MSE 3300-Lecture Note 09-Chapter 06 Mechanical Properties of Metals 1

8/18/2019 MSE 3300-Lecture Note 09-Chapter 06 Mechanical Properties of Metals 1

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MSE 3300 / 5300 UTA Spring 2015 Lecture 9 -

Lecture 9. Mechanical Properties

of Metals (1)Learning Objectives

After this lecture, you should be able to do the following:

1. Define engineering stress and strain.

2. Given an engineering stress–strain diagram, determine (a) the

modulus of elasticity, (b) the yield stress, and (c) the tensile strength and

(d) estimate the percentage elongation.

Reading

• Chapter 6: Mechanical Properties of Metals (6.1–6.5)

Multimedia• Tensile tests: https://www.youtube.com/watch?v=ZwTF_-JZgt8;

https://www.youtube.com/watch?v=67fSwIjYJ-E;

https://www.youtube.com/watch?v=K28WiL21Sgs

• Virtual Materials Science & Engineering (VMSE):

http://www.wiley.com/college/callister/CL_EWSTU01031_S/vmse/

1


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1. Concepts of Stress and Strain

2

Stress–Strain Diagram

Modulus of elasticity, the yield stress, and

the tensile strength


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Type of Loading

3

Schematic of how a load produces a deformation (strain): (a) tension, (b)

compression, (c) shear, and (d) torsion


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MSE 3300 / 5300 UTA Spring 2015 Lecture 9 - 4

Tension Tests: Stress-Strain Testing

Fig. 6.3, Callister & Rethwisch 9e.

(Taken from H.W. Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of

Materials, Vol. III, Mechanical Behavior , p. 2, John Wiley and Sons, New York, 1965.)

specimen extensometer

• Typical tensile

specimen

Fig. 6.2,

Callister &

Rethwisch 9e.

gauge

length

• Typical tensile test

machine


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• Stress [N/m2 = Pa]

• Strain [dimensionless]

Stress and Strain:

Tension and Compression Tests

5

By convention, a compressive

force (thus, stress and strain) is

taken to be negative


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• Shear stress [N/m2 = Pa]

• Shear strain [dimensionless]

Stress and Strain:

Shear and Torsional Tests

6

θ γ tan=


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∴ Stress has units:

N/m2 or lbf /in2

Engineering Stress

• Shear stress, τ :

Area, Ao

F t

F t

F s

F

F

F s

τ =F s

A o

• Tensile stress, σ :

original cross-sectional area

before loading

σ = F t A o

2f

2mNor

inlb=

Area, Ao

F t

F t



• Tensile strain: • Lateral strain:

Strain is always

dimensionless.

Engineering Strain

• Shear strain:

θ

90º

90º - θ y

∆ x γ = Δ x /y = tan θ

e = δ Lo

Adapted from Fig. 6.1 (a) and (c), Callister & Rethwisch 9e.

δ /2

Lowo

- δ eL =

L

wo

δ L /2



• Simple tension: cable

Note: τ = M / Ac R here.

Common States of Stress

o σ =

F

A

o

τ =F s

A

σ σ

M

M A o

2R

F s A c

• Torsion (a form of shear): drive shaftSki lift (photo courtesyP.M. Anderson)

A o = cross-sectional

area (when unloaded)

F F

τ



(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM

o σ = F

A

• Simple compression:

Note: compressivestructure member

(σ < 0 here).(photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (i)

A o

Balanced Rock, ArchesNational Park



• Bi-axial tension: • Hydrostatic compression:

Pressurized tank

σ < 0h

(photo courtesyP.M. Anderson)

(photo courtesy

P.M. Anderson)

OTHER COMMON STRESS STATES (ii)

Fish under water

σ z > 0

σ θ > 0


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2. Elastic Deformation

• Elastic deformation is nonpermanent: when the applied load is released, the

piece returns to its original shape (not breaking atomic bonds).

• Hooke’s Law

E [Pa]: Modulus of elasticity, or Young’s modulus

12

Linear elastic deformation Nonlinear elastic deformation


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Elastic means reversible!

