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Theories of Failure
• Failure: Every material has certain strength, expressed in terms of stress or strain, beyond
which it fractures or fails to carry the load.
• Failure Criterion: A criterion (standard/principle/measure/gauge/norm) is used to
hypothesize (imagine/assume/theory/visualize) the failure.
• Failure Theory: A Theory behind a failure criterion.
Why we need failure theories?
• To design structural components/elements and calculate margin of safety.
• To guide in materials development.
• To determine weak and strong directions.
Failure Mode
• Yielding: a process of global permanent plastic deformation. Change in the geometry of the
object.
• Low stiffness: excessive elastic deflection.
• Fracture: a process in which cracks grow to the extent that the component breaks apart.
• Buckling: the loss of stable equilibrium. Compressive loading can lead to bucking in
columns.
• Creep: a high-temperature effect. Load carrying capacity drops.
Theories OF Failure
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Four important failure theories, namely (1) maximum shear stress theory, (2) maximum
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 or Rankins’s
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.
1. In predicting failure, the limiting strength values obtained from the uniaxial testing are
used.
2. The failure theories have been formulated in terms of three principal normal stresses
(S1, S2, S3) at a point.
3. 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.
4. The three principal normal stress components S1, S2, & S3, each which can be
comprised of positive (tensile), negative (compressive) or zero value.
5. 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.
6. The material properties are usually determined by simple tension or compression tests
7. Some structural members are subjected to biaxial or triaxial stresses.
8. To determine whether a component/element will fail or not, some failure theories are
proposed which are related to the properties of materials obtained from uniaxial
tension or compression tests.
9. Ductile materials usually fail by yielding 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.
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Conservative ( traditional/old fashioned/ conventonal)
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Problem solving strategy for Failure thoery:
Syp- yield strength , Sut and Suc=ultimate tensile strength and ultimate compressive
strengths
Failure Theories
1. Failure under load can occur due to excessive elastic deflections or due to excessive
stresses.
2. Failure prediction theories due to excessive stresses fall into two classes: Failure when
the loading is static or the number of load cycles is one or quite small, and failure due
to cyclic loading when the number of cycles is large often in thousands of cycles.
Failure under static load
Parts under static loading may fail due to:
a) Ductile behavior: Failure is due to bulk yielding causing permanent deformations that are
objectionable. These failures may cause noise, loss of accuracy, excessive vibrations, and
eventual fracture. In machines, bulk yielding is the criteria for failure. Tiny areas of yielding
are OK in ductile behavior in static loading.
b) Brittle behavior: Failure is due to fracture. This occurs when the materials (or conditions)
do not allow much yielding such as ceramics, grey cast iron, or heavily cold-worked parts or
concrete.
End of Lecture……..
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The yield point, alternatively called the elastic limit, marks the end of elastic behaviour and the
beginning of plastic behaviour. When stresses less than the yield point are removed, the
material returns to its original shape. For many materials that do not have a well-defined yield
point, a quantity called yield strength is substituted. Yield strength is the stress at which a
material has undergone some arbitrarily chosen amount of permanent deformation, often 0.2
percent.
Any increase in the stress beyond the yield point causes greater permanent deformation and
eventually fracture.
A point at which Maximum load or stress required to initiate the plastic deformation of
material such point is called as Upper yield point. And a point at which minimum load
or stress required to maintain the plastic behavior of material such a point is called as Lower
yield point.
Upper yield point is the point after which the plastic deformation starts. This is due to the fact
that the dislocations in the crystalline structure start moving. But after a while, the dislocations
become too much in number and they restrict each other’s movement. This is called strain
hardening and lower yield point is the point after which strain hardening begins.
Dislocations are defects present in crystal areas where atoms are out of position (irregular
alignment).
Why the lower yield point stress value of mild steel is consider as a strength of material
instead of upper yield point stress?
Failure of mechanical component means it fail To perform it's operations efficiently for
example consider shaft which transmits rotational motion ,now when shaft is unable to
transmit motion efficiently then it will fail. Basically their are three types of failure in case of
mechanical component i.e
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1) failure due to elastic deformation
2) failure due to plastic deformation
3) failure due to fracture
When component deforms elastically it's dimensions changes and it fails. And this failure is
known as failure due to elastic deformation
When component undergoes plastic deformation it's dimension changes permanently and
failure takes place this is known as failure due to plastic deformation.
