Engineering Fracture Mechanics

Transcription

1 Engineering Frature Mehanis 76 (29) Contents lists available at SieneDiret Engineering Frature Mehanis journal homepage: Numerial modelling of magnesium die-astings using stohasti frature parameters C. Dørum a,b, *, O.S. Hopperstad b, T. Berstad a,b, D. Dispinar a a SINTEF Materials and Chemistry, Trondheim, Norway b Strutural Impat Laboratory (SIMLab), Centre for Researh-based Innovation, Department for Strutural Engineering, Norwegian University of Siene and Tehnology, Trondheim, Norway artile info abstrat Artile history: Reeived 1 June 28 Reeived in revised form 25 May 29 Aepted 1 July 29 Available online 4 July 29 Keywords: Magnesium Die-astings Dutile frature Weibull distribution Finite element analysis Quasi-stati material tests using speimens ut from a generi ast omponent are performed to study the behaviour of the high-pressure die-ast magnesium alloy AM6 under different stress states. The experimental data set is applied to establish a validated probabilisti methodology for finite element modelling of thin-walled die-astings subjeted to quasi-stati loading. The test speimens are modelled in the expliit finite element (FE) ode LS-DYNA using shell elements. The ast magnesium alloy AM6 is modelled using an elasto-plasti onstitutive model inluding a high-exponent, isotropi yield riterion, the assoiated flow law and isotropi hardening. To simulate frature, the Cokroft Latham frature riterion is adopted, and the frature parameter is hosen to follow a modified weakest-link Weibull distribution. Comparison between the experimental and predited behaviour of the ast magnesium speimens gives very promising results. Ó 29 Elsevier Ltd. All rights reserved. 1. Introdution To redue the fuel onsumption, the automotive industry strives for strutural solutions ombining low weight with low ost. In this respet, the high-pressure die-asting of light weight metals suh as aluminium and magnesium alloys has attrated attention as being a ompetitive prodution method. One of the main hallenges with this prodution method is to optimise the proess parameters with respet to the part design and the solidifiation harateristis of the alloy in order to obtain a sound asting without asting defets. Unbalaned filling and lak of thermal ontrol an ause bifilms, porosity and surfae defets due to turbulene and solidifiation shrinkage. As a result, stohasti harateristis tend to prevail in the frature behaviour of the astings. Design and prodution of thin-walled ast strutural omponents for the automotive industry involve development of alloys and manufaturing proesses, strutural design and rashworthiness analysis. To redue the lead time for development of a new produt, it is neessary to use finite element (FE) analysis to ensure a strutural design that fully exploits the material without sarifiing safety. Aurate desription of the material behaviour is essential to obtain reliable results from suh analyses. To minimise the weight of the strutural omponent while maintaining the safety in a rash situation, the dutility of the material has to be utilised without risking un-ontrolled failure. Hene, a reliable failure riterion is also required, enabling the designer to take advantage of the potential of the ast material. From the literature, it is seen that several different approahes for finite element based modelling of frature for thinwalled astings have been investigated over the last years. In a work by Okewitz et al. [1], it has been suggested to use a * Corresponding author. Address: SINTEF Materials and Chemistry, Trondheim, Norway. address: ato.dorum@sintef.no (C. Dørum) /$ - see front matter Ó 29 Elsevier Ltd. All rights reserved. doi:1.116/j.engframeh

2 C. Dørum et al. / Engineering Frature Mehanis 76 (29) Nomenlature u r 1, r 2 k r Y r Q i, C i e e W W P V V r m W f u W e1 W max W e L yield surfae prinipal stresses material parameter flow stress proportionality limit hardening parameters effetive plasti strain Cokroft Latham integral frature parameter frature probability volume saling volume saling stress Weibull modulus saling value for the frature parameter probability density funtion random number value for Weibull distributed frature parameter for element e maximum value for frature parameter loal value for frature parameter for element e length sale material model onsidering miro mehanial damage suh as the Gurson model [2]. Chen et al. [3] predited frature using a maximum plasti strain and a maximum prinipal strain riterion. Their onlusion was that both of the investigated riteria have limitations in predition of frature initiation, and that stress state should be aounted for. Altenhof et al. [4] suggested to use a pure stress based riterion. Leppin et al. [5] proposed to use the IDS riterion (Instability, Dutile and Shear frature) [6], a frature model that requires a minimum of six different material tests for alibration of an isotropi and homogeneous material. Mae et al. [7] alibrated the Bao Wierzbiki frature lous [8] for a ast aluminium alloy by using a total of 12 tests. Another approah for modelling of frature in high-pressure die-asting aluminium alloys have been explored by Mohr and Treitler [9] by using a phenomenologial frature riterion that was fitted to the experimental results of four different material tests. In previous works [1 13] by the authors of this manusript, it has been suggested to use the relatively simple phenomenologial dutile frature riterion as proposed by Cokroft and Latham [14], a riterion that for isotropi and homogeneous materials only requires one single material test to be alibrated. However, as the experimental results from these previous studies have demonstrated, and also as pointed out in the work by Gokhale and Patel [15], the measured dutility an vary signifiantly due to stohasti variations in sizes, numbers and amount of defets present in the ast material. Fig. 1 shows the geometry of the generi AM6 omponent investigated in this study, together with the orresponding gating system. The length of omponent is 4 mm and the wall thikness is approximately 2.5 mm. In previous studies [1,16], the AM6 material was haraterized using uniaxial tensile tests, uniaxial ompression tests, and plate bending Fig. 1. Illustration of generi ast omponent: length = 4 mm, thikness = 2.5 mm, width = 8 mm, and height = 4 mm.

