Engineering Materials Lecturbbe 6

7/27/2019 Engineering Materials Lecturbbe 6

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Chapter 6: Mechanical Properties

Stress and strain: What are they and why are they used instead of load and deformation

Elastic behaviour: Recoverable Deformation of small magnitude

Plastic behaviour: 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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Concepts of Stress and Strain(tension and compression)

To compare specimens of different sizes, the load is calculated per unit area.

Engineering stress:

= F /Ao

F is load applied perpendicular to specimen cross-section; A0 is cross-sectionalarea (perpendicular to the force) before application of the load.

Engineering strain: = l / lo ( 100 %)

l is change in length, lo is the original length.

These definitions of stress and strain allow one to compare test results forspecimens of different cross-sectional area A0 and of different length l0.

Stress and strain are positive for tensile loads, negative for compressive loads

Strain is always Dimensionless!

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Shear and Torsion

Shear stress: = F / AoF is applied parallel to upper and lower faces each having area A0.

Shear strain: = tan ( 100 %) is strain angle

Torsion is variation of pure shear. The shear stress in this case is a function ofapplied torque T, shear strain is related to the angle of twist, .

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To understand and describe how materials deform (elongate, compress, twist) orbreak as a function of applied load, time, temperature, and other conditions weneed to discuss standard test methods and standard language for mechanicalproperties of materials.

Tensile Testing

specimenextensometer

LOAD vs. EXTENSION PLOTS


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- During Tensile Testing,

I nstantaneous loadand displacement i s measur ed

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 areatwice that of another, you will require twice the load to produce the sameelongation.

The Force .vs. Displacement plot will be the same shape as theEng. Stress vs. Eng. Strain plot

0

200

400

600

800

0 5 10Stress(MPa)

Strain (mm/mm)

Stress vs Strain Curve

Stress vs StrainCurve

0

5

10

15

20

25

0 5 10

Force vs Displacement Curve

Force vsDisplacementCurve

Displacement (mm)

Force(

N)

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Compression Tests

A compression test is conducted in a manner similar to thetensile test, except that the force is compressive and thespecimen contracts along the direction of the stress.

Tensile tests are more common because they are easier toperform; also, for most materials used in structural

applications, very little additional information is obtainedfrom compressive tests. Compressive tests are used when amaterials behaviour under large and permanent (i.e.,plastic) strains is desired, as in manufacturing applications,or when the material is brittle in tension.

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The Engineering Stress - Strain curve

Divided into 2 regions

ELASTIC PLASTIC

Elastic means reversible! Plastic means permanent!

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Stress-Strain Behaviour: Elastic

Deformation

In tensile tests, if the deformation is elastic, the stress-strain relationship is called Hooke's law:

= E

Higher Ehigher stiffness

Units:

E: [GPa] or [psi]: in [Mpa] or [psi]: [m/m or mm/mm] or[in/in]

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Modulus of Elasticity This modulus may be thought of as stiffness, or a

materials resistance to elastic deformation.

The greater the modulus, the stiffer the material, orthe smaller the elastic strain that results from theapplication of a given stress.

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Nonlinear elastic behaviorIn some materials (e.g., concrete, grey cast iron, manypolymer ), elastic deformation is not linear, but it is stillreversible.

Definitions of E


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Elastic Deformation: Atomic scale

On an atomic scale, macroscopic elastic strain is manifested as small changes

in the interatomic spacing and the stretching of interatomic bonds. As a consequence,the magnitude of the modulus of elasticity is a measure of the resistance to separation ofadjacent atoms, i.e., the interatomic bonding forces. Modulus of elasticity is proportional to the slope of the interatomic force separationcurve at the equilibrium spacing:

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Modulus of elasticity plot

Plot of modulus of elasticity versus temperature fortungsten, steel, and aluminium. (Adapted from K. M. Rall

T. H. Courtney, and J.Wulff, Introduction to MaterialsScienceand Engineering.

With increasing temperature, themodulus of elasticity diminishes, as is shownin Figure for several metals .

As would be expected, the imposition ofcompressive, shear, or torsional stresses alsoevokes elastic behaviour.

Shear stress and strain are proportionalto each other through the expression

Where G is the shear modulus, the slope ofthe linear elastic region of the shearstressstrain curve

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Anelasticity

(time dependence of elastic deformation)

Have assumed elastic deformation is time independent

(applied stress produces instantaneous strain)

Elastic deformation takes time; can continue even after loadrelease.

This behavior is known as anelasticity.

For metals the anelastic component is normally small and is

often neglected.

However, for some polymeric materials its magnitude issignificant; in this case it is termed viscoelastic behavior

EXAMPLE PROBLEM 6.1

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Elastic Deformation: Poissons ratio

Materials subject to tension shrinklaterally. Those subject to compression, bulge. The ratio of lateral and axial strains iscalled the Poisson's ratio . The negative sign is included in theexpression so that will always be positive,since xand z will always be of opposite sign. is dimensionless

metals: n ~ 0.33ceramics: n~ 0.25polymers: n ~ 0.40

For isotropic materials, shear and elasticmoduli are related to each other andto Poissons ratio according to

EXAMPLE PROBLEM 6.2

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Stress-Strain Behaviour: Plastic

deformation

In Plastic deformation: stress is not proportional to strain

deformation is not reversible

deformation occurs by breaking and re-arrangement of atomicbonds (crystalline materials by motion of defects)

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Tensile properties: Yielding

Yield point: PWhere strain deviates from beingproportional to stress (theproportional limit)

Yield strength: yIn most cases the position of thispoint P can not be determinedprecisely. Line is drawn parallel toelastic portion of the stress-straincurve at strain offset of 0.002.