For ductile metals elastic failure is criteria of failure because ductile metals undergo elastic
deformation before failure. And elastic deformation starts at lower yield point.
As Mild steel is ductile material we consider lower yield point
Upper yield point is not constant it varies with shape of specimen and rate of loading
Lower yielding point is constant for all shapes and rate of loading because of its consistency
lower yielding point is taken as yield stress of mild steel
Upper yield point corresponds to the load that is required to initiate yielding. Lower yield point
corresponds to the min load that is required to maintain yield.
Normally we use the lower yield point to determine the yield strength of the material being
tested, cause the upper yield is momentary.
Upper yield point is the max load at which deformation starts, starting of deformation means
dislocations are started moving in the material.
So this type of phenomenon is called permanent deformation by slip ( slip mechanism).
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As the slip is taking place in the material, it offers less resistance to the material and hence
curve falls slightly ( stress is the measurement of resistance offered by the material during the
application of load).
And it reaches to some stress ( lower yiled point stress) which is the minimum stress required
to maintain the deformation in the mateial.. And at the lower yield point for the low carbon
steels ( mild steels) the stress strain cure is in some wave nature , this is because to break bonds
with impurites while dislocations are moving out of the material , hence resistance increases
and decreases periodically after that strain hardening takes place which increases resistance
slowly by increasing of dislocations in the material...
What is strain softening and strain hardening?
Work hardening, also known as strain hardening is the strengthening of a metal by plastic
deformation. This strengthening occurs because of dislocation movements and dislocation
generation within the crystal structure of the material.
Reason for Work hardening: As the deformation of the material occur in the plastic region,
the dislocation of the material increases. The dislocation interaction is repulsive in nature. As
the dislocation density increases the further deformation of the material become difficult, this
is called Work Hardening or Strain Hardening.
some materials exhibit an elevation in yield stress along with plastic strain, sometimes strain
rate or some internal variables which is known as hardening and if it shows a decrease in yield
stress with plastic strain, it is called softening.
Strain hardening is the process of increasing the hardness and strength of a metal by plastic
deformation and is a cold working process. Strain hardening is due to the increased resistance
to dislocation movement through a crystal lattice.
No crystal lattice is perfect, it has some crystallographic defects called dislocations
( Dislocation).
The dislocation movement is along the slip plane (plane of greatest atomic density and direction
is along the closest packed direction within the slip plane).Slip will occur when the shear stress
along the crystallographic plane reaches a critical value, which leads to movement of
dislocations.
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.
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The classical theory of plasticity grew out of the study of metals in the late nineteenth century.
It 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..
Plastic deformations are normally rate independent, that is, the stresses induced are
independent of the rate of deformation (or rate of loading).
Plasticity theory began with Tresca, when he undertook an experimental program into the
extrusion of metals and published his famous yield criterion discussed later on. 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.
Imp Points:
• Permanent deformation that cannot be recovered after load removal
• Hookes law (linear relation between stress and strain) not valid
• Beyond Hooke’s 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 Hooke’s 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.
Plasticity vs elasticity
Plasticity is a property of a material or a system that allows it to deform irreversibly. Elasticity
is a property of a system or a material that allows it to deform reversibly.
Elasticity is a concept directly connected with the deformation of materials. When an exterior
stress is applied to a solid body, the body tends to pull itself apart. This causes the distance
between atoms in the lattice to increase. Each atom tries to pull its neighbor as close as possible.
This causes a force trying to resist the deformation. This force is known as strain.
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If a graph of stress versus strain is plotted, the plot will be a linear one for some lower values
of strain. This linear area is the zone which the object is deformed elastically. Elastic
deformation is always reversible. It is calculated using Hooke’s law.
The Hooke’s law states that for the elastic range of the material applied stress is equal to the
product of the Young’s modulus and the strain of the material. The elastic deformation of a
solid is a reversible process, when the applied stress is removed the solid returns to its original
state.
