Fundamentals of Material Science

CHAPTER 7

Mechanical Properties

Fundamentals of Material Science Dr. Gamal Abdou

Behavior Of Material Under

Mechanical Loads = Mechanical Properties.

• Stress and strain:

• What are they and why are they used instead of load and

deformation

• Elastic behavior:

• Recoverable Deformation of small magnitude

• Plastic behavior:

• Permanent deformation We must consider which materials

are most resistant to permanent deformation?

• Toughness and ductility:

• Defining how much energy that a material can take before

failure. How do we measure them?

• Hardness:

• How we measure hardness and its relationship to material

strength

•(i) Tensile strength

•(ii) Hardness

•(iii) Impact strength

• (iv) fatigue

•(v) Creep

3

Fundamental Mechanical Properties

Comparison of Units: SI and Engineering Common

Unit SI Eng. Common

Force Newton (N) Pound-force (lbf)

Area mm2 or m2 in2

Stress Pascal (N/m2) or MPa (106 pascals) psi (lbf/in2) or Ksi (1000 lbf/in2)

Strain (Unitless!) mm/mm or m/m in/in

Conversion Factors SI to Eng. Common Eng. Common to SI

Force N*4.448 = lbf Lbf*0.2248 = N

Area I mm2*645.16 = in2 in2 *1.55x10-3 = mm2

Area II m2 *1550 = in2 in2* 6.452x10-4 = m2

Stress I - a Pascal * 1.450x10-4 = psi psi * 6894.76 = Pascal

Stress I - b Pascal * 1.450x10-7 = Ksi Ksi * 6.894 x106 = Pascal

Stress II - a MPa * 145.03 = psi psi * 6.89x 10-3 = MPa

Stress II - b MPa * 1.4503 x 10-1= Ksi Ksi * 6.89 = MPa

One other conversion: 1 GPa = 103 MPa

• Simple tension: cable

Note: = M/AcR here. Where M is the “Moment” Ac shaft area & R shaft radius

Common States of Stress

Ao = cross sectional

area (when unloaded)

FF

o

s =F

A

o

=Fs

A

ss

M

M Ao

2R

FsAc

• Torsion (a form of shear): drive shaftSki lift (photo courtesy

P.M. Anderson)

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

o

s =F

A

• Simple compression:

Note: compressive

structure member

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

OTHER COMMON STRESS STATES (1)

Ao

Balanced Rock, Arches National Park

• Bi-axial tension: • Hydrostatic compression:

Pressurized tank

s < 0h

(photo courtesy

P.M. Anderson)

(photo courtesy

P.M. Anderson)

OTHER COMMON STRESS STATES (2)

Fish under water

sz > 0

sq > 0

• Tension Test

– Strength

– Ductility

– Toughness

– Elastic Modulus

– Strain-hardening capability

• Test Specimen

– Usually solid and round

– Original Gauge length lo– Cross-sectional area Ao

Tension

4

• Tensile stress, s: • Shear stress, :

Area, A

Ft

Ft

s =Ft

Aooriginal area

before loading

Area, A

Ft

Ft

Fs

F

F

Fs

=Fs

Ao

Stress has units:

N/m2 or lb/in2

ENGINEERING STRESS

8

• Tensile strain: • Lateral strain:

• Shear strain:q/2

/2

/2 - q

q/2

/2

/2

L/2L/2

Lowo

=

Lo

L =L

wo

= tan q Strain is always

dimensionless.

ENGINEERING STRAIN

Linear: Elastic Properties

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

• Hooke's Law:

s = E s

Linear-

elastic

E

Units:

E: [GPa] or [psi]

s: in [Mpa] or [psi]

: [m/m or mm/mm] or [in/in]

F

Ao/2

L/2

Lowo

Here: The Black

Outline is Original,

Green is after

application of load

Stress-Strain: Testing Uses Standardized methods

developed by ASTM for Tensile Tests it is ASTM E8

• Typical tensile test

machine

Adapted from Fig. 6.3, Callister 7e. (Fig. 6.3 is 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.)

specimenextensometer

• Typical tensile

specimen (ASTM A-bar)

Adapted from

Fig. 6.2,

Callister 7e.

gauge length

- During Tensile Testing,

Instantaneous load and displacement is measured

These load / extension graphs depend on the size of the specimen.

E.g. if we carry out a tensile test on a specimen having a cross-sectional area

twice that of another, you will require twice the load to produce the same

elongation.

LOAD vs. EXTENSION PLOTS

The Force .vs. Displacement plot will be the same shape as the

Eng. Stress vs. Eng. Strain plot

Raw Data Obtained

Load, PfDeformation

Load

,P

(kN

)Total Elongation

Uniform Deformation

X

Maximum

Load, Pmax

Elastic

Elongation, L (mm)

• Stress-strain curves

– Linear elastic: elongation in the specimen that is

proportional to the applied load.

