Abstract This work presents the linearisation, about an arbitrary motion, of the main kinematical quantities in Solid Mechanics and, as a novel result, of the polar decomposition of the deformation. When the polar decomposition is linearised about the identity motion (undeformed configuration), one retrieves the classical result that the multiplicative polar decomposition of the deformation gradient becomes the additive decomposition of the displacement gradient additive. The setting is in affine spaces, but the covariant framework allows for a relatively straightforward extension to Riemannian manifolds.

Abstract The simulation of complex physical systems using a discretized mesh is a cornerstone of applied mechanics, but traditional numerical solvers are often computationally prohibitive for many-query tasks. While Graph Neural Networks (GNNs) have emerged as powerful surrogate models for mesh-based data, their standard autoregressive application for long-term prediction is often plagued by error accumulation and instability. To address this, we introduce MeshODENet, a general framework that synergizes the spatial reasoning of GNNs with the continuous-time modeling of Neural Ordinary Differential Equations. We demonstrate the framework's effectiveness and versatility on a series of challenging structural mechanics problems, including different elastic bodies undergoing large, non-linear deformations. The results demonstrate that our approach significantly outperforms baseline models in long-term predictive accuracy and stability, while achieving substantial computational speed-ups over traditional solvers. This work presents a powerful and generalizable approach for developing data-driven surrogates to accelerate the analysis and modeling of complex structural systems.

Abstract This study investigates the boundary layer solution in the tip region of shallow hydraulic fractures that propagate with a finite fluid lag. Owing to their proximity to the free surface, shallow hydraulic fractures develop under relatively low confining stress, conditions that promote the formation of a pronounced fluid lag. Simultaneously, stress and geometric asymmetries induce mixed-mode propagation, giving rise—under specific conditions—to a sliding zone at the crack tip. A scaling analysis shows that the tip region solution is governed by two dimensionless parameters: the dimensionless toughness K and the dimensionless stress S. The dependence of the fluid lag length and the extent of the sliding zone on these parameters is analyzed in detail. The resulting solution displays a multiscale asymptotic structure comprising tip, intermediate, and far-field regions, with the configuration of these regions varying systematically with K and S. The simultaneous presence of a fluid lag and a sliding zone substantially increases the complexity of the boundary layer behavior compared with cases where the lag is absent. Furthermore, elastic deformation of the substrate influences the far-field moment, necessitating the inclusion of a root rotation term in the far-field beam asymptotic representation. The dependences of both the far-field moment and the root rotation on the governing parameters K and S are shown to be highly nonlinear, reflecting the coupled effects of the fluid lag and the sliding zone on the overall solution structure.

Abstract The performance of bio-inspired fibrillar adhesives can be compromised by surface roughness, manufacturing imperfections or impurities. Previous studies investigated the cases of distributed defects on the array, and defects at the level of single fibrils. However, the influence of localized, macroscopic defects remains largely unexplored. Using numerical simulations of a discrete mechanical model for a fibrillar adhesive with a thick backing layer, we investigate how the size and location of a single circular defect affect the established scaling law between the adhesion force (F) and the effective compliance of the system (β), i.e, F ∝ β−1/2. We find that edge defects are generally more detrimental than central ones, as they act as pre-cracks that amplify stress concentrations at the adhesive's edge, accelerating a crack-like failure. Consequently, the established adhesion scaling law is preserved, with the defect only reducing the effective contact area. In contrast, a central defect fundamentally alters the mechanics of detachment. By transforming the contact geometry into an annulus, it promotes more uniform load sharing across the remaining fibrils and mitigates the edge-dominated failure mechanism. This change makes the adhesive strength less sensitive to the compliance of the system, as reflected by a less negative scaling exponent. The transition between these two regimes appears to occur for defects whose boundary merges with the one of the adhesive.

Abstract Magnetically actuated intelligent structures have shown great promise in various applications ranging from soft robotics to biomedical devices. One important type of structure involves embedding rigid magnets in polymer matrices, thanks to their simplicity in manufacturing and flexibility in structure design. However, accurately modeling their behavior remains challenging due to the strong nonlinear coupling between large deformation and spatially varying magnetic fields. To address this, we employ a fictitious magnetic charge method to transfer the magnetic forces and torques into surface traction under both uniform and nonuniform external magnetic actuation. We implement the proposed method in the commercial finite element software abaqus and validate it through a series of benchmark examples, which cover tuning the area of magneto-mechanical metamaterials, the shape changing of magnetic lattice structures, and the magnetic-field-driven buckling of cellular structures. Our approach enables accurate prediction of shape transformations in magnetically responsive structures, offering a practical tool for the design and optimization of future magneto-mechanical metamaterials and devices.

