So far we have talked about the different sources of non-linearity
and why is it important to account for them in our engineering analysis
now let's investigate some examples and see practically how much of a difference do they make
in this lesson we have lined up four different examples
each of which presents how different sources of non-linearity are introduced into a system
in the first example we start with a simple dog bone sample
that is made up of polyethylene material it's a very common practice to use such samples
to perform tensile testing which in turn is used for calculating material properties
we use this to see how the tensile response starts
linear at small deformations and turns non-linear as the deformation increases
in the second example we use the same dog bone but replace the material with that of a rubber
and see how material non-linearity plays a role in making the model non-linear
in the third example we move on to a closed vessel
that has a graphite gasket sandwiched between two steel plates in this example
we'll see how contact non-linearity is responsible for making a model non-linear
and in the fourth and final example we'll look at a snap fit buckle which
has all the three non-linearities and an additional case of loss of contact
let's start by looking at the behavior of polyethylene material
which is widely used in various applications due to its low weight to strength ratio
there are various grades of this material such as high or low density polyethylene
that are used in many commercial products such as bottles and caps phone and tablet cover cases
and fasteners in several assemblies
to be able to use them as part of the design the first thing to do is to calculate the
material properties of the grade of polyethylene that we are working with
we start with tensile testing of polyethylene sample
which is typically made into a dog bone structure
the sample is held rigidly on one end and it is pulled at a constant rate on the other while the
load cell records the force reaction in order to compare the effect of large deformation versus
small deformation we'll simulate two different cases the first one the sample is stretched
to a displacement of 0.1 mm the second one the sample is stretched to a displacement of 10 mm
during deformation the area of cross-section of the sample decreases due to poisson's effect
and this introduces non-linearity
let's start by plotting the force displacement curve for first case
as we can see the response is linear and can be assumed to be a linear system
but when we continue to stretch it to a displacement of 10 mm which is case 2 we
start seeing that the force displacement response deviates from the linear curve
this can be explained from monitoring the area of cross section at mid plane during the deformation
in case one the change in area of cross section is very small but as we increase the amount of
deformation the area reduces non-linearly and starts affecting the system's response
so from this example we can see why we need to use a non-linear solver to analyze a structure when it
is known to undergo large deformation in this case if we idealize this system to be linear then we'll
be over predicting the force reaction by a huge difference and this will result in faulty designs
now moving on to the next example where we will investigate the material non-linearity
to do this let's take the same dog bone geometry but test the response for two different materials
an elastomer and an aluminum alloy
elastomers are very soft materials and are commonly used in making
gaskets o-rings and in some cases as mounts for mitigating the vibration from heavy machines
due to their soft nature they tend to deform severely which naturally makes
them susceptible to large deformation however due to their non-linear material behavior
they tend to have a non-linear force response even at small deformations
to demonstrate this we perform a displacement control test
on two dog bone samples one made of elastomer and the other made of aluminum since we are operating
under small deformation the aluminum sample records a linear force displacement response
but at the same small deformation the elastomer sample records a non-linear response
even though there isn't a huge change in the area of cross-section
the elastomer sample deforms non-linearly due to its material behavior
so a non-linear solver is required to analyze a system made of such materials
for our next example let's look at a pressure vessel
a key design feature for such bezels is to retain the pressure
inside the vessel and one way to do that is to prevent any leakage of its contents
gaskets are commonly used for this purpose they are sandwiched between the vessel and its closing
cap doing so the gasket material deforms and creates a contact pressure this pressure must
be more than the pressure of the contents of the vessel to make sure that they don't escape out
in this example we take one such bezels with the geometry simplified for our demonstration purpose
in this case we have a gasket sandwiched
and the pressure is developed by tightening a set of four bolts
let's use a graphite gasket for this case as it may be approximated to be linear
at small deformations so we've eliminated or rather mitigated the effects of the
other two non-linearities so we can focus on the third non-linearity which is contact
since the graphite is arrested between the two plates
and it doesn't separate from either surface we may in theory solve this case using a linear solver
to do this we approximate the contact to be bonded
by doing so we prevent the gasket material to slide over the surfaces when it deforms
now let's get realistic and replace the bonded contact with the frictional contact
so the graphite can slide when it deforms this is closer to reality
now let's compare the force displacement response for these two cases
in case of bonded contact we do get a linear response
but when we replace it with a more realistic contact type the response is clearly non-linear
surely the second analysis is more realistic but is this difference big enough to affect
our engineering judgment there's only one way to find out let's compare the contact pressure which
is the quantity of interest for this analysis calculated from both the contact treatments
as we can see there's a 52 percent difference in the calculated contact pressures
so by assuming that the contact to be bonded we are overestimating the operational pressure
limits of the vessel this leads us to a design failure which could be catastrophic
now coming to our fourth and final example we'll look at a snap fit buckle which are commonly
used in backpacks in this case there's usually presence of all the non-linearities to some degree
but at the moment of engagement there's a sudden loss of contact
which adds a fourth type of non-linearity which is a loss in stiffness
usually when this happens it makes the model unstable
a concept that we'll discuss in a future course
a loss of stiffness is a direct indication of a physical phenomena and ignoring it would
disregard the set physics due to this reason we cannot solve such systems using linear solvers
let's go ahead and study the system using a non-linear solver
and look at the force displacement response
as you can see the response is non-linear but focus on this portion of the curve
where it shows there's a sudden drop in reaction force this is the stage where the
snap fit gets to an engaged position and starts providing the support that it was intended to
for an engineer who's trying to design one such systems
may be interested in knowing at what sliding distance does it engage
and how much force resistance does the system offer during this process
such an analysis can help them in tweaking the geometric features and even the material
selection for the assembly so it meets their design requirements
so we've seen different practical examples
and how different sources of non-linearity play a part in modifying the response of the system
there are many more such examples that can demonstrate these non-linearities
and we will explore them as we encounter them during discussions on other topics
you