Welcome to MechanicsArXiv. This repository provides a platform for researchers to share early-stage research, working or completed papers, technical reports, reviews, and tutorials in the broadly defined field of solid mechanics. Our mission is to accelerate the dissemination of scholarly work, foster collaboration, and provide open access to the latest innovations across mechanics.
Polydisperse polymer networks with irregular topologies: Mechanics of cross-link distributions
The structure of polymer networks, defined by chain lengths and connectivity patterns, fundamentally influences their bulk properties. While existing polymer network models connect chain properties to emergent network behavior, they are often limited to monodisperse networks with regular connectivities. In this work, we introduce a novel modeling framework that shifts the focus from individual polymer chains to cross-links and their connected chains as the fundamental unit of analysis. The key features of this framework are the relaxation of the cross-link junction position to satisfy local force balance and physically intuitive means for satisfying material frame indifference. By relaxing to equilibrium, the cross-link structure is allowed to deform non-affinely and its chains adopt a more energetically efficient distribution of stretches and forces than that permitted from equal stretch or equal force homogenization theories. We explore two distinct limiting behaviors for the orientation of the frame of a cross-link: (1) the free rotation limit, which assumes the cross-link rotates to minimize free energy, and (2) the frame averaging limit, which incorporates structural heterogeneity by averaging over all possible cross-link orientations. It is found that an increase in variance in monomer numbers generally leads to network softening, while in bimodal networks, the onset of strain stiffening is controlled by shorter chains and the stiffening response is modulated by the ratio of short to long chains. By deriving closed-form approximations for both limits valid in the regimes of small deformation or small polydispersity, we offer an efficient computational approach to modeling the mechanics of complex, polydisperse networks. An aim of this framework is to take a step toward the rational modeling and design of heterogeneous polymer networks with structures tailored for specific properties.
Solving the Dissipation Inequality not as a constitutive restriction
A solution procedure is formulated and solved for treating the nonlinear Dissipation Inequality as a constraint equation within continuum mechanics, and allowing for incomplete knowledge of constitutive behavior. The scheme is demonstrated in the context of the rate-dependent, elastoplastic response of a bar, resulting in a nonlinear problem of constrained optimization. Both closed form and computational results are developed. The computational solutions utilize a sequence of convex optimization problems, and are shown to be accurate. In the example considered, the approach is shown to automatically correct an (intentionally) faulty constitutive specification, resulting in the solution to be in accord with the fundamental postulates of continuum mechanics.
3D Chemo-Mechanical Simulation of Intergranular Fracture in Polycrystalline Ternary Cathodes Coupled with Damage-Dependent Dynamic Electrolyte Infiltration
Polycrystalline ternary cathodes are the dominant high-energy cathode material for commercial lithium-ion batteries, yet intergranular fracture along grain boundaries triggered by lithiation-induced strain severely deteriorates long-term cycling durability. The interconnected grain-boundary crack networks formed during charging provide penetration pathways for liquid electrolyte, creating a bidirectional feedback effect on electrochemical performance: electrolyte infiltration expands electrochemically active surfaces and alleviates kinetic limitations in early cycles, whereas sustained electrolyte-grain boundary interfacial corrosion continuously weakens interfacial mechanical strength and accelerates progressive intergranular crack propagation, forming a self-amplifying degradation loop. Most existing chemo-mechanical numerical simulations for ternary cathodes rely on oversimplified 2D microstructures or artificially predefined fully-wetted crack boundaries, which fail to capture the gradual electrochemical activation of grain-boundary cracks synchronized with dynamic intergranular damage evolution. To fill this research gap, this work establishes a fully coupled 3D finite-discrete element chemo-mechanical framework exclusively equipped with bilinear cohesive zone elements at grain boundaries to characterize intergranular fracture. A novel damage-dependent dynamic electrolyte infiltration model is innovatively proposed to quantitatively correlate local grain-boundary damage magnitude with real-time electrolyte penetration degree and surface electrochemical activity. Systematic 3D parametric simulations are conducted to decouple the individual and synergistic influences of primary particle size and charging C-rate on Li-ion concentration heterogeneity, internal von Mises stress concentration, and cumulative intergranular damage index. Simulation results reveal that intergranular cracking always constitutes the dominant degradation pathway under all tested operating conditions; low charging rates are primarily governed by primary particle size effects, while high C-rates become the decisive factor triggering extensive intergranular crack network proliferation. Compared with the two limiting idealized infiltration scenarios (no infiltration/full instantaneous 100% infiltration), the proposed damage-dependent infiltration model delivers physically realistic intermediate evolution laws for Li distribution, stress accumulation and crack propagation, and explicitly quantifies the bidirectional positive feedback between dynamic electrolyte permeation and intergranular fracture. This 3D modeling framework provides quantitative multi-scale microstructure design guidelines for suppressing grain-boundary cracking and extending the service life of high-energy polycrystalline ternary cathode materials.
