EBK MANUFACTURING PROCESSES FOR ENGINEE
EBK MANUFACTURING PROCESSES FOR ENGINEE
6th Edition
ISBN: 9780134425115
Author: Schmid
Publisher: YUZU
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Chapter 6, Problem 6.79P

(a)

To determine

The force vs. reduction in height curve in open die forging of cylinder for μ=0.2 between the flat dies and the specimen.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given:

The initial thickness of the specimen is ho=10mm .

The initial diameter of the specimen is do=25mm .

The friction coefficient is μ=0.2 .

Formula used:

The expression for the flow stress is given as,

  σf=Kεn ....... (1)

Here, σf is the flow stress, K is the strength coefficient, ε is the true strain, n is the strain hardening coefficient.

The expression for the true strain is given as,

  ε=ln(hohf)

Here, hf is the final thickness.

The expression for the final radius by equating the volume is given as,

  rf2=ro2hohf

The expression for the forging force is given as,

  F=paπrf2

Here, pa is the average pressure.

The expression for the average pressure is given as,

  pa=σf(1+2μrf3hf)

The expression for final height for 10% reduction in height in given as,

  ( h o h f h o )×100%=10%hf=0.9ho

The expression for final height for 20% reduction in height in given as,

  ( h o h f h o )×100%=20%hf=0.8ho

The expression for final height for 30% reduction in height in given as,

  ( h o h f h o )×100%=30%hf=0.7ho

The expression for final height for 40% reduction in height in given as,

  ( h o h f h o )×100%=40%hf=0.6ho

The expression for final height for 50% reduction in height in given as,

  ( h o h f h o )×100%=50%hf=0.5ho

Calculation:

For 10% reduction,

The final height can be calculated as,

  hf=0.9hohf=0.9×10mmhf=9mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm9mmrf=13.17mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 9mm)ε=0.1053

The flow stress can be calculated as,

  σf=Kεnσf=1400MPa×(0.10536)0.015σf=1353.53MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=14000MPan=0.015

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=1353.53MPa×(1+ 2×0.2×13.17mm 3×9mm)pa=1617.61MPa

The forging force can be calculated as,

  F=paπrf2F=1617.61MPa×3.14×(13.17mm)2F=880998.2N( 1MN 10 6 N)F=0.88MN

For 20% reduction,

The final height can be calculated as,

  hf=0.8hohf=0.8×10mmhf=8mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm8mmrf=13.97mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 8mm)ε=0.223

The flow stress can be calculated as,

  σf=Kεnσf=1400MPa×(0.22314)0.015σf=1368.85MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=14000MPan=0.015

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=1368.85MPa×(1+ 2×0.2×13.97mm 3×8mm)pa=1652.15MPa

The forging force can be calculated as,

  F=paπrf2F=1652.15MPa×3.14×(13.97mm)2F=1012446.15N( 1MN 10 6 N)F=1.033MN

For 30% reduction,

The final height can be calculated as,

  hf=0.7hohf=0.7×10mmhf=7mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm7mmrf=14.94mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 7mm)ε=0.356

The flow stress can be calculated as,

  σf=Kεnσf=1400MPa×(0.356)0.015σf=1378.47MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=14000MPan=0.015

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=1378.47MPa×(1+ 2×0.2×14.94mm 3×7mm)pa=1770.74MPa

The forging force can be calculated as,

  F=paπrf2F=1770.74MPa×3.14×(14.97mm)2F=1241039.604N( 1MN 10 6 N)F=1.241MN

For 40% reduction,

The final height can be calculated as,

  hf=0.6hohf=0.6×10mmhf=6mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm6mmrf=16.13mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 6mm)ε=0.51

The flow stress can be calculated as,

  σf=Kεnσf=1400MPa×(0.51)0.015σf=1385.96MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=14000MPan=0.015

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=1385.96MPa×(1+ 2×0.2×16.13mm 3×6mm)pa=1882.749MPa

The forging force can be calculated as,

  F=paπrf2F=1882.749MPa×3.14×(16.13mm)2F=1538122.08N( 1MN 10 6 N)F=1.538MN

For 50% reduction,

The final height can be calculated as,

  hf=0.5hohf=0.5×10mmhf=5mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm5mmrf=17.67mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 5mm)ε=0.69

The flow stress can be calculated as,

  σf=Kεnσf=1400MPa×(0.69)0.015σf=1392.32MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=14000MPan=0.015

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=1392.32MPa×(1+ 2×0.2×17.67mm 3×5mm)pa=2048.381MPa

The forging force can be calculated as,

  F=paπrf2F=2048.381MPa×3.14×(17.67mm)2F=2008230.344N( 1MN 10 6 N)F=2.00MN

For μ=0.2

    Reduction (in % )Forging force (in MN )
    100.88
    201.033
    301.24
    401.53
    502.00

The plot between forging force and reduction in height is shown in figure (1) below,

  EBK MANUFACTURING PROCESSES FOR ENGINEE, Chapter 6, Problem 6.79P , additional homework tip  1

  Figure (1)

(b)

To determine

The force vs. reduction in height curve in open die forging of cylinder for μ=0.4 between the flat dies and the specimen.

