Al composites during tension

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Simulations of deformation and damage processes of SiCp/Al composites during tension J.F. Zhang a,b , X.X. Zhang a,∗ , Q.Z. Wang a , B.L. Xiao a,∗ , Z.Y. Ma a a b

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e

i n f o

Article history: Received 13 April 2017 Received in revised form 28 June 2017 Accepted 25 August 2017 Available online xxx Keywords: Metal matrix composites (MMCs) Fracture Finite element (FE) analysis Interfacial strength Tensile strength Representative volume element (RVE)

a b s t r a c t The deformation, damage and failure behaviors of 17 vol.% SiCp/2009Al composite were studied by microscopic finite element (FE) models based on a representative volume element (RVE) and a unit cell. The RVE having a 3D realistic microstructure was constructed via computational modeling technique, in which an interface phase with an average thickness of 50 nm was generated for assessing the effects of interfacial properties. Modeling results showed that the RVE based FE model was more accurate than the unit cell based one. Based on the RVE, the predicted stress-strain curve and the fracture morphology agreed well with the experimental results. Furthermore, lower interface strength resulted in lower flow stress and ductile damage of interface phase, thereby leading to decreased elongation. It was revealed that the stress concentration factor of SiC was ∼2.0: the average stress in SiC particles reached ∼1200 MPa, while that of the composite reached ∼600 MPa. © 2017 Published by Elsevier Ltd on behalf of The editorial office of Journal of Materials Science & Technology.

1. Introduction Particle reinforced metal matrix composites (PRMMCs) possess high stiffness and strength, improved resistances to fatigue, wear and creep compared to unreinforced metals, which makes them the ideal structural materials for aerospace and defense applications. In-depth understanding of the deformation and fracture behaviors of PRMMCs is critical in the development of those materials [1–3]. In the past several decades, experiments and numerical simulations have been used to study the mechanical behaviors of PRMMCs [4–10]. Experimental methods are convincing to investigate the microstructure and macroscopic properties of PRMMCs [11–14]. For instance, Lloyd [15] examined the relationship between the particle size and particle fracture of SiCp/6061Al composites based on experimental observations. He found that few SiC particles cracked during tensile test when the particle size was below 10 ␮m. Guo et al. [16] used micro-pillars containing a slanted 4H-SiC(0001)/Al interface to study the interfacial properties of PRMMCs during uniaxial compression. The interfacial shear strength was found to be 133 ± 26 MPa. Due to the complex stress state in actual deformation

and fracture process, the behavior of interface of PRMMCs is still difficult to characterize using conventional experimental methods. Alternatively, numerical methods, especially the finite element (FE) methods, have been extensively used. For instance, many 2D FE models [17,18] have been used to predict the deformation and fracture properties of PRMMCs. However, traditional 2D models (of plane-stress or plane-strain types) have low accuracy to characterize the distribution of equivalent plastic strain [19]. In addition, Williams et al. [20] showed that 3D model containing spherical or ellipsoid particles cannot reproduce the stress evolution in PRMMCs accurately. Recently, 3D realistic models are receiving increasing attention in simulation investigations of metal matrix composites (MMCs). Many approaches are developed to construct 3D realistic microstructure models [19,21,22], including serial sectioning, X-ray tomography and computer modeling techniques. Serial sectioning technique [23], as a conventional method, is time consuming and requires lots of labor works. Besides, one dimension of the resulted geometrical model is usually smaller than the other two dimensions [19]. X-ray tomography technique can construct 3D real microstructure models quickly. However, the resolution is usually 1–2 ␮m [22,24,25], which is difficult to determine fine details of small particles, such as sharp particle edges and corners. Compared to serial sectioning and X-ray tomography tech-

∗ Corresponding authors. E-mail addresses: [email protected] (X.X. Zhang), [email protected] (B.L. Xiao). http://dx.doi.org/10.1016/j.jmst.2017.09.005 1005-0302/© 2017 Published by Elsevier Ltd on behalf of The editorial office of Journal of Materials Science & Technology.

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Fig. 1. Realistic microstructure based RVE model of 17 vol.% SiCp/Al composite with a size factor of 5.

Fig. 2. Applied loading conditions of 3-D FE model in y direction.

