Modeling heat flow across material interfaces and cracks using the material point method

Transcription

1 Comutatonal Partcle Mechancs manuscrt No. (wll be nserted by the edtor) Modelng heat flow across materal nterfaces and cracks usng the materal ont method John A. Narn Acceted: July 18 Abstract Heat conducton through an object wth materal nterfaces or cracks s nfluenced by heat flow across those dscontnutes. Ths aer resents a numercal artcle method for modelng such heat flow couled to comutatonal mechancs all wthn the materal ont method (MPM). In bref, MPM models contact and cracks by extraolatng multle velocty felds to a grd. To model nterfacal heat flow, MPM should smlarly extraolate multle temerature felds. Interfaces nodes that see more than one temerature feld modfy ther heat flow to reflect nterfacal hyscs. For examle, nterfaces n contact may transfer heat by erfect conducton whle searated nterfaces may block heat flow or cause reduced heat flow by convecton. After some valdaton examles, two real-world examles consder coolng an ngot wthn a crucble where coolng causes the ngot to lose contact wth the crucble walls and thermal magng of cracks wthn an oaque sold. Keywords Heat conducton, materal ont method, exlct cracks, materal contact 1 Introducton A straghtforward addton to artcle-based, materal ont method (MPM) modelng s to coule the mechancs analyss to thermal conducton. Coulng s done by trackng artcle temerature, extraolatng temerature to a background grd, and solvng the heat flow equaton on the grd along wth the momentum equaton. Havng such a feature allows MPM to model heatng mechansms caused by hyscal henomena such as Oregon State Unversty, Wood Scence & Engneerng, 112 Rchardson Hall, Corvalls, OR 97330, USA E-mal: John.Narn@oregonstate.edu volumetrc exanson or comresson, lastc, vscoelastc, or damage energy dssaton, or contact wth frcton and to fully track thermodynamc state varables (e.g., entroy, enthaly, and free energy). Standard MPM conducton methods extraolate artcle temerature to a sngle temerature feld, thereby solvng for a global temerature feld. When a roblem has materal nterfaces or cracks, however, ths sngletemerature-feld aroach cannot model the nfluence of those dscontnutes on heat flow. Ths stuaton s analogous to MPM contact methods. When an MPM code uses a sngle velocty feld, the calculatons can model only stck contact. Extendng MPM to model contact by frcton [2] or an merfect nterface [10, 11] requres use of multmateral velocty felds. Nodes that see only a sngle materal roceed by conventonal MPM whle nterface nodes that see multle materals adjust nodal momenta to reflect some modeled contact mechancs. Ths aer adots a smlar aroach for heat flow calculatons by extraolatng multle temerature felds. Sngle feld nodes roceed by conventonal conducton methods whle nterface nodes adjust nodal heat flows to model realstc heat flow across an nterface. The Heat flow equatons secton frst descrbes global conducton analyss and then descrbes how to modfy that analyss for multle temerature felds. The methods are aled both to multmateral MPM and to MPM wth exlct cracks [9]. The methods were verfed by smle calculatons across a contact nterface. Alcatons for the method are demonstrated wth two realworld examles for coolng an ngot n contact wth a crucble where thermal shrnkage may cause the ngot to lose contact wth the crucble walls and thermal magng of oaque objects wth nternal cracks. Although MPM rovdes no secfc advantages over other nu-

2 2 J. A. Narn mercal methods (such as fnte element analyss) for soluton of heat transort equatons, MPM may have advantages for modelng exlct cracks [9] and comlex contact mechancs [11, 14]. The methods n the aer allow MPM smulatons wth cracks and nterfaces to account for effects of those dscontnutes on heat transort. heat caacty) and χ (x) s a artcle bass functon for artcle (whch s tycally 1 wthn the artcle s doman and zero elsewhere [3]). Next exand the weght functon and ts gradent usng standard, soarametrc grd shae functons, N (x): w(x) = w N (x) and w(x) = w N (x) (8) 2 Heat flow equatons 2.1 Global analyss A common addton to MPM codes s a global heat conducton oton that gnores materal nterfaces and cracks. Deste ts use n many MPM codes, a full generalzed nterolaton materal ont (GIMP) [3] dervaton s rarely n the lterature [15]; t s gven here. The heat conducton equaton s ρc T t + q(x) = q s(x) (1) where ρ s densty, C s heat caacty (er unt mass), T s temerature, q s heat flux (er unt area), and q s s volumetrc heat source rate. For heat conducton, heat flux s q = k T where k s the thermal conductvty tensor. Solvng ths equaton n the MPM weak form gves ( ρc T ) t + q q s(x) w(x) dv = 0 (2) V where V s total volume and w(x) s an arbtrary weghtng functon. Usng the vector dentty: w(x) q(x) = (w(x)q(x)) w(x) q(x) (3) and the dvergence theorem, the weak form equaton becomes: ( w(x)ρ(x)c(x) T w(x) q(x) V t ) q s (x)w(x) dv + w(x)q(x) ˆn ds = 0 (4) δv where δv s the border of V and ˆn s a surface normal vector. By GIMP methods [3], artcle quanttes are exanded n a artcle bass to get: ρ(x)c(x) T dt = q s (x) = ρ C T dt χ (x) (5) q s, χ (x) (6) q(x) = q χ (x) (7) where subscrt denotes a artcle roerty (note: C s artcle heat caacty and not constant-ressure where w are nodal values of w(x) on the grd. After substtutng all