Linear Elasticity Research Articles

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'.

Topology optimization of curved thick shells using level set method and non-conforming multi-patch isogeometric analysis

We present a novel framework for topological shape optimization of curved non-conforming multi-patch and trimmed thick-shells subjected to external loads. Our method integrates the level set method (LSM) with a diffuse interface, a Hadamard shape derivative, and multi-patch isogeometric analysis (IGA) into a gradient descent algorithm to systematically capture the evolution of the shape. This integration enables us to directly manipulate CAD-compatible geometries and analysis techniques and to obtain the results as a CAD surface. The novelty lies in the utilization of multi-patch IGA models based on NURBS functions, which allows us to simultaneously maximize the stiffness and minimize the volume of the shell by searching for an optimal material distribution within its middle surface. The material is modeled under a small strain assumption in linear elasticity using a Reissner–Mindlin kinematic shell model in plane stress. The effectiveness of our approach is demonstrated on several curved conforming and non-conforming multi-patch geometries in 3D.

Study of the flexural behavior of UHPC-HPC composite beams strengthened with BFRP sheet after chloride secondary erosion

To solve the issue of insufficient durability of reinforced concrete (RC) structures, a new structure of ultra-high performance concrete (UHPC)-high performance concrete (HPC) composite beam was designed, and the flexural performance of the composite beam strengthened with basalt fiber reinforced polymer (BFRP) after pre-damage was investigated. Results indicate that main cracks are formed and developed after the secondary erosion of chloride salt, leading to three forms of failure modes of the beams: BFRP sheet fracture and concrete crushing, BFRP sheet debonding and concrete crushing and BFRP sheet debonding. The test beam failure process can be classified into three stages: linear elasticity, crack development and yield. The maximum cracking and ultimate loads increased by 27.9 % and 35.8 %, respectively. The crack load growth rate of the beam decreased after 7 days (d) of secondary chloride salt erosion. The formula for calculating the peel strength of the UHPC-HPC composite beam was modified. The calculated bending capacity of the beam agrees well with the measured value. This study provides a reference for enhancing the lifespan and calculating the flexural bearing capacity of structures in corrosive environments.

Experimental Analysis of Hyperelastic Materials Using the Vibration Method

The elaborated paper presents a series of methodologies with which the dynamic characteristics (damping coefficient, damping factor) can be determined, depending on the working conditions, of a hyperelastic material using vibration theory. These methodologies can be extended to characterize any type of hyperelastic material. The main aim of this work is to develop experimental technology and methodology to characterize this type of materials like rubber to establish a series of dynamic factors like damping factor, transmissibility at resonance, pulsation at resonance, dynamic elastic constant. These characteristics are variable, depending on composition, request, etc. In conclusion, they are not available in specialized literature as are the characteristics of linear-elastic materials. The application of numerical calculation programs in carrying out resistance calculations, in the case of structures made of such materials, is also impossible to achieve, having as an impediment the lack of knowledge of the values of the material characteristics.

On crack simulation by mixed‐dimensional coupling in GFEM with global‐local enrichments

AbstractA strategy that combines the global‐local version of the generalized finite element method (GFEM) with a mixed‐dimensional coupling iterative method is proposed to simulate two‐dimensional crack propagation in structures globally represented by Timoshenko‐frame models. The region of interest called the local problem, where the crack propagates, is represented by a 2D elasticity model, where a fine mesh of plane stress/strain elements and special enrichment functions are used to describe this phenomenon accurately. A model of Timoshenko‐frame elements simulates the overall behavior of the structure. A coarse mesh of plane stress/strain elements provides a bridge between these two representation scales. The mixed‐dimensional coupling method imposes displacement compatibility and stress equilibrium at the interface between the two different element types by an iterative procedure based on the principle of virtual work. After establishing the constraint equations for the interface, the 2D elasticity model is related to the small‐scale model by the global‐local enrichment strategy of GFEM. In such a strategy, the numerical solutions of the local problem subjected to boundary conditions derived from the global‐scale problem enrich the approximation of this same global problem in an iterative procedure. Each step of the crack propagation requires a new sequence of this iterative global‐local procedure. On the other hand, the constraint equation for the interface is defined only once. The crack representation in a confined region by the global‐local strategy avoids a remeshing that would require new constraint equations. Two numerical problems illustrate the proposed strategy and assess the influence of the analysis parameters.

