PSI - Issue 42

Sarim Waseem et al. / Procedia Structural Integrity 42 (2022) 1692–1699 Waseem et al. / Structural Integrity Procedia 00 (2019) 000–000

1693

2

dual phase media, finite element phase field models were adapted to model brittle crack growth through the variational approach to fracture based on Gri ffi th’s energy model (see Gri ffi th (1921), Francfort and Marigo (1998), Bourdin et al. (2000)). This approach boasts several advantages, including no remeshing requirements and no ad hoc criteria required to determine crack paths. More complicated models have since then been proposed, demonstrably capturing specific cases such as dynamic crack growth (see e.g. Borden et al. (2012)), ductile crack growth (see e.g. Borden et al. (2016)), anisotropic crack growth (see e.g. Schreiber et al. (2021)) among others. More recently, phase field models have been adapted to simulate fatigue growth. The basis of phase field fatigue models is the degradation of the fracture toughness with consistent variation in the elastic energy. This is accom plished through the introduction of the fatigue damage variable (see e.g. Boldrini et al. (2016), Alessi et al. (2018), Carrara et al. (2020)). Phase field fatigue models based on damage accumulation tend to solve exclusively for fixed amplitude loading cases. Wolf (1970) observed premature fracture surface contact during cyclic tensile loading, a phenomenon referred to as crack closure and its influence on the crack growth rate. Single overloads were found to significantly retard the crack growth rate in subsequent cycles. These e ff ects are generally incorporated into Paris’ Law type formulations through an increase in the K min value to K op (see e.g. Newman Jr (1984), Correia et al. (2016)). For a more detailed review of the crack closure phenomenon see Pippan and Hohenwarter (2017). While models have been developed to capture crack growth in complex loading cases (see e.g. Willenborg et al. (1971)) and they have been implemented in computational settings (see e.g. Dirik and Yalc¸inkaya (2018)), these models are di ffi cult to apply to the phase field fatigue paradigm, which centers around damage accumulation rather than the crack growth rate. This study aims to introduce a simple modeling approach to capturing overload e ff ects with the phase field method without the explicit introduction of plasticity. The coupled temperature-displacement model by Navidtehrani et al. (2021) is adapted to accommodate fatigue e ff ects. The fatigue model is based on the work done by Carrara et al. (2020), neglecting fatigue gradient damage e ff ects as shown by Kristensen and Mart´ınez-Pan˜eda (2020). An alteration to the fatigue damage accumulation inspired by Paris’ Law is proposed to accommodate crack closure e ff ects without the explicit integration of plasticity to our model. Based on the deformation history, an elastic energy threshold is introduced, below which no fatigue damage is taken. This threshold has a dependence on the maximum deformation energy of the material. The paper follows the following format. Section 2 covers the phase field theory on which our model is based as well as the additional fatigue formulations and the proposed alterations. Section 3.1 covers several overload ratio cases and the corresponding results, exploring the capability of the approach in e ff ectively capturing overload. Section 3.2 details the solution of a boundary value problem with a complex crack path which is generally di ffi cult to capture using a linear elastic brittle model.

2. Phase Field Theory and Numerical Implementation

The phase field methodology is based around the introduction of another degree of freedom known as the phase field parameter φ . This parameter describes damage in the domain with certain restrictions applied. φ ⊂ (0 , 1) where a value of 0 represents perfectly intact material and a value of 1 represents completely cracked material. It is subject to an irreversibility condition ˙ φ ≥ 0 which ensures that there can be no material recovery once the damage is done. The foundations of fracture mechanics were laid by Gri ffi th (1921), where the following energy balance was intro duced to describe crack evolution

∂ E ∂ A

∂ W s ∂ A

∂ψ e ∂ A

= 0

(1)

=

+

where E is the total energy of the system, ψ e is the elastic potential energy and W s is the work done to create new surfaces during fracture. We can consider ∂ A as the incremental change in the crack surface area. The simplified implications of Gri ffi th’s energy balance are that the work done to create two new surfaces upon fracture must be compensated by a drop in the elastic energy in the vicinity. The term ∂ W s /∂ A is referred to as the critical energy release rate or the fracture toughness of the material, designated by G c and this is a constant material parameter. This in turn implies that the elastic energy density in an area must at the very least match the fracture toughness of the material for fracture to initiate.