mechanicalproperties10-5-11

8/2/2019 MechanicalProperties10-5-11

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MY2100

Ch 7, Slide 1

Chapter 7: Mechanical Properties

Structure

Processing Properties Performance

Mechanical Properties Generally Pertain to How a Material

Responds to Forces. This Subject is Extremely Important to

Almost Every Engineer who is Using Materials or Designing

Structures.


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MY2100

Ch 7, Slide 2

Stress

To Understand and Calculate the Effects of Forces, We Define a Parameter

Referred to as Stress (s).

In One Dimension, Stress is Defined as the Applied Force (F) (Which May

be Tensile or Compressive) Divided by the Area (A) Upon Which it Acts.

F

F

A

A

F

F

F

A

Eq. 7.1

Compressive Tensile


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Ch 7, Slide 3

Units of Stress

The Units of Stress are the Same as Those of Pressure

English System:

lbs/in2, Usually Abbreviated psi

kilo-lbs/in2, Usually Abbreviated ksi (1 ksi = 1,000 lbs/in2)

Metric System:

N/m2, Called a Pascal (Pa)

Typical Stresses are in MegaPascals (MPa, 106 Pa) or GigaPascals(GPa, 109 Pa)

Useful Conversion: 1 ksi = 6.895 MPa


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Ch 7, Slide 4

Engineering Stress vs. True Stress

During Deformation, the Area of a Material Subjected to a Force

Changes Constantly.

Engineering Stress is Stress Calculated Using the Original Area of a

Material. True Stress(sT)is Stress Calculated Using the Real Time

Area of a Material. At Small Deformations, These Stresses are Similar.

AoAi

Ai Denotes Instantaneous Area

During Deformation

T

i

F

A

Eq. 7.15


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Ch 7, Slide 5

Strain

AoAi

lo lf

f o

o o

l l l

l l

In One Dimension (Tension or Compression),We Define Engineering Strain (Eq. 7.2) as:

To Quantify Deformation, or the Change in Shape of a Material UnderStress, we Define a Dimensionless Parameter Called Strain ()


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Ch 7, Slide 6

Engineering Strain vs. True Strain

AoAi

lo

li

T

i i

o o

l A ln ln

l A

In One Dimension (Tension or Compression):

i Denotes Instantaneous Dimensions.

Again, at Small Strains the

Engineering Strain and True Strain are

About the Same.

True Strain (T) Takes into Consideration the Constantly Changing Shape

of a Deforming Material.

Eq. 7.16


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Ch 7, Slide 7

Force-Length Relationships for One-Dimensional Elastic Loading

Force

Change in

Length

F = kx

Load

Unload

F

F

Dl

Dl

At Low Levels of Force, Most Materials Act Like

Springs, Deforming Elastically.

Elastic Loading: Loading that Causes a Temporary

(Recoverable) Shape Change


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Ch 7, Slide 8

Stress-Strain Relationships for One-Dimensional Elastic Loading

We Plot svs. Instead of F vs.Dl

(Note that in a Given Circumstance,s F, and Dl)

Stress

Strain

Load

Unload

s

s


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Ch 7, Slide 9

Elastic Modulus

Stress

Strain

The Slope of the Stress Strain Curve (It is a

Straight Line for Most Materials) is Called

the Elastic Modulus, or Youngs Modulus,

and is Given the Symbol E. It is a Measure

of the Stiffness of a Material (Like the

Spring Constant, k).

In Metals, Ceramics and Composites, the Elastic Modulus is Controlled

by Atomic Bond Strength (the Bonds Act like Springs). Therefore, itCannot be Changed Much by Heat Treating or Other Means. It is a

Materials Property That is only a Function of Chemical Composition.

E = s/ Eq. 7.5


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Ch 7, Slide 10

Elastic Moduli of Metals


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Ch 7, Slide 11

Elastic Moduli of Ceramics

E

GPa Million psi


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Ch 7, Slide 12

Elastic Moduli of Polymers and Biomaterials

Polymers and Biological Tissues Can Have Varying Stiffness,

Depending Upon Structure.

Polymers: Arrangement of Long Chain Molecules, and

Degree of Polymerization/Crystallization will Change the

Modulus.

