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# Relationship between Hardness and Elastic modulus?

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What is the relationship between hardness and elastic modulus? The higher hardness, the higher elastic modulus? My understanding is that hardness is a local mechanical property, and

elastic modulus is an averaged global mechanical property. Am I right about this?

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## Modulus and Hardness relationships

Great question and a topic that is commonly misunderstood!

Elastic modulus is an intrinsic material property and fundamentally related to atomic bonding. Hardness is an engineering property and for some materials it can be related to yield strength. Hardness has strong usefulness in characterization of different types of microstructures in metals and is frequently used in the context of comparing things like work-hardened and tempered metals. The classic experiment in this regard is the Jominy end-quench test . There are no apparent changes in elastic modulus in metals that have undergone different hardening treatments so the hardness is a good indication of the underlying microstructure. In metals undergoing indentation deformation, the majority of deformation is plastic and the hardness gives a good metric of plastic deformation differences between materials.

In general, if you plot the indentation (i.e. Vickers) hardness against the elastic modulus for a large range of materials (using software like CES from Granta makes this really easy since both properties are listed in the database) you will find that the two do increase together. In non-metals, a large fraction of the indentation deformation is elastic, so the two properties are not truly independent.

You can take a simplified model such as that forwarded by Sakai (and examined in additional detail in my own paper on the subject ) where the elastic and plastic deformation components are assumed to act in series, with two fundamental material parameters: an elastic modulus and a "resistance to plastic deformation". In this approach, the indentation hardness is actually related to both of these parameters, a function of both the elastic and plastic parts. The limiting behaviour then for metals is easy to understand, where the resistance to plastic deformation is relatively small such that the elastic deformation contributions to the indentation hardness are minimal and hardness is an approximate measure of plastic deformation resistance.

In most other materials, including ceramics and mineralized tissues (organic-inorganic composites) the contributions to the total deformation from elastic and plastic deformation can be similar and so the results from, for example, a series of nanoindentation tests the hardness is directly dependent on elastic modulus.

Further complicating the picture is the case of polymers, where the hardness is a time-dependent function (where the total deformation can be considered as a series sum of viscous, elastic and plastic deformation components ). In this type of case the measurement of "hardness" under different loading rates or load-holding times can actually be used to examine creep response of the material .

## Modulus - Potential Well - Hardness

Dear All,

I guess the discussion has been closed now, but couldn't contain my eagerness to throw some student thoughts :)

I'm only one day old at iMech, but have spent almost half a day browsing through various delighting discussions.

As Dr. Oyen says or known otherwise, Moduli (compliance / stiffness) are material's intrinsic properties, while "hardness" in other hand should be treated as an extrinsic materials response (/property). Let's stick to Elastic stiffness (C) for the sake of discussion, as we know C is a function of the bond strength, in turn, it depends on the potential energy (E) well. Stiffer the material, narrower is the potential well (C is dependent on the double derivative of E).

Now lets look at hardness. Going by the undergrad definition, Hardness is a measure (or response) of a material's flow resistance. Meaning, how easy (or hard) to move the dislocations in a periodic structure (like metals). This is influenced by the local structure of the material at the elasto-plastic transition and during subsequent plastic straining.

I think of a quantitative way to link the above two by using Peierls-Nabarro stress resistance (\Τau{PN}), which is to be overcome in order to move a group of dislocations resting on parallel slip planes. This thought was running in back of my mind while going through this thread. Someone must have done this, which I'm not aware of as my thesis deals with a very different subject on large plastic strains, far far away from the elasto-plastic regime, but nevertheless I've been always interested in dislocations and their dynamics :)

A question just cropped into my mind while writing this: In an indentation test (for example Vickers) is there any stress triaxiality involved?

This is important because hydrostatic component (dσ{ii}) of stress can alter the dislocation core diameter, and in turn, can change the P-N resistance (\Τau{PN}). Again, my thesis doesn't encompass indentation, rather I'm looking at plane strain compression of metals.

I might be way off ;) Cheers - Atish

## Dr. Oyen, Thanks a lot

Dr. Oyen,

Thanks a lot for the explanation! I will study it in details to learn more.

Another question about this topic is how the differences are reflected in Finite element analysis. Of course,

there is no single material parameter to reflect hardness in FEA, but hardness does play important role in

material behaviors in reality.

Thanks for the reply again,

## I would like to follow up

I would like to follow up the discussion on the relationship between elastic moduli and yield strength in materials. As pointed out by Dr. Oyen, elastic modulus is an intrinsic material property and fundamentally related to atomic bonding. The strength of materials is associated with plastic deformation mechanisms in a material and is hence structural and deformation-mechanism dependent.

In metals, we know in general that grain size, dislocation density, precipitations, et al. have a strong influence on the strength of materials with almost no change in elasticity. If the plasticity in a material is by creep, then the strength will show high strain-rate sensitivty.

In a polymeric material, internal chain structures decide both elastic and plastic properties of the material considering chains control both elastic and plastic deformation bahavior.

In a very broad viewpoint, there is a rough relationship between moduli and strengths in materials: large moduli correspond to higher strengths. I think that there is a chart in Ashby’s book on this regard (can’t remember clearly which book but I do have the impression in mind). In metallic glasses, since there are no apparent internal structures, this relationship is indeed quite significant, see http://www.nanonet.go.jp/english/mailmag/2004/014a.html (the Figure at the RHS). It may due to the fact that the onset of plasticity in metallic glasses is due to breakages of metallic bonds.

I hope that the information here, together with that by Dr. Oyen, make more sense for the discussion.

## Yujie, thanks for the

Yujie, thanks for the response!

## Quantitative results and analyses on the relation of H-Y-E

Michelle gave a great summary on the qualitative relationship between hardness (

H), elastic modulus (E), and yield strength (Y). In fact, it has been a long history to find this relation, and it is still an on-going topic. K. L Johnson (“Contact Mechanics”, page 175) proposed one of the most famous models for explaining the physical process of indentation, which is based on the rough equivalence between an expanding cavity and an indentation for the same elastoplastic material. According to Johnson’s expanding cavity model, there is the following quantitative relation betweenH,E, andY,H=2Y/3{2+ln[E/(3Ytana)]} (1),

assuming that the Poisson’s ratio is equal to 0.5, where a is the semi-angle of a conical indenter. Equation (1) shows that

His closely related toYand also related toEthrough the ratioE/Ytana. According to experiments and finite element analysis (FEA), Equation (1) is satisfied forE/Ytana < ~30; whenE/Ytana > ~30, H»3Y, which is commonly called Tabor’s relation. To know more, you might want to checkJohnson’s BookandCheng and Cheng's paper. For some FEA results, you could also seeour article(Fig. 5a). Then, in the relationship ofH-Y-E, another important issue is the relation ofY-E, which was well discussed by Yujie as in the previous post.## Gang, thanks for the input!

Gang, thanks for the input! I think it is very helpful.

Now I have a clearer picture of this topic. Will study the information to learn more.