
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

70247
10.1038/s41598-024-70247-3
Article
Insights into particle dispersion and damage mechanisms in functionally graded metal matrix composites with random microstructure-based finite element model
Naguib M. E. menaguib@zu.edu.eg

Gad S. I.
Megahed M.
Agwa M. A.
https://ror.org/053g6we49 grid.31451.32 0000 0001 2158 2757 Department of Mechanical Design and Production Engineering, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt
6 9 2024
6 9 2024
2024
14 2083512 5 2024
14 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
This study investigates the impact of Al2O3 particle volume fraction and distribution on the deformation and damage of particle-reinforced metal matrix composites, particularly in the context of functionally graded metal matrix composites. In this study, a two-dimensional nonlinear random microstructure-based finite element modeling approach implemented in ABAQUS/Explicit with a Python-generated script to analyze the deformation and damage mechanisms in AA6061-T6/Al2O3 composites. The plastic deformation and ductile cracking of the matrix are captured using the Gurson–Tvergaard–Needleman model, whereas particle fracture is modelled using the Johnson–Holmquist II model. Matrix-particle interface decohesion is simulated using the surface-based cohesive zone method. The findings reveal that functionally graded metal matrix composites exhibit higher hardness values (HRB) than traditional metal matrix composites. The results highlight the importance of functionally graded metal matrix composites. Functionally graded metal matrix composites with a Gaussian distribution and a particle volume fraction of 10% achieve HRB values comparable to particle-reinforced metal matrix composites with a particle volume fraction of 20%, with only a 2% difference in HRB. Thus, HRB can be improved significantly by employing a low particle volume fraction and incorporating a Gaussian distribution across the material thickness. Furthermore, functionally graded metal matrix composites with a Gaussian distribution exhibit higher HRB values and better agreement with experimental distribution functions when compared to those with a power-law distribution.

Keywords

Random microstructure-based model
Functionally graded metal matrix composites
Finite element method
Damage behavior
Spherical indentation
Subject terms

Aerospace engineering
Mechanical engineering
Structural materials
Computational science
Software
Zagazig UniversityOpen access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

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pmcIntroduction

Composite structures have the advantage of exhibiting different material properties at different points, offering the potential to enhance their mechanical behavior1. Functionally graded materials (FGMs) are a class of advanced structures that possess non-uniform material properties throughout the spatial domain of a material. The advancement of composite materials with graded properties has led to a revolution in the fabrication of engineered components. This is particularly true in industries such as electronics, automobiles, aviation, and biomedicine, where conventional metallic or ceramic matrix composites cannot meet design requirements2.

Functionally graded structures can be observed in nature, such as in the bio-tissues of animals and plants. Bones and dental crowns are excellent examples of functionally graded structures because they require a wear-resistant surface combined with a ductile core to withstand high contact and dynamic fatigue loading3,4. Although FGMs were initially designed for heat-resistant materials, they have been increasingly employed to control the deformation, pressure, wear, corrosion, and stress concentration by enabling smooth transitional gradients across all dimensions of a product5.

Functionally graded metal matrix composites (FGMMCs) belong to the category of metal matrix composites (MMCs) and are characterized by a continuous variation in the volume fraction (VFr) of the reinforcement along a specific direction of the matrix alloy6. FGMMCs offer a gradual or continuous transition in engineering properties at the macroscopic scale, allowing for the combination of desirable properties without the presence of mechanically weak interfaces, which is a limitation often encountered in surface coating techniques7.

A literature review revealed the existence of multiple particle distribution curves within FGMMCs; examples are:8–16, as depicted in Fig. 1. These empirical curves serve as a valuable resource for theoretical investigations, enabling the development of a controllable function that accurately represents particle distribution within the composite material. Utilizing such a controllable function facilitates a systematic approach for manipulating and analyzing the particle distribution. By fitting the experimental distribution curve to an approximated function, it is possible to obtain a mathematical model that captures the essential characteristics of particle distribution in FGMMCs. This model can be integrated into theoretical studies to enable a deeper understanding of the behavior of FGMMCs, which can serve as a powerful tool for predicting and optimizing the performance of FGMMCs in practical applications.Figure 1 Graded distribution of reinforcing phase in matrix from outer to inner surface based on experimental studies.

To assess the mechanical properties of FGMs, the indentation method is often favored owing to its simplicity and minimal specimen preparation requirements. This technique can be easily implemented multiple times on both both small-scale materials and miniaturized structures using a single specimen and a suitable load and indenter tip geometry. The use of numerical modeling to simulate the indentation process of FGMMCs can provide insight into the deformation mechanisms, stress, and strain fields within the metal matrix surrounding the reinforcement particles, as well as the interaction between the particles and the matrix17.

Reference18 conducted a study wherein graded Al7075/SiC composites were fabricated using a centrifugal casting technique with the aim of enhancing the mechanical properties and wear resistance for automotive purposes. The fabrication process involved utilizing two distinct weight fractions of particles (6.5% and 9.5%), while maintaining a constant mold rotational speed of 1300 rpm. The findings of the study indicated an improvement in the mechanical properties and wear resistance of the composites, especially in the outer region, with an increase in the weight fraction of particles owing to the influence of centrifugal forces.

In the study conducted by19, the distribution of SiC particles in aluminum-based FGMMCs used in brake rotor discs was investigated. The average particle size used in this study was 23 μm. The results demonstrated that the maximum hardness achieved after heat treatment of the Al(356)-SiC and Al(2124)-SiC FGMMCs at the outer periphery was 155 BHN and 145 BHN, respectively. Reference20 conducted a study on Al-SiC FGMMCs, where they varied the volume percentage of reinforcement. They found that, as the percentage of reinforcement increased, the density decreased. However, they observed that the hardness and wear resistance increased from the core to the cast surface. At low sliding speeds, they noted that microcracking and abrasive wear were dominant factors.

