Solid Mechanics Research Articles

On a space-time implementation of the wave equation using virtual elements

AbstractThe virtual element method (VEM) was developed not too long ago, starting with the paper (Beirão-da-Veiga et al. in SIAM J Numer Anal 51:794–812, 2013) related to elasticity in solid mechanics. The virtual element method allows to revisit the construction of different elements, however has so far not applied to space-time formulations for one-dimensional structural elements like strings, trusses and beams. Here we study several VEM elements suitable for space-time formulations that are build upon the Hamilton’s principle. It will be shown that these elements can be easily incorporated in classical finite element codes since they have the same number of unknowns. Furthermore, we show that the property of VEM to deal with non-conforming meshes is of special interest for holistic space time formulation: VEM formulations allow locally varying time discretizations (time increments) in a natural and efficient way.

The mixed displacement method to avoid shear locking in problems in elasticity

AbstractThe mixed displacement (MD) method was initially developed to mitigate geometrical locking effects in beams, plates, and shells with the intention of having intrinsically locking‐free characteristics while using equal‐order interpolation for all degrees of freedom. In other words, it is an unlocking scheme that works independent of the element shape, polynomial order, and discretization scheme. It includes additional degrees of freedom that adhere to a carefully designed differential relation that can be interpreted as a kinematic law, incorporated in a mixed sense. Certain constraints are to be enforced on these additional degrees of freedom to obtain a well‐posed system of equations. In this work, the MD method is extended for problems in solid mechanics. We present the underlying variational formulation, followed by its application to 2D solid elements. Additionally, we showcase an idea to enforce the additional constraints in a general sense. Various numerical examples, within the framework of the finite element method and isogeometric analysis, are outlined to demonstrate the performance of the MD method in the geometrically linear and geometrically nonlinear cases.

Compatibility of strains and the three-fold differentiability of the displacement field

The problem of the necessary class of smoothness of solutions to quasi-static problems of deformable solid mechanics in terms of displacements was discussed. It is shown that in order for the equations of compatibility of deformations to become identities when displacements are substituted in them, the existence of some third mixed derivatives of displacements is required. A counterexample for a linearly elastic compressible isotropic elastic medium was given. In this counterexample, the displacement field, being a doubly differentiable solution to the boundary value problem for the system of Lame equations in the entire domain, is not a solution to the displacement problem at all points in this domain.

COMPARATIVE ANALYSIS OF STRESS-STRAIN STATE OF SINGLE-ROOTED TEETH DURING INDIRECT RESTORATION OF THE CROWN WITH STUMP INLAYS OF DIFFERENT DESIGNS

The aim of our study on the biomechanics of the tooth inlay is to select the most efficient design of stump pin inlay for indirect restoration of single-rooted teeth. Materials and methods. To implement the aim of the study, the biomechanics research was conducted in the following four designs of stump inlays: 1 - traditional, 2 - with a single-stage change in the diameter of the pin along its length, 3 - with a two-stage change in the pin along its length, and 4 - with a single-stage change in the diameter of the pin and a cap in the crown of the tooth, covering the end of the root. The mathematical modeling was performed using the common software MSC NASTRAN for finite element analysis. The package, with the help of which the elastic three-dimensional models of the dentition are built and analyzed based on the finite element procedure, determines the displacement of each node of the finite element along the axis system, with three coordinates, normal and tangential stresses, as well as equivalent Huber-Mises stresses, which are calculated according to the well-known formula of deformable solid mechanics Results. The minimum values of equivalent stresses in both the pin inlay and the dentin of the root of the tooth at the level of the apical end of the pin inlay occur when using a pin inlay with a one-stage change in the diameter of the pin and a cap on the crown of the premolar. At the same time, the values of equivalent stress in dentin at the level of the apical end of the inlay pin are closest to the maximum values of equivalent stresses observed on the surface of the root of the tooth (differences do not exceed 10%). The maximum values of equivalent stresses in both the pin inlay and the dentin of the root of the tooth at the level of the apical end of the pin inlay occur when using a pin inlay of conventional design. In this case, the difference in the values of equivalent stresses in the pin compared to the inlay with a one-stage change in the diameter of the pin and a cap is 9%, and in the dentin at the level of the apical end of the inlay pin, the difference in stresses is 10%. The maximum values of equivalent stresses in both the dentin of the root of the tooth to be restored and the pin of the stump inlay occur when applying a traditionally used pin inlay. Among the various pin inlays considered, a stump inlay featuring a single-stage change in the diameter of the pin and a cap on the crown of the tooth to be restored demonstrates the highest efficiency, yielding the lowest equivalent stresses in the dentin of the tooth's root.