Dual Variational Principles for Curl Forces
Curl forces are position-dependent, non-conservative, and non-dissipative forces that, in general, cannot be derived from an ordinary potential energy. Consequently, their equations of motion do not, in general, follow from a standard variational principle. In this paper, we present a dual variational formulation for particle dynamics under curl forces. By introducing variables dual to position and velocity and an auxiliary function, we construct a pre-dual action in which the equations of motion act as constraints. Stationarity with respect to the primal variables defines a dual-to-primal mapping, whose substitution into the pre-dual action gives an action expressed entirely in terms of the dual variables. The Euler--Lagrange equations of the dual action recover both the original equations of motion and their prescribed initial conditions. We also introduce an auxiliary dual Hamiltonian that is conserved along stationary dual trajectories, although it does not represent the physical energy. The formulation is illustrated using two nonlinear curl force fields in two and three dimensions and the classical Ziegler column. These examples demonstrate that non-conservative curl-force dynamics can admit variational descriptions even in the absence of an ordinary potential energy or a conventional Lagrangian.
Making coupled-field Abaqus user elements simple
Soft materials, energy storage, and advanced manufacturing couple mechanics with transport, reaction, or damage, so their simulation must solve several interacting fields together. In Abaqus, custom coupled formulations are typically implemented through user elements (UELs), and a coupled UEL must supply the residual, an iteration matrix (ideally the consistent tangent), degree-of-freedom maps, and state updates in exactly the form the solver expects. Deriving and debugging these ingredients, especially the unsymmetric cross-field tangent blocks, separates a model on paper from a working simulation, and an implementation error is hard to distinguish from a modeling choice. We present abaqus_ufl, a Python framework in which the user declares the fields and their interpolations and selects and configures supported balance templates for the governing equations. The framework generates a self-contained Fortran UEL and evaluates the required tangent blocks with the established complex-step approximation. Correctness is addressed by layered verification, from Python derivative checks through compiled single-point tests and a lightweight assembly runtime to Abaqus benchmarks, and each demonstration reports the evidence level it actually reached. The examples progress from reproducing an established corrosion element to stabilized, mixed-order, and locally condensed formulations exercised in three-dimensional morphing.
Experimental and numerical spatio-temporal analysis of the Portevin–Le Chatelier dynamics in a Nickel based superalloy
This study examines the Portevin–Le Chatelier (PLC) effect in the nickel-based superalloy Inconel 718 through a combined experimental-computational approach using statistical indicators from nonlinear dynamical systems theory. We develop a finite element model incorporating the Kubin–Estrin–McCormick constitutive law to capture Dynamic Strain Ageing effects, accounting for machine stiffness influence, and reproduce various dynamics under both hard and soft loading conditions. Statistical analysis reveals that machine stiffness significantly affects serration morphology and dynamics, influencing mean amplitudes, stress drop periods, and dynamical indicators such as the correlation dimension and Lyapunov exponents. Comparison of simulated and experimental stress time series demonstrates chaotic behaviour for type B and C serrations across all strain rates, with no evidence of self-organised criticality in experimental data. However, simulations predict self-organised criticality at high strain rates corresponding to type A bands, consistent with literature references. Statistical indicators reveal power-law behaviour for stress drop amplitudes as a function of strain rate, with critical exponents dependent on band type. Multifractal analysis shows that simulations overestimate complexity relative to experimental observations, suggesting the need for additional internal variables and finer-scale dynamics in modelling. Additionally, multifractal analysis of spatio-temporal diagrams reveals power-law distributions of plastic strain rates with consistent critical exponents across all strain rates, demonstrating its potential for characterising PLC spatio-temporal dynamics. Statistical, dynamical, and multifractal indicators show consistent correlations, collectively capturing transitions between serration regimes and serving as reliable quantitative metrics for characterising PLC dynamics. The analysis is finally applied to the spatio-temporal strain fields measured by digital image correlation. The results demonstrate the value of multi-indicator analysis for assessing the agreement between experiment and simulation and subsequently improving constitutive model parameter identification.
Constrained asymptotic crack-tip fields of a Mooney–Rivlin sheet in plane stress: a symplectic analysis
Analytical crack-tip fields for soft sheets are known mainly for generalized neo-Hookean materials. We formulate the crack-tip problem in a radial symplectic (Hamiltonian) framework, using ξ = lnr as the evolution coordinate. The deformation on a material circle and its weighted radial traction form the canonical pair, and the constraint reaction enters through the traction. Within this framework, dominant balance for an incompressible plane-stress Mooney Rivlin sheet shows that the I2 contribution becomes a kinematic constraint near the tip. The constraint drives the local state toward uniaxial tension. The leading stretch magnitudes are independent of angle, and the opening is y2 ∼ Pr1/2sin(θ/2). The compensated areal Jacobian Jr1/4 approaches an angle independent plateau. The same field gives G = (π/2)c1P2, where c1 is the first Mooney–Rivlin coefficient. In a pure-shear strip, the grip stretch there fore fixes P, and finite-element calculations support both this parameter-free amplitude and the compensated-Jacobian prediction. The constraint also permits a regular O(r) crack-parallel motion set by specimen-scale matching. After this motion is separated, the in-plane field contains an r5/4 residual whose angular equation admits a specimen-selected homogeneous member. Core and angular refinement of the strip fields recovers both the exact-axis 5/4 exponent and its parameter-free amplitude to within 1.3%. The closed normal-opening sector gives the half-oddradial spectrum and a conserved symplectic pairing. The calculation shows how a constitutive term that is subleading in the near-tip energy can nevertheless control the local kinematics, a mechanism that can be sought in other stiffening soft networks.
