Mech Testing

NSW 700: WELDING METALLURGY

CHAPTER 12: Mechanical Testing

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12. MECHANICAL TESTING

12.1 Tensile testing

12.1.1 The engineering stress strain curve

The engineering tensile test curve is probably one of the most widely used tests to determine the mechanical

properties of metals. It is simple to perform and can provide significant engineering data on the strength of

materials. It has, however, also its limitations and the full understanding what this test can and can not, do is

essential for the discipline of mechanical metallurgy

In essence the engineering stress strain curve is obtained by pulling in tension a tensile test sample of which the

initial diameter and gauge length have been measured, until failure occurs and plotting the engineering stress

S (as the load P at any point/original cross sectional area A0) as a function of the engineering strain e which

is obtained from the relationship: extension L at any point/original gauge length L0. These relationships are given once more below in following the accepted notation for the engineering stresses S and strains e to

distinguish them from the later true stresses and strains .

Engineering stress at any point: 0A

PS (Eq 12.1.1-1)

where P is the applied load at any point and A0 is the original cross sectional area of the test sample.

Engineering strain at any point: 0L

Le

(Eq 12.1.1-2)

where L is the length increment at any point and L0 is the original gauge length of the test sample. A typical engineering stress strain curve is shown below.

Figure 12.1(a): A typical engineering stress strain curve for a relatively ductile metal or alloy



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Note particularly the definitions of the:

Offset yield strength S0.2;

The tensile strength SUTS or Smax;

The fracture stress Sf;

The uniform strain eu;

The strain to fracture ef; and

The point where so-called necking begins at eu and Smax.

Engineering tensile strength Rm or Smax The engineering tensile strength or the ultimate tensile strength (UTS) of a material is obtained from the

maximum load that the material can sustain under uniaxial loading conditions before the onset of plastic

instability. It is commonly quoted and has served a purpose for many decades as a design criterion where a

suitable safety factor was included in the design of a structure or a component to allow for the difference

between the point of yielding and the maximum tensile stress value. With the more advanced understanding of

the role of complex stresses within most structures that are not only loaded uniaxially, however, the more

rational use of basing the design rather on the strength at yielding has become generally accepted. In the SI

units the tensile strength is known as Rm.

Engineering yield strength Rp or S0.2

The offset yield strength ( or sometimes called the proof strength in the UK) is an engineering method to

define the practical onset of plastic yielding where a definite yield point is difficult to establish, as in the above

example for a ductile metal. Mostly the arbitrary offset of 0.2% plastic strain is used to define this point of

yielding as S0.2 or Rp0.2 according to the SI units whereas actual yielding (defined by the onset of the

movement of the first dislocations) has, of course, already started before at the so-called elastic limit.

Sometimes the offset plastic strain of 0.1% or even 0.5% are also used and the correct way of notification is to

include the offset strain as a subscript to the strength designation, i.e. Rp0.1 or Rp0.5 as the case may be.

Note that strictly in the SI units, the notation Rp is usually reserved for hardened and tempered steels whereas

ReL and ReH are reserved for unhardened steels with a clearly defined yield strength range.

The proportional limit is defined as the highest point on the stress strain curve where the stress is still directly

proportional to the strain, i.e. a linear or elastic proportion. The true point of yielding is called the elastic limit

and is defined as the greatest stress that the metal can withstand without any measurable permanent strain after

a complete release of the load. The determination of the elastic limit is a tedious method to perform and requires

a process of loading and unloading with measurements for permanent strain in between. The measurement of the

elastic limit is, of course, dependent on the sensitivity of the strain measuring equipment and at the ultimate point

of an infinite sensitivity, the elastic limit would coincide with the proportional limit (so-called because of the

linear proportion of the elastic strain and Youngs modulus from Hookes law in the elastic region, i.e. Sel = e E).

The term flow stress is also frequently encountered, particularly with high temperature deformation studies, and

this defines a stress level at any point in the plastic deformation portion of the stress strain curve where the

material flows under deformation.



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Some of these yield parameters are defined in the Figure below.

Figure 12.1(b): A typical engineering stress strain curve for a ductile material with the criteria for yielding defined

as: point A: elastic limit, point A: proportional limit, point B: yield strength, line CB: offset yield strength, OC: offset elongation

Note: Distinguishing between the strength level and a stress level needs to be understood.

A strength level, i.e the offset yield strength S0.2 or the UTS Smax, is a specific stress point (or sometimes called a critical point) on the stress strain curve that defines a material characteristic or

constant and is independent from the test method or the laboratory where the test was performed. In fact

it is a value that can be found even outside the laboratory in hand books.

A stress level S, however, defines any point on the stress strain curve at a given test load P and is known as a test variable. The difference between a stress (a test variable) and the strength (a material

constant) is sometimes even confused in the literature.

Finally, one also needs to distinguish between an independent test variable (for instance the load P or diameter D0 or cross sectional area A0 of the specimen, which can mostly be freely chosen) and a

dependent test variable, i.e. the stress S or engineering strain e at any point on the S e curve that arises from the application of the load P and the value of A0.

Engineering strain e

The onset of yielding (which is fundamentally defined by the onset of permanent dislocation movement under

stress) is most difficult to measure in practice and is a function of the sensitivity of the measuring instrument.

The terms uniform strain and necking are related, with the first term describing the engineering strain e

during which the gauge diameter of the tensile sample reduces uniformly as it elongates homogeneously. At the

point of necking, however, a so-called plastic instability sets in and further elongation is now concentrated

heterogeneously in only a small area of the gauge length and this area is called the necking area. The further

elongation of the sample with applied load is now primarily concentrated in this necked area until final fracture

occurs. In engineering terms, this stress at the maximum load at which necking begins, is called the tensile

strength or the Ultimate Tensile Strength UTS SUTS or Rm of the material.

A measure of ductility

Ductility is a qualitative term that is a subjective property of the material and the short comings of a general term

such as ductility, will be encountered once more in Chapter 5 in the topic of Fracture Mechanics where it will

be seen that a seemingly ductile material can also fracture in a seemingly brittle manner under the right conditions. Nevertheless, the use of the general term ductility is useful as it is very widely used in practice.



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Finding a quantitative expression for the ductility of a metal, is of interest in three areas:

To indicate to which extent a metal can be deformed during cold and hot working operations such as bending, rolling etc.

To indicate to the designer to which extent a metal in a structure will deform plastically before it fractures. A metal with a high ductility, will give ample warning before it fractures catastrophically.

Finally, the ductility of a metal can serve as an indicator of the level of impurities in the metal as the ductility is very often sensitive to these impurities.

From the above stress strain curve, the engineering strain at fracture ef is given by:

0

0f

0

fL

LL

L

Le

(Eq 12.1.1-3)

where Lf is the measured gauge length at fracture and L0 is the original gauge length. Because a significant

proportion of the fracture elongation will be concentrated in the necked area, the fracture elongation ef will be

a function of the original gauge length L0 and should be quoted always with the value of ef, i.e. for instance

ef(25mm). In line with the international SI system, the fracture elongation is usually known as A5 and would be

written as A5(25 mm).

The reduction in area in the SI units, is known as Z:

0

f0

A

AAZ

(Eq 12.1.1-4)

where A0 is the original cross sectional area of the gauge length and Af that at fracture. The reduction in area

Z does not suffer from the gauge length problem of the elongation at fracture ef and is independent from the

original gauge length L0. From this, a zero gauge length elongation value has been defined through the

constancy of volume principle, where it is assumed that practically no necking occurs because of the vanishingly

short gauge length (or conversely, that our gauge length is so short that it falls fully in the center of the necked

region):

A0L0 = AfLf (Eq 12.1.1-5)

And )Z1(

1

A

A

L

L

f

0

0

f

(Eq 12.1.1-6)

and defining the zero gauge length elongation e0:

)Z1(

Z1

)Z1(

11

A

A

L

LLe

f

0

0

0f0

(Eq 12.1.1-7)

Z1

ZAe )0(S0

(Eq 12.1.1-8)

This expression will represent the elongation at fracture of a specimen with a very short gauge length of L0 0 and e0 or A5(0) will essentially be independent of the initial gauge length.



