Two-Scale Computational Analysis of Deformation and Fracture in an Al-Si Composite Material Fabricated by Electron Beam Wire-Feed Additive Manufacturing

Abstract

Numerical simulation of deformation and fracture of an AlSi12% alloy additively fabricated by layer-by-layer electron beam melting of a wire is carried out. The microstructure of the alloy is studied by scanning and transmission electron microscopy at different resolutions. The experimental study at a length scale of several dozens of microns reveals a dendritic structure, which can be treated as a composite material consisting of aluminum arms separated by a eutectic network. The volume fraction of dendrites varies with the distance from the base plate in the build direction. The eutectics can also be thought of as a composite with an aluminum matrix reinforced by silicon particles at a scale of a few microns. Particles of different shapes are nearly equally spaced in the matrix. The eutectic and dendritic structures are taken into account explicitly in the calculations. The dynamic boundary-value problems are solved by ABAQUS/Explicit. The isotropic elastic-plastic and elastic models are used to simulate the response of aluminum and silicon. The fracture model includes a maximum distortion energy criterion formulated for the particle and matrix materials in terms of the equivalent stress and plastic strain. A two-scale approach is proposed to investigate deformation and fracture of the AlSi12% alloy. On the eutectic scale, the thermomechanical behavior of the Al matrix-silicon particle two-phase composite is simulated to obtain the homogenized properties of the eutectic composite material, which is then used at a higher scale to investigate the deformation and fracture of a two-phase dendritic structure. Residual stresses formed during cooling of the additively manufactured material were found to decrease the strength of the composite, while the strength increases with the volume fraction of dendrites.

1. Introduction

Metal additive manufacturing is a layer-by-layer process for fabrication of components/parts of intricate geometry from metals and alloys in a relatively short time period with minimal material consumption [1,2]. Within recent years, the requirements regarding the quality and time of manufacturing the products and their parts have become ever more stringent. As a result, additive manufacturing has turned into a fast-developing promising technology quite popular in the aerospace, automotive, power engineering and other industries [1,3,4]. Due to the use of additive technologies, it has become possible to reduce the weight and ensure a faster, on-demand production of critical components, e.g., mounting brackets and crankshafts, and to provide a new design of such parts as pistons, plungers and heat exchangers [1,5]. Primary air fans [6], spherical tank and rocket thrust chamber liners [7], thrust chamber assemblies, turbopump stators and fuel nozzles [8] can be produced by 3D metal printing with good quality for industrial and aerospace applications. Additive manufacturing is frequently referred to as a resource-saving production, since in addition to manufacturing new articles, it allows minimizing the material and energy consumption via reclamation of worn repairable components [9].

Aluminum alloys are lightweight structural materials extensively applied in automotive engineering, the aerospace industry, electronic communication and railway transportation due to their high strength-to-weight ratio [10,11,12,13]. Aluminum–silicon alloys, or silumines, belong to the class of casting alloys [14]. It is well known that Al-Si alloys possess high thermal conductivity and corrosion- and wear resistance, due to their active use in the aerospace, automotive and other industries [15]. The main alloying element of silumines is silicon. Depending on its content in the aluminum–silicon alloys, they are classified into hypoeutectoic (up to 12 wt.%), eutectoic (from 12 to 13 wt.%) and hypereutectoic (from 13 to 30 wt.%) alloys. An α-Al+Si eutectic alloy is formed at a temperature of 577 °C in the case where the content of silicon is 12.6 wt.% [16]. It is the composition and cooling rate of AlSi alloys which control the type of their resulting microstructure. The first type of alloys demonstrates a homogeneous distribution of the α-Al+Si eutectics in the alloy volume; the second type is characterized by the presence of aluminum dendrites having an interdendritic eutectics [17]. In order to improve the mechanical properties of these alloys, such modifiers as sodium, antimony, or strontium are added to them, because the silicon particle morphology changes from acicular to rounded particles. However, this type of structure modification has the disadvantage of high porosity [18]. Another method for refining silicon particles in silumines is their microalloying with nanoparticles of rare-earth elements, such as La and Ce [19]. Silicon particles are modified when the concentration of La is increased to 1.5 wt.%, which results, however, in the formation of brittle intermetallic particles detrimental for the final service characteristics. Suarez [20] shows that an ultrasonic treatment of Al-Si11% alloys provides coarsening of the silicon particles in the eutectics.