Discovering 3D hidden elasticity in isotropic and transversely isotropic materials with physics-informed UNets

Three-dimensional variation in structural components or fiber alignments results in complex mechanical property distribution in tissues and biomaterials. In this paper, we use a physics-informed UNet-based neural network model (El-UNet) to discover the three-dimensional (3D) internal composition and space-dependent material properties of heterogeneous isotropic and transversely isotropic materials without a priori knowledge of the composition. We then show the capabilities of El-UNet by validating against data obtained from finite-element simulations of two soft tissues, namely, brain tissue and articular cartilage, under various loading conditions. We first simulated compressive loading of 3D brain tissue comprising of distinct white matter and gray matter mechanical properties undergoing small strains with isotropic linear elastic behavior, where El-UNet reached mean absolute relative errors under 1.5 % for elastic modulus and Poisson's ratio estimations across the 3D volume. We showed that the 3D solution achieved by El-UNet was superior to relative stiffness mapping by inverse of axial strain and two-dimensional plane stress/plane strain approximations. Additionally, we simulated a transversely isotropic articular cartilage with known fiber orientations undergoing compressive loading, and accurately estimated the spatial distribution of all five material parameters, with mean absolute relative errors under 5 %. Our work demonstrates the application of the computationally efficient physics-informed El-UNet in 3D elasticity imaging and provides methods for translation to experimental 3D characterization of soft tissues and other materials. The proposed El-UNet offers a powerful tool for both in vitro and ex vivo tissue analysis, with potential extensions to in vivo diagnostics. Statement of significanceElasticity imaging is a technique that reconstructs mechanical properties of tissue using deformation and force measurements. Given the complexity of this reconstruction, most existing methods have mostly focused on 2D problems. Our work is the first implementation of physics-informed UNets to reconstruct three-dimensional material parameter distributions for isotropic and transversely isotropic linear elastic materials by having deformation and force measurements. We comprehensively validate our model using synthetic data generated using finite element models of biological tissues with high bio-fidelity–the brain and articular cartilage. Our method can be implemented in elasticity imaging scenarios for in vitro and ex vivo mechanical characterization of biomaterials and biological tissues, with potential extensions to in vivo diagnostics.

Parametric Neural Networks as Full-Field Surrogates for Material Model Calibration

The identification of material parameters occurring in material models is essential for structural health monitoring. Due to chemical and physical processes, building structures and materials age during their service life. This, in turn, leads to a deterioration in both the reliability and quality of the structures. Knowing the current condition of the building structures can help prevent disasters and extend service life. We developed a physics-informed neural network (PINN) [1] for the calibration of the linear-elastic material model from full-field displacement data measured by digital image correlation. In an offline-phase, the PINN is trained to learn a parameterized solution of the underlying parametric partial differential equation without the need for training data [2]. We demonstrate the ability of the parametric PINN to act as a surrogate in a least-squares based material model calibration. In order to quantify the uncertainty, we further use the parametric PINN with Markov Chain Monte Carlo based Bayesian inference. Even with artificially noisy data, the calibration produces good results for reasonable material parameter ranges. Especially in sampling based methods, parametric PINNs have the advantage that model evaluation is very cheap compared to, e.g., the Finite Element Method. Thus, information on the material condition can be provided in near real-time in the online-phase. Moreover, PINNs use a continuous ansatz and thereby avoid the need to interpolate sensor locations to the simulation domain. In our ongoing work, we plan to apply the parametric PINN to more complex material models, such as those for elasto-plastic materials. We also investigate the extension of the proposed method to more complex geometries. [1] M. Raissi et al.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics 378, 686-707 (2019). [2] A. Beltrán-Pulido et al.: Physics-Informed Neural Networks for Solving Parametric Magnetostatic Problems, IEEE Transactions on Energy Conversion 37(4), 2678-2689 (2022).

A New Mixed Finite Element for the Linear Elasticity Problem in 3D

A New Mixed Finite Element for the Linear Elasticity Problem in 3D

Thin elastic films and membranes under rectangular confinement

We address the deformations within a thin elastic film or membrane in a two-dimensional rectangular confinement. To this end, analytical considerations of the Navier-Cauchy equations describing linear elasticity are performed in the presence of a localized force center, that is, a corresponding Green's function is determined, under no-slip conditions at the clamped boundaries. Specifically, we find resulting displacement fields for different positions of the force center. It turns out that clamping regularizes the solution when compared to an infinitely extended system. Increasing compressibility renders the displacement field more homogeneous under the given confinement. Moreover, varying aspect ratios of the rectangular confining frame qualitatively affect the symmetry and appearance of the displacement field. Our results are confirmed by comparison with corresponding finite-element simulations.