Vitreous tissue cryopreservation using a blood vessel model and cryomacroscopy for scale-up studies: Observations and mathematical modeling

Successful long-term cryobanking of multicellular tissues and organs at deep subzero temperatures calls for the avoidance of ice cryoinjury by reliance upon ice-free cryopreservation techniques. However, the quality of the cryopreserved material is the direct result of its ability to survive a host of harmful mechanisms, chief among which is overcoming the trifecta effects of ice crystallization, toxicity, and mechanical stress. This study aims at exploring improved conditions to scale-up ice-free cryopreservation by combining DP6 as a base cryoprotective agent (CPA) solution with an array of synthetic ice modulators (SIMs). This study is conducted by integrating cryomacroscopy techniques, thermal modeling, solid mechanics analysis, and viability and contractility investigation to correlate physical effects, thermal outcomes, and cryobiology results. As an extension of previous work, this study aims at scale-up of established baseline blood vessel models, while comparing the relative toxicity and vitreous stability of 4 ml and 10 ml samples of DP6 containing either sucrose as a SIM, or the commercial synthetic ice blockers (X1000 and Z1000). Using that established protocol, the addition and removal of DP6+0.6M sucrose and DP6 + 1% X1000 + 1% Z1000 were both well tolerated in rabbit carotid and pig femoral artery models, when assessed for metabolic recovery and contractility. Using cryomacroscopy, it was demonstrated that DP6 + 0.6M sucrose provided a stable vitrification medium under marginal cooling and warming conditions that resulted in >50% survival rate. By contrast, DP6 + 1% X1000 + 1% Z1000 was subject to visible ice formation during cooling under the same thermal conditions, resulting in a significantly lower recovery of ∼20%. Thermal modeling is used in this study to verify the actual cooling and rewarming rates in the specimens, while thermomechanics analysis is used to explain why fractures were observed using cryomacroscopy when the specimens were contained in glass vials but not in plastic vials.

Latent-Energy-Based NNs: An interpretable Neural Network architecture for model-order reduction of nonlinear statics in solid mechanics

Latent-Energy-Based NNs: An interpretable Neural Network architecture for model-order reduction of nonlinear statics in solid mechanics

Simple yet effective adaptive activation functions for physics-informed neural networks

Physics-informed neural networks (PINNs) gained widespread advancements in solving differential equations, where the performance tightly hinges on the choice of activation functions that are inefficient when selected manually. To tackle this issue, we propose two straightforward yet powerful adaptive activation functions: a weighted average function that adjusts activation functions by directly manipulating their weights, and a L2-normalization function that compresses learnable parameters. These methods ensure a consistent sum of weights for each activation function, thereby enhancing optimization efficiency. We assess the performance of these approaches across a range of differential equation problems, encompassing Poisson equation, Wave equation, Burgers equation, Navier-Stokes equation, and linear/nonlinear solid mechanics problems. Through comparisons with exact solutions, we demonstrate significant improvements in convergence rate and solution accuracy. Our results underscore the efficacy of these techniques, providing a simple yet promising pathway for augmenting PINN performance across diverse applications. The source codes and software implementation are available at https://github.com/jzhange/AAF-for-PINNs.

