Showing papers on "Linear elasticity published in 2022"

Analyses of internal structures and defects in materials using physics-informed neural networks

Abstract: Characterizing internal structures and defects in materials is a challenging task, often requiring solutions to inverse problems with unknown topology, geometry, material properties, and nonlinear deformation. Here, we present a general framework based on physics-informed neural networks for identifying unknown geometric and material parameters. By using a mesh-free method, we parameterize the geometry of the material using a differentiable and trainable method that can identify multiple structural features. We validate this approach for materials with internal voids/inclusions using constitutive models that encompass the spectrum of linear elasticity, hyperelasticity, and plasticity. We predict the size, shape, and location of the internal void/inclusion as well as the elastic modulus of the inclusion. Our general framework can be applied to other inverse problems in different applications that involve unknown material properties and highly deformable geometries, targeting material characterization, quality assurance, and structural design.

Free and Forced Vibration Analysis of Two-Dimensional Linear Elastic Solids Using the Finite Element Methods Enriched by Interpolation Cover Functions

Abstract: In this paper, a novel enriched three-node triangular element with the augmented interpolation cover functions is proposed based on the original linear triangular element for two-dimensional solids. In this enriched triangular element, the augmented interpolation cover functions are employed to enrich the original standard linear shape functions over element patches. As a result, the original linear approximation space can be effectively enriched without adding extra nodes. To eliminate the linear dependence issue of the present method, an effective scheme is used to make the system matrices of the numerical model completely positive-definite. Through several typical numerical examples, the abilities of the present enriched three node triangular element in forced and free vibration analysis of two-dimensional solids are studied. The results show that, compared with the original linear triangular element, the present element can not only provide more accurate numerical results, but also have higher computational efficiency and convergence rate.

A framework based on physics-informed neural networks and extreme learning for the analysis of composite structures

Abstract: This paper presents a novel approach for solving direct problems in linear elasticity involving plate and shell structures. The method relies upon a combination of Physics-Informed Neural Networks and Extreme Learning Machine. A subdomain decomposition method is proposed as a viable mean for studying structures composed by multiple plate/shell elements, as well as improving the solution in domains composed by one single element. Sensitivity studies are presented to gather insight into the effects of different network configurations and sets of hyperparameters. Within the framework presented here, direct problems can be solved with or without available sampled data. In addition, the approach can be extended to the solution of inverse problems. The results are compared with exact elasticity solutions and finite element calculations, illustrating the potential of the approach as an effective mean for addressing a wide class of problems in structural mechanics.

Locking-Free Enriched Galerkin Method for Linear Elasticity

Abstract: Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 14 January 2021Accepted: 14 September 2021Published online: 05 January 2022Keywordslinear elasticity, enriched Galerkin method, locking-free, operator preconditioningAMS Subject Headings65N30, 65N55, 65F08, 74S05Publication DataISSN (print): 0036-1429ISSN (online): 1095-7170Publisher: Society for Industrial and Applied MathematicsCODEN: sjnaam

Anomalous elasticity of a cellular tissue vertex model

Abstract: Vertex models, such as those used to describe cellular tissue, have an energy controlled by deviations of each cell area and perimeter from target values. The constrained nonlinear relation between area and perimeter leads to new mechanical response. Here we provide a mean-field treatment of a highly simplified model: a uniform network of regular polygons with no topological rearrangements. Since all polygons deform in the same way, we only need to analyze the ground states and the response to deformations of a single polygon (cell). The model exhibits the known transition between a fluid/compatible state, where the cell can accommodate both target area and perimeter, and a rigid/incompatible state. We calculate and measure the mechanical resistance to various deformation protocols and discover that at the onset of rigidity, where a single zero-energy ground state exists, linear elasticity fails to describe the mechanical response to even infinitesimal deformations. In particular, we identify a breakdown of reciprocity expressed via different moduli for compressive and tensile loads, implying nonanalyticity of the energy functional. We give a pictorial representation in configuration space that reveals that the complex elastic response of the vertex model arises from the presence of two distinct sets of reference states (associated with target area and target perimeter). Our results on the critically compatible tissue provide a new route for the design of mechanical metamaterials that violate or extend classical elasticity.

