Nonlinear Elastic Behavior Research Articles

Mechanical relaxations of hydrogels governed by their physical or chemical crosslinks

In the field of tissue engineering, in order to restore tissue functionality hydrogels that closely mimic biological and mechanical properties of the extracellular matrix are intensely developed. Mechanical properties including relaxation of the surrounding microenvironment regulate essential cellular processes. However, the mechanical properties of engineered hydrogels are particularly complex since they involve not only a nonlinear elastic behavior but also time-dependent responses. An accurate determination of these properties at microscale, i.e. as probed by cells, becomes an essential step to further design hydrogel-based biomaterials able to induce specific cellular responses. Atomic Force Microscopy (AFM) with contact sizes of the order of few micrometers constitutes an appropriate technique to determine the origin of relaxation mechanisms occurring in hydrogels. In the present study, AFM force relaxation experiments are conducted on chemically and physically crosslinked hydrogels respectively based on a synthetic polymer, polyacrylamide and a natural polymer, a bacterial exopolysaccharide infernan, produced by the deep-sea hydrothermal vent bacterium, Alteromonas infernus. Two distinct relaxation mechanisms are clearly evidenced depending on the nature of hydrogel network crosslinks. Chemically crosslinked hydrogel exhibits poroelastic relaxations, whereas physically crosslinked hydrogel shows time-dependent responses arising from viscoelastic effects. In addition, two relaxation processes are revealed in ionic physical hydrogel originating from chain rearrangement and breaking/reforming of the ionic crosslinks. The effect of the ionic strength on both the long-term elastic modulus and relaxation times of physical hydrogels was also shown. These findings highlight that physical hydrogels with well-defined time-dependent mechanical properties could be tuned for an optimized response of cells.

A UMAP-based clustering method for multi-scale damage analysis of laminates

In this study, a novel multi-scale damage model based on the parametric finite-volume direct averaging micromechanics theory (parametric FVDAM), extended finite element method (XFEM), and a clustering algorithm, is built, to describe the progressive damage evolution of laminates, where a uniform manifold approximation and projection (UMAP)-based clustering method is specially designed and incorporated with the FVDAM and XFEM for the first time. The subvolumes in the unit cell generated by FVDAM with similar mechanical responses were included in the same cluster using the UMAP-based clustering method. The semi-analytical framework of FVDAM allows the efficient generation of unit cells. The built-in advantage in the visualization of the proposed UMAP-based clustering method allows researchers in the field of mechanics to have a clear and intuitive understanding of the clustering results. The clustered unit cell is then conjugated to the XFEM, which allows the damage information to be transmitted between different scales. Three damage mechanisms of laminates under a three-dimensional (3D) stress state are considered, which include matrix damage, fiber breakage, and delamination. Furthermore, a shear nonlinear matrix Ds for a 3D element is derived and introduced in the model, which enables the description of the unique elastic nonlinear behavior of laminates. The model is implemented using UDMGINI and USDFLD subroutines in ABAQUS, which not only allows the calculation of the load–displacement curve, but also the simulation of clear and smooth crack propagation paths. To verify the effectiveness of the model, the experiments of [±60]2s and [90]8 notched glass-fiber/epoxy laminates under tension were performed by digital image correlation (DIC) monitoring, and the results compared were in good agreement with the simulations, thus proving the effectiveness of the proposed model. The code of the UMAP-based clustering method developed in this study can be downloaded from https://github.com/Danhui-Yang/MicroMechanicsUmapClustering.

Gelatin-glucosamine hydrochloride/crosslinked-cyclodextrin metal-organic frameworks@IBU composite hydrogel long-term sustained drug delivery system for osteoarthritis treatment

