Materials Science Manual Chapter 4

7/29/2019 Materials Science Manual Chapter 4

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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------

Chapter 4

MECHANICAL PROPERTIES OF MATERIALS.

4.1. STATIC MECHANICAL PROPERTIES

The choice of a material for any particular application largely depends on its behaviour when actedupon by mechanical forces. Most materials under load undergo elastic and plastic types ofdeformation. Elastic deformation is recovered when the material is unloaded, while plasticdeformation is un-recoverable. The properties of a material are dependent on the rate of loadapplication, temperature, time, direction and magnitude of load etc. Hence we need to device a setof mechanical tests whereby a variety of properties such as strength, stiffness, ductility, toughness,hardness, etc. under specific conditions can be obtained. Mechanical tests can be classified as:(a) Static, such as in tensile, compression, bending and torsion tests,(b) Dynamic, as in impact and fatigue tests,(c) Long term, as in creep and stress relaxation tests and

(d) Non-destructive tests

The above tests are to be considered with reference to the mechanical properties they aredesigned to generate.

4.1.1 Tensile properties

The aim of the tensile test is to determine the strength, stiffness and ductility of a material. The

test gives useful information on the Ultimate tensile strength [ u], Yield strength [ y], Young's

modulus [E], Elongation to fracture [ ] and Reduction in area [ ]. Let us define a few terms beforegoing into the details of the test.

Consider a material with cross section area Ao under a tensile load P.

Engineering stress [o] is the ratio of the mechanical force [P] to the original underformed crosssection area [Ao] of the body upon which the force is acting.

oA

P= (4.1)

True stress [t] is the actual stress in a material obtained by dividing the load to the instantaneouscross-section area over which it acts.

A

Pt = (4.2)

Engineering strain [e] is a measure of deformation in a stressed material. It is expressed as apercentage of the deformation length (l-lo) to the original length lo.

100.)(

o

o

l

lle

= (4.3)

True strain [] is the instantaneous value of total strain. When a material is stressed, itsdimensions i.e. the original length and cross section area change, hence the true strain is thesummation of all instantaneous strains.

66


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=

++

+

+

=

i

i i

ii

o

o

l

ll

l

ll

l

ll

l

ll

0 1

1

2

23

1

121 ...

A

A

l

l

l

dlo

olnln === (4.4)

It can be shown that

)1ln( e+= (4.5)

And

)1( eot += (4.6)

The tensile test is a static test in which a specimen is pulled until fracture. The testing machine isusually a hydraulic or screw type device consisting of two cross heads in which the test specimen

is held. Usually one crosshead is stationary while the other is free to move thus applying a pullingload (tension) to the specimen. The applied load and specimen displacement are recorded in achart, or can be read directly on the machine's instruments. For displacement (strain)measurements the use of more sensitive instruments is often required. Mechanical or opticalextensometers, and sometimes electrical strain gauges are used.

The test specimen may be of round or rectangular cross section, the middle part being reduced toform a dog bone uniform shape that is smooth as shown in Fig.4.1.

Fig. 4.1 The dog bone shaped tensile test specimen

For round specimen, the gauge length, lo, is estimated as five or ten times the nominal diameterfor short and long specimen respectively. For non-circular specimen, the gauge length isestimated as lo = 5.65(A)

1/2 or 11.3 (A)1/2 for short and long specimen respectively. It is not usualto test non-uniform specimen. However, certain materials such as reinforcement steel bars, maybe required by standards to be tested in their application state, hence a mean cross section of thebar has to be calculated as,

LmAm

= (4.7)

where m is the mass of the specimen with length L and is the density of the material.

