Modeling Nanoindentation using the Material Point Method · 2018. 5. 4. · Modeling Nanoindentation using the Material Point Method Chad C. Hammerquist and John A. Nairn Wood Science

Modeling Nanoindentation using the Material Point Method

Chad C. Hammerquist and John A. Nairn∗

Wood Science and Engineering, Oregon State University,Corvallis, OR 97330, USA

(Received 3 January 2018; accepted 16 March 2018)

Abstract

A numerical nanoindentation model was developed using the Material Point Method, which was chosen because itcan handle both large deformations and dynamic contact under the indenter. Because of the importance of contact,prior MPM contact methods were enhanced to improve their accuracy for contact detection. Axisymmetric and full 3Dsimulations investigated the effects of hardening, strain-rate dependent yield properties, and local structure under theindenter. Convergence of load-displacement curves required small cells under the indenter. To reduce computationtime, we used an effective non-regular grid, called a tartan grid and describe its implementation. Tartan grids reducedsimulation times by an order of magnitude. A series of simulated load-displacement curves were analyzed as “virtualexperiments” by standard Oliver-Pharr methods to extract effective modulus and hardness of the indented material.We found that standard analysis methods give results that are affected by hardening parameters and strain-ratedependence of plasticity. Because these parameters are not known during experiments, extracted properties willalways have limited accuracy. We describe an approach for extracting more properties and more accurate propertiesby combining MPM simulations with inverse methods to fit simulation results to entire load-displacement curves.

Keywords: nano-indentation, simulation, hardness

I. Introduction

Humans have been using indentation to test materialproperties since the first person poked a stick into softground to see if it was firm enough to walk on. Moremodern techniques are described by Oliver and Pharr[1]who developed a method for analyzing microscale in-dentation experiments using the maximum indentationload, maximum indentation depth, and initial unload-ing stiffness. This method for analyzing nanoindenta-tion, which is generally referred to as the “Oliver-Pharrmethod,” extracts material properties from nanoinden-tation, load-displacement curves. Since then, muchwork has been done in analyzing, numerically modeling,and developing new experimental techniques. Most nu-merical modeling has used finite element analysis (FEA)[2–4]. This paper describes a new simulation methodfor modeling nanoindentation using the particle-based,Material Point Method (MPM).

MPM has been used for modeling nanoindentation ex-periments [5] and for modeling coupled with MolecularDynamics [6]. This paper describes new axisymmetricand 3D MPM simulations of nanoindentation that addedthree improvements to increase accuracy and efficiency

∗Corresponding author: [email protected]

of prior MPM simulations [5]. First, accurate modelingof nanoindentation requires that contact between theindenter and the material is well modeled. We describean improvement to the standard MPM contact algo-rithms [7–9] that more accurately detects contact basedon displacements of the two material surfaces. Second,converged nanoindentation results requires small cellsunder the indenter. We describe a mesh refinementscheme, called a “tartan” grid, that allows for refinedmesh under the indenter and larger cells elsewhere, butmaintains orthogonality of standard MPM grids. The re-tained orthogonality greatly simplifies implementationof tartan grids. Use of tartan grids significantly reducedcomputational time for axisymmetric simulations andmade converged, 3D simulations feasible. Third, allMPM simulations used dynamic code with explicit time-stepping. Several techniques were used to suppressdynamic effects and noise.

The new simulation methods led to output of load-displacement curves that faithfully represented quasi-static nanoindentation experiments. Each curve couldbe viewed as a “virtual experiment” on a materialwith precisely known material properties (e.g. rate-independent, non-linear elastic properties with variousnon-linear J2 plasticity properties). We subjected a

1

C. C. Hammerquist et al.: Modeling nanoindentation using the Material Point Method

series of such “virtual experiments” to standard Oliver-Pharr analysis methods. The results show that suchmethods are reasonable, but have limited accuracy forextraction of effective modulus or hardness. The prob-lem is that extracted results depend on plasticity prop-erties. Thus, when experiments are done on any un-known material, compromises have to be introducedinto analysis methods. A potential method to avoidsuch compromises and to measure both more materialproperties and more-accurate material properties is tocouple MPM simulations with inverse methods. By fit-ting entire load-displacement curves from experimentsto simulated load-displacement curves, in some cases itmay be possible to extract both modulus and plasticityproperties.

II. Nanoindentation Simulation Methods

MPM is a numerical method for solving continuum me-chanics problems and can handle complex geometries,large deformations, history-dependent materials, mul-tiple interfacing materials, and extreme loading con-ditions [10, 11]. It is a hybrid Eulerian/Lagrangianmethod that avoids some disadvantages of each whileretaining their advantages [12]. The continuum is dis-cretized into a set of Lagrangian material points or par-ticles that store their complete deformation history. Thematerial points interact with each other by interpolatinginformation to a background grid where the equationsof motions are solved. The use of particles allows forMPM to easily handle complex geometries by avoidingthe need for difficult meshing algorithms. This featurehas been demonstrated by modeling the cellular struc-ture of wood [13–15], complex biological structures[16, 17], and polymer foams [18]. Other MPM applica-tions include modeling of complex materials [19, 20],extreme loading conditions [21, 22], fracture [23, 24],cutting [25, 26], and fluid-structure interactions [27].MPM is recommended for nanoindentation simulationsbecause of its ability to handle the large deformations[11] expected under the indenter tip and because ofits ability to dynamically model contact [7–9]. MPMcontact methods were refined further in this work. Allsimulations used the MPM code OSParticulas (which isdevelopment version of the public domain NairnMPMcode [28]).

Figure 1: Axisymmetric MPM simulation of nanoidentationwith colors (or shades of gray) showing radial displacement. A.The full modeled specimen from an axisymmetric simulation. B.Zoomed-in view of absolute value of transverse displacementsfrom a 3D simulation using a Berkovich indenter.

