HENRY AMANKWAH, SIGRID LAUER, SURESHAN KARICHERY AND MD. ASHRAF-UL-ALAM

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1 GOBA JOURNA O URE AND AIED IENE VO NO. 2 2 : 22-2 OYRIGHT BAHUDO IENE O. TD RINTED IN NIGERIA. IN 8-7 DETERMINATION O ATIA GRAIN IZE DITRIBUTION O A INTERED META HENRY AMANKWAH IGRID AUER UREHAN KARIHERY AND MD. AHRA-U-AAM (Reeved 2 Arl 28; Revson Aeted 2 Januar 2) 22 ABTRAT The determnaton of the satal gran sze dstrbuton of a sntered metal from the sze dstrbuton estmated from a samle obtaned n the seton lane s a stereologal roblem. Ths roblem s dsussed wth referene to homothet artles (ubes of two dfferent szes) and to a sstem of three es of grans (fne and oarse ubes and oarse trangular rsm). Two models are develoed to solve the roblem one takng nto aount the sze of the grans and seton rofles onl and one that nludes shae onsderatons. The models are tested wth smle and artfal examles as well as wth smulated data. KEYWORD: ntered metal fne gran oarse gran gran sze seton rofle. INTRODUTION ntered metals are ver useful for man ratal uroses for examle n the to of drlls and saws to ut stoneware. nterng n general s the formaton onglomerate from dfferent materals to form a new materal wth dfferent roertes. One goal mght be to nrease the denst and strength of the materal. It s mortant to know the dstrbuton of grans n the sntered metal beause the roertes of the sntered metal largel deend on t. The urose of ths stud s to determne the satal gran sze dstrbuton of a sntered metal. ne t s not ossble to determne the dstrbuton n the volume t s usuall estmated b examnng a number of ross setons. Ths wa of reasonng s alled stereolog. Imagne a sntered metal tube whh onssts of grans of dfferent szes and shaes. Assume that ths tube s ut arbtrarl and that on the ross-seton well-shaed domans (seton rofles) an be dstngushed wth a mrosoe (see gure.). The two-dmensonal seton rofles var n sze and shae and from ths nformaton one an estmate the gran sze dstrbuton n the volume. onsder a stuaton where the sntered metal onssts of two knds of grans fne and oarse. et all the fne grans be ubed shaed wth dameter.7μm and let the oarse grans be ether ubes or trangular rsms wth dmensons 2μm. B dameter we mean the largest dstane of all the ross-setons. The roblem s aroahed b determnng the seton rofle dstrbuton n a ross-seton and relatng t to ossble gran dstrbutons n the volume. The man dffult s that the rato of small and large seton rofles n the area mght dffer from the rato of fne and oarse grans n the volume beause oarse grans mght aear as small seton rofles after uttng. In ths ase the robabltes of obtanng a ertan seton rofle when uttng a ertan e of grans wll la a rual role. In ths aer we develo a lnear model to omare the quanttes on a gven ross-seton to the quanttes n a gven volume. At frst two dfferent es of grans (oarse and fne ubes) are onsdered. eondl the model s extended for three es of grans nludng oarse trangular rsms. The model s then heked wth a smle and artfal examle as well as wth smulated data. 2. MODEING THE ROBEM Ths work s based on the assumtons of homogenet and sotro n the volume. Homogenet n the volume means that the rato of the volume oued b fne grans to that oued b oarse grans s onstant n ever ontrol volume that s large omared to the gran sze. Isotro means that the grans do not have an referred orentaton. These assumtons are absolutel neessar to ensure that the nformaton on the ross-seton s meanngful. Henr Amankwah Deartment of Mathemats & tatsts Unverst of ae oast ae oast Ghana grd auer Deartment of Mathemats Unverst of Kaserslautern German ureshan Karher Deartment of Mathemats Unverst of Kaserslautern German Md. Ashraf-Ul-Alam Deartment of Mathemats Unverst of Kaserslautern German

