Linear Elasticity Research Articles

Approximation of acoustic black holes with finite element mixed formulations and artificial neural network correction terms

Wave propagation in elastodynamic problems in solids often requires fine computational meshes. In this work we propose to combine stabilized finite element methods (FEM) with an artificial neural network (ANN) correction term to solve such problems on coarse meshes. Irreducible and mixed velocity–stress formulations for the linear elasticity problem in the frequency domain are first presented and discretized using a variational multiscale FEM. A non-linear ANN correction term is then designed to be added to the FEM algebraic matrix system and produce accurate solutions when solving elastodynamics on coarse meshes. As a case study we consider acoustic black holes (ABHs) on structural elements with high aspect ratios such as beams and plates. ABHs are traps for flexural waves based on reducing the structural thickness according to a power-law profile at the end of a beam, or within a two-dimensional circular indentation in a plate. For the ABH to function properly, the thickness at the termination/center must be very small, which demands very fine computational meshes. The proposed strategy combining the stabilized FEM with the ANN correction allows us to accurately simulate the response of ABHs on coarse meshes for values of the ABH order and residual thickness outside the training test, as well as for different excitation frequencies.

Thermal vapor sulfurization of Ti3C2TX MXene to TiS2/carbon nanosheet cathode for long-life and high-loading all-solid-state lithium batteries

Stable and fast operation of all-solid-state lithium batteries (ASSBs) depends on durable and sufficient interfacial contacts between electrode components, especially when the loading and content of electrode active materials are high. Here, we propose promoting ionic/electronic transportations and contacts in sulfide electrolyte-based ASSBs by using TiS2/carbon nanosheets (TiS2/C-s) with large area-to-thickness aspect ratios as the cathode, which are prepared by simple thermal vapor sulfurization of exfoliated Ti3C2Tx MXene. The elastic 2D structure of the TiS2/C-s material can substantially improve interfacial contact with the solid electrolyte, minimize the diffusion distance of lithium ions, and buffer volume/stress changes during cycling, thereby ensuring the electro-chemo-mechanical stability and fast reaction kinetics of electrodes. Consequently, a 50 wt% TiS2/C-s cathode without any conductive carbon additives demonstrates a high discharge capacity (295 mAh g−1), long-term cyclic stability (94.7 and 74.7 % capacity retention after 250 cycles at 0.1 mA cm−2 and after 2300 cycles at 1 mA cm−2, respectively), and decent rate capability (126.0 mAh g−1 at 3 mA cm−2). Moreover, stable cycling with a high areal capacity (5.95 mAh cm−2) can also be achieved.

Anisotropic parameters and bulk waves propagation in arbitrarily-anisotropic materials

To reflect the anisotropic degree of materials and to further reveal its relevance with the elastic wave propagation in materials, the Thomsen's parameters were proposed for transversely-isotropic materials with weakly anisotropy and were generalized to materials with lower symmetries. This paper derives the complete 21 anisotropic parameters of arbitrarily-anisotropic (triclinic) materials to reflect their anisotropic degree by virtue of the Christoffel equation, as the propagation of generally coupled bulk waves is considered. First, the derivation is conducted analytically on basis of the coordinate transformations to the propagation direction of general bulk waves and to the stiffness matrix of arbitrarily-anisotropic materials involved in the Christoffel equation, but does not require the concrete expanded expression of the Christoffel equation. Then, the influences of any anisotropic parameters on the anisotropy characteristics on arbitrary planes and on the dispersion characteristics along any directions of bulk waves are investigated in detail, which are reflected by the variations of the slowness/phase velocity curves and the comprehensive dispersion curves, respectively. Finally, the influential rules are illustrated and summarized with combination of numerical examples about two typical exemplified anisotropic materials. It is found that the 21 anisotropic parameters for triclinic materials, which can actually be utilized to describe the anisotropic degree for all materials, are divided into five groups according to their influential patterns on the anisotropy characteristics of bulk waves. Furthermore, the dispersion characteristics of each bulk wave mode along specific direction are affected by three anisotropic parameters.

