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We present an approach to solving problems in micromechanics that is amenable to massively parallel calculations through the use of graphical processing units and other accelerators. The problems lead to nonlinear differential equations that are typically second order in space and first order in time. This combination of nonlinearity and nonlocality makes such problems difficult to solve in parallel. However, this combination is a result of collapsing nonlocal, but linear and universal physical laws (kinematic compatibility, balance laws), and nonlinear but local constitutive relations. We propose an operator-splitting scheme inspired by this structure. The governing equations are formulated as (incremental) variational problems, the differential constraints like compatibility are introduced using an augmented Lagrangian, and the resulting incremental variational principle is solved by the alternating direction method of multipliers. The resulting algorithm has a natural connection to physical principles, and also enables massively parallel implementation on structured grids. We present this method and use it to study two examples. The first concerns the long wavelength instability of finite elasticity, and allows us to verify the approach against previous numerical simulations. We also use this example to study convergence and parallel performance. The second example concerns microstructure evolution in liquid crystal elastomers and provides new insights into some counter-intuitive properties of these materials. We use this example to validate the model and the approach against experimental observations.