Plasticity is a concept which is connected with the plastic deformation. When the plot of stress
versus strain is linear, the system is said to be in the elastic state. However, when the stress is
high the plot passes a small jump on the axes. This limit is when it becomes a plastic
deformation. This limit is known as the yield strength of the material.
Plastic deformation occurs mostly due to the sliding of two layers of the solid. This sliding
process is not reversible. The plastic deformation is sometimes known as the irreversible
deformation, but actually some modes of plastic deformation are reversible.
After the yield strength jump, the stress versus strain plot becomes a smooth curve with a peak.
The peak of this curve is known as the ultimate strength. After the ultimate strength, the
material begins to “neck” making unevenness of the density over length. This makes very low
density areas in the material making it easily breakable. Plastic deformation is used in metal
hardening to pack the atoms thoroughly.
What is the difference between Plasticity and Elasticity?
• Plasticity is the property that causes irreversible deformations on an object or a system.
Such deformations can be caused by forces and impact.
• Elasticity is a property of objects or systems that allows them to deform reversibly. Elastic
deformations can be caused by forces and impacts.
• An object must pass the elastic deformation stage in order to enter the plastic deformation
stage.
Assumptions of linear elasticity:
1) continuity of material,
2) homogenity(just one material) and isotropy (properties are the same in all directions),
3) linear elasticity (valid Hook´s law),
4) the small deformation theory,
5) static loading,
6) no initial state of stress
A solid is a continuum, it has got its volume without any holes, gaps or any interruptions. Stress
and strain is a continuous function. Homogeneous material has got physical characteristics
identical in all places (concret, steel, timber). Combination of two or more materials ( concret
+ steel) is not homogeneous material. Isotropy means that material has got characteristics
undependent on the direction – (concret, steel – yes, timber – not). Elasticity is an ability of
material to get back after removing the couses of changes (for example load) into the original
state. If there is a direct relation between stress and strain than we talk about Hooke´s law =
this is called physical linearity.
Small deformations theory:
Changes of a shape of a (solid) structure are small with aspect to its size (dimensions). Then
we can use a lot of mathematical simplifications, which usually lead to linear dependency
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Static loading:
It means gradually growing of load (not dynamic effects)
In the initial state there are all stresses equal zero. (Inner tension e.g. from the production).
All these assumptions enable to use principal of superposition which is based on linearity of
all mathematic relationship.
Saint - Venant principle of local effect
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Saint - Venant principle of local is not valid in these cases:
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 Bauschinger effect refers to a property of materials where the material's stress/strain
characteristics change as a result of the microscopic stress distribution of the material. For
example, an increase in tensile yield strength occurs at the expense of compressive yield
strength.
Bauschinger effect represents loss of isotropic behavior in strength-strain behavior produced
due to deformation produced in metallic materials. When steel is loaded in tension, it starts
deforming first elastically but later plastically. Plastic deformation occurs due to dislocation
movement. However, dislocation entangles during movement which requires more stress for
the further movement. This is known as work hardening.
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When the direction of stress is reversed, say from tensile to compressive, dislocation movement
can start at lower strength resulting in a decrease of strength in compression. This phenomena
is known as Bauschinger effect.
Bouschinger Effect is also known as strain softening.First observe the figure given below,
Region OA -This region is Elastic Region in tension.Within this reagion, if we unload the material it will follow the same path in the reverse direction i. e. From A to O.
Region OZ- This region is Elastic Region in compression. Within this region, if we unload the material it will follow the same path in the reverse direction i.e.From Z to O.
Region AB- Due to increase in load,tensile stresses overcome the bond strength. Dislocation starts moving towards grain boundary. Material starts yielding due to
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movement of these dislocations.Accumulation of dislocations near grain boundary creates a back pressure, because same type of dislocations repel each other.
Region BC- Immediate unloading will take curve from B to C. Elastic recovery takes place in this region. Length OC represents the permanent deformation of material.
Region CD- Compression of material takes place from C to D.
Region DZ- As in case of tension, back pressure opposes the movement of dislocations i. e. this back pressure resists the tensile load. Same back pressure will now assist the compressive load. Due to combined effect of compression and this back pressure, a curvature is observed from D to Z.
