KHOA HOC-CONG N G H i .
So 05/2019
Phan tfeh, danh gia sai so tren chuan nang luong theo sy phan bo ung suat cho mang mong nano chju tai bang phuong phap phan tu huu han
• ThS. NCS. LE V A N AN; PGS. TS. it HIEU GIANG - TrUdng Dai hpc Suphqm Ky thudt TP Ho Chi Minh
• PGS. TS. BUl X U A N L A M - Trudng Dqi hpc Cdng nghe TP Ho Chi Minh
• ThS. TRINH THAI Hl/NG - TrUdng Dgi hoc Giao thdng van tdi TR Ho Chi Minh
TOW TAT: Muc tieu cCia bai bao la xay dung mo hinh t o l n cho dang mang mdng ch|u t^i bPi dau dam nano Berkovich. Chung toi cli thien phuong phap phan tu huu han (PTHH) truyen thong bSng viec lam mm ludi kieu h trong phan tich PTHH n h i m mo plidng, phan ticPi, danh gia tinh hi^u qua sai so tren chuan nang luong theo sy phan bo ung suat trong mang mong va mPc dg hpi tu cua pliuong phap su dung ngon ngu* l|p trinh Matlab. Ket q u i duoc so sanli vol plian mem thuong mai Abaqus va la tien de nghien cuu phat trien, Png dyng vat li#u nano cho ung dung trong cong nghiep va dgc biet la phan tich tinh chat co hgc cua cac v^t lieu mang mdng
TLf KHOA: Matlab, nanocomposite, phan tu huu han, chuyen vi, bien dgng d^o
ABSTRACT: The objective of the paper is to build a mathematical model for the nanocomposite thin films loaded by nanoindentation with a Berkovich tip We improved the traditional finite element method by the type h refinement in finite element analysis to simulate, analyze, and evaluate the errors on the energy standard according to the stress distribution within the thin films and the convergence rate using Matlab programming. The result was compared with the commercial software Abaqus and it was the premise developing new nano-sized materials for industrial applications and especially the analysis of the mechanical properties of thin film matenals.
KEYW/ORDS: Matlab, nanocomposite, displacement, plastic deformation.
FEM,
I.DATVANDE
Phuong p h i p tao vet lorn nano dupe dung rpng rai de do do ePng. mo-dun d i n hoi, tinh ehong eao xuPe, dp dao cCia mang mong. Ket hop vpi elc g i l i p h i p tinh t o l n b i n g phaong phap PTHH (FEM), hieu q u i phan tich die tinh CO hpe cua vat lieu gia t i n g dang ke [1,2,3].Trong bai bio, ehung toi xiy dpng mo hinh toan mo phong qua
trinh chju t l i cua m I n g mong. MP hinh t o l n duac g i l i bang phuong p h i p PTHH, lap trinh Matlab Pe xac djnh trUPng ehuyen vi, p h i n bo Png suat trong mIng mPng.
Chuan sai so nang luong theo h dUOc sPdung de danh g i l do chinh xlc cua IPi gill.
2. THIET Bj DO D d CCfNG B A N G PHl/CTNG P H A P TAOVETLOMNANO
Do cPng va mo-dun dan hoi dupe xlc dinh b i n g thiet bi do dp ePng nano XP vPi mui kim cUong Berkovich.
Chieu sau an vao mang eua mui kim euong dUoe lap trinh de khong vUpt q u i 10% be day cua mang nham tranh anh huPng eiia v i t duac phu den ket q u i do dp cPng.
Tren moi mau, 10 vi trf ngau nhien dUpc do dp cPng va mo-dun d i n hoi. Gil trj dp cPng v l mP-dun dan hoi cua mau dupe lay trung binh eua eae lan do (sai lech chul'n khPng q u i 10%). TP duPng cong gia t l i va giam tai tren do thi cua thiet do dp ePng, do deo cua mIng dupe xie djnh bang cIch ehia khoang dich ehuyen cua mui kim cuong Png vPi giai doan g i l m t l i cho dieh chuyen IPn nhat. OP ePng mang duoe xae dinh bPi dupng eong gia t l i - chuyen vi trong q u i trinh dam vPi lpe t i n g dan v l phan tieh bang phUOng p h i p Oliver-Pharr [4].