2 222 HENRY AMANKWAH IGRID AUER UREHAN KARIHERY and MD. AHRA-U-AAM 2. NOTATION U: Tal sze of a gran n the volume (e.g. maxmal dameter edge length) : ze of a seton rofle n the area (maxmal dameter) d : rtal sze to dstngush between fne and oarse grans resetvel between small and large seton rofles : Intenst of fne grans.e. the mean number of grans wth sze U d er unt volume : Intenst of oarse grans.e. the mean number of grans wth sze U > d er unt volume : lanar ntenst of fne grans.e. the mean number of seton rofles er unt area whh stem from grans of sze U d.e. U (fne grans) : lanar ntenst of oarse grans.e. the mean number of seton rofles er unt area whh stem from grans of sze U > d.e. U (oarse grans) : Intenst of small seton rofles.e. the mean number of seton rofles wth sze d er unt area : Intenst of large seton rofles.e. the mean number of seton rofles wth sze > d er unt area : ( d U d ): robablt that a seton rofle s small under the ondton that the orresondng nterseted gran s fne : ( > d U d ): robablt that a seton rofle s large under the ondton that the orresondng nterseted gran s fne : ( d U > d) : robablt that a seton rofle s small under the ondton that the orresondng nterseted gran s oarse : ( > d U > d) : robablt that a seton rofle s large under the ondton that the orresondng nterseted gran s oarse An quantt wth ^ denotes an estmate or an aroxmaton for the sef quantt. 2.2 IRT MODE To get a frst dea of the roblem a two-dmensonal ase s onsdered where a ross-seton s a straght lne. Here the dea of lassfng the seton rofles wth reset to sze and usng robabltes of gettng small or large seton rofles from a ertan artle s develoed. It was qukl realzed that the extenson to the three-dmensonal ase does not ause man dffultes and that was wh the desrton of the frst model was started dretl wth the three-dmensonal ase. or smlfaton n the frst model a dstnton between onl fne and onl oarse grans was looked at wth the ntenstes and resetvel. Also the seton rofles on the ross-seton were lassfed wth reset to sze onl whh gves the ntenstes and of small and large seton rofles resetvel. A seton rofle s onsdered to be small when ts maxmum dameter s smaller than or equal to d whh s the maxmum dameter of a fne gran. Instead of the ntenst vetor ( ) of whh the omonents are resetvel the mean number of fne and the mean number of oarse grans er unt volume the lanar ntenst vetor ( ) whh s easer to handle beause t has the same dmensons as the ntenstes of the seton rofles ( ) was used. In the most general ase of an arbtrar ut through a struture wth an sotro gran dstrbuton a al length of a ertan e of gran s ts rotaton average b or the mean breadth k b lγ π where k s the number of edges l the length of the th edge and γ the angle between the surfae normal of the two faes touhng the th edge. or examle for a ube of edge length l. As a motvaton for the defnton of b 2l

3 DETERMINATION O ATIA GRAIN IZE DITRIBUTION O A INTERED META 22 one an thnk of a ube wth edge length (see gure 2.) whh s dsretzed n N ub elements. Eah element omrses of an arbtrar orented smaller ube of the same sze n whh ase the number of grans n the 2 unt volume N there are N grans. The number of seton. In one laer of unt area wth thkness N rofles n a ut through one laer also deends on the rato between the mean breadth b of the artles and the laer thkness N (ertanl the dsretzaton has to fulfll N b ) beause the ut mght not go through a gran. Ths elds b 2 N bn b. N As alread mentoned ths s the wa to omute the lanar ntenst n the most general sotro ase. All the followng examles are seal non-sotro ases the al length has been albrated to other quanttes (edge length dameter). The large seton rofles stem ertanl from oarse grans but the small seton rofle ma be art of ether a fne or a oarse artle. That s the reason wh the ntenst of small seton rofles s the sum of the lanar ntenst of multled b the robablt to get a small seton rofle from a fne gran lus the lanar fne grans ntenst of the oarse grans multled b the robablt to ut a oarse gran n suh a wa that the seton rofle s small. kewse the equaton for s obtaned. Thus we have + +. Ths sstem of two lnear equatons an be wrtten as a vetor on the left hand sde and a matrx-vetor multlaton on the rght hand sde hene or. The sum of eah olumn of the above matrx sums u to one: ertanl the robablt to ut a fne gran n suh a wa that the seton rofle s large equals zero whereas the robablt to ut a fne gran so that the seton rofle s small equals one. mlarl n the seond olumn the robabltes to ut a oarse gran n suh a wa that the seton rofle s ether small or large sum u to one that s +. uh a matrx s known as a robablt matrx. or a general desrton of the ontnuous stereologal roblem see Ne and Ohser (7) and Ne and Ohser (). Nowadas there are tehnques to measure the quanttes and n the area. The b far more dffult task to determne the robabltes an be aroahed b omuter smulaton (see eton ). If the robablt matrx s nvertble we an omute estmates for the unknown quanttes n the volume. Examle As an llustraton of the frst model let us onsder a ver smle but non-sotro examle where a struture onsstng onl of fne and oarse ubes (where the dameter D of the oarse ubes s aroxmatel three tmes that of the fne ubes d.e. D d and the volume of the struture s 8 ub unts) s ut vertall. In ths ase t s onvenent to alulate the lanar ntenstes as the rodut of the edge lengths ( for the oarse ube and for the fne ube) and the ntenstes

