Linear Elasticity Research Articles

Designing brittle fracture-resistant structures:A tensile strain energy-minimized topology optimization

This research proposes a novel method for designing fracture-resistant structures. By analyzing the relationship between tensile strain energy and phase field brittle fracture, it has been found that minimizing tensile strain energy can delay fracture and enhance resistance. Capitalizing on this insight, a new topology optimization method is proposed. This method focuses on minimizing tensile strain energy to suppress the well-documented tension-dominated fracture behavior observed in phase field brittle fracture analysis. In contrast to traditional topology optimization methods based on von Mises stress, this method generates more robust structures under tension. Furthermore, the method can incorporate stress constraints to mitigate the potential for stress concentrations arising from geometric discontinuities. Numerical results demonstrate the effectiveness of the proposed method. Using phase-field modeling, the mechanical fracture properties of the optimized structures, including peak load, failure displacement, and absorbed elastic energy before fracture, are quantified. Furthermore, experimental tests are also conducted. Both numerical simulations and experimental results are consistently show that structures designed with minimized tensile strain energy exhibit superior fracture toughness. Furthermore, the method offers significant computational efficiency compared to conventional approaches due to its reliance solely on linear elasticity analysis within the optimization process.

Phase-field based shape optimization of uni- and multiaxially loaded nature-inspired porous structures while maintaining characteristic properties

AbstractTriply periodic minimal surfaces (TPMS) are highly versatile porous formations that can be defined by formulas. Computationally based, load-specific shape optimization enables tailoring these structures for their respective application areas and thereby enhance their potential. In this investigation, individual sheet-based gyroid structures with varying porosities are specifically optimized with respect to their stiffness. A modified phase-field method is employed to establish a simulation framework for the shape optimization process. Despite constant volume and the preservation of the periodicity of the unit cells, volume redistribution occurs through displacement of the interfaces. The phase-field-based optimization process is detailed using unidirectional loading on three gyroidal unit cells with porosities of 75 %, 80 %, and 85 %. Subsequently, the gyroidal unit cell with a porosity of 85 % is shape-optimized under multidirectional loading. A subsequent experimental validation of the unidirectionally loaded cells confirms that the shape-optimized structures exhibit, on average, higher stiffness than the non-optimized structures. The highest increase of 40 % in effective modulus is achieved with the gyroid structure having a porosity of 75 %, while maintaining minimal alteration to the surface-to-volume ratio and preserving periodicity. Additionally, the experimental data show that the optimization process resulted in a shift in the linear elasticity and plasticity range. In summary, the phase-field method proves to be a valid optimization technique for complex porous structures, allowing the preservation of characteristic properties.

Analysis of the screw pile group subjected to combined V–H loading for energy application

ABSTRACT The present study investigates the behaviour of screw piles in very stiff clay under the combined vertical and horizontal loads using a three-dimensional finite element analysis. The soil is modelled using the Mohr–Coulomb elastic, perfectly plastic material, whereas the pile shaft and helices are modelled as a linear elastic material. This study examines the effect of the number of helices, helix diameter, inter-helix spacing, and pile spacing on the performance of screw piles under both vertical compression and lateral loads. The study shows that the effect of the number of helices is negligible under the combined load. The increase in the helix diameter (D h) improves the pile performance. The results also reveal that the optimum interhelix spacing is 3D h, and the optimum pile spacing is 2D h. Thus, the present study demonstrates that optimizing the geometry of the screw pile enhances its performance, making it suitable for offshore wind turbines.

Numerical analysis of the stress‐based formulation of linear elasticity

AbstractThe Beltrami–Michell equations of linear elasticity differ from the Navier–Cauchy equations, in that the primary field in former equations is the stress tensor rather than the displacement vector. Consequently, the equations can be used for circumstances where the displacement field is not of interest, for example in design, or when increased smoothness of the solution of the stress tensor is desired. In this work, we explore the stress‐based Beltrami–Michell equations for isotropic linear elastic materials. We introduce the equations in modern tensor notation and investigate their limitations. Further, we demonstrate how to symmetrise and stabilise their weak formulation, complemented by existence and uniqueness proofs. With latter at hand, we construct a conforming finite element discretisation of the equations, avoiding the need for intermediate stress functions. Finally, we present some numerical examples.