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A

lo

Ao


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Fig. 4.2. The engineering and true stress-strain curves

Consider the stress-strain curves shown in Fig.4.2. In most crystalline materials, the initial part ofthe curve (O-P) is nearly straight. The point P is called the limit of proportionality. This part of thecurve is said to obey Hooke's law, and the gradient of the curve in this part is the Young'smodulusof elasticity, [E]. The point E is called the elastic limit. In most ductile materials, a phenomenoncalled yielding is observed between points E and Y. Yielding arises from displacements within thecrystalline structure involving slip between atomic planes when a critical shear stress is reached.The stress then begins to rise till the point U is reached. Between O and U the specimen crosssection is decreasing uniformly as the stress is increased. However at the point U, localizedreduction in cross section (necking) occurs and suddenly the load (stress) in the material appearsto be falling to the point F at which the specimen fractures. In reality the stress in the material ishigher than that observed, as a smaller cross-section is actually carrying the load and not theoriginal cross section that is used to calculate the engineering stresses. The actual stress strainbehaviour can be established by constructing a true stress-true strain curve. By measuring the

actual diameter of the specimen between yielding (Y) and necking (U), and measuring thediameter at the fractured section true stresses and true strains can be calculated hence obtainingthe curve Y-U'-F'.

The following properties are often reported from a tensile test:-

(a) Yield stress [y]

o

y

yA

F= (4.8)

where Fy is the yield load and Ao is the original cross section area

Some materials do not show the yield point. In such cases, an offset orproof stress [x] has to beobtained by drawing a line parallel to the Hooke's line of stress-strain curve from a strain x, usually

0.2%. This line will intersect the - e curve at a point Q. The stress at this point, is the proofstress [x], which would give a permanent deformation of x%.

(b) Ultimate tensile strength [u]

o

uA

Fmax= (4.9)

where Fmax is the maximum load on the force elongation curve.

(c) Elongation at fracture []

100.o

of

l

ll

= (4.10)

68

O

PE

Y U

U'

F

F'

Stress(MPa)

Strain (%)

True

Engineering


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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------where lo is the original gauge length and lf is the gauge length after fracture.

(d) Reduction in area [].

100.o

fo

A

AA = (4.11)

where Af is the cross section area at the point of fracture.

(e) Young's Modulus [E]

eE

=

(4.12)

in the elastic range.

Sometimes the resilience and toughness of a material may have to be evaluated. Resilience is theability of a material to absorb energy when deformed elastically and return it when unloaded. Thisproperty is required for springs and is measured as the modulus of resilience, which is the strainenergy per unit volume required to strain a material from zero stress to the limit of proportionality,i.e. the area under the elastic part of a stress-strain curve. Toughness is simply a measure ofability to absorb energy in the plastic range, hence represented by the area under the plastic partof the stress-strain curve. These are shown in Fig. 4.3

Fig. 4.3 The stress-strain curves for resilient and tough materials

The shape of the force-elongation or stress-strain curve depends on the properties of thematerial. For a particular material, it may vary dependent on testing temperature , heat treatment,and previous history of cold work, speed of testing, size of specimen and stress distribution. Fig.4.4 and 4.5 show the effects of temperature and heat treatment on the stress strain behaviour ofcarbon steel.

Fig. 4.4 The effect of temperature on the tensile properties of steel

69

77 K

300 K

673 KStress

Strain

A- High resilience, low toughness

B- Low resilience,high toughness

Stress(MPa)

Strain (%)


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Fig. 4.5. The effect of heat treatment of steel on its tensile properties.

The shape of the fracture indicates the degree of ductility in a material. Brittle materials in tensionfracture in a plane normal to the tensile stress axis without any plastic deformation or necking. Apurely ductile material is bound to fracture along the maximum shear stress plane i.e. 45 to theaxis of tensile stress. Most materials exhibit a combined mode of fracture. The common types offracture observed under tensile loading are shown in Fig. 4.6.

Fig. 4.6 Modes of tensile fracture: 1-ductile shear, 2-brittle and 3-mixed mode.

4.1.2 The compression test

Brittle materials are often weaker in tension, but strong in compression. That is the reasoncompression testing is important for civil engineering concrete materials and cast iron, which areinherently brittle. Ductile materials are seldom tested in compression.

The test specimen dimensions depend on the type of material being tested. Length to width ratiosof 1 and 2.5 - 3 are preferred for brittle and ductile materials respectively. Longer specimen mayfail by buckling for ductile materials.

The specimen to be tested is fixed between two flat crossheads one of which remains stationarywhile the other is powered either hydraulically or mechanically as in the tensile test, to give a

compressive force, as shown in Fig. 4.7. The force [P]- deflection [h] curve is plotted by a chartrecorder, and thereafter used to generate a compression stress-strain curve.

