# Von Mises Plasticity in FEniCS

Hello All,

I am trying to do an analysis similar to the analysis done here: Elasto-plastic analysis of a 2D von Mises material — Numerical tours of continuum mechanics using FEniCS master documentation

I have made adjustments to the geometry and boundary conditions without applying any external force. However, there is a nonzero prescribed vertical displacement at the top side, and the analysis is running with very high residuals. Despite my efforts, I have been unable to resolve this issue. Any guidance or advice would be greatly appreciated.

``````from dolfin import *
import numpy as np
import warnings

E = Constant(70e3)
nu = Constant(0.3)
lmbda = E*nu/(1+nu)/(1-2*nu)
mu = E/2./(1+nu)
sig0 = Constant(250.)  # yield strength
Et = E/100.  # tangent modulus
H = E*Et/(E-Et)  # hardening modulus

mesh = UnitSquareMesh(50, 50)
# plot(mesh)

def left_right_boundary(x, on_boundary):
return (near(x[0], 0) or near(x[0], 1)) and on_boundary

def top_boundary(x, on_boundary):
return near(x[1], 1) and on_boundary

def bottom_boundary(x, on_boundary):
return near(x[1], 0) and on_boundary

# Function spaces
deg_u = 2
deg_stress = 2
V = VectorFunctionSpace(mesh, "CG", deg_u)
We = VectorElement("Quadrature", mesh.ufl_cell(), degree=deg_stress, dim=4, quad_scheme='default')
W = FunctionSpace(mesh, We)
W0 = FunctionSpace(mesh, W0e)

sig = Function(W)
sig_old = Function(W)
n_elas = Function(W)
beta = Function(W0)
p = Function(W0, name="Cumulative plastic strain")
u = Function(V, name="Total displacement")
du = Function(V, name="Iteration correction")
Du = Function(V, name="Current increment")
v = TrialFunction(V)
u_ = TestFunction(V)

# Boundary conditions
bc_top = DirichletBC(V.sub(1), 0.03, top_boundary)  # Initial displacement is 0.03
bc_bottom = DirichletBC(V.sub(1), 0, bottom_boundary)
bc_left_right = DirichletBC(V.sub(1), 0, left_right_boundary)
bc = [bc_top, bc_bottom, bc_left_right]

def eps(v):
return as_tensor([[e[0, 0], e[0, 1], 0],
[e[0, 1], e[1, 1], 0],
[0, 0, 0]])
def sigma(eps_el):
return lmbda*tr(eps_el)*Identity(3) + 2*mu*eps_el
def as_3D_tensor(X):
return as_tensor([[X[0], X[3], 0],
[X[3], X[1], 0],
[0, 0, X[2]]])

ppos = lambda x: (x+abs(x))/2.
def proj_sig(deps, old_sig, old_p):
sig_n = as_3D_tensor(old_sig)
sig_elas = sig_n + sigma(deps)
s = dev(sig_elas)
sig_eq = sqrt(3/2.*inner(s, s))
f_elas = sig_eq - sig0 - H*old_p
dp = ppos(f_elas)/(3*mu+H)
n_elas = s/sig_eq*ppos(f_elas)/f_elas
beta = 3*mu*dp/sig_eq
new_sig = sig_elas-beta*s
return as_vector([new_sig[0, 0], new_sig[1, 1], new_sig[2, 2], new_sig[0, 1]]), \
as_vector([n_elas[0, 0], n_elas[1, 1], n_elas[2, 2], n_elas[0, 1]]), \
beta, dp

def sigma_tang(e):
N_elas = as_3D_tensor(n_elas)
return sigma(e) - 3*mu*(3*mu/(3*mu+H)-beta)*inner(N_elas, e)*N_elas-2*mu*beta*dev(e)

a_Newton = inner(eps(v), sigma_tang(eps(u_)))*dxm
res = -inner(eps(u_), as_3D_tensor(sig))*dxm

def local_project(v, V, u=None):
dv = TrialFunction(V)
v_ = TestFunction(V)
a_proj = inner(dv, v_)*dxm
b_proj = inner(v, v_)*dxm
solver = LocalSolver(a_proj, b_proj)
solver.factorize()
if u is None:
u = Function(V)
solver.solve_local_rhs(u)
return u
else:
solver.solve_local_rhs(u)
return

file_results = XDMFFile("plasticity_results.xdmf")
file_results.parameters["flush_output"] = True
file_results.parameters["functions_share_mesh"] = True
P0 = FunctionSpace(mesh, "DG", 0)
p_avg = Function(P0, name="Plastic strain")

# Incremental vertical displacement applied to the top boundary
Nitermax, tol = 200, 1e-8  # parameters of the Newton-Raphson procedure
Nincr = 50
disp_increments = np.linspace(0, 0.03, Nincr+1)[1:]
results = np.zeros((Nincr+1, 2))
for (i, d) in enumerate(disp_increments):
bc_top = DirichletBC(V.sub(1), d, top_boundary)
bc = [bc_top, bc_bottom, bc_left_right]
# Rest of the Newton-Raphson procedure
A, Res = assemble_system(a_Newton, res, bc)
nRes0 = Res.norm("l2")
nRes = nRes0
Du.interpolate(Constant((0, 0)))
print("Increment:", str(i+1))
niter = 0
while nRes/nRes0 > tol and niter < Nitermax:
solve(A, du.vector(), Res, "mumps")
Du.assign(Du+du)
deps = eps(Du)
sig_, n_elas_, beta_, dp_ = proj_sig(deps, sig_old, p)
local_project(sig_, W, sig)
local_project(n_elas_, W, n_elas)
local_project(beta_, W0, beta)
A, Res = assemble_system(a_Newton, res, bc)
nRes = Res.norm("l2")
print("    Residual:", nRes)
niter += 1
u.assign(u+Du)
p.assign(p+local_project(dp_, W0))
sig_old.assign(sig)

# Record results
file_results.write(u, d)
p_avg.assign(project(p, P0))
file_results.write(p_avg, d)

``````