Dan hoi
Dich diuygii
Hinh 2.1: Duong cong gia tii • giim tai Trang t h l i dau dam Berkovich eo the duoc mo hinh hPa b i n g mpt dau d i m hinh non vPl mpt nPa gPe quay cD - 70.3°. Chu vi ngoai bien tiep xue tao vet lom ehim vao vat the dupe dien ta bang mo hinh dau dam cPng co hinh hpe don gian [5]. Neu bo qua sU tieh tu cua vat lieu P ehu vi ngoai bien doi vPi vat lieu dan hoi-deo, luong
"chim vao" v i t lieu mIng h^ (Hmh 2.2) duac xac dinh bang:
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KHOA HOC - CONG NGHE .
/ " . = 6 ^ (1) Trong do: e - Hang so phu thuPe v I o hinh dang cua
dau d i m (e = 0.72 cho hinh non, 0.75 cho parabol tron xoay va 1.00 eho dang phIng).
Hinh 2.2: Giin do gia tii va dd tii Dp s i u doe theo IPp tiep xue dupc tao ra giPa dau d i m va mau h^- h^^^-h^ I I :
n.=h,„,,-e'-Y (2) Neugpi F(d) la m p f ' h l m dien tfeh" mieu t l dien tich
dupe chieu (hole mat cit) cua dau dam P khoang each d tP be m i t den mui eua dam, dien tich tiep xue A se I I :
A = F(h^) (3) Ham dien tich la h i m hinh dang dau dam, p h i i dupe
hieu ehinh bang elc phep do dpc l i p de dp lech tP hinh dang dau d i m hinh non dupe dua vao tfnh t o l n thay eho mui Berkovich la nho nhat. Dp lech n l y eo the la kha IPn so vPi dau dam Berkovich neu cP sp bo tron x i y ra trong qua trinh mai sac. Day la phuong p h i p co b i n cho viec x l c dinh ham dien tfeh tiep xue. Pharr va cpng sp [6] da bieu dien A l l mpt ham nhUsau:
-1 = n=cC„(ftc)^"" = Ccft' + C,h + C,h=« + C,Ji'"+ • +Cgh''™ (4) De hoan t h i n h qua trinh hieu chinh ham dien tich, cle he so khic trong phuong trinh (4) p h i i dupe x l c dinh b i n g sp dieu chinh duPng eong dP lieu A doi vPi h^. Doi vPi dp lieu thuc nghiem trong nghien cPu [7], elc he so cho ham dien tich n h i n duac Va C3= 24.65, C, ^ 202.7, Cj = 0.03363, C3^ 0.9318, C^= 0.02827, C , - 0.03716, C , - 1.763, C,-0.04102,vaC3=1.881.
Khi dien tfeh tiep xue dupe xae djnh, dp cPng dupe xac dinh bang bieu thPe:
H='-^ (5) Chu y ring, viee x l c dinh dp cPng nay dUa v I o dien
tich tiep xue duPi t i e dung eua t l i , no cP the lech so vPi dp cPng truyen thong Puoc do tP dien tfeh cua vet lorn nen cPn dU neu cP sU hoi phue d i n hoi d i n g ke trong suot q u i trinh g i l m t l i . Tuy nhien, dieu nly ehi quan trong trong cae vat lieu vPi eae g i l tri E/H epc ky nho [6].
Phep do mo-dun dan hoi theo moi quan he ham dien tfeh tiep xue v l dp ePng g i l m t l i duac thong qua he thPc:
S ^ ISJ= E^VA (6)
Trong do: E_ - MP-dun d i n hoi hieu dung dUpc xae dinh bPi:
dau dam (kim cuong) va mo-dun dan h6i E v l he so Poisson V la cua vat lieu mau. Phuong trinh (6) I I mot he thPc rat tong q u i t ap dung eho bat ky dau dam doi xPng trye nao. PhUOng trinh n l y khong han che doi vPi mot hinh hpe don g i i n cu the nao, ke e l vet lom tii va co th^
I p dyng vPi tiep xue dan h6i-d#o [6].