70

Hardened

Hardened and

tempered

Annealed

Strain

Stress

1 23


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Fig. 4.7. The compression test

The compressive stress

o

compA

P= (4.13)

The strain in this case is negative, and is given as:

o

o

o h

hh

h

he

=

= (4.14)

The compressive load that causes failure is recorded as the maximum compressive load [Pmax].

Hence the compressive strength of the material [comp] is computed as,

o

compA

Pmax= (4.15)

Most brittle materials fail by severe cracking after which the specimen crumbles into pieces.Ductile materials undergo plastic flow as they are squashed between the compression plates. Thenature of deformation depends on the interfacial coefficient of friction between the material beingtested and the compression plates.

Brittle materials are stronger in compression than in tension. The compression strength of graycast iron is about 3-4 times its tensile strength. There is also some relation between the Brinellhardness and compression strength for gray iron ranging from 111 - 386 [MN/m2] tensile strengthwith hardness from 110 - 269 HBN. The ratio of compression strength to Brinell number is usuallybetween 3.4 and 4.0. The Young's modulus of elasticity in compression for gray cast iron at lowstresses is roughly 3-5% higher than in tension.

4.1.3 Bend tests.Bend tests are performed either to assess whether a material meets fabrication requirements orwhether it has the desired design strength. Hence they can be classified accordingly as:-

(a) Ductility and cracking tests, and(b) Transverse tests

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hho

P


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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------Ductility and cracking tests are used to assess the ductility of a material, and often, to verify thatthe minimum plasticity required of a material is exceeded. A common application involvesbending through 180 freely between flat faces or closely round a pre-determined radius, set onsome kind of forming tool. Another example is the face and root bend tests on butt welds, wherebythe welded seam is made to bend around a former until the first crack occurs. The angle ofdeformation at which cracking begins is recorded, otherwise the specimen is bent through 180and then inspected.

The transverse test is applied to brittle materials such as cast irons. Brittle materials are verydifficult to test in tension as they tend to be highly notch sensitive hence may break in a sectionaway from the gauge length. The use of a simple bend test (3-point. bend) or a four-point bendtest enables on to evaluate critical values of tensile stress required to cause rupture. The 3-pointbend test is designed for those materials that break at comparatively small deflections whereas the4 -point bend tests is designed particularly for materials that undergo large deflections duringtesting.

In a 3 pointbend test, the test bar rests between two supports and is loaded at mid-span so as toproduce a maximum bending moment at mid-span as shown in Fig.4.8 (a). However, this pointhas a shear force, which produces a shear stress in the plane parallel to the axis of the test bar,being maximum in the neutral plane. This test is not recommended for specimen with a small spanto width ratio.

In the 4-point bend test, the bar resting on two supports is loaded at two points, each at equaldistance from the adjacent support point and produces a maximum bending moment [Mmax] that isuniformly distributed between the two loading points as shown in Fig 4.8 (b). Unlike the three-pointbend test, the section of maximum bending moment has a zero shear force, hence the four-bendtest represents a case of pure bending stresses.The bending strength is usually the maximum bending stress (tensile) calculated as,

eZMmax= (4.16)

where Ze is the section modulus.

This is called the Flexural Strength or the Modulus of Rupture. For those materials that do notcrack, the maximum bending stress is called the Flexural Yield Strength.

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Fig. 4.8. (a) Three-point bend test and (b) Four- point bend test.

It is normal to measure the deflection occurring just before fracture so as to assess the elasticity

of the material. In a 3-point bend test, the maximum deflection [max] is given as

EI

FL

48

3

max = (4.17)

where Eis the modulus of elasticity and I is the second moment of area.

In a 4-point bend test, the maximum deflection is

EI

aLFa

24

)43( 22

max

= (4.18)

Using the above equations, the modulus [E] of the material can be calculated.

4.1.4. Torsion test

The torsion test is aimed at assessing the mechanical behaviour of a material under shearingstresses. The testing equipment comprises a twisting head with a chuck for gripping one end ofthe specimen and also measures the twisting moment or torque, and a troptometer for measuringthe angle of deformation. Circular cross-section specimen is preferred for simplicity of stresscalculations as shown in Fig 4.9.