There are various additive technologies for fabrication of metal material products depending on the heat source (electric arc/laser/electron beam) and the fed filament (powder/wire/bar stock) [21,22]. The most common additive printing of silumines is Laser Powder Bed Fusion (LPBF) [6,23,24]. The energy per layer was proposed for the first time by Rashid et al. [25] as an important input process parameter. Manufacturing of the products by the LPBF from several batches of AlSi12 powder was reported by Baitimerov et al. [26]. Due to high cooling rates, the microstructure of a eutectic alloy is characteristic for hypoeutectic silumines. A disadvantage of this method is the high porosity of the resulting product. It was demonstrated by Lykov and Baitimerov [27] that the porosity and roughness of the LPBF products could be decreased via using a powder fraction with the average particle diameter of about 17 μm and a double beam pass when printing every layer. High porosity is not the only limitation of the LPBF method. Despite the high precision (0.05–0.3 mm) [28] of printing the products with intricate inner geometry using powder-based additive technologies, an advantage of the wire-feed process is the fact that it does not require any additional preparation of the raw stock and is not so sensitive to the aggressive surrounding media compared to the powder-based technologies. To crown it all, the cost of energy in the wire-feed processes is very low. For instance, the studies reported by Jackson et al. [29] demonstrated that the power consumption in the wire-feed additive technologies is 85% lower than that in the additive powder-based process. The Wire Arc Additive Manufacturing method (WAAM) was used by Köhler et al. [30] for printing thin walls from the Al–5356 and Al–4047 alloys. Their studies demonstrated that the relative elongation to fracture of the specimens cut perpendicular to the build direction was smaller than that of the samples cut in the horizontal direction. A comparison of the microstructure of an AlSi12 alloy manufactured by the WAAM and sand- or steel mold casting was performed by Langelandsvik et al. [31]. In all cases, the porosity was lower than one percent. In contrast to the cast alloy, the WAAM AlSi12 alloy microstructure is characterized by the presence of individual beads. Hauser et al. [32] focused on varying the additive printing modes aimed at preventing the porosity of the WAAM-fabricated AlSi5 alloy. When the shielding gas feed rate is decreased, the melt pool was shown to crystallize slower, which reduces the porosity. The method of electron beam additive manufacturing demonstrates clear advantages, since the process is performed in vacuum, which allows preventing oxidation of aluminum alloys, and the power density and heat penetration depth are much higher than those in the laser-based process [7,33]. To our knowledge, there are no literature data on controlling the microstructure of the AlSi12 alloy produced by wire-feed electron beam additive manufacturing (EBAM).

Despite the wide range of approaches to modification of the AlSi12 alloy structure, the challenge of searching for the most effective, energy- and resource-saving methods has not been met yet. In this study, we perform a numerical analysis of the deformation and fracture of the EBAM-formed AlSi12 alloy using the data of the experimental microstructure investigations. We show that the EBAM-manufactured AlSi12 alloy represents a multi-level composite material. Some of the features of the deformation and fracture of particle-reinforced composites were reported in our earlier works [34,35,36]. We examined the influence of the volume fraction, spatial distribution of reinforcing particles, and strength and plasticity characteristics of the matrix and particle materials on the macroscopic response of the composite. Special attention was given to the residual stresses, which develop during the alloy manufacture and affect the mechanical behavior of the material in the subsequent service of the products thereof [37].

This work aims at a numerical-experimental study of the structure and new features of the behavior of the EBAM-manufactured AlSi12 alloy at different scale levels, which would allow making a new fundamental contribution into the materials science of eutectic silumines produced by the additive electron beam melting for their new potential future commercial applications. Additively manufactured alloys are hierarchically structured materials. Stress concentrations of different scales appearing near the interfaces between the matrix and the particles, dendrites and eutectics, printed layers and polycrystalline grains may cause plastic strain localization and fracture of the material. A multiscale analysis is required to understand the nature of these processes and to predict the thermomechanical behavior of the materials. A two-scale approach is proposed in this work for the first time, and new regularities of microstructure formation during EBAM manufacturing of AlSi12 alloys and plastic strain localization and fracture are found experimentally and numerically. In Section 2, the fabrication of EBAM samples and investigation of their microstructure are described and different volume fraction of the eutectics are identified. A two-scale simulation is mathematically formulated in Section 3.1. In Section 3.2, the deformation and fracture of the aluminum matrix–silicon particles composite is studied to derive homogenized properties of the eutectics material, and Section 3.3 is focused on the numerical simulation of thermomechanical behavior of the in-layer material with dendritic structure.

2. Experimental Evidence

2.1. Materials and Experimental Set-Up

A 1.2 mm ESAB (Elektriska Svetsnings-Aktiebolaget; Gothenburg, Sweden) wire of an AlSi12 OK Autrod 4047 alloy were used, which was deposited on a 5 mm AlMg5 alloy substrate to manufacture a thin-wall product. The chemical compositions of the AlSi12 and AlMg5 alloys are listed in

Table 1. Compositions of AlMg5 substr ate and AlSi12 wire.

