Modeling Nanoindentation using the Material Point Method

Transcription

1 Moeling Nanoinentation using the Material Point Metho Cha C. Hammerquist an John A. Nairn Woo Science an Engineering, Oregon State University, Corvallis, OR 97330, USA 1. SUPPLEMENTAL MATERIAL The ublishe aer associate with these notes concentrate on material science asects of nanoinentation by using virtual, numerical exeriments to investigate best aroaches to extracting effective moulus from nanoinentation exeriments. The simulations all use the material oint (MPM) metho an this work inclue two relatively minor enhancements to MPM secifically neee to otimize its use for nanoinentation simulations. Because those two MPM enhancements have not been reviously ublishe, this sulemental material is being rovie to ocument them. 2. CONTACT MODELING Contact etection in MPM is imrove by incluing a calculation of article ege islacements. In other wors, two material omains are only moele as in contact when i,b 0 where i,j is the istance along the normal vector from the ege of material j to noe i [1]. The challenge in imlementing this criterion is that ege locations are tyically not tracke in MPM. A metho for that calculation is neee. The calculation of i,j is starte by extraolating material oint ositions to the gri using usual mass-weight MPM extraolations : x = 1 m x S i where m i,j = m S i (1) m j Here m an x are article mass an location S i is MPM shae function for article an noe i, an the sums are for all material oints of material j. An aarent istance from extraolate article osition (x ) to noe i (at x i ) along normal vector ˆn i for interface between two contacting materials is: j = (x i,j i,j x i ) ˆn i (2) But this istance is not the actual istance neee for contact calculations. To etermine the actual istance, the value of nees to be correcte. The rocess is equivalent to fining the function ( ) for actual istance as a function of. i,j The require calculations are one by using MPM shae functions to fin as function of actual istance an then inverting the results. Imagine a 1D gri with material a aroaching noe i from the left an material b aroaching from the right (see Fig. S1). From Eqs. (1) an (2) with noe i at the origin: k=1 = m S i r (2k 1) r (2k 1) i=1 m (3) S i r (2k 1) The similar exression for material b is k=1 = m S i i,b + r (2k 1) i,b + r (2k 1) i,b n i=1 m (4) S i i,b + r (2k 1) 1

2 C. C. Hammerquist an J. A. Nairn Material a i,b Material b k: noe i 2r Figure S1: A 1D MPM gri with materials a an b aroaching noe i from left an right, resectively. The inicate lengths an i,b are actually istances from eges of materials a an b to noe i. Each article has raius r. Particles for each material tye are numbere k = 1, 2,... starting from ege of the material near noe i. Here r is article raius, an S i (x) is shae function for article on noe i when mioint of article is at x an origin (or x = 0) is at noe i. These calculations were one with GIMP shae functions that integrate gri shae functions over the current article omain. For these uneforme articles, the exlicit shae function for noe i at x = 0 is [2]: 2r (1 r ) x 2 2r x < r 1 x r S i (x) = < x < 1 r (1+r x ) 2 4r 1 r < x < 1 + r 0 x > 1 + r This evaluation of S i alies only to r x/2 where x is cell sacing or to MPM moel with one or more articles er cell. Most MPM simulations use 2 articles er cell with r = x/4. The calculations using CPDI shae functions woul not change much for these uneforme articles. The summations for inclue all articles with non-zero shae functions. The resulting functions (inverte to lot actual istance as a function of extraolate istance) are the ashe lines in Fig. S2. As execte, the extraolate istance iffers from the esire ege istance. For examle, will always be negative as the ege of material a moves from noe i 1 to noe i + 1, while actual will vary from x to + x. Similarly, the extraolate istance for material b is always ositive. The imosition of contact etection in MPM calculations requires efinition of a function i,j ( to calculate actual ege istance from extraolate aarent istance. Reference [1] roose a simle constant correction such that (5) ) i,b (x i,b x i,b ) ˆn i δ c x (6) where δ c is a constant offset (in units of cell size) use to aroximate actual searation. By analysis an test simulations, a value of δ c = 0.8 works well for MPM simulations with two articles er cell (or r = x/4). This aroach is equivalent to assuming the istance maing functions are: ( ) = i,j i,b x left or j = a 0.4 x right or j = b These linear functions are lotte in Fig. S2 an can be escribe as best fit to a line with sloe equal to one or to i,j ( ) = + δ c x/2 where δ c is the only fitting arameter. To imrove the material 2 (7)

