### Abstract

Two-phase Delaunay and regular triangular networks, with randomness per vertex, provide generic models of granular media consisting of two types of grains - soft and stiff. We investigate effective macroscopic moduli of such networks for the whole range of area fractions of both phases and for a very wide range of stiffnesses of both phases. Results of computer simulations of such networks under periodic boundary conditions are used to determine which of several different self-consistent models can provide the best possible approximation to effective Hooke's law. The main objective is to find the effective moduli of a Delaunay network as if it was a field of inclusions, rather than vertices connected by elastic edges, without conducting the computer-intensive calculations of large windows. First, we report on the dependence of effective Poisson's ratio on p for a single-phase Delaunay network with all the spring constants k assigned according to k=l^{p}. In case of two-phase media, it is found that the Delaunay network is best approximated by a system of ellipses perfectly bonded to a matrix in a symmetric self-consistent formulation, while the regular network is best approximated by a circular inclusion-matrix model. These two models continue to be adequate up to the point of percolation of holes, but the reverse situation of percolation of rigid inclusions is better approximated by the ellipses model in an asymmetric formulation. Additionally, we give results of calculation of Voigt and Reuss bounds of two-dimensional matrix-inclusion composites with springy interfaces.

Original language | English |
---|---|

Pages (from-to) | 57-72 |

Number of pages | 16 |

Journal | Acta Mechanica |

Volume | 110 |

Issue number | 1-4 |

DOIs | |

Publication status | Published - Mar 1995 |

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### ASJC Scopus subject areas

- Mechanics of Materials
- Computational Mechanics

### Cite this

*Acta Mechanica*,

*110*(1-4), 57-72. https://doi.org/10.1007/BF01215416

**Linear elasticity of planar Delaunay networks. III : Self-consistent approximations.** / Ostoja-Starzewski, M.; Alzebdeh, K.; Jasiuk, I.

Research output: Contribution to journal › Article

*Acta Mechanica*, vol. 110, no. 1-4, pp. 57-72. https://doi.org/10.1007/BF01215416

}

TY - JOUR

T1 - Linear elasticity of planar Delaunay networks. III

T2 - Self-consistent approximations

AU - Ostoja-Starzewski, M.

AU - Alzebdeh, K.

AU - Jasiuk, I.

PY - 1995/3

Y1 - 1995/3

N2 - Two-phase Delaunay and regular triangular networks, with randomness per vertex, provide generic models of granular media consisting of two types of grains - soft and stiff. We investigate effective macroscopic moduli of such networks for the whole range of area fractions of both phases and for a very wide range of stiffnesses of both phases. Results of computer simulations of such networks under periodic boundary conditions are used to determine which of several different self-consistent models can provide the best possible approximation to effective Hooke's law. The main objective is to find the effective moduli of a Delaunay network as if it was a field of inclusions, rather than vertices connected by elastic edges, without conducting the computer-intensive calculations of large windows. First, we report on the dependence of effective Poisson's ratio on p for a single-phase Delaunay network with all the spring constants k assigned according to k=lp. In case of two-phase media, it is found that the Delaunay network is best approximated by a system of ellipses perfectly bonded to a matrix in a symmetric self-consistent formulation, while the regular network is best approximated by a circular inclusion-matrix model. These two models continue to be adequate up to the point of percolation of holes, but the reverse situation of percolation of rigid inclusions is better approximated by the ellipses model in an asymmetric formulation. Additionally, we give results of calculation of Voigt and Reuss bounds of two-dimensional matrix-inclusion composites with springy interfaces.

AB - Two-phase Delaunay and regular triangular networks, with randomness per vertex, provide generic models of granular media consisting of two types of grains - soft and stiff. We investigate effective macroscopic moduli of such networks for the whole range of area fractions of both phases and for a very wide range of stiffnesses of both phases. Results of computer simulations of such networks under periodic boundary conditions are used to determine which of several different self-consistent models can provide the best possible approximation to effective Hooke's law. The main objective is to find the effective moduli of a Delaunay network as if it was a field of inclusions, rather than vertices connected by elastic edges, without conducting the computer-intensive calculations of large windows. First, we report on the dependence of effective Poisson's ratio on p for a single-phase Delaunay network with all the spring constants k assigned according to k=lp. In case of two-phase media, it is found that the Delaunay network is best approximated by a system of ellipses perfectly bonded to a matrix in a symmetric self-consistent formulation, while the regular network is best approximated by a circular inclusion-matrix model. These two models continue to be adequate up to the point of percolation of holes, but the reverse situation of percolation of rigid inclusions is better approximated by the ellipses model in an asymmetric formulation. Additionally, we give results of calculation of Voigt and Reuss bounds of two-dimensional matrix-inclusion composites with springy interfaces.

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U2 - 10.1007/BF01215416

DO - 10.1007/BF01215416

M3 - Article

VL - 110

SP - 57

EP - 72

JO - Acta Mechanica

JF - Acta Mechanica

SN - 0001-5970

IS - 1-4

ER -