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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v7.i4.70
pages 329-350

Modeling the Particle Size and Interfacial Hardening Effects in Metal Matrix Composites with Dispersed Particles at Decreasing Microstructural Length Scales

Rashid K. Abu Al-Rub
Department of Civil Engineering, Catholic University of America, Washington, DC 20064, USA


The focus of this paper is on incorporating the particle size effect and the effect of particlematrix interfacial properties on the average onset of plasticity and strain hardening rates of metal matrix composites reinforced with hard, stiff, or soft particles. In order to achieve this objective, a higher-order gradient plasticity theory that explicitly includes the effect of interfacial energy at particle-matrix interfaces is formulated within the frameworks of virtual power and thermodynamic laws. The derived higher-order gradient plasticity theory also takes into account large variations in plastic strain tensor and effective plastic strain; namely, the gradient of plastic strain, the gradient of the effective plastic strain, and the accumulation of plastic strain gradients. Moreover, unlike the majority of the existing gradient plasticity theories in the literature, it is shown that the matrix nonlocal yield condition as well as a yield-like condition for the particle-matrix interface can be directly derived from the principle of virtual power without any further constitutive assumptions. Also, in this work the interfaces dissipate energy similar to the bulk material during plastic deformation. The interfacial yield condition takes into consideration the particle type (soft, stiff, hard) through the incorporation of the particle-matrix interfacial yield strength and interfacial hardening in case of dislocation transmission across the interface (i.e. shearing of particles). The proposed higher-order gradient plasticity theory is shown to be qualitatively successful in predicting the increase in the average yield strength, strain hardening rates, and flow stress as the particle size decreases and the particle interfacial strength increases.


  1. Argon, A. S., Im, J., and Safoglu, R., Cavity formation from inclusions in ductile fracture. DOI: 10.1007/BF02672306

  2. Llorca, J., Needleman, A., and Suresh, S., An analysis of the effects of matrix void growth on deformation and ductility in metal-ceramic composites. DOI: 10.1016/0956-7151(91)90014-R

  3. Zhao, D., Tuler, F. R., and Lloyd, D. J., Fracture at elevated temperatures in a particle reinforced composite. DOI: 10.1016/0956-7151(94)90333-6

  4. Dierickx, P., Verdu, C., Reynaud, A., and Fougeres, R., A study of physico-chemical mechanisms responsible for damage of heat treated and as-cast ferritic spheroidal graphite cast irons.

  5. Xue, Z., Huang, Y., and Li, M., Particle size effect in metallic materials: A study by the theory of mechanism-based strain gradient plasticity. DOI: 10.1016/S1359-6454(01)00325-1

  6. Gall, K., Yang, N., Horstemeyer, M. F., McDowell, D. L., and Fan, J., Debonding and fracture of Si particles during the fatigue of a cast Al-Si alloy. DOI: 10.1007/s11661-999-0218-2

  7. Horstemeyer, M. F., Ramaswamy, S., and Negrete, M., Using a micromechanical finite element parametric study to motivate a phenomenological macroscale model for void/crack nucleation in aluminum with a hard second phase. DOI: 10.1016/S0167-6636(02)00165-5

  8. Hao, S., Liu, W. K., Moran, B., Vernerey, F., and Olson, G. B., Multi-scale constitutive model and computational framework for the design of ultra-high strength, high toughness steels. DOI: 10.1016/j.cma.2003.12.026

  9. Bandstra, J. P., Koss, D. A., Geltmacher, A., Matic, P., and Everett, R. K., Modeling void coalescence during ductile fracture of a steel. DOI: 10.1016/j.msea.2003.08.018

  10. Babout, L., Maire, E., and Fougeres, R., Damage initiation in model metallic materials: X-ray tomography and modeling. DOI: 10.1016/j.actamat.2004.02.001

  11. Lee, J. H., Shishidou, T., Zhao, Y. J., Freeman, A. J., and Olson, G. B., Strong interface adhesion in Fe/TiC. DOI: 10.1080/14786430500199278

