Finite element formulations of strain gradient theory for microstructures and the C0-1 patch test

Based on finite element formulations for the strain gradient theory of microstructures, a convergence criterion for the C0–1 patch test is introduced, and a new approach to devise strain gradient finite elements that can pass the C0–1 patch test is proposed. The displacement functions of several plane triangular elements, which satisfy the C0 continuity and weak C1 continuity conditions are evaluated by the C0–1 patch test. The difference between the proposed C0–1 patch test and the C0 constant stress and C1 constant curvature patch tests is elucidated. An 18-DOF plane strain gradient triangular element (RCT9+RT9), which passes the C0–1 patch test and has no spurious zero energy modes, is proposed. Numerical examples are employed to examine the performance of the proposed element by carrying out the C0–1 patch test and eigenvalue test. The proposed element is found to be without spurious zero energy modes, and it possesses higher accuracy compared with other strain gradient elements