Will Linear Elasticity Last Forever
(Credit for many illustrations is given to McGraw Hill publishers and an array of
internet search results)
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Parallel Reading
Chapter 2
Section 2.8
A Real Life Case
I wonder if this just goes on forever?
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E
ε
An engineer designed a tunnel
Supported on both sides by
Pillars of rock.
The engineer put Young’s Modulus
For the rock into the computer
And without paying much attention
Told the computer the linear
Stress strain relationship went on
Forever.
The computer confirmed that the
Tunnel would be stable even though
The weight of 2,400 ft of rock above
Was large for the support pillar size
The Pillars Suddenly Exploded in a
Chain Reaction wiping out a mile of
tunnels
The good news? The people working in the tunnel never knew what hit them
The Moral of the Story
• Linear Elasticity does not go on forever
– (of course we knew that – dip stick the
computer jocky just got careless with his
program inputs)
So What Does Happen?
• That depends on the material
One Possibility
Of Course We Hope We Have the
Sense to Back-off in Time
Loading
σ
Unloading follows the same path down
Called Elastic Behavior
ε
If We Push Beyond our Elastic
Limit we begin to create permanent
Deformations
Plastic Behavior
(we noted our brittle material tried
A little plastic behavior but then
Failed)
Some Materials Have a Bit More
Plastic Range
These are called ductile materials
They give us a lot of plastic
Deformation before they
Finally break
A Few Terms
A materials shows a nice linear stress strain relationship. Eventually that linear
Relationship starts to slip just a little and the elastic region ends.
For most engineering designs you will confine yourself to the elastic region.
Closely Related Terms
For most materials the
Elastic region starts to taper
Off a little
Where the nice straight line
Starts to taper is called the
Proportional Limit
(Hooke’s Law starts to slip
A little above this point)
Most materials will still rebound back without deformation for a short while after
The line slope tips a little. When plastic deformations start to set-in we call that
The Yield point.
Some materials don’t have a well defined plastic deformation yield point
Offset Yield Point or Proof
Stress
Start at 0.2% strain – go up at the Young’s
Modulus slope – when you hit the curve you
Call it the Offset Yield Point and use that as
The Yield Point.
More Terms
Even as the material goes
Plastic it may still take some
Additional stress.
The highest stress that the
Material will take is called
The “Ultimate Tensile Strength”
Of Course Ductile can Have Twists
Some materials suddenly yield as they go plastic
We can talk about
Yield points or limits
(except of course not
All materials have
Them)
And the develop a little more resistance as they
Stretch.
What is that Weird Dip at the End?
The Specimen Starts Necking
Lets Get that Right
The Specimen
Starts Necking
As the specimen thins
There is less area
Even though our
Stress strain curve
Assumed the area
Stayed the same.
If You Consider the Change In
Area
The line keeps on going up to the
breakpoint.
The dip is an artifact of our constant area assumption
So if We Behave Well and Design
in the Elastic Range are We OK?
Maybe not.
It turns out that repeated
Loading cycles make things
Fail at lower stress levels.
It was called metal fatigue because when it was first discovered people believed
The metal “got tired”
The drop in failure point with repeat
loadings can be seen in test results
Metal Fatigue
Can actually lower the failure
Point.
Aloha Airlines Flight 243
A regional carrier using mostly Boeing 737’s on short runs with large numbers of
Landing and take-off cycles.
The Problem of Metal Fatigue
Side split in mid air at 24,000 feet
Sucking a flight attendant out.
The top of the plane
Ripped off in the sky
Somehow the Pilots Landed the
Thing
How is Testing Done
A polished smooth test specimen is prepared
It is put in a machine that puts a bending load
On the specimen and then rotates it
Rapidly.
The specimen undergoes repeat
Cycles of tension and compression
As it rotates.
The Point at Which Failure Occurs
After 10,000,000 cycles is called
the Endurance Limit
For steel with an ultimate tensile strength
Of under 160,000 psi it is about 50%
Of the ultimate strength.
For very high strength materials it is
Usually less.
Interesting Observation
If my test specimen is nicked or
Corroded it has a lower endurance
Limit.
(If people keep re-polishing the test
Specimen the will take longer to fail)
Cracks Propagate from Points of
Weakness
Some of you have seen this when a
Rock gets flipped up and hits your
Windshield.
The stress at the end of a crack is incredible
(using math only an M.E. could love)
Some Fractures are Obvious
Others are Not
Defects can be at mineral grain boundaries
Or even at the atomic level
Once cracks start to propagate they are like a cancer – and will keep on growing
(which is why re-polishing test specimen surfaces helped)
Keeping Below Endurance Limits
For steel you can see that after
Some point the failing stress levels
Off. If you stay below this point
We say the material has infinite life.
Then there are materials like
Aluminum that don’t seem to have
A stress level that will provide
Infinite life.
Not All Materials Are Perfectly
Polished
There are class specific procedures
For adapting endurance limits to
Specific cases.
Interesting consideration – Ford is looking to reduce vehicle weight by replacing
Steel in their vehicles with aluminum.
Could Ford get into some design issues with which their experience in building
Steel vehicles has not prepared them?
Unique Materials Properties
One of the Reasons for in Depth Classes
• Classes in Steel Structures
• Classes in Concrete Structures
• Classes in Pre-Stressed and Cast
Concrete
• Classes in Soil Mechanics
• Classes in Rock Mechanics
Would Anyone Ever Try to Operate
Outside the Elastic Range
• For most types of things we try to avoid
the plastic ranges
• Plastic Ranges can be important for things
like shaping steel fenders out of sheet
metal
– Lots of shaped panels are made of steel – not
many from aluminum – any idea why?