Micro And Macro Hardness Measurements, Correlations,
AIAA 2006-97944th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, NevadaMicro and Macro Hardness Measurements,Correlations, and Contact ModelsM. M. Yovanovich Microelectronics Heat Transfer LaboratoryDepartment of Mechanical EngineeringUniversity of Waterloo, Waterloo, Ontario, Canada, N2L 3G1Brief reviews of Brinell (Meyer) and Rockwell indenters and macrohardness tests,Berkovich, Knoop and Vickers indenters and microhardness tests, and nanoindentationtests using the Berkovich indenter are given. Vickers, Brinell and Rockwell C indentationresults for Ni 200, SS 304, Zr-4, and Zr-Nb are reported and correlation equations for micro and macrohardness versus penetration depth are given. Temperature effects on yieldstrength, Brinell and Vickers hardness are given and correlation equations are presentedto account for elevated temperatures. Models are presented for calculation of the appropriate value of contact microhardness which depends on apparent contact pressure and theeffective surface roughness of the joint. Examples are given to illustrate the use of thecorrelation equations.NomenclatureAAa , Ac AVaBCcCT Ccp , cvc1c2DddVEEiErerfc(·)erfc 1(·)Ffgsurface area of a single tube, m2apparent area, contact area, m2contact radius, munloading curve correlation coefficientdimensionless contact conductance, Cc σhc /mkscorrelation coefficient, C 1degree Celsiusspecific heats at constant pressure and volume, J/kgKVickers correlation coefficient, GPaVickers size indexball diameter, mmindentation diameter or diagonal, mmVickers diagonal, mmspecimen elastic modulus, GPaindenter elastic modulus, GPareduced elastic modulus, Er 1 (1 ν 2)/E (1 νi2)/Ei, GPacomplementary error functioninverse complementary error functionindentation load, mNgas gap function DistinguishedProfessor Emeritus, Department of Mechanical Engineering, Fellow AIAA.1 of 28American Institute of Aeronautics and AstronauticsCopyright 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
gHHBHB GMHBKHKHMHRCHVhc , hg , hrhjhhf , hmaxKIgJkk1, k2kgksMm1 , m2 , mnPPPaPgPg,0PmaxPrQQjRSySTTgTg,0TrmTmt TjuWYZacceleration constant, 9.81m/s2nano, micro, macrohardness, kg/mm2, GPaBrinell hardness number, kg/mm2, GPageometric mean of minimum and maximum Brinell hardness, kg/mm2, GPaBerkovich hardness number, kg/mm2 , GPaKnoop hardness number, kg/mm2 , GPaMeyer hardness number, kg/mm2 , GPaRockwell C hardness numberVickers hardness number, kg/mm2, GPacontact, gap, radiation conductances, W/m2 · Kjoint conductance, hj hc hg hr , W/m2 · Kindentation depth, nmfinal and maximum indentation depths, nmabsolute temperature, kelvingas gap integralunit of energy, joulethermal conductivity, W/m · Ksolid thermal conductivities, W/m · Kgas thermal conductivity, W/m · Keffective thermal conductivity, ks 2k1k2/(k1 k2), W/m · Kgas gap parameter, M αβΛ, mpmean asperity slopes, effective slope, m m21 m22contact spot density, m 2contact pressure, GPaindentation load, mNapparent contact pressure, MPagas pressure, kPareference gas pressure, kPamaximum indentation load, mNPrandtl number, ν/αheat transfer rate, Wjoint heat transfer rate, Wthermal resistance, K/Wyield strength, GPacontact stiffness, dP/dh, mN/nmtemperature, Celsius and absolute, C, Kgas temperature, Kreference gas temperature, Kroom temperature, Cmelt temperature, Clocal gap thickness, µmjoint temperature drop, Kdimensionless local gap thickness, t/σenergy per unit time, wattmean plane separation, µmnormalized Brinell hardness, Z HB /HB GMGreek Symbolsαthermal diffusivity, m2/s2 of 28American Institute of Aeronautics and Astronautics
α 1 , α2αβγ ΛΛ0λµνπρσ1, σ2σσ0ψaccommodation coefficientsgas gap accommodation parameter, α (2 α1)/α1 (2 α2)/α2gas gap parameter, β 2γ/(γ 1)P rratio of specific heats, cp /cv prelative real contact area, Ar /Aafactor in nanoindentation testmolecular mean free path, nmreference molecular mean free path, nmrelative mean plane separation, Y /σmicroPoisson’s ratiopimass density, kg/m3surface roughnesses, µmpeffective joint roughness σ σ12 σ22 , µmreference effective joint roughness, 1µmthermal constriction parameter, ψ (1 lBerkovichbulkcontacteffectivegas, gapindenterKnoopMeyermean, materialmaximumminimumRockwell CrealVickersI.Introductionhenever heat transfer occurs across a mechanical joint formed by two conforming rough solids thereis a measurable temperature drop T which is associated with the joint heat transfer rate Q . TheWjoint temperature drop and the joint heat transfer rate are related to the joint conductance h and the jointjjjthermal resistance Rj by means of the following relationships:Qj hj Aa TjandQj TjRj(1)where Aa is the apparent area (nominal area) of the joint. The joint conductance and resistance are related:3 of 28American Institute of Aeronautics and Astronautics
