Elasto-dynamics

Kinematics

\(\Omega_0\) is the refrence frame at intial configuration of our strcuture. \(\Omega_t\) is the refrence frame of our structure in the deformed state. \(X\) is the the coordinates of the initial or reference configuration. \(y\) is the dispalacement with respect to the initial configuration.

The mapping between \(\Omega_0\) anf \(\Omega_t\) can be expressed as

\[x(X,t) = X + y(X,t)\]

The velocity \(u\) and acceleration \(a\) of the structure are obtained by differentiating the displacement \(y\) with respect to time holding the material coordinate \(X\) fixed

\[\begin{split}\begin{aligned} u &=& \frac{dy}{dt} \\ a &=& \frac{d^2y}{dt^2} \\ \end{aligned}\end{split}\]

The deformation Gradient \(F\) is expressed as

\[\begin{split}\begin{aligned} F &=& \frac{\partial x }{\partial X} = \nabla u + I \\ \end{aligned}\end{split}\]

the Cauchy–Green deformation tensor \(C\) is expressed as

\[\begin{aligned} C &=& F^T F \end{aligned}\]

the Green–Lagrange strain tensor \(E\) is expressed as

\[\begin{split}\begin{aligned} E &=& \frac{1}{2} (C - I ) = 0.5 \times (F^T F - I ) \\ \end{aligned}\end{split}\]

The determinant of the deformation gradient \(J\) is given by

\[\begin{split}\begin{aligned} J &=& \text{det} F \\ \end{aligned}\end{split}\]

Principal of virtual work and variational form of structural mechanics

The derivation of the weak form starts with the principal of virtual work,

\[\begin{aligned} \delta W &=& \delta W_{int} + \delta W_{ext} = 0 \end{aligned}\]

where \(W\), \(W_{ext}\) and \(W_{int}\) are the total work, external and internal work, respectively, and \(\delta\) denotes their variation with respect to the virtual displacement \(w\).

\[\delta W = \frac{d}{d \varepsilon} W(y+\varepsilon W) |_{\varepsilon=0}\]

The external virtual work includes work done by the inertial and body forces and surface tractions.

\[\delta W_{ext} = \int_{\Omega_t} w.\rho (f-a) d\Omega + \int_{(\Gamma_t)_h} w.h d\Gamma,\]

The internal virtual work is due to the internal stresses and can be expressed as

\[\delta W_{int} = - \int_{\Omega_0} \delta E : S d\Omega\]

\(S\) is is the second Piola–Kirchhoff stress tensor, which is symmetric and work-conjugate to \(E\).

the variational formulation of the structural mechanics problem:find the structural displacement \(y\) such that for all \(w\)

\[\int_{\Omega_t} w . \rho a d \Omega + \int_{\Omega_0} \delta E : S d\Omega - \int_{\Omega_t} w.\rhof d \Omega - \int_{(\Gamma_t)_h} w.h d\Gamma = 0\]

Structural Mechanics Formulation in the Reference Configuration

we first start by formulating the external virtual work in the reference configuration

\[\int_{\Omega_t} w. \rho (a-f) d \Omega = \int_{\Omega_0} w. \rho_0 (a-f) d \Omega\]

where \(\rho_0\) is the mass density in the reference configuration. the following variational formulation of the structural mechanics problem posed in \(\Omega_0\)

\[\int_{\Omega_0} w. \rho_0 a d \Omega + \int_{\Omega_0} \nabla_X w : P d\Omega - \int_{\Omega_0} w.\rho_0 f d \Omega - \int_{(\Gamma_0)_h} w. \hat{h} d\Gamma = 0\]

The variation of strain \(\delta E\) is expressed as

\[\delta E = \frac{1}{2} (F^T \nabla_X w + \nabla_x w^T F)\]

\(\nabla_X\) denotes the gradient taken with respect to the spatial coordinates of the reference configuration. Due to the symmetry of \(S\), the scalar product \(\delta E :S\) simplifies to

\[\delta E : S = \nabla_x w : P\]

where

\[P = FS\]

and using the Saint Venant–Kirchhoff model

\[S = \lambda \times tr(E) \times I + 2 \times \mu E\]

with

\[\begin{split}\begin{aligned} \lambda &=& \frac{E \nu}{((1.0 + \nu ) (1.0 - 2.0 * \nu))} \\ \mu &=& \frac{E}{ (2.0 (1.0 + \nu))} \end{aligned}\end{split}\]

recover boundary conditions are

\[\begin{split}\begin{aligned} \rho_0 (a - f ) - \nabla_X \dot P &=& 0 \text{on} \Omega_0 \\ y_i &=& g_i \text{on} (\Gamma_0)_{g_i} \\ P\hat{n} = \hat{h} \text{on} (\Gamma_0)_{h} \\ \end{aligned}\end{split}\]

Follower pressure load

\[\int_{(\Gamma_t)_h} w \dot h d\Gamma_t = - \int w \dot pn d\Gamma_t = - \int_{(\Gamma_0)_h} w \dot p J F^{-T} \hat{n} d \Gamma_0\]

Fluid Solve

Mesh deformation