R3iCe

R3iCe formulation

Install

Tutorial : run simulation

Simulation output

CTI formulation

How to build the CTI law is describe in the page : CTI demonstration

CTI formulation

Numerical implementation

Description of the Continuous Transverse Isotropic (CTI) formulation

A transverse isotropic material is a material with a different behavior in one main direction \(e_3\) and perpendicular to this direction. But the properties perpendicular to \(e_3\) are identical. Therefore the properties transverse to \(e_3\) are isotropic.

Such formulation can be suitable model for modelling single crystal behavior for hexagonal material such as ice, quartz, magnesium alloy.

Transverse isotropic representation. The vector \(e_3\) gives the principal direction.

Single crystal linear behavior

• \(\eta_1\) is the viscosity for shear parallel to the basal plane

\[S_{13}=2 \eta_1 D_{13}, \quad S_{23}=2 \eta_1 D_{23}\]

• \(\beta\) is the viscosity ratio between shear parallel a basal plane and within the basal plane

\[S_{12}=2\frac{\eta_1}{\beta} D_{12}\]

• \(\gamma\) is the viscosity ratio between compression (or traction) along the \(e_3\) axis (\(\eta'\)) and in one direction perpendicular \(e_r\)

\[S_{33}=2 \eta' D_{33}, \quad \gamma S_{rr}=2 \eta' D_{rr}\]

Therefore for a same solicitation in different direction \(S=S_{33}=S_{11}\); it gives :

\[ D_{rr}=\gamma D_{33}\]

CTI equation

Dimensionless equations

Note

To make the equation dimensionless we use :

• typical viscosity \(\eta_n \sim Pa.s^{\frac{1}{n}}\)

1. If stress boundary condition is applied :

   • typical stress \(\Sigma\sim Pa\).

   Therefore a characteristic time \(T\) is defined as :

   • \(T=\left(\frac{\eta_n}{\Sigma}\right)^{n}\)

2. If strain rate condition is applied :

   • typical strain rate : \(\dot{\varepsilon}_m\sim s^{-1}\)

   • \(T=\frac{1}{\dot{\varepsilon}_m}\sim s\)

   • \(\Sigma=\eta_n \dot{\varepsilon}_m^{\frac{1}{n}}\sim Pa\)

This is the general form for the CTI equation. The linear formulation is obtained with \(n=1\).

\[\tilde{S}=\tilde{\eta}^\star\left(2\alpha_1 \tilde{D} + 2\alpha_2 M^D Tr(MD) + \alpha_3 (M\tilde{D}+\tilde{D}M)^D\right)\]

\[\tilde{\eta}^\star=2\left(\alpha_1 tr(\tilde{D}^2) + \alpha_2 tr(M\tilde{D})^2+\alpha_3 tr(M\tilde{D}^2)\right)^{\frac{1-n}{2n}}\]

with :

• \(\alpha_1= \frac{1}{2\beta}\)

• \(\alpha_2=\frac{\gamma}{\beta}-1\)

• \(\alpha_3=1-\frac{1}{\beta}\)

Non Linear Equation

\[S=\eta^\star\left(2\alpha_1 D + 2\alpha_2 M^D Tr(MD) + \alpha_3 (MD+DM)^D\right) \]

with \(\eta^\star\) the apparent viscosity that depend of the strain rate \(D\).

\[\eta^\star=2\eta_n \left(\alpha_1 tr(D^2) + \alpha_2 tr(MD)^2+\alpha_3 tr(MD^2)\right)^{\frac{1-n}{2n}}\]

with :

• \(\alpha_1= \frac{1}{2\beta}\)

• \(\alpha_2=\frac{\gamma}{\beta}-1\)

• \(\alpha_3=1-\frac{1}{\beta}\)

Linear Equation

\[S=\eta_1 \left(2\alpha_1 D + 2\alpha_2 M^D Tr(MD) + \alpha_3 (MD+DM)^D \right)\]

with :

• \(\alpha_1= \frac{1}{\beta}\)

• \(\alpha_2=2\left(\frac{\gamma}{\beta}-1\right)\)

• \(\alpha_3=2\left(1-\frac{1}{\beta}\right)\)

Description of the orientation evolution equation

Dimensionless equations

\[\dot{\tilde{c}}=\mathbf{\tilde{W}}(\tilde{u}).c-\lambda \left[\mathbf{\tilde{D}}(\tilde{u}).c-(c^T.\mathbf{\tilde{D}}(\tilde{u}).c).c \right] + \mathcal{Mo} (c_0-c)\]

with

• \(\mathcal{Mo}=\left(\frac{\eta_n}{\Sigma}\right)^{n}\frac{1}{\Gamma_{RX}}\)

None Dimensionless Equation

\[\dot{c}=\mathbf{W}.c-\lambda \left[\mathbf{D}.c-(c^T.\mathbf{D}.c).c \right]+\frac{1}{\Gamma_{RX}}(c_0-c)\]