PSI - Issue 66

Daniele Gaetano et al. / Procedia Structural Integrity 66 (2024) 478–485 Author name / Structural Integrity Procedia 00 (2025) 000 – 000

482

5

3. Numerical results This section is devoted to the numerical results, obtained by employing the model previously described to investigate the buckling phenomena at the microscale, induced by debonding. The nonlinear stress-strain behaviour is described by means of a hyperelastic constitutive law for a Neo-Hookean solid. Assuming that C is the right Cauchy-Green tensor,  the initial shear modulus, and J the Jacobian of the transformation, the strain energy density function takes the following form, in which the compressibility of the materials is dictated by the Lamé parameter  .

1 2

1 2

( ) ( ) tr 3  − C

2

.

(12)

ln( ) J

ln( ) J

W

=

−



+



The RVE is made of a reinforcement material, embedded into a matrix phase, arranged in a unit cell Hx5H (Fig. 1), with a volume fraction / f f V H H = , in which f H is the height of the fiber and has been set equal to 1 mm. For the chosen RVE the reinforcement volume fraction is equal to 0.05.

2

1

atrix

einforcement

ohesive interface

Fig. 1. Representative Volume Element (RVE) of the hyperelastic layered composite.

In order to assess how the differences in stiffness between the two phases influence the microstructure’s behaviour, a stiffness ratio / E f m k E E = between Young’s modulus of the matrix and the fiber has been introduced. In this work, such a factor has been set equal to 1, with the matrix Young’s modulus equal to 1 GPa, meaning that a homogenous material has been adopted. The Poisson ratios have been chosen as 0.33 and 0.48 for matrix and reinforcement, respectively. The domain has been discretized by employing a mapped mesh consisting of 1080 quadrilateral elements. The cohesive parameters needed to describe the interface behaviour by means of the proposed traction-separation law are here summarized: max 7[MPa] t = , 2 107[J /m ] Ic IIc G G = = . The determination of the critical load levels and mode shapes have been carried out via a one-way coupled parametric FE model, formulated by a total Lagrangian approach. The numerical procedure is made of 2 steps: the first one is needed to evaluate the principal path step of the unit cell, whereas in the second one eigenvalue problems have been solved, which are useful for microstability analyses (Greco et al., 2021). During the former step, the shear load has been applied until different values of damage are reached, and then a compression load has been applied to induce instability. In the latter step, however, the eigenvalue problem, due to the adopted cohesive law, is a nonlinear problem. At this stage of preliminary results, linearized eigenvalue problems have been solved. They have been evaluated along the loading curve, which being the path with lower stiffness may provide a lower bound of the associated eigenvalues. A combined shear-compression loading condition has been applied to the unit cell within the first step via the periodic boundary conditions depending on the macroscopic deformation gradient. As shown in Fig. 2(A), the critical load factor is plotted against the percentage of the maximum mode II displacement jump exhibited by the cohesive interface. In particular, on the principal shear macroscopic path, six different values of the mode II displacement jump have been chosen as a fraction of the mode II critical displacement jump corresponding to the total decohesion (i.e. 5%, 30%, 80%, 90%, 95% and 97%). On such values, the compression load has been applied in order to induce instability and assess the critical load factor by employing the proposed nonlinear homogenization framework. The results are plotted with reference to the proposed stability functional formulations: the exact one (Eq. 7), and the simplified I and II formulations, reported in equations 10 and 11, respectively. Fig. 2(B) reports instead the critical mode shapes associated with the exact stability functional. It