Predicting the influence of pore characteristics on ductility of thin-walled high pressure die casting magnesium

Materials Science & Engineering A 572 (2013) 45–55

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Predicting the influence of pore characteristics on ductility of thin-walled high pressure die casting magnesium X. Sun n, K.S. Choi, D.S. Li Pacific Northwest National Laboratory, P.O. Box 999, Richland, WA 99352, USA

a r t i c l e i n f o

abstract

Article history: Received 9 July 2012 Received in revised form 8 February 2013 Accepted 13 February 2013 Available online 21 February 2013

In this paper, a two-dimensional microstructure-based finite element modeling method is adopted to investigate the effects of porosity in thin-walled high pressure die casting Mg materials on their ductility. For this purpose, the cross-sections of AM50 and AM60 casting samples are first examined using optical microscope to obtain the overall information on the pore characteristics. The experimentally quantified pore characteristics are then used to generate a series of synthetic microstructures with different pore sizes, pore volume fractions and pore size distributions. Pores are explicitly represented in the synthetic microstructures and meshed out for the subsequent finite element analysis. In the finite element analysis, an intrinsic critical strain value is used for the Mg matrix material, beyond which work-hardening is no longer permissible. With no artificial failure criterion prescribed, ductility levels are predicted for the various microstructures in the form of strain localization. Mesh size effect study is also conducted, from which a mesh size dependent critical strain curve is determined. A concept of scalability of pore size effects is then presented and examined with the use of the mesh size dependent critical strain curve. The results in this study show that, for the regions with lower pore size and lower volume fraction, the ductility generally decreases as the pore size and pore volume fraction increase whereas, for the regions with larger pore size and larger pore volume fraction, other factors such as the mean distance between the pores begin to have some substantial influence on the ductility. The results also indicate that the pore size effects may be scalable for the models with goodrepresentative pore shape and distribution with the use of the mesh size dependent critical strain curve. & 2013 Elsevier B.V. All rights reserved.

Keywords: Mg castings Ductility Microstructure Pore size Pore volume fraction Finite element analysis (FEA)

1. Introduction Mg castings have found increasing applications in lightweight vehicles because magnesium and its alloys have high specific strength and are the lightest metallic structure materials. High pressure die casting (HPDC) is generally characterized by rapid die filling, cooling, and metal solidification with considerable microstructural variability associated with the specific process parameters. Various alloy compositions have also been explored to enhance the properties of Mg casting while overcoming its weaknesses. Naming of Mg alloys generally follows the scheme of two letters indicating the primary alloying elements and two numbers indicating the approximate percentage by weight of the elements. The designations AM50 and AM60 indicate that aluminum and manganese are present in the highest concentrations. The average Al content is 5% and 6% by weight, respectively, and less than 1% Mn. AZ91 indicates 9% Al and 1% Zinc. The AM and AZ

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series are common for automotive applications [1], with the AZ series referred in strength-sensitive applications. The characteristic microstructure of HPDC AM series alloys consist of a-Mg dendrite cells and a divorced eutectic (a-Mg and b-Mg17Al12) in the interdendritic regions [2,3]. Intermetallic Mn-rich particles are also observed. Some of the a-Mg dendrite cells found in the center of a casting are prematurely solidified in the shot sleeve and carried into the die. This results in a bimodal distribution of dendrite cells sizes [3]. The microstructure closer to the die walls consists of a very fine dendrite cell size, approximately 5–7 mm, with little to no porosity observed. This ‘‘skin’’ region results from higher cooling rates and exhibits higher strength and ductility compared to the center [4]. The dendrite cells in the center are 10s of mm in diameter depending on processing parameters with externally solidified cells being the largest. As such, the skin region and thin casting have been shown to be better in improving the strength and ductility compared with the center and thick sections [5]. A third region has been observed in some castings. A porosity segregation zone (PSZ) is sometimes observed between the skin and center regions consisting of one or more bands of high porosity that follow the

