Will Linear Elasticity Last Forever (Credit for many illustrations is given to McGraw Hill publishers and an array of

internet search results)

Or Take it to the LimitOr Take it to the Limit

Parallel Reading

Chapter 2 Section 2.8

A Real Life Case

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I wonder if this just goes on forever?

An engineer designed a tunnelSupported on both sides byPillars of rock.The engineer put Young’s ModulusFor the rock into the computerAnd without paying much attentionTold the computer the linearStress strain relationship went onForever.The computer confirmed that theTunnel would be stable even thoughThe weight of 2,400 ft of rock aboveWas 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

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Loading

Unloading follows the same path down

Called Elastic Behavior

If We Push Beyond our Elastic Limit we begin to create permanent

Deformations

Plastic Behavior(we noted our brittle material triedA little plastic behavior but thenFailed)

Some Materials Have a Bit More Plastic Range

These are called ductile materials

They give us a lot of plasticDeformation before theyFinally break

A Few Terms

A materials shows a nice linear stress strain relationship. Eventually that linearRelationship 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 theElastic region starts to taperOff a little

Where the nice straight lineStarts to taper is called theProportional Limit(Hooke’s Law starts to slipA little above this point)

Most materials will still rebound back without deformation for a short while afterThe line slope tips a little. When plastic deformations start to set-in we call thatThe 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’sModulus slope – when you hit the curve youCall it the Offset Yield Point and use that asThe Yield Point.

More Terms

Even as the material goesPlastic it may still take someAdditional stress.

The highest stress that theMaterial will take is calledThe “Ultimate Tensile Strength”

Of Course Ductile can Have Twists

Some materials suddenly yield as they go plastic

And the develop a little more resistance as theyStretch.

We can talk aboutYield points or limits(except of course notAll materials haveThem)

What is that Weird Dip at the End?

The Specimen Starts Necking

Lets Get that Right

The SpecimenStarts Necking

As the specimen thinsThere is less areaEven though ourStress strain curveAssumed the areaStayed 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 repeatedLoading cycles make thingsFail at lower stress levels.

It was called metal fatigue because when it was first discovered people believedThe metal “got tired”

The drop in failure point with repeat loadings can be seen in test results

Metal FatigueCan actually lower the failurePoint.

Aloha Airlines Flight 243

A regional carrier using mostly Boeing 737’s on short runs with large numbers ofLanding and take-off cycles.

The Problem of Metal Fatigue

Side split in mid air at 24,000 feetSucking a flight attendant out.

The top of the planeRipped 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 loadOn the specimen and then rotates itRapidly.

The specimen undergoes repeatCycles of tension and compressionAs it rotates.

The Point at Which Failure Occurs After 10,000,000 cycles is called

the Endurance Limit

For steel with an ultimate tensile strengthOf under 160,000 psi it is about 50%Of the ultimate strength.

For very high strength materials it isUsually less.

Interesting Observation

If my test specimen is nicked orCorroded it has a lower enduranceLimit.

(If people keep re-polishing the testSpecimen the will take longer to fail)

Cracks Propagate from Points of Weakness

Some of you have seen this when aRock gets flipped up and hits yourWindshield.

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 boundariesOr 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 afterSome point the failing stress levelsOff. If you stay below this pointWe say the material has infinite life.

Then there are materials likeAluminum that don’t seem to haveA stress level that will provideInfinite life.

Not All Materials Are Perfectly Polished

There are class specific proceduresFor adapting endurance limits toSpecific cases.

Interesting consideration – Ford is looking to reduce vehicle weight by replacingSteel in their vehicles with aluminum.

Could Ford get into some design issues with which their experience in buildingSteel vehicles has not prepared them?

Unique Materials PropertiesOne 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?