Application of generalized logistic equation for b-value analysis in fracture of plain concrete beams under flexure

Abstract

A B S T R A C TThis work investigates the effect of finite size and evolution of fracture process zone on b-value of acoustic emission in concrete beams. It is shown that the AE b-value determined using the Gutenberg-Richter (GR) law varies with beam size and invalidates universality. Since GR law fits the frequency-magnitude curve partially, a recently proposed Maslov's generalized logistic equation (GLE) is used for b-value analysis. The existence of cut-off magnitude and nonlinearity of frequency-magnitude distribution is explained through crack interaction mechanisms occurring within the FPZ. The growth of micro and macro cracking in concrete is observed to follow logistic or sigmoid growth law rather than the power law. The b-values obtained from GR and GLE are compared and it is found that the b-value of GLE correlates well with the effective crack length during damage process of plain concrete thereby exhibiting damage compliant behavior which could be used in health monitoring of structures. As GR law was first introduced for statistical analysis of earthquakes and applied without any modification to acoustic emission https://doi.T due to its reasonable interpretation of this phenomenon [2], it is necessary to review its development in seismological studies. Therefore, the present work mostly reviews seismological and geophysical literature. Mainly, three different types of approaches are followed in literature to improve b-value estimation: (1) employing statistics based formulation by accounting nonlinearity while preserving pure GR law, (2) modifying GR law at large magnitudes by associating it with a polynomial or exponential or logarithmic cut-off and (3) suggesting an alternative model or generalizing GR law to consider the shape of distribution. By idealizing the occurrence of earthquakes as a Poisson process, the first approach with validity of GR law has yielded various statistical b-value estimates such as maximum likelihood estimate or Aki's formula [3], improved b-value [4], Weichert's formula [5], generalized Aki-Utsu β-estimator [6] and few others. An alternate approach based on robust fitting [7] has been proposed which is shown to be more efficient than maximum likelihood estimate. Cut-off on maximum magnitude approach has resulted in many of upper bound models such as tapered GR, truncated Gamma law, truncated GR, etc. The third approach has not received significant contribution except for a few which are noted further. The recent development in Tsallis nonextensive statistical mechanics has offered a generalized form of GR law [8-10] based on Tsallis entropy. A Bayesian approach [11] is also notable in the category of generalized GR law. Apart from the above mentioned contributions, an approach based on logistic equation proposed by Maslov et al. [12] is distinct due to its fractional power-law exponent. This two parameter model regards nonlinear CFD over whole magnitude range with soft cut-off for larger magnitudes. The overall shape of CFD suggest that the growth of microcracks is log...

Analysis of Acoustic Emission Characteristics and Failure Mode of Deep Surrounding Rock of Sanshandao Gold Mine

“…The change in the b -value is closely related to the brittleness of the rock. The b -value change is sensitive to the unstable growth of macro-cracks [ 45 , 46 , 47 , 48 , 49 ]. Therefore, the b -value is employed as an early warning signal of rock failure.…”

The universality of b-value and size effect in acoustic emission: Experimental investigations in quasi-brittle fracture

“…In this direction, one of the approach currently exercised is the generalization of GR law. Recent works based on generalized logistic equation [11,12], Bayesian statistics [13] and non-extensive statistical mechanics [14] are promising and needs to be explored further.…”

Acoustic Emission Monitoring of Progressive Damage of Reinforced Concrete T-Beams under Four-Point Bending

“…In the field of seismology, C. F. Richter and B. Gutenberg proposed an empirical relationship between earthquake frequency and magnitude, which is commonly known as the GR law, and the exponent of this law is known as the b-value [ 50 ]. The expression for the seismic b-value is given as: where N is the number of earthquakes whose magnitude is .…”

“…In the context of AE-based damage assessment using the -value , earlier research works were carried out to find the fracture process in rocks [ 54 , 55 ]. Later, some researchers implemented this method in the progressive damage assessment of concrete structures [ 50 ].…”