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expansion due to water-to-ice phase change" +project = ["ThermoHydroMechanics/9percentWaterFreezingExpansion/UnitSquare.prj"] +image = "Ice_strain_setup.png" ++++ + +{{< data-link >}} + +## Problem description + +The aim of this benchmark is to verify that our THM model is capable of +simulating the 9% volumetric expansion during the liquid-to-ice phase +transition. +This must be captured by the $\boldsymbol\sigma_\mathrm{I}$-stress component +of the formulation + +$$ +\begin{align*} + \boldsymbol\sigma\_\mathrm{SI} :=&\boldsymbol\sigma\_\mathrm{S}+ \boldsymbol\sigma\_\mathrm{I} \\\\\\ + =&\mathbb{C}\_\mathrm{S}:\left(\boldsymbol\varepsilon-\alpha^\mathrm{S}\_T (T-T\_0){\bf I}\right) \\\\\\ + +&\phi S\_\mathrm{I}(T) \mathbb{C}\_\mathrm{IR}: + \left(\boldsymbol\varepsilon-\boldsymbol\varepsilon\_\mathrm{S0} + - \alpha^\mathrm{I}\_T (T-T\_\mathrm{m}){\bf I} + - \alpha\_{\phi\_\mathrm{I}} S\_\mathrm{I}(T){\bf I} + \right), +\end{align*} +$$ + +where $\mathbb{C}\_\mathrm{S}$ and $\mathbb{C}\_\mathrm{IR}$ are the forth order +elasticity tensors of solid matrix and ice phase, respectively, +$\boldsymbol\varepsilon$ is the total strain, $\alpha^\mathrm{S}\_T$ and +$\alpha^\mathrm{I}\_T$ are the linear thermal expansivities of solid and +ice phases, respectively, $\phi$ is the porosity, +$\alpha_{\phi_\mathrm{I}}=0.03$ is the linear expansion coefficient due +to water-to-ice phase change and, finally, $S\_\mathrm{I}(T)$ is the +regularized ice phase indicator function ($S\_\mathrm{I}\rightarrow 1$ is +ice, and $\rightarrow 0$ is liquid water). +Then the material properties in the benchmark are to be chosen such that +$E_\mathrm{S}\ll E_\mathrm{IR}$. +In this way, and also with the absence of mechanical loading, the +deformation of a specimen --- with $\alpha_{\phi_\mathrm{I}}$ being the +strain increase in each space direction --- will occur solely due to liquid freezing. + +The material properties in the benchmark are to be chosen such that the Young's +modulus of solid is smaller than that of ice, i.e. $E_\mathrm{S}\ll E_\mathrm{IR}$. +In this way, and also with the absence of mechanical loading, the deformation of +a specimen with $\alpha_{\phi_\mathrm{I}}=0.03$ being the strain increase in +each space direction will occur solely due to liquid freezing. + +For the geometric setup we consider a fully saturated cylindrical column whose +bottom edge is supported by a rigid foundation. +Using the axis symmetry the problem is reduced from 3 to 2 dimensions with a +simple (square) computational domain and related (Dirichlet) boundary +conditions, see Figure 1, where an expected deformed configuration is also +sketched. +Again, we apply no mechanical loading to the specimen, whereas the thermal +loading is presented by temperature evolution over time. +The temperature $T$ is prescribed as a constant in $\Omega$ at each time step +and decays linearly from $+4\\,^\circ \text{C}$ to $-4\\,^\circ \mathrm{C}$ +during one hour term, as depicted in Figure 2, left. + +{{< img src="Ice_strain_setup.png" >}} +**Figure 1:** Fully saturated column expansion due to water-to-ice phase +transition: geometry (on the left) and the 2d computational setup along with the +expected deformed configuration (on the right). + +Material data used in the computations are presented in Table 1. +No liquid phase parameters are present since we don't solve for the hydraulic +equation and the temperature field is prescribed. +The two parameters that are varied are the time step increment $\Delta t$ (also +denoted as $\mathrm{dt}$ in the corresponding captions) and the parameter $k>0$ +in the Sigmoid function $S_\mathrm{I}$ which governs the "thickness" of a +temperature-related phase transition zone. +More specifically, we take $\Delta t\in\\{10\\,\mathrm{s}, 30\\,\mathrm{s}, +60\\,\mathrm{s}\\}$ and $k\in\\{2,5,20,50\\}$. +$\Omega$ is discretized with only one element (which is possible/allowed here +since the setup implies no freezing front propagation within the domain). +Finally, the FE approximation of the components of strain tensor +$\boldsymbol\varepsilon$ uses the $Q_1$-quadrilaterals. + +{{< img src="Ice_strain_MaterTable.png" >}} +**Table:** Material properties and parameters. + +## Simulation results and analysis + +Figure 2, right, depicts evolution of strain tensor components +$\varepsilon_{xx}$, $\varepsilon_{yy}$, $\varepsilon_{zz}$ computed for the +fixed parametric pair $(\Delta t, k)=(60\\,\mathrm{s}, 20)$. +All three strains behave identically and, more importantly, as expected, they +transit from 0 to the reference magnitude of 0.03 during the freezing, in +accordance to the term $\alpha_{\phi_\mathrm{I}}S_\mathrm{I}(T)$. + +{{< img src="Ice_strain_3strains.png" >}} +**Figure 2:** The prescribed temperature loading applied to the specimen (on the +left) and the induced strains evolution due to phase change (on the right). + +Figure 3 details the parametric studies for the computed $\varepsilon_{xx}$: on +the left plot, for the fixed time increment, one observes that for any +considered $k$ the corresponding strains transit up to the required value 0.03 +and, as also expected, the increase of the parameter yields a steeper and more +localized transition zone, almost mimicking the Heaviside-like behaviour at +$k=50$. +It is interesting to observe a slight downward deviation of $\varepsilon_{xx}$ +from the horizontal reference value in the post-freezing time range (that is, +when the prescribed temperature keeps on decreasing from $T_\mathrm{m}$ to +$-4\\,^\circ\mathrm{C}$). +This behaviour is physical and implies the ice contraction in such temperature +range. +The right plot of Figure 3 presents the evolution of +$\varepsilon_{xx}$---specifically, the required transit during the phase +change---for a fixed steepness-related parameter $k$ and varying time step size. +All three computational results seem identical. + +{{< img src="Ice_strain_dt-k.png" >}} +**Figure 3:** Parametric studies for the computed volumetric strain expansion in +dependence on the time step size and parameter $k$ in $S_\mathrm{I}(T)$ that governs +the thickness of the phase transition zone.