Properties of Materials

Vikram K. KuppaVikram K. Kuppa

Energy & Materials Engineering ProgramEnergy & Materials Engineering ProgramSEEBMESEEBME

University of CincinnatiUniversity of Cincinnati

866 ERC866 ERCPh: 513-556-2059Ph: 513-556-2059

Vikram.kuppa@uc.eduVikram.kuppa@uc.eduwww.uc.edu/~kuppavm

Office Hours: MWF 10-11AMOffice Hours: MWF 10-11AM

Types of Stresses

FF

Tensile

F

Bending

FF

Compressive

F

Shear

F

Stress vs Strain

stress forcearea

strain lengthlength

Representative Stress-strain curves

Young’s Modulus (E)• The slope of the stress-strain

curve in the elastic region.– Hooke’s law: E = /

• A measure of the stiffness of the material.

• Larger the value of E, the more resistant a material is to deformation.

• Note: ET = Eo – bTe-To/T where Eo and b are empirical

constants, T and To are temperatures

Units:E: [GPa] or [psi]

: dimensionless

Stress-Strain Behavior (summary)Elastic deformationReversible:

( For small strains)Stress removed material returns to original size

Plastic deformationIrreversible: Stress removed material does not return to

original dimensions.

Yield Strength (y)

• The stress at which plastic deformation becomes noticeable (0.2% offset).

• P the stress that divides the elastic and plastic behavior of the material.

True Stress & True Strain

0

0

0

strain gEngineerin

stress gEngineerin

lll

AF

• True stress = F/A• True strain = ln(l/l0)

= ln (A0/A)(A must be used after

necking)Apparent softening

True Strain t dl

lL o

L

lnL

Lo

True Stress t Load

A

Load

A0

AL A oLo

t ln 1

t 1

• The total area under the true stress-strain curve which measures the energy absorbed by the specimen in the process of breaking.

Toughness

Toughness d

Tensile properties: Ductility

The total elongation of the specimen due to plastic deformation, neglecting the elastic stretching (the broken ends snap back and separate after failure).

TextbooksEssentials of Materials Science & Engineering

Second EditionAuthors: Donald R. Askeland & Pradeep P. Fulay

Materials Science and Engineering: An IntroductionSixth Edition, Author: William D. Callister, Jr.

The Science and Engineering of MaterialsFourth Edition, Authors: Askeland and Phule (Fulay ?)

Introduction to Materials Science for EngineersSixth Edition, Author: James F. Shackelford

• 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 y.

• Toughness: The energy needed to break a unit volume of material.

• Ductility: The plastic strain at failure.

Note: materials selection is critically related to mechanical behavior for design

applications.

SUMMARY

Viscoelastic BehaviorPolymers have unique mechanical properties vs. metals & ceramics.

Why?Bonding, structure, configurations

Polymers and inorganic glasses exhibit viscoelastic behavior (time and temperature dependant behavior)

Polymers may act as an elastic solid or a viscous liquidi.e. Silly Putty (silicon rubber)

- bounces, stretches, will flatten over long times

Low Strain RateHigh extension - failure

resilient rubber ballElastic behavior rapid deformation

Very low Strain rate - FlattenFlow like a viscous fluid

PolymersPolymer : Materials are made up of many (poly) identical chemical units (mers) that are joined together to construct giant molecules.

Plastics - deformable, composed of polymers plus additives. E.g. a variety of films, coatings, fibers, adhesives, and foams. Most are distinguished by their chemical form and composition.

The properties of polymers is related to their structures, which in turn, depend upon the chemical composition. Many of these molecules contain backbones of carbon atoms, they are usually called "organic" molecules and the chemistry of their formation is taught as organic chemistry.

The most common types of polymers are lightweight, disposable, materials for use at low temperatures. Many of these are recyclable. But polymers are also used in textile fibers, non-stick or chemically resistant coatings, adhesive fastenings, bulletproof windows and vests, and so on.

Polymers

Polymer : Materials are made up of many (poly) identical chemical units (mers) that are joined together to construct giant molecules. Carbon – 1s22s22p2

It has four electrons in its outermost shell, and needs four more to make a complete stable orbital. It does this by forming covalent bonds, up to 4 of which can be formed.