exansons, the weak form equaton becomes w N (x)q(x) ˆn ds = δv + V { [w N (x) q χ (x)] + q s, χ (x)w N (x) } T χ (x)w N (x)ρ C dv (9) dt Exlotng the fact that w(x) s arbtrary, ths equaton transforms to a system of equatons for node : M C T dt S = V q G + δv V q s, S N (x)q(x) ˆn ds (10) where M and V are artcle mass and volume and S and G are GIMP shae functons [3]: S = 1 χ (x)n (x) dv (11) V V G = 1 χ (x) N (x) dv (12) V V Allowng artcle volume and heat caacty to change each tme ste (suerscrted wth (n)), the thermal energy on node (wth SI unts J) can be defned as: T = M C (n) T (13) The MPM thermal conducton equaton becomes: d T dt = where = =,q +,q (14) ( V (n) q G (n) δv ) + q s,s (n) (15) N (x)q(x) ˆn ds (16) are nodal thermal energy flows (wth SI unts J/sec or Watts). s nternal thermal flow whle,q s thermal flow at boundares due to flux boundary condtons. Unlke the corresondng MPM momentum calculaton

3 Contact Heat Flow 3 for force that deends on tracked artcle stress, artcle heat flux s not tracked on the artcle; nstead t s calculated on each tme ste usng shae functon gradents: q = k (n) T (n) = k (n) G (n) (17) where k s artcle thermal conductvty and nodal temerature,, s found from: T (n+1) = T = where + q(n) = M C (18) s nodal heat caacty (SI unts J/K) extraolated from artcles. The temerature udate on each node s +,q t = + T t (19) where nodal temerature velocty (SI unts K/sec) s T = q(n) +,q () Once temerature veloctes are found, each artcle s temerature udates by: T (n+1) where T = + T t (21) s temerature velocty extraolated to the artcle. Two otons are ossble. The frst extraolates the temerature velocty usng T = T S(n) (22) whle the second extraolates heat flux and dvdes that result by artcle heat caacty: T = 1 ρ C where = +,q S (n) (23) V S (n) (24) s volume extraolated to the node. These two methods are dentcal when all artcles have the same mass, densty, and heat caacty, but would dffer n comoste materals and when modelng hase transtons where heat caacty vares n the meltng regon. In general, the results seem smlar, but unublshed results wth hase transtons suggest the frst method (extraolaton of temerature rate) s more stable. Ths aer uses that method (and actually ether method would work because all examles used artcles wth the same mass, densty, and heat caacty). Note that artcle temerature must be udated by extraolatng temerature velocty nstead of temerature, to the artcle. A temerature extraolaton leads to artfcal conducton, even f the materal s conductvty s zero [4]. The conducton soluton s easly couled to standard MPM mechancs analyss. Coulng occurs two ways - through thermal exanson or through heat generated by varous mechansms. Thermal exanson s couled by evaluatng temerature ncrement on each artcle for nut to consttutve law calculatons that account for thermal exanson. Because of the way artcle temeratures udate, some smulatons exhbt varatons n T (n) wthn a cell. Deste these varatons, the temerature feld on the grd remans smooth and s a better descrton of the current temerature feld. As a consequence, temerature ncrement on each artcle s best calculated from T = T g (n+1) T g (n) where T g (n) = S (n) (25) rather then from T (n). To make ths calculaton ossble, artcles should track both T (n) and T g. (n) T (n) s used n conducton equatons whle T g (n) s used to fnd T or to mlement any other feature that deends on artcle temerature. Heat generated by consttutve laws may be adabatc heatng (e.g., adabatc deformaton of an elastc materal changes temerature by dt ad = (M dε)/ρ where M s the stress-temerature tensor and dε s the stran ncrement [5]) or any energy that s dssated as heat (e.g., lastc or vscoelastc energy dssaton). These heat terms can be converted to temerature changes (deendng on artcle s current heat caacty), accumulated durng consttutve law calculatons as d,ad, and then added to artcle temerature durng MPM artcle udates. In effect, ths aroach s solvng an adabatc artcle heat equaton: dt (n) ρ C dt wth soluton T (n+1) = = q s (26) + q s t = T (n) + d,ad (27) ρ C Other heat generaton mght be frctonal contact at materal nterfaces and cracks [14] or crack t heatng durng crack roagaton. These heats are often found on the grd and mlemented by addng to n conducton calculatons (total) = + j q j, (28)

4 4 J. A. Narn where the sum s over events j that each generates heat q j, on node (SI unts Watts). The above dervaton s for Cartesan coordnates. Extenson to axsymmetrc MPM s trval and detals are rovded n Ref. [15]. The only changes needed for axsymmetry are to use axsymmetrc S and G shae functons and to use artcle mass (M ) and volume (V ) on er-radan bass. 2.2 Heat flow across materal nterfaces Extendng MPM to handle heat flow at materal nterfaces requres each materal to extraolate to ndeendent temerature felds on the nodes and then to mlement contact heat flow hyscs at all nterface nodes (defned as nodes that see temerature felds from more than one materal). We defne materal secfc quanttes from sums that nclude only artcles of materal α: T,α = α,α = α,α = α M C (n) T (29) M C (30) ( ) V (n) q (n) G (n) + q(n) s, S (n) (31) where (mortantly) q (n) s heat flow that would occur on artcle n the absence of nteractons wth other materals. It s calculated from the temerature feld for the materal tye of artcle : q (n) α = k (n),α G(n) = k(n) T,α,α