Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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348 Part C Materials Properties Measurement

Bearing A

Test specimen

Specimen

Roller 1

View A View BAB

LSd

L/2

mounted in the test machine, a section of the tab is

cut or burned away, leaving the specimen free to be

tested. Strains are estimated by the machine displace-

ment, or are measured optically from the edges of the

mounting tab.

Test method I

3-point

Test method IIA

4-point-1/4 point

Test method IIB

4-point-1/3 point

Support span = L

Load span = L/3 L/3L/3

Support span = L

Load span = L/2 L/4L/4

Support span = L

L/2L/2

Fig. 7.6 Examples of fully articulating bearing in test ﬁx-

tures in test ﬁxtures designed to test advanced ceramic

composite materials (after [7.23]) 

Table 7.2 Relation between strain rate and displacement

rate for ﬂexure testing in ASTM C 1341-00 [7.23]

Test geometry I

(3-point)

D =0.167

εL

Test geometry IIA

(4-point; 1/4 point)

D =0.167

εL

Test geometry IIB

(4-point; 1/3 point)

D =0.185

εL

For bulk-size ceramics, or for other materials such

as stones or concretes, the Young’s modulus can be

measured from the displacement of a beam tested

in three- or four-point bending. To assure that the

specimen is loaded evenly, ASTM C 1341-00 [7.23]

recommends the use of fully articulating bearing in the

test ﬁxtures (Fig. 7.6). In these test ﬁxtures, one of the

bearings is ﬁxed to stabilize the test specimen. The other

bearings can both roll and rock to keep in full con-

tact with the specimen cross section. Using a constant

displacement rate, load is measured as a function of dis-

placement. The Young’s modulus E is obtained from

the linear portion of the stressing rate – straining rate

equation

ε = dε/dt =

σ/E . (7.11)

In this test conﬁguration, the strain rate is directly pro-

portional to the rate of cross-head displacement. For

the geometries shown in Fig. 7.7,Table7.2 gives the

relation between the strain rate and the displacement

rate.

Here

D is the rate of crosshead motion in mm/s,

L is the outer support span in mm, d is the specimen

thickness in mm. A similar technique is described in

ASTM C 674-88 (1999) [7.24].

In addition to tension or compression tests to de-

termine Young’s modulus and Poisson’s ratio, the shear

modulus μ can be determined directly through torsion

tests on tubes or solid round bars, ASTM E 143 [7.17].

The angle of twist in radians θ is measured as a func-

tion of applied torque T and a shear stress is calculated

from the appropriate equation. The shear modulus μ is

given by

μ = TL/ Jθ, (7.12)

Fig. 7.7 Specimen geometries for ﬂexure testing (after

[7.23])



Part C 7.1

Mechanical Properties 7.1 Elasticity 349

where L is the gage length and J is the polar moment

of inertia of the section about its center. For a cylinder

J = π



−D



32 , (7.13)

where D

is the external diameter of the cylinder and D

is the internal diameter. For a solid cylinder, D

equals

zero.

7.1.4 Dynamic Methods

of Determining Elastic Constants

Dynamic methods of measuring the elastic constants

of materials are the most commonly used for all kinds

of materials. They were ﬁrst introduced in the mid

1930s [7.33, 34]. These techniques have the advantage

that the stresses used to measure the moduli are far be-

low those at the elastic limit. They therefore do not

give rise to complex creep effects or elastic hystere-

sis. Because the stresses are inﬁnitesimal, the specimens

are not altered by the stresses so that repeated mea-

surements can be made on the same sample. A single

specimen can, therefore, be used to obtain the elastic

moduli as a function of temperature, or pressure. Spec-

imens are small and simple compared to those used to

determine elastic constants in static tests.

