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Section 8
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1 Introduction to Plasticity
8
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1

Introduction

The theory of linear elasticity is useful for modelling materials which undergo small
deformations and which return to their original configuration upon removal of load
...
With metals, significant permanent deformations will usually occur
when the stress reaches some critical value, called the yield stress, a material property
...
Permanent
deformations involve the dissipation of energy; such processes are termed irreversible, in
the sense that the original state can be achieved only by the expenditure of more energy
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It is concerned with materials which initially deform elastically, but which
deform plastically upon reaching a yield stress
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In sands and other
granular materials plastic flow is due both to the irreversible rearrangement of individual
particles and to the irreversible crushing of individual particles
...
The deformation of microvoids and the development of micro-cracks is also an important cause of plastic
deformations in materials such as rocks
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There are two broad
groups of metal plasticity problem which are of interest to the engineer and analyst
...
Analysis of problems involving small plastic strains allows one to design
structures optimally, so that they will not fail when in service, but at the same time are not
stronger than they really need to be
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The second type of problem involves very large strains and deformations, so large that the
elastic strains can be disregarded
...
In these latter-type problems, a simplified model known as perfect
plasticity is usually employed (see below), and use is made of special limit theorems
which hold for such models
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This is in marked
1

two other types of failure, brittle fracture, due to dynamic crack growth, and the buckling of some
structural components, can be modelled reasonably accurately using elasticity theory (see, for example, Part
I, §6
...
3)

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Section 8
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Materials commonly known as “plastics” are not plastic in the sense described here
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Due
to their viscosity, their response is, unlike the plastic materials, rate-dependent
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When a material undergoes plastic deformations, i
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irrecoverable and at a critical yield stress, and these effects are rate dependent, the
material is referred to as being viscoplastic
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Further advances with yield criteria and plastic flow rules were made in the years which
followed by Saint-Venant, Levy, Von Mises, Hencky and Prandtl
...
The arrival of
powerful computers in the 1980s and 1990s provided the impetus to develop the theory
further, giving it a more rigorous foundation based on thermodynamics principles, and
brought with it the need to consider many numerical and computational aspects to the
plasticity problem
...
1
...
Some of these phenomena are simplified or ignored in some of the standard
plasticity models discussed later on
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The Tension Test
Consider the following key experiment, the tensile test, in which a small, usually
cylindrical, specimen is gripped and stretched, usually at some given rate of stretching
(see Part I, §5
...
1)
...
8
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1
...
Beyond this point, permanent plastic deformations
are induced
...

In the elastic range the force-displacement behaviour for most engineering materials
(metals, rocks, plastics, but not soils) is linear
...
8
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1), the material “gives” and is said to undergo plastic flow
...
1

phenomenon is known as work-hardening or strain-hardening
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If the specimen is unloaded from a plastic state
(B) it will return along the path BC shown, parallel to the original elastic line
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The strain which remains upon unloading is the permanent plastic
deformation
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Further increases in
stress will cause the curve to follow BD
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(2) the force-displacement curve is more or less the same regardless of the rate at which
the specimen is stretched (at least at moderate temperatures)
...
1
...
First,
normalising with respect to the original cross sectional area of the tension test specimen
A0 , one has the nominal stress or engineering stress,

n 

F
A0

(8
...
1)

Alternatively, one can normalise with respect to the current cross-sectional area A,
leading to the true stress,



Solid Mechanics Part II

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F
A

(8
...
2)

Kelly

Section 8
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For very small elongations, within the
elastic range say, the cross-sectional area of the material undergoes negligible change and
both definitions of stress are more or less equivalent
...
Denoting the
original specimen length by l 0 and the current length by l, one has the engineering strain



l  l0
l0

(8
...
3)

Alternatively, the true strain is based on the fact that the “original length” is continually
changing; a small change in length dl leads to a strain increment d  dl / l and the
total strain is defined as the accumulation of these increments:
l
dl
 ln
l
l
 0
l0
l

t  






(8
...
4)

The true strain is also called the logarithmic strain or Hencky strain
...
The true
strain and engineering strain are related through

 t  ln 1   

(8
...
5)

Using the assumption of constant volume for plastic deformation and ignoring the very
small elastic volume changes, one has also {▲Problem 3}

 n

l

...
1
...
8
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2
...
8
...
2a, is of course the same as the graph of force
versus displacement (change in length) in Fig
...
1
...
A here denotes the point at which
the maximum force the specimen can withstand has been reached
...
After this point, the
specimen “necks”, with a very rapid reduction in cross-sectional area somewhere about
the centre of the specimen until the specimen ruptures, as indicated by the asterisk
...
For example,
if one unloads and re-loads (as in Fig
...
1
...
In this respect, the stress-strain curve can be
regarded as a yield stress versus strain curve
...
1



n

A





Y

A

Y



t

(a )

(b)

Figure 8
...
2: typical stress/strain curves; (a) engineering stress and strain, (b) true
stress and strain