Alloy	Al, %	Mg, %	Si, %	Mn, %	Fe, %	Cr, %	Ti, %	Zn, %	V, %	Zr, %
AlMg5	balance	5.21	0.41	0.48	0.43	0.23	0.11	0.16	-	-
AlSi12		-	13.3	-	0.11	0.08	0.03	-	0.035	0.005

.

It was shown in our previous work [38] that when printing thin-wall products by the additive process, it is reasonable to use a regime with an exponentially decreasing heat input, which allows forming the samples with minimal porosity and discontinuity defects and makes it possible to prevent spilling of the top layers. In this case, the value of the current was exponentially decreased during the layer-by-layer printing of the AlSi12 alloy (Figure 1). The other printing process parameters are presented in

Table 2. Process parameters of the electron beam additive manufacturing of a thin AlSi12 alloy wall.

Accelerating Voltage, kV	Number of Layers	Beam Spot Type	Table Motion Velocity, mm/min	Beam Spot Diameter, mm	Wire Diameter, mm
30	40	Spiral from center	540	~4	1.2

.

As a result of printing in the selected exponential regime, a thin wall was built (Figure 2).

The sections for testing were cut from the bottom, middle, and top parts of the thin AlSi12 wall as shown in Figure 2. The metallographic examination was carried out according to a standard procedure: the samples were ground on a 200–2000 grit grinding paper, polished on the diamond paste, and finally polished on a suspension. Their microstructure was examined in a TESCAN VEGA II LMU scanning electron microscope (SEM). The fine structure was studied in a JEM-2100 transmission electron microscope (TEM).

2.2. Microstructure of the Additively Manufactured Alloy

Figure 3a–c shows SEM microstructure images of the AlSi12 alloy, fabricated by the electron beam additive manufacturing process, sampled from the bottom (Figure 3a), middle (Figure 3b) and top (Figure 3c) parts of the thin-wall product. A coarse dendritic structure of solid solution of aluminum is observed inside the printed layers, which is shown by the light gray color, and in the interdendritic space there is the α-Al+Si eutectics, shown in dark gray. A TEM image in Figure 3d shows the fine structure of the eutectic region, and Figure 3e,f shows the results of the EDS mapping of this region. It has been found out that silicon particles are of a variety of shapes (rectangular, oval, rounded and elongated tooth-like particles).

Using the SEM images, the volume fraction of the aluminum dendrites and the eutectics were calculated using ImageJ software widely used by the researchers (Figure 4). In total, 5 images were used from each region, the lower, middle and upper regions of the sample (15 images altogether). It has been found out that the volume fraction of the dendrites considerably decreases with respect to the built wall, while that of the α-Al+Si eutectics increases. This is accounted for by the heat removal conditions. The dissipation of heat in the top part is slower than in the first printed layers due to vacuum conditions.

3. Computational Analysis

3.1. Formulation of the Problem

The experimental data presented in Section 2 can be treated as follows. At the scale levels selected for this study, the EBAM-manufactured AlSi12 is a two-level composite. At the scale of tens of microns, there are plastic inclusions of aluminum dendrites in a harder and stiffer matrix representing a eutectic network (Figure 3a–c). The eutectics at a lower micron scale level is in its turn determined by the transmission electron microscopy as volumetric silicon particles in an aluminum matrix (Figure 3d–f) and is, in essence, a classical metal matrix elastic-brittle particle-reinforced composite material. In the present work, a two-level numerical approach is proposed to study the regularities of deformation and fracture of the AlSi12% alloy. In the first stage, the deformation of the composite eutectic material was simulated, and by averaging with respect to the volume, the effective thermo-elastic and plastic properties and fracture characteristics of the aluminum–silicon composite were determined (Level 2). In the second stage, the obtained relationships are used as the properties of the eutectic network material for the numerical simulation of the evolution of thermal residual stresses, plastic strain localization and cracking in dendritic structures at a higher scale level (Level 1). The main task for this study is to test and perfect this procedure using the example of solving a two-dimensional problem. In the future, the approach will be used in 3D modeling, where the multiscale analysis is most relevant, since it allows reducing the computational time and data volume by a few orders of magnitude, or, in principle, solving such problems to the finest detail of structural features, which, in the case of a single-level approach, cannot be solved due to the insurmountable requirements to the hardware operating memory and computing power.

