# Copyright (C) 2009-2010 Harish Narayanan and Garth N. Wells
#
# This file is part of DOLFIN.
#
# DOLFIN is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# DOLFIN is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with DOLFIN. If not, see <http://www.gnu.org/licenses/>.
#
# Modified by Anders Logg 2011
#
# First added:  2009-09-29
# Last changed: 2011-06-28
#
# The bilinear form a(du, v) and linear form L(v) for
# a hyperelastic model.
#
# Compile this form with FFC: ffc -l dolfin -feliminate_zeros -fprecompute_basis_const -fprecompute_ip_const HyperElasticity.ufl

# Function spaces
element = VectorElement("Lagrange", tetrahedron, 1)

# Trial and test functions
du = TrialFunction(element)     # Incremental displacement
v  = TestFunction(element)      # Test function

# Functions
u = Coefficient(element)        # Displacement from previous iteration
B = Coefficient(element)        # Body force per unit volume
T = Coefficient(element)        # Traction force on the boundary

# Kinematics
I = Identity(element.cell().d)  # Identity tensor
F = I + grad(u)                 # Deformation gradient
C = F.T*F                       # Right Cauchy-Green tensor

# Invariants of deformation tensors
Ic = tr(C)
J  = det(F)

# Elasticity parameters
mu    = Constant(tetrahedron)
lmbda = Constant(tetrahedron)

# Stored strain energy density (compressible neo-Hookean model)
psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2

# Total potential energy
Pi = psi*dx - inner(B, u)*dx - inner(T, u)*ds

# First variation of Pi (directional derivative about u in the direction of v)
F = derivative(Pi, u, v)

# Compute Jacobian of F
J = derivative(F, u, du)