PSI - Issue 78

Anastasios Drougkas et al. / Procedia Structural Integrity 78 (2026) 2102–2109

2106

′ can also be calculated according to the new volume

inclusions. The new sti ff ness tensor of the deformed composite C

fractions and aspect ratios using Eq. 4. Finally, crack closure is considered by reducing the volume fraction of the cracks oriented perpendicularly to the applied compressive stress. Cementitious and other brittle materials are characterised by a crack closure stress value σ cc , ranging widely between 5% − 50% of the compressive strength f c and which does not present a clear threshold value, as discussed in Eberhardt et al. (1998). Not all cracks experience closure simultaneously due to variations in crack aspect ratio and width. To account for this uncertainty, it is assumed that the crack closure stress for individual cracks follows a skew normal distribution. For a uniaxial stress applied in the x direction equal to the crack closure stress, the value for the crack closure micro-strain ε cc can be calculated from Eq. 9 and Eq. 10 by plugging in the strain concentration tensor A cx for the cracks perpendicular to x and the stress vector containing σ cc . Thus it is assumed that the crack closure strain follows a similar as that for σ cc , with a mean value ε cc , a standard deviation of s and a skewness a . Considering that compressive stresses and strains are negative, the proportion of closed cracks p cc due to compres sion when the micro-strain in the x -oriented crack family is equal to ε c can be calculated from the expression: with T being Owen’s T function and a being the skewness parameter for the distribution. Once p cc has been calculated, the total volume of the composite is reduced by the volume of closed cracks, the volume fraction of the x -oriented cracks is modified according to p cc and all other volume fractions are finally modified based on the new reduced total volume of the composite. Based on the new volume fractions and aspect ratios of the components due to the three mechanical e ff ects, the new composite resistivity ρ ′ of the deformed composite can be handily calculated. Assuming that the composite having a resistivity in x direction equal to ρ x is subjected to a uniaxial stress σ x resulting in a normal strain ε x and a shift in resistivity to ρ ′ x , the piezoresistive gauge factor for a given loading interval is equal to: p cc = 1 − 1 s √ 2 π ε c −∞ exp − ( x − ε cc ) 2 2 s 2 dx + 2T ε c − ε cc s , a (13)

ρ ′ x − ρ x ρ x ε x

(14)

λ =

The gauge factor over a loading curve can be calculated as the slope of the linear regression model linking strain ε x and fractional change of resistivity ρ ′ x − ρ x /ρ x registered at distinct points. 3. Analysis results

3.1. Preliminary analysis

For evaluating the performance of the proposed model, the results of an analysis using nominal values typ ical of cementitious materials, corresponding to a cement mortar, will be performed. The reference values used for this analysis are presented in Table 1. The published literature is currently lacking in combined mechani cal / electrical / microstructural characterisation experimental campaigns, which are required for generating the nec essary data for the proposed model. In the absence of such a combined comprehensive campaign, the material phases are prescribed nominal values from selected sources the literature, such as Esposito and Hendriks (2016); Nezˇerka and Zeman (2012); Drougkas et al. (2023b). The Young’s modulus of the paste E m refers to the solid phase of the paste, i.e., excluding the e ff ect of porosity, thus it does not represent the apparent or bulk modulus of the hardened paste as regularly measured in the lab. The three crack families with the three di ff erent orientations are assigned permuted sets of the half-lengths a c 1 , a c 2 and a c 3 indicated in the Table, while the provided volume fraction refers to the sum of all cracks. In addition to the mechanical properties, the mechanical loading needs to be prescribed. The parameters used are presented in Table 2. The standard deviation s and skewness a of the crack closure strength distribution have been selected so that less than 0.1% of the cracks have a positive crack closure stress and that the distribution reflects the experimentally-determined bias towards smaller-width cracks noted by Wu et al. (2014). The mechanical loading is