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yielding of materials under complex stress states Plastic deformation is a non reversible process where Hookes law is no longer valid.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