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### Chapter 7:

(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

MECHANICAL PROPERTIES

- Terminology for Mechanical Properties
- The Tensile Test: Stress-Strain Diagram
- Properties Obtained from a Tensile Test
- True Stress and True Strain
- The Bend Test for Brittle Materials
- Hardness of Materials

Questions to Think About

• Stress and strain: What are they and why are they used instead of load and deformation?

- Elastic behavior: When loads are small, how much deformation occurs? What materials deform least?
- Plastic behavior: At what point do dislocations cause permanent deformation? What materials are most resistant to permanent deformation?
- Toughness and ductility: What are they and how do we measure them?
- Ceramic Materials: What special provisions/tests are made for ceramic materials?

Important Mechanical Properties from a Tensile Test

- Young\'s Modulus: This is the slope of the linear portion of the stress-strain curve, it is usually specific to each material; a constant, known value.
- Yield Strength: This is the value of stress at the yield point, calculated by plotting young\'s modulus at a specified percent of offset (usually offset = 0.2%).
- Ultimate Tensile Strength: This is the highest value of stress on the stress-strain curve.
- Percent Elongation: This is the change in gauge length divided by the original gauge length.

- Load - The force applied to a material during testing.
- Strain gage or Extensometer - A device used for measuring change in length (strain).
- Engineering stress - The applied load, or force, divided by the original cross-sectional area of the material.
- Engineering strain - The amount that a material deforms per unit length in a tensile test.

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Typical stress-strain behavior for a metal showing elastic and plastic deformations, the proportional limit P and the yield strength σy, as determined using the 0.002 strain offset method (where there is noticeable plastic deformation). P is the gradual elastic to plastic transition.

Plastic Deformation (permanent)

- From an atomic perspective, plastic deformation corresponds to the breaking of bonds with original atom neighbors and then reforming bonds with new neighbors.
- After removal of the stress, the large number of atoms that have relocated, do not return to original position.
- Yield strength is a measure of resistance to plastic deformation.

(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

- Localized deformation of a ductile material during a tensile test produces a necked region.
- The image shows necked region in a fractured sample

Permanent Deformation

- Permanent deformation for metals is accomplished by means of a process called slip, which involves the motion of dislocations.
- Most structures are designed to ensure that only elastic deformation results when stress is applied.
- A structure that has plastically deformed, or experienced a permanent change in shape, may not be capable of functioning as intended.

ultimate

tensile strength

3

necking

Strain

Hardening

Slope=E

yield

strength

Fracture

5

2

Elastic region

slope =Young’s (elastic) modulus

yield strength

Plastic region

ultimate tensile strength

strain hardening

fracture

Plastic

Region

Stress (F/A)

Elastic

Region

4

1

Strain ( ) (DL/Lo)

- Elastic Region (Point 1 –2)
- - The material will return to its original shape
- after the material is unloaded( like a rubber band).
- - The stress is linearly proportional to the strain in
- this region.

or

: Stress(psi)

E : Elastic modulus (Young’s Modulus) (psi)

: Strain (in/in)

- Point 2 : Yield Strength : a point where permanent
- deformation occurs. ( If it is passed, the material will
- no longer return to its original length.)

- Strain Hardening
- - If the material is loaded again from Point 4, the
- curve will follow back to Point 3 with the same
- Elastic Modulus (slope).
- - The material now has a higher yield strength of
- Point 4.
- - Raising the yield strength by permanently straining
- the material is called Strain Hardening.

- Tensile Strength (Point 3)
- - The largest value of stress on the diagram is called
- Tensile Strength(TS) or Ultimate Tensile Strength
- (UTS)
- - It is the maximum stress which the material can
- support without breaking.
- Fracture (Point 5)
- - If the material is stretched beyond Point 3, the stress
- decreases as necking and non-uniform deformation
- occur.
- - Fracture will finally occur at Point 5.

