Determination of residual stresses and material properties by an energy-based method using artificial neural networks

Transcription

1 296 Poceedngs of he Esonan Academ of Scences, 2012, 61, 4, Poceedngs of he Esonan Academ of Scences, 2012, 61, 4, do: /poc Avalable onlne a Deemnaon of esdal sesses and maeal popees b an eneg-based mehod sng afcal neal newoks Hongpng Jn, en Yang, and Ln Yan The Sae Ke Lab of Dgal Manfacng Eqpmen and Technolog, Mechancal Scence and Engneeng, Hazhong Unves of Scence and Technolog, han , Chna Receved 29 Jl 2011, evsed 20 Decembe 2011, acceped 4 Jana 2012, avalable onlne 20 Novembe 2012 Absac. h he help of an eneg-based mehod and dmensonal analss, an afcal neal newok model s consced o exac he esdal sess and maeal popees sng sphecal ndenaon. The elaonshps beween he wok of esdal sess, he esdal sess, and maeal popees ae nmecall calbaed hogh anng and valdaon of he afcal neal newok (ANN) model. The enable he dec mappng of he chaacescs of he ndenaon paamees o he eq-baxal nfom esdal sess and he elasc plasc maeal popees. The poposed ANN can qckl and effecvel pedc he esdal sess and maeal popees based on he load deph cve of sphecal ndenaon. Ke wods: sphecal ndenaon, esdal sess, maeal popees, fne elemen analss. 1. INTRODUCTION Deemnng he esdal sess s ccal n a wde vae of ssems becase esdal sess n maeals and sces s assocaed wh fage, cooson, wea, and fales of he ssems. The adonal mehods of measng esdal sess can be dvded no wo caegoes: descve and non-descve mehods. Descve mehods, ncldng hole-dllng and sawcng echnqes, ec., can geneall be sed o mease esdal sess qanavel who an efeence sample, b he have descve chaacescs. Non-descve mehods, ncldng X-a and neon dffacon, ec., can mease he esdal sess who descon. Howeve, he ae geneall expensve and complex. Moeove, he ae hghl sensve o sch meallgcal facos as gan sze and exe [1]. Insmened ndenaon has been sed o mease he mechancal popees of maeals [2 7], sch as hadness H, elasc modls E, san-hadenng expo- Coespondng aho, mewang@mal.hs.ed.cn nen n, and eld sengh. Dng ndenaon, a gd ndene s peneaed nomall no a homogeneos sold whle he ndenaon load F and deph h ae connosl ecoded dng one complee ccle of loadng and nloadng. Becase he ndenaon load deph cve eflecs he behavo of he maeal nde ndenaon loadng and nloadng, ndenaon can be sed o evalae maeal popees. Howeve, fne elemen model (FEM) and expemenal nvesgaons have shown ha esdal sess has a sgnfcan effec on he load deph cve. Geneal obsevaons eveal ha ensle sess ends o sech o he load deph cves, a smalle foce can psh he ndene down o he same ndenaon deph, whle compessve sess compesses he cves, and hs a lage load s eqed o acheve he same ndenaon deph [8]. Ths s he eason wh ndenaon can also be sed o evalae he esdal sess. Seveal mehodologes have been nodced o esmae esdal sess sng nsmened ndenaon. The eales effo was o examne he elaonshp beween hadness and esdal sess. The esls sgges ha hadness measemen ma be sed o

