PSI - Issue 68

Q.M. Vuong et al. / Procedia Structural Integrity 68 (2025) 887–893 Q.M. Vuong et al. / Structural Integrity Procedia 00 (2024) 000–000

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dynamic properties Jia et al. (2017) contribute as principals factors acting on frost heaving pressure. The freeze-thaw damage of rock is induced by several environmental impacts including pore ice pressure and thermal stress Coussy and Monteiro (2008); Guo et al. (2022) which are produced under thermo-hydro-mechanical (THM) condition at low temperature Sleiman et al. (2022); Tang et al. (2022). The accuracy and capability to capture the phenomenon of coupled T-H-M model are controlled by critical parameters such as unfrozen water content, ice pressure or expansion, permeability of rock matrix, and the equivalent thermal conductivity Huang et al. (2019). In the early 2010s, Bourdin, Francfort, and Marigo proposed a model, nowadays known with the name Phase Field method for fracture, with the idea of replacing sharp discontinuous crack by a smear one as a di ff usive zone in the solid under variational approach Francfort and Marigo (1998); Bourdin et al. (2000). In a decade, Phase Field method for fracture has emerged as a promising method which is able to capture crack path in a solid media. An abundance of contributions has been provided by community to improve and develop the method in order to generalize and to complete its intrinsic flaws e.g. the dependence in length scale parameter and initiation of crack Wu and Nguyen (2018); Molnar et al. (2022). This approach can be coupled with other physical phenomenon without any explicit crack tracking needed. In this study, it is proposed a THM model for low permeability rock inspired by Huang et al. (2018); Tao et al. (2021) coupled with phase field method for fracture to reproduce cracking process by freezing water in laboratory test in order to understand the potentials and the challenges of phase-field method in simulating rock fracture under frost heaving pressure. The paper is structured as follow: In section 2, the the background of phase-field approach in the context of frost driven crack is presented. Several examples are presented in Section 3, and discussed. In 1998, Frankfurt and Marigo introduced a framework based on Gri ffi th’s theory for brittle fracture, through the minimization of an energy functional Francfort and Marigo (1998) (equation 1. This approach then became popular in the computer science and mechanics community for fracture modeling after being implemented numerically, and the introduction of a continuous field ϕ ( ∈ [0 , 1]) to represents the crack Bourdin et al. (2000). The energy functional is written as E ( u ,ϕ ) = Ω ψ ( ε ( u ,ϕ )) + G c γ ( ϕ, ∇ ϕ ) dV −F ( u ) (1) where ψ ( ε ,ϕ ) is the elastic energy density function ( ε being the strain tensor) of the cracked domain and G c represents the critical energy release rate. F ( u ) represents the work of the applied forces on Ω . γ ( ϕ, ∇ ϕ ) represents the crack, which features (evolution with time, branching, shape...) are obtained by the minimisation of E ( u ,ϕ ). The flexibility of phase field method becomes clearer because this intermediate field allows for tracking crack propagation without any explicit ad-hoc Ambati et al. (2015). crack opening propagation will be driven by the mini mization principle of solid energy. Driving force based on energy split governs damage evolution partial di ff erential equations. In this sense, one can replace sharp crack Γ by integral on whole volume surface density energy γ ( ϕ, ∇ ϕ ). Among several variety of existing phase field models, the AT1 Pham et al. (2011) and PF-CZM models by Wu (2017) for brittle and quasi-brittle behaviors can be highlighted. AT1 is considered to lead to very good simulation results due to an essential yield point that accurately simulates damage initiation, but its main drawback remains its dependence on the length scale parameter ( l c ), i.e. on the material properties. PF-CZM model is proven to have a lower dependence with material properties than AT1, but the choice for its pa rameters is not always convenient to take into account all type of material and behavior. Therefore, this independence of PF-CZM model allows to simulate crack path and propagation in certain configuration that AT1 cannot, but the precision of crack initiation and its accuracy are still in question Kumar et al. (2020); Doitrand and Molna´r (2024). 2. Method 2.1. Phase Field Method for fracture