50 COlvlPOSITE MATERIALS

LlNKING NANOSCALE AND !\IACROSCALE: CALCULATIONOF THE CHANGE IN CRACK GROWTH RES IST ANCE OF

STEELS WITH DIFFERENt STATES OF CU PRECIPlT ATIONUSING A MODIFICATION OF STRESS-STRAIN ClJRVES OWING

TO DISLOCATION THEORY+

P. KIZlER, D. UHlMANN, S. SCHMAUDER

Abstract

In thc prcscnt sludy. strenh'1hening of steels due t0 Cu precipitat.:'s in the

nanometer scale (s invcstigatcd. These precipilaks lead to a signiticant tncrcase ur

thc matrix yiclJ stress duc to impeded dislocation mo\'ement. This cft"cC\ can b.:'

treated quantitativei)! on the mesomechanical level of dislocation theory. Thc

impact on the rnacroscopic failurc beha\ iour of ferritic stce! is in\'estigated in the

framework of damage theory. The presented strategy can be also applied to

related materials seience problems.

1.1l1troductioll

The linkage of modelling on the nanoseale (nm - Icngthsea1e) with such on the

macro- scale (Iengthseale of real specimens) is a current challenge in materials

science. Despite cheaper and faster computer resources it seems not mallageablc

to inOate atomistic models of crystals to the size of specimens and components.

Special probhns of crack propagation ha"e been successfully treated by

dedicated coupled atomistic-continuum mcthods (e.g. [I], [2]). Newnhcless, a

general rccipe to conneet theorctical ealeula\ions on the nanoscale with the

macroseale is actually not availablc. Instead, presently it seems more promising to

rest riet oneself to selected materials problems 2nd 10 eonnect differem lens'1h

scalcs by qualilied physical laws describing corresponding materials behaviour

[3].[4]. The temlillology of relevamlenS'lh scales together with referring physical

phenomena and methods is illustrated in Fig. I.

~ mallLlscript tu bc publishcd also in:"Nuclcar Engineering and Design". with permissiollthJll1 El~e\·i.:rScience

'Staatliche Materialptüfungsanstalt (MPA) Uni\crsity OfStuttgartPfaffeIl\\aluring 31. 0 - 7U569 Stuttgart, Germany

~18)10.9

g @§3 ~D~~~106 10.3 t

*10° m

....,...---4,r---- ....•.----;----T'----j-----j----r, -----.----_---<, .

rJo$:-co(/l~$:>­-1r:1;N

;;;r(/J

Plastic

deformation

Structllral model

Layered model

Microstrllcture

Self-consistentembedded cell

Micromechanics Mesomechanics Strllctllral mechanics

I Local approach IMicrocrack Delamination Macrocrack

Dislocations

Orowan-Model

Dislocation theory

Void formation at incilisions

Dislocation/particle- Real microstructllral modelinteraction

Dislocation movement

Moleclilar

Dynamics

Fracture ofatomic bonds

Atomic bonds

(nm) (pm) (mm) (m)Atomistics Materials Science Engineering

l·--~:~~~~--=::~~~-----~Micro Meso 11 Macro· INano simulation Dislocation Model Microstrllcture Model Mesoscopic Meso-Macro-

fracture aspects coupling

Coating Specimen,Component

(Finite element!Atomistic-coupling)

FEAT

Fig. I· Numerieal Melhoels rererring 10 eli/kren! Icnglh seal es

v.

52 COMPOSITE MATERIALS

The topie 01' the present study is the strengthening 01' a steel due to cop perpreeipitation. Finite element method (FEM) calculations can be regarded as a"Iink-mesoscale-Il-to-macroscale module". On the other hand, a "Iink-nanoscale­to-mesoscale-l module" is able to utilise data from the atomistic scale. In the

present study, such "modules" are linked by characteristic materials parameters,which stem either from direct experimental observations 01' from micro- andmesoscale calculations using dislocation theory and damage mechanics(hierarchieal approach).

2. Nanoscale information from the material

In the present work. the ferritie steel 15 NiCui\loNb 5 with a Cu content 01'0.66%is examincd as an example. At T = 240°C ami in the circumferential direetion inpiping, the material possesses a yield stress 01' about 400 MPa, see Fig. 2, and anotch impact energy 01' 100.1, see Fig. 3 [5]. The axial (L) direction with a notchimpact energy 01'200.1 in the upper shell~ see Fig. 3, is not diseussed ful1her.

