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A new, and extremely fast, computational modeling paradigm is introduced here for specific finite elasticity problems that arise in the context of soft robotics. Whereas continuum mechanics is a very classical area of study, and significant effort has been devoted to the development of intricate constitutive models for finite elasticity, we show that in the kinds of large-strain mechanics problems arising in soft robotics, many of the parameters in constitutive models are irrelevant. For the most part, the isochoric (locally volume-preserving) constraint dominates behavior, and this can be built into closed-form kinematic deformation fields before even considering other aspects of constitutive modeling. We therefore focus on developing and applying primitive deformations that each observe this constraint. It is shown that by composing a wide enough variety of such deformations that the most common behaviors observed in soft robots can be replicated. Case studies include an inflatable rubber chamber, a slender rubber rod, and a rubber block subjected to different boundary conditions. We show that this method is at least 50 times faster than the ABAQUS implementation of the finite element method (FEM). Physical experiments and measurements show that both our method and ABAQUS have approximately 10% error relative to experimentally measured displacements, as well as to each other. Our method provides a real-time alternative to FEM, and captures essential degrees of freedom for use in feedback control systems.