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contours of the die [2,3,6]. The PSZ have a negative effect on the strength and ductility of the casting and are thought to be linked to processing parameters. Dahle et al. [7] have developed a model to predict when and where the PSZ will form with different casting parameters. Despite of lightweight and high specific strength, the limited ductility for Mg castings and its highly variable nature present a critical hurdle to their more widespread usage in industry. It is well established that the various microstructural features described above (i.e., grain size, porosity, brittle eutectic phase and precipitates, etc.) can significantly influence the ductility of Mg alloys. Unfortunately, these features vary spatially throughout a casting and from specific alloy to alloys. Moreover, the specific casting process used and the processing parameters can also influence the final microstructure, hence its ductility. Generally speaking, the factors limiting the ductility of Mg castings can be categorized into two types: intrinsic and extrinsic. Intrinsic factors include features intrinsic to the specific Mg alloy such as the phase composition, grain size, morphology, volume fraction and mechanical properties of the a-Mg matrix and the eutectic b phase. Extrinsic factors come from the external processes applied to the alloy such as casting and heat treatment processes, and they include: porosity, segregation, incomplete fill, hot tear and cold shut, etc. The specific alloy design and casting process parameters determine both factors which in turn influence the ductility of the final cast. Previous studies have demonstrated that extrinsic defects (i.e., micro porosity) are the dominant ductility limiting factor for locations with porosity volume fraction exceeding a critical value [2,3]. Only when the porosity volume fraction is less than the critical value, fracture brought by the intrinsic factors, such as heterogeneous grain size and the brittleness of the b eutectic phase, is dominant. Of course, as the HPDC process is continuously being improved and process-induced defects are being minimized/eliminated, the intrinsic microstructure features may play a more important role on ductility. There have been multiple efforts reported in the open literature in correlating the bulk porosity level to ductility and fracture for Mg castings. For example, Weiler et al. [8] utilized a critical local strain model [9–11] with the areal fraction of porosity to model plastic instability resulting from internal necking due to the reduced cross section area. The model approximates the effects of multiple micropores on the fracture surface as a single void located at the center of the gauge length with equivalent areal fraction f, and predicted the relationship between the fracture strain and the areal fraction of porosity with reasonably good accuracy. Lee et al. [12,13] experimentally examined the ductility variability for HPDC AM50 at different test temperatures and found that fracture path preferentially goes through the regions of clusters of pores. They also concluded that the ductility of HPDC AM50 alloy can be increased by decreasing the regions of localized clustered pores and large gas pores in the microstructure, which may not necessarily require decreasing the global average volume fraction of the pores in the specimen. Hence, ductility of HPDC depends on both pore volume fraction and largest pore size. Realizing that the homogenization techniques ignore the non-uniform, non-random spatial arrangements of microstructure features at different length scales, Gokhale and Yang [14] presented a finite element-based modeling methodology to model the multi-length scale deformation process of Al–Si–Mg cast alloy by incorporating the various microstructural features through image processing techniques. Even though localized yielding and plastic localization have been predicted at the micro-pore level, the overall ductility of the A356 casting was not investigated from the multi-length scale simulations. The main purpose of this study is to investigate the influence of the different characteristics of extrinsic defects on the ductility of HDPC Mg by considering different pore sizes, pore volume

Fig. 1. Typical stress–strain curves for AM60 in one sample population.

fractions and pore size distributions. For this purpose, information on the pore characteristics was first obtained based on the optical microscopy of the cross-sections of AM50 and AM60 casting samples. Because of the statistical nature of the Mg casting microstructure, synthetic microstructures with the same statistical descriptors (i.e., pore size, pore volume fraction and pore size distribution) as those of the actual microstructures were constructed as the representative volume elements (RVEs) to represent the microstructures of the AM50 and AM60 casting samples. Two-dimensional (2D) plane stress finite element analyses are then performed with these synthetic RVEs to predict the ductility as well as overall stress–strain behaviors. As the Mg casting materials generally have two or three different regions (i.e., skin, center, PSZ) through the radial or thickness direction, threedimensional analysis may be required in order to accurately capture the deformation interactions between different regions for thicker castings. However, simple 2D analysis is adopted in this study as it is expected to be sufficient for the purpose of examining the influence of different pore characteristics on ductility. In the analysis, an intrinsic critical strain value is adopted for the Mg matrix material, beyond which workhardening is no longer permissible, and the ultimate ductility of the RVE is predicted as the natural outcome of the plastic localization caused by the microstructure level pore-induced inhomogeneity without an artificially imposed failure criterion. A mesh size dependent critical strain curve is also determined from a mesh size effect study, and the scalability of the results is then presented and examined with the use of the determined mesh size dependent critical strain curve. Finally, some conclusions will be made based on the obtained results and observations.

2. Experiments AM50 (Mg–4.9 wt%Al–0.39 wt%Mn–0.2 wt%Zn) and AM60 (Mg–5.7 wt%Al–0.37 wt%Mn–0.057 wt%Zn) casting samples were selected to quantify the pore characteristics (i.e., pore size, pore volume fraction, pore size distribution) of Mg casting materials. These samples are dog-bone shaped for the tensile test. The AM50 samples have cylindrical cross-section whereas the AM60 samples have rectangular cross-section. Quasi-static tensile tests (e_ ¼ 103 =s) were conducted with these tensile samples. Fig. 1

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shows some typical stress–strain curves of the tested AM60 samples within one sample population. A large variation in the ductility of Mg casting materials can be observed, which is due to the difference in pore characteristics of each sample. The current study focuses on examining the influence of various pore characteristics on the large ductility variation, rather than the accurate prediction of stress versus strain curves for the Mg casting materials. Hence, only microstructural observations of the AM50 and AM60 casting materials are discussed and illustrated in this section, as the information on pore characteristics will be used in the subsequent microstructure-based finite element analysis. For the AM50, a small portion was cut from the griping ends of each sample for microstructure analysis. Optical microscopic pictures were taken from these cut samples. Fig. 2 shows some optical microscopic pictures of the cross-sections of the AM50 casting samples. The eutectic b-phase in a-Mg matrix can be seen, though vaguely, in Fig. 2(a). As for the porosity features, it is observed that the center regions of samples have different microstructures (i.e., pore size, volume fraction, distribution, shape) for different samples as shown in Fig. 2(a), (b) and (c), illustrating the sample to sample variations of the microstructures. Due to the different solidification rates between the skin and center regions, all samples exhibit different microstructures between these two regions as shown in Fig. 2(c) and (d). Little or no porosity is typically observed in the near-skin region as compared to the center region. Porosity analysis has been performed on the cross-section micrographs of the AM50 samples to quantify the areal porosity, porosity distribution and largest pore size of each sample. It is found that, depending on location, the pore volume fraction at the sample skin varies from 0.5% to 1.5%, and the pore volume fraction at the center region of the sample varies from 2 to 6%. Overall, pores size varies from 4 mm to 100 mm depending on location, except for some very large, crescent shaped pores near the center of the samples. For the AM60 casting samples, high resolution X-ray tomography technique was adopted to generate three-dimensional

