Theory of Plasticity

28

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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 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. 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). When a material undergoes plastic deformations, i.e. irrecoverable and at a critical yield stress, and these effects are rate dependent, the material is referred to as being viscoplastic.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. The 1940s saw the advent of the classical theory; Prager, Hill, Drucker and Koiter amongst others brought together many fundamental aspects of the theory into a single framework.

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.

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 anelastic 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. 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).

A true stress strain curve provides the stress required to cause the metal to flow plastically at any strain _ it is often called a flow curve. A mathematical equation that fit to this curve from the beginning of the plastic flow to the maximum load before necking is a power expression of the type = Kn Where K is the stress at = 1.0 ; n is the strain hardening exponent (slope of a log-log plot of above equation)

Note: higher o 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, 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.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.(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).

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. For example, 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.Together with these, assumptions can be made on the type of hardening and on whether elastic deformations are significant. For example, consider the hierarchy of models illustrated in Fig. 8.1.4 below, commonly used in theoretical analyses. In (a) both the elastic and plastic curves are assumed linear. In (b) work-hardening is neglected and the yield stress is constant after initial yield. Such perfectly-plastic models are particularly appropriate for studying processes where the metal is worked at a high temperature such as hot rolling where work hardening is small. In many areas of applications the strains involved are large, e.g. in metal working processes such as extrusion, rolling or drawing, where up to 50% reduction ratios are common. In such cases the elastic strains can be neglected altogether as in the two models (c) and (d). The rigid/perfectly-plastic model (d) is the crudest of all and hence in many ways the most useful. It is widely used in analysing metal forming processes, in the design of steel and concrete structures and inthe analysis of soil and rock stability.

Generalized theory of plasticity

Yield criterion whether material deforms plastically or not Hardening rule is how material continues to deform Flow rule is what path material follows during plastic deformation to achieve new position according to hardening rule

Yielding criteria

Plastic yielding of the material subjected to any external forces is of considerable importance in the field of plasticity. For predicting the onset of yielding in ductile material, there are at present two generally accepted criteria,1) Von Mises or Distortion-energy criterion2) Tresca or Maximum shear stress criterion

Note: the difference between the two criteria are approximately 1-15%.

Comparison between maximum-shear-stress theory and distortion-energy (von Mises) theory

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.The microscopic yielding mechanism in ductile material is understood to be due to relative sliding of materials atoms within their lattice structure. This sliding is caused by shear stresses and is accompanied by distortion of the shape of the part. Thus the yield strength in shear Ssy is strength parameter of the ductile material used for design purposes. Generally used theories for Ductile Materials are:

Failure occurs when material starts exhibiting inelastic behaviorBrittle and ductile materials different modes of failures mode of failure depends onloadingDuctile materials exhibit yielding plastic deformation before failureBrittle materials no yielding sudden 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 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 are used. 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 failuredue 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 simpletension or compression testsThe mechanical members are subjected to biaxial ortriaxial stresses.To determine whether a component will fail or not, somefailure theories are proposed which are related to the properties of materials obtained from uniaxial tension orcompression tests.Initially we will consider failure of a mechanical membersubjected to biaxial stressesDuctile materials usually fail byyielding and hence the limiting strength isthe yield strength of material as determinedfrom simple tension test which is assumedthe same in compression also.For brittle materials limiting strengthof material is ultimate tensile strength intension or compression1. Max. principal stress theory or normal stress theory ((Rankines theory)

It is assumed that the failure or yield occurs at apoint in a member when the max. principal ornormal stress in the biaxial stress systemreaches the limiting strength of the material in asimple tension test.In this case max. principal stress is calculated ina biaxial stress case and is equated to limitingstrength of the material.This theory is basically applicable for brittle materials which are relatively stronger in shear and not applicable to ductile materials which are relatively weak in shear.

2. Max. principal strain theory (Saint Venants Theory)It is assumed that the failure or yielding occurs at a point in a member where the maximum principal (normal) strain in a biaxial stress exceeds limiting value of strain (strain at yield port) as obtained from simple tension test.

3. Maximum Strain energy theory (Heighs Thoery)

Failure is assumed to take place at a point in a member where strain energy per unit volume in a biaxial stress system reaches the limiting strain energy that is strain energy at yield point per unit volume as determined from a simple tension test.

4. Distortional energy theory (von-Mises theory) or (von Mises-Henckys theory)

It has been observed that a solid under hydro-static, external pressure (e.g. volume element subjected to three equal normal stresses) can withstand very large stresses. When there is also energy of distortion or shear to be stored, as in the tensile test, the stresses that may be imposed are limited. Since, it was recognized that engineering materials could withstand enormous amounts of hydro-static pressures without damage, it was postulated that a given material has a definite limited capacity to absorb energy of distortion and that any attempt to subject thematerial to greater amounts of distortion energy result in yielding failure. It is assumed that failure or yielding occurs at a point the member where the distortion strain energy (also called shear strain energy) per unit volume in a biaxial stress system reaches the limiting distortion energy (distortion energy at yield point) per unit volume as determined from a simple tension test. The maximum distortion energy is the difference between the total strain energy and the strain energy due to uniform stress.

5. Max. shear stress theory (Tresca or Guests Theory)

The failure or yielding is assumed to take place at a point in a member where the max shear stress in a biaxial stress system reaches a value equal to shear strength of the material obtained from simple tension test. This theory is mostly used for ductile materials

The Maximum Shear Stress theory states that failure occurs when the maximum shear stress from a combination of principal stresses equals or exceeds the value obtained for the shear stress at yielding in the uniaxial tensile test.

von-Mises Tresca theories

6. Octahedral shear stress theory