G (n) (32) For any number of materals, global and materal extraolatons are related by: T = α T,α and = α,α (33) In contrast, global and materal heat flows may dffer ( α q(n),α ) because q(n) and,α s are based on dfferent artcle heat flows ( vs. q (n) ) that are calculated from dfferent temerature feld gradents. The governng equaton on node for materal α s d T,α =,α dt + q(n),α + φ(n),α q(n),q (34) where,α added to account for materal nterfaces at node and φ (n),α s any addtonal heat flow that must be s the fracton of any heat flux boundary condton on node that must be aled to materal α. The temerature udate for artcle of materal α usng temerature rates for materal α s: T (n+1) α = T (n) + t,α + q(n),α + φ(n),α,α q(n),q S (35) and the back extraolaton to fnd temerature ncrement s g α =,α S(n) (36) The mlementaton of MPM heat conducton analyss that accounts for materal nterfaces s now reduced to determnng,α and φ(n),α at each nterface node. 2.3 Contact calculatons When MPM calculatons are n multmateral mode [2, 7, 11], the standard aroach to contact s to adjust momenta on all nodes n contact. In bref, whenever a node has more than one materal, mechancs calculatons determne f they are n contact [11]. If they are n contact, momenta are changed to reflect contact hyscs (e.g., frctonal contact [2, 7] or an merfect nterface [11]). When ths mechancs analyss s couled to heat flow analyss, these contact calculatons should also calculate,α and φ(n),α on the same contact nodes. Furthermore, whle contact forces can be sked when nterface are searated, heat flow calculatons are needed for nterfaces both n contact and searated. Determnaton of contact or searaton can be used to mlement dfferent heat flow characterstcs for each state. For examle, nterfaces n contact mght transfer heat by conducton, whle searated nterfaces mght use convecton. To start, consder erfect conducton at an nterface such that multmateral conducton analyss should revert to the global analyss that gnores nterfaces. In ths lmt, the artcle udates must be dentcal. Equatng Eq. (21) to Eq. (35) and solvng for erfect conducton heat flow change, q (n),α, gves q (n),α = c(n),α ( ) (n) c,α +,α φ,α,q (37) If we assume φ (n),α = c(n),α /c(n) (.e., assume that externally aled flux s logcally sread over avalable materals accordng to ther heat caacty fracton on the node), ths term becomes: q (n),α = φ,α,α (38) Next consder an nterface wth exactly two materals α and β havng contact by convecton wth

5 Contact Heat Flow 5 convecton coeffcent h (n W/(m 2 -K)). The convecton heat flow aled to each materal s: ( ),α = ha,β,α and,β = q(n),α (39) where a s nterfacal contact area on the grd for node. In other words, heat flow nduced by convecton at an nterface s added to each materal and total extra heat flow s zero. The contact area can be calculated by the same methods used when mlementng MPM contact methods that deend on contact area as descrbed n (author?) [11]; the needed area s: 2(,α a = + v(n),β ) mn(v(n),α, v(n),β ) () t where,α = V (41) α s materal volume extraolated to the grd and t s an effectve thckness of the contactng volume. For a regular grd wth equal element sdes, x = y = z, t = x or s equal to the constant cell sze. For elements wth rectangular elements, t s needed to account for nterfaces orented n dfferent grd drectons. More exlanaton and a method for fndng t are gven n Ref. [11]. Imortantly, a s an effectve contact area that reduces to an area as a functon of dstance from node to the nterface. Ths scalng s crucal for grd ndeendence of contact results as demonstrated n the Results and dscusson secton. A robust numercal mlementaton must allow for three or more materals on a sngle node. To handle ths stuaton, relace materal β wth a vrtual materal that lums all materals besdes materal α or T,β = T and,β τ (n) T,α, c(n) = v(n),α,α,,β = T,β /c(n),β,. Substtutng nto Eq. (39) and,β = c(n) elmnatng all β terms, a general convecton flow added to materal α becomes,α = ha,α (42) 1 φ,α Ths change s aled (ndvdually) to each materal on the node. For two materals, ths aroach reduces exactly to Eq. (39) and total added heat flow s zero. For more than two materals, t gves a reasonable result although the sum of,α may not be zero. The full contact heat flow algorthm s: 1. After udatng momenta on the nodes, evaluate the velocty felds at each node wth more then one materal, determne whether or not they are n contact [11], use the contact state to select contact roertes (e.g., assume heat flow by convecton wth convecton coeffcent h or equlbrated heat flow by conducton), and then reeat followng stes 2 and 3 for each materal (α) at the node (). 2. Fnd q (n),α for heat flow under the erfect conducton lmt. 3. If the contact state s usng equlbrated conducton, set,α = q (n),α. If nstead t s usng convecton, fnd a and,α by Eq. (42). Ths heat flow, however, must not exceed conducton heat flow. If,α < q (n),α, then use q(n),α ; otherwse set,α = q (n),α. 4. If mechancal analyss determnes nterfaces n contact by frcton, the frctonal sldng can be converted to heat by addng a calculated q frcton to q,q (.e., by addng φ,α q frcton to heat flow n each materal s temerature feld). Note that calculaton of,α requres knowledge of both, whch deends on, and,α, whch de-. As a consequence, MPM code to mle- ends on q (n) ment nterfacal heat flow must extraolate both global and materal temerature felds to calculate both and q (n) on each artcle. 