A. Geometry

The geometry for 2D, axisymmetric simulations [29] ofnanoindentation is shown in Fig. 1A. In axisymmetricsimulations, the indenter has a conical tip, which differsfrom most experiments that use a pyramidal Berkovichindenter. Axisymmetric simulations, however, allowmuch faster simulations and thereby more testing ofsimulation variables. Much nanoindentation analysissimilarly relies on axisymmetry for tractability [30, 31].To maximize similarity between conical indenter and aBerkovich pyramid, the angle of the cone was chosensuch that the projected contact area as a function ofindentation depth is the same for both shapes. For aBerkovich indenter with a tip angle of 65.3, the equiv-alent cone has a tip half-angle of θ = 70.3 [1, 30].

The axisymmetric block used for indentation was a

2 J. Mater. Res., Vol. 0, No. 0, 2018


rectangular block with symmetry conditions along ther = 0 plane, zero velocity grid boundary conditionson the bottom, and free surfaces on the top and rightedges. The indenter was modeled with rigid materialpoints that were slanted to accurately represent thecone angle and the indenter surface. The goal of mostsimulations was to simulate nanoindentation on a largebulk object. To achieve this goal, the size of the blockwas varied until it no longer influenced the simulations.For a nanoindentation depth of 1.5 µm, which waschosen to reach indentation load of 5 mN for simulatedmaterials, the block radius and depth had to be 0.05 mmor larger. Figure 1A shows radial displacements duringan axisymmetric simulation of a sufficiently-large block.

Although most simulations were axisymmetric, wealso ran 3D simulations with a Berkovich indenter toverify similarity to axisymmetric simulations. The 3Dsimulations used a block of material ±0.05 mm in trans-verse directions and a depth of 0.05 mm. The simula-tions modeled half the block by cutting at its mid-planeand applying symmetry boundary conditions. The topand side surfaces were stress free; the bottom surfacehad zero velocity boundary conditions. Figure 1B showsa zoomed-in view of absolute value of transverse dis-placements during a 3D simulation (absolute value wasplotted for better comparison to axisymmetric results).

B. Material Modeling

The indenter was modeled as a rigid material. All in-denter particles were pushed into the material at a pre-scribed velocity. After reaching a desired maximum in-dentation load, the indenter particles reversed directionand returned to their initial positions at a prescribedreversing velocity. For simulations with strain-rate de-pendent yield properties, the indenter load was heldconstant for selected periods of time before reversingthe particles. The rigid particle interacted with the bulkmaterial by contact mechanics. A rigid indenter wasused to allow simulations with larger time steps and toprovide accurate, non-deforming contacting surfaces.

The material being indented was modeled as a neo-Hookean, hyperelastic-plastic material using J2 plastic-ity with nonlinear and rate-dependent hardening laws.In the elastic region, this material uses the Mooney-Rivlin [32] strain energy function with G1 = G and

G2 = 0 or:

W (F, J) =G2

Tr(FFT )

J2/3−3

+K2

12(J2−1)− ln J

(1)

where F is the deformation gradient, J = det(F) isthe relative volume change, G = E/(2(1+ ν)), K =E/(3(1− 2ν)), and E are the low strain shear, bulk,and tensile moduli, respectively, and ν is low-strainPoisson’s ratio. The Cauchy stress for this material is:

σ =K2

J −1J

I+G

J5/3dev(FFT ) (2)

The non-linear hardening law used in all simulationswas

σy = (σy0 + kh||εp||nh)

1+ C ln

˙||εp||

ε0p

!!

(3)

whereσy0 is initial yield stress, kh and nh are hardeningparameters, εp is plastic strain, and C and ε0

p describestrain rate dependence of the yield stress. When C = 0,this law is rate-independent with power-law harden-ing. Nanoindentation experiments commonly hold ata constant maximum load before unloading. If mod-eled with a rate independent material, this hold periodwould not be evident in the load-displacement curves.Experiments show, however, that indenter displacementincreases while the load is held constant [33–35], re-sulting in a flat spot on top of load-displacement curves.While this flat spot or “creep” could be thermal drift,most of it is material dependent and can be describedby a logarithmic model [33]. When C 6= 0, the loga-rithmic term in Eq. (3) allowed us to investigate strainrate effects. This term came from the Johnson-Cook[36] hardening law (the temperature-dependent termin Johnson-Cook law [36] was omitted because all sim-ulations were isothermal).

Contact between the rigid indenter and the plasticmaterial was modeled using the contact methods de-scribed in Ref. [8]. Because contact is crucial to nanoin-dentation simulations, the contact methods used hereincluded a new option to improve detection of contact.Contact in MPM is modeled by extrapolating plastic ma-terial a and rigid material b to separate velocity fieldson the grid [7, 8]. Any node that “sees” both materials

J. Mater. Res., Vol. 0, No. 0, 2018 3


models contact mechanics in these simulations by thefollowing three steps:

1. Find Contact Normal: The first step is to find a normalvector from material a to b using

n i ||n i ||=

−g i,b ||g i,b|| ≥||g i,a ||

100Ωi,ag i,a −ΩI ,bg i,b otherwise

(4)

whereΩi, j is “domain” of material j and g i, j is “domain”gradient of material j found by extrapolating domainsof particles to node i on the grid using gradient shapefunctions [8]. For 3D simulations, the “domain” is theparticle volume, but for axisymmetric simulations, the“domain” is the area of the material point in the r-zplane and not its volume [8]. The first option is usingthe rigid material volume gradient (RMVG) option inRef. [8]. The second option switches to the averagevolume gradient (AVG) option in Ref. [8] for nodes inwhich the domain gradient of the rigid material is verysmall [37]. Axisymmetric simulations made use of sym-metry to set normal vector for nodes along the r = 0symmetry plane to be n i = (0,1). Similarly, nodes onthe 3D mid-plane of symmetry enforced normal vectorsto have zero component perpendicular to that symmetryplane.

