PSI - Issue 61

Bekir Kaçmaz et al. / Procedia Structural Integrity 61 (2024) 130–137 Author name / Structural Integrity Procedia 00 (2024) 000–000

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It is well documented that the discretized models based on classical CDM do not converge to physically meaningful results since these models predict that dissipated energy approaches to zero with mesh refinement although physically, finite amount of energy is dissipated during the formation of crack surfaces, Bazant et al. (1984), Lasry and Belytschko (1988). Non-local and gradient extended formulations have been very instrumental to overcome this non-physical response, Peerlings et al. (2001). Among gradient extended formulations, implicit gradient damage model (in the text called as conventional implicit gradient damage and abbreviated as CIGD) has been used to couple damage and various constitutive models due to its e ff ectiveness stemming from strong non-locality, Peerlings et al. (1996); Geers (2004). Although the pathological mesh dependency could be suppressed with CIGD formulation, it has been observed that the resulting damage distributions tend to widen artificially and in some cases the predicted damage propagation directions are not correct, Geers et al. (1998); Simone et al. (2004). Di ff erent strategies have been adopted to address these shortcomings. As compared to CIGD model, transient gradient damage (TGD) model is equipped with an evolving internal length scale and it successfully suppresses the artificial di ff usion of damage to neighbouring intact material points, Geers et al. (1998). In TGD model, an additional continuity equation emerges that implies an extended set of nodal variables. However, as detailed in Saroukhani et al. (2013), the additional continuity equation can be circumvented through reformulation of TGD model and therefore an e ffi cient implementation is realizable. It was also shown that spurious damage growth can be avoided by using a weighted average of both local and non-local variables to describe the damage process. By setting the weighting factor of non-local term larger than unity, so-called over non-local models of integral and gradient types were formulated which e ff ectively solve the spurious growth problem, Luzio and Bazant (2005); Poh and Swaddiwudhipong (2009); Nguyen (2011). Apart from these remedies, higher order continuum descriptions have also been exploited to address aforemen tioned problems of continuum damage mechanics and problems associated with regularized models. A particularly interesting one that is based on micro-morphic framework Forest (2009) was put forward and has been extended by Poh and his co-workers recently, Poh and Sun (2017); Sarkar et al. (2019); Xu et al. (2020). The framework describes the micro processes through a morphic variable and its gradient in an average sense and a micro-force balance emerges that resembles to the Helmholtz equation of implicit gradient damage model. Furthermore, as opposed to TGD model, an evolving length scale that diminishes with damage was proposed by this model. This is motivated by the fact that as one of the cracks dominates the failure process, the interaction between micro-cracks weakens and they tend to close. The resulting thermodynamically consistent model is called as localizing implicit gradient damage (LIGD) model and its e ff ectiveness has been demonstrated both for quasi-brittle and ductile failure. As far as quasi-brittle failure predictions is concerned, presented models are almost exclusively two-dimensional and therefore, so far, experimental set-ups that allow reduced dimensional models with planar crack surfaces have been addressed, Shedbale et al. (2021). The performance of the LIGD model in case of three-dimensional set-ups with mixed loading scenarios has not been assessed yet. To address this gap, this paper aims at assessing the capabilities of LIGD model in case of truly three-dimensional cracks with curved surfaces. To this end, LIGD model is embedded within a tetrahedra element formulation and implemented through the user element (UEL) subroutine of finite element software Abaqus, Systemes (2013). An experimental study from the literature which is forced to fail under combined bending and torsion is modeled and the predictions are compared with measurements. In the next section, underpinnings of the LIGD model and the weak forms of the governing balance equations are recapitulated. Afterwards, element formulation is briefly presented and the experiment is considered in a separate sub-section including comparisons between LIGD and CIGD model predictions. Based on these findings, this paper is closed with some conclusive comments and potential extensions that can be pursued.

2. Localizing Implicit Gradient Damage Formulation

The formulation proposed by Poh and his co-workers Poh and Sun (2017) is based on the micro-morphic theory Forest (2009) and introduces micro-fluctuation enhanced equivalent strain ˜ ϵ . The di ff erence between the macroscopic equivalent strain ϵ which is going to be defined in Section 3, and ˜ ϵ is a measure of heterogeneity of the strain field and can be conveniently used to couple the two scales; i.e., micro and macro. A closer look at the failure of quasi-brittle materials reveals that, distributed micro-cracks which were initially active tend to close with the emergence of a macroscopic crack. Since the closure of micro-cracks is associated with