scholarly journals The distortion tensor of magnetotellurics: a tutorial on some properties

2016 ◽  
Vol 47 (2) ◽  
pp. 85-99 ◽  
Author(s):  
Frederick E. M. Lilley
Keyword(s):  
Distortion Tensor

Constitutive Equations for Hyperelastic Materials Based on the Upper Triangular Decomposition of the Deformation Gradient

2018 ◽  
Vol 24 (6) ◽  
pp. 1785-1799 ◽  
Author(s):  
Y. Q. Li ◽  
X.-L. Gao
Keyword(s):  
Energy Density ◽  
Constitutive Equations ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Deformation Gradient ◽  
Density Functions ◽  
Triangular Decomposition ◽  
Deformation Gradient Tensor ◽  
Hyperelastic Materials ◽  
Distortion Tensor

The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.

Computer programANIZCfor the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals

2003 ◽  
Vol 36 (1) ◽  
pp. 160-162 ◽  
Author(s):  
András Borbély ◽  
Juliana Dragomir-Cernatescu ◽  
Gábor Ribárik ◽  
Tamás Ungár
Keyword(s):  
Computer Program ◽  
Single Crystals ◽  
Programming Language ◽  
Slip Plane ◽  
Anisotropic Medium ◽  
Mechanical Equilibrium ◽  
Strain Anisotropy ◽  
Diffraction Contrast ◽  
Contrast Factor ◽  
Distortion Tensor

The computer programANIZChas been developed using the Pascal programming language for the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals. The contrast factor is obtained numerically by integrating the angular part of the distortion tensor in the slip plane. The distortion tensor is calculated by solving the sextic equation provided by the mechanical equilibrium of a single dislocation in an infinite anisotropic medium. The contrast factors can be used for the interpretation of strain anisotropy as obtained from peak profile measurements made on either single crystals, textured polycrystals or powders.

scholarly journals TWO-CONNECTION RENORMALIZATION AND NON-HOLONOMIC GAUGE MODELS OF EINSTEIN GRAVITY

2010 ◽  
Vol 07 (05) ◽  
pp. 713-744 ◽  
Author(s):  
SERGIU I. VACARU
Keyword(s):  
Riemannian Manifolds ◽  
Einstein Gravity ◽  
De Sitter ◽  
Renormalization Procedure ◽  
Gravitational Fields ◽  
Gauge Models ◽  
Renormalization Group Equations ◽  
Distortion Tensor ◽  
Perturbative Quantum ◽  
New Framework

A new framework for perturbative quantum gravity is proposed following the geometry of non-holonomic distributions on (pseudo)Riemannian manifolds. There are considered such distributions and adap-ted connections, also completely defined by a metric structure, when gravitational models with infinite many couplings reduce to two-loop renormalizable effective actions. We use a key result from our partner work arXiv: 0902.0911 that the classical Einstein gravity theory can be reformulated equivalently as a non-holonomic gauge model in the bundle of affine/de Sitter frames on pseudo-Riemannian space–time. It is proven that (for a class of non-holonomic constraints and splitting of the Levi–Civita connection into a "renormalizable" distinguished connection, on a base background manifold, and a gauge-like distortion tensor, in total space) a non-holonomic differential renormalization procedure for quantum gravitational fields can be elaborated. Calculation labor is reduced to one- and two-loop levels and renormalization group equations for non-holonomic configurations.

scholarly journals Кинетика развития паттернов макролокализации пластического течения металлов

2018 ◽  
Vol 60 (7) ◽  
pp. 1358
Author(s):  
Л.Б. Зуев ◽  
С.А. Баранникова ◽  
Б.С. Семухин
Keyword(s):  
Plastic Deformation ◽  
Acoustic Emission ◽  
Plastic Strain ◽  
Plastic Flow ◽  
Strain Localization ◽  
Plastic Distortion ◽  
Localization Zone ◽  
Distortion Tensor ◽  
The Relationship

AbstractThe features of the macroscopic inhomogeneity of plastic deformation in the form of autowaves with a pulsating amplitude are analyzed, and data on the localization of sources of acoustic emission at different stages of plastic flow in the stretching of fcc mono- and polycrystals are presented. The relationship between the local components of the plastic distortion tensor in the strain localization zone is traced. The role of acoustic phenomena accompanying the localization of plastic strain in the development of the process of plastic deformation is considered.

The effect of local distortions on time‐domain electromagnetic measurements

Geophysics ◽  
2004 ◽  
Vol 69 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Andreas Hördt ◽  
Carsten Scholl
Keyword(s):  
Magnetic Field ◽  
Electric Field ◽  
Time Domain ◽  
Step Response ◽  
Model Parameters ◽  
Difficult Case ◽  
Storage Site ◽  
Transient Electromagnetic ◽  
Electromagnetic Measurements ◽  
Distortion Tensor

Based on the time‐domain integral equation, we derive expressions for the effect of an anomalous body close to the receiver or close to the transmitter on transient electromagnetic measurements. Similar to magnetotellurics, the distortion of electric fields at late times can be described by a constant distortion tensor relating the secondary electric field to the primary field components that would be obtained in the absence of the body. The distortion of a single electric field transient is a static shift only for particular configurations over a layered half‐space. In the general case, the perturbation is time dependent because the direction of the total electric field vector varies with time. The theory nicely explains spatial variations in electric field transients measured during a high‐redundancy long‐offset transient electromagnetics (LOTEM) survey over an underground gas storage site. An inversion example with synthetic data illustrates how distortion can be corrected. The elements of the distortion tensor are determined simultaneously with the model parameters. Ambiguity is reduced by a regularization of the distortion parameters. In the example, the background model is recovered well, even for the difficult case where only one transmitter is used. The distortion of the magnetic field time derivatives caused by bodies close to the receiver is proportional to the time derivative of the primary electric step response. The distortion is generally not limited to early times and cannot be neglected in general. Transmitter overprint effects resulting in static shifts of vertical magnetic field time derivatives may also be understood from the theory.

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