2. Detect Contact: The second step is to determine if thetwo materials are in contact. A necessary condition isthat the materials are approaching each other or that(v i,b − v i,a) · n i < 0, but this condition is not sufficient.Contact detection is improved by also requiring thatedges of material domains are in contact or that di,b −di,a ≤ 0 where di, j is the distance along the normalvector from the edge of material j to node i [8]. Thecalculation of di, j is done by extrapolating materialpoint positions to the grid. For a material j, we canfind an “apparent” distance from extrapolated particleposition (x i, j [8]) to node i (at x i) using:

d(ex t)i, j = (x i, j − x i) · n i (5)

but, this distance will not equal the desired edge dis-tance di, j . An accurate use of edge positions in contact

detection requires definition of a function di, j(d(ex t)i, j )

to calculate actual edge distance from extrapolated “ap-parent” distance. Reference [8] proposed a simpleconstant correction that is equivalent to linear mapping

functions:

di, j(d(ex t)i, j ) =

(

d(ex t)i,a + 0.4∆x j = a

d(ex t)i,b −0.4∆x j = b

(6)

For these nanoindentation simulations, we instead im-plemented non-linear mapping functions given by:

di, j(d(ex t)i, j )

∆x=

1−2

−d(ex t)i,a

1.25∆x

0.58

j = a

−1+ 2

d(ex t)i,b

1.25∆x

0.58

j = b

(7)

These functions were determined by explicit calcula-

tions of d(ex t)i, j as a function of di, j for either convected

particle domain integration shape functions (CPDI in-troduced in [38]) or undeformed generalized interpola-tion material point shape functions (uGIMP, introducedin [39]), inverting the results, and fitting to a powerlaw. The calculation process and plots of calculationsare given in the supplemental material.

3. Add Contact Forces: For any multimaterial node de-tected to be in contact, the last step in rigid materialcontact is to apply contact force to material a by chang-ing its extrapolated momentum. For contact with rigidmaterials, this task first finds momentum change re-quired for material a to “stick” to the rigid material as∆p i,a = mi,a(v i,b − v i,a) where mi,a is extrapolatedmass of material a. This momentum change impliesnormal and tangential contact forces to “stick” to therigid material of

f n =(∆p i,a · n i)

∆tn i and f t =

(∆p i,a · t i)

∆tt i (8)

where∆t is time step and t i is unit vector in the tangen-tial direction of motion. These simulations used contactwith Coulomb friction. For this contact law, the finalmomentum change applied to material a is:

∆p′i,a =

∆p i,a ft < −µ fn(∆p i,a · n i)(n i −µt i) otherwise

(9)

whereµ is the coefficient of friction [7, 8]. The first formis “stick” conditions while the second is for frictional“slip.” The final change in momentum implies a contactforce on node i of f i,c = ∆p′i,a/∆t. We summed allnodal contact forces to track the indenter load on thenon-rigid material.

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All simulations used explicit MPM methods, whichmeans the time step must follow ∆t <∆x/w where wis wave speed of the material. For modeling to be possi-ble in a reasonable time frame, we used loading ratesmuch higher than in typical experiments and relied ontwo techniques to mitigate or eliminate dynamic effects.First, the prescribed indenter velocity was chosen suchthat indenter displacement as a function of time wassigmoidal in both the loading and unloading phases.This approach helped reduce stress waves induced byimpulses associated with constant-velocity loading andits abrupt shift to constant-velocity unloading. Second,we used a new MPM velocity update scheme calledXPIC(2) [40]. This update scheme suppresses noiseand high frequency velocity waves not appropriate inquasi-static simulations without causing over-damping.Figure 2A gives a typical simulated load-displacementcurve by cross-plotting simulated load and applied sig-moidal displacement. The sigmoidal speed varied from0 to 1.3 m/sec, which for the simulated materials wasless than 0.1% of its wave speed. Loading faster in-troduced dynamic effects while loading slower gaveidentical results. We thus used the maximum possiblespeed such that sigmoidal loading and XPIC(2) updateswere sufficient to provide results equivalent to noise-free, quasi-static loading.

C. MPM Grid

Simulation output provides a virtual nano-indentationexperiment for indenter load (from contact forces) ver-sus displacement (from position of rigid indenter parti-cles). We varied cell sizes to determine necessary spatialresolution for convergence. We found, not surprisingly,that necessary grid cell size scales with depth of inden-tation, i.e., deeper indentation can use lower resolution.For the range of depths studied, we found that 100 nmcells, which corresponds to 50 nm particles when usingtwo particles per cell in each dimension, gave convergedresults.

Most MPM modeling uses a regular grid of equally-sized cells. Regular grids are used because of simplicityafforded to some calculations and because they areneeded in problems with large movement through thegrid [19, 21, 26, 41]. The literature on mesh refine-ment in MPM is sparse. A few examples can be found inRefs. [27, 42, 43]. The nano-indentation problem mod-eled here is an excellent candidate for mesh refinement.

Displacment (µm)

Load

(mN

)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0

1

2

3

4

5

6

Unloading

(hmax, Pmax)

hf

Loading

A

Displacement (µm)

Load

(mN

)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0

1

2

3

4

5

6

0.5 µs

hold time 0 µs

1 µs2 µs

3 µs

B

Figure 2: A. Cross plot of load output vs. displacement inputfor typical simulated indenter load as a function of displacementfor both loading and unloading. B. Simulated load-displacementcurves for a strain-rate dependent material with C jc = 0.1 fordifferent hold times.

We know that throughout the simulation, we need arefined mesh only under the indenter tip. All regions re-mote from the indenter can be modeled with sufficientaccuracy by using larger grid cells and material points.

