采用數(shù)值分析的壓痕法識(shí)別陶瓷材料斷裂韌性的精度問題研究

王家梁,馬德軍,孫 亮,肖富君

(1.裝甲兵工程學(xué)院機(jī)械工程系,北京 100072;2.總裝備部南京軍事代表局,江蘇 南京 210024)

采用數(shù)值分析的壓痕法識(shí)別陶瓷材料斷裂韌性的精度問題研究

王家梁1,馬德軍1,孫 亮1,肖富君2

(1.裝甲兵工程學(xué)院機(jī)械工程系,北京 100072;2.總裝備部南京軍事代表局,江蘇 南京 210024)

采用虛擬裂紋閉合法的基本原理對(duì)基于數(shù)值分析的壓痕法識(shí)別陶瓷材料斷裂韌性的精度問題進(jìn)行研究,分析比較了3種典型基于數(shù)值分析的壓痕法計(jì)算公式對(duì)陶瓷材料斷裂韌性的測(cè)試精度．結(jié)果表明,Tang公式和Lee公式所測(cè)斷裂韌性值與理論計(jì)算值的最大誤差均隨比功增加而增大,其最大誤差分別為193.39%和81.99%;Hyun公式除對(duì)彈性模量E=70 GPa、We/Wt=0.7的材料測(cè)試誤差高達(dá)-75.59%之外,其余材料范圍測(cè)試誤差均處于30.65%之內(nèi);相比Tang公式和Lee公式,Hyun公式明顯具有較高的測(cè)試精度．本文工作為合理使用壓痕法準(zhǔn)確獲得陶瓷材料斷裂韌性值以及為下一步研究建立具有測(cè)試精度高、適用范圍廣的新型斷裂韌性測(cè)試方法提供了一定的理論基礎(chǔ)．

數(shù)值分析;壓痕法;陶瓷材料;斷裂韌性;比功

壓痕法作為陶瓷材料斷裂韌性的常用測(cè)試方法,其相比于表面裂紋彎曲法(SCF)[1-2]、山形切口梁法(CNB)[3-4]、單邊預(yù)裂紋梁法(SEPB)[5-6]、單邊切口梁法(SENB)[7-8]具有試樣尺寸小、測(cè)試效率高、無(wú)損等明顯優(yōu)勢(shì),一直以來(lái)受到國(guó)內(nèi)外學(xué)者的廣泛關(guān)注[9-15]．以往傳統(tǒng)壓痕法的理論基礎(chǔ)和公式建立系基于半解析與半經(jīng)驗(yàn)方法的聯(lián)合,在導(dǎo)出半解析半經(jīng)驗(yàn)公式過程中往往對(duì)應(yīng)力場(chǎng)的分析過于簡(jiǎn)單,導(dǎo)致斷裂韌性計(jì)算結(jié)果偏差較大．因此,近年來(lái),部分學(xué)者考慮采用有限元數(shù)值分析的方法得到被壓材料的應(yīng)力場(chǎng),從而試圖準(zhǔn)確地建立材料壓入響應(yīng)與斷裂韌性的內(nèi)在關(guān)系．

目前,有限元數(shù)值分析方法計(jì)算陶瓷材料斷裂韌性值主要有2種途徑:第一種途徑系通過疊加原理(superposition principle)間接計(jì)算獲得[16-18],即被壓材料在壓頭載荷作用下,裂紋尖端的應(yīng)力強(qiáng)度因子等于無(wú)裂紋時(shí)該壓頭載荷在裂紋處產(chǎn)生的內(nèi)應(yīng)力反作用于沒有壓頭載荷時(shí)裂紋表面所對(duì)應(yīng)的應(yīng)力強(qiáng)度因子;第二種途徑系通過粘結(jié)單元(cohesive elements)直接計(jì)算獲得[19-24],即通過粘結(jié)單元設(shè)置合理的斷裂準(zhǔn)則用以模擬兩部分之間的黏性連接,將其預(yù)先設(shè)置于裂紋開裂面可以模擬材料的斷裂．近幾年,基于上述2種數(shù)值分析方法獲得的壓痕公式形式較多,但究竟選用何種公式能夠獲得準(zhǔn)確的斷裂韌性值至今尚無(wú)定論．

