Theory of Plasticity
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Theory of Plasticity
M.Tech Lecture Notes
Dr V S Reddy , Associate Professor
GRIET Hyderabad
Department of Civil Engineering
What is Plasticity?
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. Almost all real
materials will undergo some permanent deformation, which remains
after 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.
Elastic deformations are termed reversible; the energy expended in
deformation is stored as elastic strain energy and is completely
recovered upon load removal. 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. The classical theory of plasticity is
concerned with materials which initially deform elastically, but
which deform plastically upon reaching a yield stress. In metals
and other crystalline materials the occurrence of plastic
deformations at the micro-scale level is due to the motion of
dislocations and the migration of grain boundaries on the
micro-level. 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.
Similarly, compression of bone to high stress levels will lead to
particle crushing. The deformation of microvoids and the
development of micro-cracks is also an important cause of plastic
deformations in materials such as rocks. Plastic deformations are
normally rate independent, that is, the stresses induced are
independent of the rate of deformation (or rate of loading).
Imp Points: Permanent deformation that cannot be recovered after
load removal Hookes law (linear relation between stress and strain)
not valid Beyond Hookes law to failure is Plastic behaviour Tensile
test to study plastic behaviour Elastic properties may be of
interest, but these are measured ultrasonically much more
accurately that by tension testing Plasticity theory deals with
yielding of materials under complex stress states Plastic
deformation is a non reversible process where Hookes law is no
longer valid. One aspect of plasticity in the viewpoint of
structural design is that it is concerned with predicting the
maximum load, which can be applied to a body without causing
excessive yielding. Another aspect of plasticity is about the
plastic forming of metals where large plastic deformation is
required to change metals into desired shapes.
True Stress, True Strain, Engineering Stress, and Engineering
Strain
Engineering stressis the appliedloaddivided by the original
cross-sectional area of a material. Also known as nominal
stress.True stressis the appliedloaddivided by the actual
cross-sectional area (the changing area with respect to time) of
the specimen at that loadEngineering strainis the amount that a
material deforms per unit length in a tensile test. Also known as
nominal strain.True strainequals the natural log of the quotient of
current length over the original length as given byEq4.
(Eq1)=P
A0
engineeringstressPload
A0cross-sectional area of specimen beforedeformationhas taken
place
Across-sectional area of specimen at which the load is
applied
totalelongation
L0original value of the gage length
Lsuccessive values of the length as it changes
(Eq2)t=P
A
true stress
(Eq3)=
L0
engineering strain
(Eq4)t= lnL
L0
true strain
True stress and strain are often not required. When theyield
strengthis exceeded, the material deforms. The component has failed
because it no longer has the original intended shape. Furthermore,
a significant difference develops between the two curves only
whenneckingbegins. But when necking begins, the component is
grossly deformed and no longer satisfies its intended use.
True stress continues to increase afterneckingbecause, although
theloadrequired decreases, the area decreases even more.
What is Yield Strength, Y The yield strength is the engineering
stress at which the material begins to undergo permanent plastic
deformation. When a lower stress is applied, the material will
deform under load, but will return to its original geometry when
the load is removed. This point is observed as the departure of the
stress-strain curve from a perfectly linear relationship. Because
this point is difficult to determine accurately, a rule called the
0.2% criterion is used. According to the 0.2% criterion, the yield
strength, Y, occurs at the point where the stress-strain curve
deviates from a straight line by 0.2% (0.002 strain).
The flow curve
True stress-strain curve for typical ductile materials, i.e.,
aluminium, show that the stress - strain relationship follows up
the Hookes law up to the yield point, o. Beyond o, the metal
deforms plastically with strain-hardening. This cannot be related
by any simple constant of proportionality. If the load is released
from straining up to point A, the total strain will immediately
decrease from 1 to 2. by an amount of /E. The strain 1-2 is the
recoverable elastic strain. Also there will be a small amount of
the plastic strain 2-3 known as an elastic behavior which will
disappear by time.(neglected in plasticity theories.) Usually the
stress-strain curve on unloading from a plastic strain will not be
exactly linear and parallel to the elastic portion of the curve. On
reloading the curve will generally bend over as the stress pass
through the original value from which it was unloaded. With this
little effect of unloading and loading from a plastic strain, the
stress-strain curve becomes a continuation of the hysteresis
behavior. (But generally neglected in plasticity theories.)A
stress-strain curve when referring to the true stress-strain curve,
is called as flow-stress curveA true stress-strain curve is called
flow curve as it gives the stress required to cause the material to
flow plastically to certain strain.
What is Bauschinger effect
If specimen is deformed plastically beyond the yield stress in
tension (+), and then in compression (-), it is found that the
yield stress on reloading in compression is less than the original
yield stress. a > The dependence of the yield stress on loading
path and direction is called the Bauschinger effect. (however it is
neglected in plasticity theories and it is assumed that the yield
stress in tension and compression are the same). In most materials,
plastic deformation in one direction will affect subsequent plastic
response in another direction. Amaterial that is pulled in tension,
for example, shows a reduction in compressive strength.This effect
is calaled as the Bauschinger effect.
What is Strain Hardening?In the plastic region, thetrue
stressincreases continuously i.e when a metal is strained beyond
the yield point, more and more stress is required to produce
additional plastic deformation and the metal seems to have become
more stronger and more difficult to deform. This implies that the
metal is becoming stronger as the strain increases. Hence, it is
called name"Strain Hardening". Strain hardening reduces ductility
and increases brittleness.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.