A measure of resistance to plasticdeformation


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Tensile properties: YieldingStress

Strain

In some materials (e.g. low-carbon steel), the stressvs. strain curve includes two yield points (upper andlower). The yield strength is defined in this case asthe average stress at the lower yield point.

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Tensile StrengthIf stress = tensile strength is maintainedthen specimen will eventually break

FractureStrength

NeckingStress,

Strain, For structural applicationsYield stress,y , usually more importantthan tensile strength. Once yield stress has been passed, structurehas deformed beyond acceptable limits.

Tensile strength =

max. stress

(~ 50 - 3000 MPa)


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Tensile properties: Ductility

Defined bypercent elongation

(plastic tensile strain at failure)Or

Percent reduction in area

Ductility is a measure of the deformation at fracture. i.e. thecapacity to undergo plastic deformation

100l

llEL%

0

0f

100A

AARA%

0

f0

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Mechanical Properties of Metals

The yield strength and tensile strength vary with priorthermal and mechanical treatment, impurity levels etc.

Whereas elastic moduli are relatively insensitive to theseeffects.

The yield and tensile strengths and modulus of elasticitydecrease with increasing temperature, ductility increaseswith temperature.


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Resilience, Ur

Ability of a material to store (elastic) energy Energy stored best in elastic region

If we assume a linearstress-strain curve thissimplifies to

Adapted from Fig. 6.15,Callister 7e.

y dUr0

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Energy to break a unit volume ofmaterial or it is a measureof the ability of a material to absorb energy up to fracture Approximate by the area under the entire stress-strain

curve.

Toughness

Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy

very small toughness(unreinforced polymers)

Engineering tensile strain,

Engineeringtensilestress,

small toughness (ceramics)large toughness (metals)

Adapted from Fig. 6.13,Callister 7e.


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True Stress & Strain

Note: Stressed Area changes when sample is deformed (stretched)

True stress = load divided by actualarea or instantaneous area in the necked-down region (Ai):

Sometimes it is convenient to use true strain defined as

iT AF oiT ln

1ln

1

T

TAdapted fromFig. 6.16,Callister 7e.

If no volume changeoccurs duringdeformation,Aili = A0l0 and the true and

engineering stress andstress are related as

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Elastic Recovery During Plastic Deformation

If a material is deformed plastically andthe stress is then released, the materialends up with a permanent strain. If the stress is reapplied, the materialagain responds elastically at the beginningup to a new yield point that is higher thanthe original yield point. The amount of elastic strain that it willtake before reaching the yield point iscalled elastic strain recovery.

Adapted fromFig. 6.16,Callister 7e.It is an important factor in

forming products (especially sheetmetal and spring making)


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Hardness (I)Hardness is a measure of the materials resistance to localized plasticdeformation (e.g. dent or scratch)Large hardness means:

--resistance to plastic deformation or cracking in compression.--better wear properties.

Variety of hardness tests have been designed based on the indenter shape (Brinell,Rockwell, Vickers, etc.).

e.g.,10 mm sphere

apply known force measure sizeof indent afterremoving load

dD

Smaller indents

mean largerhardness.

increasing hardness

mostplastics

brassesAl alloys

easy to machinesteels file hard

cuttingtools

nitridedsteels diamond

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Hardness (II)

Both tensile strength andhardness may be regarded as degreeof resistance to plastic deformation.

Hardness is proportional to thetensile strength but note that theproportionality constant is differentfor different materials


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N

y

working

N

y

working

Design uncertainties mean we do not push the limit!

Typically as engineering designers, we introduce aFactor of safety,N OftenNis

between1.5 and 4

Example: Calculate a diameter, D, to ensure that yield doesnot occur in the 1045 carbon steel rod below subjected to a

working loadof 220,000 N. Use a factor of safety of 5.

Design or Safety Factors

1045 plaincarbon steel:

y = 310 MPaTS= 565 MPa

F= 220,000N

D

Lo

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Solving:

2

2

0

2

2

2 2

6

2 3 2

2

310

5

220000

4

310220000

5

4

220000 4 5 m310 10

4.52 10 m

6.72 10 m 6.72 cm

Yworking

working

Nm

N

NFA D

MNmN

D

D

D x

D x D

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Stress and strain: These are size-independentmeasures 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 alarge elastic modulus (Eor G).

Toughness: The energy needed to break a unit

volume of material. Ductility: The plastic strain at failure.

Summary

Plastic behavior: This permanent deformationbehavior occurs when the tensile (or compressive)

uniaxial stress reaches y.

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Engineering Materials Lecturbbe 6