Chapter 6 - 1

• 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 does permanent deformation occur? What materials are most resistant to permanent deformation?

• Toughness and ductility: What are they and how do we measure them?

Mechanical Properties

• Hardness: What is hardness of a material and how do we measure it?

Chapter 6 -2

How materials deform as a function of applied load Testing methods and language for mechanical properties of materials.

Str

es

s,

(M

Pa

)

Strain, (mm / mm)

Chapter 6 - 3

Elastic means reversible!

Elastic Deformation

1. Initial 2. Small load 3. Unload

F

bonds stretch

return to initial

F

Linear- elastic

Non-Linear-elastic

Chapter 6 - 4

Plastic means permanent!

Plastic Deformation (Metals)

F

linear elastic

linear elastic

plastic

1. Initial 2. Small load 3. Unload

p lanes still sheared

F

elastic + plastic

bonds stretch & planes shear

plastic

Chapter 6 -

Forces and Responses

• Tensile – applied loads “pull” the sample

Chapter 6 - 6

Linear Elastic Properties

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

• Hooke's Law:

= E

Linear- elastic

E

F

Fsimple tension

test

Chapter 6 -

Tensile Forces

¾ inch

½ inch

8 ½ inches

Failure ZoneGripping Zone

Gripping Zone

Chapter 6 -

Forces and Responses

Compression: applied loads “squeeze” the sample

Chapter 6 - 9

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

elastic modulus) depends on bond strength of metal

Adapted from Fig. 6.7, Callister 7e.

Chapter 6 - 10

• Elastic Shear modulus, G:

G

= G

Other Elastic Properties

• Special relation for isotropic materials:

2(1 )EG

Chapter 6 - 11

Poisson's ratio,

• Poisson's ratio, :

Units:E: [GPa] or [psi]: dimensionless

L

-

L

metals: ~ 0.33ceramics: ~ 0.25polymers: ~ 0.40

Chapter 6 -

Plastic Deformation

Chapter 6 - 13

(Ultimate) Tensile Strength, TS

Adapted from Fig. 6.11, Callister 7e.

y

strain

Typical response of a metal

F = fracture or

ultimate

strength

Neck – acts as stress concentrator

eng

inee

ring

TS s

tres

s

engineering strain

• Maximum stress on engineering stress-strain curve.

Chapter 6 -

Ductility

Measures how much the material can be stretched before fracture

High ductility: platinum, steel, copperGood ductility: aluminumLow ductility (brittle): chalk, glass, graphite

Chapter 6 -

• Plastic tensile strain at failure:

Engineering tensile strain,

Engineering tensile stress,

smaller %EL (brittle if %EL<5%)

larger %EL (ductile if %EL>5%)

%RA

Ao A

f

Ao

x100

Lo LfAo Af

%EL

Lf L

o

Lo

x100

Ductility (%EL and %RA)

Adapted from Fig. 6.13, Callister 7e.

Chapter 6 - 16

• The ability to absorb energy up to fracture =the total area under the strain-stress curve up to fractureUnits: the energy per unit volume, e.g. J/m3

Toughness

Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy

very small toughness (unreinforced polymers)

Engineering tensile strain,

E ngineering

tensile

stress,

small toughness (ceramics)

large toughness (metals)

Adapted from Fig. 6.13, Callister 7e.

Chapter 6 -

Toughness

• Impact (toughness) – applied loads “hit” the sample

• Impact (charpy, dart)

Chapter 6 - 18

Resilience, Ur• Ability of a material to store energy

– Energy stored best in elastic region

If we assume a linear stress-strain curve this simplifies to

Adapted from Fig. 6.15, Callister 7e.

yyr2

1U

y dUr 0

Chapter 6 - 19

True Stress & Strain

• True stress

• True Strain

iT AF

oiT ln

1ln

1

T

T

Adapted from Fig. 6.16, Callister 7e.

Chapter 6 -

Elastic Solid

• Stress-strain• What happens when force is removed?

– Recovery

Chapter 6 - 21

Elastic Strain Recovery

Adapted from Fig. 6.17, Callister 7e.

Chapter 6 - 22

Hardening

• Curve fit to the stress-strain response:

T K T n

“true” stress (F/A) “true” strain: ln(L/Lo)

hardening exponent:n = 0.15 (some steels) to n = 0.5 (some coppers)

• An increase in y due to plastic deformation.

large hardening

small hardeningy 0

y 1

Chapter 6 -

Stress-Strain Diagram

Strain ( ) (L/Lo)

41

2

3

5

Str

ess

(F

/A)

Elastic Region

PlasticRegion

StrainHardening Fracture

ultimatetensile strength

Slo

pe=

E

Elastic region slope =Young’s (elastic) modulus yield strengthPlastic region ultimate tensile strength strain hardening fracture

necking

yieldstrength

UTS

y

εEσ

ε

σE

12

y

ε ε

σE

Chapter 6 - 24

Hardness• Resistance to permanently indenting the surface.• Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties.

e.g., 10 mm sphere

apply known force measure size of indent after removing load

dDSmaller indents mean larger hardness.

increasing hardness

most plastics

brasses Al alloys

easy to machine steels file hard

cutting tools

nitrided steels diamond

Adapted from Fig. 7.18.

Chapter 6 - 25

Hardness Testers

Chapter 6 - 26

Hardness: Measurement

• Rockwell– No major sample damage– Each scale runs to 130 but only useful in range

20-100. – Minor load 10 kg– Major load 60 (A), 100 (B) & 150 (C) kg

• A = diamond, B = 1/16 in. ball, C = diamond

• HB = Brinell Hardness– TS (MPa) = 3.45 x HB– TS (psi) = 500 x HB

Chapter 6 - 27

Conversion of Hardness Scales

Also see: ASTM E140 - 07 Volume 03.01Standard Hardness Conversion Tables for Metals Relationship Among Brinell Hardness, Vickers Hardness, Rockwell Hardness, Superficial Hardness, Knoop Hardness, and Scleroscope Hardness

Chapter 6 -

Chapter 6 - 29

Variability in Material Properties

• Elastic modulus is material property• Critical properties depend largely on sample flaws

(defects, etc.). Large sample to sample variability. • Statistics

– Mean

– Standard Deviation 2

1

2

1

n

xxs i

n

n

xx n

n

where n is the number of data points

Chapter 6 -

Property Variability and Design/Safety FactorsAverage and standard deviation

Chapter 6 - 31

• Design uncertainties mean we do not push the limit.• Factor of safety, N

Ny

working

Often N isbetween1.2 and 4

• Example: Calculate a diameter, d, to ensure that yield does not occur in the 1045 carbon steel rod below. Use a factor of safety of 5.

Design or Safety Factors

4

0002202 /d

N,

5

Ny

working

1045 plain

carbon steel: y = 310 MPa

TS = 565 MPa

F = 220,000N

d

Lo

d = 0.067 m = 6.7 cm

Chapter 6 - 32

• 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).

• Toughness: The energy needed to break a unit volume of material.

• Ductility: The plastic strain at failure.

Summary

• Plastic behavior: This permanent deformation behavior occurs when the tensile (or compressive) uniaxial stress reaches y.