800 ----------------Material: 15 N,CuMo:-<b 5 I T = 240 oe

600&: ~---

400Cf) Cf)~

U5 200

Re 0.2 = 399 MPa

Rm = 579 MPa

Ä5 = 22.5 10-2 mimZ = 53.5 10'2 mim

E = 193000 MPa

0.00 0.05 0.10 0.15

Strain / -

0.20 025

Fig.2. Stress-strain curve ofthe material 15 NiCuMoNb 5 at T = 240°C [5]

A very similaI' melt 01' 15 "iiCuMoNb 5 with a Cu content 01'0.64% wasexamined in the initial state and in an aged state after 57000 hours 01'operation at350°C, which causes an impOl1ant inerease in strength: At T = 240°C, the yieldstress and the ultimate strength 01' the material in the aged state are increased by100 MPa and by 72 MPa, respecti\ely, due to the formation and growth 01'copper

COMPOSITE MATERIALS 53

precipitates. Transmission electroll microscopy (TEM) images of these materialsare dcpicted in Figs. 4 and 5

250 ~,--------------~Material. 15 NiCuMoNb 5 / T = 240°C

300200-100

o-200

50

150

>- 200Cl(fiCQ)

Öcu0. 100

,~

.r:ü-0Z

o 100

Temperature / oe

Fig.3. Notch impact energy of [he material 15 NiCuMoNb 5 at T = 240°C [5]

•

Fig.4. TEM image of 15 NiCuMoNb 5. initialstate, with paltick size distribution [6]

5-1 COMPOSITE MATERIALS

~~'"

Fig.5. TEM image llf 15

NiCuMoNb 5, agcd state. withpanieie sizc distribution [6]

Thc precipitated copper pal1iclcs wcre cvaluated by size and counted bydigital image processing and the results arc included in thc tigures. The histogrambars 01' thc figures display the total number 01' detccted panicles versus theirradius. The data and tigures stem from a detailed materials characterisation bySchick et al. [6].

Already in the initial state. a considerable tfaction 01' the 0.64% Cu hasprecipitated. This tfaction 01' copper panicles can be estimatcd from thedistribution 01' particle sizes, weighted by their volume, ami amounts to -40%.The matrix strcngth due to solution hardening [7] is negligibly affccted by thesmall content 01' dissolved Cu (60 < 8 MPa). Therefore, the initial state can betreated like an aged steel containing 0.26% Cu, whieh has completelypreeipitated.

3. [\Icsoscalc Thcory

COMPOSITE MATERV\LS 55

Materials ean be strengthened by il11peding the 1110vel11entof dislocations bygeol11etrical obstacles such as precipitates. Dispersion strenh1hening of a materialleaves the elastic modulus unaffected but inereases the yield stress and all stressvalucs beyond the yield stress in the stress-strain relationship cr(E) by an amount

01' :',cr such that cr = cro + I'>cr. where cro is the yield stress 01' the unstrengthenedstate. The case 01'precipitation strengthening 01'Fe by Cu particles is a pal1icularone. Ün one hand, the plastic behaviour 01'a bee crystal (e.g. Fe) differs from that01' fce (e.g. Cu) or hcp slructures. On the olher hand, unlike tor other obstacles intypical steels such as the carbide par- ticles, the Cu preeipitates are softer than theembedding Fe matrix. Thus, a disloeation is not strictly blockeel like in theOrowan moelel but elissoeiates into seetions insiele anel outsiele the preeipitatewhe're elifferent energies are encountereel anel the movement is impeded, but notblclckeel [8]. This difference causes a restliction tor the eutting of preeipitates byelislocations. This resistanee together with the obstacle spaeing lonns the basis forthe eün'esponding fomlula baseel on a theory 01'Russell allClBrown [8] to elescribe

the incrcase in yielel stress I'>cr (= 2.51'>T as assumeel tor Fe [8]) due to eutting 01'precipitates. whieh are sotter than the embeelding matrix:

with

p< log~ log~P ro rp~l-=----+--E E log"- log"-

ro ro

whae G = shear 1110elulusofthe matrix: b = burgers vector ofthe dislocation; D =elistance betw'cen the precipitates; E~ anel P~ = energies per unit length 01' adislocatiün in the intinite matrix anel particle. respectively; rpp• = precipitateradius; r, and r" = üuter and inner cut-off raelii. respectively, useel to caleulate theenergy ofthe dislocation.

