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Problems
for Chapter 8
Theory and
Implementation of the Finite Element Method
8.6. Advanced
Element Formulations
8.6.1.
Volumetric
locking can be a serious problem in computations involving nonlinear
materials. In this problem, you will
demonstrate, and correct, locking in a finite element simulation of a
pressurized hypoelastic cylinder.
8.6.1.1.
Set up an input
file for the example hypoelastic finite element code described in Section
8.3.9 to calculate the deformation and stress in a hypoelastic pressurized
cylinder deforming under plane strain conditions. Use the mesh shown in the figure, with
appropriate symmetry boundary conditions on  and . Apply a pressure of 20 (arbitrary units)
to the internal bore of the cylinder and leave the exterior surface free of
traction. Use the following material
properties: . Plot a graph of the variation of the radial
displacement of the inner bore of the cylinder as a function of the applied
pressure. Make a note of the
displacement at the maximum pressure.
8.6.1.2.
Edit the code
to reduce the number of integration points used to compute the element
stiffness matrix from 9 to 4. (modify
the procedure called `numberofintegrationpoints’). Repeat the calculation in 8.6.1.1. Note the substantial discrepancy between
the results of 8.6.1.1 and 8.6.1.2 this is caused by locking. The solution in 5.2, which uses reduced
integration, is the more accurate of the two.
Note also that using reduced integration improves the rate of
convergence of the Newton-Raphson iterations.
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8.6.2.
Modify the
hypoelastic finite element code described in Section 8.3.9 to use selective
reduced integration. Check your code
by (a) repeating the calculation described in problem 8.6.1; and (b) running
a computation with a mesh consisting of 4 noded quadrilateral elements, as
shown in the figure. In each case,
calculate the variation of the internal radius of the cylinder with the
applied pressure, and plot the deformed mesh at maximum pressure to check for
hourglassing. Compare the solution
obtained using selective reduced integration with the
8.6.3.
Run the simple
demonstration of the B-bar method described in Section 8.6.2 to verify that
the method can be used to solve problems involving near-incompressible
materials. Check the code with both
linear and quadratic quadrilateral elements.
8.6.4.
Extend the
B-bar method described in Section 8.6.2 to solve problems involving
hypoelastic materials subjected to small strains. This will require the following steps:
8.6.4.1.
The virtual
work principle for the nonlinear material must be expressed in terms of the
modified strain measures and defined in Section 8.6.2. This results in a system of nonlinear
equations of the form
which must be solved using the Newton-Raphson
method. Show that the Newton-Raphson
procedure involves repeatedly solving the following system of linear
equations for corrections to the displacement field
where
is defined in Section 8.6.2.
8.6.4.2.
Modify the
hypoelastic code provided to compute the new form of the stiffness matrix and
element residual. You will find that
much of the new code can simply be copied from the small strain linear
elastic code with the B-bar method
8.6.4.3.
Test the code
by solving the problems described in 8.6.1 and 8.6.2.
8.6.5.
Extend the
B-bar method described in Section 8.6.2 to solve problems involving
hypoelastic materials subjected to small strains, following the procedure
outlined in the preceding problem.
8.6.6.
In this problem
you will extend the B-bar method to solve problems involving finite
deformations, using the hyperelasticity problem described in Section 8.4 as a
representative example. The first step
is to compute new expressions for the residual vector and the stiffness
matrix in the finite element approximation to the field equations. To this end
·
New variables
are introduced to characterize the volume change, and the rate of volume
change in the element. Define
Here,
the integral is taken over the volume of the element in the reference
configuration.
·
The deformation
gradient is replaced by an approximation ,
where n=2 for a 2D problem and n=3 for a 3D problem, while J=det(F).
·
The virtual
velocity gradient is replaced by the approximation
The virtual work equation is replaced by
8.6.6.1.
Verify that
8.6.6.2.
In calculations
to follow it will be necessary to calculate . Find an expression for .
8.6.6.3.
The virtual
work equation must be solved for the unknown nodal displacements by
Newton-Raphson iteration. Show that,
as usual, the Newton-Raphson procedure involves repeatedly solving the
following system of linear equations for corrections to the displacement
field
and derive expressions for and .
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