445.204

Introduction to Mechanics of Materials

(재료역학개론)

Chapter 4: Introduction to mechanical

properties of solids

Myoung-Gyu Lee, 이명규

Tel. 880-1711; Email: myounglee@snu.ac.kr

TA: Chanmi Moon, 문찬미

Lab: Materials Mechanics lab.(Office: 30-521)

Email: chanmi0705@snu.ac.kr

What we learn from this chapter

- Elastic and plastic deformation- Uniaxial mechanical responses- Measure stress-strain curves and calculate engineering and true

stress-strain relations under uni-axial loading

2

Elasticity vs. Plasticity

Elastic means reversible!

2. Small load

F

d

bonds stretch

1. Initial 3. Unload

return to initial

F

d

Linear-elastic

Non-Linear-elastic

3

Elasticity vs. Plasticity

Plastic means permanent!

F

δlinear elastic

linear elastic

δplastic

1. Initial 2. Large load 3. Unload

planesstill sheared

F

δelastic + plastic

bonds stretch & planes shear

δplastic

4

Tensile Strength

- Tensile test: most convenient and simple test to characterize the materials’ mechanical properties

- A cylindrical or sheet type sample having length L and cross-sectional area A is anchored at one end and subjected to a load P

L

d

P

A0

Strength of materials loaded in tension (no necking but fracture; brittle materials)

0

ff

A

Pσ =

fσ

fP

0A

Ultimate Tensile Stress

Load at fracture

Initial cross-sectional area

5

Tensile Stress, or Stress

- For the specimen loaded by an axial force P with the initial cross-sectional area, A0, the tensile stress is defined as

0A

Pσ =

- Tensile stress: force per unit area acting on a plane transverse to the loading (uniaxial)

- Stress (or Engineering Stress) = applied load/original cross-sectional area

- Unit: N/m2 = Pa, lbs/in2 = psi

Tensile Stiffness or Young’s modulus

- Stiffness vs. strength?Strength (강도): material propertyStiffness (강성): includes geometrical effect

- K : a proportionality representing a constant or “Stiffness”(unit: N/m, lb/in)

- Stiffness is a function of both “material” and sample “shape “- When K is a “constant” or the displacement and applied load is linear, the

relation is known as “Hooke’s law”

δKP =Pδ

Axial Load

Displacement

Strain, Hooke’ law

- Strain (ε): deformation normalized by initial specimen length or “length change” per “unit length”

- Rewrite the Hooke’s law by using stress and strain measurement

0L

δε = 0Lδ

Length before deformation

Displacement

εδσ EL

EA

P===

00

- A constant “E” is named “Young’s modulus (영률)” or “modulus of elasticity”

Strain, Hooke’ law

L

AEK =

AE

PL=δ

Question:

Derive the two equations

Young’s modulus

σ

Linear-elastic

E

ε

MetalsAlloys

GraphiteCeramicsSemicond

Polymers Composites/fibers

E(GPa)

0.2

8

0.6

1

Magnesium,Aluminum

Platinum

Silver, Gold

Tantalum

Zinc, Ti

Steel, NiMolybdenum

G raphite

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

46

10

20

406080

10 0

200

600800

10 001200

400

Tin

Cu alloys

Tungsten

<100>

<111>

Si carbide

Diamond

PTF E

HDP E

LDPE

PP

Polyester

PSPET

C FRE( fibers) *

G FRE( fibers)*

G FRE(|| fibers)*

A FRE(|| fibers)*

C FRE(|| fibers)*

Based on data in Table B.2,Callister & Rethwisch 8e.Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.

Strain, Hooke’ law

- Hookean materials: materials that obey the Hooke’s law

- Hookean materials = Linear elastic materials

Poisson ratio

- A negative (or positive) strain occurs if the tensile (or compressive) strain is applied in the longitudinal direction

- The lateral contraction (or extension) accompanying a longitudinal extension (or contraction) is called “Poisson’s ratio”

L

D

εεν −= Lε

DεStrain along loading

Strain along lateral direction

Lε

Dε

εL

ε

-ν

Bulk modulus: how to express the materials compressibility?

- Typical values of Poisson’s ratio: Ceramic ~0.2; Metals ~0.3; Plastics ~0.4; Rubber ~0.5

- As material becomes brittle, the “Poisson’s ratio” decreases, and it increases as the materials become softer …in general

- Bulk modulus: modulus of compressibility

VV

pK

∆−= p

VHydrostatic pressure (정수압)Volume

( )ν213 −=

EK

- For isotropic materials (will be shown later)

Bulk modulus: how to express the materials compressibility?

Q) If Poisson’s ratio approaches 0.5, what happens?

Q) What if Poisson’s ratio is larger than 0.5 or if Poisson’s ratio is negative ?