Osteoarthritis (OA) is a disease of articular cartilage degradation and inflammation of the joint capsule. Combining anti-inflammatory therapy with nutritional supplement is an effective means for the treatment of OA. In this study, we prepared gelatin (Gel)-glucosamine hydrochloride (GH) mixed crosslinked-cyclodextrin metal-organic framework (G-GH/CL-CD-MOF) composite hydrogel. Polyethylene glycol diglycidyl ether was the crosslinking agent of GH and Gel to solve the problem of poor mechanical properties and water solubility at 37 °C. CL-CD-MOF was fabricated through a simple one-step chemical reaction to crosslink the hydrophilic hydroxyl groups in CD-MOF with diphenyl carbonate. Electron microscopy and x-ray diffraction analysis of CL-CD-MOF showed perfect porous morphology with a chaotic internal structure. CL-CD-MOF@IBU was prepared by immersing CL-CD-MOF in high-concentration ibuprofen (IBU) solution. CL-CD-MOF@IBU was uniformly dispersed in Gel and GH mixed solution to prepare G-GH/CL-CD-MOF@IBU composite hydrogel long-term sustained drug delivery system. The compression curve of G-GH/CL-CD-MOF composite hydrogel showed a non-linear elastic behavior. The cyclic loading–unloading compression showed that the shape of the G-GH/CL-CD-MOF composite hydrogel can be kept intact under 50% strain. On the day 14, the G-GH/CL-CD-MOF@IBU composite hydrogel was degraded by 87.1%, 61% of IBU was released. G-GH/CL-CD-MOF@IBU exhibited good biocompatibility during co-culture with MC3T3-E1 cells. Briefly, the certain mechanical properties, sustained drug release behavior, and good biocompatibility of G-GH/CL-CD-MOF@IBU composite hydrogel showed that it has potential application in OA treatment of long-term sustained nutritional supplement and anti-inflammatory synchronously.

Nonlinear-elastic characterization of cement-based materials under uniaxial stress: A comparison between ultrasonic and resonance techniques

The nonlinear-elastic behavior of concrete and other cement-based materials has shown promising potential for determining quasistatic stresses using nondestructive testing (NDT) techniques. However, given concrete’s intrinsic complex behavior, the topic is not fully understood. To investigate this problem, two groups of NDT-techniques have emerged, one based on ultrasonic wave propagation and the other one based on resonance vibration. In this research, we first present a unified theoretical description of both techniques using the 1-D classical nonlinear parameter βG. Then, we consider a mortar specimen and characterize its material nonlinearity by measuring βG with both techniques simultaneously. Our results show that both the torsional frequency of vibration and the S-wave speed increase, with good repeatability, as compressive stress increases. Moreover, resonance vibration yielded a βG value almost four times greater than ultrasound, even though the theory predicts equal results. A final section discusses a possible physical explanation.

Investigation of a nonlinear piezoelectric energy harvester with advanced electric circuits with the finite element method

In this contribution, a system simulation method based on the finite element method (FEM) is applied to simulate a strongly coupled bimorph piezoelectric vibration-based energy harvester (PVEH) with various nonlinear, non-ideal and active circuits: The standard circuit, the synchronized switch harvesting on inductor circuit and the synchronized electric charge extraction circuit are considered. Furthermore, nonlinear elastic behavior of the piezoelectric material is taken into account and harmonic base excitations of different magnitudes at a fixed frequency are applied. The holistic FEM-based system simulation approach solves the complete set of piezoelectric equations together with the equation of the electric circuit such that all electromechanical coupling phenomena are taken into account. This fully coupled numerical analysis enables the detailed evaluation of the influences of the electric circuits on the vibrational behavior and the harvested energy of the PVEH with respect to the magnitude of base excitation. Results from literature on the efficiency of electric circuits are confirmed and interactions between mechanical and electrical nonlinearities of PVEHs are revealed.Article highlightsSystem simulations of a mechanically and electrically nonlinear piezoelectric energy harvester are performed using only one software tool.The infuence of electric circuits on the vibration behavior and the effciency of an energy harvester are investigated in detail.Interactions between mechanical and electrical nonlinearities of an energy harvester are revealed.

Origin of the non-linear elastic behavior of silicate glasses

For small tension the response of a solid to an applied stress is given by Hooke’s law. Outside this linear regime the relation between stress and strain is no longer universal and at present there is no satisfactory insight on how to connect for disordered materials the stress-strain relation to the microscopic properties of the system. Here we use atomistic computer simulations to establish this connection for the case of silicate glasses containing alkali modifiers. By probing how in the highly non-linear regime the stress-strain curve depends on composition, we are able to identify the microscopic mechanisms that are responsible for the complex dependence of stress on strain in these systems, notably the presence of an unexpected quasi-plateau in the tangent modulus. We trace back this dependence to the mobility of the modifiers which, without leaving their cage or modifying the topology of the network, are able to relieve the local stresses. Since the identified mechanism is general, the results obtained in this study will also be helpful for understanding the mechanical response of other disordered materials.