3 2234 C. Dørum et al. / Engineering Frature Mehanis 76 (29) tests. The results from the uniaxial tensile tests showed that the satter in elongation at frature is quite large. The poorest area is the outlet side, where values of effetive plasti strain at frature as low as 2 3% were measured. The best areas were found to be the 8 mm flange in front of the gates, where values of effetive plasti strain at frature as high as 22% were measured [1]. An overview of the tensile elongation distribution is provided in Fig. 2. The experimental results also revealed different tensile and ompressive behaviour for magnesium alloys. For details, it is referred to Dørum et al. [16]. However, sine the reported strength differene is relatively small, this has not been further examined here. The omponents were ast of magnesium alloy AM6 at Hydro s Researh Centre in Porsgrunn, Norway with a Bühler SC42D 42-ton old hamber dieasting mahine. A reent study on aluminium die-astings [13] indiated that identifiation of frature parameters for ast materials depends signifiantly on the size and/or the geometry of the material test speimens. In this study, the stress strain behaviour and frature harateristis of die-ast magnesium alloy AM6 are investigated. Uniaxial tension tests, plane-strain tension tests, nothed tension tests, and shear tests are arried out using speimens ut from generi ast omponents. This allows studying the influene of speimen geometry (size effets) and stress state on the observed behaviour. The frature surfaes are investigated using sanning eletron mirosope (SEM). A new approah for FE modelling of frature in astings is developed. The material behaviour is desribed by an elasto-plasti model inluding a high-exponent, isotropi yield riterion, the assoiated flow law and isotropi hardening. Frature is modelled by the Cokroft Latham riterion [14], assuming the frature parameter to follow a modified weakest-link Weibull distribution [17]. Shear frature (due to shear band loalisation) has not been aounted for in the present work. 2. Material tests Four types of tests were performed for the ast AM6 alloy: uniaxial tension, plane-strain tension, nothed tension and shear tests. The different types of tests were arried out using speimens ut from some seleted positions in the ast U-profile, as illustrated in Fig. 3. With this proedure, ast material ut from idential positions in the omponent is subjeted to different states of stress and strain. The tests were arried out in a hydrauli testing mahine under displaement ontrol. Fore and displaement/strain were ontinuously measured. The displaement rate was adjusted to obtain a strain rate approximately equal to s 1. All tests were arried out at ambient temperature. Uniaxial tension speimens were ut from the inlet wall and the outlet wall in the longitudinal diretion, and from the 8 mm web of the asting in both the longitudinal and transverse diretion. The geometry of the tensile speimens is Fig. 2. Dutility map for ast AM6 omponent, whih illustrates the distribution of effetive plasti strain at frature throughout the omponent [12]. Fig. 3. Position of speimens ut from the generi ast omponent.

4 C. Dørum et al. / Engineering Frature Mehanis 76 (29) shown in Fig. 4a. The strain in the length diretion was measured by an extensometer with 25 mm gauge length. The experimental engineering stress strain urves for the AM6 asting are shown in Fig. 5, while Cauhy stress versus logarithmi plasti strain urves are provided in Fig. 6 for different parts of the omponent. It is seen that the differene between tests in the longitudinal and transverse diretions is small, both with respet to stress strain behaviour and dutility. From the tests in the longitudinal diretion, it is observed that there are marked differenes in dutility with position in the asting, while the stress strain behaviour is generally muh less affeted. In partiular, the material in a Ø5 R6 8 b R4 6 R Ø R8 R15 R27 R d Ø R 1 7 Fig. 4. Material test speimens: (a) uniaxial tension, (b) plane-strain tension, () shear, and (d) nothed tension.

