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  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    G FE FORMULATIONS FOR PLASTICITY

    1

    These slides are designed based on the book:

    Finite Elements in Plasticity Theory and Practice, D.R.J. Owen and E. Hinton,

    1970, Pineridge Press Ltd., Swansea, UK.

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    2

    Course Content:

    A INTRODUCTION AND OVERVIEW

    Numerical method and Computer-Aided Engineering; Physical

    problems; Mathematical models; Finite element method.

    B REVIEW OF 1-D FORMULATIONS

    Elements and nodes, natural coordinates, interpolation function, bar

    elements, constitutive equations, stiffness matrix, boundary

    conditions, applied loads, theory of minimum potential energy; Plane

    truss elements; Examples.

    C PLANE ELASTICITY PROBLEM FORMULATIONS

    Constant-strain triangular (CST) elements; Plane stress, plane

    strain; Axisymmetric elements; Stress calculations; Programming

    structure; Numerical examples.

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    3

    D ELASTIC-PLASTIC PROBLEM FORMULATIONS

    Iterative solution methods,1-D problems, Mathematical theory of

    plasticity, matrix formulation, yield criteria, Equations for plane stress

    and plane strain case; Numerical examples

    E PHYSICAL INTERPRETATION OF FE RESULTS

    Case studies in solid mechanics; FE simulations in 3-D; Physical

    interpretation; FE model validation.

    .

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    4

    OBJECTIVE

    To learn the use of finite element method for

    the solution of problems involving elastic-

    plastic materials.

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    Terikan

    0.0 0.1 0.2 0.3 0.4 0.5

    Tegasan (

    MP

    a)

    0

    200

    400

    600

    Ujikaji A

    Ujikaji B

    5

    Stress-strain diagram

    P

    P

    lo+llo

    Ao

    Effects of high straining rates

    Effects of test temperature

    Foil versus bulk specimens

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    6

    Elastic-Plastic Behavior

    Characteristics:

    (1) An initial elastic material

    response onto which a plastic

    deformation is superimposed after

    a certain level of stress has been

    reached

    (2) Onset of

    yield governed

    by a yield

    criterion

    (3) Irreversible on

    unloading

    (4) Post-yield deformation occurs

    at reduced material stiffness

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    7

    Monotonic stress-strain behavior Engineering stressoA

    PS

    A

    P

    oo

    o

    l

    l

    l

    lle

    l

    dld

    ol

    l

    l

    dlln

    eS 1

    eA

    Ao 1lnln

    Typical low carbon steelTrue stress

    Engineering strain

    True or natural strain

    For the assumed constant

    volume condition,

    Then;

    oo lAlA

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    8

    Plastic strains

    Elastic region Hookes law:

    E

    Terikan

    0.0 0.1 0.2 0.3 0.4 0.5

    Tegasan (

    MP

    a)

    0

    200

    400

    600

    Ujikaji A

    Ujikaji B

    Engineering stress-strain diagram

    log)1(loglog E

    Plastic region:

    npK

    logloglog nK

    True stress-strain diagram

    SS316 steel

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    9

    PLASTIC STRAIN, p

    0.0 0.1 0.2 0.3 0.4 0.5 0.6

    ST

    RE

    SS

    ,

    (M

    Pa

    )

    0

    100

    200

    300

    400

    500

    600

    700

    PLASTIC STRAIN, p

    0.01 0.1 1

    ST

    RE

    SS

    ,

    (M

    Pa)

    100

    1000

    = Kn

    log K = 2.8735n = 0.1992

    r2 = 0.9772

    199.03.747 p

    Plastic strains - Example

    Non-linear /Power-law

    SS316 steel npK

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    10

    Ideal Materials behavior

    Elstic-perfectly plastic

    perfectly plastic

    Elstic-linear hardening

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    11

    Simulation of elastic-plastic problem

    Requirements for formulating the theory to model elastic-plastic

    material deformation:

    explicit stress-strain

    relationship under elastic

    condition.

    a yield criterion

    indicating the onset

    of plastic flow.

    stress-strain relationship

    for post yield behavior

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    12

    Elastic-plastic problem in 2-D

    In elastic region:

    klijklij C

    jkiljlikklijijklC

    , are Lam constants

    jiif

    jiifij

    0

    1

    Kroneckers delta:

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    Elastic case for plane stress and plane strain (Review)

    13

    x

    y

    xy

    u

    x

    v

    y

    u v

    y x

    Strain

    Stress

    x

    y

    xy

    D

    Hookes Law

    21

    2

    1 0

    1 01

    0 0 v

    vE

    D vv

    12

    1 0

    1 01 1 2

    0 0

    v vE

    D v vv v

    v

    Elasticity matrix

    Plane stress

    Plane strain

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    14

    Yield criteria:

    Stress level at which plastic deformation begins

    kf ij k() is an experimentally determined function of hardening parameter

    The material of a component subjected to complex loading will start to

    yield when the (parametric stress) reaches the (characteristic stress) in

    an identical material during a tensile test.

    Other parameters:

    Strain

    Energy

    Specific stress component (shear stress, maximum principal stress)

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    15

    Yield criteria:

    Stress level at which plastic deformation begins

    kf ij

    Yield criterion should be independent of coordinate axes orientation,

    thus can be expressed in terms of stress invariants:

    kijkij

    ijij

    ii

    J

    J

    J

    3

    1

    2

    1

    3

    2

    1

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    16

    Plastic deformation is independent of hydrostatic pressure

    kJJf 32 ,

    kkijijij 3

    1

    are the second and third invariant of deviatoric stress: 32 , JJ

    +

    deviatoric hydrostatic

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    17

    Yield Criteria

    For ductile materials:

    Maximum-distortion (shear) strain energy criterion (von-Mises)

    Maximum-shear-stress criterion (Tresca)

    For brittle materials:

    Maximum-principal-stress criterion (Rankine)

    Mohr fracture criterion

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    18

    Maximum-distortion-energy theory

    (von Mises)

    22132

    32

    2

    21 2 Y

    22

    221

    2

    1 Y

    +

    deviatoric hydrostatic

    For plane problems:

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    19

    Maximum-normal-stress theory

    (Rankine)

    ult

    ult

    2

    1

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    20

    Tresca and von Mises yield criteria

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    21

    Tresca and von Mises yield criteria

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    22

    Strain Hardening behavior

    The dependence of stress level for further plastic deformation,

    after initial yielding on the current degree of plastic straining

    Perfect plasticity

    - Yield stress level does not depend on current

    degree of plastic deformation

    Strain Hardening models

  • FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    23

    Isotropic strain hardening

    -Current yield surfaces are a uniform expansion of the original yield curve,

    without translation

    -For strain soften material, the yield stress level at a point decreases with

    increasing plastic strain, thus th