A data-driven approach to characterizing nonlinear elastic behavior of soft materials

The Autoprogressive (AutoP) method is a data-driven inverse method that leverages finite element analysis (FEA) and machine learning (ML) techniques to build constitutive relationships from measured force and displacement data. Previous applications of AutoP in tissue-like media have focused on linear elastic mechanical behavior as the target object is infinitesimally compressed. In this study, we extended the application of AutoP in characterizing nonlinear elastic mechanical behavior as the target object undergoes finite compressive deformation. Guided by the prior of nonlinear media, we modified the training data generated by AutoP to speed its ability to learn to model deformations. AutoP training was validated using both synthetic and experimental data recorded from 3D objects. Force–displacement measurements were obtained using ultrasonic imaging from heterogeneous agar–gelatin phantoms. Measurement on samples of phantom components were analyzed to obtain independent measurements of material properties. Comparisons validated the material properties found from neural network constitutive models (NNCMs) trained using AutoP. Results were found to be robust to measurement errors and spatial variations in material properties.

Non-linear elastic behavior and constitutive model of coal during compression and its application

Non-linear elastic behavior and constitutive model of coal during compression and its application

Feasibility of Using Shear Wave Ultrasonic Probes as Pump-Wave Sources in Concrete Microcrack Detection and Monitoring by Nonlinear Ultrasonic Coda Wave Interferometry.

This paper represents a first attempt to study the feasibility of using shear wave (SW) ultrasonic probes as pump-wave sources in concrete microcrack detection and monitoring by Nonlinear Ultrasonic Coda Wave Interferometry (NCWI). The premise behind our study is that the nonlinear elastic hysteretic behavior at microcracks may depend on their orientation with respect to the stationary wave-field induced by the pump-wave source. In this context, the use of a SW probe as a pump-wave source may induce the nonlinear elastic behavior of microcracks oriented in directions not typically detected by a conventional longitudinal pump-wave source. To date, this premise is hard to address by current experimental and numerical methods, however, the feasibility of using SW probes as a pump-wave source can be experimentally tested. This idea is the main focus of the present work. Under laboratory conditions, we exploit the high sensitivity of the CWI technique to capture the transient weakening behaviour induced by the SW pump-wave source in concrete samples subjected to loading and unloading cycles. Our results show that after reaching a load level of of the ultimate stress, the material weakening increases as a consequence of microcrack proliferation, which is consistent with previous studies. Despite the lack of exhaustive experimental studies, we believe that our work is the first step in the formulation of strategies that involve an appropriate selection and placement of pump-wave sources to improve the NCWI technique. These improvements may be relevant to convert the NCWI technique into a more suitable non-destructive testing technique for the inspection of microcracking evolution in concrete structures and the assessment of their structural integrity.

Discrete element method analysis of non-linear stiffness for granular media

This contribution considers the nonlinearity of stiffness in granular materials using 3D DEM simulations, with a primary focus being placed on three issues. First, it assesses stress-dependent stiffness and density-dependent stiffness based on DEM simulations of monotonic triaxial tests. The modified formulation of hyperbolic relation based on the DEM data is generated to capture stiffness degradation, which can take the various types of an S-shaped curve due to the initial mean effective stress and the sample density. The use of various void ratio correction functions is made to remove the effect of density on the stiffness curve. DEM results reveal that inclusion of the coordination number in both forms of the void ratio function serve a better role in unifying the stiffness dataset from samples with different densities. Second, the framework of kinematic modified yielding points proposed by Jardine (1992) that identifies three main zones (i.e. linear elastic behaviour, non-linear elastic behaviour and elasto-plastic behaviour) provides a benchmark for the gap between the dynamic stiffness and the static stiffness to be studied in an effective manner. A key observation is that the dynamic shear stiffness tends to increase to a peak value before experiencing of a decrease in magnitude. Third, the investigation into the nature of energy dissipation during triaxial shearing tests is made. Several DEM simulations were performed to assess effects of the coefficient of uniformity (Cu) on the non-linear behaviour of granular materials, arriving at some key observations that: (i) a higher amount of work should be input for samples with coarse particles to attain a particular strain level; and (ii) more energy dissipation is observed for samples with a lower Cu value.

Short-term resilient behaviour and its evolution with curing in cold in-place recycled asphalt mixtures

This study evaluates the short-term resilient behaviour of cold in-place recycled (CIR) asphalt mixtures with bitumen emulsion when they are more similar to non-cohesive granular materials than to conventional hot mixtures. Gyratory compacted CIR specimens were manufactured using different proportions of bituminous emulsion and water, and dynamic triaxial tests were conducted at different curing times. The resilient moduli were obtained experimentally and fitted using three numerical models. Characteristic nonlinear elastic behaviour and an increase in stiffness with curing time were observed. Water loss during curing and stiffness increase were found to be related. Mixtures with 2.50% residual bitumen and 2.75% added water showed the best short-term stiffness evolution.

