Eigenstrain tomography combines diffraction-based strain measurement with elasticity theory to reconstruct full three-dimensional residual stress fields within solids. Notwithstanding a number of recent examples, the uniqueness of such reconstructions has not yet been clearly established. In this paper, we examine the underlying inverse problem in detail and construct explicit counterexamples demonstrating non-uniqueness for a recent implementation of x-ray eigenstrain tomography involving reconstruction from a single measured component of strain. We follow on to explore minimum conditions for well-posedness and conclude that the full elastic strain tensor within an isotropic sample can be uniquely reconstructed from three measured components; specifically the three shear components, or the three diagonal components. We further prove two key results related to eigenstrain reconstruction in a general sense; 1. That any possible residual stress field can be generated by a diagonal eigenstrain and 2. That residual stress fields exist that cannot be generated by isotropic eigenstrains. Together, these findings establish rigorous minimum experimental and computational requirements for well-posed eigenstrain tomography techniques and inverse eigenstrain problems in general.

Controlling the crystal phase and lattice mismatch of semiconductors offers a powerful route to engineer electronic and optical properties of heterostructures. As a consequence, semiconductors in the wurtzite phase are increasingly sought after, superseding the thermodynamically favored cubic zinc blende phase. Empirical atomistic modeling, required for large scale simulations of heterostructures and their properties, relies heavily on valence force field (VFF) methods to find the equilibrium atomic positions in an alloy. For zinc blende crystals, VFF models are well-established. In the case of wurtzite, VFF parameters are frequently adopted without rigorous analysis, despite subtle but critical differences from the zinc blende structure. Such an approach can compromise accuracy in describing material properties, since the structural differences between zinc blende and wurtzite directly influence electronic and optical characteristics. Based on the analytical VFF model by Tanneret al(2019Phys. Rev. B100094112), and using structural similarities between wurtzite and [111]-oriented zinc blende crystals, we guide the development of a wurtzite VFF without introducing additional parameters. Our framework utilizes analytic expressions and minimization routines to project zinc blende models onto wurtzite systems. Beyond elastic tensors, we train the model to reproduce bond length asymmetries and band gaps by using output of the VFF model in density functional theory (DFT) electronic structure calculations. Applied to wurtzite III-N compounds and BN, the model accurately reproduces targeted observables but also properties it has not been trained on, including the internal parameteru. We further validate the model on highly mismatched alloys such as (B,Ga)N and (B,In,Ga)N, exhibiting good agreement between VFF and DFT results when using identical supercells in these calculations.

Abstract For solids and structures composed of architected metamaterials, detailed micro-level numerical modeling becomes a critical bottleneck due to both memory and processor requirements. For periodic metamaterials, computational homogenisation provides an attractive alternative, whereas multi-scale continuum theories provide an appropriate framework for capturing size effects stemming from the metamaterial architecture. This article focuses on asymptotic computational homogenisation for three-dimensional strain-gradient elasticity (SGE) by assessing its numerical performance for periodic unit cells. After revisiting the variational formulation of SGE, the derivation of homogenised fourth- and sixth-order elasticity tensors is accomplished, leading to solving micro-level corrector problems with periodic boundary conditions. Regarding verification of the corresponding numerical implementation, two stabilisation strategies are assessed: a global-constraint formulation and a Tikhonov regularisation—the latter avoids additional Lagrange multipliers and turns out to be both efficient and stable. The workflow is implemented by combining the finite element software COMSOL with MATLAB LiveLink for obtaining the homogenised (meta)material tensors. As a virtual validation phase, the homogenised constitutive tensors are involved in three-dimensional SGE simulations to solve macroscopic boundary-value problems of lattice structures via isogeometric analysis. This phase is accomplished within the open-source GeoPDEs-software via user-defined subroutines developed for SGE to obtain conforming Galerkin approximations. Verification and validation for the homogenisation approach cover p - and h -convergence studies, mesh-type comparisons between hexahedral and tetrahedral elements, sensitivity to unit-cell volume fraction, and a comparison between the global-constraint formulation and the Tikhonov regularisation. Results show that higher-order basis functions essentially accelerate convergence, mesh type differences become negligible for sufficiently rich approximation spaces, and gradient moduli peak at an intermediate range of volume fractions. Simulations for cantilever lattice beams confirm that SGE predicts bending deflections more accurately than classical elasticity due to the size effect phenomenon present in bending and shear deformations.