5 2236 C. Dørum et al. / Engineering Frature Mehanis 76 (29) mm web, longitudinal, position 1 8 mm web, longitudinal, position 2 Uniaxial ompression, longitudinal 4 8 mm web, transverse, position 1 8 mm web, transverse, position s [MPa] 2 s [MPa] e e 4 Inlet wall, longitudinal, position 1 Inlet wall, longitudinal, position 2 4 Outlet wall, longitudinal, position 1 Outlet wall, longitudinal, position s [MPa] 2 s [MPa] e e Fig. 5. Experimental engineering stress strain urves from uniaxial tensile tests. the outlet flange is less dutile than the material in the inlet flange and in the web. Furthermore, the material in position 1 is more dutile than the material in position 2 for the 8 mm web. The satter in dutility between dupliate tests is signifiant, and is aused by asting defets that vary from one tensile test speimen to another in a stohasti manner. For omparison, the engineering stress strain urve obtained from uniaxial ompression test in a previous work [16] is shown together with the data from uniaxial tensile tests using speimens ut from the web in longitudinal diretion, see Fig. 5. No signifiant strength differene is observed. Plane-strain tension tests were arried out using speimens ut from the 8 mm web of the U-profiles. The speimen is taken in the longitudinal diretion, and the geometry is illustrated in Fig. 4b. The longitudinal displaement over the entral region was measured by an extensometer with 55.5 mm gauge length. Fig. 7 shows the measured fore displaement urves. There is signifiant satter in the dutility for speimens mahined from the same position in the astings, i.e. between dupliate tests. With respet to spatial variations, the ast material in position 1 is again more dutile than the material in position 2. The fore level is onsistent between the dupliate tests, and is not signifiantly affeted by position. The variations in displaement at frature show that the dutility in ast omponents an vary from asting to asting, even for astings produed in a well-ontrolled manner. Shear tests were arried out using speimens ut from the 8 mm wide web of the U-profiles. The speimen geometry is shown in Fig. 4. The speimens were aligned with the longitudinal diretion of the AM6 omponents. The longitudinal displaement over the entral region of the shear speimen was measured by an extensometer with 55.5 mm gauge length. The measured fore displaement behaviour is provided in Fig. 8. It is seen that there is no signifiant differene in the fore displaement behaviour and dutility of speimens ut from positions 1 and 2. This ould indiate that the ast material

6 C. Dørum et al. / Engineering Frature Mehanis 76 (29) mm web Outlet wall Inlet wall σ [MPa] ε p Fig. 6. Cauhy stress versus logarithmi plasti strain urves for AM F [kn] mm web, position 1 8 mm web, position Fig. 7. Measured fore displaement behaviour from plane-strain tension tests. is less sensitive to the spatial distribution of asting defets when subjeted to shear loading. However, it should be kept in mind that the volume of strained material is onsiderably smaller in the shear test than in the uniaxial and plane-strain tension tests. Nothed tension test speimens were ut from the 8 mm web of the asting, in the transverse diretion. The geometry of the speimens is shown in Fig. 4d. With respet to possible size effets, the ritial area where frature is likely to initiate is signifiantly smaller in the nothed tensile speimen ompared with the ritial area for frature in the ordinary uniaxial tensile speimen (Fig. 4a). However, the state of stress is not dramatially hanged, even if some transverse stress will be introdued. The displaement over the noth was measured by an extensometer with 25 mm gauge length. Fig. 9 shows the measured fore displaement urves from the nothed tensile tests. Frature ours at maximum fore. Signifiant differenes are observed in dutility of the ast material with position in the asting, supporting the results from the uniaxial tensile tests. Material ut from position 1 in the 8 mm web is onsistently more dutile than the material ut from position 2 when subjeted to tensile loading.

7 2238 C. Dørum et al. / Engineering Frature Mehanis 76 (29) Experiment, position 1 Experiment, position F [kn] Fig. 8. Measured fore displaement behaviour from shear tests. 4 3 F [kn] mm web, position 1 8 mm web, position Fig. 9. Measured fore displaement behaviour from nothed tensile tests. 3. Metallurgial onsiderations The mehanial properties of an alloy depend on the defets that may be present in the matrix. These defets ould be point, line, surfae or volume defets. Among these defets, volume defets (porosity, seondary phases or inlusions) are known to be the most signifiant ones and may affet the mehanial properties dramatially. Inlusions, basially oxides, play an important role in asting operations. Partiularly in high-pressure die-asting operations, with asting speeds of minimum 15 m/s up to 4 m/s, the liquid metal advanes into the mould in jets that introdue the surfae oxide to beome inorporated into the melt. However, the oxide inlusions annot exist in melts as a single, beause the only way they an beome inorporated into the liquid is by entrainment ation [18]. During suh a simple folding ation, the two non-wetted oxide surfaes ome in ontat to form a bifilm that ats as a rak in the asting. Therefore, in high-pressure die-astings, the asting will have a spatial distribution of asting defets. The size and population of these defets are ritial sine they at as the initiation points for porosity and also as stress risers. As seen from Fig. 1, pitures