Hygrothermal effect of bio-inspired helicoid laminate plate for strengthening damaged RC beam

This study investigates the influence of moisture and Thermo-hygroscopic influence on bio-inspired helicoidally laminated composite plates (BHLC) for strengthening damaged reinforced concrete (RC) beams. Borrowing inspiration from nature’s design principles, BHLC plates represent a novel approach to rehabilitation. An analytical approach based on deformation compatibility is employed, assuming constant shear and normal stresses across the adhesive layer within the framework of linear elasticity. BHLC plates are implemented to restrict crack propagation within the RC beam. The analytical methodology considers equilibrium and deformation compatibility across all components: the concrete beam, BHLC plate, and adhesive layer. A parametric study explores the sensitivity of interfacial stresses to parameters such as laminate and adhesive stiffness, plate thickness, and helicoidal layup configuration. The results reveal a significant influence of these factors on the magnitude of maximum interfacial shear and normal stresses. This study establishes a foundation for future analyses of damaged RC beams strengthened with BHLC plates, potentially leading to advancements in rehabilitation methodologies and improved crack propagation modeling in RC beams.

From Phantoms to Patients: Improved Fusion and Voxel-Wise Analysis of Diffusion-Weighted Imaging and FDG-Positron Emission Tomography in Positron Emission Tomography/Magnetic Resonance Imaging for Combined Metabolic-Diffusivity Index (cDMI).

Hybrid positron emission tomography/magnetic resonance imaging (PET/MR) opens new possibilities in multimodal multiparametric (m2p) image analyses. But even the simultaneous acquisition of positron emission tomography (PET) and magnetic resonance imaging (MRI) does not guarantee perfect voxel-by-voxel co-registration due to organs and distortions, especially in diffusion-weighted imaging (DWI), which would be, however, crucial to derive biologically meaningful information. Thus, our aim was to optimize fusion and voxel-wise analyses of DWI and standardized uptake values (SUVs) using a novel software for m2p analyses. Using research software, we evaluated the precision of image co-registration and voxel-wise analyses including the rigid and elastic 3D registration of DWI and [18F]-Fluorodeoxyglucose (FDG)-PET from an integrated PET/MR system. We analyzed DWI distortions with a volume-preserving constraint in three different 3D-printed phantom models. A total of 12 PET/MR-DWI clinical datasets (bronchial carcinoma patients) were referenced to the T1 weighted-DIXON sequence. Back mapping of scatterplots and voxel-wise registration was performed and compared to the non-optimized datasets. Fusion was rated using a 5-point Likert scale. Using the 3D-elastic co-registration algorithm, geometric shapes were restored in phantom measurements; the measured ADC values did not change significantly (F = 1.12, p = 0.34). Reader assessment showed a significant improvement in fusion precision for DWI and morphological landmarks in the 3D-registered datasets (4.3 ± 0.2 vs. 4.6 ± 0.2, p = 0.009). Most pronounced differences were noted for the chest wall (p = 0.006), tumor (p = 0.007), and skin contour (p = 0.014). Co-registration increased the number of plausible ADC and SUV combinations by 25%. The volume-preserving elastic 3D registration of DWI significantly improved the precision of fusion with anatomical sequences in phantom and clinical datasets. The research software allowed for a voxel-wise analysis and visualization of [18F]FDG-PET/MR data as a "combined diffusivity-metabolic index" (cDMI). The clinical value of the optimized PET/MR biomarker can thus be tested in future PET/MR studies.