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Another method to avoid the complications of the gauge length dependence of the total strain to fracture ef or

A5, is to use the uniform strain eu until the onset of necking. This term correlates quite well with stretcher

forming operations although the engineering stress strain curve is often quite flat in this region and it is,

therefore, often difficult to find the exact point of the uniform strain on this curve.

The modulus of elasticity

The slope of the initial linear portion of the stress strain curve is the modulus of elasticity or Youngs modulus according to Hookes law:

el

el

e

SE (Eq 12.1.1-9)

where Sel is the engineering elastic stress at any point on the linear portion of the curve and eel is the

corresponding engineering elastic strain at that same point. The elastic modulus E is also a measure of the

stiffness of the material and is very little dependent on the microstructure of the material. This is basically

because it is derived from the interatomic forces of the lattice atoms and these are not affected by the macro-effects of the microstructure, such as the presence of precipitates, the grain size etc. The elastic modulus is, however, sensitive to the temperature and generally reduces slightly with an increase in temperature.

The modulus of resilience UR Another parameter that is sometimes obtained from the elastic portion of the stress strain curve, is the modulus of

resilience UR of the material. This parameter is the ability of a material to absorb energy when deformed

elastically and to return this energy when it is again unloaded. The modulus of resilience UR is usually

measured as the elastic strain energy per unit volume required to stress the material to the elastic limit and is

obtained from the area under the elastic portion of the stress strain curve:

E2

S

2

)E/S(S

2

eSU

2elelelelel

R

E2

SU

2el

R (Eq 12.1.1-10)

The toughness of a material

The toughness of a material is its ability to absorb energy in the plastic strain range and is a parameter that is

particularly difficult to define from a stress strain curve, as will be seen later in Chapter 4 where Fracture

Mechanics will be covered. In many components such as train couplings, crane hooks, chains etc, it is of interest

to know whether a component will be able to withstand an occasional load exceeding its yield strength without

fracturing catastrophically in a rapid or brittle manner. Where pre-existing flaws in a component may be present

that provide sites for a stress concentration, fracture may even occur at a nominal stress below the yield strength.

This forms the basis for Linear Elastic Fracture Mechanics. In both cases the concept of yield before break should apply.



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The toughness of a material is actually represented by the total area under the stress strain curve and not only

the elastic portion as with the modulus of resilience. This total area is a measure of the total work per unit

volume done on the material before fracture will occur and is often known as UT. This is still a simplistic way of

measuring the fracture toughness of a material as uniaxial stress conditions very seldom exist in real life (for

instance at the tip of a crack) and that the generally accepted way is by measuring the fracture toughness as KIC

for plane strain conditions

Figure 12.1(c): Comparison of the typical stress strain curves for a steel with a high toughness (the structural steel)

and one with a low toughness (the spring steel) with an indication of the modulus of resilience UR for the two steels

also shown as the cross hatched areas under the elastic portions of the curves.

Note that the spring steel is required to have the larger modulus of resilience UR due to its function, but it has

the lower toughness if compared to the structural steel. The toughness of a material is, therefore, a parameter

that is affected by both the strength and ductility of the steel whereas the resilience is affected primarily by

only the strength (or elastic limit) of the steel.

Many attempts have been made to integrate the area underneath a stress strain curve and some approximations

have been proposed as follows:

For a ductile material such as structural steel:

UT SUTS ef or 2

e)SS(U fUTS2.0T

(Eq 12.1.1-11)

where S0.2 is the 0.2% offset yield strength and SUTS the tensile strength of the material.

For a spring steel: 3

eS2U fUTST (Eq 12.1.1-12)

At most these equations are only rough guidelines and attention to the published values of the fracture toughness

KIC should rather be given as the latter is based on sound fundamental principles of the stress intensity KI (or

KII or KIII as the case may be) at the tip of a notch or crack and not only the nominal uniaxial stress S applied

to the structure or component.



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12.1.2 The true stress strain curve

The true stress strain curve versus the engineering stress and strain

The engineering stress strain curve is a convenient and easy measurement of some very useful engineering data

on the strength of materials. It has a major shortcoming, however, in that it is entirely based on the original

dimensions of the specimen whereas the constancy of volume principle, dictates that these dimensions must

change as the specimen elongates. The true stress strain curve overcomes this by relating the stress to the

instantaneous cross section and the strain to the instantaneous gauge length at every point during the tensile

test. In an engineering stress strain curve, the significant effect of necking on the cross sectional area is also

ignored with the consequence that the engineering stress appears to fall after necking has started. During necking,

however, the material must continue to work harden and the actual stress during this stage should continue to

rise.

A true stress strain curve is also known as the flow curve of the material as it is based on the actual plastic flow

properties of the material.

The true strain during uniform strain

Up to the point of necking during which uniform or homogeneous deformation occurs, the volume of the gauge

length may be assumed to be constant, i.e. A0 L0 = Ax Lx where Ax and Lx are the cross sectional area and the

gauge length at any point x up to the point of necking.

The true strain at any point will be given by the summation of the actual strains at every point along the stress

strain curve, as follows:

etc......L

LL

L

LL

L

LL

2

23

1

12

0

01

0L

Lln

L

dL (Eq 12.1.2-1)

D

Dln2

A

Aln

L

Lln 00

0

(Eq 12.1.2-2)

From the engineering strain equation:

1L

L

L

LL

L

Le

0

x

0

0x

0

(Eq 12.1.2-3)

and 0

x

L

L)1e( and

0

x

L

Lln)1eln( (Eq 12.1.2-4)

or = ln(e + 1) (Eq 12.1.2-5)

This equation is valid until the point of necking is reached and means that the true diameter (or the cross

sectional area Ax) need not be measured at every point but can be calculated from the engineering strain e if the

starting values of L0 and A0 are known. Beyond the point of necking where the strain becomes

inhomogeneous, however, the true diameter of the necked region needs to be measured at every point until

fracture occurs.



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The true stress

The true stress at any point on the stress strain curve is given by:

x

x

A

P (Eq 12.1.2-6)

Using the expression for the engineering stress:

0

x

A

PS (Eq 12.1.2-7)

one can rewrite the expression for the true stress as;

x

0

0

x

x0

0x

A

A

A

P

AA

AP and

0

x

x

0

L

L

A

A

Therefore: )1e(SL

LS

0

x (Eq 12.1.2-8)

Therefore: = S (e + 1) (Eq 12.1.2-9)

The true stress strain curve

The true stress strain curve is shown schematically below for the case where the calculation of the true stresses

and true strains according to the above equations, were used throughout to fracture.

Figure 12.1(d): The schematic difference between the engineering and the true stress strain curves.

Note that because of the corrective factor of (e + 1) that is applied to the engineering stress S to convert it to the

true stress , the true stresses are higher than the engineering stresses. This also applies to the true strain at

fracture f or point b. The stress analysis after necking has started, is not necessarily a simple one as a necked region introduces some elements of a mild notch with its triaxial stress distribution, into the test. Often the true

stress strain curve is simply taken as a linear extrapolation from the point a where necking has started, to point

b where fracture occurs.



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The true maximum stress u at the maximum load Pmax The true stress u at the point where true uniform elongation u ends and necking starts, corresponds to the true tensile strength. Assume that the cross sectional area of the specimen is Au at this point and that the load

has reached the maximum value of Pmax.

