CHAPTER 6Mechanical Behavior

Mechanical testing machines can be automatedto simplify the analysis of the mechanical per-formance of materials in a variety of productapplications. (Courtesy of MTS Systems Cor-poration.)

Load cell

Grip

Grip

Specimen

Crosshead

Gage length

Figure 6-1 Tensile test.

100

50

00 1 2 3 4 5

Fracture

Loa

d (1

03 N)

Elongation, mm

Figure 6-2 Load-versus-elongation curve ob-tained in a tensile test. The specimen was alu-minum 2024-T81.

500

00 0.02 0.04 0.06 0.08 0.10

400

300

200

100

Figure 6-3 Stress-versus-strain curve obtained bynormalizing the data of Figure 6–2 for specimengeometry.

500

00 0.002 0.004 0.006 0.008 0.010

400

300

200

100

Yield strength

Figure 6-4 The yield strength is defined relative tothe intersection of the stress–strain curve with a“0.2% offset.” This is a convenient indication ofthe onset of plastic deformation.

Elastic recovery

500

00 0.020.01

400

300

200

100

Figure 6-5 Elastic recovery occurs when stress is removed from a speci-men that has already undergone plastic deformation.

3

2

1 5

4 Strain

Stress

Figure 6-6 The key mechanical properties obtained from a ten-sile test: 1, modulus of elasticity, E ; 2, yield strength, Y.S.;3, tensile strength, T.S.; 4, ductility, 100 × εfailure (note thatelastic recovery occurs after fracture); and 5, toughness =∫

σdε (measured under load; hence, the dashed line is verti-cal).

Figure 6-7 Neck down of a tensile test specimenwithin its gage length after extension beyondthe tensile strength. (Courtesy of R. S. Wort-man)

Engineering or true strain (in./in. or m/m) × 10–2

Eng

inee

ring

or

true

str

ess

(psi

) ×

103

140

130

120

110

100

90

80

70

60

50

40

30

20

10

10 20 30 40 50 60 70 80 90 1000

Fracture

Fracture

True stress–strain curve

Engineering stress–strain curve

Figure 6-8 True stress (= load divided by actual area in the necked-down region) continues to rise to the point of fracture, in con-trast to the behavior of engineering stress. (After R. A. Flinn/P.K. Trojan: Engineering Materials and Their Applications,2nd Ed., Copyright c© 1981, Houghton Mifflin Company, usedby permission.)

StressHigh strength, low ductility, low toughness

High strength, high ductility,high toughness

Low strength,high ductility,low toughness

Strain

Figure 6-9 The toughness of an alloy depends on a combination of strength and ductil-ity.

Stress

Upper yield point

Lower yield point

Strain

Figure 6-10 For a low-carbon steel, the stress-versus-strain curve includes bothan upper and lower yield point.

z

y

x

(a) Unloaded

(b) Loaded

σ

σ

εx

εzν = –

Figure 6-11 The Poisson’s ratio (ν) characterizes the contraction per-pendicular to the extension caused by a tensile stress.

z

y

x

(a) Unloaded (b) Loaded

τ

Dy

zo

Figure 6-12 Elastic deformation under a shear load.

500

400410

480

300

200

100

00.02 0.04 0.06 0.08 0.100

0.0043

1000

0.005 0.0100

1000

2000

0.005 0.0100

Figure 6-13 The brittle nature of fracture in ceramics is illustrated by these stress–strain curves,which show only linear, elastic behavior. In (a), fracture occurs at a tensile stress of 280 MPa.In (b) a compressive strength of 2100 MPa is observed. The sample in both tests is a dense,polycrystalline Al2O3.

F

h

b

F2

F2

L

Point of fracture

Modulus of rupture = MOR= 3FL/(2bh2)

Figure 6-14 The bending test that generates a modulus ofrupture. This strength parameter is similar in magnitudeto a tensile strength. Fracture occurs along the outermostsample edge, which is under a tensile load.

c

1 2Figure 6-15 Stress (σm) atthe tip of a Griffith crack.

0

32,000–40 C (–40 F)

23 C (73 F)

93 C (200 F)

149 C (300 F)

28,000

24,000

20,000

16,000

12,000

8,000

4,000

0

240

200

160

120

80

40

0 2 4 61 3 5 7Strain (%)

Tens

ile s

tres

s (M

Pa)

Tens

ile s

tres

s (p

si)

Figure 6-16 Stress-versus-strain curves for a polyester engineer-ing polymer. (From Design Handbook for Du Pont Engi-neering Plastics, used by permission.)