Simple and effective mechanical cloaking

We show theoretically that essentially perfect elastostatic mechanical cloaking of a circular inclusion in a homogeneous surrounding medium can be achieved by means of a simple cloak comprising three concentric annuli, each formed of a homogeneous isotropic linear elastic material of prescribed shear modulus. Importantly, we find that the same combination of annuli will cloak any possible mode of imposed deformation or loading, for any randomly chosen admixture of imposed compression, pure shear and simple shear, without the need to re-design the cloak for different deformation modes. A full range of circular inclusions can be cloaked in this way, from soft to stiff. In consequence, we suggest that an inclusion of any arbitrary shape can also be cloaked, by first enveloping it in a stiff circle, then cloaking the combined structure with three annuli as described. Given that a single inclusion can be fully cloaked in this way, even at near field close to the cloaking perimeter, it also follows that multiple such neutral inclusions arranged with arbitrarily high packing fraction in a surrounding medium can also be cloaked. We confirm this by direct simulation. This indicates a possible route to fabricating composite materials with the same global mechanical response as a counterpart homogeneous material, and with uniform strain and stress fields outwith the cloaked inclusions.

Stress-deformation analysis of the cracked elastic body

A new analytical plane Elastic Fracture Mechanics (LEFM) model of the effective elastic moduli exhibited by an isotropic elastic cracked solid in tensile and compressive stress fields is presented. The effective Young’s modulus and Poisson’s ratio E¯,v¯, respectively, relate the stresses applied to the cracked solid with its macroscopic deformation. In the case of a compressive stress field the effective elastic moduli depend solely on properly oriented cracks that are either open or closed. This model is then used for the stress and displacement estimations around a circular axisymmetric hole under internal pressure and external hydrostatic stress. The creation of the hole alters the stresses in its neighborhood, and induces a deterioration of the elastic moduli. In this manner, the effective elastic moduli depend on the radial distance from the hole’s boundary. The validation of the algorithm was done against Kirsch’s analytical solution of the axisymmetric hole problem for nearly zero density of cracks as well as for large values of the friction angle. The model predicts a lower tangential stress concentration at the wall of the hole compared to that predicted by the constant linear elasticity solution and a maximum stress concentration that is not necessarily located at the wall of the hole. This apparent enhancement of rock strength adjacent to the wall is more pronounced for larger concentrations of open cracks.

Bending-driven patterning of solid inclusions in lipid membranes: Colloidal assembly and transitions in elastic 2D fluids.

Biological or biomimetic membranes are examples within the larger material class of flexible ultrathin lamellae and contoured fluid sheets that require work or energy to impose bending deformations. Bending elasticity also dictates the interactions and assembly of integrated phases or molecular clusters within fluid lamellae, for instance enabling critical cell functions in biomembranes. More broadly, lamella and other thin fluids that integrate dispersed objects, inclusions, and phases behave as contoured 2D colloidal suspensions governed by elastic interactions. To elucidate the breadth of interactions and assembled patterns accessible through elastic interactions, we consider the bending elasticity-driven assembly of 1-10 μm solid plate-shaped Brownian domains (the 2D colloids), integrated into a fluid phospholipid membrane (the 2D fluid). Here, the fluid membranes of giant unilamellar vesicles, 20-50 μm in diameter, each contain 4-100 monodisperse plate-domains at an overall solid area fraction of 17 ± 3%. Three types of reversible plate arrangements are found: persistent vesicle-encompassing quasi-hexagonal lattices, persistent closely associated chains or concentrated lattices, and a dynamic disordered state. The interdomain distances evidence combined attractive and repulsive elastic interactions up to 10 μm, far exceeding the ranges of physio-chemical interactions. Bending contributions are controlled through membrane slack (excess area) producing, for a fixed composition, a sharp cooperative multibody transition in plate arrangement, while domain size and number contribute intricacy.