Fig.6. Cutting 01'Cu precipitates in Fe by dislocations [9]

56 COMPOSITE MATERIALS

Th~ fommla points out the impact 01' the distance 0: The dislocations arepoorly impeded when they can bow strongly between largely distant obstacles andthus 101111shaq) angles at the pre~ipitate surt~lces to promote their culting. In thecase 01' Cu in Fe, the numerical val lies amollnt to P~/E~ ~ 0.6, b ~ 0.248 nm. ru =2.5b, G = 83 GPa, and 1',~ 1000 1'"[8]. In the idealized case with homogeneouslydistributed particles. 0 depends on rpp,as 0 ~ 1.77 rpp, I' -" where I' = atolllicpercelllage 01' Cu. Using this O(rpr,) la\\', ß0 can be directly caIculated as afunction 01' the panicle size. For the present Cu concentrations, these functionspeak 101'rpl'laround 1.3 nm, see Fig. 7. In real melts like in the present study,neither sizes nor distances 01' precipitates are identical, see Figs. 4 and 5, and theabov~ equation linking 0 lI'ith rpp,and I' does not hold. A detailed treatment 01'

uncqual distributions 01' obstacles like discussed by, e.g .. Nembach [9] wouldexcced th~ scope of the presem study. Precipitation m~ans diffusion-drivengrowth 01' pal1icles at thc expense 01' emptying the sun'ounding region from Cu,,,'" also S,>i,son et al. [10]_

300

(ii'200CL2~ 100

2.04.0 [ ]6.0rppt nm

Fig.7, Graph ofthe Russell­ßrown t,mnula: dependence of

ß0 Irom precipitate radius 1'"""pIOIl~dll)r tll'O differem Cuconcentrations. Venicallines:

a\'erage panicle sizes tor thc tll'Odiscussedmaterial states as

derived from Figs. 4 and 5.

Therelore. nucleation 01' small panicles next to biggcr ones is improbable. Thisjustilies to assume that panicles groI\' in mutually exclusive spheres and ratherclaim a\ erage distallccs than distribute randomly. Thus. in a simple approachawrage panicle sizes alld corresponding distances according to the abo\'e O(r"",)Ja\\' are ~ssullled. FlOm the panicle size distributions in Figs. 4 and 5. the etTecti\-eCu cOlltem. average panicle sizes al1d distances were derived and amount [00.26%. 2,6 11mal1<.l80 11m laI' the initial state and 0.66%, 2.8 nm and 60 nm tor

the agcd state. respectively. lnsening these vallIes into the cited Rllssell-Browntom111layidds ß0 = 144 MPa for the initial and 224 MPa tor the aged state. seeFig. 7. ka,-ing a total increase in yield stress of 80 MPa. This value has to be

Cü~IPOSITE MATERIALS 57

r~duced by the contribution from solution hardening which vanishes during

ageing when the rernaining dissolved copper atoms precipitate to fom1 particles.

Thc solution hardening [7] tor the initial state amounts to approxil11ately 8 MPa.

1ea\ing a total increase in yield stress of 72 MPa tor the aged state when all Cu

atoms are precipitated.

·t i\licro / I\lacroscale Thcory

,<~. i{o,'].:..., .••..; I

\:-: ,/ :~,~:oo~~.-' --_n"-a -. -- 1--

- -- '2:'..'- ..:.=-_~-=-I

Fig.8. Model ofCT specimen

showing geometry ancl Finite Elementdiseretization

CASTEM 2000 [11] is a multi-purpose tinite element code developed at the CEA.