2.4 Heat flow across cracks Analyss of heat flow across cracks s nearly dentcal to heat flow at materal nterfaces excet searate temerature felds for each materal are relaced by searate temerature felds for each sde of the crack. In MPM wth cracks (whch s called CRAMP [9]), each artcle node ar s assgned a crack velocty feld, v(, ) = 1 or 0 deendng whether a lne from the artcle to the node crosses a crack (1) or does not cross a crack (0). For a sngle crack, each node wll have at most two velocty felds (0 and 1). The CRAMP method can be extended to handle two nteractng cracks by allowng u to four velocty felds on each node (or v(, ) = 0 to 3) corresondng to lnes that cross no cracks (0), one crack (1), a second crack (2), or both cracks (3) [13]. When modelng heat flow across cracks, the crack velocty feld secfc thermal terms become: T,j =,j,j,j = = = M C (n) S (n) δ j,v(,) (43) M C δ j,v(,) (44) ( ) V (n) q (n) G (n) + q(n) s, S (n) δ j,v(,) (45) V δ j,v(,) (46)

6 6 J. A. Narn where j s crack temerature feld j on node and δ s the Kronecker delta. The crack heat flow term,,j, ncludes yet another artcle heat flow, q (n), that s calculated each tme ste usng q (n) = k (n) T,v(,),v(,) G (n) (47) In other words, t extraolates temerature gradent from each node usng the temerature arorate for that artcle-node ar. The crack contact calculatons determne f crack surfaces are n contact and heat flow calculatons are aled to each crack temerature feld nstead of each materal temerature feld. Crack heat flow calculatons are dentcal to materal calculatons n secton 2.3 excet they use crack terms nstead of materal terms. The artcle udates n Eqs. (35) and (36) relace α wth the arorate velocty feld v(, ). To mlement these calculatons, the extraolatons must evaluate both and q (n) on each artcle. 2.5 Combnng cracks and materal nterfaces To combne heat flow across both cracks and materal nterfaces, temerature felds have to be arranged n a herarchcal structure. The aroach used here was to allow each node to have multle crack velocty felds (u to two for a sngle crack or u to four to handle two nteractng cracks). Each crack velocty feld may have 1 to m materal temerature felds, where m s the number of materals n the smulaton. In ths arrangement, an analyss may encounter three tyes of nodes requrng addtonal heat flow calculatons nodes wth multle materals wthn a sngle crack velocty feld, nodes wth multle crack velocty felds each havng only a sngle materal, and nodes wth multle crack velocty felds contanng multle materals. The frst two are handled exactly as descrbed n sectons 2.2 and 2.4. The last one requres secal treatment: 1. Heat flow at materal nterfaces are handled frst. Because materals are wthn a crack velocty feld, these calculatons relace global values (,, and ) wth the corresondng crack temerature feld values (,j, c(n),j, and q(n),j ) and then roceed as descrbed n secton Heat flow at cracks are handled second. The change to the crack temerature feld heat flow s calculated exactly as descrbed n secton 2.4. When done, however,,α q(n),j /c(n),j s added to each materal temerature feld (.e., the added heat flow s sread over materal temerature felds accordng to ther thermal mass fracton wthn the crack velocty feld). For all nterface nodes, artcle udates use the materal temerature feld (see Eq. (35)). For ths udate to work on nodes havng multle crack velocty felds wth only a sngle materal, the total crack heat flow n such felds, j, should be coed to that one materal, α (.e., q,α (n) =,j + q(n),j ). Fnally, for MPM code to mlement combned nterfacal and crack heat flow, t must extraolate global, materal, and crack temerature felds to calculate, q (n), and q (n) on each artcle. 3 Results and dscusson The algorthm resented s fully 3D and could be added to any code that models 3D contact mechancs and/or 3D cracks. The verfcaton examles gven here were all 2D or axsymmetrc. The frst roblem was a 2D smulaton, but models a 1D roblem. Consder a str of length L = 100 mm from x 0 = 50 mm to x 1 = +50 mm and wdth mm wth MPM background cell sze of mm (and, lke all smulatons n ths aer, four artcles er cell). The str was comrsed of two searate sotroc materals, but wth dentcal (arbtrarly-selected) thermal roertes: k = 00 W/(m K), C = 1000 J/(kg K), and ρ = 1 g/cm 3. All calculatons were done n the MPM code OSPartculas [12], whch fully coules mechancal and thermal calculatons. Because thermal conducton s tycally much slower than stress waves, these calculatons used a low-modulus, zero-exanson materal (E = 0.01 MPa, ν = 0.33, and thermal exanson coeffcent = 0) to allow a larger tme ste for exlct ntegratons. At tme zero, all artcles were set to temerature T 0 = 0 C and boundary condtons set T = T 0 = 0 C at x 0 and T = T 1 = 100 C at x 1. The nterface was at x = 0 and was modeled as conducton when n contact, but convecton (wth convecton coeffcent h) when searated. To nduce contact or searaton, the two materals were ether ushed together or ulled aart by 0.5 mm before the heat flow reached the nterface. A Fourer seres soluton to ths roblem for conducton n the absence of an nterface s [6]: T (x, t) = T 0 + T ( ξ + 2 n ( 1) n nπ 2 e λ n t sn(nπξ) ) (48) where T = T 1 T 0, ξ = (x x 0 )/L, and λ n = (nπ/l) k/(ρc).

