1. Mechanical Behavior: Part I Dr. Aaron L. Adams, Assistant Professor
Mechanical Engineering Department Alabama A&M University Fall 2014, Lecture 15

2. Learning Objectives…. Relevant Reading for this Lecture...
• Stress and strain: What are they and why are
they used instead of load and deformation?
• Elastic behavior: When loads are small, how much
deformation occurs? What materials deform least?
• Plastic behavior: At what point does permanent deformation occur? What materials are most resistant to permanent deformation?
• Toughness and ductility: What are they and how
do we measure them?
Relevant Reading for this Lecture...
Chapter 7, Pages

3. ALL MATERIALS AND STRUCTURES MUST WITHSTAND MECHANICAL LOADS
WE MUST UNDERSTAND HOW TO MANAGE THEM!

4. Why Study??? Need to know how a material will act under a given load.
Airplane wings, car axles, fork lift, etc.
Must design for given loads to prevent failure
Moore, OK 2013 F F

5. MECHANICAL BEHAVIOR Field that addresses how materials or structures respond to mechanical or thermal loads Two fundamental responses A material will either:
Deform 1
Fracture 2 2 5

6. When something separates into pieces.
What is deformation?
Shape Change
Elastic
Plastic
What is fracture?
When something separates into pieces.
TIME INDEPENDENT DEFORMATION
Elastic deformation is recovered immediately upon unloading. It is analogous to the stretching of atomic bonds. In the elastic regime, stress σ is usually linearly proportional to strain ε. Hooke’s law applies: σ = Eε where E is the modulus of elasticity.
Plastic deformation is permanent and is not recovered upon reloading. It commences at the proportional limit. At this point the material is said to yield. Yielding is characterized by the yield strength σo. After yielding, σ is not linearly proportional to ε. Other empirical relationships have been developed to describe the relationship between σ and ε.
Materials that can sustain lots of plastic deformation prior to failure are called “ductile.”
Materials that fail with little or no plastic deformation are called “brittle.”
TIME-DEPENDENT DEFORMATION
Creep: deformation that occurs with time. Usually occurs at high homologous temperatures (i.e., T/Tmp ≥ 0.4).
Ductile
Brittle 4 6

7. Important Terms
Stress, σ: Measure of an applied mechanical load, normalized to account for area Strain, ε: The amount of deformation induced by a stress Types of Stress/Strain: Engineering, True, Tensile, Compressive, Torsion, and Shear Modulus of Elasticity, E: The ratio of stress to strain when deformation is elastic Poisson’s Ratio, ν: Ratio of lateral and axial strains

8. What is stress? “STRESS” Noun Pressure exerted on a material object.
Refers to the distribution of applied forces.
A material will respond to applied stress.

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

10. ENGINEERING STRESS seng = Ft Ao Ft Fs F t = Ao • Tensile stress, s:
original cross-sectional area
before loading
seng = Ft Ao
Area, Ao
• Shear stress, t:
Area, Ao Ft Fs F t = Ao
Stress has units:
N/m2 = Pa (or lb/in2)
Useful relation: 145 psi = 1 MPa
CONSIDER NORMALIZED FORCE – WHY???
Stress = applied load (force) / cross-sectional area
Volume is conserved during deformation

11. COMMON STATES OF STRESS
• Simple tension: cable A o
= cross-sectional
area (when unloaded) F o
seng = F A
Distribution of force per unit area
Ski lift (photo courtesy P.M. Anderson)
seng

12. OTHER COMMON STRESS STATES (I)
• 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)

13. OTHER COMMON STRESS STATES (II)
Moore, OK 2013
Interior
Exterior
Force was applied perpendicular to the foundation resulting in the house being sheared from the foundation
High winds resulted in a torsional force being applied to the lamp post y θ
• Shear:
• Torsion:

14. Another definition for stress
TRUE STRESS
Engineering Stress = seng = force / original area = F/Ao o  F A
Deformation is considered to be a constant volume process
RECALL:
THEREFORE o  F Ao A'
Is the final area, the same as the initial area? NO!
Another definition for stress
True Stress = true = force / instantaneous area = F/A'
Volume is conserved during deformation

15. This implies distortion of an object! (aka shape change)
What is strain?
“STRAIN”
Noun
To pull or stretch something to an extreme or damaging degree. Occurs in response to a stress.
This implies distortion of an object! (aka shape change)

16. Engineering Strain L w e = d L - d e = w q g = Dx/y = tan
• Tensile strain:
• Lateral strain: d /2 L o w e = d L o - d e L = w o
To maintain constant volume d L /2
• Shear strain: q
90º-q y x
g = Dx/y = tan
90º
Strain is always
Dimensionless
BUT
It is often reported as % strain.
% strain is a unit. This affects calculations!
Adapted from Fig. 7.1 (a) and (c), Callister & Rethwisch 4e.