For nano-indentation simulations, we used a gridscheme termed a “tartan” grid. In a tartan grid, oneor more “regions of interest” are modeled with a high-resolution, regular grid with equally-sized elements. Forthe nano-indentation problem, the one region of interestunder the indenter was modeled with 100 nm cellsdetermined above as needed for convergence. Outsideregions of interest, the grid cell sizes were allowed toincrease, thus forming a tartan-like pattern.

A tartan grid maintains orthogonal grid lines, whichgreatly simplifies its implementation compared to arbi-trary background grid cell sizes. The changes neededfor a tartan grid depend on the MPM shape functionsbeing used and are easily implemented for CPDI shapefunctions [38]. In brief, CPDI shape functions for bothCartesian grids [38] and axisymmetric grids [29] de-pend only on current size of the particle and elementshape functions evaluated at the corners of current par-

J. Mater. Res., Vol. 0, No. 0, 2018 5


ticle domain. Thus any current CPDI code automati-cally supports a tartan grid provided the shape functioncalculations recognize variable particle sizes and vari-able element sizes when finding the shape functionsat the corners. Classic MPM shape functions [44] simi-larly support tartan grids with little modification, butsuch shape functions are never recommended for ac-curate simulations. The changes required for uGIMPshape functions are more substantial. Implementationof uGIMP shape functions is typically done by analyti-cal solution to the GIMP integral as function of particleposition in a cell and that integral may involve inte-gration over neighboring cells [39]. For a regular grid,the analytical functions are the same for all particles.Implementation of a tartan grid would not be difficult,but would require re-evaluation of all integrals allowingfor variable size particles and variations in sizes of cellssurrounding each node.

All simulations here used tartan grid and CPDI shapefunctions. The grid spacing outside the region of in-terest was increased with a linear ratio: ∆x t = nR∆x ,where n is the number of cells away from the region ofinterest, R = 2 is the size ratio, and∆x = 100 nm is thecell size in the region of interest. For 2D, axisymmetricsimulations, we verified that tartan grid results gavevirtually identical results as a regular grid with equal-sized cells. By using a tartan grid, however, we couldreduce the number of particles by an order of magni-tude, which also reduced simulation time by an orderof magnitude. Full 3D simulations using our resources(Dell servers with 32, 3 GHz, Xeon processors) and tar-tan grids took 10+ hours. 3D grids without tartan gridsin a reasonable time would require significantly moreprocessors. The supplemental material illustrates thetartan grids used along with some additional details.

Besides shape function calculations, tartan grids alsorequire modification of any contact calculations thatdepend on grid size [9]. For example the calculationof material separation described above depends on celllength or ∆x . For regular grid with square (2D) orcubic (3D) cells, grid dimensions are the same in all di-rections and ∆x is the appropriate factor. For a regulargrid with equally-sized rectangular (2D) or rectangularcuboid (3D) cells, ∆x needs to change to h⊥ which isan effective cell-size normal to the contacting plane [8].For a tartan grid, ∆x (and some other contact calcula-tions), must be changed further to account for sizes ofall cells surrounding node i. For these nano-indentation

simulations, all contacting nodes were located in theregion of interest that had equally-sized elements. Thecontact calculation methods explained above for a regu-lar mesh could therefore be used without modification.The changes for general contact in a tartan grid will bein a future publication.

III. Results and Discussion

Nanoindentation simulations described above predictthe load-displacement curve for nanoindentation exper-iments on any material with known constitutive lawin any specimen with known geometry (typical curvein Fig. 2A). The constitutive law used here was forneo-Hookean, hyperelastic-plastic materials, but thesimulation method could be used with other materialmodels such as models including thermal effects, rate-dependence, and brittle failure processes. If the ma-terial and indenter have comparable properties, thesimulations could explicitly model the indenter ratherthan assume a rigid indenter. One application of suchsimulations would be to vary material properties untilsimulations and experiments agree. Although this useof simulations has potential future uses (especially fornanoindentation on materials with complex morphol-ogy such as wood cells walls [34]), another applicationof simulations is to validate conventional methods for in-terpreting nanoindentation experiments. In brief, thesesimulations can be treated as accurate and low-noisevirtual experiments on a material with precisely knownproperties and on various geometries of the indented ob-ject. In this section, we subject such virtual experimentsto various analysis methods to see if those methodscorrectly extract the input material properties.

A. Analysis of Load-Displacement Curves

The Oliver-Pharr method to interpret nanoindentationexperiments is based on nanoindentation of an isotropicmaterial in an infinite half space. We can apply theirmethods to results from simulations on an isotropicbulk with length and width large enough to representan infinite half space. In the Oliver-Pharr method, theunloading portion of the curve is modeled with a powerlaw:

P(h) = α(h−h f )m (10)

where P is the contact force, h is the indenter depth,h f is the final depth (i.e., the depth at which the load

6 J. Mater. Res., Vol. 0, No. 0, 2018


returns to zero), and α and m are fitting parameters.The measured modulus of the material is deduced fromthe initial slope of the unloading curve at h = hmax(denoted as S = dP(hmax )/dh = αm(hmax −h f )

m−1)and the maximum projected area of contact, Ac , be-tween the indenter and the material using:

E1−ν2

=1β

pπ

2Sp

Ac(11)

where β is an empirical parameter used to account forall physical processes that might affect S (e.g., indentershape or properties of the indented material) [30]. Theβ parameter was chosen in early work to be unity, but isnow recognized as larger. Oliver and Pharr [30] recom-mend β = 1.05 as good choice for a range of materials.The material’s hardness can also be determined fromthe maximum load and contact area:

H =Pmax

Ac(12)

Calculation of E or H relies on measurement of Ac .For the axisymmetric cone used here, a geometric cal-culation of projected contact area gives:

Ac(h) = πh2 tan2(70.3) (13)

(note cone angle 70.3 was chosen to give identicalA(h) function as Berkovich indenter with angle 65.3).Unfortunately, one cannot use hmax in this equationbecause the measured displacement combines both pen-etration into the material and “sink-in” deflection ofthe surface away from the indenter [30]. The standardanalysis method accounts for “sink-in” by correctinghmax to give an actual contact depth, hc , using:

hc = hmax − εPmax

S=

(m− ε)hmax + εh f

m(14)

and then finding contract area from Ac(hc) [30]. Theparameter ε depends on material properties and ge-ometry of the indented material. Calculations for anisotropic, elastic material in an infinite half space showthat ε depends on m from the unloading curve andvaries from 0.74 to 0.79 for typical values. For sim-plicity, it was suggested that ε= 0.75 provides a goodoverall value, although a function to find ε from mfor isotropic, elastic, infinite, half space is an availableoption [30].