Reference21 conducted an investigation into the effects of particle segregation ratio and particle distribution on the VFr and particle size (PS) of reinforcement in aluminum based FGMMCs with in-situ primary Si/Mg2Si particles. The study involved varying parameters, such as the mold temperature and high pouring temperature. The fabricated FGMMCs showed a graded dispersion of primary Si and Mg2Si particles in the inner region, with diameters ranging from 70 to 30 μm and 30 to 18 μm, respectively. Additionally, the volume fractions of Mg2Si and primary Si exhibited a gradient distribution ranging from VFr of 14.8 to 27.7%. The highest hardness values were observed in the middle portion of the inner layer of the fabricated FGMMCs, ranging from Rockwell hardness (HRB) 72.0 to 75.0. Conversely, the outer layer exhibited the lowest hardness value of approximately HRB 56.0 63.0.

Reference22 enhanced the resistance to contact damage in a graded material system comprising silicon nitride (Si3N4) ceramic (higher modulus) and oxynitride glass (lower modulus) by employing a unidirectional gradient of the elastic modulus from the contact surface to the interior. They achieved a 30% reduction in the maximum tensile stresses outside the Hertzian contact circle compared with the monolithic material (Si3N4) of the graded system.

Reference23 investigated the effectiveness of two-scale modeling for analyzing the mechanical properties of FGMs and evaluated aluminum as the supporting matrix and silicon carbide as particle inclusions. The mechanical properties of the material were derived at the macroscale level using a full model, and at the microscale level using a representative volume element in two-scale modeling. The results demonstrated the usefulness and reliability of the two-scale model in reducing the numerical computational time. As the VFr of the inclusion increases, the deviation in the stress values decreases, providing evidence supporting the theory that FGMs offer the advantage of smoother stress distributions.

Reference17 conducted a study on the indentation behavior of FGMMCs, revealing that the number of layers, compositional gradient exponent, and random particle dispersion have a significant impact on the properties of material. Increasing the number of layers resulted in noticeable increases in indentation depths, whereas increasing the compositional gradient exponent led to higher mean residual stresses. Conversely, at a specific layer number, an increase in the compositional gradient exponent decreased the mean residual stresses and strains owing to an increase in the ceramic VFr. Random particle dispersion influences the central indentation depth and deformed surface profiles, resulting in non-uniform levels and distributions of residual stress and strain.

To the best of our knowledge, no computational FE model has been developed thus far to quantitatively analyze the evolution of multiple damage mechanisms during the indentation of FGMMCs. The development of such a model is crucial for predicting material behavior under various conditions by systematically analyzing failure mechanisms and simulating responses to stress and strain over time. These models are essential for optimizing the material performance because they enable the tailoring of microstructures to enhance durability and resilience, thus reducing the need for extensive physical testing. They manage complex geometries and loading conditions, providing a virtual testing environment that identifies potential failure points and predicts the real-world performance. Integrating finite element damage models into research and development not only improves cost efficiency and innovation but also deepens material understanding and facilitates advancements in material engineering across various industries. The objective of this study is to address the gap in stress-state-dependent quantitative damage growth analysis in FGMMCs by investigating the indentation of AA6061-T6/Al2O3 FGMMCs. To achieve this, a 2D microstructure-based FE model is employed to closely approximate the actual microstructure and consider all three potential failure modes. The hardness assessment procedure outlined in the ASTME18-22 standard24 is incorporated into finite element analysis (FEA). To capture the damage mechanisms in the AA6061-T6 matrix, the Gurson–Tvergaard–Needleman (GTN) model is employed. This model represents the behavior of the matrix considering void growth and coalescence. Simultaneously, the Johnson–Holmquist II (JH2) model is integrated into the FE model to represent Al2O3 particle cracking. The matrix-particle interfacial behavior is further modeled using the cohesive zone method (CZM). The analysis is conducted using ABAQUS/Explicit FE software, complemented by Python for scripting and preprocessing25,26.

Problem formulation

Figure 2 depicts the configuration of an AA6061-T6/Al2O3 FGMMCs with a spherical indenter. The composite structure is simulated as a rectangle with a width (W) of 5mm and height (H) of 2.5mm. In accordance with the ASTM E18-22 standard24, the diameter of indenter (Dind) and applied force (F) were determined as 1.558mm and 980 N, respectively. The HRB was computed using the following equation1 HRB=130-h0.002

where h represents the permanent penetration in millimeters (mm).Figure 2 Model configuration of FGMMCs.

Material constitutive models

This study focuses on investigating a composite material consisting of AA6061-T6 alloy, which is chosen as the base matrix because of its extensive range of engineering and structural applications7. The composite is reinforced with Al2O3 particulates, which are the most commonly used reinforcement for PRMMCs27 with volume fractions of 10% and 20% and an average size of 100μm. In general, composite materials can fail because of one of the three major failure modes. These modes include plastic deformation and ductile cracking of the matrix, fracture of the reinforcement particles, and decohesion at the matrix-particle interface. The following subsections provide a detailed description of each failure mode.

Plastic deformation and ductile cracking of the matrix

A three-stage mechanism involving void nucleation, growth, and coalescence is responsible for the ductile damage observed in the metals. These voids can be initiated by inclusions or microcracks. Subsequently, these voids undergo enlargement owing to the accumulation of plastic strain. The GTN model28 is commonly employed for the theoretical analysis of this phenomenon in porous metals. The GTN model is a modified version of the Gurson model29 and its yield function is mathematically expressed as2 φ(σ,f)=σqσy2+2q1f∗cosh3q2p2σy-1+q3f∗2=0,

where the non-dilatational strain energy is denoted by φ, and constants q1, q2, and q3 are introduced by Tvergaard30 to account for void interactions. Von Mises and flow stresses of the intact material are represented by σq and σy, respectively.

To accurately capture the rapid reduction in stress-carrying capacity resulting from void coalescence, Tvergaard–Needleman28 introduced the parameter f∗, known as the effective porosity. This parameter acts as a modeling tool and is mathematically defined as follows:3 f∗=ff≤fcfc+fu∗-fcfF-fcf-fcfc≤f≤fF,fu∗f≥fF

The GTN model incorporates several parameters to describe void behavior. The critical void volume fraction (VVF) at the beginning of coalescence is denoted by fc and fu∗=1q1 represents the VVF at which the material loses its stress-bearing capacity. Meanwhile, fF represents the VVF at which the material experiences complete failure and controls the element deletion process. The increase in the VVF owing to void nucleation and growth is considered. The function describing the effective porosity can be expressed as follows:4 df=dfn+dfg.

