PSI - Issue 68

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

889

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

3

2.2. Thermo-hydro-mechanical Model for frost heaving pressure

In cold region where severe environmental impact can e ff ect negatively on structural resistance. Huang et al. (2018) has proposed a coupled model to account for the thermal e ff ect as well as the impact of micro pressure. Tao et al. (2021) developed this model by proposing a new equivalent thermal conductivity representing the competition between dry state and wet state of rock:

λ e = K e f λ sat , i + K e u λ sat , l + (1 − K e f − K e u ) λ dry

(2)

λ sat , i , λ sat , l represent saturated thermal conductivity in function of unfrozen water content ( S L , S C ) and λ dry is thermal conductivity at dry state. K e f , K e u are the Kersten numbers under frozen and unfrozen conditions. Micro pressure induced by the change of water-ice content in rock matrix. Among the di ff erent models that can be found in literature, the proposed by Huang et al. (2018) is adopted in the present work. Cauchy stress tensor for heaving pressure e ff ect is modeled in equation 3 and stress expansion applied on the notch is presented in equation 4:

σ = C ( ε − ε θ ) − [ n (1 − S L ) p i + nS L p l ] δ i j

(3)

p i =   

E i 

0 . 029 

− 1

1 + 2 n + (1 − 4 n ) v s 2(1 − n )

1 − 2 v i

1 E s

+ 1 . 029

T − T m ≤ 0

(4)

T − T m > 0

0

n represents the porosity in the rock material, E s , E i are the Young’s modulus and v s , v i the Poison ratio, set depended on ( S L , S C ), S L represents respectively unfrozen water content and S C represents ice content in rock micro-structure. In fully saturated configuration as in experimental condition, p i is the e ff ect of ice in micro-pore and the influence of expelled water from rock matrix ( p l ) is ignored.

3. Application

In this study, the phase field method’s ability to simulate cracking by frost heaving pressure is investigated, by using a finite element model defined in the Abaqus software. The considered model is presented in 1b: it represent a notched rock, in which water is introduced, and thus frozen Vicentini et al. (2024). To model that problem, two parts are used: one for the rock, and one for the ice. The thermo-mechanical and failure problem is introduced in the Abaqus finite element software using two layer of UEL subroutines Vasikaran et al. (2020). The ice expansion during frost is introduced by a UEXPAN one, as presented in the flowchart (fig. 1c). A 2D configuration is used, with 4-node bilinear element (fig. 1a) with coarse mesh near border of the geometry and a finer mesh where crack is supposed to happen. No-friction contact is set between the two parts.

3.1. Comparison of decomposition energy schemes

To prevent cracking appears in compression zone, three classical decomposition scheme of energy are compared, i.e., the one proposed by Amor et al. (2009), Miehe et al. (2010) for strain decomposition and Wu (2017) for stress decomposition. The rock Young modulus is E = 10 GPa, and its Poisson ratio is 0.12. Last, G c = 1 N / m 2 . Results are presented in figure 2. In fig. 2, it cab be observed that Amor’s split scheme does provide a consistent results, as it account for shear stresses σ 12 , leading to damage along the notch. Wu’s and Miehe’s split scheme are almost identical and demonstrate excellent correspondence with experimental observations in mix-mode fracture.