Period doubling and quadrupling of wrinkles in film/substrate bilayers: a nonlinear symplectic analysis
Under compression, a stiff film bonded to a soft substrate can progress from sinusoidal wrinkling to period doubling and quadrupling before localization. We develop an analytical-depth Rayleigh–Ritz method for this finite-amplitude cascade. Hamiltonian–Stroh modes represent the homogeneous depth response, nonlinear product rates enrich the finite state, and signed Floquet spaces test secondary stability through the constitutive second variation. For µf/µs = 100, the method gives a period-doubling threshold T24 = 0.179 and a retained-space period-four neutral point T45 = 0.244. A fixed-grid lattice calculation gives corresponding neutral crossings at 0.176 and 0.243 and produces similar period-two and period-four patterns. The two transitions arise through repeated subharmonic coupling: the primary wrinkle couples the signed half-wavenumber pair, and the doubled state couples the quarter-wavenumber pair. Across modulus ratios from 14 to 375, the primary onset strain varies nearly ninefold, whereas the doubling boundary remains near 0.18. Published incompressible finite-element calculations show the same weak dependence of the secondary boundary on stiffness contrast. The method therefore follows two successive wavelength multiplications with out substrate-depth nodal unknowns.
Constrained minimization problems and FFT-based solvers: application to local Dirichlet boundary conditions and contact mechanics
The present paper focuses on a recent and active research axis to overcome the limitations of FFT-based solvers in order to apply various types of boundary conditions (BCs) and not only periodic BCs. A complete framework based on discrete trigonometric transforms (DTTs) of various types, possibly combined with discrete Fourier transform, has been proposed recently to account for any type of BCs defined per face of the unit-cell and per component of the displacement or traction vector. A parallel implementation by Amouzou-Adoun et al. [1] recently proved its robustness and versatility. However, this approach is not able to account for a mix of BCs on a same face (for example Dirichlet BC on a part of a face and Neumann BC on the other part of the same face), neither to prescribe Dirichlet BC to points defined inside the domain, nor to define kinematic relations between displacement on different nodes. Starting from the displacement-based approach together with the use of DTTs, as proposed by Amouzou-Adoun et al. [1], simple modifications are proposed to account for all these questions. The modified solver, first introduced from the viewpoint of discrete local equations, is then discussed from the perspective of constrained minimization with equality constraints and the introduction of Lagrange multipliers. Simulations of a compact tension-like specimen are performed and validated. To go further, the algorithm proposed for equality constraints is then extended to account for inequality constraints to simulate contact mechanics. For the sake of validation, results obtained with a rigid spherical indenter with frictionless contact are compared to the Hertz theory for small ratios (indentation depth/sphere radius). Comparison between the small strain and finite strain context demonstrates increasing discrepancies when increasing this ratio. It is believed that the proposed methodology will significantly expand the field of applications of FFT-based solvers.
Energetics of Nucleation in Finitely Deformed, Phase-Transforming Soft Solids
Classical nucleation theory describes the rate at which stable nuclei form within a metastable parent phase by crossing a free-energy barrier set by competing bulk and interfacial energies. In an elastic material, a pre-existing stress state modifies this barrier through an elastic contribution to the bulk driving force. This contribution is well characterized for linear elastic materials, but the corresponding finite-deformation result for soft solids remains less developed. The gap is computationally significant: in simulations that sample candidate nuclei throughout a stressed body, direct evaluation of the elastic contribution to free-energy change would require solving a new nonlinear elasticity boundary-value problem for each possible nucleus. Here, we derive an asymptotic expansion of the equilibrium elastic potential energy change for a hyperelastic body before and after formation of a small transformed region. The expansion is with respect to the amplitude of an isotropic transformation strain, while the pre-existing deformation and stress may be finite. At leading order, the elastic contribution to the formation energy is determined entirely by the known untransformed equilibrium fields, with additional terms accounting for stiffness contrast between the parent and transformed phases. Incorporating this into classical nucleation theory yields the stress-shifted transformation temperature, critical radius, and nucleation barrier. Representative results are shown for a compressible neo-Hookean solid under hydrostatic, uniaxial, and equibiaxial loading; tensile stresses promote nucleation and compressive stresses suppress it when transformation strain is expansive. Comparison with the corresponding linear-elastic result shows that finite-deformation effects can substantially change the predicted energy barrier at moderate stretches.