Compression Test
A compression test will lead to similar results as the tensile stress
...
If one plots the true stress versus true strain curve for both tension and
compression (absolute values for the compression), the two curves will more or less
coincide
...
If one were to use the nominal stress and strain,
then the two curves would not coincide; this is one of a number of good reasons for using
the true definitions
...
In fact the yield point in this case will be significantly less
than the corresponding yield stress in tension
...
The effect is illustrated in Fig
...
1
...
The solid line depicts the
response of a real material
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Section 8
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1
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However, it is
not an issue provided there are no reversals of stress in the problem under study
...
That is, a stress state  xx   yy   zz   p has negligible effect on
the yield stress of a material, right up to very high pressures
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8
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3

Assumptions of Plasticity Theory

Regarding the above test results then, in formulating a basic plasticity theory with which
to begin, the following assumptions are usually made:
(1)
(2)
(3)
(4)
(5)

the response is independent of rate effects
the material is incompressible in the plastic range
there is no Bauschinger effect
the yield stress is independent of hydrostatic pressure
the material is isotropic

The first two of these will usually be very good approximations, the other three may or
may not be, depending on the material and circumstances
...
After large plastic deformation however, for example in rolling,
the material will have become anisotropic: there will be distinct material directions and
asymmetries
...
For example, consider the hierarchy of models
illustrated in Fig
...
1
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In (a) both the
elastic and plastic curves are assumed linear
...
1

yield stress is constant after initial yield
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In many areas of applications the strains
involved are large, e
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in metal working processes such as extrusion, rolling or drawing,
where up to 50% reduction ratios are common
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The rigid/perfectly-plastic model
(d) is the crudest of all – and hence in many ways the most useful
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



Y

Y



0



0

(a) Linear Elastic-Plastic

(b) Elastic/Perfectly-Plastic





Y

Y



0



0

(c) Rigid/Linear Hardening

(d) Rigid-Perfectly-Plastic

Figure 8
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4: Simple models of elastic and plastic deformation

8
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4

The Tangent and Plastic Modulus

Stress and strain are related through   E in the elastic region, E being the Young’s
modulus, Fig
...
1
...
The tangent modulus K is the slope of the stress-strain curve in the
plastic region and will in general change during a deformation
...
1
...
1

K






d p

d e

d
E


Figure 8
...
5: The tangent modulus

After yield, the strain increment consists of both elastic,  e , and plastic, d p , strains:

d  d e  d p

(8
...
8)

The stress and plastic strain increments are related by the plastic modulus H:

d  H d p

(8
...
9)

1 1 1
 
K E H

(8
...
10)

and it follows that {▲Problem 4}

8
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5

Friction Block Models

Some additional insight into the way plastic materials respond can be obtained from
friction block models
...
8
...
6
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Then there
is movement – although the amount of movement or plastic strain cannot be determined
without more information being available
...
1
...
8
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6b
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1


unload


Y

permanent
deformation

(a)



(b)

Figure 8
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6: (a) Friction block model for the rigid perfectly plastic material, (b)
response of the rigid-perfectly plastic model

The linear elastic perfectly plastic model incorporates a free spring with modulus E in
series with a friction block, Fig
...
1
...
The spring stretches when loaded and the block
also begins to move when the stress reaches Y, at which time the spring stops stretching,
the maximum possible stress again being Y
...

E



Y

Figure 8
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7: Friction block model for the elastic perfectly plastic material

The linear elastic plastic model with linear strain hardening incorporates a second,
hardening, spring with stiffness H, in parallel with the friction block, Fig
...
1
...
Once the
yield stress is reached, an ever increasing stress needs to be applied in order to keep the
block moving – and elastic strain continues to occur due to further elongation of the free
spring
...

Upon unloading, the block “locks” – the stress in the hardening spring remains constant
whilst the free spring contracts
...

The slope of the elastic loading line is E
...
1
...


Solid Mechanics Part II

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Section 8
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1
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1
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Give two differences between plastic and viscoelastic materials
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A test specimen of initial length 0
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0101 m
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015 m?
3
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1
...

4
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8
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10
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Which is larger, H or K? In the case of a perfectly-plastic material?
6
...

e

Solid Mechanics Part II

p

250



n

Kelly

Section 8
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x 10
5
...
5
E  70 GPa and
 4
n4
b  800 MPa is strained
3
...
02
3
and is subsequently
2
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Find the
1
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5
assuming isotropic
0
0
0
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04 0
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08
0
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12 0
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16 0
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2

hardening
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]
(i)

8

7
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What is the elastic modulus?
(i)
What is the yield stress?
(ii)
(iii) What are the tangent and plastic moduli?
Draw a typical loading and unloading curve
...
Draw the stress-strain diagram for a cycle of loading and unloading to the rigid plastic model shown here
...
What is the permanent deformation after complete removal of the load?
[Hint: split the cycle into the following regions: (a) 0    Y1 , (b) Y1    3Y1 , (c)
3Y1    4Y1 , then unload, (d) 4Y1    3Y1 , (e) 3Y1    2Y1 , (f) 2Y1    0

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