Heterogeneous and homogenized models for predicting the indentation response of particle reinforced metal matrix composites



In this study, three-dimensional heterogeneous and homogenized finite element models are used to predict the indentation response of particle reinforced metal matrix composites (PRMMCs). The matrix is assumed to have elasto-plastic behavior whereas the particles (uniform in size and spherical in shape) are assumed to be harder than the matrix, and possess linear elastic behavior. The particles (25 % by volume) are randomly distributed in the metal matrix. Two modeling approaches are used. In the first approach, the PRMMC is fully replaced by an equivalent homogenous material, and its material properties are obtained through homogenization using representative volume element approach under periodic boundary conditions. In second approach, a small cubical volume under the indenter is modeled as heterogeneous material with randomly distributed particles, whereas the remaining domain is assigned equivalent material properties obtained through homogenization. The elastic material properties obtained through simulations are found within Hashin–Shtrikman bounds. A suitable size cubical volume consisting of heterogeneities under the indenter is established by considering different cubical volumes so as to capture the actual indentation response. The simulations are also carried out for different particle sizes to establish a suitable particle size. These simulations show that the second modeling approach yields harder indentation response as compared to first modeling approach due to the local particle concentration under the indenter.


Metal matrix composites RVE Micro-mechanics Indentation response Nonlinear finite element analysis 



Diameter of indenter


Diameter of spherical particles


Distance from particle center to RVE boundary


Center to center distance of particles


Shear modulus of phase i


Strength coefficient


Bulk modulus of phase i


Length of cubic RVE


Strain hardening exponent


Periodic boundary conditions


Representative volume element


Average strain energy density


Displacement field in i-direction


Volume fraction of phase i

\(x_{1} ,x_{2} ,x_{3}\)

Cartesian coordinates


True stress


True plastic strain

\(\sigma_{ij} ,\varepsilon_{ij}\)

Local stress and strain field

\(\left\langle {\sigma_{ij} } \right\rangle ,\left\langle {\varepsilon_{ij} } \right\rangle\)

Macroscopic stress and strain



This work is performed under the doctoral fellowship program of Ministry of Human Resource Development (MHRD), New Delhi, Government of India, and it does not receive any funding from any other source.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringIndian Institute of Technology RoorkeeRoorkeeIndia

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