Region ZE- Due to further increase in compressive load, material starts yielding in compression. Again a back pressure is created. Now this back pressure will resist the compressive load but will assist the tensile load.
Region EF - Represents removal of compressive load.
Region FG- Again we apply a tensile load.
Region GA- Due to combined effect of tensile load and back pressure created during compression, a curvature can be observed here also.
These curvatures represents the Strain softening and this effect is know as Bouschinger Effect.
The stress-strain behaviour of steel in compression is identical to that in tension.
However, if the steel is stressed into the inelastic range in uniform tension, unloaded, then
subjected to uniform compression in the opposite direction, it is found that and the stress-strain
curve becomes nonlinear at a stress much lower than the initial yield strength [Fig.].This is
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referred to as the ‘Bauschinger effect’.In this case, the hysteresis loop is also more pronounced.
In inelastic deformation processes involving continual reversal of stress (such as metal
working, high intensity reversed seismic loading, etc), the Bauschinger effect is very important
and cannot be ignored. In other cases, where there is in general no more than one stress reversal,
the Bauschinger effect can safely be neglected.
Structural members are likely to subjected to reversal of stresses. While the mild steel in
compression behaves same as like in tension upto the yield point. However actual behavior is
different and indicates an apparently reduced yield stress in compression. This occurs only
when change in direction of strain changes. The divergence from ideal path is called
Bauschinger effect.
End of Lecture…..
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Most metals can be regarded as isotropic. After large plastic deformation however, for example
in rolling, the material will have become anisotropic: there will be distinct material directions
and asymmetries.
Theories of ductile failure (yielding)
Yielding is a shear stress phenomenon. That means materials yield because the shear stresses
on some planes causes the lattice crystals to slide like a deck of cards. In pure tension or
compression, maximum shear stresses occur on 45-degree planes – these stresses are
responsible for yielding and not the larger normal stresses.
The best predictor of yielding is the maximum distortion energy theory (DET). This theory
states that yielding occurs when the Von Mises stress reaches the yield strength. The more
conservative predictor is the maximum shear stress theory (MST), which predicts yielding to
occur when the shear stresses reach Sy/2.
Note that in static loading and ductile behavior, stress concentrations are harmless as they only
create small localized yielding which do not lead to any objectionable dimensional changes.
The material “yielding” per se is not harmful to materials as long as it is not repeated too many
times.
Theories of brittle failure There are two types of theories for brittle failure. The classical theories assume that the material
structure is uniform. If the material structure is non-uniform, such as in many thick-section
castings, and that the probability of large flaws exist, then the theory of fracture mechanics
predicts the failure much more accurately.
An important point to remember is that brittle materials often show much higher ultimate
strength in compression than in tension. One reason is that, unlike yielding, fracture of brittle
materials when loaded in tension is a normal stress phenomenon. The material fails because
eventually normal tensile stresses fracture or separate the part in the direction normal to the
plane of maximum normal stress.
In compression the story is quite different. When a brittle material is loaded in compression,
the normal stress cannot separate the part along the direction normal to the plane of maximum
normal stress. In the absence of separating normal stresses, shear stresses would have to do the
job and separate or fracture the material along the direction where the shear stresses are
maximum.
In pure compression, this direction is at 45 degrees to the plane of loading. Brittle materials,
however, are very strong in shear. The bottom line is that it takes a lot more compressive normal
stress to create a fracture.
We only discuss these theories for a 2D state of stress – 3D is similar but is more formula-
based.
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Failure Theories for Isotropic Materials: Strength and stiffness are independent of the
direction. Failure in metallic materials is characterized by Yield Strength.
1. Maximum principal stress theory.
2. Maximum principal strain theory.
3. Quadratic or Distortional Energy Theory.
What is the definition of Failure?
Obviously fracture but in some components yielding can also be considered as failure, if
yielding distorts the material in such a way that it no longer functions properly
Which stress causes the material to fail?
Usually ductile materials are limited by their shear strengths.
While brittle materials (ductility < 5%) are limited by their tensile strengths.