We present a general, constructive procedure to find the basis for tensors of arbitrary order subject to linear constraints by transforming the problem to that of finding the nullspace of a linear operator. The proposed method derives from well-established representation-theoretic foundations and utilizes standard numerical linear algebra techniques that are highly optimized and well-behaved. Our primary applications are in mechanics where modulus tensors and so-called structure tensors can be used to characterize anisotropy of functional dependencies on other inputs such as strain. Like modulus tensors, structure tensors are defined by their invariance to transformations by symmetry group generators but have more general applicability. The fully automated method is an alternative to classical, more intuition-reliant methods such as the Pipkin–Rivlin polynomial integrity basis construction. We demonstrate the utility of the procedure by: (a) enumerating elastic modulus tensors for common symmetries and (b) finding the lowest-order structure tensors that can represent all common point groups/crystal classes. Furthermore, we employ these results in two calibration problems using neural network models following classical function representation theory: (a) learning the symmetry class and orientation of a hyperelastic material given stress–strain data and (b) representing strain-dependent anisotropy of the stress response of a soft matrix-stiff fiber composite in a sequence of uniaxial loadings. These two examples demonstrate the utility of the method in model selection and calibration by: (a) determining structural tensors of a selected order across multiple symmetry groups and (b) determining a basis for a given group that allows the characterization of all subgroups. Using a common order in both cases allows sparse regression to operate on a common function representation to select the best-fit symmetry group for the data.

Cloaking based on the coordinate transformation theory presents an alternative avenue for developing an earthquake protection method by precisely controlling seismic waves, distinct from the conventional band structure engineering method that attenuates seismic waves within band gaps. Seismic cloaks have remained elusive due to the lack of formal invariance in the Navier equations underlying the operation of the cloak. This challenge is primarily attributed to the elastic tensor breaking the minor symmetry and requiring polar and chiral characteristics that no known materials exhibit. Here, we propose physically realizable seismic cloaks based on a discrete transformation elasticity theory by designing uniform lattice-based polar metamaterials capable of exhibiting polar and chiral elastic tensors. Numerical simulations demonstrate excellent cloaking performance under different types of seismic wave loads including Rayleigh waves, longitudinal (P) waves and transverse (S) waves, and confirm the validity of the proposed cloaks over a wide frequency band.

ABSTRACT Understanding the behavior of wave propagation in viscoelastic materials is essential across numerous scientific and engineering fields. It improves our ability to predict and interpret the dynamics of viscoelastic materials and wave interactions in complex environments. This paper presents a tensor‐based framework for analyzing and visualizing frequency‐dependent wave propagation in three‐dimensional linear viscoelastic media. Fractional‐order damping is incorporated through combining elastic and viscous tensors, leading to complex stiffness models governed by fractional calculus. These models are integrated into the frequency‐wavenumber domain formulation of the Green's function, which captures anisotropic elasticity and directional attenuation. While it is well established that damping shifts wave poles into the complex plane, the present framework embeds this effect directly into a frequency‐dependent Green's function formulation extended to fractional viscoelasticity. It demonstrates that viscoelastic damping regularizes the Green‐Christoffel operator by removing real‐direction singularities. Furthermore, polar decomposition is applied to the Green's function tensors, uniquely separating them into stretch (amplitude, damping) and rotation (phase, direction) components. This dual spectral‐geometric representation separates amplitude (stretch) from direction‐dependent phase evolution encoded in the unitary rotation tensor that reflects complex polarization. Overall, this framework offers a structured method for visualizing viscoelastic wave propagation and supports anisotropic material characterization.

