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Continuum Plasticity
Largely from Mechanical Metallurgy,
by G. E. Dieter, McGraw Hill, SI Metric Edition
Chapters 3 and 8
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Stress-strain response
Stress,s = P/A0
Strain, e =d/L0
Thes-e response will
de end on Tem .
composition, strain
rate, heat treatment,
state of stress, history,etc.
E,sy, UTS, eu, RA
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Total Strain,e
=e
elalstic +eplastic
= s/E + eplastic
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True Stress-True Strain Curve
s =s(1+e)
e=ln(1+e)
Also known as the flow curve.
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Strain Hardening Exponent, n
s=Ken
K is known as strengthening coefficient
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Work Hardening Data
Stainless Steel 0.45-0.55
70/30 Brass 0.49Copper (annealed) 0.3-0.54
Aluminum 0.15-0.25
Iron 0.05-0.15
0.05% C steel 0.26
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Necking
Volume of the specimen is constant during plasticdeformation: AL =A0L0
Initial strain hardening more than compensates for
reduction in area. Engineering stress continues to raise
with en ineerin strain.
A point is reached where decrease in area > increase in
strength due to strain hardening. Deformation gets
localized. A decreases more rapidly than the load duestrain hardening. Further elongation occurs with
decreasing load.
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Considres Criterion
Necking begins when the increase in stress due todecrease in the cross-sectional area is greater thanthe increase in load bearing capacity of the specimendue to work hardening.
= = + =-
Volume preservation -dA/A = dL/L =de
Combining, ds/de = s. Necking begins at a point
where rate of strain hardening is equal to the stress.
In terms of engineering values, ds/de = 0, at max.s!!
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Plasticity
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Yield Criteria
In uniaxial loading, plastic flow begins when s
= s0, the tensile yield stress
When does yielding begin when a material is
subjected to an arbitrary state stress?
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Von Mises Yield Criterion Yielding would occur when J2 exceeds some
critical value J2=k2
Yielding in uniaxial tension: s1=s0, s2=s3=0
2
132
322
212
1
J
s0/3 = k
2/12
13
2
32
2
210 2
1
2/12222220 )(6)()()(2
1xzyzxyxzzyyx
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12 1
2 412 6k2
Pure shear (torsion test): s1=-s3=t, s2=0
At yielding:
t=k
Yield stress in pure tension is higher!s0/ t = 3
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Energy Equivalence
Hencky (1924) showed that Von Misesyield criterion is equivalent to assuming
yielding occurs when the distortion energy
reaches a critical value.
, 0
U0 1
2E
(x2 y
2 z2 )
E
(xy yz zx )
1
2G(xy
2 yz2 zx
2)
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In terms of principal stresses:
)(221
3123212322210 EU
In terms of invariants of the stress tensors:
1 2 2E
Expressing in terms of bulk modulus (K) representing vol.
change and shear modulus (G) representing distortion
GK
GKE
3
9
GK
GK
26
23
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The first term on the RHS is dependent on
change in volume and the second on
distortion.
)3(6
1
182
21
21
0 IIGK
IU
For a uniaxial state of stress,
213232221,0 )()()(12
1
GU dist
20,0 2
12
1
GU dist
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Tresca (Max. Shear Stress) Criterion
Yielding occurs when the max. shear stressreaches the value of shear yield stress in theuniaxial tension test.
Max shear stress,
=
2/)( 31max
,
Tresca criterion: (s1-s3) = s0 Pure shear: s1 =-s3 = k; s2 = 0
(s1-s3) = 2k= s0 k= s0 /2
Predicts the same stress for yielding in uniaxialtension and in pure shear
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Less complicated mathematically than vonMises criterion
Often used in engineering design
Doesnt considers2 Need to know apriori the max. and min.
principal stresses General form:
0649636274 6242
222
332 kJkJkJJ
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Yield Locus
2/1
21323222102
1
For a biaxial plane-stress condition (s2 =0);
the von Mises criterion can be expressed as
Equation of an ellipse whose major semi-axis is
2s0 and minor semi-axis is (2/3)s0
2031
23
21
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2/
Von Mises and Tresca predict the same yield stress
for uniaxial and balanced biaxial stress loading.
Max. difference (15.5%) for pure shear case.
0
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Anisotropy in Yielding
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Yield Surface
The yield criteria can be represented
geometrically by a cylinder oriented at equalangles to the s1, s2, & s3 axes.
A state of stress which gives
a oint insi e the c lin er
represents elastic behavior.
Yielding begins when the
state of stress reaches thesurface of the cylinder.
MN, the cylinder radius is
the deviatoric stress.
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The cylinder axis, OM, which makes equal
angles with the principal axes represents the
hydrostatic component of the stress tensor.