Stabilization-free virtual element method for plane elasticity

Abstract: We present the construction and application of a first order stabilization-free virtual element method to problems in plane elasticity. Well-posedness and error estimates of the discrete problem are established. The method is assessed on a series of well-known benchmark problems from linear elasticity and numerical results are presented that affirm the optimal convergence rate of the virtual element method in the L2 norm and the energy seminorm.

Calibration and Validation of a Linear-Elastic Numerical Model for Timber Step Joints Based on the Results of Experimental Investigations

Abstract: The paper is dedicated to the numerical analysis of a single-step joint, enabling the prediction of stiffness and failure modes of both single- and double-step joints. An experimental analysis of the geometrically simplest version, the single-step joint, serves as a reference for the calibration of the subsequent finite element model. The inhomogeneous and anisotropic properties of solid timber make detailed modelling computationally intensive and strongly dependent on the respective specimen. Therefore, the authors present a strategy for simplified but still appropriate modelling for the prediction of local failure at certain load levels. The used mathematical approach is based on the linear elasticity theory and orthotropic material properties. The finite element calculations are performed in the environment of the software Abaqus FEA. The calibrated numerical model shows a good conformity until first failures occur. It allows for a satisfactory quantification of the stiffness of the connection and estimation of the force when local failure begins and is, therefore, recommended for future, non-destructive research of timber connections of various shapes.

Role of Elastic Upper Limit in Shakedown Study for Granular soils

Abstract: The shakedown study for determining plastic shakedown limit is desirable for preventing geotechnical infrastructure failure under cyclic loads. The plastic shakedown limit is conceptually clear but difficult to determine experimentally since the permanent strain accumulation response is a gradual process, and the criterion for determining the plastic shakedown limit is sensitive and variable limited by lack of further understanding of mechanisms during shake down process. Motivated by the need to clarify the mechanism of shakedown limit, a new concept of elastic upper limit of quasi-elasticity extension is hypothesized in this paper. Three-stage slow cyclic triaxial tests are carried out on three types of sand to clarify this new concept. The results show that the quasi-elastic curve extends as the cyclic stress level increases until it reaches the elastic upper limit and then the deviation occurs. The objective of this study is to explore the role of elastic upper limit (the elastic extension domain limit) in the shakedown theory. It is found that the derived elastic upper limit is close to the plastic shakedown limit, especially at high levels of confining pressure. A new criterion is proposed to determine the plastic shakedown limit based on the linear dependence of the degree of elasticity on the cyclic stress level. The introduction of the concept of elastic upper limit provides new insights into shakedown study for granular soils.

Generalized Multiscale Finite Element Method for piezoelectric problem in heterogeneous media

Abstract: In this paper, we study multiscale methods for piezocomposites. We consider a model of static piezoelectric problem that consists of deformation with respect to components of displacements and a function of electric potential. This problem includes the equilibrium equations, the quasi-electrostatic equation for dielectrics, and a system of coupled constitutive relations for mechanical and electric fields. We consider a model problem that consists of coupled differential equations. The first equation describes the deformations and is given by the elasticity equation and includes the effect of the electric field. The second equation is for the electric field and has a contribution from the elasticity equation. In previous findings, numerical homogenization methods are proposed and used for piezocomposites. We consider the Generalized Multiscale Finite Element Method (GMsFEM), which is more general and is known to handle complex heterogeneities. The main idea of the GMsFEM is to develop additional degrees of freedom and can go beyond numerical homogenization. We consider both coupled and split basis functions. In the former, the multiscale basis functions are constructed by solving coupled local problems. In particular, coupled local problems are solved to generate snapshots. Furthermore, in the snapshot space, a local spectral decomposition is performed to identify multiscale basis functions. Our approaches share some common concepts with meshless methods as they solve the underlying problem on a coarse mesh, which does not conform heterogeneities and contrast. We discuss this issue in the paper. We show that with a few basis functions per coarse element, one can achieve a good approximation of the solution. Numerical results are presented.

