lo l types of strain - wordpress.com · manufacturing processes for engineering materials, 5th ed....

Manufacturing Processes for Engineering Materials, 5th ed. Kalpakjian • Schmid© 2008, Pearson EducationISBN No. 0-13-227271-7

Types of Strain


FIGURE 2.1 Types of strain. (a) Tensile. (b) Compressive. (c) Shear. All deformation processes in manufacturing involve strains of these types. Tensile strains are involved in stretching sheet metal to make car bodies, compressive strains in forging metals to make turbine disks, and shear strains in making holes by punching.

(c)(b)(a)

lo

l

lo

l

a

b

g

e=l! lolo

γ=ab

Engineering Strain:

Shear Strain:


Tensile-Test

FIGURE 2.2 (a) Original and final shape of a standard tensile-test specimen. (b) Outline of a tensile-test sequence showing different stages in the elongation of the specimen.

(a) (b)

Originalgagelength, lo

Fracture

l f

tan-1 E

PlasticElasticStress

UTS

YFracture

Strain (for lo = 1)l

eueo ef0

Offset

Af

Ao

Uniform elongation

Neck

Total elongation

Y

Post-uniform elongation

lo

le

lu

l f


Mechanical Properties

TABLE 2.1 Typical mechanical properties of various materials at room temperature. See also Tables 10.1, 10.4, 10.8, 11.3 and 11.7.

Elongation Poisson’sE (GPa) Y (MPa) UTS (MPa) in 50 mm (%) Ratio (!)

METALS (WROUGHT)Aluminum and its alloys 69-79 35-550 90-600 45-5 0.31-0.34Copper and its alloys 105-150 76-1100 140-1310 65-3 0.33-0.35Lead and its alloys 14 14 20-55 50-9 0.43Magnesium and its alloys 41-45 130-305 240-380 21-5 0.29-0.35Molybdenum and its alloys 330-360 80-2070 90-2340 40-30 0.32Nickel and its alloys 180-214 105-1200 345-1450 60-5 0.31Steels 190-200 205-1725 415-1750 65-2 0.28-0.33Stainless steels 190-200 240-480 480-760 60-20 0.28-0.30Titanium and its alloys 80-130 344-1380 415-1450 25-7 0.31-0.34Tungsten and its alloys 350-400 550-690 620-760 0 0.27NONMETALLIC MATERIALSCeramics 70-1000 — 140-2600 0 0.2Diamond 820-1050 — — — —Glass and porcelain 70-80 — 140 0 0.24Rubbers 0.01-0.1 — — — 0.5Thermoplastics 1.4-3.4 — 7-80 1000-5 0.32-0.40Thermoplastics, reinforced 2-50 — 20-120 10-1 —Thermosets 3.5-17 — 35-170 0 0.34Boron fibers 380 — 3500 0 —Carbon fibers 275-415 — 2000-5300 1-2 —Glass fibers (S, E) 73-85 — 3500-4600 5 —Kevlar fibers (29, 49, 129) 70-113 — 3000-3400 3-4 —Spectra fibers (900, 1000) 73-100 — 2400-2800 3 —Note: In the upper table, the lowest values for E, Y , and UTS and the highest values forelongation are for the pure metals. Multiply GPa by 145,000 to obtain psi, and MPa by145 to obtain psi. For example 100 GPa = 14,500 ksi, and 100 MPa = 14,500 psi.


Loading & Unloading; Elongation

FIGURE 2.3 Schematic illustration of loading and unloading of a tensile-test specimen. Note that during unloading the curve follows a path parallel to the original elastic slope.

Elastic recovery

Permanentdeformation

Strain

Unload

Load

Str

ess

FIGURE 2.4 Total elongation in a tensile test as a function of original gage length for various metals. Because necking is a local phenomenon, elongation decreases with gage length. Standard gage length is usually 2 in. (50 mm), although shorter ones can be used if larger specimens are not available.

