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Multiscale computational techniques play a major role in solving problems related to viscoelastic composite materials due to the complexities inherent to these materials. In the present work, a numerical procedure for multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks is proposed in which the (global scale) homogenized viscoelastic incremental constitutive equations have the same form as the local scale viscoelastic incremental constitutive equations, but the homogenized tangent constitutive tensor and the homogenized incremental history dependent stress tensor depend on the amount of damage accumulated at the local scale. Furthermore, the developed technique allows the computation of the full anisotropic incremental constitutive tensor of solids containing evolving cracks (and other kinds of heterogeneities) by solving the micromechanical problem only once. The procedure is basically developed by relating the local scale displacement field to the global scale strain tensor and using first order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify and demonstrate the model capabilities. A two-scale analytical solution for a functionally graded elastic material subject to dynamic loads is also derived in order to verify the multiscale computational model and additional code verification is also performed. Even though the presented model has been implemented in an explicit time integration algorithm, it can be especially useful when the global scale problem is solved by an implicit finite element algorithm, which requires the knowledge of the global tangent constitutive tensor in order to assemble the corresponding stiffness matrix.