Domains of applications are 2D- and 3D-structural mechanics, fluid mechanics.

thermal amI magnelic analyses. It can be uscd not only für fracture meehanical

calculations but also lor the simulation of crack initiation and tor local approach

of ductile crack ~~'o" th. In the field of struetural mechanies. C ASTEM 2000

otTers sp~cial features for elastic-plastic fraeture mechanics. It also ineludes local

approach features tor modelling the nucleation, growth and coalescence of voids

in a mdal matrix. whieh torm the undcrlying mechanism for ductile fracture.

Void nuclcation oceurs cssentially at the second phase particles. These second

phase particles arc \'lnS with a size of severall11icromcters. If the load c\)J1tinues

to inCl"case. void growth will directly folio,,' I"rom cavity fonnation. Void

coa1esccnce is implicitly included in thc !,)rmulation 01" the Roussclier model [12]:

ClS damagc incr('ase:-; for increasing \'alue:; of lond. the stress~s drop wirh

ill~rc:asillg straills, simulating void coalesccllce. Thc fol!o\\-illg parameters I~"(. l~.,[) a11(t cr, tor the Rousselier model are chosen to til numcrically the experimen­

lally cktermined J. curve (initial state) tor the 20% side grooved CT~O eOll1pollnd

specimen (see Fig. 3): initial ,"oid volume fraction I;, = 0.011. critical \"()id \'olume

friKtion r:. = 0.05. the constants orthe Rousselier model D = 2. cr, = -145 MPa and

the half ofthe element length near the crack tip 01" I, = O.lmm. Due to sYll1metryconditions it was surticient to use a FE

mesh "'ith an initial crack len~1h 01'

27.5 mm tor one half 01' the specimen

only. see Fig. 8.

58 COMPOSITE MATERIALS

Fig.9. Experimental true stress-strainrelations tor material 15 NiCuMoNb5 at T = 240°C in the material states

"initial"' and "aged" [5]

Required input parameters 1'01' CASTH,I 2000 are, besides the FE-mesh, theboundary conditions, the load hislOry and the stress-strain relation 01' the mateIial1'01' the relevant temperature including the threshold to plastic defol111ation.characterized by the yield stress. Nomlally. the required stress-strain relations arederived from smooth tensile specimens and the Rousselier parameters aredetennined trom fits to experimental values obtained from notched tensilespecimens and compact tension (CT) specimens. Altematively. r., and I, can beobtained metallographically from image analyses 01' material cross sections [13].The increase in yield stress and ultimate strength due to different material statesare traditionally detennined from experiments. too. For the 20 elastic-plasticdisplacement controlled plane strain calculat ions including geometrical non­linearities, quadrilateral finite elements with reduced integration were lIsed.

5. Conncction 01' Lcngth Scalcs

J7ig. 9 shows experimentally derived true stress-strain relations tor the material 15NiCuMoNb 5 (WB 36) in the initial and the agcd state [5]. They differ by 100MPa at the onset ofthe cur.'es and by 70 MPa at lligher strains.

1OC:i . __ ._ h _I

:1 Aged State -.--==:=J'" "'''1 ------;:::=:c----~ ""'// tnitialState i I

~.::oIV I~ .ocir -r Ao = 72 MPa,

g "'l~.!10= 100MPa !1/)"",

~"., I,1- ..-. ,- .. - .. ~ .-~ __..-I

oeo 0 os ~~n Ii v", Cll0 C ~~ 0.30

Strain 1-

These experimental data are in good agreement with the value 01' 72 M Pa

obtained in section 3 where the increase in yield stress (ßcr) was predicted frommesoscopic dislocation theory making lIse 01' nanoscale structllral intormation.This enables one to accurately predict changes in the mechanical behaviour.expressed in the stress-strain curves based on nanoscale data. independent frommechanical experiments.

Fig. 10 shows a comparison 01' calclliated and experimentally obtaineddllctile crack growth curves [5]. Such cur.·es visualize the resistance 01' a CTspccimen against crack growth under monolOnically increasing tensile load. Thecalculated curve up to 2 mm 1'01' the "initial" material state agrees weil with the

