445.204
Introduction to Mechanics of Materials
(재료역학개론)
Chapter 4: Introduction to mechanical
properties of solids
Myoung-Gyu Lee, 이명규
Tel. 880-1711; Email: [email protected]
TA: Chanmi Moon, 문찬미
Lab: Materials Mechanics lab.(Office: 30-521)
Email: [email protected]
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 ?