Ultrasonic Pulse Technique

Two types of dynamic tests are commonly conducted to

measure the elastic constants. In one an ultrasonic pulse

Table 7.3 ASTM standard test methods employed to determine elastic constants by the resonance technique

Speciﬁcation number Speciﬁcation title

ASTM C 215 [7.25] Test method for fundamental transverse, longitudinal and torsional

frequencies of concrete specimens

ASTM C 623 [7.26] Test method for Young’s modulus, shear modulus and Poisson’s ratio for

glass and glass–ceramics by resonance

ASTM C 747-93

(1998) [7.27]

Test method for moduli of elasticity and fundamental frequencies of carbon

and graphite materials by sonic resonance

ASTM C 848-88

(1999) [7.28]

Standard test method for Young’s modulus, shear modulus, and Poisson’s

ratio for ceramic whitewares by resonance

ASTM C 885-87

(2002) [7.29]

Standard test method for Young’s modulus of refractory shapes by sonic

resonance

ASTM C 1198-01 [7.30] Standard test method for dynamic Young’s modulus, shear modulus and

Poisson’s ratio for advanced ceramics by sonic resonance

ASTM E 1875 [7.31] Standard test method for dynamic Young’s modulus, shear modulus, and

Poisson’s ratio by sonic resonance

ASTM E 1876-01 [7.32] Standard test method for dynamic Young’s modulus, shear modulus, and

Poisson’s ratio by impulse excitation of vibration

is sent through the specimen and its time of ﬂight meas-

ured. The elastic constant determined by this technique

depends on the kind of pulse used for the measurement.

If a longitudinal pulse is used, then the Young’s modu-

lus is deﬁned by the density of the specimen ρ and the

longitudinal velocity of the pulse v

[7.2]

E = ρv

. (7.14)

The velocity of the pulse is measured directly from the

time to transit a ﬁxed distance in the specimen. For

a specimen of a well-deﬁned thickness L the distance

traveled by the wave is usually 2L, when the pulse-echo

techniques is used to measure the velocity.

If an ultrasonic shear pulse v

is used for the mea-

surement then the shear modulus μ can be obtained

from the density and the velocity measurement [7.2]

μ =ρv

(7.15)

where v

is the velocity of the shear pulse.

The accuracy of these measurement techniques de-

pends on the dimensions of the specimen and on the

end-face parallelism, the coupling technique and the

signal-to-noise ratio. The accuracy of these techniques

is typically ±0.1% of the mean measurement. Despite

its accuracy and simplicity, the material test commu-

nity has not adapted this technique. It is used, however,

in ASTM C 1419-99a [7.35] as a standard method of

obtaining approximate values of Young’s modulus in

refractory materials at room temperature.

Part C 7.1

350 Part C Materials Properties Measurement

Free-Vibration Techniques

The second type of dynamic test employs freely vibrat-

ing rods to determine the elastic constants. The rods can

have a rectilinear or circular cross section. Specimens

are excited mechanically, either by an impulse or con-

tinuously through the rod supports. Then, the resonant

frequency for the rod is measured and the density ρ or

mass m and the specimen dimensions are used to de-

termine the elastic constants. The bars can be excited

in such a way as to favor measurement of the shear

modulus or the Young’s modulus; equations have been

developed to calculate both these quantities. The deriva-

tions of these equations are complex, especially the

equation for standing ﬂexural vibrations, and so are not

addressed in this review. A discussion of the equations

used in the resonance techniques and references to the

earlier work in this ﬁeld can be found in the article by

Tefft [7.36]. The impulse technique and the continuous

resonance technique have both been adapted by a num-

ber of ASTM committees for the purpose of measuring

Out-of-plane

flexure

In-plane

flexure

Flexure

node line

Flexure

node line

Flexure

node line

Flexure

node line

X1 = out-of-plane impulse point

P1 = out-of-plane contact sensor points

M1 = out-of-plane microphone sensor point

X2 = in-plane impulse point

P2 = in-plane contact sensor points

M2 = in-plane microphone sensor point

Fig. 7.8 A schematic diagram of the kind of specimen used to mea-

sure elastic moduli by the free-resonance technique (ﬂexure testing)

(after [7.32])

elastic constants (Table 7.3). The techniques proposed

by these committees are very similar to one another.