The rate of void growth (dfg) can be mathematically represented as a function of plastic volume change when the material is considered plastically incompressible.5 dfg=(1-f)dεiiP,

where the trace of the plastic strain-rate tensor is expressed by dεiiP. The nucleation of voids is strongly influenced by plastic strain, particularly under hydrostatic tension31. This phenomenon can be mathematically represented as follows6 dfn=Andε¯mp,

7 An=fNSN2πe-0.5ε¯p-εNSN2ifp≥0,0ifp<0

Variable p represents the hydrostatic stress, and fN represents the volume fraction of nucleated voids. Additionally, εN signifies the mean equivalent plastic strain for void nucleation and sN represents the standard deviation of the void distribution. The macroscopic plastic work rate can be equivalent to the rate of matrix plastic dissipation, which in turn determines the rate of equivalent plastic strain dε¯p according to the following equation8 dε¯p=σ:dεp(1-f)σy.

Table 1 presents the elastoplastic and GTN model constants employed for the AA6061-T6 matrix derived from previous studies.

Fracture of reinforcement particles

The JH2 model constitutive model was initially employed to simulate the mechanical response of materials exhibiting brittle fractures, particularly ceramic materials. Based on the foundational principles outlined in32, the JH2 model incorporates the mechanisms of softening and pressure-dependent strength, material damage, and fracture; it also accounts for significant residual strength after fracture, bulking, and sensitivity to the loading rate. The JH2 model characterizes the behavior of materials in their damaged state by considering three distinct states: intact, damaged, and fractured. In this context, the model employs an analytical function to express the normalized equivalent stress in the damaged state. The generic form of this function is as follows9 σ∗=σi∗-D(σi∗-σf∗)=σ/σHEL,

σi∗, σf∗, D, and σHEL are defined as follows: σi∗ denotes the normalized intact equivalent stress; σf∗ represents the normalized fracture stress; D is the damage factor, which varies between 0 and 1.0; and σHEL denotes the equivalent stress at the Hugoniot Elastic Limit (HEL). The critical point signifies the net compressive stress, which considers both hydrostatic pressure and deviatoric stress components. This is the point at which a one-dimensional shock wave with uniaxial strain exceeds the elastic limit of the material33. Lastly, σ denotes the actual equivalent stress.

The normalized intact strength (σi∗) can be determined using the following equation:10 σi∗=A(P∗+T∗)N1+C·lnε˙∗.

where A,  C,  and N are the material constants. P∗ represents the normalized pressure defined as the ratio of the actual hydrostatic pressure (P) to the hydrostatic pressure at the HEL (PHEL). T∗ represents the normalized maximum tensile hydrostatic pressure, which is defined as the ratio of the maximum tensile hydrostatic pressure that the material can withstand (T), to PHEL. ε˙∗ represents the dimensionless strain rate, defined as the ratio of the actual equivalent strain rate (ε˙) to the reference strain rate (ε˙0=1.0s-1).

Similarly, the equation for the normalized fracture strength (σf∗) can be expressed as11 σf∗=B(P∗)M(1+C·lnε˙∗)≤SFMAX,

where B,  C,  and M are the material constants. SFMAX represents the ultimate value of the normalized fracture strength (σf∗), providing additional flexibility in the definition of the fracture strength.

According to the JH2 model, the softening process in brittle materials can be described by Eq. (9), which allows for the gradual softening of the material as the plastic strain increases. The softening process continues until the material is fully damaged (Dp=1). The expression describing the cumulative damage resulting from fracture is expressed as follows12 Dp=∑ΔεPεfP=∑ΔεPd1(P∗+T∗)d2,

where ΔεP represents the accumulated plastic strain during the integration cycle. Function εfP=f(P) represents the plastic strain necessary for fracture under a constant pressure P. The parameters d1 and d2 correspond to the damage factors associated with εfP.

When a material reaches a certain threshold of plastic deformation or damage, it enters a failure state described by fluid-like behavior33. In this state, the material loses its strength and cannot withstand the stress. Both the hydrostatic pressure and deviatoric stress become zero. The relationship between hydrostatic pressure P and volumetric strain μ is described by a polynomial equation of state (EOS), which consists of two distinct stages: an elastic stage and a plastic damage stage. The mathematical expressions for these stages are as follows13 P=K1·μ+K2·μ2+K3·μ3Dp=0P=K1·μ+K2·μ2+K3·μ3+ΔP(0<Dp≤1),

Here, K1, K2, and K3 are constants and μ is defined as the ratio of the current density (ρ) to the initial density (ρ0). For tensile pressure (μ<0), Eq. (13) is replaced by P=K1·μ. The incremental pressure (ΔP) is added when the material fractures owing to bulking energy.

The reduction in incremental internal elastic energy is transformed into potential internal energy via an incremental increase in hydrostatic pressure ΔP. As the fracture progresses, the shear and deviatoric stresses diminish owing to the decrease in the equivalent plastic flow stress σ. The elastic internal energy related to these shear and deviatoric stresses can be expressed mathematically as14 U=σ26G,

where G denotes the rigidity modulus. The incremental energy loss is given by15 ΔU=UD(t)-UD(t+Δt),

The quantities UD(t) and UD(t+Δt) are determined using Eq. (14). The change in energy, ΔU, is primarily converted into incremental fracturing energy, ΔF. An approximate expression for this energy conversion is given by16 ΔU=UD(t)-UD(t+Δt),

where β is a fraction satisfying (0≤β≤1), which represents the extent of energy transformation. The JH2 model constants used for the Al2O3 particles are summarized in Table 1.Table 1 Material constitutive models parameters utilized in this study.