Fig. 3. (a) Unit cell model, (b) RVE model and (c) tensile strain-stress responses of 17 vol.% SiCp/Al composite.

niques, computer modeling technique is more efficient and exhibits adjustable resolution for describing real microstructures. Several studies have been conducted on the mechanical behaviors of PRMMCs based on 3D realistic models [22,26]. The characterization of interfacial properties is a concerned problem

in these studies. Cohesive zone method is commonly applied in modelling composite interface [27]. However, in the cohesive zone method, the fracture surface coincides with the cohesive element boundary, which limits its application and accuracy [28]. Zhang et al. [26] used the cohesive zone models in Abaqus. Since

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Fig. 4. (a) Tensile strain-stress responses of 17 vol.% SiCp/Al composite predicted by (b) coarse mesh, (c) fine mesh and (d) very fine mesh.

2. Experimental and computational approaches 2.1. Material preparation and experimental procedures

Fig. 5. Effects of interfacial strength on stress-strain response of 17 vol.% SiCp/Al composite.

17 vol.% 7 ␮m SiCp/2009Al composite was used in this study. This composite was prepared by vacuum hot pressing at 580 ◦ C and then hot rolled into 6 mm thick plates at 480 ◦ C. Subsequently, the composite plates were T4 treated: solution treating at 500 ◦ C for 120 min, followed by water quenching and natural aging for 4 d. Tensile specimens with a thickness of 3 mm and a gauge length of 20 mm were machined from the rolled composite plates with the tensile axis parallel to the rolling direction. Uniaxial tensile tests were performed at a constant strain rate of 1.6 × 10−3 s−1 and room temperature. 2.2. Microscopic structural characteristics of SiCp/Al

the cohesive element cannot be cut in the Abaqus, no particle crossed the representative volume element (RVE) surface in their models, which deviated from real state and introduced computational errors. Ferguson et al. [29] employed different bonding ratios between the matrix and the reinforcements rather than the cohesive zone method. However, their study was carried out in simplified microstructure models with spherical particles and it is difficult to use this method in realistic models. In this work, the damage and failure initiation during uniaxial tension of a SiCp/Al composite were studied by using a 3D RVE based finite element model (RVE-FEM) and a 3D unit cell based finite element model (UC-FEM) with an interface phase 50 nm in thickness. Effects of interfacial strength on the mechanical properties of the composite were investigated. The stress evolutions in the particles, the interfaces and the matrix during deformation were examined to assess the load transfer from the matrix to the particles. In addition, the particle size effect caused by quenching hardening was taken into account in this study.

A RVE with 3D realistic microstructure was used in this work. Method of constructing realistic microstructure models of PRMMCs can be found in Ref. [30]. To study the effect of interfacial properties of the composite, we generated an interface phase with a thickness of 50 nm, which represented the reinforcement/matrix interface. The RVE of the 17 vol.% SiCp/Al composite is shown in Fig. 1. The size factor ı of the RVE was 5. The definition of ı is: ı = 2L/D

(1)

where D is the mean diameter of the particles and L is the side length of the microstructure domain. In this work, the RVE was meshed using unstructured 4-nodes tetrahedrons by TetGen [31]. The mesh density was controlled by the maximum tetrahedron volume constraint. 2.3. Constitutive behaviors In the SiCp/Al composites, the matrix was described by an elasto-plastic constitutive law with ductile damage. To model the elasto-plastic deformation behavior of the 2009Al matrix, we

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Fig. 6. (a) Comparison between computational and experimental strain-stress curves of 17 vol.% SiCp/Al composite, and corresponding evolution of equivalent plastic strain during tension at (b) ε = 0.0028, (c) ε = 0.005 and (d) ε = 0.05.

expressed the relation between flow stress  and plastic strain εp as [32]:

 

 n

 = ref f εp = y + K εp

(2)

in which  is the flow stress,  ref is the reference stress, f is the non-dimensional function determined from the uniaxial tensile test,  y is the yield stress, εp the equivalent plastic strain, and K

and n are the parameters describing work-hardening. The material parameters for 2009Al were obtained from uniaxial tensile tests:  y = 308 MPa, K = 408 MPa and n = 0.45. To model the fracture behavior of the 2009Al matrix, a ductile criterion was used to predict the onset of damage. In this model, damage initiates when the plastic strain reaches the fracp ture strainεf . To simplify the damage model, a linear softening type

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Fig. 7. Crack morphologies of RVE model from different angle of view for 17 vol.% SiCp/Al composite of fracture surface: (a) oblique view; (b) left view; (c) front view; (d) oblique view.

the tensile yield strength  y of the matrix with the magnitude [39,40,42]:

 y = ˇ

Fig. 8. SEM fractograph of 17 vol% SiCp/2009Al composite.