2 H. Jn e al.: Resdal sesses and maeal popees 297 chaaceze esdal sess n maeals [8,9]. Howeve, he change n he hadness b he esdal sess s less han 10% of s nsessed vale, and he effecs of compessve sess ae ofen no as lage as hose of ensle sess. In addon, he effecs of esdal sess on he ndenaon pofle ae elavel lage, e.g., he esdal sess can change he sae of ple-p o snk-n, whch cases a lage eo n he aea measemen, and hese eos esl n a lage eo n he esdal sess measemen becase deemnaon of he hadness eqes measemen of he ndenaon aea. Sesh and Gannakopolos [8] sed he dffeence n he conac aea of sessed and nsessed specmens ndened o he same deph o deemne he esdal sess. Based on hs mehod, Lee and Kwon [10] developed a new mehod o esmae esdal sess b analsng he sface sess effec on conac pesse n ems of shea plasc dng deph-conolled ndenaon. Compason wh he esls of convenonal saw-cng ess showed ha he ndenaon es cold be effecvel and easl sed fo he assessmen of esdal sess. Howeve, he mehods fo esdal sess deemnaon eqe an nsessed efeence sample and he esdal ndenaon aea needs o be meased. Chen e al. [11], Zhao e al. [12], and Yan e al. [13] poposed alenave mehods o mease he esdal sess and pefecl elasc plasc popees b sng dmensonal analss and evese analss fo concal mcondenaon, especvel. The mehods do no eqe measemen of a conac ads and a efeence sess-fee maeal. Howeve, he accac of sch appoaches s hghl dependen on he pecson of he cve o sface fng pocede as well as he obsness of he evese analss algohm. X and L [14] fond ha he ao of elasc ecove dsplacemen of nanondenaon o he maxmm peneaon deph has a lnea elaonshp o he ao of esdal sess o eld sess. Theefoe, he pesened an empcal model fo esdal sess deemnaon fom he elasc ecove dsplacemen of nanondenaon. Howeve, fo ve sof maeals, hs ao s bascall ndependen of he esdal sess. Ths, hs model ma be sable fo he deemnaon of esdal sess onl n ve had maeals. Based on he pemse ha elasc nloadng esponses dng ndenaon ae fll ndependen of esdal sess, ang e al. [15] deved a fomla fo deemnng he esdal sess sng shap ndenaon and he eneg mehod. Ths model assmes ha hee s no vaaon n he nden angle afe he emoval of he load. Howeve, s known ha he sface pofle ndegoes a lage change dng he nloadng sage de o he elasc ecove of he maeal. Fo hs eason, hs assmpon wold oveesmae he esdal sess. Dean e al. [16] fond ha he peak ndenaon load s fal sensve o he pesence of esdal sess and he sed nanondenaon o mease esdal sesses n sface laes. The poned o ha hs echnqe s well sed fo he mappng of esdal sesses ove he sface of a componen becase nanondenaon nvolves nvesgaon of elavel small volmes of maeal. The also aged ha sng hadness as a meased paamee o deemne esdal sess wll case dffcles, snce he sensv of hadness s lowe and less conssen becase he nflence of plep on he aea of conac s lage. Swadene e al. [17] obseved ha sphecal ndenaon s moe sensve o sess effecs han shap ndenaon. Thence, he poposed wo mehods of deemnng baxal esdal sess sng sphecal ndenaon. One s based on he fac ha he conac pesse a he onse of eldng s affeced b he esdal sess, whch can be analsed b Hez conac mechancs, and hen he baxal sess can be deemned va a closed fom analcal solon. The ohe s based on he empcal obsevaon ha he cves epesenng he mean conac pesse vess nomalzed conac ads ae vecall shfed as compaed o he nsessed maeal b an amon ve close o he magnde of he esdal sess. In spe of he seemngl exensve leae on how o ncopoae esdal sess b sng ndenaon, hee ae seveal shocomngs n hese mehods. The ehe eqe measng he ndenaon aea accael, o onl can be sed o pacla maeals, e.g., pefecl elasc plasc maeals. In addon, some assmpons of he model esl n lage eos n he esdal sess measemen. De o he