Study on the Mechanical and Permeability Evolution of a Composite Rock Mass in an Aquifuge Under Hydraulic Coupling

The lithology and composition type of an aquifuge in overburden play a crucial role in influencing the crack evolution and permeability changes of the aquifuge. This study utilized the high-temperature and high-pressure rock triaxial seepage test system to conduct triaxial compression tests on mudstone, sandstone, and their combined rock samples. The mechanical characteristics and permeability evolution of each lithology law during the failure were investigated. Furthermore, computed tomography (CT) scanning technology was utilized for the three-dimensional (3D) reconstruction and theoretical permeability calculation of single and combined rock samples. The results indicated that the stress–strain curves for single and combined rock samples exhibited similar patterns, which were divided into four stages: pore compaction, linear elasticity, yield deformation, and post-peak residual deformation. The peak strength of rock samples positively correlated with confining pressure. Permeability trends for mudstone and sandstone exhibited an “N”-type pattern characterized by “slow decrease–gradual stabilization–sudden increase–rebound decrease”, while the permeability of mudstone–sandstone combined rock followed a “U”-type pattern of “initial decrease–stabilization–subsequent increase”. Notably, the permeability of the combined rock samples was significantly lower compared to the single rock samples. The failure mode indicated that fractures in a single rock sample transversed the entire sample, whereas failures in the combined rock samples were confined to the mudstone component. This observation accounted for the differences in the permeability changes between the rock sample types. Additionally, the theoretical permeability results from the 3D reconstruction correlated with the experimental results.

Ab initio calculations of second-, third-, and fourth-order partial and inner elastic constants of diamond.

By means of ab initio calculations, a unified framework is presented to investigate the effect of internal displacement on the linear and nonlinear elasticity of single diamond crystals. The calculated linear and nonlinear elastic constants, internal strain tensor and internal displacement in single diamond crystals are compatible with the available experimental data and other theoretical calculations. The complete set of second-, third- and fourth-order elastic constants and internal strain tensor not only offer a better insight into the nonlinear and anisotropic elasticity behaviors, but also shows us the basic internal mechanical response of diamond. This study provides a route to calculate the nonlinear internal and external elasticity response in a nonprimitive lattice.

Stability Analysis of Construction Factors for Partially Cable-Stayed Bridges with Multiple Towers and High Piers

Partially cable-stayed bridges have the characteristics of continuous rigid-frame bridges and cable-stayed bridges, making them a novel composite bridge system. This study focuses on the construction project of a multi-tower high-pier curved partially cable-stayed bridge to investigate the bridge’s stability during construction. The Midas/Civil software was used to establish a model for key construction stages of the bridge, considering structural linear elasticity and geometric nonlinearity. The study examines the impact of static wind loads, asymmetric construction of the main girder, closure sequence, and the load and detachment of the hanging basket on the bridge’s stability during construction. The results indicate that static wind loads have a significant impact on structural geometric nonlinearity, with a maximum reduction of 4.99%. Asymmetric construction at both ends of the main girder can cause structural instability and should be avoided. The geometric nonlinearity stability coefficient for the hanging basket load decreased by 10.83% during the maximum no-cable stage and by 7.84% during the cable stage, significantly affecting the stability during construction. A bridge closure sequence of side-span, secondary midspan, and midspan provides the most stable condition during the construction phase. The results of this study can inform the construction of similar partially cable-stayed bridges.

Mesh topology-based spurious pressure stabilization in 3D finite elasticity using Voronoi tessellations

AbstractIn this paper, we present a mesh topology-based stabilization approach to suppress spurious pressure modes in 3D nearly-incompressible finite elasticity. The focus lies on a mixed formulation with lowest-order approximation for the displacement and pressure fields. Motivated by the fact that the popular H1/P0 element does not fulfill the inf-sup condition, all possible local spurious pressure modes are derived on a patch of elements. The nullspace method is used to determine all spurious pressure solutions. From this, the topological requirements of the finite element mesh are established. We conclude that no more than four elements are allowed to intersect in the same vertex to overcome local checkerboarding. To fulfill this requirement, we employ non-degenerate 3D Voronoi diagrams with several different site distributions. These result in random, centroidal, and honeycomb Voronoi meshes. The resulting convex polyhedral elements are discretized by a polyhedral mixed finite element based on the lowest possible interpolation pair. The numerical examples illustrate that spurious pressure modes do not occur for any degree of mesh refinement as long as the topological mesh requirements are met. Furthermore, it is shown that the numerical inf-sup test is passed. By violating the topological requirements, it is shown that a stable pressure field cannot be guaranteed and the checkerboard phenomenon is provoked.