Mechanical flexibility in molecular crystalline materials represents a compelling paradigm shift from the long-held perception of crystals as inherently brittle solids. Herein, we demonstrate a brittle-to-elastic transition by subtle molecular modification in a pair of structurally analogous aromatic amides; N-[(4-methoxyphenyl)methyl]formamide (N4MFA, Crystal 1) and N-benzylformamide (NBFA, Crystal 2). Despite their close structural similarity, Crystal 1 exhibits brittle fracture under minimal stress, whereas Crystal 2 shows 1D elastic flexibility with reversible bending. Structural, computational, and mechanical analyses reveal that this contrast arises from substituent-controlled supramolecular packing. In Crystal 1, the methoxy (-OCH3) group promotes dense, anisotropic packing, leading to rigidity and fracture under stress. Removing the substituent in Crystal 2 enhances isotropy, π-π stacking, and interlocked packing, enabling reversible strain during elastic bending. Nanoindentation, energy framework, and elastic tensor analyses confirm this transition: Crystal 2 shows near-isotropic stiffness (Emax/Emin=1.65) and interconnected energy networks, whereas Crystal 1 exhibits pronounced anisotropy (Emax/Emin=3.95) and 1D cohesion. Hirshfeld surface analysis supports more balanced contacts in the elastic crystal. This work establishes a direct structure-mechanical correlation, showing that minor chemical modifications can tune flexibility and provide insights to guide the development of adaptive crystalline materials.

In this contribution, we investigate the stability of boron phosphide in various low dimensional forms ranging from the 3D bulk, the 2D slab model to the 1D single- and multi-walled zigzag nanotubes. A variety of energetic and geometric parameters including relaxation, cohesive and formation energies, polarisability, piezoelectric and elastic tensors components, and equilibrium lattice parameters have been reported. All arrangements are confirmed to exhibit a relatively wide band gap with properties dependent of geometric parameters. A connection between the 2D phonon modes and those of the 1D zigzag nanotubes has been established. Comparisons between IR and Raman of the single- and multi-walled nanotubes reveal that symmetry reduction leads to more active modes. By contrast, we found that angles and bond lengths only slightly deviate from those of the 1D single-walled nanotubes. By increasing the number of walls, the low frequency phonon modes become softer and shift toward lower wavelengths while high frequency phonon modes become harder with a blue shift owing to possible mechanical distortions that could occur between walls. These outcomes are expected to guide and motivate both experimentalists and theorists to design and optimize new emerging low dimensional inorganic materials for next generation nanodevices. All computational modeling has been performed based on the density functional theory methodology with the B3LYP hybrid functional as implemented in the CRYSTAL23 program. The electronic wave-functions of the periodic 3D, 2D and 1D boron phosphide ground state are expressed with Bloch functions which are constructed as linear combination of Gaussian local type functions. Let us recall that the mode frequencies at the center of the Brillouin zone are obtained from the diagonalization of the mass-weighted Hessian matrix of the second derivatives of the total energy per cell with respect to atomic displacements. Therefore, IR and Raman spectra of all arrangements are simulated using the Coupled Perturbed Hartree-Fock or Kohn-Sham CPHF/KS approach.

The mechanical properties of molecular crystals are fundamentally important in their industrial applications across pharmaceuticals, agrochemicals, energetic materials and other areas. Despite this, complete measurement or even computational prediction of elastic tensors for molecular crystals is anything but commonplace. The absence of rapid, reliable and broadly applicable methods in this endeavor frequently leads chemists to rely on intuitive ideas and examination of pairwise intermolecular interactions such as hydrogen- or halogen-bonds in order to rationalize the mechanical behavior of molecular crystals. Such perspectives are widespread in contemporary literature, but the extent to which these notions yield reliable and quantifiable insight is itself relatively unexplored. We propose a simple approximation, the Equilibrium Pairwise Model (EPM), compatible with any method to predict intermolecular interaction energies, that directly and efficiently yields an estimate of the complete elastic tensor. The protocol can be performed for any given molecular crystal structure, even those directly from experiment (i.e., without geometry optimization), and is guaranteed by construction to yield a positive-definite result─in contrast to conventional methods where computing valid elastic tensors for molecular crystals can prove challenging even for well-established and otherwise accurate model chemistries. We examine the accuracy of this protocol, along with other classical and contemporary methods, against experiment and periodic (plane-wave) density functional theory calculations to assess their reliability and accuracy. Through examination of the failures and successes, we aim to provide chemical insight into the kinds of materials where the model and, more broadly, thinking based on pairwise intermolecular interactions can reliably explain mechanical or other material properties and where they should be avoided.