Free Resonance by Impulse Excitation

A full description of this technique is given in ASTM

designation E 1876-01. The following brief description

has been abstracted from the designation. A schematic

diagram of the kind of specimen used to measure elastic

moduli by free resonance is shown in Fig. 7.8. The ratio

of the specimen length to the minimum cross-section

dimension should be 20–25 for ease of calculation.

For shear modulus measurement on rectangular bars,

a width-to-thickness ratio of ﬁve or greater is recom-

mended. The specimen supports should be located at the

fundamental nodal points 0.224L from each end. The

specimen is then impacted with a small impulser, such

as that shown in Fig. 7.9. The vibration is then picked up

by a microphone sensor located at the antinode. Alter-

natively, a contact sensor located near the node lines can

also be used to pick up the vibration. The impact sites

and the location of the sensors are indicated in Fig. 7.8.

The Young’s modulus is calculated from the funda-

mental vibrational frequency f

, the specimen mass m,

width b, length L, and thickness t, and a correction

factor T

[7.32]

E = 0.9465







. (7.16)

The correction factor is a complex algebraic term that

depends on the Poisson’s ratio, the thickness and length

of the test specimen. For specimen lengths 20 times

greater than the thickness, T

is independent of the Pois-

son’s ratio to a good approximation. In this limit, the

following equation may be used for T

[7.32]

=1.000+6.585



t/L



. (7.17)

The procedure for determining the Young’s modulus

from rods with circular cross sections is similar to those

Flexible polymer rod

Steel ball

Fig. 7.9 A typical impulser for small specimens (after

[7.32])

Part C 7.1

Mechanical Properties 7.1 Elasticity 351

for rectangular cross sections, but the equations used are

different. These may be found in ASTM E 1876 [7.32].

Young’s modulus can also be obtained from rectangular

specimens subjected to longitudinal vibrations. Here the

specimen is impacted on one end and the vibrations are

picked up from the other. The equations for the calcula-

tion and the details of the procedure are given in ASTM

E 1876.

The dynamic shear modulus can be obtained from

rectangular specimens of a similar geometry to that used

to obtain the Young’s modulus. As with the ﬂexural

specimens, the specimen supports are placed along the

node lines, but the location of these lines, Fig. 7.10,dif-

fer substantially from those of the ﬂexural tests. The

location of the support lines, the impulse point and the

sensor points are shown in Fig. 7.10. Once the funda-

mental torsional resonant frequency f

is determined,

the dynamic shear modulus can be determined from the

following equation [7.32]

G =

4Lm f



B/(1+ A)



, (7.18)

where B and A are correction factors that can be found

in ASTM E 1876.

Free Resonance by Continuous Excitation

The continuous resonance technique differs from the

impulse technique only in the way the resonant fre-

quencies are excited. Therefore, the equations used to

calculate the shear modulus and Young’s modulus are

identical to those used for the impulse technique. One

conﬁguration of test specimen is shown in Fig. 7.11.

A rectangular specimen is suspended with two loops

of thread or ﬁne wire. The exciting transducer sends

a sonic vibration down one of the supports and the sig-

nal is detected through the other support. The resonant

vibration is detected as an enhancement of the signal as

the frequency of the input signal is varied. Care has to be

taken to separate the ﬂexural resonant modes from the

torsional resonant modes and the various resonant har-

monics from each other. Suggestions as to how this can

be achieved are given in ASTM E 1875. A second con-

ﬁguration for specimen support is shown in Fig. 7.12.

Locating the supports at the ﬂexural or torsional nodes

tends to favor the primary harmonic for each of these

modes. Nevertheless, a proper identiﬁcation of the reso-

nant frequencies requires movement of the transducers

along the total specimen length to identify the nodes and

antinodes. The suspension method has been used to de-

termine the elastic moduli as a function of temperature,

from cryogenic [7.37] to high temperatures [7.38].