Model-Informed Joint Material-Structural Optimization of Hard-Magnetic Soft Materials
This work develops a model-informed framework for predictive analysis and optimal design of hard-magnetic soft materials (hMSMs). These materials undergo contact-free, field-driven deformation, making them attractive for soft robotics, adaptive structures, and bio-inspired systems. Accurate prediction requires effective structure--property relations, while optimal design requires simultaneous control of structural density, magnetic particle distribution, and remanent magnetization direction. To address these issues, this work makes two main contributions. First, classical rigid-inclusion relations, a Hill self-consistent relation, and constrained-kinematics models are placed into a unified effective shear-modulus framework for particle-filled elastomers. With one default control relation, seven shear-modulus relations are combined with three strain-energy density functions to obtain 21 constitutive models. The results show that the strain-energy density form has a relatively small effect for the actuation problems considered, whereas the effective shear-modulus relation can significantly affect deformation when magnetic material overlaps with highly deforming regions. Experimental stress--strain data are then used to select a representative shear-modulus relation, with the Mooney relation giving the best overall agreement. Second, using the selected constitutive model, a joint material--structural optimization framework is developed for simultaneous design of structural density, magnetic particle volume fraction, and remanent magnetization direction. Rotational, translational, and restorative examples show that the framework handles different active design fields, objectives, and single- or multi-load-case formulations, producing non-intuitive hMSM designs with prescribed deformation responses. The framework is implemented in the open-source \texttt{CEADpx/top\_optim} repository.
Bifurcation reordering programs snap-through symmetry in folded elastic ribbons
Snap-through in slender elastic structures is often viewed as a sudden transition between stable configurations, yet the pathway taken during this transition can differ fundamentally. A structure may snap while preserving symmetry, or first lose symmetry and pass through an asymmetric state before reaching its final configuration. What selects between these pathways remains less well understood, especially when the geometry and loading are themselves symmetric. Here, we show that snap-through symmetry can be programmed by reordering competing bifurcations in folded elastic ribbons. We study an elastic ribbon with two localized folds placed symmetrically about the midpoint and show that varying the fold position changes the relative order of two instabilities: a symmetry-breaking pitchfork bifurcation and a saddle-node bifurcation on the symmetry-preserving branch. When the pitchfork bifurcation occurs first, the ribbon loses symmetry before snapping and follows an asymmetric pathway. Conversely, when the saddle-node bifurcation occurs first, the ribbon loses stability while remaining on the symmetric branch, resulting in a symmetry-preserving transition. Combining experiments, discrete differential geometry simulations and numerical continuation, we map this exchange in bifurcation ordering and construct a phase diagram that predicts the switch between asymmetric and symmetric snap-through regimes. A reduced-order double-mass von Mises truss model captures the same mechanism as a generic competition between symmetry-breaking and symmetry-preserving instabilities. These results establish bifurcation reordering as a geometric mechanism for programming snap-through pathways in slender elastic structures, offering a design principle for multistable systems, morphing structures and instability-based mechanical devices.
Modeling Stress-Enhanced Photo-Degradation in Amorphous Glassy Polymers
The emergence of renewable, degradable polymers in everyday applications necessitates the prediction of their mechanical behavior during service. In a recent contribution, Alkhoury et al. (2024) characterized changes in microstructural and mechanical behavior resulting from stress-enhanced photo-degradation of cellulose acetate, a renewable and degradable glassy amorphous polymer used in various consumer products. In this work, we complement the existing set of experiments and develop a thermodynamically consistent constitutive model informed by both micro- and macro-scale information. The outcome of this work enables the prediction of the mechanical behavior of photo-degrading glassy amorphous polymers while subjected to stresses. This can help engineers design sustainable devices while ensuring product integrity.
A non-local constitutive model for the Mullins effect in filled elastomers
Filled rubber-like materials are widely used in engineering applications and are well known to exhibit the Mullins effect. In this work, an established local constitutive model from the literature is extended to a non-local setting to resolve the mesh dependence inherent to the local approach. Non-local effects are incorporated using two separate approaches: (i) a Helmholtz-type equation governing a non-local soft volume fraction, and (ii) a Laplacian term introduced directly into the soft volume fraction local evolution law. In both formulations, an additional governing partial differential equation arises and is solved numerically in Abaqus using an analogy with the heat equation. The two approaches yield different results, leaving the choice between them to be guided by experimental findings. The details of the implementation, along with the code developed in this work are also provided.
Harnessing disorder to decouple extension and shear in kirigami metamaterials
Kirigami transforms stiff sheets into compliant, shape-morphing structures via periodic cut patterns. These repeating motifs rotate with stretch, so that extension in general leads to shear, and anisotropic properties cannot be tuned independently. Biological tissues evade this through controlled disorder, such as graded fiber orientations in skin and hierarchical anisotropy in myocardium. Here, we show that engineered disorder is tunable design degree of freedom: stochastic kirigami accesses a continuous and far broader region of mechanical response than periodic patterns, including programmable anisotropy with near-complete suppression of extension–shear coupling. Because disordered patterns lack a simple parameterization, we navigate this space with a geometry-aware graph neural network (GNN) that maps cut topology to full nonlinear, bidirectional stress–strain response, coupled to a genetic algorithm for inverse design of kirigami patterns that reproduce target stress–strain responses in two perpendicular loading directions. The GNN is trained an order of magnitude faster and more accurately than image-based models. Fabricated elastomer samples reproduce the targeted nonlinear, anisotropic responses, closing the loop from design to physical component. By turning disorder into a control axis for directional stiffness, this work points toward architected materials that stretch without parasitic shear, enabling tissue-interfacing devices matched to the anisotropy of living tissue.
