Document Type : Research Article

**Authors**

Department of Mechanical Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

**Abstract**

This study aims to analyze the linear elastic behavior of an aluminum matrix

nanocomposite reinforced with SiC nanoparticles. Once, a representative volume element was considered

for the nanocomposite with a cuboidal inclusion. The elastic moduli of the matrix and the inclusion were

the same, but it contained eigenstrain. The stress and the strain fields were obtained for the inclusion

and the aluminum by Galerkin vector method. The stress and the strain fields in the inclusion problem

were in a good agreement with the results in the literature. A similar representative volume element was

considered for the nanocomposite with a cuboidal inhomogeneity. The elastic moduli of the matrix and

the inhomogeneity were different, but it did not have any eigenstrain. For the calculation of the Eshelby

tensor and the elastic fields for the inhomogeneity problem, the equivalent inclusion method (EIM) was

applied. In the EIM, the uniform and equivalent eigenstrain were considered. The stress and the strain

fields within the inhomogeneity and the matrix were obtained. Results showed that the stress and the

strain in the cuboidal inclusion were less than the cuboidal inhomogeneity due to the difference between

the matrix and the reinforcement materials.

nanocomposite reinforced with SiC nanoparticles. Once, a representative volume element was considered

for the nanocomposite with a cuboidal inclusion. The elastic moduli of the matrix and the inclusion were

the same, but it contained eigenstrain. The stress and the strain fields were obtained for the inclusion

and the aluminum by Galerkin vector method. The stress and the strain fields in the inclusion problem

were in a good agreement with the results in the literature. A similar representative volume element was

considered for the nanocomposite with a cuboidal inhomogeneity. The elastic moduli of the matrix and

the inhomogeneity were different, but it did not have any eigenstrain. For the calculation of the Eshelby

tensor and the elastic fields for the inhomogeneity problem, the equivalent inclusion method (EIM) was

applied. In the EIM, the uniform and equivalent eigenstrain were considered. The stress and the strain

fields within the inhomogeneity and the matrix were obtained. Results showed that the stress and the

strain in the cuboidal inclusion were less than the cuboidal inhomogeneity due to the difference between

the matrix and the reinforcement materials.

**Highlights**

[1] D. Vollath, D.V. Szabó, Synthesis and properties of nanocomposites, *Advanced Engineering Materials*, 6(3) (2004) 117-127.

[2] P.M. Ajayan, L.S. Schadler, P.V. Braun, *Nanocomposite science and technology*, Wiley-VCH, 2003.

[3] Y. Yang, J. Lan, X. Li, Study on bulk aluminum matrix nano-composite fabricated by ultrasonic dispersion of nano-sized SiC particles in molten aluminum alloy, *Materials Science and Engineering: A*, 380(1–2) (2004) 378-383.

[4] C. Borgonovo, D. Apelian, M.M. Makhlouf, Aluminum nanocomposites for elevated temperature applications, *JOM*, 63(2) (2011) 57-64.

[5] R. Casati, Vedani, M., Metal Matrix Composites Reinforced by Nano-Particles—A Review, Metals, 4 (2014) 65-63.

[6] J. Bernholc, D. Brenner, M. Buongiorno Nardelli, V. Meunier, C. Roland, Mechanical and electrical properties of nanotubes, *Annual Review of Materials Research*, 32(1) (2002) 347-375.

[7] I. Ovidko, A. Sheinerman, Elastic fields of inclusions in nanocomposite solids, *Reviews on Advanced Materials Science*, 9 (2005) 17-33.

[8] J. Qu, M. Cherkaoui, *Fundamentals of Micromechanics of Solids*, Wiley, Hoboken, New Jersey, 2006.

[9] T. Mura, *Micromechanics of Defects in Solids*, 2nd ed., Springer, Netherlands, 1987.

[10] L. Tian, Rajapakse, R.K.N.D., Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity, *ASME Journal of Applied Mechanics *74, 568–574, (2007).

[11] J.A. Zimmerman, E.B. WebbIII, J.J. Hoyt, R.E. Jones, P.A. Klein, D.J. Bammann, Calculation of stress in atomistic simulation, Modelling and Simulation in Materials Science and Engineering, 12(4) (2004) S319.