Study of the isoscalar scalar bcu¯d¯ tetraquark Tbc with lattice QCD

We present a lattice QCD study of the elastic S-wave DB¯ scattering in search of tetraquark candidates with explicitly exotic flavor content bcu¯d¯ in the isospin I=0 and JP=0+ channel. We use four lattice QCD ensembles with dynamical u/d, s, and c quark fields generated by the MILC collaboration. A nonrelativistic QCD Hamiltonian, including improvement coefficients up to O(αsv4), is utilized for the bottom quarks. For the rest of the valence quarks we employ a relativistic overlap action. Five different valence quark masses are utilized to study the light quark mass dependence of the DB¯ scattering amplitude. The finite volume energy spectra are extracted following a variational approach. The elastic DB¯ scattering amplitudes are extracted employing Lüscher’s prescription. The light quark mass dependence of the continuum extrapolated amplitudes suggests an attractive interaction between the B¯ and D mesons. At the physical pseudoscalar meson mass (Mps=Mπ) the DB¯ scattering amplitude has a subthreshold pole corresponding to a binding energy of −39(−6+4)(−18+8) MeV with respect to the DB¯ threshold. The critical Mps at which the DB¯ scattering length diverges and the system becomes unbound corresponds to Mps*=2.94(15)(5) GeV. This result can hold significant experimental relevance in the search for a bound scalar Tbc tetraquark, which could well be the next “doubly heavy” bound tetraquark to be discovered with only weak decay modes. Published by the American Physical Society 2024

A porous elastomer with a cavity array for three-dimensional plantar force sensing

Human gait is an important indicator for diagnosis of various movement-related disorders. Three-dimensional (3D) plantar force monitoring enables real-time recording of contact forces in both the normal and tangential directions in each step, providing useful clues for identification of irregular gait patterns or gait-related health issues. Most flexible sensors for gait monitoring focus on contact forces in the normal direction, being limited to detect forces in the tangential direction. Herein we have developed a porous elastomer with a cavity array (PECA) to measure 3D contact forces for gait analysis. The sensor consists of two fabric electrodes and a dielectric layer of porous Ecoflex elastomer containing a 5 × 5 cavity array, forming four parallel-plate capacitor units in a soft construction. Therefore, a 3D force in any arbitrary direction generates four unique capacitive electrical signals. Importantly, the PECA includes a well-designed combination of micrometer-scale pores and millimeter-sized cavities, dramatically improving the detection sensitivity and linear elasticity range. We have demonstrated high sensitivities of 0.41 N−1 and 0.23 N−1 in the normal and tangential directions, respectively. The flexible PECA-based sensors have been tested to sensitively distinguish various standing postures and different stride lengths for gait analysis, promising reliable translation into areas such as healthcare monitoring, sports training, and rehabilitation engineering.

Numerical Analysis of the Cylindrical Shell Pipe with Preformed Holes Subjected to a Compressive Load Using Non-Uniform Rational B-Splines and T-Splines for an Isogeometric Analysis Approach

In this paper, we implement the finite detail technique primarily based on T-Splines for approximating solutions to the linear elasticity equations in the connected and bounded Lipschitz domain. Both theoretical and numerical analyses of the Dirichlet and Neumann boundary problems are presented. The Reissner–Mindlin (RM) hypothesis is considered for the investigation of the mechanical performance of a 3D cylindrical shell pipe without and with preformed hole problems under concentrated and compression loading in the linear elastic behavior for trimmed and untrimmed surfaces in structural engineering problems. Bézier extraction from T-Splines is integrated for an isogeometric analysis (IGA) approach. The numerical results obtained, particularly for the displacement and von Mises stress, are compared with and validated against the literature results, particularly with those for Non-Uniform Rational B-Spline (NURBS) IGA and the finite element method (FEM) Abaqus methods. The obtained results show that the computation time of the IGA based on the T-Spline method is shorter than that of the IGA NURBS and FEM Abaqus/CAE (computer-aided engineering) methods. Furthermore, the highlighted results confirm that the IGA approach based on the T-Spline method shows more success than numerical reference methods. We observed that the NURBS IGA method is very limited for studying trimmed surfaces. The T-Spline method shows its power and capability in computing trimmed and untrimmed surfaces.