0 1 2 3 4 5 6 87 9 10

80

70

60

50

40

30

20

10

0

0 50 100 150 200 250

Elo

ngation (

%)

Gage length (in.)

mm

1100-O Aluminum

Mild steel Structural silicon steel

Magnesium

Copper

Hig

h-p

urity alum

inum-annealed

BrassMonel


True Stress and True Strain

TABLE 2.2 Comparison of engineering and true strains in tension

e 0.01 0.05 0.1 0.2 0.5 1 2 5 10! 0.01 0.049 0.095 0.18 0.4 0.69 1.1 1.8 2.4

True stress

True strain

σ=PA

ε= ln(llo

)= ln

(AoA

)= ln

(Do

D

)2

= 2ln(Do

D

)


True Stress - True Strain Curve

FIGURE 2.5 (a) True stress--true strain curve in tension. Note that, unlike in an engineering stress-strain curve, the slope is always positive and that the slope decreases with increasing strain. Although in the elastic range stress and strain are proportional, the total curve can be approximated by the power expression shown. On this curve, Y is the yield stress and Yf is the flow stress. (b) True stress-true strain curve plotted on a log-log scale. (c) True stress-true strain curve in tension for 1100-O aluminum plotted on a log-log scale. Note the large difference in the slopes in the elastic and plastic ranges. Source: After R. M. Caddell and R. Sowerby.

True strain ( )

Tru

e s

tre

ss

K

Yf

Y

00 1 1 f

= K n

Fracture

(a)

n

Log

Lo

g

(b)

(c)

n = 0.25

K = 25,000

0.0001 0.001 0.01 0.1 11

10

100

Y

Tru

e s

tre

ss (

psi x 1

03)

n

1

True strain ( )


Power Law Flow Rule

TABLE 2.3 Typical values for K and n in Eq. 2.11 at room temperature.

Material K (MPa) nAluminum, 1100-O 180 0.20

2024-T4 690 0.165052-O 210 0.136061-O 205 0.206061-T6 410 0.057075-O 400 0.17

Brass, 7030, annealed 895 0.4985-15, cold rolled 580 0.34

Bronze (phosphor), annealed 720 0.46Cobalt-base alloy, heat treated 2070 0.50Copper, annealed 315 0.54Molybdenum, annealed 725 0.13Steel, low carbon, annealed 530 0.26

1045 hot rolled 965 0.141112 annealed 760 0.191112 cold rolled 760 0.084135 annealed 1015 0.174135 cold rolled 1100 0.144340 annealed 640 0.1517-4 P-H, annealed 1200 0.0552100, annealed 1450 0.07304 stainless, annealed 1275 0.45410 stainless, annealed 960 0.10

Note: 100 MPa = 14,500 psi.

Flow rule:

σ= Kεn

K = Strength coefficient

n= Strain hardening exponent


True Stress-True Strain for Various Materials

FIGURE 2.6 True stress-true strain curves in tension at room temperature for various metals. The point of intersection of each curve at the ordinate is the yield stress Y; thus, the elastic portions of the curves are not indicated. When the K and n values are determined from these curves, they may not agree with those given in Table 2.3 because of the different sources from which they were collected. Source: S. Kalpakjian.

Copper, annealed

2024-T36 Al

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

40

60

80

100

120

140

160

1801200

1000

800

600

400

0

True strain ( )

Tru

e s

tress (

psi x 1

03)

MP

a

304 Stainless steel

70–30 Brass, as received

70–30 Brass, annealed

1020 Steel

1100-O Al

1100-H14 Al

6061-O Al

2024-O Al

8650 Steel

1112 CR Steel

4130 Steel

200

20


Idealized Stress-Strain Curves

FIGURE 2.7 Schematic illustration of various types of idealized stress-strain curves. (a) Perfectly elastic. (b) Rigid, perfectly plastic. (c) Elastic, perfectly plastic. (d) Rigid, linearly strain hardening. (e) Elastic, linearly strain hardening. The broken lines and arrows indicate unloading and reloading during the test.

(b) (c)

=Y + Ep!

(e)

Y

Y + Ep( -Y/E )

Y

(d)

(a)

Y= Y

tan-1 E

tan-1 Ep

Strain

Str

ess = EP

Y= Y

Y/E

= E

Y/E

E

FIGURE 2.8 The effect of strain-hardening exponent n on the shape of true stress-true strain curves. When n = 1, the material is elastic, and when n = 0, it is rigid and perfectly plastic.