On semi-analytical Stochastic Boundary Element Method and its application to eigenproblem of thin elastic plate immersed into a fluid

Abstract: The main objective in this work is to study an application of the B-spline approximations in the Least Squares Method recovery of the structural response for boundary value problems in some linear elastic systems. These responses were approximated before using polynomial bases in the Stochastic perturbation-based Boundary Element Method and now are replaced with second or third order B-spline functions. A majority of such an approach is that the resulting expected values and standard deviations of structural response are obtained using analytical calculus of probability integrals, so that neither statistical nor expansion methods are necessary. Numerical illustration delivered in this paper includes fundamental eigenfrequencies of elastic thin homogeneous and isotropic plate immersed into Newtonian fluid subjected to various boundary conditions at its external edges.

The universal program of linear elasticity

Abstract: Universal displacements are those displacements that can be maintained, in the absence of body forces, by applying only boundary tractions for any material in a given class of materials. Therefore, equilibrium equations must be satisfied for arbitrary elastic moduli for a given anisotropy class. These conditions can be expressed as a set of partial differential equations for the displacement field that we call universality constraints. The classification of universal displacements in homogeneous linear elasticity has been completed for all the eight anisotropy classes. Here, we extend our previous work by studying universal displacements in inhomogeneous anisotropic linear elasticity assuming that the directions of anisotropy are known. We show that universality constraints of inhomogeneous linear elasticity include those of homogeneous linear elasticity. For each class and for its known universal displacements, we find the most general inhomogeneous elastic moduli that are consistent with the universality constrains. It is known that the larger the symmetry group, the larger the space of universal displacements. We show that the larger the symmetry group, the more severe the universality constraints are on the inhomogeneities of the elastic moduli. In particular, we show that inhomogeneous isotropic and inhomogeneous cubic linear elastic solids do not admit universal displacements and we completely characterize the universal inhomogeneities for the other six anisotropy classes.

Mechanics of transient semi-flexible networks: Soft-elasticity, stress relaxation and remodeling

Abstract: Networks of semi-flexible (or athermal) filaments cross-linked by flexible chains are found in a variety of biopolymers such as soft connective tissues, the cell’s cytoskeleton or the wall of plant cells. They can also be synthetized in the lab to create liquid crystal elastomers-like gels as well as tissue mimetics. While the elasticity of these networks has been explored, the visco-elastic response that originate from the existence of reversible and dynamic cross-links is still poorly understood. We here develop a model for these networks by taking a multiscale, statistical mechanics approach where the network is decomposed into its most basic building blocks: elastic rods (to describe semi-flexible filaments) and the flexible chains used to cross-link them. The topology of this assembly is represented by a hairy rod model for which we express the non-affine kinematics, and evolution equations for both cross-linkers and rods conformation. The mechanical response of this hairy rod is then expressed by an elastic potential that is built as a function of the basic elasticity of its components. The resulting model is able to capture salient features of the mechanics of such networks, including nonlinear elasticity (and in particular a liquid crystal-like soft-elastic response), creep and stress relaxation, as well as rate- and history-dependent network remodeling. The theory can thus be potentially used to better understand the rich response of these complex, yet ubiquitous networks and guide their development in the laboratory.