Torsion

Flexure

node line

Flexure

node line

Torsion

node lines

X3 = Torsion impulse point

P3 = Torsion contact sensor points

M3 = Torsion microphone sensor point

Fig. 7.10 A schematic diagram of the kind of specimen used to mea-

sure elastic moduli by the free resonance technique (torsion testing)

(after [7.32])

Exciting

transducer

Detecting

transducer

Fig. 7.11 A schematic diagram of the kind of experimental

set up used to determine elastic moduli by the continuous

resonance technique (after [7.31])

Driver

Pickup

Driver

Pickup

Fig. 7.12 An alternate conﬁguration for determining the elastic

moduli by the continuous resonance technique. Here the supports

are directly under the resonance nodes (after [7.31])

Part C 7.1

352 Part C Materials Properties Measurement

Comparing the Static and Dynamic Methods

From a historical perspective, the static methods of

determining elastic constants were the ﬁrst to be devel-

oped. Dynamic methods were only discovered in the

1930s [7.33, 34] and required some time to become

known and then popular. Standards for the static meth-

ods of determining elastic constants were developed in

the 1950s. For example, ASTM E 111 was ﬁrst issued

in 1955. The dynamic techniques required a rather com-

plex theoretical analysis and a numerical evaluation in

order to know the resonant frequencies as a function of

specimen shape, density and dimensions. Also, mod-

ern electronics had to have been developed before the

measurements could be made to determine resonant fre-

quencies. It was not until the late 1950s that progress

was sufﬁcient that the dynamic techniques could be de-

velopedinsuchawayastobeeasytouseanduseful.

The ﬁrst ASTM designation for measurement of elastic

constants by resonance was C 623 in 1969, i. e., about

10 years after the technique was developed.

Ide [7.33] recognized the advantages of dynamic

techniques over static techniques of determining elastic

constants as early as 1935. He stated,

Dynamic methods for measuring Young’s modu-

lus of elasticity possess advantages over the usual

static methods. The dynamic modulus is obtained

for minute alternating stresses far below the elastic

limit, which do not give rise to complex creep ef-

fects or to elastic hysteresis. This accurately fulﬁlls

the assumptions of the mathematical theory.

This opinion about the inherent superiority is still held

today. In a study by Smith et al. [7.11], the authors

used dynamic test techniques to determine the Young’s

modulus of an Inconel alloy to an accuracy of bet-

ter than 1% of the mean value. The static techniques

could only achieve an accuracy of between 5 and 7%

on the same alloy. These authors recommend one of

the dynamic techniques (impulse excitation) for rou-

tine quality-control work because of their accuracy and

simplicity. The continuous excitation method was rec-

ommended for more complete elastic characterization

studies of materials. A major advantage of the impulse

technique is that a nodal analysis is not required to de-

termine the resonance frequencies. They feel that the

static method is not desirable for quality-control work,

because it is a labor-intensive technique that is time-

consuming.

There are fundamental reasons for differences be-

tween the dynamic and static techniques [7.39]. Static

tests are isothermal because there is plenty of time

for heat conduction during the test. Ultrasonic tests

take place so fast that the conditions are adiabatic and

the sample’s temperature actually changes during the

test. Resonant tests are usually adiabatic, but this de-

pends on the resonant frequency and the size of the

specimen. Dislocation bowing and other sources of in-

ternal friction always take place during static tests and

never during ultrasonic tests. Again, resonant tests may

or may not be affected by these phenomena. In fact,

tests in which the frequency is changed are used to

characterize these effects. Static tests tend to explore

higher stresses than dynamic tests. Therefore, higher-

order elastic behavior is implicitly included in static

measurements.

If the application is highly loaded and static in na-

ture, such as in a building, ship, or bridge, the static

methods for measuring modulus are more appropriate

despite the uncertainty levels mentioned above. If the

application is dynamic in nature, such as ballistic pen-

etration or elastic wave propagation, one should use

dynamic methods to evaluate the elastic constants.

