### Document

```Universit
à degli Studi di Bari
Università
MECHANICS
3
prof. ing. Livio Quagliarella
The body
The body is a system modelled as:
point
Kinetics study
rigid body
Static study
deformable body
Stress strain description
Body mass
2/45
Introduction
• Knowledge of a material’s properties, and how it behaves
–Materials are selected for specific applications dependent on their
properties and characteristics
• The behaviour of a material under load is markedly different
depending on whether the material response is elastic or
plastic
Rheology
Different materials deform differently under the same state of stress. The
material response to a stress is known as rheology.
Ideal materials fall into one of the following categories:
o Elasticity.
o Viscosity.
o Plasticity.
Mechanical analogues
Stress-Strain relations
σ = Cε
Elastic
σs = η
Viscous
dεs
dt
slip = 0 if
σ s < μσ n
Plastic
The behavior of real materials is better described by combining
simple models in series or parallel. For example:
A visco-elastic
(or Maxwell) solid:
dashpot
A visco-plastic
(or Bingham) material:
An elasto-plastic
(Prandtl) material:
spring
Elasticity
The one-dimensional stress-strain relationship may be written as:
σ = Cε n
where C is an elastic constant.
Note that:
o The response is instantaneous.
o Here the strain is the infinitesimal strain.
The material is said to be linear elastic if n=1.
Hooke’s law:
σ = Cε
In three dimensions, Hooke’s law is written as:
σ ij = Cijkl ε kl
where Cijkl is a matrix whose entries are the stiffness coefficients.
It thus seems that one needs 81(!) constants
in order to describe the stress strain
relations.
σ 11 = C1111ε11 + C1112ε 12 + C1113ε 13 + C1121ε 21 + C1122ε 22 + C1123ε 23 + C1131ε 31 + C1132ε 32 + C1133ε 33
σ 12 = C1211ε 11 + C1212ε 12 + C1213ε13 + C1221ε 21 + C1222ε 22 + C1223ε 23 + C1231ε 31 + C1232ε 32 + C1233ε 33
σ 13 = C1311ε 11 + C1312ε 12 + C1313ε13 + C1321ε 21 + C1322ε 22 + C1323ε 23 + C1331ε 31 + C1332ε 32 + C1333ε 33
σ 21 = C2111ε11 + C2112ε 12 + C2113ε13 + C2121ε 21 + C2122ε 22 + C2123ε 23 + C2131ε 31 + C2132ε 32 + C2133ε 33
σ 22 = C2211ε 11 + C2212ε 12 + C2213ε 13 + C2221ε 21 + C2222ε 22 + C2223ε 23 + C2231ε 31 + C2232ε 32 + C2233ε 33
σ 23 = C2311ε11 + C2312ε 12 + C2313ε 13 + C2321ε 21 + C2322ε 22 + C2323ε 23 + C2331ε 31 + C2332ε 32 + C2333ε 33
σ 31 = C3111ε 11 + C3112ε12 + C3113ε 13 + C3121ε 21 + C3122ε 22 + C3123ε 23 + C3131ε 31 + C3132ε 32 + C3133ε 33
σ 32 = C3211ε11 + C3212ε 12 + C3213ε13 + C3221ε 21 + C3222ε 22 + C3223ε 23 + C3231ε 31 + C3232ε 32 + C3233ε 33
σ 33 = C3311ε11 + C3312ε12 + C3313ε13 + C3321ε 21 + C3322ε 22 + C3323ε 23 + C3331ε 31 + C3332ε 32 + C3333ε 33
The case of isotropic materials
o A material is said to be isotropic if its properties are independent of
direction.
o In that case, the number of non-zero stiffnesses (or compliances) is
reduced to 12, all are a function of only 2 elastic constants.
Young modulus:
E=
σy
εy
The case of isotropic materials
Poisson’s ratio:
ν =−
εx
εy
Poisson’s ratio of incompressible isotropic materials equals 0.5;
real materials are compressible and their Poisson ratio is less than 0.5.
Elasticity
• For the tensile bar, the external load has been assumed to be
low enough that the bar will resume its initial shape once the
• This state of elastic
σ
deformation is possible
only when the external
Elastic limit
limits
• In the elastic range, the
stress-strain curve is linear
ε
Plasticity
• If the loading is increased, it will reach a certain limit whereby
elastic deformation would end and plastic deformation would
start
A yield strength or yield point is the
material property defined as the
• This limit is known as the
stress
at which a material begins to
elastic limit and beyond this σ
deform plastically.
point the material is said to
have yielded
σy Elastic limit
the elastic limit
–Permanent or irreversible
deformation
• Stress corresponding to
yielding is called the yield
strength, denoted by σy
E
Point of
yielding
ε
Elastic vs. Plastic Behavior
• If the strain disappears when
the stress is removed, the
material is said to behave
elastically.
