Section 5.2
5.2 Plasticity I: Hypoelastic-Plastic Models
The two main types of classical plasticity model for large strains are the hypoelasticplastic model and the hyperelastic-plastic model. The first of these is discussed in this
section.
5.2.1
Hypoelasticity
In the hypoelastic-plastic models of plasticity, the elastic response is assumed to be
hypoelastic. As discussed in §4.1.4, hypoelastic materials are characterised by
constitutive relations of the form
σ& = f (σ, d)
(5.2.1)
and, in mutiaxial problems, this implies that the response cannot be expressed in terms of
an elastic strain energy function. Thus the response is path-dependent and dissipation
may occur even though the material is supposed to be elastic. However, the idea with the
hypoelastic-plastic models is that the elastic strains are assumed to be relatively small so
that any error in the conservation of energy is very small, and so can be neglected.
The stress-rate σ& in the hypoelastic equation 5.2.1 must be objective. Thus the material
derivative of the Cauchy stress can not be used, but any of the many objective rates, for
example the Jaumann, Green-Naghdi, Truesdell, etc., can be used.
A large class of hypoelastic materials is encompassed in the linear relation between
objective stress-rate and the rate of deformation:
σ ∇ = C∇ : d
(5.2.2)
where σ ∇ is an objective stress rate and C ∇ is the corresponding fourth order tensor of
elastic moduli, which may itself depend on the stress, in which case it must be an
objective function of the stress. For a given finitely deformed state, the (small)
increments in stress and strain are linearly related and are recovered upon unloading.
However, for finite deformations, the work done in a closed path may not be zero.
The elastic modulus tensor C ∇ is also called the tangent modulus. It possesses the
minor symmetries due to the symmetry of d and σ ∇ . It is usually assumed to possess also
the major symmetries.
Example
Consider the following hypoelastic constitutive equation:
σ ∇J = CσJ : d
(5.2.3)
where σ ∇J is the Jaumann stress-rate, Eqn. 3.5.20, σ& − wσ + σw . The Truesdell stressrate σ ∇T , defined by Eqn. 3.5.22, σ& − lσ − σl T + tr (d )σ , is then
Solid Mechanics Part III
360
Kelly
Section 5.2
σ ∇T = σ ∇J + wσ − σw − lσ − σl T + tr (d )σ
= σ ∇J − dσ − σd T + tr (d )σ
ˆ +σ ⊗ I : d
= CσJ − C
(
(5.2.4)
)
ˆ : d = dσ + σd . Thus 5.2.3 can be expressed as σ ∇T = CσT : d with the Truesdell
where C
ˆ +σ ⊗ I .
tangent modulus given by CσJ − C
It is interesting to note that, if CσJ is constant, then σ ∇T is not. Further, if CσJ has the
major symmetries, then σ ∇T does not (since σ ⊗ I does not). For this reason one often
uses the Kirchhoff stress τ = Jσ rather than the Cauchy stress. For example, the Jaumann
rate of the Kirchhoff stress is τ ∇J = τ& − wτ + τw . Then, with J& = Jtr (d) ,
(
σ ∇T = J −1 τ ∇J − Jdσ − Jσd T
ˆ :d
= J −1CτJ − C
(
)
)
(5.2.5)
≡ CσT : d
Now if CσT has the major symmetries, then so does CτJ .
5.2.2
Hypoelastic – Plastic Model
Additive Decomposition of the Rate of deformation
The rate of deformation is now decomposed additively into elastic and plastic parts
according to
d = de + d p
(5.2.6)
The elastic response is then
(
σ ∇ = Cσe : d − d p
)
(5.2.7)
The yield condition is
f (σ, α ) = 0
(5.2.8)
where α represents any other variable(s) besides σ upon which f depends.
The flow rule is
d p = λ&G (σ, α )
Solid Mechanics Part III
361
(5.2.9)
Kelly
Section 5.2
The tensor G is usually expressed in the form G = ∂g (σ, α ) / ∂σ , where g is the plastic
potential; λ is the plastic multiplier. For associative flow-rules, f = g . The variables α
are assumed to follow an evolution law of the form
α& = λ&A(σ, α )
(5.2.10)
The loading and unloading conditions may be expressed as
λ& ≥ 0,
f ≤ 0, λ&f = 0
(5.2.11)
These are known as the Kuhn-Tucker conditions. The first of these states that the
plastic multiplier rate is non-negative; for plastic loading, λ& > 0 , otherwise λ& = 0 . The
second states that the stress must lie on or inside the yield surface. The last condition
assures that the stress state remains on the yield surface during plastic flow. This last
condition can also be expressed in rate form, f& = 0 . This is known as the consistency
condition:
∂f
∂f
f& =
: σ& +
: α& = 0
∂σ
∂α
(5.2.12)
Isotropic Materials
From the objectivity requirement, the yield function f (σ, α ) in Eqn. 5.2.8 must be an
objective scalar function of σ . This implies that f must be a function only of the
invariants of σ . Thus the form f (σ ) necessarily represents the yield function of an
isotropic material. For anisotropic materials, one must use a different stress measure, for
example the yield function could be expressed in term of the PK2 stress, f (S ) = 0 .
Consider then an isotropic material, with f = f (I 1 , I 2 , I 3 ) , where I i are the invariants of
σ . Then it can be shown that {▲Problem 1}
σ
∂f
∂f
=
σ
∂σ ∂σ
(5.2.13)
that is, the tensors σ and ∂f / ∂σ are coaxial.
Consider now a formulation in terms of the Jaumann stress-rate. From 5.2.13, it follows
that {▲Problem 2}
∂f
∂f
: σ& =
: σ ∇J
∂σ
∂σ
(5.2.14)
The consistency condition 5.2.12 can now be expressed in terms of the Jaumann stress
rate:
Solid Mechanics Part III
362
Kelly
Section 5.2
∂f
∂f
f& =
: σ ∇J +
: α& = 0
∂σ
∂α
(5.2.15)
The plastic multiplier can now be solved for, using the hypoelastic relation 5.2.7, the flow
rule 5.2.9, the evolution equation 5.2.10 and 5.2.15 {▲Problem 3}:
∂f
: Cσe J : d
∂σ
λ& =
∂f
∂f
−
:A+
: Cσe J : G
∂α
∂σ
(5.2.16)
Substituting this expression into the flow rule 5.2.9 and using the hypoelastic relation
5.2.7 then leads to {▲Problem 4}
σ
∇J
=C
σJ
: d,
C
σJ
σJ
= Ce
⎛ ∂f
⎞
: G ⊗ ⎜ : Cσe J ⎟
σ
∂
⎝
⎠
−
∂f
∂f
σJ
:A+
: Ce : G
−
∂α
∂σ
(C
σJ
e
)
(5.2.17)
The tensor CσJ is the elasto-plastic tangent modulus. It consists of an elastic
component Cσe J and a component which results from plastic flow.
When the flow is associative, so that G = ∂f / ∂σ , the elasto-plastic modulus possesses
the major symmetries. The plasticity equations can also be based upon other stress-rates;
for example, the Truesdell stress-rate. For the same reasons as discussed earlier, although
CσJ has the major symmetries for associative flow rules, the Truesdell modulus will not.
If, however, the equations 5.2.6-17 are formulated in terms of the Kirchhoff stress, then
the Truesdell modulus will possess the major symmetries (in fact, all the relevant relations
above are valid for the Kirchhoff stress, with σ simply being replaced by τ ). Note that,
if the elastic strains are small and plastic deformations are isochoric (volume preserving),
then J ≈ 1 and τ ≈ σ .
Small Strains
In the case of small strains, in the above relations one replaces d with ε& and decomposes
the small strain rate according to ε& = ε& e + ε& p . The stress rate is the time derivative of the
Cauchy stress, since objective stress-rates are not now a consideration. The hypoelastic
relation is σ& = C : ε& − ε& p and the remaining relations follow, for example the elastoplastic tangent modulus is given by 5.2.17b with Cσe replaced with the small-strain elastic
modulus tensor C. The small-strain formulation is valid for anisotropic elastic moduli.
(
5.2.3
)
J2 Flow Theory
In the J2 Flow Theory, one assumes the material is isotropic, plastic flow is independent
of the hydrostatic pressure, plastic flow is incompressible and the yield surface used is the
Solid Mechanics Part III
363
Kelly
Section 5.2
Von Mises yield surface. An effective stress is used to generalise the uniaxial behaviour
to multiaxial stress states.
First to be examined is the isotropic hardening model.
Isotropic Hardening
It is useful to express the equations 5.2.6-17 now in terms of the Kirchhoff stress and the
Jaumann stress-rate. In that case, one again has Eqn. 5.2.6, d = d e + d p , with the
hypoelastic Eqn. 5.2.7 now reading
(
)
(5.2.18)
s :s −Y = 0
(5.2.19)
τ ∇J = CτeJ : d − d p
The Von Mises yield function can be expressed as
f (τ , α ) = 3 J 2 − Y =
3
2
where J 2 is the second invariant of the deviatoric Kirchhoff stress and s = devτ . The
term
3
2
s : s acts as an effective stress σ̂ .
The (associative) flow rule 5.2.9 can be expressed as (see the Appendix to this section for
details of the differentiation involved here)
d p = λ&G (τ ),
G (τ ) =
3s
∂f
=
∂τ 2 32 s : s
(5.2.20)
The only variable α necessary in the model is the scalar accumulated effective plastic
strain:
α ≡ εˆ p ,
εˆ p = ∫ dεˆ p = ∫ εˆ& p dt ,
dεˆ p =
2 p
dε : dε p
3
(5.2.21)
The proposed evolution law for α is simply (Eqn. 5.2.10 with the function A = 1 )
(= εˆ& )
α& = λ&
p
(5.2.22)
dY (εˆ p )
dεˆ p
(5.2.23)
With the definition of the plastic modulus being
H (εˆ p ) =
the consistency condition 5.2.12 is, from 5.2.19,
∂f
f& =
: τ& − H εˆ p εˆ& p = 0
∂τ
( )
Solid Mechanics Part III
364
(5.2.24)
Kelly
Section 5.2
The plastic multiplier rate is now given by Eqn. 5.2.16,
∂f
: CτeJ : d
∂τ
λ& =
∂f
∂f
p
: CτeJ :
H εˆ +
∂τ
∂τ
(5.2.25)
( )
Finally, the elasto-plastic tangent modulus, Eqn. 5.2.17, is
⎛ τJ ∂f ⎞ ⎛ ∂f
τJ ⎞
⎜ Ce : ⎟ ⊗ ⎜ : Ce ⎟
∂τ ⎠ ⎝ ∂τ
⎠
CτJ = CτeJ − ⎝
∂f
∂f
p
τJ
: Ce :
H εˆ +
∂τ
∂τ
τ ∇J = CτJ : d,
( )
(5.2.26)
It is useful to decompose the response into volumetric and deviatoric components. First
express the elastic moduli as in Eqns. 4.1.19,
⎡ 1
⎤
CτeJ = κ I ⊗ I + 2 μ ⎢I − I ⊗ I ⎥
3
⎣
⎦
(5.2.27)
where I = δ imδ jn e i ⊗ e j ⊗ e m ⊗ e n . Note then that {▲Problem 5}
CτeJ :
∂f
∂f
= 2μ ,
∂τ
∂τ
∂f
∂f
: CτeJ :
= 3μ
∂τ
∂τ
(5.2.28)
so that the tangent modulus 5.2.26b can be expressed as
(4 / 3)μ ∂f ⊗ ∂f
⎡ 1
⎤
CτJ = κ I ⊗ I + 2 μ ⎢I − I ⊗ I ⎥ −
⎣ 3
⎦ 1 + H / 3μ ∂τ ∂τ
(5.2.29)
Kinematic Hardening
To accommodate kinematic hardening, one introduces as another hardening parameter the
back stress α . The yield condition 5.2.8 is now (in terms of the Kirchhoff stress)
(
)
f τ, α , εˆ p =
3
2
(s − α ) : (s − α ) − Y (εˆ p ) = 0
(5.2.30)
The flow rule is
d p = λ&G (τ, α ),
G (τ , α ) =
∂f
=
∂τ 2
3(s − α )
3
2
(s − α ) : (s − α )
(5.2.31)
The evolution equation for the back-stress is, for example using a linear hardening rule
and the Jaumann stress-rate,
Solid Mechanics Part III
365
Kelly
Section 5.2
(
α ∇J = cd p = cλ&G
)
(5.2.32)
Note that the use of the Jaumann rate in the back-stress evolution law can lead to
physically unreasonable stress oscillations in simple shear for large deformations
(Nagtegaal and DeJong, 1981). However, this behaviour is not significant provided the
elastic strains are not too large. A formulation based upon the Green-Naghdi strain can
eliminate such stress oscillations (Johnson and Bammann, 1984).
5.2.4
Drucker – Prager Model
The yield criterion for a Drucker-Prager material is a modification of the Von Mises
function so as to incorporate a plastic response due to a hydrostatic pressure:
f (σ, α ) = 3 J 2 + αI 1 − Y = 0
(5.2.33)
where I 1 = trσ = σ : I . The associated flow rule is then
d p = λ&G (σ ),
G (σ ) =
∂f
3s
=
+ αI
∂σ 2 32 s : s
(5.2.34)
where s = devσ . A suitable non-associative flow-rule might be
d p = λ&G (σ ),
5.2.5
G (σ ) =
∂g
3s
=
, g = 3J 2
∂σ 2 32 s : s
(5.2.35)
Elastic Moduli and Objectivity
It was mentioned above that objectivity requires that a yield function of the form f (σ )
necessarily represents an isotropic material. Objectivity also places restrictions on the
elastic moduli. First, consider a constant modulus tensor CτeJ . Objectivity of a
constitutive equation requires that
(τ )
∇J *
( )
= CτeJ : d e
or
(
*
(5.2.36)
)
(5.2.37)
]
(5.2.38)
Qτ ∇J Q T = CτeJ : Qd e Q T
so, using the index notation,
(τ ) = [Q
∇J
ij
pi
Qqj Qrk Qsl (CτeJ ) pqrs ((d e )kl )
In order that objectivity be satisfied, one must have
Solid Mechanics Part III
366
Kelly
Section 5.2
(C )
τJ
e
ijkl
( )
= Q pi Qqj Qrk Qsl CτeJ
pqrs
(5.2.39)
which shows that, if the modulus is constant, it must be isotropic.
If one requires an anisotropic elastic modulus, one must use a formulation based on a
configuration other than the spatial configuration, as discussed in the next sub-section.
5.2.6
Corotational Stress Formulation
The models discussed thus far have two drawbacks:
1. the yield function must be an isotropic function of the stress
2. if the elastic moduli are constant, then the moduli must be isotropic
To overcome these restrictions to isotropic materials, one can formulate a plasticity model
in terms of, for example, the corotational stress.
The corotational stress is defined by Eqn. 3.5.12,
σ U = R T σR
(5.2.40)
Using a formulation based on the Kirchhoff stress, the corotational Kirchhoff stress is
τ U = R T τR
(5.2.41)
Recall from §3.5.3 that the PK2 stress S is the pull-back of the Kirchhoff stress τ ,
#
S = χ *−1 (τ ) = F −1 τF − T ; the corotational stress τ U is the pull-back of τ but with respect
to R and not F: τ U = χ *−1 (τ )F = R = R −1 τR − T .
#
Define now the corotational rate of deformation:
d U = R T dR
(5.2.42)
d U = d eU + d Up
(5.2.43)
with the decomposition
Note that the corotational stress and rate of deformation are insensitive to rigid body
rotations Q to the current configuration:
(τ U )* = (R T τR )* = (QR )T (QτQ T )(QR ) = τ U
(d U )* = (R T dR )* = (QR )T (QdQ T )(QR ) = d U
(5.2.44)
The corotational stress-rate is
Solid Mechanics Part III
367
Kelly
Section 5.2
⋅
T
& T τR + R T τ& R + R T τR
&
τ& U = R τR = R
(5.2.45)
A rigid body rotation Q to the current configuration also results in {▲Problem 6}
(τ& U )* = τ& U
(5.2.46)
With the corotational stress-rate objective, the elastic response given by
τ& U = CτUe : d eU
(5.2.47)
Note that rigid body rotations to the current configuration leave τ& U and d eU unchanged
and so CτUe remains unchanged also. Thus, unlike the isotropic restriction 5.2.39, there is
no objectivity restriction here on the form of CτUe and so CτUe can represent in general an
anisotropic response.
Another way of looking at this is as follows: for an anisotropic material and using a fixed
coordinate system, the components of an elastic modulus tensor C will in general change
as the material rotates. With the corotational formulation, however, the axes rotate with
the material, and so material rotation has no effect on CτUe .
The remainder of the formulation follows as before, for example the yield condition
would be f (τ U , α U ) = 0 and the flow rule d eU = λ&G (τ U , α U ) . The evolution law for the
α variable(s) is α& = λ&A(τ , α ) . Again, the scalar yield function may now represent
U
U
U
U
in general anisotropic material behaviour. The loading and unloading conditions are
again given by 5.2.11.
The plastic multiplier rate is now (see 5.2.16)
∂f
: CτUe : d U
∂τ U
λ& =
∂f
∂f
:A+
: CτUe : G
−
∂α U
∂τ U
(5.2.48)
and
⎛ ∂f
⎞
: G ⊗ ⎜⎜
: CτUe ⎟⎟
⎝ ∂τ U
⎠
−
∂f
∂f
τ
−
:A+
: C Ue : G
∂α U
∂τ U
(C
τ
5.2.7
1.
∇J
τ
= CU : dU ,
τ
τ
C U = C Ue
τ
Ue
)
(5.2.49)
Problems
The derivatives of the invariants of the stress tensor are (see Eqn. 1.11.33)
Solid Mechanics Part III
368
Kelly
Section 5.2
2.
3.
4.
5.
∂I 3
∂I1
∂I 2
= I,
= I1 I − σ,
= σ 2 − I1σ + I 2 I = I 3σ −1
∂σ
∂σ
∂σ
Use these relations to show that, for the yield function f = f (I 1 , I 2 , I 3 ) , the tensors
σ and ∂f / ∂σ are coaxial.
Use the fact that σ and ∂f / ∂σ are coaxial, and Eqn. 1.9.3h,
(
)
(
)
A : (BC ) = B T A : C = AC T : B , to show that
∂f
∂f
: σ& =
: σ ∇J
∂σ
∂σ
where σ ∇J = σ& − wσ + σw .
Derive Eqn. 5.2.16
Use the index notation to verify that
C : (A : C : d ) : G = (C : G ) ⊗ (A : C) : d
where A, d, G are second order tensors and C is a fourth-order tensor. Hence
derive the elasto-plastic modulus 5.2.17b.
Use the relations 1.9.62 and 1.9.64 to show that, for an arbitrary tensor A,
CτeJ : A = κ (trA )I + 2 μdevA
A : CτeJ : A = κ (trA ) + 2 μdevA : devA
2
with CτeJ given by 5.2.27. Hence, using 1.9.31, show that, for a deviatoric tensor A,
CτeJ : A = 2 μA
A : CτeJ : A = 2 μA : A
Finally, use 5.2.20,
3s
∂f
=
∂τ 2 32 s : s
to derive Eqns. 5.2.28,
6.
∂f
∂f
: CτeJ :
= 3μ
∂τ
∂τ
The application of a rigid body rotation Q to the current configuration results in a
change to the corotational stress rate 5.2.45,
(τ& U )* = R& *T τ * R * + R *T τ& * R * + R *T τ * R& *
T
⎛ ⋅ ⎞
⎛ ⋅ ⎞
T
T
= ⎜⎜ QR ⎟⎟ QτQ T (QR ) + (QR ) ⎜ QτQ T ⎟(QR ) + (QR ) QτQ T
⎜
⎟
⎝
⎠
⎝
⎠
(
)
(
⋅
T
)⎛⎜⎜ QR ⎞⎟⎟
⋅
⎝
⎠
⋅
& TQ + QTQ
& = 0 to show that (τ& )* = R T τR = τ& .
Use the relation Q Q = Q
U
U
5.2.8
Appendix to §5.2
Differentiation of the Von Mises Yield Function
The Von Mises yield criterion is f = J 2 − k = 0 where J 2 = 12 sij sij . Using the product
rule of differentiation, in index notation:
Solid Mechanics Part III
369
Kelly
Section 5.2
∂ J2
∂σ ij
=
∂
1
2
s mn s mn
∂σ ij
=
=
=
=
=
=
=
1
2
1 ∂ (s mn s mn )
2 ∂σ ij
1
1
2
s pq s pq
1
2
1
2
s mn
s pq s pq
∂s mn
∂σ ij
∂ (σ mn − 13 σ kk δ mn )
∂σ ij
s mn
2 s pq s pq
s mn
2 s pq s pq
s mn
2 s pq s pq
(δ
mi
δ nj − 13 δ ki δ kj δ mn )
(δ
mi
δ nj − 13 δ ij δ mn )
sij − 13 δ ij s mm
2 s pq s pq
sij
=
2 s pq s pq
sij
2 J2
(5.2.50)
In tensor notation:
∂ J2
∂σ
=
∂
1
2
s:s
∂σ
=
=
1
2
1 ∂ (s : s )
s : s 2 ∂σ
1
1
2
1
2
1
2
s:s
s:
∂s
∂σ
∂ (σ − 13 (trσ )I )
∂σ
2s : s
s
: (I − 13 I ⊗ I )
=
2s : s
s − 13 (trs )I
=
2s : s
s
s
=
=
2s : s 2 J 2
=
Solid Mechanics Part III
s
370
:
(5.2.51)
Kelly