LectureNote 4-Strain Hardening

Mechanical Behavior of Materials (MATL 362) Lecture Note (4) 1/3/2010

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With elastic deformation, the strains are proportional to the stress. A definite level of stress must be

applied before any plastic deformation occurs. As the stress is further increased, the amount of

deformation increases, but not linearly. After plastic deformation starts, the total strain is the sum of the

elastic strain (which still obeys Hook’s law) and the plastic strain. (Because the elastic part of the

strain is usually much less than the plastic part, it will be neglected and symbol “ε” will signify

the true plastic strain).

Some materials show increased stress during plastic

flow, with a phenomenon called strain hardening.

The ability of a crystalline material to plastically

deform largely depends on the ability for

dislocation to move within a material. Therefore,

impeding the movement of dislocations will result

in the strengthening of the material. There are a

number of ways to impede dislocation movement,

which include:

controlling the grain size (reducing continuity

of atomic planes)

strain hardening (creating and tangling

dislocations)

alloying (introducing point defects and more

grains to pin dislocation)

The terms strain-hardening and work-hardening are used interchangeably to describe the

increase of the stress level necessary to continue plastic deformation.

The term flow stress is used to describe the stress necessary to continue deformation at any stage

of plastic strain

Mathematical descriptions of true stress-true strain curve are needed in engineering analyses that involve

plastic deformation, such as

Prediction of energy absorption in automobile crashes,

Design of dies for consist stamping parts, and

Analysis of stresses around the crack

http://en.wikibooks.org/wiki/Image:Elastic_strainharden_Solid_Mechanics.png

http://en.wikibooks.org/wiki/Image:Elastic_strainharden_Solid_Mechanics.png


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The strength of strain hardening materials continually increases with increasing deformation. At

necking the load carrying capacity starts to decrease (load carrying ability decreases, strength which is a

stress continues to increase). The load carrying capacity decreases because the effect decreasing cross-

section area overcomes the effect of increasing material strength due to strain hardening

The simplest model is one with no work-hardening. The flow stress, σt, is independent of

strain, so

σt = Y, (1)

Where Y is the tensile yield strength (see Figure 1a). For linear work-hardening

(Figure1b),

σt = Y + Aε. (2)

It is more common for materials to work-harden with a hardening rate that decreases with

strain. For many metals a log–log plot of true stress versus true strain is nearly linear

Figure 1 Mathematical approximations of the true stress–strain curve

In this case, a power law,

nt K , (3)

is a reasonable approximation (Figure 1c). A better fit is often obtained with


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n

t oK (4)

This expression is useful where the material has undergone a prestrain of εo. (See Figure

1d)

Still another model is a saturation model (Figure 1e),

1 exp At o

(5)

Equation (5) predicts that the flow stress approaches an asymptote, σo, at high strains.

This model seems to be reasonable for a number of aluminum alloys.

At the end of this lecture you should be able to answer the following questions:

1. At what value of engineering strain does necking start?

2. What is the value of true strain at the onset of necking?

3. Is the strain at the onset of necking related to the strength coefficient K?

4. Is the strain at the onset of necking related to the strain hardening exponent n?

The most commonly used expression is the simple power law [Equation (3)]. Typical

values of the exponent “n” are in the range from 0.1 to 0.6. Table 1 lists K and n for

various materials. As a rule, high-strength materials have lower n-values than

low-strength materials. Figure 2 shows that, the exponent, n, is a measure of the

persistence of hardening.

Table.1. Typical values of n and K


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The flow curve of many metals in the region of uniform plastic deformation can be

expressed by the simple power curve relation

nt K

Then ln ln lnt K n

Determination of K and n for a power law hardening model by plotting

log σ versus log ε

So the true stress– strain relation plots as a straight

line on log–log coordinates as shown in Figure 3.

The exponent, n, is the slope of the line. The

preexponential, K, can be found by extrapolating to ε

= 1.0. K is the value of σt at this point. The level of

n is particularly significant in stretch forming

because it indicates the ability of a metal to distribute

the strain over a wide region. The value of n is then

taken as the slope of the linear portion of the curve:

ln

ln

t t

t

d dn

d d

(6)

Figure 3 A plot of

the true stress–

strain curve on

logarithmic scales.

Because σ = Kεn, ln σ

= ln K + n ln ε.

The straight line

indicates that σ =

Kεn holds. The slope

is equal to n and K

equals the intercept

at ε = 1.


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As a tensile specimen is extended, the level of true

stress σt rises but the cross-sectional area carrying the

load decreases.

At the onset of necking, the force is maximum:

At the ultimate point: 0

dF

de

Then 0dF

Using tF A

Differentiating gives

0t tAd dA (7)

Since the volume, AL, is constant, one can write that:

0ldA Adl

dA dLd

A L

Rearranging terms,

t t t

tt

dAd d or

A

d

d

(8)

Equation (8) simply states that the maximum load is reached when the rate

of work hardening is numerically equal to the stress level. The above equation is valid at the ultimate point for any true stress-true strain relation.


