PSI - Issue 78

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

2105

with e 1 , e 2 and e 3 being the Cartesian unit vectors and I 1 , I 2 and I 3 being the depolarisation factors for an ellipsoid with half-lengths a 1 , a 2 and a 3 . The depolarisation factors are calculated according to:

1 2  1 2  1 2 

∞

a 1 a 2 a 3

dx

I 1 =

3 / 2  ( x + a 2 3 / 2  ( x + a 2 3 / 2  ( x + a 2 a 1 a 2 a 3 a 1 a 2 a 3

( x + a 2

2 3 )

0

1 )

2 )( x + a

∞

I 2 =

dx

(7)

( x + a 2

2 3 )

0

2 )

1 )( x + a

∞

dx

I 3 =

2 2 )

( x + a 2

0

3 )

1 )( x + a

Finally, the electrical resistivity tensor of the composite is equal to: ρ =    σ m +     i ω i N i    ·    ω m I +  i ω i  σ i − ρ − 1  − 1 · N i    − 1    − 1

(8)

2.4. Analysis procedure

In the proposed methodology, a mechanically-driven electromechanical coupled problem is solved. Three mecha nisms are considered a ff ecting the properties of the composite due to imposed mechanical strain: a) shift of inclusion aspect ratio (AR), b) shift of matrix and inclusion volume fraction (VF), and c) closure of cracks due to compression (CC). Since all inclusions are spherical or aligned with the axis of mechanical loading, it is not necessary to account for reorientation of the inclusions in the deformed composite. Initially, the sti ff ness tensor of the cementitious composite is calculated in the undeformed state according to Eq. 4 and the resistivity is calculated using Eq. 8. Next, the composite is subjected to a macroscopic stress vector σ . The strain vector of the composite is calculated according to: ε = σ · C − 1 (9) Next, the strain vector at each phase of the composite (matrix and inclusion families) is calculated according to: ε m = A m · ε ε i = A i · ε (10) The properties of the deformed composite are calculated next. Firstly, the shift in aspect ratio is considered by calculating the new inclusion half-lengths. These new half-lengths of an ellipsoid subjected to normal strains ε x , ε y and ε z are equal to:

a ′ 1 = (1 + ε x ) · a 1 a ′ 2 = (1 + ε y ) · a 2 a ′ 3 = (1 + ε z ) · a 3

(11)

Secondly, the new volume V ′ i of an ellipsoid or cube-shaped composite with initial volume V subjected to a set of normal strains ε x , ε y and ε z is equal to: V ′ = (1 + ε x ) · (1 + ε y ) · (1 + ε z ) · V (12) Applying Eq. 12 to the composite as a whole and to the inclusions, the new volume fractions for all components can be calculated, the volume of the matrix being the di ff erence between the total volume and the volume of the