We consider the system of viscoelasticity with second-order gradients and nonconvex energy in several space dimensions. After establishing global existence of weak solutions, we study the asymptotic limits as the viscosity tends to zero or as the coefficient of the second-order gradient vanishes. In the latter problem and for the two-dimensional case, we also prove a stability result for the solutions in the regularity class and establish a rate of convergence.The study is based on an a priori estimate capturing the dissipative structure of the problem.