain localized regions, but the current commercial Finite Element tools cannot model the micro rotations. Therefore, a user defined element must be developed to include the rotational degree of freedom. The micropolar approach is proven to be effective to model the grain rations in present paper. More suitable than other classical soil or sand constitutive models, the selected non-associated sand model in present paper is capable of describing well the contraction and shear dilatancy behaviors of sand. Then the model has been enhanced by means of micropolar technique, in this way, the reasonable strain localization phenomena in laboratory tests could be predicted well. There are always the mesh dependent problems for traditional simulations of the strain localization phenomena in finite element analysis. It can be found in present paper that the mesh independent numerical solutions are obtained by means of micropolar technique. Furthermore, the micropolar approach can obviously improve convergence difficulties in finite element analyses.

Strain Analysis of Transfer Faults in Extending Regions: Inferences from Micropolar Theory

<p>Transfer faults are generally identified as transversely oriented discrete faults linking normal fault segments in extensional tectonic settings.  The presence of the transfer faults in fault networks provides displacement transfer between the normal faults. The role and tectonic significance of transfer faults in overall extensional deformation of the upper crust is however not known very well. Micropolar theory extended by J-2 plasticity facilitates evaluation of a deforming medium in which cataclastic flow takes place with respect to each component of deformation. In this study, a series of experiments based on the Micropolar theory are performed, using fault-slip patterns, to better understand interplay among dip angle of normal and transfer faults connecting to each other, angle of linkage, and extensional direction. Synthetic linkage cases are created systematically considering various orientation of both faults sharing common stretching direction.<br>Our findings reveal that in orthogonal and oblique linkage cases, 3D strain field is mostly observed; a few cases exhibit plane strain. All cases are subjected to simple shearing. In cases of orthogonal linkage, extensional direction is predominantly oblique to the strike of the normal faults. Many of these cases have no block rotation (microrotation) independent from macrorotation. No particular relationship between changing dip amount of faults and direction of extension is observed. In cases of oblique linkage, (sub)orthogonal direction of extension appear in nearly half of experiments, especially those including normal faults dipping less than 60˚. The frequency of non-zero microrotation is seen apparently more than that in orthogonal linkage cases.<br>The study represents that structural togetherness of the transfer and normal faults essentially can accommodate complete micropolar strain in a region. This further suggests that not only the normal faults but the transfer faults should also be considered as major primary structural elements in extending domains.</p>

Torsional Characteristics of Carbon Nanotubes: Micropolar Elasticity Models and Molecular Dynamics Simulation

Efficient application of carbon nanotubes (CNTs) in nano-devices and nano-materials requires comprehensive understanding of their mechanical properties. As observations suggest size dependent behaviour, non-classical theories preserving the memory of body’s internal structure via additional material parameters offer great potential when a continuum modelling is to be preferred. In the present study, micropolar theory of elasticity is adopted due to its peculiar character allowing for incorporation of scale effects through additional kinematic descriptors and work-conjugated stress measures. An optimisation approach is presented to provide unified material parameters for two specific class of single-walled carbon nanotubes (e.g., armchair and zigzag) by minimizing the difference between the apparent shear modulus obtained from molecular dynamics (MD) simulation and micropolar beam model considering both solid and tubular cross-sections. The results clearly reveal that micropolar theory is more suitable compared to internally constraint couple stress theory, due to the essentiality of having skew-symmetric stress and strain measures, as well as to the classical local theory (Cauchy of Grade 1), which cannot accounts for scale effects. To the best of authors’ knowledge, this is the first time that unified material parameters of CNTs are derived through a combined MD-micropolar continuum theory.

Modeling of orientational polarization within the framework of extended micropolar theory

In this paper the process of polarization of transversally polarizable matter is investigated based on concepts from micropolar theory. The process is modeled as a structural change of a dielectric material. On the microscale it is assumed that it consists of rigid dipoles subjected to an external electric field, which leads to a certain degree of ordering. The ordering is limited, because it is counteracted by thermal motion, which favors stochastic orientation of the dipoles. An extended balance equation for the microinertia tensor is used to model these effects. This balance contains a production term. The constitutive equations for this term are split into two parts, one , which accounts for the orienting effect of the applied external electric field, and another one, which is used to represent chaotic thermal motion. Two relaxation times are used to characterize the impact of each term on the temporal development. In addition homogenization techniques are applied in order to determine the final state of polarization. The traditional homogenization is based on calculating the average effective length of polarized dipoles. In a non-traditional approach the inertia tensor of the rigid rods is homogenized. Both methods lead to similar results. The final states of polarization are then compared with the transient simulation. By doing so it becomes possible to link the relaxation times to the finally observed state of order, which in terms of the finally obtained polarization is a measurable quantity.

Density–orientation coupling for a microcontinuum approach to nematic liquid crystals subject to electric field

A microcontinuum description of compressible liquid crystals is examined accounting for a constitutive model based on mass microdensity. As a first point, we discuss the effectiveness of the micropolar theory on compressible continua, which is limited to static problems. Then, by a micromorphic representation of mass density, we show the consistence of some classical constitutive models for compressible nematic liquid crystals and remark their connection with the microinertia tensor. After the analysis of a constitutive micropolar model, we discuss a static problem for a layer of compressible nematic liquid crystal in a planar configuration. The effects of an applied electric potential are considered remarking the coupling of density distribution with the molecular orientation.