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### CHE 333 Class 11

Mechanical Behavior of Materials

Elastic Deformation.

Consider a metal rod fixed at one end.

At the other end a load can be applied

by some manner. When a small amount

of load is applied, if the length of the metal

rod was measured it would be longer. If the

load is removed, and the rod measured

again, it would return to the original

length. It is said that the deformation

was recovered. This type of deformation

is ELASTIC, that is all recovered on load removal.

It was also found that the extension of the rod

was directly proportional to the load applied.

Load extension data would be as shown

in the diagram.

Service loads should be ELASTIC

Load

Extension

Plastic Deformation

Load

Following elastic deformation, the load

extension curve is no longer linear,

as shown in the diagram. After the linear

elastic portion, a non linear region starts

which indicates the start of PLASTIC

deformation. If the load is removed at

a point after plastic deformation is initiated

the metal rod will not return to the same

length as the initial length. It will be longer

by the amount of plastic deformation. The

new increase length is the plastic

deformation. In this case all the deformation

was not recovered. The elastic portion is

recovered but not the plastic deformation.

The load removal curve decreases parallel

to the elastic deformation line.

Load Removal

Final length

after load removal

Extension

Stress Strain Curves

Stress

The load extension data can be transformed

into Stress Strain data by normalising

with respect to material dimensions.

The stress is the load divided by the

original cross sectional area.

s = L/A

s – stress , units MPa, or psi or ksi

L – load applied

A – original cross sectional area

The strain is the increase in length

normalised by the original length.

e = Dl/l

e – strain – dimensionless (in/in)

Dl – increase in length

l – original length

Strain is often given in percent so x100

As the normalisations are by constants

the shapes of the curves stays the same.

Strain rate is e/t. Most materials are strain

rate sensitive that is their mechanical behavior

depends on the rate of deformation.

Strain

Hooke’s Law and Young’s Modulus

Stress

Hooke’s Law is concerned with

Elasticity.

s = Ee

Stress is proportional to strain,

But only in the elastic region.

This is the “elasticity” or elastic

Modulus of materials, sometimes

Called “Young’s Modulus”.

Metal Youngs Mod

106psi

Aluminum 11

Gold 16

Copper 28

Iron (BCC) 41

Yield Stress

Strain

Yield Stress Ultimate Tensile Stress

The Yield Stress is at the

onset of plastic deformation.

The Ultimate Tensile Stress is

the maximum stress during

the stress strain test.

Manufacturing between YS and UTS

The strain to failure can be

measured from the stress strain

data,

The 0.2% yield stress is used for

materials such as steel as the

yield point is sometimes difficult

to determine. At 0.2% strain a line

is drawn parallel to the elastic

portion of the data until it intersects

the plastic portion of the data. The

stress level at this point is the 0.2%

yield stress. (0.002 strain)

Stress

Ultimate Tensile Stress

0.2% YS

Yield Stress

Strain at Failure

Strain

Brittle Behavior

Stress

Failure at this stress

Brittle materials exhibit little on

no plastic deformation region.

Only elastic deformation is found.

The energy of failure is then the

area under the stress stain curve,

which for a brittle material is the

area of a right angel triangle,

or half base multiplied by the height.

Or half the strain at failure multiplied

by the stress at failure.

Plastic deformation adds a considerable

amount of energy to the failure process.

Ceramics and martensitic steels show

this behavior.

Energy of failure is the area under the

stress strain curve. For brittle materials

it is half the strain multiplied by the

failure stress.

Strain

Reduction of Area

At the UTS, for metals local deformation starts, and thereafter the deformation is concentrated

locally. This causes a “NECK” to occur shown above along with the crack at failure.The

cross section is reduced at the failure point compared to the region outside the neck. One

measure of “DUCTILITY” besides elongation at failure is “reduction of area”

ROA = final cross sectional area/ original cross sectional area

Cup and Cone Failure

Final failure in round bar is often

characterized for a ductile material

as a “Cup and Cone” failure. An

example is shown. The fracture starts

in the interior of the material and spreads

internally until only a small annulus of material

remains. This then shears at 45o to the

applied stress. The more ductile the material

the larger the shear lip.

Sheet Tensile Sample

A sheet material tensile sample is shown above. ASTM has standard dimensions. At either

end is a grip area, and in the center is the gauge length which is a narrower section to ensure

failure outside the grip area effects. The thickness and width of the sample need to be known

to calculate the stress data and the original length to calculate the strain at failure.

Failed Sample Metal

A failed sample is compared to a new untested sample. Note the failure is at 45o to the

applied stress. The local deformation in this case is very near the failure point. ROA

Data would be very difficult in this case. Elongation at failure would be more useful

Failed Sample - Polymer

A failed polymer sample has a large elongation at failure in comparison to the metal sample.

Sample is 0.5 in wide to provide a scale.

Polymer Stress Strain Curve

Stress

Strain

Polymers generally have low elastic modulus and long elongations to failure compared to

Metals.

Homework

- Draw a stress strain curve for a ductile material indicating yield stress, UTS, strain to failure.
- Draw the stress strain curve for a brittle material.
- Briefly describe strain rate sensitivity.

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