4 22 HENRY AMANKWAH IGRID AUER UREHAN KARIHERY and MD. AHRA-U-AAM 27 8 thus. 8 rom the seton rofles on the ross-seton we an omute the ntenstes 2. 2 ne t s mossble to obtan a large seton rofle from a fne gran whereas. Beause the struture s ut vertall t s not ossble to get a small seton rofle whh stems from a oarse gran. o and. Hene the equatons for the estmates ) of the unknown quanttes ) read as follows:. ( One mmedatel sees that the estmates are n fat exat n ths ase. 2. NOTATION 2 In addton to the notatons ntrodued before we need the followng: ( : Intenst of oarse ubes.e. the mean number of ubes wth sze U > d er unt volume : Intenst of oarse trangular rsms.e. the mean number of trangular rsms wth sze U > d er unt volume : lanar ntenst of oarse ubes.e. the mean number of seton rofles er unt area whh stem from ubes wth sze U > d.e. U ( oarse ube) : lanar ntenst of oarse trangular rsms.e. the mean number of seton rofles er unt area whh stem from trangular rsms wth sze U > d.e. U ( oarse trangular rsm) n : Intenst of small n-noded seton rofles.e. the mean number of n-noded seton rofles wth sze d er unt area n n : Intenst of large n-noded seton rofles.e. the mean number of n-noded seton rofles wth sze > d er unt area n n : ( d n orner nodes U d ): robablt that a seton rofle s small and has n orner nodes under the ondton that the orresondng nterseted gran s fne. : ( > d n orner nodes U d ): robablt that a seton rofle s large and has n orner nodes n under the ondton that the orresondng nterseted gran s fne n : ( d n orner nodes U ube > d ): robablt that a seton rofle s small and has n orner nodes under the ondton that the orresondng nterseted gran s a oarse ube n : ( > d n orner nodes U ube > d ): robablt that a seton rofle s large and has n orner nodes under the ondton that the orresondng nterseted gran s a oarse ube n : ( d n orner nodes U trangular rsm > d ): robablt that a seton rofle s small and has n orner nodes under the ondton that the orresondng nterseted gran s a oarse trangular rsm ( ) ( ) ( )

5 DETERMINATION O ATIA GRAIN IZE DITRIBUTION O A INTERED META 22 : ( d ): robablt that a seton rofle s large and > d n orner nodes U ( trangular rsm) n > has n orner nodes under the ondton that the orresondng nterseted gran s a oarse trangular rsm An quantt wth a ^ denotes an estmate or an aroxmaton for the sef quantt. 2. EOND MODE The seond model s an extenson of the frst model and not onl sze lassfes a artle or a seton rofle but also the shae. Hene the lanar ntenst vetor s extended b a thrd omonent. To be rese the lanar ntenst of the oarse grans s dvded nto for oarse ubes and for trangular rsms. uttng a trangular rsm arbtrarl ma result n seton rofles wth to nodes the ubes an rodue addtonall sx-noded seton rofles. lassfng the seton rofles wth reset to sze and shae results n the ntenstes of small and large n-noded seton rofles n. Also the robablt matrx has to be n n and where { } extended. rst of all t has a thrd olumn for the robabltes to get ertan seton rofles from the trangular rsm and seondl t has eght rows now. On eah row there are robabltes to get one of the eght es of seton rofles from the three es of grans. To solve ths over-determned sstem of equatons one needs to use a generalzed nverse. Examle 2 To llustrate the seond model a ver smle artfal examle as shown n gure 2. s onsdered. In ths struture of volume 27 ub unts (see gure 2. a) a ube wth dameter D s ether oued b one oarse ube or b twent-seven fne ubes or b two oarse trangular rsms and the ntenstes take the values After multlng eah omonent wth the edge length of the orresondng gran we get After a vertal ut one onl observes four-noded small seton rofles wth the ntenst ( ). The large seton rofles are ether trangles or squares wth the ntenstes ( 2) and resetvel. Beause n ths ase of vertal uttng the fne ubes an rodue onl small four-noded seton rofles the onl robablt n the frst olumn of the robablt matrx that s non-zero s and thus. kewse the onl e of seton rofle we get from the oarse ubes s a large square so and the onl e of seton rofle from the trangular rsms s a large trangle so. All the other entres n the robablt matrx are zeros. Hene the equatons for the estmates of the lanar ntenstes read

