Mechanics of Materials II(3)

8/14/2019 Mechanics of Materials II(3)

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Mechanics of Materials II

UET, Taxila

Lecture No. (3)


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Typical stress-straincurves resulting from

tensile tests on somemetals are

shown in next Figures


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Tensile test curves for various metals


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Typical stress-strain curves for hard drawn

wire, note the reduction in strain values


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Brass tension test


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Aluminium alloy tensile test


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Brittle polymer tensile test


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Eye Glass tensile test


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When u & y is not valid?

In some loading cases,e.g. buckling of struts,

neither the yield stressnor the ultimate

strength is a realisticcriterion for failure of

com onents.


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Load factor

In such cases it isconvenient to replace the

safety factor, based on

stresses, with a differentfactor based on loads.


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Definition of load factor

Theload factor

is therefore defined as:load at failure /allowable

working load


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This is particularlyuseful in applications

of the so-calledplastic limit designprocedures.


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Temperature stresses

When the temperatureof a component is

increased or decreased

the material Respective

expands or contracts


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If this expansion or

contraction is notresisted in any way

then the processes

take place free of

stress


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If, however, the changes

In dimensions arerestricted then stresses

termed as

Temperature stresseswill be set up within the

material


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Consider a bar of

material with a linearcoefficient of expansion

. Let the original

length ofthe bar Land let the temperature

increase be t


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If the bar is free to expand

the change in length wouldbe given by

L = L tThen the new length L will

be:L= (L + L ) = L+ L t

= L (1 + t)


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Compressive thermal stresses

If this extension weretotally prevented, then a

compressive stress wouldbe set up equal to thatproduced when a bar oflength: L ( 1 + t) iscompressed through a

distance of L t. I thi th b i


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In this case the bar experience

a compressive thermal strain

equal to:


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In most cases tisvery small compared

with unity so that:


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But E = /

Thus = E

= t


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This is the stress set

up owing to totalrestraint onexpansions

or contractions caused

by a temperature rise,

or fall t


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In the former case

the stress iscompressive,

in the latter case thestress is tensile.


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Partial Prevention

If the expansion orcontraction of the bar is

partiallypreventedthen the stress set up

will be less than thatgiven by the equation

above. l ll b f d


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Its value will be found in asimilar way to thatdescribed above exceptthat instead of being

compressed through thetotal free expansion

distance ofL tit will becompressed through some

proportion of this distance.


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The new mode will bedepending on the

amount of restraint.


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Assuming some

fraction n of(L t)is allowed, then the

extension which isprevented is:

(1 - n) L t.

Thi ill d


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This will produce acompressive strain, asdescribed previously, ofmagnitude:


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Or approximately


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The stress set up will then be:

Etimes

= (1-n) E t

Thus for example if one


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Thus, for example, if one-third of the free expansion

is prevented the stress setup will be two-thirds of

that given by theequation:

= t

Stress concentrations &


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Stress concentrations &

stress concentration factorIf a bar of uniform cross-section is subjected to an

axial tensile or compressiveload the stress is assumed

to be uniform across thesection.

H


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However,in the presence of anysudden change of

section, hole, sharpcorner, notch, keyway,

material flaw, etc., thelocal stress will rise

significantly

Th ti f thi t t


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The ratio of this stress to

the nominal stress atthe section in the

absence of any of theseso-calledstress

concentrationsistermed as thestress

concentration factor


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stress concentration factor

SCF =Local stress/nominalstress

T h


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Toughness

Toughnessis defined as:the ability of a material to

withstand cracks,In other words to prevent

the transfer or propagationof cracks across its sectionhence causing failure.


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Types of toughness of materials

Two distinct types oftoughness mechanism

exist and in each case it isappropriate to considerthe crack as a very highlocal stress concentration.


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First Toughness Type

The first type ofmechanism relates

particularly to ductilematerials which aregenerally regarded

as tough Thi i b th


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This arises because the

very high stresses atthe end of the crack

produce local yieldingof the material and

local plastic flow at thcrack tip.


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This has the action of blunting thesharp tip of the crack and hencereduces its stress concentration effect

considerably (Fig. 1.15).


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High stress concentration

factor at crack tip (notch tip)

rea o oca y e ng o


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rea o oca y e ng omaterial reducing the stress-

concentration effect


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Second toughness mechanism

The secondmechanism refers to

fibrous, reinforced orresin-basedmaterials which have

weak interfaces


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Examples for second mode of toughness

Typical examplesare glass-fibrereinforced

materials and

wood In the second mechanism of


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In the second mechanism oftoughness

It can be shown thata region of localtensile stress always

exists at the front of apropagating crack.

Al


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Also

and provided that theadhesive strength of the

fibre/resin interface isrelatively low (one-fifth

the cohesive strength ofthe complete material)

T il t h i


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Tensile stress mechanism

This tensile stress opens upthe interface and producescrack sink, i.e. it blunts the

crack by effectivelyincreasing the radius at the

crack tip, thereby reducingthe stress-concentration

effect as appears in next fig This principle is used stop


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This principle is used stop,or at least delay, crack

propagation in engineeringcomponents when a

temporary "repair" iscarried out by drilling a

hole at the end of a crack,again reducing its stress-

concentration effect Toughness mechanism-type 2.


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Toughness mechanism type 2.


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C d F ti


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Creep and Fatigue

In the precedingparagraphs it has been

suggested that failureof materials occurs

when the ultimatestrengths have been

exceeded Pl ti D f ti


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Plastic Deformation

excessive deformation, ascaused by plasticdeformation beyond theyield point, can beconsidered as a criterion

for effective failure ofcomponents.

This chapter would not be


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This chapter would not becomplete, therefore,

without reference to certainloading conditions under

which materials can fail atstresses much less than the

yield stress, namely creepand fatigue.

D fi iti f


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Definition of creep

Creep is the gradualincrease of plastic

strain in a materialwith time at constantload.

Mechanics of Materials II(3)

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