FORMULATION OF A MODE I CRACK PROPAGATION MODEL BY ANALYZING THE STRESS FIELD AT THE CRACK TIP

One of the phases in the development of materials is knowledge of their possible degradation. At this stage, taking into account the influence of temperature variation remains essential for certain materials. The objective of this work is to take this aspect into account by formulating a mode I crack propagation model by studying the stress field in the vicinity of the crack tip. To do this, starting from Hookes law generalized in linear elasticity, we use the asymptotic expressions in displacements of the crack tip of Moes et Al. in order to obtain the stress field model. To validate the model, a comparison is made with data from the literature. The observations show a very high agreement of results. The advantage of this model is to take into account the role of temperature in the degradation of materials by analyzing crack propagation. It will be used in our future work.

2D numerical modeling of crack propagation using SFEMD method of a LEHI material

Numerical methods today play a useful and important role in solving various problems related to fracture mechanics including modeling crack propagation, fretting fatigue and cohesion... Furthermore, these methods have been widely used to solve problems in linear and nonlinear fracture mechanics in cases of elastic, plastic fracture problems. The evaluation of stress intensity factor in 2D and 3D geometries, thus these techniques widely used for non-standard crack configurations. The objective of this work is study the effects of the crack length and the number of structural elements on the two crack parameters such as the stress intensity factor KI and KII and the contour integral (J). As well as presenting a numerical modeling of crack propagation, for a LEHIM (Linear Elastic Homogeneous Isotopic Material) with mechanical characteristics by the method SFEMD (Stretching Finite Element Method Developed), the model chosen with quadratic elements with 4 nodes (CPE4). However, this method is based on the effect of the number of contours and the number of elements around the crack tip on the variation of the stress intensity factors. In addition, the crack propagation criterion (MCSC) was used, a computer program was created in FORTRAN language to develop and evaluate the stress intensity factors and the contour integral (J). Several examples of crack lengths a = 0.7, 1.4, 2.1, 2.8 and 3.5 mm were used. Additionally, the number of items has been changed several times. The stress intensity factors of modes I and II and direction angles (α) are calculated to solve the problem using the ABAQUS finite element code. The results obtained by the SFEMD method and the results of the analytical method are very close.

Dynamic modeling and simulation of hard-magnetic soft beams interacting with environment via high-order finite elements of ANCF

Hard-magnetic soft (HMS) beams made of soft polymer matrix embedded with hard-magnetic particles can generate large and fast deformation under magnetic stimulation. Dynamic modeling and simulation of HMS beams interacting with complex environment are challenging in terms of computational accuracy and efficiency. This paper presents a method for high-order modeling and efficient computation of HMS beams. The major contribution of the method is a new three-node HMS beam element of absolute nodal coordinate formulation (ANCF), which applies to two material models of nonlinear and linear elasticities (i.e. neoHookean and St. Venant-Kirchhoff) coupled with magnetic energy. To improve the efficiency of the method, the paper presents how to derive the generalized internal forces and their Jacobians via invariant tensors, and how to determine the generalized external forces to model dynamic loads and interactions including gravity, hydrodynamics in fluids, and frictional contact in pipelines. Afterwards, the paper gives both static and dynamic equations with Rayleigh damping and discusses the numerical algorithms. Finally, the paper makes a comparison of static analysis and the experimental observation to validate the accuracy of the proposed modeling method. The paper also discusses the dynamic simulations, including forced vibration, swimming motion, crawling locomotion, and navigating motion to demonstrate the predictive capability and efficacy of the proposed method for dynamic problems.

Elastic solution of surface-loaded one-dimensional hexagonal quasicrystal layered elastic media

This study aims to investigate the elastic response of a two-dimensional, surface-loaded substrate coated by a multi-layered system consisting of one-dimensional hexagonal quasicrystal materials. A linear elasticity theory for quasicrystals is adopted to simulate phonon and phason elastic fields. First, a set of governing differential equations for each layer is established in terms of the phonon and phason displacements. The general solution for a generic layer is constructed through the Fourier transform method in a closed form in the transform space. This solution is then utilized to formulate a layer stiffness equation. Adopting a direct stiffness method along with continuity along the layer interfaces, a stiffness equation for the multi-layered coated substrate is derived. An efficient quadrature is adopted to evaluate all involved integrals resulting from Fourier integral inversion. After verification with benchmark cases, the capability of the derived solutions are demonstrated by investigating the influence of various parameters such as layer thickness, layer arrangement, and loading conditions on the predicted response. Finally, applications of the developed multi-layer scheme to tackle a functionally graded quasicrystal layer are elucidated.