Dendrite-growth resistance in solid state electrolytes
We address dendrite growth resistance in solid electrolytes in all-solid-state Li-ion batteries. First, we introduce a dendrite-growth model in the electrical, ionic-flow, temperature, and stress fields in the solid-state electrolyte. The model analysis shows that the applied electrode potential generates field singularities at the dendrite/crack tip, driven by electrochemical ion-deposition pressure and the thermal-conductivity jump at the dendrite/electrolyte interface. An extended J- integral is introduced to characterize the stress and temperature-gradient singularities and associated driving forces of the dendrite/crack tip in the heat-generating medium. These singularities are then used to construct universal threshold-current-density diagrams of dendrite/crack growth in solid electrolytes. From the universal diagram, the resistance to dendrite/crack growth is defined. This resistance depends on the dendrite/crack length, film thickness, and the properties of the electrochemically active solid-state electrolyte. These properties include ion conductivity, thermal conductivity, thermal expansion coefficient, elastic moduli, and toughness against dendrite/crack growth. The analyses show that the driving force from thermal stresses induced by Joule heating of the ionic current in the electrolyte is particularly significant for short dendrite/crack structures. The polarity of the temperature gradient in the electrolyte is found to be a primary factor in determining the threshold current density for short- dendrite/crack growth. The analysis offers methods for testing the dendrite/crack toughness of the electrochemically active solid-state electrolyte. It also guides the design of battery-cell packing sequences to maximize resistance to dendrite/crack formation and growth along the cooling path.
Joule-heating-induced temperature gradient
Electro-chemo-mechanical embrittlement/toughening
Impedance-Guided Programmable Transmission of Localized Deformation in Modular Soft Metamaterials
Soft metamaterials provide a promising platform for robotics, biomedical devices, and flexible electronics. The localized mechanical responses by nonuniform excitation are ubiquitous in soft materials, yet their controlled transmission across assemblies remains largely overlooked in metamaterial design, which critically constrains nontrivial functionalities with end-to-end and long-range deformation transmission. Here, we introduce an impedance-guided design framework that enables programmable transmission of localized deformation in modular soft metamaterials, achieving behaviors unattainable by intuitive design. By establishing a nonlinear model considering position-dependent interactions and integrating the concept of mechanical impedance within metamaterials, we regulate assembly-level transmission solely through unit-cell topology optimization. The resulting framework enables effective synthesis of module families, allowing both homogeneous and heterogeneous assemblies to be custom-built with markedly enhanced transmission characteristics. Leveraging the highly combinatorial and extensible design space, we physically realize diverse on-demand displacement manipulation architectures, including obstacle-bypassing modular soft-metamaterial assemblies, defect-tolerant soft gripping, and embodied signal processing. Beyond deformation programming, the reconfigurability and reassemblability of these soft modules can embed electric logic signals, enabling energy-efficient and low-latency information processing through compliant-switch-controlled mechanical LED displays and wearable finger-motion-sensing controllers. Our method provides fundamental insights into localized deformation transmission in modular soft metamaterials and establishes a scalable route toward embodied-intelligence material systems, particularly for soft-metamaterial-centric actuation, sensing, and collective computing.
Modeling damage and fracture in additively manufactured polymeric triply periodic minimal surface lattices
Architected triply periodic minimal surface (TPMS) lattices offer superior specific energy absorption, toughness, fatigue strength, and tunability. While recent advancements have established rate-dependent viscoplastic constitutive models to capture the complex nonlinear deformation response of additively manufactured polymeric TPMS structures, predicting fracture and the resulting structural failure remains a significant challenge. We address this by performing systematic experiments on unit cells and lattices of various sizes under tension, compression, and non-monotonic loading. The experiments inform the development of a new constitutive model that captures the damage and fracture behavior of polymeric TPMS lattices. We first implement a high-fidelity viscoplastic deformation constitutive model from Ma et al. (2026) into finite element software Abaqus/Explicit via a user material subroutine. We then propose a damage initiation criterion for amorphous polymers based on stored elastic energy and equivalent plastic strain. The damage model is implemented in Abaqus using gradient-damage framework following Konale and Srivastava (2025). The damage model and numerical simulation capability are quantitatively and qualitatively validated using experimental results for a unit cell under non-monotonic loading and lattices under tension. The proposed damage model and simulation capability enable in silico design of architected polymer structures.
