Linear Elasticity Research Articles

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.

Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems

AbstractImmersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods.

Numerical Investigation of Solitary Wave Attenuation by a Vertical Plate-Type Flexible Breakwater Constructed Using Hyperelastic Neo-Hookean Material

This study conducted numerical investigations on solitary wave attenuation by a vertical plate-type flexible breakwater constructed using hyperelastic neo-Hookean material. The wave attenuation performance and elastic behaviors of the flexible breakwater were discussed systematically by considering the effects of three prominent factors: mass coefficient, stiffness coefficient, and Poisson’s ratio. It is indicated that more compressible and flexible materials are beneficial for enhancing efficiency in mitigating solitary wave energy and protecting the structure from damage. In addition, the performance of the hyperelastic neo-Hookean material model was compared with that of a linear elastic isotropic material model coupled with linear and nonlinear geometry analysis (LGEOM and NLGEOM) by evaluating several key targets: wave reflection coefficient, transmission coefficient, horizontal tip displacement, and wave load. Our findings revealed that the hyperelastic neo-Hookean material model showed almost the same predictions as the linear elastic isotropic material model with NLGEOM, but significantly diverged from that with LGEOM. The linear elastic isotropic material model with LGEOM cannot capture the nonlinear variations in structural geometry and stress–strain relationship, resulting in the underestimation and overestimation of horizontal tip displacement under moderate and extreme wave loads, respectively. Moreover, it underestimates the damage inflicted by solitary waves due to inaccurately predicted wave reflection and transmission.

Point collocation with mollified piecewise polynomial approximants for high‐order partial differential equations

AbstractThe solution approximation for partial differential equations (PDEs) can be substantially improved using smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of cell‐wise defined piecewise polynomials with a smooth mollifier of certain characteristics. The properties of the mollified basis functions are governed by the order of the piecewise functions and the smoothness of the mollifier. In this work, we exploit the high‐order and high‐smoothness properties of the mollified basis functions for solving PDEs through the point collocation method. The basis functions are evaluated at a set of collocation points in the domain. In addition, boundary conditions are imposed at a set of boundary collocation points distributed over the domain boundaries. To ensure the stability of the resulting linear system of equations, the number of collocation points is set larger than the total number of basis functions. The resulting linear system is overdetermined and is solved using the least square technique. The presented numerical examples confirm the convergence of the proposed approximation scheme for Poisson, linear elasticity, and biharmonic problems. We study in particular the influence of the mollifier and the spatial distribution of the collocation points.

Thermal buckling analysis of nano composite laminated and sandwich beams based on a refined nonlocal zigzag model

A refined nonlocal zigzag model for thermal buckling analysis of nano composite laminated and sandwich beams is proposed in this study based on a refined zigzag theory and Eringen’s nonlocal theory. Firstly, present model satisfies the stress-free and continuity conditions a priori by introducing the piecewise-linear zigzag functions and a preprocessing, such that the transverse shear correction factors are not needed. In the preprocessing, accurate and continuous transverse shear stresses are obtained with the aid of the general mixed variational principle, which can be solved simultaneously with other stresses in the governing equations. This is quite different from the previous post-processing. Subsequently, thermal buckling problems of nano composite laminated and sandwich beams are analytically solved in simply supported boundary conditions. The degenerated results without small effect indicate that the non-dimensional critical loads and critical temperatures have a good agreement with the 3D elasticity solutions and previous results, which demonstrate the accuracy and reliability of present model. Moreover, it is observed that the small effect of the critical temperatures can be effectively captured by Eringen’s differential constitutive law (EDCL), which shows the small effect decreases the critical temperature by weakening the stiffness of the beam. Finally, the effects of different thermal expansion coefficients, laminations, geometric sizes and beam theories are discussed. The results show that present model is robust in the arbitrary layouts for both of composite and sandwich structures, which may have some referential significance to Micro-Electro-Mechanical Systems (MEMS) sensors and actuators.

Analytic solution for two dimensional beam problems: Pure displacement boundary conditions

The present manuscript delineates the derivation of strong solutions for the linear elasticity problem in a two dimensional rectangular beam. The materials under consideration can exhibit either isotropic or orthotropic properties. Additionally, the analysis is not restricted to slender beams because the ratio between the length and the height of the beam can be arbitrary. The boundary conditions fall into the Dirichlet category, implying that both horizontal and vertical displacements are specified on all four surfaces. The sole requirement is that these surface displacements are continuous functions, although they may not necessarily be smooth. Since the displacements at the surfaces can be arbitrary, there is no need to consider approximations, such as those concerning local (small) boundaries in slender beams. Nonetheless, it is demonstrated that an equivalent principle to the Saint-Venant principle exists for pure displacement boundary conditions. The partial differential equations are solved through the separation of variables method, leading to the identification of two solution types, encompassing both cosine and sine Fourier series. Particular emphasis is placed on evaluating the convergence of these solutions. For two distinct and general examples, it is confirmed that the solutions indeed exist.