Abstract Residual stress measurement at different depths is critical for structural deformation assessment and fatigue life prediction. In this work, a numerical modeling approach based on acoustoelastic theory was developed to investigate the propagation behavior of critically refracted longitudinal (LCR) waves under a prestress gradient. A prestress field varying along the thickness direction was introduced to analyze its influence on LCR wave velocity and time of flight. The equivalent elastic response tensor under gradient prestress conditions was derived using the acoustic tensor formulation incorporating Lamé and Murnaghan constants. By correlating time of flight variations with known stress depth distributions, a relationship between LCR wave frequency and penetration depth was established. Four-point bending experiments were conducted to validate the proposed model. The results show that the approach effectively characterizes variations in LCR wave penetration depth under prestress conditions, providing a theoretical and numerical basis for ultrasonic evaluation of residual stress at different depths.

Abstract Stress-induced anisotropy is prevalent in the subsurface and has a significant impact on seismic wave propagation. Employing the framework of acoustoelastic theory, we establish velocity-stress formulations in the first-order form tailored for media exhibiting stress-induced anisotropy. A numerical implementation based on high-order staggered-grid finite-difference is proposed for wavefield simulation. The study demonstrates that isotropic media subjected to overburden stress exhibit an effective elastic stiffness tensor with VTI (Vertical Transverse Isotropy) symmetry, and the degree of anisotropy scales proportionally with the applied stress level. Numerical simulation results indicate that as the overburden stress increases, the wavefront morphology gradually transitions from circular (isotropic) to elliptical (anisotropic). This research provides a theoretical foundation for understanding seismic wave propagation in stressed media and for subsequent inversion of stress parameters.

Abstract Two critical questions in brittle rock mechanics are how rocks developed localized strains and to what extent internal stress heterogeneity controls this localization and subsequent macroscopic failure. Definitive answers have not yet emerged, but would provide insight into rock fracture mechanics as relevant to hydrocarbon extraction and sequestration. Here, we use synchrotron X‐ray tomography (XRT) and 3D X‐ray diffraction (3DXRD) during uniaxial and triaxial tests on Nugget and Bentheimer sandstones to examine strain and stress localization prior to mechanical failure. 3DXRD was used to measure intra‐granular lattice strains which were used to compute elastic stress tensors of each grain. Digital volume correlation (DVC) was applied to XRT images to determine the strain field in the sample. Both samples featured marked spatial heterogeneity, localization, and temporal persistence of elevated stresses and strains during their mechanical deformation toward failure. Both samples featured a majority of grains with at least one principal stress component that was tensile, a signature of the influence of heterogeneity on stress transmission. Measurements further revealed that compressive stress orientations and statistics evolved in a similar manner to those of inter‐particle forces in loose granular materials, with triaxially‐compressed rock exhibiting enhanced grain stress heterogeneity compared to uniaxially‐compressed rock. Our results complement recent work by others who employed XRT and scanning 3DXRD to study triaxially‐compressed sandstone, but extend those results to uniaxial compression, sandstones of varied porosity, and grain stress measurements throughout the 3D full extent of the samples rather than in a single layer examined with scanning 3DXRD.

Abstract This work derives an O(h3) electroelastic plate theory for thin dielectric sheets from a three-dimensional variational formulation, incorporating material nonlinearity and Maxwell stresses. The theory captures wrinkling behavior in the presence of mechanical traction and externally applied electric fields. The resulting two-dimensional formulation is specialized to an isotropic, incompressible material obeying reflection symmetry about the sheet mid-surface, and field-dependent explicit expressions are obtained for the elasticity tensor, electroelastic coupling coefficient, and permittivity. The strong-form equations are then derived from the two-dimensional variational formulation by setting the first variation of the potential to vanish. These equations are employed to analyze the wrinkling response of a stretched rectangular sheet subjected to a uniform electric field about a biased reference state. Approximate analytical solutions are obtained via linearization of the plate equations for small slopes, to understand the influence of electric field intensity and applied stretch on the onset and amplitude of wrinkling. The results show that increasing voltage raises the critical loads and wrinkle amplitude while reducing wrinkle count, thereby establishing voltage as a tunable parameter to control wrinkling.

