There is a generally well known rule of thumb linearlyrelating hardness and ultimate tensile strength valid for a large range of metal alloys (see Materials Science in Design and Engineering by Pieter van Mourik, Jaap van Dam and Stephen J. Picken, Delft NL, 2012, 468 p.). In case of magnesium alloys the proportionality constant of this rule of thumb may deserve a special attention in view of the grain size effects possibly related to the hcp lattice of Mg.

Yield strength in MPa roughly equals  three times of hardness, if using HV scale, and after aging if the alloy can be age hardened.

But this can be overruled if you put high density dislocations into them, like worm even cold deformation with no further annealing. I can make AZ80 as hard as 120 HV(usually 85 at peak hardness), yet their yield strength only at about 220.

Hardness and strength has direct relationship. T his is also related to the grain size of the alloys. Decreasing the grain size will increase the strength and hardness of the alloys (HALL PICTH THEOREM). Consult material science by Askeland  Phule for further explanation

I am afraid the empirical relationships that may well work for other systems do not for Mg alloys. My personal experimental observation is this.

The reasons are numerous. Just to mention a few, the grain size, the orientations of the grains if texture is present, inclusion content (if high), the alloy compositions in many cases, the processing metod ... I have been struggling with this relationship over the past 6 months personally. I have been dealing with over twenty different new compositions. Alas, with no reliable result.

The relationship between hardness and strength definitely depends on the microstructure of the material. The nature of this microstructre dependence is usually explained through different work hardening characteristics of materials with different grain structures. Texture also plays a very important role here. You can find some useful discussions on the subject in the following paper:

http://www.sciencedirect.com/science/article/pii/S0921509315003846

F. Khodabakhshi et al.  Hardness−strength relationships in fine and ultra-fine grained metals processed through constrained groove pressing. Mater Sci Eng A (2015) 636: 331-339.

Let's put the Mg aspect aside for a moment. We must keep in mind that hardness is not really an intrinsic property of a material. Rather, the hardness value depends more on the technique and attributes of the material (e.g. strain hardening, microstructure) than on the fundamental physical properties of the material. Hardness is simply a measure of the materials resistance to localized plastic deformation. Conventional hardness measurements (e.g. Vickers, Brinell, Rockwell) are all influenced by the elastic recovery of the material under test. Empirically we have observed some relationships, in particular for steels and cast irons, which serve as estimators of a materials strength (tensile, yield). For example, the ultimate tensile strength of steel (in ksi) is approximately half the Brinell (BHN), and for many materials the Vickers (VHN) is roughly three times the yield strength (in kgf/mm2). ASTM E140 can be used to convert between scales. On the other hand, instrumented indentation testing can allow the measurement of elastic modulus using the Oliver-Pharr method. Still, only empirical relationships can be used to translate between hardness and tensile strength.

Now, let's come back to Mg. It is well known Mg alloys have limited available slip systems and different CRSS for compressive and tensile twinning. They are also susceptible to Portevin-LeChatelier (PLC) instabilities or inhomogeneous yielding and exhibit an asymmetrical response to compressive and tensile loading. These attributes are certainly complicating factors which must be unraveled to develop an empirical relationship between hardness and tensile strength.

To add to earlier answers, the general empirical relationship (for  steels, cast iron, brass and most metals) between tensile strength (TS) and Brinell hardness (HB: Brinell hardness) is TS (MPa) = K x HB and K is close to 3; where HB is obtained with a standard indenter with 3000 kgf load.

And the general empirical relation between Vickers Hardness (HV) and yield stress (YS) is HV ~ 3 YS. But the HV number obtained as kgf/mm2 needs to be multiplied by 9.81 to get HVN in MPa and then divide HV/3 gives YS.

A word of caution that such empirical correlations between hardness and strength depends upon specific test data and cannot be extrapolated to include other materials not tested.

Please see References:  (a) Martin Gaško, Gejza Rosenberg, Materials Engineering - Materiálové inžinierstvo 18 (2011) 155-159; (b) "General relationship between strength and hardness", P. Zhang, S.X. Li, Z.F. Zhang, Materials Science and Engineering A 529 (2011) 62– 73; (c) "On the tensile strength and hardness relation for metals", Joseph Datsko, Laura Hartwig, Brian McClory, Journal of Materials Engineering and Performance, December 2001, Volume 10, Issue 6, pp 718-722 and (d) http://www.calce.umd.edu/TSFA/Hardness_ad_.htm#6 

In case of strongly textured Mg alloys having  limited slip systems, the empirical relation may not be that straight forward; possibly because the localised deformation by an indenter may not reflect the strength behaviour correctly.

There is a study on Mg alloy AZ 19, where Caceres has found Vickers hardness = 0.3 flow stress (please see:- C.H. Caceres, 2002, Hardness and yield strength in cast Mg-Al alloys, AFS Transaction, 110: 1163-1169).

Also please: (a) "Microhardness mapping and the hardness-yield strength relationship in high-pressure diecast magnesium alloy AZ91", C.H. C´aceres, ET AL., Materials Science and Engineering A 402 (2005) 258–268, and (b) Acta Materialia 51 (2003) 3293–3307.

Generally the relation in most materials that

Hv= 3 yield strength 

sometimes this relation is OK up to 5 yield strength depending on the grain size and you must be care in the case of some materials like aluminum as their are no yield and so you must find the proof strength. Also the relation mention above can be more accurate in theoretical calculation of strenght from the dislocation queations 

Hardness is measured on the surface and the bulk may have defects (casting defects) and hence a relationship between tensile strength and hardness may not be predictable for a particular alloy as we may not get the same casting even with the same chemical composition again. Especially alloys containing intermetallic phases and bi-films.

For light weight materials such as Aluminium, there exists a linear relation between UTS and Hardness number. Aluminium alloys also obey this statement, whereas this may not hold good for casting process since it may contain defects such as blow holes etc., so measurement in the defective region may prove the linear relationship wrong. Also, for composite materials very large number of data of Hardness is required for the prediction of relationship. This may be true for the similar materials like Mg.

http://www.sciencedirect.com/science/article/pii/S0921509315003846

http://www.sciencedirect.com/science/article/pii/S0921509315001665

I. Shapiro (1963) established that the contact hardness (load/area of the contact flat) of freely compressed pure magnesium spheres is 32.5 kg/mm^2 (compares extremely well with the handbook value of indentation hardness and is approximately twice the ultimate tensile strength of pure magnesium). Our extensive single-sphere compression tests have shown that these relations hold good for pure copper and pure nickel also.  

Generally, Tabor's relationship used by many researchers suggests the correlation between the microhardness hardness values and yield strength as given by:

Yield strength (MPa) = Hardness (MPa)/3 = 9.81*Hardness in HV/3.

Other than Tabor's model, other models have been used to estimate the yield strength using harness values such as Dao model, Marsh model, and Johnson model.

Please find the attached file for more information about these models.

Dear All,

Actual mine is not an answer but rather a question. Is there a relationship for HV(GPa) obtained via DFT modelling such as CASTEP or QE? Say for example I got my hardness Vickers for TiAl to be equal 2.6GPa. Can the same procedure be used to convert to HVN(kgf/mm2) and visa versa for both micro and macro Vickers hardness measurement? I am trying to compare the experimental hardness values and the DFT predicted and see if there is any correlation factors.

Thanking you all in advance,

Regards - Bongani