7.1.5 Instrumented Indentation

as a Method of Determining

Elastic Constants

Up to this point, the discussion has involved what

may be regarded as standard test methods. The meth-

ods are standard in the sense that they were developed

to evaluate the elastic constants of materials that are

of a macroscopic size. The materials are uniform and

can be considered to behave as a continuum. Chem-

ically and microstructurally the materials are uniform

throughout. Most standard materials are ﬁrst manufac-

tured and then are machined to the size and shape of

the part. The elastic constants of the materials are the

same in the part as in the bulk. In the case of mater-

ials that are manufactured directly into parts (H-beams,

motor castings, etc.), the parts are large enough that

pieces can be removed to measure their property, and

the property measured in this way is assumed to be

identical to the property in the part. For certain appli-

cations, however, removal of pieces from the part is not

an easy option; the parts may be too small to have pieces

removed from them in such a way as to be representa-

tive of the part on the whole. Also, the piece removed

may be too small to be tested by standard means. Ma-

terials that ﬁt this category include multilayers, and

interlayers in electronic components, component parts

of microelectromechanical systems (MEMS), and ther-

Part C 7.1

Mechanical Properties 7.1 Elasticity 353

mal barrier coatings in the stators and blades of gas

turbines. For these components, the elastic constants of

the materials that make up the part have to be meas-

ured by techniques that are not the standard techniques

discussed earlier in this paper. Often, the properties

of the materials have to be measured in situ, that is,

while the material is still in place in the part. (Materials

scientists are currently developing small test ﬁxtures

and methods of testing specimens μm in size. The

specimens are obtained directly from components, so

that the properties measured will be representative of

those components.) Instrumented indentation testing is

a technique that offers the possibility of determining the

elastic constants in situ and currently is being used ex-

tensively for this purpose. In the rest of this section, this

technique is brieﬂy reviewed and the advantages and

disadvantages of the technique are discussed. A good

review article for the instrumented indentation appara-

tus may be found in [7.40]. The technique is discussed

in greater detail in [7.41] and in Sect. 7.3.4 of this

volume.

Instrumented Indentation Test Apparatus

A schematic diagram of an instrumented indentation ap-

paratus is shown in Fig. 7.13. The apparatus consists of

three parts: a force actuator to apply the load, a dis-

placement sensor to measure the depth of penetration

and an indenter to penetrate the solid. Forces for these

instruments have been generated electromagnetically,

electrostatically or piezoelectrically [7.40]andthelevel

of force is often determined by the voltage or current

used to generate the force. Displacements are measured

by a variety of gages: capacitive sensors, linear variable

differential transformers and laser interferometers.

The indenters are generally made of diamond, hav-

ing several geometries. In general, four-sided pyramid

indenters are not used for nanoindenters, because they

usually end in a ﬂat-ended chisel edge that interferes

with the method at very light loads. Three-sided inden-

ters are used since they can be sharpened to a point.

The most common three-sided indenter is the Berkovich

indenter, which has the same depth to area as the

commonly used Vickers hardness indenter. The center-

line to face angle for the Berkovich indenter is 65.3

◦

A cube-corner pyramidal indenter, so called because the

three faces of the indenter are at 90

◦

to one another,

is also commonly used for nanoindentation studies

(Sect. 7.3.4). The centerline to face angle for the cube-

corner indenter is 35.3

◦

. This indenter angle is more

acute than that for the Berkovich indenter. Hence, the

stresses at the tip of the indenter are much higher than

Machine

compliance C

Displacement, h

Specimen

Indenter

Displacement

sensor

To electronics

Force P

Force

actuator

Fig. 7.13 A schematic diagram illustrating the parts to an

instrumented indentation tester (after [7.40])

the stresses at the tip of the Berkovich indenter and

the tendency for crack formation is greater. The third

type of indenter used for instrument indentation studies

is the spherical indenter. This has the advantage over

the others that the initial contact of the indenter to the

solid is elastic, which would make it ideal for the de-

termination of elastic constants. Unfortunately, small

spherical indenters – a micron or less in diameter –

are difﬁcult to make, which is why the Berkovich in-

denter is the indenter of choice for most small-scale

testing [7.40].