本文基于虛擬裂紋閉合法[25]的基本原理對(duì)基于數(shù)值分析的壓痕法識(shí)別陶瓷材料斷裂韌性的有限元模型進(jìn)行計(jì)算,以此為基礎(chǔ),分析比較3種典型基于數(shù)值分析的壓痕法計(jì)算公式(Tang公式[17]、Lee公式[21]、Hyun公式[24])對(duì)陶瓷材料斷裂韌性的測(cè)試精度,以期為合理使用此類方法測(cè)試陶瓷材料斷裂韌性提供理論依據(jù)和技術(shù)基礎(chǔ)．

1 幾種典型基于數(shù)值分析的壓痕法介紹

大量研究表明[13-15,17,26-28],壓痕法測(cè)試陶瓷材料斷裂韌性過程中,材料沿壓痕對(duì)角線方向會(huì)形成2種裂紋開裂形式,即徑向裂紋(Radial Crack,RC)和半硬幣狀裂紋(Half-Penny Crack,HPC),如圖1．

2008年,Tang等[17]針對(duì)Vickers壓頭壓入陶瓷材料測(cè)試其斷裂韌性建立了三維有限元模型,通過量綱分析和疊加原理,分別建立了針對(duì)HPC和RC的2種開裂形式的陶瓷材料斷裂韌性KIC計(jì)算公式,從而獲得了基于數(shù)值分析的壓痕法測(cè)陶瓷材料斷裂韌性的Tang方法,其具體公式如下.

對(duì)于HPC裂紋開裂形式:

圖1 壓痕法測(cè)試陶瓷材料斷裂韌性的2種裂紋開裂形式

Fig.1 Schematic diagram of indentation method to determining the fracture toughness of ceramic materials

(1)

對(duì)于RC裂紋開裂形式:

(2)

其中:E和σy分別為材料彈性模量和屈服強(qiáng)度,c和a分別為實(shí)際裂紋開裂半長(zhǎng)和壓痕對(duì)角線半長(zhǎng),l=c-a,多項(xiàng)式系數(shù)取值如表1．

表1 Tang公式各系數(shù)取值

2012年,Lee等[21]采用cohesive interface elements對(duì)四棱錐剛性壓頭壓入脆性材料的壓頭幾何形狀、壓入載荷、cohesive參數(shù)、被壓材料參數(shù)分別對(duì)裂紋系統(tǒng)的影響進(jìn)行研究,并基于Lawn-Evans-Marshall模型提出了由壓入載荷、裂紋長(zhǎng)度、硬度、彈性模量和壓頭幾何形狀識(shí)別脆性材料斷裂韌性的Lee方法,其具體公式如下

(3)

Hd=Pmax/2a2，

(4)

(5)

其中：Ci(v)=χijvj,i=0、1、2,j=0、1,Lee公式各系數(shù)χij的取值如表2．

表2 Lee公式各系數(shù)取值

2015年,Hyun等[24]基于Lee等[21]的cohesive模型,檢驗(yàn)了材料參數(shù)對(duì)裂紋尺寸的影響,并提出了通過材料屈服應(yīng)變、泊松比、楊氏模量、壓入載荷和裂紋長(zhǎng)度識(shí)別材料斷裂韌性的Hyun方法,其具體公式如下:

(6)

(7)

(8)

(9)

fi(v)=gijvj,i, j=0,1,2

(10)

其中:Hyun公式各系數(shù)取值如表3,公式(10)各系數(shù)取值如表4．

表3 Hyun公式各系數(shù)取值

表4 公式(10)各系數(shù)取值

2 數(shù)值計(jì)算

為了分析比較3種典型基于數(shù)值分析的壓痕法計(jì)算公式(Tang公式[17]、Lee公式[21]、Hyun公式[24])對(duì)陶瓷材料斷裂韌性的測(cè)試精度,本文通過商用有限元Abaqus軟件[29]建立陶瓷材料Vickers壓入有限元模型．由于Vickers壓頭是面角為136°的正四棱錐,根據(jù)對(duì)稱性結(jié)合裂紋開裂方向去Vickers壓頭的1/4建立模型,如圖2,其被壓材料取相應(yīng)的1/4模型,整個(gè)被壓材料尺寸為4 000 μm×4 000 μm×2 000 μm(如圖3),壓頭最大壓入深度hm=5 μm．