of damage evolution was chosen. The total damage displacement of the matrix (from the initiation of damage to its completion) was assumed to be 15 nm [33]. Other parameters for the 2009Al matrix include the density (2.7 g cm3 ), Young’s modulus (75 GPa) and the Poisson’s ratio (0.33) [34]. According to Lloyd [15] and Kennedy et al. [35,36], when the SiC particles size were below 10 ␮m, few SiC fractured. Therefore, in this work, we assume that the SiC particles are elastic without brittle fracture. The parameters for modeling SiC particles include the density (3.2 g cm3 ), the Young’s modulus (427 GPa) and the Poisson’s ratio (0.17) [37]. According to Davidson and Regener’s work [38], we assume that the interfacial properties are similar to those of the matrix. To simplify the material models, we assume that the interface is ideally elasto-plastic having the same density, Young’s modulus and Poisson’s ratio with the matrix. The interfacial strength was inversely determined, which will be discussed in Section 3.3.

6|CTE||T |

Vf b 1 − Vf d

(3)

where ˇ is a constant with a value of 2.7 [39],  and b are the shear modulus and the Burgers vector of the matrix, respectively, CTE is the difference in the coefficients of thermal expansion between the matrix and the particles, T is the temperature change during quenching, Vf is the particle volume fraction and d is the particle size. The uniaxial stress-strain relation of the matrix after quenching, accordingly, can be expressed as:

 

 = ref f εp + y

(4)

2.5. Loads and boundary conditions The model was subjected to uniaxial tension by defining a loading surface on one side and a fixed surface on the opposite side. The tensile displacement of the loading surface along the Y direction was increased linearly from 0 to the peak of 0.7 ␮m. Fig. 2 illustrates the loading process along the Y direction. The effective stress and strain of the model can be obtained from the following equations [22]:

N eff =

f 1 rf

A l+U l

(5)

2.4. Quench hardening

εeff = ln

The dislocation density in the metal matrix increases due to thermal mismatch between the two phases when the composites cool down from high temperature, e.g. water quenching. This leads to quenching hardening effect [17,18,39–41], which increases

where frf is the reaction force at the nodes on the loading surface, N is the number of the nodes on the loading surface, A is the area of that surface, l is the length of the RVE edge, and U is the displacement of the loading surface.

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(6)

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Fig. 9. Load transfer in 17 vol.% SiCp/Al composite: (a) average stress evolution in different phases during tension; (b) evolutions of phase stress concentration factors of SiC, matrix and interface.

Fig. 10. Elastic and plastic zone distribution of (a) matrix and (b) interface at ε = 0.04 for 17 vol.% SiCp/Al composite.

To characterize the load transfer between the matrix and the reinforcement, from the relationship between volume and stress, the average stress of one phase can be obtained as [43]: j =

m Vi i 1m 1

Vi

(7)

where j can be the reinforcement, the matrix or the interface phase. If j is SiC, m is the number of elements in all SiC particles,  i and Vi are the stress and volume of each element in the SiC particles, respectively. To characterize the load transfer, the phase stress concentration factor can be expressed as [44]: Rj =

j eff

(8)

where j can be the reinforcement, the matrix or the interface phase.  j is the average longitudinal stress of one phase, and  eff the effective longitudinal stress of the RVE. The stress concentration factors of SiC particles, interface and matrix are denoted as RSiC , RInterface , RMatrix , respectively. 3. Computational results and discussion 3.1. Effects of microstructure model For the purpose of comparison, a UC-FEM was also investigated as shown in Fig. 3(a). The SiC particle in the unit cell is of cube shape with the length of the particle diagonal being 7 ␮m. The predicted strain-stress curves for the RVE-FEM and UC-FEM are shown in Fig. 3. It is obvious that the two models had similar elastic modulus and yield strengths. However, the flow stress and

the elongation predicted by the UC-FEM were significantly overestimated. It is known that the UC-FEM usually lead to significant inaccuracies [43,45]. For example, the UC-FEM overestimated the flow stress as shown by Zhang et al. [43]. The flow stress and elongation predicted by the RVE-FEM were slightly underestimated. This computational accuracy can be improved by using RVE with larger domain size, with raising the computational cost.

3.2. Effects of mesh size It is known that FE based damage models are influenced by mesh size [46]. To assess the mesh size effect, three different mesh sizes were investigated. The mesh size was controlled by the maximum tetrahedron volume constraint (a) in TetGen [31]. a was 1, 0.01, and 0.001, for the coarse mesh model, fine mesh model and very fine mesh model in this work, respectively. However, because the interface phase was thin, the elements of interface or near the interface have smaller mesh size than other parts. As shown in Fig. 4, the finer mesh model predicted lower flow stress. It was caused by the inhomogeneous deformation of the matrix and interface. When the mesh was refined, some severely deformed fine elements reached the ductile damage stage earlier than the coarse elements, which caused the elements near these fine elements to sustain more stress. The more stress the elements sustained, the earlier they reached the ductile damage stage. Therefore, the finer mesh model had lower flow strength. Generally, coarse mesh size reduces the computational accuracy and fine mesh size leads to high computational cost. On the balance of computational accuracy and cost, the fine mesh model with the maximum element volume constraint of 0.01 was selected in the following analysis of this work.

