Modeling of the Plastic Behavior of Inhomogeneous Media

1984 ◽  
Vol 106 (4) ◽  
pp. 295-298 ◽  
Author(s):  
M. Berveiller ◽  
A. Zaoui
Keyword(s):  
Integral Equation ◽  
Strain Rate ◽  
The Self ◽  
Plastic Strain Rate ◽  
Plastic Behavior ◽  
Consistent Model ◽  
Self Consistent ◽  
Common Framework ◽  
Strain Rate Field ◽  
General Method

The general method which work for the case of linear properties are taken as a pattern for the derivation of the local total strain rate field of an inhomogeneous material (such as a polycrystal, a composite, or multiphase material) submitted to a uniform macroscopic elastic-plastic strain rate; the localization procedure leads to an integral equation, the solution of which is shown to yield the overall behavior. This method allows to locate within a common framework all the classical approaches, namely Taylor’s, Lin’s ones, the self-consistent scheme…and to suggest possible extensions. Some significant results are reported, concerning strain-hardening, yield stress, and texture devolopment of FCC polycrystals, according to simplified, one-site self-consistent model.

scholarly journals Comparison of finite-element and homogenization methods for modelling the viscoplastic behaviour of a S2–columnar-ice polycrystal

2000 ◽  
Vol 30 ◽  
pp. 115-120 ◽  
Author(s):  
Jacques Meyssonnier ◽  
Armelle Philip
Keyword(s):  
Finite Element ◽  
Strain Rate ◽  
Transversely Isotropic ◽  
Uniform Strain ◽  
Upper And Lower Bounds ◽  
Uniform Stress ◽  
Consistent Model ◽  
Self Consistent ◽  
Polycrystalline Ice ◽  
Homogenization Methods

AbstractThe main homogenization schemes used to model the behaviour of polycrystalline ice are assessed by studying the particular case of a two-dimensional polycrystal which represents natural S2–columnar ice. The results of the uniform-stress, uniform-strain-rate and one-site self-consistent models are compared to finite-element computations. The comparisons were made using the same model of grain, described as a continuous transversely isotropic medium, in the linear and non-linear cases. The uniform-stress and uniform-strain-rate models provide upper and lower bounds for the macroscopic fluidity which are too far from each other to be useful when a degree of anisotropy relevant to ice is considered. Although the self-consistent model gives a weak representation of the interaction between a grain and its surroundings, due to the strong anisotropy of the ice crystal, the resulting macroscopic behaviour is found to be acceptable when compared to the results from finite-element computations.

Quantitative estimation of incompatibility stresses and elastic energy stored in ferritic steel

2008 ◽  
Vol 41 (5) ◽  
pp. 854-867 ◽  
Author(s):  
A. Baczmanski ◽  
P. Lipinski ◽  
A. Tidu ◽  
K. Wierzbanowski ◽  
B. Pathiraj
Keyword(s):  
Elastic Energy ◽  
Ferritic Steel ◽  
Quantitative Estimation ◽  
Second Order ◽  
The Self ◽  
X Ray Diffraction ◽  
Near Surface ◽  
Consistent Model ◽  
Self Consistent ◽  
Euler Space

Plastic incompatibility second-order stresses were determined for different orientations of a polycrystalline grain, using X-ray diffraction data and results of the self-consistent elasto-plastic model. The stresses in cold rolled ferritic steel were determined both in as-received and under tensile loaded conditions. It has been shown that the Reuss model and the self-consistent model applied to near surface volume provide the best approaches to determine diffraction elastic constants. For the first time, the elastic energy in an anisotropic material (arising from plastic incompatibilities between grains having various lattice orientations) has been determined. The second-order incompatibility stresses and stored elastic energy are presented in Euler space.