• The largest stress for which
this occurs is called the
elastic limit.
• When the strain does not
stress is removed, the
material is said to behave
plastically.
2 - 14
Elastic-Plastic Stress-Strain Curves
Ultimate tensile strength (UTS), often shortened to tensile strength
(TS) or ultimate strength is the maximum stress that a material can
withstand while being stretched or pulled before breaking.
E = tan α =
α
σ
ε
True Stress-Strain Curve
tensione di rottura
curva nominale
G=
tensione di rottura
tensione
τ
G=
di snervamento
γ
Limite di snervamento
G=
E
2(1 + ν )
τ
γ
curva reale
G=
limite elastico
τ
G=
γ
limite di elasticità
lineare
Modulo di Young
E = tan α =
α
E
2(1 + ν )
σ
ε
16/62
STRESS-STRAIN TESTING
• Typical tensile specimen
• Typical tensile
test machine
Callister 6e.
specimen
extensometer
gauge (portion of sample with
=
length reduced cross section)
• Other types of tests:
--compression: brittle
materials (e.g., concrete)
--torsion: cylindrical tubes,
shafts.
Stress-Strain Test
2 - 18
G=
G=
τ
γ
Stress-Strain Diagram:
Ductile Materials
E
2(1 + ν )
α
E = tan α =
2 - 19
σ
ε
Stress-Strain Diagram: Brittle Materials
2 - 20
YOUNG’S MODULI: COMPARISON
Metals
Alloys
1200
1000
800
600
400
E(GPa)
200
100
80
60
40
109 Pa
Graphite
Composites
Ceramics Polymers
/fibers
Semicond
Diamond
Si carbide
Tungsten
Al oxide
Molybdenum Si nitride
Steel, Ni
<111>
Tantalum
Si crystal
Platinum
<100>
Cu alloys
Zinc, Ti
Silver, Gold Glass-soda
Aluminum
Carbon fibers only
CFRE(|| fibers)*
Aramid fibers only
AFRE(|| fibers)*
Glass fibers only
Magnesium,
Tin
GFRE(|| fibers)*
Concrete
GFRE*
20
10
8
6
4
2
1
0.8
0.6
0.4
0.2
Eceramics
> Emetals
>> Epolymers
CFRE*
GFRE( fibers)*
Graphite
Polyester
PET
PS
PC
CFRE( fibers)*
AFRE( fibers)*
Epoxy only
PP
HDPE
PTFE
LDPE
Wood(
grain)
13
TENSILE STRENGTH: COMPARISON
Metals/
Alloys
Tensile strength, TS (MPa)
5000
3000
2000
1000
300
200
100
40
30
20
Graphite/
Ceramics/ Polymers
Semicond
Composites/
fibers
C fibers
Aramid fib
E-glass fib
Steel (4140)qt
Diamond
W (pure)
Ti (5Al-2.5Sn)a
Steel (4140)a
Si nitride
Cu (71500)cw
Cu (71500)hr Al oxide
Steel (1020)
Al (6061)ag
Ti (pure)a
Ta (pure)
Al (6061)a
Si crystal
<100>
Glass-soda
Concrete
Graphite
AFRE(|| fiber)
GFRE(|| fiber)
CFRE(|| fiber)
TS(ceram)
~TS(met)
~ TS(comp)
>> TS(poly)
Room T values
Nylon 6,6
PC PET
PVC
PP
HDPE
Based on data in Table B4,
Callister 6e.
fiber)
a = annealed
fiber)
hr
= hot rolled
fiber)
ag = aged
cd = cold drawn
cw = cold worked
qt = quenched & tempered
AFRE, GFRE, & CFRE =
aramid, glass, & carbon
fiber-reinforced epoxy
fiber)
composites, with 60 vol%
fibers.
wood(|| fiber)
GFRE(
CFRE(
AFRE(
LDPE
10
wood(
19
1
Viscoelasticity
Some features that are observed in polymeric materials that
do not seem to be noticeable in metals or ceramics
1. Mechanical properties depend on Temperature
2. Mechanical properties depend on Strain Rate
3. Creep (noticed in metals at high temperatures)
4. Stress Relaxation
5. Hysteresis
Comportamento viscoso
La viscosità (μ) quantifica la resistenza dei fluidi allo
scorrimento e dipende dal tipo di fluido e dalla temperatura; la
viscosità si misura in Pascal per secondo, (Pa·s).
Vi sono materiali solidi che sia pure in maniera assai minore
rispetto ai fluidi mostrano fenomeni di scorrimento detto
appunto scorrimento viscoso cui si associa il comportamento
elastico, o plastico o elasto-plastico.