8 C. Dørum et al. / Engineering Frature Mehanis 76 (29) Fig. 1. Pitures of frature surfae from sanning eletron mirosope (SEM). using sanning eletron mirosope (SEM) show that the frature surfae has a high density of rak-like pores. By loser examination, it an be onfirmed that the rak-like pores are all oxides. It is well known that in the presene of defets or stress risers, the omponents may frature at stresses far away from their nominal theoretial limits. Fig. 6 is a perfet example to suh phenomena. Even within the groups (longitudinal and transverse diretion, outlet and inlet side) there is a huge satter of elongation and maximum stress values. It is important to note that even the proof stress hanges onsiderably. Here, the oxides (or bifilms) at as a sort of strengthening meha-

9 224 C. Dørum et al. / Engineering Frature Mehanis 76 (29) nism in the matrix whih is very similar to the behaviour of metal matrix omposites. It is also interesting to note that there is one sample in Fig. 6 that fratures even before reahing the proof stress. This pre-mature frature is another example of the presene of defets (most probably bifilms) in the asting. 4. Material model and parameter identifiation The experimental work by Kelley and Hosford [19] showed that the yield surfae of textured magnesium alloys takes omplex shapes. Their onlusions were that the investigated polyrystalline materials were very anisotropi and that the anisotropy inreased with inreasing levels of texture. In addition, the yield surfaes were neither elliptial nor entred at the origin due to the diretionality of the twinning mode, resulting in a different behaviour in tension and ompression. Aording to Cazau and Barlat [2]; if the internal shear mehanism of plasti deformation is sensitive to the sign of the stress, then the marosopi yield funtion ought to be represented by an odd funtion of the prinipal values of the stress deviator. Lou et al. [21] investigated the hardening evolution of AZ31B sheet. They onstruted a model of deformation mehanisms on the basis of predominantly basal slip for initial tension, twinning for initial ompression, and untwinning for tension following ompression. Staroselsky and Anand [22] developed a rystal-mehanis based model for polyrystalline hp materials applied to magnesium alloy AZ31B. Previous work has shown that the fine-grained HPDC AM6 material an be regarded as isotropi and that the differene in tension and ompression behaviour is relatively small [16]. Thus, the ast magnesium alloy AM6 is modelled using an elasto-plasti onstitutive model inluding a high-exponent, isotropi yield riterion, the assoiated flow law and isotropi hardening. It should be noted that the hoie of yield riterion is not based on any experimental data in the literature supporting this hoie for an hp material and that the hosen yield surfae is not able to desribe any asymmetri behaviour (different behaviour in tension and ompression). Frature is modelled by element erosion when a frature riterion is reahed. The model has been implemented in the expliit finite element ode LS-DYNA [23]. The high-exponent isotropi yield riterion [24,25] is written in the form uðr 1 ; r 2 Þ¼ðr 1 Þ 2k þðr 2 Þ 2k þðr 1 r 2 Þ 2k ¼ 2r 2k Y where r 1 and r 2 are prinipal stresses in plane stress and k is a material parameter. For k = 1 the high-exponent yield surfae redues to the lassial von Mises yield surfae. In this work, the exponent k is evaluated against the available experimental data. The flow stress r Y is defined by the isotropi hardening rule r Y ¼ r þ X2 i¼1 Q i ð1 expð C i e e ÞÞ where e e is the effetive plasti strain, r is the proportionality limit, and Q i and C i are hardening parameters. Using a least squares method, the hardening parameters were determined from the Cauhy stress versus logarithmi plasti strain urves in Fig. 6 and are given in Table 1. This implies that any variation in flow stress with position in the asting was not aounted for in the FE simulations. An unoupled ontinuous disontinuous approah to desribe frature is adopted (see e.g. Mediavilla et al. [26]). This means that the influene of damage evolution on the material behaviour is negleted and there is no material softening before initiation of frature. Crak propagation is desribed by element erosion when a frature riterion is fulfilled within the element. The Cokroft Latham riterion [14] was used in the simulations, i.e. Z W ¼ maxðr 1 ; Þde e 6 W ð3þ where r 1 is the maximum prinipal stress and W is the ritial value of the integral W. Hene, frature ours when W = W. Heneforth, W will be referred to as the frature parameter, while W will be denoted the Cokroft Latham integral. It is seen that frature annot our when the maximum prinipal stress is ompressive and that neither stresses nor strains alone are suffiient to ause frature. Furthermore, the frature strain inreases with dereasing stress triaxiality (in the shear tests, the stress triaxiality is signifiantly redued ompared to the uniaxial tension test). As the Cokroft Latham frature riterion is based upon only one parameter, a single material test is suffiient for the alibration. In the present study, the frature parameter W was independently identified from uniaxial tension tests, plane-strain tension tests, nothed tension tests, and shear tests. In this way, the validity of the Cokroft Latham frature riterion for different deformation modes (or stress states) and statistial effets on frature may be evaluated. If the Cokroft Latham frature riterion aurately desribes frature for ast magnesium alloys, one would expet that the value of ð1þ ð2þ Table 1 Work hardening parameters for AM6 obtained from uniaxial tensile tests. Alloy r (MPa) Q 1 (MPa) C 1 Q 2 (MPa) C 2 AM6 Web

