Macroscopic Stiffness Research Articles

Fluctuations and failure in granular materials: theory and numerical simulations

We consider a dense aggregate of elastic, frictional particles isotropically compressed and next uniaxial strained at constant pressure. We show how failure can be predicted if fluctuations in the kinematics of contacting particles are introduced. We focus on the second order work and the possibility that at some stressed states it becomes negative under proper perturbations. Our analysis involves both a theoretical model and numerical simulations based upon the distinct element method (DEM). The theoretical model deals with contacting particles with incremental relative displacements that deviate from the average deformation in order to ensure their equilibrium. Because of this, the macroscopic stiffness tensor of the aggregate, that relates increments in stress with increments in strain, does not have the major symmetry. Consequently, in the hardening regime, we predict stressed states in which the second order work vanishes. The model seems transparent, and it makes clear and illustrative the role played by the fluctuations introduced in the kinematics of contacting particles in relation to the vanishing of second order work in an aggregate of compressed particles. The comparison with numerical simulations data supports the model.Graphical Statistical representation of the aggregate: conditional average.

Anisotropic Elastic Strain-Gradient Continuum from the Macro-Scale to the Granular Micro-Scale

A multi-scale framework is constructed for the computation of the stiffness tensors of an elastic strain-gradient continuum endowed with an anisotropic microstructure of arbitrarily-shaped particles. The influence of microstructural features on the macroscopic stiffness tensors is demonstrated by comparing the fourth-order, fifth-order and sixth-order stiffness tensors obtained from macro-scale symmetry considerations to the stiffness tensors deduced from homogenizing the elastic response of the granular microstructure. Special attention is paid to systematically relating the particle properties to the probability density function describing their directional distribution, which allows to explicitly connect the level of anisotropy of the particle assembly to local variations in particle stiffness and morphology. The applicability of the multi-scale framework is exemplified by computing the stiffness tensors for various anisotropic granular media composed of equal-sized spheres. The number of independent coefficients of the homogenized stiffness tensors appears to be determined by the number of independent microstructural parameters, which is equal to, or less than, the number of independent stiffness coefficients following from macro-scale symmetry considerations. Since the modelling framework has a general character, it can be applied to different higher-order granular continua and arbitrary types of material anisotropy.

(Invited) Advancing Understanding of Composite Polymer Electrolytes with LLZO Nanofibers

Ceramic Li6.55Al0.2La3Zr2O12 (LLZO) is one of the most attractive electrolyte materials for solid-state batteries that combined with metallic Li anodes holds the promise for safer and more energetically dense battery, but its performance is limited by the increases in electrolyte-electrode interfacial resistances upon cycling. The combination of Li+-conducting ceramics and polymers offers a new pathway to create better electrolytes with both high ionic conductivity and good (electro)mechanical interfacial properties. We optimized two-step annealing processing conditions for the fabrication of low temperature cubic-LLZO nanofibers and Li2CO3 removal from the surface of the nanofibers. Using industrially relevant roll-to-roll electrospinning and slot-die coating, we fabricated thin composite membranes with reproducible thickness down to 50 microns. We investigated PEO based composite polymer electrolyte (CPE) with a high, 50 wt% loading of Al-doped LLZO nanofibers in comparison with nanofiber-free PEO electrolyte. XPS measurements show that LLZO is not present at the composite electrolyte surface, and solid electrolyte interphase (SEI) formation is dictated solely by PEO and LiTFSI when reacting with Li metal. Electrochemical SEI formation, studied by cyclic voltammetry, shows SEI formation is identical with and without up to 75 wt% LLZO in the electrolyte. Galvanostatic cycling with lithium symmetric cells shows that the critical current density (CCD) can be tripled by including 50 wt% LLZO, but half cell cycling reveals this comes at the cost of CE. Varying the LLZO loading shows that even a small amount of LLZO drastically lowers the CE, from 88% at 0 wt% LLZO to 77% at just 2 wt% LLZO. Mesoscale modelling reveals that the increase in CCD cannot be explained by an increase in the macroscopic or microscopic stiffness of the electrolyte; only the microstructure of the LLZO nanofibers in the PEO-LiTFSI matrix slows dendrite growth by presenting physical barriers that the dendrites must push or grow around. This tortuous lithium growth mechanism around the LLZO is corroborated with mass spectrometry imaging. Electrochemical impedance spectroscopy (EIS) analysis showed comparable bulk ionic and interfacial resistances with and without nanofibers, indicating that PEO dominates Li+ transport and interfacial chemistry. The total Li-ion conductivity of the composite is still governed by the polymer matrix due to high interfacial resistance between the garnet particles and the PEO/LiTFSI matrix. This work highlights important elements to consider in the design of CPEs for high-efficiency lithium metal batteries.