Nei materiali a comportamento viscoso la deformazione
dipende dal valore del carico, dalla sua velocità di
applicazione, dalla temperatura e dal tempo di applicazione.
Non esiste un valore di soglia; il materiale scorre fino a quando
il carico è applicato.
24/62
viscous behavior
Alcune proprietà dei materiali viscoelastici sono :
• scorrimento viscoso" (o creep), in presenza di un carico applicato
costante, la deformazione cresce con il tempo;
• rilassamento se si mantiene costante la deformazione, lo sforzo
decresce con il tempo;
• la rigidità del materiale dipende dalla velocità di applicazione del
carico;
• in presenza di un carico ciclico, si verifica il fenomeno dell’isteresi
(un ritardo periodico), con conseguente dissipazione (sotto forma di
calore) di energia meccanica;
Dipendono dalla temperatura ed iniziano a manifestarsi per:
1
T > Tf
3
T f è la temperatura di fusione
dove Tf
25/62
Creep
• Take a tension specimen made from a polymer and and
put on a series of constant stresses on it.
• We observe
Creep: Progressive strain (deformation) over time at
constant stress (load), usually at high temperatures
General behavior: the creep curve
•
Three creep regimes are obtained:
I. Primary or transient creep
(decreasing strain rate)
II. Secondary or
(constant strain rate)
III. Tertiary or
accelerated creep
(increase strain rate)
Stress Relaxation
• Think of a polymer specimen loaded with a constant
strain.
| Note that both linear elastic
and viscous fluid behaviors
are present.
| Note that there seems to be
some residual stress at the
end, i.e. the material does
release of
tension
not completely recover.
There is both elasticity and
plasticity.
Stress Relaxation: Progressive loss of stress (load) over
time under constant strain (deformation), usually at high
temperatures
Curva sforzo-deformazione
solido viscoso
Al variare della temperatura si ha
un’altra famiglia di curve
29/62
Hysteresis
• Polymers often don’t load and unload on the same line on
the stress-strain curve.
• The difference in areas under those curves represents
energy loss (often to heat).
• This means that polymers can have inherent energy
damping.
• This means plastic springs may not be as good an idea as
plastic dampers.
Hysteresis
viscoelastoplastico
viscoelastico
•
•
•
•
Very complex behavior!
Difficult to model.
Great sensitivity to temperature.
Great sensitivity to strain rate.
31/62
stiffness
STRUCTURE
PROPERTIES
STIFFNESS
DEPENDS ON :
• MATERIAL
• SHAPE AND DIMENTIONS
• CONSTRAINTS TYPE AND DISTRIBUTION
32/62
osso
Bending stifness
33/62
osso
torsion stifness
34/62
Stress concentration: Hole
Discontinuities of cross section may result
in high localized or concentrated stresses.
K=
σ max
σ ave
2 - 35
Stress concentration: Hole
σ0
ρt
a
x
x’
2a
σ0
Per il materiali duttili, capaci di considerevole deformazione plastica,
la presenza di fori, intagli ecc. non causa concentrazione degli sforzi
tali da portare a rottura, perché ogni concentrazione degli sforzi viene
limitata dalla deformazione plastica.
36/62
Stress concentration: Fillet
chamfer
2 - 37
Fatigue Failure
It has been recognized that a metal subjected
to a repetitive or fluctuating stress will fail at a
stress much lower than that required to cause
failure on a single application of load. Failures
are called fatigue failures.
Fatigue failure is characterized by three stages
Crack Initiation
Crack Propagation
Final Fracture
Jack hammer component,
shows no yielding before
fracture.
Crack initiation site
Fracture zone
Propagation zone, striation
Crank shaft
Gear tooth failure
Fatica
Hip stem prosthesis
41/62
Sollecitazioni dinamiche - Fatica
500x
42/62
1000x
Fatigue Failure – Type of Fluctuating Stresses
Alternating stress
σa =
most dangerous
σmax
σmin
2
Mean stress
σm =
σmax + σmin
2
Sollecitazioni dinamiche - Fatica
breaking
yield
Endurance limit
2
1
La rottura per fatica avviene in modo fragile anche per materiali
duttili. Le prove di fatica vengono condotte a vari valori di tensione
e viene registrato il numero di cicli per causare la rottura. Per il
dimensionamento degli organi meccanici si usano diagrammi
ottenuti da tali prove in cui in ascissa è il numero di cicli e in
ordinata la tensione massima.
44/62
Fatigue
• Fatigue properties are shown
on S-N diagrams.
