trial trial
be,n+1 = »e,± n± — n± ; = (A.13)
„ n± — n±
‚ ln »trial
e,±
±=1 ±=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 ∆p so that
7
A.4 INCREMENTAL KINEMATICS
Fn + 1
Fe, n + 1
Fn
Fe, n trial
Fe, n + 1
time = tn+1
F∆ 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 ∆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 ∆p (A.14)
e,n+1
Note that this incremental multiplicative decomposition and the correspond
ing
three con¬gurations “ current state and unloaded con¬gurations at incre
ments n and n + 1 “ are analogous to the global decomposition F = F e F p
shown in Figures A.2 and A.3. In e¬ect, the previous unloaded state is taking
the place of the initial reference con¬guration, 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 justi¬ed by recalling that
in small strain plasticity, for instance, the incremental plastic strain ∆µp is
given by a ¬‚ow rule as the gradient of a certain stress potential φ. If the
material is isotropic, this potential must be a function of the invariants of
the stress and therefore its gradient will be colinear 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 ∆p are related as,
ln »trial = ln »e,± + ln »∆p,± (A.15)
e,±
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 »∆p,± .
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 de¬ned by a yield
surface φ, which is a function of „ and a hardening parameter given by the
Von Mises equivalent strain µp as,
¯
√
φ(„ , µp ) = „ : „ ’ 2/3 (σy + H µp ) ¤ 0
¯ ¯ (A.16)
where the deviatoric and pressure components of the Kirchho¬ stress tensor
are,
3
„ = „ + pI; „= „±± n± — n± (A.17a,b)
±=1
Observe that the principal directions of „ 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 stretchbased energy function described in
Section 5.6, Equation (5.103), the principal components of the deviatoric
Kirchho¬ tensor are,
2
„±± = 2µ ln »e,± ’ 3 µ ln J (A.18)
Substituting for »e,n+1 from Equation A.15 gives,
2
trial
„±± = 2µ ln »trial ’ 3 µ ln J
trial
„±± = „±± ’ 2µ ln »∆p,± ; (A.19a,b)
e,±
Note that „ 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 ’2µ ln »∆p,± is the correction necessary to ensure
that the yield condition (A.16) is satis¬ed.
In small strain theory the incremental plastic strain is proportional to
the gradient of the yield surface. We can generalize this ¬‚ow rule to the
large strain case by making the logarithmic incremental plastic stretches
proportional to the gradient of φ as,
‚φ
ln »∆p,± = ∆γ (A.20)
‚„±±
where the gradient of the von Mises yield surface gives a unit vector ν in
the direction of the stress as,
‚φ „
= √ ±± = ν± (A.21a,b)
‚„±± „ :„
The use of Equation (A.20) to evaluate the incremental plastic stretches
that ensure that „ lies on the yield surface is known as the return mapping
algorithm.
A.5.2 THE RADIAL RETURN MAPPING
Substituting Equations (A.20“1) into (A.19a) gives,
trial
„±± = „±± ’ 2µ∆γ ν± (A.22)
This equation indicates that „ is proportional to „ trial and is therefore
known as the radial return mapping (see Figure A.5). As a consequence of
this proportionality, the unit vector ν can be equally obtained from „ trial ,
that is,
trial
„±± „±±
ν± = √ =√ (A.23)
trial trial „ :„
„ :„
and therefore the only unknown in Equation (A.22) is now ∆γ.
In order to evaluate ∆γ we multiply Equation (A.22) by ν± and enforce
the yield condition (A.16) to give after summation for ± and use of (A.23),
√
√
„ : „ = „ trial : „ trial ’ 2µ∆γ = 2/3 (σy + H µp,n + H∆¯p ) (A.24)
¯ µ
In small strain analysis the increment in von Mises equivalent strain is given
as ∆¯2 = 2 ∆µp : ∆µp . Again replacing small strains by logarithmic strains
µp 3
gives,
3
2
∆¯2 (ln »∆p,± )2
µp = (A.25)
3
±=1
10 LARGE INELASTIC DEFORMATIONS
„3 
φ(„,µ p,n+1) = 0
2µ∆γ

φ(„,µ p,n) = 0 „0trial
n+1
„0 + 1
v n
„1 „2
FIGURE A.5 Radial return.
and substituting for ln »∆p,± from Equation (A.20) yields,
∆¯p =
µ 2/3 ∆γ (A.26)
Substituting this expression into Equation (A.24) enables ∆γ to be evaluated
explicitly as,
trial
φ(„ , µp,n )
¯
±
if φ(„ trial , µp,n ) > 0
¯
2
∆γ = (A.27)
2µ + 3 H
if φ(„ trial , µp,n ) ¤ 0
0 ¯
Once the value of ∆γ is known, the current deviatoric Kirchho¬ stresses
are easily obtained by reexpressing Equation (A.22) as,
2µ∆γ trial
Jσ±± = „±± = 1’ „±± (A.28)
„ trial
√
where the notation „ trial = „ trial : „ 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,
µp,n+1 = µp,n +
¯ ¯ 2/3 ∆γ (A.29)
In addition the current plastic deformation gradient tensor F p , or more pre
cisely C’1 = F ’1 F ’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 ’1 F gives,
e
C ’1 = F ’1 F ’T
p,n+1 p,n+1 p,n+1
= F ’1 F e,n+1 F T ’T
e,n+1 F n+1
n+1
= F ’1 be,n+1 F ’T (A.30)
n+1 n+1
where the elastic left Cauchy“Green tensor is,
3
»2 n± — n±
be,n+1 = (A.31)
e,±
±=1
and the elastic stretches are,
ln »e,± = ln »trial ’ ∆γ ν± (A.32)