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images of the tensile samples with the actual porosities mapped on them. Fig. 3 shows some optical microscopic pictures and X-ray tomography of the cross-sections of gauge area of AM60 casting samples. Note here that the AM60 samples are machined from various locations of a series of complex shape thin-walled casting structures. Similar to the cylindrical AM50 samples, the AM60 samples also exhibit different microstructures between center and skin regions, which may be more clearly observed with X-ray tomography shown in Fig. 3(b). Pore volume fractions in the two cross-section micrographs shown in Fig. 3(a) were estimated to be about 1.6% and 3.0%, respectively. The pore size appears to reach up to 100 mm. The quantified pore size and volume fraction values for the AM50 and AM60 samples are used as the characteristic input for the subsequent finite element-based modeling method through a series of synthetic microstructures generated with statistical microstructure reconstruction technique. Following that, twodimensional plane stress models are employed to investigate the influence of pore size, volume fraction and size distributions on the predicted ductility of the Mg castings.

3. Microstructure-based finite element method Continuum-damage mechanics (CDM) models, homogenization method and other statistical mechanics methods have been used quite successfully in predicting the elastic properties and yield strength of heterogeneous materials in the open literature. For example, Gurson model [15], a classical CDM-based plasticity model for ductile fracture, is typically used to predict void nucleation and growth in metallic materials containing microvoids, and ductile failure for a material point is determined by comparing the pore volume fraction with its critical value [16]. However, ductility prediction based on a smeared, homogenized microstructure can only provide first order approximation resulted from the overall pore volume fraction, and cannot take

Fig. 2. Cross-section micrographs of AM50 casting samples: (a) center region of sample #1, (b) center region of sample #2, (c) center region of sample #3 and (d) skin region of sample #3.

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Fig. 3. (a) Cross-section micrographs of an AM60 casting sample and (b) reconstructed computerized tomography showing different cross-sections of an AM60 sample and (c) representation of W–T–L coordinates. The cross-section images in (a) and (b) have the dimensions of width  6 mm; thickness  2 mm.

into consideration of pore size, pore morphology, and pore spatial distribution in a microstructure. Yet multiple previous studies have concluded that these are all important factors controlling the overall ductility of Mg castings. In this section, we present a microstructure-based finite element modeling method to predict ductility of Mg castings that takes explicit considerations of all the microstructure level features mentioned above. The methodology is similar to the plastic strain localization theory used to predict the ductility of multi-phase advanced high strength steels [17–19], and in this case, the micro-pores serve as the sources of the microstructural inhomogeneity triggering the strain localization, leading to final failure of RVE.

3.1. Synthetic microstructures Based on the pore characteristics quantified in the previous section, synthetic microstructures of the Mg castings with statistically the same characteristics were generated. These synthetic microstructures are used as the base RVEs for the subsequent two-dimensional finite element analysis in investigating the effects of pore characteristics on ductility. Note that the generated synthetic microstructures are pixel-based image, and each pixel can be generally considered to be one element in the finite element model. Round pores are randomly distributed inside of the microstructure to make up different overall pore volume fractions from 1% to 4%. Different pore sizes are considered, ranging from 4 mm to 40 mm in diameter. Uniform or variable pore size distributions are also considered in this study. The models with variable pore size distribution were generated such that they have an average pore size with 750% variation in pore size based on a uniform density function. For example, for the case with average pore size of 40 mm, the largest and smallest pore sizes in the model are 60 mm and 20 mm, respectively, with a uniform distribution in between. Very large-size pores (i.e.,

