PSI - Issue 10

A. Kyriazopoulos et al. / Procedia Structural Integrity 10 (2018) 97–103 A. Kyriazopoulos et al. / Structural Integrity Procedia 00 (2018) 000 – 000

99 3

Eq.(1) (as reported by Tsallis (2009)) introduces an invariant under permutation entropic form based on q i p using a simple formulation as:   1 W q q i i S F p    (3)

The simplest expression for

is a linear function, thus it may easily concluded that:

1 i i S C C p     2 1 w q q

(4)

As expected, in order for the entropy to quantify the disorder of a system it must satisfy that C 1 + C 2 =0, hence Eq.(4) becomes as follows:   1 1 1 W q q i i S C p     (5)

B C k q  

In order for the Sq to approach BG entropy 1

must be fulfilled. The traditional BG entropy S 1 is said to be

1

additive since for a system composed of two (probabilistically) independent subsystems the entropy S 1 of the sum coincides with the sum of the entropies. Contrary to that the entropy Sq (when q is not equal to 1) is non-additive. For a system composed of two statistically independent systems, A and B, the Tsallis entropy satisfies:

 B A B q S A B S S S A S B k              1 , q q q q q

(6)

which is referred to as pseudoadditivity. The last term on the right hand side of Eq.(6) brings the origin of non extensivity. Therefore the value of q indicates the degree of non-additivity in the described system and q <1, q =1, q >1 correspond to super-additivity, additivity and sub-additivity conditions, respectively. This is the fundamental principle for non-extensive statistical physics. All other quantities, concerned with thermodynamics, may be derived from it. The PSC relaxation recorded during the 3PB tests is analysed using non-extensive statistical physics in order to study the behavior of the Tsallis parameters as well as the background level of the electric signal emission and con clude whether they constitute potential indices for monitoring the loading level of the specimen, targeting at the prediction of the upcoming fracture of a cement mortar beam. According to the experimental data and the conducted analysis it is clearly concluded that the variation of the entropic index q obtains a well deterministic form enabling the estimation of the upcoming event of fracture. In this work, cement mortar specimens are subjected to five repetitive Three-Point Bending (3PB) loading/ unload ing tests. It was verified that the PSC values during each next loading/unloading process were lower. The PSC relaxation during the time interval for which the mechanical load was maintained constant at its high level followed a q-exponential decrease, according to the Non Extensive Statistical Physics (NESP) Tsallis’ entropy (Tsallis (2009)). Similar experiments have been conducted on marble and amphibolite specimens, involving other loading protocols and clearly show compatible results (Vallianatos and Triantis (2012)).

2. Materials and the experimental protocol

The cement-mortar specimens used were made of ordinary Portland cement, sand and water at a ratio (per weight) 1:3:0.5, respectively. Prismatic specimens (square cross section 40 mm x 40 mm, length 100 mm) were prepared and cured for 90 days in order to attain 95% of their strength. Preliminary 3PB tests were conducted and the force causing fracture was found about 4.5 kN. All the mechanical tests were carried out using an Instron DX-300 electromechanical