The influence of uneven frost heave and thermal conditions on the deformation and damage of slab track in seasonally frozen regions

To ensure the secure operation of high-speed trains in seasonally frozen regions, this paper investigates the deformation and failure patterns of slab tracks subjected to uneven frost heave and environmental issues. Utilizing our team’s previous research findings, finite element analysis software was employed to construct a mechanical transfer calculation model for the subgrade and slab track. This model was used to explore the relationship between uneven frost heave and slab track deformation at different time periods, along with the track structure’s damage behavior. In addition, comprehensive temperature field tests were conducted on the China railway track system (CRTS) III slab track, probing temperature variations, warping deformations, and stress distributions within the track structure. Finally, the interaction of uneven frost heave and thermal loads unveiled the interlayer debonding and stress characteristics of slab track structures across different months in seasonally frozen regions. The results revealed that the greater amplitude of frost heave and the smaller wavelength led to a higher change rate of rail irregularity. The uneven lateral deformation of the subgrade surface due to the sunny-shady slopes effect resulted in significant rail level irregularity of slab tracks. The combined impact of short-wavelength frost heave and temperature gradient loads emerged as the primary cause of macroscopic damage and stiffness degradation of concrete materials within slab track structures. These findings will help to understand the mechanical behavior of slab tracks in cold regions, thereby improving the quality of high-speed railway operations.

A novel 3D printed compliant ball-end grinding tool with crystal structure: Feasibility and performance analysis

During grinding complex components with difficult-to-machine materials, compliance grinding tools often experience shortened life and weakened performance due to intense temperature rise. This paper presents a novel solution strategy of preparing crystal-structured compliance tools using multi jet fusion (MJF), and five structures were selected to evaluates its feasibility and grinding performance. The macro and micro elastic deformation of the tools were analyzed by combining finite element and experiment. The results show that the crystal structure provides similar macroscopic stiffness of the tools and introduces microscopic compliance anisotropy. The crystal structure resulted in a reduction of the grit pull-out ratio by more than 82.60 %, while also alleviating the adhesion of flexible substrates adhesion and the prevention of abrasive layer fall-off, leading to prolonged tool life. Furthermore, compliant grinding tools with a crystal structure achieved comparable material removal capacity, lower surface roughness, and higher energy efficiency. In addition, they used only 29.06 % to 36.22 % of the material consumed in manufacturing. Notably, the material removal rate (MMR) of the star-structure tool can reached 94.72 % of the solid-structure tool. The diamond-structure tool exhibited the lowest ground Ra, accompanied by a reduction in grinding temperature of 19.4 °C.

Micro–macro approach of anisotropic damage: A semi-analytical constitutive model of porous cracked rock

A semi-analytical constitutive model of a porous cracked rock is proposed based on the micro–macro approach within a thermodynamic framework. The representative volume element (RVE) of a porous cracked rock is divided into two subproblems, P1 and P2. The porosity evolution equation of a spherical pore is proposed for P1, whereas P2 is an inelastic problem containing numerous anisotropic penny-shaped cracks. First, the relationship between microscopic and macroscopic strains is established through homogenization. This relationship is introduced into the macroscopic free energy equation. A crack propagation criterion based on the Griffith release rate is proposed, and the evolution equations of the shear and tensile crack densities are proposed based on the surface energy function. Subsequently, the Gaussian numerical integration method is used to obtain a semi-analytical explicit expression of the macroscopic stiffness tensor. Finally, based on the proposed crack propagation criterion, a strength-prediction method is developed. A series of uniaxial and triaxial experimental data for different rocks from other references is used to validate the proposed constitutive model and strength prediction method. The results obtained using the proposed model agree well with the experimental curves obtained from uniaxial tension and triaxial compression tests performed on different rocks. The macroscopic mechanical behavior and evolution of microscopic parameters for different rocks with an increase in confining pressure are effectively demonstrated by the proposed model. Strength prediction method can accurately describe the failure strength of different rocks. Furthermore, a sensibility analysis of model parameters revealed that initial porosity e0 and dilation coefficient ζ have a deteriorating effect on the failure strength, and material constant η has an enhancing effect on the failure strength.