– Engineering stress: the ratio of the applied load P,

to the original cross-sectional area, Ao, of the

specimen.

• Engineering stress equation: σ = P/Ao

• Engineering strain equation: e = (l-lo)/lo

• Yield Stress: the stress at which permanent

(plastic) deformation occurs.

• Permanent (plastic) deformation: stress and

strain are no longer proportional.

• Ultimate tensile strength (UTS): the maximum

engineering stress

Tension

The Engineering Stress - Strain curve

Divided into 2 regions

ELASTIC PLASTIC

Tensile Testing

(a)(b)

18

Ductile vs Brittle Failure

Very

Ductile

Moderately

DuctileBrittle

Fracture

behavior:

Large Moderate

(%EL)=100%

Small

• Ductile

fracture is usually

desirable!

• Classification:

Ductile:

warning before

fracture, as increasing

is required for crack

growth

Brittle:

No

warning

19

Ductile vs. Brittle Failure

cup-and-cone fracture brittle fracture

Elastic and Plastic behavior

All materials deform when subjected to an external load.

Up to a certain load the material will recover its original

dimensions when the load is released. This is known as

elastic behavior.

The load up to which the materials remains elastic is the

elastic limit. The deformation or strain produced within the

elastic limit is proportional to the load or stress. This is

known as Hook’s Law , Stress Strain or Stress = E*Strain.

E is known as the Elastic Modulus.

When the load exceeds the elastic limit, the deformation

produced is permanent. This is called plastic deformation.

Hook’s law is no longer valid in the plastic region.

Tensile Properties

In many other metals and alloys the yield point is not

distinct (Curve 2, Fig. b). In such cases, a line parallel to the

linear region is drawn at a strain = 0.002 (0.2%) and its

intercept on the plastic region is taken as the yield stress

(Fig. b). This is called 0.2% Proof stress.

The stress at the maximum load is called ultimate tensile

strength (UTS).

The strain up to UTS is the uniform plastic strain. Beyond

this the cross sectional area reduces and necking takes

place.

The fracture strain ef = (Lf - Lo)/Lo, where Lf is the length

after fracture, is taken as the measure of Ductility.

Tensile Properties

EL = Elastic limit, up to which Hook’s Law (Stress Strain)

is valid. The material comes back to original shape when theload is released.

Elastic limit is difficult to determine. The proportional limit,

PL, the load at which the curve deviates from linearity, is

taken as the elastic portion.

The slope of the linear region is the Young’s Modulus or

Elastic Modulus (E).

Loading beyond PL produces permanent or plastic

deformation. The onset point of plastic deformation is known

as Yield stress (YS).

In some materials like mild steel the yield point is prominent

(Curve 1 in Fig. b)

Poisson’s ratio

A tensile force in the x direction produces an extension along

thatand

The

axis while it produces contraction along the transverse yz axis.

ratio of the lateral to axial strain is the Poisson's ratio, .

For most metals it is around 0.33

y z

= = x

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

10

• Hooke's Law:

s = E

• Poisson's ratio, :

metals: ~ 0.33

ceramics: ~0.25

polymers: ~0.40

= L

L

1-

F

Fsimple tension test

s

Linear- elastic

1

E

Units:

E: [GPa] or [psi]

: dimensionless

LINEAR ELASTIC PROPERTIES

• Elastic Shear

modulus, G:

12

1

G

= G

• Elastic Bulk

modulus, K:

P= -KVVo

P

V

1-K

Vo

• Special relations for isotropic materials:

P

P P

M

M

G =

E

2(1 ) K =

E

3(1 2)

simple

torsion

test

pressure

test: Init.

vol =Vo.

Vol chg.

= V

OTHER ELASTIC PROPERTIES

True-Stress and True-Strain

• True-stress: ratio of the load, P, to the

instantaneous cross-sectional area, A, of the

specimen.

• True-strain: the sum of all the instantaneous

engineering strains.

– True-stress equation: σ = P/A

– True-strain equation: e = ln(l/lo)

True Stress-Strain Curve

Since the engineering stress-strain curve is based on

original area, it descends after maximum load as the load

bearing ability of the sample decreases due to reduction in

area.

The true stress-strain curve (blue) however, continues

to go up till fracture as it is based on the actual area.

True Stress & StrainNote: Stressed Area changes when sample is deformed

(stretched)

• True stress

• True Strain

iT AF=s

oiT llln=

=

s=s

1ln

1

T

T

Adapted from Fig. 6.16,

Callister 7e.

• An increase in sy due to plastic deformation.

22

• Curve fit to the stress-strain response:

s

large hardening

small hardeningu

nlo

ad

relo

ad

sy 0

sy 1

sT = C T

n

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

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

HARDENING

Construction of Stress-Strain Curves

• The stress-strain curve can be

represented by the equation: σ = Ken

– K = strength coefficient

– n = strain hardening exponent

• Specific energy: energy-per-unit

volume of the material deformed.