4100 mm) are not considered in the current study as they occur much less frequently compared to smaller pores. For each set of pore characteristics (i.e., pore size, pore volume fraction, pore size distribution), three different synthetic microstructures are generated and then subjected to tensile loading along both horizontal and vertical directions, predicting six different stress–strain curves for each case of interest. Note here we consider loading horizontally or vertically on the same synthetic microstructure as two models with different random pore distributions. Fig. 4 shows some examples of the synthetic microstructures which are the images with 500  500 pixels. For example, Fig. 4(a) and (b) shows the synthetic microstructures with the same pore size of 20 mm, but with different pore volume distributions, and Fig. 4(b) and (d) shows the synthetic microstructures with the same pore volume fraction of 2% and pore size of 20 mm, but with different pore size distributions. Note that the microstructure in Fig. 4(b) has the uniform pore size of 20 mm whereas the one in Fig. 4(d) has the average pore size of 20 mm with the pore sizes uniformly distributed between 10 mm and 30 mm. 3.2. Finite element models The 1 mm  1 mm finite element models were then generated based on the synthetic microstructures. Three different pore volume fractions (i.e., 1, 2, 4%) with five different pore sizes (i.e., 4, 12, 20, 28, 40 mm in diameter) are considered with the two different pore size distributions (i.e., uniform and variable pore size distribution) as discussed above [20–23]. Note that, for each case, six different simulations are conducted by applying tensile loading in the horizontal and vertical directions onto the three different synthetic microstructures with the same pore characteristics. Fig. 5 shows an example of the finite element models which is developed based on the synthetic microstructure in Fig. 4(a).

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The pores (i.e., 20 mm size) are explicitly meshed out in the synthetic microstructure as shown in the figure. Uniform size/ shape meshes of 2 mm are used in generating the finite element models with an intention to examine only the effects of different pore sizes/volume fractions/distributions by minimizing any possible effects from the usage of different size/shape meshes. 2D plane stress elements are adopted to predict the stress–strain behaviors considering the thin-walled nature of the casting. Since the models are generated based on synthetic microstructures, multi-point constraints are applied to the edges such that all the edges of the model remain straight during the tensile loading process [17]. The overall engineering stress and strain values are determined by dividing the reaction force of the model in the loading direction with the initial area (i.e., the initial height or width for the plane-stress model) and dividing the displacement of the loaded edge with the initial length of the model, respectively.

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In the finite element analysis, the matrix material outside the pores is assumed to be uniform with elastic modulus of 45 GPa and initial yield strength of 140 MPa, and a constant plastic hardening rate of 1065 MPa is assumed beyond yielding [8]. An intrinsic critical strain value of 14% (in terms of the equivalent plastic strain) is used for the Mg matrix [9] of 2 mm size mesh, beyond which no work hardening is present. Mesh size dependence of the critical strain will be discussed in the discussion section. With this modeling method, no artificial failure criterion is prescribed, and the failure/ductility of the model (i.e., representative volume element) is predicted in the form of strain localization induced by the pores. During the deformation process, strain localizations occur within the critical regions (i.e., regions with clustered pores and large-size pores) in the model and coalesce, leading to a final failure band. For example, Fig. 6 shows a failure mode in the form of plastic strain localization predicted under the tensile loading in the horizontal direction. Note that porosity volumetric evolution due to loadinginduced pore nucleation and growth is not considered in the model. This is because brittle fracture of eutectic b-phase in a-Mg matrix within the critical regions is reported as the primary crack propagation mechanism for Mg casting materials [3]. Typical stress–strain curves (i.e., Fig. 1) indicate that Mg castings generally experience sudden and brittle fracture without noticeable necking, and most post-mortem fracture surface analyses show no evidence of pore elongation or aspect ratio change, which is a pre-requisite for loading-induced pore growth (volumetric evolution) and coalescence.

4. Influence of pore characteristics on predicted ductility

Fig. 4. Examples of synthetic microstructure (size of 1 mm  1 mm ) of Mg castings: (a) pore size 20 mm and volume fraction 1%, (b) pore size 20 mm and volume fraction 2%, (c) average pore size 20 mm and volume fraction 1% and (d) average pore size 20 mm and volume fraction 2%.

In this section, the effects of various pore characteristics are examined based on the stress–strain curves predicted for the cases of interest. As an example, Fig. 7 shows the predicted stress–strain curves for the models with 1% pore volume fraction and five different pore sizes with uniform pore size distribution. Note that six different simulations were conducted for each pore size case in order to obtain statistically meaningful average and standard deviation of the strength and ductility for each case. As shown in Fig. 7, six different simulations for the same pore size result in different stress–strain curves (of the same color) due to the differences in their detailed distributions with respect to the tensile loading direction. The strain value at the ultimate tensile strength (UTS) is determined from the stress–strain curve to represent the ductility of each case as the necking behavior after

Fig. 5. A finite element model developed based on a synthetic microstructure.

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Fig. 6. Typical predicted failure mode in the form of plastic strain localization.

Fig. 7. Stress–strain curves for the models with 1% pore volume fraction and uniform pore size distribution.

the UTS is not observed from the tensile tests of these Mg casting materials. Note here that the 2D plane stress model can well represent the deformation of a thin sheet or a thin film, which fails through in-plane strain localization with minimum thickness effects, but will under-predict the ultimate ductility of thicker materials due to the lack of consideration of the throughthickness links resisting overall sample failure through strain localization. 4.1. Effects of pore size Fig. 8(a) summarizes the predicted effects of pore size on ductility for three different pore volume fractions. Note that both averages and variations are presented in this chart for each case. For volume fraction of 1%, the predicted ductility (and UTS as well) monotonously decreases as the pore size increases as shown in the figure. It is also interesting to observe that, for this low pore volume fraction, the predicted ductility variations for the cases

