The non-linearity in a model is introduced when its stiffness changes during deformation
in this lesson we'll discuss what are the different physical factors
that can potentially change the stiffness of a system and make it non-linear
typically there are three main factors that can change the stiffness of a system
they are large deformations contact non-linearities and material non-linearities
there are other special cases which can trigger a change in overall stiffness but we'll discuss
them towards the end let's start by focusing on the first factor which is the large deformations
as the name suggests when a structure undergoes large deformations it tends
to introduce non-linearity this is because when the deformation is large enough to change the
shape of the structure it triggers changes to geometrical aspects such as wall thicknesses
area of cross section and may even change the direction of loading
all these changes can affect the load distribution
and in turn introduce a non-linear displacement response for an applied force
take for instance an aluminum bar that's under tensile or compressive loading as the load
increases the bar undergoes large deformation in this process its area of cross section increases
so as the bar deforms and the applied load continues to be distributed over a larger area
this leads to a non-linear force displacement response especially at large deformation
note that the area of cross section changes at small deformations too so the response
is non-linear all along but at small deformations the change in area of cross section is negligible
so the force displacement response at those loads can be approximated to be a linear response
now we may have the luxury of making approximations with a solid aluminum bar
but if we shift our focus to a similar loading in case of a thin hollow structure such as a soda can
which is made of the same aluminum material the mechanics change
the soda can crushes under large compressive loads and as you can notice there's a
significant change in its shape and therefore cannot be approximated as a linear response
these two examples clearly show how large deformations alone can introduce
non-linearity in the system response of course there's more to it than just these two examples
and we'll discuss it in more detail in the next section but for the current discussion
these two examples convey the message of how large deformations can make a model non-linear
now let's move on to the second factor which is contact non-linearity
to understand about contact let's take a few steps back in time and imagine how
the great pyramids of giza in egypt were built
the pyramids were built using over 2 million huge blocks of limestone that
weigh about 25 to 80 tons each a section of the historians believe that the construction workers
used to drag these blocks from a distance over a slope to place them in their current location
now let's imagine a group of workers who are trying to move a block from its starting place
when they start pulling it the block may not start moving from the get go
instead they continue to pull it harder until it begins to slide
now let's look at the force displacement response of this system even though the
workers were applying some force initially there was no gross sliding so the displacement is zero
once the applied force overcomes the static frictional force of the block
it begins to slide and if the workers are say moving at a constant speed the block
undergoes a displacement that's linearly increasing with the force
but even though there's a linear response you must note that initially there was no displacement
so there's a jump in the response and this discontinuity introduces non-linearity
now let's take another scenario if the workers are dragging these blocks for
say a mile the frictional force of the block need not remain the same throughout this distance
it may be easier to drag it for the first quarter of the mile if the surface is wet but
it may increase over the next quarter as the water dries up and making it harder to drag the stone
so if we plot the frictional force as a function of the distance of sliding the response is no
longer linear this is again another classic case of how contact introduces non-linearities
there are many more industrial applications which are more relatable in present times
where contacts exist and introduce non-linearities but not all contact types introduce non-linearity
for instance when two metallic parts are welded together then they are in contact but there is
no relative motion between them in such a case the contact does not introduce a non-linearity
but when the same two parts are held together by a bolted plate mechanism then the surfaces can
slide over each other to a small degree even this small motion can make the system non-linear
having said that if this bolted joint happens to be part of a huge assembly where there is small
relative motion between the parts is irrelevant then it can be idealized as a linear contact
so the analyst must first determine whether this detail is important for their study and based on
that plan on whether the contact should be non-linear sliding is not the only source
of contact non-linearity loss of contact is also considered as a contact non-linearity
say a block that's resting on a surface is lifted upwards it loses contact with the ground and this
is also considered as a non-linearity as there is no longer a transfer of forces between the two
so to sum it up contacts between parts of an assembly can introduce non-linearity in a model
a general recommendation to determine whether the contact is non-linear in nature
is to see if there is any relative motion possible between the two bodies that cannot be neglected
both in normal and tangential directions if there is then that contact introduces non-linearity
if there isn't then it can be approximated as a linear contact
we'll have a more detailed review of contact in a future course
but for the scope of the current section we may end this discussion
now moving on to the third source of non-linearity which is material non-linearity this is introduced
by mechanical behavior of the specific materials used for different parts in case of materials we
generally look at their stress strain response to see whether the material is linear or non-linear
most metals tend to have a linear behavior at least at small strains and most soft
materials such as rubbers and biological tissues tend to have a non-linear behavior
and generally undergo large deformations
in such cases it's straightforward to pick a linear or non-linear solver
however a material non-linearity in general is associated with a physical behavior
and ignoring it in the analysis may result in incorrect predictions
for instance as mentioned earlier most metals can be approximated to have a linear behavior
at small deformations however if we extend this approximation to larger strains as well
then in an application such as coal forging we will ignore the effects of metal plasticity which
introduces non-linearity in the material behavior and end up with incorrect results
this is because the key physics that's resulting in permanent deformation during cold forging
is the metal plasticity and assuming a linear material behavior will disregard metal plasticity
and hence our metal samples don't undergo any permanent changes in their shape
which is incorrect now let's take a different example consider the installation of
a rubber o-ring and see how much internal pressure it can support
this can be done by measuring the contact pressure generated due to the deformation of the o-ring
the minimum value of contact pressure is nothing but the maximum internal
pressure that can be supported by this assembly without any leakages
in the earlier case we considered the non-linear behavior of rubber
in calculating the contact pressure but if we approximate it as a linear measure
and use the tangential stiffness for material stiffness we can still calculate the contact
pressure but the predicted value is very different from our earlier prediction as a result the
design engineer is either underestimating or overestimating the sealing capacity of the
assembly by making a linear approximation which is a clear case of design failure
so in summary if a special material behavior is of interest
for a given application or if the material is undergoing large deformations then using
the appropriate non-linear material behavior is highly recommended this is a very broad discussion
so we will discuss more details in a future course
apart from these three factors there are other aspects that may introduce
non-linearities in a model some instances include delamination material failure or snap
fit assemblies all such instances introduce a loss in stiffness which makes the model
non-linear
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