(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

The stress-strain curve for an aluminum alloy.

Stress-strain behavior found for some steels with yield point phenomenon.

Yield Strength: Comparison

Room T values

a = annealed

hr = hot rolled

ag = aged

cd = cold drawn

cw = cold worked

qt = quenched & tempered

Tensile Strength, TS

- After yielding, the stress necessary to continue plastic deformation in metals increases to a maximum point (M) and then decreases to the eventual fracture point (F).
- All deformation up to the maximum stress is uniform throughout the tensile sample.
- However, at max stress, a small constriction or neck begins to form.
- Subsequent deformation will be confined to this neck area.
- Fracture strength corresponds to the stress at fracture.

- Region between M and F:
- Metals: occurs when noticeable necking starts.
- • Ceramics: occurs when crack propagation starts.
- • Polymers: occurs when polymer backbones are aligned and about to break.

In an undeformed thermoplastic polymer tensile sample,

- the polymer chains are randomly oriented.
- When a stress is applied, a neck develops as chains become aligned locally. The neck continues to grow until the chains in the entire gage length have aligned.
- The strength of the polymer is increased

Tensile Strength: Comparison

Room T values

Based on data in Table B4, Callister 6e.

a = annealed

hr = hot rolled

ag = aged

cd = cold drawn

cw = cold worked

qt = quenched & tempered

AFRE, GFRE, & CFRE =

aramid, glass, & carbon

fiber-reinforced epoxy

composites, with 60 vol%

fibers.

VMSE

http://www.wiley.com/college/callister/0470125373/vmse/strstr.htm

http://www.wiley.com/college/callister/0470125373/vmse/index.htm

Tensile Testing of Aluminum Alloy

Convert the change in length data in the table to engineering stress and strain and plot a stress-strain curve.

Ductility, %EL

Ductility is a measure of the plastic deformation that has been sustained at fracture:

A material that suffers very little plastic deformation is brittle.

• Another ductility measure:

- • Ductility may be expressed as either percent elongation (% plastic strain at fracture) or percent reduction in area.
- %AR > %EL is possible if internal voids form in neck.

Toughness

• Energy to break a unit volume of material

• Approximate by the area under the stress-strain

curve.

21

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Linear Elastic Properties

s = Ee

• Hooke\'s Law:

n = ex/ey

• Poisson\'s ratio:

metals: n ~ 0.33

ceramics: n~0.25

polymers: n~0.40

Modulus of Elasticity, E:

(Young\'s modulus)

Units:

E: [GPa] or [psi]

n: dimensionless

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Axial (z) elongation (positive strain) and lateral (x and y) contractions (negative strains) in response to an imposed tensile stress.

36

- True stress The load divided by the actual cross-sectional area of the specimen at that load.
- True strain The strain calculated using actual and not original dimensions, given by εt ln(l/l0).

- The relation between the true stress-true strain diagram and engineering stress-engineering strain diagram.
- The curves are identical to the yield point.

Example 2: Young’s Modulus - Aluminum Alloy

From the data in Example 1, calculate the modulus of elasticity of the aluminum alloy.

Example 2: Young’s Modulus - Aluminum Alloy - continued

- Use the modulus to determine the length after deformation of a bar of initial length of 50 in.
- Assume that a level of stress of 30,000 psi is applied.

Young’s Moduli: Comparison

Graphite

Ceramics

Semicond

Metals

Alloys

Composites

/fibers

Polymers

E(GPa)

Composite data based on

reinforced epoxy with 60 vol%

of aligned carbon (CFRE),

aramid (AFRE), or glass (GFRE)

fibers.

Example 3: True Stress and True Strain Calculation

Compare engineering stress and strain with true stress and strain for the aluminum alloy in Example 1 at (a) the maximum load. The diameter at maximum load is 0.497 in. and at fracture is 0.398 in.