Model dendritic structures with varied volume fractions of aluminum dendrites at Level 1 (Figure 5a–c) and the structures of an aluminum matrix–silicon particles composite at Level 2 (Figure 5d–f) were designed. The eutectic structures shown in Figure 5a–c were designed manually by simplifying the experimental images shown in Figure 3a–c to two-phase pixel maps. The composites shown in Figure 5d–f were designed relying on the following considerations. Different volume fractions of dendrites (VFD) in the bottom, medium and top parts of the printed sample mean that the eutectics contains different volume fractions of silicon particles (VFS). For instance, if a composite material was formed from the fed wire with the aluminum and silicon volume fractions of 86% and 14%, respectively, which is shown in Figure 5a, where the volume fraction of pure aluminum in the form of dendrites is 56%, then 30% of aluminum remained in the eutectics. Therefore, the eutectic material contains 30% of aluminum and all 14% of silicon; in other words, the volume fraction of silicon in the eutectics is on the order of 32% (Figure 5d). Accordingly, the volume fraction of silicon in the eutectics in the medium and top parts of the sample are 26% and 20%, respectively (Figure 5e,f). To make it rigorous, the structures at Level 2 had the same morphology corresponding to that observed experimentally in one of the regions in the middle part of the sample (Figure 3d), and only the volume fraction was varied by a uniform change of the distance between the particles.

The pixel maps shown in Figure 5 were transformed by an in-house program into orphan voxel meshes written in ABAQUS input files (*.inp). The model dendritic structures (Figure 5a–c) have a size on the order of 100 × 100 µm and are discretized by a uniform straight-line grid containing 1000 × 1000 square finite elements. The numerical models of the eutectic structures shown in Figure 5d–f are approximated by the 430 × 430, 480 × 480 and 550 × 550 grids with a 5 nm spatial step. The computational domains have been discretized for the interfaces between the Al matrix and Si particles or the eutectics material and Al dendrites to lie along the finite-element grid nodes. The properties of silicon or aluminum are prescribed in the adjacent elements on either side of the interface. The conditions of a perfect mechanical contact are met at the interfaces. The two-dimensional dynamic boundary-value problems on loading of the composite are solved in a plane-stress formulation using ABAQUS/Explicit (2019, Dassault Systèmes, Velizi, France). In order to describe the thermoelastic behavior of the aluminum matrix at Level 2 and that of the aluminum dendrites at Level 1, the Duhamel–Neumann relations are used [39]

$${\dot{\sigma }}_{ij}=-\dot{P}{\delta }_{ij}+{\dot{S}}_{ij}=K({\dot{\epsilon }}_{kk}-3\alpha \dot{T}){\delta }_{ij}+2\mu ({\dot{\epsilon }}_{ij}-{\dot{\epsilon }}_{kk}{\delta }_{ij}/3-{\dot{\epsilon }}_{ij}^p)$$

(1)

where ${\sigma }_{ij}$ and $S_{ij}$ are the components of stress and stress deviator tensors, $P$ is the pressure, ${\epsilon }_{ij}$ and ${\epsilon }_{ij}^p$ are the components of the total and plastic deformation, ${\delta }_{ij}$ is the Kronecker delta, $K$ and $\mu$ are the bulk and shear moduli, $\alpha$ is the thermal expansion coefficient, $T$ is the temperature, the upper dot indicating a time derivative.

Using the flow rule, ${\dot{\epsilon }}_{ij}^p=\dot{\lambda }{S}_{ij}$, associated with the yield criterion in the form of ${\sigma }_{eq}-\phi \left({\epsilon }_{eq}^p\right)=0$, the strain hardening of the aluminum and eutectics is assumed to be given by the following:

$$\phi \left({\epsilon }_{eq}^p\right)={\sigma }_s-\left({\sigma }_s-{\sigma }_0\right)\exp \left(-{\epsilon }_{eq}^p/{\epsilon }_r^p\right)$$

(2)

where $\lambda$ is the scalar factor identically equal to zero in the elastic region, ${\sigma }_s$ and ${\sigma }_0$ are the ultimate stress and the yield point, respectively, ${\epsilon }_r^p$ characterizes the strain hardening degree. The equivalent stress and accumulated plastic strain are given by

$${\sigma }_{eq}=\sqrt{{\sigma }_{xx}{}^2+{\sigma }_{yy}{}^2+3{\sigma }_{xy}{}^2-{\sigma }_{xx}{\sigma }_{yy}}$$

(3)

$${\epsilon }_{eq}^p=\frac{\sqrt{2}}{3}{\displaystyle {\int }_0^t\sqrt{{\left({\epsilon }_{xx}^p-{\epsilon }_{yy}^p\right)}^2+{\left({\epsilon }_{zz}^p-{\epsilon }_{yy}^p\right)}^2+{\left({\epsilon }_{xx}^p-{\epsilon }_{zz}^p\right)}^2+6\left({\epsilon }_{xy}^p{}^2+{\epsilon }_{yz}^p{}^2+{\epsilon }_{xz}^p{}^2\right)}dt}$$

(4)

The thermoelastic response of silicon particles is described by Equation (1), where ${\dot{\epsilon }}_{ij}^p=0$. In aluminum and in the eutectic material, fracture is assumed as soon as the accumulated plastic strain reaches a certain critical value, as is shown:

$${\epsilon }_{eq}^p={\epsilon }_f^p$$

(5)

Silicon is taken to be elastically brittle. In order to take into account the particle fracture, a maximum equivalent stress criterion of the Huber type is used, which includes the type of the stress state into consideration, as is shown:

$${\sigma }_{eq}=C_{ten}, \mathrm{if} {\epsilon }_{kk}> 0.$$

(6)

The fractured finite element does not resist shear or bulk tension, but does resist bulk compression.