The UTS in engineering terms is defined as:

0

maxu

A

PS (Eq 12.1.2-10)

and u

maxu

A

P (Eq 12.1.2-11)

Eliminate Pmax from these two equations:

u

0uu

A

AS and u

u

0

A

Aln

(Eq 12.1.2-12)

Therefore: u = Su exp(u) (Eq 12.1.2-13)

True fracture stress f Because the true fracture stress f should strictly be corrected for the triaxial state of the stresses in the necked region (which is not a simple problem) the calculated true fracture stresses are generally only approximations

and are prone to error. This, therefore, is a parameter that is not used very frequently in practice.

The true fracture strain f The true strain at fracture would be given by: f = ln(A0/Af) where Af is the specimen area at fracture, i.e. after necking. This parameter represents the maximum strain that the specimen may undergo before it fractures

and should correspond to the engineering strain at fracture ef. Because the earlier equations to convert an

engineering strain to a true strain, are not valid beyond the point where necking starts, one may not use this

conversion directly. It is true, however, that the reduction in area Z is closely related to the true strain at

fracture f through the following equation:

)Z1(

1lnf

(Eq 12.1.2-14)

The true uniform strain u This is the true strain based on the load Pmax until necking starts and it may be calculated either from the length

at the end of uniform strain load Lu or from the cross sectional area at this point Au as the constancy of

volume equation of A0L0 = AuLu still holds up to this point, even with subsequent necking, and:

)1eln(A

Aln u

u

0u (Eq 12.1.2-15)



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The true local necking strain n This would be the incremental strain required to deform the specimen from the onset of necking to the point of

fracture, i.e. n = (f - u) and is also given by:

f

un

A

Aln (Eq 12.1.2-16)

where Au and Af are the respective cross sectional areas at the end of uniform strain and at fracture.

The effect of the initial gauge length L0 on ductility determinations in the tensile test

As stated before, the quantitative measurement of the ductility of a metal is quite difficult to define. A seemingly

ductile metal may also fracture rapidly in a brittle mode under certain conditions. Nevertheless, the term

ductility, even if applied rather loosely, is ingrained and has some use in engineering design. Attempts at

determining the ductility of metals via the stress strain curve have, therefore, been undertaken.

The measured elongation in a tensile test depends on the dimensions of the specimen and, in particular, on the

gauge length L0 chosen. The total strain to fracture therefore, consists of two components, i.e. the uniform

strain u and the incremental necking strain n that occurs between the onset of necking and fracture.

The uniform strain u depends to a large degree on the metallurgical microstructure of the material which is again dependent on the strain hardening exponent n (see later) as well as the shape of the specimen gauge

length, its shoulder profiles etc. The variation of the locally measured strain at an infinitely small gauge length is

shown schematically below.

Figure 12.1(e): Schematic representation of the localized elongation with position along the gauge length of a tensile

specimen with a necked region.

This gauge length effect is shown below for a specimen where the gauge length was varied from about 25 mm to

230 mm.



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Figure 12.1(f): Effect of gauge length on the measured elongation during a tensile test.

The extension of a tensile specimen at the point of fracture, can be given by:

0un0ff LeLLLL (Eq 12.1.2-17)

where Ln is the necking extension up to fracture and euL0 is the uniform extension obtained from the engineering strain eu.

The tensile engineering elongation at fracture ef is then given by:

u0

n

0

0ff e

L

L

L

LLe

(Eq 12.1.2-18)

This clearly proves that the total engineering elongation ef at fracture, is a function of the initial gauge length L0

with a shorter gauge length L0 leading to greater total elongation ef.

12.1.3 Standardised specimen dimensions

Because of the above dependence of the engineering elongation on the gauge length L0, specimen sizes have

been standardised through ASTM A370: Standard Methods and Definitions for Mechanical Testing of Steel products and ASTM E8: Standard Methods of Tension Testing of Metallic Materials.

Location of specimens: The orientation and location of specimens from a product need to be defined carefully

and is covered in the above ASTM Standards. This particularly true in wrought products where rolling or forging

direction needs to be considered but also in steels that had been cooled from a higher temperature. The cooling

rate at the surface of a cooling product is always faster than in the interior and the microstructure of steels (but not

only steels) is particularly sensitive to cooling rate. For instance, specimens cut from near the surface of a casting

will be stronger than for one cut from the interior.

On long bar products of round, hexagonal or square shape, ASTM A370 recommends that the specimens be cut

from the area halfway between the surface and the center of the bar. For forgings, ASTM E8 recommends that

specimens be taken from the thickest part of the forging.



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Size and shape of specimens:

Figure 12.1(g): (Top) ASTM A370 nomenclature of tension specimens and (bottom) specimen dimensions for round

and flat specimens.

Note that in round specimens the diameter to gauge length ration is maintained at 4:1 for other specimen sizes in order to be able to compare elongation values with each other;

Specimen with a 50 mm gauge length from thin flat products. These are defined as sheet, thin plate, flat wire and strip, band, hoop etc. with a nominal thickness of between 0.13 to 16 mm ;

Specimen with a 200mm gauge length from flat products, such as plate, shapes and other flat products.

In flat products with a thickness greater than 16 mm, round specimens are preferably machined from the flat product.

Curved specimens: Curved specimens generally result from large pipes cut in the hoop direction or other

irregular shapes. These pose a particular challenge in the production of gas or liquid line pipe as the straightening

or flattening of the curved specimen introduces an oppositely signed strain into the specimen which introduces the

Bauschinger effect and often leads to lower measured strength values. The Bauschinger effect is found where a

specimen is partly strained in tension to a certain flow stress 1 and the stress is then relaxed. There after the

stress is reapplied but now in compression and yielding then takes place at a stress 2 which is significantly

lower than the original 1.

This is illustrated schematically below where the difference in yield strength is shown by the drop (1 2).



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Figure 12.1(h): Illustration of the Bauschinger effect. The specimen was strained in tension along line OA first and at

point A the stress was relaxed according to line AD. Upon reapplication of the tensile stress the specimen will yield

again at point A (or A) and continue along the extension of the original curve. The stress was then reversed and the specimen restrained in compression and it now deformed along line DB and not line DAA as was the case without a stress reversal.

In line pipe formation from flat plate, the pipe is first formed into an U and then in a second step into an O

before welding. This introduces a tensile strain after fabrication on the outer skin. After fabrication, a hoop

section is usually cut from the pipe to obtain a tensile test piece in the hoop direction (where the higher stresses

will be in operation). If this curved specimen is then flattened for tensile testing, a compressive strain is now

applied onto the outer skin and this will affect the tensile test curve through the Bauschinger effect.

Barbas law: This makes it imperative that ductility comparisons can only be done on geometrically equivalent specimens and various equations have been proposed to find the geometrical equivalence. As a general rule one

may use Barbas law that predicts that if the ratio ( 0A )/L0 is the same between two differently sized

specimens, then geometrical equivalence has been achieved.

Comparing elongation of a round with a flat specimen: ASTM A370 Section VI, provides an empirical

conversion equation for steels:

a

rflL

A47.4ee

(Eq 12.1.3-1)

where efl is the measured engineering elongation of a flat specimen with cross sectional area A and gauge

length L, er is the engineering elongation of a standard round specimen with 50 mm gauge length and 12.5 mm

diameter and a is an empirical constant and has been found to be 0.4 for many grades of carbon and low alloy

steels and is 0.13 for annealed austenitic stainless steel. This equation may not be used for flat specimens below

0.64 mm thick and is also not valid for non-ferrous materials.

2

1



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12.1.4 Tensile testing machine characteristics

In many laboratories the elastic deflection of the cross head and load cell of the testing machine is disregarded as

it often is considered to be much smaller than that of the specimen. This is, however, not always true and in many

instances the elastic stiffness K of the machine needs to be considered. The effect is shown schematically in the

Figure below.

Figure 12.1(i) Schematic sketch of the effect of elastic deformation by the cross head of the machine. (i) before loading,

(ii) after loading in a very stiff machine and (iii) actual elongation with elastic deformation by the machine of M

decreasing the actual elongation.