• A member may fail due to
fatigue at stress levels
significantly below the ultimate
strength if subjected to many
• When the stress is reduced
below the endurance limit,
fatigue failures do not occur
for any number of cycles.
2 - 45
improve the fatigue strength
Per migliorare la resistenza a fatica è possibile:
• indurre uno stato di compressione sulla superficie; ad esempio
eseguire lavorazioni di pallinatura;
• effettuare trattamenti di indurimento superficiale; in qualche caso,
i rivestimenti duri (ad es. l’anodizzazione dura), possono ridurre
la vita a fatica, anche del 40-50%;
• migliorare la finitura superficiale del pezzo;
• ottimizzare la geometria: aumentando i raggi di raccordo, evitando
intagli e spigoli interni;
• utilizzare materiali appropriati.
46/62
Friction and Wear
No relative movement
Quando due corpi sono premuti
l’uno contro l’altro nascono delle
pressioni di contatto e uno stato
di tensione
quando i due corpi strisciano
tra loro, a causa delle
microasperità si ha l’usura
relative movement
47/62
Tribology
(dal greco tribos: strofinare o scorrere)
Tribology comprises the science and technology of interacting
surfaces in relative motion; that is, friction, lubrication and wear.
Tribology is a vast and interdisciplinary subject, ranging from the
fundamental physics of surface contact and adhesion to the
application of advanced materials and lubricants to solve practical
industrial friction and wear problems.
What is Friction
• Force tangential to the
interface of two
contacting bodies = Ff.
–Dynamic and static
–Dynamic produces heat
Ff = μ N
Friction Force
Normal Force
Assumptions:
Ff independent of contact area, μ = constant
Real Contact Area
Schematic illustration of the interface of two bodies in contact showing real
areas of contact at the asperities. In engineering surfaces, the ratio of the
apparent-to-real areas of contact can be as high as
4 to 5 orders of magnitude.
More Complicated Models Exist
• Contact Mechanics
In actuality, as N increases, contact
area increases, thereby affecting μ.
μ is a non-linear function of N. What
else might μ vary with??
Definition of Surface Wear
• Wear - Damage to a solid surface involving progressive
loss of material due to contact and relative motion with
another surface. 13 types of wear!!
• Erosion – Damage to a solid surface involving
progressive loss of material due to mechanical interaction
between that surface and a fluid, impinging liquid or solid
particles. 5 kinds of erosion
Types of Friction
• Kinetic friction occurs when force is applied to an
object and the object moves.
• Examples:
Sliding Friction: pushing an object across a surface
Rolling Friction: between wheels and a surface
Fluid Friction: opposes the motion of objects traveling through a fluid
(air or water)
Affecting Friction
• To reduce the amount of friction, apply a lubricant between two
surfaces.
• Motor oil, wax, and grease are examples.
• Friction can also be reduced by rolling, rather than pushing, an
object.
Affecting Friction
• Friction increases as surfaces are made rougher.
• Friction increases when the force between two objects is
increased.
LUBRICATION
Why do we need it?
y
y
lower the friction
prevent wear
↓ wear
↓ friction
remove heat and
contamination
DUCTILITY, %EL
• Plastic tensile strain at failure:
Engineering
tensile
stress, σ
L − Lo
x100
%EL = f
Lo
smaller %EL
(brittle if %EL<5%)
larger %EL
(ductile if
%EL>5%)
Lo
Ao
Af
Lf
Callister 6e.
Engineering tensile strain, ε
• Another ductility measure:
%AR =
Ao − A f
x100
Ao
• Note: %AR and %EL are often comparable.
--Reason: crystal slip does not change material volume.
--%AR > %EL possible if internal voids form in neck.
20
HARDNESS
• Resistance to permanently indenting the surface.
• Large hardness means:
--resistance to plastic deformation or cracking in compression.
--better wear properties.
e.g.,
10mm sphere
apply known force
(1 to 1000g)
d
D
most
plastics
measure size
of indent after
brasses easy to machine
Al alloys steels
file hard
cutting
tools
Smaller indents
mean larger
hardness.
nitrided
steels
diamond
increasing hardness
Adapted from Fig. 6.18, Callister 6e. (Fig. 6.18 is adapted from G.F. Kinney, Engineering Properties
and Applications of Plastics, p. 202, John Wiley and Sons, 1957.)
28
La Resilienza
la resilienza indica la capacità di
un materiale di
assorbire energia in caso di urto,
ovvero di
sopportare gli urti
i materiali fragili assorbono poca energia
i materiali duttili assorbono molta energia
pendolo di Charpy
59/62
TOUGHNESS
È l’energia necessaria per la
frattura di un materiale sotto
un carico statico.
È rappresentata dall’area
sottesa alla curva reale σ-ε
della prova di trazione.
61/62
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