Equations (1)–(6) are integrated into ABAQUS/Explicit by the VUMAT user subroutine.

Two types of problems are solved: M—mechanical loading and TM—thermomechanical loading. In the former case (M), the kinematic tension of the composites from the initial undeformed state is modeled. In the latter case (TM), the tensile loading was applied after the composite cooling from 350 °C to a room temperature of 23 °C. In the cooling stage (C—cooling), the temperature, which is the same in every finite element of the computational domain, linearly decreases. Cooling of the fully crystallized material is simulated. Despite the fact that a dynamic formulation of the problem is used, the constitutive equations do not take into account the strain rate and temperature sensitivity of the material. The mass velocity gradually increased during tension and the temperature decreased rather slowly during cooling in order to minimize the dynamic effects associated with the formation and propagation of elastic waves near the matrix–particle interfaces due to the difference between the elastic moduli and thermal expansion coefficients of silicon and aluminum.

The values of thermo-elastic parameters $\xi =K,\mu ,\alpha$ for the eutectic material at Level 1 were calculated as

$${\xi }^{Eut}={\xi }^{Al}\left(1-f\right)+{\xi }^{Si}f$$

(7)

where $f$ is the volume fraction of silicon particles at Level 2.

The parameters of strain hardening function (2) for the eutectic material at Level 1 were approximated by the averaged plastic response of the composites at Level 2 during their cooling (C) or tension (M):

$${\phi }^{Eut}\left({\epsilon }_{eq}^p\right)=〈{\sigma }_{eq}〉\left(〈{\epsilon }_{eq}^p〉\right)$$

(8)

where $〈{\sigma }_{eq}〉={\displaystyle \sum _{k=1,N}{\sigma }_{eq}^k}{S}^k/{\displaystyle \sum _{k=1,N}{S}^k}$ and $〈{\epsilon }_{eq}^p〉={\displaystyle \sum _{k=1,N}{\epsilon }_{eq}^p{}^k}/N$ are the equivalent stress and accumulated plastic strain averaged over the entire computational domain, $N$ is the number of computational cells, $S^k$ is the local volume of the k-th cell.

The fracture strain of the eutectic material ${\epsilon }_f^p$ at Level 1 was taken to be equal to $〈{\epsilon }_{eq}^p〉$ in (8), at which the main crack formed in the composites at Level 2.

3.2. Aluminum–Silicon Composites at the Eutectic Scale and Homogenization

The goal of this section is twofold. Firstly, the regular features of plastic strain localization and crack nucleation and propagation in EBAM-manufactured AlSi12 at a scale level of the metal matrix composite (Level 2, Figure 5) were investigated separately for the cases of its mechanical loading from the initially undeformed state (M) and cooling (C). The emphasis is on the study of the effects from the volume fraction of ceramic particles. Secondly, a spatial homogenization of the simulation results was performed and the effective plastic properties and fracture characteristics of the composite material of the eutectics for the cases of the triaxial (C) and uniaxial (M) deformation were determined to be used at a higher scale level of the dendrite structure (Level 1, Figure 5). The model structures of the eutectic material with the experimentally determined volume fractions of 32%, 26% and 20% are given in Figure 5d–f. Let us discuss certain special features of the formation of the regions with tensile volumetric stresses during all-round and uniaxial compression of composites during their cooling (C) and tension (M) in the case of strain localization in the vicinity of the plastic matrix–elastic particles interface.

Under the triaxial loading, i.e., after cooling (C), the particles are entirely in the volumetric loading conditions, since aluminum, possessing a higher thermal expansion coefficient than that of silicon, compresses them on all sides. The regions of both volumetric compression and tensile residual stresses form in the matrix. For a more detailed description of the staging character of the deformation of composites subjected to cooling followed by tension, see our earlier publication [40]. When the particle volume fraction in the composite decreases, the value of compressive stresses increases both in the particles and in the matrix (blue-color regions in Figure 6). In order to interpret this finding, let us single out one of the particles from the composite central part together with the matrix material surrounding it (dashed rectangular boxes in Figure 6f). In the case of the minimal silicon volume fraction of 20%, the particle is surrounded by the matrix material only, while for the silicon volume fractions of 26% and 32%, the same composite volume contains both the matrix and the adjacent silicon particles. Since it is due to the compressing material of the matrix that the particles are experiencing compressive volumetric stresses, when the volume of the matrix around an individual particle is increased, it would be subjected to the larger-value compressive stresses. When the particle volume fraction in the composite is increased, the volumetric tension regions in the matrix become more pronounced, which is due to shorter distances between the particles, and rounded regions with stronger induced tensile stresses formed at a certain distance from every particle (red-color regions in Figure 6d–f). Note that the size of the volumetric compression regions in the matrix material around every particle decreases (blue-color regions in Figure 6d–f).