Note that the actual elongation L is decreased by the amount M = FK where F is the applied load and K is the spring constant or stiffness of the machine. To model the effect of machine stiffness, consider a tensile test as shown schematically in the Figure below.

Figure 12.1(j): A schematic diagram of a tensile test with the machines elastic behaviour represented by a spring that deforms elastically.



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Assume that the elastic deformation of the machine includes that of its load cell and cross head as represented by

an imaginary spring with a spring constant of K. Consider, furthermore, that the cross head of the machine

moves at a constant speed of s = dL/dt so that the cross head displacement in a time t is given by (s t). The

total elastic displacement of the machine is given by (K F) where F is the applied force on the specimen and

the plastic elongation of the specimen is given by (p L0) where p is the plastic strain and L0 is the original gauge length of the specimen. The total displacement (s t) of the cross head is then given by:

s t = K F + p L0 (Eq 12.1.4-1)

The plastic strain p and the plastic strain rate p of the specimen is then given respectively by:

0

pL

}KFt)dt/dL{( and

0

pL

)}dt/dF(Kdt/dL{

dt

d

(Eq 12.1.4-2)

Note, therefore, that the plastic strain p and the plastic strain rate p are dependent not only on the cross head

speed dL/dt but also on the instantaneous load F and the instantaneous loading rate dF/dt respectively where

both will vary with the applied stress along the stress strain curve. Experimental values of machine stiffness have

produced values ranging from 740 kg/mm up to 2970 kg/mm.

Note from the above equations that the instantaneous loading rate dF/dt varies more sharply during elastic

loading which occurs over a smaller elongation than during the later plastic loading which occurs over a larger

elongation. The stiffness of the machine, therefore, plays a more significant role in determining the yield

strength, particularly where a yield drop occurs as in low carbon steels. It is mostly for this reason that the

determination of the elastic Youngs modulus E from a conventional stress strain curve is not very accurate unless a machine with a very high stiffness had been used. At the UTS, however, dF/dt = 0 and the measured

plastic strain rate equals the actual strain rate p at this point.

12.1.5 An equation to describe the true stress strain curve

The Hollomon equation

Many attempts have been made throughout the years to express the true stress strain curve mathematically, with

varying success. The equation that appears, however, to possibly come the closest is the so-called Hollomon

equation:

= K n (Eq 12.1.5-1)

where K is a constant (called the strength coefficient) and n is called the strain hardening exponent. A log-

log plot of this equation should provide a straight line as shown below.

Figure 12.1(k): A schematic log-log plot of the strain hardening equation.



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The slope of this straight line is n and the value of the strength coefficient K is found at the interpolation of = 1, (or at Z = 0.63) as shown above.

Note: The Hollomon equation is an empirical equation and has no theoretical basis. It is also very seldom that

a metal will show a single straight log-log line over the entire plastic deformed area and it is, therefore, common

to rather find the value of n at relatively low plastic strains where a straight line on the {log vs log } plot is

usually found. In spite of these shortcomings, it still remains a very useful mathematical tool to analyse the curve.

The value of the strain hardening coefficient n may have various values as shown below.

Figure 12.1(l): The various values that the strain hardening coefficient n may assume, as shown for the plastically

deformed region of the curve.

Where n = 0, a perfectly plastic solid is found (i.e. no work hardening) and where n = 1 a perfectly elastic

solid is found (i.e. a fully elastic material without any plastic deformation). For most metals the value of n

appears to be between 0.10 and 0.50.

The strain hardening coefficient n as a material constant

The strain hardening coefficient n is known as a material constant and is, therefore, affected by the alloy

content and the microstructure of the steel but not by the external test conditions. Values for n have been

determined extensively for steels as well as for other metals and some values are shown below in the Table:

Table: Typical values of K and n for some metals

Metal Condition n K (MPa)

Fe 0.05%C steel Annealed 0.26 77

SAE 4340 Annealed 0.15 93

Fe 0.6%C steel Quenched and tempered at 540C 0.10 228

Fe 0.6%C steel Quenched and tempered at 705C 0.19 178

Copper Annealed 0.54 47

70/30 Brass Annealed 0.49 130

The value of n appears to decrease with alloying element additions for steels and the following empirical

equation has been developed for low Carbon mild steels for this purpose:

n = 0.28 0.2%C 0.25%Mn 0.044%Si 0.039%Sn 1.2%Nf (Eq 12.1.5-2)

where the element additions are in wt% and Nf = free Nitrogen content in wt%.



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In general it has been found that grain size has only a small effect on the value of n and the following empirical

relationship has been proposed for low Carbon mild steels as a function of the ferrite grain size:

)D10(

5n

2/1GS

(Eq 12.1.5-3)

where n thus increases with an increase in ferrite grain size DGS where the grain size is measured by the

intercept length method. The evidence for this relationship is, however, not very conclusive and generally it is

accepted that the value of n does not vary significantly with ferrite grain size in steels.

The rate of strain hardening d/d From the equation one may arrive at the rate of strain hardening d/d as follows:

d

d

)(lnd

)(lnd

)(logd

)(logdn (Eq 12.1.5-4)

or

n

d

d (Eq 12.1.5-5)

Note: that the strain hardening exponent n is not the same as the rate of strain hardening d/d (also called

the work hardening rate) as the latter is given by the instantaneous slope of the curve at any point whereas n is a constant.

Alternative true stress true strain equations

Alternative true stress true strain curve equations have been proposed from time to time as deviations from the

Hollomon equation at low strains (< 10-3

) and at high strains (> 1) are often found. For instance, it is often

found that the log-log plot leads to two straight lines with different slopes and it has been proposed in such cases

that the following equation should apply:

= K (0 + )n (Eq 12.1.5-6)

where 0 is considered to be the strain hardening that the sample has received prior to the tensile test.

Another common relationship is the so-called Ludwik equation:

= 0 + Kn (Eq 12.1.5-7)

where 0 is the yield strength and K and n have the same meaning as with the Hollomon equation. This equation is probably more acceptable than the original Hollomon equation as the latter implies that at zero true

strain the stress must be zero, yet this is not true. The value of 0 is, of course the yield strength and if we

accept this as the elastic limit, the value of 0 is obtained at the intercept between the elastic portion and the plastic portion of the true stress strain curve and this has been shown to be at:



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n1

1

n0 E

K

(Eq 12.1.5-8)

In the case of austenitic stainless steels such as the 18/8 variety, the true stress strain curve has been found to

best fit the following equation:

= Kn + exp(K1)exp(n1) (Eq 12.1.5-9)

where exp(K1) proportional limit and n1 is the slope of the deviation of stress from the Hollomon equation

plotted against .

Some authors have also argued (quite convincingly) that the true strain in any of the above equations, should be

the plastic strain only and that the elastic strain should be subtracted:

p = total el = total (0/E) (Eq 12.1.5-10)

12.1.6 Instability in uniform strain

A condition to find the point where necking starts, was proposed by Considere and is based on the assumption that the onset of necking is at the point of maximum load Pmax.

Pmax = u Au (Eq 12.1.6-1)

and dP = u dA + Au d = 0 (Eq 12.1.6-2)

as Pmax is at its maximum where dP = 0.

or the instability condition is: uu

d

A

dA

(Eq 12.1.6-3)

From the constancy of volume relationship Lu Au = L0A0 at the last point of uniform elongation:

Au dL + Lu dAu = 0 (Eq 12.1.6-4)

and

d

A

dA

L

dL

uu

(Eq 12.1.6-5)

and from the instability condition above:

ud

d

(Eq 12.1.6-6)

The point of necking at maximum load Pmax can, therefore, be found from the true stress strain curve where it has

a subtangent of unity (i.e. d/d = u/1) or where the rate of stain hardening d/d equals the stress u.