The bonds can be either single bonds, ie one electron donated by each participating element, or double bonds (2 e- from each), or triple bonds (3 from each)

C X1

X2

X4

X4

Xi can be any entity ex H, O, another C, or even a similar monomer

C X1

X2

X4

X4

Polymers – many repeating units

C X1

X2

X4

X4 + C X1

X2

X4

X4 +…

CCCC CAnd so on… if the bonds can keep getting formed, entire string-like structures (strands, or chains) of the repeating units are created. C is the most common element in polymers. Occasionally, Si may also participate in such bonding.

Classes of PolymersThermoplastics:

Consist of flexible linear molecular chains that are tangled together like a plate of spaghetti or bucket

of worms. They soften when heated.

Thermosets:

Remain rigid when heated & usually consist of a highly cross-linked, 3D network.

Elastomers:

Consist of linear polymer chains that are lightly cross-linked. Stretching an elastomer causes chains to partially untangle but not deform permanently

(like the thermoplastics). Of all the materials, polymers are perhaps the most versatile, not only because the properties can be drastically modified by simple chemistry, but the behavior is also

dependent on the architecture of the chains themselves.

From proteins to bullet-proof jackets to bottles, polymers are INDISPENSIBLE to life as we know it

Illustration©

2003

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Lear

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used

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a) & b) 3 dimensional models,

c) Is a simpler 2-D representation

backbone

side-group

Chain Conformations

Polymer Synthesis - I

Addition in which one “mer” is added to the structure at a time.

This process is begun by an initiator that "opens up" a C=C double bond, attaches itself to one of the resulting single bonds, & leaves the second one dangling to repeat the process

Polymer Synthesis - II

Condensation in which the ends of the precursor molecules lose atoms to form water or alcohol, leaving bonds that join with each other to form bits of the final large molecules. An example is shown in the Detail - the formation of nylon.

Molecular weight distribution

The degree of polymerization (DP) = no. of monomers per polymer. It is determined from the ratio of the average molecular weight Mw of the polymer

to the molecular weight of the repeat unit (MRP).DP = Mw / MRP

where Mw = fi Mi : Mw = weight average molecular weightMn = xi Mi : Mn = number average molecular weight

Mi = mean molecular weight of each range fi = weight fraction of polymer having chains within that range

xi = fraction of total number of chains within each range

Molecular Weight Distributions

M n xiM ii

M w wiM ii xiM i

2

i

xi ni

nii

number fraction

Degree of Polymerization

nn Mn

m ; nw Mw

m m "mer" molecular weight

Degree of polymerization & molecular weightDegree of polymerization (DP)- number of monomers per polymer chain, ie no. of repeat units.

Obviously, the weight (either in AMU, or in g/mol) is the same for each repeat unit. Then, the total weight of the polymer chain, ie its molecular weight is :-

mol. Wt. = N.Mm

where N is the number of monomers in that chain, ie the DP; Mm is the weight of the monomer.

In a polymer sample synthesized from monomers by either condensation or addition polymerization, one always has a distribution of DPs amongst the resulting chains.

So let us consider that we have 100 monomers. Let the weight of each monomer be 1g/mol (in reality, this is Hydrogen !) Let us see some ways in which we can arrange this:1)1 chain of N=100, ie mol. Wt. = 1002)2 chains of N=50 each, ie mol. Wt. = 503)10 chains of N=10 each, ie mol. Wt. = 104)3 chains, 2 of N=25, and 1 of N=50

Degree of polymerization & molecular weight3 chains, 2 of N=25, and 1 of N=50. Now, to calculate the average molecular weight, we have two methods:1) Take the simple numerical average, ie

(25+25+50)/3.0 = (2x25 + 1x50)/3.0 = 33.33. This value is according to the number fraction of each type of chain (1/3 of the chains are of N=50, and 2/3 have N = 25)

2) Take the average according to the weight fraction of each chain. What is the total weight ?

Mtotal=100Wfraction

50 = 50/100, ie ½ , Wfraction25=2*25/100 = 1/2

So, taking weight fractions, we get the average molecular weight as Mw = 50*1/2 + 25*1/2 = 25+12.5 = 37.5

So, numerical fractions, and weight fractions for mol. Wt. give different answers!Mn = SUM(niMi)/Sum(ni) , where ni = no. of chains of length Mi

Mw = SUM(wiMi), where wi = weight fraction of chains of length Mi.