Our baseline simulation used E = 2.0 GPa, ν= 0.3,ρ = 1.2 g/cm3, σy0 = 30 MPa, kh = 100 MPa,

nh = 0.5, and C = 0. We first interpreted the virtualload-displacement curves using the Oliver-Pharr simpli-fication that β = 1.05, but different options for findingε. References [1] and [45] suggest that power-law fitsto only the top 33% (or even top 10%) of unloadingcurves are sufficient for determining unloading param-eters. To test this suggestion, the power law in Eq. (10)was fit to various fractions of the unloading curves us-ing R [46]. Initial parameters were obtained by fittinga linear model to the log-transformed data. These pa-rameter estimates were then refined to fit the curvesby nonlinear least squares using R’s built-in functionoptim(). All power law fits to our simulated curves wereessentially perfect with R2 values always greater than0.99999. Nevertheless, due to nature of power laws,the fitting parameters depended on fraction fit — mvaried from 1.05 to 1.32; α varied from 33.5×103 to1460×103 N/mm; h f varied from 1.08 to 1.16 µm.

Figure 3 plots percent error in effective modu-lus (Ee f f = E/(1− ν2) compared to input Ee f f =2.198 GPa) extracted from simulated curves by threedifferent methods for finding ε. First, notice that resultsvary widely when fraction fit is less than 60%. Becausethis region is fitting a narrow range in h, the power lawresults are not reliable. The results for 60% and higherare reliable and consistent; the variations below 60%are not meaningful. The solid curve (square symbols)assumed ε= 0.75. The extracted modulus for 60% ormore fit is about 5% too high and the error increasedslightly as more of the unloading curve was fit. Thedashed line (diamond symbols) found ε from relationin Ref. [30]. For the range of m seen in this data, thecalculated ε ranged from 0.77 to 0.86, but increasing εcaused larger errors compared to actual modulus.

Rather then find ε from a relation that was derivedfor an elastic material [30], the last method was to“measure” Ac from simulation output; i.e., find ε for anplastic material by numerical methods. All particles incontact with the indenter were in compression (in thedepth direction), while the first surface particle at theedge of contact was in tension. The radial position ofthis tensile particle equals the radius of the projectedcontact area or hc tanθ (±25 nm or half a particle size).From this simulated hc and fits to unloading curves,the simulation results corresponded to ε ranging from0.711 to 0.736 or always lower than 0.75. Using thesenumerically determined ε’s, the extracted modulus isslightly closer to the correct result.

J. Mater. Res., Vol. 0, No. 0, 2018 7


Percent Unloading Fit

Inte

rpre

ted

Mod

ulus

Erro

r (%

)

0 10 20 30 40 50 60 70 80 90 100 0

1

2

3

4

5

6

7

8

β = 1.05

ε = 0.75

ε from simulation

ε = f(m)

Figure 3: The percent error in effective modulus extrapolatedfrom simulated nanoindentation curves as a function of thepercentage of the unloading curves used in the analysis.Thethree curves are for three methods for choosing ε in the analysismethod. All curves used β = 1.05.

This baseline simulation analysis suggests that mod-ulus calculation should analyze at least 60% of the un-loading curve and ε is best found from measured hc .Because measuring hc may not be possible, a value ofε = 0.75 is as good, or preferable to, finding ε fromthe relation for an elastic material in Ref. [30]. An ob-servation that fitting less than 60% of an experimentalunloading curve, R2 > 0.99 still does not support aclaim that fitting parameters are reliable.

Even when ε is numerically determined from knownmaterial properties (i.e., using simulation results), theextracted modulus is still 5% or more higher than actualmodulus of the material. This discrepancy can be inter-preted as use of an incorrect value for the β parameter.In other words, by changing β , the extracted Ee f f canbe made to exactly match the input value. Using thereasonable ε = 0.75, the results when using 60% ormore of the unloading curve give exact results if β isvaried from 1.111 to 1.118. This higher β is consistentwith prior calculations. Larsson et al. [45] conductedfinite element experiments for linear elastic materials,and found β dependent on ν with β = 1.14 at ν= 0.3.Oliver and Pharr [30] mentioned that β is higher formaterials with strain hardening. References [34] and[47] also claimed β should be higher and closer to1.09. Our simulations, which are consistent with theseprevious findings, show the β depends on both elasticand hardening properties of the material. When doingnanoindentation on an unknown material, the correctvalue of β will not be known. Choice of values between1.05 and 1.14 will therefore have uncertainties in the

Scaling Factor

Eef

f (%

Err

or)

0 1 2 3 4 5 -4

-2

0

2

4

6

8

10

nhard

khard & nhard

khard

yield

A

Scaling Factor

H (G

Pa)

0 1 2 3 4 5 0.08

0.12

0.16

0.20

0.24

0.28

0.32

nhard

khard & nhard

khard

yield

B

Figure 4: Effect of scaling the hardening parameters on pa-rameters extracted from load-displacement curves. A. Percenterror in calculated effective modulus. B. Calculated hardness.All curves used ε= 0.75 and β = 1.115.

range of ±5%.