3. XAY DLTNG M O HINH TOAN 3.1. Md hinh toan khi mang chju tai Mo hinh ehiu t l i ciia mang nhu Hinh 2.2, t l i se tang dan cho den khi dp sau cua mui dam dat 120nm. Bien dang d^o dupc phan tieh PTHH. Doi vPl v i t lieu dang mang ehju t l i CO the duoc chia thanh 3 nhPm: Vat lieu phi tuyen, eau true phi tuyen tfnh va ea hai dang phi tuyen tren.
d bai b i o nay, ehung toi xet vPi vat lieu phi tuyen P trang t h l i bien dang deo theo cong thPc:
{K}{Au}'={A/}' (8) VPi:
[K] - Ma tran dp ePng;
{Au} -Vector chuyen vj eiia nut;
{ A / } - V e c t o r t a i ; 1-BUPe lap thp i
De kiem tra trang t h l i tPi han trong qua trinh gia t l i se dPOc cap nhat tPng buPc theo eong thPe (8).
MP-dun d i n hoi E v l he so Poisson's v ciia vat lieu
Hinh 3.1: Ma tran do ciing khdng doi - Phuong phap do dp ePng khong doi sP dyng giai thuat lap cho viec gia tai n h i m tinh chuyen vj va bien dang sau moi bUPe gia t l i v l kiem tra dp bien dang moi sau moi buPc.
{Au}' = [K]-'{Af}' (9) { A e } ' = [ B ] { A u } ' (10) Trong dd: [B] - Ma tran dao ham trong quan he dong
hpe giPa ehuyen vi - bien dang.
Neu vat lieu bien dang deo bao gom dan hoi va deo:
{Aei'^IAE---}' -i-{AeP}' (11) Quan he giPa Png sul't va bien dang dan hoi diroc
tinh thong qua ma tran Png xP C:
{Ao}' = [C]{Ae^}' (12) Spgia tang Png suat dUOe kiem tra thong qua dieu
kien giPi han chly deo.
{ o } i = { o } - ' | A a } ' (13) Neu Png suit van duPi trang thai giPi han nay
{bien dang dan hoi) tiep tue gia tai bang viee sP dung laiePng thPe{8).
Ngpoc lai, neu Png suat vUpt qua giPi han (F > 0), Png suit du can dupe ein bang lai de dat tPi gia trj gan b i n g vpi glPi han (F = 0). Khi do, vat lieu bj ehiy deo va giPi han chay dupe tinh theo Vermeer [7].
(14)
(..^)=AA[ci-.)(g)" +.(££)']
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KHOA HOC - CONG NGHE .
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VPi Ak la bupc tang bien dang deo (o II he so phu thupc v I o t h P i gian.
VPi 0) = 0, phuong trinh dUpc goi la ham tuPng minh (Explicit) va (0 = 1 thl goi II h i m an (Implicit). SP dyng h i m ti/Png minh vPI Png suat lay tP bPPc trPPc co = 0, do docPngthPe (14) dupc Viet thanh:
{AEP}= A A g ) ' " ' (15) Bi^n dang deo dupe tinh dpa tren nhPng Png suit
"dP'tai cic nut v l Png suat n l y duac cpng vao buPc tiep theo. Luc tao ra dupe tinh theo phuong p h i p bien dang deo Viseo.
3.2.Tinh chu^n sai so nang lUdng
Thong thuPng, phuong phap tfnh so cy the d day la phpong phap PTHH, ket q u i thu dupe cP khic biet so vPi thpe te. Viee tinh ehuan sai so ve n i n g luong cho ta nhPng danh gia thpe tien ve sp khae biet va hieu q u i eua phuong phap so vpi thpe nghiem.
3.2.1. Tinh chuan sai so theo h
Tinh chuan sai so bang phuong phap tang mat do phan tP theo h thi khi I p dung cho bai t o l n co mat dp phan tP IPn khong dUa ra duoc ket q u i chfnh xlc nhu mong muon. De khae phue dieu nay, eo the so sinh sai so chuan nang lupng P hai mat do cP luPi min nhat ke nhau theo Png suit tuong png 0,^1 va o^j(h| v l h^ kich thuPc luPithPI vathP2vPih^<h,).
^ = iLi'^ia- ''hi)^ D-'(a^- a,,)d£i]"= (16) Suyra:e^-(llii,J^-Su,„l^)"^ ^ (17) Ki"oe' Ks'^e • '-^" ^^^ '^ chuan nang lupng bien dang lUPi 1 v l 2 tuong Png.
Va tong sai so nang lupng:
E = {'LTJlEef (18) TP do, tacP the tinh duoc chuan sai so n i n g luong:
fll»ft2lP-|l"MlP
lluftzIP (19)
Cong thPc nay dupe dung cho cac bai toan khong cP diem di biet.
