1. 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

2. Elastic Deformation d F F d 1. Initial 2. Small load 3. Unload
bonds
stretch
return to
initial F d
Linear-
elastic
Non-Linear-
Elastic means reversible!

3. Plastic Deformation (Metals)
1. Initial
2. Small load
3. Unload p
lanes
still
sheared F d
elastic + plastic
bonds
stretch
& planes
shear
plastic F d
linear
elastic
plastic
Plastic means permanent!

4. Comparison of Units: SI and Engineering Common
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
N*4.448 = lbf
Lbf* = N
Area I
mm2* = 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 * = Pascal
Stress I - b
Pascal * 1.450x10-7 = Ksi
Ksi * x106 = Pascal
Stress II - a
MPa * = psi
psi * 6.89x 10-3 = MPa
Stress II - b
MPa * x 10-1= Ksi
Ksi * 6.89 = MPa
One other conversion: 1 GPa = 103 MPa

5. we can also see the symbol ‘s’ used for engineering stress
• Tensile stress, s:
original area
before loading
Area, A F t s = A o 2 f m N or in lb
• Shear stress, t:
Area, A F t s = A o
 Stress has units: N/m2 (Mpa) or lbf/in2
we can also see the symbol ‘s’ used for engineering stress

6. 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.

7. Common States of Stress
• Simple tension: cable A o
= cross sectional
area (when unloaded) F o s = F A s
Ski lift (photo courtesy P.M. Anderson)
• Torsion (a form of shear): drive shaft M A o 2R F s c o t = F s A
Note: t = M/AcR here.
Where M is the “Moment” Ac shaft area & R shaft radius

8. OTHER COMMON STRESS STATES (1)
• Simple compression: A o
Balanced Rock, Arches
National Park
(photo courtesy P.M. Anderson)
Canyon Bridge, Los Alamos, NM o s = F A
Note: compressive
structure member
(s < 0 here).
(photo courtesy P.M. Anderson)

9. OTHER COMMON STRESS STATES (2)
• Bi-axial tension:
• Hydrostatic compression:
Fish under water
Pressurized tank
(photo courtesy
P.M. Anderson)
(photo courtesy
P.M. Anderson) s z
> 0 q
s < 0 h

10. Engineering Strain: w e = d L - d e = w q q g = Dx/y = tan
• Tensile strain: d /2 L o w
• Lateral strain: e = d L o - d e L = w o
Here: The Black Outline is Original, Green is after application of load
• Shear strain:
Adapted from Fig. 6.1 (a) and (c), Callister 7e. q
Strain is always
Dimensionless! x q
g = Dx/y = tan y
90º - q
90º
We often see the symbol ‘e’ used for engineering strain

11. Stress-Strain: Testing Uses Standardized methods developed by ASTM for Tensile Tests it is ASTM E8
• Typical tensile test machine
• Typical tensile specimen (ASTM A-bar)
Adapted from Fig. 6.2,
Callister 7e.
gauge
length
specimen
extensometer
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.)

12. LOAD vs. EXTENSION PLOTS
During Tensile Testing, Instantaneous load and displacement is measured
LOAD vs. EXTENSION PLOTS
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.
The Force .vs. Displacement plot will be the same shape as the
Eng. Stress vs. Eng. Strain plot

13. The Engineering Stress - Strain curve
Divided into 2 regions
ELASTIC
PLASTIC

14. Linear: Elastic Properties
Units:
E: [GPa] or [psi]
: in [Mpa] or [psi]
: [m/m or mm/mm] or [in/in]
• Modulus of Elasticity, E:
(also known as Young's modulus)
• Hooke's Law:
s = E e F A o d /2 L Lo w s
Linear-
elastic E e
Here: The Black Outline is Original, Green is after application of load

15. 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 N
Experiences elongation of: 0.43 mm
Determine Young’s Modulus if all the deformation is recoverable

16. Solving:

17. Poisson's ratio, n eL e - n e n = - • Poisson's ratio, n: Units:
metals: n ~ 0.33 ceramics: n ~ 0.25 polymers: n ~ 0.40
Units:
n: dimensionless
 > density increases
 < density decreases (voids form)

18. Other Elastic Properties
simple
torsion
test M t G g
• Elastic Shear
modulus, G:
t = G g
• Elastic Bulk
modulus, K:
pressure
test: Init.
vol =Vo.
Vol chg.
= DV P
P = - K D V o
• Special relations for isotropic materials:
2(1 + n) E G =
3(1 - 2n) K
E is Modulus of Elasticity
 is Poisson’s Ratio

