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.

Classical Theory of Plasticity

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. The deformation of microvoids and the development of

micro-cracks is also an important cause of plastic deformations. Plasticity theory began with

Tresca in 1864, 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 etc.

Classification of Plasticity Problems

There are two broad groups of metal plasticity problem which are of interest to the engineer and

analyst.

1. Small Plastic strains The first involves relatively small plastic strains, often of the same order as the elastic strains

which occur. Analysis of problems involving small plastic strains allows one to design structures

optimally, so that they will not fail when in service, but at the same time are not stronger than they

really need to be. In this sense, plasticity is seen as a material failure.

2. Large Plastic Strains The second type of problem involves very large strains and deformations, so large that the elastic

strains can be disregarded. These problems occur in the analysis of metals manufacturing and

forming processes, which can involve extrusion, drawing, forging, rolling and so on. In these

latter-type problems, a simplified model known as perfect plasticity is usually employed. The

word perfect means that the material in this model does not strain-harden, that is the yield strength is used which is independent of the amount of plastic strain.

The Bauschinger Effect

If one takes a new sample and loads it in tension into the plastic range, and then unloads it and

continues on into compression, one finds that the yield stress in compression is not the same as the

yield strength in tension, as it would have been if the specimen had not first been loaded in

tension. In fact the yield point in this case will be significantly less than the corresponding yield

stress in tension. This reduction in yield stress is known as the Bauschinger effect. The effect is

two extreme cases which are used in plasticity models; the first is the isotropic hardening model,

in which the yield stress in tension and compression are maintained equal, the second being

kinematic hardening, in which the total elastic range is maintained constant throughout the

is not an issue provided there are no reversals of stress in the problem under study.

Stress AnalysisDr. Maaz akhtar

illustrated in Figure-1. The solid line depicts the response of a real material. The dotted lines are

deformation. The presence of the Bauschinger effect complicates any plasticity theory. However, it

Figure-1: Baushinger Effect

Assumptions of Plasticity Theory

For basic plasticity theories following assumptions are usually made:

(3) There is no Bauschinger effect (4) The material is isotropic

(5) Yield stress is independent of hydrostatic pressure

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 Figure 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 structures.

Figure-2: Stress-strain curves for different processes

(1) Response is independent of rate effects (2) Material is incompressible in the plastic range

Stress and strain are related through in the elastic region, E being the Youngs modulus. The tangent modulus K is the slope of the stress-strain curve in the plastic region and in general

change during a deformation. At any instant of strain, the increment in stress is related to the increment in strain through .

Figure-3: Stress-strain curve for tangent modulus (K)

Elastic-Perfectly Plastic deformation Behavior

Plastic deformation does not change volume. For a metal under uniaxial stress, the two transverse

plastic strains are equal, related to the longitudinal plastic strain as

In applications like metal forming, the plastic strain is so large that elastic strain is negligible.

Thus, we may neglect elastic strain, and identify the net strain entirely with plastic strain in rigid,

perfectly plastic model. When the stress is within the yield strength, , the material is rigid, and the strain does not change. For now we wish to include elasticity. Even though a

metal is capable of arbitrarily large deformation, in many situations the plastic strain is small, on

the order of elastic strain. For example, the plastic deformation of the metal can be constrained by

elastic surroundings. When plastic strain and elastic strain are comparable, we need to include

both in the model. This model is called elastic, perfectly plastic model. The net strain is the sum

of elastic strain and plastic strain:

Strain Hardening and Flow Rule

Work hardening, also known as strain hardening or cold working, 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. Many non-brittle metals with a

reasonably high melting point as well as several polymers can be strengthened in this fashion.

Dislocations on intersecting slip planes permit both elastic interactions and dislocation reactions to

contribute to work hardening.

Figure-4: Stress-strain curve for rolling operation

The stress-strain curve (Flow curve) in the region of uniform plastic deformation does not increase

proportionally with strain. The material is said to work harden (i.e., strain harden). Empirical

relationships attributed to Ludik and Holloman can be used to describe the shape of plastic stress-

strain curve. It has general form

or

where, is stress, is yield stress, is strain, and are different strength coefficients and is the strain hardening exponent (n=0 for perfectly plastic solids, n=1 for perfectly elastic solids &

n=0.1-0.5 for most metals).