3.2.2. Hieu ehinh theo phep ngoai suy cua Richardson (ngoai suy theo h)
Dua theo quy t i e Runge, chuan sai so xac dinh theo so lupng ehuan nang lupng u duoetinh nhUsau:
U - Uj, = ch*^ + 5(h) (20) VPi U - Ket qua nang lUpng ehinh xlc, U^ - Ket q u i
nang lupng bang FEM theo mat dp luPi h, c - Hang so dpc lap vPi lUPi h, k - Cap dp chfnh xae eiia phUOng p h i p v l 5(h) la mpt so rat nho so vPi c^h". Trong truPng hpp nay, dp rpng eua luPi dupe g i l m dan kem vpi loai b6 bpt he so rat nho 5(h) de rim ra g i l trj gan dung eiia c va tP do tinh dupe ehuan sai so.
Thpe te, day la phuong phap ngoai suy ba diem theo chul'n nang lupng gan dung U^^ ll'y tP chuan nang lupng U_ thu dPpe tP sp bien thien h:
u-t-S (21)
Hieu s6U^-U la gia tri chuan sai so nang luong [7].
TPeong thPc (21) ehung ta eo the g i l trj chfnh xac n i n g lUong bien dang U, toe dp hpi ty k va he so C.
4 . V i D U A P D U N G
Kfeh thuPc: Vat can phii (de): Ban kinh 5mm, chieu cao 5mm.
M i n g mong: Ban kfnh 5 mm, chieu cao 1200nm.
Indenter: Loai hinh non (doi xuTig) d i m xuong m I n g toi da 120 nm ~ 10% ehieu day mIng de tranh I n h huPng cita de den ket q u i xlc dinh tinh chat cO hpe eOa mang. VI vay, m i t de va mat ben eo djnh.
Hmh 4.1: Mo hinh mang chju tii Bang phan tfeh PTHH vpi kieu lam mm h ta thu dUOe cic ket q u i sau:
- Moi quan he gia t l i vpi ehuyen vi:
Hinh 4.2: Do thj gia tii - chuyen vi TU Hinh 4.2, moi quan he giPa gia tai va chuyen vj, khi lpe tac dung tang lam dp chuyen vj cua mIng. DuPng eong chuyen vj II ham phi tuyen co g i l tri IPn nhat II 120nm. Su phi tuyen eua ehuyen vj phu hpp vPi dien tieh tiep xue thay doi theo lpe d i m t i n g dan.
- Phan bo Png suat mang mong:
Hinh 4.3: lfng suat cua mang khi chiu til TP Hmh 4.3. ehung tPi thl'y rang ufng sul't IPn nhat khi mIng chju tai la theo phuong phip tuyen eua be mat dau dam, tai nhPng vi tri nay se hinh thanh vet nPt va lan truyen ben trong lam eho ca'u mang vi vP neu Ipc d i m du IPn.
- D i n h g i l sai so ehuan nang lupng va toe do hpi tu:
Chon phan tPQ4 vPi dp min khic nhau eho moi tuPi, bac phan t P 3 d e d i n h gia sai so chuan n i n g luong va toe 121
KHOA HOC - CONG NGHE _
do hpi tu CO duoc ket q u i nhU Bdng 4.1. LuPi eang mjn, bac t u do IPn ning luong bien dang theo ehuyen vj nho, su that thoat nang lapng be, he lUc cing can b i n g va toe do hpi tu eang nhanh (Hinh 4.4).
Bing 4.1. Ket qui chuan sai so'theo Richardson Mat do
1 2 3 2 3 4 3 4 5
S^bac tir do
8262 20402 24642 20402 24642 26862 24642 26862 30S02
chuyen vi theo liToi) | 0-0347 0.0237 0.0201 0,0237 0.0201 0.0172 0.0201 0 0172 0.0117
0.0869
0.0332
0.0335 0.05225
0.0206
0.0241 k
1.445
1,1720
1.0506
Tii Hmh 4.4 ta thl'y nhom luPi 1,2,3 eo bac t u do tang dan tPIUPi 1 (8262) den luPi 3 (24642) v l nang lUpng bien dang theo chuyen vj giam tP 0.0237 den 0.0201, toe dp hpi tu IPn (1.445). Con nhPm lUPi mjn hon 2,3,4 va 3,4,5 nang luong bien dang theo ehuyen vi eo g i l trj nam l l n ein g i l tri 0 va toe dp hPI tu be.