But, wi = niMi/SUM(niMi) ie the weight of that polymer (i), divided by total weight.

So, in the previous example, W50 = 50/100, W251 = 25/100, W25

2 = 25/100

Degree of polymerization & molecular weightSuppose we want to find out the average population of each state.* We can go to each senator of each state and find out what the population of

their state is, and then divide that number by 100. This number is the number-average population for each state. This is exactly

similar to the Mn that we calculated earlier, ie no. av. Mol. wt.. Problem ?Yes, of course. What do we do about say, CA and AK ?

Now, senators are busy, so we ask congressmen from each state. Then, we take the value that each congressman/congresswoman gives us, and then divide by the number of congresscritters. What value do we get ? Certainly one different from our earlier attempt ! Problem ?

Now the value is much higher than before. This is exactly similar to the Mw that we calculated earlier, ie to weight av. mol. Wt.

Is this value MUCH more representative (eh eh !) of the average population of each state ? Well, not really. But at least, it is an average.

We learn about these differences, because different measurement techniques measure different averages, and the ratio of Mw to Mn, called the Poly Dispersity Index (PDI) often determines properties.

* taken from “Polymer Physics” by M. Rubinstein & R. H. Colby, 1st edition, OUP

• Polymer = many mers

• Covalent chain configurations and strength:

Direction of increasing strengthBranched Cross-Linked NetworkLinear

secondarybonding

C C C C C CHHHHHH

HHHHHH

Polyethylene (PE)

mer

ClCl ClC C C C C C

HHH

HHHHHH

Polyvinyl chloride (PVC)

mer

Polypropylene (PP)CH3

C C C C C CHHH

HHHHHH

CH3 CH3

mer

Polymer Architecture

Structure of polymers strongly affects their properties; e.g., the ability of chains to slide past each other (breaking Van der Waals bonds) or to arrange themselves in regular crystalline patterns. Some of the parameters are: the extent of branching of the linear polymers;the arrangement of side groups. A regular arrangement (isotactic) permits the greatest regularity of packing and bonding, while an alternating pattern (syndiotactic) or a random pattern (atactic) produces poorer packing which lowers strength & melting temperature.

Polymer Architecture - II

Stereoisomerism

CC

HH

R H

CC

HH

R H

CC

HH

R H

CC

HH

R H

CC

HH

R H

CC

HH

R H

CC

HH

H R

CC

HH

R H

CC

HH

H R

CC

HH

R H

CC

HH

R H

CC

HH

H R

CC

HH

R H

CC

HH

R H

CC

HH

R H

Isotactic

Syndiotactic

Atactic Can’t Crystallize

Isomerism – different structures, but same chemical composition

Polymer Architecture - Schematics

Random

Alternating

Branched

If you have some red beads and some black beads, how can you make polymers out of them ?

Blocky

We have discussed polymers comprised of a single kind of a monomer, ie just one repeating entity. However, this is not unique: we can synthesize polymers that consist of different repeating units, and such polymers are called copolymers

The combination of different mers allows flexibility in selecting properties, but the way in which the mers are combined is also important. Two different mers can be alternating, random, or in blocks along the backbone or grafted on as branches.

Polymer Architecture - III

Thermoplastic & Thermosetting Polymers

• Thermoplastics: --little cross-linking --ductile --soften w/heatingEx: grocery bags, bottles• Thermosets: --large cross-linking (10 to 50% of mers) --hard and brittle --do NOT soften w/heating --vulcanized rubber, epoxies, polyester resin, phenolic resinEx: car tyres, structural plastics

cross-linking

VulcanizationIn thermoset, the network is inter-connnected in a non-regular fashion. Elastomers

belong to the first category. Polyisoprene, the hydrocarbon that constitutes raw natural rubber, is an example. It contains unsaturated C=C bonds, and when vulcanizing

rubber, sulfur is added to promote crosslinks. Two S atoms are required to fully saturate a pair of –C=C— bonds and link a pair of adjacent molecules (mers) as indicated in the

reaction. Without vulcanization, rubber is soft and sticky and flows viscously even at room temperature. By crosslinking about 10% of the sites, the rubber attains mechanical

stability while preserving its flexibility. Hard rubber materials contain even greater sulfur additions.