  12. Tan, H., Huang, Y., Liu, C., and Geubelle, P. H., The effect of nonlinear interface debonding on the constitutive model of composite materials. DOI: 10.1615/IntJMultCompEng.v4.i1.100

  13. McVeigha, C., Vernereya, F., Liu, W. K., and Brinson, L. C., Multiresolution analysis for material design. DOI: 10.1016/j.cma.2005.07.027

  14. McVeigha, C., Vernereya, F., Liu, W. K., Morana, B., and Olson, G. B., An interactive micro-void shear localization mechanism in high strength steels. DOI: 10.1016/j.jmps.2006.08.002

  15. Lloyd, D. J., Particle reinforced aluminum and magnesium matrix composites. DOI: 10.1179/095066094790150982

  16. Rhee, M., Hirth, J. P., and Zbib, H. M., A superdislocation model for the strengthening of metal matrix composites and the initiation and propagation of shear bands. DOI: 10.1016/0956-7151(94)90206-2

  17. Zhu, H. T. and Zbib, H. M., Flow strength and size effect of an Al-Si-Mg composite model system under multiaxial loading. DOI: 10.1016/0956-716X(95)00033-R

  18. Nan, C.-W. and Clarke, D. R., The influence of particle size and particle fracture on the elastic/plastic deformation of metal matrix composites. DOI: 10.1016/1359-6454(96)00008-0

  19. Kiser, M. T., Zok, F. W., and Wilkinson, D. S., Plastic flow and fracture of a particulate metal matrix composite. DOI: 10.1016/1359-6454(96)00028-6

  20. Kouzeli, M. and Mortensen, A., Size dependent strengthening in particle reinforced aluminium. DOI: 10.1016/S1359-6454(01)00327-5

  21. Ashby, M. F., The deformation of plastically non-homogenous alloys. DOI: 10.1080/14786437008238426

  22. Chen, S. and Wang, T. C., Size effects in the particle-reinforced metal-matrix composites. DOI: 10.1007/BF01182158

  23. Fleck, N. A. and Willis, J. R., Bounds and estimates for the effect of strain gradients upon the effective plastic properties of an isotropic two phase composite. DOI: 10.1016/j.jmps.2004.02.001

  24. Aifantis, K. E. and Willis, J. R., The role of interfaces in enhancing the yield strength of composites and polycrystals. DOI: 10.1016/j.jmps.2004.12.003

  25. Stolken, J. S. and Evans, A. G., A microbend test method for measuring the plasticity lengthscale. DOI: 10.1016/S1359-6454(98)00153-0

  26. Huang, H. and Spaepen, F., Tensile testing of free-standing Cu, Ag and Al thin films and Ag/Cu multilayers. DOI: 10.1016/S1359-6454(00)00128-2

  27. Shrotriya, P., Allameh, S. M., Lou, J., Buchheit, T., and Soboyejo, W. O., On the measurement of the plasticity length scale parameter in LIGA nickel foils. DOI: 10.1016/S0167-6636(02)00273-9

  28. Haque, M. A. and Saif, M. T. A., Strain gradient effect in nanoscale thin films. DOI: 10.1016/S1359-6454(03)00116-2

  29. Espinosa, H. D., Prorok, B. C., and Peng, B., Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension. DOI: 10.1016/j.jmps.2003.07.001

  30. Simons, G., Weippert, C., Dual, J., and Villain, J., Size effects in tensile testing of thin cold rolled and annealed Cu foils. DOI: 10.1016/j.msea.2005.10.060

  31. Stelmashenko, N. A., Walls, M. G., Brown, L. M., and Milman, Y. V., Microindentation on W and Mo oriented single crystals: An STM study. DOI: 10.1016/0956-7151(93)90100-7

  32. DeGuzman, M. S., Neubauer, G., Flinn, P., and Nix, W. D., The role of indentation depth on the measured hardness of materials. DOI: 10.1557/PROC-308-613

