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.
Within the framework of Eshelbian mechanics and linear elasticity, we develop a kinematic and energetic formulation for treating different classes of defects. Central to the approach is a defect map, through which interface discontinuities and bulk material inhomogeneities are described within a common notation. The driving force is derived as an energy-release rate, leading to a single identity in which defect-dependent mechanical effects, material inhomogeneity, and boundary/interface contributions are explicitly separated. For affine modes of defect evolution—translation, rotation, and dilatation—the formulation recovers the corresponding classical integrals as special cases. In particular, the analysis shows that path independence is not automatic, but follows under the assumptions required for the balance of the different contributions to the driving force. Finally, a prototypical one-dimensional example is discussed to describe the physical meaning of the deduced analytical expression of the total driving force.
This article discusses two axisymmetric contact problems for a piecewise homogeneous space. Space consists of two dissimilar half-spaces with an interphase disc-shaped crack, onto one of the edges of which an absolutely rigid stamp is pressed, taking into account static friction. The radius of stamp is less than or equal to the radius of the crack. By using discontinuous solutions of the axisymmetric theory of elasticity, the governing equation of the problems is obtained both in the form of one singular integral equation of the second kind with respect to the reduced unknown contact pressure in the first case and a system of singular integral equations with respect to reduced normal contact pressure and dislocations of the displacements points of the crack edges in the second case. The solution of the governing equations in both cases is constructed using the numerical-analytical method of mechanical quadratures. A numerical calculation was carried out and patterns of change of important physical and mechanical characteristics for problems were identified depending on the physical, mechanical and geometric characteristics.
Helically poled magnetoelastic tubes enable programmable extension, torsion, and inflation under magnetic loading, making them attractive for soft robotic and tubular actuation systems. Building on a previously developed variational formulation, explicit closed-form expressions are derived for the effective axial, torsional, inflation, and magnetoelastic coupling stiffnesses in the thin-tube limit. A consistent asymptotic reduction transforms the three-dimensional nonlinear magnetoelastic problem into a compact Cosserat-type model with analytically identifiable moduli that depend explicitly on the poling angle, elastic anisotropy, and magnetoelastic parameters. Unlike classical piezomagnetic formulations, the present approach captures geometry-induced coupling arising from helical magnetization and establishes a direct link between three-dimensional field equations and reduced-order structural behavior. The resulting expressions clarify how helicity governs chiral and non-chiral actuation pathways: azimuthal magnetic fields activate extension and inflation only in the presence of helicity, whereas axial magnetic loading induces deformation even for purely axial poling, with the torsional response depending on field orientation. The formulation recovers known limiting cases and yields physically consistent predictions across a range of material and geometric parameters. Order-of-magnitude estimates based on experimentally reported properties of magnetoactive elastomers further indicate that the predicted responses lie within realistic deformation regimes. The closed-form stiffness relations provide a rigorous and computationally efficient framework for rapid parametric analysis and support the systematic design of magnetically actuated soft tubular devices.
This note discusses the balance of angular momentum and the balances of director momentum in a small deformation model of a deformable Cosserat continuum. It is shown that for small deformation elastic response, the deformable Cosserat model in Rubin [14] and the theory of microstructure in Mindlin [8] both satisfy the balance of angular momentum and both predict the same response for the same given strain energy function. This result corrects a discussion in Rubin [14] (Appendix C) related to equations for the microstretch theory in Eringen [3] and the microstructure theory in Mindlin [8].
Flexoelectricity represents a higher-order electromechanical coupling phenomenon whereby strain gradients within a material can induce electric polarization. Flexoelectricity is observed in hard dielectric ceramics as well as a wide range of soft matter. This contribution establishes a rigorous and comprehensive framework for finite-deformation flexoelectricity in generalized Toupin’s electroelasticity. Following the variational principle, we present the equilibrium equations, boundary conditions, and constitutive relations for nonlinear flexoelectricity. The existing theory of flexoelectricity at small deformations is reproduced via linearization of the governing equations. As an application example, we derive an exact solution for the finite-deformation problem of a flexoelectric bar. Our exact solution demonstrates remarkable distinctions between nonlinear flexoelectricity and linear flexoelectricity. To underpin the theoretical framework with numerical examples, we establish a mixed finite element formulation for nonlinear flexoelectricity and investigate the finite-deformation problem of flexoelectric solids with a circular cavity. We meticulously highlight the impact of nonlinear effects on electric potential, electric polarization magnitude, and circumferential stress concentration. Our results advance higher-order electroelasticity from both theoretical and computational perspectives, revealing the significance of nonlinearities, hence offering novel insights for the design of soft flexoelectric devices.