X-ray 3D imaging–based microunderstanding of granular mixtures: Stiffness enhancement by adding small fractions of soft particles

This research focuses on performing ultrasound propagation measurements and micro-X-ray computed tomography (µXRCT) imaging on prestressed granular packings prepared with biphasic mixtures of monodisperse glass and rubber particles at different compositions/fractions. Ultrasound experiments employing piezoelectric transducers, mounted in an oedometric cell (complementing earlier triaxial cell experiments), are used to excite and detect longitudinal ultrasound waves through randomly prepared mixtures of monodisperse stiff/soft particles. While the fraction of the soft particles is increasing linearly from zero, the effective macroscopic stiffness of the granular packings transits nonlinearly and nonmonotonically toward the soft limit, remarkably via an interesting stiffer regime for small rubber fractions between 0.1 ≲ ν ≲ 0.2. The contact network of dense packings, as accessed from µXRCT, plays a key role in understanding this phenomenon, considering the structure of the network, the chain length, the grain contacts, and the particle coordination. While the maximum stiffness is due to surprisingly shortened chains, the sudden drop in elastic stiffness of the mixture packings, at ν ≈ 0.4, is associated with chains of particles that include both glass and rubber particles (soft chains); for ν ≲ 0.3, the dominant chains include only glass particles (hard chains). At the drop, ν ≈ 0.4, the coordination number of glass and rubber networks is approximately four and three, respectively, i.e., neither of the networks are jammed, and the chains need to include particles from another species to propagate information.

Multiscale stiffness of human emphysematous precision cut lung slices.

Emphysema is a debilitating disease that remodels the lung leading to reduced tissue stiffness. Thus, understanding emphysema progression requires assessing lung stiffness at both the tissue and alveolar scales. Here, we introduce an approach to determine multiscale tissue stiffness and apply it to precision-cut lung slices (PCLS). First, we established a framework for measuring stiffness of thin, disk-like samples. We then designed a device to verify this concept and validated its measuring capabilities using known samples. Next, we compared healthy and emphysematous human PCLS and found that the latter was 50% softer. Through computational network modeling, we discovered that this reduced macroscopic tissue stiffness was due to both microscopic septal wall remodeling and structural deterioration. Lastly, through protein expression profiling, we identified a wide spectrum of enzymes that can drive septal wall remodeling, which, together with mechanical forces, lead to rupture and structural deterioration of the emphysematous lung parenchyma.

Understanding the Influence of Li7La3Zr2O12 Nanofibers on Critical Current Density and Coulombic Efficiency in Composite Polymer Electrolytes.

Composite polymer electrolytes (CPEs) are attractive materials for solid-state lithium metal batteries, owing to their high ionic conductivity from ceramic ionic conductors and flexibility from polymer components. As with all lithium metal batteries, however, CPEs face the challenge of dendrite formation and propagation. Not only does this lower the critical current density (CCD) before cell shorting, but the uncontrolled growth of lithium deposits may limit Coulombic efficiency (CE) by creating dead lithium. Here, we present a fundamental study on how the ceramic components of CPEs influence these characteristics. CPE membranes based on poly(ethylene oxide) and lithium bis(trifluoromethanesulfonyl)imide (PEO-LiTFSI) with Li7La3Zr2O12 (LLZO) nanofibers were fabricated with industrially relevant roll-to-roll manufacturing techniques. Galvanostatic cycling with lithium symmetric cells shows that the CCD can be tripled by including 50 wt % LLZO, but half-cell cycling reveals that this comes at the cost of CE. Varying the LLZO loading shows that even a small amount of LLZO drastically lowers the CE, from 88% at 0 wt % LLZO to 77% at just 2 wt % LLZO. Mesoscale modeling reveals that the increase in CCD cannot be explained by an increase in the macroscopic or microscopic stiffness of the electrolyte; only the microstructure of the LLZO nanofibers in the PEO-LiTFSI matrix slows dendrite growth by presenting physical barriers that the dendrites must push or grow around. This tortuous lithium growth mechanism around the LLZO is corroborated with mass spectrometry imaging. This work highlights important elements to consider in the design of CPEs for high-efficiency lithium metal batteries.

