Theory of Plasticity – lecture 01, Adam Paul Zaborski
Lecture 01 - introduction
Introductory remarks
What is the matter?
Theory of plasticity represents a necessary extension of the theory of elasticity. Like the
theory of elasticity, it is concerned with the analysis of stresses and strains in the structure.
There is a train of concepts:
 loading forces
 internal forces (stresses)
 mechanical properties of material
 material strains
 material deformations.
Besides elastic response of material, we often observe irreversible deformations. This happens
during some technological processes of shaping, like forging, rolling, (plastic) bending etc. In
these cases, the strains are enormous. But plastic strains arise also at cross sections of
structural members and structural foundations. The scope of our considerations will be limited
to the structural materials as steel, concrete and soil.
This time, however, the analysis:
 is not limited to the elastic range,
 furnishes more realistic estimates of load capacity of the structure,
 it provides a better understanding of the structural elements reaction to the forces.
Better knowledge of material behavior, the more exact will be the design and better will be
the structure.
From the uniaxial tensile test, we know that material can sustain much greater stress level
than elastic one. It would be not economic not to take into consideration these additional
material capacities.
Scope
There are three main regions of our interest:
 formalization of experimental observations of the macroscopic behavior of a deformable
solid (phenomenological description, schematization of stress-strain curve, physical
nonlinearity)
 general description of mechanical dependence between strains and stresses (including
some “weird” phenomena like work kinematic or isotropic/anisotropic hardening or strain
softening during loading, unloading or reloading)
 some basic methods of nonlinear problem solution: linearization, rough estimate, limit
analysis
Historical remarks
1870 – Tresca (plastic extrusion of metals), A. J. C. B. de Saint-Venant (first constitutive
relations), Levy (the same in 3D)
1904 – M. T. Huber (effort hypothesis), 1913 – von Mises (yield criteria)
1924, 1928, 1930 – works of Prandtl, Reuss, von Mises on general equations of plasticity
1938 – Melan (plastic adaptation theorem)
1951 – Drucker’s material stability postulate, Drucker, Prager, Hill (theorems of limit
analysis)
1953 – Koiter (plastic inadaptation)
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Theory of Plasticity – lecture 01, Adam Paul Zaborski
Plastic behavior in simple tension and compression
Different stages of the test
 elastic range
 elastic unloading and reloading
 elastic limit (yield point, initial yield stress)
 plastic flow
 unloading and reloading elastic again, but:
o residual strains are observed
o a greater stresses in elastic regime due to strain or work hardening
o strain softening of compressed concrete
 reversed loading: Bauschinger effect
What is news?
 partial or total irreversibility of some processes
 dependence of the material behavior on process history
 it is clear that the description of real mechanical properties of material should be
evolutionary (material constant are not really constant)
Hooke model
This is the simplest, linearly elastic model of the solid. This is one-to-one correspondence
between stress and strain. From linearity and unstressed natural state follows the principle of
additivity (superposition) of solutions. For instance, when the structure loading increases by,
let’s say 5 times, the deflections are also 5 times greater.
Real materials have elastic properties, at least partially. The transition from elastic state to
inelastic one can be mild, gradual (hard steel) or distinct (mild steel). Real diagrams obtained
during tests are too complex to be used for calculations. So, usually, the real diagrams are
simplified for the calculation purposes and presented in these schematic forms.
Perfect elastic-plastic solid






Perfectly elastic-plastic model without hardening (Prandtl, 1928) and with hardening
In this model, we assume that the plastic strains are limited to such value that elastic strains
we cannot neglect. Depending on material behavior beyond the yield point, material plasticity
can be ideal (without hardening) or with hardening.
The yield point for the material without hardening is a constant value. As a rule, we assume
the yield point in compression the same as in tension. Usually, we neglect the loop of
hysteresis and assume that the unloading and reloading processes are elastic.
For the material with hardening, yielding point depends on prior stage of the process. Often,
the greater tensile strain (beyond the yield point), the smaller is value of yielding stress in
compression. This is so-called Bauschinger effect.
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Theory of Plasticity – lecture 01, Adam Paul Zaborski
In both cases, the strain depends on process history. For loading, we can write the equations in
the form:
E
 
 0
(   0 E )
(   0 E )
,
 E (   0 E )

 
0 

 0  E1    E  (   0 E )



Models of perfectly plastic solid
If the plastic strains are much greater than elastic ones, the last can be neglected. This is the
case of technological processes of plastic forming.






Perfectly plastic material without and with hardening
As before, actual state depends on the history of process. Again, in the case of hardenng we
can observe Bauschinger effect. Equations of loading:
 0 ,
or
   0  E1 .
Models of exponential scheme
Nonlinear exponential schemes for the materials with hardening:
 exponential plastic hardening
  k n ,
 

 0   0




n
0  n  1
 rigid-plastic materials with exponential hardening (Ludwik, 1909)
   0  k n
 elastic-plastic exponential hardening

E    0 E 
n

   0 E 
k
 
 elastic-plastic exponential hardening (Ramberga-Osgooda, 1943)


 
 k 
E
E
n
n  1
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Theory of Plasticity – lecture 01, Adam Paul Zaborski


n=1


n=0




Models with nonlinear hardening
Models of asymptotic ideal plasticity
There are the schemes of the type   f   : f  0    :
 two parameter scheme of hyperbolic tangent (Prager, 1938)