Fig. 8. (a) Effects of pore size and (b) effects of pore volume fraction on the predicted strain at UTS.

with larger pore sizes are larger as compared to the cases with smaller pore sizes. Note that, with this low pore volume fraction, the model can have only a few large pores in it (for example, 40 mm pore case), hence the ductility may be governed by the relative locations of these large pores, leading to the larger variation in the predicted ductility for the cases of larger pore

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sizes. For example, if two or three out of a few pores happen to locate very close to one another in the model, the stress fields around these pores may interact and induce strain localization very early in the deformation process and become the major driver for the ultimate failure of the model. For the pore volume fractions of 2% and 4%, the change in ductility as a function of pore size is different from that of the pore volume fraction of 1%: average ductility initially decreases as the pore size increases, but above the pore size of 20 mm it increases slightly. At about 40 mm, the average ductility for the three pore volume fractions are almost the same, indicating the diminishing influence of pore volume fraction for large pores. The authors attribute this different behavior to the fact that, as the pore volume fraction for larger pores (i.e., 420 mm) increases above 2%, the mean distance between the pores in the models begins to influence the strain distributions of the model, hence the predicted material ductility in this region. More specifically, if the model has only 1% pore volume fraction with lager size pores, it is very possible that the model has only a few pores in it and the ductility can be governed by the relative locations of the pores in the model as well as pore size as mentioned above. However, if the model has relatively higher pore volume fractions (for example, 42%) even with larger size pores, the model can have those pores more evenly distributed in the model than those with lower pore volume fractions, which leads to lower mean distance between the pores in the model. These evenly distributed pores may help the RVE to have relatively uniform strain distribution during the loading process and to carry the load evenly for a relatively longer period. The effects of mean distance between the pores are well observed from the cases with the pore size of 40 mm in Fig. 8(a), where the cases with the three different pore volume fractions show a similar level of ductility.

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pore size distribution, the synthetic microstructures are generated to have the pore sizes uniformly distributed between  50% and þ50% of the specified average pore size as mentioned before. As examples, the microstructures with variable pore size distribution are illustrated in Fig. 4(c) and (d). Fig. 9 compares the predicted results for the variable pore size distribution cases with those for the uniform pore size distribution cases. As shown in the figure, in general, the predicted ductility for the variable pore size distribution cases are almost identical with that for the uniform pore size distribution cases except for two cases. In Fig. 9(a), the cases of pore size 4 mm and 40 mm with pore volume fraction 4% show some noticeable difference in ductility between the uniform and variable pore size distribution cases. Results in Fig. 9(b) further exemplify these exceptions, particularly for the case with pore size of 4 mm. Besides these two exceptions, the results in Fig. 9 generally indicate that the average pore size can be the controlling factor of ductility and the pore size distributions examined in this study do not have overwhelming influence on ductility. In this study, only the uniform density function is adopted for the variable pore size distribution. However, it is experimentally observed that one or two very large pores may exist in the material and these very large pores may act as the primary failure source and result in a low ductility. In order to investigate the effects of these extremely large size pores on ductility, the density function with other types of distributions are needed in generating the synthetic microstructures. Characterization of general pore distribution in a casting material is beyond the scope of the current work.

4.2. Effects of pore volume fraction Fig. 8(b) shows the predicted effects of pore volume fraction on ductility for the pore sizes of 4, 20 and 40 mm. For pore sizes of 4 and 20 mm, the predicted ductility decreases monotonically with increasing pore volume fraction. However, for pore size of 40 mm, the predicted average ductility stays almost flat: increases slightly for the pore volume fraction from 1% to 2%, then decreases slightly from 2% to 4%. This indicates again that, for this large pore case, the pore volume fraction does not have significant influence on the predicted ductility. As explained in the previous section, possible reasons for this behavior are related to the effects of the mean distance between the pores in the model. The results in this figure suggest that, for larger size pores (i.e., 440 mm), further reducing the pore volume fraction, for example, below 2% does not help improving the ductility since the ductility is predominantly governed by the pore size with their relative locations in the model. Based on the observation on the results in Fig. 8(a) and (b), it appears that, for the regions with lower pore volume fractions and lower pore size, the ductility is governed by the pore size (with their relative locations) and pore volume fraction whereas, for the regions with larger pore volume fraction (i.e., 2%) and larger pore size (i.e., 420 mm), the mean distance between the pores (i.e., the distribution features of the pores) begins to have significant influence on the ductility. 4.3. Effects of statistical distribution of pore size In order to examine the effects of statistical pore size distribution, simulations with variable pore size distribution are also performed with the average pore size of 4, 12, 20, 28, 40 mm, and different pore volume fraction ranging from 1% to 4%. For this

Fig. 9. Effects of variable pore size distribution: (a) cases for different pore sizes with pore volume fraction of 1% and 4% and (b) cases for different pore volume fraction with pore size of 4 mm and 40 mm.