An Engineering Prediction Model for Stress Relaxation of Polymer Composites at Multiple Temperatures.

This study develops an engineering prediction model for stress relaxation of polymer composites, allowing the prediction of stress relaxation behaviour under a constant strain, over a range of temperatures. The model is based on the basic assumption that in the stress relaxation process the reversible strain is transformed to irreversible strain continuously. A strain-hardening model is proposed to incorporate nonlinear elastic behaviour, and a creep rate model is used to describe the irreversible deformation in the process. By using stress relaxation data at different temperatures, under different strains, the dependence on temperature and initial strain of the model parameters can be established. The effectiveness of the proposed model is verified and validated using three polymer composite materials. The performance of the model is compared with three commonly used stress relaxation models such as the parallel Maxwell and Prony series models. To ease the use of the proposed model in realistic structural problems, a user subroutine is developed, and the stress relaxation of a plate structure example is demonstrated.

Ice-shedding-induced vibration of conductors with active vibration control

A method is proposed for attenuating the vibration of suspended conductors. The idea involves the application of vibration control that contributes to reducing substantially the conductor rebound height following ice shedding. The theoretical background of the method is developed, and its application to vibration following ice shedding is tested by numerical simulations. A simplified model of a conductor with vibration damper is constructed, which simulates the dynamic behaviour of the conductor with damper at the location where the damper is attached. The model takes into account the nonlinear elastic material behaviour of the conductor together with the effects of a vibration damper with active control. Such vibration control may successfully attenuate even high-amplitude vibration that follows ice shedding. Simulation results provide to what extent the vibration amplitude is reduced due to the application of active control. After further improving the method, it may also be adaptable to decrease the amplitude of forced vibrations which are induced by wind and characterized by high amplitude.

Postseismic Survey of a Historic Masonry Tower and Monitoring of Its Dynamic Behavior in the Aftermath of Le Teil Earthquake (Ardèche, France)

ABSTRACTOn 11 November 2019, an Mw 4.9 earthquake struck the middle Rhône valley (South-East France) producing moderate to severe damage in the town of Le Teil and its surroundings. This unexpected event stressed the vulnerability of the French cultural built heritage to a moderate seismic hazard. Commonly applied to modern civil engineering structures, passive seismic methods are still lacking on historic constructions to understand properly the different factors driving their dynamic behavior. In this article, the results of a two-month seismic monitoring survey carried out shortly after the Le Teil mainshock in a historic masonry tower are presented and discussed. Located only 5 km south of the epicenter, the Gate Tower of Viviers (eleventh century) was instrumented with four highly sensitive seismic nodes. Ambient vibrations, as well as aftershocks and quarry blasts from the nearby Le Teil quarry, were recorded and used in the analysis. Through vibration-based analysis, the article addresses three relevant aspects of the dynamic response of ancient masonry structures. We discuss first the differences in the building’s response induced by the three reported types of vibrations, focusing on the particular signal characteristics of shallow aftershocks and quarry blasts. Then, we apply the Random Decrement Technique (RDT) to track the dynamic behavior variations over two months and to discuss the role of the environmental conditions in the slight fluctuations of the structural modal parameters (natural frequencies, damping coefficients) of unreinforced masonry structures. We also show evidence of the nonlinear elastic behavior under both weak seismic and atmospheric loadings. The correlation between the presence of heterogeneities in the construction materials and the nonlinear threshold supports the relevance of such types of monitoring surveys as a valuable tool for future modeling works and conservation efforts.