AA6061-T6 (GTN model) References30,34,35	Value	Al2O3 particles (JH2 model) References36,37	Value	
E	70 GPa	A	0.93	
Sy	275 MPa	B	0.31	
Sult	361 MPa	G	155 GPa	
q1	1.5	T	0.6 GPa	
q2	1	N	0.6	
q3	2.25	HEL	10.5 GPa	
f0	0.001	PHEL	4.5 GPa	
fc	0.02	M	0.6	
fF	0.06	C	0.0	
εN	0.03	σi,max	12.2 GPa	
SN	0.1	σf,max	1.3 GPa	
fN	0.0005	K1	193 GPa	
–	–	K2	0.0 GPa	
–	–	K3	0.0 GPa	
–	–	εf,min	0.0	
–	–	εf,max	1.2	
–	–	d1	0.005	
–	–	d2	1.0	
–	–	FS	0.2	
–	–	β	1.0	
–	–	ρ	3890 kg/m3	

Cohesive zone modeling (CZM) for matrix and particle interfaces

The failure process of an interface surface involves three fundamental components: initiation of damage, progression of damage, and overall debonding resulting from substantial damage. The cohesive zone method (CZM) is utilized to describe the potential debonding occurring at the interface between the particles and matrix. This study characterizes the CZM through a bilinear relationship between traction (T) and separation (Δ). The initiation of damage is governed by the quadratic stress failure criterion. This criterion postulates that damage is initiated when a quadratic interaction function reaches a unit value38. Mathematically, this criterion can be expressed as17 TnTn02+TsTs02+TtTt02=1,

where Tn0, Ts0, and Tt0 represent the peak values of the nominal stresses during deformation along specific directions: normal to the interface, the first shear direction, and the second shear direction, respectively.

In the traction-separation model, it is assumed that the scalar damage variable D changes from 0 to 1 as the material undergoes further loading after damage initiation. The damage variable D exerts the following influence on the stress components:18 Tn=(1-D)σn∗σn∗≥0σn∗σn∗<0,

19 Ts=(1-D)σs∗,

20 Tt=(1-D)σt∗,

The stress components σn∗, σs∗, and σt∗ are obtained by utilizing the elastic traction-separation behavior for the current strain before damage initiation. To characterize the progression of damage within the interface under combined normal and shear deformations, it is helpful to introduce an effective displacement Δm. The effective displacement is defined as follows21 Δm=Δn2+Δs2+Δt2,

The nominal separations in the normal (Δn) and in-plane shear (Δs, Δt) directions govern the evolution of damage variable D through the following expression22 D=0Δmmax≤Δm0ΔmfΔmmax-Δm0ΔmmaxΔmf-Δm0Δm0<Δmmax<Δmf,1Δmmax>Δmf

where Δm0 and Δmf characterize the effective separations at the initiation of damage and complete failure, respectively, and Δmmax represents the peak value of the effective displacement attained during the loading history. The fracture energy, Gc, which is a measure of the area enclosed by the traction-separation displacement curve, can be obtained as follows23 Gc=12Tn0Δmf.

The CZM constants used for the AA6061-T6 and Al2O3 interfaces in this study are listed in Table 2.Table 2 Parameters employed for CZM of matrix/particle interface.

Parameter	Knn, Kss, Ktt39	Tn040	Ts0, Tt040	Gc40	
Value	2×107MPa·mm-1	4.57 GPa	2.93 GPa	15 J/m2	

Computational model

The FE model and its validation are described in the following subsection. To investigate the damage mechanisms related to the failure modes of AA6061-T6/Al2O3 composites, a Python-generated script is used. This script enables the implementation of indentation-hardness test specifications to assess the resulting microstructure. Using this script, it is possible to generate particles with different distributions and volume fractions throughout the FGMMCs structure.

Mathematical equations representing the different failure modes outlined in this study have been compiled and integrated into the ABAQUS FE software. The analysis process is summarized in the flowchart presented in Fig. 3, providing an overview of the steps involved in the current study.Figure 3 Flowchart of the present analysis.

Finite element model

The analysis considered various factors such as geometric considerations, material nonlinearities, and nonlinear deformations caused by the contact between a rigid indenter and a metal matrix surface. The analysis employed a 2D FE model with 4-node bilinear plane strain quadrilateral elements (CPE4R). To capture the damage behavior and failure modes precisely, a refined mesh with an element size of 0.01mm is employed throughout the microstructure. This resulted in an average of 146,700 elements and 156,000 nodes per microstructure (Fig. 4). In accordance with the findings of a previous investigation41, the optimal particle shape utilized in this study is circular with a diameter (d) of 100μm

To optimize computational efficiency, the FE model employs a perfectly rigid indenter, and a fixed lower boundary is chosen for the structure to maintain stability. A Coulomb friction coefficient of 0.142,43 is applied at both the indenter-matrix and particle-matrix interfaces to model frictional interactions. For quasi-static analysis in the ABAQUS software, the force was gradually applied in a smooth step44, as depicted in Fig. 5.Figure 4 Finite element mesh for the studied model.

Figure 5 Applied force as a smooth step in ABAQUS.

Validation of the model

The accuracy of the FE model is validated by comparing its predicted HRB values with those obtained experimentally for the corresponding PRMMCs reported by45. The measurement procedure defined in ASTM E18-2224 is employed in the experimental study. This method initially involves applying a minor load to establish a reference position, followed by the application of a major load for a specified time interval. Subsequently, the major load is removed and the difference in the penetration depth of the indenter is measured and used to calculate the HRB value.

For both volume fractions of 10% and 20%, at least ten random base models were considered in the experimental study, as the Al2O3 particles were randomly dispersed within the structure (refer to Fig. 6). The average HRB value was used for the comparison.

Table 3 presents a comparison between the random distribution of the proposed model and experimental data. The results indicate that the error ranges from 0.635 to 1.811% when the present random distribution is compared with the experimental data. This low percentage error indicates that the HRB values obtained from the FE model for volume fractions of 0%, 10%, and 20% are very close to the corresponding experimental values. The comparison demonstrates good agreement between the predicted and experimental results. This minimal error percentage error suggests that the HRB values derived from the FE model for volume fractions of 0%, 10%, and 20% closely align with the corresponding experimental values. The comparison demonstrates strong agreement between the predicted and experimental results. For further details about the model and its validation, please refer to the previous work41.Figure 6 Representation of microstructure of AA6061-T6/Al2O3 composite with various VFr.