Another flow equation known as Ramberg-Osgood law, works upto ultimate tensile strength is

given by:

where, is Ramberg Osgood strength coefficient, is Ramberg Osgood strain hardening

coefficient, is offset yield stress, E is Youngs modulus and is

Viscoelastic and Viscoplastic Behavior of Solids

Materials having rate dependent deformation are commonly known as viscoelastic material.

Although the viscoelastic materials can suffer irrecoverable deformation, they do not have any

critical yield or threshold stress, which is the characteristic property of plastic behaviour. In many

forming processes the deformation rates are small enough to consider the material behavior to be

independent of strain rate and to use an elastoplastic material model. For high strain rates this

assumption leads to faulty results. In a tensile test the yield stress is seen to increase with higher

strain rates.

Figure-5: Stress-strain curve showing increase in yield strength

with higher strain rate

Polymers and certain metallurgical alloys show softening behavior immediately after reaching the

yield point. At larger strains the softening is followed by hardening. The complete stress-strain

behavior is strain rate dependent, but the initial yield stress is constant.

Figure-6: Stress-strain behavior is strain rate dependent

with constant initial yield

Failure Theories (Yield Criteria)

All engineering materials are classified as ductile or brittle. Generally a ductile material is one in

which gross plastic deformation is greater than 5%, while brittle material having plastic

deformation less than 5%. Ductile materials show large plastic deformation and observed necking

prior to fracture but brittle materials fails suddenly without proper indication. Both materials

behave in different manner; hence they have different failure criteria. One single theory cannot be

used to predict failure in both the materials. Permanent deformation in ductile material is observed

when load reaches the yield point. Any further load will strain hardened the material. In brittle

materials yield point is not clear (determined by offset method) and fracture usually occurs near to

ultimate tensile strength (UTS). Hence, for ductile material failure criteria is based on Yield

Strength while for brittle material failure criteria is based on ultimate tensile strength,

(1) Failure Theories for Ductile Material

Many theories are present which gives failure criteria for ductile materials are discussed below:

(a) Maximum Shear Stress Theory

It is also known as Tresca criteria states that yielding begins when maximum shear stress at a point

reaches the maximum shear stress at yield under uniaxial tension or compression. For multiaxial

state of stress shear stress is obtained by:

,

,

Maximum shear stress is largest, if then

Consider an element from uniaxial tension subjected to yield, hence,

Therefore, using Mohrs circle equation principal stresses will be equal to:,

,

Maximum shear stress gives,

If the principal stresses are unordered, yielding under multiaxial state of stress occurs for any one

of the following conditions:

For two dimensional system (let , above equations become

Above results when plotted on two dimensional principal plane, it gives a hexagon as shown

below:

Figure-7: Tresca Yield Locus

At point A which makes -45 with x-axis the shear stress is found to be

.

(b) Maximum Distortion Energy Theory

This theory is also known as Von-Mises theory or Octahedral shear stress theory. It states that

yielding occurs when the distortional strain energy at a point equals to the distortional strain

energy at yield under uniaxial tension or compression. Consider are principal stresses and are principal strains, then total strain energy is given by:

From Hookes law, stress-strain relationships are given by:

,

,

Total stress is the sum of hydrostatic and deviatoric states of stress as shown in figure-8.

Figure-8: Total strain energy as su of hydrostatic and deviatoric state of stress

Energy stated with hydrostatic state of stress gives:

Also, , hence strain energy for hydrostatic stat of stress become:

Hence strain energy for deviatoric state of stress can be determine by

A

Consider an element from uniaxial tension subjected to yield, hence,

,

Comparing deviatoric relations, we get

For plane stress condition above relation simplified to

Above relation is an equation of ellipse, which gives for plane stress condition an elliptical shape

yield surface as shown in figure-9.

Figure-9: Von-Mises Yield Locus

At point A and , hence

, which gives

.

Comparison of Tresca and Von-Mises failure criteria tells that Tresca criteria is more conservative

and gave 15% less yield surface area (figure-10).

A

Figure-10: Comparison of Von-Mises and Tresca Yield criteria

(c) Strain Energy Density or Total Strain Energy Criteria

The strain energy density criteria proposed by Beltrami states that yielding occurs when the strain

energy density at a point equals the strain energy density at yield in uniaxial tension or

compression. Total strain energy is given by:

For a uniaxial state of stress at yield principal stresses can be given by:

,

By comparing both expressions for , we get the following relationship:

For plane stress condition above equation simplified and gives following relation:

Shape of yield surface for strain energy density criteria is an ellipse in principal stress space that

depends on the Poissons ratio. Assume a special case where Poissons ratio is zero above equation

represents the circle equation

and yield surface will be a circular region.

