PSI - Issue 80

Riccardo Giacometti et al. / Procedia Structural Integrity 80 (2026) 219–231 R. Giacometti, N. Grillanda, V. Mallardo / Structural Integrity Procedia 00 (2023) 000–000

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(a)

(b)

Fig. 2: (a) Reference volume element (RVE) for running bond masonry texture, (b) kinematics of the RVE for each strain rate (membrane strains ϵ 11 and ϵ 22 at the top, shear strain ϵ 12 and micro-rotation ω 3 at the bottom).

with reference to a plastic strain rate tensor E applied to the full RVE: v G → P = Ey G + Ω y P − y G (3) in which the symbol y denotes the spatial coordinates of a point in the local reference system of the RVE, and Ω is the skew-symmetric tensor that contains a micro-rotation ω 3 . Such a micro-rotation works here as a rigid in-plane rotation of each individual block. Through Eq. (3) it is possible to fully identify the kinematics of the RVE starting from a plastic strain tensor and a micro-rotation (Fig. 2(b)). However, a kinematic field is kinematically admissible only if the associated flow rule is satisfied. In other words, a Mohr-Coulomb frictional law (expressed in kinematic quantities) must be imposed along the joints: n and ∆ v j s are respectively the normal and the tangential component of the velocity jump along the j-th interface, whereas µ is the friction coe ffi cient. It is quite easy now to express the velocity jumps in terms of plastic strain rates and micro-rotations. By combining Eqs. (3) and (4) we obtain the following conditions: µ 2 | 2 ϵ 12 − ϵ 11 β + 2 ω 3 | ≤ ϵ 22 − ϵ 12 β 2 + ω 3 β 2 (5a) µ 2 | 2 ϵ 12 + ϵ 11 β + 2 ω 3 | ≤ ϵ 22 + ϵ 12 β 2 − ω 3 β 2 (5b) µ | ϵ 12 − ω 3 | ≤ ϵ 11 (5c) where β = b / a is the aspect ratio of the brick. Eqs. (5) are expressed in matrix from through the matrix P in the LP problem (2). ∆ v j n ≥ µ ∆ v j s for j = 1...5 (4) where ∆ v j Local mesh refinement is applied to optimize the mechanism representation and minimize the dependence on the adopted mesh. In this procedure, the limit analysis problem is iteratively defined and solved while the mesh is refined at each iteration in correspondance of elements undergoing plastic strain rates. Once the analysis is performed on an initial mesh, a load factor and a set of plastic strain rates is obtained. The following steps are then followed. • The L 2 -norm of plastic strain rates is computed to identify the elements that need to be refined. Any element is refined if the L 2 -norm of its plastic strain rates is higher than a tolerance value. The refinement is conducted through subdivision into four elements with equal area. 2.3. Mesh refinement