Hinh 4.4: Quan he gida sai so chuan ning lupng va bac tudo phan tu ngoai suy theo Richardson De them do tin cay, chung toi ll'y so lieu chuyen vi nut khi tinh t o l n bang phan mem Matlab de so sinh vP mo hinh mp phong trong phan mem thuong mai Abaqus v l dupe ket q u i nhu sau:
19 28 37 46 55 64 73 82 91 100109 S6 biroc tinh (il MATLAB ABAQUS
Hinh 4.5: Bd thj quan he chuyen vi vi so budc tinh (i) Quan sit Hmh 4.5 cho thay g i i n do t l i - ehuyen vj duoc lap trinh tinh toan b i n g code Matlab vPi mo hinh 122
2D tuong tU mo hinh m I n g chju tai dupc mo phdng bang phan mem thuong mai ABAQUS la trung nhau tai diem dau va diem cuoi chuyen vj.Tai mpt so vi tri dUPng cong chuyen vj cua Matlab khong triing vPi Abaqus la do chung toi chi lay mot lUpng nhP diem nut ehuyen vj eiia Matlab de so sinh vPi t o l n bp diem nut ehuyen vi trong Abaqus. Mae khae, viee ehia luPi cua Abaqus duac thpe hien bang gPi mo-dun kin va tat nhien no khie vPi hinh dang luPi ma chung toi lap trinh b i n g code Matlab.
Tuy nhien, ket qua ciia mo hinh t o l n m l chung tpi xiy dung tuong dong vPi mo hinh mo phong b i n g Abaqus da chPng minh mo hinh bai toan phii hop. Dieu nay eo y nghia vo eiing quan trpng trong viec nghien cPu va phat trien bai t o l n vpi nhieu tham so khac nhau.
5. K^T L U A N
- Chung toi d l tien hanh x i y dpng mo hinh toln va tfnh t o l n mo phong mang mong 2D ehiu t l i . Cle ket qui tinh t o l n mo phong bang FEM vPi Matlab dupe so sanh, d i n h g i l so vPi phan mem Abaqus. Ket q u i thu dUpc la d i n g tin cay.
- D i n h gia dp ehfnh xlc cua IPI g i l i bang viee tinh chuan sai so theo h. Ket q u i cho thl'y m i t do lUpi eang min thi n i n g lUpng bien dang theo chuyen vj eing nho, tPe la that t h o i t n i n g lUpng cang be nen d i m bao nguyen ly can bang.
-TP mo hinh b l i t o l n nay se dUpc Png dung rpng rai trong nghien cPu che tao vat lieu mPl, die biet la nghien ePu ve tinh ehat co hoe ciia vat lieu cP cau true nano vdi nhieu tham so khac nhau.
Tai lieu tham khao
[1]. Feindt, Jared Alexander (2018), "An Improved Numerical Method for Assessing Cell Elasticity from Atomic Force Microscopy Nanoindentation Data" by Lehigh Preserve, pp.5-22.
[2]. S. Saravanan, K. Kaviya, M. Ravichandran, P.
Senthilkumar (2016},/nfernaf/ona/Jouma/o/"Tec/inoChem Research Stress-Strain Analysis of AA6063-5AND7.5 Wt. % TIC Nano Composites. ISSN:2395-4248, pp.127-132.
[3]. Bhattacharya, A. K. and Nix, Int. J. (1998), Solids Struct, Hibbitt, Karlsson and Sorensen Inc., ABAQUS, Version 6.3, User's Manual, Pawtueket, Rl. 24,881,9.
[4]. Oliver, W.C. and G.M. Pharr (1992), An Improved Technique for Determining Hardness and Elastic-Modulus Using Load and Displacement Sensing Indentation Expriment, )ouna\ of Materials Research 7(6): pp.n3-123.
[5]. l.N. Sneddon (1951), Fourier Transforms, pp.450- 467 (MeGrav^'-Hill, New/York).
[6]. H. Gao and T-W. Wu, J. Mater (1993), Res. 8,3229.
Ngay nhan bai: 14/4/2019 Ngay chap nhan dang: 27/4/2019 NgUdi phSn bien: PGS.TS. Le HOu Sam
T S . U Van Toan