圖2 1/4對(duì)稱的Vickers壓頭

Fig.2 Schematic diagram of the Vickers indenter 1/4 symmetry

圖3 1/4對(duì)稱的Vickers壓入有限元模型

Fig.3 Finite element model of the Vickers indentation 1/4 symmetry

定義壓頭為理想彈性壓頭,其彈性模量和泊松比分別為Ei=1 141 GPa,νi=0.07;定義被壓材料與金剛石Vickers壓頭的平面應(yīng)變彈性模量之比η=[E/(1-0.22)]/[1 141/(1-0.072)],被測(cè)陶瓷材料為均勻、各向同性的率無(wú)關(guān)固體,壓入比功[30]取值為:We/Wt=0.3～0.7,彈性模量取值分別為E=70 GPa、200 GPa、400 GPa和600 GPa,相應(yīng)地η=0.063 5、0.181 7、0.363 4和0.545 1,應(yīng)變硬化指數(shù)n=0(陶瓷材料為低硬化水平),相應(yīng)的屈服強(qiáng)度取值σy=1.4～40 GPa,ν=0.2．壓頭與被壓材料間的摩擦系數(shù)取定值0.16[31]．根據(jù)接觸核心區(qū)網(wǎng)格精細(xì),遠(yuǎn)場(chǎng)非核心區(qū)網(wǎng)格稀疏的原則,最終,RC和HPC裂紋開裂模型的壓頭均劃分為12 000個(gè)C3D4單元,被壓材料劃分為12 000個(gè)C3D8R單元和4 500個(gè)C3D4單元,其網(wǎng)格劃分如圖4．收斂性分析表明,兩模型網(wǎng)格分析誤差均不超過0.5%．

圖4 有限元模型的網(wǎng)格劃分

有關(guān)裂紋開裂面的建模,可通過被壓材料沿壓頭對(duì)角線截面的對(duì)稱性實(shí)現(xiàn),即開裂面不作對(duì)稱處理,未開裂面仍按對(duì)稱設(shè)置．其具體裂紋面幾何形狀通過虛擬裂紋閉合法[25]計(jì)算得到接近真實(shí)情況的等KIC值半橢圓面作為裂紋開裂面．定義名義壓痕對(duì)角線半長(zhǎng)a為壓頭達(dá)到最大壓入深度hm時(shí)對(duì)應(yīng)的理論壓痕對(duì)角線長(zhǎng)度的一半．建立裂紋開裂半長(zhǎng)c與名義壓痕對(duì)角線半長(zhǎng)a之比c/a=1.25、1.5、2.25、3、4.5和6的6種模型．其中,當(dāng)c/a=1.25、1.5、2.25時(shí),裂紋為RC開裂形式;當(dāng)c/a=3、4.5、6時(shí),裂紋為HPC開裂形式．

3 結(jié)果分析

對(duì)圖5～7基于數(shù)值分析壓痕法的3種數(shù)值公式理論誤差進(jìn)行分析可知,Tang公式和Lee公式所測(cè)斷裂韌性值與理論計(jì)算值的最大誤差均隨比功增加而增大,其最大誤差分別為193.39%和81.99%;Hyun公式除對(duì)彈性模量E=70 GPa、We/Wt=0.7的材料測(cè)試誤差高達(dá)-75.59%之外,其余材料范圍測(cè)試誤差均處于30.65%之內(nèi)(E=70 GPa、We/Wt=0.7的材料超出了Hyun公式的適用范圍),較其他壓痕法計(jì)算公式測(cè)試精度最高．

顯然,采用疊加原理的Tang公式誤差明顯高于采用粘結(jié)單元的Lee公式和Hyun公式,其主要原因在于,疊加原理適用于線彈性材料,雖然陶瓷材料具有較高的屈服強(qiáng)度和較小的應(yīng)變硬化指數(shù),但是將具有彈塑性參數(shù)的陶瓷材料近似用線彈性方法進(jìn)行疊加計(jì)算,從而得到的斷裂韌性值必然存在一定偏差．而粘結(jié)單元仿真結(jié)果的準(zhǔn)確性依賴于單元最大拉應(yīng)力σmax、網(wǎng)格尺寸e、粘結(jié)過程區(qū)(Cohesive Process Zone)LCZ的有效設(shè)定,雖不適用于較短裂紋開裂形式(如RC裂紋開裂形式),但相比于疊加原理間接計(jì)算KIC的方法顯然更為準(zhǔn)確．此結(jié)論可進(jìn)一步通過實(shí)驗(yàn)進(jìn)行驗(yàn)證．

