PSI - Issue 79

Umberto De Maio et al. / Procedia Structural Integrity 79 (2026) 386–393

389

  

  

(

)

  

  

 

∫

∫

∫

(2)

e

T u n σ

T u q σ

W ds m

J

W

ds

W

ds

=

−∇

+

+

−∇



(

) (

)

c +

c −

r

r

Γ

∂Ω − ∪ ∂Ω −

∂

i ω

where W is the strain energy density, ∇ u is the displacement gradient and σ is the stress tensor. m and q are unit outward normal to crack face c ∂Ω and internal material interface boundaries i ω ∂ (see Figure 1 1), while the double brackets symbol   ⋅ indicates the jump of the enclosed quantity, defined as the difference between the values on the positive and negative sides of the interface. With the J-integral components computed, the crack kinking angle is determined by maximizing the energy release rate, as per the MERR criterion:

J

( ) G J θ =

( ) θ

( ) sin , θ

2

(3)

cos

arctan

J

+

θ ⇒ =

1

2

J

1

A special case is handled for cracks that nucleate within the bulk material, far from any pre-existing notch or crack. In such scenarios, a J-integral evaluation is not reliable. Therefore, the propagation direction is instead determined by a local, stress-based criterion, assuming the crack grows orthogonal to the direction of the maximum principal stress evaluated at the initiation point. 3. Results This section presents the validation of the proposed ALE-driven cohesive framework. The model is first employed to analyze the influence of the mesh size on the global structural response of a concrete specimen and then, a comparison with both experimental and numerical results is provided to assess the effectiveness of the proposed model in capturing the nonlinear process of the cracking phenomena in multiphase materials. A notched concrete beam, already experimentally and numerically analyzed by (Choi et al., 2025), containing a single circular granite inclusion is simulated. The geometry and boundary conditions are illustrated in Figure 2. The elastic properties assigned to the constituent materials are as follows: the concrete matrix is defined with a Young’s modulus E m = 25 GPa and a Poisson’s ratio ν m = 0.2, while the granite inclusion is stiffer, with E g = 70 GPa and the same Poisson’s ratio ν g = 0.2).

Figure 2. Geometry and boundary conditions of the tested specimen.

To investigate potential mesh dependence, a sensitivity analysis was conducted. This study focused specifically on the discretization of this critical zone highlighted in Figure 2 with the dashed blue line. Four distinct unstructured meshes were generated using a Delaunay tessellation, as depicted in Figure 3. These meshes feature progressively increasing refinement; the maximum element size was systematically reduced, ranging from 20 mm for the coarsest mesh (Mesh 1) down to 3 mm for the finest mesh (Mesh 4). This study is crucial for assessing the convergence of the solution and selecting a mesh density that provides accurate results without incurring excessive computational costs. It is important to note that, for all four mesh configurations, a hybrid kinematic approach was employed to optimize performance. The moving mesh formulation was constrained exclusively to the central region of the beam, directly ahead of the initial notch (indicated by the dashed blue rectangle in Figure 2). This is the critical area where damage and crack propagation are expected. The remaining part of the domain, which is not expected to fracture, was modelled using a standard, computationally cheaper Lagrangian description.