7 Contact Heat Flow 7 Temerature (C) Contact MPM Analytcal Poston (mm) Fg. 1 Temerature rofle n a bar wth two materals n contact at x = 0 mm. All materal onts started at T = 0 C and the results are after 500 ms wth T set to 0 C at x = 50 mm and 100 C at x = +50 mm. The sold black lne s the MPM result. The dotted red lne s the analytcal soluton. Temerature (C) Searated h=3x10 4 h=10 4 h=0 h= h=10 5 Analytcal Poston (mm) Fg. 2 Temerature rofle n a bar wth two materals searated by 0.5 mm at x = 0 mm. All materal onts started at T = 0 C and results are after 500 ms wth T set to 0 C at x = 50 mm and 100 C at x = +50 mm. The sold lnes are MPM results for varous values of nterfacal convecton coeffcent h (n W/(m 2 -K)). The dashed red lne s the analytcal soluton for erfect conducton. Fgure 1 shows results after 500 ms wth the nterface ushed nto contact. Because contact was modeled as conducton, the results should reduce to erfect conducton. The MPM soluton (sold black lne) s nearly dentcal to the analytcal soluton (dotted red lne). Fgure 2 shows the analogous results when the nterface was searated. Ths case develoed a temerature dscontnuty at the nterface. When h = 0 no heat s transferred across the nterface and all temerature rse was to the rght of the nterface (x > 0). The left sde materal correctly stayed exactly at zero. As h ncreased, the dscontnuty got smaller and the results aroached the analytcal soluton for erfect conducton. Notce that the h = (whch used very large h) does not exactly match the analytcal soluton, whch s understood by lookng at exected nterface nodes. Fgure 3 shows materal onts around an nterface near grd lne 0 when n contact (Fg. 3A) or searated (Fg. 3B). The crcled nodes show all nodes that would be nterface nodes when usng GIMP shae functons (and non-gimp methods used n early MPM aers [16] should be avoded because they actvate fewer nodes). When n contact, the nodes at -1, 0 and +1 are all nterface nodes that adjust ther heat flow and ths arrangement s able to exactly revert to erfect conducton for large h. The nodes at ±2 are not nterface nodes, but they are suffcently far from the nterface that global ( ) and materal (q (n) ) heat flows would be dentcal. Thus, when n contact all nodes can exactly recover erfect conducton for large h. In contrast, when searated (Fg. 3B), only the nodes at 0 are nterface nodes. Although those nodes can revert to conducton for very hgh h, the nodes at ±1 have only a sngle materal feld. Because those nodes are close to an nterface, they wll have a slght dscreancy between ther global ( ) and materal (q (n) ) heat flows (due to use of (n),α nstead of T, whch dffer near an nterface). Because ther heat flow must be based on the materal temerature feld (to be correct for lower h), these nodes cannot recover the erfect conducton lmt at hgh h. Correctng the analyss to revert to erfect conducton lmt at hgh h s relatvely unmortant because 1) the dscreancy from erfect conducton s tycally small and 2) when h s suffcently hgh, the smulaton can smly be run usng standard conducton that gnores nterfaces. Nevertheless, a smle correcton seems to work. Realzng that heat flow at snglemateral nodes ±1 cell from an nterface may have slghtly reduced heat flow, one oton s to comensate for that by allowng the,α on nterface nodes to slghtly exceed the conducton lmt q (n),α. Fgure 4 magnfes the dscontnuty regon for very hgh h as a functon of the excess heat flow allowed over conducton. As the allowed excess ncreased to 1% the smulatons got very close to the analytcal soluton. Increasng beyond 1% dd not mrove the agreement and by 4%, the excess heat flow caused a temerature nstablty. Although allowng 1% excess allowed hgh-h smulatons to aroach erfect conducton, what are the consequences on that excess heat flow when h s lower? Fgure 5 shows the temerature dscontnuty at the nterface n the searated 1D bar examle as a functon of h for ether 1% excess heat flow (sold lne) or 0% excess heat flow (dashed lne). For low h, the added excess had no effect (because convecton flow s always less than the conducton lmt), but at hgh h, the 1% excess allowed the dscontnuty to aroach the cor-

8 8 J. A. Narn A 50 ΔT (C) 30 B % Excess h (W/(m 2 K)) 0% Excess C Fg. 5 Temerature dscontnuty at the materal nterface for a bar usng condtons dentcal to Fg. 2 as a functon of h and for 0% (dashed lne) or 1% (sold lne) excess heat flow allowed at nterface nodes. Fg. 3 Materal onts (sold crcles) on to of a background grd for cases when the materal nterface between black and gray materals s A. n contact. B. searated. C. searated but the nterface s near the mdont of background cells. The red emty crcles ndcate nterface nodes n the MPM calculatons that can adjust heat flow to account for the nterface. Temerature (C) % 1% Analytcal Poston (mm) 0% Contact ga Fg. 4 Temerature rofle n a bar usng condtons dentcal to Fg. 2 for very hgh h (effectvely nfnte) as a functon of the excess heat flow allowed at nterface nodes. The lot magnfes the regon near the nterface. The dashed red lne s the analytcal soluton for erfect conducton at