In this paper, we present what appears to be the first simulation model for grasping of flexible bodies based on the three-dimensional elastic constitutive relations and Newton's Second Law for solids known as the Navier-Cauchy equations. We give an overview of the most important equations for strain, stress, and elasticity tensors based on which we outline the format of the Navier-Cauchy equations of motion in the general anisotropic case. We then specifically study the equations for homogeneous isotropic bodies. We formulate a numerical scheme based on finite differences for solving the equations. Finally, we present preliminary experimental work and outline future directions.