chapter 6: behavior of material under mechanical loads = mechanical properties. stress and strain:...

Chapter 6: 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

Elastic means reversible!

Elastic Deformation1. Initial 2. Small load 3. Unload

F

bonds stretch

return to initial

F

Linear- elastic

Non-Linear-elastic

Plastic means permanent!

Plastic Deformation (Metals)

F

linear elastic

linear elastic

plastic

1. Initial 2. Small load 3. Unload

planes still sheared

F

elastic + plastic

bonds stretch & planes shear

plastic

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

Stress has units: N/m2 (Mpa) or lbf/in2

Engineering Stress:

• Shear stress, :

Area, A

Ft

Ft

Fs

F

F

Fs

= Fs

Ao

• Tensile stress, :

original area before loading

Area, A

Ft

Ft

=Ft

Ao2f

2m

Nor

in

lb=

we can also see the symbol ‘s’ used for engineering stress

Geometric Considerations of the Stress State

The four types of forces act either parallel or perpendicular to the planar faces of the bodies.

Stress state is a function of the orientations of the planes upon which the stresses are taken to act.

• 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 F

A

o

FsA

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 F

A

• Simple compression:

Note: compressivestructure member( < 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

< 0h

(photo courtesyP.M. Anderson)

(photo courtesyP.M. Anderson)

OTHER COMMON STRESS STATES (2)

Fish under water

z > 0

> 0

• Tensile strain: • Lateral strain:

• Shear strain:

Strain is alwaysDimensionless!

Engineering Strain:

90º

90º - y

x = x/y = tan

Lo

L L

wo

Adapted from Fig. 6.1 (a) and (c), Callister 7e.

/2

L/2

Lowo

We often see the symbol ‘e’ used for engineering strain

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

The Engineering Stress - Strain curveDivided into 2 regions

ELASTIC PLASTIC

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

• Hooke's Law:

= E

Linear- elastic

E

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

• Typical Engineering Issue in Mechanical Testing:– Aluminum tensile specimen, square X-Section

(16.5 mm on a side) and 150 mm long– Pulled in tension to a load of 66700 N– Experiences elongation of: 0.43 mm

• Determine Young’s Modulus if all the deformation is recoverable

230

0

66700 244.99516.5*10

0.43 0.00344125

Because we are to assume all deformation is

recoverable, Hooke's Law can be assumed:

244.9950.00344

71219.6 71.2

NF MPaA

mmLL mm

MPaE E

E MPa GPa

Solving:

Poisson's ratio, • Poisson's ratio, :

Units:: dimensionless

> 0.50 density increases

< 0.50 density decreases (voids form)

L

-

L

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

• Elastic Shear modulus, G:

G

= G

Other Elastic Properties

simpletorsiontest

M

M

• Special relations for isotropic materials:

2(1 )EG

3(1 2)

EK

• Elastic Bulk modulus, K:

pressuretest: Init.

vol =Vo. Vol chg. = V

P

P PP = -K

VVo

P

V

K Vo

E is Modulus of Elasticity is Poisson’s Ratio

Looking at Aluminum and the earlier problem:

71.22 1 2 1 0.33

26.8

71.23 1 2 3 1 2 0.33

69.8

HB

HB

GPaEG

G GPa

GPaEK

K GPa

MetalsAlloys

GraphiteCeramicsSemicond

PolymersComposites

/fibers

E(GPa)

Based on data in Table B2,Callister 7e.Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.

Young’s Moduli: Comparison

109 Pa

0.2

8

0.6

1

Magnesium,Aluminum

Platinum

Silver, Gold

Tantalum

Zinc, Ti

Steel, NiMolybdenum

Graphite

Si crystal

Glass -soda

Concrete

Si nitrideAl oxide

PC

Wood( grain)

AFRE( fibers) *

CFRE*

GFRE*

Glass fibers only

Carbon fibers only

Aramid fibers only

Epoxy only

0.4

0.8

2

4

6

10

20

40

6080

100

200

600800

10001200

400

Tin

Cu alloys

Tungsten

<100>

<111>

Si carbide

Diamond

PTFE

HDPE

LDPE

PP

Polyester

PSPET

CFRE( fibers) *

GFRE( fibers)*

GFRE(|| fibers)*

AFRE(|| fibers)*

CFRE(|| fibers)*

• Simple tension:

FLo

EAo

L

Fw o

EAo

• Material, geometric, and loading parameters all contribute to deflection.• Larger elastic moduli minimize elastic deflection.