10 C. Dørum et al. / Engineering Frature Mehanis 76 (29) the frature parameter would be approximately the same for all of the tests. However, as the ast material is very inhomogeneous, due to the spatial distribution of asting defets, it was expeted that size effets ould be quite signifiant. The uniaxial tensile test speimens failed before the point of diffuse neking for the AM6 alloy, and, aordingly, the stress and strain fields are uniform up to frature. Hene, the frature parameter is obtained as the area under the workhardening urve, sine for uniaxial tension, the Cauhy stress r equals the maximum prinipal stress r 1 and the logarithmi plasti strain e p is equal to the effetive plasti strain e e. The resulting values of W are listed in Table 2 for the inlet flange, the outlet flange and the web. The dramati variation in dutility is refleted in the large variations of W with position but also between uniaxial tension tests at the same position. For the other types of tests, the stress and strain fields are not uniform and the frature parameter W has to be estimated via FE analysis. To this end, eah of the material tests was simulated numerially without aounting for frature. With this proedure, the value of the Cokroft Latham integral W ould be monitored throughout the gauge area and the frature parameter ould be identified for the investigated material tests. It was assumed that W is equal to the maximum value of W within the speimen s gauge area at maximum load. To evaluate the shape of the yield surfae, the material tests were simulated with different values of the material parameter k, defining the shape of the yield surfae. It is notied that k = 1 gives the von Mises riterion, while k? 1 results in the Tresa riterion. Here, simulations were run with k = 1 and 4. Shell (or membrane) elements were used to model the speimens, sine plane stress prevails in all test speimens. Homogeneous material properties were assumed for the various parts of the ast material, adopting the hardening parameters given in Table 1. The deformation of the speimens was applied smoothly but muh faster than in the experiments. It was heked that the kineti energy and its variation were both negligible throughout, and thus the simulations an be onsidered quasi-stati. The FE mesh of the plane-strain tension test speimen onsists of 26,644 shell elements, giving a harateristi element size of.6 mm in the gauge area. A omparison between the experimental and predited fore displaement behaviour is provided in Fig. 11. It seen that fore displaement behaviour is well desribed in the simulation. Better agreement with the experiments is obtained with k = 4, but the influene of this parameter is small. Also shown in Fig. 11, is the maximum value of the Cokroft Latham integral W over the speimen s gauge area. The dashed vertial lines in the figure indiate the displaement at frature in two of the tests, while the orresponding horizontal dashed lines point towards the estimated values of W. The alulated frature parameters are listed in Table 2. It is seen that the uniaxial and plane-strain tension tests give frature parameters of the same order for the web of the U-profile, although the frature parameters for the plane-strain tension test is somewhat higher. Table 2 Cokroft Latham frature parameters W (MPa). Uniaxial tension test Plane-strain tension test Shear test Nothed tensile test Inlet wall Web Outlet wall Web Web Web Experiment Simulation, k = 1 Simulation, k = 4 W, k = 4 F/A, W [MPa] Fig. 11. Comparison of the experimentally measured and numerially predited fore (F/A ) together with the predited Cokroft Latham integral W as funtions of displaement (w) for plane-strain tension tests for AM6.

11 2242 C. Dørum et al. / Engineering Frature Mehanis 76 (29) Experiment Simulation, k = 1 Simulation, k = 4 W, k = 4 F/A, W [MPa] Fig. 12. Comparison of the experimentally measured and numerially predited fore (F/A ) together with the predited Cokroft Latham integral W as funtions of displaement (w) for shear tests for AM6. The shear test speimens were modelled by 8675 shell elements. The harateristi element size in the gauge area is.1 mm. The predited fore displaement urve and the maximum value of the Cokroft Latham integral versus displaement are illustrated in Fig. 12. As for the simulation of the plane-strain tension test, k = 4 gives exellent agreement with the experimental results. The estimated frature parameters W are listed in Table 2, where it has been assumed that frature ours at peak fore. By omparing the values of the frature parameter obtained from the shear tests with those obtained from the uniaxial and plane-strain tension tests, it is seen that the former values are about three times higher for AM6. This indiates that the Cokroft Latham riterion is not generally valid when assuming homogeneous material properties. Furthermore, the frature mode may hange depending on the material test; uniaxial tension, plane strain tension, nothed tension and in-plane shear. However, it should be realled that the volume of material tested in the shear tests is only a small fration of the volume tested in uniaxial and plane-strain tension. Thus, owing to statistial effets, the probability of testing a part of the material having a defet of a given size and orientation is muh smaller in the shear tests, whih should lead to inreased dutility. It an therefore be assumed that the maximum W value obtained from these shear tests represent a good estimate on the maximum dutility that an be ahieved in a flawless material. The nothed tensile test speimen was modelled using 72 shell elements, giving a harateristi element size of.8 mm. Fig. 13 shows a omparison of the experimental and numerial fore displaement behaviour and the predited maximum value of the Cokroft Latham integral W versus displaement. The agreement is again exellent, and it is observed that the shape of the yield surfae seems to be of little importane. The estimated values for W are ompiled in Table 2. Somewhat unexpetedly, the maximum value of W in the nothed tensile tests is a bit higher than the maximum value obtained in the shear tests. Regarding transferability of damage parameters between the different types of material tests, is should be noted that the geometry of the nothed tensile speimen was hosen to give a stress strain ondition in the narrowed setion of the speimen whih was similar to uniaxial tension. Thus, the inreased values for W estimated from this test ompared to those obtained from uniaxial tensile tests indiated that size effets are important with respet to alibration of damage parameters for the investigated material. Another approah to investigate the size effet ould be taken by omparing tensile and shear tests having the same volume of material in the gauge area in whih frature an take plae. However, this would require that the gauge area were uniformly strained in both the uniaxial tensile test and the shear test. For the uniaxial tensile test speimen used in this work, the volume of the gauge area is equal to (1.3 mm 25 mm mm 6 mm) 2.5 mm (thikness) = 81 mm 3. For the in-plane shear test, the material in the narrowed setion of the speimen is not uniformly strained. But the volume in whih frature is likely to initiate is less than 5 mm (length of the narrowed setion) 2 mm (approximately width of narrowed setion) 2.5 mm (thikness) = 25 mm 3. As an illustration of the volume where frature may initiate, Fig. 14 shows ontour plots of the integral W at the point of frature for the most dutile of the different speimens. It should be noted that due to the non-homogeneity of the mirostruture, the most ritial element may not neessarily be loated in the region with the highest values for W. 5. A probabilisti approah to frature modelling By omparing the values of the frature parameter obtained from the nothed tensile tests and the shear tests with those obtained from the uniaxial and plane-strain tension tests, it is seen that the former values are signifiantly higher. This ould

