Direct current electric potential in an anisotropic half-space with vertical contact containing a conductive 3D body

For a quadripole‐quadripole array, in which current is sequentially injected into the ground by two perpendicular dipoles, an apparent resistivity can be defined in terms of the vectorial cross product of the two electric fields measured at the receiver site. Transform equations are derived (for horizontally layered media) which relate this apparent resistivity to the apparent resistivities obtained with conventional dipole‐dipole and Schlumberger arrangements. To evaluate the method, two mathematical models are used. The first model is a half‐space with an “alpha conductivity center,” and the second model is a half‐space with a vertical contact. For an idealized quadripole‐quadripole array, simple expressions are found for the apparent resistivity, which is shown to be independent of the orientation of the current quadripole. Theoretical anomalies calculated for the quadripole‐quadripole array are compared with those obtained for a dipole‐quadripole array. It is shown that whereas the apparent resistivity map for the dipole‐quadripole array varies greatly with the azimuth of the source dipole, the results obtained with the quadripole‐quadripole array consistently display a remarkable resemblance to the assumed distribution of true resistivity. This is especially true when the current quadripole is placed at a large distance from the surveyed area.

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Determination of Crack-Initiation in Fracture Toughness Testing Using an Experimental Key-Curve Methodology

10.1115/pvp2021-61812 ◽

2021 ◽

Author(s):

S. Pothana ◽

G. Wilkowski ◽

S. Kalyanam ◽

J. K. Hong ◽

C. J. Sallaberry

Keyword(s):

Fracture Toughness ◽

Crack Initiation ◽

Crack Growth ◽

Power Law ◽

Direct Current ◽

Electric Potential ◽

Crack Mouth Opening Displacement ◽

Ductile Tearing ◽

Curve Method ◽

Unloading Compliance

Abstract A new approach was implemented to confirm the start of ductile tearing relative to assessments by other methods such as direct-current Electric Potential (d-c EP) method in coupon specimens. This approach was developed on the Key-Curve methodology by Ernst/Joyce and is similar to the ASTM E-1820 Load Normalization procedure used to determine J-R curves directly from load versus Load-Line Displacement (LLD) record of the test specimen. It is consistent with Deformation Plasticity relationships for fully plastic behavior. Using this Experimental Key-Curve method, crack initiation can be determined directly from load versus LLD data or load versus Crack-Mouth Opening Displacement (CMOD) obtained from a fracture test without the need for additional instrumentation required for crack initiation detection. It is based on the fact that plastic deformation of homogeneous metals at the crack tip follows a power-law function until the crack tearing initiates. Crack tearing initiation is determined at the point where the power-law fit to the load versus plastic part of CMOD or LLD curve deviates from the total experimental load versus plastic-CMOD or LLD curve. The procedure for fitting of the data requires some care to be exercised such that the fitted data is beyond the elastic region and early small-scale plastic region of the Load-CMOD or Load-LLD curve but include data before crack initiation. An iterative regression analysis was done to achieve this, which is shown in this paper. The iterative fitting in this region typically results with a coefficient of determination (R2) values that are greater than 0.990. This method can be either used in conjunction with other methods such as direct-current Electric Potential (d-c EP) or unloading-compliance methods as a secondary (or primary) confirmation of crack tearing initiation (and even for crack growth); or can be used alone when other methods cannot be used. Furthermore, when using instrumentation methods for determining crack-initiation such as d-c EP method in a fracture toughness test, it is good to have a secondary confirmation of the initiation point in case of instrumentation malfunction or high signal to noise ratio in the measured d-c EP signals. In addition, the Experimental Key-Curve procedure provides relatively smooth data for the fitting procedure, while unloading-compliance data when used to get small crack growth values frequently has significant variability, which is part of the reason that JIC by ASTM E1820 is determined using an offset with some growth past the very start of ductile tearing. In this work, the Experimental Key-Curve method had been successfully used to determine crack tearing initiation and demonstrate the applicability for different fracture toughness specimen geometries such as SEN(T), and C(T) specimens. In all the cases analyzed, the Experimental Key-Curve method gave consistent results that were in good agreement with other crack tearing initiation measuring method such as d-c EP but seemed to result in less scatter.

