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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

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

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    23

    Normality rule of associated plasticity

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

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

    Notations:

    24

    0

    FQK

    0 FQ K

    Nonlinear problem:

    Thus, iterative solution is required

    Linear elastic problem:

    0

    fH

    0f H

    For 1-D (1 variable) problem:

    0 fH

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

    MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS