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A Fracture Mechanics-Based Approach to Estimating the Fracture Stress and Strain of Cast Al7%Si0.3%Mg Alloys

A Fracture Mechanics-Based Approach to Estimating the Fracture Stress and Strain of Cast Al7%Si0.3%Mg Alloys
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  A Fracture Mechanics-Based Approach to Estimating the Fracture Stress and Strain of Cast Al-7%Si-0.3%Mg Allos Murat Tiryakioğlu 1 , James T. Staley 2  and John Campbell 3 1  Robert Morris Uniersity, !ittsbur"h, !# 1$1%&, US# 2  'urham, (C 2))%$, US# 3  Uniersity o* +irmin"ham, d"baston +1$ 2TT, U- !e"ords#  Castin" de*ets, *rature, porosity, /ork hardenin". Abstract.  A fracture-mechanics approach has been followed to estimate the fracture stress of Al-7%Si-0.3%Mg alloy castings as a function of size and shape of pores in castings. racture strain was predicted by using the !oce e"uation for wor# hardening. $stimated fracture stress and strain aluescompared well with e&perimental findings. 'he methodology and results are discussed in the paper. Introduction (n cast Al-7%Si-Mg alloys) mechanical properties are affected to a great e&tent by the processdesign. Mechanical properties are strongly influenced by the presence of defects in the structure)which not only result in large ariation in properties) particularly tensile strength *'S+ and elongationto fracture *el  +) but can significantly decrease fatigue life ,)/. ield stress *S+) howeer) isalmost unaffected by the presence of defects ,3)1/. 2onse"uently) samples with the same alloycomposition and processing history but with different defect sizes and morphologies can be e&pectedto hae appro&imately the same S but different 'S and el  . 'he goal of this wor# was to apply a method based on fracture mechanics principles toaluminum alloy castings so that fracture stress and strain of castings with arying defect sizes andmorphologies can be estimated. $&perimental data for an Al-7wt.%Si-0.30wt.%Mg alloy fromliterature ,1/ were used. A hybrid model based on load limit and fracture mechanics was utilized toestimate fracture stress) and wor# hardening characteristics were used to estimate fracture strain. A Model for the Effect of Structural Defects on Tensile Properties 'he effect of defect size on fracture stress can be analyzed in three different regions ,/. 'his is presented schematically in ig. . 'he first range addresses small defects where the fracture stresscan be calculated by a net section strength) or load limit) analysis. 'his analysis considers only thecontribution of reduced cross-section area following4 +f -* +df *'S5  −σ=σ  .*+where σ 5  is the limit stress *M6a+) σ 'S*df+  is the true stress at the tensile strength of a defect-freematerial and  f is the fraction of the cross sectional area of tensile specimen coered by the defect.5arger defects) egion 3) obey the residual strength ersus defect size relationship of linear elastic fracture mechanics *5$M+. 'his region is characterized by the e"uation4  b8  (c5$M π=σ .*+  igure . Schematic illustration of the effect of defect size on fracture stress and the three distinctregions where the effect of defect size is characterized using different criteria ,/. where  K   Ic  is the fracture toughness of the material *M6a m 9 +) Y   is the boundary correction factor atthe ma&imum depth point) b  *m+ of an idealized elliptical crac#. 'he intermediate region correspondsto the transition elastic-plastic range between the two e&treme criteria. :ucci et al  .,);/) in their study on the stress corrosion crac#ing of 707-'; specimens) used an empirical e"uation thatincorporates the two e&treme criteria) and found it to be as effectie as the more rigorous elastic- plastic fracture mechanics *$6M+ analyses. A similar approach was followed in this inestigation)and the egion  was represented as4 "5$M p5$6M f +f -*  σ+−σ=σ .