圖5 c/a為1.25～6時(shí)不同η下的 (KIC-Tang-KIC-FEM)/KIC-FEM與We/Wt關(guān)系

圖6 c/a為1.25～6時(shí)不同η下的 (KIC-Lee-KIC-FEM)/KIC-FEM與We/Wt關(guān)系

圖7 c/a為1.25～6時(shí)不同η下的 (KIC-Hyun-KIC-FEM)/KIC-FEM與We/Wt關(guān)系

4 實(shí)驗(yàn)驗(yàn)證

本文實(shí)驗(yàn)選用的被測(cè)陶瓷材料為按照國(guó)際標(biāo)準(zhǔn)ISO14577-1[33]要求制備的SRM2100和Fused Silica壓入標(biāo)準(zhǔn)試樣塊．其中,SRM2100試樣為美國(guó)National Institute of Standards & Technology 提供的陶瓷斷裂韌性標(biāo)準(zhǔn)參考材料,其斷裂韌性標(biāo)準(zhǔn)值為4.572±0.228 MPa·m1/2;Fused Silica試樣為中國(guó)寶山鋼鐵股份有限公司提供的納米壓入儀用標(biāo)準(zhǔn)樣品(GSB 03—2496-2008),其斷裂韌性標(biāo)準(zhǔn)值為0.798±0.023 MPa·m1/2[34]．應(yīng)用先期獲得國(guó)家發(fā)明專利授權(quán)的高精度宏/微觀儀器化壓入儀[35]和金剛石Vickers壓頭對(duì)上述2種陶瓷材料進(jìn)行儀器化壓入實(shí)驗(yàn)．考慮到不同壓入載荷對(duì)被壓材料壓痕形貌的影響,對(duì)SRM2100試樣選用最大壓入載荷Pm=100 N進(jìn)行彈性模量和斷裂韌性測(cè)試,而對(duì)于Fused Silica試樣,由于其在壓入載荷較大時(shí)容易形成橫向裂紋(Lateral Crack)[28,36]從而影響彈性模量的識(shí)別精度,故選用最大壓入載荷分別為Pm=0.25 N和Pm=2 N進(jìn)行彈性模量和斷裂韌性測(cè)試．為保證測(cè)試結(jié)果的準(zhǔn)確性,每種材料實(shí)驗(yàn)重復(fù)進(jìn)行10次,取有效數(shù)據(jù)進(jìn)行計(jì)算．圖8和圖9分別為SRM2100試樣10次實(shí)驗(yàn)的Vickers壓痕形貌圖及載荷-位移曲線圖．圖10和圖11分別為Fused Silica試樣10次實(shí)驗(yàn)的Vickers壓痕形貌圖及載荷-位移曲線圖．其中, Fused Silica試樣在P=0.25 N時(shí)有1組實(shí)驗(yàn)數(shù)據(jù)由于載荷-位移曲線擬合指數(shù)較為奇異而舍去．

對(duì)已進(jìn)行了儀器化Vickers壓入測(cè)試的試樣利用光學(xué)顯微鏡分別量取Vickers壓痕實(shí)際對(duì)角線半長(zhǎng)a′=(a1+a2)/2和Vickers壓痕對(duì)角線方向上的裂紋開裂半長(zhǎng)c=(c1+c2+c3+c4)/4,如圖12,從而可以分別獲得2種材料的儀器化壓入有效測(cè)試結(jié)果,如表5～7所列．其中,材料彈性模量E由文獻(xiàn)[30]所提方法獲得,屈服強(qiáng)度σy由有限元數(shù)值方法通過圖13中所示關(guān)系獲得,Hn=Pm/A(hm)被定義為名義硬度,即最大壓入載荷Pm與金剛石Vickers壓頭橫截面積A(hm)之比．