  33. Ma, Q. and Clarke, D. R., Size dependent hardness in silver single crystals. DOI: 10.1557/JMR.1995.0853

  34. Poole, W. J., Ashby, M. F., and Fleck, N. A., Micro-hardness of annealed and workhardened copper polycrystals. DOI: 10.1016/1359-6462(95)00524-2

  35. McElhaney, K.W., Valssak, J. J., and Nix,W. D., Determination of indenter tip geometry and indentation contact area for depth sensing indentation experiments. DOI: 10.1557/JMR.1998.0185

  36. Lim, Y. Y. and Chaudhri, Y. Y, The effect of the indenter load on the nanohardness of ductile metals: An experimental study of polycrystalline work-hardened and annealed oxygenfree copper. DOI: 10.1080/01418619908212037

  37. Elmustafa, A. A. and Stone, D. S., Indentation size effect in polycrystalline F.C.C. metals. DOI: 10.1016/S1359-6454(02)00175-1

  38. Swadener, J. G., George, E. P., and Pharr, G. M., The correlation of the indentation size effect measured with indenters of various shapes. DOI: 10.1016/S0022-5096(01)00103-X

  39. Aifantis, E. C., On the microstructural origin of certain inelastic models. DOI: 10.1115/1.3225725

  40. Aifantis, E. C., The physics of plastic deformation. DOI: 10.1016/0749-6419(87)90021-0

  41. Arsenlis, A. and Parks, D. M., Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. DOI: 10.1016/S1359-6454(99)00020-8

  42. Voyiadjis, G. Z. and Abu Al-Rub, R. K., Nonlocal Continuum Damage and Plasticity: Theory and Computation.

  43. Lasry, D. and Belytschko, T., Localization limiters in transient problems. DOI: 10.1016/0020-7683(88)90059-5

  44. Zbib, H. M. and Aifantis, E. C., On the localization and postlocalization behavior of plastic deformation, I.

  45. Zbib, H. M. and Aifantis, E. C., On the localization and postlocalization behavior of plastic deformation, II.

  46. Zbib, H. M. and Aifantis, E. C., On the localization and postlocalization behavior of plastic deformation, III.

  47. de Borst, R. and Mühlhaus, H.-B., Gradientdependent plasticity formulation and algorithmic aspects. DOI: 10.1002/nme.1620350307

  48. de Borst, R., Sluys, L. J., Mühlhaus, H.-B., and Pamin, J., Fundamental issues in finite element analysis of localization of deformation.

  49. Pamin, J., Gradeitn-dependent plasticity in numerical simulation of localization phenomena.

  50. de Borst, R. and Pamin, J., Some novel developments in finite element procedures for gradient-dependent plasticity. DOI: 10.1002/(SICI)1097-0207(19960730)39:14<2477::AID-NME962>3.0.CO;2-E

  51. Bammann, D. J., Mosher, D., Hughes, D. A., Moody, N. R., and Dawson, P. R., Using spatial gradients to model localization phenomena.

  52. Abu Al-Rub, R. K. and Voyiadjis, G. Z., A direct finite element implementation of the gradient plasticity theory. DOI: 10.1002/nme.1303

  53. Eringen, A. C. and Edelen, D. G. B., On nonlocal elasticity. DOI: 10.1016/0020-7225(72)90039-0

  54. Pijaudier-Cabot, T. G. P. and Bazant, Z. P, Nonlocal damage theory. DOI: 10.1061/(ASCE)0733-9399(1987)113:10(1512)

  55. Bazant, Z. P. and Pijaudier-Cobot, G., Nonlocal continuum damage, localization instability and convergence. DOI: 10.1115/1.3173674

  56. Fleck, N. A., Muller, G. M., Ashby, M. F., and Hutchinson, J. W., Strain gradient plasticity: Theory and experiment. DOI: 10.1016/0956-7151(94)90502-9

  57. Fleck, N. A. and Hutchinson, J. W., A phenomenological theory for strain gradient effects in plasticity. DOI: 10.1016/0022-5096(93)90072-N