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Figure 12.1(m): The geometrical representation of the necking criterion where the true strains are used.

By introducing the engineering strain e rather than the true strain the construction becomes somewhat simpler.

uu

0

u

u

0 )e1(de

d

L

L

de

d

)L/dL(

)L/dL(

de

d

d

de

de

d

d

d

Therefore: )e1(de

d

u

u

(Eq 12.1.6-7)

Figure 12.1(n): Consideres construction using the engineering strain instead of the true strain



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Note that in this construction, the engineering strain e is plotted at the bottom against the true stress on the

vertical axis. In this construction the point of the end of uniform strain u need not be estimated from the curve as was the case in the upper Figure where the true strain was plotted. Here all one needs to do is to find the

point A of the negative engineering strain = 1 on the horizontal axis and draw a straight line to point C where the straight line will form a tangent to the stress strain curve. Note that this straight tangent should cross

the true stress axis at point D where OD equals the engineering stress at necking or Rm or the UTS.

Finally, one may also substitute the Hollomon equation at the point of plastic instability and find the two

equalities that were also derived above:

u

und

d and u

d

d

(Eq 12.1.6-8)

Therefore: u = n (Eq 12.1.6-9)

It has also been shown that this relation ship of u = n is independent of the power law used and will also apply if equations other than the Hollomon equation have been used.

12.1.7 The yield point phenomenon and strain ageing

The stress strain curve

The yield point phenomenon is particularly of interest in Fe C and Fe C N alloys where the interstitially dissolved C and N atoms may segregate to dislocations, thereby effectively locking them against movement until a certain breakaway stress is reached. Once these dislocations have torn themselves loose from the cloud of C or N atoms, the stress required to move the dislocation is now lower than the breakaway stress and a so-called upper yield point and lower yield point is observed in these alloys.

Figure 12.1(o): Typical yield point behaviour of a low Carbon steel



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Note the definition of the upper yield point, the lower yield point and the yield point elongation during which

so-called Lders bands at about 45 to the tensile axis of the specimen, form heterogeneously across the specimens gauge length, progressively covering almost the entire gauge length. As is evident from the above Figure, upon straining, these steels will elongate elastically first until a sharp yield point, the upper yield point, is

reached. Then a sudden drop in load will be observed with a subsequent fluctuation about some seemingly

constant value of the load before it rises again steadily as for a normal stress strain curve. Typically, the first

Lders band will form at a stress concentration such as an inclusion or a notch and more than one can form at the

same time. Where more than one Lders band is formed during the yield point elongation, the lower yield stress

will fluctuate, each time dropping slightly as a new band forms. The Lders bands are also sometimes called

stretcher strains or even Hartmann lines.

Figure 12.1(p): Lders bands formed on a SAE 1008 low carbon steel with a tensile elongation just beyond the upper yield point.

Static strain ageing after deformation

Strain ageing is used to describe the strengthening behaviour of a low Carbon steel that had been deformed

through the yield point (often called temper rolling or skin pass rolling) and that regains its strain ageing

strengthening at relatively low temperatures after a certain amount of time. This is shown quite clearly below for

an Fe 0 0.03%C steel that had been cold deformed by about 4% strain to take it past the initial locking of the

dislocations by the Carbon atoms and then annealed for different times at 60C where the return of the yield point was determined as the Carbon atoms diffused back to the dislocations.

Figure 12.1(q): Strain ageing of an 0.03%C steel after straining first to about 4% and then ageing at 60C for different times.



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Note the following:

The initial strain of 4% past the initial yield point had removed the yield point phenomenon completely (see bottom curve for no ageing) and the curve continues where the initial stress strain curve had been

interrupted at 4% strain.

Ageing for only a few minutes at 60C re-introduces the yield point once more with the upper yield

strength increasing with further ageing at 60C. This is proof of the high diffusion mobility of C and N

atoms in -Fe at these relatively low temperatures.

After about 126 hours little further strain ageing occurred and basically all the dislocations had been filled with C and N atoms.

Both the upper yield strength and the yield elongation appear to increase as the extent of dislocation locking increases until a point of saturation is reached.

Dynamic strain ageing during deformation

Dynamic strain ageing occurs while the deformation proceeds and mostly leads to a serrated stress strain curve.

As the C and N atoms must now move with the already moving dislocations (due to the plastic strain) and

keep up with them, dynamic strain ageing is sensitive to the temperature and the rate of straining . Note from the Figure below where straining at 25C leads only to static strain ageing even after a static annealing

treatment of 2 hours at 200C but at 200C during deformation, dynamic strain ageing has set in and causes the serrated flow curve.

Figure 12.1(r): Curve (i) Dynamic strain ageing at 200C as compared to curve (ii) that was statically strain aged at 25C in an Fe 0 0.03%C steel. Note that the static specimen was also statically aged at 200C for 2 hours before resuming deformation at 25C.

Dynamic strain ageing generally has the following characteristics:

It is accompanied by a high rate of work hardening (see Figure below);

The flow stress during dynamic strain ageing has a negative strain rate dependence, i.e. if the strain rate increases, the flow stress will decrease;

The stress strain curve is serrated as the C atoms repeatedly capture the dislocations but are almost immediately torn off again by the strain moving the dislocations; and

The hardness (or stress at a given plastic strain at a constant strain rate) goes through a maximum as a function of the temperature.



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As the deformation temperature is increased beyond a certain point, the serrations will disappear.

Figure 12.1(s): Stress strain curves for an Fe 0.03%C steel as function of temperature where dynamic stain ageing

occurs between about 100 and 200C

12.1.8 Effect of strain rate

The effect of rate of strain application during a tensile test may have an influence on the strength values found

and normally the strain rate is prescribed in the ASTM specifications. In most cases will the yield strength

increase if the strain rate is increased and this is something to consider in industry where rapid processing is

often necessary for purposes of productivity.

The strain rate is defined as: dt

d and is measured in s-1. (Eq 12.1.8-1)

The increase in flow or yield strength, measured at various strains on the stress strain curve, is shown below as a

function of the strain rate of testing.

Figure 12.1(t): The true yield stress at various strains on the stress strain curve as a function of the testing strain

rate for a low C steel at room temperature.



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Note in particular that the flow stress for the low C steel is only strain rate sensitive at strain rates in excess of

about 10-3

to 10-2

s-1

when the tests are performed at room temperature. At higher temperatures this strain rate

sensitivity is far more pronounced, as will be seen later when the topic of creep will be covered.

12.1.9 Effect of temperature

The flow stress and ductility of metals normally vary significantly with the temperature of testing although the

general characteristics of the stress strain curve remain the same. Generally the flow stress increases as the

temperature is lowered and conversely, the flow stress decreases as the temperature is increased. This is

illustrated for stainless steel AISI 304 in the Figure below.

Figure 12.1(u) Tensile stress strain curves for AISI 304 stainless steel (Left) at low temperatures and (Right) at

elevated temperatures.

The strength of metals at elevated temperature is of particular importance for the metallurgical engineer

as the strength performance may be influenced very significantly by optimising the microstructure of the

alloy. Some general yield strengths and tensile elongations, as a function of temperature, are shown

below for a number of commercial alloys designed for elevated temperature use.

Figure 12.1(v): (i) The tensile yield strength and (ii) the elongation of various commercial alloys at elevated

temperature MAR-M200 is a Ni based alloy, MAR-M509 is a Cobalt based alloy, 304 is AISI 304 stainless steel,

7075 T6 is an Al Mg Zn alloy and 1015 is a low Carbon SAE steel..



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Note the peak in yield strength of the MAR-M200 Ni based alloy at about 700C. This is due to the

strengthening precipitate - Ni3Al. The ductility does not always behave predictably and the existence of a ductility trough in some alloys is usually related to some microstructural effects.