The shear stress increases from zero in the axis to a maximum at the periphery. The maximum

shear stress, max is given as

73

Bending momentdiagrams.

L/2L/2

a a

L

LOAD

LOAD

(a)

(b)


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J

DMt

2max = (4.19)

where Mt is the torsional moment, D is the diameter and J is the polar moment of inertia of the

specimen section.

Fig. 4.9. Torsion in a solid cylindrical specimen

For a solid cylindrical specimen,

32

4DJ

= (4.20)

The troptometer is used to determine the angle of twist usually expressed in radians.

If L is the test length of the specimen, then the shear strain at a radius ris given as

L

r= (4.21)

During the test, measurements are made of the twisting moment M t and the angle of twist, . Atorque -twist diagram is obtained. This can be easily converted into a shear stress [] - shear strain[] diagram. It is observed that in elastic materials, the shear stress is proportional to the shearstrain. The constant of proportionality is the modulus of elasticity in shear, or the Modulus of rigidity[G].

Hence,

= G (4.22)

Substituting equations 4.19 and 4.21 into equation 4.22, then

J

LMG

t= (4.23)

Torsional failures are different from tensile failures in that there is little localized reduction in areaor elongation. A ductile material tends to fracture by shear, along one of the planes of maximumshear stress, which is generally normal to the longitudinal axis. A brittle material fails along a planeperpendicular to the direction of maximum tensile stress. In this case, the direction makes anangle of 45 to the axis, resulting into a helical fracture.

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L

dAr

2a


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4.2. DYNAMIC PROPERTIES

4.2.1. The impact strength

The capacity of a material to withstand shock loads without fracture is called toughness. Impacttests are used to give an indication of the relative toughness of a material.

Materials that are not tough tend to fail in brittle manner, with no plastic deformation. Brittle-cleavage type of fracture results from either of the following:(a) a tri-axial state of stress, resulting from high stress concentration around notches, grooves,holes, sharp changes of cross section etc(b) at low temperature, where interatomic forces are higher, fewer point and line defects in thecrystalline structure and inter-crystalline bonding (grain boundaries) are strong, thus allowing verylittle plastic flow. Fracture is often trans-crystalline.(c) a high strain rate or rapid rate of load application prevents the time dependent plastic flow.(d) component size.(e) metallurgical variables such as chemical composition, grain size, impurity distribution anddegree of cold work.(f) welding effects, such as slag inclusions, porosity, heat affected zones (HAZ), hot cracks,hydrogen induced cracking etc.(g) fatigue, due to repeated loading giving rise to cumulative micro cracking before final brittlefracture.

Notched-bar impact tests are employed to study the simultaneous effect of stress concentration,temperature and high speed of load application on materials. Two types of specimen are widelyemployed. The Charpy bar specimen (USA) and the Izod specimen (UK). The Charpy specimenhas a square cross section and contains a central notch of V or U shape. It is then supported as abeam in a horizontal position, with the load being applied from a swinging pendulum to the mid-span of the beam on the side opposite to the notch. The Izod specimen is either circular or squarein cross section, and contains a V-notch near one end. The specimen is clamped vertically like a

cantilever and the pendulum is made to strike the free end. Fig.4.10 shows these two impact testschematically.

Fig. 4.10. Standard methods for impact tests

75

IMPACT

LOAD

IMPACT

LOAD

(a) Charpy V- notch

(b) Izod


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Fig. 4.11. Brittle ductile transition curves

The energy required to fracture a specimen is read directly on the calibrated dial of the impact

tester. The impact strength [ak] can be determined as the ratio of impact energy required to causefracture [Ef] to the cross section area [A];

A

Ea

f

k = (4.24)

For ferrous metals and some other materials, the tests are normally conducted over a wide rangeof temperatures so as to establish the ductile to brittle transition temperature. Typical transitioncurves are shown in Fig 4.11. Material A has a higher transition temperature (TA) than material B(TB). Material C shows no transition. It is always desired that a material should not exhibit brittleto ductile transition within its service temperature range.