Table 3 HRB validation results.

Al2O3VFr	Experimental45	Present model random distribution	Error %	
0%	61±0.4	61.387	0.635	
10%	69±0.2	68.601	0.578	
20%	75±0.9	73.642	1.811	

Results and discussions

The main objective of this section is to explore the effect of Al2O3 particle distribution on the mechanical characteristics and damage mechanisms of the AA6061-T6/Al2O3 composite. To achieve this objective, this study analyzes the effects of the Al2O3 particle distribution across the FGMMCs structure on the mechanical properties of the composite.

An analysis is conducted to investigate the indentation behavior of structures composed of FGMMCs under the influence of a spherical indenter. The structure under examination is modelled to exhibit a variation in the VFr of the Al2O3 particles throughout its thickness, as depicted in Fig. 2.

FGMMCs are modelled using the linear rule of mixtures, which is a commonly employed approach17. This modelling technique assumes a linear relationship between the volume fractions of the Al2O3 reinforcement and AA6061-T6 matrix material. The sum of the volume fractions of the reinforcement and matrix material is equal to one and is expressed as24 Vrein+Vmat=1,

Vrein and Vmat represent the volume fractions of Al2O3 and AA6061-T6 matrix materials, respectively. In the investigated FGMMCs structures (Fig. 2), the Al2O3 particles are distributed randomly throughout the matrix. This irregularity in the arrangement of the reinforcement particles reflects the actual conditions observed in the FGMMCs.

The investigation focused on studying the variation in VFr of Al2O3 particles across the thickness of FGMMCs using two approaches. The first approach is the power law approach46, whereas the second approach utilized a Gaussian distribution47.

The first approach (power law) is commonly employed in the theoretical modeling of FGMMCs structures17,48,49. In this approach, the VFr of Al2O3 particles at any position y across the thickness is assumed to be determined by the equation25 VFr(y)=0.3×yHn.

Here, n represents the non-negative compositional gradient exponent, which determines the variation profile of the Al2O3 particles. Different values of n result in distinct particle distribution profiles and overall particle volume fractions. Position y indicates the location within the thickness of the FGMMCs structure measured from its bottom surface. H represents the total thickness of the FGMMCs structure.

To enable meaningful comparisons between PRMMCs and FGMMCs, two total particle volume fractions are considered, 10% and 20%. To determine the volume fractions, the compositional gradient exponent n is estimated to be 2.65 for a total VFr of 10%, whereas a value of 0.65 is estimated for n when the total VFr is 20%. Figure 7 shows the distribution curves obtained using the estimated values.Figure 7 Compositional profiles of the FGMMCs using power law.

Figure 8 shows the samples representing FGMMCs resulting from the application of the power law approach. A sample with a total VFr of 10% (n =2.65) is depicted in Fig. 8a, whereas a sample with a total VFr of 20% (n =0.65) is illustrated in Fig. 8b.Figure 8 The distribution of Al2O3 particles across the FGMMCs structure thickness using power law.

The second approach (Gaussian distribution) used to vary the Al2O3 particle distribution within the thickness of the FGMMCs structure is based on experimental work, specifically the study conducted by11–13,16 (see Fig. 1). In this approach, the VFr distribution at any position y is determined by fitting the experimental distribution curves. The best fit is achieved using a Gaussian distribution function47, which is represented by the following formula:26 VFr(y)=A0exp-(y-x0)22σG2.

The constants A0, x0, and σG determine the shape of the Gaussian curve, and consequently, the particle volume fraction. These constants are estimated using trial methods to obtain specific volume fractions, these constants are estimated through trial methods. For a total VFr of 10%, the estimated values of A0, x0, and σG are 0.365, 2.215, and 0.295, respectively. Similarly, for a total VFr of 20%, the constants are estimated to be A0 =0.365, x0 =1.95, and σG =0.65. The distribution curves for VFr values of both 10% and 20% using a Gaussian distribution are shown in Fig. 9.Figure 9 Compositional profiles of the FGMMCs using Gaussian distribution.

Representative samples of FGMMCs structure with these VFr variations obtained using the Gaussian approach are shown in Fig. 10. Figure 10a depicts a structure with a total VFr of 10%, whereas Fig. 10b shows a structure with a total VFr of 20%.Figure 10 The distribution of Al2O3 particles across the FGMMCs structure thickness using Gaussian distribution.

HRB results

Nonlinear FE analyses are performed to explore the influence of the power law and Gaussian distributions in FGMMCs on the indentation behavior of AA6061-T6/Al2O3 FGMMCs. Specifically, two total volume fractions of Al2O3 particles are considered: 10% and 20%. This is achieved by varying the compositional gradient exponents (n =0.65and2.65) for the power law distribution and by adjusting three parameters (A0, x0, and σG) for the Gaussian distribution.

Ten samples with random reinforcement distributions are analyzed for each VFr and distribution, and the calculated indentation depths are used to determine the HRB values. Figure 11 presents the HRB results for the pure AA6061-T6 matrix as a reference value and the average HRB values for the AA6061-T6/Al2O3 composites with volume fractions of 10% and 20% Al2O3 particles. In addition to the HRB of the pure matrix (0% VFr), the HRB results are shown three times in Fig. 11 for both the 10% and 20% particle VFr. The first set of results corresponds to the random distribution of Al2O3 particles throughout the thickness without any variation in particle concentration, labeled as PRMMCs. The other two sets of HRB results show the effects of particle variation through the thickness using the power law and Gaussian distribution, labeled FGM-1 and FGM-2, respectively.