12 C. Dørum et al. / Engineering Frature Mehanis 76 (29) F/A, W [MPa] 2 1 Experiment Simulation, k = 1 Simulation, k = 4 W, k = Fig. 13. Comparison of the experimentally measured and numerially predited fore (F/A ) together with the predited Cokroft Latham integral W as funtions of displaement (w) for nothed tensile tests for AM6. mean that the Cokroft Latham riterion is not valid, or it ould be an evidene of a size effet, sine the speimens of the two groups of tests have different gauge areas. Owing to statistial effets, the probability of testing a part of the material having a asting defet of a given size and orientation is muh smaller when the gauge area is small, whih should lead to inreased dutility. It is therefore probable that the differenes in frature parameters obtained from the different tests are (at least partly) due to a size effet. Zhou and Molinari [27,28] propose a miro-raking model for brittle materials (eramis) onsidering the stohasti distribution of internal defets. The model introdues of a Weibull distribution of the loal strength of ohesive elements. Thus, the probability of introduing a weak ohesive element inreases with the ohesive element size. Inspired by this idea, the frature parameter W of a finite element is assumed to follow a modified weakest-link Weibull distribution in the urrent study. The Weibull distribution [17] gives the frature probability P(r) of a material volume under effetive tensile loading, i.e. PðrÞ ¼1 exp V r m ð4þ V r where V is the volume, V is the saling volume, r is the saling stress, and m is the Weibull modulus. Sine ast magnesium alloys are not brittle materials, the use of a ritial frature stress is not justified. Instead, the Cokroft Latham dutile frature riterion is adopted, and the frature probability of a material volume is reast as PðWÞ ¼1 exp V m W ð5þ V W where W is the saling value for the frature parameter. Consequently, the orresponding probability density funtion for the loal material dutility reads f ðw Þ¼ dp dw ¼ m m 1 V W exp V m W ð6þ W¼W W V W V In the finite element model, the frature parameter is assumed to follow the modified weakest-link Weibull distribution, represented by Eqs. (5) and (6). The volume V is then the volume of the element, whih implies that the dutility of an element dereases with inreasing size. Hene, large elements are more likely to fail than small elements sine they have higher probability of ontaining defets. In pratie, the Weibull distribution of the frature parameter is ahieved by using a random number generator and inverse transform sampling. A random number u is generated from the standard uniform distribution. A trial value of the frature parameter for element e, denoted W e1, is first alulated suh that PðWe1 Þ¼u. As mentioned earlier, it is assumed that the maximum value of W obtained from the shear tests represents a reasonable estimate of the maximum dutility W max of the ast material. Thus, the loal value of the frature parameter for element e is taken as W e ¼ minðwe1 ; Wmax Þ. Now, the frature parameters whih are determined from the experimental results represent marosopi frature of the ast material. To avoid simulating miro-raks when the element size is small, a length sale has to be W

13 2244 C. Dørum et al. / Engineering Frature Mehanis 76 (29) Fig. 14. Contour plots of the integral W: (a) uniaxial tension, (b) plane-strain tension, () shear, and (d) nothed tension. introdued. To this end, the frature parameter W e is taken as the minimum value of the frature parameter for all elements within a given radius L emanating from the entre of atual element. 6. Numerial study The Weibull parameters were alibrated using frature parameters obtained from the uniaxial tension tests. The saling volume is taken as the volume of the gauge region of the speimen, i.e. V =3mm 5mm 2.5 mm. The shape of the Weibull distribution is very sensitive to the parameters. With a relatively low number of experimental parallels, as in this work,

14 C. Dørum et al. / Engineering Frature Mehanis 76 (29) the shape of the Weibull distribution depends on the amount of experimental data. To irumvent this problem, it was assumed that the experimentally obtained frature parameters follow a normal distribution. By first alulating the mean value and the standard deviation of W, the Weibull parameters m and W ould then be alibrated to give the best fit to this normal distribution. Table 3 shows the Weibull parameters for the inlet wall, the outlet wall and the 8 mm web of the generi ast AM6 omponent. Finally, the non-loal radius L was set to.5 mm in all simulations to avoid simulating miroraks. Table 3 Weibull parameters. V (mm 2 ) m W (MPa) W max Inlet wall Web Outlet wall (MPa) 3 Experimental Numerial 2 s [MPa] e Fig. 15. Comparison of the experimentally measured and numerially predited (six parallels) engineering stress strain urves for uniaxial tensile tests. 3 Experimental Numerial F/A, W [MPa] Fig. 16. Comparison of the experimentally measured and numerially predited fore displaement urves for plane-strain tension tests.

15 2246 C. Dørum et al. / Engineering Frature Mehanis 76 (29) To validate the proposed probabilisti approah for modelling of frature in magnesium die-astings, the uniaxial tensile tests, plane-strain tension tests, shear tests and nothed tensile tests with speimens ut from the 8 mm web were simulated in LS-DYNA. The uniaxial tension test speimens were modelled by 72 shell elements (i.e. a harateristi element size equal to 1. mm), the FE-models for the other test speimens were as desribed in Setion 4. Six repeated simulations were arried out for the uniaxial tensile tests, plane-strain tension tests and shear tests, while eight parallels were arried out for the nothed tensile tests. In eah simulation, a different distribution of the frature parameter was obtained as explained above. The results from the simulations are illustrated in Figs together with the experimental results. As seen in Fig. 15, the predited engineering stress strain urves from simulations of the uniaxial tensile tests are very similar to the experimental one. Furthermore, both the slight inrease in dutility (in terms of W values) for the plane-strain test and the large inrease in dutility for the shear tests and the nothed tensile tests are aptured in the numerial preditions. For the simulations of the shear tests, frature ourred when the frature parameter reahed the maximum value W max in four of the simulations. Consequently, the predited satter is very small, similar to what was observed from the experimental tests. It should be noted that also in one of the simulations of the nothed tensile tests, frature ourred due to the max- 2 Experimental Numerial 16 F/A, W [MPa] Fig. 17. Comparison of the experimentally measured and numerially predited (six parallels) fore displaement urves for shear tests. 3 Experimental Numerial 2 F/A [MPa] Fig. 18. Comparison of the experimentally measured and numerially predited (eight parallels) fore displaement urves for nothed tensile tests.