B. Hardening parameters

This section explores the effect of changing materialplasticity properties on the modulus and hardness ex-tracted with the Oliver-Pharr method. The baselineproperties were from the previous section. These addi-tional simulations varied yield stress, σy0, hardeningmodulus, kh, and hardening exponent, nh, about thebaseline values. Four sets were run: vary only kh, varyonly nh, vary kh and nh together, and vary onlyσy0. Foreach set, the variable parameter was scaled by factors0.25, 0.5, 1.5, 3, and 5 relative to baseline value whileall other parameters were held at their baseline value.Based on results in previous section, the effective mod-uli and hardness were extracted by analyzing 75% ofthe unloading curve and using ε= 0.75 and β = 1.115.

The percent error in the calculated effective modulifor these simulations as function of scaling factor areshown in Fig. 4A. The yield stress had the smallest effectwhile strain hardening parameters had larger effects.When kh and nh were varied together, the results in-creased almost as much as varying nh alone suggesting

8 J. Mater. Res., Vol. 0, No. 0, 2018


that nh is the dominate hardening parameter affectingthe analysis.

The errors in measured effective moduli varied from-2.9% to 8.6%. Because these errors occurred by vary-ing material properties other than modulus, we suggestthe Oliver-Pharr should never expect better than about±10% accuracy for a material with unknown harden-ing properties. Improving accuracy of nanoindentationexperiments would require analysis methods that canaccount for both a material’s stiffness and its hardeningproperties. One option would be to use inverse methodsand vary all material properties until the full, simulatedload-displacement curve matches experimental results.Because this process would use the full curve, it hasmore potential to detect effects of material hardening(the loading portion of the curve depends on harden-ing parameters more then the unloading portion). Thisapproach was used with some success for analysis ofnanoindentation experiments on materials with gradi-ent properties [5]. Their inverse methods could findEe f f , K , and n by inverse methods. They found resultswere insensitive to σy0, which is consistent with aboveobservation that varying σy0 had the smallest effect onthe unloading curve. The simulations in Ref. [5] werelower resolution and had more noise. The simulationshere with tartan grid and methods for reducing noiseshould lead to improved inverse analysis of nanoinden-tation experiments.

The calculated hardness H for these simulations rela-tive to baseline values and as function of scaling factorare shown in Fig. 4B. Unlike modulus, the H value forany material is not an input to simulations and there-fore extracted H cannot be compared to a known value.Experiments for hardening are mostly qualitative andexpress some number based on final indent size relativeto indentation pressure. A simulated material’s hard-ness should therefore vary with change in hardeningproperties and results in Fig. 4B track expectations. Themeasured H increases as σy0 or kh increases. The varia-tion with nh is also expected. For power-law hardeningwith constant σy0 and kh, hardening curves for variousnh intersect when ||εp|| = 1 (or 100% plastic strain).Because all simulations show that ||εp|| remained lessthan 1 for all material points (except sometimes a fewpoints directly under the indenter tip), increasing nhwill shift the hardening curve closer to an elastic-plasticmaterial with lower hardness. Con versely, decreasingnh will shift the curve to a material with higher yield

stress and higher hardness. The dependence of H on nhfollows this expected trend. The curve that varied khand nh again shows that nh is the dominant hardeningparameter affecting indentation experiments.

C. Strain-rate dependent materials

The goal of simulations with time-dependent strainhardening (i.e., when C 6= 0 in Eq. (3)) was to mimictypical experiments where indenter load is held constantfor some time period before unloading. Because load isheld constant, we had to switch from indenter velocitycontrol to load control simulations. The rigid parti-cles used in these simulations were constant-velocityparticles that effectively have infinite mass. Because ap-plying loads to infinite-mass particles would not affecttheir velocity, we implemented a PID (Proportional, In-tegral, Derivative, controller, see [48]) feedback loop tocontrol indenter velocity. This loop compared a targetforce function to the current contact force, which wasdetermined by summing all contact forces on the rigidindenter. The feedback control modified indenter veloc-ity to achieve the desired force function. As in velocitycontrol simulations, the PID loop was based on a targetload as a function of time that was sigmoidal both inthe loading and unloading phase to prevent oscillations.To simulate experimental procedures, the target loadfunction had a hold phase at constant maximum loadfor variable amounts of time.

These simulations used the same baseline propertiesas above, but now varied C from 0.01 to 0.1. The ref-erence strain rate parameter, or strain rate at whichhardening is equal to baseline hardening, was set toε0

p = 106 sec−1. This reference rate was set to 1/(load-ing time) for the simulations, where loading time was1 µs. Because the loading rate (or maximum displace-ment over loading time of about 1 m/s) was much lessthan 1% of the material’s wave speed, these simula-tions can be considered as quasi-static results wherethe only rate effects would be rate-depending yieldingof the material. In other words, although the simula-tions were much faster than experiments, the time axiscan be viewed as reduced time relative to the referencestrain rate parameter. Predictions at slow rates withreal materials could be made by rescaling the time axisto any new reference strain rate. For positive C , par-ticles loaded below the reference rate will see drop inyield stress while those at higher rates will see higher

J. Mater. Res., Vol. 0, No. 0, 2018 9


yield stress. Figure 2B gives simulated nanoindentationcurves with C = 0.1 and various hold times at maximumload. The PID load control worked well, giving simu-lations that were relatively free from noise and thatcontained a stable hold phase at constant maximumload. The quasi-static character of these simulations isconfirmed by small oscillations in the plateau region(and some of the oscillations may be caused by PID con-trol parameters). We looked at the effect of hold timefor a strain-rate dependent yield material on both thehardness and the effective modulus extracted from simu-lated results by the Oliver-Pharr method (and assumingε= 0.75 and β = 1.115). Figure 5A shows measuredhardness compared to baseline hardness (horizontalline for C = 0). Increasing C caused the hardness todrop and the drop increased with C . The drop in hard-ness occurred because plastic strain rates under theindenter were always less than the chosen referencestrain rate. As a result, the strain-rate term causes adrop in yield stress or a material with lower hardness.As a function of time, the hardness dropped further.During the hold phase the plastic strain rate is very lowleading to further drop in yield stress.