*3+where σ $6M  is the fracture stress based on elastic-plastic fracture mechanics) and  p  and q  are weighte&ponents.'he boundary correction factor) Y  ) can be found by ,7/4            −+= 73.0 7 c;;.-e&p;;.-;1.0 .*1+where  R  is the radius of the circle with the same area as the defect) and c  is the length of crac# e&tending from the pore. $". 1 can be used for defect morphologies ranging from a small crac# around a pore to an embedded circular crac#) and has been used effectiely in cast aluminum alloys,7/. Since defect shapes are "uite different from ideal circular or elliptical pores)  R  and c  werecalculated as *ab+ 0.  and b-R ) respectiely) where a  is half-width and b  half-length of the crac#. Application of the Model to Experimental Data 'he data of Surappa et al  .,1/ were used to alidate the model for fracture stress. 'S was conertedto fracture stress by assuming conseration of olume during the tensile test. Since el   in these alloysis almost always e"ual to uniform strain ,</) σ   = 'S*> el  +. 'o estimate fracture strain) the !ocee"uation ,?/ was used for the true-stress-true strain because it has been demonstrated to be anappropriate descriptor of  the plastic deformation of Al-7%Si-Mg alloys ,0/@egion  Defect Size    F  r  a  c  t   u  r  e   S  t  r  e  s  s Load limit egion 3egion  LEFMEPFM        εε−σ−σ−σ=σ  ∞∞ 00 e&p+* .*+where σ ∞  is saturation stress) ε  is true strain) σ 0  is stress at ε =0 and ε 0  is an empirical parameter denoted by !oce ,?/ as characteristic strain. $".  can be rearranged by substituting ln*> el  + for ε 4 -el 0 0  −      σ−σσ−σ= ε−∞∞ .*;+Since stress-strain cures are not aailable) σ ∞ ) σ 0  and ε 0  hae to be estimated. σ 0  was assumed to be0.?;7S) which is a good appro&imation for Al-7%Si-Mg alloys ,/. σ ∞  was selected such that the best fit for $". ; is obtained and the boundary condition at σ =S is met simultaneously. ence)       −σ−σ−=ε ∞∞ S?;7.0 Sln$S 0 .*7+where  E   is the modulus of elasticity *=7; B6a for A3; alloy ,/+. 'he defect-free fracture stresswas ta#en as 3; M6a. Cswalt and Maloit ,3/ listed the fracture toughness of an underaged A37specimen with appro&imately the same S and an el   of 0.; as 3 M6a. m 9 . 'hat alue was usedfor the 5$M calculations. esults 'he results of application of the model for fracture stress to the data of Surappa et al  . are presentedin ig. . Dote that the model estimates almost a perfectly linear relationship and that there is a goodagreement between the model and e&perimental data. Also all data points seem to fall into egion )where elastic plastic fracture mechanics determine the defect size - fracture stress relationship. :estfit was obtained with a  p  of 1.?1 and a q  of .3. 'he estimated fracture stress alues were used to calculate el   of samples using $". ;.esults are shown in ig. 3. :est fit was obtained with a σ ∞  of 37 M6a) which yielded a ε 0  of 0.073 using $". 7. Dote in ig. 3 that there is a good agreement between the e&perimental and predicted results) although the model slightly oerestimates at high  f   alues. Discussion 'he fracture mechanics-based model effectiely predicts the fracture stress. 'he defect size-fracturestress relationship seems to be dominantly in the $6M region. 'his helps e&plain why Surappa et al  . stated that the load limit approach could not e&plain the drop in fracture stress with increasing  f   .errera and 8ondic ,1/) on the other hand) used a fracture mechanics approach in their study on anAl-0wt.%Si-0.1wt.%Mg) and concluded that linear elastic fracture mechanics were not sufficient tofully e&press the effect of defects on fracture) which is consistent with the results of this study.  igure . Application of the fracture mechanics based model to fracture stress estimation in Al-7wt.%Si-0.3wt.%Mg alloy castings.igure 3. $&perimental alues of Surappa et al  . and the predicted alues for the elongation tofracture of Al-7wt.%Si-0.3wt.%Mg alloy castings.Surappa et al  . also suggested that an empirical relationship can be established betweenfracture stress and  f   by linear regression) and that el   and 'S can be estimated if defect sizes are#nown. 'he model used in this study) as shown in ig. ) follows almost a linear relationship.Surappa et al  .) howeer) did not proide a model. 2Eceres ,/ deeloped such a model thatcombines wor# hardening characteristics of the alloy and the load limit approach) and applied it tothe data of Surappa et al  .. 2Eceres assumed that fracture was strain-critical *as opposed to thestress-critical approach followed in this study+. (n his model) fracture ta#es place once the strainnear the pore reaches the fracture strain of the defect-free material. 2Eceres did not use any factor  0.000.00.010.0;0.0<0.-00.-0.-10.-;0.000.0-0.00.030.010.0 f        e         l       F Surappa et alia predicted   07300330370.000.00.00.030.010.0 f    L Surappa et alia   LEFM   F  ! L "#$f %.&% ' ( LEFM  f #.#)    F  r  a  c   t  u  r  e   S   t  r  e  s  s   "   M   P  a   '  for stress concentration. Fespite its important contribution) 2EceresG model could e&press the effectof f on fracture stress and strain only "ualitatiely.A different approach from that followed by 2Eceres was deeloped in this study. racturemechanics and the wor# hardening characteristics of the alloy were used to estimate the fracturestrain) once the fracture stress was estimated. 'he !oce e"uation has been shown to effectiely predict fracture strain) once fracture stress is estimated following the fracture mechanics approach. *onclusions .A model based on elastic plastic fracture mechanics was used successfully to predict the fracturestress of Al-7wt.%Si-0.3wt.%Mg alloy castings based on the size and morphology of thestructural defects. 'he model combined the load limit approach and linear elastic fracturemechanics..Hithin the region of the e&perimental data) defect size I fracture stress relationship wasdominated by elastic plastic fracture mechanics.3.Cnce fracture stress is estimated) the !oce e"uation can be used to effectiely predict fracturestrain.  eferences .J. 2ampbell) 2.H. Dyahumwa) D.. Breen) in  Adances in Aluminum 2asting 'echnology) ASM (nternational *??<+) p.. .D.. Breen) J. 2ampbell@ Mater. Sci. $ng. A) !ol. A37 *??3+) p.;.3.J. 2ampbell@ Castings  *C&ford@ :utterworth einemann) ??+. 1.M.8. Surappa) $. :lan#) J.2. Ja"uet@ Scrip. Metall.) !ol. 0 *?<;+) p.<...J. :ucci) .5. :razill) F.C. Sprowls) :.M. 6onchel) 6.$. :retz@ 6roc. (nt. 2onf.on atigue)2orrosion 2rac#ing) rac. Mech. K ail. Analysis) ASM) Salt 5a#e 2ity) ?<) p.;7.;.F.A. 5u#asa#) .J. :ucci) $.5. 2olin) :.H. 5if#a@ (n AS'M S'6 31 *??+) p.0.7.:. S#allerud) '. (eland) B. Lrge#rd@ $ng ract Mech. !ol. 11 *??3+ p.<7. <.F.5. Mc5ellan@ J. 'est. $al. !ol. < *?<0+) p.70.?.$. !oce@ Metallurgia) !ol.  *?+ p.?.0.M. 'irya#ioNlu) J.'. Staley) J. 2ampbell@ J Mater Sci 5et. !ol. ? *000+ p.7?..M. 'irya#ioNlu@ 6h.F. 'hesis) Oniersity of :irmingham) 00..F.J. 5loyd@ (nt Mater e. !ol. 31 *??1+ p..3.8.J. Cswalt) A. Maloit@ AS 'ransactions *??0+ p. <;.1.A. errera) !. 8ondic@ 6roc. (nt. 2onf. on Solidification and 2ast Metals) 'he Metals Society) Sheffield) ?77) p.1;0..2.. 2Eceres@ Scrip Metall Mater. !ol. 3 *??+ p.<.
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