complex of he conac poblem as well as hgh nonlnea of maeals nvolved n ndenaon, closed fom solons fo he ndenaon cve ae no eadl avalable. Based on he FEM, he polnomal fncon of ndenaon was esablshed b sng he dmensonal analss mehod, b s hghl dependen on he pecson of he cve o he sface fng pocede. Afcal neal newoks (ANNs) ae nvesal appoxmaons, whch have been mahemacall poved o be able o appoxmae an connos nonlnea fncon abal well as long as he conan a leas one hdden lae. Models of ANN have been sccessfll consced o solve man engneeng poblems. As ANNs do no eqe a complcaed mahemacal model, he have been wdel appled o solve some complcaed poblems of ndenaon. Seveal eseaches [18 23] adoped ANNs o chaaceze maeal popees sng nsmened ndenaon. In hs pape, he shape of he load deph cve s analsed. Based on he eneg mehod and he dmensonal analss mehod, he daabase elaonshp

3 298 Poceedngs of he Esonan Academ of Scences, 2012, 61, 4, beween he eneg of esdal sess, esdal sess, and maeal popees s consced. Revese analss sng he ANN appoach o chaaceze he mechancal popees s esablshed fom he fne elemen smlaon esls of sphecal ndenaons on elasc plasc san-hadenng maeals. The esdal sess and maeal popees can be exaced fom ndenaon cves sng he ANN appoach. 2. THEORETICAL ANALYSIS In an ndenaon es, an ndene s pessed no he specmen s sface o each a pese maxmm deph o a pese maxmm foce, and hen he ndene s emoved. Dng hs pocede, s assmed ha he ndened sface s sbjeced o an eq-baxal n-plane esdal sess and ha he esdal sess s nfom ove he deph of he ndenaon. In addon, assmng he nloadng pocess s naffeced b he esdal sess, he ndenaon has an elasc plasc loadng and elasc nloadng esponse. Seveal sdes have fond ha he shape of he loadng and he nloadng cves of he sessed sample devaes fom he deal shape of he nsessed sample [8,10,14], as shown n Fg. 1. A sessed sample sbjeced o ndenaon, f compaed o an nsessed one, s naall expeced o exhb a dffeen mechancal esponse n ems of he ndenaon cve and mpn geome. If he nal sae of he esdal sess s pedomnanl ensle, he maeal geneall sas eldng a a lage deph appled o he ndene fo a gven ndenaon load, as compaed o he case of an nsessed specmen; vce vesa, f s compessve. In ohe wods, fo a gven ndenaon deph, he esdal Fg. 1. Indenaon load deph cves fo specmens wh and who esdal sess. compessve sess ends o ncease he cvae and he maxmm peneaon load wll be lage (Fg. 1, cve OCD), wheeas a esdal ensle sess edces he cvae and esls n a lowe ndenaon foce (Fg. 1, cve OEF). Also he amon of maeal snkn/ple-p a he conac bonda ns o o be ve sensve o he pesence of nal sess [24,25]: namel, snk-n nceases when he maeal s sbjeced o ensle sess and ple-p nceases n he compessve sess case. Consde he case of a dffeen esdal sess. If he esdal deph s he same as ha of he nsessed sample, shold ncease he ndenaon deph fo he esdal compessve sess, o decease he ndenaon deph fo he esdal ensle sess. Howeve, n case of esdal compessve sess, shold exend he cve OC and BA n Fg. 1 hogh o nesec a pon G. Tha s, f he ndenaon deph s h 1 when he sample s sbjeced o he esdal compessve sess, afe complee nloadng he esdal deph wll be h 0. Smlal, fo he case of esdal ensle sess, he loadng cve OE nesecs he cve AB a he pon H. Tha s, when he ndenaon deph s h 2 f he nal sae of self-sess s ensle, hen he esdal deph wll be h 0 afe complee nloadng. Dng