Vulcanization

• Molecular weight Mw: Mass of a mole of chains.

• Tensile strength (TS): --often increases with Mw.

--Why? Longer chains are entangled (anchored) better.• % Crystallinity: % of material that is crystalline. --TS and E often increase with % crystallinity. --Annealing causes crystalline regions to grow. % crystallinity increases.

crystalline regionamorphous region

smaller Mw larger Mw

Molecular weight, Crystallinity and Properties

“Semicrystalline” Polymers

~10 nm spacing

Oriented chains with long-range order

Amorphous disordered polymer chains in the “intercrystalline” region

Mechanical Properties of Polymers

Elasticity of PolymersRandom arrangement = High Entropy Stretched = Low Entropy

Entropy is a measure of randomness: The more ordered the chains are, the lower is the entropy. Spontaneous processes always tend to increase the entropy, whichmeans that after stretching, the chains will tend to return to a high-entropy state

Viscosity of Polymers

Elastic Deformation

creep

Cross-linking stops the sliding of chains

random

Slow Deformation

Low entropy state

Elastic

ViscousViscoelastic

VISCOELASTIC RESPONSE

Viscoelasticity: T DependenceTemperature & Strain Dependence:

Low T & high strain rates = rigid solids

High T & low strain rates = viscous

Rubber-like ElasticDeformation

Slow relaxation

Glassy (Elastic-high modulus)

Leathery(Elastic-low modulus)

Thermoplastic (uncrosslinked)

Tg Tm

Mod

ulus

of e

last

icity

Temp.

Rubbery Plateau Elastic at high strain rateViscous at low strain rate

medium times

Long times

Crosslinked Branched

Effect of crosslinkingThermoset

Heavy Crosslinking

ElastomerLight crosslinking

Effect of crystallinity

Tg Tm

Log

Mod

. Of E

last

icity

amorphous

50 % Crystalline

100 % crystalline

Tm

Log

Mod

. Of E

last

icity

ThermoplasticNo crosslinking

Tg

Branched polymer

Crystals act like crosslinksStrain Induced Crystallization in NR

Viscoelasticity: Structure Dependence

• Compare to responses of other polymers: --brittle response (aligned, cross linked & networked case) --plastic response (semi-crystalline case)

TENSILE RESPONSE: ELASTOMER (ex: rubberband)

initial: amorphous chains are kinked, heavily cross-linked.

final: chains are straight,

still cross-linked

0

20

40

60

0 2 4 6

(MPa)

8

x

x

x

elastomer

plastic failure

brittle failure

Deformation is reversible!

• Decreasing T... --increases E --increases TS

--decreases %EL

• Increasing strain rate...

--same effects as decreasing T. 20

40

60

80

00 0.1 0.2 0.3

4°C

20°C

40°C

60°C to 1.3

(MPa)

Data for the semicrystalline polymer: PMMA (Plexiglas)

T & STRAIN RATE: THERMOPLASTICS(ex: plastic bottles or containers)

• Stress relaxation test:

Er(t)

(t)o

--strain to and hold.--observe decrease in stress with time.

• Relaxation modulus:

• Data: Large drop in Er for T > Tg.

(amorphouspolystyrene)

103

101

10-1

10-3

105

60 100 140 180

rigid solid (small relax)

viscous liquid (large relax)

transition region

T(°C)Tg

Er(10s) in MPa

TIME-DEPENDENT DEFORMATION

time

straintensile test

ot( )

Time-Temperature Superposition

Log Time

Log

Rel

axat

ion

Mod

ulus

Rel

axat

ion

Mod

ulus

Hi T

Lo T

Relaxation Modulus

time

Stre

ss,

10 s

10

L

fixed LLo

Er(0)= E, Young’s ModulusEr( )= 0

Glass-like elasticity

Rubber-likeelasticity

Fluid-likeViscous

Viscoelstic modulus

Modulus of elasticity        E r (10s) = (10) fixed

Relaxation Modulus