6 22 HENRY AMANKWAH IGRID AUER UREHAN KARIHERY and MD. AHRA-U-AAM 2 and one an mmedatel see that the estmates 2 2 equal the real quanttes ( ). IMUATION O THE ROBABIITIE The task of determnng the robabltes of gettng dfferent es of seton rofles from a ube or trangular rsm was aroahed b omuter smulaton. The method used was based on the followng stes:. onstruton of a artle (ube or trangular rsm) of sze a 2. Random rotaton of ths artle usng Euler angles. Generaton of the seton rofles b ntersetng the rotated artle wth randoml hosen lanes. lassfaton of the seton rofles wth reset to sze and shae and storng the nformaton. alulaton of the robabltes usng the stored nformaton tes 2 and are reeated for a large number of tmes and from the stored nformaton of the seton rofles the requred robabltes were alulated. The flow hart n gure. gves a better dea of the smulaton roedure. A more detaled dsusson of the stes s resented as follows: The reresentaton of a artle entered at the orgn b ts n orner nodes ( ) z x 2 n K n artesan oordnates n the subroutne of the smulaton rogram s self-exlanator. The oordnates ) ( z x 2 n K of the rotated artle b the Euler angles ( ) ϕ ψ are obtaned from z x z x os sn sn os os sn sn os os sn sn os ψ ψ ψ ψ ϕ ϕ ϕ ϕ where the Euler angles are random varables. Multlng ths ombnaton of matres wth a oordnate vetor erforms frst a rotaton around the z-axs b the angleψ seondl a rotaton around the reentl obtaned new x-axs b the angle and fnall a rotaton around the new z-axs bϕ. The requrement for sotro s fulflled b usng the followng ondtons:. The random angles ϕ ψ are ndeendent of eah other 2. The onts ( ) ϕ gven n sheral olar oordnates are unforml dstrbuted on the unt shere whenever (a) ϕ s unforml dstrbuted on the nterval ) 2 [ π and (b) ξ os s unforml dstrbuted on [-]. The angles ψ are unforml dstrbuted on ) 2 [ π uh a random rotaton of a gven artle an be erformed b the smulaton rograms. rom eah rotated artle a seres of seton rofles s generated b ntersetng t wth lanes. These lanes are arallel to the x--lane and the z-oordnate s hosen randoml n ( ) mn z max z where ( ) mn z max z s the mnmum (maxmum) z-oordnate of the

7 DETERMINATION O ATIA GRAIN IZE DITRIBUTION O A INTERED META 227 rotated artle. Ths roedure ensures that a real seton rofle s generated whereas the nterseton set of an arbtrar lane wth a non-rotated artle mght be emt. uttng a ube (or trangular rsm) arbtrarl mght result n seton rofles wth to ( to ) orner nodes. The number of orner nodes and the sze whh s the largest dstane between two orner nodes are stored for eah seton rofle. The robablt of obtanng an n-gon (or an n- noded seton rofle) wth sze smaller (or larger) than d (whh s the rtal sze) s the number of that e of seton rofles dvded b the total number of seton rofles that has been lassfed. gure.2 and gure. show the frequen dstrbuton of n-gons of a ube and a trangular rsm of sze where a total of seton rofles have been lassfed. Wth these smulaton results t s eas to alulate the desred robabltes. or examle the robablt of obtanng a trangular seton rofle whh s smaller than. from the trangular rsm s alulated as follows: One ntegrates over the * urve from to. and dvdes ths b the total number of seton rofles that have been lassfed. The followng tables show the robabltes of gettng an n-gon from a ube resetve a regular trangular rsm of sze. Table.: robablt of gettng an n-gon wth sze d n n n d from a ube of sze. n n { } Table.2: robablt of gettng an n-gon wth sze d from a retangular rsm of sze. D n n n n { }

8 228 HENRY AMANKWAH IGRID AUER UREHAN KARIHERY and MD. AHRA-U-AAM EVAUATION O MODE UING IMUATED DATA. IMUATION TO GET DATA Beause of lak of realst data a smulaton of arbtrar uts through the two dfferent strutures of grans that was arred out. In both strutures of gure. the ntenstes of eah e of gran resetve lanar ntenstes an be easl omuted: (a) (b) The struture s rotated randoml and then ut b lanes that are arallel to the x--lane. Eah tme the total area of nterseton and the number of seton rofles of eah e (lassfaton wth reset to sze as n frst model or wth reset to sze and shae as n seond model wth rtal sze 77. / d ) are stored. The seton rofle ntenstes an be alulated from ths nformaton..2 EVAUATION O THE MODE Usng the smulated robabltes as exlaned n the revous seton the two models an be heked onsderng the frst model on struture (a) then from the smulaton and olvng the equatons for the estmates of the lanar ntenstes results n The relatve error s gven b. % % : e e Now onsderng the seond model on struture (a) from the smulaton

