A Fully Coupled Thermomechanical Phase Field Method for Modeling Cracks with Frictional Contact
Abstract
:1. Introduction
2. Phase Field Method for Coupled Thermomechanical Problems
2.1. Regularized Variational Framework
2.2. Phase Field Approximation
3. Governing Equations and Corresponding Weak Forms
4. Stress Tensor and Thermal Conductivity Calculation for Different Contact Conditions
4.1. Stress Tensor Updates in the Phase Field Modeling
4.2. Thermal Conductivity Tensor Updates in Phase Field Modeling
Four Thermal Conductance Contact Models
5. Numerical Examples
5.1. Square Plate with (a) Central Horizontal and (b) Inclined Cracks
5.2. Squared Domain with an Internal Slant Frictional Crack
5.3. The Frictional Sliding Problem under Different Thermomechanical Coupled Conditions
5.4. Propagation of an Inclined Crack in a Rectangular Plate
6. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Conflicts of Interest
Nomenclature
Symbol | Description | Unit |
T | Temperature | |
E | Young’s modulus | |
k | Thermal conductivity | |
Specific heat capacity | J/kgC | |
Density | ||
Heat conductivity | ||
Thermal expansion coefficient | /C | |
Coefficient of thermal contact conductivity | W/mC | |
Vickers hardness | ||
Heat flux on the contact surface | ||
Critical stress | ||
Softening stiffness. | ||
Critical separation | m | |
Chemical bonding conductance | ||
Surface contact conductance | ||
Gas conductance | ||
Applied heat flux | ||
Damage length parameter | ||
Griffth’s critical energy | ||
Strain energy history | ||
Stored elastic energy | ||
Fractured surface energy | ||
Bulk energy density | ||
Crack energy density |
Abbreviations
Abbreviation | Description |
PF | Phase Field |
VMDG | Variational Multiscale Discontinuous Galerkin |
FCM | Fully Coupled Model |
ADM | Adiabatic Model |
PDM | Pressure Dependent Model |
UCM | Uncoupled Model |
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Thermal Conductance Models | Thermal Expansion Coefficient | ||
---|---|---|---|
Pressure dependent model (PDM) | |||
Uncoupled model (UCM) | |||
Fully conductive model (FCM) | |||
Adiabatic contact model (ACM) |
Bulk Material | ||
---|---|---|
Young’s modulus | E | 10,000 MPa |
Poisson’s ratio | 0.3 | |
Density | ||
Thermal expansion coefficient | ||
Bulk thermal conductivity | ||
Interface thermal conductivity (for adiabatic case) |
Poisson’s ratio | |
Young’s modulus | |
Thermal conductivity | |
Thermal conductance coefficient | |
Vickers hardness | |
Thermal constant | |
Frictional coefficient |
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Wan, W.; Chen, P. A Fully Coupled Thermomechanical Phase Field Method for Modeling Cracks with Frictional Contact. Mathematics 2022, 10, 4416. https://doi.org/10.3390/math10234416
Wan W, Chen P. A Fully Coupled Thermomechanical Phase Field Method for Modeling Cracks with Frictional Contact. Mathematics. 2022; 10(23):4416. https://doi.org/10.3390/math10234416
Chicago/Turabian StyleWan, Wan, and Pinlei Chen. 2022. "A Fully Coupled Thermomechanical Phase Field Method for Modeling Cracks with Frictional Contact" Mathematics 10, no. 23: 4416. https://doi.org/10.3390/math10234416
APA StyleWan, W., & Chen, P. (2022). A Fully Coupled Thermomechanical Phase Field Method for Modeling Cracks with Frictional Contact. Mathematics, 10(23), 4416. https://doi.org/10.3390/math10234416