COMPOSITE MATERV\LS 59

uisplayeu experimental behaviour alld proofs the successflll prediction t"om theFEM program. Keeping the Rousselier parameters constant but changing thestrcss-strain relationship 01'the material from "initial" to "aged" means to alter theFEM prof,'Tam input data by increasing the yield stress by ilcr, either fromexperimental results or, as in the present analysis, from theoretical results. Now,the CllnSequences tür the behaviour 01' speeimens can be studied Ilumerically. Forthe described CT specimen, the increase in stren,,>1hby ilcr leads to a decrease inthe cOITesponding JK-values, see lower curve in Fig. 10, which means that withincrcased yield strenglh and nolV stress a high load indllces crack grolVth rat herthan plastic defonnation. This agrees with experimental results for CT speeimens01' an almost identical melt where a J.-integral 01' 150 Nimm eorresponded to aninerease in craek growth from 1.0 to 1.4 nm1 for the "initial" and the "ageu"states. respectively [6].like für the ealeulations shoIVn in Fig. 10.

300

Steel15 NiCuMoNb 5250

Caleulations InitialState

~I\~EXP

~\l'

..6"' .

;; ~ AgedSt,:te11,bi

Ductile Crack Growth 6a I mm

E ,

.§ 2001z 1

:::: ,50J'" '~ ~oJc ,'7 j-, ,

50 .:

lo I

-02;i 0 025 0.50 O.7~ 1

6. Conclusiolls

l25 l~O l75 2 2.25 2:0

Fig.IO. Craek t,~'owth eurvesfrom experiment at T ~ 240°Cand FEM ea1culations [5]

The present study demonstrates how to suecessfully bridge the gap between nano­scale infom1ation and maeroscale meehanieal behaviour using meso- and miero­meehanieal thcoretical analyses. The cOlmecting materials parameter between thetheories is L'lcr.

The suggested strate,,'Y offers several furt her opportunities for usingaltemative "modules" on the different length seales, aiming to import furtherknolVledge from modelling into materials science. Some possibilities shall bementioncd in the 1'olloIVing:

60 COMPOSITE MATERIALS

Fig.ll. Seheme 1'01' the linkage between the nanoseale and the maeroseale

Nanoseale: 'vlonle Carlo simulations 01' thc fÖnnation 01' obstacles (e.g. [10], [14])

and Moleeu]ar Dynamies simulations 0[' panieie! dislocation interactions.

Link I: Cakulated panieie sizes ami distailCes as weil as the Orowan angle and

in-situ shear modulus ti'om simulations ot· dislocation pinning as input tal' the

Orowan strenglhening formula, as tin' as the Orowan··model [15].[ 16] isapplieab1e.

\1cs()scak For other systems than Fe(Cu), suitable strengthening models ean be

applied, Itlr instanee those 01' Orowan (tal' non-shcarable panieies) [15],[ 16],

hiedel (laI' shearable paliicles) [17] 01' Nan and Clarke (Iilr large panicles, even\\hen fracturing) [18].

Link 2a: T\\o ot' the Rou;sellier-parameters (~, and U ean also be clerived from

metallo- graphieal instcad of purcly numerical il1\estigations [13].

l.ink 2b: A eurrent challenge is to cleriw absolute yielcl stress values from

moclelling 011 the nano- 01' mesoscale. Such valucs might serve as FEM input at

this point.

Microscalc: To simulate ductilc crack resistance is an outstanding task 1'01' a FEM

program, but müre common structural meehanieal effeets such as crack opening

displacement (COD) values, 01' eollapse loacls can be clerivecl as weil. Sueh results

linally eorresponcl tü thc rele,·ant materials experiments on the l11acroscale.

,\ckno\\ledgement

Thc support by the Gerl11an Bundesministeriul11 für Bildung, Wissensehatt, Für­

schung und Technologie (BMBF) under ~'Tant No 1501029 is gratefully

acknolVledgecl.

Refcrcnces

COMPOSITE MATERIALS 61

[I] Kohlhoff, S. and Schmaueler. S .. Atomislic SimIlImioll o(Mulerials: Bevond

puir pOlelll/uls. ed. by V. Vitek and DJ. Srolo\itz, Plenum, New York

(1989) p. 411.

[2] Kohlhoff, S., Gumbsch, P., and Fischmeister. H., Cruck propagulioll in b.e.e.ervsluls sllldied \\ilh a combilledjillire-e!elllem ulld uroillisrie mode!. Phi!.

Mag. A 64 (1991) 851.