A robust and versatile parallel FFT-based mechanical solver for general non-periodic and periodic boundary conditions
General boundary conditions are implemented within a fast Fourier transform framework for linear and non-linear mechanical problems using small or finite transformation formulations. In the context of parallel computing (distributed memory), we present a framework that enables the combination of periodic and non-periodic (Dirichlet or Neumann) boundary conditions. Taking advantage of the link between non-periodic boundary conditions and the symmetries of the relevant components of the fluctuation displacement and stress fields, discrete trigonometric transforms are employed to adapt the classical Moulinec–Suquet fast Fourier transform approach. The present study employs an original displacement-based fixed-point algorithm in combination with a convergence acceleration method in order to solve boundary value problems. Finite difference approaches are used to build the discrete Green operators associated with a pre-conditioner (reference material), whose choice depends on the loading type and the small or finite transformation frameworks. The newly developed double tetrahedron scheme is employed to investigate non-periodic problems. Outcomes are compared to those of the classical hexahedral scheme. The robustness and computational efficiency of the presented parallel solver is demonstrated through numerical experiments of non-trivial loading scenarios (tension, bending, normal-mixed loading, torsion-bending), complex and densely discretized microstructures and diverse behavior laws (elasticity, isotropic plasticity, crystal plasticity), within small and finite transformation frameworks.
Discrete Trigonometric and Fourier transforms
Morphology-Property Interplay in Chemo-Mechanics of Ion-Intercalation Active Particles
Morphology, material property, and mechanical constraint jointly govern the chemo-mechanical behavior of ion-intercalation particles, yet their coupled effects remain insufficiently understood. Here we establish a thermodynamically consistent single-particle framework and combine analytical solutions with multiphysics simulations to determine how these factors regulate lithiation and stress generation. We study hollow spherical, cylindrical, and ellipsoidal particles with isotropic or transversely isotropic material properties under fully constrained, inner-free, or unconstrained boundary conditions. We show that the transient lithiation pathway and the associated stress and strain fields are governed not by morphology, property, or constraint alone, but by their coupled interaction: isotropic particles are sensitive to the mechanical constraint, whereas transversely isotropic particles exhibit persistent heterogeneous lithiation dominated by anisotropic diffusivity. Flux decomposition analysis reveals that the mechanical contribution to Li flux is negligible in spheres but dominant in ellipsoids. Correlation analysis further shows that Li concentration and volumetric strain exhibit strong anti-correlation in unconstrained particles but weak correlation under full constraints. Bayesian optimization of hollow ellipsoids identifies Pareto-optimal morphologies that balance lithiation capacity against peak tensile stress. These results provide a unified framework for the morphology-property interplay in intercalation particles and offer morphology design principles for chemo-mechanical stability.
Spatial Variation of Stiffness and Edge Beam Participation in RC Box Culverts
Reinforced concrete (RC) box culverts exhibit a non-uniform structural response under two patch localized loading due to plate action and boundary elements at the opening that are not accounted for in conventional 2-D models for design and analysis purposes. This paper addresses the spatial variation in the stiffness of RC box culverts with and without edge beams, using linear-elastic finite-element analysis. Stiffness was derived from the load-deflection responses at various locations along the culvert length. The results show that the stiffness depends on the load location, indicating that the culvert system does not behave as a uniform structural element. The presence of the edge beam leads to local stiffness amplification near the opening, which gradually diminishes towards midspan. The reaction distributions also indicate that the edge beam acts primarily as a boundary stiffener and that the global load-transfer mechanism remains unchanged. The contribution of the edge beam was isolated using a relative stiffness parameter that exhibits a consistent exponential-decay trend across different spans. Based on these observations, a span sensitive formulation was developed to describe this behavior as a function of distance, and it showed good agreement with finite-element results (R2 = 0.979). The proposed framework offers a physically interpretable representation of edge-beam stiffness participation in an indeterminate structure and adds to a better understanding of the load distribution behavior in the culvert with and without edge beams.
Mechanistic understanding of sodium storage from operando stress measurements in hard-carbon thin-films
Hard carbon is a promising anode material for sodium-ion batteries, yet the sodium-storage mechanism in hard carbon remains debated. Here, operando substrate-curvature measurements on hard-carbon thin films were used to probe the volume-expansion-induced stress during sodiation and desodiation. The volume-expansion-induced stress rises steeply in the higher-voltage sloping region during initial sodiation, then it increases at a slower rate in the low-voltage plateau region, followed by a second steep slope below approximately 0.025 V at the end of sodiation. These regimes in stress evolution are consistent with intercalation-dominated storage in the sloping region, pore filling or adsorption in the plateau region, and a plating process at low potentials. The stress relaxation profile evolves asymmetrically during desodiation, indicating that sodium removal from hard carbon is not a simple reversal of sodium insertion into it. The irreversibility in sodiation-desodiation profiles of hard carbon is associated with sodium retention and/or (micro)structural rearrangement.