Finite element analysis of 2.5D packaging processes based on multi-physics field coupling for predicting the reliability of IC components

The 2.5D packaging technology is a high–performance method for electronic packaging. This study addresses the reliability issues of 2.5D packaging during the manufacturing process. A multi–physics field coupling Finite Element Method (FEM) has been developed, combined with sub–modeling techniques, to investigate the curing of underfill adhesive, the curing of Epoxy Molding Compound (EMC), and the reflow soldering between the interposer and substrate in a 2.5D packaging entity during various manufacturing procedures. The focus is on the thermo–mechanical–chemical behavior of viscoelastic components within the packaging structure, as well as the viscoplastic characteristics of the micro solder balls and microbumps. A systematic analysis is conducted on the warpage deformation and stress distribution of the 2.5D packaging at crucial time points. The results demonstrate that after curing, the overall warpage of the packaging exhibits a ‘concave’ warpage profile. Additionally, as the thickness of the EMC above the chip increases, the warpage value of the packaging also increases. The warpage value defined by linear elasticity is larger than that defined by viscoelasticity. The maximum Von Mises stress value in the key areas of the submodel is greater than the maximum Von Mises stress value in the corresponding key areas of the global model. After reflow soldering, the stress concentration in the micro solder balls occurs at the edge of the micro solder ball array. The maximum stress values for each component of the packaging are observed in the interface areas between the components. Packaging components that undergo the curing process have notably higher warpage and Von Mises stress values than those that do not undergo the curing process. The simulation method established in this study can accurately predict the warpage deformation and stress distribution state of 2.5D packaging, providing significant engineering application value for process optimization and reliability enhancement of 2.5D packaging in the production process.

Concurrent experimental testing and computational micromechanical modeling of a UD NCF GFRP composite ply

The current research work presents a detailed and thoroughly validated computational micromechanical modeling methodology to study the damage initiation and propagation in a uni-directional (UD) glass fiber-reinforced non-crimp fabric (NCF) composite ply. Under the applied transverse tension and compression loads, the effect of various microscale material and geometrical parameters on the ply level stress–strain behavior is studied. To this end, along with the distinctly modeled individual microscale constituents of the UD NCF composite ply (axial fibers, backing fibers, and matrix), the generated RVE (Representative Volume Element) model consists of manufacturing-induced defects and variabilities such as voids as well as the backing fiber’s out-of-plane waviness. The fiber/matrix interface failure behavior in the generated RVE model is simulated using cohesive zone formulations that follow bi-linear traction-separation law. Whereas the hydrostatic pressure-dependent non-linear stress–strain response followed by the fracture of the epoxy matrix is captured using the linear Drucker-Prager plasticity model coupled with the stress triaxiality-based failure criterion. The backing fibers stress–strain and failure behavior is modeled using the linear elastic and isotropic material model in conjunction with the maximum stress criterion.The proposed numerical methodology is thoroughly validated both qualitatively and quantitatively by conducting detailed experiments on a neat epoxy as well as at the composite laminate level. Upon comparing the experimental and computational results, the observed knee or slope change in the bi-linear stress–strain response of the UD NCF composite ply under transverse tension is attributed to the loss of the load-carrying ability of the axial fiber bundle due to fiber/matrix interface debonding followed by the ply splitting. Under the application of transverse compression load, the composite ply failure behavior is governed by the formation of a dominant matrix plastic shear band in the axial fiber bundles, which is in turn triggered by the fiber/matrix interface failure. Under the applied loads in the matrix-dominated direction, the presented research work provides a detailed insight into the microscale damage initiation and propagation in a UD NCF composite ply and its consequent influence on the macroscale stress–strain response.