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J.F. Zhang et al. / Journal of Materials Science & Technology xxx (2017) xxx–xxx Table 1 Experimental and computational mechanical properties of 17 vol.% SiCp/2009Al composite.

Experimental Computational

E (GPa)

 y (MPa)

 u (MPa)

εf

103 102

374 371

595 581

0.058 0.055

3.3. Effect of interfacial strength To study the effect of interfacial strength, four FE models based on RVE with different interfacial strength (300, 500, 600 and 700 MPa) were investigated. The stress-strain curves with different interfacial strength are presented in Fig. 5. The predictions showed that weaker interfaces promoted earlier interface damage, which decreased the strength and the total elongation of the composite. However, a very strong interface could be taken as another reinforcing phase, which led to a high strength of composite. When the interfacial strength was 600 MPa, the simulation results match the experiment results well, as shown in Fig. 5. While the yield strength of the matrix was around 300 MPa, the interface had a much higher strength than the matrix. According to Fleck et al. [47], in the deformation of plastic crystals containing hard particles, local strain gradients are generated between particles especially in the interface zone, which leads to a higher flow stress in the zone. Therefore, the interface zone should have a higher strength than the matrix. It should be noted that the interfacial strength is strongly related to the microstructure and mechanical properties of the metal matrix. For instance, via finite element modeling with the cohesive zone model, Su et al. found that the interfacial strength was 326 MPa for the extruded 7 vol.% SiCp/7A04 composites [21], while the interfacial strength was 400 MPa for the solution heat treated and water quenched counterpart [26]. Guo et al. [16] found that the shear interfacial strength for the 4H-SiC(0001)/Al pillars cooling from 700 ◦ C was about 133 MPa, equaling to an interfacial strength of about 230 MPa. In this study, considering that the matrix was 2009Al and the SiCp/2009Al composite was T4 heat treated, the interfacial strength of 600 MPa was in the reasonable range of the reported values for the interfacial strength of the SiCp/Al composites. 3.4. Model validation To validate the computational model, the predicted and measured mechanical properties of the 17 vol.% SiCp/2009Al composite are listed in Table 1. The predicted mechanical properties were obtained from the RVE based FE model with interfacial strength of 600 MPa and the maximum element volume constraint of 0.01. The predicted strain-stress curve of the 17 vol.% SiCp/2009Al composite is shown in Fig. 6. As shown in Table 1, the experimental and computational elasticity modulus (E), yield stress ( y ), ultimate strength ( u ) and elongation (εf ) matched well. As shown in Fig. 6(a), the computational strain-stress curve agreed well with the experimental strain-stress curve. Fig. 6 shows the evolution of the equivalent plastic strain field during tensile deformation. During the elastic stage, as shown in Fig. 6(b), some matrix zones had already reached the plastic stage. Around the yield point of the 17 vol.% SiCp/2009Al, as seen in Fig. 6(c), the plastically deformed matrix zones expanded. Some of these matrix zones merged together to form a strain concentration layer. When the load raised up to the peak point of the strain-stress curve, as shown in Fig. 6(d), the plastically deformed matrix zones expanded further and the equivalent plastic strain in the strain concentration layer grew. Near the SiC particles some interface zones cracked, therefore, the load bearing of SiC particles dropped sharply.