The generator of the yield surface is the line
parallel to OM. If stress state characterized
by (s , s , s ) lies on the yield surface, sodoes (s1+H, s2 +H, s3 +H)
Von Mises criterion is represented by a
right circular cylinder whereas the Trescacriterion is represented by a regular
hexagonal prism.
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Normality
Drucker (1951): The total plastic strain vector,
must be normal to the yield surface.
Net work has to be expended during the plastic
deformation of a body. So the rate of energy
The is the incremental plastic strain vector
and must be normal to the yield surface.
Because of the normality rule, the yield locus is
always convex.
0pijijdPijd
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Hardening Models How does the yield surface change during plastic
deformation? Isotropic Hardening: The yield surface expands
uniformly, but with a fixed shape (e.g. ellipse in the caseof von Mises solid) and fixed center.
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Bauschinger Effect
The lowering ofyield stress when
deformation in one
direction is followedby deformation in
opposite direction.
Bauschinger 1881
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Kinematic HardeningThe yield surface does not change its shape and size,
but simply translates in the stress space in the direction
of its normal.
Accounts for theBauschinger effect.
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Invariants of Stress and Strain
Useful to simplify the representation of a
complex state of stress or strain by means of
respective invariant functions in such a way
that the flow curve is unaltered.
Most fre uentl used invariant functions:
Effective stress, , and effective strain,
2/1213232221
2/1213
232
221
)()()(3
2
)()()(2
1
ddddddd
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Pij
Pij
ijij
ddd
SSJ
3
2
2
33 2
1
Sij = sij
Numerical constants are
Elasticij
Totalij
Plasticij
321 22 dddd
nK
,
Uniaxial tension,
Power law hardening:
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Plastic Stress-Strain Relations
In elastic regime, the stress-strain relations areuniquely determined by the Hookes law
In plastic deformation, the strains also depend
on the history of loading.It is necessary to determine the differentials or
increments of plastic strains throughout the
loading path and then obtain the total strain byintegration.
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Example
A rod, 50 mm long, is extended to 60 mmand then compressed back to 50 mm.
On the basis of total deformation:
5060 dLdL 6050 LL
365.02.1ln250
60
60
50
L
dL
L
dL
On an incremental basis:
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Proportional Loading
A particular case in which all the stressesincrease in the same ratio, i.e.,
321
ddd 321
Plastic strains are independent of the loading path
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Two general categories of plastic stress-strain
relationships. Incrementalorflow theories relate stresses
to plastic strain increments.
Deformation ortotal strain theories relatethe stresses to totalplastic strains. Simpler
mathematically.
Both are the same for proportional loading!
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Levy-Mises Equations
Ideal plastic solids where elastic strains are
negligible.
Consider yielding under uniaxial tension:
3/and,0,0 1321 m
Constant vol. condition:3
;
3
2 132
111
m
2
1
2
1
321
2
22
d
d
ddd
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Generalization:
At any instant of deformation, the ratio of
the plastic strain increments to the current
deviatoric stresses is constant.
d
ddd
3
3
2
2
1
1
n erms o ac ua s resses,
etc.Recall,
)(
2
1
3
23211 dd
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Using the effective strain concept to
evaluate l, which yields
dd 3
2
)(
2
13211
dd
d 3
1 321 22 dddd
)(2
1
)(2
1
1233
3122
d
d
ddijij
2
)(1 zyxx E
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Drawback: Only plastic strains are considered.
Evaluation
)(
2
1
)(2
1
)(
2
1
1233
3122
3211
dd
dd
dd
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Prandtl-Reuss Equations
Proposed by Prandtl (1925) and Ruess(1930) for Elastic-Plastic Solid
Considers elastic strains as well
Recall,
Pij
Eij
Tij ddd
E
1
EE
ijkkijEij
E
d
E
d
1
ijkk
ij
d
Ed
E
3
211
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Combining with the plastic strain increments
(given by Levy -Mises Equation)
Use these with a yield criterion and the equation
ijijkk
ijTij
dd
Ed
Ed
2
3
3
211
material (e.g. power-law hardening) to calculate
the strain increment for an increment in load.
The complete solution must also satisfy
equilibrium, e-d relations, and the BCs.
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SummaryGeneral Theory of Plasticity requires the following
1) A yield criterion, which specifies the onset of plasticdeformation for different combinations of applied load.
e.g. von Mises and Tresca
2) A hardening rule, which prescribes the work hardening
the progression of plastic deformation.
Isotropic or kinematic: power-law hardening
3) A flow rule which relates increments of plasticdeformation to the stress components.
e.g. Levy-Mises or Prandtal-Reuss