Determining the Elastic Modulus

by Indentation

A schematic representation of the indentation load P as

a function of displacement h is shown in Fig. 7.14. The

loading of the indenter is due to a mixture of elastic and

plastic deformation. The initial portion of the loading

is primarily due to elastic deformation in the vicinity

of the indentation [7.41]. As the penetration increases,

so does the plastic contribution to the deformation. An

Part C 7.1

354 Part C Materials Properties Measurement

Load, P

max

Unloading

max

Displacement, h

Fig. 7.14 An illustration of a load displacement curve for

a single loading cycle on an instrumented indentation tester

(after [7.40])

estimate for the Young’s modulus is obtained from the

slope of the unloading curve S, at the point of maximum

load using the following formula, which is founded in

elastic contact theory [7.40,42,43]

√

2β

√

, (7.19)

where A is the area of contact β is a constant that de-

pends on the geometry of the indenter [7.44], and E

the reduced elastic modulus. The reduced elastic mod-

ulus is used to take account of the fact that deformation

occurs in both the target material and the indenter. To

calculate the Young’s modulus of the target material, an

estimate of the Poisson’s ratio of the material must be

made. This estimate results in an error of only 5% in the

determination of E. The value of β for the Berkovich

and the cube-corner indenters is 1.034 [7.40].

The slope of the unloading curve S is known as the

contact stiffness, and can be obtained by using a power

relation of the load P to the penetration depth h devel-

Standard Current stage Purpose

ISO 14577-1:2002 Ed. 1 Published Test method

ISO 14577-2:2002 Ed. 1 Published Veriﬁcation and calibration

of testing machines

ISO 14577-3:2002 Ed. 1 Published Calibration of reference blocks

ISO 14577-4 Ed. 1 Under development Test methods for coatings

ISO 14577-5 Ed. 1 Under development Indentation tensile properties

Table 7.4 International

Organization for Stan-

dardization ISO 14577,

Metallic materials – In-

strumented indentation test

for hardness and materials

parameters. Work is also

underway in the ASTM on

similar standards

Surface profile

after load removal

Surface profile

under load

Original surface

profile

Indenter profile

Fig. 7.15 A schematic representation of a section through

a conical indentation illustrating the various parameters

used to obtain (7.18) through (7.21)(after[7.40])

oped by Oliver and Pharr [7.45]

P = BA(h −h

)

, (7.20)

where B and m are empirical ﬁtting parameters and h

is the ﬁnal displacement after complete unloading. The

contact stiffness is determined by differentiating (7.20)

with regard to h and evaluating the result at h = h

max

S = Bm(h

max

−h

)

m−1

. (7.21)

Equation (7.21) does not always provide an adequate

description of the unloading curve in which case a ﬁt

is usually applied just to the upper 25% to 50% of the

data [7.40].

All that is needed to characterize Young’s modulus

by (7.19) is the contact area A at maximum load P

max

The contact area is usually expressed in terms of the

depth of the circle of contact h

which deﬁnes the depth

of contact for a conical indentation (Fig. 7.15). The ge-

ometry of the area of contact is discussed in [7.41] with

regard to the penetration depth h

and the area of con-

tact A. For a Berkovich indenter, the two are related by

the following equation

A =24.5h

. (7.22)

Part C 7.1

Mechanical Properties 7.2 Plasticity 355

The parameter h

is given by the following formula

=h −ε

. (7.23)

The parameter ε is a constant that depends on the

indenter geometry. Its value can be obtained through

elastic or elastic–plastic analyzes of the indentation pro-

cess. For the Berkovich indenter, the parameter ε has

a value of about 0.75 [7.45].

With the determination of the area of contact, all

of the parameters in (7.17) are known and the Young’s

modulus can in principle be calculated. In 2002, the

technique was issued as a test standard by the Inter-

national Organization for Standardization (ISO): ISO

14577-1 through ISO 14577-3 (Metallic materials – In-

strumented indentation test for hardness and materials

parameters). Two additional parts to the standard are

still under consideration. The parts of this standard and

their purpose are listed in Table 7.4.

