# Strain hardening exponent Question

#### kelvin490

Gold Member
For some metals and alloys the region of the true stress–strain curve from the
onset of plastic deformation to the point at which necking begins may be approximated
by σ=Kεn where n is strain hardening exponent. I wonder whether this equation can be applied to engineering stress-strain curve or just true stress-strain curve? The problem is engineering stress-strain curve would decrease after the point of ultimate stress because of necking, which does not occur in true stress-strain curve.

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#### Greg Bernhardt

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

#### dav2008

Gold Member
I think the simple answer is that you know the relationship between engineering and true stress/strain so you could "plug in" the expressions for true stress and true strain into the strain hardening equation and obtain it in terms of engineering stress/strain.

Of course you still have to consider when engineering stress/strain actually make sense to use. Maybe someone else can chime in here? I believe engineering stress/strain is really for unaxial loading?

#### Astronuc

Staff Emeritus
For some metals and alloys the region of the true stress–strain curve from the
onset of plastic deformation to the point at which necking begins may be approximated
by σ=Kεn where n is strain hardening exponent. I wonder whether this equation can be applied to engineering stress-strain curve or just true stress-strain curve? The problem is engineering stress-strain curve would decrease after the point of ultimate stress because of necking, which does not occur in true stress-strain curve.
Great question.

Here is a good reference - http://books.google.com/books?id=jcD_K-WOS1kC&pg=PA128&lpg=PA128&dq=hollomon equation&source=bl&ots=EL4qurlmj2&sig=XNRhNsfgpF4tvoFwrawWSejMHv8&hl=en&sa=X&ei=Bq9eVNbbFsSBiwLMiYGIBw&ved=0CB0Q6AEwADgU#v=onepage&q=hollomon equation&f=false [Broken]
Mechanical Properties of Engineered Materials - If one looks on pages 125-128, one will find a good discussion of true-stress/true-strain and engineering-stress/engineering-strain, and one sees two different mathematical relationships/curves.

It's also important to understand how to apply the 'Hollomon' equation - $\sigma = K \epsilon^n$

Here is another good reference - http://www.colorado.edu/engineering/CAS/courses.d/Structures.d/IAST.Lect05.d/IAST.Lect05.pdf

H. J. Kleemola, M. A. Nieminen
On the strain-hardening parameters of metals
Metallurgical Transactions
August 1974, Volume 5, Issue 8, pp 1863-1866

The Hollomon equation is one of several empirical relationships used to characterize ductility/formability of a metal. The relationship relates true-stress with true plastic strain, not total strain which is elastic + plastic. There are a number of good questions in this forum on relating stress and strain for isotropic, anisotropic, orthotropic materials, so we probably should do a FAQ on the subject of mechanics of materials.

As dav2008 indicted, the testing tend to be uniaxial, and often for anisotropic/orthotropic materials, one will find uniaxial testing in two directions. Sometimes, testing will involve biaxial tests, e.g., pressurized tubes. Then one has to adapt the results to 'real' situations that can be biaxial or triaxial.

For most engineering applications, materials are used in the elastic range, with plenty of margin to YS. But there are transient situations, e.g., crashes, where materials must behave predictably under plastic deformation. And we haven't even touched on creep, which is a consideration in high temperature systems, such as power generation.

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#### kelvin490

Gold Member
Great question.

The Hollomon equation is one of several empirical relationships used to characterize ductility/formability of a metal. The relationship relates true-stress with true plastic strain, not total strain which is elastic + plastic.

Thank you. I wonder why you said the relationship relates true-stress with true plastic strain. Do you mean true strain should be counted from the yield point in
the Holloman-Ludwig equation? That would make the true strain zero at the yield point but the true stress is not zero at this point. Or we need to substitute the "extra" true stress after the yield point into the equation?

#### Astronuc

Staff Emeritus

Thank you. I wonder why you said the relationship relates true-stress with true plastic strain. Do you mean true strain should be counted from the yield point in
the Holloman-Ludwig equation? That would make the true strain zero at the yield point but the true stress is not zero at this point. Or we need to substitute the "extra" true stress after the yield point into the equation?
If it mentions strain, as in a stress-strain curve, then strain is the total strain, which includes the elastic+plastic strain. The point is that I found conflicting information in the published literature, and I'd like to reconcile the conflict.

The pure elastic strain stops at the proportional limit. In the elastic range, Hooke's law describes the relationship between stress and strain, namely σ = E ε, where E is the elastic modulus. I have seen reference to the Hollomon equation, given by σ = K εn, where ε is described as the true strain or true plastic strain. Depending on which value of ε, total or plastic strain, the n exponent would be different. I plan to dig into this more, since I'm curious about what Hollomon intended.

I suppose one can give a piecewise continuous model where the elastic range defines the stress-strain relationship below the proportional limit, and the Hollomon (or similar) expression defines the true stress-strain above the proportional limit or yield strength.

#### Astronuc

Staff Emeritus
I went back through the literature, and the literature indicates that the Hollomon equation uses true plastic strain.

The original work was published by J. H. Hollomon, “Tensile Deformation,” Transactions of the Metallurgical Society of AIME, Vol. 162, 1945, pp. 268-290.

It's possible this journal resides in the university library.

#### kelvin490

Gold Member
I went back through the literature, and the literature indicates that the Hollomon equation uses true plastic strain.

The original work was published by J. H. Hollomon, “Tensile Deformation,” Transactions of the Metallurgical Society of AIME, Vol. 162, 1945, pp. 268-290.

It's possible this journal resides in the university library.
Thanks for updating. I have found that the paper can also be downloaded here: http://wenku.baidu.com/view/e8f7ed1910a6f524ccbf851a.html

I've read the part mentioning the equation. It haven't indicate whether the strain should be counted from elastic limit, so I suppose it is not set zero at the point of elastic limit. The strain is counted from beginning but the equation only applied to the plastic region (i.e. for true strain after the elastic limit).

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