1 Theories of Failure • Failure: Every Material Has Certain Strength

Home , Buckling, Elasticity (physics), Plasticity (physics)

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

 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.

Conservative ( traditional/old fashioned/ conventonal)

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……..

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

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).

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.

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.

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 9

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

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.

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

12 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

13 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…..

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.

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.

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

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.

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.

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’.

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

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

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:

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

Parameters in uniaxial tension

End of lecture…………….