Criteria for determining the transition temperature is based on the transition energy absorbed,changes in the fracture appearance (measure of the degree of crystallinity) or a transition inductility, as measured by the contraction of the root of the notch. The transition temperature istherefore selected as:-(a) the temperature at which 40% of the energy required to fracture at room temperature isattained, or(b) the temperature where 50% crystallinity is obtained, or(c) the temperature where 1% lateral contraction of the notch is attained.

It has been observed that a transition based on fracture appearance always occurs at a highertemperature than if based on a ductility or energy criterion.

4.2.2. Metal Fatigue.

Failures occurring under conditions of dynamic loading are called fatigue failures. Fatigue results ina brittle fracture with no gross deformation. A fatigue failure can be recognized by the appearanceof the fracture surface, which consists of a smooth region, due to the rubbing action as the crack ispropagated, and a rough region, where the cross section would no longer carry the load, hencefailure is in a ductile manner. The progress of the fracture is denoted by a series of rings, referredto as beach marks which progress from the point of crack initiation. Fatigue failures are oftenobserved to arise at sections of high stress concentration such as sharp corners or notches.

The fatigue life is the number of load cycles, which a material can undergo under some stressloading condition before sustaining fracture. It consists of three distinct periods, namely: (i)

initiation, (ii) propagation, and (iii) fast fracture. The initiation period is largely governed by surfacecondition and mechanisms that tend to damage the surface. Fatigue cracks normally start at the

76

A

B

C

T TA B

Temperature

Energy

absorbed


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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------surface. A rough surface already is considered to be initially cracked, hence initiation is completeeven before application of a stress cycle. The period of crack propagation depends on thematerial properties such as toughness, and its ability to arrest the crack growth. This is usually thelongest and most dominant in fatigue life.

There are three basic factors necessary to cause fatigue failure:-(1) a maximum tensile stress of sufficiently high magnitude(2) a large enough variation or fluctuation in the applied stress and(3) a sufficiently large number of cycles of the applied stress.

Other factors that may alter the conditions for fatigue are stress concentration (geometry),corrosion, temperature, metallurgical structure, residual stress, combined stresses and surfaceroughness.

Fig. 4.12. The S-N curves for ferrous and non-ferrous metals

Fatigue testing provides to engineers, adequate information on the behaviour of materials underrepeated loads. In this type of test, a specimen is subjected to a fluctuating load until it fractures.The number of cycles to fracture is recorded. The test is repeated for progressively lower stressamplitudes until a stress is reached where the material sustains an infinitely large number of cycleswithout fracture. This stress is called the endurance strength of the material. The data of stressand fatigue life (number of cycles to failure) is used to construct the S-N curve as shown in Fig.4.12.

Fig. 4.13. Types of stress cycles for fatigue loading.

a - completely reversed, zero mean stress; b - Alternating, tensile mean stress ; c - Alternating,compressive mean stress; d - Pulsating tension and e - Pulsating compression.

77

Stress

amplitude

Log (Number of cycles to failure)

Non-ferrous

Ferrous

Endurancelimits

Stress

0

Tensile

Compressiv

e

b

c

d


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Fig. 4.14. The Modified Goodmans Diagram

Failure due to fatigue is largely dependent on the type of stress cycle. Normally, tensile stressesare more damaging than compressive stresses, because it is the tensile part of the stress cyclethat causes crack propagation. Different stress cycles are illustrated in Fig. 4.13.

For design purposes, the S-N curve is inadequate when we have to deal with static stressessuperimposed on alternating stresses etc. Most fatigue test machine employ the stress cycle (a) or(d). The influence of tensile mean stress can be taken into account by making use of the modifiedGoodman's diagram shown in Fig.4.14.

The construction of the Goodman's diagram can be easily done once the endurance strength [e]and the tensile strength [u] for the material are known.

4.3. LONG TERM (TIME DEPENDENT) MECHANICAL BEHAVIOUR.

4.3.1. Creep.

Creep is the slow continuous deformation with time at a constant load. In metals, creep occurs atelevated temperatures T> 0.3 to 0.4 Tm, where Tm is the absolute melting temperature. Inpolymers, creep is observed at ambient temperatures, and mechanical properties of such materialsare sensitive to strain rate and temperature.

