Nanoindentation. M. R. VanLandingham, Review of instrumented indentation, J. Res. Natl. Inst. Stand. Technol. 108, (2003).

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1 Nanoindentation References Nanoindentation, nd Ed., Antony C. Fiscer-Cripps, Springer, 010. Introduction to Contact Mecanics, nd Ed., Antony C. Fiscer-Cripps, Springer, 007. Contact Mecanics, Kennet L. Jonson,,1985 M. R. VanLandingam, Review of instrumented indentation, J. Res. Natl. Inst. Stand. Tecnol. 108, (003). W. C. Oliver, G. M. Parr, Measurement of ardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to metodology, J. Mater. Res., 19(1) (004) PY500 Nanoscience 59

2 Nanoindentation Touc te surface of a material to determine mecanical properties Origins in Mos ardness scale of 18 based on scratcing, numbered 1-10 Followed by Brinell, Rockwell, Vickers and Knoop indentation Distinguising features of nanoindentation: Displacement is continuously measured wit resolution in nanometers Load is continuously measured in nanonewton to millinewtons Contact area is inferred from known geometry and properties of indenting tip Small amplitude oscillatory response is continuously measured PY500 Nanoscience 60

3 Nanoindentation Properties measured: Hardness Elastic modulus (..Poisson s ratio, compliance tensor,..) Strain ardening exponent Fracture tougness (brittle) Viscoelastic properties (Storage and Loss Modulus, complex viscosity, ) Adesion properties (interfacial fracture tougness, ) Isolated micro/nanoscopic defect and defect population beaviour Coupled parameters: Raman signal, electrical, etc PY500 Nanoscience 61

4 Instrumentation One dimensional simple armonic oscillator PY500 Nanoscience 6

5 Basic Measurement Berkovic 3-sided pyramid Warning on notation differences from contact mecanics conventions: For load (ie. force) nanoindentation sometimes uses L instead of P For displacement, nanoindentation usually uses instead of d PY500 Nanoscience 63

6 Elastic-plastic Deformation Te elastic limit of deformation of most materials is a few percent, ten Hooke s Law fails Plasticity deals wit beaviour beyond tis limit. Rigid plastic Elastic plastic Elastic strain ardening plastic PY500 Nanoscience 64

7 Yield Criterion Transition from elastic to plastic beaviour in ductile materials is called te yield point. Under te simple tension test, tis occurs at te yield stress 0 Te pysical conditions for yield in metals are: Hydrostatic pressure does not cause yield in solids (empirical result) For an isotropic material, te yield condition must be independent of coordinate system Yield conditions are based on invariants of te stress deviator PY500 Nanoscience 65

8 Stress tensor decomposition Define te mean normal stress Te stress tensor can be decomposed into a sum of a mean and deviation xx xy xz m 0 0 xx m xy xz ij xy yy yz 0 m 0 xy yy m yz xz yz zz 0 0 m xz yz zz m Total (I) m 3 Hydrostatic (H) xx yy zz Deviatoric (J) A pysical interpretation of tis decomposition can be made wen a body is subject to an arbitrary stress: Te mean or ydrostatic component is associated wit volume cange ie. Dilation or compression wit sape self-similarity Te deviatoric component is associated wit sape cange wile preserving volume By definition, a fluid is a body tat, at rest, cannot sustain a deviatoric component of stress (tis appens at long times) PY500 Nanoscience 66

9 Decomposition of stress PY500 Nanoscience 67

10 Principle stresses and von Mises stress In 3D, te following comprise te stress invariants. Ie. Quantities invariant under coordinate transformation. I1 xx yy zz I xx yy yy zz zz xx xy yz zx I 3 xx yy zz xx yz yy zx zz xy xy yz zx Te principle stresses in 3D are I1 1 I1 3I cos 3 3 I1 I1 3I cos I1 4 3 I1 3I cos I 9I I 7I arccos I 3 1 I von Mises Stress: Te condition for plastic yield based on strain energy density ij See Barber E I 3 I ( ) ( ) ( ) PY500 Nanoscience 68