In the uniaxial tension case (M), the plastic flow localizes near the curvilinear matrix–particle interfaces (red-color regions in Figure 7a–c). As the silicon volume fraction is increased in the composite, the plastic flow becomes more intensive. At the average accumulated strain value of 2%, its maximum values in the strain localization regions can be as high as 10%. Note that in between the particles, there are low-strain zones, where the matrix deforms elastically (blue-color regions in Figure 7a–c). Figure 7d–f shows the fracture behavior of the composite. The crack-formation mechanism is as follows. The equivalent stress in the hard silicon particles reaches a critical value of 170 MPa, and fracture occurs in these local regions of the tensile volumetric stress concentration. A new interface between the fractured material and aluminum matrix is formed, and an additional stress concentration is developed in its vicinity with a high plastic strain localization. As soon as its intensity reaches a critical value for aluminum ${\epsilon }_f^p=35\%$, the matrix starts cracking. The resulting crack propagates along the X axis, i.e., perpendicularly to the loading direction Y. In the case where the particle fraction is 26%, the site of crack nucleation is different from those with the particle volume fractions of 20% and 32%, which is due to the stress and plastic strain redistribution upon variation of the distances between the silicon particles. The crack is finally formed earlier in the composite with the highest particle volume fraction of 32% (Figure 7d), which is due to a higher strain localization in this composite.

According to the calculations performed for the cases of triaxial (C) and uniaxial (M) loading, the stressed state was averaged over the entire computational domain using Formula (8). The respective curves are presented in Figure 8. It has been found out for both cases, (C) and (M), that the higher the silicon volume fraction, the larger the effective values of the yield stress and the strain hardening coefficient of the composite. Comparing Figure 8a,b, it is readily seen that the strain resistance of the composites during cooling is lower than that under tension. This is due to the fact that in the case of cooling (C), all boundaries of the computational domain are free from loading. Therefore, the presence of stress concentration and plastic strain localization is totally controlled by the differences in the thermal expansion coefficients and the elastic moduli of aluminum and silicon: if the material was homogeneous, the stresses would have been equal to zero. In the case of (M), the mechanical response of the composite is due to the kinematics of uniaxial deformation as such, in addition to the difference between the thermo-mechanical properties of the matrix and the particles. The calculated flow curves were approximated by the exponents (black-color line in Figure 8) that will be a function of the isotropic hardening ${\phi }^{Eut}\left({\epsilon }_{eq}^p\right)$ of the eutectic material at Level 1. The constants of strain-hardening functions and the averaged thermoelastic properties of the eutectic material are listed in

Table 3. Effective properties of the eutectic material with different volume fractions of silicon particles during tension and cooling.

Volume Fraction of Si, %	ρ, g/cm3	K, GPa	µ, GPa	α, 10−6 °C−1	σS, MPa		σ0, MPa		εpr, %		εpf, %
					C	M	C	M	C	M	M
20	2.63	72	34	18	35	68	25.7	30.5	0.6	5	5.8
26	2.6	74	36	17	36	66	26	31	0.9	4.5	5.6
32	2.58	76	38	16	38	65.6	25.6	31.3	1.3	4	5.5

. The values of thermal expansion coefficients (TECs), bulk and shear moduli and the eutectic material density were calculated via Formula (7). The TEC of the eutectics is smaller than that of aluminum, while its elastic moduli are larger than those of aluminum. The critical value of plastic strain for ${\epsilon }_f^p$ was taken to be equal to the deformation at which the main crack formed in the composites (Figure 7d–f). The resulting values will be used in Section 3.3 as the properties of the eutectic network material in numerical modeling of the deformation and fracture behavior of the dendritic structures under thermomechanical loading.