3 MODELING NANOINDENTATION USING MPM - SUPPLEMENTAL MATERIAL 1.0 Ege Distance ( i,j /Δx) Linear Left Material "a" Power Law Right Material "b" Shae Functions Extraolate Distance ( (ext) i,j /Δx) Figure S2: Calculation of i,j as a function of for left an right materials aroaching noe i. The black otte curves show exlicit calculations using GIMP shae functions. The re soli lines show the linear functions (Eq. (7)) an the ower-law functions (Eq. (8)) to aroximate the shae function calculations. searation calculation, we relace the constant correction with new maing functions given by: i,j ( ) x 1 2 = x left or j = a i,b 1.25 x right or j = b (8) These two maing functions are comare to shae function calculations in Fig. S2. Although the linear correction generally works well, it was easy to imlement non-linear corrections instea. For nanoinentation simulations in this aer, all of which een strongly on contact mechanics, we use the more accurate ower law functions. This aroach gave small, but noticeable, imrovements in the results. Note that the above maing functions een on article r or ifferent linear offsets (δ c ) an ifferent non-linear functions woul be neee for MPM simulations using ifferent sizes of articles. We use the above aroach that foun δ c when r = x/4 to investigate how it changes with ifferentlysize articles. For one article er cell or r = x/2, δ c = 1.07 is the recommen offset. For two or more articles er cell r < x/4, the recommene δ c eens only weakly on article size ecreasing from δ c = 0.8 for two articles er cell to δ c = 0.72 for six article er cell. Because of these finings, an MPM simulation that varies size of articles with all r < x/4 coul likely etect contact well with a single value for δ c. We i not investigate changes in the non-linear functions neee for more avance contact calculations at ifferent article sizes MPM Tartan Gri For nano-inentation simulations, we use a gri scheme terme a tartan gri as illustrate in Fig. S3. In a tartan gri, one or more regions of interest are moele with a high-resolution, regular gri with equally-size elements. For the nanoinentation roblem, the one region of interest uner the inenter was moele with 100 nm cells etermine above as neee for convergence. Outsie regions of interest, the gri cell sizes were allowe to increase, thus forming a tartan-like attern. 3

4 C. C. Hammerquist an J. A. Nairn Figure S3: An MPM tartan gri with orthogonal backgroun cells for nanoinentation simulations. The region of interest has a regular gri with equal size elements. The gri cell size increases linearly with istance from the region of interest. Note that a tartan gri maintains orthogonal gri lines. This tye of gri greatly simlifies imlementation, esecially when using CPDI shae functions [3] (see text of aer). To simlify contact calculations, which all occur in the region of interest, the cells within the region of interest are all the same size. Outsie the region of interest, the cell size increases as a function of istance from the region of interest. We imlemente two cell scaling methos. The first is a linear scaling where x n = nr x (9) where n is the number of cells away from the region of interest, R is a secifie ratio, an x is the constant cell size in the region of interest (see Fig. S3). Alternatively, the cell size can increase geometrically such that x n = R n x (10) where R is now ratio of cell size for element n to size of the revious element. For this simulations, the linear metho worke well an converge simulations use R = 2 an x = 100 nm. A tartan gri was articularly effective or nanoinentation simulations. Virtually the comlex stress states occur uner the inenter an that region is an area of interest with a regular gri. The stresses outsie that are vary more slowly an are easily hanle with large articles. Figure S4 shows comression force vs. time for the same roblem one with a regular gri (black line) an a tartan gri (re line). The results are essentially suerosable. A tartan gri can have general alicability in MPM coes (although may not always be as effective as it is for nanoinentation simulations). Furthermore, a more comlete imlementation of tartan gri methos woul require some aitional etails that were not neee for nanoinentation simulations: 1. Shae functions: what ever shae functions are use, they have to be caable of accounting for variable element sizes aroun any given noe an variable size articles. In articular, using of GIMP shae functions woul require re-evaluation of many stanar functions use in current MPM coes. Use of CPDI shaes functions (as one here) nees no changes excet for coe to be able to fin corners of articles within a gri with variably-size articles an cells. 4

5 MODELING NANOINDENTATION USING MPM - SUPPLEMENTAL MATERIAL Figure S4: Nanoinentation force as a function of time uring loaing an unloa as calculate using a regular gri (black line) an a tartan gri (re line). 2. Contact calculations: contact calculations een on article size an as mentione above, contact etection eens on article size as well. If contact occurs outsie areas of interest, the use of tartan gris nees customization to account for variable element an article sizes. For nanoinentation simulations, all contact was within the regular gri, area of interest. 3. Particle integrations: Sometimes MPM calculations look at integrations over articles such as average article stress or integration insie a J-integral contour to account for ynamic stress states [4]. These integrations nee to account for variable article sizes. For examle, the average stress shoul be foun from a volume-weighte average stress. REFERENCES 1. J. A. Nairn: Moeling imerfect interfaces in the material oint metho using multimaterial methos. Comut. Moel. Eng. Sci. 1, 1 15 (2013). 2. S. G. Barenhagen an E. M. Kober: The generalize interolation material oint metho. Comuter Moeling in Engineering & Sciences. 5, (2004). 3. A. Saeghira, R. M. Brannon, an J. Burghart: A convecte article omain interolation technique to exten alicability of the material oint metho for roblems involving massive eformations. International Journal for Numerical Methos in Engineering. 86, (2011). 4. Y. Guo an J. A. Nairn: Calculation of j-integral an stress intensity factors using the material oint metho. Comuter Moeling in Engineering & Sciences. 6, (2004). 5