[12] J.-L. Tsai, S.-H. Tzeng, Y.-T. Chiu, Characterizing elastic properties of carbon nanotubes/polyimide nanocomposites using multi-scale simulation, *Composites Part B: Engineering*, 41(1) (2010) 106-115.

[13] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, *The Royal Society*, 1957, pp. 376-396.

[14] Y.P. Chiu, On the Stress Field Due to Initial Strains in a Cuboid Surrounded by an Infinite Elastic Space, *Journal of Applied Mechanics*, 44(4) (1977) 587-590.

[15] L. Wu, S. Du, The Elastic Field Caused by a Circular Cylindrical Inclusion—Part I: Inside the Region x12 + x22 < a2, .. < x3 < . Where the Circular Cylindrical Inclusion is Expressed by x12 + x22 . a2, .h . x3 . h, *Journal of Applied Mechanics, *62(3) (1995) 579-584.

[16] L. Wu, S.Y. Du, The Elastic Field Caused by a Circular Cylindrical Inclusion—Part II: Inside the Region x12 + x22 > a2, .. < x3 < . Where the Circular Cylindrical Inclusion is Expressed by x12 + x22 . a2, .h . x3 . h, *Journal of Applied Mechanics*, 62(3) (1995) 585-589.

[17] W.C. Johnson, Y.Y. Earmme, J.K. Lee, Approximation of the Strain Field Associated With an Inhomogeneous Precipitate—Part 1: Theory, *Journal of Applied Mechanics*, 47(4) (1980) 775-780.

[18] W.C. Johnson, Y.Y. Earmme, J.K. Lee, Approximation of the Strain Field Associated With an Inhomogeneous Precipitate—Part 2: The Cuboidal Inhomogeneity, *Journal of Applied Mechanics*, 47(4) (1980) 781-788.

[19] J.D. Eshelby, The Elastic Field Outside an Ellipsoidal Inclusion, Proceedings of the Royal Society of London. *Series A. Mathematical and Physical Sciences*, 252(1271) (1959) 561-569.

[20] H. Ma, X.-L. Gao, Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory, *Acta mechanica*, 211(1-2) (2010) 115-129.

[21] G.J. Rodin, Eshelby’s inclusion problem for polygons and polyhedra, *Journal of the Mechanics and Physics of Solids *44, 1977-1995, (1996).

[22] H. Nozaki, M. Taya, Elastic fields in a polyhedral inclusion with uniform eigenstrains and related problems, *Journal of Applied Mechanics, Transactions ASME *68, 441-452, (2001).

[23] J. Waldvogel, The Newtonian potential of homogeneous polyhedra, *Zeitschrift für Angewandte Mathematik und Physik *30, 388-398, (1979).

[24] B.N. Kuvshinov, Elastic and piezoelectric fields due to polyhedral inclusions, *International Journal of Solids and Structures *45, 1352-1384., (2008).

[25] Y.P. Chiu, On the stress field due to initial strains in a cuboid surrounded by an infinite elastic space, *Journal of Applied Mechanics, Transactions ASME *44, 587-590, (1977).

[26] J.K. Lee, W.C. Johnson, Calculation of the elastic strain field of a cuboidal precipitate in an anisotropic matrix, *physica status solidi *(a), 46(1) (1978) 267-272.

[27] S. Liu, Q. Wang, Elastic Fields due to Eigenstrains in a Half-Space, *Journal of Applied Mechanics*, 72(6) (2005) 871-878.

[28] G.S. Pearson, D.A. Faux, Analytical solutions for strain in pyramidal quantum dots, *Journal of Applied Physics*, 88(2) (2000) 730-736.

[29] F. Glas, Elastic relaxation of truncated pyramidal quantum dots and quantum wires in a half space: An analytical calculation, *Journal of Applied Physics*, 90(7) (2001) 3232-3241.

[30] A.V. Nenashev, A.V. Dvurechenskii, Strain distribution in quantum dot of arbitrary polyhedral shape: Analytical solution, *Journal of Applied Physics*, 107(6) (2010) 064322.

[31] C.B. Carter, M.G. Norton, *Ceramic Materials: Science and Engineering*, Springer, 2007.

[32] M. Takahiro, K. Yoshitake, Y. Taku, S. Naoki, A. Jun, Superconductivity in carrier-doped silicon carbide, *Science and Technology of Advanced Materials*, 9(4) (2008) 044204.