= K n

n = 10 < n < 1n = 0K

0 1

True strain (P)

Tru

e s

tress


Temperature and Strain Rate Effects

FIGURE 2.9 Effect of temperature on mechanical properties of a carbon steel. Most materials display similar temperature sensitivity for elastic modulus, yield strength, ultimate strength, and ductility.

0 200 400 600

(°C)

0 200 400 600 800 1000 1200 1400

Temperature (°F)

Str

ess (p

si 3 1

03)

120

80

40

0

Str

ess (

MP

a)

600

400

200

0

Elongation

Elastic modulus

200

150

100

50

0

Ela

stic m

odulu

s (

GP

a)

Elo

ngation (

%)

0

20

40

60

Tensile strength

Yield strength

200

100

50

10

1

2

4

6

8

10

20

30

40

10-6 10-4 10-2 100 102 104 106

Strain rate (s-1)

Tensile

str

ength

(psi x 1

03)

800°

600°

400°

200°

30°C

MP

a

1000

°

FIGURE 2.10 The effect of strain rate on the ultimate tensile strength of aluminum. Note that as temperature increases, the slope increases. Thus, tensile strength becomes more and more sensitive to strain rate as temperature increases. Source: After J. H. Hollomon.


Typical Strain Rates in Metalworking

Process True Strain Deformation Speed (m/s) Strain Rate (s!1)Cold Working

Forging, rolling 0.1-0.5 0.1-100 1! 103

Wire and tube drawing 0.05-0.5 0.1-100 1! 104

Explosive forming 0.05-0.2 10-100 10! 105

Hot working and warm workingForging, rolling 0.1-0.5 0.1-30 1! 103

Extrusion 2-5 0.1-1 10!1 ! 102

Machining 1-10 0.1-100 103 ! 106

Sheet-metal forming 0.1-0.5 0.05-2 1! 102

Superplastic forming 0.2-3 10!4 ! 10!2 10!4 ! 10!2

TABLE 2.4 Typical ranges of strain, deformation speed, and strain rates in metalworking processes.


Effect on Homologous Temperature

FIGURE 2.11 Dependence of the strain-rate sensitivity exponent m on the homologous temperature T/Tm for various materials. T is the testing temperature and Tm is the melting point of the metal, both on the absolute scale. The transition in the slopes of the curve occurs at about the recrystallization temperature of the metals. Source: After F. W. Boulger.

00.2 0.4 0.6 0.8 1.00

0.05

0.10

0.15

0.25

0.20

CopperSteelAluminum304 StainlessTitaniumRene 41Mo-T2C

Str

ain

ra

te s

en

sitiv

ity e

xp

on

en

t (m

)

T

TmHomologous temperature,


Strain Rate EffectsC

Material Temperature, !C psi ! 103 MPa mAluminum 200-500 12-2 82-14 0.07-0.23Aluminum alloys 200-500 45-5 310-35 0-0.20Copper 300-900 35-3 240-20 0.06-0.17Copper alloys (brasses) 200-800 60-2 415-14 0.02-0.3Lead 100-300 1.6-0.3 11-2 0.1-0.2Magnesium 200-400 20-2 140-14 0.07-0.43Steel

Low carbon 900-1200 24-7 165-48 0.08-0.22Medium carbon 900-1200 23-7 160-48 0.07-0.24Stainless 600-1200 60-5 415-35 0.02-0.4

Titanium 200-1000 135-2 930-14 0.04-0.3Titanium alloys 200-1000 130-5 900-35 0.02-0.3Ti-6Al-4V" 815-930 9.5-1.6 65-11 0.50-0.80Zirconium 200-1000 120-4 830-27 0.04-0.4" at a strain rate of 2! 10#4 s#1.Note: As temperature increases, C decreases and m increases. As strainincreases, C increases and m may increase or decrease, or it may becomenegative within certain ranges of temperature and strain.Source: After T. Altan and F.W. Boulger.