Meshfree method for bending and free vibration analysis of laminated plates using the Reissner’s mixed variational theorem

Free vibration and bending behavior of laminated plates are investigated using a combination of Reissner’s mixed variational theorem and the meshless radial point interpolation method. By introducing Reissner’s mixed variational theorem, a refined transverse shear stress field is derived based on three-dimensional (3D) elasticity equations. The efficiency and accuracy of this model are demonstrated through numerical examples. It is found that accurate prediction of shear stresses significantly improves the prediction of natural frequency for composite laminated plates. The difference between shear stresses derived from 3D equilibrium equations and that from constitutive equations is significant.

A Method for Analyzing Transverse Stress in Link Slabs of Simply Supported Steel–Concrete Composite Bridges

The cracking of link slabs in jointless bridges presents significant challenges due to the complexity of their stress conditions. This study focused on analyzing the transverse stresses in link slabs of jointless steel–concrete composite bridges. Utilizing linear elasticity theory and partial differential equations of plates, the deflection and stress distribution functions for the link slabs were determined. The validity of these analytical solutions was confirmed through comparisons with finite element models and load tests. Results from both the load tests and the finite element model indicate that the upper face of the girder end link slabs experiences maximum tensile stresses in both transverse and longitudinal directions. The stress values obtained from the analytical method align well with these results, showing that the total stress, when considering transverse stresses, reaches 107% of the longitudinal stresses alone. Furthermore, a 40% reduction in longitudinal girder spacing or a 50% increase in girder end length can lead to link slab stresses of 128% and 145% of the longitudinal stresses, respectively. This finding suggests that even loads lower than those designed based solely on longitudinal stresses can result in cracking. Therefore, it is recommended that transverse stresses be considered in the design of link slabs for jointless bridges. Relying solely on conventional longitudinal stress analyses may underestimate actual stress conditions and contribute to the formation of cracks.

I=2 ππ s -wave scattering length from lattice QCD

The I=2 ππ elastic s-wave scattering phase shift is measured by lattice quantum chromodynamics with Nf=3 flavors of the Asqtad-improved staggered fermions. The lattice-calculated energy-eigenvalues of ππ systems at one center of mass frame and some moving frames using the moving wall source technique are utilized to secure phase shifts by Lüscher’s formula. Our computations are fine enough to obtain threshold parameters: scattering length a, effective range r, and shape parameter P, which can be extrapolated at the physical point by next-to-leading order in chiral perturbation theory, and our relevant next-to-next-to-leading order predictions from expanding nuclear physics with lattice quantum chromo dynamics collaboration’s works are novelly considered as the systematic uncertainties. Our outcomes are consistent with Roy equation determinations, newer experimental data, and lattice estimations. Numerical computations are performed with a coarse (a≈0.12 fm, L3T=32364), two fine (a≈0.09 fm, L3T=40396) and a superfine (a≈0.06 fm, L3T=483144) lattice ensembles at four pion masses of mπ∼247 MeV, 249 MeV, 275 MeV, and 384 MeV, respectively. Published by the American Physical Society 2024

Role of viscoelasticity in the adhesion of mushroom-shaped pillars

The contact behaviour of mushroom-shaped pillars has been extensively studied for their superior adhesive properties, often inspired by natural attachment systems observed in insects. Typically, pillars are modeled with linear elastic materials in the literature; in reality, the soft materials used for their fabrication exhibit a rate-dependent constitutive behaviour. Additionally, conventional models focus solely on the detachment phase of the pillar, overlooking the analysis of the attachment phase. As a result, they are unable to estimate the energy loss during a complete loading-unloading cycle.
This study investigates the role of viscoelasticity in the adhesion between a mushroom-shaped pillar and a rigid flat countersurface. Interactions at the interface are assumed to be governed by van der Waals forces, and the material is modeled using a standard linear solid model. Normal push and release contact cycles are simulated
at different approaching and retracting speeds.
Results reveal that, in the presence of an interfacial defect, a monotonically increasing trend in the pull-off force with pulling speed is observed. The corresponding change in the contact pressure distribution suggests a transition from short-range to long-range adhesion, corroborating recent experimental and theoretical investigations.
Moreover, the pull-off force remains invariant to the loading history due to our assumption of a flat-flat contact interface. Conversely, in the absence of defects and under the parameters used in this study, detachment occurs after reaching the theoretical contact strength, and the corresponding pull-off force is found to be rate
independent. Notably, the hysteretic loss exhibits a peak at intermediate detachment speeds, where viscous dissipation occurs, which holds true in both the presence and absence of a defect. However, the presence of a defect shifts the region where the majority of viscous dissipation takes place.