We construct a convergent recurrence scheme for a solution of the three-dimensional Neumann type boundary value problem of the elasticity theory when on the boundary of a homogenous anisotropic elastic body the stress vector is prescribed. By the potential method, the boundary value problem is reduced to the second kind Fredholm integral equation with zero index generated by the single layer potential. The corresponding boundary integral operator has a six-dimensional null-space. Therefore, the nonhomogeneous integral equation possesses a solution if the right-hand side vector-function meets the necessary and sufficient solvability conditions, i.e., it is orthogonal to the space of rigid displacements, which are solutions to the corresponding homogeneous adjoint integral equation. First, we reduce the boundary integral equation to the equivalent first kind Fredholm integral equation with symmetric non-negative compact operator. Afterwards, we construct a uniquely solvable modified boundary integral equation with symmetric compact positive operator having the following property: if the above mentioned necessary conditions are satisfied then the solution of the modified boundary integral equation is a particular solution of the original integral equation. Further, we construct a recursive sequence of vector-functions which converges to the unique solution of the modified boundary integral equation in appropriate Bessel-potential spaces of functions defined on the boundary. Using these approximations as densities of the single layer potential, we construct a sequence converging to a particular solution (a particular displacement vector) of the Neumann type boundary value problem in the appropriate Sobolev-Slobodetskii spaces of functions defined in the region occupied by the elastic body. The general solution of the Neumann type boundary value problem can be constructed by adding an arbitrary rigid displacement vector to the obtained particular solution. Evidently, the corresponding strain and stress tensors are defined uniquely.
We introduce the elastic fluctuation tensor to quantify the stochastic fluctuation of the apparent stiffness of finite microstructural volume elements. Typically, in computational homogenization using volume elements of finite size, the apparent stiffness converges to the effective stiffness as the volume element size tends to infinity, such that the material can be approximated as homogeneous on the macroscale. For volume elements of finite size, the apparent stiffness fluctuates on the macroscale. In thermal conductivity homogenization, the fluctuations can be quantified using the fourth-order fluctuation tensor, which computes as the infinite-volume limit of the apparent conductivity covariance, rescaled with the volume. The fluctuation tensor for linear elasticity is of tensor order eight. We show that this fluctuation tensor inherits the symmetry of its ensemble. For instance, rotational statistical symmetry of the ensemble leads to isotropy of the elastic fluctuation tensor. Using results from group representation theory, we define efficient representations of the eighth-order fluctuation tensor for various microstructure symmetry classes and discuss the physical meaning of individual components for the statistically isotropic case. We furthermore leverage symmetry to mitigate numerical errors, thereby reducing the expense of computing the fluctuation tensor. As an example material, we consider polypropylene reinforced by fibers and spherical inclusions. Additionally, we examine polycrystalline copper microstructures. By numerically computing the elastic fluctuation tensor, we confirm theoretical asymptotic convergence rates and symmetry properties. For many of the considered statistically isotropic microstructures, the fluctuations of isotropic stiffness components, which are often the only fluctuations reported, are negligible compared to isotropic fluctuations of the anisotropic stiffness components. Therefore, the full fluctuation tensor must be considered when quantifying the uncertainty of stochastic homogenization.
Classical homogenization is insufficient for finite-sized structures as it does not account for crucial scale-size effects. We develop a thermodynamically consistent model and subsequent two-scale expansion homogenization framework for strain-gradient elasticity that derives scale-dependent effective models complimenting the findings of [58] enabling prediction of finite-size effects in architected metamaterials. Our key result is that the homogenized mechanical properties depend not only on the micro-geometry and volume fraction but also on the absolute size of the underlying constituents. Hence, the homogenized coefficients are not constant (as in classical homogenization) but rather functions of the microstructural size. Numerical validation confirms that the homogenized coefficients converge to the classical ones as the scale-size effects become vanishingly small, providing a critical tool for designing architected materials.
Regarding a shell as the natural restriction of the Euclidean frame bundle to an embedded surface, and further restricting its deformation so that the surface tangent planes be preserved, a first-order (simple) material behaviour leads automatically to a theory that includes strain gradients. Using the principle of virtual work, the weak and strong forms of the shell equilibrium equations and boundary conditions are derived. A comparison with other higher gradient theories is carried out and commented upon. An important outcome of the proposed geometric framework is that the gradient of curvature does not emerge as an independent contribution in the resulting higher-gradient shell formulation.
We study the equation of one-dimensional quasistatic nonlinear viscoelasticity of strain-rate type with Dirichlet boundary conditions, in the particular case that the underlying dissipation geometry (provided by the viscosity) is comparable to the Bhattacharya metric on probability densities. We establish a global existence result for weak solutions, with an approach based on a spatial discretization allowing us to work directly with the Riemannian metric associated to the viscosity. Strong convergence of spatially discrete solutions is shown directly – this is possible thanks to Lipschitz estimates achieved locally on energy sublevels enabled by an explicit derivation of the stretching of tangent vectors under the flow in the discrete setting and the relationship to the Bhattacharya metric. We furthermore prove gradient-flow representations for the solutions: they are curves of maximal slope and, under a global convexity hypothesis on the energy sublevels, we prove they satisfy a metric evolutionary variational inequality.