Numerical study of liquid sloshing in three-dimensional flexible cylindrical tank under multi-directional seismic excitation

The present study aims to evaluate the effect of the flexibility of three-dimensional (3D) cylindrical tank on liquid sloshing inside the tank as well as the influence of the hydrodynamic forces on the structure's deformation. A ground-supported cylindrical flexible tank filled with water and subjected to seismic excitation is investigated using a numerical coupling methodology to take into account the fluid–structure interactions. The Navier–Stokes equations in the fluid domain are solved using the finite volume method, while the finite element method is used to solve the linear-elastic equations of the structure. The arbitrary Lagrangian–Eulerian formulation is applied for the moving mesh in two-phase flow system. The simulations are conducted using free open-source software: OpenFOAM for fluid dynamics and FEniCS for solid mechanics, both utilize the preCICE library for data exchanging and fluid–structure coupling. The numerical methodology is validated by the experimental and numerical results given in literature. A multi-directional earthquake ground motion is then considered as external loading, and the elasticity of the tank wall is varied to investigate the effect on the fluid sloshing. It has been shown that the more flexible the tank walls are, the greater will be both the sloshing height and the structural deformations during an earthquake event. Several flow field information and structural responses have been provided, such as the sloshing height, hydrodynamic pressure, structure deformations, and structural stress distribution. The potential elephant's foot buckling phenomenon at the lower part of tank can also be predicted.

An Elastohydrodynamic model of the slot-die coating process

Abstract The slot-die coating process plays an important role in the industry, as it is employed in many different fields. The characteristics of the final application are determined by the flow between the die and the roller. This research paper aims to develop a mathematical model of such flow that takes into account the roller deformations caused by high pressure values reached by the coating fluid. This Elastohydrodynamic model is made up of a coupling between the mathematical model of the flow and the mathematical model of the roller deformations. Model resolution is undertaken numerically by deforming the flow domain according to the roller deformations using computational fluid dynamics (CFD) and computational solid mechanics (CSM) techniques. For its part, the finite volume method (FVM) is used to perform the flow model analysis while the finite element method (FEM) is employed to deal with roller deformations. The results obtained from this model give information on the flow pressure distribution, coating gaps, meniscus position, extent of roller deformations in the coating flow and the influence of different operating conditions. The information obtained from this study is valuable for industrial applications, as it gives insights into the coating process that can help manufacturers to define a suitable combination of operating parameters in order to obtain coating applications that meet quality and performance requirements.

High‐Performance Flexible Silicon Nanowire Field Effect Transistors on Plastic Substrates

AbstractInorganic semiconductor nanowires, known for their exceptional electronic properties and mechanical flexibility, are widely regarded as the ideal 1D channel materials for creating high‐performance flexible electronics. In this work, the integration of ordered arrays of silicon nanowire (SiNW) field effect transistors (FETs) directly onto flexible plastic substrates is showcased. The self‐aligned crystalline SiNW multi‐channels are first grown through an in‐plane solid–liquid–solid mechanism on rigid substrates, and then efficiently transferred in‐batch onto flexible polyethylene terephthalate (PET) plastics. The FETs constructed on these transferred SiNW channels exhibit outstanding performance, with a high on/off current ratio of >105, a low subthreshold swing of 175 mV dec−1, and remarkable mechanical stability that can endure an extremely small bending radius of 0.5 mm for 1000 cycles. Furthermore, inverter logics are also successfully demonstrated on plastic substrates, highlighting a prominent routine for scalable integration of high‐quality SiNW channels in the development of low‐cost, high‐performance flexible displays and wearable electronics.

Zero-Strain Sodium Nickel Ferrocyanide as Cathode Material for Sodium-Ion Batteries with Ultra-Long Lifespan.

Prussian blue analogues are recognized as one of the most promising cathode materials for sodium-ion batteries (SIBs) owing to the open 3D framework structure with large interstitial sites and developed Na-ion diffusion channels. However, high content of anion vacancy and poor structural stability have hampered the prospect of their application. In this work, sodium nickel ferrocyanide (Na1.34Ni[Fe(CN)6]0.92, NNHCF) is proposed as cathode material for SIBs. N-coordinated Ni-ion can boost NNHCF to possess less Fe[(CN)6]4- defect, low Na-ion migration barrier, and more negative formation energy compared with Na0.91Cu[Fe(CN)6]0.77 (NCHCF), thus exhibiting more active sites, fast electrochemical kinetics behavior and great structure stability. It is confirmed that NNHCF undergoes a solid solution mechanism without phase evolution for reversible Na-ion intercalation/deintercalation, employing Fe associated with C atom as redox center for charge compensation. Therefore, NNHCF contributes a high initial energy density of 180.94 Wh·g-1 at 10 mA·g-1, excellent rate capability, superior cycling stability with ultra-long lifespan of 13000 cycles, and low fading rate of 0.0027% per cycle at 500 mA·g-1. This work sheds light on the construction of low-defect PBA cathodes with outstanding dynamics and stability.