According to the results presented in Fig. 11, the HRB results exhibited a consistent pattern. Specifically, when examining volume fractions of 10% and 20%, it is apparent that the HRB values of PRMMCs are greater than those of the pure AA6061-T6 matrix (0% VFr). Additionally, it can be observed that the HRB values of the FGMMCs using the power law distribution (labeled FGM-1) are higher than those achieved with the PRMMCs, and the HRB values of FGMMCs obtained through the Gaussian distribution (labeled FGM-2) are higher than those obtained through the power law distribution.Figure 11 The variations of HRB for different Al2O3 particles VFr and variation through the FGMMCs thickness.

Table 4 shows a comparison between the HRB results and their enhancement over the pure AA6061-T6 matrix for all Al2O3 particle volume fractions and variations through the structure thickness. It is shown that the HRB is enhanced for a 10% VFr over the pure matrix by 11.7%, 15.6%, and 17.8% for PRMMCs, FGM-1, and FGM-2, respectively. For 20% VFr, the enhancements are 19.8%, 25.2, and 25.7 for PRMMCs, FGM-1, and FGM-2, respectively. It is noted from the results in Table 4 that the hardness values in the case of the Gaussian distribution are greater than in the case of the power law distribution as a result of the convergence of the particles in the part closest to the surface of the samples, which leads to an increase in the resistance of the samples to penetration due to the increase in the hardest material in the upper part of the samples, this difference in results appears as a result of the difference in the distribution according to the shape of the functions, which is clearly shown in Figs. 7 and 9, as well as the distribution in samples representing the distributions in Figs. 8 and 10.

The results highlight the importance of FGMMCs, because the HRB values obtained for FGMMCs with a Gaussian distribution and a particle VFr of 10% are comparable to the HRB values obtained for PRMMCs with a particle VFr of 20%. The difference between the HRB results is approximately 2%. This means that with a low particle VFr and changing the particle variation across the material thickness, that is, by a Gaussian distribution, HRB can be enhanced significantly.

As particle VFr increases, it becomes evident that the HRB values also increase. Notably, the results obtained for both distributions of FGMMCs with a particle VFr of 20% exhibited minimal differences. This observation indicates that it is challenging to achieve a significant difference in the HRB values through particle distribution for FGMMCs with higher particle volume fractions such as 20%.Table 4 HRB results for various particles VFr and variation across the MMCs thickness.

VFr	HRBPRMMCs	Enhancement % over pure matrix	HRB FGMMCs Power law	Enhancement % over pure matrix	HRB FGMMCs Gaussian	Enhancement % over pure matrix	
10%	68.6	11.7	71.0	15.6	72.3	17.8	
20%	73.6	19.8	76.9	25.2	77.2	25.7	

The load-displacement and surface profiles curves of representative specimens are shown in Fig. 12. The indentation depth decreased significantly as VFr increased. Notably, distinct variations in the results are evident for all representative samples with a VFr of 10%, as illustrated in Fig. 12a,c. Conversely, for a VFr of 20%, the curves for the two representative samples of FGMMCs are nearly identical, as depicted in Fig. 12b,d.Figure 12 Effects of VFr of Al2O3 particles on load-displacement curves and deformed indentation surfaces.

Residual von-Mises stress and plastic strain

Figure 13 illustrates the effect of VFr on the distributions of residual von-Mises stress and effective plastic strain within the FGMMCs structure. As VFr decreases, the levels of residual stress and strain in the representative samples decrease. It is worth noting that the regions of the effective plastic strain are localized beneath the indenter and propagate at an angle of nearly 45∘.

The stress concentration is influenced by the relative distances between the particles and indenter as well as between the particles themselves. Consequently, as VFr increases, FGMMCs becomes more susceptible to stress concentration. This is because the permissible area for particle movement is reduced with higher particle VFr. Moreover, the particles located beneath the indenter experienced the most significant effects, as shown in Fig. 13b,d.

The range of von-Mises stress varies from 1010 MPa to 1565 MPa for a particle VFr of 10% (n =2.65), and for a particle VFr of 20% (n =0.65), with a range of 1282 MPa to 3215 MPa, as shown in Table 5. The effective plastic strain ranges from 1.566 to 2.41 for a VFr of 10% (n =2.65), and 1.507 to 2.524 for VFr of 20% (n =0.65) (see Table 5).Figure 13 Effect of power law distribution on residual von-Mises stress and effective plastic strain during loading.

Figure 14 presents the effects of the Gaussian distribution on the distributions of the residual von-Mises stress and effective plastic strain within the FGMMCs structure for both 10% and 20% volume fractions. Similar to previous observations, localized stress concentration areas beneath the indenter and the propagation angle of the effective plastic strain are evident in these results (see Fig. 14b,d).

By analyzing the distribution of von-Mises stress for the representative samples (Fig. 14a,b), it is observed that the stress levels are higher for 10% VFr. This can be attributed to the chosen Gaussian distribution, which results in a higher concentration of particles in the region directly below the indenter compared with the distribution with a 20% VFr. This can be observed by noting the positions of the peak values in the curves, as depicted in Fig. 9.

The range of von-Mises stress for FGMMCs with a Gaussian distribution is 1401 MPa to 2446 MPa for a particle VFr of 10% and 1196 MPa to 2228 MPa for a particle VFr of 20%, as shown in Table 6. The effective plastic strain ranged from 1.615 to 2.865 for the 10% particle VFr and 1.331 to 2.474 for the 20% particle VFr (see Table 6). These values indicate the extent of plastic deformation experienced by the matrix material in the FGMMCs structures with Gaussian distributions.Figure 14 Effect of Gaussian distribution on residual von-Mises stress and effective plastic strain during loading.

A comparison of the von-Mises stress results between FGMMCs and PRMMCs showed that there is a significant increase in the von-Mises stress values in the case of FGMMCs. For instance, when comparing the average von-Mises stress values in both cases, it is found that with 10% VFr, the von-Mises stress values increased by 64% and 117% for FGMMCs with power law and Gaussian distributions, respectively. Similarly, with 20% VFr, the average value of the von-Mises stress increased by 81% and 57% for both the power law and Gaussian distributions, respectively.

