Civil Engineering Reference
In-Depth Information
15
Nonlinear Problems
SDQWDUHL
(Everything flows)
Aristotle
15.1
INTRODUCTION
So far we have discussed problems where there is a linear relationship between applied
loading and displacement, or between applied flow and temperature/potential. The
system of equations
>
^`^
F
(15.1)
T
u
corresponds to a linear analysis, if {
u
} is a linear function of {
F
}.
The linearity of (15.1) is only guaranteed if certain assumptions are made when
deriving the system of equations. These assumptions are:
1.
The relationships between flux and temperature/potential or stresses and strains are
linear
Matrix
>
T
is not affected by changes in geometry of the boundary that occurs during
loading
2.
3.
Boundary conditions do not change during loading
Indeed, we have implicitly relied on these assumptions to be true in all our previous
derivations of the theory.
An example where the first assumption is violated is elasto- or visco-plastic material
behaviour (this is generally referred to as material nonlinear behaviour). The second one
is violated if displacements significantly change the boundary shape (large displacement
problems). Finally, the third no longer holds true for contact problems, where either the
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