Shrinkage and deformation of material extrusion 3D printed parts during sintering: Numerical simulation and experimental validation

Material extrusion 3D printing (ME3DP) combined with sintering is a low-cost additive manufacturing technique for fabricating components of difficult-to-print metals, such as copper, aluminum, and ceramics. However, the sintering process includes complex material science, such as volumetric shrinkage and free structural bending, to identify the relative density and deformation of a 3D printed sample. The prediction of the relative density and deformations during the sintering process provides information to the design engineer to optimize the design of the CAD model before sintering. In this study, a phenomenological model based on constitutive equations was developed to predict the density and structural deformation during the sintering process of pure copper components fabricated by ME3DP using metal injection molding feedstock. The densification rate was determined using shrinkage estimation with an isothermal stairway heating cycle in a vertical dilatometer. Furthermore, different sets of experiments were performed with a load on the probe with long isothermal heating cycles at 850, 900, 950, 1000, and 1050 °C in a vertical dilatometer to estimate the axial viscosity of the copper. The constitutive equations were solved using the solid mechanics module with user-defined creep in COMSOL Multiphysics by considering isotropic assumptions. Two types of geometries, cube and overhanging I section, were used to predict shrinkage and deformation during the sintering process. The developed model successfully predicted the relative density based on shrinkage and structural deformation owing to gravity during the sintering process.

Treatment of inelastic material models within a dynamic ALE formulation for structures subjected to moving loads

AbstractThis article showcases the development of a dynamic Arbitrary Lagrangian Eulerian (ALE) formulation to account for inelastic material models within a finite element framework. Such a formulation is commonly utilized in research domains like fluid mechanics, fluid‐structure interaction, quasi static remeshing techniques, and quasi static load movement. The work at hand describes the application of the ALE formulation to efficiently analyse structures subjected to moving loads in the field of transient inelastic solid mechanics. In particular, structures such as pavements, gantry crane girders etc., which are subjected to moving loads, can be numerically simulated, and their transient response in the relevant region around the load can be obtained without relying on moving loads. The focus of this article is to facilitate the treatment of history variables stemming from inelastic material models. Of particular interest is the advection procedure required to transport the history variables through the mesh, as the material appears to flow through it. The mathematical framework necessary to treat this advection process is described in detail, considering a nonlinear viscoelastic material model on a neo‐Hookean base at finite deformations. Then, four methods for numerically achieving the advection are implemented within a transient finite element ALE formulation. These methods are compared against each other, and additionally with the conventional Lagrangian method for validation. The results demonstrate satisfactory agreement with conventional simulation methods, while offering a significant improvement in terms of computation speed. With the work at hand, the dynamic response of inelastic materials subjected to moving loads can be numerically simulated in a computationally efficient manner.

Directional Transport in Hierarchically Aligned ZSM-5 Zeolites with High Catalytic Activity.

Zeolites, the most technically important crystalline microporous materials, are indispensable cornerstones of chemical engineering because of their remarkable catalytic properties and adsorption capabilities. Numerous studies have demonstrated that the hierarchical engineering of zeolites can maximize accessible active sites and improve mass transport, which significantly decreases the internal diffusion limits to achieve the desired performance. However, the construction of hierarchical zeolites with ordered alignments and size-controlled substructures in a convenient way is highly challenging. Herein, we develop a facile procedure using two common structure-directing agents, tetrapropylammonium hydroxide (TPAOH) and tetraethylammonium hydroxide (TEAOH), to synthesize hierarchically aligned ZSM-5 (Hie-ZSM-5) crystals with a-axis alignment substructures of controllable size. The control of the substructure size (α) in the range of 10-60 nm and the corresponding similarity (r = α/β, where β is the size of Hie-ZSM-5) ranging from 0.004 to 0.033 can be tuned by varying the Si/Al ratios (40-120). A systematic investigation of the overall crystallization process, using time-dependent XRD, SEM, TEM, and solid-state magic-angle spinning NMR (13C, 27Al, 29Si) methods, enable us to construct a solid mechanism for the generation of Hie-ZSM-5. Most importantly, directional transport in the unique structures of Hie-ZSM-5 efficiently enhances mass diffusion, as well as catalytic activity and stability. These findings improve our understanding of the zeolite crystallization process and inspire novel methods for the rational design of hierarchical zeolites.