A study on partial pivot ACA boundary element method for elasticity problems

The boundary element method (BEM) employing the partial pivot adaptive cross approximation (PACA) algorithm has been observed to experience convergence failures and reduced solution accuracy when solving elasticity problems, especially at large scales. To address this issue, this paper proposes an improved algorithm for both 2D and 3D elasticity problems.Our investigations revealed that when the normal of an element is parallel to the coordinate axes and the element is in the same plane as the source point, zero entries with regular distributions will appear in the coefficient matrix. This results in the premature satisfaction of the convergence criterion of the PACA algorithm, leading to the omission of some rows and columns of the coefficient matrix. To address this issue, this paper proposes improvements to the PACA algorithm through the Coordinate Rotation Method (CRM) and the Component Traversal Method (CTM), and validates the effectiveness of these two improved methods.Furthermore, we investigate reasons behind the decrease in accuracy when employing the CTM to solve large-scale problems. We attribute this to the oscillatory nature of the estimated relative error and propose the modified PACA (M-PACA) algorithm. M-PACA not only maintains the advantages of the PACA algorithm in reducing storage space and accelerating coefficient matrix generation but also addresses its limitations, ensuring more accurate and reliable solutions for elasticity problems.

Highly elastic 2D covalent organic framework films of interwoven single crystals.

Highly elastic 2D covalent organic framework films of interwoven single crystals.

Comparative FEM study on intervertebral disc modeling: Holzapfel-Gasser-Ogden vs. structural rebars.

Introduction: Numerical modeling of the intervertebral disc (IVD) is challenging due to its complex and heterogeneous structure, requiring careful selection of constitutive models and material properties. A critical aspect of such modeling is the representation of annulus fibers, which significantly impact IVD biomechanics. This study presents a comparative analysis of different methods for fiber reinforcement in the annulus fibrosus of a finite element (FE) model of the human IVD. Methods: We utilized a reconstructed L4-L5 IVD geometry to compare three fiber modeling approaches: the anisotropic Holzapfel-Gasser-Ogden (HGO) model (HGO fiber model) and two sets of structural rebar elements with linear-elastic (linear rebar model) and hyperelastic (nonlinear rebar model) material definitions, respectively. Prior to calibration, we conducted a sensitivity analysis to identify the most important model parameters to be calibrated and improve the efficiency of the calibration. Calibration was performed using a genetic algorithm and in vitro range of motion (RoM) data from a published study with eight specimens tested under four loading scenarios. For validation, intradiscal pressure (IDP) measurements from the same study were used, along with additional RoM data from a separate publication involving five specimens subjected to four different loading conditions. Results: The sensitivity analysis revealed that most parameters, except for the Poisson ratio of the annulus fibers and C01 from the nucleus, significantly affected the RoM and IDP outcomes. Upon calibration, the HGO fiber model demonstrated the highest accuracy (R2 = 0.95), followed by the linear (R2 = 0.89) and nonlinear rebar models (R2 = 0.87). During the validation phase, the HGO fiber model maintained its high accuracy (RoM R2 = 0.85; IDP R2 = 0.87), while the linear and nonlinear rebar models had lower validation scores (RoM R2 = 0.71 and 0.69; IDP R2 = 0.86 and 0.8, respectively). Discussion: The results of the study demonstrate a successful calibration process that established good agreement with experimental data. Based on our findings, the HGO fiber model appears to be a more suitable option for accurate IVD FE modeling considering its higher fidelity in simulation results and computational efficiency.

Effect of margin designs and loading conditions on the stress distribution of endocrowns: a finite element analysis

BackgroundMargin designs and loading conditions can impact the mechanical characteristics and survival of endocrowns. Analyzing the stress distribution of endocrowns with various margin designs and loading conditions can provide evidence for their clinical application.MethodsThree finite element analysis models were established based on the margin designs: endocrown with a butt-joint type margin (E0), endocrown with a 90° shoulder (E90), and endocrown with a 135° shoulder (E135). The E0 group involved lowering the occlusal surface and preparing the pulp chamber. The E90 group created a 90° shoulder on the margin of model E0, measuring 1.5 mm high and 1 mm wide. The E135 group featured a 135° shoulder. The solids of the models were in fixed contact with each other, and the materials of tooth tissue and restoration were uniform, continuous, isotropic linear elasticity. Nine static loads were applied, with a total load of 225 N, and the maximum von Mises stresses and stress distribution were calculated for teeth and endocrowns with different margin designs.ResultsCompared the stresses of different models under the same loading condition. In endocrowns, when the loading points were concentrated on the buccal side, the maximum von Mises stresses were E0 = E90 = E135, and when there was a lingual loading, they were E0 < E90 = E135. In enamel, the maximum von Mises stresses under all loading conditions were E0 > E90 > E135. In dentin, the maximum von Mises stresses of the three models were basically similar except for load2, load5 and load9. Compare the stresses of the same model under different loading conditions. In endocrowns, stresses were higher when lingual loading was present. In enamel and dentin, stresses were higher when loaded obliquely or unevenly. The stresses in the endocrowns were concentrated in the loading area. In enamel, stress concentration occurred at the cementoenamel junction. In particular, E90 and E135 also experienced stress concentration at the shoulder. In dentin, the stresses were mainly concentrated in the upper section of the tooth root.ConclusionStress distribution is similar among the three margin designs of endocrowns, but the shoulder-type designs, especially the 135° shoulder, exhibit reduced stress concentration.