Continuum Plasticity

7/31/2019 Continuum Plasticity

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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

Continuum Plasticity

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