Tension

Compression

120

15,000

10,000

5,000

0

5,000

10,000

15,000

100

80

60

40

20

0

20

40

60

80

100

12010 6 28 4 0

Strain (%)

Stre

ss (

MP

a)

Stre

ss (

psi)

4 82 6 10

60% relative humidity(2.5% moisture content)

Dry as molded(0.2% moisture content)

Figure 6-17 Stress-versus-strain curves for a nylon 66 at 23◦C showing theeffect of relative humidity. (From Design Handbook for Du Pont Engi-neering Plastics, used by permission.)

Metal atoms

0

+

–

Tensile test specimen

a

Bon

ding

for

ce

Loa

d

Elongation

0

+

–

a

Bon

ding

ene

rgy

Stre

ss

Strain

a

Figure 6-18 Relationship of elastic deformation to the stretching of atomicbonds.

Slip plane

(a) (b)

Figure 6-19 Sliding of one plane of atoms past an adjacent one. Thishigh-stress process is necessary to plastically (permanently) deforma perfect crystal.

Slip plane

(a) (b) (c)

(d) (e) (f)

Figure 6-20 A low-stress alternative for plastically deforming a crys-tal involves the motion of a dislocation along a slip plane.

Figure 6-21 Schematic illustration ofthe motion of a dislocation un-der the influence of a shear stress.The net effect is an increment ofplastic (permanent) deformation.(Compare Figure 6–21a with Fig-ure 4–13.)

b

(c)

(b)

b

(a)

(a) (b)

I

II

Goldie

Goldie

Goldie

Goldie

Goldie

Goldie

Goldie

Goldie

Figure 6-22 Goldie the caterpillar illustrates (a) how difficult it is to move alongthe ground without (b) a “dislocation” mechanism. (From W. C. Moss,Ph.D. thesis, University of California, Davis, Calif., 1979.)

Slip plane (low atomic density)

Slip plane (high atomic density)

Slip distance

Slip distance

(a)

(b)

Figure 6-23 Dislocation slip is more difficult along(a) a low-atomic-density plane than along (b)a high-atomic-density plane.

(0001)k1120l =(0001)[1120] (0001[1210]) (0001)[2110]

{111}k110l =(111)[110] (111)[101] (111)[011](111)[110] (111)[101] (111)[011](111)[110] (111)[101] (111)[011](111)[110] (111)[101] (111)[011]

(a) Aluminum

(b) Magnesium

Figure 6-24 Slip systems for (a) fcc aluminum and (b) hcp magnesium. (Compare toFigure 1–18.)

Direction of “attempted” dislocation motion

Figure 6-25 How an impurity atom generates a strain field in acrystal lattice, causing an obstacle to dislocation motion.

Normal to slip plane

Slip direction

F

A

Figure 6-26 Definition of the resolved shear stress, τ , which di-rectly produces plastic deformation (by a shearing action) asa result of the external application of a simple tensile stress, σ .

Indentor

Specimen surface(a)

Load

(b)

(c)

Figure 6-27 Hardness test.The analysis of indenta-tion geometry is summa-rized in Table 6.10.

0 0

40

80

120

160

160 180 200 240 280260220 300

600

200

800 65–45–12,annealed

60–40–18, annealed

Tensile strengthYield strength

100–70–03, air-quenched

80–55–06, as cast

Grade 120–90–02, oil quenched

1000

1200

400

0 500 10000

Tensile strength, T.S. (MPa)

Bri

nell

hard

ness

num

ber,

BH

N

100

200

300

400

(a)Hardness, BHN

Stre

ngth

, MP

a

Stre

ngth

, ksi

(b) Tensile properties of ductile iron versus hardness

Figure 6-28 (a) Plot of data from Table 6.11. A general trend of BHN with T.S. is shown.(b) A more precise correlation of BHN with T.S. (or Y.S.) is obtained for given familiesof alloys. [Part (b) from Metals Handbook, 9th Ed., Vol. 1, American Society for Met-als, Metals Park, Ohio, 1978.]

Time

Figure 6-29 Elastic strain induced in an alloy at room temperature is independent of time.

Furnace

Constant load

Figure 6-30 Typical creep test.

Primarystage

Secondary stage

Finalstage

Fracture

Time

Elastic ( instantaneous) deformation

Figure 6-31 Creep curve. In contrast to Figure 6–29, plastic strainoccurs over time for a material stressed at high temperatures(above about one-half the absolute melting point).

(a) (b)

Climb

= Vacancy

Figure 6-32 Mechanism of dislocation climb. Obviously,many adjacent atom movements are required to pro-duce climb of an entire dislocation line.