The force required to hold the specimen at a given stretch is
recorded, Fig. 8.1.1. If the material is a metal, the deformation
remains elastic up to a certain force level, the yield point of the
material. Beyond this point, permanent plastic deformations are
induced. On unloading only the elastic deformation is recovered and
the specimen will have undergone a permanent elongation (and
consequent lateral contraction). In the elastic range the
force-displacement behaviour for most engineering materials
(metals, rocks, plastics, but not soils) is linear. After passing
the elastic limit (point A in Fig. 8.1.1), further increases in
load are usually required to maintain an increase in displacement;
this phenomenon is known as work-hardening or strain-hardening. In
some cases the force-displacement curve decreases, as in some
soils; the material is said to be softening. If the specimen is
unloaded from a plastic state (B) it will return along the path BC
shown, parallel to the original elastic line. This is elastic
recovery. What remains is the permanent plastic deformation. If the
material is now loaded again, the force-displacement curve will
re-trace the unloading path CB until it again reaches the plastic
state. Further increases in stress will cause the curve to follow
BD.
Assumptions of Plasticity Theory
In formulating a basic plasticity theory the following
assumptions are usually made:(1) the response is independent of
rate effects(2) the material is incompressible in the plastic
range(3) there is no Bauschinger effect(4) the yield stress is
independent of hydrostatic pressure(5) 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.
What is yield criterion?In case the stress is un-axial and the
yield point can readily be determined. But what if there are
several stress acting at a point in different direction The
criteria for deciding which combination of multi-axial stress will
cause yielding are called criteria.True elastic limit The lowest
stress at which dislocations move. This definition is rarely used,
since dislocations move at very low stresses, and detecting such
movement is very difficult. Proportionality limit Up to this amount
of stress, stress is proportional to strain (Hooke's law), so the
stress-strain graph is a straight line, and the gradient will be
equal to the elastic modulus of the material. Elastic limit (yield
strength) Beyond the elastic limit, permanent deformation will
occur. The lowest stress at which permanent deformation can be
measured. This requires a manual load-unload procedure, and the
accuracy is critically dependent on equipment and operator skill.
For elastomers, such as rubber, the elastic limit is much larger
than the proportionality limit. Also, precise strain measurements
have shown that plastic strain begins at low stressesOffset yield
point (proof stress) THis is the most widely used strength measure
of metals, and is found from the stress-strain curve as shown in
the figure to the right. A plastic strain of 0.2% is usually used
to define the offset yield stress, although other values may be
used depending on the material and the application. Upper yield
point and lower yield pointSome metals, such as mild steel, reach
an upper yield point before dropping rapidly to a lower yield
point. The material response is linear up until the upper yield
point, but the lower yield point is used in structural engineering
as a conservative value.
Theories of Failure
In the case of multidimensional stress at a point we have a more
complicated situation present. Since it is impractical to test
every material and every combination of stresses , a failure theory
is needed for making predictions on the basis of a materials
performance on the tensile test., of how strong it will be under
any other conditions of static loading.The theory behind the
various failure theories is that whatever is responsible for
failure in the standard tensile test will also be responsible for
failure under all other conditions of static loading.
Failure occurs when material starts exhibiting inelastic
behaviorBrittle and ductile materials different modes of failures
mode of failure depends on loadingDuctile materials exhibit
yielding plastic deformation before failureBrittle materials no
yielding sudden failure
Four important failure theories, namely (1) maximum shear stress
theory, (2) maximum principal or normal stress theory, (3) maximum
strain energy theory, and (4) maximum distortion energy theory. Out
of these four theories of failure, the maximum normal stress theory
is only applicable for brittle materials, and the remaining three
theories are applicable for ductile materials.
Following are the important common features for all the
theories.In predicting failure, the limiting strength (Syp or Sut
or Suc) values obtained from the uniaxial testing is used. Since
stress and strain are tensor qualities they can be described on the
basis of threeprincipal directions, in the case of stress these are
denoted by The failure theories have been formulated in terms of
three principal normal stresses (S1, S2, S3) at a point. For any
given complex state of stress (sx, sy, sz, txy, tyz, tzx), we can
always find its equivalent principal normal stresses (S1, S2, S3).
Thus the failure theories in terms of principal normal stresses can
predict the failure due to any given state of stress. The three
principal normal stress components S1, S2, & S3, each which can
be comprised of positive (tensile), negative (compressive) or zero
value. When the external loading is uniaxial, that is S1= a
positive or negative real value, S2=S3=0, then all failure theories
predict the same as that has been determined from regular
tension/compression test.The material properties are usually
determined by simple tension or compression testsThe mechanical
members are subjected to biaxial or triaxial stresses.To determine
whether a component will fail or not, some failure theories are
proposed which are related to the properties of materials obtained
from uniaxial tension or compression tests. Initially we will
consider failure of a mechanical member subjected to biaxial
stressesDuctile materials usually fail byyielding and hence the
limiting strength is the yield strength of material as determined
from simple tension test which is assumed the same in compression
also. For brittle materials limiting strength of material is
ultimate tensile strength intension or compression
Theories of failure or yield criteria
This theory says that: Yielding occurs when the maximum shear
stress in the material reaches the value of the shear stress at
yielding in a uniaxial tension (or compression) test.
Yielding will occur when the maximum shear stress reaches the
values of the maximum shear stress occurring under simple
tension.The maximum shear stress in multi-axial stress = the
maximum shear stress in simple tension