We solve a homogeneous damage under nonisothermal conditions. The size of the cubic model is 1$\times$1$\times1$ mm. Detailed model description can refer the latest published benchmark book “ThermoHydroMechanicalChemical Processes in Fractured Porous Media: Modelling and Benchmarking” (Chapter 9.7 – A PhaseField Model for Brittle Fracturing of ThermoElastic Solids).
A unconfined compression test was applied as a comparison. The thermal expansion test was implemented by imposing a temperature increase to the domain while the top surface of the model was held in place. The temperature loading was chosen to achieve the same compressive load as that imposed in the unconfined compression test.
Results show PhaseField evolution in the thermomechanical case can follow the mechanical case, and both solutions correspond to the analytical solution:
The analytical solution is: $$d = \dfrac{G\textrm{c}}{G\textrm{c}+4\epsilon \psi_\textrm{e}^+}$$ where due to negative (elastic) volume strains only the deviatoric energy drives the phase field.
$$ \begin{equation} \psi_\textrm{e}^+ = \mu \mathbf{\epsilon}^\textrm{D} : \mathbf{\epsilon}^\textrm{D} = \frac{2\mu}{3} \left(\frac{u(1+\nu)}{L} \right)^2 \end{equation} $$
for mechanical case, and
$$ \begin{equation} \psi_\textrm{e}^+ = \mu \mathbf{\epsilon}^\textrm{D} : \mathbf{\epsilon}^\textrm{D} = \frac{2\mu}{3} [\alpha \Delta T(1+\nu)]^2 \end{equation} $$
for thermomechanical case. Where the Poisson’s ratio is evolving with the degradation of the shear modulus
$$ \begin{equation} \nu(d) = \frac{3K2Gd^2}{2(3K+Gd^2)} \end{equation} $$
This article was written by XingYuan Miao. If you are missing something or you find an error please let us know.
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Last revision: September 7, 2022
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