Higher-Order Terms in the Current-Current Theory of Weak Interactions

1971 ◽  
Vol 3 (2) ◽  
pp. 559-566 ◽  
Author(s):  
R. S. Willey ◽  
J. M. Tarter
Keyword(s):  
Weak Interactions ◽  
Current Theory ◽  
Higher Order ◽  
Higher Order Terms

Using the Multi-Parameter Fracture Mechanics for more Accurate Description of Stress/Displacement Crack Tip Fields

2013 ◽  
Vol 586 ◽  
pp. 237-240 ◽  
Author(s):  
Lucie Šestáková
Keyword(s):  
Stress Distribution ◽  
Fracture Mechanics ◽  
Boundary Conditions ◽  
Displacement Field ◽  
Singular Term ◽  
Crack Tip ◽  
Higher Order ◽  
Precise Description ◽  
Accurate Knowledge ◽  
Higher Order Terms

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.

The Skyrme model and higher order terms

1990 ◽  
Vol 235 (1-2) ◽  
pp. 141-146 ◽  
Author(s):  
Luc Marleau
Keyword(s):  
Higher Order ◽  
Skyrme Model ◽  
Higher Order Terms

scholarly journals Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I

2020 ◽  
pp. 237-249
Author(s):  
L. V Stepanova
Keyword(s):  
Series Expansion ◽  
Digital Image ◽  
Power Series Expansion ◽  
Crack Tip ◽  
Higher Order ◽  
Accurate Estimation ◽  
Principal Stresses ◽  
Isochromatic Fringe ◽  
Higher Order Terms ◽  
Fringe Patterns

This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

scholarly journals FRACTIONAL TREND OF THE VARIANCE IN CAVALIERI SAMPLING

2011 ◽  
Vol 19 (2) ◽  
pp. 71 ◽  
Author(s):  
Marta García-Fiñana ◽  
Luis M Cruz-Orive
Keyword(s):  
Fractional Derivatives ◽  
Real Data ◽  
Current Theory ◽  
Continuous Derivative ◽  
General Representation ◽  
Higher Order Terms ◽  
Measurement Function ◽  
Constant Distance ◽  
Derivatives Of ◽  
Maclaurin Formula

Cavalieri sampling is often used to estimate the volume of an object with systematic sections a constant distance T apart. The variance of the corresponding estimator can be expressed as the sum of the extension term (which gives the overall trend of the variance and is used to estimate it), the 'Zitterbewegung' (which oscillates about zero) and higher order terms. The extension term is of order T2m+2 for small T, where m is the order of the first non-continuous derivative of the measurement function f, (namely of the area function if the target is the volume). A key condition is that the jumps of the mth derivative f (m) of f are finite. When this is not the case, then the variance exhibits a fractional trend, and the current theory fails. Indeed, in practice the mentioned trend is often of order T2q+2, typically with 0 <q <1. We obtain a general representation of the variance, and thereby of the extension term, by means of a new Euler-MacLaurin formula involving fractional derivatives of f. We also present a new and general estimator of the variance, see Eq. 26a, b, and apply it to real data (white matter of a human brain).

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