1(2)hj AaThermal joint conductance hj is a complex microgeometrical, thermal, and physical parameter whichdepends on the thermophysical properties of the contacting solids, the thermal properties of the substancein the microgaps, and the mechanical load applied to the joint.The models for predicting the joint conductance are based on three separate models: (1) a microgeometricmodel that describes the surface roughness features of the contacting surfaces and the resultant contact, (2)the mechanical interaction of the contacting asperities, and (3) the thermal constriction/spreading resistancesat the microcontacts and the heat transfer across the microgaps by conduction only or conduction andradiation when the gap substance is transparent to radiation.There are mechanical models for conforming rough surfaces whose contacting asperities deform (i) elastically, (ii) plastically, or (iii) elastoplastically.Several microgeometrical, mechanical and thermal models have been developed over four decades bynumerous researchers for the many types of joints that occur in the microelectronics and aerospace industries.Comprehensive reviews of the various types of models are presented in Chapter 41 and Chapter 162.This paper will be restricted to the joint formed by two conforming rough surfaces as depicted in Fig. 1.The joint of interest can be modeled as an equivalent joint formed by the contact of an equivalent roughsurface and an ideal, smooth, flat surface as shown in Fig. 1.In this paper the plastic contact model will be examinedin some detail because it can be applied to many practicalproblems encountered in industry. The plastic contact modelrequires knowledge of the microhardness of the contacting asperities. It will be shown that this important physical parameter appears in the contact and gap components of the jointconductance.The paper will review the different types of indentation testsusing different types of indenters which result in a measureof the resistance of the material to penetration of the indenter. The resistance to penetration of the indenter is called thehardness of the material.The indentation tests fall into three types of indentations:(i) macroindentations as determined by the Brinell (Meyer) andRockwell indenters, (ii) microindentation as determined by theBerkovich, Knoop and Vickers indenters, and (iii) nanoindentation as determined by the Berkovich indenter.Correlation equations will be presented for the Vickers mi- Figure 1. Typical Joints Between Conformcrohardness measurements as a function of the Vickers diagonal ing Rough Surfaces.or the penetration depth. It will be shown that the Vickers microhardness correlation equations for different materials depend on two correlation coefficients which areclosely related to the Brinell hardness. The correlation equations and a mechanical model will be used topredict the appropriate value for the contact microhardness given the effective joint roughness and mechanical load. The Brinell hardness and the Vickers microhardness are temperature dependent and correlationequations will be presented for three metals to illustrate this fact.Rj II.Contact, Gap, Radiation, and Joint ConductancesSteady heat transfer across a joint formed by the static contact of two conforming rough surfaces is givenby the relationship:Qj hj Aa Tj(3)4 of 28American Institute of Aeronautics and Astronautics
