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Problems
for Chapter 8
Theory and
Implementation of the Finite Element Method
8.3. The Finite
Element Method For Hypoelastic Materials
8.3.1.
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 50 (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 showing the variation of the radial displacement of the inner
bore of the cylinder as a function of the internal pressure. HEALTH
WARNING: The example code uses fully integrated elements and will give
very poor results for large values of n.
8.3.2.
Solve the
following coupled nonlinear equations for x
and y using Newton-Raphson
iteration.
8.3.3.
Consider the
simple 1D bar shown in the figure.
Assume that the boundary conditions imposed on the bar are identical
to those for the 1D elastic bar discussed in Section 8.1.5, so that is the only nonzero displacement component
in the bar. Assume that the material
has a hypoelastic constitutive equation, which has a nonlinear volumetric and
deviatoric response, so that the stress-strain relation has the form
where
8.3.3.1.
Show that the
virtual work principle can be reduced to
8.3.3.2.
Introduce a 1D
finite element interpolation for and following the procedure outlined in Section
8.1.5 to obtain a nonlinear system of equations for nodal values of . Show that the nonlinear equations can be
solved using a Newton-Raphson procedure, by repeatedly solving a system of
linear equations
where
denotes a correction to the current
approximation to . Give expressions for and .
8.3.3.3.
Extend the
simple 1D linear elastic finite element code described in Section 8.1.5 to
solve the hypoelasticity problem described here.
8.3.3.4.
Test your code
by (i) calculating a numerical solution with material properties ,
n=2, and loading b=0, . and (ii) calculating a numerical solution
with material properties ,
n=10, and loading b=10, .
Compare the numerical values of stress and displacement in the bar with the
exact solution.
8.3.4.
Calculate the
tangent moduli for the hypoelastic material described in
the preceding problem.
8.3.5.
In this problem
you will develop a finite element code to solve dynamic problems involving a hypoelastic material with the
constitutive model given in Section .
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
8.3.5.1.
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 .
8.3.5.2.
These 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 hypoelastic code and the Newmark elastodynamic code
provided.
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8.3.5.3.
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. Would you expect the vibration frequency of
the end of the bar to increase or decrease with n? Test your intuition by
running a few simulations.
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