• Ductility: extent of plastic deformation that

the material undergoes before fracture.

• Two measures of ductility:

– Total elongation: (lf-lo)/lo x 100%

– Reduction of Area: (Ao-Af)/Ao x 100%

Ductility

• Plastic tensile strain at failure:

20

Engineering tensile strain,

Engineering

tensile

stress, s

smaller %EL

(brittle if %EL<5%)

larger %EL

(ductile if

%EL>5%)

• Another ductility measure:

%AR =

Ao A f

Ao

x100

• Note: %AR and %EL are often comparable.

--Reason: crystal slip does not change material volume.

--%AR > %EL possible if internal voids form in neck.

Lo LfAo

Af

%EL =

L f Lo

Lo

x100

Adapted from Fig. 6.13,

Callister 6e.

Mechanical Properties of some commonly used materials

Material E, GPa YS, MPa UTS, MPa %Elong. Poisson's

ratio

C steel 207 220 - 250 400 - 500 23 0.30

Stain less steel 193 515 850 10 0.30

Alloy steels 207 860 1280 12 0.30

Al 70 34 90 40 0.33

Al alloys 72 - 85 250 -500 300 - 550 10 -20 0.34

Ti 103 170 240 30 0.34

Ti alloy 114 1100 1170 10 0.34

Mg 45 25 - 40 50 – 60 8 – 10 0.35

Mg alloys 45 220 290 15 0.35

Ni 204 148 460 47 0.31

Ni super alloy 207 517 930 - 0.21

Al2O3 380 550 - 0.16

PET 2.7 - 4 60 70 30-300 0.39

Hardness

Hardness can be defined as resistance

indentation or resistance to scratch.

to deformation or

Hardness

Indentation Scratch Rebound

Indentation hardness is of particularand is most commonly used.

interest to engineers

Indentation hardness can be measured by differentmethods.

Classified based on how it is measured.

The following are the hardness test methods

• Rockwell hardness test

• Brinell hardness

• Vickers

• Knoop hardness

• shore

Hardness Measurement Methods

Hardness

• Resistance to permanently (plastically) indenting the surface of a

product.

• Large hardness means:

--resistance to plastic deformation or cracking in compression.

--better wear properties.

e.g., Hardened 10

mm sphere

apply known force measure size of indentation 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

Rockwell Hardness

In this type of test, depth of indentation at a constant

load is taken as the measure of Hardness.

A minor load of 10 kg is first applied for good contact

between the indenter and the sample surface.

The major load is then applied and the depth of indentation

is recorded on a dial gage in terms of an arbitrary number.

The dial consists of 100 divisions, each division representing

a penetration depth of 0.002 mm.

Rockwell Hardness

Indenter and Hardness Scale

Two types of indenters – 120 diamond cone called Brale

indenter and 1.6 and 3.2 mm diameter steel balls

Combination of indenter and major load gives rise to

different hardness scales.

C - Scale – Brale indenter + 150 kg load, designated as RC.

Range is RC 20 – RC 70. Used for hard materials like hardened

steels.

B-Scale – Steel ball indenter + 100 kg load, written as RB.

Range is RB 0 to RB 100.

Minor loads in RC and RB scales are 10 kg and 3 kg

respectively.

Microhardness

Sometime hardness determination is needed over a very

small area.

For example, hardness of carburised steel surface,

coatings or individual phases or constituents of a material.

The load applied is much smaller compared to

macrohardness.

The indentation is very small. An optical microscope is

used to observe it. Sample preparation is needed.

Two methods are used for microhardness testing.

Microhardness

Vickers Microhardness

This is same as Vickers hardnessload is much smaller so as to cover

The applied load range is 1 – 100

except that the applieda small area.

g.

Hardness: Common Measurement Systems

Callister Table 6.5

HB = Brinell Hardness

TS (psia) = 500 x HB

TS (MPa) = 3.45 x HB

Comparing

Hardness

Scales:

Inaccuracies in Rockwell / Brinell hardness

measurements may occur due to:

An indentation is made too near a specimen edge.

Two indentations are made too close to one another.

Specimen thickness should be at least ten times the

indentation depth.

Allowance of at least three indentation diameters

between the center on one indentation and the

specimen edge, or to the center of a second

indentation.

Testing of specimens stacked one on top of another is

not recommended.

Indentation should be made into a smooth flat surface.

Impact Tests

Toughness of metals is the ability to

withstand impact.

