elastic_phasefield#

For elastic_phasefield behaviour displacement \(\underline{\textbf{u}}\), concentration \(c\) and order parameter \(\phi\) are the degrees of freedom of the system. Finding \(\phi (X, t), c (X, t), \vect u(X, t))\), \(\forall (X, t>0)\) will be done by solving the system of equation.

(591)#\[c(X, 0) = c_0(X)\]

(592)#\[\phi(X, 0) = \phi_0(X)\]

(593)#\[\forall (t>0, \phi^*(X), c^*(X))\]

(594)#\[\begin{split}\left\{ \begin{array}{lll} \displaystyle{ \int_V (\pi \phi^* - \vect{\xi} . \nabla \phi^*)dv + \int_{\partial V} \zeta \phi^* ds = 0 } & \mbox{(1)} \\ \\ \displaystyle{ \int_V (\dot{c}c^* - \vect{J} . \nabla c^*)dv + \int_{\partial V} j c^* ds = 0 } & \mbox{(2)} \\ \\ \displaystyle{ \int_V (-\ten \sigma:\nabla \vect{u}^* - \vect f . \vect{u}^*)dv + \int_{\partial V} \vect t . \vect{u}^* ds = 0 } & \mbox{(3)} \\ \end{array} \right.\end{split}\]

The free energy density for the coupled phase field/diffusion/mechanical problem can be approximated by the Ginzburg-Landau coarse-grained free energy functional, which contains a chemical free energy density \(f_{ch}(c, \phi)\), an elastic free energy density \(f_e(\phi, c, \ten \epsilon)\) and a gradient term.

(595)#\[f(\phi, \nabla \phi, c, \ten{E}^e) = f_{ch}(\phi, c) + f_e(\phi, c, \ten{E}^e) + \frac{\alpha}{2} \nabla \phi . \nabla \phi\]

In addition to the chemical state laws, which is defined in the phasefield_phic behavior, the strain-stress relationship in the homogeneous effective medium obeys Hooke’s law as follows

(596)#\[\ten \sigma = \frac{\partial f}{\partial \ten \varepsilon} = \tenf{C}(\phi,c):(\ten{E} - \ten{E}^\star(\phi,c))\]

where \(\ten{E}\) and \(\ten \sigma\) are respectively the macroscopic strain and Cauchy stress quantities.

The effective elasticity tensor \(\ten{\ten C}\) and the effective eigenstrain \(\ten{E}^\star\) due to phase transformation are presented in the <HOMOGENIZATION> section.

Stored Variables

prefix	size	description	default
`sig`	T-2	Cauchy stress	yes
`eto`	T-2	Total (small deformation) strain	yes
`eel`	T-2	Elastic strain	yes
`defo_tr`	T-2	Effective eigenstsrain	no
`dC`	V	Concentration gradient	yes
`J`	V	Concentration flux	yes
`dphi`	V	Order parameter gradient	yes
`xi`	V	Microstress	yes
`pi`	S	Internal microforce	yes
`C`	S	the concentration	yes
`phi`	S	Order parameter	yes

***behavior elastic_phasefield \(~\,\) **energy <ENERGY> \(~\,~\,\) ... \(~\,\) **kinetics \(~\,~\,\) *mobility COEFFICIENT \(~\,\) **chemical_interpolating_function val \(~\,\) **mechanical_interpolating_function val \(~\,\) **homogenization <HOMOGENIZATION> \(~\,\) **phase1 \(~\,~\,\) *elasticity1 <ELASTICITY> \(~\,~\,\) ... \(~\,~\,\) *eigen_coeff1 double \(~\,\) [*delta1 double ] \(~\,\) [*c_ref1 double ] \(~\,\) **phase2 \(~\,~\,\) *elasticity2 <ELASTICITY> \(~\,~\,\) ... \(~\,~\,\) *eigen_coeff2 double \(~\,\) [*delta2 double ] \(~\,\) [*c_ref2 double ]

**mechanical_interpolating_function

Defines the polynomial degree of interpolating function. Tree choices are availables.

CODE	DESCRIPTION
\(0\)	\(h_u(\phi) = \phi\)
\(1\)	\(h_u(\phi) = \phi^2 ( 3 - 2 \phi)\)
\(2\)	\(h_u(\phi) = \phi^3 ( 6 \phi^2 - 15 \phi + 10)\)

**homogenization

This option will be detailed in the <HOMOGENIZATION> section.

**phase1

Definition of the material elastic parameters and eigenstrain induced by variation of concentration.

**phase2

Identical as **phase1

The eigenstrain in the phase \(i\) is defined as follow

(597)#\[\ten{\epsilon}_i^* = ({\tt eigen\_coeff_i} + {\tt delta_i} (c- {\tt c\_ref_i} )) \ten 1,\qquad {\rm where}\qquad i = \{1,2\}\]

***behavior elastic_phasefield
 **energy kim
  *phase1
   c1 0.7
   b1 0.0
   k1 1.
   D1 0.1
  *phase2
   c2 0.3
   b2 0.0
   k2 1.
   D2 0.1
  *interface
   energy 1.
   thickness 0.25
   zeta 0.05
   ENER 0.5
 **kinetics
  *mobility 1.
 **chemical_interpolating_function 1.
 **mechanical_interpolating_function 1.
 **homogenization Khachaturyan
 **phase1
  *elasticity1
   young 70000.
   poisson 0.3
  *eigen_coeff1 0.000
  *delta1 0.0015
  *c_ref1 0.
 **phase2
  *elasticity2
   young 70000.
   poisson 0.3
  *eigen_coeff2  0.000
  *delta2 0.0015
  *c_ref2 0.
***return