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The overarching goal of this work is to develop an accurate, robust, andstable methodology for finite deformation modeling using strong-formperidynamics (PD) and the correspondence modeling framework. We adopt recentlydeveloped methods that make use of higher-order corrections to improve thecomputation of integrals in the correspondence formulation. A unified approachis presented that incorporates the reproducing kernel (RK) and generalizedmoving least square (GMLS) approximations in PD to obtain higher-ordergradients. We show, however, that the improved quadrature rule does not sufficeto handle correspondence-modeling instability issues. In Part I of this paper,a bond-associative, higher-order core formulation is developed that naturallyprovides stability. Numerical examples are provided to study the convergence ofRK-PD, GMLS-PD, and their bond-associated versions to a local counterpart, asthe degree of non-locality approaches zero. It is shown that thebond-associative approach improves the robustness of RK-PD and GMLS-PDformulations, which is essential for practical applications. The higher-order,bond-associated model can obtain second-order convergence for smooth problemsand first-order convergence for problems involving field discontinuities, suchas curvilinear free surfaces. In Part II of this paper we use our unified PDframework to: (a) study wave propagation phenomena, which have provenproblematic for the state-based correspondence PD framework; (b) propose a newmethodology to enforce natural boundary conditions in correspondence PDformulations, which should be particularly appealing to coupled problems. Ourresults indicate that bond-associative formulations accompanied by higher-ordergradient correction provide the key ingredients to obtain the necessaryaccuracy, stability, and robustness characteristics needed forengineering-scale simulations.

A unified, stable and accurate meshfree framework for peridynamic correspondence modeling. Part I: core methods