Biological Tissues: Density, Water Content, and

Arrangement of the Ligaments (or Tubules, Cell Walls, etc.)

will Affect Modulus.

Therefore, for These Materials, E is not a Constant. It Can Have a

Range of Values, and May be Varied via Processing.


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Ch 7, Slide 13

Elastic Moduli of Polymers

E

GPa Million psi


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Ch 7, Slide 14

Importance of Elastic Modulus in Mechanical Design

The Elastic Modulus is the Only Materials Property Contributing

to the Stiffness of an Engineering Component. (Other Factors

Relate to Design, Such as Component Shape, Assembly, Loading,

etc.) Therefore, E Controls How Much a Component will

Deflect, Bend, or Extend, When it is Loaded Elastically.


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Ch 7, Slide 15

Example: Stiffness and Deflection of a Beam

FL

d

EI

FL

3

3

d

Where I is the Moment of Inertia, Related to the Shape of the

Cross Section. For a Rectangular Beam, I = bh3/12.

Notice that E is the only Materials Parameter.h

b

You will find in Your Mechanics of Materials Course that the Deflection

(d) of an End-Loaded Beam can be Calculated as Follows:


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Ch 7, Slide 16

In Class Practice Problem 1

Calculate the Deflection That Will Occur when

a Weight of 150 lbs is Placed at the End of theBeam Shown Below if the Beam is Made of:

steel (E = 30,000,000 psi)

low density polyethylene (E = 30,000 psi)

10 feet

5 inches

3 inches

EI

FL

3

3

d

3

12

bhI


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Ch 7, Slide 17

Poissons Ratio

lo

do

lf

df

z

y

z

x

For Most Metals and Ceramics, ~ 1/3.

xy

z

Eq. 7.8

When a Material Undergoes Uniaxial Elastic Deformation, its

Dimensions Also Change in Directions Normal to the Direction of

Applied Stress. Poissons Ratio () is the Ratio of the Lateral Strain to

the Axial Strain.


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Ch 7, Slide 18

Other Types of Stresses

s = F/A is Only Valid for One Dimensional Loading of Rods or

Bars Along their Axes. In General, Stresses are in 3-D, and theStress at a Point in a Material is Described by 6 Numbers

Rather Than 1.

There are Two Basic Types of Stresses. Normal Stresses () act Perpendicular to a Plane Defined

Within a Material. We Have Only Considered Normal

Stresses Thus Far.

Shear Stresses (t) act Parallel to a Plane Defined Within aMaterial.


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Ch 7, Slide 19

Normal Stress vs. Shear Stress

F

F

A

A

F

F

F

A

Normal Stress

A

F

Shear Stress


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Ch 7, Slide 20

Normal Strain vs. Shear Strain

Normal Strain () Shear Strain (g)

F

F

F

Fq

g = tanq for Small Shear Strains.

If the Loading is Elastic,

t = Gg, (Eq. 7.7) where G is the

Shear Modulus.

For a Material with IsotropicElastic Properties,

E = 2G(1 + ) Eq. 7.9


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Ch 7, Slide 21

Shear Stresses Are Always Present

F

F

It is Important to Note that Evenif Normal Loads are Applied to a

Material, Shear Stresses are

Present on Some Planes within

the Material. In SimpleCompression or Tension, the

Maximum Shear Stress Occurs on

Planes Inclined 45o from the

Loading Axis.

tmax


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Ch 7, Slide 22

Mechanical Behavior of Metals-Elastic Deformation

At Low Stresses, Metals Exhibit Elastic(Recoverable) Deformation

Stress

Strain

s = E

Load

Unload

s

s


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Ch 7, Slide 23

Mechanical Behavior of Metals - Plastic Deformation

Stress

Strain

Load

Unload

s

s

p

l

At Sufficiently High Stresses, Metals

Undergo Plastic (Permanent) Deformationin Addition to Elastic Deformation.