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5. Discussions on mesh size dependence of critical strain In this section, the well-known issue of mesh size dependency in finite element-based localization analysis is discussed. A concept on introducing the intrinsic length scale of the model

Fig. 10. Mesh size dependent critical strain curve with a concept on the scalability of pore size effect.

system will then be presented, and the scalability of pore size effect will also be discussed. 5.1. Mesh size effect One of the critical issues in using conventional continuum plasticity theory (for the AM50 matrix here) at the micro scale to predict ductility is that neither the theory itself, nor its finite element implementation has any intrinsic length scale in the formulations [24]. In other words, in addition to its intended representation of a 1 mm  1 mm microstructure with a pore size of 20 mm, Fig. 5 can also represent a 0.1 mm  0.1 mm microstructure with a pore size of 2 mm. Therefore, the predicted stress–strain curves and ductility levels with the same set of material property input would yield the same results for the two finite element models, which contradicts the experimental observations that larger pores lead to lower ductility. Given this lack of intrinsic length scale, Hutchinson [24] cautioned the application of conventional plasticity to submicron sized void growth predictions, and advocated the application of strain gradient plasticity theory [25] to account for the gradient effects that emerge in deformation phenomena at the micron scale. In the finite element based ductility predictions in this study, since each pore is explicitly resolved, the strain gradient is naturally controlled by the microstructure level inhomogeneity in the scale of pore size and pore spacing. In order to incorporate an intrinsic length scale to the finite element based ductility model, we introduced mesh size dependent critical strain values for different mesh sizes, beyond which no work hardening will be permissible. Fig. 10 shows an example of mesh size dependent critical strain curve determined and used in this study.

Fig. 11. Models (size of 1 mm  1 mm) used to examine the mesh size effects: (a) mesh size 34.4 mm, (b) mesh size 10.1 mm and (c) mesh size 2.5 mm.

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For example, for the models with the mesh size of 2 mm, a critical strain level of 14% will be used as the material input for the Mg matrix, and for the models with the mesh size of 35 mm, a critical strain level of 4.3% needs to be used for the Mg matrix. The development process for this mesh size dependent critical strain curve will be illustrated in the next section. It is also noted that a concept on the scalability of pore size effects is illustrated in Fig. 10, which will be introduced later. The concepts of using strain gradient plasticity theories or mesh-size-dependent failure strain for reducing the mesh size effects are well demonstrated in Li and Karr [26] and Li and Wierzbicki [27] through theoretical derivations, simulations and experiments. 5.2. Development of mesh size dependent critical strain master curve In this study, the simulations were started with the 1 mm  1 mm microstructures with mesh size of 2 mm. An intrinsic critical strain of 14% is used for the Mg matrix in the models [9,28]. Mesh size of 2 mm with the critical strain of 14% is therefore used as the baseline for the development of the mesh size dependent critical strain master curve, as will be discussed below. A simple model with a large central hole, rather than a synthetic microstructure model, is used for the development of a critical strain master curve with the intention to exclude any possible effects from the pore distribution on ductility. Fig. 11 shows some example models with a large hole in it. As shown in the figure, they have the same diameter hole in the same size model, but with different mesh sizes. The same critical strain of 14% is first adopted in these hole-models with different mesh sizes to examine the mesh size effect. Fig. 12(a) shows an example of the mesh size effect based on the same critical strain of 14%, where the stress–strain curves predicted by different mesh size models are plotted. As expected, even with the same critical strain values, models with coarser mesh predict later onset of plastic strain localizations, resulting in higher ductility as compared to the models with finer mesh. Recently, Sun et al. [29] also discussed the rationale from the intrinsic localized deformation’s perspective on the implementation of mesh size dependent ductility in engineering simulations. With an intention to eliminate the effects from different mesh size and obtain the same ductility from different mesh size models, different critical strain values are determined for different mesh sizes. For this purpose, the critical strain value input in each model is adjusted such that the predicted ductility (i.e., strain at UTS) based on the adjusted critical strain value will be the same as that of the baseline case. The mesh size dependent critical strain master curve obtained from this method is plotted in Fig. 10. In principle, experimental results should be used as a baseline case in obtaining the mesh dependent critical strain values. In this study, however, the mesh size of 2 mm with the critical strain of 14% is used as the baseline case, since the goal is to understand the trends of mesh size dependent critical strains without tensile experimental results for samples with controlled micron-size central holes. Fig. 12(b) shows some examples of the predicted stress–strain curves after the calibration of critical strain values for each model, where the three predicted curves have the same strains at UTS. Note that, in addition to the calibrated nature of the mesh dependent critical strain curve presented in Fig. 10, the mesh size dependent critical strength can also be qualitatively explained with the deformation gradient theory. Since the intrinsic length scale l is a constant for a specific material, smaller mesh (comparable with l) will be able to capture most of the strengthening induced by geometrically necessary dislocations (GND) generated by the non-uniform straining. On the other hand, when

Fig. 12. (a) Effects of mesh size based on the same critical strain and (b) stress– strain curves based on the adjusted critical strains.