Abstract Hydrogels, characterized by a factorial microscopic structure, generally exhibit homogeneous deformation under mechanical and chemical loads in the absence of directional reinforcement. However, curvilinear geometries, such as cylindrical specimens, may exhibit inhomogeneous deformation under torsion. This paper explores the intriguing phenomenon where a hydrogel cylinder undergoes radial inhomogeneous deformation, accompanied by axial elongation during twisting. The study begins by presenting a thermodynamic framework that captures the chemo-mechanical interactions, encompassing the governing and constitutive equations. The free energy density is then specified for hydrogel, followed by a Legendre transformation to incorporate the incompressibility condition with new form of free energy density. After a detailed geometric analysis, we identify the factors that triggers inhomogeneity. Finally, an equilibrium analysis elucidates the effects of key parameters on the coupled twist-extension and radial inhomogeneous deformation of a hydrogel cylinder under torsion.

Abstract As a fundamental force, friction exerts a profound influence on various aspects of our daily lives across multiple disciplines. To understand why adhesive friction is associated with the contact area, here we investigate the generic sliding of elastic solids adhered to a rigid surface by considering re-attachment/healing. We then reveal multiple adhesive fronts closely aligning along the interface with the number of these regions generally increasing with the contact area. These adhesive fronts exhibit rich dynamics and their accumulation along an interface can aid each other through re-attachment/healing in friction, apparently resulting in the increase in the calculated shear-off force with the contact area. Based on these findings, we propose a generalized law of adhesive friction. Our analysis further suggests that accumulating adhesive fronts possibly to a critical area density along the interface can trigger crack-like propagation of individual fronts at high velocities, which potentially bridges the gap between tribology and fracture mechanics. We discuss the relevance of this work to earthquake mechanics, which might provide a unified framework that captures key aspects of fault behavior. We expect that this work can provide a fundamental understanding of healing-mediated interfacial phenomena in diverse systems spanning biology, geology, and engineering.

Abstract An ultra-high strain gradient field usually exists near dislocations, which breaks the inversion symmetry in dielectrics and induces electric polarization in the material through the flexoelectric effect. Flexoelectricity is an electromechanical coupling phenomenon commonly found in dielectric material, and this flexoelectric effect in anisotropic materials also exhibits complex and significant directional dependencies. In this article, anisotropic strain gradient theory and mathematical methods are used to establish precise constitutive and governing equations for cubic flexoelectric materials. The framework incorporates four constitutive tensors: the second-order reciprocal dielectric susceptibility, the fourth-order elastic tensor, the fourth-order flexoelectric tensor, and the sixth-order elastic tensor. Both open-circuit and short-circuit conditions are considered to define the bounds of the governing equations. By applying Green's function, we address the fundamental nonsingular dislocation problem and derive the displacement and elastic distortion tensors for screw and edge dislocations in cubic flexoelectric materials. Our results not only provide solutions to straight dislocation problems in flexoelectric cubic materials but also can serve as a basis for tuning flexoelectricity in materials by dislocation engineering.

Abstract This study, motivated by applications in nuclear engineering, examines the fluid-induced vibrations of a flexible inner cylinder concentrically positioned within a rigid outer cylinder, separated by a quiescent Newtonian viscous fluid. Building on our previous work, which focused on forced motions, we extend the theoretical formulation to account for vibrations induced by fluid forces. A new expression for the linear fluid force is derived, introducing a fluid transfer function that depends on key dimensionless numbers such as the aspect ratio, radius ratio, and Stokes number. The vibration frequency and viscous decay rate are predicted by coupling this force with an Euler–Bernoulli beam model and using a modal decomposition based on a complex angular frequency. A computational approach is also developed, integrating an explicit partitioned coupling scheme into the open-source triocfd software. This method enables efficient numerical simulations of the fluid–structure interaction, ensuring internal coupling without excessive data exchange. Comparisons between theoretical predictions and numerical results show excellent agreement for pinned-pinned and clamped-clamped configurations across specific Stokes number and density ratio values. Additionally, the validity of the proposed theory and numerical simulations is further confirmed by comparisons with experimental data from the literature, showing strong consistency for a clamped-free configuration. A parametric study including variations of the Stokes number is eventually also performed, demonstrating the robustness and applicability of the theoretical and numerical approaches.

Abstract The nanoscale contact behavior of an elastic layer with a finite thickness is significantly influenced by the effects of not only surface but also layer thickness. Using the elasticity theory based on the surface energy density as well as the Fourier transformation technique, the two-dimensional frictionless contact problem of a rigid nanoindenter with a flat or cylindrical shape on a finite elastic layer is investigated. Integral solutions of stresses and displacements in the finite elastic layer are obtained, which take into account the effects of both surface and layer thickness. It is found that the surface effect results in a hardening of the finite elastic layer, which is basically due to the opposite action of the surface-induced normal traction to the externally applied compression. The decrease of layer thickness induces an increase of the normal stress and a decrease of the normal displacement in the contact region because of the constraint of the rigid substrate to the layer deformation. An interesting finding is also unveiled that the surface effect could effectively eliminate the bulging behavior in a very thin layer. Furthermore, an indentation hardness including the coupling effects of surface and thickness in the finite elastic layer is predicted, which increases with either an enhancing surface effect or a decrease of the layer thickness. The present research should be helpful for designs and performance evaluations of layer-substrate structures in practical engineering, such as coatings and microelectronic devices.