11 Von Mises (Distortion Energy) Criterion Te Von Mises condition states tat yield occurs wen te second invariant of te stress deviator J reaces a critical value k (eg. see Dieter p. 76): J k were To relate tis pysical condition to te stress in a simple tension test consisting of uniaxial tension (assume we are aligned to principal directions for simplicity): 1 0; 3 0 6k 1 J ( ) ( ) ( ) k Te uniaxial yield stress or yield stress in tension For an arbitrary stress, in terms of 1 tis uniaxial yield stress we ave: ( ) ( ) ( ) Or for an arbitrary coordinate system tis becomes: ( xx yy ) ( yy zz ) ( zz xx ) 6( xy yz zx) PY500 Nanoscience 69

12 Distortion Strain Energy Elastic strain energy by work-energy teorem U For an elemental cube under tension 1 1 U Pdu t ( xx A )( exxdx ) 1 P t t (use simple case of P~) Peak load, displace. 1 1 ( xxexx )( Adx) ( xxexx ) dv Elastic strain energy density (per unit volume) is 1 1 xx 1 U0 ( xxexx ) exxe E (NB. Poisson effect does not appear as no extra lateral force applied) Applying Hooke s law In general: U 0 U 0 1 ( ijeij ) For sear: U0 ( e ) e xy xy xy xy 1 1 E E ( xx yy zz ) ( xx yy yy zz zz xx ) ( xy yz zx) PY500 Nanoscience 70

13 In terms of principle stresses Distortion Strain Energy U 1 ( ) ( ) E U0 I1 I(1 ) E In terms of (total) stress invariants or I 1 U ( I 3 I ) Kb 6 ydrostatic deviatoric E Kb 3 3(1 ) Bulk modulus, Slide U 0 ( I1 3 I) distortion 1 ( ) ( ) ( ) 1 Example: Uniaxial tension 1 1 0; 3 0 U 0 distortion ( 1 ) ( 3) ( 3 1) 3 J PY500 Nanoscience 71 Ie. same as te von Mises criterion

14 Hardness For indentation, te original motivation was to measure plastic properties troug a parameter called ardness Hardness is te resistance to penetration by te indenter tat leaves permanent deformation. For ardness defined as P A proj It is found (Tabor) tat H C 0 Te constant C is te constraint factor and as a value of: ~3 for ig E 0 ratio materials eg. metals ~1.5 for low E 0 ratio materials eg. polymers PY500 Nanoscience 7

15 Hertz (sperical) inelastic contact Maximum sear stress in te contact stress field of two solids of revolution occurs below te surface on te axis of symmetry r,, z Are principle stresses on tis axis r On te z-axis, so critical principle stress difference for von Mises stress equation is z r Slide 56 along z-axis: r z 1 a 1 (1 ) 1 tan z 1 p0 p 0 a z (1 z a ) and p (1 z a ) z r were p 0 0 4a (1 ) R Is max value of 0.6 p at 0.48a 0 ( 0.3) Solid lines: Hertz contact p.8k Von Mises P yield (Slide 70) yield p R (1.6) R 1 (1 ) 1 (1 ) yield PY500 Nanoscience 74

16 Sarp Indentation Probes Generally nanoindentation is performed wit sarp pyramidal indenters a d Ratio of contact radius wit dept of penetration is constant giving geometric similarity Scale invariant deformation wit constant strain Value of strain is about 8% for Vickers and Berkovic tips Singular geometry at te tip is important practical matter PY500 Nanoscience 75

17 Instrumented Indentation Bulycev, Alekin, Sorsorov (1970 s), Oliver, Hutcings and Petica (1983) Tabor, essential pysics of unloading of elastic-plastic materials after indentation: 1. Plastic deformation is not recovered: All is elastic. Under conical indentation, residual impression elastically recovers dept, but diameter remains uncanged. It was sown tat for any smoot body of revolution, te initial unload slope is given dp E d * A proj Aproj 1 Projected area of contact * E E1 E Determine A proj wit a sape function for te tip for some dept: eg. or r E was cosen based off of TEM replication data as best coice * 1 A proj dp d r H t Pt A proj PY500 Nanoscience 76 P P t A r t C e B NB: d