  58. Fleck, N. A. and Hutchinson, J. W, Strain gradient plasticity.

  59. Fleck, N. A. and Hutchinson, J.W., A reformulation of strain gradient plasticity. DOI: 10.1016/s0022-5096(01)00049-7

  60. Nix,W. D. and Gao, H., Indentation size effects in crystalline materials: A law for strain gradient plasticity. DOI: 10.1016/S0022-5096(97)00086-0

  61. Gao, H., Huang, Y., Nix, W. D., and Hutchinson, J. W., Mechanism-based strain gradient plasticity—I. Theory. DOI: 10.1016/S0022-5096(98)00103-3

  62. Gao, H. and Huang, Y., Taylor-based nonlocal theory of plasticity. DOI: 10.1016/S0020-7683(00)00173-6

  63. Hwang, K. C., Jiang, H., Huang, Y., Gao, H., and Hu, N., A finite deformation theory of strain gradient plasticity. DOI: 10.1016/S0022-5096(01)00020-5

  64. Gurtin, M. E., On the plasticity of single crystals: Free energy, microforces, plastic-strain gradients. DOI: 10.1016/S0022-5096(99)00059-9

  65. Gurtin, M. E., A gradient theory of singlecrystal viscoplasticity that accounts for geometrically necessary dislocations. DOI: 10.1016/S0022-5096(01)00104-1

  66. Gurtin, M. E., On a framework for smalldeformation viscoplasticity: Free energy, microforces, strain gradients. DOI: 10.1016/S0749-6419(01)00018-3

  67. Gurtin, M. E., A gradient theory of smalldeformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. DOI: 10.1016/j.jmps.2004.04.010

  68. Gurtin, M. E. and Anand, L., A theory of strain-gradient plasticity for isotropic, plasticity irrotational materials. Part I: Small deformations. DOI: 10.1016/j.jmps.2004.12.008

  69. Gudmundson, P., A unified treatment of strain gradient plasticity. DOI: 10.1016/j.jmps.2003.11.002

  70. Abu Al-Rub, R. K., Voyiadjis, G. Z., and Bammann, D. J., A thermodynamic based higherorder gradient theory for size dependent plasticity. DOI: 10.1016/j.ijsolstr.2006.08.034

  71. Voyiadjis, G. Z. and Abu Al-Rub, R. K., Nonlocal gradient-dependent thermodynamics for modeling scale dependent plasticity. DOI: 10.1615/IntJMultCompEng.v5.i3-4.110

  72. Eringen, A. C., Theory of micropolar elasticity.

  73. Mindlin, R. D., Micro-structure in linear elasticity. DOI: 10.1007/BF00248490

  74. Cosserat, E. and Cosserat, F., Theorie des corp deformables.

  75. Acharya, A. and Bassani, J. L., Lattice incompatibility and a gradient theory of crystal plasticity. DOI: 10.1016/S0022-5096(99)00075-7

  76. Acharya, A. and Beaudoin, A. J., Grain-size effect in viscoplastic polycrystals at moderate strains. DOI: 10.1016/S0022-5096(00)00013-2

  77. Bassani, J. L., Incompatibility and a simple gradient theory of plasticity. DOI: 10.1016/S0022-5096(01)00037-0

  78. Bammann, D. J., A model of crystal plasticity containing a natural length scale. DOI: 10.1016/S0921-5093(00)01614-2

  79. Clayton, J. D., McDowell, D. L., and Bammann, D. J., Modeling dislocations and disclinations with finite micropolar elastoplasticity. DOI: 10.1016/j.ijplas.2004.12.001

  80. Abu Al-Rub, R. K. and Voyiadjis, G. Z., A physically based gradient plasticity theory. DOI: 10.1016/j.ijplas.2005.04.010

  81. Abu Al-Rub, R. K. and Voyiadjis, G. Z., A finite strain plastic-damage model for high velocity impacts using combined viscosity and gradient localization limiters, Part I: Theoretical formulation. DOI: 10.1177/1056789506058046