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As long as

tt

d

d

deformation will occur uniformly along the test bar. However, once

the maximum load is reached

tt

d

d

, the deformation will localize. Any region that

deforms even slightly more than the others will have a lower load-carrying capacity and

the load will drop to that level. Other regions will cease to deform, so deformation will

localize into a neck. Figure 4 is a graphical illustration.

Figure.4 The condition for necking in a

tension test is met when the true stress, σt,

equals the slope, dσt/dε, of the true stress–

strain curve.

If a mathematical expression is assumed for the stress–strain relationship, the limit of

uniform elongation can be found analytically. For example, with power-law hardening,

equation (3),

1,n ntt

dK and nK

d

Substituting into equation (8) gives the relation at the ultimate point:


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1n nK nK Which, simplifies to

n Then, the true strain at the ultimate point:

u n (9)

So, the strain at the start of necking equals n. Uniform elongation in a tension test

occurs before necking. Therefore, materials with a high n value have large uniform

elongations. Since the ultimate tensile strength is simply the engineering stress at

maximum load, the power-law hardening rule can be used to predict it.

The rest of this page is for you to predict the engineering

ultimate tensile strength σu for a power law model

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The true fracture strain εf is the true strain based on the original area A0 and the area after fracture Af

ln of

f

A

A

This parameter represents the maximum true strain that the material can withstand before fracture and is

analogous to the total strain to fracture of the engineering stress-strain curve. Since the equation ε= ln

(1+e) is not valid beyond the onset of necking, it is not possible to calculate εf from measured values of

εf. However, for cylindrical tensile specimens the reduction of area Ra is related to the true fracture

strain by the relationship

1ln

1f

Ra

Based on the above very useful equation, one can also derive the area reduction in terms of true strain:

1 expRa

The true uniform strain eu is the true strain based only on the strain up to maximum load. It may be

calculated from either the specimen cross-sectional area Au or the gage length Lu at maximum load. The

uniform strain is often useful in estimating the formability of metals from the results of a tension test.

ln ou

u

A

A

The local necking strain εn is the strain required to deform the specimen from maximum load to fracture.

ln un

f

A

A


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True stress & true strain of

There are several special limitations to the compression test to which attention

should be directed:

1- The difficulty of applying a truly concentric or axial load.

2- The relatively unstable character of this type of loading as contrasted with tensile loading.

There is always a tendency for bending stresses to be set up and for the effect of

accidental irregularities in alignment within the specimen to be accentuated as loading

proceeds.

3- The long specimen may be suffering from buckling rather than compression stresses.

4- Friction between the heads of the testing machine or bearing plates

and the end surfaces of the specimen due to lateral expansion of the

specimen. This may alter considerably the results that would be

obtained if such a condition of test were not present.

5- The relatively larger cross sectional areas of the compression-test

specimen, in order to obtain a proper degree of stability of the piece.

This results, in the necessity for a relatively large-capacity testing

machine or specimens so small and therefore so short that it is difficult

to obtain from them strain measurements of suitable precision.

Modes of Deformation in Compression Testing

Problems with compression testing:

(a) Friction at the ends prevents

spreading, which results in barreling;

(b) Buckling of poorly lubricated

specimens can occur if the height-to-

diameter ratio, h/d, exceeds about 3;

(c) Without any friction at the ends

buckling can occur if h/d is greater

than about 1.5.


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Compression Stress-Strain relationship

Elastic Behavior As in tension loading up to elastic limit

Plastic Behavior

If friction can be neglected, the uniaxial compressive stress corresponding to an applied

compressive force F is:

o

F

A

Engineering stress

t

F

A

True stress

Based on the law of constancy of volume:

22 2 2 o oo o

D hD h D h then D

h

2

4t

o o

Fh

D h

The true strain is:

ln oh

h

If friction cannot be neglected, the relation become more complicated even assuming

constant coefficient of friction (Theory of metal forming). You can expect that, if

substantial friction is present, the average stress, required to deform the cylinder is

greater than σt

The Method to overcome the effect of Friction

A number of specimens of different initial height/diameter ratios (ho/Do) are compressed

under constant conditions of lubrication. The results are extrapolated in the manner

illustrated in the figure below.


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At D/h = 0 the stress-strain behavior can be found for an infinitely long specimen, in which the

friction effects would be negligible.

How you predict Engineering Stress-Strain curve in compression from the true

stress-true strain curve in tension

The very important assumption here is:

The true stress-strain curve in tension is the same in compression. We can

apply the following equations:

1 ln 1t e and e


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Stress-strain relation in compression for ductile material

Each point σt, ε on the true stress-true strain curve corresponds to a point σ, e on the engineering

stress-strain curve

Documents

LectureNote 4-Strain Hardening