the mesh size is much larger than l, localized strengthening from GND will be averaged over a much larger volume for the material integration point for the element, resulting in a lower ultimate strength level. The critical strain curve for different mesh sizes, described by the master curve in Fig. 10, also bears striking similarities with the indentation size effect (ISE) reported by various researchers in the last two decades where measured indentation hardness (i.e., strength) for a given material is higher for smaller indenter tip radius or indentation depth [30,31]. To this end, the authors are currently exploring possible methods in quantifying the mesh size dependent master curve by correlating indentation pop-in load with various indenter radii. With this master curve concept, it will also be possible to computationally link with the other sizedependent phenomena such as Hall–Petch effect and the particle size effects on strengthening in metals by a given volume fraction of hard particles. 5.3. Scalability of the ductility prediction with the critical strain master curve In previous sections, six different simulations were conducted to obtain a statistically meaningful average values for each case of interest, which turned out to be quite time-consuming. If the effect of pore size is truly scalable, one can use one set of microstructure models with randomly distributed pores to investigate the effect of pore size on ductility with the help of the critical strain master curve, as discussed below. To test the effectiveness of the mesh size dependent critical strain master curve derived earlier, we examine the scalability of

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Fig. 13. Scalability of pore size effect: scaled by a model with original pore size of (a) 4 mm, (b) 12 mm, (c) 20 mm, (d) 28 mm and (e) 40 mm.

the pore size effect with the use of the master curve. Fig. 10 also illustrates this concept on the scalability. Here, VF, MS and PS represent the pore volume fraction, mesh size and pore size, respectively. Models #2 and #3 have the same microstructure as Model #1 since they are the scaled-up and -down versions of Model #1. In these three models, Model #2 is half the size of Model #1, while Model #3 is twice the size of Model #1. Due to the change of model size, the mesh size and pore size in the model are also proportionally scaled up or down while keeping the pore volume fraction and morphology the same. It can be then considered that Models #1, #2 and #3 have the same pore volume fraction but with different pore sizes. In order to consider the

effects of different mesh sizes in these three models, different critical strain values corresponding to the different mesh sizes need to be obtained from the mesh size dependent critical strain master curve as illustrated in Fig. 10. The scaled-up or -down models in this way are expected to predict the ductility close to the average values obtained from each set of six simulations with the corresponding pore size. Here, the scalability of the pore size effect is examined based on the models with 1% pore volume fraction and uniform pore size distribution. For this purpose, one model is selected from each set of three models (i.e., 1 mm  1 mm size and 2 mm mesh size model) having the same pore size. Then, the selected models

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are appropriately scaled up or down to represent the models with the different pore sizes. Different critical strains are accordingly adopted in these scaled models. Fig. 13 compares the ductility predicted by the scaled models with the average ductility predicted by the six simulations presented before. Here, the normalized ductility is compared in order to exclude the effects of pore distributions in the selected models for scaling. As shown in Fig. 13(b), (c) and (d), the predictions based on the scaled models from the ones with the original pore sizes of 12, 20 and 28 mm compare well with the averaged ductility from the six simulations, except for the case with the smallest pore size 4 mm. In Fig. 13(a) and (e), however, the predictions based on the scaled models from the ones with the original pore sizes of 4 and 40 mm do not compare well with the averaged ductility. Note here that the models with the original pore sizes of 4 and 40 mm have mostly square or rectangular shaped pores due to the 2 mm mesh size and relatively uneven pore distribution due to the small number of pores in the model, respectively, as compared to the relatively circular shaped pores and even pore distribution in other models with the original pore sizes of 12, 20 and 28 mm. Therefore, the different pore shapes and distributions in the models with the original pore sizes of 4 and 40 mm are the possible sources for the discrepancy shown in Fig. 13(a) and (e). Based on the observations in this section, it appears that, within the ranges of pore size and volume fractions considered in this study, the pore size effects are scalable for the models with good representative pore shapes and distributions.

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of relatively thin materials with the pores evenly distributed throughout the whole thickness. However, in reality, due to the existence of different regions (i.e., skin, center) with different pore characteristics in Mg casting materials, three-dimensional modeling work may be needed to predict the accurate correlation between the microstructure/defect features and the fracture behaviors of Mg castings. To this end, three-dimensional microstructures based on X-ray tomography measurements are considered in the authors’ future work. The influence of extreme distributions of pore size and sample size effects should also be considered in future studies.

Acknowledgments Pacific Northwest National Laboratory is operated by Battelle Memorial Institute for the US Department of Energy under Contract no. DE-AC05-76RL01830. This work was funded by the Department of Energy Office of FreedomCar and Vehicle Technologies under the Automotive Lightweighting Materials Program managed by Mr. William Joost. The authors would also like to acknowledge the help of Dr. Tamas Varga and Alexa Chua in conducting the X-ray tomography and microscopy. X-ray tomography was performed at Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the Department of Energy’s Office of Biological and Environmental Research.

