PSI - Issue 77

Muhammad Jahanzeb Zia et al. / Procedia Structural Integrity 77 (2026) 111–118 Muhammad Jahanzeb Zia et al. / Structural Integrity Procedia 00 (2026) 000–000

114

4

(a)

(b)

t=0.14mm

53.1°

10mm 2.5 mm

20 mm

10mm

Fig. 1. (a) Specimen geometry; (b) Ply position along with thickness and fibre direction.

2.2. Damage criteria and evolution The Hashin damage criterion is one of the most widely utilized failure theories for fibre-reinforced polymer composites, as it effectively differentiates between distinct damage mechanisms within a laminate. In contrast to traditional criteria such as Tsai-Hill and Tsai-Wu, which provide a single failure index, the Hashin formulation identifies four separate modes of failure: fibre tension, fibre compression, matrix tension, and matrix compression. Each of these modes is described by a mathematical expression that correlates the applied stress components with the corresponding material strengths. Owing to its balance between physical and computational efficiency, the Hashin criterion has been integrated into finite element platforms for the simulation of composite damage initiation and evolution (Hashin, 1980). The four modes of failure are described as follows. Fibre tensile failure: ( 11 ⁄ ) 2 +( 12 ⁄ ) 2 =1 (1) Fibre compressive mode: ( ₁₁ / ₜ )² + ( ₁₂ / ₐ )² = 1 (2) Matrix tensile mode: ( ₂₂ / ₜ )² + ( ₁₂ / ₐ )²= 1 (3) Matrix compression mode: ( ₂₂ /(2 ))² + [( ₜ /(2 ))² − 1] ( ₂₂ / ₜ ) + ( ₁₂ / ₐ )² = 1 (4) The Benzeggagh-Kenane (BK) damage evolution law, originally developed for mixed-mode delamination, has also been extended to describe progressive damage in matrix and fibre failure within laminated composites. In this context, the BK formulation defines the critical energy release rate as a function of the mode mixing between normal and shear stresses, thereby governing the evolution of damage once initiation is predicted by a failure criterion such as Hashin: = +( − )( /( + ) η (5) For matrix cracking, the BK law captures the interaction between transverse tensile and shear stresses, while for fibre breakage, it accounts for the combined effects of longitudinal tension or compression with shear loading. The mixed-mode fracture toughness is expressed through a power-law relationship, where the material parameter dictates the relative contribution of shear to the overall fracture energy. By incorporating this evolution law, finite element