Shear stress and shear strain

- Deformation that distorts a square grid but remains length in the loading direction: shear deformation (전단변형)

- Normal stress (or strain) vs. Shear stress (or strain)

A

P=τ

PA

Load applied transverselyArea

- Shear strain (γ)

τ

H

δ

γδγ ≈=H

tanHγ

HeightAngle change in the right angle

- Shear stress (τ)

Hooke’s law in shear deformation

- The same linear relationship between shear stress and shear strain is maintained under shear deformation

- G is called as “Shear Modulus”

xyxy Gγτ =

- For isotropic materials (will be derived later)

( )v

EG

+=

12

τG

γ

Stress-strain curve: engineering

- Stress-strain curve: graphical representation of mechanical properties of materials (the most useful way of expressing mechanical properties of materials)

- Plot eng. stress (σe or S) vs. eng. Strain (εe or e)

00

,LA

Pee

δεσ ==

Stress-strain curve

Stress-strain curve

Yield strength

Room temperaturevalues

Based on data in Table B.4,Callister & Rethwisch 8e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

Yiel

d st

reng

th,σ

y(M

Pa)

PVC

Har

d to

mea

sure

, si

nce

in te

nsio

n, fr

actu

re u

sual

ly o

ccur

s be

fore

yie

ld.

Nylon 6,6

LDPE

70

20

40

6050

100

10

30

200

300400500600700

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

cer

amic

mat

rix a

nd e

poxy

mat

rix c

ompo

site

s, s

ince

in te

nsio

n, fr

actu

re u

sual

ly o

ccur

s be

fore

yie

ld.

HDPEPP

humid

dryPC

PET

¨

Stress-strain curve: engineering

UTS or Tensile strength

Ductile Brittle

Ductile

Brittle

Stress-strain curve: engineering

Stress-strain curve: engineering

- Beyond the yield stress, the materials show 1) strain hardening, 2) a maximum stress (UTS), and 3) (strain) softening after UTS

- During plastic flow, there is a substantial area reduction and the actual stress becomes larger; i.e., if there is another measurement utilizing real area, then this stress should be larger than the engineering stress

- The Engineering stress-strain curves include the geometric effect

- UTS is the point the necking initiates; that is why the yield stress is sometimes preferred to the UTS in designing structures with ductile metals

- Once necking occurs, the stress state is no more uniform throughout the specimen, while the stress state becomes really complex inside the necking zone

Tensile strength (UTS)

Si crystal<100>

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

Tens

ilest

reng

th, T

S(M

Pa)

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

PPPC 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 fibC fibers

Aramid fib

Based on data in Table B.4,Callister & Rethwisch 8e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & temperedAFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%fibers.

Room temperaturevalues

Tensile test: equipment

Adapted from Fig. 6.2,Callister & Rethwisch 8e.

gauge length

Tensile test: test specimens

And, more …

Tensile test

Cross head moving direction in the tension mode.Moving Speed is controllable.

Universal Testing Machine(UTM)

Load cellFore measuring sensor

Wedge action gripSpecimen fixture

Extensometer: Measuring the change of length at the gage region

Wedge action Grip

Tensile test: Digital Image Correlation

Digital Image Correlation(DIC)- Tracking technology by correlating two or more images- Measure the strain field of whole specimen- Equipped with camera system and analyzing software.

Camera system - One camera for 2D analysis.- Two or more cameras are needed for 3D analysis.- There must be some angle between cameras like human eyes

Analyzing software- Calculate physical position of cameras and

specimen- Tracking the position change and shape

change during the test mathematically.28

Tensile test: Digital Image Correlation

Mechanical Test with prepared specimen• Taking gray scale image with two or more cameras.• By comparing footages of different camera,

physical position can be calculated in 3-D

Analysis with software• Trace position of same or

resemble patternthrough different time step images

• Displacement (or strain) is calculated based on the tracking data

Tensile test: Digital Image Correlation

* Image by CY Kim

Appendix (부록)

Ref. Callister

• Plastic tensile strain at failure:

Ductility

• Another ductility measure: 100xA

AARA%

o

fo -=

x 100L

LLEL%

o

of −=

LfAo Af

Lo

Adapted from Fig. 6.13, Callister & Rethwisch 8e.

Engineering tensile strain, ε

E ngineering tensile stress, σ

smaller %EL

larger %EL

32

• 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

Adapted from Fig. 6.13, Callister & Rethwisch 8e.

very small toughness (unreinforced polymers)

Engineering tensile strain, ε

E ngineering tensile stress, σ

small toughness (ceramics)

large toughness (metals)

33

Elastic Strain Recovery (springback)

Adapted from Fig. 6.17, Callister & Rethwisch 8e.

Stre

ss

Strain

3. Reapplyload

2. Unload

D

Elastic strainRecovery ~ springback

1. Load

σyo

σyi

34

Hardness

• Resistance to permanently indenting the surface.• Large hardness means:

-- resistance to plastic deformation or cracking incompression.

-- 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

35

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 (psia) = 500 x HB• TS (MPa) = 3.45 x HB

36

Hardness: Measurement

Table 6.5

37

Hardening

• Curve fit to the stress-strain response:

σT = K εT( )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 σy due to plastic deformation.σ

ε

large hardening

small hardeningσy0

σy1

38

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

Ny

working

σ=σ

Often N isbetween1.2 and 4

• Example: Calculate a diameter, d, to ensure that yield doesnot occur in the 1045 carbon steel rod below. Use a factor of safety of 5.

Design or Safety Factors

220,000Nπ d2 / 4( )

5

Ny

working

σ=σ 1045 plain

carbon steel: σy = 310 MPa TS = 565 MPa

F = 220,000N

d

L o

d = 0.067 m = 6.7 cm39

Questions ?