In situ elastic constant determination of unidirectional CFRP composites via backwall reflected multi-mode ultrasonic bulk waves using a linear array probe

Abstract: The determination of elastic constants of a transversely isotropic carbon fiber reinforced plastic (CFRP) composite is the prerequisite of mechanical strength calculation, material degradation evaluation, ultrasonic non-destructive testing for main load-bearing CFRP structures both after manufacturing and in service. Conventional ultrasonic methods request extra sample preparation, water immersion condition or special designed goniometric devices to rotate the test structures, making the elastic constants measurement expensive, inconvenient and not in situ. In this paper, a novel ultrasonic method that enables in situ determination of elastic constants via backwall reflection method (BRM) using a linear array probe is proposed. The BRM can achieve single sided measurement of ultrasonic travel times in various directions including the fiber direction using different pairs of transmitter and receiver array elements. Both quasi longitudinal and quasi shear waves are captured via the mode conversions from the multiple surface reflections. All elastic constants are determined through the particle swarm optimization by minimizing the sum of the squared deviations between the BRM measured and theoretically calculated multi-mode bulk wave travel times in the fiber orthogonal and parallel planes. This method is experimentally verified on a 4.45 mm thick unidirectional T700 carbon fiber/epoxy CFRP composite. The BRM measured Young's modulus in the fiber direction agrees well with that measured by tensile test, with a small deviation of −4.58%. This work proves that the proposed method is single sided, easy to operate, without the necessity of sample preparation, water immersion and extra rotation device, and can determine all elastic constants with high precision, which is therefore promising for in situ applications.

Carrera unified formulation (CUF) for the shells of revolution. Numerical evaluation

Abstract: Here, higher order models of elastic shells of revolution are developed using the variational principle of virtual power for 3-D equations of the linear theory of elasticity and generalized series in the coordinates of the shell thickness. Following the Carrera Unified Formulation (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. As a result, all the equations of the theory of elasticity were transformed into the corresponding equations for the expansion coefficients in a series in terms of the coordinates of the shell thickness. All equations for shells of revolution of higher order are developed and presented here for cases whose middle surfaces can be represented analytically. The resulting equations can be used for theoretical analysis and calculation of the stress-strain state, as well as for modeling thin-walled structures used in science, engineering, and technology.

Solution of the Thermoelastic Problem for a Two-Dimensional Curved Beam with Bimodular Effects

Abstract: In classical thermoelasticity, the bimodular effect of materials is rarely considered. However, all materials will present, in essence, different properties in tension and compression, more or less. The bimodular effect is generally ignored only for simple analysis. In this study, we theoretically analyze a two-dimensional curved beam with a bimodular effect and under mechanical and thermal loads. We first establish a simplified model on a subarea in tension and compression. On the basis of this model, we adopt the Duhamel similarity theorem to change the initial thermoelastic problem as an elasticity problem without the thermal effect. The superposition of the special solution and supplement solution of the Lamé displacement equation enables us to satisfy the boundary conditions and stress continuity conditions of the bimodular curved beam, thus obtaining a two-dimensional thermoelastic solution. The results indicate that the solution obtained can reduce to bimodular curved beam problems without thermal loads and to the classical Golovin solution. In addition, the bimodular effect on thermal stresses is discussed under linear and non-linear temperature rise modes. Specially, when the compressive modulus is far greater than the tensile modulus, a large compressive stress will occur at the inner edge of the curved beam, which should be paid with more attention in the design of the curved beams in a thermal environment.

A cell-based smoothed finite element method for finite elasticity

Abstract: Abstract In this study, we present a displacement based polygonal finite element method for compressible and nearly-incompressible elastic solids undergoing large deformations in two dimensions. This is achieved by projecting the dilatation strain onto the linear approximation space, within the framework of volume averaged nodal projection method. To reduce the numerical integration burden over polytopes, a linear strain smoothing technique is employed to compute the terms in the bilinear/linear form. The salient features of the proposed framework are: (a) does not require derivatives of shape functions and complex numerical integration scheme to compute the bilinear and linear form and (b) volumetric locking is alleviated by adopting the volume averaged nodal projection technique. The efficacy, convergence properties and accuracy of the proposed framework is demonstrated through four standard benchmark problems.

Rayleigh and Stoneley waves in linear elasticity

Abstract: We construct microlocal solutions of Rayleigh and Stoneley waves in isotropic linear elasticity with the density and the Lamé parameters smooth up to a curved boundary or interface. We compute the direction of the microlocal polarization and show a retrograde elliptical motion of these two type of waves.