<p>Conventional peridynamic correspondence material models suffer from zero-energy mode instability. A stabilization strategy is proposed for correspondence models of hyperelastic materials in this paper. For isotropic strain energy density functions described by the principal invariants of the right Cauchy-Green deformation tensor, the first invariant and other invariants are computed using two distinct nonlocal approximations of the tensor. One of the two nonlocal approximations is obtained directly, the other is derived from the deformation gradient tensor. To avoid nonzero internal forces in the undeformed configuration caused by representation inconsistency of the tensor, an alternative approximation of the deformation gradient tensor is introduced. The force vector state is derived from the Fréchet derivative of the strain energy density function. Neither artificial parameters nor bond-associated tensors are required. The present strategy is applicable to a wide range of hyperelastic material models. To validate the effectiveness of the proposed approach, an eigenvalue analysis of the global tangent stiffness matrix of the system is conducted. Additionally, numerical examples are presented to examine the convergence behavior of the approach and its application to fracture of hyperelastic materials.</p>