Useful Linear Elastic Relationships

F

Ao/2

L/2

Lowo

• Simple torsion:

2MLo

ro4G

M = moment = angle of twist

2ro

Lo

Resilience, Ur

• Ability of a material to store (elastic) 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

(at lower temperatures, i.e. T < Tmelt/3)

Plastic (Permanent) Deformation

• Simple tension test:

engineering stress,

engineering strain,

Elastic+Plastic at larger stress

permanent (plastic) after load is removed

p

plastic strain

Elastic initially

Adapted from Fig. 6.10 (a), Callister 7e.

• Stress at which noticeable plastic deformation has occurred.

when p 0.002

Yield Strength, y

y = yield strength

Note: for 2 inch sample

= 0.002 = z/z

z = 0.004 in

Adapted from Fig. 6.10 (a), Callister 7e.

tensile stress,

engineering strain,

y

p = 0.002

Tensile properties Yielding Strength

Most structures operate in elastic region, therefore need to know when it ends

Some steels (lo-C)

Yield strength

-- Generally quoted

Proof stress

Proportional Limit

Room Temp. values

Based on data in Table B4,Callister 7e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered

Yield Strength : ComparisonGraphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

Yie

ld s

tren

gth,

y

(MP

a)

PVC

Har

d to

mea

sure

,

sin

ce in

te

nsi

on

, fr

act

ure

usu

ally

occ

urs

be

fore

yie

ld.

Nylon 6,6

LDPE

70

20

40

6050

100

10

30

200

300

400500600700

1000

2000

Tin (pure)

Al (6061) a

Al (6061) ag

Cu (71500) hrTa (pure)Ti (pure) aSteel (1020) hr

Steel (1020) cdSteel (4140) a

Steel (4140) qt

Ti (5Al-2.5Sn) aW (pure)

Mo (pure)Cu (71500) cw

Har

d to

mea

sure

, in

ce

ram

ic m

atr

ix a

nd

ep

oxy

ma

trix

co

mp

osi

tes,

sin

cein

te

nsi

on

, fr

act

ure

usu

ally

occ

urs

be

fore

yie

ld.

HDPEPP

humid

dry

PC

PET

¨

Tensile Strength, TS

• Metals: occurs when noticeable necking starts.• Polymers: occurs when polymer backbone chains are aligned and about to break.


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

• TS is Maximum stress on engineering stress-strain curve.

Tensile Strength : Comparison

Si crystal<100>

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

Ten

sile

str

engt

h, T

S

(MP

a)

PVC

Nylon 6,6

10

100

200300

1000

Al (6061) a

Al (6061) agCu (71500) hr

Ta (pure)Ti (pure) aSteel (1020)

Steel (4140) a

Steel (4140) qt

Ti (5Al-2.5Sn) aW (pure)

Cu (71500) cw

LDPE

PP

PC PET

20

3040

20003000

5000

Graphite

Al oxide

Concrete

Diamond

Glass-soda

Si nitride

HDPE

wood ( fiber)

wood(|| fiber)

1

GFRE(|| fiber)

GFRE( fiber)

CFRE(|| fiber)

CFRE( fiber)

AFRE(|| fiber)

AFRE( fiber)

E-glass fib

C fibersAramid fib

Room Temp. valuesBased on data in Table B4,Callister 7e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & temperedAFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%fibers.

• Plastic tensile strain at failure:


Ductility

• Another ductility measure: 100xA

AARA%

o

fo -=

x 100L

LLEL%

o

of

Engineering tensile strain,

Engineering tensile stress,

smaller %EL

larger %ELLf

Ao AfLo

Lets Try one (like Problem 6.29)Load (N) len. (mm) len. (m) l

0 50.8 0.0508 0

12700 50.825 0.050825 2.5E-05

25400 50.851 0.050851 5.1E-05

38100 50.876 0.050876 7.6E-05

50800 50.902 0.050902 0.000102

76200 50.952 0.050952 0.000152

89100 51.003 0.051003 0.000203

92700 51.054 0.051054 0.000254

102500 51.181 0.051181 0.000381

107800 51.308 0.051308 0.000508

119400 51.562 0.051562 0.000762

128300 51.816 0.051816 0.001016

149700 52.832 0.052832 0.002032

159000 53.848 0.053848 0.003048

160400 54.356 0.054356 0.003556

159500 54.864 0.054864 0.004064

151500 55.88 0.05588 0.00508

124700 56.642 0.056642 0.005842

GIVENS:

Leads to the following computed Stress/Strains:

e stress (Pa) e str (MPa) e. strain

0 0 0

98694715.7 98.694716 0.000492

197389431 197.38943 0.001004

296084147 296.08415 0.001496

394778863 394.77886 0.002008

592168294 592.16829 0.002992

692417257 692.41726 0.003996

720393712 720.39371 0.005

796551839 796.55184 0.0075

837739398 837.7394 0.01

927885752 927.88575 0.015

997049766 997.04977 0.02

1163354247 1163.3542 0.04

1235626755 1235.6268 0.06

1246506488 1246.5065 0.07

1239512374 1239.5124 0.08

1177342475 1177.3425 0.1

969073311 969.07331 0.115

0

2 20

0

use m if F in Newtons; in if F in lb

results in Pa (MPa) or psi (ksi)

f

FA

A

and

ll

Leads to the Eng. Stress/Strain Curve:Engineering Stress Strain

0

200

400

600

800

1000

1200

1400

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Strain (m/m)

Str

ess

(MP

a)

Magenta Line Model:

.002*

.0021 to .0065

m E

m E

T. Str. 1245 MPa

Y. Str. 742 MPa%el 11.5%

F. Str 970 MPa

E 195 GPa (by regression)

TOUGHNESS

High toughness = High yield strength and ductility

Dynamic (high strain rate) loading condition (Impact test)1. Specimen with notch- Notch toughness2. 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 energyDuctile fracture: elastic + plastic energy

very small toughness (unreinforced polymers)

Engineering tensile strain,

Engineering tensile stress,

small toughness (ceramics)

large toughness (metals)


Elastic Strain Recovery – After Plastic Deformation


It is an important factor in forming products (especially sheet metal and spring making)

True Stress & StrainNote: Stressed Area changes when sample is

deformed (stretched)• True stress

• True Strain

iT AF

oiT ln

1ln

1

T

T


Strain Hardening

• Curve fit to the stress-strain response:

T K T n

“true” stress (F/Ai) “true” strain: ln(Li /Lo)

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

• An increase in y due to continuing plastic deformation.

large Strain hardening

small Strain hardeningy 0

y 1

Earlier Example: True Properties

T. Stress T. Strain

0 0

98.74328592 0.000492

197.5875979 0.001003

296.5271076 0.001495

395.571529 0.002006

593.9401363 0.002988

695.1842003 0.003988

723.9956807 0.004988

802.525978 0.007472

846.1167917 0.00995

941.8040385 0.014889

1016.990761 0.019803

1209.888417 0.039221

1309.764361 0.058269

1333.761942 0.067659

1338.673364 0.076961

1295.076722 0.09531

1080.516741 0.108854

Necking Began

We Compute:

T = KTn

to complete our True Stress vs. True Strain plot (plastic data to necking)

• Take Logs of both T and T

• ‘Regress’ the values from Yielding to Necking

• Gives a value for n (slope of line) and K (its T when T = 1)

• Plot as a continuation line beyond necking start

Stress Strain Plot w/ True ValuesEngineering Stress Strain

& True Stress Stain

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Strain (m/m)

Str

ess

(MP

a)

n 0.242 K 2617 MPa

Variability in Material Properties

Critical properties depend largely on sample flaws (defects, etc.). Most can exhibit Large sample to sample variability.

• Real (reported) Values, thus, are Statistical measures usually mean values.

– Mean

– Standard Deviation 2

1

2

1

n

xxs i

n

n

xx n

n

where n is the number of data points (tests) that are performed

Ny

working

Ny

working

• Design uncertainties mean we do not push the limit!• Typically as engineering designers, we introduce a Factor of safety, N Often N is

between1.5 and 4

• Example: Calculate a diameter, d, to ensure that yield does not occur in the 1045 carbon steel rod below subjected to a

working load of 220,000 N. Use a factor of safety of 5.

Design or Safety Factors

1045 plain carbon steel: y = 310 MPa

TS = 565 MPa

F = 220,000N

d

Lo

Solving:

2

20

2

2

2 26

2 3 2

2

310

5220000

4

310220000

54

220000 4 5 m310 10

4.52 10 m

6.72 10 m 6.72 cm

Yworking

working

Nm

NNF

A D

MNmN

D

D

D x

D x D

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

Hardness: Common Measurement Systems

Callister Table 6.5

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.

Correlation Between Hardness and Tensile Strength

Both measures the resistance to plastic deformation of a material.

HB = Brinell HardnessTS (psia) = 500 x HBTS (MPa) = 3.45 x HB

• 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.

chapter 6: behavior of material under mechanical loads = mechanical properties. stress and strain:...

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