PSI - Issue 78

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

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Being spherical, the orientation of pores and aggregates is irrelevant in a homogenisation context. To account for the expected random orientation of the cracks, and to maintain model simplicity, three families of cracks ( cx , cy and cz ) are considered oriented along the three Cartesian directions, x , y and z . Each family is considered to occupy one third of the total volume fraction of the cracks. Orientation bias, such as that caused by extrusion in 3D printing, can be handily introduced by altering the relative proportion of each crack family. The composite material and the inclusion families are illustrated in Fig. 1b. All material phases are considered homogenous and isotropic, although anisotropy can be prescribed with minimal changes to the modelling strategy. The inclusions are considered homogeneously dispersed in and perfectly bonded with the surrounding matrix. For deriving the mechanical properties of the cementitious composite, the plain Mori-Tanaka scheme is employed, described in Mori and Tanaka (1973). According to this approach, the dilute estimate of the i -th inclusion family is equal to: T i =  I + S i · ( C m ) − 1 · ( C i − C m )  − 1 (1) where I is the identity tensor, C i and C m are respectively the sti ff ness tensors of the inclusion and the matrix and S i is Eshelby’s tensor, found in Mura (1987). The strain concentration factor of the matrix is equal to: A m =    ω m I +  i ω i T i    − 1 (2) where ω m and ω i are the volume fractions of the matrix and the i -th inclusion family. The strain concentration tensor of the i -th inclusion family is equal to: A i = T i · A m (3) Finally, the sti ff ness tensor of the cementitious composite is equal to: C = C m +  i ω i ( C i − C m ) · A i (4) 2.3. Electrical homogenisation The electrical conductivity of the cementitious composite is derived by employing a self-consistent Mori-Tanaka scheme. The plain Mori-Tanaka scheme produced results close to the Hashin-Shtrikman lower bound, namely values that greatly underestimate the true electrical conductivity. In the self consistent scheme, the inclusions are considered embedded in the composite medium instead of simply in the matrix. This approach produces more realistic results at the cost of nonlinearity of the problem, which therefore needs to be solved numerically. In the adopted approach, the conductivity contribution tensor of the i -th family of inclusions is equal to: N i =  P i + ( σ i − σ m ) − 1  − 1 (5) where σ m and σ i are the electrical conductivity tensors of the matrix and i -th family of inclusions respectively and P i is the interaction tensor, calculated according to the expression: P i = I 1 e 1 ⊗ e 1 + I 2 e 2 ⊗ e 2 + I 3 e 3 ⊗ e 3 (6) 2.2. Mechanical homogenisation