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Nonlinear Finite Element Methods
Assignment for summer term 2020
The problem of creep of a thick-walled pipe under internal pressure p is considered as
sketched in Figure 1. The pressure rises linearly up to its final value pmax and is then
hold until tf as shown in Figure 2. Plain strain εzz = 0 conditions are assumed.
Figure 1: Thick-walled pipe
axisymmetric conditions, the only non-vanishing equilibrium condition is
0 =∂(rσrr)∂r − σφφ . (1)
Therein, σrr and σφφ refer to the stress components with respect to a polar coordinate
system. The weak form of Eq.
(2)
with stresses and strains written in Voigt notation as. (3)
Therein, the only non-vanishing displacement component is ur(r) as the displacement
in radial direction. The boundary conditions for the problem in Figure 1 are σrr(r =
a) = −p and σrr(r = b) = 0, respectively.
1
The linear visco-elastic behavior of the material is described by the equations
σ = C · ε + σov (4a)σ˙ov = Q dev (˙ε) −1Tσov (4b)
wherein C is the isotropic (long-term) elastic stiffness matrix, expressed by Young’s
modulus E and Poisson ration ν. The evolution of the overstress σ
ov (as internal state
variable) is governed by the modulus Q and a characteristic time scale T.
Create a program (MatLab/Octave/Python) which solves this static FEM problem.
The program has to be verified by comparisons with known analytical solutions and a
convergence study shall be performed (see below).
2 Details
The following list gives a brief overview of the features which have to be implemented:
• quasi-static conditions: δW = δuˆT·hFˆint − Fˆexti
• linear shape functions for ur(r):[N] = 12(1 − ξ),12(1 + ξ)T
in Ω = {ξ ∈ [−1, 1]} (5)
• quadrature with 1 Gauss point per element
• local mesh refinement closer to the interior of the pipe: he(r = a) = 12
h
e
(r = b)
(h
e
: element size), see code sniplet in appendix
• time integration with Euler backward method (EB) or modified Euler method
(EM)
• variable number of elements and time increment ∆t
• Newton-Raphson method with convergence criteria
0.005 kuˆk∞ (with k◦ˆk∞ denoting the infinity norm, i. e. the maximum component
by amount of the column vector ◦ˆ)
The particular material parameters (E, ν, Q, T), loading parameters (pmax, tL, tf),
time integration scheme (EM/EB) and geometric properties a and b to be implemented
depend on your variant as given in Table 1. Each student has to work on the variant
that corresponds to the last digit of her or his matriculation number.
3 Workflow
1. Theory
• Discretize the weak form (2) in space (i. e. in r).
• Identify the B matrix to be defined as ε = B · uˆ
e
for the shape functions in
Eq. (5).
• Identify the vectors of internal and external nodal forces Fˆ e
int and Fˆ
ext, respectively.
• Discretize the constitutive equations (4) in time and compute the algorithmically
consistent material tangent stiffness.
2. Implementation in MatLab/Octave/Python:
• Implement B and [N] into an element routine to compute Fˆ e
int (Hint: The
Jacobian of the element is identical to the FEM of rods considered in the
exercises.)
• Develop the main program which assembles total nodal forces for each time
increment and performs the Newton-Raphson scheme.
• Note that the material routine requires internal state variables (the overstresses
σ
ov) for which memory has to be allocated and which have to be
passed through main program and element routine.
3. Verification:
a) According to classical theory of elasticity, the exact solution of the considered
boundary value problem Eqs.(6)
compare basic course Engineering Mechanics B. In a first step, use a material
routine for the purely elastic case (corresponding formally to Q = 0). Perform
a convergence study with respect to the number of elements and verify that
your FEM solution converges towards the exact solution (6). Verify that the
Newton-Raphson method converges within a single iteration for the linear
problem.
Table 1: Assignment of parameters
var E [MPa] ν Q [MPa] T [s] a [mm] b [mm] pmax [MPa] tL [s] tf
[s]
1 EM 200 000 0.20 100 000 1 50 100 140 2 10
2 EM 70 000 0.25 35 000 2 40 80 50 4 20
3 EM 70 000 0.30 35 000 3 60 120 50 6 30
4 EM 200 000 0.30 100 000 1 30 60 140 2 10
5 EM 100 000 0.30 50 000 2 40 80 70 4 20
6 EB 200 000 0.20 100 000 3 50 100 140 6 30
7 EB 70 000 0.25 35 000 1 40 80 50 2 10
8 EB 70 000 0.30 35 000 2 60 120 50 4 20
9 EB 200 000 0.30 100 000 3 30 60 140 6 30
0 EB 100 000 0.30 50 000 4 40 80 70 8 40
b) In the next step, perform a convergence study with respect to number of
elements and time increments ∆t for the visco-elastic model. Identify the
necessary number of elements and the required ∆t.
4. Results:
• Extract the distributions of ur(r), σrr and σφφ at final loading t = tf
from
• Extract the time history of the widening of the pipe ur(r = b, t) for t ∈ [0, tf
].
Verify, that the visco-elastic solution relaxes towards the elastic solution (6).
4 Documentation
In addition to the program code, a short technical documentation is to be created (in
hard-copy form) containing:
1. a brief overview over the implemented theory
2. an overview over the structure of the program (routines, files, . . . ), in text form or
graphically
3. a short user’s manual answering the following questions:
• How to start the program?
• Where does the program get its input from?
• What output does the program generate and where does it store it to?
4. verification: results requested in section 3.3 and 3.4
5 Remarks
• The successful completion of the task is a prerequisite to be admitted to the final
examination.
• The deadline for the assignment is Friday, July 10, 2020 when the program has
to be sent to Geralf.Huetter@imfd.tu-freiberg.de and the documentation has to
be submitted in hard-copy form at secretary of the institute (room WEI-130).
The program has to be presented individually before the examination. Details
on the mode of presentation will be decided and published, depending on the
circumstances of teaching at that time.
Appendix