Furthermore, an increase in the average values of effective plastic strain is also observed. When considering a 10% VFr of particles, the effective plastic strain increased by 84% and 90% for the power law and Gaussian distribution, respectively. Similarly, with 20% VFr, the effective plastic strain for power law and Gaussian distribution increased by 35% and 37%, which indicates the tendency to damage for FGMMCs over PRMMCs.

Damage analysis of FGMMCs

This subsection presents an analysis of the existing and potential damage locations within the AA6061-T6/Al2O3 FGMMCs structure for three potential failure modes: (i) matrix damage, (ii) particle fracture, and (iii) interfacial decohesion. The results of the damage parameters associated with each failure mode are analyzed. (i) Matrix damage:

For VFr of 10%, no damage is observed in any sample for both FGMMCs distribution approaches. Fig. 15a,c shows the representative samples for this VFr, which help predict the failure mechanisms in the AA6061-T6 matrix. As shown in Table 5, the range of the void volume fraction (VVF) for a VFr of 10% (n =2.65) is between 0.00121 and 0.00398, with an average value of 0.00216. Considering the results of VVF for the Gaussian FGMMCs distribution, the range of VVF is between 0.00115 and 0.01254, with an average value of 0.00299 for a VFr of 10% (see Table 6).

Based on the results, it is observed that the matrix damage mode occurred when VFr is 20% for both the power law and Gaussian distributions of the FGMMCs. Out of the ten samples studied for each distribution, only one sample in each case exhibited clear damage. These two samples (S2andS7) are shown in Fig. 15b,d. For a VFr of 20% (n =0.65), the range of the VVF is between 0.00229 and 0.06, with an average value of 0.012353 (Table 5). These findings indicated that a higher VFr leads to a higher possibility of void nucleation and growth. For a VFr of 20% Gaussian FGMMCs distribution, the range is between 0.00162 and 0.06, with an average value of 0.00913 (see Table 6). The higher possibility of matrix damage in these cases is attributed to the increase in the number of particles, which reduces the distance between them owing to the limited space available for particle movement.Figure 15 Void volume fraction in AA6061-T6 matrix for power law and Gaussian during loading.

(ii) Particle fracture:

The JH2 damage parameter for the particles is found to be zero for both volume fractions (10% and 20%) in the two types of FGMMCs with the power law and Gaussian distributions. The distribution of the JH2 yield stress parameter for the particles is shown in Fig. 16, which indicates critically loaded particles that are susceptible to damage.

For samples with a power law distribution (represented in Fig. 16a,b), the range of the yield stress parameter is between 3958 MPa and 4971 MPa, with an average value of 4309 MPa for a VFr of 10% (n =2.65), as shown in Table 5. For a VFr of 20% (n =0.65), the range is between 3994 MPa and 5207 MPa, with an average value of 4488 MPa (see Table 5).

For samples with a Gaussian distribution (represented in Fig. 16c,d), the range of the yield stress parameter is between 4104 MPa and 5651 MPa, with an average value of 4530 MPa for a VFr of 10% (see Table 6). For a VFr of 20%, the range is between 4040 MPa and 4993 MPa, with an average value of 4517 MPa. These findings provide insight into the distribution of the yield stress parameter, highlighting the potential susceptibility of critically loaded particles to damage in both the power law and Gaussian FGMMCs.Figure 16 Yield stress distribution in Al2O3 particles for power law and Gaussian during loading.

(iii) Interfacial decohesion:

For samples with a power law distribution and 10% VFr (n =2.65), there are three samples (S2,S5,andS6) in which complete interfacial decohesion is observed. One of these samples (S6) is shown in Fig. 17a. Two samples (S4andS8) exhibited interfacial decohesion when the VFr is 20% (n =0.65), with one of the samples (S4) shown in Fig. 17b.

Regarding the Gaussian distribution, interfacial decohesion is observed in three samples (S3,S6,andS7) with a VFr of 10%, and one sample (S7) with a VFr of 20%. Figure 17c,d shows one sample for each VFr. These findings indicate that interfacial decohesion is a prevalent phenomenon in FGMMCs regardless of the distribution type and VFr.Figure 17 Spatial distribution of matrix/particle decohesion damage parameter (CSDMG) for FGMMCs during Loading.

Comparative analysis of the power law and Gaussian distribution

To compare the two distribution approaches for FGMMCs (the power law and Gaussian distributions), several factors are considered. First, the HRB values are examined, which indicated that the Gaussian distribution approach is the preferred method, particularly for a VFr of 10%. Additionally, to obtain a fair understanding of the preferred distribution approach, the damage incurred by the three different modes in the FGMMCs is studied. Because damage does not occur uniformly across all modes, a normalization procedure is conducted for each damage parameter50. For each VFr, the damage parameter, or the parameter indicating the occurrence of damage, is normalized based on the results obtained from the samples distributed according to both the power law and Gaussian approaches. The following normalization formula of50 is used27 Xnorm=X-XminXmax-Xmin.

Here, Xnorm represents the normalized value, X denotes the set of values used for normalization, Xmin represents the minimum values within X, and Xmax represents the maximum values within X. By employing this normalization process, a fair comparison between the damage parameters obtained from the two distribution approaches is achieved, allowing for a complete evaluation of the preferred distribution approach for FGMMCs. Figure 18 presents the normalized values obtained for each distribution approach and VFr.

Based on the analysis of Fig. 18a,c,e for a VFr of 10%, it can be concluded that FGMMCs with a Gaussian distribution are more susceptible to damage, particularly interfacial decohesion (as depicted in Fig. 18e). This is because the higher susceptibility to damage in the Gaussian distribution of samples with a particle VFr of 10% can be attributed to the fact that 60% of these samples exhibited more than 40% damage. In contrast, for samples following a power law distribution, the percentage of samples that experienced damage is lower (approximately 30%). Therefore, for a VFr of 10%, the Gaussian distribution offers higher hardness, but is also more susceptible to damage, although it does not necessarily ensure the occurrence of such damage.