Mechanical and elastic properties of materials are among the most fundamental quantities for many engineering and industrial applications. Here, we present a formulation that is efficient and accurate for calculating the elastic and bending rigidity tensors of crystalline solids, leveraging interatomic force constants and long-wavelength perturbation theory. Crucially, in the long-wavelength limit, lattice vibrations induce macroscopic electric fields, which further couple with the propagation of elastic waves, and a separate treatment on the long-range electrostatic interactions is thereby required to obtain elastic properties under the appropriate electrical boundary conditions. A cluster expansion of the charge-density response and dielectric screening function in the long-wavelength limit has been developed to efficiently extract multipole and dielectric tensors of arbitrarily high order. We implement the proposed method in a first-principles framework and perform extensive validations on silicon, Na Cl , Ga As and rhombohedral Ba Ti O 3 as well as monolayer graphene, hexagonal BN , Mo S 2 , and In Se , obtaining good to excellent agreement with other theoretical approaches and experimental measurements. Notably, we establish that multipolar interactions up to at least octupoles are necessary to obtain the accurate short-circuit elastic tensor of bulk materials, while higher orders beyond octupole interactions are required to converge the bending rigidity tensor of two-dimensional crystals. The present approach greatly simplifies the calculations of bending rigidities and will enable the automated characterization of the mechanical properties of novel functional materials.

Additively manufactured polymer lattices are increasingly used in biomedical and structural applications due to their tunable mechanical properties and architectural similarity to biological materials. However, accurately resolving their anisotropic elastic response remains challenging due to fabrication inconsistencies, energy loss mechanisms, and differences between static and dynamic characterization techniques. In this study, a dynamic technique, resonant ultrasound spectroscopy (RUS) was applied to a stereolithography-fabricated body-centered tetragonal (BC-Tetra) lattice composed of a polyurethane-like resin. Elastic constants were extracted from both experimental and model (FEA) eigenfrequencies using a particle swarm optimization (PSO) scheme with modified parameter tuning to improve exploration of the non-convex inversion space. Comparison of inverted elastic tensors showed strong agreement for the axial stiffness,C33 and shear-related coefficients C44​ and C66, , while in-plane stiffness constants C11​ and C12​ and the axial coupling term C13 exhibited the greatest variance, reflecting inversion sensitivity and the limited number of resonances below the continuum cutoff. Engineering moduli derived from RUS were internally consistent and in-plane values agreed closely with FEA predictions, but quasi-static measurements of the in-plane moduli E1=E2 and out-of-plane modulus E3 were 20% and 32% lower, respectively. This divergence highlights fundamental differences in compliance across loading regimes: quasi-static compression is strongly influenced by strut bending, resin pooling, and boundary effects, whereas RUS probes free-free global vibrational modes that enforce affine deformation at the scale of the entire lattice, suppressing local compliance mechanisms and yielding higher effective moduli. Our study is an effort to test the boundaries of RUS for high-loss polymer lattices and to develop practices that could eventually reduce operator dependence.

This study investigates the structural, mechanical, electronic, optical, and thermodynamic properties of FrHF 3 (H = Mg, Ca) perovskites using Density Functional Theory (DFT) with the Generalized Gradient Approximation (GGA) and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional. The structural and mechanical stability of these phases is confirmed by their negative formation enthalpy, tolerance factor, and positive phonon dispersion curves, indicating their potential for experimental synthesis. The compounds exhibit direct band gaps, with an increase from 5.932 eV (FrMgF 3 ) to 6.47 eV (FrCaF 3 ), influencing their optoelectronic behavior. The electronic structure was analyzed using partial (PDOS) and total (TDOS) density of states to assess electron localization across different energy bands. Additionally, the ELATE program was employed to evaluate elastic tensors and generate 2D and 3D representations. High bulk and Young's moduli suggest promising industrial applications for FrHF 3 (H = Mg, Ca). The analysis of mechanical properties, including Pugh's ratio, Poisson's ratio, and Cauchy pressure, reveals that FrMgF 3 is ductile (Poisson's ratio = 0.31), whereas FrCaF 3 is brittle (Poisson's ratio = 0.25). Both compounds exhibit anisotropic characteristics. Optical properties, including dielectric function, absorption coefficient, optical conductivity, loss function, refractive index, reflectivity, extinction coefficient, transmittance, and absorbance, were examined. The materials demonstrate strong UV absorption, 327,109.25 cm −1 for FrMgF 3 and 302,462.55 cm −1 for FrCaF 3 , as well as high optical conductivity, with peak values of 6.66 (1/fs) and 6.30 (1/fs) for FrMgF 3 and FrCaF 3 , respectively. Additionally, their significant absorption index makes them suitable for applications in sensors, photodetectors, optoelectronics, and other optical technologies. Debye temperatures exceeding 200 K indicate high sound velocities, elevated melting points, and low minimum thermal conductivities, suggesting potential use in thermal barrier coatings (TBCs). Thermodynamic properties, such as specific heat capacity and entropy, were evaluated across a temperature range of 0–1000 K.