Figure 6-33 Variation of the creepcurve with (a) stress or (b) tem-perature. Note how the steady-state creep rate (ε̇) in the secondarystage rises sharply with tempera-ture (see also Figure 6–34).

Increasing T

(a)Time

(b)Time

0.5 1.0 1.5 2.0

T (K)2000 1500 1000 500

1T

× 1000 (K–1)

High-temperature laboratory data

Service temperaturerange

Slope = – QR

Figure 6-34 Arrhenius plot of ln ε̇ versus 1/T , where ε̇ is the secondary-stage creep rate and T is the absolute temperature. The slopegives the activation energy for the creep mechanism. Extensionof high-temperature, short-term data permits prediction of long-term creep behavior at lower service temperatures.

Time

t

Figure 6-35 Simple characterization of creep behavior is ob-tained from the secondary-stage strain rate (ε̇) and the timeto creep rupture (t ).

10–40

–50

–60

–70

–80

–100

–120

–140

–150

102 103 104 105

Rupture time, h

Constant temperature curves

Stre

ss, k

si

540˚C(1000˚F)

595˚C(1100˚F)

650˚C(1200˚F)

705˚C(1300˚F)

Figure 6-36 Creep rupture data for the nickel-based superalloy Inconel 718. (From Met-als Handbook, 9th Ed., Vol. 3, American Society for Metals, Metals Park, Ohio,1980.)

Figure 6-37 Arrhenius-type plot ofcreep-rate data for several poly-crystalline oxides under an appliedstress of 50 psi (345 × 103 Pa).Note that the inverse temperaturescale is reversed (i.e., temperatureincreases to the right). (From W.D. Kingery, H. K. Bowen, and D.R. Uhlmann, Introduction to Ce-ramics, 2nd Ed., John Wiley &Sons, Inc., New York, 1976.)

Temperature ˚C

1000/T, K–1

120010

1

10–1

10–2

10–3

Stra

in r

ate

at 5

0 ps

i, in

./in.

/hr

10–4

10–5

10–6

0.70 0.60 0.50 0.400.65 0.55 0.45 0.35

1400 1600

ZrO2 – compression

ZrO2 – compression

Al2O3 – compression

Al2O3 – compression

BeO, MgO –compression

Al2O3 – tension

MgO – tension

MgO – tension

MgO –compression

1800 2000 2200 2400

5

4

3

2

1

00.001 0.01 0.1 1 10

Time (hours)

Stra

in (

%)

100 1,000 10,000

13.8 MPa (2.000 psi)

6.9 MPa (1.000 psi)

Figure 6-38 Creep data for a nylon 66 at 60◦C and 50% relative humidity.(From Design Handbook for Du Pont Engineering Plastics, used by per-mission.)

60500 600 700

T = 585˚C

T (˚C)

70

80

90

100

110

120

130

Plotting gives

T

DLL0

Tg Ts

Figure 6-39 Typical thermal expansion measure-ment of an inorganic glass or an organic poly-mer indicates a glass transition temperature,Tg, and a softening temperature, Ts.

Tg Tm

Temperature

Liquid

Glass

Crystal

Supercooled liquidV

olum

e (p

er u

nit m

ass)

Figure 6-40 Upon heating, a crystal undergoesmodest thermal expansion up to its meltingpoint (Tm), at which a sharp increase in spe-cific volume occurs. Upon further heating, theliquid undergoes a greater thermal expansion.Slow cooling of the liquid would allow crys-tallization abruptly at Tm and a retracing ofthe melting plot. Rapid cooling of the liquidcan suppress crystallization producing a su-percooled liquid. In the vicinity of the glasstransition temperature (Tg), gradual solidifi-cation occurs. A true glass is a rigid solid withthermal expansion similar to the crystal butan atomic-scale structure similar to the liquid(see Figure 4–23).

Area, A

Slab of viscous material

Force, F

Velocity gradient,

Figure 6-41 Illustration of terms used to define viscosity, η, in Equation 6.19.

20

15

10

5

00 500 1000 1500

Annealing point

Annealing range

Melting range

Working range

T (˚C)

Softening point

Annealing point

Figure 6-42 Viscosity of a typical soda–lime–silica glass from room temperatureto 1500◦C. Above the glass transition temperature (∼ 450◦C in this case),the viscosity decreases in the Arrhenius fashion (see Equation 6.20)

(a) Above Tg.

(b) Air quench surface below Tg.

(c) Slow cool to room temperature.

Tg

T0

0

Tension

Surfacecompressivestress =

Compression

0

Tension

Compression

0

Tension

Compression

T

Tg

T0

T

Tg

T0

T

RTsource ofstrength

Figure 6-43 Thermal and stress profiles occurring during the pro-duction of tempered glass. The high breaking strength of thisproduct is due to the residual compressive stress at the materialsurfaces.