[33] G.L. Harris, INSPEC, Properties of Silicon Carbide, INSPEC, *Institution of Electrical Engineers*, 1995.

[34] Y.C. Fung, X. Chen, P. Tong, Classical and Computational Solid Mechanics (Second Edition), World Scientific Publishing Company Pte Limited, 2016.

[35] K. Zhou, L.M. Keer, Q.J. Wang, Semi-analytic solution for multiple interacting three-dimensional inhomogeneous inclusions of arbitrary shape in an infinite space, *International Journal for Numerical Methods in Engineering*, 87(7) (2011) 617-638.

[36] X.L. Gao, M.Q. Liu, Strain gradient solution for the Eshelby-type polyhedral inclusion problem, *Journal of the Mechanics and Physics of Solids*, 60(2) (2012) 261- 276.

[37] S.H. Pourhashemi, *Analysis of the linear elastic behaviour of the aluminum matrix nanocomposite reinforced with silicon carbide nanoparticles*, Shahid Bahonar University of Kerman, (2015).

**Keywords**

[1] D. Vollath, D.V. Szabó, Synthesis and properties of nanocomposites, *Advanced Engineering Materials*, 6(3) (2004) 117-127.

[2] P.M. Ajayan, L.S. Schadler, P.V. Braun, *Nanocomposite science and technology*, Wiley-VCH, 2003.

[3] Y. Yang, J. Lan, X. Li, Study on bulk aluminum matrix nano-composite fabricated by ultrasonic dispersion of nano-sized SiC particles in molten aluminum alloy, *Materials Science and Engineering: A*, 380(1–2) (2004) 378-383.

[4] C. Borgonovo, D. Apelian, M.M. Makhlouf, Aluminum nanocomposites for elevated temperature applications, *JOM*, 63(2) (2011) 57-64.

[5] R. Casati, Vedani, M., Metal Matrix Composites Reinforced by Nano-Particles—A Review, Metals, 4 (2014) 65-63.

[6] J. Bernholc, D. Brenner, M. Buongiorno Nardelli, V. Meunier, C. Roland, Mechanical and electrical properties of nanotubes, *Annual Review of Materials Research*, 32(1) (2002) 347-375.

[7] I. Ovidko, A. Sheinerman, Elastic fields of inclusions in nanocomposite solids, *Reviews on Advanced Materials Science*, 9 (2005) 17-33.

[8] J. Qu, M. Cherkaoui, *Fundamentals of Micromechanics of Solids*, Wiley, Hoboken, New Jersey, 2006.

[9] T. Mura, *Micromechanics of Defects in Solids*, 2nd ed., Springer, Netherlands, 1987.

[10] L. Tian, Rajapakse, R.K.N.D., Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity, *ASME Journal of Applied Mechanics *74, 568–574, (2007).

[11] J.A. Zimmerman, E.B. WebbIII, J.J. Hoyt, R.E. Jones, P.A. Klein, D.J. Bammann, Calculation of stress in atomistic simulation, Modelling and Simulation in Materials Science and Engineering, 12(4) (2004) S319.

[12] J.-L. Tsai, S.-H. Tzeng, Y.-T. Chiu, Characterizing elastic properties of carbon nanotubes/polyimide nanocomposites using multi-scale simulation, *Composites Part B: Engineering*, 41(1) (2010) 106-115.

[13] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, *The Royal Society*, 1957, pp. 376-396.

[14] Y.P. Chiu, On the Stress Field Due to Initial Strains in a Cuboid Surrounded by an Infinite Elastic Space, *Journal of Applied Mechanics*, 44(4) (1977) 587-590.

[15] L. Wu, S. Du, The Elastic Field Caused by a Circular Cylindrical Inclusion—Part I: Inside the Region x12 + x22 < a2, .. < x3 < . Where the Circular Cylindrical Inclusion is Expressed by x12 + x22 . a2, .h . x3 . h, *Journal of Applied Mechanics, *62(3) (1995) 579-584.

[16] L. Wu, S.Y. Du, The Elastic Field Caused by a Circular Cylindrical Inclusion—Part II: Inside the Region x12 + x22 > a2, .. < x3 < . Where the Circular Cylindrical Inclusion is Expressed by x12 + x22 . a2, .h . x3 . h, *Journal of Applied Mechanics*, 62(3) (1995) 585-589.

