PSI - Issue 28

Mamadou Abdoul Mbacké et al. / Procedia Structural Integrity 28 (2020) 1431–1437 Mamadou Abdoul MBACKE/ Structural Integrity Procedia 00 (2019) 000–000

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Paris and Erdogan (Paris and Erdogan 1963). The second method, more recent is based on the association of damage mechanics and cohesive zone models . A damage variable is then added and its evolution is governed by the updating of the energy release rate in accordance with the load cycles. Many studies are available on the extendibility of the cohesive zone models to take into account cyclic fatigue. Most of models relating the damage accumulation to the number of cycles are based on phenomenological laws that evolves with the cycle number increasing. Turon et al. (Turon et al. 2007) were the firsts to propose a model linking cohesive zone modeling and fracture mechanics. The damage variation proposed is given as follows:       0 2 , ' 0 1 1 , , f tot tot m f cz D D D f w w N a            (1) Harper and Hallett (Harper and Hallett 2008) developed a model with a new method for the strain energy release rate assessment. The damage evolution is given by:   , , 1 , , s f u tot tot m f D D D f w w N         (2) Later, Moroni and Pirondi (Moroni and Pirondi 2011, Moroni and Pirondi 2012) proposed a model, this later is a variant of the Turon model. Therefore, the damage evolution proposed by Turon is derived from damage mechanics framework, however, the Pirondi and Moroni Model links damage mechanics to fracture mechanics (Pirondi, Giuliese and Moroni 2017). The damage evolution is formulated as follows:   1 , , m cz D f G G N a      (3) Bak et al. (Bak et al. 2016) develop another method based on linking a cohesive zone model for quasi-static crack growth and a Paris law-like model described as a function of the energy release rate for the crack growth rate during cyclic loading. The energy release rate is evaluated using J-Integral. Damage variation in the model is given by the following formulae:   , , D F F f G N                        (4) In the present work, the model of Turon et al. (Turon et al. 2007) is implemented by means of user subroutines in the Abaqus FE package. DCB test simulations are performed and the results are compared to available experimental data. The comparison shows the capability of the modeling to predict the adhesively bonded composites lifetime. 2. Available options and selection of integration method In this section, a brief survey on the methods used to incorporate a fatigue-cohesive zone model in finite element software is presented. These models are not natively available, then, many efforts are required for their implementation. Jiang et al. (Jiang, Gao and Srivatsan 2009) implemented an irreversible cohesive zone model in order to predict the effect of overload and loading mode on fatigue crack growth. Khoramishad et al. (Khoramishad et al. 2010) improved the bi-linear cohesive zone model by its association with a strain-based fatigue damage model in order to simulate the progressive damage of adhesive joints. The improved model is also implemented in Abaqus through a user subroutine. Abaqus offers various options to integrate a material behavior law. This goes from native model modification to new element formulation. The UMAT (User MATerial) subroutine is widely used, especially for fatigue cohesive zone models integration (Walander et al. 2016, Rocha et al. 2020). This method consists on an implicit formulation of the constitutive equations of the model or explicit (in the case of a VUMAT: Vectorized User MATerial), note that this case is less present in the literature. This method requires to choose a suitable integration algorithm. The more the model is complex, the more the choice of the integration algorithm will be complicated. This method is considered as stable but it presents some drawbacks as the necessity of the tangent stiffness formulation. This requires many efforts for formulation and extensive testing for validation. The second option is the UEL (User ELement) subroutine. It consists on the formulation of a new element including the spatial discretization and the behavior law integration. This method is also used for fatigue cohesive zone models implementation (Monteiro et al. 2019). The method is robust and stable but present more difficulties than UMAT solution. The last solution consists on the adding of field variables in order to control material parameters of models already available in Abaqus. The USDFLD subroutine is used by some researchers (Moroni and Pirondi 2011, Moroni and Pirondi 2012, Ebadi-Rajoli et al. 2020) for fatigue cohesive zone models implementation by controlling the native cohesive zone models available in Abaqus. The use of this subroutine seems to be more pragmatic comparatively to UMAT or UEL subroutines. The results obtained by the