Estimating Brown-Korringa constants for fluid substitution in multimineralic rocks

Geophysics ◽  
2013 ◽  
Vol 78 (3) ◽  
pp. L27-L35 ◽  
Author(s):  
Gary Mavko ◽  
Tapan Mukerji
Keyword(s):  
Solid Phase ◽  
The Self ◽  
Fluid Substitution ◽  
Gassmann’S Equation ◽  
Consistent Model ◽  
Additional Constant ◽  
Self Consistent ◽  
The Individual ◽  
Pore Lining ◽  
Consistent Formulation

Brown and Korringa extended Gassmann’s equations for fluid substitution in rocks to allow for arbitrarily mixed mineralogy. This extension was accomplished by adding just one additional constant—replacing the mineral bulk modulus with two less intuitive constants. Even though virtually all rocks have mixed mineralogy, the Brown and Korringa equations are seldom used because values for the constants are unknown. We estimate plausible values for the Brown-Korringa constants, based on effective medium models. The self-consistent formulation is used to describe a rock whose mineral and pore phases are randomly distributed ellipsoids—a plausible representation of randomly mixed mineral grains, as with dispersed clay in sandstone. Using the self-consistent model, the two constants are predicted to be nearly identical, justifying the use of an average mineral modulus in Gassmann’s equations. For small contrasts in mineral stiffness, the Brown-Korringa constants are approximately equal to the Voigt-Reuss-Hill average of the individual mineral bulk moduli. In a second approach, a multilayered spherical shell model is used to describe a rock where a particular solid phase preferentially coats grains or lines pores. In this case, the constants can differ substantially from each other, demonstrating the need for the Brown-Korringa equation. A third model represents weak pore-lining or pore-filling clay within an arbitrary pore geometry. The clay-fluid mix can be replaced exactly with an average fluid or “mud.” When the nonclay minerals have similar moduli, then the replacement of the clay-fluid mix causes the Brown-Korringa equation to revert to Gassmann’s equation.

scholarly journals A generalization of the equations of the self-consistent field for two-electron configurations

1937 ◽  
Vol 160 (903) ◽  
pp. 588-604 ◽  
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Configuration Interaction ◽  
The Self ◽  
Angular Function ◽  
Radial Functions ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
General Method

The most successful general method so far devised for dealing with many- electron atoms is th a t of the self-consistent field (abbreviated in what follows to “ s. c. f.” ). If greater accuracy is required than is obtainable with the method as ordinarily used (either with or without exchange), either the so-called “ configuration interaction ” must be taken into account —usually a very laborious procedure—or else more complicated (varia­tional) methods must be used, which must be designed separately for each particular case, and in which the concept of each electron being assigned to its own “ orbit” is usually abandoned. It would seem desirable, therefore, to have, if possible, some general method which will increase the accuracy of the calculations without taking into account configuration interaction, and which will still allow the conceptual features of the s. c. f. method (i. e. the assignment of “ orbits” ) to be retained. In this paper such a method is developed for the case of two-electron configurations in Russell-Saunders coupling. The method consists in assuming a form for the wave function which is similar to that used in the s. c. f. method, except that the proper spatial symmetry is allowed for (which is not so in the case of the s. c. f. equations without exchange), and further, an adjustable function of Θ, the angle between the radii vectores to the two electrons, is inserted as a multiplying factor. The usual varia­tional method is then applied, and yields differential equations for the two radial functions which are similar to those of the ordinary theory, together with an equation for the angular function.

scholarly journals Modelling Secular Variation in the Southwest Pacific for the last 400 years

2021 ◽  
Author(s):  
◽  
Maha Ali Alfheid
Keyword(s):  
Magnetic Field ◽  
High Resolution ◽  
Geomagnetic Field ◽  
Dipole Model ◽  
The Self ◽  
Pacific Region ◽  
Spherical Cap ◽  
Southwest Pacific ◽  
Consistent Model ◽  
Self Consistent

<p>A spherical cap harmonic analysis (SCHA) model has been used to derive a high-resolution regional model of the geomagnetic field in the southwest Pacific region over the past 400 years. Two different methods, a self-consistent and the gufm1 dipole method, have been used to fill in gaps in the available data.  The data used in the analysis were largely measurements of the magnetic field recorded in ships logs on voyages of exploration in the region. The method chosen for the investigation used a spherical cap of radius 𝜃₀ = 50° centered at co-latitude and longitude of (115°, 160°). The results of each method used for SCHA are presented as contour plots of magnetic field declination, inclination and intensity and are compared with similar plots for a global model, gufm1. The root mean square misfit of the self- consistent and gufm1 dipole model to the actual data were around 2900 nT and 23000 nT respectively.  Overall, the results suggest that the self-consistent model produces a more reliable model of the geomagnetic field within the area of interest than does the gufm1 dipole model. With more data included the self-consistent model could be further improved and used to develop a high-resolution mathematical model of the geomagnetic field in the southwest Pacific region.</p>

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