Example 3 SOLUTION

Mechanical Behavior - Ceramics

- The stress-strain behavior of brittle ceramics is not usually obtained by a tensile test.
- It is difficult to prepare and test specimens with specific geometry.
- It is difficult to grip brittle materials without fracturing them.
- Ceramics fail after roughly 0.1% strain; specimen have to be perfectly aligned.

The Bend Test for Brittle Materials

- Bend test - Application of a force to the center of a bar that is supported on each end to determine the resistance of the material to a static or slowly applied load.
- Flexural strength or modulus of rupture -The stress required to fracture a specimen in a bend test.
- Flexural modulus - The modulus of elasticity calculated from the results of a bend test, giving the slope of the stress-deflection curve.

(c)2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

The stress-strain behavior of brittle materials compared with that of more ductile materials

(a) The bend test often used for measuring the strength of brittle materials, and (b) the deflection δ obtained by bending

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Schematic for a 3-point bending test.

Able to measure the stress-strain behavior and flexural strength of brittle ceramics.

Flexural strength (modulus of rupture or bend strength) is the stress at fracture.

See Table 7.2 for more values.

MEASURING ELASTIC MODULUS

• Room T behavior is usually elastic, with brittle failure.

• 3-Point Bend Testing often used.

--tensile tests are difficult for brittle materials.

• Determine elastic modulus according to:

23

MEASURING STRENGTH

• 3-point bend test to measure room T strength.

• Typ. values:

• Flexural strength:

Si nitride

Si carbide

Al oxide

glass (soda)

700-1000

550-860

275-550

69

300

430

390

69

Data from Table 12.5, Callister 6e.

24

Stress-Strain Behavior: Elastomers

3 different responses:

A – brittle failure

B – plastic failure

C - highly elastic (elastomer)

--brittle response (aligned chain, cross linked & networked case)

--plastic response (semi-crystalline case)

- Hardness test - Measures the resistance of a material to penetration by a sharp object.
- Macrohardness - Overall bulk hardness of materials measured using loads >2 N.
- Microhardness Hardness of materials typically measured using loads less than 2 N using such test as Knoop (HK).
- Nano-hardness - Hardness of materials measured at 1–10 nm length scale using extremely small (~100 µN) forces.

Hardness

- Hardness is a measure of a material’s resistance to localized plastic deformation (a small dent or scratch).
- Quantitative hardness techniques have been developed where a small indenter is forced into the surface of a material.
- The depth or size of the indentation is measured, and corresponds to a hardness number.
- The softer the material, the larger and deeper the indentation (and lower hardness number).

Hardness

• Resistance to permanently indenting the surface.

• Large hardness means:

--resistance to plastic deformation or cracking in

compression.

--better wear properties.

Adapted from Fig. 6.18, Callister 6e. (Fig. 6.18 is adapted from G.F. Kinney, Engineering Properties and Applications of Plastics, p. 202, John Wiley and Sons, 1957.)

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Conversion of Hardness Scales

Also see: ASTM E140 - 07

Volume 03.01

Standard Hardness Conversion Tables for Metals Relationship Among Brinell Hardness, Vickers Hardness, Rockwell Hardness, Superficial Hardness, Knoop Hardness, and Scleroscope Hardness

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Correlation between Hardness and Tensile Strength

- Both hardness and tensile strength are indicators of a metal’s resistance to plastic deformation.
- For cast iron, steel and brass, the two are roughly proportional.
- Tensile strength (psi) = 500*BHR

Summary

• Stress and strain: These are size-independent

measures of load and displacement, respectively.

• Elastic behavior: This reversible behavior often

shows a linear relation between stress and strain.

To minimize deformation, select a material with a

large elastic modulus (E or G).

• Plastic behavior: This permanent deformation

behavior occurs when the tensile (or compressive)

uniaxial stress reaches sy.

• Toughness: The energy needed to break a unit

volume of material.

• Ductility: The plastic strain at failure.

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