Linear Elasticity Research Articles

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.

Atomic simulation of phosphorus segregation in nickel grain boundaries and its impact on deformation mechanisms

The segregation behavior of phosphorus on the grain boundary (GB) structure and mechanical response has been studied in the Ni–P system. Fourteen twist and tilt bi-crystals have been selected from the Olmsted dataset. Then, hybrid Monte Carlo/Molecular dynamics simulations (MC/MD) are used to obtain the equilibrium GB at 300 K and 1000 K, respectively. The present study indicates that solute phosphorus dissolved in the nickel matrix is inhomogeneous segregated to all of the GBs. GB characters including GB free volume and energy are found to correlate with segregation. The participated phase with Ni3P composition was observed only at 1000 K and for the lower energy GBs. Furthermore, we performed simulations of tensile deformation for the GBs with and without segregation. The twist and tilt GBs initially exhibit linear and non-linear elasticity behavior, respectively. The twist GBs demonstrate an increased yield strength with the help of phosphorus segregation. Besides, the thermal annealing process at 1000 K promotes uniform segregation at GB, leading to further strength enhancement. Conversely, the strength is weakened in segregated tilt GBs. This work illustrates the relationships among composition, GB structure, and mechanical properties.

A method for predicting deformation field of deep foundation pit considering spatial effect

Urban underground spaces have experienced significant development and utilization due to the rapid progress of urban construction and continuous advancements in construction technology. The number of foundation-pit projects immediately adjacent to buildings, structures, and pipelines has been increasing. Ensuring the safety of construction and surrounding facilities, it is essential to predict deformation caused by foundation pit excavations. In this study, a simple calculation method was proposed for predicting deformation of an envelope structure and surface settlement caused by deep foundation excavation with spatial effects. Firstly, the principle of minimum potential energy was applied and the lateral displacement equation of the envelope structure was derived, considering spatial deformation effects. Secondly, based on linear elasticity theory, a surface settlement expression was formulated utilizing the plane strain boundary. Finally, the stratigraphic loss method was used to establish a connection between horizontal envelope deformation and surface settlement, allowing for calculations of surface settlement at any position behind the envelope structure. The effectiveness and practicability of the proposed method were demonstrated by comparison of two existing foundation-pit engineering examples.

The method of developing the complex stress tensor by basic states for the construction solutions of spatial boundary problems of the linear elasticity theory

The method of developing the complex stress tensor by basic states for the construction solutions of spatial boundary problems of the linear elasticity theory

A Critical Comparison of Shape Sensing Algorithms: The Calibration Matrix Method versus iFEM.

Two shape-sensing algorithms, the calibration matrix (CM) method and the inverse Finite Element Method (iFEM), were compared on their ability to accurately reconstruct displacements, strains, and loads and on their computational efficiency. CM reconstructs deformation through a linear combination of known load cases using the sensor data measured for each of these known load cases and the sensor data measured for the actual load case. iFEM reconstructs deformation by minimizing a least-squares error functional based on the difference between the measured and numerical values for displacement and/or strain. In this study, CM is covered in detail to determine the applicability and practicality of the method. The CM results for several benchmark problems from the literature were compared to the iFEM results. In addition, a representative aerospace structure consisting of a twisted and tapered blade with a NACA 6412 cross-sectional profile was evaluated using quadratic hexahedral solid elements with reduced integration. Both methods assumed linear elastic material conditions and used discrete displacement sensors, strain sensors, or a combination of both to reconstruct the full displacement and strain fields. In our study, surface-mounted and distributed sensors throughout the volume of the structure were considered. This comparative study was performed to support the growing demand for load monitoring, specifically for applications where the sensor data is obtained from discrete and irregularly distributed points on the structure. In this study, the CM method was shown to achieve greater accuracy than iFEM. Averaged over all the load cases examined, the CM algorithm achieved average displacement and strain errors of less than 0.01%, whereas the iFEM algorithm had an average displacement error of 21% and an average strain error of 99%. In addition, CM also achieved equal or better computational efficiency than iFEM after initial set-up, with similar first solution times and faster repeat solution times by a factor of approximately 100, for hundreds to thousands of sensors.

Quantifying Domain Uncertainty in Linear Elasticity

Quantifying Domain Uncertainty in Linear Elasticity

Efficient and accurate uncertainty quantification in engineering simulations using time-separated stochastic mechanics

A robust method for uncertainty quantification is undeniably leading to a greater certainty in simulation results and more sustainable designs. The inherent uncertainties of the world around us render everything stochastic, from material parameters, over geometries, up to forces. Consequently, the results of engineering simulations should reflect this randomness. Many methods have been developed for uncertainty quantification for linear elastic material behavior. However, real-life structure often exhibit inelastic material behavior such as visco-plasticity. Inelastic material behavior is described by additional internal variables with accompanying differential equations. This increases the complexity for the computation of stochastic quantities, e.g., expectation and standard deviation, drastically. The time-separated stochastic mechanics is a novel method for the uncertainty quantification of inelastic materials. It is based on a separation of all fields into a sum of products of time-dependent but deterministic and stochastic but time-independent terms. Only a low number of deterministic finite element simulations are then required to track the effect of (in)homogeneous material fluctuations on stress and internal variables. Despite the low computational effort the results are often indistinguishable from reference Monte Carlo simulations for a variety of boundary conditions and loading scenarios.