Finite Element Modeling of the Structural Response of a Reinforced Concrete (RC) Box Culvert with Edge Beam and Skewness
Reinforced concrete (RC) box culverts are buried in structures installed under highways for various purposes, primarily for water flow. Usually, non-skewed culverts are preferred, but in areas with topography that require skewed culverts, cast-in-place RC culverts may be considered. Researchers have extensively investigated the structural performance of the non-skewed culverts. Limited research studies are available to understand the structural behavior of RC-skewed culverts. Therefore, this study examined the impact of a skew angle in the top slab on the structural response of an RC culvert provided with an edge beam. Finite element modeling (FEM) was employed to simulate RC culverts using a shell element model with varying skew angles of 0, 15, 30, 45, and 60 degrees in Abaqus. A solid model was introduced to confirm the shell model’s limitation of capturing localized failure in the edge beam. It is observed that the culvert’s edge beam with adequate shear reinforcement is essential to maintain its ultimate capacity. The study establishes a finite-element-based approach for evaluating load-deflection response in skewed culvert systems, supported by solid-element modeling to examine localized effects. The results provide a consistent basis for comparison across configurations, with emphasis on relative response trends rather than absolute prediction of ultimate capacity.
The discrete origin of configurational forces
We develop a discrete formulation of configurational forces for open subsystems of a finite particle system. Membership in an open subsystem may change by particle admission or deletion. We first use zero-one membership functions to formulate standard balances for mass, standard momentum, angular momentum, and kinetic energy: continuous terms are evaluated on fixed-membership subintervals, but admission and deletion enter through jump-transfer sums. We then introduce a configurational momentum balance for open subsystems and configurational contact-force resultants associated with the ordered separation between a subsystem and its complement. To describe membership changes geometrically, we introduce an auxiliary smooth description based on a scalar membership field on the Euclidean space of particle placements. For each particle, the level set through the current particle position is used as a local membership cut, and the gradient at that position defines a normal to the cut. We assign an observed velocity to these level sets; its tangential component can be changed without altering the smooth membership value or its rate. We require the working associated with membership transfer to be invariant under such tangential changes, and we refer to this invariance as intrinsicity. If this invariance is imposed on a working involving only standard contact-force assignments and standard momentum transfer, then an artificial normality condition is imposed on the standard contact-force assignment. We therefore add configurational contact-force assignments. For a particle with assigned mass, standard momentum, standard velocity, and specific configurational momentum, we impose the intrinsicity requirement and obtain that the specific configurational momentum is the negative of the standard velocity. For nondissipative membership transfer, we use a mechanical energy imbalance to identify the particlewise membership tension: the combined standard-plus-configurational cut-force assignment equals the difference between the membership interaction-energy increment and the kinetic energy, multiplied by the cut normal. After imposing localization hypotheses for contact-force assignments, current representative volumes, and energy increments, we obtain the continuum relation $\bu=-\bv$ for the specific configurational momentum and the spatial inertial Eshelby relation $\C=\rho(\psi-\tfrac12|\bv|^2)\sidem-\T$, where $\C$ is the spatial configurational stress, $\T$ is the Cauchy stress, $\rho$ is the mass density, $\psi$ is the specific free energy, $\bv$ is the standard velocity, and $\sidem$ is the identity tensor.
A new perspective in linear Cauchy Elasticity: variational minimum principles for statics, dynamics, and heterogeneous materials
A variational minimum principle for linear elastodynamics of a possibly heterogeneous material without a stored energy function is developed. It involves a change of variables to dual fields, and results in a degenerate elliptic Euler-Lagrange system, even when the primal elastodynamics is hyperbolic. Uniqueness assertions for the dual dynamic and static problems and implications of the degenerate ellipticity are sketched. Some implications pertaining to heterogeneous materials and ones with indefinite elastic moduli are discussed.
Driven Odd Elasticity in Passive Mechanical Metamaterials
We present a mechanical mechanism leveraging passive mechanical components, i.e. chiral gears and a square lattice metamaterial, to demonstrate driven odd elasticity in a mechanical metamaterial. The mechanism couples tension and shear in a non-reciprocal way, resulting in an odd shear modulus. The emergence of this odd shear modulus enables non-conservative work in a standard quasistatic strain cycle, and further enables the non-Hermitian skin effect in dynamics. Our results demonstrate that odd elasticity can be achieved in mechanical structures using passive elements without electronic components coupled with feedback or robotic control systems.
Odd Elasticity Metamaterial non-Hermitian
Conformal Mechanics of Serpentine Meshes for Millimeter-Scale Neural Probes
The mechanical reliability of implantable neural probes depends on their ability to accommodate large deformations during implantation and chronic operation while limiting stress concentrations within conductive interconnects. Despite extensive studies of serpentine interconnects in stretchable electronics, their mechanics under loading conditions relevant to cylindrical neural probes remain poorly understood. This work investigates the mechanics of two stretchable neural interface architectures, Single Serpentine Traces and Interconnected Mesh networks, designed with gold (Au) interconnects encapsulated in polyimide (PI) and embedded within a soft elastomeric substrate. Finite element analysis and experiments are used to characterize their response under stretching, conformal wrapping onto cylindrical neural probes, and bending of the wrapped structure. Deformation in both architectures is accommodated primarily through bending and rotation of the serpentine segments, with localized metal yielding initiating at strains of approximately 5.5–8.25% depending on loading direction and topology. During conformal wrapping onto probes with radii as small as 0.4 mm, the Single Serpentine Traces architecture remains below the Au yield stress, reaching a maximum stress of 145 MPa, whereas the Interconnected Mesh develops localized yielding at interconnection junctions with peak stresses exceeding 215 MPa. Subsequent bending produces only modest stress amplification, although the Interconnected Mesh exhibits approximately twofold greater stress sensitivity than the Single Serpentine Traces. The results demonstrate that rotational compliance governs stress redistribution, while geometric constraints at interconnection junctions control failure initiation. These findings reveal a fundamental tradeoff between electrical connectivity and stress concentrations and provide mechanics guidelines for conformal neural probes and related stretchable bioelectronic systems.
