PSI - Issue 39

Fabrizio Greco et al. / Procedia Structural Integrity 39 (2022) 638–648 Author name / Structural Integrity Procedia 00 (2019) 000–000

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In (Pulatsu et al. (2016)), the discrete element method (DEM) proposed by Cundall (Cundall and Strack (1979)) has been used for investigating the uniaxial tensile behavior of quasi-brittle materials. In the DEM, the geometry of the computational domain is depicted as a discontinuous systemmade up of polyhedral blocks. The interaction between the blocks is governed by contact constitutive models reproducing softening behavior because of crack nucleation and propagation. This strategy permits effectively capturing the overall material behavior both in terms of carrying capacity and cracking mechanism. Therefore, the use of modeling strategies based on refined geometric representation for the numerical models offers a fresh perspective on the development of novel approaches able to ensure reliable predictions of the fracture behavior of quasi-brittle materials. This is especially important in the case of masonry structures, in which the geometric arrangement of units and mortar joints highly affects the overall mechanical and fracture behavior (Greco et al. (2018b), Greco et al. (2020b), Greco et al. (2021b)). The present work proposes a refined FE modeling strategy based on the use of a Diffusive Interface Modeling (DIM) approach for the nonlinear analysis of brick masonries at the micro-scale, i.e. , the scale in which masonries can be view as a combination of bricks and mortar joints. The DIM is an inter-element fracture approach that consists of the insertion of cohesive interface elements along all the internal boundaries of the given volumetric finite element mesh (Xu and Needleman (1994), De Maio et al. (2019a), De Maio et al. (2019b), De Maio et al. (2020a), De Maio et al. (2020b), De Maio et al. (2021)). Specifically, the bulk finite elements of bricks and mortar joints behave as linear elastic, whereas cohesive interface elements reproduce the inelastic response because of crack propagation mechanisms. In particular, the model comprises (physical) brick/mortar interfaces and (mathematical) mortar/mortar interface elements equipped with mixed-mode traction-separation laws. The DIM offers several advantages, such as a unified reproduction of nucleation and propagation of cracks. However, it can suffer from convergence issues during numerical simulation when many cohesive elements undergo softening behavior simultaneously (García et al. (2015), Lazarus et al. (2015)). To address such an issue, an effective strategy base on the introduction of a certain form of imperfection in terms of cohesive properties of the embedded interface elements is used. To assess the reliability of the proposed model, numerical simulations of direct tensile tests of brick masonry couplets are performed. In this context, comparison results between the proposed DIM model and simplified models based on the use of a single layer of interface elements placed along with theoretical crack trajectories are proposed. 2. Theoretical background and numerical implementation 2.1. Cohesive/volumetric finite element formulation Figure 1 shows a two-dimensional masonry composite structure occupying the region Ω , which consists of a combination of mortar joints and bricks, denoted with m Ω and b Ω , respectively, regarded as continuous linear elastic phases. The boundary Γ of Ω comprises two subsets denoted as D Γ and N Γ , where Dirichlet and Neumann boundary conditions are assigned, respectively. Such a scheme is discretized in several finite elements once that the geometry is introduced in a traditional FE program. Once that the mesh configuration is defined, interface elements are inserted in the numerical model. Specifically, (physical) brick/mortar ( int bm Γ ) and (mathematical) mortar/mortar ( int mm Γ ) interfaces are used. This assumption is consistent with experimental observations denoting that the crisis of the masonry can occur because of mortar cracking or debonding phenomena along with brick mortar interfaces (Pluijm (1997)). Under the action of body force f on Ω and surface forces t on N Γ , in presence of constraints on D Γ , by assuming quasi-static loading conditions and small deformations, the following cohesive/volumetric variational formulation is valid, i.e., find U ∈ u such that

∫

∫

∫

∫

∫

(1)

δ C ε ε ⋅

m C ε ε δ ⋅

( ) t u u § ¨ § ¨ δ ⋅

δ f u

δ t u

δ Γ ∀ ∈ u

d

d

d

d

d

U

Ω +

Ω +

Γ = ⋅

Ω + ⋅

int

b

int mm

\ Ω Γ

Ω

Γ

Γ

\ Ω Γ m

int

int

b

N