Proceedings in Applied Mathematics and Mechanics, 18 March 2021 Advancements in multi-phase unsaturated porous media fracture Yousef Heider 1, ∗ , WaiChing Sun 2 , and Bernd Markert 1 1 Institute of General Mechanics, RWTH Aachen University. Templergraben 64, 52062, Aachen, Germany 2 Columbia University, Department of Civil Engineering and Engineering Mechanics, 614 SW Mudd, NY, NY10027, USA Key words Hydraulic fracturing, Desiccation cracking, Phase-field modeling, Machine learning The focus in this research is on introducing an accurate and stable numerical modeling framework and to compare the numer- ical results with experimental data from the literature for desiccation-induced fracturing of unsaturated porous materials. The macroscopic modeling approach is based on combined continuum porous media mechanics and a diffusive phase-field method (PFM). In the case of unsaturated porous media, one has to deal with more than one pore fluid (e.g. water and air). In this, the mechanical behavior can be expressed via using the Bishop’s effective stress principle, which considers the total stress, the capillary pressure (air pressure minus water pressure), and saturation degree. The onset of desiccation-induced fracturing is driven by the capillary pressure, which leads to degradation not only in the effective stresses but also in the capillary pressure itself. In this contribution, we discuss via a numerical example the effect of considering the air pressure variation on the cracking process and numerical stability. Besides, we briefly discuss the potential inclusion of machine-learning material laws, such as for retention curves and anisotropic permeability, using, e.g., deep recurrent neural networks (RNN) to improve the model accuracy. Copyright line will be provided by the publisher 1 Mathematical modelling In the last few decades, the phase-field method has emerged as a powerful tool that has been applied by the current authors, among other research groups, in fields like fracture mechanics [1–4] or for modeling of phase-change materials [5, 6]. With focus in this contribution on unsaturated porous media fracture, the material under consideration ϕ is composed of immiscible three phases, i.e. one solid phase ϕ S and two fluid phases (liquid and gas) with ϕ F = ϕ L ∪ ϕ G . Following the continuum porous media approach summarized in [1], one defines the volume fractions n α and the partial and intrinsic densities ρ α and ρ αR , respectively, yielding ρ α = n α ρ αR . To this, the saturation condition reads n S + n F =1 with n F = n L + n G . Besides, the saturation degree is expressed as s β := n β /n F with β ∈{L,G}, where the variation of saturation degree and the related relative permeability factors κ β r in this work are considered functions of the capillary pressure (p c ) according to van Genuchten (VG) model, i.e. s β = s β (p c ) and κ β r = κ β r (p c ). The variation of the solid volume fraction can be expressed in terms of the initial solidity n S 0 and the solid displacement u S as n S ≈ n S 0 (1 − div u S ). In addition, it is assumed that the fluid consistuents are barotropic, in which ρ βR is defined in terms of the the compressibility κ β and the effective pressure p β . Within a small strains framework, a Lagrangean description is assigned to the solid motion via displacement u S and velocity v S and, thus, the linearised solid strain tensor ε S := 1 2 (grad u S + grad T u S ) is considered. Moreover, the Eulerian description is utilized for the fluid motion via the seepage velocity w F = v F − v S . Having incompressible solid phase and linear-elastic solid matrix with μ S and κ S as the shear and bulk moduli, the Bishop’s effective stress for intact porous material is expressed as σ ′ = σ + p I = σ + p G I − s L p c I with p c = p G − p L and σ ′ =2μ S ε D + κ S (tr ε) I . (1) Employing the approach of PFM of brittle fracture, a scalar-valued phase-field variable d S is introduced to distinguish between the cracked (d S =1), the intact (d S =0) and the diffusive (0 <d S < 1) states of ϕ S , see, e.g. [1, 3] for references. Based on this, a degradation function of the elastic energy is defined, i.e. G (d S ) := [(1 − η)(1 − d S ) 2 + η], with 0 <η<< 1 as a small residual stiffness parameter. Having G c as the Griffith’s critical fracture energy and l c > 0 as the regularization factor, the total potential energy ψ in terms of the fracture energy ψ S frac (d S , ∇d S ) and the the free energy ψ(ε,p c ) is expressed as ψ(ε,p c ,d S , ∇d S ) := G (d S ) ψ + + ψ − + ψ S frac with ψ S frac (d S , ∇d S ) := Gc 2 lc  (d S ) 2 + l c 2 |∇d S | 2  . (2) The latter formulation proceeds from the assumption of crack degradation under tension and shear (+) and not under com- pression (−). With 〈(  )〉 ± = 1 2  (  ) ±|(  )|  as the Macauley brackets, the total stress reads σ = G (d S )  2μ S ε D +  κ S (tr ε) I + χ (p G − p L ) I  +  +  κ S (tr ε) I + χ (p G − p L ) I  − − p G I . (3) * Corresponding author: e-mail heider@iam.rwth-aachen.de, phone +49 241 80 98282, fax +49 241 80 92231 Copyright line will be provided by the publisher