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Bleustein-Gulyaev Waves in an Inhomogeneous Layered Piezoelectric Half-Space

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.306-308.1223 ◽

2006 ◽

Vol 306-308 ◽

pp. 1223-1228

Author(s):

Fei Peng ◽

Hua Rui Liu

Keyword(s):

Fourier Transform ◽

Phase Velocity ◽

Half Space ◽

Electric Potential ◽

Piezoelectric Layer ◽

Field Equations ◽

Transform Method ◽

Coupled Field ◽

Coupled Field Equations ◽

The Fourier Transform

The propagation of Bleustein-Gulyaev (BG) waves in an inhomogeneous layered piezoelectric half-space is investigated in this paper. Application of the Fourier transform method and by solving the electromechanically coupled field equations, solutions to the mechanical displacement and electric potential are obtained for the piezoelectric layer and substrate, respectively. The phase velocity equations for BG waves are obtained for the surface electrically shorted case. When the layer and the substrate are homogenous, the dispersion equations are in agreement with the corresponding results. Numerical calculations are performed for the case that the layer and the substrate are identical LiNbO3 except that they are polarized in opposite directions. Effects of the inhomogeneities induced by either the layer or substrate are discussed in detail.

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ELECTRIC POTENTIAL NEAR A SPHERICAL BODY IN A CONDUCTING HALF‐SPACE

Geophysics ◽

10.1190/1.1440212 ◽

1971 ◽

Vol 36 (4) ◽

pp. 763-767 ◽

Cited By ~ 12

Author(s):

David B. Large

Keyword(s):

Point Source ◽

Electric Current ◽

Half Space ◽

Electric Potential ◽

Spherical Body ◽

Classical Potential

An extensive summary of classical potential solutions has been given recently by Van Nostrand and Cook (1966). This note presents a solution for the potential due to a point source of electric current placed on the earth’s surface in the vicinity of a buried spherical body of arbitrary resistivity. The analysis follows the procedure suggested by Van Nostrand and Cook and is similar to that used recently by Merkel (1969, 1971).

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Analysis of direct current flowing inside a linear increasing conductivity half-space

2010 12th International Conference on Optimization of Electrical and Electronic Equipment ◽

10.1109/optim.2010.5510434 ◽

2010 ◽

Author(s):

Ovidiu Centea ◽

Iosif Vasile Nemoianu ◽

Emil Cazacu ◽

Veronica Paltanea ◽

Gheorghe Paltanea

Keyword(s):

Half Space ◽

Direct Current

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Depth of investigation of collinear electrode arrays over homogeneous anisotropic half‐space in direct current methods

Geophysics ◽

10.1190/1.1441214 ◽

1981 ◽

Vol 46 (5) ◽

pp. 768-780 ◽

Cited By ~ 12

Author(s):

B. B. Bhattacharya ◽

M. K. Sen

Keyword(s):

Half Space ◽

Direct Current ◽

Inverse Relationship ◽

Electrode Arrays ◽

Definition Of ◽

Depth Of Investigation

The definition of depth of investigation as suggested by Evjen (1938) [subsequently used by Roy and Apparao (1971) also for the study of depth of investigation of electrode arrays in direct current methods for homogeneous isotropic earth] has been used to study the depth of investigation of various collinear electrode arrays for a homogeneous anisotropic half‐space. It has been shown that some simple transformations are to be applied to the expressions of normalized depth of investigation characteristic (NDIC) of the same arrays over homogeneous isotropic earth to obtain normalized depth of investigation characteristic of various arrays placed over homogeneous anisotropic earth. The novelty of anisotropy is that the depth of investigation of collinear electrode arrays over homogeneous anisotropic half‐space bears an inverse relationship with the coefficient of anisotropy and also depends upon array length and dip of the plane of stratification. The effect of the coefficient of anisotropy is most pronounced for horizontally stratified anisotropic earth and is independent of it for vertically stratified anisotropic earth—entirely consistent with the concept of the “;paradox of anisotropy.” The depth of investigation of all the collinear arrays for inclined stratification lies somewhere between the values obtained for horizontal and vertical stratifications.

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Relationship of the self‐potential Green’s function to solutions of controlled source direct‐current potential problems

Geophysics ◽

10.1190/1.1440945 ◽

1979 ◽

Vol 44 (11) ◽

pp. 1879-1881 ◽

Cited By ~ 5

Author(s):

David V. Fitterman

Keyword(s):

Green’S Function ◽

Green's Function ◽

Direct Current ◽

Streaming Potential ◽

The Self ◽

Simple Relationship ◽

Controlled Source ◽

Source Mechanisms ◽

Vertical Contact ◽

Self Potential

This note presents a simple relationship between the self‐potential (SP) Green’s function and the solution of the controlled‐source direct‐current (dc) potential problem which allows a simplified means of determining the SP Green’s function. An example of its application to the vertical contact problem will be presented. The case of a streaming potential source mechanism will be considered, although any of the SP source mechanisms described by Nourbehecht (1963) could be substituted.

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