# Electromechanical Applications

(Difference between revisions)
 Revision as of 13:49, 28 April 2011 (view source)Amir (Talk | contribs) (→Theory)← Older edit Revision as of 13:55, 28 April 2011 (view source)Amir (Talk | contribs) (→Theory)Newer edit → Line 28: Line 28: h(\mathbf{\varepsilon},\mathbf{p},\nabla\mathbf{p},\mathbf{E},v) =  (v^2 + \eta_\kappa)\left[U(\nabla \mathbf{p}) + W(\mathbf{p},\mathbf{\varepsilon}) - \frac{\varepsilon_0}{2}|\mathbf{E}|^2 -\mathbf{E}\cdot\mathbf{p}\right] + \chi(\mathbf{p}) \quad  \quad  \quad \quad  \quad \quad (3) \quad \text{Impermeable Crack} h(\mathbf{\varepsilon},\mathbf{p},\nabla\mathbf{p},\mathbf{E},v) =  (v^2 + \eta_\kappa)\left[U(\nabla \mathbf{p}) + W(\mathbf{p},\mathbf{\varepsilon}) - \frac{\varepsilon_0}{2}|\mathbf{E}|^2 -\mathbf{E}\cdot\mathbf{p}\right] + \chi(\mathbf{p}) \quad  \quad  \quad \quad  \quad \quad (3) \quad \text{Impermeable Crack} [/itex] [/itex] + + where the energy functions $U$, $W$ and $\chi$ are stated as + + $+ U(p_{i,j})= \frac{a_0}{2}(p^2_{1,1} + p^2_{1,2} + p^2_{2,1} + p^2_{2,2}) +$ + + $+ W(p_i, \varepsilon_{jk})= &~-\frac{b_1}{2}(\varepsilon_{11}p^2_{1} + \varepsilon_{22}p^2_{2}) - \frac{b_2}{2}(\varepsilon_{11}p^2_{2} + \varepsilon_{22}p^2_{1}) - {b_3}(\varepsilon_{21} + \varepsilon_{12})p_{1}p_{2} \\ + &~+\frac{c_1}{2}(\varepsilon^2_{11} + \varepsilon^2_{22}) + {c_2}\varepsilon_{11}\varepsilon_{22} + \frac{c_3}{2}(\varepsilon^2_{12} + \varepsilon^2_{21}), +$ + + $+ \chi(p_i) = &~\alpha_1(p^2_{1} + p^2_{2}) + \alpha_{11}(p^4_{1} + p^4_{2}) + \alpha_{12}(p^2_{1}p^2_{2}) + \alpha_{111}(p^6_{1} + p^6_{2}) + \alpha_{112}(p^2_{1}p^4_{2} + p^2_{2}p^4_{1}) \\ + & + \alpha_{1111}(p^8_{1} + p^8_{2}) + \alpha_{1112}(p^6_{1}p^2_{2} + p^6_{2}p^2_{1}) + \alpha_{1122}(p^4_{1}p^4_{2}), +$ + + where the combination of energy functions $\chi$ and $W$ is the + total Landau-Devonshire energy density, $a_0$ is the scaling + parameter of the domain wall energy, $b_i (i=1,2,3)$ are the + constants of the coupling terms between strain and polarization and + $c_i (i=1,2,3)$ are the elastic constants.

## General Description

We have provided the electro-mechanical applications in KRATOS for the simulations of ferroelectric and piezoelectric materials. The code has been developed mainly to study the crack propagation in these materials but it can be effectively used for other purposes. Here we start from a brief theory of these materials and then we move to the numerical implementation in KRATOS. The theory of ferroelectric materials is presented in our published paper which can be accessed in the following link.

