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The prediction of inelastic processes like plastic deformations and cracks within the microstructure of modern man-made materials by realistic, yet simple and efficient continuum models remains a major task in material modelling. For this purpose, gradient-extended standard dissipative solids represent one of the most promising model classes, which is also formulated and applied in this work to investigate microscopic failure mechanisms in three exemplary three-dimensional composite microstructures. The model combines geometrically nonlinear isotropic elastoplasticity with an isotropic damage model with gradient-extension. For the numerical treatment, a variational constitutive update algorithm based on the exponential map is applied. The model is used to provide insight into the microscopic failure of a brittle woven composite material, a particle-reinforced plastic and a carbon fiber reinforced composite. The influence of different microstructural and material parameters on the overall failure behavior is characterized. Adaptive meshing is used to enable a refined numerical resolution of the cracked regions.