PSI - Issue 61

Onur Ali Batmaz et al. / Procedia Structural Integrity 61 (2024) 305–314 Onur Ali Batmaz et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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Following the initiation, damage progresses through the gradual degradation of the transverse and shear moduli within any flagged element, as governed by the matrix damage variable d m . This variable ranges from 0 (indicating an undamaged state) to 1 (representing complete damage) along the line A-C in Figure 2(a). The explicit formula for d m given by d m = [δ e f q (δ eq −δ e0 q )] [δ eq (δ e f q −δ e0 q )] ⁄ where δ e0 q , δ eq and δ e f q are the equivalent displacements at the initiation, progression and complete damage states, respectively. The equivalent displacement at the completely damaged state is calculated by equating the dissipated energy (i.e. the area under the curve) to the fracture toughness, G c , of the respective mode, i.e. δ e f q =2G c σ e0 q ⁄ . G c is considered equivalent to the Mode-I fracture toughness (G I,c ) for a tensile matrix failure (σ N >0), and approximated by the relation G II,c /cos( 0 ) (Maimí et al., 2007) for a compressive matrix failure (σ N <0). As the damage evolves, stiffness degradation is implemented by incorporating degraded values of transverse and shear moduli into the compliance tensor S, as described in detail by Matzenmiller et al. (1995). 2.2.2. Delamination Delamination is simulated using the built-in cohesive zone model (CZM) in ABAQUS. A bilinear traction response to the respective separation at the interfaces is utilized in this study, as depicted in Figure 2(b). Considering the multiaxial nature of transverse impact loading, the interaction of opening and shear fracture modes is taken into account (mixed-mode CZM). Prior to delamination initiation, the region enforces a linear elastic response with penalty stiffness values of K N and K S for the opening and shear modes, respectively. K N and K S are determined using the methodology presented by Turon et al. (2018, 2010) and the corresponding numerical values can be found in Table 1. The delamination initiation is predicted using the quadratic nominal stress criterion by Chang and Springer (1986), expressed as ሺ 〈T I 〉 T o,I ⁄ ሻ ʹ ൅ ሺ T II T o,II ⁄ ሻ ʹ ൅ ሺ T III T o,III ⁄ ሻ ʹ =1 where T i ( = I, II, III ) are the surface tractions of corresponding modes. The separation at the damage initiation is determined by the relation δ o =T o /E o where the corresponding initiation traction is calculated as T o =(T o2 ,I +T o2 ,II +T o2 ,III ) 1/2 . After predicting the damage initiation within a cohesive element, a linear softening behavior is employed during the damage propagation by degrading the penalty stiffnesses with the cohesive damage variable D, defined as D= [δ c (δ−δ 0 )] [δ(δ c −δ 0 )] ⁄ . The effective separation δ is c alculated by δ=√〈δ I 〉 2 +δ I 2 I +δ I 2 II , and the critical separation indicating the point of fracture, is determined by δ c =2 /T o . The mixed-mode fracture toughness is computed using the Benzeggagh and Kenane (B-K) criterion,

(5)

where η is the mixed-mode interaction parameter.

Initiation criterion

FI =1

Propagation criterion

(a) (b) Figure 2. Constitutive response of (a) composite material including matrix damage model and the (b) mixed-mode cohesive zone model for delamination damage.