Theory of Elasticity and Plasticity

This open resource is a collection of academic course of post graduation program for M. Tech (Structural Engineering) as per the syllabus of  Dr. B.A.T University, Lonere, Raigad (m.s), India prepared by Dr. Mohd. Zameeruddin,  Associate Professor, of MGM's College of Engineering, Nanded for use in the out-of-class activity. The content covers both theoretical and analytical studies. There are six lessons as part of this document, and each deals with an aspect related to Theory of Elasticity and Plasticity

Module 1: Analysis of Stresses and Strains

Module 2: Stress-Strain Relationship

Module 3: Stress Concentration Problems

Module 4: Torsion

Module 5: Plasticity

Module 6: Yield Criteria & Yield Surface

What is Plasticity?

Plasticity, ability of certain solids to flow or to change shape permanently when subjected to stresses of intermediate magnitude between those producing temporary deformation, or elastic behavior, and those causing failure of the material, or rupture (yield point). It is concerned with materials which initially deform elastically, but which deform plastically upon reaching a yield stress.

For example, rolling steel into a particular shape (like rebar for construction) involves plastic deformation, since a new shape is created

 In other words;

1. The quality or state of being plastic especially: capacity for being molded or altered.

2. The ability to retain a shape attained by pressure deformation.

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

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. Two other types of failure, brittle fracture, due to dynamic crack growth, and the buckling of some structural components, can be modelled reasonably accurately using elasticity theory

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. A simplified model known as perfect plasticity is usually employed, and use is made of special limit theorems which hold for such models [Solid Mechanics Part II, Kelly].

Introduction to Plasticity Theory

Plasticity theory began with Tresca in 1864, when he undertook an experimental program into the extrusion of metals and published his famous yield criterion. 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. The arrival of powerful computers in the 1980s and 1990s provided the impetus to develop the theory further; giving it a more rigorous foundation based on thermodynamics principles, and brought with it the need to consider many numerical and computational aspects to the plasticity problem [Solid Mechanics-II, Kelly].

The theory of plasticity is the branch of mechanics that deals with the calculation of stresses and strains in a body, made of ductile material, permanently deformed by a set of applied forces. The theory is based on certain experimental observations on the macroscopic behaviour of metals in uniform states of combined stresses.

The observed results are then idealized into a mathematical formulation to describe the behaviour of metals under complex stresses. Unlike elastic solids, in which the state of strain depends only on the final state of stress, the deformation that occurs in a plastic solid is determined by the complete history of the loading. The plasticity problem is, therefore, essentially incremental in nature, the final distortion of the solid being obtained as the sum total of the incremental distortions following the strain path [J. Chakrabarty].

What do you mean about plastic deformations?

Plastic deformations are normally rate independent, that is, the stresses induced are independent of the rate of deformation (or rate of loading) [Solid Mechanics Part II, Kelly].

Viscoelastic behavior:

The material response has both elastic and viscous components. Due to their viscosity, their response is, unlike the plastic materials, rate-dependent. The viscoelastic materials can suffer irrecoverable deformation; they do not have any critical yield or threshold stress, which is the characteristic property of plastic behavior [Solid Mechanics Part II, Kelly].

Viscoplastic behavior:

The material undergoes plastic deformations, i.e. irrecoverable and at a critical yield stress, these effects are rate dependent [Solid Mechanics Part II, Kelly].

Observations from Standard Tests

To determine the plastic properties of metals, experiments are performed on the tension and compression of flat or cylindrical specimens and the deformation of thin-walled cylindrical tubes under the action of tensile force, torque, and internal pressure— that is, experiments permitting independent recording of forces and deformations. The deformation of a given material is characterized by a stress-strain diagram [Solid Mechanics Part II, Kelly].

The Tension Test [Solid Mechanics Part II, Kelly].

In the tensile test, usually cylindrical, specimen is gripped and stretched. The force required to hold the specimen at a given stretch is recorded, as shown in Figure 1. 

