<< Предыдущая стр. 52(из 54 стр.)ОГЛАВЛЕНИЕ Следующая >>
trial
trial trial
be,n+1 = О»e,О± nО± вЉ— nО± ; = (A.13)
П„ nО± вЉ— nО±
в€‚ ln О»trial
e,О±
О±=1 О±=1

Invariably this state of stress will not be compatible with the assumption
that no further permanent strain takes place during the increment. In-
troducing the incremental permanent deformation gradient F в€†p so that
7
A.4 INCREMENTAL KINEMATICS

Fn + 1

Fe, n + 1
Fn
Fe, n trial
Fe, n + 1

time = tn+1

Fв€† p
Fp, n time = tn
time = 0

Fp, n + 1

FIGURE A.4 Multiplicative decomposition at increments n and n + 1.

F p,n+1 = F в€†p F p,n (see Chapter 3, Exercise 2), the following multiplica-
tive decomposition for F trial is obtained,
e,n+1

F trial = F e,n+1 F в€†p (A.14)
e,n+1

Note that this incremental multiplicative decomposition and the correspond-
ing
three conп¬Ѓgurations вЂ“ current state and unloaded conп¬Ѓgurations at incre-
ments n and n + 1 вЂ“ are analogous to the global decomposition F = F e F p
shown in Figures A.2 and A.3. In eп¬Ђect, the previous unloaded state is taking
the place of the initial reference conп¬Ѓguration, the incremental permanent
strain replaces the total inelastic strain, and F trial replaces F .
e,n+1
We now make the crucial assumption that the incremental permanent
strain must be colinear with the current stresses and therefore with the cur-
rent elastic deformation gradient. This can be justiп¬Ѓed by recalling that
in small strain plasticity, for instance, the incremental plastic strain в€†Оµp is
given by a п¬‚ow rule as the gradient of a certain stress potential П†. If the
material is isotropic, this potential must be a function of the invariants of
the stress and therefore its gradient will be co-linear with the stress tensor.
As a consequence of this assumption, Equation (A.10b) applies in this incre-
mental setting and therefore the principal stretches of F trial , F e,n+1 , and
e,n+1
8 LARGE INELASTIC DEFORMATIONS

F в€†p are related as,
ln О»trial = ln О»e,О± + ln О»в€†p,О± (A.15)
e,О±

Moreover, the principal directions of stress at increment n + 1 can be eval-
uated directly from btrial before F e,n+1 is actually obtained.
e,n+1
Further progress cannot be made without a particular equation for ln О»в€†p,О± .
This is given by the particular inelastic constitutive model governing the
material behavior. One of the simplest possible models that works well for
many metals is the von Mises plasticity theory.

A.5 VON MISES PLASTICITY

A.5.1 STRESS EVALUATION
Von Mises plasticity with linear isotropic hardening is deп¬Ѓned by a yield
surface П†, which is a function of П„ and a hardening parameter given by the
Von Mises equivalent strain Оµp as,
ВЇ
в€љ
П†(П„ , Оµp ) = П„ : П„ в€’ 2/3 (Пѓy + H Оµp ) в‰¤ 0
ВЇ ВЇ (A.16)
where the deviatoric and pressure components of the Kirchhoп¬Ђ stress tensor
are,
3
П„ = П„ + pI; П„= П„О±О± nО± вЉ— nО± (A.17a,b)
О±=1

Observe that the principal directions of П„ are those given by the trial tensor
btrial .
e,n+1
In Von Mises plasticity theory, as in many other metal plasticity mod-
els, the plastic deformation is isochoric, that is, det F p = 1. Under such
conditions J = Je , and the hydrostatic pressure can be evaluated directly
from J as in standard hyperelasticity. If we now assume that the elastic
stress is governed by the simple stretch-based energy function described in
Section 5.6, Equation (5.103), the principal components of the deviatoric
Kirchhoп¬Ђ tensor are,
2
П„О±О± = 2Вµ ln О»e,О± в€’ 3 Вµ ln J (A.18)
Substituting for О»e,n+1 from Equation A.15 gives,
2
trial
П„О±О± = 2Вµ ln О»trial в€’ 3 Вµ ln J
trial
П„О±О± = П„О±О± в€’ 2Вµ ln О»в€†p,О± ; (A.19a,b)
e,О±

