Download Concepts of stress and strain

Chapter 6: Mechanical properties of metals
Outline
Introduction
Concepts of stress and strain
Elastic deformation
• Stress-strain behavior
• Elastic properties of materials
Plastic deformation
•
•
•
•
Yield and yield strength
Ductility
Resilience
Toughness
Concepts of stress and strain
Tension, compression, shear, and torsion
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Stress-strain testing
Tension tests
σ= F
• engineering stress
A0
• engineering strain
ε=
li − l 0
l0
=
Δl
l0
extensometer
specimen
Gauge length
Compression tests
Shear stress
Shear and torsional tests
• Shear stress
τ = F = Gγ
A0
• Shear strain
γ = Δx/y = tan θ
Δx
90º - θ
y
90º
Geometric considerations of the stress state
2
Elastic deformation
1. Initial
2. Small load
3. Unload
bonds
stretch
return to
initial
δ
F
F
Linearelastic
Elastic means reversible!
Non-Linearelastic
δ
Linear elastic properties
Modulus of Elasticity, E:
(also known as Young's modulus)
Hooke's Law:
σ=Eε
σ
F
E
ε
Linearelastic
F
simple
tension
test
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3
Stress-strain behavior
Stress-strain for linear elastic deformation
Stress-strain for non-linear elastic deformation
Stress-strain behavior(continue)
Modulus of elasticity E is proportional to (dF/dr)r
Slope of stress strain plot (which is proportional to the elastic
modulus) depends on bond strength of metal
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Other elastic properties
M
τ
Elastic Shear
modulus, G:
G
simple
torsion
test
γ
τ=Gγ
M
Elastic Bulk
modulus, K:
P=- K
P
P
ΔV
ΔV
Vo
P
Vo
K
P
pressure
test: Init.
vol =Vo.
Vol chg.
= ΔV
Special relations for isotropic materials:
E
E
G =
K =
2(1 + ν)
3(1 − 2ν)
Comparison of Young’s moduli
1200
1000
800
600
400
E(GPa)
200
100
80
60
40
Diamond
Tungsten
Molybdenum
Steel, Ni
Tantalum
Platinum
Cu alloys
Zinc, Ti
Silver, Gold
Aluminum
Magnesium,
Tin
Si carbide
Al oxide
Si nitride
Carbon fibers only
CFRE(|| fibers)*
<111>
Si crystal
Aramid fibers only
<100>
A FRE(|| fibers)*
Glass -soda
Glass fibers only
GFRE(|| fibers)*
Concrete
GFRE*
20
10
8
6
4
2
1
0.8
0.6
0.4
0.2
CFRE*
GFRE( fibers)*
Graphite
Polyester
PET
PS
PC
CFRE( fibers) *
AFRE( fibers) *
Epoxy only
PP
HDP E
PTFE
Wood(
grain)
LDPE
5
Comparison of yield strength
Metals/
Alloys
200
Ti (5Al-2.5Sn)a
W (pure)
Cu (71500)cw
Mo (pure)
Steel (4140)a
Steel (1020)cd
Al (6061)ag
Steel (1020)hr
Ti (pure)a
Ta (pure)
Cu (71500)hr
100
70
60
50
40
30
Al (6061)a
20
10
Tin (pure)
¨
dry
PC
Nylon 6,6
PET
PVC humid
PP
HDPE
Hard to measure,
700
600
500
400
300
in ceramic matrix and epoxy matrix composites, since
in tension, fracture usually occurs before yield.
Yield strength, σy (MPa)
1000
Steel (4140)qt
Hard to measure,
since in tension, fracture usually occurs before yield.
2000
Graphite/
Ceramics/ Polymers Composites/
fibers
Semicond
LDPE
Elastic properties of materials
Poisson’s ratio
ν =−
ε
εx
=− y
εz
εz
metals: ν ~ 0.33
ceramics: ν ~ 0.25
polymers: ν ~ 0.40
Units:
E: [GPa] or [psi]
ν: dimensionless
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Plastic Deformation (Metals)
1. Initial
2. Small load
bonds
stretch
& planes
shear
3. Unload
planes
still
sheared
δplastic
δelastic + plastic
F
F
Plastic means permanent!
linear
elastic
linear
elastic
δplastic
δ
Plastic deformation: yield and yield strength
Yielding
Proportional limit
Yield strength
7
Tensile strength
TS
σy
engineering
stress
Tensile strength: the stress at the
maximum on the engineering
stress-strain curve
Metals: occurs when noticeable
“necking” starts
Ceramics: occurs when crack
propagation starts
Polymers: occurs when polymer
backbones are aligned and about
to break
Typical response of a metal
engineering strain
Plastic (permanent) deformation
(At lower temperature: T< Tmelt/3)
Simple tension test:
Elastic+Plastic
at larger stress
engineering stress, σ
Elastic
initially
permanent (plastic)
after load is removed
εp
engineering strain, ε
plastic strain
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Elastic and plastic deformations
Stress-strain relations under uniaxial loading
Ceramics
Polymeric material
Ductility
Ductility: a measure of the degree of plastic
deformation that has been sustained at fracture
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Mechanic properties of typical metals
Resilience
Resilience: is the capacity of a material to absorb energy when it is
deformed elastically and then, upon unloading, to have this energy
recovered.
Resilience
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Toughness
Energy to break a unit volume of material
Approximate by the area under the stress-strain curve
Engineering
tensile
stress, σ
small toughness (ceramics)
large toughness (metals)
very small toughness
(unreinforced polymers)
Adapted from Fig. 6.13,
Callister 7e.
Engineering tensile strain,
ε
Brittle fracture: elastic energy
Ductile fracture: elastic + plastic energy
Summary
• 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).
• Plastic behavior: This permanent deformation
behavior occurs when the tensile (or compressive)
uniaxial stress reaches σy.
• Toughness: The energy needed to break a unit
volume of material.
• Ductility: The plastic strain at failure.
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