PSI - Issue 26

Andronikos Loukidis et al. / Procedia Structural Integrity 26 (2020) 277–284 Loukidis et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction and theoretical background Estimating the structural integrity and the remaining service life of structural elements and structures is important, especially when human activity is involved. In order to achieve that many non-destructive testing techniques have been proposed and tested. Among them, the Pressure Stimulated Currents (PSC) technique (Stavrakas et al., 2004) which is based on the detection of weak electrical signals generated in quasi-brittle materials when subjected to mechanical loading. The PSCs are related to the creation, growth and concretion of microcracks to macrocracks, in the bulk of the specimen. The crack network forms a charged system due to the development of local polarizations and thus an electric charge flow is generated (Varotsos et al., 1982, 1999). Experimental results have shown that the PSC technique has been applied successfully for the detection of the impending failure in brittle materials such as marble (Stavrakas et al., 2003), amphibolite (Triantis et al., 2007) and cement based specimens under compression (Kyriazopoulos et al., 2011; Triantis et al., 2012), subjected to various loading protocols. PSCs in materials under mechanical stress are considered non-linear processes under non-equilibrium conditions, involving a wide range of spatiotemporal scales, exhibiting fractality, long range interaction and memory effects (Hirata, 1987; Meredith et al., 1990; Cox and Meredith, 1993; Benson et al., 2008; Vallianatos et al., 2011, 2012). In order to describe such phenomena, Tsallis based on the principle of entropy, introduced a generalization of the Boltzmann-Gibbs (BG) statistical physics known as non-extensive statistical physics (NESP) (Tsallis, 1988, 2009a,b). The Tsallis entropy q S for the case of a variable x with probability distribution function   p x , is defined as where B k is Boltzmann’s constant, w is the total number of microstates and q is the entropic index that reflects the degree of non-additivity in a physical system representing the case of many non-independent, long-range interacting subsystems and memory effects (Tsallis, 1988, 2009a,b). According to Eq. 1 the entropic form, based on   p x , must be invariant under permutation (Tsallis, 2009a). The simplest formulation which holds, as reported in Tsallis (2009a), is:   1 w q q i i S H p    , with ( ) H x being a continuous function. The simplest expression of ( ) H x is a linear function, hence, 1 2 1 w q q i i S C C p     . Due to its entropic nature, q S quantifies as a measure of disorder, resulting to 1 2 0 C C   , and thus,   1 1 1 w q q i i S C p     , is obtained from the above equations. In order for the q S to approach the Boltzmann-Gibbs entropy of classical statistical physics the condition   1 / 1 B C k q   must be fulfilled, resulting to Eq.1. It should be noted, that when 1 q  , Eq.1 gives the classical Boltzmann – Gibbs entropy formulation, 1 ln w BG B i i i S k p p     . As already mentioned, the index q represents the degree of non-additivity in the described system, when 1 q  , 1 q  and 1 q  correspond to sub-additive, additive and super-additive systems, respectively (Tsallis, 2009a). The non-additivity of a system encompassing two statistically independent sub-systems A and B can be described by the following relation           1 q q q q q B q S A B S A S B S A S B k      , where the last term expresses the non-additivity of the physical system, due to long-range interactions. Thus, for a super-additive system,       q q q S A B S A S B    while, for a sub-additive one,       q q q S A B S A S B    . 1.1. PSC relaxation signal It is experimentally verified (Anastasiadis et al., 2004; Triantis et al., 2007; Kyriazopoulos et al., 2011;Triantis et al., 2012; Stergiopoulos et al., 2015) that a steep increase of the compressive stress from an initial level i  to a higher 1 1 , 1   w  q B q i    i S k  p q     (1)