CFD Modeling of HBI/scrap Melting in Industrial EAF and the Impact of Charge Layering on Melting Performance.

The melting of scrap and hot briquetted iron (HBI) in an AC electric arc furnace (EAF) is simulated by an advanced 3D computational fluid dynamics (CFD) model that captures the arc heating, the scrap/HBI melting process, and the solid collapse mechanisms. The CFD model is used to simulate a scenario where charge layering and EAF power profiles are provided by a real EAF operation. CFD simulation of the EAF operation shows proper prediction of the charge melting when compared with standard industry practice. Namely, the CFD model predicts a 32.5%/67.5% ratio of solid/liquid steel at the beginning of refining, which approaches the 30%/70% ratio used in standard practice. Based on this prediction, the melting rate in the CFD results differs by 8.3% from actual EAF operation. The impact of charge layering on melting is also investigated. CFD results show that distributing charge material into a greater number of layers in the first bucket (10 layers as compared to 4) enhances the melting rate by 12%. However, including dense material at the bottom of the furnace deteriorates melting performance, reducing the impact of the number of layers of the charge. The CFD platform can be used to optimize the use of HBI/scrap in real EAF operations and to determine best recipe practices.

Numerical simulation of transient thermal stresses in medium- and high-temperature latent heat storage systems with different fins: Incorporation of experimental stress cracking

Latent heat storage systems play an essential role in energy applications by improving the utilization efficiency and increasing the system flexibility. In this study, a pilot-scale medium-to-high-temperature latent heat storage system was constructed, and experimental heat storage tests were conducted. The temperature distribution of the experimental system was verified through numerical simulations that coupled the physical fields of the fluids, heat transfer, and solid mechanics. Also, the changing dynamic behavior of the stresses in the heat storage process related to the experiments was carefully studied. It was discovered that there were stress concentrations where the heat exchanger tubes touched the casing, with the highest level being over 370.00 MPa. Experimental stress rupture of the shell occurred in the horizontally arranged heat-exchange tubes, whereas vertically arranged tubes significantly reduced the stress at the contact surfaces by 24.70 % according to simulations. Simulation studies on various influencing factors indicated a more significant impact of spiral fins on the system stress. Increasing the preheating temperature by 30 °C reduced stress on the front surface by 109.67 MPa; each 20 °C increase in heating temperature raised the stress by 223.30 MPa, whereas the effect of heat transfer fluid speed was minimal. Regarding the heat-exchange tube dimensions, increasing the height increased the tube body stress by 25.74 MPa, whereas increasing the thickness reduced the stress by 24.20 %. This study comprehensively analyzed the stress dynamics and mitigation strategies of the high-temperature latent heat storage system under various conditions, emphasizing the importance of heat-exchange tube orientation, temperature control, and fin design for optimizing system performance and durability.

Delaying cyclization of polyacrylonitrile by boric acid for sulfurized poly(acrylonitrile) cathode materials