loadng, he aea enclosed b he loadng cves epesens he ndenaon load eneg, e.g. he aea of OAN. hen compleng nloadng, he aea enclosed b he nloadng cves epesens he elasc ecoveable eneg, e.g. he aea of BAN. hen he specmen s fee of sess, he ne aea enclosed b he loadng and nloadng cves epesens he eneg los n plasc defomaon, e.g. he aea of OAB. e wll assme eqvalence n he plasc eneg fo ndens when he esdal deph s he same. Ths, when he ndenaon deph s h 1 wh he esdal compessve sess o h 2 wh he esdal ensle sess, he plasc eneg s he same as ha of he ndenaon deph h 0 wh fee sess becase he have he same esdal ndenaon deph h 0. In ohe wods, he plasc eneges of all of hem ae n he aea of OAB n all cases he esdal deph s h. 0 Becase he aea of OGAB o OHB as shown n Fg. 1 s he sm of he eneg n plasc defomaon and he eneg conbon of he esdal sesses, he aea of OGA o OAH s he eneg conbon of he esdal sesses. Dmensonal analss has been sccessfll sed o analse ndenaon esponse. Based on dmensonal analss and FEM, Cheng and Cheng [26] pesened seveal scalng elaonshps ha povde a new nsgh no he shape of ndenaon cves. Fo he sphecal ndenaon of elasc plasc maeal, dng he loadng pocede he ndenaon load F ms be a fncon f L of he followng nne ndependen paamees: Yong s modls E, Posson s ao ν of he elasc plasc sold, Yong s

4 H. Jn e al.: Resdal sesses and maeal popees 299 modls E, Posson s ao ν of he elasc ndene, he eld sengh, he san-hadenng exponen n, he esdal sess, he ndenaon deph h, and he ndene ads R,.e.: L F = f ( E, ν, E, ν,,, n, h, R). (1) Usng he edced Yong s modls: ν 1 ν = +. (2) E E E Eqaon (1) can be edced o L F = f ( E,,, n, h, R). (3) B applng he Π heoem n dmensonal analss, Eq. (3) can be expessed as F E R =Π 2 α h,, n,, h whee he dmensonless fncon Π α elaes he ndenaon esponse o he mechancal popees and ndenaon paamees. The wok done b ndenaon s (4) h h 2 3 E = Fdh = 0 0 h Π αdh = hπβ,, n. (5) Becase nloadng akes place afe loadng when he ndene eaches he maxmm ndenaon deph h, he ndenaon nloadng load F s a fncon f of en ndependen paamees: F = f ( E, ν, E, ν,,, n, h, h, R). (6) Smlal, F can be expessed as: 2 E h R F = hπγ,, n,,. h h The wok done b maeal ecove s (7) Ths, n wo dffeen ndenaon dephs, e.g. h 1 and h, 0 he ao of he wok done b esdal sess o he wok done b ndenaon load s E E 3 3 h1πψ 1,, n h0πψ0, n 1 p1 p0 = = h1πβ1,, n E E =Πχ,, n. (10) Combnng Eqs (5), (8), (9), and (10) leads o E 1 =Π1,, n, 1 E 1 =Π2,, n. 1 (11) (12) Ths, he wo nvesal dmensonless fncons, Π 1 and Π, 2 can be sed o elae he ndenaon esponse o he mechancal popees. 3. FINITE ELEMENT SIMULATION In hs wok, elasc plasc ndenaon was smlaed b sng he ABAQUS sofwae on HP woksaons [27]. The ndene was modelled as a sphee wh a ads of mm, and he specmen was eaed as a bod of evolon. The mesh sed o model he specmen s shown n Fg. 2. The bonda condons wee defned as he olle bonda along he axs of smme, whle he boom bonda condon can also be defned as a olle bonda. The model was compsed of CAX3 mesh nea he conac egon and 4400 CAX4R mesh fhe awa fom he conac egon o ense nmecal accac. A he maxmm load, he mnmm nmbe of conac elemens n he conac zone was no less han 25 n each FEM com- h h 2 3 E = Fdh= h γdh h,, n. h Π = Π h ϕ (8) Based on Eqs (5) and (8), he wok done b esdal sess and plasc defomaon s 3 E p = = hπψ,, n. (9) Fg. 2. The fne elemen mesh.