Thermo-elastic properties of hydrated epoxy-graphene nanocomposites from ensemble-based molecular dynamics simulations
Epoxy-based materials are inherently hygroscopic, absorbing moisture from the environment, which can significantly alter their short and long-term performance. The presence of graphene is often considered as a potential candidate to act as a microscopic barrier, mitigating the adverse effects of hydration on the matrix. This study investigates the impact of hydration on the glass transition and elastic mechanical properties of epoxy resins and their graphene nanocomposites, focusing on water content up to 5 wt%. Using large-ensemble molecular dynamics simulations, we analyze the temperature-driven glass transition and mechanical response of both neat epoxy and epoxy-graphene systems under varying hydration levels. Our results reveal a distinct threshold at 3 wt% water content: below this, hydration primarily reduces the glass transition temperature, while mechanical properties remain unaffected. Beyond 3 wt%, however, the mechanical properties deteriorate, highlighting a non-linear sensitivity to water uptake. Furthermore, we emphasize the critical role of ensemble size in ensuring the reliability of molecular dynamics predictions for such heterogeneous systems. Our simulations demonstrate that ensembles substantially larger than current state-of-the-art standards are necessary to achieve converged distributions of the predicted mechanical properties, particularly in highly heterogeneous hydrated epoxy-graphene nanocomposites. These findings provide novel insights into the hydration behavior of epoxy-based materials and underscore the potential of graphene to enhance their environmental resistance. This work also advances the understanding of structure-property relationships in polymer nanocomposites, offering guidance for the design of more robust materials in humid environments.
First-Order Compatible-Strain Mixed Quadrilateral Finite Elements for $2$D Nonlinear Elasticity
Compatible-strain mixed finite elements (CSMFEs) use the differential complex of nonlinear elasticity to construct discretizations that preserve the underlying topological structure. Existing developments of CSMFEs have focused on simplicial meshes, for which efficient formulations have been developed for first- and second-order elements and successfully applied to compressible and incompressible nonlinear elasticity problems. In this paper, we develop two compatible-strain mixed formulations for quadrilateral elements applicable to compressible and incompressible nonlinear elasticity. For quadrilateral elements, the Piola transformation preserves vector fields tangent and normal to element edges only for special classes of elements, such as rectangles, parallelograms, and trapezoids. Consequently, it cannot be used to construct compatible shape functions for general quadrilateral elements. To overcome this limitation, the shape functions are computed directly in the physical space using numerical integration. N\'ed\'elec shape functions of the first kind are used to construct the interpolation of the displacement gradient. We also introduce a new class of shape functions for the stress tensor that is compatible with the displacement and displacement-gradient discretizations. The formulation for compressible elasticity follows the framework previously developed for first- and second-order simplicial CSMFEs, whereas a new formulation is proposed for incompressible elasticity. The incompressible formulation employs element-level condensation of the pressure field and therefore does not increase the number of global degrees of freedom relative to the corresponding compressible formulation. These developments extend the compatible-strain mixed finite element framework from simplicial to general quadrilateral meshes for both compressible and incompressible nonlinear elasticity. Numerical examples demonstrate that the proposed quadrilateral elements can successfully solve problems that require second-order simplicial CSMFEs while using fewer degrees of freedom.
Interpenetrating Gyroid Lattice - Foam Composites with Synergistically Enhanced Stiffness and Energy Absorption for Impact Mitigation
Advanced lightweight structural materials require both high stiffness and energy-absorption capacity for demanding protective applications. Low-density polymer foams offer excellent specific energy absorption but have inadequate stiffness and structural stability for load-bearing applications. Stiff structural materials, on the other hand, provide load-bearing capacity at the expense of deformability and energy absorption capacity. Here, we overcome this trade-off by creating lattice–foam interpenetrating phase composites (IPCs) via in-situ foaming that integrate stochastic polymeric foam with elastomeric gyroid lattices featuring robust interfacial integration and precisely controlled architecture and density profile. This hybrid design couples the high specific energy absorption of foam with the tunable stiffness and structural integrity of architected lattices, leading to synergistically improved mechanical performance that neither standalone constituent achieves. Introducing a density gradient to IPC enables spatially programmed deformation and gradient-governed load transfer. Under impact loading, the IPCs exhibit dynamic synergy beyond the response of each individual component, markedly reducing transmitted peak stress and acceleration while simultaneously increasing energy absorption at low strain levels. The gyroid lattice–foam IPCs offer a versatile strategy for designing lightweight structural materials that integrate load-bearing capacity with impact-mitigation capability.