Optical Investigation of the Limits of Modeling the Nonlinear Elastic Behavior of PA6 Using Linear Elastic Material Models

One of the most critical issues during polymer finite element simulations is the selection of the proper material models. The widely used and accepted multilinear material models require load case-specific material tests, which are time and cost demanding. Data for these characteristics must be acquired by standardized measurements. On the other hand, the parameters required to create a linear elastic material model in most cases are easy to obtain, and the establishment of the model is a shorter process. This research is aimed to provide information to engineers about the possibility of modeling the nonlinear elastic materials by using linear elastic material models and about the limits of such models. To create the most accurate material models, laboratory measurements were performed on polyamide (PA6) material, which is a widely used raw material in the industry. Test specimens were manufactured to obtain material constants according to the ISO 527-2 standard, and for validating the effectiveness of the applied material models, three different tensile specimens were created, which were tested under quasi-static loading in the elastic region. A comprehensive finite element investigation was performed, and the numerical results were then compared to laboratory measurements using the GOM Aramis digital image correlation (DIC) system. By comparing the optically measured strain data to the numerical results, it was determined that the nonlinear elastic materials can be modeled using linear elastic models in a well identifiable strain range with sufficient accuracy.

Non linear elasticity of foam films made of SDS/dodecanol mixtures.

Foam film elasticity plays a significant role in film drainage and film stability and is thus expected to influence foam dynamical properties. It strongly depends on the foaming solution composition and differs from the interface elasticity measured in unconfined geometries. We use a deformable frame to deform an assembly of five films and we measure the tension and extension of each film. This provides a simple and accurate determination of the film elasticity, in the linear and non-linear regimes, for a set of SDS/dodecanol mixtures, at various concentrations. We show that the non-linear elastic behavior is well reproduced by Mysel's model coupled with a Langmuir coadsorption isotherm for a large range of chemical compositions.

A Data-Driven Approach to Characterizing Nonlinear Elastic Behavior of Soft Materials

A Data-Driven Approach to Characterizing Nonlinear Elastic Behavior of Soft Materials

Numerical Analysis of A Nonlinear Elastic Composite Leaf Spring

In this study, numerical analysis of a composite spring with a nonlinear elastic force-displacement relationship was performed. Due to the lightness and durability of composite materials, their use in the automotive industry is increasing and production methods allow for a high performance spring design. The behavior of a pair of composite leaf springs working in tandem with the cross-section profiles of the leaf springs to be obtained by using the involute curve of the spur gear pairs under load was examined in detail by numerical method, and the desired behavior was obtained. In the analyzes performed using Ls-Dyna software, samples were produced and many mechanical tests were performed for the determination of material parameters. The results obtained from these tests were used as the required material parameters in numerical analysis. The leaf spring thickness that will give the desired non-linear elastic behavior under 400 N load used in numerical analysis has been determined and successfully targeted force-elongation behavior has been created.

Nonlinear elastic analysis of 2D materials of arbitrary symmetries with application to black phosphorus

Murnaghan’s polynomial based nonlinear elastic constitutive model has been previously applied to 2D materials of hexagonal symmetry. We present a general approach for determining the nonlinear elastic constants of 2D materials of arbitrary symmetries and any constitutive polynomial order. The methodology, which is based on ray sampling of the strain energy density in strain space, is independent of the energy calculation method. The ray based methodology is verified by evaluating the elastic constants of graphene which is a hexagonally symmetric 2D material previously considered in the literature. The methodology is then applied to determine the elastic constants of black phosphorus, an orthorhombic 2D material whose comprehensive nonlinear elastic behavior has not been previously considered in the literature. The energy calculations are carried out using plane-wave density functional theory. Detailed convergence analyses are performed to assess the accuracy of the nonlinear elastic constants. The linearized mechanical properties of black phosphorus are obtained from the elastic constants for comparison with the results published in the literature.

A phenomenological constitutive model for predicting both the moderate and large deformation behavior of elastomeric materials

Constitutive models that describe nonlinear elastic mechanical behavior are indispensable in the design of engineering components fabricated of elastomeric materials. The main drawbacks of the existing models include complexities in calibrating the model to experimental data and erroneous descriptions of multi-axial response. These challenges motivate researchers to continually formulate new or improved models with better predictive capabilities. In this work, we propose a new polynomial-type phenomenological model with linear and logarithmic dependence on the first and the second invariants respectively. Model parameters were obtained by utilizing the Levenberg-Marquardt algorithm to fit the model expression to the strain energy density data calculated from the experimental data of uniaxial tension loading. The predictive performance of our proposed model in comparison to that of three popular existing models was determined by utilizing three sets of classical experimental data from the literature with varying deformation ranges. Quantities used to relate the predicted and the experimental data include the coefficient of determination, relative errors, and average relative error (for overall behavior). The computations demonstrated that the proposed model exhibits superior predictive capabilities for the entire deformation range whilst requiring minimum efforts in obtaining its parameters, thus, exhibits the desirable features of a phenomenological hyperelastic model.