Hyperelastic beam

dolfinx
1

Hi, I’m solving the case of a 2d Mooney-Rivlin beam subjected to a distributed shear on the free edge x=6. I report my code that works correctly.

 from dolfin import *
import fenics as fe
import time

start = time.time()

parameters["form_compiler"]["cpp_optimize"] = True
ffc_options = {"optimize": True, \
               "eliminate_zeros": True, \
               "precompute_basis_const": True, \
               "precompute_ip_const": True}

mesh = RectangleMesh(Point(0.0, 0.0), Point(6.0, 1.5), 250, 50, "crossed")

V = VectorFunctionSpace(mesh, "Lagrange", 2)
left = CompiledSubDomain("near(x[0], side) && on_boundary", side=0.0, tol=10e-15)  # Code C++

c = Expression(("0.0", "0.0"), degree=2)
t = Expression(("0.0", "-1.0"), degree=2)


du = TrialFunction(V)            # Incremental displacement
v = TestFunction(V)             # Test function
u = Function(V)                 # Displacement from previous iteration

D = mesh.topology().dim()
print(D)
neumann_domain = MeshFunction("size_t", mesh, D-1)
neumann_domain.set_all(0)
CompiledSubDomain("near(x[0], side) && on_boundary", side=6.0, tol=10e-15).mark(neumann_domain, 1)
ds = Measure("ds", subdomain_data=neumann_domain)


bcs = DirichletBC(V, c, left)

d = u.geometric_dimension()
I = Identity(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
J = sqrt(det(C))
I1 = tr(C)
I2 = 0.5 * (I1**2 - tr(C*C))
c1 = Constant(630)
c2 = Constant(-1.2)
c = Constant(100)
d = 2 * (c1 + 2 * c2)

#MR
psi = c*(J-1)**2 - d*ln(J) + c1*(I1-3) + c2*(I2-3)

# E, nu = 10**3, 0.3
# mu, lmbda = Constant(E/(2*(1 + nu))), Constant(E*nu/((1 + nu)*(1 - 2*nu)))

# Neo-Hookean
# psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2

# Total potential energy
Pi = psi*dx - dot(t, u)*ds(1)

F = derivative(Pi, u, v)

J = derivative(F, u, du)
problem = NonlinearVariationalProblem(F, u, bcs, J)
solver = NonlinearVariationalSolver(problem)
prm = solver.parameters
prm['newton_solver']['absolute_tolerance'] = 1E-6
prm['newton_solver']['relative_tolerance'] = 1E-6
# prm['newton_solver']['maximum_iterations'] = 25
# prm['newton_solver']['relaxation_parameter'] = 1.0

prm['newton_solver']['linear_solver'] = 'petsc'

solver.solve()

L2 = inner(u, u) * dx
H10 = inner(grad(u), grad(u)) * dx
energynorm = sqrt(assemble(psi*dx))
# H1 = inner(F, P) * dx
L2norm = sqrt(assemble(L2))
H10norm = sqrt(assemble(H10))

print("L2 norm = %.10f" % norm(u, norm_type="L2"))
print("H1 norm = %.10f" % norm(u, norm_type="H1"))
print("H10 norm = %.10f" % norm(u, norm_type="H10"))
print("Running time = %.3f" % float(time.time()-start))

I decide to plot the Cauchy stress tensor, using this lines of code:

import matplotlib.pyplot as plt
F = I + grad(u)
C = F.T * F
J = sqrt(det(C))
I1 = tr(C)
I2 = 0.5 * (I1**2 - tr(C*C))
S = (2*c1 + 2*c2*I1)*I - 2*c2*C.T + (2*c*(J-1)*J - d)*inv(C.T)
s=1/J*F*S*F.T
pp=plot(s[1,1],mode='color',cmap='viridis')
plt.colorbar(pp)

That’s the component \sigma_{yy}:
FENICSsyy

I’m not able to understand if the non-zero stresses that I have in correspondence of the free edge x=6 are correct or they are due to an error in the code or a problem in the mesh (if I refine they remain).
Could you help me?
Thank you

2

You cannot expect that \sigma_{yy}=0 on this edge, you should rather have \sigma_{nn}=0 where n is the direction of the normal in the deformed configuration

3

Thank you very much.

4

One more thing, referring to what you say, is it correct to say that the component xx of the first Piola-Kirchhoff stress tensor is zero on the right side (since its the normal stress in the reference configuration)?

5

Yes this is correct.

6

Thank you very much.