19. Looking at Aluminum and the earlier problem:

20. Young’s Moduli: Comparison
Graphite
Ceramics
Semicond
Metals
Alloys
Composites
/fibers
Polymers
0.2 8
0.6 1
Magnesium,
Aluminum
Platinum
Silver, Gold
Tantalum
Zinc, Ti
Steel, Ni
Molybdenum G
raphite
Si crystal
Glass -
soda
Concrete
Si nitride
Al oxide PC
Wood( grain)
AFRE( fibers) *
CFRE
GFRE*
Glass fibers only
Carbon
fibers only A
ramid fibers only
Epoxy only
0.4
0.8 2 4 6 10 00
1200
Tin
Cu alloys
Tungsten
<100>
<111>
Si carbide
Diamond
PTF E
HDP
LDPE PP
Polyester PS
PET C
FRE( fibers)
FRE( fibers)*
FRE(|| fibers)*
E(GPa)
Based on data in Table B2,
Callister 7e.
Composite data based on
reinforced epoxy with 60 vol%
of aligned
carbon (CFRE),
aramid (AFRE), or
glass (GFRE)
fibers.
109 Pa

21. Useful Linear Elastic Relationships
• Simple tension:
• Simple torsion: a = 2 ML o p r 4 G
M = moment
= angle of twist
2ro Lo d = FL o E A d L = - n Fw o E A F A o d /2 L Lo w
• Material, geometric, and loading parameters all
contribute to deflection.
• Larger elastic moduli minimize elastic deflection.

23. 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 y r 2 1 U e s @
Adapted from Fig. 6.15, Callister 7e.

24. Plastic (Permanent) Deformation
(at lower temperatures, i.e. T < Tmelt/3)
• Simple tension test:
Elastic+Plastic
engineering stress, s
at larger stress
Elastic
initially
permanent (plastic)
after load is removed ep
engineering strain, e
plastic strain
Adapted from Fig (a),
Callister 7e.

25. Yield Strength, sy y = yield strength sy
• Stress at which noticeable plastic deformation has
occurred.
when ep  0.002
tensile stress, s
engineering strain, e sy p
= 0.002
y = yield strength
Note: for 2 inch sample
 = = z/z
 z = in
Adapted from Fig (a),
Callister 7e.

26. Tensile properties Yielding Strength Some steels (lo-C) Proof stress
Most structures operate in elastic region, therefore need to know when it ends
Yield strength
-- Generally quoted
Proportional Limit
Some steels (lo-C)
Proof stress

27. Yield Strength : Comparison
Graphite/
Ceramics/
Semicond
Metals/
Alloys
Composites/
fibers
Polymers
Yield strength, s y
(MPa)
PVC
Hard to measure ,
since in tension, fracture usually occurs before yield.
Nylon 6,6
LDPE 70 20 40 60 50
100 10 30 2 00 3 4 5 6 7
Tin (pure) Al
(6061) a ag Cu
(71500) hr Ta
(pure) Ti
Steel
(1020) cd
(4140) qt
(5Al-2.5Sn) W
Mo (pure) cw
Hard to measure,
in ceramic matrix and epoxy matrix composites, since
in tension, fracture usually occurs before yield. H
DPE PP
humid
dry PC
PET
Room Temp. values
Based on data in Table B4,
Callister 7e.
a = annealed
hr = hot rolled
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered

28. Tensile Strength, TS TS engineering stress strain engineering strain
• TS is Maximum stress on engineering stress-strain curve. y
strain
Typical response of a metal
F = fracture or
ultimate
strength
Neck – acts as stress concentrator
engineering TS
stress
engineering strain
Adapted from Fig. 6.11, Callister 7e.
• Metals: occurs when noticeable necking starts.
• Polymers: occurs when polymer backbone chains are
aligned and about to break.

29. Tensile Strength : Comparison
Si crystal
<100>
Graphite/
Ceramics/
Semicond
Metals/
Alloys
Composites/
fibers
Polymers
Tensile
strength, TS
(MPa)
PVC
Nylon 6,6 10
100
200
300
1000 Al
(6061) a ag Cu
(71500) hr Ta
(pure) Ti
Steel
(1020)
(4140) qt
(5Al-2.5Sn) W cw L
DPE PP PC
PET 20 30 40
2000
3000
5000
Graphite
Al oxide
Concrete
Diamond
Glass-soda
Si nitride H
wood
( fiber)
wood(|| fiber) 1
GFRE
(|| fiber)
( fiber) C
FRE A
FRE( fiber)
E-glass fib
Aramid
fib
Room Temp. values
Based on data in Table B4,
Callister 7e.
a = annealed
hr = hot rolled
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered
AFRE, GFRE, & CFRE =
aramid, glass, & carbon
fiber-reinforced epoxy
composites, with 60 vol%
fibers.