6. Conclusions References In this paper, a two-dimensional microstructure-based finite element modeling method is used to investigate the influence of different pore characteristics in Mg casting materials on their ductility. For this purpose, microstructure analysis using optical microscopy has been first performed with AM50 and AM60 casting samples to obtain the general information on the pore characteristics (i.e., pores size, pore volume fraction and pore size distribution) of the Mg casting materials. The resulted pore characteristics are then used to generate a series of synthetic microstructures with different pore sizes, volume fractions and size distribution features. Pores are explicitly represented in these synthetic microstructures and meshed out for the subsequent finite element analysis. In the finite element analysis, an intrinsic critical strain value of 14% is used for the Mg matrix material, beyond which there is no work hardening, and no artificial failure criterion is prescribed in the model. The ductility levels are predicted in the form of strain localization in the model for various microstructures. The results show that, for the regions with lower pore size and lower volume fraction, the ductility generally decreases as the pore size and pore volume fraction increase whereas, for the regions with larger pore size and larger pore volume fraction, other factors such as the mean distance between the pores appear to have some substantial influence on the ductility. Mesh size effect study has been also performed, from which a mesh size dependent critical strain curves could be determined. A concept of scalability of pore size effects is then presented and examined with the use of the determined mesh size dependent critical strain curve. The results on this scalability study show that the pore size effects may be scalable for the models with goodrepresentative pore shape (i.e., close to circle shape) and distributions (i.e., relatively evenly distributed pores) based on the use of the mesh size dependent critical strain master curve. We noted that two-dimensional plane stress finite element modeling method adopted in this study may be used for the study

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

B.L. Mordike, T. Ebert, Mater. Sci. Eng. 302A (2001) 37–45. J. Song, S.M. Xiong, M. Li, J. Allison, J. Alloys Compd. 477 (2009) 863–869. G. Chadha, J.E. Allison, J.W. Jones, Metall. Mater. Trans. 38A (2007) 286–297. Z. Shan, A.M. Gokhale, Mater. Sci. Eng. 361A (2003) 267–274. H. Hu, M. Zhou, Z. Sun, N. Li, J. Mater. Process. Technol. 201 (2008) 364–368. A.K. Dahle, Y.C. Lee, M.D. Nave, P.L. Schaffer, D.H. St. John, J. Light Met. 1 (2001) 61–72. A.K. Dahle, D.H. St. John, Acta Mater. 47 (1999) 31–41. J.P. Weiler, J.T. Wood, R.J. Klassen, E. Maire, R. Berkmortel, G. Wang, Mater. Sci. Eng. 395A (2005) 315–322. C.H. Ca´ceres, Scr. Metall. Mater. 32 (1995) 1851–1856. C.H. Ca´ceres, B.I. Selling, Mater. Sci. Eng. 220A (1996) 109–116. A.K. Ghosh, Acta Metall. 25 (1977) 1413–1424. S.G. Lee, G.R. Patel, A.M. Gokhale, A. Sreeranganathan, M.F. Horstemeyer, Mater. Sci. Eng. 427A (2006) 255–262. S.G. Lee, G.R. Patel, A.M. Gokhale, A. Sreeranganathan, M.F. Horstemeyer, Scr. Mater. 53 (2005) 851–856. A.M. Gokhale, S. Yang, Metall. Mater. Trans. 30A (1999) 2369–2381. A.L. Gurson, J. Eng. Mater. Technol. 99 (1977) 2–15. ABAQUS, Analysis User’s Manual, Version 6.10, 2010. X. Sun, K.S. Choi, W.N. Liu, M.A. Khaleel, Int. J. Plasticity 25 (2009) 1888–1909. X. Sun, K.S. Choi, A. Soulami, W.N. Liu, M.A. Khaleel, Mater. Sci. Eng. 526A (2009) 140–149. K.S. Choi, W.N. Liu, X. Sun, M.A. Khaleel, Acta Mater. 57 (2009) 2592–2604. W. Wang, J.L. Murray, S.Y. Hu, L.Q. Chen, H. Weiland, J. Phase Equilib. Diffus. 28 (2007) 258–264. D.S. Li, H. Garmestani, M. Baniassadi, S. Ahzi, M.M. Reda Taha, D. Ruch, J. Comput. Theor. Nanosci. 7 (2010) 1462–1468. F.J. Vernerey, W.K. Liu, B. Moran, G. Olson, J. Mech. Phys. Solids 56 (2008) 1320–1347. J.W. Swadener, M.I. Baskas, M. Nastasi, in: Proceedings of the 11th International Conference on Fracture, Turin, Italy, March 20–25 2005. J.W. Hutchinson, Int. J. Solids Struct. 37 (2000) 225–238. N.A. Fleck, J.W. Hutchinson, J. Mech. Phys. Solids 41 (1993) 1825–1857. Y. Li, D.G. Karr, Int. J. Plasticity 25 (2009) 1128–1153. Y. Li, T. Wierzbicki, in: Proceedings of the SEM Annual Conference, Albuquerque, New Mexico, USA, June 1–4 2009. M.K. Surappa, E. Blank, J.C. Jaquet, Scr. Metall. 20 (1986) 1281–1286. X. Sun, A. Soulami, K.S. Choi, O. Guzman, W. Chen, Mater. Sci. Eng. 541A (2012) 1–7. R. Rodrı´guez, I. Gutierrez, Mater. Sci. Eng. 361A (2003) 377–384. Y. Huang, X. Feng, G.M. Pharr, K.C. Hwang, Modell. Simul. Mater. Sci. Eng. 15 (2007) S255–S262.