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
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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. CommonForce 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
= FsAo
• Tensile stress, :
original area before loading
Area, A
Ft
Ft
= FtAo
2f
2mNor
inlb=
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 shaft Ski lift (photo courtesy P.M. Anderson)
(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM
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]
(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 0.00344125Because 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
PP P
P = -K VVo
PV
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
26.871.2
3 1 2 3 1 2 0.33
69.8
HB
HB
GPaEG
G GPaGPaEK
K GPa
MetalsAlloys
GraphiteCeramicsSemicond
Polymers Composites/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
46
10
20
406080
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:
FLoEAo
L
Fw oEAo
• 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.
yyr 21U
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
Variability in Material PropertiesCritical 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 21
2
1
n
xxs i
n
nxx 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
3105
220000
4
310220000
54
220000 4 5 m310 104.52 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.