  We introduce and analyze an explicit formulation of fractional powers of the
Lam\'e-Navier system of partial differential operators. We show that this
fractional Lam\'e-Navier operator is a nonlocal integro-differential operator
that appears in several widely-used continuum mechanics models. We demonstrate
that the fractional Lam\'e-Navier operator can be obtained using compositions
of nonlocal gradient operators. Additionally, the effective form of the
fractional Lam\'e-Navier operator is the same as the operator obtained as a
particular choice of parameters in state-based peridynamics. We further show
that the Dirichlet-to-Neumann map associated to the local classical
Lam\'e-Navier system posed in a half-space coincides with the square root power
of the Lam\'e-Navier operator for a particular choice of elastic coefficients.
  We establish basic analysis results for the fractional Lam\'e-Navier
operator, including the calculus of positive and negative powers, and explore
its interaction with the H\"older and Bessel classes of functions. We also
derive the fractional Lam\'e-Navier as the Dirichlet-to-Neumann map of a local
degenerate elliptic system of equations in the upper half-space. We use an
explicit formula for a Poisson kernel for the extension system to establish the
well-posedness in weighted Sobolev spaces. As an application, we derive the
equivalence of two fractional seminorms using a purely local argument in the
extension system, and then use this equivalence to obtain well-posedness for a
variational Dirichlet problem associated to the fractional Lam\'e-Navier
operator.