16 C. Dørum et al. / Engineering Frature Mehanis 76 (29) Fig. 19. Distribution of frature parameters W from simulations of tensile tests that gave (a) lowest and (b) highest predition of elongation at frature. a 3 Experimental Numerial b 3 Experimental Numerial F/A [MPa] 2 1 F/A [MPa] Fig. 2. Predited fore displaement urves from simulations with harateristi element length equal to (a).4 mm and (b).2 mm. imum value W max of the frature parameter. The distribution of W values from the numerial simulations of uniaxial tensile tests that predited the lowest and the highest value of elongation at frature is illustrated in Fig. 19a and b, respetively. To examine the mesh sensitivity of the proposed approah to modelling of frature in high-pressure die-astings, the nothed tensile tests were simulated numerially with two refined meshes in addition to the simulations presented in the previous setion. The nothed speimens were modelled with harateristi element size equal to.4 mm and.2 mm. For omparison, the nothed uniaxial tensile test was modelled with harateristi element size equal to.8 mm. Thus, three different meshes were used in the mesh sensitivity study. By omparing the predited fore displaement behaviour from these simulations, illustrated in Fig. 2, with the preditions shown in Fig. 18 (element size equal to.8 mm) it is seen that the predited response (and the satter) is very similar for all of the hosen mesh densities. Based on this rather limited mesh sensitivity study, it appears that the proposed stohasti frature modelling approah is rather mesh size insensitive. 7. Conluding remarks The quasi-stati behaviour of high-pressure die-ast magnesium alloy AM6 has been studied. Experiments were performed with speimens of different geometry and size of the gauge region. The experimental data were used to develop a validated probabilisti methodology for finite element modelling of thin-walled die-astings subjeted to quasi-stati loading. The main results are summarised in the following: 1. The dutility of the speimens ut from the astings depends on the position in the asting. There are also signifiant variations in dutility when omparing the measured harateristis of speimens ut from different astings ast under equal asting onditions. Thus, as a result of unstable flow of the liquid magnesium in the mould avity, the mehanial properties of the asting are onsidered to be of stohasti nature.

17 2248 C. Dørum et al. / Engineering Frature Mehanis 76 (29) An elasti plasti model inluding a high-exponent, isotropi yield riterion, the assoiated flow law and isotropi hardening, was used to model the ast material, and was shown to provide good preditions of the fore displaement behaviour for the investigated material tests. 3. The frature parameter for the Cokroft Latham riterion was identified independently from different material tests. It was found that the identified values varied signifiantly. The frature parameters determined from shear tests and nothed tensile tests were signifiantly higher than those identified from uniaxial and plane-strain tension tests. The dutility inreased with dereasing volume of the gauge region of the test speimen. This indiates that size effets are signifiant in alibration of frature models for die-astings. 4. By ombining the Cokroft Latham frature riterion and the Weibull statistial distribution funtion, the frature parameter was defined as a stohasti Weibull distributed parameter. Repeated numerial simulations of the material tests were arried out, giving preditions very similar to the experimental behaviour. 5. Simulations of nothed tensile tests with FE-models of different mesh densities predited very similar fore displaement behaviour. Thus, the proposed stohasti frature modelling approah seems to be relatively mesh size independent. Aurate numerial predition of the mehanial apaity (espeially in terms of dutility) of astings requires that the inhomogeneous distribution of defets is inluded. To provide more details about the frature mehanisms, possible shear band formation ould be investigated by optial mirographs of the investigated materials. A oupling of die-asting proess simulations and the urrent approah should be investigated to establish a deterministi-stohasti approah that an model both the variations in dutility depending on the material s position in the asting as well the stohasti aspets. Aknowledgements The authors would like to aknowledge the support through the EU projet NADIA, and Hydro Aluminium and the Researh Counil of Norway for their support through SIMLab, the Centre for Researh-based Innovation. Referenes [1] Okewitz A, Shendera C, Sun D-Z. Deformation and frature behaviour of magnesium strutural omponents. In: Kainer KU, editor, Magnesium alloys and their appliations; September 2. p [2] Gurson AL. Continuum theory of dutile rupture by void nuleation and growth: part I yield riteria and flow rules for porous dutile media. J Engng Mater Tehnol 1977;99:2 15. [3] Chen X, Wu S-R, Wagner DA, Hu W. Study of die ast magnesium omponents for rash safety. 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