5 300 Poceedngs of he Esonan Academ of Scences, 2012, 61, 4, paon. The mesh was well esed fo convegence and was deemned o be nsensve o fa-feld bonda condons. In all fne elemen compaons, he ndene was modelled as a gd bod; he vale of he fcon coeffcen beween meallc sfaces and damond s n he ange of 0.10 o 0.15 as epoed b Tabo [28]. A common consan vale of 0.15 fo fcon beween he damond ndene and meallc maeals was sed hogho hs sd; and lage defomaon FEM compaon was pefomed. The eq-baxal compessve o ensle sess was added o he model b pescbng an nal sess n he fne elemen model. All smlaons wee pefomed n wo dffeen load nload ndenaon ccles, he ndenaon deph was 30 µm and 50 µm, especvel. The ndened maeal was modelled as a homogeneos elasc plasc von Msses sold wh soopc hadenng. Te sess and e san ae elaed va he followng eqaon: = Eε, ε E. (13) n n = ( E ) ε, ε E. To cove a wde vae of elasc plasc maeals, Yong s modls E = 10, 50, 90, 130, 170, 210 GPa and eld sengh = 200, 300, 500, 700, 1000, 1800, 2100, 3000 MPa wee sed. The vales of E wee vaed ove he ange 100 E 1000, whch coves mos meals. Posson s ao s no an mpoan faco n he ndenaon expemen [26], and fo mos engneeng maeals ν 0.3. The san-hadenng exponen was vaed fom 0 o 0.5; fo mos meals n s beween 0.1 and 0.5. The load deph cve of he ndenaon was obaned b ecodng he eacon foces on he ndene and he coespondng appled dsplacemens. The focs of hs sd was he ndenaon load deph behavo; heefoe analss of load deph cves was sed n ndenaon echnqes o povde a measemen of he wok. So he pope paamees wee se n solon conols of ABAQUS o ense he seps of loadng and nloadng wee geae han 20. In hs sd, boh loaded and nloaded ndenaons wee sded o esablsh how he esponse of sphecal ndenaon was nflenced b he esdal sess and maeal popees. Resdal ndenaon deph h and maxmm load F m wee obaned fom he load deph cves fo each gven h. The oal wok and evesble wok wee obaned b negang he loadng and nloadng cves, especvel. The dmensonless fncon can be nmecall deved fom exensve fne elemen smlaons. The sfaces denoed b he fncons Π 1 and Π 2 ae shown n Fg. 3 and Fg. 4. The llsae he vaaons of and wh espec o E,, and n fo dffeen ndenaon dephs. I can be seen fom Fg. 3a and Fg. 4a ha nceases wh n, b deceases wh nceasng E (o I s eas o ndesand ha when E (o ). ) nceases o n deceases, he maeal s sofe, so he elasc ecove of he maeal s smalle. As he ndenaon deph nceases, he wok done b loadng nceases fase han he wok done b nloadng. So deceases wh nceasng ndenaon deph. Fom Fg. 3b and Fg. 4b we can see ha he effecs on of compessve sess ae ofen no as lage as ensle sess, especall n he case of smalle E. hen E s smalle, he effecs on of n ae lage n he case of ensle sess. Fges 3b and 4b ae smla becase he change n he amon of and s appoxmae n he case of 30 µm and 50 µm. If we oban he fncon elaonshp of o, and E,,, n especvel, hen E, and n can be deemned. I can be, (a) (b) Fg. 3. Vaaon of (a) / and (b) / based on he ndenaon deph of 30 µm.