3.3. Thermomechanical Behavior of the Dendritic Structures

Let us discuss the behavior of thermomechanical deformation and fracture of the EBAM-manufactured AlSi12 at the dendritic structure level (Level 1, Figure 5). The effective properties of the eutectic material having different silicon volume fractions (VFS) were determined at a lower scale level of the particle-reinforced composite (Table 3). The calculations of deformation of the composites given in Figure 5 under two loading types, mechanical (M) and thermomechanical (TM) loading, were performed. The dendrite volume fractions (VFD) in the bottom, middle and top parts of the alloy were 56%, 47% and 31%, respectively. Accordingly, the fractions of the eutectic network in these structures were equal to 44%, 53% and 69%, wherein 32%, 26% and 20% of silicon, respectively, were present. The simulation results are given in Figure 9, Figure 10 and Figure 11. Let us examine the stress–strain state of the AlSi12 alloy after its cooling from the temperature of 350 ℃ to a room temperature of 23 °C on the example of a structure cut from the middle part of the sample having a dendrite volume fraction of 47% and a silicon volume fraction in the eutectics of 26% (Figure 9). At Level 2, a bulk compression of the elastic particles and a prevailing bulk tension of the plastic matrix were demonstrated during cooling of the aluminum matrix–silicon particles composites. The situation at Level 1 of the dendritic structure is quite contrary. The plastic inclusions of aluminum dendrites experience exclusively tensile stresses (red-color regions in Figure 9a), while the stiffer eutectic network, acting as the matrix, is prevailingly undergoing bulk compression (blue-color regions in Figure 9d). There are small-sized regions of tensile stresses (red-color regions in Figure 9d) appearing between the closely located dendrites. Interestingly, the equivalent stress in certain areas of the dendritic inclusions and the eutectic network is at its maximum (deep blue color in Figure 9b,e), but along with this, the pressure is equal to zero (green color in Figure 9a,d). This indicates that the regions do not undergo any bulk deformations, i.e., they are found in the conditions of pure shear during the macroscopic triaxial compression of the dendritic structure. It is prevailingly the parts of the dendritic branches immediately adjacent to the eutectic network which experience pure shear, while the pure shear in the eutectic network itself occurs, obviously, between the regions of bulk tension and compression.

The calculations have shown that both the dendrites and the eutectic material undergo plastic deformation. In the eutectic network, the plastic strain localizes near the curvilinear boundaries of the dendrites, with the largest plastic strain values observed in between the closely spaced adjacent aluminum inclusions and between the inclusions and the free surface (Figure 9f). The material in these areas predominantly experiences either bulk tension or simple shear (Cf. Figure 9d,f). A plastic flow in the dendrites develops at a later time; therefore, the plastic strain localization degree in them by the end of cooling is twice lower than that in the eutectics. The maximum values are mainly observed near the interface concavities (Figure 9c).

Figure 10 presents the macroscopic stress–strain curves of the dendritic structures under cooling followed by tension along X (red-color curves, TM) and under tension along X without taking pre-cooling into account (blue curves, M). Along the ordinate axis, the value of the equivalent stress (9) averaged over the entire computational domain is put. The deformation on the abscissa axis is the relative elongation of the domain along the X axis (Figure 5) $\epsilon =(L-L_0)/L_0$, where L₀ and L are the initial and current sample lengths. The residual strain after cooling of the sample is –0.5%. For the sake of comparison of the initial states in the M and TM cases, the flow curves for TM are shifted rightwards by the value of the residual strain. Therefore, the flow curves in both types of the problems start from the same point on the abscissa axis. The value of ${\mathsf{\epsilon }}_{\mathrm{f}}$ shown in the flow curves is the deformation of the sample preceding a sharp stress drop due to the crack propagation in the dendritic structure (Figure 11d–f).

Based on the calculation results, two principal conclusions can be drawn. Firstly, the cooling-induced residual stresses in all of the VFD cases decrease the macroscopic strength of the AlSi12 alloy under tension, and result in a nearly twofold decrease in its total elongation to fracture (Figure 10). This is attributed to the preliminary plastic yielding during cooling. Secondly, in the M case, as VFD increases, the strain ${\mathsf{\epsilon }}_{\mathrm{f}}$ exponentially decreases, while in the TM case, on the contrary, it increases (Figure 10b). A decrease of ${\mathsf{\epsilon }}_{\mathrm{f}}$ in the M case occurs due to two reasons. Firstly, when VFD is increased, the plastic strain in the eutectic material accumulates faster (Figure 10c); therefore, the critical accumulated strain value ${\epsilon }_f^p$ is attained earlier. This is due to the fact that the number of dendrites has increased, they are distributed closer to each other, and the stress concentration also increases, resulting in a faster plastic deformation accumulation. Secondly, ${\epsilon }_f^p$ determined at Level 2 decreases with an increase in VFD (Table 3). An increase in ${\mathsf{\epsilon }}_{\mathrm{f}}$ in the TM case is attributed to a nonlinear increase in the maximum equivalent plastic strain (Figure 10c), which is due to the fact that the specific plastic strain localization pattern formed in the cooling stage under triaxial compression affects the character of strain localization during the subsequent uniaxial tension. As a result, the areas with the largest plastic strain values in the TM case are located in other places of the structures with varied VFDs than those in the M case. The plastic deformation in these loci in the case of TM accumulates faster, and the cracks nucleate earlier at smaller VFDs. In the M case, the uniaxial kinematic deformation conditions inherently induce a quasi-homogeneous distribution of the plastic strain localization regions in the eutectic network, whose maximum value linearly increases with VFD until it exceeds the critical value. Therefore, ${\mathsf{\epsilon }}_{\mathrm{f}}$ decreases with an increase in VFD in the M case and increases in the TM case (Figure 10a,b). It is worth noting that these simulated results obtained at the mesoscale level of dozens of microns reflect the mechanical properties of the material inside a single printed layer and cannot be directly compared with the macroscopic experimental stress–strain curves. At the macroscale, at least three additional morphological factors should be taken into account in simulations: layered structure of the printed sample, microstructure of the material in between the layers and polycrystalline crystallographic anisotropy. The multiscale analysis requires additional experiments including an EBSD analysis and nanoindentation to solve an inverse computational problem.