9 and olvng the equatons usng nv wth default settngs n MATAB gves for whh the relatve error s gven b 2%.7% e e. The seond model whh uses more detaled nformaton about the seton rofles gves muh better values for. Ths s lear beause lookng at the frequen dstrbuton of seton rofles on a smulated ross-seton n gure.2 and gure. and ang attenton to the dfferent salng one an see that muh more nformaton about the fne grans has been used. U to ths ont the evaluaton of the models has been done usng the struture n gure. (a). mulatng ross-setons of a struture dd not show an sgnfant dfferene. A smulaton of uts through a larger struture was not ossble beause of the lmt of omuter memor. onsderng the seond model on struture (b) then from the smulaton DETERMINATION O ATIA GRAIN IZE DITRIBUTION O A INTERED META 22

10 2 HENRY AMANKWAH IGRID AUER UREHAN KARIHERY and MD. AHRA-U-AAM and an be alulated. olvng the sstem where the relatve error s gven b for the estmates ( ) results n e e e % 2%..% Unfortunatel the models ould not be tested wth realst data; however wth the results from the smulatons and the errors n a range below % one an sa that the models seem to work qute well for ths ll-osed roblem. It should be noted however that the examles onsdered are of non-sotro gran dstrbutons where the general al length the rotaton average b has been albrated to other quanttes (edge length dameter).. ONUION It ould be sad fnall that the models do exress a relatonsh between the quanttes on the ross-seton and the quanttes n the volume under the assumtons of homogenet and sotro. An extenson of the seond model to dfferent knds of grans s ossble beause ever knd of gran s treated searatel. It was however not ossble to hek the models n more detal due to lak of realst data for the seton rofles and the quanttes n the volume. oures of error mght be from the use of estmates or aroxmatons. Also the strong homogenet and ansotro that mght our n realt wll ause roblems. REERENE Gauther G. and o.. Morhologal egmentaton of uttng Tools. Mros. Mronal. Mrostrut Gauther G. and o.. egmentaton of Gran Boundares n W-o ermets. Ata tereol Mehnert K. and o. 8. Testng tereologal Methods for the Estmaton of atal ze Dstrbuton b means of omuter- mulated Gran truture. Materals ene & Engneerng A Ne M. and Ohser J.. The tereologal Unfoldng roblem for stems of Homothet artles. attern Reognton. 2: -. Ohser J. and Muklh. 2. tatstal Analss of Mrostruture n Materals ene. John Wle and ons. Ohser J. and Ne M. 7. tereolog of ub artles: Varous Estmators for the ze Dstrbuton. Journal of Mroso. 87. t Voss K. 82. requenes of n-olgons n lanar etons of olhedrons. Journal of Mroso. 28. t

11 DETERMINATION O ATIA GRAIN IZE DITRIBUTION O A INTERED META 2 AENDIX gure.: ross-seton examnaton gure 2.: Unt volume dsretzed nto N ub elements eah ontanng an arbtrar orented ube gure 2.2: (a) A ube of dameter D (b) A ross-seton s oued b one oarse ube or b 27 fne ubes.

12 22 HENRY AMANKWAH IGRID AUER UREHAN KARIHERY and MD. AHRA-U-AAM gure 2.: (a) A ube of dameter D s ether oued b one oarse ube or b 27 fne ubes or b two trangular rsms. (b) A ross-seton gure.: A flow hart of the smulaton roedure

13 DETERMINATION O ATIA GRAIN IZE DITRIBUTION O A INTERED META 2 gure.2: A frequen dstrbuton of n-gons for a ube of sze. gure.: A frequen dstrbuton of n-gons for a regular trangular rsm of sze. gure.: (a) A gven ube of volume ub unts wth a dameter D s ether oued b one oarse ube or b 27 fne ubes. (b) A gven ube of volume 27 ub unts wth a dameter D s ether oued b one oarse ube or b 27 fne ubes or b 2 oarse trangular rsms.

14 2 HENRY AMANKWAH IGRID AUER UREHAN KARIHERY and MD. AHRA-U-AAM gure.2: A frequen dstrbuton of seton rofles on smulated ross-seton (from oarse grans). gure.: A frequen dstrbuton of seton rofles on smulated ross-seton (from fne grans).