6 H. Jn e al.: Resdal sesses and maeal popees 301 (a) (b) Fg. 4. Vaaon of (a) / and (b) / based on he ndenaon deph of 50 µm. seen fom Fg. 3 and Fg. 4 ha he elaonshp beween o and E,, and n s a nonlnea fncon. De o he complex of he fncons, s dffcl o ge an analc fncon o descbe he elaonshp; a al and eo pocede s nomall necessa. Moeove, he accac of sch an appoach s hghl dependen on he pecson of he sface fng pocede as well as he obsness of he evese analss algohm. 4. ARTIFICIAL NEURAL NETORKS The flowcha n Fg. 5 olnes he pocess of deemnng he esdal sess and maeal popees b sng ANNs. Fne elemen smlaons ae sed o oban he elaonshp beween he dmensonless fncons and he esdal sess and maeal paamees. To fll an/es he ANNs, he dmensonless fncons ae newoks nps and he esdal sess and maeal paamees ae newoks op. The nps o ANN P ae ( h003) ( h003) ( h003) ( h003) P =, (14) ( h005) ( h005) ( h005) ( h005) whee he sbscps h 003 and h 005 epesen he ndenaon dephs of 30 µm and 50 µm, especvel. The ops fom he newok T ae T E. = n (15) Fg. 5. Flowcha fo ANN. Back-popagaon mllae feed-fowad ANN s ceaed b sng he Neal Newok Toolbox n Malab [29]. I compses he np lae, he hdden lae, and he op lae. The nmbe of neons n he np and op laes of he ANN s dencal o he nmbe of np and op paamees, especvel. Howeve, he nmbe of neons n he hdden laes of he neal newok s calbaed dng he anng and valdaon pocess. The angen sgmod ansfe fncon s sed n he ANN. The sfaces descbed b fncons Π 1 and Π 2 as shown n Fg. 3 and Fg. 4 ae sed o seve as he anng and valdang daa ses fo ANN models. Tanng s he pocess of epeaed applcaons of he back-popagaon algohm nl he eo becomes accepable o some ohe ceon s acheved. Snce he dffeence beween he anng daa s lage, hs wll esl n a

7 302 Poceedngs of he Esonan Academ of Scences, 2012, 61, 4, lage ANN pedcon eo. Ths, s necessa o nomalze he expemen daa befoe applng hem o he newok. The nomalzng eqaon s x k x x = x x max mn mn, (16) whee x s he eal vale of he vaable befoe nomalzaon, x mn and x max ae he mnmm and maxmm vales of he vaable x. The ae nomalzed o he vales x. k The anng algohm of gaden descen wh momenm s sed n o sd o an he ANNs. The daa fo anng and valdaon of ANN wee obaned nmecall hogh 280 fne elemen smlaons. O of he 280 ses of np and op daa, 240 ses wee andoml assgned as anng daa whle he emanng 40 ses wee sed fo valdaon pposes. The mean sqae eo (MSE) of he newok ops and he age vales ae sed as he newok pefomance ndcao. Dng anng, he leanng le s sed o eavel adjs he weghs and bases of he newok n ode o move he newok ops close o he age vales b mnmzng he newok pefomance ndcao. In an effo o moe popel eflec nemedae vales of ndenaon, he nmbes of neons n hee hdden laes ae 200, 100 and RESULTS AND DISCUSSION Afe he ANNs ae sccessfll aned and esed, he ANN model maps he fnconal elaonshp beween he dmensonless fncon and he mechancal paamees. In ode o examne he accac of he ANN appoach, seveal nmecal expemens of ndenaon wee pefomed. The esls of esng condced on 35 expemenal daa ses ae pesened n Fg. 6. I shold be noed ha hese ses of fne elemen esls wee no sed n he anng and valdaon pocess descbed n Secon 4. I can be obseved fom Fg. 6 ha he poposed ANN model pedced he esdal sess and maeal popees easonabl accael. The maxmm eo s less han 9% beween he ognal np daa and evese analss fo all examned, less han 13% fo n, and less han 4% fo E. Thee ae poenall man soces of eos n phscal expemens, whch esl n calclaon eos of he esdal sess and maeal popees. Theefoe, we wll nvesgae he sensv of hs mehod fo eos n he meased paamees sch as and. Fs, of he ndenaon deph of 30 µm s gven 2% eo, whle he ohe paamees ae nchanged. The poposed ANN model s condced o oban he esdal sess and maeal popees. The esls ae smmazed n Fg. 7a. The maxmm eo s lage han 30% fo, 25% fo n, and 46% fo E. Smlal analss of s examned. Compang he np daa wh hose denfed fom ANN n Fg. 7b, we can fnd ha mos evese analss esls ae fal accae of san-hadenng exponen and. Howeve, he eo of E s lage. Ths means ha E obaned fom ANN algohm ae moe sensve o he meased paamees eos compaed o and n. Compang Fg. 7a and Fg. 7b, we can fnd ha he sensv fo he eo of s lage han he eo of. Ths s becase he elevance of and s enhanced, so he eo of has edced he mpac on E and n. Fg. 6. Idenfcaon of E /, /, and n fo ANN.