the nterface. rect lmt of zero. In bref, t aears reasonable to always allow 1% excess heat flow. Makng ths excess a smulaton arameter would allow t to be adjusted for dfferent roblems f needed. The roer excess would be determned by comarng hgh-h smulaton results to a smulaton that gnores nterfaces. All revous examles had the nterface close to a grd lne n the background grd, but t s mortant to verfy heat flow calculatons are ndeendent of nterface locaton wthn that grd. Fgure 3C shows an nterface through mdonts of background cells. The crcled nodes show that twce as many nterface nodes are resent for such an nterface comared to an nterface near a grd lne (c.f., Fg. 3B). If contact heat flow calculatons do not account for total number of nterface nodes, these two nterface locatons would gve dfferent results. Ths ssue s dentcal to contact force calculatons n MPM [11] and solved by roer choce of contact area a n Eq. (41), whch should be read as an effectve contact area. When the nterface s near a grd lne (Fg. 3B),,α = v(n),β v(n) /2, where s total volume extraolated to node. The effectve contact area reduces to a = /t, whch s equal to the hyscal contact area. When the nterface s through mdonts of background cells (Fg. 3C),,α 7v(n) /8 and,β v(n) /8, where materal α s the one that surrounds the node [11]. The resultng effectve contact area, a = /(2t ), now correctly assgns half the contact heat flow to each nterface node n Fg. 3C where the nterface s affectng twce as many nodes. Fgure 6 comares results for an nterface near a grd lne to one through mdonts of background cells for h = W/(m 2 -K). The results are nearly dentcal. The dashed lne, however, shows results for an nterface through mdonts but usng the hyscal contact area ( /t ) nstead of the effectve contact area a. Because the uncorrected area s about twce a, the added heat flow s two tmes too hgh leadng to a temerature dstrbuton corresondng to doublng of h. For realstc smulatons, esecally when nterfaces move durng calculatons, grd ndeendence requres use of a. The overlang, h = curves show that the results are

9 Contact Heat Flow 9 Temerature (C) h = 3X10 4 Analytcal Near Grd Lne Unscaled A Near Cell Mdont h = (both) Poston (mm) Fg. 6 Temerature rofle n a bar usng condtons dentcal to Fg. 2 for h = W/(m 2 -K) and h = for smulatons wth the nterface near a grd lne or through mdonts of background cells. The dashed blue lne s for an nterface through mdonts of cells but usng hyscal nstead of effectve contact area. The dotted red lne s the analytcal soluton for erfect conducton at the nterface. ndeendent of nterface locaton and are dentcal to the analytcal soluton. Although less correcton was needed when the nterface was near the mdonts of cells (because of the hgher number of nterface nodes), each calculaton agreed well wth the erfect conducton lmt by allowng the recommended 1% excess heat flow. To valdate heat flow analyss across cracks, smulatons n Fg. 2 were reeated wth the materal nterface beng relaced by a crack wthn a sngle materal or by a crack art way along a materal nterface n multmateral mode. The frst tested heat flow at cracks nstead of materal nterfaces whle the second tested roblems that combne cracks and multmateral mode. These new results were dentcal to the results n Fg. 2 (therefore no new lot s shown). Fgure 7 shows temerature dscontnuty at an nterface for the smulaton n Fg. 2 wth h = W/(m 2 K), but now as a functon of nterfacal searaton. For nterfaces n contact (searaton < 0), the modelng assumed conducton and T was always zero. For searated nterfaces, T ncreased as a functon of searaton and aroached the h = 0 lmt (dotted red lne) for searaton of more than two cells n the background grd. A searaton of more than two cells corresonds to materals that are suffcently far aart that the smulaton no longer has nterface nodes. When there are no nterface nodes, no calculatons are done to adjust heat flow and thus heat transfer across the ga dros to zero. In other words, a searaton of more than two cells results n numercal searaton and thermal solaton of the materals. Because of numercal searaton, the modelng descrbed here works best for nterfaces and cracks wth small searatons (less than the background cell sze). ΔT (C) h = 0 h = 3X Interface Searaton (cells) Fg. 7 Temerature dscontnuty at a materal nterface for a bar usng condtons dentcal to Fg. 2 wth h = W/(m 2 K) as a functon of searaton between the two materals. The dotted red lne s temerature dscontnuty when h = 0. Whle t s hyscally reasonable for effectve convecton coeffcent h to decrease as searaton ncreases, the searaton deendence aarent n Fg. 7 s controlled by background grd cell sze and not hyscs of heat transort. Ths roerty could be changed by allowng the nterfacal heat flow (see Eq. (42)) to be any functon of,,α, and δ (where δ s nterfacal oenng dslacement avalable n both contact [11] and exlct crack [9, 10] calculatons). For examle, decreased heat flow as a functon of searaton could be modeled by allowng h to decrease as searaton ncreases. But no functon can comensate for loss of thermal contact after