30. Ductility x 100 L EL % - = • Plastic tensile strain at failure: e s Lfo f - =
• Plastic tensile strain at failure:
Engineering tensile strain, e E
ngineering
tensile
stress, s
smaller %EL
larger %EL Lf Ao Af Lo
Adapted from Fig. 6.13, Callister 7e.
• Another ductility measure:
100 x A RA % o f - =

31. Lets Try one (like Problem 6.29)
Load (N)
len. (mm)
len. (m) l
50.8
0.0508
12700
50.825
2.5E-05
25400
50.851
5.1E-05
38100
50.876
7.6E-05
50800
50.902
76200
50.952
89100
51.003
92700
51.054
102500
51.181
107800
51.308
119400
51.562
128300
51.816
149700
52.832
159000
53.848
160400
54.356
159500
54.864
151500
55.88
124700
56.642
GIVENS:

32. Leads to the following computed Stress/Strains:
e stress (Pa)
e str (MPa)
e. strain
0.005
0.0075
0.01
0.015
0.02
0.04
0.06
0.07
0.08
0.1
0.115

33. Leads to the Eng. Stress/Strain Curve:
T. Str.  1245 MPa
F. Str  970 MPa
Y. Str.  742 MPa
%el  11.5%
E  195 GPa (by regression)

34. TOUGHNESS
Is a measure of the ability of a material to absorb energy up to fracture
High toughness =
High yield strength and ductility
Important Factors in determining Toughness:
1. Specimen Geometry & 2. Method of load application
Dynamic (high strain rate) loading condition (Impact test)
1. Specimen with notch- Notch toughness
2. Specimen with crack- Fracture toughness
Static (low strain rate) loading condition (tensile stress-strain test)
1. Area under stress vs strain curve up to the point of fracture.

35. Toughness • Energy to break a unit volume of material
• Approximate by the area under the stress-strain
curve.
very small toughness
(unreinforced polymers)
Engineering tensile strain, e E
ngineering
tensile
stress, s
small toughness (ceramics)
large toughness (metals)
Adapted from Fig. 6.13, Callister 7e.
Brittle fracture: elastic energy Ductile fracture: elastic + plastic energy

36. Elastic Strain Recovery – After Plastic Deformation
It is an important factor in forming products (especially sheet metal and spring making)
Adapted from Fig. 6.17, Callister 7e.

37. True Stress & Strain
Note: Stressed Area changes when sample is deformed (stretched)
True stress
True Strain
Adapted from Fig. 6.16, Callister 7e.

38. Strain Hardening
• An increase in sy due to continuing plastic deformation. s e
large Strain hardening
small Strain hardening y 1
• Curve fit to the stress-strain response: s T = K e ( ) n
“true” stress (F/Ai)
“true” strain: ln(Li /Lo)
hardening exponent:
n =
0.15 (some steels)
0.5 (some coppers)

39. Earlier Example: True Properties
T. Stress
T. Strain
Necking Began

40. We Compute:
T = KTn 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

41. Stress Strain Plot w/ True Values
K  2617 MPa

42. 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
All samples have same value
Because of large variability must have safety margin in engineering specifications
where n is the number of data points (tests) that are performed

43. Design or Safety Factors
• Design uncertainties mean we do not push the limit!
• Typically as engineering designers, we introduce a Factor of safety, N
Often N is
between
1.5 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.
1045 plain
carbon steel: s y
= 310 MPa
TS = 565 MPa
F = 220,000N d L o

44. Solving:

45. 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 d D
Smaller indents
mean larger
hardness.
increasing hardness
most
plastics
brasses
Al alloys
easy to machine
steels
file hard
cutting
tools
nitrided
diamond

46. Hardness: Common Measurement Systems
Callister Table 6.5

47. Comparing Hardness Scales:

48. 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.

49. Correlation Between Hardness and Tensile Strength
Both measures the resistance to plastic deformation of a material.

50. HB = Brinell Hardness
TS (psia) = 500 x HB
TS (MPa) = 3.45 x HB

51. 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 sy.
• Toughness: The energy needed to break a unit
volume of material.
• Ductility: The plastic strain at failure.