A refined model for functionally graded graphene nanoplatelets reinforced composite beams under thermal loads

Graphene is highly regarded as a promising material for various engineering purposes due to its remarkable physical characteristics. However, existing higher-order shear deformation theories may struggle to accurately capture the interlaminar mechanical response of functionally graded graphene nanoplatelets (FG-GNP) reinforced composite beams subjected to thermal loads. This arises from the pronounced discrepancies in material properties and thermal expansion coefficients among the various layers, making the interlaminar mechanical behavior of composite structures exceedingly complex under thermal loading. As a result, a meticulous analysis of interlaminar stresses becomes necessary for the structural assessment of GNP-reinforced composite beams under temperature variations. To overcome this limitation, we propose an advanced model incorporating transverse normal thermal deformation, tailored specifically for FG-GNP reinforced composite beams subjected to thermal loads. The developed model primarily boasts two advantages. Firstly, although transverse normal deformation is introduced into the displacement field, no additional displacement variables are increased, thereby maintaining model simplicity and efficiency. Secondly, the model achieves a high-precision interlaminar shear stress field taking into account the interlaminar continuity conditions and temperature effects. Furthermore, the elimination of second-order derivatives of in-plane displacement parameters from the transverse shear stress components leads to a simplification in the finite element implementation. Based on the proposed model, a simple three-node beam element is constructed. The 3D elasticity solutions and the results from alternative modeling frameworks are used to assess the performance of the developed model. Additionally, a comprehensive investigation is conducted to explore the effects of various parameters on the deformations and stresses of GNP-reinforced composite beams.

Generalized weak Galerkin finite element method for linear elasticity interface problems

Generalized weak Galerkin finite element method for linear elasticity interface problems

Vectfem: a generalized MATLAB-based vectorized algorithm for the computation of global matrix/force for finite elements of any type and approximation order in linear elasticity

In this paper, we introduce a new vectorized MATLAB-based algorithm for efficient serial computation of global matrix/force arising from finite element method (FEM) for meshes of any type and approximation order in linear elasticity. Because for-loops in MATLAB are very slow, we propose a modified process that takes advantage of vectorization and sparse assembly to achieve good performance while using the same memory as the standard algorithm. For this purpose, by using good programming practices, the implementation of this scheme is succinctly described and can be integrated into any MATLAB package dealing with FEM. Specifically, attention is paid to the calculation of the triplet (row index, column index, matrix components) as well as the assembly of the global stiffness matrix, mass matrix and force vector. Additionally, an extension of the proposed approach for Mindlin plate theory and functionally graded materials is outlined. Finally, the accuracy of this strategy is verified on selected numerical tests after comparing the obtained results with those of ABAQUS. In terms of performance, the study conducted on a set of meshes considering the standard algorithm and two other well-known MATLAB vectorized algorithms revealed that: (i) for a 2D beam problem meshed with P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{1}$$\\end{document}-triangle elements, a speedup of about 8 and 15 is achieved with sparse and fsparse, respectively. (ii) for a 3D plate problem meshed with P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{1}$$\\end{document}-tetrahedral elements, a speedup of about 4 and 8 is achieved with sparse and fsparse, respectively. When compared to ABAQUS performance, the proposed scheme results in a computational time that is about five times smaller.

Two-Dimensional Linear Elasticity Equations of Thermo-Piezoelectric Semiconductor Thin-Film Devices and Their Application in Static Characteristic Analysis

Two-Dimensional Linear Elasticity Equations of Thermo-Piezoelectric Semiconductor Thin-Film Devices and Their Application in Static Characteristic Analysis

A locking-free mixed enriched Galerkin method of arbitrary order for linear elasticity using the stress–displacement formulation

In this paper, we develop an arbitrary-order locking-free enriched Galerkin method for the linear elasticity problem using the stress–displacement formulation in both two and three dimensions. The method is based on the mixed discontinuous Galerkin method in Wang et al. (2020), but with a different stress approximation space that enriches the arbitrary order continuous Galerkin space with some piecewise symmetric-matrix valued polynomials. The enriched Galerkin method maintains locking-free property of the mixed discontinuous Galerkin method in Wang et al. (2020), but has fewer degrees of freedom for commonly used and appropriately high orders. We present some numerical examples in two and three dimensions to demonstrate the effectiveness of the proposed method.