From the examination of Fig. 18b,d,f, it is evident that the Gaussian distribution exhibits lower susceptibility to damage. Hence, for a VFr of 20%, a Gaussian distribution is a favorable choice to achieve both high hardness and improved resistance against damage. Considering these results, decisions regarding the distribution approach should consider the specific requirements of the application and evaluate the desired hardness against an acceptable level of potential damage.Figure 18 Comparison between the two approaches of FGMMCs distribution.

Table 5 Output values for FGMMCs using power law distribution.

Samples	FGMMCs using Power law distribution	
10%VFr(n=2.65)	20%VFr(n=0.65)	
von−Mises	PEEQ	VVF	Particleyield	CZMdamage	von−Mises	PEEQ	VVF	Particleyield	CZMdamage	
S1	1010	1.838	0.0014	3958	0.25259	1485	1.7888	0.00232	4283	0.48826	
S2	1115	2.182	0.00142	4302	1	1386	1.565	0.06	4140	0.5955	
S3	1401	2.196	0.00378	4323	0.38791	1651	2.158	0.01041	4179	0.48805	
S4	1346	2.41	0.00398	4314	0.32125	2034	1.767	0.00229	4991	1	
S5	1400	1.566	0.00121	4971	1	1282	1.507	0.01117	3994	0.55584	
S6	1454	1.757	0.00199	4438	1	3215	2.089	0.0033	5207	0.57758	
S7	1565	1.756	0.0017	4519	0.56296	1746	2.514	0.0053	4613	0.85727	
S8	1131	1.867	0.00268	3999	0.29048	1475	2.524	0.01737	4146	1	
S9	1350	1.736	0.00207	4171	0.34019	1891	1.567	0.0063	4501	0.98032	
S10	1307	2.139	0.00133	4101	0.35345	1920	1.539	0.00507	4830	0.86073	
Min.	1010	1.566	0.00121	3958	0.25259	1282	1.507	0.00229	3994	0.48805	
Max.	1565	2.41	0.00398	4971	1	3215	2.524	0.06	5207	1	
Avg.	1307.9	1.9447	0.00216	4309.6	0.551	1808.5	1.90188	0.013	4488.4	0.741	

Table 6 Output values for FGMMCs using Gaussian distribution.

Samples	FGMMCs using Gaussian distribution	
10%VFr(n=2.65)	20%VFr(n=0.65)	
von−Mises	PEEQ	VVF	Particleyield	CZMdamage	von−Mises	PEEQ	VVF	Particleyield	CZMdamage	
S1	1645	1.735	0.00124	4455	0.5005	1547	1.775	0.00212	4674	0.62322	
S2	1765	1.96	0.00184	4429	0.49425	1356	2.006	0.004	4455	0.4851	
S3	1722	1.822	0.00415	4610	1	1411	2.277	0.00479	4353	0.37578	
S4	1413	1.917	0.00216	4104	0.64823	1343	2.266	0.00345	4058	0.8111	
S5	1894	1.671	0.01254	4519	0.4018	1464	1.331	0.00162	4223	0.54484	
S6	1401	2.863	0.00134	4355	1	1741	1.867	0.00381	4888	0.5326	
S7	1676	1.615	0.0012	4411	1	1547	1.752	0.06	4590	1	
S8	1489	1.648	0.00115	4260	0.55083	1196	1.76	0.00218	4040	0.38482	
S9	2446	1.927	0.00176	5651	0.69258	1811	1.739	0.00239	4993	0.64814	
S10	1841	2.865	0.00248	4510	0.80866	2228	2.474	0.00694	4896	0.58603	
Min.	1401	1.615	0.00115	4104	0.4018	1196	1.331	0.00162	4040	0.37578	
Max.	2446	2.865	0.01254	5651	1	2228	2.474	0.06	4993	1	
Avg.	1729.2	2.0023	0.003	4530.4	0.71	1564.4	1.9247	0.01	4517	0.6	

Conclusion

A novel finite element model based on a random microstructure is proposed to investigate the deformation and damage behavior of functionally graded metal matrix composites subjected to indentation loading. This study focuses on the influence of the volume fraction and distribution percentage of Al2O3 particles on the thickness of the composite structure. To the best of our knowledge, this study represents the first investigation into the influence of particle volume fraction and distribution on the damage mechanisms of AA6061-T6/Al2O3 functionally graded metal matrix composites while considering various damage modes occurring during the indentation process. The model incorporated the Gurson-Tvergaard-Needleman (GTN) damage model to simulate the elastoplastic behavior and damage of the matrix, the JH2 cracking model to represent particle fracture, and the cohesive zone model (CZM) to account for matrix-particle interfacial decohesion. The model was employed in ABAQUS/Explicit software using a Python script. The significant findings of this study are as follows:The finite element model effectively captures the three primary damage mechanisms in PRMMCs, particularly in FGMMCs, thereby providing valuable insights into optimizing their performance.

The hardness value (HRB) increases as the particle volume fraction and size increase. Additionally, the probability of fracture for the Al2O3 particles increases with an increase in the volume fraction.

Al2O3 particles enhance the resistance of FGMMCs to deformation, although their effectiveness is influenced by the particle volume fraction.

PRMMCs can be represented using a variation in the particle concentration throughout the thickness of the structure, resulting in FGMMCs. FGMMCs exhibit a higher hardness value (HRB) than traditional Metal Matrix Composites (MMCs).

This study demonstrates that with a volume fraction of 10% particles and a Gaussian distribution for particle variation through the structure thickness, the HRB is approximately equivalent to the HRB achieved with a volume fraction of 20% particles in MMCs without any variation in particle distribution through the thickness.

FGMMCs with a Gaussian distribution yield a higher HRB than those with a power-law distribution and exhibit closer alignment with the experimental distribution functions.

Author contributions

The manuscript titled [Insights into particle dispersion and damage mechanisms in functionally graded metal matrix composites with random microstructure-based finite element model] was authored by M.E. Naguib., who wrote the main manuscript text and prepared the figures. The manuscript was further reviewed by S.I. Gad., M. Megahed., and M.A. Agwa. to ensure its quality and accuracy.

Funding

Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

Data availability

The datasets used during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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