8 H. Jn e al.: Resdal sesses and maeal popees 303 (a) (b) Fg. 7. Sensv of E /, /, and n fo ANN: (a), (b). 6. CONCLUSIONS An eneg-based mehod and dmensonal analss of ndenaon paamees wee appled o consc an afcal neal newok model n ode o exac he esdal sess and maeal popees based on sphecal ndenaon. The elaonshps beween he wok of esdal sess, he esdal sess, and maeal popees wee nmecall calbaed hogh anng and valdaon of he ANN model. The enable he dec mappng of he chaacescs of he ndenaon paamees o he esdal sess and he elasc plasc maeal popees. The poposed ANN can qckl and effecvel pedc he esdal sess and maeal popees based on he load deph cve of sphecal ndenaon. I was fond ha E obaned fom he ANN algohm was moe sensve o he eos of he meased paamees compaed o and n. ACKNOLEDGEMENTS Ths wok was sppoed b gans fom he Majo Sae Basc Reseach Developmen Pogam of Chna (973 Pogam) (No. 2009CB724306) and he Naonal Naal Scence Fondaon of Chna (No ).

9 304 Poceedngs of he Esonan Academ of Scences, 2012, 61, 4, REFERENCES 1. L, J. Handbook of Measemen of Resdal Sesses. The Famon Pess, Llbn, Pehca, J. B., Hchngs, R., and Olve,. C. Hadness measemen a peneaon dephs as small as 20 nm. Phlos. Mag. A, 1983, 48, Olve,. C. and Pha, G. M. An mpoved echnqe fo deemnng hadness and elasc modls sng load and dsplacemen sensng ndenaon expemens. J. Mae. Res., 1992, 7, Tnvs, K., Bsso, E. P., O dowd, N. P., and Banne, H. P. Deemnaon of he mechancal popees of meallc hn flms and sbsaes fom ndenaon ess. Phlos. Mag. A, 2002, 82, Lo, J., Ln, J., and Dean, T. A. A sd on he deemnaon of mechancal popees of a powe law maeal b s ndenaon foce-deph cve. Phlos. Mag., 2006, 86, Fenandes, J. V., Annes, J. M., Sakhaova, N. A., Olvea, M. C., and Menezes, L. F. Yong s modls of hn flms sng deph-sensng ndenaon. Phl. Mag. Le., 2010, 90, Khan, M. K., Hanswoh, S. V. Fzpack, M. E., and Edwads, L. A combned expemenal and fne elemen appoach fo deemnng mechancal popees of almnm allos b nanondenaon. Comp. Mae. Sc., 2010, 49, Sesh, S. and Gannakopolos, A. E. A new mehod fo esmang esdal sesses b nsmened shap ndenaon. Aca Mae., 1998, 46, LaFonane,. R., Paszke, C. A., Kohonen, M. A., and L, C. Y. Resdal sess measemens of hn almnm meallzaons b connos ndenaon and x-a sess. J. Mae. Res., 1991, 6, Lee, Y. H. and Kwon, D. Resdal sess n DLC/S and C/S ssems: applcaon of a sess-elaxaon model o he nanondenaon echnqe. J. Mae. Res., 2002, 17, Chen, X., Yan, J., and Kalsson, A. M. On he deemnaon of esdal sess and mechancal popees b ndenaon. Mae. Sc. Eng. A, 2006, 416, Zhao, M. H., Chen, X., Yan, J., and Kalsson, A. M. Deemnaon of naxal esdal sess and mechancal popees b nsmened ndenaon. Aca Mae., 2006, 54, Yan, J., Chen, X., and Kalsson, A. M. Deemnng