The model samples undergo fracture in the eutectic network as soon as the critical value of ${\epsilon }_f^p$ is attained. The main crack propagates along the boundaries of aluminum dendrites at an angle of 60 degrees to the mechanical load application, thus dividing the sample in two parts. The patterns of plastic strain localization in the eutectics before the crack nucleation are different for different structures. For the structure of the upper part of the EBAM sample with VFD 31%, a system of crossing plastic strain localization bands is observed, which are arranged at an angle of 60 degrees to the axis of tension throughout the entire volume of the material (Figure 9a). In the cases of structures with VFD 47% and 56%, the strain is mainly localized in thin interlayers of the eutectic network near the dendrites. Note that the higher the VFD, the larger the number of plastic strain localization sites. A uniaxial load application gives rise to a redistribution of the regions of localization formed after cooling primarily around the dendrites (Cf. Figure 9f and Figure 11b).

4. Conclusions

A numerical-experimental study of deformation and fracture of the AlSi12 alloy fabricated by the wire-feed electron beam additive manufacturing has been performed. With the use of the methods of transmission and scanning electron microscopy, it was experimentally demonstrated that this alloy can be presented as a two-level composite material. At the level of tens of microns (Level 1), individual aluminum dendrites surrounded by the eutectic network are observed. The eutectic material at the micron level (Level 2) represents separate silicon particles distributed in the aluminum matrix. In accordance with the experimental data, model structures of aluminum–silicon composite were generated at Levels 1 and 2, and a two-level numerical modeling of deformation and fracture of the AlSi12 alloy was performed. The effective thermo-elastic-plastic properties of the aluminum–silicon composites and their fracture characteristics have been determined by spatial averaging at Level 2, and used as the eutectic material properties at Level 1 for investigating the thermomechanical deformation and fracture of composites. In order to identify the role of residual stresses caused by cooling (RS), two loading scenarios were considered: M—tension and TM—tension after cooling. The results obtained allow drawing the following conclusions:

• The volume fractions of dendrites (VFD) were found to be 56%, 47% and 31% in the bottom, middle and top parts of the printed sample in the building direction, respectively. In the eutectics, the particles are of spherical, ellipsoidal or toothed shapes and about 400 nm in size on average.
• After cooling of the composite, the silicon particles are entirely subjected to bulk compressive stresses, while the matrix is experiencing both bulk compression and tension. The higher the volume fraction of silicon particles, the larger the value of compressive stresses in the particles and tensile stresses in the matrix, and the earlier the main crack forms in the composite during its tension.
• During cooling of the composites, the aluminum dendrites experience bulk tension, while the eutectic matrix undergoes both bulk tension and compression. Fracture occurs in the eutectic network of the dendritic structures, where the main crack propagates along the dendrite boundaries at an angle of 60 degrees to the loading application direction.
• The residual stresses (RSs) change the plastic strain localization behavior and detrimentally affect the macroscopic strength of the composites subjected to tension. With increasing VFD, the crack develops earlier in the M case and later in the TM case, which is due to nonlinear accumulation of localized plastic strain in the latter case.

It should be noted that the model presented is a two-level simulation including consideration at micro- and mesoscale levels. It is a first step of a multiscale analysis and can predict the thermomechanical behavior of the in-layer structure with different volume fraction of the eutectics at given properties of the aluminum and silicon. To predict the macroscale mechanical behavior of the EBAM-printed material, a computation analysis through many spatial scales is required, including a solution of the inverse problems. To this end, additional experiments on REM, SEM, TEM, XRD and EBSD analysis as well as nanoindentation and thermomechanical tests should be performed. This work is in progress and the results will be published in our next papers.