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Ch 7, Slide 24

Yield Strength

Stress

Strain

The Stress at Which a Material Starts to Deform Permanently is

Called its Yield Strength (sy). (Sometimes it is Called the

Proportional Limit, Since Below sy, s is Proportional to .)For Engineering Design, Yield Strength is the Most Important

Strength Parameter. If the Stress is Kept Below sy, theDeformation is Elastic.

sy


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Ch 7, Slide 25

Quantification of Yield Strength for Metals

Stress

Strain

Stress

Strain

Many Metals Begin to Yield Gradually, and it is not Possible to

Define a Yield Strength or Proportional Limit Objectively.Because of this, and the Desire to have Reproducible Test

Procedures that do not Depend on Subjective Operator Input

from a Curve, the Yield Strength is Almost Always Defined as a

Yield Strength at 0.2% Offset.

Idealized s- Plot Real s- Plot


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Ch 7, Slide 26

Determination of Offset Yield Strength

A Line is Drawn Parallelto the Elastic Portion of

the s- Plot, But

Offset from the Origin

by 0.2% (at = 0.002).

The Intersection of That

Line with the s Curve

is Taken as the Yield

Strength of the Material.


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Ch 7, Slide 27

Other Yielding Phenomena in Metals

Some Moderate Strength Steels Yield and then Experience a Load Drop (Stress

Decrease) Followed by Localized Deformation at Constant Stress Prior to Continued

Increase in Stress vs. Strain. Such Materials are Said to Have Two Yield Points, and syis Taken as the Lower Yield Point.


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Ch 7, Slide 28

Strain Hardening of Metals

Metals Strain

Harden; That is,They Get Stronger

Upon Plastic

Deformation.

Compare syo (Initial

Yield Strength) with

syi (Yield Strength

Upon Unloading and

Reloading of the

Material).


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Ch 7, Slide 29

Strain Hardening

Here, sT is True Stress, T is True Plastic Strain, and K and n are

Material Parameters. n is called the Strain Hardening Exponent.

Typical n values for Metals are Between 0.1 and 0.5.

T Tn K

Strain Hardening Can be Easily Quantified for Most Metals at

Low Temperatures. In the Plastic Region, Most Metals at Low

Temperatures Obey a Power Law Relating Stress to Strain:

Eq. 7.19


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Ch 7, Slide 30

The Tensile Test

Engineering

Stress

Engineering Strain

Yield Strength (sy)

Tensile Strength (TS)

Tensile Strength is the

Highest Engineering StressThat May Be Supported by a

Material.

x

Point of

Fracture

Tensile Test: Application of Uniaxial Stress to a Material Until the Point of

Breakage (Fracture)


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Ch 7, Slide 31

Note to Current or Future Designers

Be Very Careful When You Ask Someone for, or Look Up, a

Materials Strength. As We Have Seen, There are Two

Strengths that are Determined by a Tensile Test - the Yield

Strength and the Tensile Strength. You Have to Specify Which

Strength You Care About, and Too Often People will Give You theTensile Strength.

The Yield Strength is the Stress above which a Part Loses its Shape,

and is Therefore the Strength that Should be Used in Design.


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Ch 7, Slide 32

Necking of Materials

At the Tensile Strength, Metals Begin to Fail by Necking (Localized Deformation). A Plot

of True Stress vs. True Strain Would Show Increasing Stress All the Way to Fracture.


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Ch 7, Slide 33

Ductility

Ductility: A Measure of a Materials Ability to Deform Permanently Without

Fracture. It is Measured by Measuring the Specimen After a Tensile Test.

When a Specimen Thins Down and Fractures, it gets Longer and its Cross-

Sectional Area Decreases. The Permanent Changes in Length (%EL) or Area

(%RA) Indicate Ductility.

Percent Elongation (%EL):

Percent Reduction in Area

(%RA):


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Ch 7, Slide 34

Metal Mechanical Properties and Engineering Design

The Mechanical Properties Typically Needed for Engineering Design

Are:

1. Elastic modulus (E)

2. Yield strength (sy)

3. Percent Elongation (%EL)

These Properties are Reported in Handbooks, But All May Be Obtained

From Tensile Testing.

Yield Strength and Percent Elongation May Be Altered via Processing,Whereas Elastic Modulus Depends Only on the Chemical Makeup of a

Material (Metals and Ceramics).