7.2 Plasticity

Plasticity, or permanent deformation, is one of the most

useful mechanical properties of materials. It permits

forming of parts, and provides a signiﬁcant degree of

safety in use. The ability to specify plastic properties

measured by the standard methods described in this

chapter provides industry with a valuable means of

controlling the manufacture of materials. These mea-

sures also provide the analyst with data for prediction

and modeling. However, the current phenomenological

basis for plasticity requires a large number of measure-

ments and the development of a more physical basis

is badly needed. In this chapter, we brieﬂy discuss the

physical basis for plasticity and then describe at length

standard methods for measuring plastic properties. Al-

though most industrial needs are being met by existing

standards, the appearance of new materials and extreme

applications of traditional materials are driving the de-

velopment of new standard measurement methods.

In the context of this section, plasticity refers to

the permanent strain that occurs after the elastic limit

and before the onset of localization that determines

a material’s ultimate strength. Plasticity in materials is

known to be due to the creation, movement, and anni-

hilation of defects (dislocations, vacancies, etc.) while

the forces between atoms remain elastic in nature. It is

also well known that these defects often move at stresses

below the elastic limit, a process called microplastic-

ity, and that the ultimate strength can be determined

by purely plastic processes. Nevertheless, from an en-

gineering point of view, it is entirely appropriate to

consider elasticity (Sect. 7.1), plasticity (Sect. 7.2), and

ultimate strength (Sect. 7.4) as separate phenomena.

7.2.1 Fundamentals of Plasticity

The onset of plasticity at the yield strength signals an

end to purely elastic deformation, i. e., the elastic limit.

Many designs are elastic and the maximum working

(or allowable) load may be chosen as the load when

the yield strength is reached in some part of the de-

sign. Plasticity may also be viewed as a margin of safety

before the ultimate strength has been reached. For this

reason, the maximum plastic strain or ductility may as-

sure the user of a certain level of safety if the yield

strength is exceeded by accident. Ductility also per-

mits the manufacturer to form materials without failure,

and workhardening (the increase in stress with plastic

strain needed to continue plastic deformation) assures

that the part will be even stronger when it is ﬁnished.

The standards that exist today were created in an effort

to standardize the meaning and measurement of these

important plastic properties.

These standards now also serve the analytical com-

munity that seeks to predict plastic behavior under

conditions of use, forming, accidents, etc. This commu-

nity uses the theory of plasticity to make its predictions.

While there is elastic strain proportional to applied

pressure (through the bulk modulus) and the fracture

strength depends on the applied pressure, plastic de-

formation is practically independent of the applied

pressure. There is also little or no volume change as-

sociated with plasticity. The plastic behavior depends

strongly on the deviatoric or shear components of the

stress tensor and results in a shape change rather than

a volume change. These observations have led to the

development of the mathematical or phenomenological

theory of plasticity. In this theory, plastic yielding be-

gins at a critical value of the second invariant of the

stress tensor (von Mises criterion) or at the maximum

shear stress (Tresca criterion). If the von Mises crite-

rion is plotted in a space of principal stresses, it appears

as a cylinder whose axis lies along the σ

=σ

lo-

cus. In the σ

/σ

plane, perpendicular to the σ

-axis,

the cut through the cylinder appears as an ellipse and is

Part C 7.2

356 Part C Materials Properties Measurement

Fig. 7.16 Model of an edge dislocation oriented so that

the atoms are viewed parallel to the dislocation line (CD)

in Fig. 7.18. (Model courtesy of R. deWit [7.48])

known as the von Mises ellipse. Inside the ellipse, all

strain is assumed to be elastic. When the combination

of stresses touches the ellipse, plastic deformation be-

gins to take place. Plastic strain usually results in strain

hardening and this may change the size, shape, or po-

sition of the von Mises ellipse. After yielding, it may

no longer be elliptical. The description of this theory in

any more detail is beyond the scope of this article. The

reader is directed to one of the earliest and most im-

portant treatise on this subject [7.46] and to one of the

most recent texts [7.47]. The theory of plasticity has had

great success in the prediction of plastic deformation

Slip

vector

Fig. 7.18 The slip that produces an edge dislocation has

occurred over the area ABCD. The boundary CD of the

slipped area is the dislocation; it is perpendicular to the

slip vector

Fig. 7.17 Model

of a screw dis-

location viewed

perpendicular

to the disloca-

tion line CD

in Fig. 7.19.