Note that П„ trial is the state of stress that is obtained directly from btrial
e,n+1
under the assumption that no further inelastic strain takes place during the
9
A.5 VON MISES PLASTICITY

increment, and the term в€’2Вµ ln О»в€†p,О± is the correction necessary to ensure
that the yield condition (A.16) is satisп¬Ѓed.
In small strain theory the incremental plastic strain is proportional to
the gradient of the yield surface. We can generalize this п¬‚ow rule to the
large strain case by making the logarithmic incremental plastic stretches
proportional to the gradient of П† as,
в€‚П†
ln О»в€†p,О± = в€†Оі (A.20)
в€‚П„О±О±
where the gradient of the von Mises yield surface gives a unit vector ОЅ in
the direction of the stress as,
в€‚П† П„
= в€љ О±О± = ОЅО± (A.21a,b)
в€‚П„О±О± П„ :П„
The use of Equation (A.20) to evaluate the incremental plastic stretches
that ensure that П„ lies on the yield surface is known as the return mapping
algorithm.

A.5.2 THE RADIAL RETURN MAPPING
Substituting Equations (A.20вЂ“1) into (A.19a) gives,
trial
П„О±О± = П„О±О± в€’ 2Вµв€†Оі ОЅО± (A.22)

This equation indicates that П„ is proportional to П„ trial and is therefore
known as the radial return mapping (see Figure A.5). As a consequence of
this proportionality, the unit vector ОЅ can be equally obtained from П„ trial ,
that is,
trial
П„О±О± П„О±О±
ОЅО± = в€љ =в€љ (A.23)
trial trial П„ :П„
П„ :П„
and therefore the only unknown in Equation (A.22) is now в€†Оі.
In order to evaluate в€†Оі we multiply Equation (A.22) by ОЅО± and enforce
the yield condition (A.16) to give after summation for О± and use of (A.23),
в€љ
в€љ
П„ : П„ = П„ trial : П„ trial в€’ 2Вµв€†Оі = 2/3 (Пѓy + H Оµp,n + Hв€†ВЇp ) (A.24)
ВЇ Оµ

In small strain analysis the increment in von Mises equivalent strain is given
as в€†ВЇ2 = 2 в€†Оµp : в€†Оµp . Again replacing small strains by logarithmic strains
Оµp 3
gives,
3
2
в€†ВЇ2 (ln О»в€†p,О± )2
Оµp = (A.25)
3
О±=1
10 LARGE INELASTIC DEFORMATIONS

П„3 -
П†(П„,Оµ p,n+1) = 0

2Вµв€†Оі

-
П†(П„,Оµ p,n) = 0 П„0trial
n+1
П„0 + 1
v n

П„1 П„2

FIGURE A.5 Radial return.

and substituting for ln О»в€†p,О± from Equation (A.20) yields,

в€†ВЇp =
Оµ 2/3 в€†Оі (A.26)

Substituting this expression into Equation (A.24) enables в€†Оі to be evaluated
explicitly as,
trial
пЈІ П†(П„ , Оµp,n )
ВЇ
пЈ±
if П†(П„ trial , Оµp,n ) > 0
ВЇ
2
в€†Оі = (A.27)
2Вµ + 3 H
if П†(П„ trial , Оµp,n ) в‰¤ 0
0 ВЇ
пЈі

Once the value of в€†Оі is known, the current deviatoric Kirchhoп¬Ђ stresses
are easily obtained by re-expressing Equation (A.22) as,
2Вµв€†Оі trial
JПѓО±О± = П„О±О± = 1в€’ П„О±О± (A.28)
П„ trial
в€љ
where the notation П„ trial = П„ trial : П„ trial has been used.
In order to be able to move on to the next increment, it is necessary to
record the current state of permanent or plastic deformation. In particular,
the new value of the von Mises equivalent strain emerges from (A.26) as,

Оµp,n+1 = Оµp,n +
ВЇ ВЇ 2/3 в€†Оі (A.29)

In addition the current plastic deformation gradient tensor F p , or more pre-
cisely Cв€’1 = F в€’1 F в€’T , will be needed in order to implement Equation (A.12)
p p p
11
A.5 VON MISES PLASTICITY

during the next increment. Noting that F p = F в€’1 F gives,
e

C в€’1 = F в€’1 F в€’T
p,n+1 p,n+1 p,n+1

= F в€’1 F e,n+1 F T в€’T
e,n+1 F n+1
n+1

= F в€’1 be,n+1 F в€’T (A.30)
n+1 n+1

where the elastic left CauchyвЂ“Green tensor is,
3
О»2 nО± вЉ— nО±
be,n+1 = (A.31)
e,О±
О±=1

and the elastic stretches are,
ln О»e,О± = ln О»trial в€’ в€†Оі ОЅО± (A.32)
 << Предыдущая стр. 52(из 54 стр.)ОГЛАВЛЕНИЕ Следующая >>