進(jìn)一步根據(jù)Tang公式、Lee公式以及Hyun公式計(jì)算上述2種材料的斷裂韌性值,結(jié)果見表8、表9．各種方法所測(cè)2種材料的斷裂韌性值與其標(biāo)準(zhǔn)值的相對(duì)誤差見表10,其中SRM2100試樣的標(biāo)準(zhǔn)值KIC-S=4.572±0.228 MPa·m1/2,Fused Silica試樣的標(biāo)準(zhǔn)值KIC-S=0.798±0.023 MPa·m1/2．

圖8 SRM2100試樣10次實(shí)驗(yàn)的Vickers壓痕形貌

圖9 SRM2100試樣10次實(shí)驗(yàn)的載荷與位移曲線

由表10可以看出,SRM2100和Fused Silica 2種材料應(yīng)用Tang公式獲得的斷裂韌性均值與其標(biāo)準(zhǔn)值的相對(duì)誤差分別為74.97%和44.86%,Lee公式相對(duì)誤差分別為24.89%和24.06%,Hyun公式相對(duì)誤差分別為11.76%和21.05%,上述公式的實(shí)驗(yàn)測(cè)試結(jié)果相對(duì)誤差與理論分析誤差分布趨勢(shì)一致且均符合其理論誤差范圍．

為了進(jìn)一步驗(yàn)證上述3種壓痕法測(cè)試誤差結(jié)論,本文另外對(duì)中國(guó)建筑材料科學(xué)研究總院陶瓷科學(xué)研究院提供的4種典型陶瓷材料(Si3N4、ZTA、ZrO2、Al2O3)進(jìn)行斷裂韌性儀器化壓入實(shí)驗(yàn),其Vickers壓痕形貌圖如圖14,測(cè)試結(jié)果見表11．其中,試樣密度分別為3.21 g/cm3、6.02 g/cm3、4.28 g/cm3和3.95 g/cm3,采用高純超細(xì)粉料(原料純度≥99.9%)通過等靜壓成型方法制得．

由表11測(cè)試結(jié)果可以看出,3種壓痕法所測(cè)4種陶瓷材料的斷裂韌性值各不相同且差異較大,但各方法所得測(cè)試結(jié)果大小分別符合圖5～7各方法理論誤差范圍;同時(shí),不同壓痕法對(duì)4種材料的測(cè)試結(jié)果符合各自理論誤差隨比功的分布趨勢(shì)．以上實(shí)驗(yàn)結(jié)果充分說明,本文對(duì)3種基于數(shù)值分析壓痕法測(cè)試陶瓷材料斷裂韌性KIC值的精度分析是準(zhǔn)確客觀的,同時(shí),Hyun公式相比其他壓痕法計(jì)算公式測(cè)試精度較高．

5 結(jié) 論

本文采用虛擬裂紋閉合法的基本原理對(duì)基于數(shù)值分析的壓痕法識(shí)別陶瓷材料斷裂韌性精度問題進(jìn)行研究,分析比較了3種典型基于數(shù)值分析的壓痕法計(jì)算公式對(duì)陶瓷材料斷裂韌性的測(cè)試精度和所測(cè)材料的適用范圍,結(jié)論如下.

(1)Tang公式和Lee公式所測(cè)斷裂韌性值與理論計(jì)算值的最大誤差均隨比功增加而增大,其最大誤差分別為193.39%和81.99%;

圖10 Fused Silica試樣10次實(shí)驗(yàn)的Vickers壓痕形貌

圖11 Fused Silica試樣10次實(shí)驗(yàn)的載荷與位移曲線

圖12 Vickers壓痕及裂紋開裂半長(zhǎng) 圖13 基于有限元數(shù)值方法計(jì)算的陶瓷材料σy/Hn與We/Wt關(guān)系

Fig.12 Schematic diagram of the Vickers impression based and crack half-length Fig.13 The relationship of ceramic materials betweenσy/HnandWe/Wton the finite element numerical method

表5 最大壓入載荷為100 N時(shí),SRM2100試樣的儀器化壓入測(cè)試結(jié)果

表6 最大壓入載荷為0.25 N時(shí),Fused Silica試樣的儀器化壓入測(cè)試結(jié)果

表7 最大壓入載荷為2 N時(shí),Fused Silica試樣的儀器化壓入測(cè)試結(jié)果

注:①We/Wt、E、σy為表6獲得的實(shí)驗(yàn)數(shù)據(jù);② 有2組實(shí)驗(yàn)數(shù)據(jù)由于裂紋開裂尺寸c/a過小而舍去.