17. Stress-Strain Testing
• Typical tensile test machine
• Typical tensile specimen
Adapted from Fig. 7.2,
Callister & Rethwisch 4e.
gauge
length
specimen
extensometer
Adapted from Fig. 7.3, Callister & Rethwisch 4e. (Fig. 7.3 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.)

18. How? Conservation of volume
True Strain
TENSION o  F A
ENGINEERING STRAIN is just NORMALIZED EXTENSION
Engineering Strain = e = change in length/original length = /o
WHAT if we NORMALIZE BY the CURRENT LENGTH?
True Strain = t = change in length/instantaneous length = /
How? Conservation of volume
True strain is preferred for computational purposes
True strain is equivalent to engineering strain at low strains

19. True Stress & Strain True stress: True strain:
Deformation is a constant volume process.
Area changes as sample stretched.
True stress:
True strain:
Adapted from Fig. 7.16, Callister & Rethwisch 4e.

20. Example of Engineering vs. True Strain
Consider an metal bar with an original length of 10 cm. The bar is stretched to a final length of 20 cm in the following 3 steps:
10 cm to 12 cm
12 cm to 15 cm
15 cm to 20 cm
What is the engineering strain? What is the true strain?
(compare summation and integration)
Engineering
strain
True
strain

21. Common Features on Stress-Strain Curves
ELASTIC
PLASTIC
Ultimate tensile strength or
UTS
Elastic limit
Stress (Force/Area)
Elastic Modulus E
YIELD STRESS
Fracture stress, σf Δσ Δε
Strain
Total strain
(Elastic strain + Plastic strain)
Plastic strain
0.2% STRAIN, p=0.002
Elastic strain (at fracture) 8 21

22. Common Features on Stress-Strain Curves
ASSUMING PRE-YIELD DEFORMATION
(stress < YP)
Common Features on Stress-Strain Curves
ELASTIC
PLASTIC
Elastic limit
Stress (Force/Area)
Elastic Modulus E
If stress is below the elastic limit then only ELASTIC deformation will occur. Δσ Δε
Strain
Elastic strain 6

23. Common Features on Stress-Strain Curves
ASSUMING DEFORMATION
BEYOND THE YP BUT below the UTS
Common Features on Stress-Strain Curves
ELASTIC
PLASTIC
Elastic limit
Flow stress, f
Stress (Force/Area)
Elastic Modulus E
If stress is above the elastic limit then BOTH ELASTIC and PLASTIC deformation will occur.
YIELD STRESS Δσ Δε
Strain
Plastic deformation
Total strain
(Elastic strain + Plastic deformation)
0.2% STRAIN, p=0.002
Elastic strain recovered upon unload
Total strain, total 9

24. It is an ELASTIC mechanical property
Young’s Modulus, E
It is an ELASTIC mechanical property
Linear value (dependence) on the stress-strain (slope)
Material dependent value, why?
 = Ee
Material
Tmp (°C)
E (GPa)
G (GPa) ν Al
660 69 25
0.33 Cu
1085
110 46
0.34
Steel
207 83
0.30 W
3410
407
160
0.28
Al2O3
2020
393
---
0.22
Si3N4
1900
304
Glass
~1500
0.23
PVC
0.38
Nylon 6,6
269
0.39 E E
Measure of bond stretching.
For a given stress, Al elastically deforms 3X more than steel

25. Be careful – watch out for the %
A common mistake when solving for Young’s modulus is recognizing that strain and percent strain are not the same.
For example: What is Young’s modulus for a material that that as been loaded elastically to σ = 50 MPa at 0.1% strain (i.e., 0.001).
First, 0.1% is less than 0.2%, so it is safe to assume the material is in the linear (elastic) region
Ds = 50 MPa – 0 MPa
Ds = 50 MPa – 0 MPa s e
= 500 MPa
= 50,000 MPa
s = Ee → E=
De = – 0.0
De = 0.1 – 0.0
Correct! Be sure to not use % to get the correct E. Why its important to have a ‘ball park’ idea for these values – previously, you were off by a factor of 100!
Wrong! Strain does not use % but the true value

26. Elastic Modulus E=2G(1+)
If the specimen is deformed elastically in tension, we have measured the Young’s modulus, E
If the specimen is sheared, we have measured the shear modulus, G F
Tensile
E=2G(1+)
What is this term?

