Define Ogden strain energy density function

Question

Hello,

I was studying the tutorials about Hyperelasticity and eigenvalues solvers, but I could not figure out how to define the strain energy density function in terms of the eigenvalues of the left Cauchy-Green deformation tensor (as it is required by Ogden model). Is it possible? If it is, can someone show me some lines of code with an example?

Thank you for your attention,
Eduardo

Answer 1

Hi, as far as I know this is not possible in a simple way. The problem is already at the first step of FEniCS chain - there is no operand for eigenvalues in UFL. I suggest you register this as a feature request. Alternatively you could try to make use of the following approach

from dolfin import *

mesh = UnitSquareMesh(10, 10)
V = VectorFunctionSpace(mesh, 'CG', 2)
u = interpolate(Expression(('x[0]', 'x[0]*x[1]'), degree=2), V)

F = Identity(2) + grad(u)
C = F*F.T

# Eigenvalues are roots of characteristic polynomial
# e**2 - tr(A)*e + det(A) = 0
def eig_plus(A): return (tr(A) + sqrt(tr(A)**2-4*det(A)))/2

def eig_minus(A): return (tr(A) - sqrt(tr(A)**2-4*det(A)))/2

# Check
S = FunctionSpace(mesh, 'CG', 2)

f0 = project(tr(C), S) 
f = project(eig_plus(C)+eig_minus(C), S)
print (f0.vector()-f.vector()).norm('l2')

f0 = project(det(C), S)
f = project(eig_plus(C)*eig_minus(C), S)
print (f0.vector()-f.vector()).norm('l2')

Comments

Thank you! I will try this approach.