Fracture mechanism and constitutive model considering post-peak plastic deformation of marble under thermal–mechanical action

Temperature plays an important impact on rock mechanical properties. In this paper, the mechanical properties, fracture mechanism and constitutive model of marble under 20–120 °C and 15 MPa are studied. The results show marble deformation can be divided into four stages: compaction, linear elasticity, crack propagation and post-peak failure. Stress–strain curve is not obviously affected by temperature. Macroscopic fracture characteristics change from shear failure to tensile mixed failure with temperature increasing. With the increase of temperature, the strength of marble tends to decrease, indicating that temperature increase has a weakening effect on marble, and there are temperature-sensitive areas of 20–60 °C and temperature sub-sensitive areas of 60–120 °C. The elastic modulus of marble decreases and Poisson’s ratio increases with increasing temperature. The energy evolution law of marble under different temperature is basically the same, which shows that before crack initiation, the energy dissipation is less, and after the damage and yielding occurs, the energy dissipation increases quickly. The energy dissipation in the failure process is mainly used for crack initiation-connection-penetration, as well as plastic deformation caused by friction and slip of cracks, and the plastic deformation and energy dissipation have good linear characteristics. The statistical damage constitutive model based on three-parameter Weibull distribution function can effectively reflect the characteristics of post-peak plastic deformation and strain softening. The weakening effect of marble at 20–120 °C is related to its internal moisture excitation. With the increase of temperature, water is stimulated to absorb and attach to the original relatively dry interface, which plays a role in lubrication. The relative motion friction resistance between solid particles or crack surfaces decreases, which leads to crack initiation and friction energy consumption reduction, changes the specific surface energy of rocks and weakens the strength of marble. The results provide a theoretical basis for predicting and evaluating the long-term stability and safety of surrounding rock of underground deep engineering in complex environment with high ground temperature and high geo-stress.

Numerical analysis of the fungibility of disclination dipole by dislocation array or dislocation monopole

Disclinations have been discovered and applied to a wide variety of materials, underscoring the importance of their analysis. Previous studies have demonstrated that a disclination dipole can be represented by a dislocation array in a deformation schematic and by a dislocation monopole in a stress state. Based on these observations, we propose that the elastic field of a disclination dipole can be substituted with that of a dislocation array or dislocation monopole, allowing the development of the disclination theory grounded in dislocation theory. In this study, we conducted a quantitative analysis of the elastic strain–energy of disclination dipoles, dislocation arrays, and dislocation monopoles using linear elasticity theory and molecular dynamics (MD) simulation. While linear elasticity theory facilitates a quantitative evaluation of the elastic field, it fails to account for out-of-plane deformation of the material. Therefore, MD analysis, which considers out-of-plane deformation, was employed to obtain versatile results applicable to a broader range of materials. Our findings from both theoretical analyses indicate that, in terms of strain–energy, a disclination dipole with a large separation can be represented by a dislocation array, whereas a dipole with a small separation can be represented by a dislocation monopole. This study newly performs strain–energy analysis of dislocations and disclinations using these two theoretical approaches.

Neural network-augmented differentiable finite element method for boundary value problems