(2) Failure Theories for Brittle Material

Failure of brittle materials is characterized by ultimate tensile strength. Failure criteria for brittle

materials are discussed below.

(a) Maximum Principal Stress Criteria

This criteria is proposed by Rankine states that yielding occurs at a point when the maximum

principal stress reaches the value equals to maximum principal stress at yield in uniaxial tension or

compression. According to this theory failure occurs when

For plane stress condition Rankine theory can be written as

or

Maximum principal stress criteria for plane stress condition gives failure surface that represents a

square.

Figure-11: Yield Surface for Rankine criteria

(b) Maximum Principal Strain Criteria

It is also known as St. Venants criteria states that yielding occurs when maximum principal strain equals to the maximum principal strain at yield under uniaxial tension or compression. Principal

strains in terms of principal stresses are given by:

,

,

For uniaxial tension case at yield gives the following relation:

For plane stress condition above equations for becomes,

Also,

Figure-12: Yield Surface for St. Venants criteria

(c) Mohrs Failure Criteria The Mohr Theory of Failure, also known as the Coulomb-Mohr criterion or internal-friction

theory, is based on the famous Mohr's Circle. Mohr's theory is often used in predicting the failure

of brittle materials. Mohr's theory suggests that failure occurs when Mohr's Circle at a point in the

body exceeds the envelope created by the two Mohr's circles for uniaxial tensile strength and

uniaxial compression strength. This envelope is shown in the figure below,

Figure-13: Yield Surface (shaded) for Mohrs criteria

The left circle is for uniaxial compression at the limiting compression stress of the material.

Likewise, the right circle is for uniaxial tension at the limiting tension stress. The middle Mohr's

Circle on the figure-13 (dash line) represents the maximum allowable stress for an intermediate

stress state. Equation for Mohrs-Coulomb failure criteria for plane stress condition is given by,

Graphically, Mohr's theory requires that the two principal stresses lie within the shaded zone

depicted in Figure-14. Also shown on the figure is the maximum stress criterion (dashed line).

This theory is less conservative than Mohr's theory since it lies outside Mohr's boundary.

Figure-14: Yield Surface (shaded) for Mohrs criteria

There are some more yield criteria such as Drucker-Pager yield criteria, Hills criteria etc, that follows similar method of determination of yield surface with minor changes in their equations,

used to develop yield criteria.

Problem-1

When the loads that act on the hub of a flywheel reach their working values, the nonzero stress

components at the critical point in the hub where yield is initiated are ,

and . The load stress-strain are linear so that the factor of safety can be applied to either the loads or stress components. The flywheel material has a yield stress equals

.

a) Assuming material follows Tresca yield criteria, determine factor of safety against yield.

b) Assuming material follows Von-Mises yield criteria, determine factor of safety against yield.

c) Determine which criteria is more conservative

Solution

Principal stresses are determined using equation of two-dimensional Mohrs circle,

a) Tresca Criteria:

b) Von-Mises Criteria:

c) Tresca criteria is more conservative as it predicts yielding at smaller loads. Answer

Problem-2

A foundation of a machine is made by gray cast iron. The most critical stress condition at a point

in the part is shown below. Find the factor of safety in the foundation, such that ,

.

Solution:

The gray cost iron is brittle material. For solution we apply maximum principal stress theory.

Maximum compressive stress in the part = 20 ksi

So from compressive stress point of view,

Maximum tensile stress in the part = 10 ksi

So from compressive stress point of view,

Thus the factor of safety of the part will be smaller value i.e. Answer

Problem-3

A thin-wall tube with closed ends is subjected to a maximum internal pressure of 35 MPa in

service. The mean radius of the tube is 30 cm. If the tensile yield strength is 700 MPa, what

minimum thickness must be specied to prevent yielding? Consider failure based on Tresca criteria.

Solution:

Hoop stresses,

Longitudinal stresses,

Yielding occurs when ,

Answer

Problem-4 A circular shaft of tensile strength 350 MPa is subjected to a combined state of loading defined by

bending moment (M=8 kN.m) and torque (T=24kN.m). Calculate the required shaft diameter (d) in

order to achieve a factor of safety of 2. Use Tresca criteria.

Solution:

Since

&

(Bending and Torsion equations)

From Mohrs circle equation,

Putting all the values into the above relation, we get

Note: If Von-Mises failure criteria is used we have following relationship,

Hence, we can easily derive the following expression