AME 60645
Mechanical Behavior of Materials
Fall 2016
DEFORMATION SOLUTIONS
Given/Resultant Conditions
#l&
#A &
• geometry and dimensions, ε = ln% ( = ln% o ( = etc.
$ A'
$ lo '
• applied/resultant forces, F = s·A, pressures, torques, work, u = volume·
σ ⋅ dε , etc.
¯­
Applied Stress State
Friction (Slab Method)
Forging: F = pavg·2aw
pavg = Y " ⋅ (1+ µa/ h) , p = Y " ⋅ exp[2µ(a − x) /h ] (sliding)
pavg = Y " ⋅ (1+ a /2h ) , p = Y " ⋅ [1+ (a − x) /h ] (sticking)
Rolling: F ≅ Y # ⋅ w R ⋅ Δh (low friction)
& µ R ⋅ Δh )
+ (high fiction)
F ≅ Y # ⋅ w R ⋅ Δh ⋅ ((1+
2 ⋅ havg +*
'
h
p = Yf" ⋅ ⋅ exp[µ(Ho − H )] (entry zone)
ho
h
p = Yf" ⋅ ⋅ exp[µH ] (exit zone)
ho
% R (
R
H =2
⋅ tan−1''
⋅ φ **
hf
h
& f
)
µ⋅ cot α
−1] (die only)
Extrusion: p = Y ⋅ [1+ tanα / µ] ⋅ [R
p ≅ 1.7 ⋅Y ⋅ lnR (die only)
p ≅ Y ⋅ (1.7⋅ ln R + 2L /Do ) (die & container)
¯­
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Stress
Transformation (Rotation of Axes)
• Mohr’s Circle
• Tensor Rotation
¯­
Principal Stresses and Strains
#σ1
&
%
(
%
σ2
( and
%
(
σ 3'
$
#ε1
&
%
(
% ε2
(
%
(
ε 3'
$
Drawing: σ d = Y ⋅ [1+ tanα / µ] ⋅ [1− (A f / Ao ) µ⋅ cot α ]
σ d = Y ⋅ [1+ α / µ] ⋅ ln(A f / Ao )⋅ Φ (inhomog.)
¯­
¯­
Effective Stress and Strain
Material Properties (Plastic)
s = Y perfectly plastic, “ideal”
n
n
σ = K ⋅ ε strain hardening (power law) σ = K ⋅ ε
Tresca (maximum shear stress):
σ = σ 3 − σ 1 (if s3 > s 2 > s 1)
ε = 2/ 3⋅ (ε1 − ε 3 )
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ε
0
€¯­
#σ
&
xx σ xy σ xz
%
(
σ ij = %σ yx σ yy σ yz (
%$σ zx σ zy σ zz ('
€
∫
von Mises (distortion energy):
1/ 2
1
σ=
(σ 1 − σ 2 ) 2 + (σ 2 − σ 3 )2 + (σ 3 − σ 1 )2 ]
[
2
1/ 2
2
ε=
(ε1 − ε2 ) 2 + (ε2 − ε3 )2 + (ε 3 − ε1 )2 ]
[
3
note: strain rate effects could also be included here
¯­
Yield Criteria
yielding occurs when… σ = Y = 2⋅ k (Tresca)
σ = Y = 3 ⋅ k (von Mises)
Y = yield strength, k = shear yield strength, Y’ = 2Y/ 3 = plane e Y
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Dislocations
motion analogy:
edge:
screw:
W.D. Callister, Jr., Materials Science and Engineering, 2nd Ed., John Wiley and Sons, New York, 1991.
AME 60645: Mechanical Behavior of Materials (R.K. Roeder)
Plastic Anisotropy: Deep Drawing
W.F. Hosford and R.M. Caddell, Metal Forming, 2nd Ed., Prentice-Hall, Inc., Edgewood Cliffs, NJ, 1993.
AME 60645: Mechanical Behavior of Materials (R.K. Roeder)
Plastic Anisotropy: Deep Drawing
W.F. Hosford and R.M. Caddell, Metal Forming, 2nd Ed., Prentice-Hall, Inc., Edgewood Cliffs, NJ, 1993.
AME 60645: Mechanical Behavior of Materials (R.K. Roeder)
Recovery, Recrystallization and Grain Growth
cold worked brass
early recrystallization
(3 s at 580°C)
partial recrystallization
(4 s at 580°C)
full recrystallization
(8 s at 580°C)
grain growth
(15 min at 580°C)
grain growth
(10 min at 700°C)
AME 60645: Mechanical Behavior of Materials (R.K. Roeder)
Recovery, Recrystallization and Grain Growth
W.D. Callister, Jr., Materials Science and Engineering, 2nd Ed., John Wiley and Sons, New York, 1991.
AME 60645: Mechanical Behavior of Materials (R.K. Roeder)
Strengthening Mechanisms: Steel Structures
Adapted from: Metals Handbook: Metallography, Structures and Phase Diagrams,
Vol. 8, 8th edition, T. Lyman Ed., American Society for Metals, Materials Park, OH, 1973.
AME 60645: Mechanical Behavior of Materials (R.K. Roeder)
Strengthening
Mechanisms:
Steel Structures
AME 60645: Mechanical Behavior of Materials (R.K. Roeder)
Superplastic Forming
Titanium alloy aircraft panel produced by diffusion bonding followed by superplastic
expansion using internal pressure. Courtesy of Rockwell International Corp.
Complex shapes formed from Zn-22%Al sheet metal using superplastic forming.
Courtesy of D.S. Fields, IBM Corp.
W.F. Hosford and R.M. Caddell, Metal Forming, 2nd Ed., Prentice-Hall, Inc., Edgewood Cliffs, NJ, 1993.
AME 60645: Mechanical Behavior of Materials (R.K. Roeder)