[3] Zoheli, T., Oden. J., anel Rodin, G .. Hierarehieul Modcllillg o!Helemgeneolls

Bodies. Compur .. IJelhods App!. Mech. Engrg. 138 (1996) 273.[-1] 13lh U.S. National Congress of Applieel Mechanics (USNCAM), June 21.­

26., 1998, Uni\trsity o\' Florida. Abstract Book, ISS'\ 0-9652609

[~] Beyer. 1-1.. Diem. 1-1.anel [skluth, B.1988, Jl'erbro!!liIllers/lc!lIIlIgell ~1iI1I

R"hr-l'ersugen\\'c'lslIch RoRV(B)(T20.2), PrÜ\'ungsbcricht NI'. 815 508/2

vom 21. Dez. 1988. Staatliche MakrialprÜfungsanstalt (MPA) Universität

Stuttgar1.

[6J Schick. M., Wiedemann. 1.. and Willer, D., Cllrersllehu//gell ~lIr sieherheilstee/mischen BC1h'r!lfl1g \'Oll gescln1"l'ij)lcll KO/llPOl1L'/I!L'i/ (lllS J+'erks/{?!!

15NiCII.lfoNh .' im lfi//hlid u/I(di,' Ziihigkeir\'(/h//ul/llll' IIl1ler Berriehs­

heol1SlJI'IIc!lIIlIg BMU-f'orhuliell SR22J9. Staatliche ~[aterialprÜfungsansta[t

(MPA) Universität Stuttgart(1997).

[7] Friede!. L Oll die enlml'.l' o(vihrarioll ,,(di.,loCil/iolls. Phi!. Mag. A 45

(1982) 271.

[8] Russell, K.C. allel Brown. !..M., A dispersioll srrellg/hc'lIillg IIIUde! hused ondi/!i!rem elusric IIIUdllii uPI,lied to rhe Fe-CII Sl's/e//l, Acta Met. 20 (1972)969 97-1.

[9] Nembach, E .. PllI'/icie Slrellgrhellillg o(Me/u!., ul/(I Allol'.\·, John Wiley &Sons, lnc., New York (1997) pp. 37.

[10] Soisson F., Barbu. A .. and Mar1in. G., kfume Curlu Sill/ula/ions o(Cop!,er

PrecljJila/ion ill Dilule fmll-Coppu' Allol'S during Therll/al Ageing al1llIInder E1ecl/'01l /rradialion. Acta \-Iat. 44 (9) (1996) 3789-3800.

[11] CASTEM 2000. Code de ca leu I pour I'analyse cle struclures par la methodedes elements tinis. Guide el·utilisation. Commissariat nl'Energie Atomique

DRN/DMT /SE\!T/LAMS, F - 91191 Gif-sur- Yvette. France (1996).

[12] Rousselier, G .. Dllclile Fracrure .Ifudels (/lid /heir Porelll/(r! in Loeal

Approach orFracl/ire. Nuc!. Eng. Design 105 ([987) 97-111.

62 COMPOSITE MATERIALS

[l3] Seidenfuß, M., Untersuchungen =ur Beschreibung des Versagensverhaltensmit Hilfe von Schiidiglllzgsmodellen wn Beispiel des Werkstoff, 20 MnMoNi55, Thesis, Staatliche Materialprüfungsanstalt (MPA) Universität Stuttgart(1992).

[14] BMBF-Vorhaben 1998; BMBF Research Project "Monte Carlo Simulation ofNuc/eatiul1 am! Growth ofCu Precipitates ofSted", StaatlicheMaterialprüfungsanstalt (MPA) Universität Stuttgart, in progress.

[15] Orowan, E., Symposiulll UI1fntemal Stresses in Meta!s al1d Al/oys. Thenstiriwte o{Metals, London (1948) p. 451.

[16] Bacon, DJ., Kocks, U., and Scattergood, R., Phi!. Mag. 28 (1973) 1241.

[17] Friede!. J., Dislocations, Pergamon, New York ( 1964).

[t8] Nan, C.W. and Clarke. D.R., Modeliing tlw Elasric-Plastic De!iJr/zwtion o{AI/AI;O, Partieulate Composites, J. Am. Ceral11. Soc, 80 (1997) 237.