TABLE 2.5 Approximate range of values for C and m in Eq. (2.16) for various annealed metals at true strains ranging from 0.2 to 1.0.

m= Strain-rate sensitivity exponent

C = Strength coefficient

σ=Cεm


Effect of Strain Rate Sensitivity on Elongation

FIGURE 2.12 (a) The effect of strain-rate sensitivity exponent m on the total elongation for various metals. Note that elongation at high values of m approaches 1000%. Source: After D. Lee and W.A. Backofen. (b) The effect of strain-rate sensitivity exponent m on the post uniform (after necking) elongation for various metals. Source: After A.K. Ghosh.

0.20.1 0.3 0.4 0.5 0.6 0.7010

102

103

To

tal e

lon

ga

tio

n (

%)

m

Ti–5A1–2.5SnTi–6A1–4VZircaloy–4High- and ultra-high-carbon steels

00.060.040.020

10

30

20

50

40

m

Po

st-

un

ifo

rm e

lon

ga

tio

n (

%)

A-K SteelHSLA SteelC.R. Aluminum (1100)2036-T4 Aluminum3003-O Aluminum5182-O Aluminum5182-O Al (150C)70-30 BrassZn-Ti Alloy

(a) (b)


Hydrostatic Pressure & Barreling

FIGURE 2.13 The effect of hydrostatic pressure on true strain at fracture in tension for various metals. Even cast iron becomes ductile under high pressure. Source: After H.L.D. Pugh and D. Green.

5

4

3

2

1

0 300 600 900

MPa

00 20 40 60 80 100 120

Tru

e s

train

at fr

actu

re (

f)

Zinc

Copper

Aluminum

MagnesiumCast iron

Hydrostatic pressure (ksi)

FIGURE 2.14 Barreling in compressing a round solid cylindrical specimen (7075-O aluminum) between flat dies. Barreling is caused by friction at the die-specimen interfaces, which retards the free flow of the material. See also Figs.6.1 and 6.2. Source: K.M. Kulkarni and S. Kalpakjian.


Plane-Strain Compression Test

FIGURE 2.15 Schematic illustration of the plane-strain compression test. The dimensional relationships shown should be satisfied for this test to be useful and reproducible. This test gives the yield stress of the material in plane strain, Y’. Source: After A. Nadai and H. Ford.

b

wh

w > 5h

w > 5b

b > 2h to 4h

Yield stress in plane strain:

Y ! =2"3Y = 1.15Y


Tension & Compression; Baushinger Effect

FIGURE 2.16 True stress-true strain curve in tension and compression for aluminum. For ductile metals, the curves for tension and compression are identical. Source: After A.H. Cottrell.

TensionCompression

0

4

8

12

16

Tru

e s

tress (

ksi)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

True strain ( )

MP

a

100

50

0

FIGURE 2.17 Schematic illustration of the Bauschinger effect. Arrows show loading and unloading paths. Note the decrease in the yield stress in compression after the specimen has been subjected to tension. The same result is obtained if compression is applied first, followed by tension, whereby the yield stress in tension decreases.

Y

2Y

P

Tension

Compression


Disk & Torsion Tests

FIGURE 2.18 Disk test on a brittle material, showing the direction of loading and the fracture path. This test is useful for brittle materials, such as ceramics and carbides.

Fracture

P

P

σ=2Pπdt

FIGURE 2.19 A typical torsion-test specimen. It is mounted between the two heads of a machine and is twisted. Note the shear deformation of an element in the reduced section.

l

l

r

r

r r

t

τ=T

2πr2t γ=rφl


Simple vs. Pure Shear

FIGURE 2.20 Comparison of (a) simple shear and (b) pure shear. Note that simple shear is equivalent to pure shear plus a rotation.

3

1

3

3

1 = - 32

3

11

(a) (b)


Three- and Four-Point Bend-Tests

FIGURE 2.21 Two bend-test methods for brittle materials: (a) three-point bending; (b) four-point bending. The shaded areas on the beams represent the bending-moment diagrams, described in texts on the mechanics of solids. Note the region of constant maximum bending moment in (b), whereas the maximum bending moment occurs only at the center of the specimen in (a).

(a) (b)

Maximumbendingmoment

σ=McI


Hardness Tests

FIGURE 2.22 General characteristics of hardness testing methods. The Knoop test is known as a microhardness test because of the light load and small impressions. Source: After H.W. Hayden, W.G. Moffatt, and V. Wulff.