Theories of Failure or Yield Criteria
It is known from the results of material testing that when bars of ductile materials are
subjected to uniform tension, the stress-strain curves show a linear range within which
the materials behave in an elastic manner and a definite yield zone where the materials
undergo permanent deformation.
In the case of the so-called brittle materials, there is no yield zone. However, a brittle
material, under suitable conditions, can be brought to a plastic state before fracture
occurs.
In general, the results of material testing reveal that the behavior of various materials
under similar test conditions, e.g. under simple tension, compression or torsion, varies
considerably.
In the process of designing a machine element or a structural member, the designer has
to take precautions to see that the member under consideration does not fail under
service conditions. The word ‘failure’ used in this context may mean either fracture or
permanent deformation beyond the operational range due to the yielding of the member.
We know that the state of stress at any point can be characterized by the six rectangular
stress components—three normal stresses and three shear stresses. Similarly, the state
of strain at a point can be characterised by the six rectangular strain components.
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When failure occurs, the question that arises is: what causes the failure? Is it a particular
state of stress, or a particular state of strain or some other quantity associated with stress
and strain? Further, the cause of failure of a ductile material need not be the same as
that for a brittle material.
Any one of the above or some other factors might have caused the yielding.
Further, as pointed out earlier, the factor that causes a ductile material to yield might be
quite different from the factor that causes fracture in a brittle material under the same
loading conditions.
Consequently, there will be many criteria or theories of failure. It is necessary to
remember that failure may mean fracture or yielding. Whatever may be the theory
adopted, the information regarding it will have to be obtained from a simple test, like
that of a uniaxial tension or a pure torsion test. This is so because the state of stress or
strain which causes the failure of the material concerned can easily be calculated.
The critical value obtained from this test will have to be applied for the stress or strain
at a point in a general machine or a structural member so as not to initiate failure at that
point.
There are six main theories of failure. Another theory, called Mohr’s theory, is slightly
different in its approach
Significance of the Theories of Failure The mode of failure of a member and the factor that is responsible for failure depend
on a large number of factors such as the nature and properties of the material, type of
loading, shape and temperature of the member, etc.
We have observed, for example, that the mode of failure of a ductile material differs
from that of a brittle material.
While yielding or permanent deformation is the characteristic feature of ductile
materials, fracture without permanent deformation is the characteristic feature of brittle
materials.
Further, if the loading conditions are suitably altered, a brittle material may be made to
yield before failure.
Even ductile materials fail in a different manner when subjected to repeated loadings
(such as fatigue) than when subjected to static loadings
Any rational procedure of design of a member requires the determination of the mode
of failure (either yielding or fracture), and the factor (such as stress, strain and energy)
associated with it.
If tests could be performed on the actual member, subjecting it to all the possible
conditions of loading that the member would be subjected to during operation, then one
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could determine the maximum loading condition that does not cause failure. But this
may not be possible except in very simple cases.
Consequently, in complex loading conditions, one has to identify the factor associated
with the failure of a member and take precautions to see that this factor does not exceed
the maximum allowable value. This information is obtained by performing a suitable
test (uniform tension or torsion) on the material in the laboratory.
In discussing the various theories of failure, we have expressed the critical value
associated with each theory in terms of the yield point stress σy obtained from a uniaxial
tensile stress.
This was done since it is easy to perform a uniaxial tensile stress and obtain the yield
point stress value. It is equally easy to perform a pure torsion test on a round specimen
and obtain the value of the maximum shear stress τy at the point of yielding.
Consequently, one can also express the critical value associated with each theory of
failure in terms of the yield point shear stress τy.
In a sense, using σy or τy is equivalent because during a uniaxial tension, the maximum
shear stress τ at a point is equal to 1/2 σ; and in the case of pure shear, the normal
stresses on a 45° element are σ and –σ, where σ is numerically equivalent to τ.
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Use Of Factor Of Safety In Design
In designing a member to carry a given load without failure, usually a factor of safety N is
used. The purpose is to design the member in such a way that it can carry N times the actual
working load without failure.
