A weak form of the peridynamic (PD) method derived from the classical Galerkin framework by substituting the traditional derivatives into the PD differential operators is proposed. The attractive features of the proposed weak form of PD method include the following: (1) a higher-order approximation than the non-ordinary state-based peridynamic (NOSB-PD) in the strain construction; (2) the NOSB-PD is demonstrated as a special case of the weak form of the PD method; (3) as an extension of the NOSB-PD, the zero-energy mode oscillations in the weak form of the PD can be significantly reduced by introducing higher-order PD derivatives. In addition, a series of numerical tests are conducted. The results show the following: (1) the three proposed stabilization items containing higher-order PD derivatives have a better accuracy and stability than the traditional items of the NOSB-PD. In particular, the stress point stabilization item is preferred since it has the highest accuracy and efficiency and does not introduce any additional parameters; (2) the weak form of PD method is very suitable in dealing with the crack propagation and bifurcation problems.
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