[17] W.C. Johnson, Y.Y. Earmme, J.K. Lee, Approximation of the Strain Field Associated With an Inhomogeneous Precipitate—Part 1: Theory, *Journal of Applied Mechanics*, 47(4) (1980) 775-780.

[18] W.C. Johnson, Y.Y. Earmme, J.K. Lee, Approximation of the Strain Field Associated With an Inhomogeneous Precipitate—Part 2: The Cuboidal Inhomogeneity, *Journal of Applied Mechanics*, 47(4) (1980) 781-788.

[19] J.D. Eshelby, The Elastic Field Outside an Ellipsoidal Inclusion, Proceedings of the Royal Society of London. *Series A. Mathematical and Physical Sciences*, 252(1271) (1959) 561-569.

[20] H. Ma, X.-L. Gao, Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory, *Acta mechanica*, 211(1-2) (2010) 115-129.

[21] G.J. Rodin, Eshelby’s inclusion problem for polygons and polyhedra, *Journal of the Mechanics and Physics of Solids *44, 1977-1995, (1996).

[22] H. Nozaki, M. Taya, Elastic fields in a polyhedral inclusion with uniform eigenstrains and related problems, *Journal of Applied Mechanics, Transactions ASME *68, 441-452, (2001).

[23] J. Waldvogel, The Newtonian potential of homogeneous polyhedra, *Zeitschrift für Angewandte Mathematik und Physik *30, 388-398, (1979).

[24] B.N. Kuvshinov, Elastic and piezoelectric fields due to polyhedral inclusions, *International Journal of Solids and Structures *45, 1352-1384., (2008).

[25] Y.P. Chiu, On the stress field due to initial strains in a cuboid surrounded by an infinite elastic space, *Journal of Applied Mechanics, Transactions ASME *44, 587-590, (1977).

[26] J.K. Lee, W.C. Johnson, Calculation of the elastic strain field of a cuboidal precipitate in an anisotropic matrix, *physica status solidi *(a), 46(1) (1978) 267-272.

[27] S. Liu, Q. Wang, Elastic Fields due to Eigenstrains in a Half-Space, *Journal of Applied Mechanics*, 72(6) (2005) 871-878.

[28] G.S. Pearson, D.A. Faux, Analytical solutions for strain in pyramidal quantum dots, *Journal of Applied Physics*, 88(2) (2000) 730-736.

[29] F. Glas, Elastic relaxation of truncated pyramidal quantum dots and quantum wires in a half space: An analytical calculation, *Journal of Applied Physics*, 90(7) (2001) 3232-3241.

[30] A.V. Nenashev, A.V. Dvurechenskii, Strain distribution in quantum dot of arbitrary polyhedral shape: Analytical solution, *Journal of Applied Physics*, 107(6) (2010) 064322.

[31] C.B. Carter, M.G. Norton, *Ceramic Materials: Science and Engineering*, Springer, 2007.

[32] M. Takahiro, K. Yoshitake, Y. Taku, S. Naoki, A. Jun, Superconductivity in carrier-doped silicon carbide, *Science and Technology of Advanced Materials*, 9(4) (2008) 044204.

[33] G.L. Harris, INSPEC, Properties of Silicon Carbide, INSPEC, *Institution of Electrical Engineers*, 1995.

[34] Y.C. Fung, X. Chen, P. Tong, Classical and Computational Solid Mechanics (Second Edition), World Scientific Publishing Company Pte Limited, 2016.

[35] K. Zhou, L.M. Keer, Q.J. Wang, Semi-analytic solution for multiple interacting three-dimensional inhomogeneous inclusions of arbitrary shape in an infinite space, *International Journal for Numerical Methods in Engineering*, 87(7) (2011) 617-638.

[36] X.L. Gao, M.Q. Liu, Strain gradient solution for the Eshelby-type polyhedral inclusion problem, *Journal of the Mechanics and Physics of Solids*, 60(2) (2012) 261- 276.

[37] S.H. Pourhashemi, *Analysis of the linear elastic behaviour of the aluminum matrix nanocomposite reinforced with silicon carbide nanoparticles*, Shahid Bahonar University of Kerman, (2015).

Autumn 2017

Pages 149-158