numercal searaton. Such wde searatons become a roblem of modelng heat flow through vods rather than across nterfaces. Vod modelng wll requre new methods. One aroach mght be to fll vods wth a gas havng dfferent thermal conductvty roertes and then modelng contact between gas and sold at materal nterface by the methods resented here. Two fnal examles consdered real-world examles that llustrate roblems where nterfaces or cracks mght affect heat flow. Vacuum arc remeltng (VAR) s a technologcally-mortant metals rocess [8]. In bref, a cylndrcal ngot s remelted by electrcal currents nto a cylndrcal crucble. If the rocess can be suffcently controlled, the re-soldfed ngot can have fewer murtes and better roertes. Because VAR s an mortant and exensve rocess, any modelng methods that can hel control t would be benefcal. Full VAR modelng requres many features such as hase transtons, recrystallzaton knetcs, Lorentz forces nduced from electrc currents, and more. A ossble MPM aroach to ths roblem wll be n a future ublcaton. Ths aer consders only a VAR-related examle for coolng a cylndrcal ngot wthn a crucble under con-

10 10 J. A. Narn dtons where thermal contracton of the ngot causes t to searate from the crucble walls. Because VAR s conducted under vacuum, such searaton s lkely to affect coolng effcency. Consder a cylndrcal ngot (radus mm; length 100 mm) fully surrounded by a crucble. Mechancal and thermal roertes of ngot () and crucble (c) were set to E = E c = 1 MPa, ν = ν c = 0.33, α = 10 6 C 1, α c = 0 C 1, k = W/(m K), k c = W/(m K), C = C c = 1000 J/(kg K), and ρ = ρ c = 1 g/cm 3. Most mortantly, the ngot s thermal exanson coeffcent was hgher than the crucble s such that the ngot wll ull away from the crucble as t cools. The axsymmetrc smulaton used mm cells, started wth all materal onts at 50 C, and gradually heated them to 100 C. The heatng was done to nduce nterfacal ressure such that all surfaces start n full contact. Gradual heatng (relatve to the materal s wave seed) was done to mnmze dynamc stress effects. After reachng 100 C, two dfferent coolng methods were used. Frst, the entre crucble was mmersed n a thermal bath at 0 C and crucble outer surfaces were modeled by heat flux boundary condtons on surface artcles of = h surf (T T res ) (see Eq. (16)) where h surf = 10 5 W/(m 2 K), T s artcle temerature, and T res = 0 C s the reservor temerature. The average temeratures n the ngot as a functon of tme after reachng 100 C for varous values of h across searated ngot/crucble nterfaces are lotted n Fg. 8. At frst, all coolng was close to the erfect conducton lmt (h = curve), but lower h gave slghtly slower coolng because thermal gradents nduced some loss of contact at the ngot/crucble nterface. After the average temerature droed below the ntal temerature of 50 C, the entre ngot searated from the crucble and the coolng rate sgnfcantly slowed. For h = 0, heat transfer stoed and therefore coolng stoed. As h ncreased, coolng contnued but coolng rates deended strongly on h. Fgure 9 shows corresondng results where the bottom half of the crucble was mmersed n a 0 C reservor whle the to half remaned n a 100 C reservor. The erfect condton lmt (h = curve) gave the exected result that average temerature aroached 50 C wth a steady state temerature gradent from cold sde to hot sde. The results wth nterfacal heat transfer, however, gave non-ntutve results. The coolng curves show regons of reduced coolng rates and regons of hgher coolng rates. These regons are determned by whch arts of the nterface were n contact. When the hotter walls on the to half lost contact, the coolng rate was hgher, but when the colder walls on the bottom half lost contact, the coolng rate slowed. <Temerature> (C) h= h=10 5 h=10 4 h=10 6 h= Tme (ms) Fg. 8 Average temerature n an ngot ntally heated from 50 to 100 C followed by mmersng the crucble n a 0 C reservor as a functon of coolng tme for varous values of h (n W/(m 2 -K)). <Temerature> (C) h=10 6 h=0 h=10 5 h=10 7 h= Tme (ms) Fg. 9 Average temerature n an ngot ntally heated from 50 to 100 C followed by the bottom half of the crucble beng mmersed n a 0 C reservor whle the to half remaned n a 100 C reservor as a functon of coolng tme for varous values of h (n W/(m 2 -K)). Transtons between slow and fast coolng ndcate dynamc moton of the ngot wthn the crucble. The role of dynamc contact between ngot and crucble on heat flow would be dffcult to model by anythng other than numercal methods such as those roosed here. The second real-world examle modeled cracks. One method for detectng nternal cracks n sold, oaque objects s by thermal magng [1]. In bref, an object s heated by varous methods and the surface temerature s maged usng an nfrared camera. If cracks affect heat flow, the number and locaton of cracks wll alter the temerature dstrbuton. The heat-flow methods descrbed here could hel to nterret such exerments. Fgure 10 shows two examles of the effect of cracks on temerature dstrbutons durng heat flow. The mm sotroc materal (E = 1 MPa, ν = 0.33, α = 0, k = W/(m K), C = 1000 J/(kg K), and ρ = 1 g/cm 3, thckness = 1 mm) started at 0 C. The bottom edge was heated usng constant heat-flux boundary condton of = 10 7 W/m 2, the to edge