A Physics-Informed General Convolutional Network for the Computational Modeling of Materials With Damage

Abstract Despite their effectiveness in modeling complex phenomena, the adoption of machine learning (ML) methods in computational mechanics has been hindered by the lack of availability of training datasets, limitations on the accuracy of out-of-sample predictions, and computational cost. This work presents a physics-informed ML approach and network architecture that addresses these challenges in the context of modeling the behavior of materials with damage. The proposed methodology is a novel physics-informed general convolutional network (PIGCN) framework that features (1) the fusion of a dense edge network with a convolutional neural network (CNN) for specifying and enforcing boundary conditions and geometry information, (2) a data augmentation approach for learning more information from a static dataset that significantly reduces the necessary data for training, and (3) the use of a CNN for physics-informed ML applications, which is not as well explored as graph networks in the current literature. The PIGCN framework is demonstrated for a simple two-dimensional, rectangular plate with a hole or elliptical defect in a linear-elastic material, but the approach is extensible to three dimensions and more complex problems. The results presented in this article show that the PIGCN framework improves physics-based loss convergence and predictive capability compared to ML-only (physics-uninformed) architectures. A key outcome of this research is the significant reduction in training data requirements compared to ML-only models, which could reduce a considerable hurdle to using data-driven models in materials engineering where material experimental data are often limited.

Use of the integration by part to derive beam/plate equilibrium equations from elasticity with applications to beams

This article proposes a rigorous mathematical justification of the derivation of the governing equations of any beam and plate theory by the three three-dimensional (3D) elasticity equations. This is done by the application of the integration by part theorem via weight residual-like methods. The Fis polynomials (subscripts i and s denote the i-direction and the generic s-term of the displacement expansion uis, respectively) related to a given beam/plate theory are used as weight functions for each of the displacement unknown. To show how the method works a number of simple problems are discussed. These are related to the equilibrium equations and boundary conditions of bending beams according to shear deformation theory. Three methods for obtaining these equations are compared in some detail. The first two are the well-known methods used in mechanics of deformable bodies: (1) use of the Principle of Virtual Displacements; (2) use of the six cardinal ‘Newton’ equations of statics. The third method coincides with the one proposed and rigorously justified in this paper which is (3) the direct integration of the equations of 3D elasticity. Beam theories based on Lagrange polynomials are used as well. The latter makes it possible to immediately develop theories for layered beams and obtain so-called ‘layer-wise’ models, which are only applied here to the case of sandwich structures. Three appendixes discuss: A – The problem in which the derivative of one of the unknowns appears among the unknowns, as is the case with the Euler–Bernoulli theory; B – The extension to dynamics; C-Simple derivation of constitutive equations and related closed-form solution via Navier method.

Screw dislocation dipoles in niobium: combination of STM observations and atomistic simulations

We recently developed an experimental device that allows us to observe the slip traces under stress at the atomic scale. Here, we report experimental results obtained at the latter scale on Nb single crystals making it possible to observe dislocation dipoles (DD), which are evidenced by two slip traces formed by emerging moving dislocations of opposite Burgers vectors ending very close to each other. The geometry and stability of the DD were fully characterized in the framework of linear anisotropic elasticity theory and by atomistic simulations. This allows us to calculate a local opposite stress impeding dislocation motion of the dislocations of the dipole.

Minimization of peak stresses with the shape derivative.

This article is concerned with the minimization of peak stresses occurring in linear elasticity. We propose to minimize the maximal von Mises stress of the elastic body. This leads to a non-smooth shape functional. We derive the shape derivative and associate it with the Clarke sub-differential. Using a steepest descent algorithm, we present numerical simulations. We compare our results to the usual [Formula: see text]-norm regularization and show that our algorithm performs better in the presented tests.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.