A thermo-mechanically coupled finite deformation model for freezing-induced damage in soft materials

TL;DR

This work addresses thermo-mechanical damage during cryopreservation of soft tissues by developing a fully coupled finite-deformation framework that simultaneously resolves heat transfer, phase change, and damage evolution. The authors integrate a phase-field model for damage with a phase-field description of freezing, linked through a Kröner–Lee multiplicative decomposition of the deformation gradient and a consistent thermodynamic formulation, solved in FEniCS with a staggered scheme. Key contributions include explicit constitutive choices (slightly compressible neo-Hookean elasticity, AT2 damage, and a double-well/latent-heat phase-change energy), and demonstrations across benchmark problems and four representative cases showing how boundary conditions and geometry affect ice formation, stresses, and damage. The framework offers a robust, extensible tool for predicting freezing-induced injury in soft tissues and guiding improved cryopreservation strategies, with potential extensions to nucleation, water transport, rate-dependent damage, and tissue microstructure.

Abstract

In the U.S., approximately 17 patients die each day awaiting an organ transplant, a crisis driven by the inability to store organs long-term via methods like cryopreservation. A primary failure mechanism is the severe thermo-mechanical damage tissues experience during freezing. A predictive understanding of this damage is hindered by the complex interplay between heat transfer, phase change, and large deformation mechanics. Motivated by this fundamental problem, we present a fully coupled, thermo-mechanical phase-field framework for modeling damage evolution in fluid-saturated soft materials under cryogenic conditions. The theoretical framework integrates heat transfer with solid-liquid phase transition, finite deformation nonlinear elasticity, and progressive mechanical damage. The governing equations are solved using \texttt{FEniCS} finite element package. The presentation will detail the theoretical framework and showcase representative simulations that capture the spatiotemporal evolution of temperature, freezing phase field, stress, and damage fields during representative freezing protocols. The developed framework serves as a powerful tool for understanding the fundamental mechanisms of freezing-induced injury and for designing improved cryopreservation strategies.

Figures (24)

• Figure 1: Thermal stresses induced cracking upon freezing a sample of bovine liver. Experimental image courtesy of Prof. Ram Devireddy's Bioengineering/Bioheat Transfer Laboratory at the Louisiana State University, Baton Rouge, LA.
• Figure 2: Schematic of the continuum body.
• Figure 3: (a) Double-well potential $g(\phi)$; (b) smooth interpolation function $p(\phi)$; (c) schematic of thermal free energy at different temperatures.
• Figure 4: (a) Schematic of the setup to study uniaxial loading of a nearly incompressible, plane strain neo-Hookean sample. The domain was discretized natively inFEniCSusing 40,000 structured quadrilateral elements. (b) First Piola Kirchhoff stress, $P_{22}$, obtained from simulation matches the analytical results.
• Figure 5: Schematic of problem to study planar freezing front propagation. The domain was discretized natively in FEniCS using 160,000 structured quadrilateral elements.