At low temperatures, the strain is only dependent on the acting stress, i.e.

= f () (4.25)

At elevated temperatures, in addition to stress, creep strain depends also on temperature and time,i.e.

= f ( ,t,T) (4.26)

Creep tests are performed by loading a specimen in tension or compression at a constant loadinside a furnace maintained at a constant temperature. The deformation (extension or contraction)is measured as a function of time. Data thus obtained is used to construct a characteristic creep

curve as shown in Fig. 4.15.

78

Fatigue failure

Mean stress

Stressamplitude

0

+e

e

u

Maximum stress line

Minimum stress line

Mean stress line


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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------In the first stage of creep, many dislocations climb away from obstacles, slip, and contribute todeformation of the metal. Eventually, the rate at which dislocations climb away from obstaclesequals the rate at which dislocations are blocked by other imperfections. This leads to secondstage or steady-state, creep. The slope of the steady-state portion of the creep curve is the creeprate.

t

=

(4.27)

Eventually, during third-stage creep, necking begins, the stress increases, and the specimendeforms at an accelerated rate until failure occurs. The time required for failure to occur is therupture time. Either a higher stress or a higher temperature reduces the rupture time andincreases the creep rate (Fig.4.16).

The creep strength (creep) is the constant nominal stress that will cause a specified creep rate at aconstant temperature. It is often described in the format,

/t/T = creep (4.28)

where is the creep strain, t is the time in hrs, and T the temperature in Celsius. For example,1/10000/650 = 300, means that a creep stress of 300 MPa produces a creep strain of 1% in 10000hrs at 650 oC.

Fig. 4.15. A typical creep curve showing the three stages of creep: I primary, II secondary and III tertiary.

Fig. 4.16. The effect of temperature and stress on the creep behaviour

At constant temperature, the strain rate varies with the stress according to a power law,

nB = (4.29)

where n is called the creep exponent (n = 3......8).

79

o

Strain

I

II III

Time

t

e

Increased stress

Strain

Time

Increased temperature

Constant TempConstant Stress


15/21

Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------Creep is said to be diffusion controlled and therefore proceeds according to the Arrhenius law,

=RT

QCexp (4.30)

where R is the universal gas constant, (equal to 8.31 kJ/mol.K), Q is the activation energy forcreep (J/mol), T is the absolute temperature, and C is a constant.

Combining the above two equations 4.29 and 4.30, the general equation for creep becomes,

=RT

QA n exp (4.31)

where A is the creep constant, n and Q are constants, which vary with the material and can bedetermined experimentally.

4.3.2. Stress relaxation.

Stress relaxation is a time dependent decrease in stress in a body, which is constrained to a fixeddeformation, e.g. bolts in flanged connections. Stress relaxation tests are often used to estimatere-tightening time in high temperature bolted joints. They may also be used to estimate the timerequired to relieve residual stress by thermal treatments. The relaxation time is defined as the timetaken for the stress to relax to half its original value.

Fig. 4.17. Stress relaxation curves

The resulting stress vs time curves (Fig. 4.17) are called stress relaxation curves, and may beused to estimate the relaxation period, i.e. the time required for the stress to reach a critical value(Fig. 4.18).

Consider a bolt, which is tightened into a rigid component so that the initial stress in the shank is i.Since the length of the shank is constant, the total strain in it is constant. The total strain tot. iscomposed of the elastic strain el and the creep strain cr which causes stress relaxation, hence atany time,

tot = el + cr (4.32)

But, el =E

, and at constant temperature, the creep strain rate obeys a power law,

80

i

T1

T2

T3

T1


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ncr Bt

=

(4.33)

Differentiating equation 4.32 with respect to time,

ttt

creltot

+

=

Hence,

nBtE

+

=1

0 (4.34)

Integrating,

=t

i

n BEt0

1

hence,

BEtnn

i

n)1(

1111

=

(4.35)

If the bolt has to be retightened when the stress has fallen to2

i, then from equation (4.35), the

time for re-tightening is given by

1

1

)1(

12

=n

n

rBEn

t

(4.36)

Fig. 4.18. Bolt re-tightening

4.4. HARDNESS TESTING

81

Creep strain

Elastic strain

Time

Stra

in

i/E


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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------Hardness is a measure of resistance to deformation or resistance to indentation. There are threebasic principles employed in hardness testing: (i) scratch hardness (ii) indentation and (iii) reboundor dynamic hardness. Of these, only indentation hardness is of importance to engineers.

