International Journal of Fracture, 69 (1995), 201-228.


Milan Jirásek and Zdenek P. Bazant
Northwestern University
Evanston, Illinois 60208, U.S.A.


This paper deals with determination of macroscopic fracture characteristics of random particle systems, which represents a fundamental but little explored problem of micromechanics of quasibrittle materials. The particle locations are randomly generated and the mechanical properties are characterized by a triangular softening force-displacement diagram for the interparticle links. An efficient algorithm, which is used to repetitively solve large systems, is developed. This algorithm is based on the replacement of stiffness changes by inelastic forces applied as external loads and makes it possible to calculate the exact displacement increments in each step without iterations and using only the elastic stiffness matrix. The size effect method is used to determine the macroscopic fracture energy and the effective process zone size for different combinations of the basic microscopic characteristics - microscopic fracture energy and coefficient of variation of microstrength. Some general trends are revealed and discussed.

Summary and Conclusions

Random particle systems characterized by a particle interaction law with softening characteristics realistically simulate fracture of quasibrittle materials. They exhibit large zones of distributed cracking and the particle structure acts as a localization limiter. The particle system reflects the microstructure of aggregate materials such as concrete. However, because of the nature of their fracture behavior, particle systems can also be used for convenient simulation of fracture of quasibrittle materials that do not consist of well-defined particles, in which case the particle size reflects the spacing of dominant inhomogeneities of the quasibrittle material.

Calculation of macroscopic fracture parameters from given microstructural properties represents a fundamental problem of micromechanics of brittle and quasibrittle materials. As proposed before, the macroscopic fracture energy can be obtained according to the size effect method. The method can also be used to determine the dependence of the mean macro-fracture energy on the microscopic properties, such as the coefficient of variation of the microstrength of the interparticle links, and the microductility of these links.

The size effect method can further be used to determine the effective process zone size of a particle system and its dependence on the aforementioned microscopic properties.

The ratio of macro-fracture energy to the fracture energy of idealized square lattice of particles for straight-line fracture parallel to a lattice line varies approximately from 1 to 3. The effective process zone size varies approximately from 0.5 to 2 times the average interparticle distance.

At constant average microstrength and microductility, the macro-fracture energy is proportional to the average particle spacing. It decreases with increasing coefficient of variation of microstrength (or micro-fracture energy) and increases with increasing microductility. The latter increase however weakens with increasing coefficient of variation of microstrength. It follows that in order to manufacture a quasibrittle material of high fracture energy, the material properties should be as uniform as possible and the size of inhomogeneities as large as possible while at the same time the microductility should be maximized (e.g., by inhibiting sudden formation of large microcracks).

An efficient solution algorithm for the response of the particle system is required. Using a bilinear force-displacement diagram for each link, such an algorithm is obtained by (1) using variable loading step from one change of status of interparticle link to another, and (2) replacing changes of the stiffness matrix due to particle link breaks by inelastic forces, which makes it possible to use the initial elastic stiffness matrix.

The particle simulation should be helpful especially for materials such as sea ice plates in which the dominant inhomogeneity spacing is so large that it would require specimens larger than feasible for laboratory tests.

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EPFL / July 1, 1996 /