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    1: Plastic deformation

    Stefan Jonsson

    2013

    Technological and true definitions

    Le

    ln

    L

    L

    L

    dLL

    LdL

    d

    0A

    Ps

    0

    A

    P

    L0 + L = L

    P P

    A0

    A

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    Elastic and plastic volumechange

    000 LA

    P P zzyyxxel

    V

    V

    0

    000 LAALV

    Volume is unaffected by plastic deformation

    because no atoms are added/removed.

    True & technological relations

    0 LLLL

    000 LLL

    )1(0

    0

    0000esL

    LLsLA

    LP

    LAA

    ALP

    A

    P

    e

    s

    .

    variables with the wanted ones.

    Remember V = constant

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    True and technological curves

    400

    450

    150

    200

    250

    300

    350

    s

    ,,

    [MPa]

    true valuesSSAB, Tunnplt

    es

    < e

    > s

    0

    50

    100

    0 2 4 6 8 10 12 14 16 18 20 22

    e , , [%]

    tech valuesDomex DD 200, Rolling directionstrain rate: 5E-03 (1/s)

    Addition of strains

    toteLLLL

    eee

    0

    03

    2

    23

    1

    12

    0

    01321

    tot

    L

    L

    L

    L

    L

    L

    L

    LL

    dL

    L

    dL

    L

    dL

    L

    dL

    3

    0

    3

    2

    2

    1

    1

    0

    321

    Only true strains can be added linearly

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    Elastic and plastic deformation

    petot

    pe

    y

    y

    p e

    Elastic and plastic deformation

    y

    Elastic deformation,

    full recoverable

    Elastic & plastic

    deformation,

    partly recoverable

    p eAs long as there is a stress,

    there is an elastic deformation!

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    Necking

    P2 P2

    P1 P1

    A B t0

    t1

    t2

    xy

    z

    m03:i

    n

    A

    B

    t1 t2

    1

    t0

    Deformation is

    homogeneous until

    onset of necking

    y

    1

    n

    A

    B

    t1

    t2

    t0

    When a neck is

    formed, deformation

    becomes highly

    localized

    Instability criterion for straining

    ddAdA

    ddAddA

    dAdA

    dP

    P=[(y)]A[(y)] V=A()L() ?

    dydddyddyddydydy

    0

    AddA

    Ld

    dL

    Ad

    dA

    Ld

    dL

    Ad

    dA

    Ld

    dV

    AdA

    For homogeneous

    0

    dy

    dA

    d

    d

    dy

    dP

    0

    dy

    d0

    d

    d

    Hom. Deform. Necking

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    Macro- & microscopic deform.

    P P

    deformation

    is only an

    average over

    the sample

    volume.

    Much

    information is

    hidden.

    The Bauschinger effect

    soft

    , ||

    f

    b

    easier

    permanent softening

    f

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    Definitions

    o po n s

    on the

    stress

    s

    Yield stressProportionality

    (Ultimate)

    Tensile strength

    curvee

    0.2%homogeneousdeformation

    necking fracture

    Lders strain

    Propagating y

    er s an s

    Inhomogeneous

    deformation Upper & lower

    yield stress

    u

    L

    Continues with

    normal plastic

    deformationL

    l

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    Strain rate sensitivity

    )( fm

    logln

    m

    )(logloglog fm

    ogn Increasedstrain rate

    Elongation v.s. m-value

    011

    md

    m

    f

    /

    )(

    1

    Deformation

    continues outside

    the neck

    Grain boundary

    sliding gives high

    m-values

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    Superplasticity

    Grain boundary sliding

    Limited to a range in T and strain rate

    Total strain energy

    ddLPdLPdW

    200

    250

    300

    350

    400

    450

    ,

    [MPa]

    2

    1

    dWV

    0

    50

    100

    150

    0 5 10 15 20

    , [%]

    Per volume unit!!