where Tj is the overall temperature drop across the joint, hj is the joint conductance, and Aa is theapparent area of the joint. If the microgaps are occupied by a gas which is transparent to radiation, thenthe joint conductance consists of three components such thathj hc hg hr(4)where hc is the contact conductance, hg is the gap conductance, and hr is the radiation conductance. If thetemperature level of the joint is below 600 C, the radiation conductance is negligibly small relative to thecontact and gap conductances, and, therefore, the joint conductance is given byhj hc hg(5)which applies to many joints of interest to the microelectronics industry. If the joint is in an environmentwhere the gas pressure is much smaller than one atmosphere, then the joint conductance is related to thecontact conductance, andhj hc(6)This simple relation is applicable to many joints of interest to the aerospace industry when radiation isnegligible2.The contact conductance for the joint formed by two conforming rough surfaces is given by followinggeneral relationship1,4:2nakshc (7)ψ( )where n denotes the contact spot density, a is the mean contact spot radius, and the effective thermalconductivity of the joint is1,42k1k2ks (8)k1 k2The thermal conductivities of the contacting asperities are k1 and k2, respectively. The constriction/spreadingresistance parameter for isothermal contact spots is1 4ψ( ) (1 )1.5(9)pwhere the relative contact spot size is Ar /Aa and the total real contact area is Ar .The general relationship for the contact conductance applies to all joints formed by two conforming roughsurfaces whose asperities have Gaussian height distributions with respect to mean planes associated with eachsurface shown in Fig. 1. The asperities are also assumed to be randomly distributed in the contact plane. Thecontact conductance relation was developed by Cooper, Mikic and Yovanovich3 and its applicable to jointswhere the contacting asperities deform (i) elastically, (ii) plastically, or (iii) elastoplastically. The Cooper,Mikic and Yovanovich model (CMY)3 is based on the plastic deformation of the contacting asperities. Thismodel has been shown to be applicable to most joints when the appropriate value of the contact microhardnessis used.The CMY model consists of the following relationships for the microgeometry of the joint1,3,4: Ar1λ erfc (10)Aa221 m 2 exp( λ2 ) n (11)16 σerfc(λ/ 2)r 2 8 σλλa experfc (12)π m22 2 1 m λ na exp (13)24 2π σ5 of 28American Institute of Aeronautics and Astronautics
The effective joint roughness parameters are defined asqσ σ12 σ22m qm21 m22(14)when the surface asperities heights are Gaussian and they are randomly distributed in the plane of contact.The RMS surface roughnesses are σ1 and σ2. The mean asperity slopes are m1 and m2 . The importantmicro-geometric parameter which appears in all relations is the relative mean plane separation which isdefined asYλ (15)σwhere Y is the mean planes separation. This parameter depends on the apparent contact pressure P andthe mode of asperity deformation.The gap conductance model for the microgaps occupied by a gas was developed by Yovanovich andco-workers5,7,8. Its given as an integral1,5,7: Z exp (λ u)2 /2kg 1kg hg (16)du Igσ 2π 0u M/σσwhere kg is the thermal conductivity of the gas. The dimensionless local gap thickness is defined as u t/σ.The complex gas gap rarefaction parameter is defined as1,5,7M αβΛ(17)2 α1 2 α2 α1α2(18)whereα β Λ 2γ(γ 1)P r TgPg,0Λ0Tg,0Pg(19)(20)The accommodation parameter α depends on the thermal accommodation parameters α1, α2 in a complexmanner1,8. The thermal accommodation coefficients must be determined by experiments for the particulargas-solid combination1,8.The gas parameter β depends on the ratio of specific heats γ cp /cv and the Prandtl number P r. Themolecular mean free path Λ depends on the reference value Λ0 and the gas temperature Tg and the gaspressure Pg , as well as the reference gas temperature Tg,0 and reference gas pressure Pg,0 .Negus and Yovanovich7 gave the following correlation equations for the gap integral:Ig fgλ M/σ(21)In the range 2 λ 4:fg 1.063 0.0471 (4 λ)fg 1 0.06 (σ/M )0.81.68[ln (σ/M )]0.84for 0.01 M/σ 1for 1 M/σ The correlation equations have a maximum error of approximately 2%.6 of 28American Institute of Aeronautics and Astronautics