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Ch 7, Slide 35

Tensile Test

EngineeringStress

Engineering Strain

Yield Strength (sy)

Tensile Strength (TS)

E (Slope of

Elastic Curve)

plastic

Percent Elongation(%EL) is plastic x 100

fracture

The Tensile Test DataMay be Curve-Fitted

to sT = KTn


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Ch 7, Slide 36


f o

o o

l l l

l l

A

F s = E

in GPa


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Ch 7, Slide 37


A Steel Rod, 1 Inch in Diameter, Yields at a Force of 200,000 lbs

a) Determine its Yield Strength

b) Determine the Load-Carrying Capacity of a Wire with a

0.125 Inch Diameter, Which is Made From the Same Steel.

A

F


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Ch 7, Slide 38

Hardness Testing

Tensile Tests are Expensive (Require Machining of Material and

Specialized Equipment) and Destructive (The Sample is Deformed Until

it Fractures).

A Simple Method for Estimating a Metals Strength is to Measure its So-

Called Hardness.

Hardness: A Measure of a Materials Resistance to Permanent

Deformation via Surface Indentation

Hardness Testers Force Small, Specially Shaped Indenters into a MaterialUsing a Specified Force. The Depth or Size of the Resulting Indentation

is Converted into a Hardness Number.


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Ch 7, Slide 39

Hardness Testers and Corresponding Indenter Geometries


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Ch 7, Slide 40

Hardness and Strength

Hardness Measures a Materials Resistance to Penetration by anIndenter (Permanent Deformation). This Resistance to Penetration is

Controlled by the Materials Yield Strength and Early Stages of Strain

Hardening.

Therefore, Hardness Testing is a Quick, Non-Destructive Method for

Qualitative Evaluation of a Materials Strength. Hardness Tests are

Very Useful for Quality Control and Non-Destructive Inspection.

For a Given Metallic Material, It is Sometimes Possible to Determinewith a Direct Relationship Between Hardness and Strength.


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Ch 7, Slide 41

Hardness - Strength Correlations for Brass, Cast Iron and Steel

Each Metal Has Its Own

Curve Relating Hardness

to Strength

Fig. 7.31


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Ch 7, Slide 42

Plastic Deformation and Dislocations

li

How Can we Change the Length of a Crystal Permanently, Without Disrupting

its Crystal Structure?


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Ch 7, Slide 43

Dislocations (Chapter 5)

All Crystalline Materials Contain Dislocations


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Ch 7, Slide 44

Dislocation Motion and Plastic Deformation (Atomic Scale)

Movement of the Dislocation From Left to Right, In Response to at

Shear Stress, Has Sheared the Crystal Along a Plane of AtomsCalled the Slip Plane. This Results in a Permanent Shape Change

(Plastic Deformation).


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Ch 7, Slide 45

Dislocation Motion and Plastic Deformation (Macroscopic Scale)

Dislocation Motion

on Large Numbers ofSlip Planes in

Different Grains

Leads to Measurable

Shape Changes inMetals.

Most metals are

Very Ductile (May

Undergo SubstantialPlastic Deformation).

i i i f i i ?


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Ch 7, Slide 46

Why are Dislocations Responsible for Plasticity?

To Shear an Entire Slip Plane Simultaneously, All Atomic Bonds Across the

Plane Would Have to be Broken at the Same Time. This Would Require a Very

High Shear Stress. To Move a Dislocation, Only the Bonds in One Row of

Atoms Must Be Broken. This is an Easier Process, But it Leads to the Same

Permanent Shape Change as Would Simultaneous Shear of the Slip Plane.

C ill Di l i A l


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Ch 7, Slide 47

Caterpillar-Dislocation Analogy

A Dislocation Causes Deformation Using the Same Strategy a Caterpillar

Employs to Crawl. To Travel Forward, the Caterpillar Moves Only Moves aFew Legs at a Time. This Takes Less Energy and Coordination than Moving All

Legs in Concert with Each Other, and Accomplishes the Same Goal.

Figure 8.3

M h i l P ti f C i


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Ch 7, Slide 48

Mechanical Properties of Ceramics

Ceramics are Usually Compounds of Metals and Non-Metals (e.g.,

Al2O3, MgO, SiO2, Si3N4, SiC).