(Model cour-

tesy of R.

deWit [7.48])

of materials. However, because of its phenomenological

basis, it requires an extensive number of measurements

and assumptions to accurately describe the behavior of

real materials undergoing complex deformation paths.

There are several physical phenomena that re-

sult in plasticity: movement of dislocations [7.49],

movement of atoms (Nabarro–Herring creep or Coble

creep) [7.50], phase transformations (TRIP)[7.51],

twinning (TWIP)[7.52], or clay plasticity [7.53]. The

predominant mechanism is movement of dislocations

and a brief description of how this results in plastic

behavior is appropriate here.

Dislocations as a source of plastic deformation in

crystals were postulated in the 1930s independently by

Taylor, Orowan, and Polanyi [7.54]. Quantum mechan-

ics was being developed at that time and at least one

of these three was searching for a quantum of plastic

deformation [7.55]. They sought something that could

explain the low yield strength as compared to the high

theoretical stress to slide a whole atomic plane over

another. It also had to be consistent with the behav-

ior of slip steps and etch pits with increasing plastic

strain. They proposed new kinds of crystal defects in the

crystal lattice, edge and screw dislocations (Figs. 7.16

and 7.17)[7.48], that could concentrate the applied

stress on a row of atoms.

Under the action of a stress much less that the theo-

retical shear stress, dislocations could move (Figs. 7.18

and 7.19), resulting in plastic deformation of a crystal

and the formation of slip steps.

The evidence supporting this at the time was

etch pits which resulted from stress-enhanced disso-

lution around the dislocation. Transmission electron

microscopy has now well established dislocations as the

primary source of plasticity (Fig. 7.20).

Part C 7.2

Mechanical Properties 7.2 Plasticity 357

Slip

vector

Fig. 7.19 The slip that produces a screw dislocation has oc-

curred over ABCD. The screw dislocation AD is parallel to

the slip vector

For signiﬁcant plasticity to occur in a crystal, the

movement and interaction of 10

–10

km of disloca-

tion length must take place in every cubic millimeter

of material [7.56]. Furthermore, most materials of

commercial importance are polycrystalline and contain

enormous numbers of crystals that may be randomly

or nonrandomly oriented. Thus, the quantitative predic-

tion of plastic behavior in commercial materials based

on an understanding of dislocation mechanics is a difﬁ-

cult problem and has not yet been successful enough to

replace the phenomenological approach described pre-

viously and embodied in current analytical schemes.

1 μm

Fig. 7.20 Dislocations in niobium as viewed in the trans-

mission electron microscope

This situation is beginning to change [7.57–60]andmay

ultimately lead to great simpliﬁcations in what actually

must be measured.

7.2.2 Mechanical Loading Modes

Causing Plastic Deformation

As noted in the previous section, plasticity is extremely

insensitive to the mean stress (hydrostatic tension or

pressure). Thus, there is no test using purely hydro-

static tension or pressure to measure plasticity. That is

not to say there is no interest in the effect of super-

imposed hydrostatic tension or pressure. This invariant

of the stress tensor will greatly affect the fracture be-

havior of most materials. Bridgman used superimposed

hydrostatic pressure to suppress fracture in tensile bars

and achieve much larger elongations to fracture and

reductions in area [7.61]. He noted that plastic yield-

ing in tension occurred at roughly the same value of

shear stress as yielding in compression. Furthermore,

workhardening behavior was unaffected by pressure.

Since the effect of hydrostatic tension or pressure is

mainly on the factors that determine strength, it will not

be discussed in this section on plasticity.

Due to the fact that plasticity responds to the devi-

atoric components of the stress tensor, most tests are

designed to develop signiﬁcant levels of shear stress.

In Fig. 7.21, commonly used mechanical modes of load-

ing are shown that may be used to measure the plastic

properties of materials. There are standard test methods

Shear

Compression

Pure

shear

3-point

bending

4-point

bending

Torsion

Biaxial/

multiaxial

tension or

compression

Initial

shape

Fig. 7.21 Modes of mechanical loading used to measure

plastic properties of materials

Part C 7.2