Load, P Hardness number

Shape of indentation

Side view Top view Test Indenter

60 kg

150 kg

100 kg

100 kg

60 kg

150 kg

100 kg

500 kg

1500 kg

3000 kg

25 g–5 kg

1–120 kg L 136°

HK = 14.2P

L2

HV = 1.854P

L2

HB = ( D)(D - D2 - d2 )

2P

t = mm

t = mm

120°

L

b

t L/b = 7.11

b/t = 4.00

d d

D

= 100 - 500t

HRA

HRC

HRD

E

Rockwell

Knoop

Brinell

Vickers

A

C

D

B

F

G

HRE

= 130 - 500t

HRB

HRF

HRG

Diamond cone

10-mm steel or tungsten carbide ball

Diamond pyramid

Diamond pyramid

- in. diameter 16

1

steel ball

- in. diameter 8

1

steel ball


Hardness Test Considerations

FIGURE 2.23 Indentation geometry for Brinell hardness testing: (a) annealed metal; (b) work-hardened metal. Note the difference in metal flow at the periphery of the impressions.

(a) (b)

d

d

FIGURE 2.25 Bulk deformation in mild steel under a spherical indenter. Note that the depth of the deformed zone is about one order of magnitude larger than the depth of indentation. For a hardness test to be valid, the material should be allowed to fully develop this zone. This is why thinner specimens require smaller indentations. Source: Courtesy of M.C. Shaw and C.T. Yang.

FIGURE 2.24 Relation between Brinell hardness and yield stress for aluminum and steels. For comparison, the Brinell hardness (which is always measured in kg/mm2) is converted to psi units on the left scale.

Carbon steels,annealed 5.3

1

3.2

3.4

1

1

Aluminum, cold worked

Carbon steels,cold worked

MPa

100 200 300 400 500

250

200

150

100

50

0

Hard

ness (

psi x 1

03)

10 20 30 40 50 60 70 80

Yield stress (psi x 103)

Brinell

hard

ness (

kg/m

m2)

150

100

50

0


Fatigue

FIGURE 2.26 Typical S-N curves for two metals. Note that, unlike steel, aluminum does not have an endurance limit.

0

500

400

300

200

100

0

80

60

40

20

1045 Steel Endurance limit

Number of cycles, N

(a) (b)

psi x 1

03

Number of cycles, N

Str

ess a

mp

litu

de

, S

(M

Pa

)

0 0

2

4

6

psi x 1

03

8

10

20

30

40

50

60

Diallyl-phthalate

Nylon (dry)

P

olycarbonate

Polysulfone PTFE

103 104 105 107 106 108 109 1010 103 104 105 106 107

2014-T6 Aluminum alloy

Phenolic

Epoxy

Str

ess a

mp

litu

de

, S

(M

Pa

)

FIGURE 2.27 Ratio of fatigue strength to tensile strength for various metals, as a function of tensile strength. 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 50 200

0 200 400 600 800 1000 1200

MPa

Endura

nce lim

it/tensile

str

ength

Titanium

Steels

Cast irons Copper alloys

Aluminum alloys

Wrought magnesium alloys

Cast magnesium

alloys

Tensile strength (psi x 103)

100 150

0.1

0


Creep & Impact

FIGURE 2.28 Schematic illustration of a typical creep curve. The linear segment of the curve (constant slope) is useful in designing components for a specific creep life.

Primary

Time

Tertiary

Strain

Instantaneousdeformation

Rupture

Secondary

Specimen(10 x 10 x 75 mm)

Pendulum

Pendulum

Notch

Specimen(10 x 10 x 55 mm)

(a) (b)

FIGURE 2.29 Impact test specimens: (a) Charpy; (b) lzod.


Residual Stresses

FIGURE 2.30 Residual stresses developed in bending a beam made of an elastic, strain-hardening material. Note that unloading is equivalent to applying an equal and opposite moment to the part, as shown in (b). Because of nonuniform deformation, most parts made by plastic deformation processes contain residual stresses. Note that the forces and moments due to residual stresses must be internally balanced.

