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Problems
for Chapter 8
Theory and
Implementation of the Finite Element Method
8.5. The Finite
Element Method For Plastic Materials
8.5.1.
Set up the
demonstration viscoplastic finite element code described in Section
8.5.7 to calculate the stress-strain
relation for the viscoplastic material under uniaxial tension. Mesh the specimen with a single 8 noded
brick element, using the mesh shown in the figure. Apply the following boundary constraints
to the specimen:  at node 1; at node 2, at nodes 3 and 4. Apply a uniform traction whose magnitude
increases from 0 to 20 (arbitrary units) in time of 2 units on face 2 of the
element.
8.5.1.1.
Run a simulation with the following material
parameters:
and plot a graph showing the variation of traction with
displacement on the element
8.5.1.2.
Modify the boundary conditions so that only the constraint is enforced at node 2. Run the code to attempt to find a solution
(you will have to abort the calculation).
Explain why the Newton
iterations do not converge.
8.5.1.3.
Repeat 1.1 with
,
the correct boundary conditions, and with a maximum traction of 18
units. In this limit the material is
essentially rate independent. Compare
the predicted traction-displacement curve with the rate independent limit.
8.5.2.
Modify the
viscoplastic finite element program to apply a constant (nominal) uniaxial
strain rate to the specimen described in the preceding problem, by imposing
an appropriate history of displacement on the nodes of the mesh. Test the code by plotting a graph showing
uniaxial stress-v-strain in the specimen, for material parameters (in this limit the material is essentially a
power-law creeping solid with constant flow stress) with an applied strain
rate of . Compare the numerical solution with the
exact solution.
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8.5.3.
Modify the
viscoplastic finite element program so that instead of using a power-law
function to represent the variation of flow-stress with accumulated plastic strain ,
the flow stress is computed by interpolating between a user-defined series of
points, as indicated in the figure (the flow stress is constant if the
plastic strain exceeds the last point).
Test your code by using it to calculate the stress-strain relation for
the viscoplastic material under uniaxial tension. Use the mesh described in Problem 1, and use
material parameters (this makes the material essentially rigid
and rate independent, so the stress-strain curve should follow the
user-supplied data points).
8.5.4.
Modify the
viscoplastic finite element code described in Section 8.5.7 to solve problems
involving a rate independent, power-law isotropic hardening elastic-plastic
solid, with incremental stress-strain relations
and a yield criterion
Your
solution should include the following steps:
8.5.4.1.
Devise a method
for calculating the stress at the end of a load increment. Use a fully implicit computation, in which
the yield criterion is exactly satisfied at the end of the load
increment. Your derivation should
follow closely the procedure described in Section 8.5.4, except that the
relationship between and must be calculated using the yield
criterion, and you need to add a step to check for elastic unloading.
8.5.4.2.
Calculate the
tangent stiffness for the rate independent solid, by
differentiating the result of 4.1.
8.5.4.3.
Implement the
results of 4.1 and 4.2 in the viscoplastic finite element code.
8.5.4.4.
Test your code
by using it to calculate the stress-strain relation for the viscoplastic
material under uniaxial tension. Use
the mesh, loading and boundary conditions described in Problem 1, and use
material properties
8.5.5.
Modify the
viscoplastic finite element code described in Section 8.5.7 to solve problems
involving a rate independent, linear kinematic hardening elastic-plastic
solid, with incremental stress-strain relations
and a yield criterion and hardening law
Your
solution should include the following steps:
8.5.5.1.
Devise a method
for calculating the stress at the end of a load increment. Use a fully implicit computation, in which
the yield criterion is exactly satisfied at the end of the load
increment. Your derivation should
follow closely the procedure described in Section 8.5.4, except that the
relationship between and must be calculated using the yield criterion
and hardening law, and you need to add a step to check for elastic unloading.
8.5.5.2.
Calculate the
tangent stiffness for the rate independent solid, by
differentiating the result of 8.5.5.1.
8.5.5.3.
Implement the
results of 8.5.5.1 and 8.5.5.2 in the viscoplastic finite element code.
8.5.5.4.
Test your code
by using it to calculate the stress-strain relation for the viscoplastic
material under uniaxial tension. Use
the mesh, loading and boundary conditions described in Problem 1, and use
material properties
8.5.6.
In this problem
you will develop a finite element code to solve dynamic problems involving viscoplastic materials. Dynamic problems for nonlinear materials
are nearly always solved using explicit Newmark time integration, which is
very straightforward to implement. As
usual, the method is based on the virtual work principle
8.5.6.1.
By introducing
a finite element interpolation, show that the virtual work principle can be
reduced to a system of equations of the form
and
give expressions for .
8.5.6.2.
To implement
the finite element method, it is necessary to calculate the stress in the solid. Idealize the solid as a viscoplastic
material with constitutive equations described in Section 8.5.1. Since very small time-steps must be used in
an explicit dynamic calculation, it is sufficient to integrate the
constitutive equations with respect to time using an explicit method, in
which the plastic strain rate is computed based on the stress at the start of
a time increment. Show that the
stress at time can be expressed in terms of the stress at time t,
the increment in total strain during the time interval and material properties as
8.5.6.3.
The equations
of motion can be integrated using an explicit Newmark method using the
following expressions for the acceleration, velocity and displacement at the
end of a generic time-step
A lumped mass matrix should be used to speed up
computations. Note that the residual
force vector is a function of the displacement field in
the solid. It therefore varies with time, and must be re-computed at each
time step. Note that this also means
that you must apply appropriate constraints to nodes with prescribed
accelerations at each step. Implement this algorithm by combining appropriate
routines from the static viscoplastic code and the Newmark elastodynamic code
provided.
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8.5.6.4.
Test your code
by simulating the behavior of a 1D (plane strain) shown in the figure. Assume that the bar is at rest and stress
free at t=0, and is then subjected
to a constant horizontal traction at for t>0. Fix the displacements for the node at and apply at . Take the magnitude of the traction to be 2
(arbitrary units) and use material properties . Take in the Newmark integration, and use 240 time
steps with step size 0.01 units. Plot
a graph showing the displacement of the bar at as a function of time.
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