Abstract Metal three-dimensional printers utilizing powder bed fusion (PBF) hold promise for the development of novel ultrasonic transducers with complex geometries especially designed by topology optimization. Since the printed metal components exhibit anisotropy, an anisotropic elasticity tensor is required for finite element analysis and optimization. In this study, we fabricate test pieces of SUS316L stainless steel, which is commonly used in PBF and ultrasonic transducers. We estimate all elements of the stiffness tensor by utilizing the mechanical resonant characteristics of the test pieces. Anisotropic Young’s moduli and resonance quality factors are measured using a laser Doppler velocimeter for the longitudinal vibration mode that is noncontact-driven via electromagnetic acoustic transduction. Poisson’s ratios and shear moduli are estimated using a point excitation source driving the longitudinal and torsional modes, respectively. The obtained anisotropic elasticity tensor is verified using the bending vibration frequency and the vibration distribution of the PBF-fabricated ultrasonic tool.

Summary A notation for third-order elastic constants (TOECs) for solids of cubic symmetry is proposed. Based upon Walpole’s formulation of linear elasticity, we show that there are six distinct sixth-rank tensors of elasticity. This leads naturally to a diagonalizable basis for the six-dimensional space of cubic TOECs. The formulation provides simple relationships between the stiffness coefficients and the compliance coefficients for each of the six TOECs of cubic symmetry. It also provides a natural partition of stress and strain into hydrostatic and deviatoric parts at the linear and quadratic level. Relations for the isotropic Voigt and Reuss averages of cubic TOECs reduce to simple expressions in terms of the proposed six-dimensional basis for cubic TOECs. The six new cubic TOECs map to the set of three isotropic Voigt and Reuss averaged TOECs in subsets of one, two and three. The Voigt and Reuss isotropic TOECs are also the unique minimizers of the “distance” between elastic moduli of isotropic and cubic symmetries, providing a simple means to estimate the orientation averaged elastic constants of cubic crystals. An analysis of 59 cubic crystals reveals that the nonlinear contribution to hydrostatic stress is typically negative, which concurs with material stiffening behavior under hydrostatic pressure. The framework introduces a nonlinear anisotropy measure analogous to Zener’s index and establishes quantitative metrics for assessing departure from isotropy in TOECs.

We develop a method to fit high-temperature Gibbs free energy data for the development of interatomic potentials for atomic systems. The approach is based on Hamiltonian thermodynamic integration, enabling the identification of suitable potential parameters such that the system's free energy matches a specified target. The method can be readily combined with conventional fitting techniques for properties such as elastic tensors and liquid pair distribution functions. We validate the effectiveness of the approach using the Uhlenbeck-Ford model and embedded-atom method potentials for pure Ni phases and binary Fe1-xOx liquids under high-pressure and high-temperature conditions. Our framework provides an efficient strategy for incorporating free energy into interatomic potential fitting.

In this paper, from the microcosmic angle, the natural supramolecular structure in the wood cell wall of the pure wood has been established. The horizon or opinion on the positional relation and distribution mode among the cellulose microfibrils (CMFs) is presented. The arrangement of the matrix (hemicellulose and lignin) and systematical description of the supramolecular structure about cellulose have been studied. We sought to clarify the mechanical roles of cellulose and matrix polysaccharides by developing a model based on polymer physics that recapitulates aspects of assembly and tensile mechanics of cell walls. Next, we discuss the formulation, solution, and steps of using asymptotic homogenization theory to solve the effective elastic tensor of online viscoelastic composite materials with periodic microstructures and pores. The established supramolecular structure model has been simulated by using molecular dynamics simulation software. The theoretical closed-form solution and numerical solution are studied comparatively. Finally, experimental results were contrasted to the academic calculated value.