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Abstract
Cohesive zone models rely on the formulation of a cohesive constitutive law. The latter describes the relation between displacement and traction in a cohesive element at an integration point. Cohesive constitutive laws in the presence of opening and shearing modes are less studied in comparison with those formulated for a single mode, particularly when the mode mixity changes. The mode mixity at an integration point is determined by the load history at the point. In this study, a formulation of the cohesive constitutive law is proposed for a mixed mode loading condition with the ability to deal with the variation in mode mixity. The proposed law is constructed incrementally and takes into account the load history. The validation is performed by simulating delamination in carbon fibre/epoxy composites in the mixed-mode bending test that is commonly used to characterize the inter-laminar fracture toughness. Although the mode mixity is fixed in this test at the specimen level, it varies locally at the element level. Cohesive constitutive laws proposed in the literature predict macroscopic delamination behaviour that is dependent on the strength of the interface, while, according to the analysis of linear elastic fracture mechanics, the dependence is expected to be only on the fracture toughness. Predictions with the current formulation, where the cohesive law is updated incrementally, show low sensitivity to the interface strength. The structural response simulated with it had a good agreement with the analytical solution of linear elastic fracture mechanics.
Highlights
Laminated materials can experience de-cohesion or delamination under different loading conditions, from simple single mode to more complex mixed mode conditions
The developed formulation of the mixed-mode cohesive constitutive law is tested by simulating mixed-mode delamination in fibre-reinforced composites
The incremental mixed-mode cohesive law was implemented into the cohesive element as a user-defined material (UMAT) subroutine in FE software Abaqus Implicit
Summary
Laminated materials can experience de-cohesion or delamination under different loading conditions, from simple single mode to more complex mixed mode conditions. From Eqs 1–3, it can be seen that once the test set-up and specimen geometry are fixed, the structural response of the specimen is only determined by the inter-laminar fracture toughness and has no dependency on the inter-laminar strength This is not the case when the MMB test was simulated with the existing mixed-mode cohesive constitutive law (Turon et al, 2010; Harper et al, 2012). The constitutive law is constructed incrementally based on the deformation, traction and mode mixity at the integration point obtained from the previous simulation step In this way the effect of loading history is included in the cohesive law. At the time of t+Δt, with mode mixity B(t+Δt), if the integration point is loaded till traction is zero, the total SERR dissipated is equal to the fracture toughness Gc(t+mΔt) obtained from Eq 22. D(ijt+Δt) δ(j t+Δt) +δ1j〈−δ(1t+Δt)〉 δ(i t+Δt) +δ1i〈−δ(1t+Δt)〉 δ(mtf+Δt) −δ(mt0+Δt) δ(mt+Δt) 3 δ (mtf+Δt) δ (mt0+Δt)
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