The important micro-geometric parameter λ appears in the contact and gap conductances. In generalthis parameter lies in the range: 2 λ 4. When the contact pressure is light, e.g., P 0.27 MPa and thecontacting surfaces are hard metals, then λ will be near 4. On the other hand when the contact pressure ishigh, e.g., P 13.5 MPa, and the contacting surfaces are soft, then λ will be near 2.To determine the value of λ its necessary to assume a mode of deformation of the contacting asperitiessuch as plastic deformation and to apply a force balance at the joint to obtain a relationship for λ.A force balance gives the following relationship:F P Aa NXHc,iAr,i(22)i 1where P is the apparent contact pressure, Aa is the apparent or nominal contact area, Hc,i is the microhardness of the ith contact spot whose real area is Ar,i , and there are N microcontacts in the apparent area. Weassume that there is a mean microhardness value such that Hc,i Hc for all contact spots; therefore,NXHc,iAr,i Hci 1NXAr,i Hc Ar(23)i 1where Ar is the total real contact area. From the force balance and the geometric relation Ar /Aa forinteraction of two Gaussian surfaces, we have PAr1λ erfc (24)HcAa22From the foregoing relation we can write λ 2 erfc 1 2PHc (25)This important relationship will be used to obtain relationships for the dimensionless contact conductancewhich is defined asσ hc1exp( λ2 /2)Cc f(λ) (26) qm ks 1.54 2π1 12 erfc(λ/ 2)The relationships for Cc and λ were combined and a numerical method were used to obtain numerical valuesof Cc for values of P/Hc. The following simple power-law relation was recommended by Yovanovich4: 0.95PCc 1.25(27)HcThis explicit relationship agrees with the theoretical values to within 1.5% in the range 2 λ 4.75 andin the range 10 6 P/Hc 2.3 10 2. Numerical quadrature is required to find values of λ for givenvalues of P and Hc. The following approximations can be used to calculate λ:Yovanovich Approximation (1981) 0.547Pλ 1.184 ln 3.132HcSong-Yovanovich Approximation (1988) 0.5Pλ 1.363 ln 5.589Hc7 of 28American Institute of Aeronautics and Astronautics
Ranges of application of the approximations are2 λ 4.7510 6 andP 2 10 2HcAntonetti Power-Law Approximation (1983) Pλ 1.53Hc 0.097can be used to obtain values quickly; however, its less accurate than the other two approximations. Itshows more clearly than the other approximations that λ is a relatively weak function of the relative contactpressure P/Hc.The foregoing short reviews of the contact and gap conductances for joints formed by the mechanicalcontact of conforming rough surfaces show that the dimensionless parameter P/Hc is very important. Sincethe apparent contact P is known, it is necessary to obtain values of the contact microhardness Hc for a givenjoint having the effective surface roughness σ/m.A.Iterative Model for Calculation of Contact MicrohardnessThe iterative model for calculating the contact microhardness Hc and the relative contact pressure P/Hc isbased on a set of equations and the given joint parameters: (σ/m, c1 , c2, P, Hb).The set of equations are1,4,6: 2P 1 (i)λ 2 erfc Hc r 2 σ 8 λλ (ii)a experfc mπ22(28) (iii) dV 2π a c2 dV (iv) Hc c1 d0The parameter d0 1 µm is introduced for convenience. The set of equations is based on the area equivalenceof the Vickers projected area and the contact spot area, therefore, AV d2V /2 Ac πa2 . Also we assumethat Hc HV .The iteration process begins with an initial guess for Hc . It has been demonstrated that when the initialguess is based on the known bulk hardness, i.e., Hc Hb, convergence occurs after 2 to 3 iterations dependingon the convergence criterion6 .To avoid the numerical calculation of the inverse complementary error function in the first equation, theapproximation of Yovanovich4 was used to calculate the inverse complementary error function.B.Explicit Relationship for Relative Contact PressureSong8 and Song and Yovanovich9 examined the iterative model and found that when the Song and Yovanovichapproximation for erfc 1 (·) is used, the iterative process leads to the following explicit relationship: 1/(1 0.071c2) PP (29)cHc1.62c1 (σ/σ0 m) 2where σ0 1µm. This relationship shows how the important parameter P/Hc depends on the joint parameters (σ/m, c1, c2, P ).In the subsequent sections it will be shown how the contact hardness can be calculated and how it isrelated to the macro, micro and nano-hardness. However, before this can be done its important to presentbrief reviews of hardness indenters and hardness tests.8 of 28American Institute of Aeronautics and Astronautics