12.1.10 The notched tensile test

The necked region as mild notch

It was stated before that the necked region during non-uniform strain is really a mild notch where a triaxial

stress distribution exists. A triaxial stress distribution will have both radial stresses r and tangential stresses

t which will raise the value of the longitudinal or axial stress x required to cause plastic flow. The average stress at the neck which is found by dividing the axial tensile load by the minimum cross sectional area of the

specimen at the neck, is higher than the stress that would be required to cause flow if only simple tension would

apply as in a uniaxial stress distribution. This is shown schematically below.

Figure 12.1(w): The schematic stress distribution at the neck of a tensile tested specimen. R is the radius of the neck

with a minimum diameter of a.

Bridgeman provided a corrective equation to allow for the triaxial stress distribution in the necked region, with

the assumptions:

The contour of the neck has regular radius R and is approximated by the arc of a circle;

The cross section of the neck remains circular;

The strains are constant over the cross section of the neck; and

The Von Mises criterion for plastic yielding applies



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According to Bridgeman the uniaxial flow stress corresponding to that that would have existed in the tensile test if no necking had occurred:

)}]R2/a1}{ln(a/)R21[{(

)ave(x

(Eq 12.1.10-1)

This correction generally lowers the calculated true stress at fracture f from that calculated without the correction.

Notched tensile tests

Ductility measurements on smooth and regular tensile specimens do not always reveal a potential weakness in the

metallurgical structure because of the absence of a triaxial stress distribution in such an unnotched specimen.

Notch sensitivity tests are, therefore, often required and these are done on a pure comparison basis, i.e.

comparing a notched with an unnotched specimen tested under the same conditions and made from exactly the

same material.

Definition: The tendency for reduced ductility in the presence of a triaxial stress distribution and steep stress

gradients, is called the notch sensitivity of the material.

A notched tensile test is usually done to reveal the notch sensitivity of the material. The dimensions of the notch

are prescribed in various standards and are typically a 60 notch with a root radius of 0.025 mm or less. The depth of the notch is usually chosen to be such that the remaining cross sectional area of the specimen after the

notch has been introduced, is about one half of the unnotched specimen. In the notched specimen the notch

strength is defined by the maximum load divided by the original cross sectional area at the notch.

The notch sensitivity is determined by the Notch Sensitivity Ratio or NSR:

)unnotchedmax(

)notchedmax(

S

SNSR (Eq 12.1.10-2)

If the NSR < 1, then the material is notch brittle. The reduction in area at the notch Znotch may also be

measured as a parameter of ductility.

In general and as a rule of thumb, the Notch Sensitivity Ratio NSR of a material will decrease (i.e. it becomes

more notch brittle) as the metallurgical structure increases in strength and hardness. This is because a harder

material will generally, restrict the flow of material at the root of a notch more than in a ductile material and

plastic flow will take place less easily and fracture will occur more easily in the harder material.

The sensitivity of the notched strength to the metallurgical structure, is shown below where it should be noted that

the conventional elongation measurement in the bottom Figure, was unable to detect any potential notch

sensitivity in the specimens that had been tempered deliberately in the temper embrittlement range, i.e in the

range 330 to 480C and it was only apparent with the notched tensile strength results in the top Figure.



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Figure 12.1(x): The tensile properties of notched and unnotched specimens of a low alloy steel that was quenched and

tempered through the range that causes temper embrittlement.



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12.2 The hardness test

12.2.1 Brinell hardness

The Brinell hardness test

Basically all hardness tests measure the resistance to plastic deformation of the material through an indentation

method. Because plastic or permanent deformation takes place under the indentation, the hardness test

inherently is also a measure of the work hardening of the material.

The Brinell hardness test (ASTM E 10) is a simple test in which a hardened steel ball of a pre-determined

diameter D is placed under load P for a certain time onto the surface of the material of which the hardness is

to be measured, and the size of the indentation is measured afterwards. This is shown schematically below.

Figure 12.2(a): (i) The schematic arrangement of the Brinell hardness tests with (ii) the measurement of the

projected diameter d of the indentation and (iii) the geometry of the indentation.

The Brinell hardness number is defined as:

Dt

P

}dDD}{2/D{

PBHN

22

(Eq 12.2.1-1)

where d is the diameter of the indentation as measured on the surface and t is the depth of the indentation,

both measured on the material after the test, as shown above. In practice, a typical load of 3000 kg with a

hardened steel ball with a minimum hardness of 850 HV (Vickers hardness) and of diameter 10 mm, is used

for medium hard materials such as most steels and cast irons and the load is normally applied for about 30

seconds. With a softer material such as Al and its alloys, the load may be decreased to 500 kg and for a

very high hardness, a Tungsten ball is usually used to avoid plastic deformation of the ball itself during the

test. The diameter of the indentation ball may also be varied typically between 5 and 10 mm although a 10

mm ball is the standard. Although the above equation may be used in arriving at a Brinell number, these

values are usually read of suitable tables based on the above equation and are given in the units kg/mm2.



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It is normally recommended that the load/ball ratio be selected to give an indentation of between 2.5 to about

6 mm which will provide an indentation diameter d of between 25 and 60% of the balls diameter. The indentation should always be measured in two perpendicular directions and the mean value taken as the reading.

The maximum range in which the Brinell test will provide reasonably reliable hardness measurements, is

normally between BHN 16 for soft Al and BHN 630 for hardened steel.

Matters to care for in the Brinell test

The indentation ball diameter: The Brinell hardness test has some shortcomings, with the main one being that the indentation load P via the above expression does not give a mean pressure on the

indentation as the surface area of the indentation is accommodated in the equation and not the

projected area. This brings about that the Brinell number may not be fully independent of the

diameter of the indentation ball D or the load P. Converting the above equation to a geometric one in

which the included angle is used (see Figure (iii) above) the BHN is given by:

)}cos1(D)2/{(

PBHN

2 (Eq 12.2.1-2)

In order to obtain a standard BHN number independent of the ball diameter D, the value of the

included angle 2 must stay constant and the ball diameter varied according to the ratio:

ttanconsD

P

D

P22

2

21

1 (Eq 12.2.1-3)

Maintaining this ratio may not always be convenient and the BHN from one material to another may,

therefore, vary with load and ball diameter.

Measurement of the indentation: The indentation diameter d is usually measured by an optical microscope and should be measured to the nearest 0.01 mm. The plastic deformation around the

indentation, may also lead to some measurement errors if so-called ridging or :sinking takes place.

Figure 12.2(b): (i) Ridging type (ii) sinking type and (iii) flat type indentation of the Brinell test.



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Note that in the ridging type indentation, the ridges diameter d extends above the surface of the work piece and in the sinking type, the edge of the indentation is below the surface of the work piece.

Cold worked metals and decarburised steels are most likely to show ridging and fully annealed and

lightly case hardened steels will often show the sinking type of indentation. Both ridging and sinking

bring some uncertainty to the measurement of the true indentation diameter d and should be avoided,

i.e. when ridging is present, the apparent diameter of the indentation is greater than the true value and

when sinking occurs, the apparent diameter is smaller than the true diameter.

Anisotropy in the material: When the material to be tested is anisotropic, i.e. its grain structure varies with direction due to rolling, for instance, the indentation will not be round. Here the mean value of

four readings at 45 to each other on each indentation and also some readings in the various directions of rolling must be used.

The thickness of the material: The Brinell test may not be used on very thin sheet material as the plastic deformation under the indentation may protrude onto the work table below the specimen. A rule

of thumb is that the work piece should have a thickness of at least 10 times the depth of the

indentation.

Flatness of the surface: Surfaces with a radius of curvature of less than 25 mm should not be tested by the Brinell method otherwise the shape of the indentation becomes uncertain.