Figure 5B shows the effects of C and hold time onpercent error in the extracted effective modulus. Ee f ferror increased with zero hold time and the increase waslarger for higher C . The increase was caused by changein strain hardening properties of the material. As seen inFig. 4A, the change in Ee f f with changes in hardening-law properties are more convoluted than changes inhardness. For all values of C , the extracted Ee f f de-creased with time and perhaps reached a plateau. As aconsequence, experiments on materials with strain-ratedependent hardening properties should use sufficienthold time to reach plateau, or at least consistent, resultsfor both hardness and modulus.

D. Shape effects

Many methods for extracting Ee f f from experimentsand analyzing effects of material hardness are based onindentation of a homogeneous infinite half-space, or agood approximation of such a space. If this assumptionis significantly violated, the standard nanoindentationanalysis methods cannot be trusted. Reference [34]presents an experimental method to correct for shapeeffects. They propose modeling total compliance Ct of

Hold Time (µs)

Har

dnes

s (G

Pa)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.08

0.09

0.10

0.11

0.12

0.13

0.14

0.15

C = 0.1

C = 0.05

C = 0.01

C = 0.0A

Hold Time (µs)

Eef

f (%

Err

or)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 -2 0

2

4

6

8

10

12

14

16

18

C = 0.1

C = 0.05

C = 0.01

C = 0.0

B

Figure 5: Effect of C jc and hold times on parameters extractedfrom load-displacement curves. A. Calculated hardness. B.Precent error in calculated effective modulus. All curves usedε= 0.75 and β = 1.115.

the indentation system using:

Ct =1

Ee f fp

Ac+ Cm + Cs (15)

where Cm is compliance of the machine and Cs is anyadditional compliance of the indentation specimen dueto deviation from being a homogeneous, infinite half-space. The term Ee f f

p

Ac is proportional to slope Sin Eq. (11). The machine compliance should be fairlyconstant (and zero for simulations), but the structuralcompliance could vary significantly with location of theindentation on a specimen. For example, indentationon one material near an interface with another materialor indention near a free edge would both see Cs changeas a function of distance to the interface or the edge. Inprinciple, by running experiments as a function of in-dentation location, one could measure Cs allowing oneto separate material Ee f f from machine and specimencompliance effects. Reference [34] proposes that theeffect of some structural demarcation should be a func-tion of the inverse of its distance to the nanoindentationsite.

To investigate the strategy from Ref. [34], we con-ducted three numerical experiments using MPM sim-

10 J. Mater. Res., Vol. 0, No. 0, 2018


d

E=1 E=1

d

E=2E=2 E=2d

A B C

Figure 6: Axisymmetric simulation geometries used to inves-tigate the effect of structure on shape. d is the distance fromindentation tip to a key structural change. A. Low stiffness ma-terial surrounded by high stiffness material (moduli in GPa).B. High stiffness material surrounded by low stiffness, C. Thinmaterial.

ulations as illustrated in Fig. 6. First, the indentationblock was divided into two materials with the materialunder the indenter have E = 1.0 GPa while the remain-der of the block had E = 2.0 GPa. The variable d wasthe distance from indenter tip to the material interface.Second, the two materials were switched such that thestiffer material was under the indenter. Third, the en-tire block had E = 2.0 GPa but the depth of the blockwas varied — d was now distance from the indentertip to the rigid, zero-velocity surface on the bottom ofthe block. The plasticity properties used the base-lineproperties from section III.A.

The calculated Ee f f from simulated load-displacement curves for the three different structuralgeometries are plotted in Fig. 7A. Not surprisingly,Ee f f varies significantly when d is small or when theindented specimen has significant deviations from ahomogeneous, infinite half-space. Also note that theeffect persists to a rather large d. The measured Ee f fdid not return to input material properties until d wasmore than 20 times the indentation depth, which was1.5 µm (see Fig. 7A). It would be a misconception tothink that because nanoindentation tips and depthsare small that the method is not altered by evenrelatively-remote structural features.

Reference [34] discusses two ways to correct for struc-tural compliance in nanoindentation experiments. Onemethod requires measurement of the contact area fromthe indentations while the second requires multiple in-dentations in the same area with different maximumloads. Here we used a third method that was madepossible because we know the correct modulus. Weassumed the structural effects should be described by

d (mm)

Eef

f (G

Pa)

0.000 0.005 0.010 0.015 0.020 0.025 0.030 1.0

1.5

2.0

2.5

3.0

3.5

Indentation depth

1 to 2 GPa

2 to 1 GPa

2 GPa vs. Thickness

A

1/d (mm-1)

1/E

eff (

GP

a-1)

0 100 200 300 400 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 to 2 GPa

2 to 1 GPa

2 GPa vs. Thickness

B

Figure 7: The effect of distance to a structural change on effec-tive modulus extracted from simulated nanoindentation curves.The horizontal dashed lines are the actual effective moduli. PlotsA and B show the same date plotted two different ways.

some function of d, which can be formulated as:

1Em(d)

=1

Ee f f+ f (d) (16)

where Em(d) is the measured modulus from simulationof material with known Ee f f and f (d) is some non-constant function. By examining simulation results,f (d) appears well represented by a linear function of1/d. In other words, plots of 1/Em(d) as a function of1/d are close to linear with the intercept being closeto 1/Ee f f (see Fig. 7B). Extrapolating to d →∞ (or1/d → 0) gives an estimate for Ee f f with an error of10% for the geometry A (“1 to 2 GPa”) in Fig. 6, 2%for geometry B (“2 to 1 GPa”) and 2% for geometry C(“2 GPa vs. thickness”). The larger error for geometryA was likely because the material under the indenterhad E = 1.0 GPa while the data was interpreted usingβ determined in section III.A for a material with E =2.0 GPa.