Figure 1: force-displacement curve for the tension test

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. 1), the material “gives” and is said to undergo plastic flow. Further increases in load are usually required to maintain the plastic flow and an increase in displacement; this phenomenon is known as work-hardening or strain-hardening.

        In some cases, after an initial plastic flow and hardening, 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.

        The strain which remains upon unloading is the permanent plastic deformation. If the material is now loaded again, the force-displacement curve will retrace 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 (on most metals) 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).

Nominal and True Stress and Strain

Nominal stress is the ratio tension force F with respect to the original cross sectional area of the tension test specimen A₀. Mathematically

σ_n = F/ A₀

True stress is the ratio tension force F with respect to the current cross sectional area of the tension test specimen A. Mathematically

σ_n = F/ A

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.

Engineering strain is the change in length observed in terms of original specimen length by l₀ and the current length by l.

ε = (l-l₀)/l

The true strain is based on 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:

The true strain is also called the logarithmic strain or Hencky strain. Again, at small deformations, the difference between these two strain measures is negligible. The true strain and engineering strain are related through

ε_t  = ln(1+ε)

Using the assumption of constant volume for plastic deformation and ignoring the very small elastic volume changes, one can say

σ = σ_n (l/l₀)

The stress-strain diagram for a tension test can now be described using the true stress/strain or nominal stress/strain definitions, as in Fig. 2.

Figure 2: typical stress/strain curves; (a) engineering stress and strain, (b) true stress and strain

Compression Test [Solid Mechanics Part II, Kelly].

A compression test showed similar results as the tensile stress. The yield stress in compression is approximately the same as (the negative of) the yield stress in tension. If a plots is prepared between the true stress versus true strain curve for both tension and compression (absolute values for the compression), the two curves found to be more or less coincide. This indicates that the behavior of the material under compression is broadly similar to that under tension. If plots are prepared between the nominal stress and strain, then the two curves would not coincide; this is one of a number of good reasons for using the true definitions.

The Bauschinger Effect

If a virgin sample is subjected to loads 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 illustrated in Fig. 3.

Fig. 3: The Bauschinger effect

The solid line depicts the response of a real material. The dotted lines are 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 deformation.

Assumptions of Plasticity Theory

From the above test results then, in formulating a basic plasticity theory with which to begin, 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. 4 below, commonly used in theoretical analyses.

Figure 4: Simple models of elastic and plastic deformation

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 analyzing metal forming processes, in the design of steel and concrete structures and in the analysis of soil and rock stability.

A general theory of plasticity specifies following criterion:

1. Yield criteria
2. Hardening rule, and
3. Flow rule

Yield Criteria

It is a hypothesis that defines the limits at which there is an onset of plastic deformation in a material under possible combination of stresses.

For Example Von-Mises yield criteria, Tresca’s yield criteria

Hardening Rule

It is the description of how a yield surface changes with plastic deformation. This depends on stress, strain, etc. The various forms of hardening are isotropic hardening, kinematic hardening and intermediate hardening

(a) Isotropic Hardening

Isotropic Hardening

In this the center axis and shape of yield surface do not change but the size of yield surface changes proportionately with the stress (Odqvist, 1933).

Kinematic Hardening

In this the center axis and shape of yield surface do not change but there is shift in the center axis. This rule is used to model Bauschinger effect (Prager, 1955). This is applicable for large deformations.

Kinematic hardening

Intermediate Hardening

It is the combination of isotropic and kinematic hardening models resulting in modification of loading surface due to simultaneous translation and expansion or contraction.

Mixed Hardening

Flow Rate:

The strain can be decomposed in to elastic strain (ε_e) and plastic strain (ε_p). The plastic strain causes the permanent deformation. The constitutive law of equivalent plastic strain is called the flow rate. The flow rate is used to estimate amount of plastic deformation in a material and also shows how the plastic strain evolves.

    Flow rule describes the relationship between the loading function (f) and the stress-strain behaviour of a strain hardening materials. When the yield surface is reached the material is in the state of plastic flow upon further loading as shown in figure.

Flow Rate

dε_p = dλ (δf/δσ₀)

where dλ is hardening parameter having value > 0