(b)

Expert Solution
Check Mark

Explanation of Solution

Given:

The initial thickness of the specimen is ho=10mm .

The initial diameter of the specimen is do=25mm .

The friction coefficient is μ=0.4 .

Formula used:

The expression for the flow stress is given as,

  σf=Kεn ....... (1)

Here, σf is the flow stress, K is the strength coefficient, ε is the true strain, n is the strain hardening coefficient.

The expression for the true strain is given as,

  ε=ln(hohf)

Here, hf is the final thickness.

The expression for the final radius by equating the volume is given as,

  rf2=ro2hohf

The expression for the forging force is given as,

  F=paπrf2

Here, pa is the average pressure.

The expression for the average pressure is given as,

  pa=σf(1+2μrf3hf)

The expression for final height for 10% reduction in height in given as,

  ( h o h f h o )×100%=10%hf=0.9ho

The expression for final height for 20% reduction in height in given as,

  ( h o h f h o )×100%=20%hf=0.8ho

The expression for final height for 30% reduction in height in given as,

  ( h o h f h o )×100%=30%hf=0.7ho

The expression for final height for 40% reduction in height in given as,

  ( h o h f h o )×100%=40%hf=0.6ho

The expression for final height for 50% reduction in height in given as,

  ( h o h f h o )×100%=50%hf=0.5ho

Calculation:

For 10% reduction,

The final height can be calculated as,

  hf=0.9hohf=0.9×10mmhf=9mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm9mmrf=13.17mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 9mm)ε=0.1053

The flow stress can be calculated as,

  σf=Kεnσf=650MPa×(0.10536)0.064σf=562.81MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=650MPan=0.064

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=562.81MPa×(1+ 2×0.4×13.17mm 3×9mm)pa=782.43MPa

The forging force can be calculated as,

  F=paπrf2F=782.43MPa×3.14×(13.17mm)2F=426135.02N( 1MN 10 6 N)F=0.42MN

For 20% reduction,

The final height can be calculated as,

  hf=0.8hohf=0.8×10mmhf=8mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm8mmrf=13.97mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 8mm)ε=0.223

The flow stress can be calculated as,

  σf=Kεnσf=650MPa×(0.22314)0.064σf=590.479MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=14000MPan=0.015

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=590.479MPa×(1+ 2×0.4×13.97mm 3×8mm)pa=865.445MPa

The forging force can be calculated as,

  F=paπrf2F=865.44MPa×3.14×(13.97mm)2F=530349.456N( 1MN 10 6 N)F=0.53MN

For 30% reduction,

The final height can be calculated as,

  hf=0.7hohf=0.7×10mmhf=7mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm7mmrf=14.94mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 7mm)ε=0.356

The flow stress can be calculated as,

  σf=Kεnσf=650MPa×(0.356)0.064σf=608.42MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=650MPan=0.064

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=608.42MPa×(1+ 2×0.4×14.94mm 3×7mm)pa=954.69MPa

The forging force can be calculated as,

  F=paπrf2F=954.69MPa×3.14×(14.97mm)2F=671793.22N( 1MN 10 6 N)F=0.67MN

For 40% reduction,

The final height can be calculated as,

  hf=0.6hohf=0.6×10mmhf=6mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm6mmrf=16.13mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 6mm)ε=0.51

The flow stress can be calculated as,

  σf=Kεnσf=650MPa×(0.51)0.064σf=622.58MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=650MPan=0.064

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=622.58MPa×(1+ 2×0.4×16.13mm 3×6mm)pa=1068.90MPa

The forging force can be calculated as,

  F=paπrf2F=1068.90MPa×3.14×(16.13mm)2F=873244.25N( 1MN 10 6 N)F=0.87MN

For 50% reduction,

The final height can be calculated as,

  hf=0.5hohf=0.5×10mmhf=5mm

The final radius can be calculated as,

  rf2=ro2hohfrf2= ( 12.5mm )2×10mm5mmrf=17.67mm

The true strain can be calculated as,

  ε=ln( h o h f )ε=ln( 10mm 5mm)ε=0.69

The flow stress can be calculated as,

  σf=Kεnσf=650MPa×(0.69)0.064σf=634.74MPa

Refer to table 2.2 “Typical values of strength coefficient K and strength hardening exponent n ” for annealed Ti-6Al-4V is,

  K=650MPan=0.064

The average pressure can be calculated as,

  pa=σf(1+ 2μ r f 3 h f )pa=634.74MPa×(1+ 2×0.4×17.67mm 3×5mm)pa=1232.91MPa

The forging force can be calculated as,

  F=paπrf2F=1232.918MPa×3.14×(17.67mm)2F=1208752.21N( 1MN 10 6 N)F=1.2MN

For μ=0.4

    Reduction (in % )Forging force (in MN )
    100.42
    200.52
    300.67
    400.87
    501.20

The plot between forging force and reduction in height is shown in figure (2) below,

  EBK MANUFACTURING PROCESSES FOR ENGINEE, Chapter 6, Problem 6.79P , additional homework tip  2

  Figure (2)

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Chapter 6 Solutions

EBK MANUFACTURING PROCESSES FOR ENGINEE

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