It has been observed that one can associate different factors for failure according to the
particular theory of failure adopted. Consequently, one can use a factor appropriately reduced
during the design process.
Let X be a factor associated with failure and let F be the load. If X is directly proportional
to F, then designing the member to safely carry a load equal to NF is equivalent to designing
the member for a critical factor equal to X/N.
However, if X is not directly proportional to F, but is, say, proportional to F2, then designing
the member to safely carry a load to equal to NF is equivalent to limiting the critical factor to
√X /N .
Hence, in using the factor of safety, care must be taken to see that the critical factor
associated with failure is not reduced by N, but rather the load-carrying capacity is increased
by N.
As remarked earlier, when a factor of safety N is prescribed, we may consider two ways of
introducing it in design:
(i) Design the member so that it safely carries a load NF.
(ii) If the factor associated with failure is X, then see that this factor at any point in the member
does not exceed X/N.
But the second method of using N is not correct, since by the definition of the factor of
safety, the member is to be designed for N times the load. So long as X is directly proportional
to F, whether one uses NF or X/N for design analysis, the result will be identical. If X is not
directly proportional to F, method (ii) may give wrong results.
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The Flow Curve
• True stress-strain curve for typical ductile materials, i.e., aluminium, show that the stress -
strain relationship follows up the Hooke’s law up to the yield point, σo.
• Beyond σo, 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 ε1 to ε2. by an amount of σ/E.
• The strain ε1-ε2 is the recoverable elastic strain. Also there will be a small amount of the
plastic strain ε2-ε3 known as inelastic behaviour 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.)
• 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. 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).
• A true stress – strain curve provides the stress required to cause the material to flow plastically
at any strain is often called a ‘flow curve’.
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Note: higher σo means greater elastic region, but less ductility (less plastic region).
True stress and true strain
• The engineering stress – strain curve is based entirely on the original dimensions of the
specimen means This cannot represent true deformation characteristic of the material.
• The true stress – strain curve is based on the instantaneous specimen dimensions.
True strain or natural strain (first proposed by Ludwik) is the change in length referred to the
instantaneous gauge length.
The true stress is the load divided by the instantaneous area.
What is Strain Hardening?
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.
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),
further increases in load are usually required to maintain an increase in displacement; this
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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.
Two important observations concerning the above tension test are the following:
(1) After the onset of plastic deformation, the material will be seen to undergo negligible
volume change, that is, it is incompressible.( assumption of plasticity)
(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).
Nominal and True Stress and Strain
There are two different ways of describing the force F which acts in a tension test. First,
normalizing with respect to the original cross sectional area of the tension test specimen Ao ,
one has the nominal stress or engineering stress,
Alternatively, one can normalize with respect to the current cross-sectional area A, leading to
the true stress,
in which F and A are both changing with time. 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. Similarly, one can describe the deformation in
two alternative ways. Denoting the original specimen length by lo and the current length by l,
one has the engineering strain
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Alternatively, the true strain accounts for the fact that the “original length” is continually
changing; a small change in length dl leads to a strain increment dε = dl / l and the total strain
is defined as the accumulation of these increments:
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Resilience and Toughness
Ability of absorb energy in the elastic range and release it when stress is removed is called
Resilience. High carbon steel has high resilience
Ability to absorb energy in plastic range is called Toughness. Spider silk has high toughness.
Too little carbon content leaves (pure) iron quite soft, ductile, and weak. Carbon contents
higher than those of steel make a brittle alloy commonly called pig iron.
• Flow rule is what path material follows during plastic deformation to achieve new
position according to hardening rule
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
material’s 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.
Brittle and ductile materials – different modes of failures – mode of failure – depends on
loading
Ductile materials – exhibit yielding – plastic deformation before failure
Brittle materials – no yielding – sudden failure
Multi-axial stress state – six stress components – one representative value
Define effective / equivalent stress – combination of components of multi-axial stress state
Equivalent stress reaching a limiting value – property of material – yielding occurs – Yield
criteria
Yield criteria define conditions under which yielding occurs
Single yield criteria – doesn’t cater for all materials
Material yielding depends on rate of loading – static & dynamic
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Parameters in uniaxial tension
End of lecture…………….