The analytical stress-strain relation with heterogeneous parameters is derived for the heterogeneous brittle materials under a uniaxial extensional load,in which the distributions of the elastic modulus and the failure strength are assumed to be statistically independent.This theoretical solution gives an approximate estimate of the equivalent stress-strain relations for 3-D heterogeneous materials.In one-dimensional cases it may provide comparatively accurate results.The theoretical solution can help us to explain how the heterogeneity influences the mechanical behaviors.Further,a numerical approach is developed to model the non-linear behavior of three-dimensional heterogeneous brittle materials.The lattice approach and statistical techniques are applied to simulate the initial heterogeneity of heterogeneous materials.The load increment in each loading stage is adaptively determined so that the better approximation of the failure process can be realized.When the maximum tensile principal strain exceeds the failure strain,the elements are considered to be broken,which can be carried out by replacing its Young's modulus with a very small value.A 3-D heterogeneous brittle material specimen is simulated during a full failure process.The numerical results are in good agreement with the analytical solutions and experimental data.
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