Download Solution Manual 3rd Ed. Metal Forming: Mechanics and Metallurgy CHAPTER 1-3 PDF

TitleSolution Manual 3rd Ed. Metal Forming: Mechanics and Metallurgy CHAPTER 1-3
Tags Stress (Mechanics) Deformation (Mechanics) Yield (Engineering)
File Size587.4 KB
Total Pages11
Document Text Contents
Page 10

3-2 During a tension test the tensile strength was found to be 340 MPa. This was
recorded at an elongation of 30%. Determine n and K if the approximation σ =K 
applies.
Solution: n = εmax load = ln(1+emax load) = ln(1.3) = 0.262.
σmax load = smax load (1+emax load) = 340(1.3) = 442 MPa. But also σmax load = K(.0.262)0.262 =
0.704K. K = 442/0.704 = 627 MPa.
*
3.3 Show that the plastic work per volume isσ 1ε 1 /(n+1for a metal stretched in tension to

ε 
1 if  σ =k ε .

Solution: w = ∫σ1dε1 = ∫k ε1
ndε1 = k ε1

n+1/(n+1) = k ε1ε1
n/(n+1) =σ 1ε 1 /(n+1

3.4 For plane-strain compression (Figure 3.11)
a. Express the incremental work per volume, dw, in terms of σ  and d ε and

compare it with dw = σ1dε1 + σ2dε2 + σ3dε3.
 b. If σ =k ε n, express the compressive stress, as a function of σ1, K and n.
Solution: a. With εy = 0 and σx = 0, dw = σ3dεz. σy = σz/2, σx =0,
σ = {[(σy - σz)

2 +(σz – σx)
2 +(σx – σy)

2]/2}1/2 = {[(-σz/2)
2 +(-σz)

2 + (-σz/2)
2]/2}1/2 = (3/4)σz

de=[(2/3)(d ε 
 x 

2
+d ε 
 y

2
+d ε 
 z

2
)]
1/2
={(2/3)[(−d ε 

 x 
)
2
+0+d ε 

 z

2
]}
1/ = (4/3)1/2dεz

σ d  = (3/4)σz(4/3)
1/2dεz = (σzdεz

 b. σ  z=(4/3)
1/2
σ =(4/3)

1/2
k ε 
n
=(4/3)

1/2
k (4/3)

n /2
= (4/3)(n+1)/2en.

3.5 The following data were obtained from a tension test:
Load Min. Neck true true corrected

dia. radius strain stress true stress
(kN) (mm) (mm) σ (MPa) σ  (MPa)
0 8.69 ∞ 0 0 0
27.0 8.13 ∞ 0.133 520 520
34.5 7.62 ∞
40.6 6.86 ∞
38.3 5.55 10.3
29.2 3.81 1.8

a. Compute the missing values
 b. Plot both σ and σ  vs. ε on a logarithmic scale and determine K and n.
c. Calculate the strain energy per volume when ε = 0.35.

Solution: a)
Load Min. Neck true true a/R corrected

dia. radius strain stress true stress
(kN) (mm) (mm) σ (MPa) σ  (MPa)
0 8.69 ∞ 0 0 0 0
27.0 8.13 ∞ 0.133 520 0 520
34.5 7.62 ∞ 0.263 754 0 654
40.6 6.86 ∞ 0.473 1099 0 1099

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