J. Mater. Res., Vol. 0, No. 0, 2018 11


E. Full 3D simulation and effective indentershapes

We repeated the base-line simulations in section III.Ausing a full 3D model with a pyramidal Berkovich inden-ter. A simulation of the same size block extended to 3Dusing a tartan grid and using slightly lower resolutionunder the indenter (150 nm cells) needed about ninemillion particles and ran in about two days. A simula-tion with 100 nm cells under the indenter is feasible, butrequired about 30 million particles and would take sev-eral weeks. The results at 150 nm resolution looked ac-ceptable. As seen in Fig. 1, the transverse displacementunder the indenter in 3D resembles the correspondingradial displacement in axisymmetric simulations. The3D simulation does reveal asymmetry in the displace-ment caused by asymmetry in the Berkovich indentershape along the symmetry plane. Despite asymmetryeffects in 3D stress state, the global load-displacementcurves for the full 3D simulations were almost identicalto curves from axisymmetric simulations. Comparedto axisymmetric simulations, the 3D simulation hadabout 3% higher peak load, had the same hardness,and had about 7% higher extracted Ee f f . The smalldifferences persist even if compared to axisymmetricsimulations with 150 nm cells, which means they arelikely caused by indenter shape effects. The overallsimilarities between axisymmetric and 3D results arereassuring as much nanoindentation analysis is carriedout using axisymmetric methods [1–3, 30].

While a Berkovich indenter is not conical, its agree-ment with conical simulations can be partially explainedby the concept of an effective indenter discussed inRef. [49]. When an indenter is being pressed into a sur-face, any plastic deformation will blunt or smooth theindenter shape seen by the purely elastic portion of thematerial. During the unloading, the deformed regionand the indenter forms a blunted “effective indenter”under which the pressure is approximately constant. Forthe power law in Eq. (10) describing the unloading por-tion of the curve, a elastic material should have m = 1for a flat punch and m = 2 for a cone or Berkovichindenter. In nanoindentation experiments (and simula-tions), this parameter is generally between these two. Inour axisymmetric simulations, we found m≈ 1.25 (pro-vided at least 60% of unloading curve was fit), which isconsistent with the concept of an effective indenter. Ref-erence [49] suggests the effective indenter shape can be

Shap

eEff

. Sha

peIs

osur

face

3DAxisymmmetric

Figure 8: Comparison of the actual indenter shapes, effectiveindenter shapes, and an isosurface of vertical compressive stressof 30 MPa under the indenter. The vertical axis for the shapeswere made five times larger and all the geometries are flippedupside down, and the flat surface was added for better viewingperspective.

found by measuring the difference between the final in-dentation shape and the indenter. Because simulationsgive full details for indentation shape, this proposed ef-fective indenter shape can be calculated. The effectiveshapes for both our axisymmetric and 3D simulationsare visualized in Fig. 8. These shapes are fairly bluntfor both cases, which is consistent with m decreasingtoward value for a flat punch and partially explainssimilarities of the results.

For another view, we looked at an isosurface of con-stant compression stress of 30 MPa in the depth direc-tion. These isosurfaces do not correspond well to theshape of the effective indenters, as would be expectedfrom a uniform pressure under an indenter. As pointedout in Ref. [50], it is not actually possible to have uni-form pressure under an indenter of any shape. Despitethe lack of axisymmetry of the Berkovich indenter, thestress isosurface shown in Figure 8 looks similar to theaxisymmetric case, which illustrates again that indentershape effects are blunted by plasticity and by dispersionof the stress distribution. Both effective indenter shapesand stress isosurfaces help explain why axisymmetricsimulations are good approximations to 3D simulationswith a pyramidal Berkovich indenter.

12 J. Mater. Res., Vol. 0, No. 0, 2018


IV. Conclusions

These new simulations have demonstrated that MPM iswell equipped for the modeling of nanoindentation. Inparticular, MPM handles well the large deformation andcontact that occurs under the indenter tip. For goodsimulations results, the region under the indenter tip re-quires high resolution and that need was met efficientlyby using a tartan grid. To achieve noise-free curveswith no dynamic artifacts, the indenter displacementshould be sigmoidal and the MPM calculations shoulduse noise reduction methods now available in XPIC(m)[40].

All virtual experiments show that stiffness and hard-ness extracted from load-displacement curves dependon the hardening parameters of the material. If thehardening parameters depend on strain rate, the ex-tracted stiffness and hardness will also depend on holdtime at maximum load. Because one typically does notknow the hardening parameters of a material studied bynanoindentation, these numerical observations suggesta limit on the accuracy achievable by using Oliver-Pharrmethods. A potential approach to achieving higher ac-curacy is by using numerical analysis. For example, aniterative inverse approach coupled with MPM, as donein [5], could be used to solve for material parametersby matching load-displacement curves to experiments.If that analysis considers the full curve (i.e., both load-ing and unloading), such an inverse approach may findboth stiffness and hardening properties of an unknownmaterial. A note of caution — the load-displacementcurves are a convolution of elasticity, plasticity, and lo-cal geometry and have been shown to not necessarilybe unique [51]. Furthermore, if the material being in-dented is strain-rate dependent, resulting in the loaddisplacement curves with a flat top, then this strain-rate dependance needs to be modeled as well. MPMcould be an effective tool for carrying out inverse meth-ods aimed at extracting additional and more accuratematerial properties from nanoindentation experiments.

Acknowledgements: The work was funded by the Na-tional Institute of Food and Agriculture (NIFA) of theUnited States Department of Agriculture (USDA), GrantNo.: 2013-34638-21483

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