III.Micro and Macro Hardness IndentersThe several micro and macrohardness indenters will be reviewed in this section.The macrohardness indentation testers are the Brinell (Meyer) and Rockwell, and the microhardnessindenters are the Berkovich, Knoop and Vickers.There are 6 types of hardness indentation tests used to determine the macrohardness and microhardnessof materials such a metals, ceramics, and plastics. The indenters are much harder than the specimen, andthey are smooth balls, cones with hemispherical tips, and pyramidal having three or four faces.A brief description of the various types of indenters are given below. More details regarding the nano,micro, and macro-indenters can be found in the texts10 14 .A.Brinell and Meyer MacrohardnessThe Brinell and Meyer macrohardness are determine by the same indentation test. A hard smooth ball ofdiameter D(mm) is pressed into a smooth flat surface under a known load F (N) for duration of 30 to 60seconds depending on the whether the metal is hard or soft as shown in Fig. 2. After removal of the ball, thediameter of the indentation is measured by means of an optical microscope. The diameter of the indentationd(mm) is the average value of 2 measurements, i.e., d (d1 d2)/2 where d1 and d2 are the measureddiameters which are perpendicular to each other.Brinell Hardness Number. The Brinell hardness number(BHN) is expressed as the load divided by the actual area ofthe indentation. Therefore,BHN 2FF 22πDtπD D D d(30)where the penetration depth, defined as the distance from theoriginal surface to the maximum indentation depth, is D D2 d2t (31)2The relative indentation size is recommended to lie in therange: d/D 0.25 0.6. If the load is given in kgfand the indentation diameter is given in mm, then theBrinell hardness has units of kgf/mm2 .These are theunits used in handbooks and older texts. Today, its morecommon to give the Brinell hardness in units of MPa orGPa. In this paper the Brinell hardness will be denoted asHB .Meyer Hardness Number. The Meyer hardness number(MHN) is based on the same indentation test, however, theMeyer hardness number is expressed as the indentation loaddivided by the projected area of the indentation. Therefore,MHN 4F Pmπd2Figure 2. Brinell and Vickers Indenters andIndentations.(32)where Pm is the mean contact pressure. The Meyer hardness is said to be a true representation of thehardness of the material. In this paper the Meyer hardness will
g acceleration constant, 9.81m/s2 H nano, micro, macrohardness, kg/mm2,GPa HB Brinell hardness number, kg/mm2,GPa HBGM geometric mean of minimum and maximum Brinell hardness, kg/mm2,GPa HBK Berkovich hardness number, kg/mm2,GPa HK Knoop hardness number, kg/mm2,GPa HM Meyer hardness number, kg/mm2,GPa HRC Rockwell C hardness
load, the indentation hardness test can be divided into macro (also called gen-eral or universal) and micro hardness testing. For macro hardness testing, the test loads are 1 kgf (9.81 N) or larger, while micro hardness testing covers the load range from 1 gf to 1 kgf. The required surface condition depends on the type of test and load used. For
This standard covers hardness conversions for metals and the relationship among Brinell hardness, Vick-ers hardness, Rockwell hardness, Superficial hardness, Knoop hardness, Scleroscope hardness and Leeb hardness. ASTM E10 (Brinell) This standard covers the Brinell test method as used by stationary, typically bench-top machines. This
Macro Dilemmas in Political Science, Heinz Eulau explains, "The fancy terms 'micro' and 'macro' have come to mean large and small or individual and aggregate or part and whole. . . . Once micro and macro had been attached to persons or groups . . . [i]t was only a small step to insist on 'bridg-ing' the micro-macro gap."9 This .
Hardness Testing Micro-Vickers Hardness HM-Series Page 565 Rockwell, Rockwell Superficial, Brinell Page 569 Portable Hardness Tester Page 574 Hardness Test Blocks Page 577 564 The prices listed are suggested retail prices (valid until 31st May 2015). All products to be sold to commercial customers. Therefore VAT is not included.
A Leeb’s Hardness Tester measures the hardness of sample material in terms of Hardness Leeb (HL), which can be converted into other Hardness units (Rockwell B and C, Vicker, Brinell and Shore D). 1.3. Notation of Leeb’s Hardness When measuring the hardness of a sample materi
1. Define Hardness. 2. Applications of Rockwell Hardness A Scale, B-Scale, C-Scale. 3. Type of Indentor used in the Three Different Scales of Rockwell Hardness Test. 4. Different Types of Hardness Testing Methods. 5. Size of the Ball to be used in Ball Indentor of Rockwell Hardness Test. 6. Di ameters of the different Balls used in Brinell Hardness Test.
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