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    Elastic strain energy

    Eheigthwidth

    W elV

    2

    250

    300

    350

    400

    450

    ,[MPa]

    0

    50

    100

    150

    0 5 10 15 20

    , [%]

    Plastic strain energy

    400

    450

    400

    450

    0

    50

    100

    150

    200

    250

    300

    350

    0 5 10 15 20

    , [%]

    ,

    [MPa]

    0

    50

    100

    150

    200

    250

    300

    350

    0 5 10 15 20

    , [%]

    ,

    [MPa]

    pl

    V

    pl

    V

    el

    V

    tot

    V WWWW

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    Forcesonabody

    2

    3Face 3

    1

    Face 2

    Face 1

    F

    FN

    F

    F2

    T

    F = FT + FN = F1 + F2 + F3

    Forcesoneach surface produces 1normalstress

    and2perpendicular shear stresses

    Stress components

    2

    333

    333231

    232221

    131211

    ij

    1

    11

    22

    Firstindex: Forcedirection

    Secondindex: Faceindex

    33

    2322

    131211

    ij

    ij = ji32 32

    23

    23Symm.

    6independentelementsofthestresstensor

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    Normalstrains,iiMechanical equilibrium,i.e.no

    accelerationsarecreated by

    11

    3

    11

    22

    applied stresses.

    11

    22 2233

    1 2

    33

    03322110

    VV

    33

    0332211

    Normalplasticstrain shall not

    produce volume change.

    Positiveshear stress;12=21

    2

    3

    12-plane

    Neutral direction

    1

    Mechanical equilibrium,i.e.no

    rotationsarecreated byapplied

    shear stresspairShear deformationcannot

    produce volume change.

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    Negativeshear stress;12=21

    2

    3

    12-plane

    Neutral direction

    1

    Positiveshear stress;13=31

    2

    3Neutral direction

    1

    13-plane

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    Negativeshear stress;13=31

    2

    3Neutral direction

    1

    13-plane

    Positiveshear stress;23=32

    2

    3

    Neutral direction

    1

    23-plane

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    Negativeshear stress;23=32

    2

    3

    Neutral direction

    1

    23-plane

    Shear strains;ij=ji

    1 2

    3

    NOTE!ij=ji

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    Definitionofstrain tensorDisplacements u1,u2andu3are

    found bymoving inx1,x2andx3

    i

    j

    j

    iij

    x

    u

    x

    u

    2

    1

    Thetensor comp.isacombination

    oftwo dis lacements u andu

    Thetensor comp.isthe

    displacement,uifound

    bymoving inxi

    i=j

    i

    1 2

    3

    found bymoving inxjandxi

    ji

    j

    i

    i

    j

    i

    j

    j

    iij

    x

    u

    x

    u

    x

    u

    x

    u

    2

    1

    2

    1

    Engineering shear, y

    Pure shear ySimple shear

    x2

    1

    x

    1

    Produced bydislocation slip

    21 jiijij

    ijjiiji

    j

    j

    iij

    x

    u

    x

    u

    2

    1

    2

    1

    2

    1

    This is only half of the

    engineering shear!!

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    Strain components, ij

    0

    02

    1

    00

    00

    3231

    2321

    33

    22

    332313

    232212

    ij

    5 Independent strains: 2 3

    Symm. Symm. Symm.

    AngularVolumetric

    03322110

    VV

    5 independent slip systems must be activated to produce a general deformation.

    Fragmentation of grains reduces this number locally.

    e orma one orma on

    Stress states

    y

    x

    h

    tanh

    cos

    yy

    xx

    xy

    xy

    2cos2sin2

    cossin2sincos 22

    xy

    xxyy

    xyyyxx

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    Mohrs circle for stresses (xx ,xy)

    R (, )

    (1 ,0)(2 ,0)

    ( ,-x )

    22

    2sinxy

    R

    R

    R

    yyxx

    yyxx

    )(2

    1

    )(2

    1

    2

    1

    2

    2

    2

    22cos

    xy

    yyxx

    yyxx

    R

    R

    Rotationbetween principaldirections

    z

    z

    =90 =45 0

    y

    y

    z

    y

    2=180 2=90 20

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    45 direction isbestforone principal

    stress

    zz zz zz

    =45

    zz zz zz

    2=90

    2

    zz

    General stress state

    31

    123

    z

    zz

    yyxx

    yzzy

    zx

    yx xy

    xz

    2max

    x

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    20

    123

    Hydrostatic

    pressure

    hydrostatic

    pressure