表8 3種壓痕法對(duì)SRM2100試樣斷裂韌性的測(cè)試結(jié)果

Tab.8 The measurement results of fracture toughness for SRM2100 with three indentation method

序號(hào)KIC-TangKIC-LeeKIC-Hyun17.985.965.3428.355.815.2137.905.745.1447.935.484.9158.515.855.2568.175.975.3677.775.605.0287.425.434.8797.995.765.16108.155.634.98平均8.005.715.11

表9 3種壓痕法對(duì)Fused Silica試樣斷裂韌性的測(cè)試結(jié)果

Tab.9 The measurement results of fracture toughness for Fused Silica with three indentation method

序號(hào)KIC-TangKIC-LeeKIC-Hyun10.451.020.6620.441.000.6430.451.020.6540.461.050.6750.410.920.5960.420.950.6170.430.970.6280.420.950.61平均0.440.990.63

(2)Hyun公式除對(duì)彈性模量E=70 GPa、We/Wt=0.7的材料測(cè)試誤差高達(dá)-75.59%之外,其余材料范圍測(cè)試誤差均處于30.65%之內(nèi);相比Tang公式和Lee公式,Hyun公式明顯具有較高的測(cè)試精度.

圖14 4種典型陶瓷材料在Pmax=100 N時(shí)的Vickers壓痕形貌

表10 3種壓痕法所測(cè)SRM2100和Fused Silica試樣斷裂韌性值與其標(biāo)準(zhǔn)值的相對(duì)誤差

表11 3種基于數(shù)值分析壓痕法對(duì)其他幾種陶瓷材料斷裂韌性的測(cè)試結(jié)果

Tab.11 The measurement results of fracture toughness for other ceramic materials with three indentation method

試樣We/WtKIC-TangKIC-LeeKIC-HyunSi3N40.4276.985.024.84ZTA0.4018.786.196.39ZrO20.3928.135.905.88Al2O30.3787.705.516.08

(3)本文采用的虛擬裂紋閉合法計(jì)算陶瓷材料斷裂韌性值簡(jiǎn)潔、有效,為下一步研究建立具有測(cè)試精度高、適用范圍廣的新型斷裂韌性測(cè)試方法提供了一定的理論基礎(chǔ)．

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(責(zé)任編輯 蘇曉東)

Study on Accuracy Problem of Indentation Method Based on the Numerical Analysis to Evaluating the Fracture Toughness of Ceramic Materials

WANG Jia-liang1, MA De-jun1, SUN Liang1, XIAO Fu-jun2

(1.Department of Mechanical Engineering, Academy of Armored Force Engineering, Beijing 100072, China;2. Nanjing Military Representative Bureau, General Armament Ministry, Nanjing 210024,China)

By using the virtual crack closure technique, the accuracy problem of indentation method, which is based on the numerical analysis to evaluating the fracture toughness of ceramic materials, is studied. The accuracy of three typical formulas established by numerical analysis are analyzed and compared. The results show that, for Tang and Lee formulas, the errors between measured and theoretical values increase withWe/Wtincreasing, and the maximum errors are 193.39% and 81.99%, respectively. The maximum errors of the Hyun formula are all within 30.65% except the material withE=70 GPa andWe/Wt=0.7( errors up to -75.59%). Compared with Tang and Lee formulas, Hyun formula has obviously higher test accuracy. This work provides a theoretical basis for reasonably using indentation method to obtain the values of ceramic materials' fracture toughness, and lays a foundation for establishing a new testing method with high accuracy and wide range of application.

numerical analysis; indentation method; ceramic materials; fracture toughness; the ratio of unloading work to total loading work

1004-8820(2015)04-0277-12

10.13951/j.cnki.37-1213/n.2015.04.009

2015-04-23

軍隊(duì)科研計(jì)劃項(xiàng)目(2014CJ011)．

王家梁(1986- )，男，陜西咸陽(yáng)人，博士研究生，研究方向?yàn)椴牧狭W(xué)性能測(cè)試技術(shù).

TQ174.75

A