Calculation of Deformation and Concentration

MaRDI QIDQ6684557

simultaneously compute the coupled evolution of the deformation and the concentration poro-visco-elastic material

Description

For given external bulk force density, external surface force density, fixed boundary values and external chemical potential.

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List of mathematical expressions with quantities

Poro-Visco-Elastic Diffusion Boundary Condition	$M(\mathrm{\nabla }\chi ,c)\mathrm{\nabla }{\partial }_c\mathrm{\Phi }(x,\mathrm{\nabla }\chi ,c)\cdot \nu =\kappa (x)({\mu }_e(t,x)-{\partial }_c\mathrm{\Phi }(x,\mathrm{\nabla }\chi ,c))$
$M$ symbol represents Hydraulic Conductivity
$\chi$ symbol represents Mechanical Deformation
$\kappa$ symbol represents Fluid Intrinsic Permeability (Porous Medium)
$\mu$ symbol represents External Chemical Potential
$c$ symbol represents Concentration
Poro-Visco-Elastic Diffusion Equation	$\dot{c}(t,x)=-\mathrm{\nabla }\cdot (M(x,\mathrm{\nabla }\chi (t,x),c(t,x))\mathrm{\nabla }{\partial }_c\mathrm{\Phi }(x,\mathrm{\nabla }\chi (t,x),c(t,x)))$
$M$ symbol represents Hydraulic Conductivity
$\mathrm{\Phi }$ symbol represents Free Energy Density
$\chi$ symbol represents Mechanical Deformation
$c$ symbol represents Concentration
$t$ symbol represents Time
Poro-Visco-Elastic (Dirichlet Boundary)	$\chi (t,x)={\chi }_D(t,x)$
$\chi$ symbol represents Mechanical Deformation
${\chi }_D$ symbol represents Mechanical Deformation (Boundary Value)
$x$ symbol represents Spatial Variable
Poro-Visco-Elastic (Neumann Boundary)	$({\partial }_{\mathrm{\nabla }\chi }\mathrm{\Phi }(x,\mathrm{\nabla }\chi (t,x),c(t,x))+{\partial }_{\mathrm{\nabla }\dot{\chi }}\zeta (x,\mathrm{\nabla }\dot{\chi }(t,x),\mathrm{\nabla }\chi (t,x),c(t,x)))\nu -{\mathrm{\nabla }}_s\cdot ({\partial }_{D^2\chi }H(x,D^2\chi (t,x))\nu )=g(t,x)$
$\mathrm{\Phi }$ symbol represents Free Energy Density
$\chi$ symbol represents Mechanical Deformation
$\nu$ symbol represents Unit Normal Vector
$\zeta$ symbol represents Viscous Dissipation Potential
$c$ symbol represents Concentration
$g$ symbol represents Surface Force Density
$x$ symbol represents Spatial Variable
Poro-Visco-Elastic Quasistatic Equation	$-\mathrm{\nabla }\cdot ({\partial }_{\mathrm{\nabla }\chi }\mathrm{\Phi }(x,\mathrm{\nabla }\chi (t,x),c(t,x))+{\partial }_{\mathrm{\nabla }\dot{\chi }}\zeta (x,\mathrm{\nabla }\dot{\chi }(t,x),\mathrm{\nabla }\chi (t,x),c(t,x))-\mathrm{\nabla }\cdot {\partial }_{D^2\chi }H(x,D^2\chi (t,x)))=f(t,x)$
$H$ symbol represents Hyperstress Potential
$\mathrm{\Phi }$ symbol represents Free Energy Density
$\chi$ symbol represents Mechanical Deformation
$\zeta$ symbol represents Viscous Dissipation Potential
$c$ symbol represents Concentration
$f$ symbol represents External Force Density
$t$ symbol represents Time
$x$ symbol represents Spatial Variable

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