We present a complete analytical solution for the stress field inside a homogeneous, inside a homogeneous, linearly elastic solid sphere subjected to a concentrated normal load applied on its surface. Starting from the three-dimensional linearized elastodynamic equations, the displacement and stress fields are derived using scalar and vector potential representations combined with spherical harmonic expansions. All expansion coefficients are determined explicitly by enforcing the traction boundary conditions. The static elastic solution is obtained rigorously as the long-time limit of the dynamical formulation. Closed-form expressions for all components of the stress tensor are provided, enabling direct evaluation of the principal stresses and their differences throughout the interior of the sphere. The analytical solution is further generalized to arbitrary loading positions by means of rotational transformations, allowing systematic treatment of multiple concentrated loads through superposition.
In this note we consider a two-dimensional semi-discrete dislocation energy and propose a simple and physically motivated construction for the grain boundary between two crystal grains with a small orientation difference. In the case of a general Bravais lattice, the energy of this construction matches the logarithmic scaling predicted by Read and Shockley.
We consider a circular cylindrical elastic membrane tube which is initially at rest and subjected to a given constant axial extension. We then consider the acceleration waves arising from the boundary condition defined for the axial velocity at one end of the tube. Using the singular surface theory, we study the propagation of acceleration waves in the context of nonlinear elasticity. The temporal evolution and propagation speeds are determined for a general incompressible elastic material. We deduce the conditions which determine whether the amplitude of the longitudinal acceleration wave blows up or not. Considering some well-known examples of strain-energy function, our study demonstrates that different constitutive models behave very differently and an amplitude blow-up may occur depending on the magnitude of the initial stretch for some elastic materials. On the other hand, we show that when the state of the medium ahead of the longitudinal acceleration wave is its natural state, an amplitude blow-up does not occur.
The deformation of microstretch elastic solids is described by the displacement vector, the microrotation vector and the microstretch function. This paper presents a theory of prestressed microstretch elastic solids. First, the nonlinear theory of microstretch continua is used to establish the equations governing the infinitesimal deformation superposed on large deformations. Then, the linear theory of microstretch elastic solids with initial stresses and couple stresses is derived. The continuous dependence of solutions upon body loads and initial data is investigated.
Stretching, drilling, and bending are the independent deformation modes of a thin shell, each of which has an individual energy content. When the energy content of a mode vanishes, that mode is neutral. We characterize all neutral modes of deformation of minimal surfaces into minimal surfaces. A hierarchy is found among these: a stretching neutral mode (which is an isometry) is also drilling neutral, and a drilling neutral mode is also bending neutral. Thus, all isometries of a minimal surface are globally neutral and give rise to soft elasticity. More generally, all minimal surfaces can be classified relative to a reference one in terms of three energy contents, which can be given in closed form.
The study of the mechanisms guiding the spider in prey capture and gathering information through web vibration is one of the main objectives of current research in this field. Recent studies have provided partial insights, primarily through the development of a mechanical model of the spider web represented as a continuous fiber membrane with an appropriate pre-tensile stress state and undergoing small deformations. To date, all analytical results available in the literature – both in modelling and in the inverse problem of prey identification – have been restricted to axially symmetrical webs. In this work, we move beyond the axial symmetry assumption and address the more realistic case of vertical asymmetry, a characteristic of almost all vertical orb webs. This generalization introduces non-trivial challenges in mechanical and mathematical modelling. We establish the well-posedness of the statical response for a class of asymmetrical orb webs in a suitable weighted Sobolev space and we investigate the dynamically forced problem via eigenfunction expansion.
A general approach is presented for inverting fourth-order material tensors expressed as a sum of products of rank-two tensors such as the identity tensor and structural tensors. With reference to transversely isotropic and orthotropic materials, the procedure exploits identities involving different tensor products of the structural tensors to obtain three expressions of the elasticity tensor in order to look for the one that can simplify as much as possible the fully intrinsic evaluation of the compliance tensor. The sets of elastic constants referred to the intrinsic expression of the elasticity and compliance tensor are mutually related and are expressed in an arbitrary reference frame, not necessarily aligned with the planes of material symmetry. Finally, to foster the use of the tensorial approach in numerical applications, we also derive explicit expressions relating elastic or compliance moduli both with the entries of the material stiffness matrix, since they are determined experimentally, and with engineering constants, since they are more often used in practice due to their direct mechanical meaning.
