Phase Field Models for Thermal Fracturing and Their Variational Structures

It is often observed that thermal stress enhances crack propagation in materials, and conversely, crack propagation can contribute to temperature shifts in materials. In this study, we first consider the thermoelasticity model proposed by M. A. Biot (1956) and study its energy dissipation property. The Biot thermoelasticity model takes into account the following effects. Thermal expansion and contraction are caused by temperature changes, and conversely, temperatures decrease in expanding areas but increase in contracting areas. In addition, we examine its thermomechanical properties through several numerical examples and observe that the stress near a singular point is enhanced by the thermoelastic effect. In the second part, we propose two crack propagation models under thermal stress by coupling a phase field model for crack propagation and the Biot thermoelasticity model and show their variational structures. In our numerical experiments, we investigate how thermal coupling affects the crack speed and shape. In particular, we observe that the lowest temperature appears near the crack tip, and the crack propagation is accelerated by the enhanced thermal stress.

[1]  Gilles A. Francfort,et al.  Revisiting brittle fracture as an energy minimization problem , 1998 .

[2]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[3]  H. Rogers,et al.  Hydrogen Embrittlement of Metals , 1968, Science.

[4]  J. Jakowiec A model for heat transfer in cohesive cracks , 2017 .

[5]  K. Ramesh,et al.  Study of Crack Interaction Effects Under Thermal Loading by Digital Photoelasticity and Finite Elements , 2020 .

[6]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[7]  T. Takaishi,et al.  Irreversible phase field models for crack growth in industrial applications: thermal stress, viscoelasticity, hydrogen embrittlement , 2021, SN Applied Sciences.

[8]  T. T. Nguyen,et al.  On the choice of parameters in the phase field method for simulating crack initiation with experimental validation , 2016, International Journal of Fracture.

[9]  S. Freiman Effects of chemical environments on slow crack growth in glasses and ceramics , 1984 .

[10]  Jean-Jacques Marigo,et al.  Morphogenesis and propagation of complex cracks induced by thermal shocks , 2013 .

[11]  J. Lions,et al.  Inequalities in mechanics and physics , 1976 .

[12]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[13]  Sandeep Kumar Dwivedi,et al.  Hydrogen embrittlement in different materials: A review , 2018, International Journal of Hydrogen Energy.

[14]  B. Bourdin,et al.  The Variational Approach to Fracture , 2008 .

[15]  C. Augarde,et al.  Thermoelastic fracture modelling in 2D by an adaptive cracking particle method without enrichment functions , 2019, International Journal of Mechanical Sciences.

[16]  Jean-Jacques Marigo,et al.  Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments , 2009 .

[17]  M. A. Kouchakzadeh,et al.  3D dynamic coupled thermoelastic solution for constant thickness disks using refined 1D finite element models , 2018, Applied Mathematical Modelling.

[18]  Christian Miehe,et al.  Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids , 2015 .

[19]  Y. Lai,et al.  Three-dimensional analysis for transient coupled thermoelastic response of a functionally graded rectangular plate , 2011 .

[20]  M. Biot Thermoelasticity and Irreversible Thermodynamics , 1956 .

[21]  M. A. Kouchakzadeh,et al.  Analytical Solution of Classic Coupled Thermoelasticity Problem in a Rotating Disk , 2015 .

[22]  H. Lord,et al.  A GENERALIZED DYNAMICAL THEORY OF THERMOELASTICITY , 1967 .

[23]  Yunteng Wang,et al.  3-D X-ray computed tomography on failure characteristics of rock-like materials under coupled hydro-mechanical loading , 2019 .

[24]  Thomas J. Mackin,et al.  Thermal cracking in disc brakes , 2002 .

[25]  P. M. Naghdi,et al.  A re-examination of the basic postulates of thermomechanics , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[26]  Marc Duflot,et al.  The extended finite element method in thermoelastic fracture mechanics , 2008 .

[27]  Selda Oterkus,et al.  Ordinary state-based peridynamic modelling for fully coupled thermoelastic problems , 2018 .

[28]  Masato Kimura,et al.  Phase Field Model for Mode III Crack Growth in Two Dimensional Elasticity , 2009, Kybernetika.

[29]  A. Karma,et al.  Phase-field model of mode III dynamic fracture. , 2001, Physical review letters.

[30]  B. Bourdin,et al.  Numerical experiments in revisited brittle fracture , 2000 .

[31]  Tinh Quoc Bui,et al.  Simulation of dynamic and static thermoelastic fracture problems by extended nodal gradient finite elements , 2017 .

[32]  Chuanzeng Zhang,et al.  A novel meshless local Petrov–Galerkin method for dynamic coupled thermoelasticity analysis under thermal and mechanical shock loading , 2015 .

[33]  Y. Nara,et al.  Effects of humidity and temperature on subcritical crack growth in sandstone , 2011 .

[34]  M. Kimura,et al.  Numerical investigation of shape domain effect to its elasticity and surface energy using adaptive finite element method , 2018 .