Surface finish: The measurement of the indentation diameter d is best done on a finely machined, ground or polished surface. As the indentation is usually visible with the naked eye, the Brinell test

should not be used where the presence of the indentations may later be aesthetically unpleasing or may

weaken the structure.

Spacing of indentations: As plastic deformation occurs at and near an indentation, they should not be spaced closer than about three times the indentation diameter to each other and should also not be

closer than the same distance from the edge of a sample.



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Brinell hardness testing machines

A wide variety of Brinell hardness testing machines are available on the market, ranging from hydraulic to

dead weight, from stationary to portable, from laboratory to automatic production machines, etc. A typical

hydraulic Brinell hardness testing machine is shown below.

Figure 12.2(c): A hydraulic Brinell hardness testing machine

12.2.2 Meyer hardness

The Meyer hardness test

The Meyer hardness test is very similar to the Brinell test and actually uses the same hardness testing machine,

with the important difference that Meyer proposed that rather than measuring the surface area the projected

area should be measured as this would accommodate the shortcoming of the traditional Brinell test on its

dependence on the load and indenter diameter. In the Meyer hardness number HM the relationship between

load P and the projected area of the indentation, therefore, rather uses the mean pressure and provides the

following relationship:

2M r

PH

(Eq 12.2.2-1)

or if the diameter of the indentation d is used:

2M d

P4H

(Eq 12.2.2-2)

The Meyer hardness HM also has the units of kg/mm2, the same as the Brinell hardness number.



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Because of the difference in defining the area of the indentation, the Meyer hardness is less sensitive to the

load P and diameter D of the indenter and should be a more representative and fundamental hardness

measurement technique. For a cold worked material, the Meyer hardness is essentially constant with load

whereas the Brinell hardness decreases as the load increases. For an annealed material, the Meyer

hardness will increase continuously with load whereas the Brinell hardness will first increase and then

decrease with load.

For various reasons, however, the Meyer hardness is seldom used in industry.

Meyer did propose the following empirical rule for the choice of load and indentation diameter d:

P = k d m (Eq 12.2.2-3)

Where d is the indentation diameter, m is a material constant related to the materials work hardening exponent n (see earlier) and k is a material constant that expresses the materials resistance to deformation.

For fully annealed materials, the value of m is about 2.5 while m 2 for strain hardened materials and the relationship to the Hollomon strain hardening coefficient n is given by the following empirical rule:

m n + 2 (Eq 12.2.2-4)

Indentation by an indenter

It is worthwhile to understand the plastic flow of a metal more fully as it is deformed by an indentation load P.

Although the indenter deforms the material at the surface plastically and leaves an indentation behind, there is

an underlying volume of material surrounding this plastic zone that will only be stressed elastically, as shown

below.

Figure 12.4(d): (i) The elastic and plastic zones underneath an indenter and the typical deformed grid pattern in

(ii) soft clay and (iii) in steel which shows typical compressive strain field lines (contours close to each other) at

the surface of the indenter with elastically stressed contours further below..



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Note that the plastically deforming zone is not entirely free to strain in any direction and is constrained by the

surrounding elastically stressed material, almost as in a closed-die forging operation where the flow of the work

piece is constrained by the die itself. The mean compressive stress required to cause plastic flow underneath the

indenter in a hardness test is, therefore, greater than in an ordinary compression test without any constraint.

Various theories have been proposed to account for the effect of this constraint factor in the Brinell and Meyer

hardness tests with one of these plastic/elastic analyses predicting that the mean pressure between the indenter

and the indentation pm 3 0 where 0 is the flow stress of the material and the factor 3 is the constraint factor

12.2.3 Rockwell hardness

The Rockwell hardness test

As with the Brinell and the Meyer hardness tests, the Rockwell hardness test (ASTM E18) relies on the

principle of an indentation but now under a load increment (and not an absolute or fixed load as in the

Brinell and Meyer tests) and the incremental depth of the indentation is measured as representing the

Rockwell hardness of the material.

Figure 12.2(e): The principle of the Rockwell hardness test with the 120 spherical-conical diamond indenter shown on the right for testing the hardness of steels and other relatively hard materials.

As shown above, the indenter typical for steels is a 120 spherical-conical diamond although a ball shaped indenter may also be used for softer materials. The load is applied in two steps, i.e. first a lower or minor load

which defines the zero indentation depth and then a higher or major load is applied for about 5 to 10

seconds. Thereafter the major load is removed whilst maintaining the minor load and the incremental

indentation depth is measured. This test, therefore, differs in principle from the Brinell and the Meyers

hardness tests



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The main reason for the two step process, is to remove any surface effects of the specimen (such as a hard

oxide layer) and also back lash on the hardness testing machine itself.

The Rockwell hardness HR is read off suitable tables (or directly from a graduated dial gauge) according to

an applicable hardness scale with over 30 scales that represent different combinations of load with indenter

(diamond or steel ball) and size of the indenter. Examples here would be the Rockwell C scale with the

hardness expressed as HRC or the Rockwell B scale with the hardness expressed as HRB and many others.

The HRC and HRB are probably the most commonly used scales on most metals and alloys.

Selecting the most appropriate Rockwell scale

On steels, Brass and most other metals either the HRC (for the harder materials) or HRB (for the softer

materials) will be sufficient. On very thin sheet materials or metals other than steel or Brass, however, some of

the other Rockwell hardness scales may be more appropriate. Factors that must be considered in the selection

of the Rockwell scale, include:

Type of material;

Specimen thickness;

Test location; and

Scale limitations

As an example, if a hardened steel or a WC drill bit has to be tested for hardness, the choice is already limited

to a diamond indenter. This leaves only 6 possible scales, i.e. Rockwell C, A, D, 45N, 30N or 15N. The

next step is to examine each scale for its accuracy, its sensitivity and its repeatability for the particular

purpose in mind.

To check whether the thickness of the specimen is large enough for the particular scale to be used, the

following empirical equations may be used to estimate the depth of indentation and to keep the specimen

thickness at least 10 times above that.

For a diamond indenter: Depth of indentation (mm) = (100 HRC) 0.002

For a ball shaped indenter: Depth of indentation (mm) = (130 HRB) 0.002

If eventually a choice between two equally appropriate Rockwell scales is arrived at, choose the one with the

heavier load as this will spread the load over a more representative area in the specimen.

Table: Selection table for the Rockwell hardness test

Scale

symbol

Indenter Major

load

kg

Typical application

A Diamond 60 Cemented carbides, this steel and shallow case hardened steel

B 1.588 mm steel ball 100 Copper alloys, soft steels, Al alloys, malleable iron

C Diamond 150 Steel, hard cast irons, pearlitic malleable iron, deep case

hardened steel, and all other materials harder than 100 HRB

D Diamond 100 This steel and medium case hardened steel and pearlitic

malleable iron

E 3.175 mm steel ball 100 Cast iron and Al and Mg alloys, bearing metals

F 1.588 mm steel ball 60 Annealed Cu alloys, thin soft sheet steel



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Table continued G 1.588 mm steel ball 150 Phosphor bronze, Cu - Be alloys, Malleable irons, with an upper

limit of HRG = 92 to avoid flattening of the steel ball

H 3.175 mm steel ball 60 Al, Zn and Pb

K 3.175 mm steel ball 150 Bearing materials and other soft or thin metals and use the

smallest ball and heaviest load

L 6.35 mm steel ball 60 Bearing materials and other soft or thin metals and use the


M 6.35 mm steel ball 100 Bearing materials and other soft or thin metals and use the


P 6.35 mm steel ball 150 Bearing materials and other soft or thin metals and use the


R 12.70 mm steel ball 60 Bearing materials and other soft or thin metals and use the


S 12.70 mm steel ball 100 Bearing materials and other soft or thin metals and use the


V 12.70 mm steel ball 150 Bearing materials and other soft or thin metals and use the


A Rockwell hardness testing machine

Although a number of variations exist on the market, most use the principles demonstrated in the following cut

away view.