Tensile

Compressive

(a)

(b)

(c)

(d)

a b c

o

a

d

o

e

f


Distortion due to Residual Stress

FIGURE 2.31 Distortion of parts with residual stresses after cutting or slitting: (a) rolled sheet or plate; (b) drawn rod; (c) thin-walled tubing. Because of the presence of residual stresses on the surfaces of parts, a round drill may produce an oval-shaped hole because of relaxation of stresses when a portion is removed.

(c)

(a)

Before After

Before After

Before After

(b)


Elimination of Residual Stresses

FIGURE 2.32 Elimination of residual stresses by stretching. Residual stresses can be also reduced or eliminated by thermal treatments, such as stress relieving or annealing.

Stress

(a) (b)

Compressive

Tensile

t

c

(c)

Yt

c

!t

!c

Tension

Y

(d)

Compression

2Y


State of Stress in Metalworking

FIGURE 2.33 The state of stress in various metalworking operations. (a) Expansion of a thin-walled spherical shell under internal pressure. (b) Drawing of round rod or wire through a conical die to reduce its diameter; see Section 6.5 (c) Deep drawing of sheet metal with a punch and die to make a cup; see Section 7.6.

(a)

(b)

(c)

!1

!1

!1 !

1

!1 !

1

!3

!3

!3

!3

!3

!3

!2

!2

!2

!2


Strain State in Necking

FIGURE 2.34 Stress distribution in the necked region of a tension-test specimen.

a

Distribution ofaxial stress

Distribution ofradial or

tangential stress

Maximum axial stress

Average axial true stress

= True stress inuniaxial tension

R1

2

3 Correction factor due to Bridgman:

σσav

=1!

1+2Ra

"#$1+

a2R

%&


States of Stress

FIGURE 2.35 Examples of states of stress: (a) plane stress in sheet stretching; there are no stresses acting on the surfaces of the sheet. (b) plane stress in compression; there are no stresses acting on the sides of the specimen being compressed. (c) plane strain in tension; the width of the sheet remains constant while being stretched. (d) plane strain in compression (see also Fig. 2.15); the width of the specimen remains constant due to the restraint by the groove. (e) Triaxial tensile stresses acting on an element. (f) Hydrostatic compression of an element. Note also that an element on the cylindrical portion of a thin-walled tube in torsion is in the condition of both plane stress and plane strain (see also Section 2.11.7).

(a)

(c)

(e)

(b)

(f)

(d)

p

p

pp

p

p

!1

!1

!3

!1

!1

!3

!2

!2

!1 !

3

!3

!3

!3

!1

!1

!3

!1

!3


Yield Criteria

Tension

Y

Compression

Maximum-shearstress criterion

Distortion-energycriterion

Tension

Compression

Y

!1 !1!1

!3

!3

!3 !1

!3

!3

!1

!1!1 !1

!1

!3

!3

!3

!3

FIGURE 2.36 Plane-stress diagrams for maximum-shear-stress and distortion-energy criteria. Note that 2 = 0.

Maximum-shear-stress criterion:

Distortion-energy criterion:

σmax!σmin = Y

(σ1!σ2)2+(σ2!σ3)2+(σ3!σ1)2 = 2Y 2


Flow Stress and Work of Deformation

FIGURE 2.37 Schematic illustration of true stress-true strain curve showing yield stress Y, average flow stress, specific energy u1 and flow stress Yf.

Yf

Y

Y

Tru

e s

tress

u1

True strain (!)

!100

Flow stress:

Specific energy

Y =Kεn1n+1

u=Z ε

0σdε


Ideal & Redundant Work

FIGURE 2.38 Deformation of grid patterns in a workpiece: (a) original pattern; (b) after ideal deformation; (c) after inhomogeneous deformation, requiring redundant work of deformation. Note that (c) is basically (b) with additional shearing, especially at the outer layers. Thus (c) requires greater work of deformation than (b). See also Figs. 6.3 and 6.49.

(a)

(b)

(c)

Total specific energy:

Efficiency:

utotal = uideal +u f riction+uredundant

η=uidealutotal

lo l types of strain - wordpress.com · manufacturing processes for engineering materials, 5th ed....

Documents