eqbaxal esdal sess and mechancal popees fom he foce-dsplacemen cves of concal mcondenaon. J. Eng. Mae. Technol., 2007, 129, X, Z. H. and L, X. Esmaon of esdal sesses fom elasc ecove of nanondenaon. Phlos. Mag., 2006, 86, ang, Q., Ozak, K., Ishkawa, H., Nakano, S., and Ogso, H. Indenaon mehod o mease he esdal sess ndced b on mplanaon. Ncl. Insm. Mehods B, 2006, 242, Dean, J., Aldch-Smh, G., and Clne, T.. Use of nanondenaon o mease esdal sesses n sface laes. Aca Mae., 2011, 59, Swadene, J. G., Talja, B., and Pha, G. M. Measemen of esdal sess b load and deph sensng ndenaon wh sphecal ndenes. J. Mae. Res., 2001, 16, Sknne, A. J. and Boghon, J. Q. Neal newoks n compaonal maeals scence: anng algohms. Model. Sml. Mae. Sc., 1995, 3, Hbe, N. and Tsakmaks, C. Deemnaon of consve popees fom sphecal ndenaon daa sng neal newoks. Pa I: The case of pe knemac hadenng n plasc laws. J. Mech. Phs. Sold, 1999, 47, Hbe, N. and Tsakmaks, C. Deemnaon of consve popees fom sphecal ndenaon daa sng neal newoks. Pa II: Plasc wh nonlnea soopc and knemac hadenng. J. Mech. Phs. Sold, 1999, 47, Tho, K. K., Swaddwdhpong, S., L, Z. S., and Ha, J. Afcal neal newok model fo maeal chaacezaon b ndenaon. Model. Sml. Mae. Sc., 2004, 12, Swaddwdhpong, S., Tho, K. K., L, Z. S., Ha, J., and Oo, N. S. B. Maeal chaacezaon va leas sqaes sppo veco machnes. Model. Sml. Mae. Sc., 2005, 13, Haj-Al, R., Hoan-Kee, K., Sa,. K., Saxena, A., and Rao, T. Nonlnea consve models fom nanondenaon ess sng afcal neal newoks. In. J. Plas., 2008, 24, Bolshakov, A., Olve,. C., and Pha, G. M. Inflences of sess on he measemen of mechancal popees sng nanondenaon: Pa II. Fne elemen smlaons. J. Mae. Res., 1996, 11, Ts, T. Y. and Pha, G. M. Sbsae effecs on nanondenaon mechancal pope measemens of sof flms on had sbsaes. J. Mae. Res., 1999, 14, Cheng, Y. T. and Cheng, C. M. Scalng elaonshps n concal ndenaon of elasc pefecl plasc solds. In. J. Solds Sc., 1999, 36, Abaqs Analss Uses Manal: Veson 6.8. Smla, Tabo, D. The Hadness of Meals. Claendon Pess, Oxfod, Malab Malab 2010a. The Mahoks Incopoae.

10 H. Jn e al.: Resdal sesses and maeal popees 305 Enega-baasl-meeodga jääkpngee ja maejalde omadse määamne ehsneovõke kasades Hongpng Jn, en Yang ja Ln Yan On kasad enega- ja mõõmelse analüüs meeode abl konseed ehsneovõkde mdeled jääkpngee nng maejalde elassee ja plassee omadse määamseks sfäälse ndeno sssepessmse eel. Maejaldes ekkvae jääkpngee ja maejalde omadse ennsamne on ndeneemskõveae abl eljesks jõd-ndeneemssügavs kel nng lhsal eosaav. Pakd ehsneovõkde mdel kalbeemse posed võmaldab enevae maejalde omadse, nag ühlane võdkaheeljelne jääkpnge, elasssmoodl, voolep ja kalesmseg, ose kaadsams. Eelmand ehsneovõkde mdel on võdlems äpne ja saldsvääne.