The formation of shear-band/fracture networks from a constitutive instability: Theory and numerical experiment

A mono-shear-band bifurcation analysis is extended to the formation of shear-band network affecting a finite brittle body. This analysis, along with the results of numerical simulations, suggests the following description of the bifurcation process. It starts when the hardening modulus h reaches a critical value h(max) which has proved to be the same as that previously obtained from mono-band analysis. The deformation pattern is penetrative at this stage and presents two conjugated sets of dense, parallel intermittent bands with accelerated and decelerated inelastic deformation. At the next stage the response of the material outside the bands with accelerated deformation becomes elastic ( elastic unloading). The size of the elastic zones rapidly grows and the spacing l between the "active'' localization bands ( incipient fractures) correspondingly increases to a value defined by the constitutive and stress-state parameters. Both the analytical solution and numerical models show that lambda is very sensitive to h: lambda = infinity at h = h(max) and l tends to the bend thickness when h -> h(min) < h(max). If h reduces rapidly below h(min), the deformation "jumps'' into the post-localization state and the material becomes completely crushed. Thus there exists only a narrow range of h values for which the deformation bifurcation, and hence the formation of regular band/fracture network, is possible. The obtained analytical solutions show how the band spacing depends on other constitutive parameters and on the stress-state.