We propose a new representation of three-dimensional orthotropic elasticity tensors based on the harmonic decomposition. We derive a spectral characterization of fourth-order harmonic tensors in terms of second-order harmonic eigentensors subject to a cofactor constraint. Under orthotropic symmetry, this constraint can be solved explicitly, yielding a new, compact parametrization of the three-dimensional orthotropic elasticity tensor in terms of an orthotropy basis, six scalar moduli, and three angular parameters. The proposed parametrization is then used to characterize higher material symmetries, namely tetragonal, cubic, transversely isotropic, and isotropic classes. We also derive explicit conditions for positive definiteness, both in the general orthotropic case and under spherical–deviatoric uncoupling. A practical reconstruction procedure is provided for a given orthotropic elasticity tensor. Finally, the usefulness of the representation is illustrated through a numerical compliance-minimization example, highlighting its potential for computational applications in anisotropic elasticity and material design and optimization.
Angular shear deformations are based solely on the angular variation of line elements and not on their extension. To describe kinematics, a material cube is considered that transforms into an oblique prism, where two opposite faces are equal and parallel. Using the displacement components of three vertices of the fundamental edges, positioned along the reference axes, the displacement field of the body has been obtained. Then, the deformation tensors have been determined and some general properties for this class of shear deformations have been highlighted. Using a Lagrangian analysis, the boundary value problem for the equilibrium of a cube subjected to a purely angular shear deformation has been formulated. Subsequently, Piola-Kirchhoff and Cauchy stress tensors, together with the normal and tangential Cauchy stress vector components acting on the inclined faces of the deformed cube, have been evaluated.
Stress concentration in notched components under torsional loading can significantly affect structural strength, durability, and fatigue life. In this study, the optimal profile of the right-angle notch is explored with the objective of minimizing stress concentration. Based on the principle of minimum potential energy and complex analysis, the optimal notch profile is derived analytically and identified as an astroid. An advanced boundary element method (BEM) is employed to verify the validity of the theoretical solution and to provide an effective numerical framework. The numerical results indicate that, compared with the circular notch and the hyperelliptic notch, the astroid notch is more effective in reducing stress concentration. In addition, a zero radius of curvature at the notch root does not necessarily lead to severe stress concentration or stress singularity. These findings offer further insight into notch-induced stress amplification and the mechanical behavior of geometric discontinuities.
We clarify a few issues that could possibly be misunderstood in our paper [J. Elast. 158(2), 25 (2026) https://doi.org/10.1007/s10659-026-10201-8 ].
The aim of this article is to provide a concise and coherent summary of existing results on the geometric properties of the planar Euler elastica. No new governing equation or solution family is proposed; the emphasis is on presenting and synthesising established results. Taking the classical equilibrium equation for a planar, inextensible and unshearable elastic rod under terminal loads as a starting point, the article collects closed-form expressions in terms of Jacobi elliptic functions for the tangent angle, curvature, intrinsic coordinates, and internal force components. The elliptic modulus k is used as the organising parameter, enabling the inflectional and non-inflectional families, the homoclinic separatrix, and the straight and circular limiting cases to be described within a common notation. The roles of the solution parameters k , λ , C , α , and the coordinate constants are clarified in terms of shape, scale, phase, orientation, and position. Symmetries, periodicity, bounds, and characteristic points are summarised, while inflection, compression, and shear points are interpreted geometrically and mechanically. Conditions for self-intersections and self-touching points are presented, and threshold values of k are related to changes in topology. Formulae for characteristic dimensions are also collected.
