A Mesomechanical Model for Brittle Deformation Processes: Part II

1989 ◽  
Vol 56 (1) ◽  
pp. 57-62 ◽  
Author(s):  
Dragoslav Sumarac ◽  
Dusan Krajcinovic
Keyword(s):  
Closed Form ◽  
Plane Stress ◽  
Inelastic Strain ◽  
The Self ◽  
Crack Interaction ◽  
Brittle Deformation ◽  
Closed Form Solutions ◽  
Consistent Model ◽  
Self Consistent ◽  
Deformation Processes

The approximate, closed-form solutions for the inelastic strain and compliances are derived for some simple plane stress and plane strain cases. The computations are performed for a model-ignoring crack interaction as well as for the case of the self-consistent model.

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 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>

Prediction of Biot’s coefficient from rock-physical modeling of North Sea chalk

Geophysics ◽  
2008 ◽  
Vol 73 (2) ◽  
pp. E89-E96 ◽  
Author(s):  
Casper Olsen ◽  
Kathrine Hedegaard ◽  
Ida L. Fabricius ◽  
Manika Prasad
Keyword(s):  
Aspect Ratio ◽  
North Sea ◽  
Free Parameter ◽  
The Self ◽  
P Wave ◽  
Biot’S Coefficient ◽  
Aspect Ratios ◽  
Consistent Model ◽  
Self Consistent ◽  
Water Saturated

We predict Biot’s coefficient for North Sea chalk based on density and P-wave velocity for water-saturated chalk. We compare three effective medium models: Berryman’s self-consistent model, the isoframe model, and the bounding-average method (BAM). The self-consistent model is used with two combinations of aspect ratios. In one combination, the aspect ratio is equal for pores and grains. In the other combination, the aspect ratio for grains is kept constant close to 1 and the aspect ratio for pores varies. All the models include one free parameter that determines the stiffness of the rock for a fixed porosity. This free parameter is compared with Biot’s coefficient to discuss whether the free parameter is related to pore-space compressibility for North Sea chalk. We also discuss how consistent the models are between P-wave modulus and shear modulus for dry and water-saturated chalk. The acoustic velocity and the density data for dry and water-saturated chalk are all laboratory data. The isoframe model and the BAM model predict Biot’s coefficient with a smaller error than the self-consistent model does. The free parameter in the isoframe model and the BAM model is related to Biot’s coefficient. The free parameter in the self-consistent model is related only to Biot’s coefficient for water-saturated chalk when the aspect ratios for pores and grains are equal. The isoframe and the BAM model are generally more consistent for chalk than the self-consistent model is.

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