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With crack initiating and growing, the stress that RVE can bear dropped, which accelerated further growth of the crack. As shown in Fig. 7, the crack grew along the strain concentration layer aforementioned in Fig. 6(c). The SEM micrograph in Fig. 8 shows the morphology of an actual fracture section. Ductile fracture was observed in metal matrix. Some SiC particles can be observed at the bottom of metal matrix dimple. In the numerical results, as shown in Fig. 7(d), ductile fracture of metal matrix and debonding of the interface were observed. Not only the strain-stress curves, but also the fracture morphology obtained from experiments and computations matched well. Therefore, it is feasible to use this RVE based FE model with aforementioned constitutive laws to predict the strain-stress response and the fracture behavior. This model can also be applied to study other metal (e.g. magnesium) matrix composites through using the corresponding constitutive models. 3.5. Stress evolution in SiC particles SiC particles in the SiCp/2009Al composites play important roles in load bearing. Therefore, it is important to study the stress evolution in SiC particles. According to Eq. (7), the stress evolution in SiC particles was obtained. As shown in Fig. 9(a), the average stress in SiC particles reached ∼1200 MPa, while the stress of the composite reached ∼600 MPa. Therefore, the load bearing of SiC particles plays an important role in increasing the elastic modulus and flow stress of the material. Fig. 6(b) clearly shows the evolution of heterogeneous deformation in the matrix. These heterogeneous deformation lead to earlier crack initiation, which is a main reason for causing a relatively lower elongation of the composites. Different from SiC particles, the average stress in the matrix and the interface was less than that of the composite. As shown in Fig. 9(b), RSiC was ∼2.0 and RMatrix was around 0.8 whereas RInterface was ∼0.75. Both RMatrix and RInterface decreased slightly with increasing the strain of the composite. The above phenomenon suggested that load was transferred to SiC particles. It is obvious that RSiC decreased during the elastic stage of the model and increased during the plastic stage of the model. The evolutions of phase stress concentration factors were coincident with the results measured via energy dispersive synchrotron X-ray diffraction by Roy et al. [44]. As shown in Fig. 9(a), the average longitudinal stress of the interface is lower than that of the matrix and the composite. The matrix bore more stress, about 50 MPa, than the interface. When the yield stress of the interface was set as 600 MPa, the average longitudinal stress of the interface is ∼300 MPa. The inhomogeneous deformation of the composite contributed to this phenomenon. As shown in Fig. 10, the interface and the matrix still had some elastic zones when the composite strain was ε = 0.04. The elastic zones of the interface were present on the center of some surfaces. The plastic zones of the interface were present around the corners and the edges of SiC particles. The elastic zone of the matrix was present in the zone which was near the elastic zone of the interface. The elastic zone distribution ratio of the matrix was less than that of interface. Because in the matrix, the inhomogeneous deformation influences less. 3.6. Limitations of the present FE model This RVE-FEM shows great feasibility in simulating the deformation and fracture process of SiCp/2009Al composites. It is proven that this model is superior to the UC-FEM. However, there are still some limitations that affect the applications of RVE-FEM: (1) We assume that the coefficient of thermal expansion (CTE) mismatch due to quenching affects the entire matrix zone.

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However, previous investigations [17,48,49] showed that the CTE mismatch affected zone is smaller than the entire matrix zone when the particle volume fraction is lower than 25%. (2) The enhancement of PRMMCs due to geometrical mismatch of metal matrix during deformation was not taken into consideration. Because of the inhomogeneous deformation of PRMMCs during tension, local strain gradients are generated among particles. Therefore, the geometrically necessary dislocations cause hardening of the matrix [50]. (3) The fracture of the SiC particles was not taken into consideration. According to Lloyd [15], when the SiC particles size was smaller than 10 ␮m, during the tension of PMMCs, the number of SiC particle cracks was small but still could be observed. Therefore, the assumption of no cracking of SiC introduces errors into the present simulation results. 4. Conclusions In this work, a RVE-FEM with a consideration of the interface was developed to predict the elastoplastic and fracture behavior of a 17 vol.% SiCp/2009Al composite. The interface was modeled by introducing an interface phase in the RVE. Quenching hardening effects and ductile damage of the matrix and the interface were taken into consideration. Based on the simulation results, the following conclusions can be made: (1) According to experimental and numerical results, the RVE-FEM developed in this work is more feasible than the UC-FEM. The UC-FEM cannot describe the interaction between different particles. The 3D finite element model based on the RVE with realistic microstructure can describe the microscale deformation of PRMMCs more accurately. (2) Because the metal matrix and interface phase used the displacement type damage evolution, the mesh size affects the simulation results slightly. Finer mesh model predicts lower flow stress. When the maximum tetrahedron volume constraint was 0.01, the RVE-FEM have sufficient accuracy and reasonable compute cost. (3) Lower interface strength leads to lower flow stress, earlier ductile damage in the interface phase, and lower elongation. According to the contrast between simulation and experiment, 600 MPa was the advisable interface strength for the 17 vol.% SiCp/2009Al composite. (4) The average longitudinal stress in SiC particles reached 1200 MPa, while that in the composite reached 600 MPa. The stress concentration factors of the SiC, the matrix and the interface were ∼2.0, ∼0.8, and ∼0.75, respectively. The stress concentration factor of SiC decreased during the elastic stage and increased during the plastic stage of the composite. Acknowledgments This work was supported financially by the National Key R&D Program of China (No. 2017YFB0703104) and the National Natural Science Foundation of China (Nos. 51671191 and 51401219).

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