27. Poisson’s ratio is another elastic constant.
Since deformation is a constant volume process
loaded F x y z
unloaded
A material that is deformed axially, will contract laterally.
metals: n ~ 0.33 ceramics: n ~ 0.25 polymers: n ~ 0.40
Measure of the lateral contraction of a material during deformation relative to the axial expansion.
Poisson’s ratio is another elastic constant.

28. Plastic (Permanent) Deformation
RECALL
(at lower temperatures, i.e. T < Tmelt/3)
• Simple tension test:
Elastic+Plastic
at larger stress
engineering stress, s
Elastic
initially
permanent (plastic)
after load is removed ep
plastic strain
engineering strain, e
Adapted from Fig (a),
Callister & Rethwisch 4e.

29. 0.2% Offset Yield Stress
On a stress-strain curve, the onset of plastic deformation is difficult to measure accurately.
As a result, we define the yield stress as the stress at a strain offset of (i.e., the 0.2% offset strain).
Proportional Limit ???
0.002
ε (mm/mm, in/in, etc.)
σ (N/mm2, lbs/in2, etc.)
0.2% offset yield strength
Line parallel to the initial path of the σ-ε curve E σy
Just means that the proportional limit varies whereas the offset is more statistically consistent.

30. 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
200
300
400
500
600
700
1000
2000
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 temperature values
Based on data in Table B.4,
Callister & Rethwisch 4e.
a = annealed
hr = hot rolled
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered

31. Also called UTS (i.e., ultimate tensile strength)
Tensile Strength, TS
Also called UTS (i.e., ultimate tensile strength)
• 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. 7.11, Callister & Rethwisch 4e.
• Metals: occurs when noticeable necking starts.
• Polymers: occurs when polymer backbone chains are
aligned and about to break.

32. Tensile Strength: Comparison
(71500)
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 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 temperature values
Based on data in Table B4,
Callister & Rethwisch 4e.
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.

33. Ductility x 100 L EL % - = • Plastic tensile strain at failure: Lf Ao
Adapted from Fig. 7.13, Callister & Rethwisch 4e.
Engineering tensile strain, e E
ngineering
tensile
stress, s
smaller %EL
larger %EL Lf Ao Af Lo
• Another ductility measure:
100 x A RA % o f - =

34. Toughness • Energy required 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. 7.13, Callister & Rethwisch 4e.
Brittle fracture: elastic energy Ductile fracture: elastic + plastic energy

35. Summary Stress and strain are measures of strength and deformability.
Stress and strain depend on material and stress state.
Elastic properties are measures of resistance to bond distortion (stretching, bending, etc.).
Elastic modulus
Shear modulus
Poisson’s ratio
Know that stress and strain are related (s = E e).
Plastic deformation occurs beyond the proportional limit.
Permanent deformation

36. EXAM #2 COVERS LECTURES # 10 - 15 Things to Study for Exam #2
L10 - Chap 6 Fick’s Laws
What are Fick’s Laws of Diffusion? Can you define the terms and units? (Know how to apply to solve mathematical problems)
How can the rate of diffusion be predicted for some simple cases?
How does diffusion depend on structure and temperature?
L11 - Chap 6 Diffusion in Solids
How does diffusion occur?
How is activation energy calculated? (Know how to apply to solve mathematical problems)
How is diffusion used in processing of materials?
L12 - Chap 10 Phase Diagrams
What is a phase?
What is thermodynamic equilibrium?
What three components need to be used (established) to define the equilibrium?
How to determine the composition and fraction of a phase in a two-phase regime (Lever Rule).
L13 - Chap 10 Microstructural development
How can we use a phase diagram to predict microstructure?
What is the consequence of solidification in an alloy (coring)?
L14 - TTT Curves
Understand how time at temperature influences microstructure.
Understand the TTT diagram – what is nucleation vs. growth
In steels, if you cool to quickly you form martensite. Why? What is tempering.
L15 - Mechanical Props #1
What are the basic equations for eng. stress-strain and elasticity? (Know how to apply to solve mathematical problems) 36