Abdollahi A, Arias I. Phase-field modeling of the coupled microstructure and fracture evolution in ferroelectric single crystals. Acta Mater (2011), doi:10.1016/j.actamat.2011.03.030

### Theory

The total electro-mechanical enthalpy for a ferroelectric body occupying a region is stated as:

$H[\mathbf{u},v,\mathbf{p},\phi] = \int_\Omega h(\mathbf{\varepsilon}(\mathbf{u}),\mathbf{p},\nabla\mathbf{p},\mathbf{E}(\phi),v)~\mathrm{d}\Omega + G_c\int_{\Omega} \left[\frac{(1-v)^2}{4\kappa} + \kappa|\nabla{v}|^2\right]~\mathrm{d}\Omega - \int_{\Gamma_{N,\mathbf{u}}} \mathbf{t}\cdot\mathbf{u}~{\rm d}S + \int_{\Gamma_{N,\phi}} \omega\phi~{\rm d}S \quad \quad \quad \quad \quad (1)$

where $\mathbf{t}$ and ω are the tractions and surface charge density respectively, and $\Gamma_{N,\mathbf{u}}$ and ΓN are the parts of the boundary of the domain $\partial\Omega$ where mechanical and electrical Neumann boundary conditions are applied. $\mathbf{\varepsilon}$ is the strain tensor associated with the mechanical displacement $\mathbf{u}$, $\mathbf{\varepsilon} = 1/2(\nabla\mathbf{u} + \nabla^{T}\mathbf{u})$, $\mathbf{p}$ is the polarization, $\mathbf{E}$ is the electric field defined as $\mathbf{E} = -\nabla\phi$, where φ is the electric potential. Gc is the critical energy release rate or the surface energy density in Griffith's theory. The scalar field v provides a diffuse representation of the fracture zone, κ is a positive regularization constant to regulate the size of the fracture zone. The electro-mechanical enthalpy density h considering permeable and impermeable cracks follows

$h(\mathbf{\varepsilon},\mathbf{p},\nabla\mathbf{p},\mathbf{E},v) = (v^2 + \eta_\kappa)\left[U(\nabla \mathbf{p}) + W(\mathbf{p},\mathbf{\varepsilon})\right] + \chi(\mathbf{p}) - \frac{\varepsilon_0}{2}|\mathbf{E}|^2 -\mathbf{E}\cdot\mathbf{p} \quad \quad \quad \quad \quad \quad \quad \quad (2) \quad \text{Permeable Crack}$

$h(\mathbf{\varepsilon},\mathbf{p},\nabla\mathbf{p},\mathbf{E},v) = (v^2 + \eta_\kappa)\left[U(\nabla \mathbf{p}) + W(\mathbf{p},\mathbf{\varepsilon}) - \frac{\varepsilon_0}{2}|\mathbf{E}|^2 -\mathbf{E}\cdot\mathbf{p}\right] + \chi(\mathbf{p}) \quad \quad \quad \quad \quad \quad (3) \quad \text{Impermeable Crack}$

where the energy functions U, W and χ are stated as

$U(p_{i,j})= \frac{a_0}{2}(p^2_{1,1} + p^2_{1,2} + p^2_{2,1} + p^2_{2,2})$

Failed to parse (syntax error): W(p_i, \varepsilon_{jk})= &~-\frac{b_1}{2}(\varepsilon_{11}p^2_{1} + \varepsilon_{22}p^2_{2}) - \frac{b_2}{2}(\varepsilon_{11}p^2_{2} + \varepsilon_{22}p^2_{1}) - {b_3}(\varepsilon_{21} + \varepsilon_{12})p_{1}p_{2} \\ &~+\frac{c_1}{2}(\varepsilon^2_{11} + \varepsilon^2_{22}) + {c_2}\varepsilon_{11}\varepsilon_{22} + \frac{c_3}{2}(\varepsilon^2_{12} + \varepsilon^2_{21}),

Failed to parse (syntax error): \chi(p_i) = &~\alpha_1(p^2_{1} + p^2_{2}) + \alpha_{11}(p^4_{1} + p^4_{2}) + \alpha_{12}(p^2_{1}p^2_{2}) + \alpha_{111}(p^6_{1} + p^6_{2}) + \alpha_{112}(p^2_{1}p^4_{2} + p^2_{2}p^4_{1}) \\ & + \alpha_{1111}(p^8_{1} + p^8_{2}) + \alpha_{1112}(p^6_{1}p^2_{2} + p^6_{2}p^2_{1}) + \alpha_{1122}(p^4_{1}p^4_{2}),

where the combination of energy functions χ and W is the total Landau-Devonshire energy density, a0 is the scaling parameter of the domain wall energy, bi(i = 1,2,3) are the constants of the coupling terms between strain and polarization and ci(i = 1,2,3) are the elastic constants.