Elastic Deformation

2. Small load

F

δ

bonds

stretch

1. Initial 3. Unload

return to

initial

F

δ

Linear-

elastic

Non-Linear- elastic


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Mechanical Properties• Slope of stress strain plot (which is proportional to the

elastic modulus) depends on bond strength of metal

Fig. 6.7, Callister & Rethwisch 9e.

Attraction

Repulsion

r 0


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Elastic Deformation:

Shear Stress and Strain• Elastic deformation is nonpermanent: when the applied load is released, the

piece returns to its original shape.

• Hooke’s Law

G [Pa]: Shear Modulus

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• Shear stress [N/m2 = Pa]

• Shear strain [dimensionless]

θ γ tan=

θ

90º

90º - θ y

∆ x

γ = Δ x /y = tan θ


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Poisson's ratio, ν

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Axial (z) elongation (positive strain)

and lateral (x and y) contractions

(negative strains) in response to an

imposed tensile stress

• Relationship among elastic

parameters for isotropic materials


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Poisson's ratio, ν

• Poisson's ratio, ν :

Units:

E : [GPa] or [psi]

ν : dimensionless

ν > 0.50 density increases

ν < 0.50 density decreases(voids form)

eL

e

-ν

eν = - L

e

metals: ν ~ 0.33

ceramics: ν ~ 0.25

polymers: ν ~ 0.40


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Table 6.1: Elastic and Shear

Moduli and Poisson’s Ratio

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Metals

Alloys

Graphite

Ceramics

Semicond

PolymersComposites

/fibers

E (GPa)

Based on data in Table B.2,

Callister & Rethwisch 9e.

Composite data based on

reinforced epoxy with 60 vol%

of aligned

carbon (CFRE),

aramid (AFRE), or

glass (GFRE)

fibers.

Young

s Moduli: Comparison

109 Pa

0.2

8

0.6

1

Magnesium,

Aluminum

Platinum

Silver, Gold

Tantalum

Zinc, Ti

Steel, Ni

Molybdenum

G raphite

Si crystal

Glass - soda

Concrete

Si nitride Al oxide

PC

Wood( grain)

AFRE( fibers) *

CFRE *

GFRE*

Glass fibers only

Carbon fibers only

A ramid fibers only

Epoxy only

0.4

0.8

2

4

6

10

20

4 0

6 08 0

10 0

200

6008 00

10 001200

400

Tin

Cu alloys

Tungsten

Si carbide

Diamond

PTF E

HDP E

LDPE

PP

Polyester

PS PET

C FRE( fibers) *

G FRE( fibers)*

G FRE(|| fibers)*

A FRE(|| fibers)*

C FRE(|| fibers)*


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Summary

1. Concept of stress and strain

2. Mechanical testing: tensile, compression, shear and

torsional tests

3. Elastic deformation: Modulus of elasticity

4. Poisson’s ratio

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Linear Elastic Properties

• Modulus of Elasticity, E :(also known as Young's modulus)

• Hooke's Law:

σ = E e σ

Linear-elastic

E

e

F

F simpletensiontest


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• Elastic Shear

modulus, G :

τ

G γ

τ = G γ

Other Elastic Properties

simple

torsion

test

M

M

• Special relations for isotropic materials:

2(1 + ν )

EG =

3(1 - 2ν )

EK =

• Elastic Bulkmodulus, K:

pressure

test: Init.vol =V o.

Vol chg.

= ΔV

P

P P P = -K

Δ VVo

P

Δ V

K Vo


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• Simple tension:

δ = FLoE Ao

δ L = - ν Fw oE Ao

• Material, geometric, and loading parameters all

contribute to deflection.

• Larger elastic moduli minimize elastic deflection.

Useful Linear Elastic Relationships

F

Aoδ /2

δ L /2

Lo wo

• Simple torsion:

α = 2MLoπ ro

4G

M = momentα = angle of twist

2r o

Lo

MSE 3300-Lecture Note 09-Chapter 06 Mechanical Properties of Metals 1

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