Sulfurized poly(acrylonitrile) (SPAN) has become one of the most promising cathode materials in rechargeable Lithium-Sulfur (Li-S) batteries due to their inhibited polysulfide dissolution, high sulfur utilization and excellent electrochemical performance via unique solid–solid conversion mechanism. However, the sulfur content of SPAN (<50 %) is still one of the significant reasons that restrict the electrochemical performance and practical applicability. In this paper, a small amount of boric acid (HBO) was introduced into the precursor of SPAN cathode materials easily before heat-treating, to increase the sulfur content and improve the electrochemical performance. Meanwhile, the results of first principles calculation suggested that the increase of sulfur content of SPAN was derived from the delayed cyclization of polyacrylonitrile (PAN) by HBO. We also discovered when the additive amount of HBO was 1.5 % of PAN mass, the SPAN cathode material possessed high sulfur content up to 54.24 wt%, and delivered prominent reversible electrochemical performances of 714 mAh/g at 0.2C and 734 mAh/g at 0.1C with high Coulombic efficiency. In summary, we proposed a simple and easy method to improve sulfur content and introduced a new strategy for the design of high-sulfur content and high-performance SPAN cathode materials.

Slowed Singlet Exciton Fission Enhances Triplet Exciton Transport in Select Perylenediimide Crystals.

Singlet fission (SF) materials used in light-harvesting devices must not only efficiently produce spin-triplet excitons but also transport them to an energy acceptor. N,N'-Bis(2-phenylethyl)-3,4,9,10-perylenedicarboximide (EP-PDI) is a promising SF chromophore due to its photostability, large extinction coefficient, and high triplet yield, but the energy transport mechanisms in EP-PDI solids are minimally understood. Herein, we use transient absorption microscopy to directly characterize exciton transport in EP-PDI crystals. We find evidence for singlet-mediated transport in which pairs of triplet excitons undergo triplet fusion (TF), producing spin-singlet excitons that rapidly diffuse. This interchange of singlet and triplet excitons shuttles triplets as far as 205 nm within the first 500 ps after photoexcitation. This enhanced transport comes at a cost, however, as it necessitates favoring triplet recombination and thus requires fine-tuning of SF dynamics to balance triplet yields with triplet transport lengths. Through numerical modeling, we predict tuning the ratio of SF and TF rate constants, kSF/kTF, to between 1.9 and 3.8 allows for an optimized triplet transport length (425-563 nm) with minimal loss (7-10%) in triplet yield. Interestingly, by adjusting the size of EP-PDI crystals, we find that we can subtly tune their crystal structure and thereby alter their SF and TF rates. By slowing SF within small EP-PDI crystals, we are able to boost their triplet transport length by ∼20%. Although counterintuitive, our work suggests slowing SF by introducing moderate structural distortions can be preferential when optimizing triplet exciton transport, provided singlet exciton transport is not significantly hindered.

Physics-Informed Neural Network for Solving a One-Dimensional Solid Mechanics Problem

Our objective in this work is to demonstrate how physics-informed neural networks, a type of deep learning technology, can be utilized to examine the mechanical properties of a helicopter blade. The blade is regarded as a one-dimensional prismatic cantilever beam that is exposed to triangular loading, and comprehending its mechanical behavior is of utmost importance in the aerospace field. PINNs utilize the physical information, including differential equations and boundary conditions, within the loss function of the neural network to approximate the solution. Our approach determines the overall loss by aggregating the losses from the differential equation, boundary conditions, and data. We employed a physics-informed neural network (PINN) and an artificial neural network (ANN) with equivalent hyperparameters to solve a fourth-order differential equation. By comparing the performance of the PINN model against the analytical solution of the equation and the results obtained from the ANN model, we have conclusively shown that the PINN model exhibits superior accuracy, robustness, and computational efficiency when addressing high-order differential equations that govern physics-based problems. In conclusion, the study demonstrates that PINN offers a superior alternative for addressing solid mechanics problems with applications in the aerospace industry.

Continuum Mechanics Applied to the Biomechanics of Motion.

Continuum Mechanics is the discipline that studies the motion and deformation of material bodies, and is at the basis of both Solid Mechanics and Fluid Mechanics. This works aims at highlighting how a basic education in modern Continuum Mechanics can be of fundamental importance not only in the Biomechanics of Tissue, but also in the Biomechanics of Movement. The latter is largely based on Rigid Body Mechanics, which can be considered as a simple particular case of the general theory of Continuum Mechanics. As an example, a straightforward methodology for the extraction of the angular velocity from motion analysis data is illustrated. The method is based on a quantity that is more primitive than the angular velocity: the spin tensor.