Scaling of the failure stress of homophase and heterophase three-dimensional spring networks

Abstract

This paper concentrates on the scaling of the failure stress of a three-dimensional spring network as a function of its volume. In particular, the influences of the geometry and the local structure are examined. Both homophase disordered three-dimensional structures and composite systems are studied, containing a more or less ordered slab. The structures are generated by starting with a node distribution. A molecular-dynamics-based algorithm uses void volume spheres, which all have Lennard-Jones interacting outer surfaces. The generated distributions of nodes form the basis of a procedure to interconnect the nodes with springs. In the calculation of the failure stress the total elastic energy is described by two-body central force, three-body bond bending, and four-body torsion contributions. The areas under uniaxial compression are varied in the range of 0.64--5.76 \ensuremath{\mu}${\mathrm{m}}^{2}$, and the height h ranges between 0.80 and 6.4 \ensuremath{\mu}m. It is found that the failure stress at constant base area could be described by ${\ensuremath{\sigma}}_{\mathrm{fail}}\ensuremath{\propto}[\mathrm{log}(h/\ensuremath{\xi}){]}^{\ensuremath{-}1/\ensuremath{\mu}},$ where \ensuremath{\xi} represents the correlation length within the sample the (logarithm is to the base e). The values of \ensuremath{\mu} are effective values. Only within the same kind of failure mechanism and microstructure does the exponent \ensuremath{\mu} become more or less universal. Actually, the modulus \ensuremath{\mu} appears to depend on the system size, but in all cases thin samples are stronger than thick samples under uniaxial compression, and the failure stress increases with increasing coordination number. The failure stress of heterophase materials differs considerably in our calculations from that of homophase materials. The composite materials exhibit an increase in strength by a factor of 4, in comparison to the disordered structures of the same size. The actual failure stress of the composite material depends critically on the layering effect of the disordered region near the ordered phase.