Theory of Plasticity – lecture 3, Adam Paul Zaborski
Stress-strain relations
Introductory remarks
From previous considerations, it is clear we need the relations between strains and stresses for
various mechanical states of the continuum:
 in elastic range
 in elastic-plastic range
 in the case of plastic hardening.
Total strain theory (Hencky-Ilyushin theory)
The total strain theory, also called the theory of small elastic-plastic strains, has been
formulated by Hencky in 1924 and developed by Iliushin in 1943. The generalization of the
elasticity theory can be made supposing the relationships between stress and strain tensors:
 principal directions of stress and strain state agree (they are the same),
 medium stress is proportional to medium strain; the coefficient of proportion is such as in
the low of the volume change (Hooke’s),
 the intensity of stress is a function of the intensity of strain, specified during experiments.
The basic relations are of the form:
D   D , A 
1
A
3K
or, in tensorial form:
1 2
 kk ,
E
where the function     i ,   1 2G  determines the plastic strain advancement. For the
perfect plasticity (without the plastic hardening), the function can be determined from yield
criterion. Multiplying the first equation by itself, we get:
eij eij   2 sij sij
and defining the intensities of stress and strain:
eij   sij ,  kk 
i 
3
2
sij sij ,  i 
2
3
eij eij
we have, introducing the Huber’s criterion  i   0  :

i   i ,    i 
i
For unloading processes, the elastic formulae are valid:
3
2
2
2 2
3
2
3
2
3
2
eij eij
0
1 ~
sij  sij ,
e~ij  eij 
2G
where the swung dash means the point from which the unloading process begins.
Flow theory of Levy-Mises
The oldest theory of Levy from 1870 neglects the elastic strain, supposing perfectly plastic
material with the proportionality between deviators of stress and (small) strain rates:
D  D ,
or, in tensorial form:
eij  s ij .
Theory of Plasticity – lecture 3, Adam Paul Zaborski
The above equation resembles the rheological constitutive equation and hence the name. The
unknown function  has to be determined from the yield criterion. Mises modified Levy’s
original proposition with Tresca’s criterion. The Huber’s criterion is more convenient for
calculations.
Proceeding as before, we get:
s ij eij  s ij s ij ,
and from the definition of the stress intensity and from the yield criterion:
i 
3
2
 sij sij  23  i2  23  02
sij sij
we have:

3 sij eij 3 W
,

2  02
2  02
where W is the density of strain power or the power per unit volume.
The equation in the incremental form is:
3s de
deij  kl 2 kl sij .
2 0
The elastic strain is neglected, so:
 in the volume change low E   , it leads to the incompressibility condition,
 yield criterion describes the neutral surface in the space of stress,
 the neutral surface in the strain space reduces to the point: every movement is active one
 the passive processes correspond with the rigid solid behavior.
Mathematically, the two above theories are similar. In the cases of eliminated strain and its
derivatives, the results from both theories are the same.
Flow theory of Prandtl-Reuss
Adding instantaneous elastic strain to the L-M equation, Prandtl (1924) and Reuss (1930)
generalized the theory:
D 
1
D  D
2G
eij 
1
sij  s ij
2G
or, in index notation:
.
Unknown function  is to be determined from the yield criterion:
 i   0  0   i2   02 
eij sij  sij sij 
3 2s s
2 ij ij
 0  sij sij  0
1
sij sij  sij sij
2G
actually we get:

eij s ij

3eij s ij
.
s ij s ij
2 02
The term in the numerator is the density of plastic strain power, because from the
decomposition of strain into elastic and plastic part follows the the second part vanishes:


sij
eij s ij  s ij eijp  eije  s ij eijp  s ij
 s ij eijp .
2
G

0
22
3 0
Zeroing of the second term is due to: sij sij 
 sij sij  0 .
As before, the time elimination leads to the incremental form of the theory:
Theory of Plasticity – lecture 3, Adam Paul Zaborski
dsij
3s kl dekl
sij .
2G
2 02
The incremental form of equation for passive processes is:
dsij  2Gdeij
deij 

Jeśli rozwiązanie problemu z zastosowaniem teorii H-I jest łatwiejsze, to takie rozwiązanie
może być sprawdzone poprzez weryfikację równań teorii P-R.
Comparison of the theories, discussion
The classical theories: total strain of H-I and flow of L-M and P-R can be derived as particular
case of some more general tensorial theory of Hohenemser-Prager (1932):
 1 sij   2 sij   3 eij   4 eij  0 ,
where 3 coefficients are constant and one is function of strain. Hence, despite of linear
tensorial form, the stress-strain relation is nonlinear.
More general form proposed Reiner (1945):
 ij   0 ij   1ij   2 ik kj
where  ij is the rate of strain: elastic, elastic-plastic and rigid-plastic as well.
In the flow theories the state of stress depends on the trajectory of the process while in the
total strain theory – not. In some cases the theories are convergent:
eij sij  sij sij  sij sij (H-I, time derivative)
1
sij sij  sij sij (P-R)
2G
Because sij s ij  0 , the right sides are identical when:
    0 .
In another form:
eij  sij  sij (H-I)
eij sij 
1
sij  sij (P-R)
2G
After subtraction and trivial case   1 2G removal, we get:
sij  0 or sij  sij0
eij 
The processes where the deviatoric components of the stress and the strain increase
proportionally are called the simple processes. The verification of this condition is
burdensome. It is much easier to check if the loading process is simple, when – according to
Ilyushin theorem – every loading increases proportionally.
For the simple processes the results from the classical theories are similar.
Cycle in the theory of plasticity
Cycle is the process after which the variables return to the original values. In theory of
plasticity, due to irreversible phenomena, there is no cycle, except some particular situations,
as at the drawing below.
For this reason, the theory of plasticity uses the concept of a quasi-cycle. This is the process
where only independent variables return to its original values. For instance, there are the
quasi-cycles of stress and of strain.
Theory of Plasticity – lecture 3, Adam Paul Zaborski




Plastic potential
The equations of the total strain theory
eij   sij
may be written in the form:
eij 
 I 2 s
2 sij
and, similarly, for the flow theories:
eij  sij , eij  sij 
1
sij
2G
as:
e p ij 
 I 2 s
2 sij
.
Before of generalization of the above formulae, we prove the following theorem: if I  ij  is
an invariant of the tensor T , then the object:
I  kl  I  ij   ij I  ij 
 kl 


aik a jl   ij aik a jl
 kl
 ij  kl
 ij
is the 2nd order tensor.
We may write the constitutive equations as:
g  ij 
g  ij 
p
eij  
, and e ij  
.
 ij
 ij
The function g  ij  is plastic potential. The analogy with the elastic potential is obvious. For
the perfectly plastic materials the plastic potential cannot be expressed in stress but its
existence is not questioned.
In 9-dimensional Prager’s space the plastic potential is the vector:
ε   grad g  ij , ε p   grad g  ij  .
Assuming the plastic potential to be zero or some constant value, we determine in the space a
surface perpendicular to the strain or strain rate vector.
Associated flow rule
The flow rule is some necessary kinematic assumption on the direction of the loading path. If
for the yield criterion in the form:
f  ij   const
the following relation holds:
g  ij   f  ij 
Theory of Plasticity – lecture 3, Adam Paul Zaborski
we call it the associated flow rule.
p
d  ij  
p
d  ij
płaski
f
  ij
gładki
p
d ij
(a )
(b )
 ij
 ij
potencjał plastyczny
g ( ij )  f ( ij )  const
(c )
 ij
p
 ij ,  ij
naroże
(d )
 ij
p
d  ij
p
d ij
In classical flow theory of P-R with the Huber’s yield criterion we recognize the associated
flow rule, because:
1
g  ij   I 2 s  ij   f HMH  ij  ,
2
what it is well confirmed by experiments, especially when the principal directions do not
change.
Theory of Plasticity – lecture 3, Adam Paul Zaborski
toct , d goct
p
1, d 1
p
dij
p
dij = sij
ij
sij
f(ij) = k
p
oct, d oct
0
0
3, d 3
p
2, d 2
p
płaszczyzna dewiatorowa
płaszczyzna hydrostatyczna
Stowarzyszone prawo płynięcia dla warunku plastyczności H-M-H
1, d 1
p
dij
p
1- 3 = 0
1 - 2 = 0
A
2 - 3 = 0
O
2, d 2
p
2 - 1 = 0
1 - 3 = 0
A
1 - 2 = 0
3 - 2 = 0
3, d 3
p
3- 1 = 0
Stowarzyszone prawo płynięcia dla warunku TG; przyrosty odkształceń plastycznych
(normalne i jako granica gładkiej powierzchni)
Theory of Plasticity – lecture 3, Adam Paul Zaborski
1
d1m,0,1
d2m,1,0
A
d60,m ,1
d30,1, m
O
2
d51, m,0
3
d41,0, m
Stowarzyszone prawo płynięcia dla warunku Mohra-Coulomba
Drucker’s stability postulate
P
ΔP2
ΔP1
f
Displacements of flat shell
In some cases, we observe the stability zone as well as some instable behavior. The similar
reasoning we adopt to the material point. We add to the state of stress  ij(0) some additional
state of stress  ij and then we unload this extra stress.
Ti
Fi
u i ,  ij ,  ij
Ti  Ti
Fi  Fi
u i  u i ,  ij   ij ,  ij  ij
We call the material stable in the Drucker’s sense if the following postulates are valid::
1. The work of the extra loading onto the displacement changing is positive.
2. The work of the extra load during the loading and unloading is non-negative.
It has to be emphasized that the work concerns only the extra load onto the extra
displacement.
In mathematical notation:
 Ti u i dA   Fi u i dV  0 (stability in small)
A
V
Theory of Plasticity – lecture 3, Adam Paul Zaborski
 Ti u i dA   Fi u i dV  0 (stability in the cycle)
A
V
The stability postulates are more restrictive than the thermodynamics principles.
In index notation the work change is:
dW *   ij   ij(0) d ijp .
The Drucker’s postulate is the assumption of non-negative complementary work:
 ij   ij(0) d ijp  0 .
This is the great Drucker’s postulate. In the limit, we can write:
d ij d ijp  0 ,
the small Drucker’s postulate. The small postulate follows from the great, but not vice versa.
The Drucker’s postulates presentation in 9-dimensional Prager’s space is easy to explain.


dε p
σ  σ0


dε p
dσ
The limit surface for stable material must be convex. From the small postulate follows that the
vector of plastic strain increase must be perpendicular to the limit surface.
 ij 
 ij 
W
  ij
W
  ij
ij
ij
O
W=const
przestrz.
napr.
przestrzeń
odkszt.
O
Wconst
The direction normal to the flow surface is given by the gradient of the yield criterion:
ε   grad g  ij , ε p   grad g  ij  .
The same form of potential gradient and the yield criterion means that for the unconditionally
stable materials the flow rule is the associated flow rule.
Summing up, from Drucker’s postulate follow the conclusions:
1. The strain energy and complementary energy exist and are always positively definite, with
agreement with thermodynamical principles
2. The stress (strain) is perpendicular to the surface of constant potential energy
(complementary energy)
3. The surfaces of constant potential energy in the strain space and the complementary energy
in the stress space are convex
4. The stress-strain relations are reversible (reciprocal).
5. For elastic materials both forms of energy are independent of the load path, possess the
total differential and depend on the original and final states only.
Theory of Plasticity – lecture 3, Adam Paul Zaborski
Isotropic hardening
σ
σ

τ
Plastic isotropic hardening
There is no the Bauschinger’s effect. The yielding surface increases proportionally. The
hardening depends on non-decreasing function of plastic strain. There are two propositions.
F.K.G.Odqvist (1933):
 i  f 1 I p 
where the Odqvist’s parameter is the length of the path in the space of plastic strain:
~
t
~
t
I p     dt   e e dt 
p
ij
p
ij
0
~
ep
p p
ij ij
0

deijp deijp
0
Taylor-Quinney (1931):
 i  f 2 W p 
where the plastic strain work (dissipation energy):
~
ep
~
ep
W  W    ij d    ij de 
p
d
p
ij
0
p
ij
0
~
ep
s
ij
deijp
0
is non-decreasing variable also.
Kinematic hardening
The yielding surface displaces as rigid solid (in 9-dimensional Prager’s space). If the initial
surface is determined by the equation:
F  ij   C
Prager (1935):
F  ij   ij   C ,
where (Melan 1938) there are two possibilities:
d ij  cd ijp , c  const
or:
Ziegler (1959):
(the shape conserved).
d ij  d ij .
d ij   ij   ij d
Theory of Plasticity – lecture 3, Adam Paul Zaborski
σ2
p.p. wg Zieglera
p.p. wg Melana
σ1
początkowa p.p.
Variants of the kinematic hardening
Mixed and anisotropic hardening
The mixed hardening is a combination of isotropic and kinematic hardening.
The anisotropic hardening consists on the active surface change. There are a few kinds of
such hardening:
 general type
 with independent mechanisms
 qualitative change of yielding surface (plastic corners)
σ1
σ1
τ
σ
σ2
σ3 σ2
σ3
Different mechanisms of anisotropic hardening.