Classical numerical methods such as finite element method (FEM) face limitations due to their low efficiency when addressing large-scale problems. As a novel paradigm, the physics-informed neural network (PINN) has demonstrated significant potential to solve partial differential equations. However, conventional PINNs utilize meshless control at discrete sampling points, which limits their ability to effectively handle complex boundaries. Moreover, catastrophic failure may occur in the deep energy method (DEM, a specific type of PINN). To handle these challenges, this study proposes a Neural Network-augmented Differentiable Finite Element Method (NNDFEM) by combining PINN and finite element approximation. In NNDFEM, the neural network backend solely predicts nodal variables. Derivatives and complex boundary conditions can be well handled by the finite element frontend. The governing equation over the domain, Dirichlet, and Neumann boundary conditions are directly enforced on the finite element frontend. Thus, losses of boundary conditions in PINN are rendered unnecessary. The overfitting problem in DEM is also significantly mitigated. Fully connected neural network (FCNN), modified FCNN, and graph-convolutional network are tested as backends. NNDFEM circumvents nodal force calculation and matrix assembly in FEM. Functional losses of linear elasticity, finite strain nonlinear elasticity, heat conduction, and flow in porous media are validated. A systematic exploration unveils the role of 3D finite element mesh. For large-scale problems, a multi-fidelity learning strategy is employed. Thus, the three-dimensional case with over three million degrees of freedom trains well in two minutes. Benefiting from the fast inference of the neural network backend, the forward pass is 8,550 times faster than FEM.

Effect of Three-Dimensional auxetic honeycomb core on behavior of sound transmission loss in shallow sandwich cylindrical shell

The primary objective of this research is to examine the sound transmission loss (STL) in a shallow sandwich cylindrical shell featuring a 3D auxetic honeycomb core. Initially, the 3D elasticity theory was employed by applying the state vector method and extracting both local and global transfer matrices to calculate STL relations for the cylindrical shell, including the auxetic honeycomb core. Subsequently, boundary conditions were applied to calculate the unknowns, eventually leading to a relationship for calculating STL within the structure. The derived equations were numerically solved using MATLAB software. The validity of the results obtained using this method was examined by comparing them with the findings of other researchers. Moreover, a comparison was conducted involving a large ratio of the curvature radius to thickness, considering both the auxetic honeycomb core and aluminum with equal mass. The results demonstrate a significant increase in STL when utilizing this auxetic honeycomb core compared to a material with the same mass. Specifically, at a frequency of 2 Hz, a significant enhancement of about 29.44 % in STL is observed when increasing the core thickness from 10.39 mm to 20.39 mm. Furthermore, STL results have been obtained for various thicknesses, radius of curvature, and incident angles.

A wearable, self-sustainable, and wireless plantar pressure and temperature monitoring system for foot ulceration prognosis and rehabilitation

Continuous foot temperature and pressure monitoring are desirable for prevention and care of foot health conditions such as Diabetic foot ulceration (DFU). However, monitoring these parameters simultaneously is complex, and mostly rely on sensors with undisrupted power supply integration. Herein, we propose high-performance self-sustainable smart footwear capable of monitoring foot pressure and temperature at various locations for real-time applications. The footwear consists of three modules of an integrated electromagnetic generator (EMG) and triboelectric (TENG) self-powered pressure sensors operated by very elastic 3D printed thermoplastic polyurethane (TPU) springs formed by stacking circular plates that allow TENG pressure sensors to be embedded within. Additionally, Laser-induced graphene (LIG) temperature sensors are used for monitoring foot temperature. The embedded EMG can deliver high output power during various activities: walking (38 mW), running (154 mW), and jumping (75 mW), sufficient to power signal processing and transmission circuitries for wireless monitoring of foot temperature and pressure. TENG-based pressure sensors and LIG temperature sensors (LTS) embedded in this design allow mapping of foot pressure (0.2 V/kPa at 1 Hz) and foot temperature (0.55 Ω °C−1) continuously. Finally, footwear was assembled successfully to demonstrate wireless self-sustainable multisensory smart footwear for monitoring foot pressure and temperature to prevent and rehabilitate diabetic foot ulcers.

Nonlinear mechanics of phase-change-induced accretion

In this paper, we formulate a continuum theory of solidification within the context of finite-strain coupled thermoelasticity. We aim to fill a gap in the existing literature, as the existing studies on solidification typically decouple the thermal problem (the classical Stefan’s problem) from the elasticity problem, and often limit themselves to linear elasticity with small strains. Treating solidification as an accretion problem, with the growth velocity correlated with the jump in the heat flux across the boundary, it presents an initial boundary-value problem (IBVP) over a domain whose boundary location is a priori unknown. This IBVP is solved numerically for the specific example of radially inward solidification in a spherical container. Several parametric studies are conducted to compare the numerical results with the rigid cases in the literature and gain insights into the role of elastic deformations in solidification.