Scratch hardness is used by mineralogists, and is measured according to Moh's scale [Talc = 1,annealed copper = 3, Martensite = 7, Diamond = 10].

In dynamic hardness, an elastic ball or hammer is dropped onto the surface to be tested, andhardness is expressed as the energy of the impact. The shore Scleroscope is a typical example ofa dynamic hardness tester, which measures hardness in terms of height of rebound of thedropping hammer.

The three common techniques available for indentation hardness testing, namely, the Brinell,Rockwell and Vickers methods will be described in more detail.

4.4.1. Brinell hardness

Fig. 4.19. Principle of the Brinnel test

The Brinell hardness test consists of a steel ball indentor, applied onto the surface to be tested bya load whose magnitude depends on type and thickness of the metal. The Brinell hardnessnumber (BHN) is expressed as the load divided by the surface area of the indentation

[ ] DtP

dDDD

PBHN =

=

)(2

22 [kg/mm2] (4.37)

where,

P = applied load, kgD = diameter of indentor ball, mmd = projected diameter of indentation, mmt = depth of the impression, mm

From Fig. 4.19, it can be seen that d=Dsin.

Hence,

)cos1(2

2

=

D

PBHN

(4.38)

82

P

d

D

t


18/21


In order to obtain the same BHN with a non standard load or loading ball diameter, the angle 2 must remain constant. Hence the load and ball diameter must be varied in the ratio,

kD

P

D

P

D

P

D

P

i

i =====22

3

3

22

2

21

1 ...(4.39)

The constant k is 30 for steel and cast iron, 10 for copper alloys and, magnesium alloys, and 2.5for Aluminium and its alloy.

The Brinell hardness value is used to estimate the ultimate tensile strength of some steels.

UTS = 3.45 x BHN [N/mm2] (4.40)

4.4.2. Vickers hardness

The Vickers test consists of indenting the surface to be tested using a diamond pyramid indenterwith an included angle of 136 between opposite faces, into the smooth surface to be tested. Thisangle approximates the most desirable ratio of indentation diameter to ball diameter in the Brinelltest. The Vickers hardness number (VHN) is defined as:

VHN = Load/Projected surface area of indentation

Practically, this area is calculated from microscopic measurements of the lengths of the diagonalsof the nearly square impression. The VHN may be determined from the following equation:-

22854.1

2sin2

d

P

d

P

VHN =

=

(4.41)

where,P = applied load,. kg

d = average length of diagonals, mm

= angle between opposite faces of the diamond pyramid, 136

The Vickers method is widely accepted because it covers a wider range of applications, that is, itcan be used from very soft materials (VHN=5) to extremely hard materials (VHN=1500). Theindenting loads are selected on the basis of specimen size and material hardness from the rangeof 1 to 120 kgf. The method is also commonly employed in micro-hardness testing where loads assmall as 5g are used. The values of hardness in VHN are nearly equal to those in BHN up toabout 400.

Fig. 4.20. Types of Vickers indentations: (a) perfect, (b) barreled and (c) pin-cushioned.

A perfect indentation is square. However, most practical indentations are barrel-shaped or pincushioned as illustrated in Fig.4.20. Therefore the diagonal measurements are not so accurate.

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a bc


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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------The barrel shaped indentation is found in cold worked metals while annealed metals tend to givethe pin cushioned type.

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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------4.4.3. Rockwell tests

This is perhaps the most widely used method for hardness testing. The Rockwell method is widelyaccepted due to speed, freedom from operator error, high sensitivity, and the small size ofindentation. In this test, the depth of the indentation at a constant load is taken as a measure ofhardness. The harder the material, the less the penetration or depth of indentation. Calibration isusually carried out using standard blocks of known hardness.