11 Contact Heat Flow A. Surface Temerature (C) One Crack Four Cracks No Cracks Poston (mm) Fg. 11 Surface temerature from the thermal felds n Fg. 10 for one crack (dashed black lne), four cracks (sold blue lne), or no crack (dotted red lne). 4 Conclusons B. Fg. 10 Temerature dstrbuton n an object wth A. one crack or B. four cracks after 500 ms of heat flux aled to the bottom surface. The mage lghtness ndcates temerature from C (dark) to 110 C (lght) and 7 C er contour. was cooled by convecton usng = h surf (T T res ) where h surf = 10 5 W/(m 2 K) and T res = 0 C, and the sdes had zero heat flux. To maxmze crack effects, convectve heat flow across cracks surfaces used h = 0. The temerature dstrbuton was evaluated at the end of a 500-ms heatng erod. The MPM model used mm cells. The frst smulaton had a sngle crack n the mddle of the object, whch, as shown n Fg. 10A, nfluenced the temerature feld. Thermal magng can only observe surface temerature. Fgure 11 shows a dro on the to surface temerature above the crack comared to temerature dstrbuton that would occur wth no cracks. The length of the crack could be estmated from ths dro, but the detals would deend on dstance of the crack to the surface and on the convecton coeffcent for heat flow across the crack surface. The second smulaton used four arbtrarly szed and laced cracks. The temerature feld was sgnfcantly altered (see Fg. 10B) and the surface temerature showed hghly convoluted effects of cracks on surface temerature dstrbuton. For examle, the temerature above the last crack on the rght was hgher than average. That crack was too far below the surface to have contnued effect on surface temerature. It s unlkely the surface temerature alone can unquely locate nternal cracks, but modelng couled wth exerments could lead to dervaton of a dstrbuton of equvalent cracks that are consstent wth observed surface temerature varatons. Ths aer descrbes a new MPM feature for modelng heat flow across materal nterfaces, cracks lanes, or both. The method works best for nterfaces wth small searatons (less than one cell sze n the background grd). The method has otental for analyss of realworld roblems that coule mechancal resonse wth heat flow across nterfaces or cracks where temerature changes may cause nterfaces or crack lanes to dynamcally lose or regan contact. Comlance wth ethcal standards Fundng: Ths work was made ossble by the endowment for the Rchardson Char n Wood Scence and Forest Products. The author also thanks Rgel Woodsde, Paul Kng, and Kevn Gartner for helful dscussons. Conflct of nterest: The corresondng author states that he has no conflcts of nterest. References 1. Almond, D.P., Peng, W.: Thermal magng of comostes. Journal of Mcroscoy 1(2), (01) 2. Bardenhagen, S.G., Gulkey, J.E., Roessg, K.M., Brackbll, J.U., Wtzel, W.M., Foster, J.C.: An mroved contact algorthm for the materal ont method and alcaton to stress roagaton n granular materal. Comuter Modelng n Engneerng & Scences 2, (01) 3. Bardenhagen, S.G., Kober, E.M.: The generalzed nterolaton materal ont method. Comuter

12 12 J. A. Narn Modelng n Engneerng & Scences 5, (04) 4. Brackbll, J., Kothe, D., Ruel, H.: FLIP: A lowdssaton, artcle-n-cell method for flud flow. Comuter Physcs Communcatons 48(1), (1988) 5. Carlson, D.E.: Lnear thermoelastcty. In: C. Truesdell (ed.) Mechancs of Solds, vol. II, Srnger-Verlag, New York (1984) 6. Guenther, R.B., Lee, J.W.: Partal Dfferental Equatons of Mathematcal Physcs and Integral Equatons. Dover Books, New York (1996) 7. Lemale, V., Hurmane, A., Narn, J.A.: Materal ont method smulaton of equal channel angular ressng nvolvng large lastc stran and contact through shar corners. Comuter Modelng n Eng. & Sc. 70(1), (10) 8. Maurer, G.E.: Prmary and secondary melt rocessng sueralloys. In: J.K. Ten, T. Caulfeld (eds.) Sueralloys, Suercomostes and Suerceramcs, Academc Press, New York (1989) 9. Narn, J.A.: Materal ont method calculatons wth exlct cracks. Comuter Modelng n Engneerng & Scences 4, (03) 10. Narn, J.A.: Numercal mlementaton of merfect nterfaces. Comutatonal Materals Scence, (07) 11. Narn, J.A.: Modelng of merfect nterfaces n the materal ont method usng multmateral methods. Comuter Modelng n Eng. & Sc. 92(3), (13) 12. Narn, J.A.: Materal ont method (NarnMPM) and fnte element analyss (NarnFEA) oen-source software. htt://osudocs.forestry.oregonstate.edu (17) 13. Narn, J.A., Amene, Y.: Modelng nteractng and roagatng cracks n the materal ont method: Alcaton to smulaton of hydraulc fractures nteractng wth natural fractures (17). In rearaton 14. Narn, J.A., Bardenhagen, S.G., Smth, G.S.: Generalzed contact and mroved frctonal heatng n the materal ont method. Comuter Partcle Mechancs 5(2), (18). DOI /s Narn, J.A., Gulkey, J.E.: Axsymmetrc form of the generalzed nterolaton materal ont method. Int. J. for Numercal Methods n Engneerng 101, (15) 16. Sulsky, D., Chen, Z., Schreyer, H.L.: A artcle method for hstory-deendent materals. Comut. Methods Al. Mech. Engrg. 118, (1994)