TOUGHNESS

High toughness = High yield strength and ductility

Dynamic (high strain rate) loading condition (Impact test)

1. Specimen with notch- Notch toughness

2. Specimen with crack- Fracture toughness

Is a measure of the ability of a material to absorb energy up to

fracture

Important Factors in determining Toughness:

1. Specimen Geometry & 2. Method of load application

Static (low strain rate) loading condition (tensile stress-strain test)

1. Area under stress vs strain curve up to the point of fracture.

• Energy to break a unit volume of material

• Approximate by the area under the stress-strain curve.

Toughness

Brittle fracture: elastic energy

Ductile fracture: elastic + plastic energy

very small toughness (unreinforced polymers)

Engineering tensile strain,

Engineering

tensile

stress, s

small toughness (ceramics)

large toughness (metals)

Adapted from Fig. 6.13,

Callister 7e.

Izod test

Strikes at 167 Joules.

Test specimen is held

vertically.

Notch faces striker.

Charpy impact test

Strikes form higher

position with 300 Joules.

Test specimen is held

horizontally.

Notch faces away from

striker.

ExamplesEx.1. A 15 mm long and 13 mm diameter sample shows the

following behavior in a tensile test. Load at 0.2% offset – 6800

kg, maximum load – 8400 kg, fracture occurs at 7300 kg,

diameter and length after fracture – 8 mm and 65 mm

respectively. Find the standard mechanical properties.

Solution: Ao = (13)2/4 = 132.7 mm2, Af = (8)2/4 = 50.3 mm2

= 620 MPax 9.8)/132.7 = 620 N/mm2UTS = Pmax/Ao = (84000.2% proof stress = (6800 x 9.8)/132.7 = 502 N/mm2 = 502 MpaBreaking stress = (7300 x 9.8)/132.7 = 539 Mpa

%elongation = 100*(Lf – Lo)/Lo = 100 x (65 – 50)/50 = 30%% area reduction = 100*(Af – Ao)/Lo = 100(132.7 – 50.3)/132.7 =

62%

ExamplesEx.2. A metal experiences a true strain of 0.16 at a true stress

of 500 MPa. What is the strain hardening exponent of the

metal? K = 825 MPa. What will be the true strain at a stress of600 MPa?

Solution: n = (logs - logK)/log = (log 500 – log 825)/log 0.16= 0.271

s= Kn

()0.271 , strain = 0.3Strain at 600 MPa: 600 = 825

Quiz

1. Define hardness. What is Mohs scale of hardness?

2. Why it is necessary to specify load-indenter combination in

Rockwell hardness test?

3. How is Brinell hardness measured. Show that BHN varies

as P/D2 where P is the load and D is the indenter diameter.

4. Why is the included angle between opposite faces of theVickers indenter 136?

5.6.

7.

8.

9.

What

What

What

What

is

is

is

is

microhardness? Why sometime it is necessary?

engineering stress and strain?

Hook’s law?elastic and proportional limit?

How is the elastic modulus measured from the stress-strain

curve?10. What is yield stress?

Quiz

11. What is 0.2% proof stress?

12.13.

14.

15.

How is the ductility measured?

WhatWhat

What

is

is

is

ductile and brittle behavior?resilience? What is toughness?

true stress and strain. Deduce the relationship

between true and engineering stress ad strain.16. Why does the engineering stress-strain curve peak and

drop where as the true stress-strain curve keep on going up?a flow curve?

shear stress and strain

Poisson's ratio?

17.18.

19.

20.

What

What

What

What

is

is

is

are structure-sensitive and structure insensitive

properties?21. What is Poisson's ratio?

Quiz22. A 15 mm long and 120 mm dia cylindrical rod is subjected

to a tensile load of 35 kN. It must not experience either plastic

deformation or a diameter reduction of more than 0.012 mm.

Which of the listed materials is suitable for such a requirement

and why? Al (E= 70 GPa, YS = 250 MPa, = 0.33), Ti (E= 105

GPa, YS = 850 MPa, = 0.36), Steel (E= 205 GPa, YS = 550MPa, = 0.27), Mg (E= 45 GPa, YS = 170 MPa, = 0.35).

23. A metal experiences a true strain of 0.1 at a true stress of

415 MPa. What is the strain hardening exponent of the metal?

K = 1035 MPa. What will be the true strain at a stress of 600MPa?

Quiz24. The following data were obtained in a tensile test of a low-

carbon steel of diameter 12 mm and gage length 50 mm.

Plot Engineering and True stress-strain curve and find thetensile properties.

Load, kN Elongation, mm Load, kN Elongation, mm

2 0.0041 25.2 0.51

4 0.0082 28 1.52

6 0.0132 30 2.03

8 0.0183 34 3.05

10 0.0226 38.4 4.57

12 0.0267 40 6.60

14 0.0310 40.4 7.62

16 0.0351 40.8 12.7

18 0.0391 40.2 14.7

20 0.0445 38.6 15.7

22 0.0485 36.4 17.8

24 0.0518 32.4 19.3