  82. M&uuml;hlhaus, H. B. and Aifantis, E. C., A variational principle for gradient principle. DOI: 10.1016/0020-7683(91)90004-Y

  83. Valanis, K. C., A gradient theory of internal variables. DOI: 10.1007/BF01171416

  84. Fremond,M. and Nedjar, B., Damage, gradient of damage and principle of virtual power. DOI: 10.1016/0020-7683(95)00074-7

  85. Polizzotto, C. and Borino, G., A thermodynamics- based formulation of gradientdependent plasticity. DOI: 10.1016/S0997-7538(98)80003-X

  86. Polizzotto, C., Unified thermodynamic framework for nonlocal/gradient continuum theories. DOI: 10.1016/S0997-7538(03)00075-5

  87. Mentzel, A. and Steinmann, P., On the continuum formulation of higher order gradient plasticity for single and polycrystals.

  88. Liebe, T., Menzel, A., and Steinmann, P., Theory and numerics of geometrically non-linear gradient plasticity. DOI: 10.1016/S0020-7225(03)00030-2

  89. Shizawa, K. and Zbib, H. M., A thermodynamical theory of gradient elastoplasticity with dislocation density tensor. I: Fundamentals. DOI: 10.1016/S0749-6419(99)00018-2

  90. Svendsen, B., Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations. DOI: 10.1016/S0022-5096(01)00124-7

  91. Voyiadjis, G. Z., Abu Al-Rub, R. K., and Palazotto, A. N., Non-local coupling of viscoplasticity and anisotropic viscodamage for impact problems using the gradient theory.

  92. Voyiadjis, G. Z., Abu Al-Rub, R. K., and Palazotto, A. N., Thermodynamic formulations for non-local coupling of viscoplasticity and anisotropic viscodamage for dynamic localization problems using gradient theory.

  93. Begley, M. R. and Hutchinson, J. W, The mechanics of size-dependent indentation. DOI: 10.1016/S0022-5096(98)00018-0

  94. Shu, J. Y. and Fleck, N. A., The prediction of a size effect in micro-indentation. DOI: 10.1016/S0020-7683(97)00112-1

  95. Yuan, H. and Chen, J., Identification of the intrinsic material length in gradient plasticity theory from micro-indentation tests. DOI: 10.1016/S0020-7683(01)00121-4

  96. Abu Al-Rub, R. K. and Voyiadjis, G. Z., Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments. DOI: 10.1016/j.ijplas.2003.10.007

  97. Abu Al-Rub, R. K. and Voyiadjis, G. Z., Determination of the material intrinsic length scale of gradient plasticity theory. DOI: 10.1007/978-94-017-0483-0_21

  98. Abu Al-Rub, R. K., Prediction of micro- and nano indentation size effect from conical or pyramidal indentation. DOI: 10.1016/j.mechmat.2007.02.001

  99. Voyiadjis, G. Z. and Abu Al-Rub, R. K., Gradient plasticity theory with a variable length scale parameter. DOI: 10.1016/j.ijsolstr.2004.12.010

  100. Aifantis, K. E. and Willis, J. R., Scale effects induced by strain-gradient plasticity and interfacial resistance in periodic and randomly heterogeneous media. DOI: 10.1016/j.mechmat.2005.06.010

  101. Fredriksson, P. and Gudmundson, P., Sizedependent yield strength of thin films. DOI: 10.1016/j.ijplas.2004.09.005

  102. Abu Al-Rub, R. K., Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals. DOI: 10.1016/j.ijplas.2007.09.005

  103. Nye, J. F., Some geometrical relations in dislocated crystals. DOI: 10.1016/0001-6160(53)90054-6

  104. Voyiadjis, G. Z., Deliktas, B., and Aifantis, E. C., Multiscale analysis of multiple damage mechanisms coupled with inelastic behavior of composite materials. DOI: 10.1061/(ASCE)0733-9399(2001)127:7(636)

  105. Edelen, D. G. B. and Laws, N., On the thermodynamics of systems with nonlocality. DOI: 10.1007/BF00251543

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