The indenter is either a conical shaped diamond cone (brale) with an apex angle of 120 or ahardened steel ball 1/16 inch diameter. The brale is used for testing harder materials.

Fig. 4.21. The Rockwell method

The test procedure starts by application of a preload of 10kg. A major load is then applied and thedepth of indentation is automatically recorded on a dial gage in terms of arbitrary hardnessnumbers (HR). Major loads of 60, 100 and 150 kg are used. Hardened steel is tested on the Cscale using the 150 kg load. The useful hardness range is 20 - 70 HRC. Softer materials aretested on scale B with the 1/16 inch diameter steel ball and a 100 kg major load. The range in thisscale is 0 - 100 HRB. The A scale uses the brale, and a 60 kg major load. This scale is applicable

to a wider range of materials, from annealed brass to cemented carbides.

The Rockwell hardness is calculated from the equation:-

c

hhkHR

o)( 1

= (4.42)

where, as shown in Fig. 4.21,k = 0.2 for brale, 0.26 for the steel ballh1= depth of penetration in mm of indenter after removal of major loadho = depth of penetration in mm of indenter under application of the pre-loadc = the value for scale divisions (0.002mm)

4.4.4. The shore scleroscope

The scleroscope tests are dynamic, using the rebound energy as a measure of hardness. Thesetests are applicable to even very thin materials made of very hard materials. The testinginstruments are usually portable and simple, hence tests can be made quite rapidly.

The shore scleroscope consists of a small diamond pointed hammer, 2.5g, which is allowed to fallfreely from a height of about 250 mm down a glass tube graduated into 140 equal parts. Theheight of the first rebound is taken as an index of hardness. A rebound of 100 is equivalent to thehardness of martensitic high carbon steel.

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P

Po

h

P= Major Load

Po = Pre-load

h1ho


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Chapter 4: The Mechanical Properties of Materials---------------------------------------------------------------------------------------------------------------------------------------------REVIEW PROBLEMS

1. A 3-mm diameter beryllium wire with a modulus of elasticity of 250 GPa is 2500 cm long.Calculate the length of the wire when a force of 20 kN acts on the wire.

2. A square 25.4 mm x 25.4 mm tensile bar with a 50.8 mm gage length is pulled to failure.The final distance between the gage marks is 69.8 mm and the final dimensions at thefracture are 20.8 mm x 20.8 mm. Calculate the % elongation and % reduction in area.

3. Discuss the importance of the impact test on steel and other temperature and notch sensitive materials.

4. Describe three indentation hardness measuring techniques. Which of the three is simplerand easier to adapt for mass production?

5. A three point bend test is performed on a block of silicon carbide that is 10 cm long, 1.5 cmwide, and 0.6 cm thick, and is resting on two supports 7.5 cm apart. The sample breakswhen a deflection of 0.09 mm is recorded. The flexural modulus for silicon carbide is 480GPa. Assume that no plastic deformation occurs. Calculate

(i) The force that caused the fracture and(ii) The flexural strength.

6. A Brinell hardness measurement, using a 10 mm diameter indenter and a 500 kg load,produces an indentation of 4.5 mm on an aluminium plate. Determine the Brinell hardnessnumber (HB) of the metal.

7. When a 3000 kg load is applied to a 10 mm diameter ball in a Brinell test of a steel, anindentation of 3.1 mm is produced. Estimate the tensile strength of the steel.

8. The activation energy for self diffusion in copper is 49,300 cal/mol. A copper specimen

creeps at 0.002 in./in. h when a stress of 15,000 psi is applied at 600 C. If the creep rateof copper is dependent on self diffusion, determine the creep rate if the temperature is 800C.

9. FCC metals are often recommended for use at low temperatures, particularly when anysudden loading of the part is expected. Explain.

10. (i) Briefly state the factors that would promote fatigue failure in a metal.

(ii) Why is it risky to use scrap metal from materials that may have been subjected tocyclic loading during their life time?

(iii) A steel whose tensile strength is 800 GPa has been tested in fatigue, and found to

have an endurance strength of 350 GPa. Sketch the modified Goodmans diagramfor this material. What maximum alternating stress is allowable when a meanstress of 100 GPa is applied to this steel?

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Materials Science Manual Chapter 4