As Discussed in Chapter 3, Ceramics Have Complicated Crystal

Structures, and Ionic Bonding. Together, These Make Dislocation

Motion Difficult at Room Temperature, and Therefore Ceramics are

Brittle Materials (Do Not Undergo Plastic Deformation).

C i f S S i i i C i


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Ch 7, Slide 49

Comparison of Stress-Strain Relationships Between Metals and Ceramics

Stress

Strain

Metals: At the Yield Stress,

Dislocations Start to Move.

This Causes Plastic Deformation

and Makes Metals Tough.

Elastic Plastic

Ceramics: Dislocations Cannot

Move at Low Temperatures,

so We get no Plasticity; Ceramics

are Elastic Until Fracture.

Stress

Strain

Elastic

FractureFracture

T i l St St i B h i F C i


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Ch 7, Slide 50

Typical Stress-Strain Behavior For Ceramics

Note the Magnitude of the Strain

at Fracture for These Materials

(


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Ch 7, Slide 51

Fracture Strength of Ceramics

As Will Be Shown in Chapter 9, the Fracture Strength (Stress

Required for Crack Extension) of Ceramics is Usually Determined

by Tiny Defects on their Surfaces.

Ceramic Fracture Strengths Typically Exhibit a Large Degree of

Variability, Because of the Probabilities of Defects with Different

Sizes Being in Different Locations. Fracture Strengths of Ceramics

are Therefore Usually Presented as Probability Distributions Using

Weibull Statistics.

M h i l B h i f P l


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Ch 7, Slide 52

Mechanical Behavior of Polymers

Brittle Polymers (Like Polystyrene - Cheap Drink Cups) Behave

Just Like Ceramics.

Non-Brittle Polymers (Like Polyethylene (Milk Jugs) or

Rubber) Behave in a Way that is Completely Different from

Metals or Ceramics.

T i l St St i C


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Ch 7, Slide 53

Typical Stress-Strain Curves

Polystyrene

Polyethylene

Rubber

Note That the Rubber Curve is Elastic but Nonlinear.

Again, Note the

Magnitude of the Strain.

For Rubber, >> 1.

Mechanical Properties of Ductile Polymers


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Ch 7, Slide 54

Mechanical Properties of Ductile Polymers

(Slope of Elastic Portion

of the Curve is E.)

PlasticElastic

Deformation Behavior of Ductile Amorphous Polymers


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Ch 7, Slide 55

Deformation Behavior of Ductile Amorphous Polymers

Unlike Metals, the

Necked RegionPropagates Along the

Gauge Length of the

Polymer

Linear Polymers which are Semi-Crystalline and Not Heavily

Crosslinked Exhibit Extensive Necking During Plastic Deformation.

Necking Mechanisms in Polymers


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Ch 7, Slide 56

Necking Mechanisms in Polymers

Fig 8.28

The Necking Which Occurs in These Materials is Not the Same as the Necking

Associated with Failure in Metals. In the Polymer Necks, the MacromoleculesAlign Along the Loading Axis During Necking. This Alignment Strengthens the

Polymer.

Influence of Temperature on Polymer Strength


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Ch 7, Slide 57

Influence of Temperature on Polymer Strength

Stress-Strain Plot for PMMA atVarious Temperatures

The Strength, Stiffness and Ductility of All

Materials Change with Temperature. For

Polymers, These Changes are Often

Substantial within Relatively ModestTemperature Ranges.

Compare the Temperature Range Here (40 F to

140 F) with Seasonal Temperature Changes in

the United States.



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Ch 7, Slide 58


Using the Data on the Graph,

Estimate the Elastic Modulus of

Polymethylmethacrylate (PMMA)

at 4 C, 30 C and 60 C.

Homework for Chapter 7


59/59

Homework for Chapter 7

C & R 7.2, 7.3, 7.4, 7.5, 7.9, 7.10, 7.12, 7.15(a-e), 7.24

Solutions will be Posted in WebCT

Use Solutions ONLY for Checking Your Answer. If You Have

Trouble Arriving at a Correct Answer, Please Come See Me for

Help.

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