Abstract A general third-gradient one-dimensional beam formulation is introduced through energy postulation. This novel formulation incorporates the effects of non-rigid (or deformable) boundary constraints by using the spring analogy. The Euler–Lagrange equations for equilibria and expressions for boundary conditions are derived. For certain specific cases, analytic solutions are found by reducing the problem to that of finding a single descriptor that accounts for the rotation of the cross-section. Based upon the obtained analytical solutions, parametric studies are conducted by varying the stiffness coefficients and boundary spring coefficients. It is found that the deformable boundaries, represented by the boundary springs, modulate the highly nonlinear response of these curvature gradient beams. Both numerical and analytic solutions are compared, revealing exotic shapes when various external actions, such as couples, double couples, and boundary terms, are activated. Finally, an example is presented of a microstructure whose homogenized limit is representative of curvature gradient beams. It is shown that the conceived continuum model describes the behavior of such a microstructure subjected to concentrated boundary couple with high accuracy. This study, therefore, represents a significant advancement in the general formulation for higher-gradient energy.
Dell et al. (2026) studied this question.