σ and the current yield stress as by oid26884

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									EN0175                                                                                   10 / 26 / 06


Plastic material behavior


Yield condition:   σ = σY

Plastic loading:   σ = σ Y , dσ > 0


                            σ
                   σY
                   σY0

                                          E
                                                   E
                                      1
                                               1
                                                                                  ε
                          0 εP                     εE
We will now denote the initial yield stress as   σ Y 0 and the current yield stress as

σ Y ; see above figure.


Decompose strain into elastic & plastic parts

                                              ε = εE +εP
In incremental form:
                                                       dσ
                                              dε E =
                                                        E
                                       ⎧ dσ
                                       ⎪ h , if σ = σ Y , dσ > 0
                                       ⎪
                                dε P = ⎨
                                       ⎪0, otherwise
                                       ⎪
                                       ⎩
     dσ
h=        is the tangent modulus of σ − ε P curve (which is obtained from experiments or
     dε P
fitting an assumed mathematical curve to experimental data).



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EN0175                                                                            10 / 26 / 06




Perfectly plastic: h = 0




                           σY0

                                                                       εP
Linear work hardening: h = const.




                                                           h
                            σY 0                   1


                                                                       εP
                                                       N
                                 ⎛ Eε P ⎞
                                 ⎜ σ ⎟
Power law hardening: σ Y = σ Y 0 ⎜1 +    ⎟
                                 ⎝    Y0 ⎠




                           σY 0
                                                                       εP
How to generalize this idea to 3D?

For elastic part,

       σ                                 1 +ν       ν            σ ij σ kk
                                                                       '

εE =                ⇒         ε    E
                                       =      σ ij − σ kk δ ij =     +     δ ij
                                                                 2μ 9 K
                                  ij
       E                                   E        E

dε ij = dε ij + dε ij
            E       P




                                                  1          1
                                       dε ij =
                                           E
                                                    dσ ij +
                                                       '
                                                               δ ij dσ kk
                                                 2μ         9K
The plastic strain is typically modeled by the Levy-Mises theory,


                                                  dε ij = σ ij dλ
                                                      P     '



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EN0175                                                                                  10 / 26 / 06




The yield condition:

                                      3 ' '
                               σe =     σ ijσ ij = σ Y (von Mises stress)
                                      2

shows that   σ e generalizes the 1D stress to an effective stress in 3D.
To utilize the measured plastic stress-strain behavior in 1D, we also need an effective measure of
plastic strain in 3D. To see how this is generalized, consider strain energy,

                    (             )
dw = σ ij dε ij = σ ij dε ij + dε ij = dw E + σ ij dε ij = dw E + dw P
                           E       P             '     P




                                            2                              3 dε P
dw P = σ ij dε ij = σ e dε P = σ ijσ ij dλ = σ e2 dλ
         '      P                '   '
                                                             ⇒     dλ =
                                            3                              2 σe

       3 ' dε
dε ij = σ ij P
    P

       2    σe
Self consistency requires,

                                          3 ' dε 3 ' dε     3 2
                             dε ij dε ij = σ ij P ⋅ σ ij P = dε P
                                 P     P

                                          2    σe 2     σe  2
Therefore,

                                                  2 P P
                                          εP =      dε ij dε ij
                                                  3


Summary:

Generalization from 1D to 3D,

                                       3 ' '               2 P P
                               σ→        σ ijσ ij , dε P →   dε ij dε ij
                                       2                   3
1D plastic stress-strain law

                                           dε = dε E + dε P
                                                        dσ
                                              dε E =
                                                         E
                                       ⎧ dσ
                                       ⎪ h , if σ = σ Y , dσ > 0
                                       ⎪
                                dε P = ⎨
                                       ⎪0,       otherwise
                                       ⎪
                                       ⎩

is generalized to 3D plastic stress-strain law


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EN0175                                                                                          10 / 26 / 06



                                            dε ij = dε ij + dε ij
                                                        E       P




                                                 1          1
                                      dε ij =
                                          E
                                                   dσ ij +
                                                      '
                                                              δ ij dσ kk
                                                2μ         9K

                                     ⎧ 3 ' dσ e
                                     ⎪ 2 σ ij hσ , if σ e = σ Y , dσ e ≥ 0
                                     ⎪          e
                            dε ij
                                P
                                    =⎨
                                     ⎪0,             otherwise
                                     ⎪
                                     ⎩


A few remarks:

1) Yield conditions:

                            3 ' '
Mises condition:    σe =      σ ijσ ij = σ Y
                            2

Tresca condition:   σ I , σ II , σ III , Max(σ I − σ II , σ I − σ III , σ II − σ III ) = σ Y


Representation of yield conditions in stress space:



                           σ III                                              1
                                                                                    (1, 1, 1)
                                                                                3
                                                     v
                                                    σ
                                                                           σ II


                                                                 σI
                                         Yield

This is called the yield surface. A perspective view along the
                                                                       1
                                                                          (1, 1, 1) direction would show
                                                                        3
the projection of the von Mises yield surface as a circle




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EN0175                                                                                     10 / 26 / 06




                        2
                          σY
                        3                          von Mises condition

                                                 Tresca condition
The Tresca yield condition corresponds to the inscribed hexagon inside the von Mises circle. The
two yield conditions are actually quite close to each other.


2) J 2 - flow theory

The three invariants of stress σ ij are sometimes denoted as


I1 = σ kk , I 2 = σ ijσ ij , I 3 = det (σ ij ) (3 invariants of σ ij )
                 1
                 2
The three invariants of deviatoric stress   σ ij are sometimes denoted as:
                                              '



       '             1 ' '
                     2
                                              ( )
J1 = σ kk = 0 , J 2 = σ ijσ ij , J 3 = det σ ij
                                              '
                                                     (3 invariants of    σ ij )
                                                                           '



The Levy-Mises flow rule can be written as

                    ∂J 2
dε ij = σ ij dλ =
    P      '
                          dλ
                    ∂σ ij
                        '




where J 2 acts as a potential for plastic deformation. Therefore, this theory is also called the J 2 -

flow theory.




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