  
0

arctanh

E
 0 
 three parameter scheme of Ylinen
Et 
0  
d
E
d
 0  c
or

1
E



c  1  c  0 ln 1 

0



 ,


0  c  1
 three parameter scheme of Życzkowski




E 1 
 0




n
,
n  0



c1
n0



Schemes of asymptotic plasticity
Are the schematic models, due to considerable simplification of real properties, useful? Even
the simplest models, reasonably applied, are helpful in consideration and solution of particular
technical problems, giving practically useful solutions. If the change of structural element
geometry can be neglected, the limit loading of the element is the same in both cases of the
perfectly plastic material: with elastic part and without.
Repetition of some concept from theory of elasticity
Decomposition of tensor of stress or strain
Any tensor can be written as a sum of so-called mean tensor (isotropic tensor) and deviator:
or,
 t xx
t xy

t yy
 t yx

t zy
 t zx
symbolically: T  A  D,
t xz   t m
 
t yz    0

t zz   0
0   t xx  t m
 
tm
0    t yx

0
t m   t zx
T  A  D , T  A  D .
0
t xy
t yy  t m
t zy


t yz  ,

t zz  t m 
t xz
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Theory of Plasticity – lecture 01, Adam Paul Zaborski
In the above formulae tm represent the mean of values from principal diagonal and deviator is
traceless tensor. The stress tensor is the sum of mean stress tensor and stress deviator. The
strain tensor is the sum of volumetric strains and deviatoric strain tensor or shear tensor. The
sum of elements from principal diagonal is the first invariant of the tensor (it is independent
on transformation of coordinates set), so the deviator has only two invariants: second and
third.
This decomposition of tensor has a profound sense.
Hydrostatic pressure
If D  0, A  0 , we have only isotropic part of tensor (any direction is principal); if  m  0 we
have the state of hydrostatic pressure. Structural materials do not damage under hydrostatic
pressure. An easy experiment consists on dropping a stone or piece of chalk into the sea and
we observe that nothing happens.
Two forms of Hooke law
Hooke law can by written in two forms of constitutive equations:
 the law of volume change (one equation) :
A  3KA
 the law of shape change (5 independent equations):
D  2 D
Volume change of infinitesimal representative element
In general case:
V  V1  V0  (1   x )(1   y )(1   z )  1  1  1   x   y   z   x  y   x  z   y  z   x  y  z
V   x   y   z  3 m  I 1
so, in the case of deviatoric strains:
V  V1  V0  (1   x   m )(1   y   m )(1   z   m ) 1   x   y   z  3 m  0
It means that deviatoric part of strain tensor does not change the volume.
Intensity of internal energy
e  12  ij  ij  12 (aij  s ij )(eij  d ij )    12 aij s ij  12 eij d ij  ev  e s
The intensity of internal elastic energy can be decomposed onto intensity of energy of (pure)
shape change and intensity of energy of (pure) volume change.
Safety conditions of work under complex loading
From the strength of materials, we may assess the safe work of material under complex
loading through some particular hypotheses. Namely, there are two main hypotheses for
plastic material as steel:
(i) Coulomb-Tresca-Guest hypothesis of extreme shearing stress
(ii) Huber-Mises-Hencky hypothesis of energy intensity of shape change.
Both hypotheses respect the fact that material do not damage under hydrostatic pressure.
Dependence on history of process – an example
Let us consider the set of vertical bars supporting loaded plate. We assume that:
 all bars have the same Young modulus
 the bars have different yield stress,  02   01 .
 as follows from the figure, all bars have the same strains
We will consider loading, unloading and reloading of the structure.
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Theory of Plasticity – lecture 01, Adam Paul Zaborski
A1
2
A2
2
A2
2
A1
2
Bar model of material
During whole process the balance of forces gives the equation, P  N1  N 2 ,
 the equation of strain compatibility is:    1   2 ,
 we adopt the material data: A1  A2  A , E1  E 2  E ,  01   02 .
a) Elastic loading
This stage lasts till reaching yield stress in bars 1.
1   2 
 01
E
,
E1  E 2
  1   2   01 ,
P   1 A1   2 A2   01 A
 a  P A   01
b) Elastic-plastic loading
The stage lasts till yielding of bars 2:

 02
E
,  1   01 ,  2   02
P   01 A1   02 A2
b 
 01 A1   02 A2
A
c) Plastic flow
In the stage strains increases without increase of loading. The strains are undetermined.
d) Unloading till P  0
Process of unloading is elastic, so:
 01 A1   02 A2  EA  0
it follows:

 01 A1   02 A2
EA
 1   01  E 
 01   02 A2
,  2   02  E 
A
 02   01 A1
A
.
Because we assumed  01   02 , it is clear that the bars I are compressed and the bars 2 –
tensioned. Due to lack of external load, we call these stresses residual.
e) Elastic-plastic compression strains
It arises, if  odc  2
N1   01 A1 ,
 01
E
, in that case the bar forces are:
N 2   02  2 01 A2
and equivalent stresses are:
e 
 02  2 01 A2   01 A1
A
,
e a
that is the flow of the bar 1 occurs for smaller value of load due to its previous compression
under load P  0 . Consequently, the set of bars models Bauschinger effect.
f) Plastic flow of all bars under compression
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Theory of Plasticity – lecture 01, Adam Paul Zaborski
It occurs, when  odc  2
 02
E
.
P   01 A1   02 A2 ,
the equivalent stress:
e  
 01 A1   02 A2
A
  b .
g) Reloading
Depending of the point of start, plastic flow of the bars 1 begins earlier then for the first time.
The plastic flow of bars 2 occurs at the same load. The strains-stresses curve for the loading
cycle is presented on the figure below..

b
f
c
a

g
h
d
e
Loading, unloading and reloading of the model
The set of bars with different yielding conditions qualitatively models real material, which has
the slip surface of different strength and exhibits the Bauschinger effect.
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