18 Elastic Flat Punc Unloading Cylindrical flat punc elastic solution for rigid punc: 1 d E P v P * ae a Slide * E E dp d a ae E E * * * A proj dp d E * A True for all axially symmetric indenters (Oliver-Parr) Also from diagram dp d Pt e PY500 Nanoscience 77

19 Sarp Punc: Doerner & Nix Stage 1 Doener and Nix (1986) key observation: Initial slope of unloading curve for Berkovic indenter is linear Due to combination of indenter sape and deformation geometry Stage Key abstraction: Can treat as unloading of cylindrical flat punc (initially) Stage 3 dp d E * A proj PY500 Nanoscience 79

20 a Doerner Nix r P P t B r fp e fp r e a p Measure: Peak displacement t, unload displacement r Consider te imaginary flat punc of radius a Te linear unload slope implies unloading of te flat punc to te surface and a e fp p r fp A r p r fp is value to be plugged into tip sape function. t C e D a e fp Berk Aproj 4.5 * 1 Eg. p E 4.5 p dp d H p P t PY500 Nanoscience 80

21 Oliver and Parr Sneddon: Load on axi-symmetric indenter of revolution indented into elastic alf space: m P Is elastic displacement in surface,m Are constants based on materials and geometry, Eg. m=1 flat cyclinder, m= cones, m=1.5 speres/paraboloids P P t B Unloading of elastic-plastic materials: Under indentation residual impression recovers only dept, not diameter dp d E * A proj A r p C rp e D a ep t PY500 Nanoscience 81

22 Oliver and Parr To extract elastic modulus: * S E A proj S dp d f At any point: Measure at te peak: c s ( Pt, t, St ) t t - f Caracterize te projected contact area via te tip sape function at te peak: A proj F( ) c were c t s So must determine te surface deflection s at te contact perimeter.. Problem of elastic contact PY500 Nanoscience 8

23 Oliver and Parr Tis depends on indenter geometry. For cone (Sneddon): s ( t f ) From sape of elastic surface outside of contact region u ( r a) z just Te quantity ( is ) is te elastic component of te displacement t f From te elastic load vs. displacement relation for a cone indenter (Sneddon) P ( ) t t f S t s Pt S t ( ) Conical indenter PY500 Nanoscience 83

24 An atomic limit for nanoindentation How do we extend deformation to te smallest of lengt scales? Do te deformation mecanics scale? Wat does an atomic scale indentation crater look like? Is it stable? PY500 Nanoscience 84

25 FIM Field Evaporation Indenter (tip) engineering by field evaporation Imaging : Voltage 4.5 kv Field ion microscope Single Atom Tip Field Evaporation : Voltage 5. kv W He PY500 Nanoscience 85

26 Uses of a field evaporation point probe Mesoscopic and atomic level control: Surface forces STM/AFM tip definition Coerent LE e- source Ultra-brigt M ~ d / R tip Nanoindentation: geometric BC Nanoforming: limits Low energy e- olograpy Additive: W, Na, Cr, Ni, etc. Contacts in nanoelectronics X-ray flas radiograpy PY500 Nanoscience 86

27 Size and symmetry limits on indentation Wat are te size limits to fabrication? Do te laws of plasticity work at te nanoscale? Cu 001 surface 50 nm Carrasco et. al., Pys. Rev. B 68 (003) Minimum size Surprising sapes PY500 Nanoscience 87

28 Nanoindentation defect processes PY500 Nanoscience 88

29 Nanoindentation defect processes PY500 Nanoscience 90

30 More information See: G.L.W. Cross, J. Pys. D., 39 p. R1-R4 (006). G. L. W. Cross, et. al., Nature Materials, 5 p (006). Rowland, H. et. al., Science, 30 p (008) PY500 Nanoscience 91