Figure 12.2(f): Cut away view of a typical Rockwell hardness tester



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12.2.4 Vickers hardness as well as the Vickers Microhardness test

The Vickers hardness test

This hardness test relies once again on the principle of a diamond indentation made under load and measuring

the diagonal width of the square indentation made by a square diamond indenter with an included angle of

136 between opposite faces. The included angle of 136 and size of the diamond indenter was chosen with reference to the Brinell test from which this was the most optimal included angle to diameter ratio. Because of

the shape of the indenter, this hardness test is often called the Diamond Pyramid Hardness test or DPH

although the acronym for Vickers hardness VHN or VPH are also used. The Vickers hardness is given by:

22 L

P854.1

L

)2/sin(P2DPH

(Eq 12.2.4-1)

where the included angle = 136, L is the average length of the diagonals of the indentation and P is the applied load in kg. The shape of the diamond indenter is shown below.

Figure 12.2(g): Diamond pyramid indenter used in the Vickers hardness test with the length of the diagonal D (or

L in the formula above) shown.

The HV hardness values are usually read of tables that provide a hardness factor that must then be multiplied

by the load used, for example: If the average diagonals are measures as 40.3 m, the Vickers table gives the hardness factor as 1.142. If a load of 500 kg was used, the Vickers hardness is given by 1.142 x 500 = 571

HV.



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Advantages and disadvantages of the Vickers hardness test

The Vickers hardness test has reached wide acceptance as it measures basically all hardnesses on a single

scale from DPH = 5 to DPH = 1500. With the Brinell and the Rockwell hardness tests the scale must be

changed which makes it sometimes difficult to compare different materials with each other through their

hardnesses. Furthermore, because the diamond shaped indentations are geometrically similar no matter what the

load is, the Vickers hardness should, therefore, not be as dependent on the load as with the other two tests.

One problem to avoid is the earlier ridging and sinking that may also be encountered with the Vickers

hardness test, as shown schematically below.

Figure 12.2(h): (i) Perfect indentation, (ii) sinking and (iii) ridging in the Vickers hardness test

Surface preparation of the specimen for a Vickers hardness test is more stringent than in the case of the

Rockwell test and at very low loads of about 100 grams, a metallographically polished surface needs to

prepared. Except for very low loads, the Vickers hardness value (and the Knoop hardness, used for very hard

or ceramic materials) is reasonably independent of the load, as shown below:

Figure 12.2(i): The relationship of hardness number versus load for Vickers and Knoop hardness tests



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Vickers hardness tester

A typical Vickers hardness testing machine is shown below.

Figure 12.2(j): A typical Vickers microhardness testing machine.

Vickers Microhardness

The Microhardness testing of materials is basically done via the Vickers hardness test on a laboratory bench

apparatus also equipped with an optical microscope to position the indenter very accurately within the selected

area of the microstructure. It is well suited to determine the hardness of different constituents in a

microstructure or to determine the hardness gradients in case hardened, surface hardened or nitrided steels.

Polished sample surfaces are usually necessary for microhardness testing.

If the hardness of a hard and specific micro-constituent such as a carbide particle in the example above, needs to

be determined, the Knoop hardness is appropriate as its very oblong indentation makes it possible to place two

indentations quite close to each other. The depth and area of a Knoop indentation is typically only about 15%

of the equivalent Vickers indentation and this makes the Knoop test also applicable to test the hardness of thin

surface layers such as hard Chrome plating etc.



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12.2.5 Other hardness tests

A number of less common hardness test methods exist, which include:

The Scleroscope hardness tester which is a light and portable instrument in which a diamond tipped hammer is released to drop onto the component and the height of rebound is measured as an indication

of the hardness. Although not as accurate as some of the other tests above, it is very convenient to test a

number of large components on the shop floor such as castings or forgings.

The Durometer is a hand held instrument to test the hardness of very soft materials such as rubber and plastics and also operates on the principle of an indentation.

The scratch hardness tests are very crude but long standing comparative tests in which the well known Moh hardness scale for minerals and the so-called file hardness test for steels, are used. The latter

consists of standard files heat treated to a hardness between 67 and 70 HRC and the comparative test

consists of drawing the file across the component. If the file does not bite the hardness of the component is designated as file hard.

Ultrasonic hardness testing is a technique well suited to the automatic testing of a large number of components moving past on a conveyor belt with up to 1200 parts per hour that can be tested. It

consists of a stylus with a Vickers diamond tip or a Rockwell indenter and uses a light indentation

load of maximum 800 grams. The principle of measuring the indentation rests on measuring the

natural resonant frequency of the stylus which consists of a magnetostrictive metal rod with the

indenter attached to its tip. As the indenter rests on the component, the resonant frequency changes and

this is calibrated to the specified hardness of the component being measured.

12.2.6 Hardness conversion relationships

It is often required to compare one hardness test value with another one from a different type of hardness test.

Such conversion tables are available in the ASM Metals Handbook and other reference books. It is important

to accept, however, that these conversion tables are based on purely empirical grounds, as there is no

fundamental relationship between the different tests. The most reliable conversion tables are probably those to

convert between Rockwell, Brinell and Vickers hardnesses for steels with a hardness above 240 Brinell

hardness.

12.2.7 Relationship between hardness and the tensile strength

Relationship with the UTS of a metal

As the determination of hardness of a metal requires minimal preparation of the specimen and also is a very

quick test, various attempts have been made to correlate hardness with other mechanical properties. As the

hardness test involve plastic deformation of the component by the indenter, at best it should correlate with the

ultimate tensile strength of a metal, which is measured after work hardening and not with the yield strength

which is measured before work hardening.

For heat treated plain carbon and low alloy steels, a useful correlation between the Brinell hardness and the

UTS of the steel, has empirically been determined as:

UTS (MPa) = 3.4 BHN (Eq 12.2.7-1)



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Relationship with yield strength

In spite of the fact that a hardness test measures the resistance of a metal to plastic deformation by an indenter,

some attempts have been made to correlate hardness with the yield strength of a metal, with varying success.

This varying success is not surprising as the fundamentals of the two determinations (hardness and yield

strength) are vastly different and may be compared at best, only in an empirical way.

Here the Meyer hardness test was found to correlate best as is shown below for Meyer hardness values

obtained from different load applications compared to the flow stress obtained from compression tests.

Figure 12.2(k): Comparison of the flow stress curve from compression tests with the hardness measurements from

Meyer hardness determinations for mild steel and Cu..

The above correlation was based on an elastic-plastic analysis of an indentation that concluded that the true

strain during an indentation is given by:

D

d2.0 (Eq 12.2.8-2)

where d is the diameter of the indentation produced by an indenter with diameter D. By varying the ratio d/D

through load variations, the true stress true strain curve was reasonably approximated.

In another empirical comparison, the following expression was obtained for the correlation of the tensile flow

stress 0 and the Vickers DPH value:

2m

0 )1.0(3

DPH

(Eq 12.2.8-3)

where 0 is the flow stress in kgf/mm2, DPH is the Vickers hardness pyramid number and m is the Meyer

exponent in Meyers law of (m = n + 2) where n is the Holloman work hardening exponent.

Although these empirical and semi-empirical correlations of hardness with flow or yield strength have provide

some acceptable correlations, other areas of failure are also numerous and it is best to approach this type of

correlation that is not based on sound fundamentals with some caution on any material where it has not

withstood the test of time.



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12.3 Fracture testing

12.3.1 Charpy impact testing

The Charpy impact testing machine

Since many decades, the so-called impact test (of which the Charpy test is the most well known) has been used

to assess the propensity of a metal to brittle fracture. Although this type of three point bend test has been

superseded by the more scientifically reliable fracture mechanics approach (see later) it is still a useful test for

comparative purposes as it is cost effective and easy to perform. Because i

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