RigidM

odul

us o

f ela

stic

ity

(log

sca

le)

Leathery

Temperature

Tg Tm

Rubbery

Viscous

Figure 6-44 Modulus of elasticity as a func-tion of temperature for a typical thermo-plastic polymer with 50% crystallinity. Thereare four distinct regions of viscoelastic be-havior: (1) rigid, (2) leathery, (3) rubbery,and (4) viscous.

(Courtesy of Tamglass, Ltd.)

(a) (b) (c)

Break pattern of three states of glass used in commercial and consumer applications. (a)Annealed. (b) Laminated. (c) Tempered. (From R. A. McMaster, D. M. Shetterly, and A.G. Bueno, “Annealed and Tempered Glass,” in Engineered Materials Handbook, Vol 4, Ce-ramics and Glasses, ASM International, Materials Park, Ohio, 1991.)

50% amorphous/50% crystalline(see Figure 6-44)

100% crystalline

100% amorphous

Mod

ulus

of e

last

icit

y (l

og s

cale

)

Temperature

Tg Tm

Figure 6-45 In comparison to the plot of Fig-ure 6–44, the behavior of the completely amor-phous and completely crystalline thermo-plastics falls below and above that for the50% crystalline material. The completelycrystalline material is similar to a metal orceramic in remaining rigid up to its meltingpoint.

… …

… …

C

H

H

C

H

H

C

S

H

C

S

CH3

C

H

H

C

H

H

C

H

H

C C

H CH3

C

H

H

C C

H

mer

CH3

C

H

H

C

H

H

C

H

C C

H

H

C

H

H

C

H

H

C C

H

C

H

H

C C

H CH3 CH3CH3

Figure 6-46 Cross-linking produces a network structure by the for-mation of primary bonds between adjacent linear molecules.The classic example shown here is the vulcanization of rubber.Sulfur atoms form primary bonds with adjacent polyisoprenemers. This is possible because the polyisoprene chain moleculestill contains double bonds after polymerization. [It should benoted that sulfur atoms can themselves bond together to form amolecule chain. Sometimes, cross-linking is by an (S)n chain,where n > 1.]

Mod

ulus

of e

last

icit

y (l

og s

cale

)

Temperature

Heavy cross-linking

Increasing cross-linking

Slight cross-linking

No cross-linking

Tm

Figure 6-47 Increased cross-linking of a thermoplastic polymer produces in-creased rigidity of the material.

Mod

ulus

of e

last

icit

y (l

og s

cale

)

Temperature

Rubbery region

Tg Troom Tm

Figure 6-48 The modulus of elasticity versus temperature plotof an elastomer has a pronounced rubbery region.

(a)

(b)

……

Figure 6-49 Schematic illustration of the uncoiling of (a) an ini-tially coiled linear molecule under (b) the effect of an externalstress. This indicates the molecular-scale mechanism for thestress versus strain behavior of an elastomer, as shown in Fig-ure 6–50.

High strain modulus(due to covalent bonding)

Low strain modulus(due to secondary bonding)

Loading

Unloading

Ehigh

Elow

Figure 6-50 The stress–strain curve for an elastomer is an example ofnonlinear elasticity. The initial low-modulus (i.e., low-slope) regioncorresponds to the uncoiling of molecules (overcoming weak, sec-ondary bonds), as illustrated by Figure 6–49. The high-modulus re-gion corresponds to elongation of extended molecules (stretching pri-mary, covalent bonds), as shown by Figure 6–49b. Elastomeric de-formation exhibits hysteresis; that is, the plots during loading and un-loading do not coincide.

0

1011

1010

109

0100

Phenolic (mineral-filled)

PMMA

Nylon-6 (dry)DTUL (indicated oncurve by ‘x’)

Epoxy – 400Phenolic – 375PMMA – 200Nylon 6 – 150

Epoxy (novolac-mineralfilled)

200 300 400 500 600Temperature, ˚F

Dyn

amic

ela

stic

mod

ulus

G, d

ynes

/cm

2

Figure 6-51 Modulus of elasticity versus temperature for a variety of commonpolymers. The dynamic elastic modulus in this case was measured in a tor-sional pendulum (a shear mode). The DTUL is the deflection temperatureunder load, the load being 264 psi. This parameter is frequently associatedwith the glass transition temperature. (From Modern Plastics Encyclopedia,1981–82, Vol. 58, No. 10A, McGraw-Hill Book Company, New York, Octo-ber 1981.)