A method to determine the bonded-particle model parameters for simulation of ores

The bonded-particle model (BPM) is commonly used in the numerical analysis of ore samples. To improve the accuracy of simulating the mechanical process of ore process of ore crushing in a crusher, the parameters of the BPM for the ore must be calibrated. In this study, a calibration method was proposed for the scientific determination of the parameters of the BPM for ore undergoing uniaxial compression. First, physical tests and simulations were conducted to determine the mechanical response (uniaxial compressive strength and macroscopic stiffness) of ore during uniaxial compression. Then, the sensitivity of the mechanical response to the values of microscopic parameters was tested using a Plackett‒Burman design. Next, the microscopic parameters with the greatest impact on the response were identified, and the range of parameters that met the target response was determined using a steepest ascent design; Second, a second-order model of the mechanical response was established using the sensitive parameters by combining a Box‒Behnken design with response surface methodology to obtain the optimal BPM parameters. Simulation tests showed that the normal stiffness per unit area, critical shear stress, and bonded disk radius had significant effects on the uniaxial compressive strength (UCS) and macroscopic stiffness (MS). To verify the validity of the proposed calibration method, laboratory tests were conducted. The consistency of the simulation results with experimental results indicated that response surface methodology with the Plackett‒Burman design, steepest ascent design, and Box‒Behnken design can be an effective method for calibrating the BPM of ores.

Reflections on the physics of slow dynamics in rocks

Slow Dynamics is a peculiar property exhibited by many sedimentary rocks and was first reported in the mid 1990s. Slow dynamics refers to the process of changing the elastic state of a rock, softening its macroscopic stiffness with application of an AC acoustic drive. What makes slow dynamics unique is that the elastic state of the rock recovers as log(time) once the acoustic drive is turned off. Many creep-like mechanisms have been proposed to explain the effect, notably it resembles recoverable/reversible DC creep. The most common explanation is due to the effects of water, or a form of it, in the spaces between grains. However, it has been exceedingly difficult to pin down the real physics. One experiment with Berea sandstone at high vacuum and temperatures showed that slow dynamics still persists even at these conditions. Followup experiments are in progress, using a pure quartz Fontainebleau sandstone to avoid any effects of clays. Another experiment is planned which will mimic conditions on Mars where water in any form is of great interest to planetary geologists. [Work supported by the Office of Basic Sciences, DOE and by Laboratory Directed Research and Development funding.]

Equivalent stiffness characteristics of gradient egg-box structure based on unit-cell finite element method

Multilayer egg-box structures can efficiently combine various forms of gradient structures by adjusting the stacking mode of different egg-box veneers. To analyze the equivalent stiffness of a gradient egg-box structure, we propose two feasible stacking combination modes. Subsequently, using the unit-cell finite element method, we establish an efficient solution model for the equivalent stiffness of the structure and discuss the effects of different structural parameters upon the model. Finally, we construct an optimization model for the equivalent stiffness of the structure and optimize the structural parameters of its macroscopic stiffness characteristics.

Strain stiffening elastomers with swelling inclusions.

Inhomogeneously swollen elastomers are an emergent class of materials, comprising elastic matrices with inclusion phases in the form of microgel particles or osmolytes. Inclusion phases can undergo osmotically driven swelling and deswelling over orders of magnitude. In the swollen state, the inclusions typically have negligible Young's modulus, and the matrix is strongly deformed. In that regime, the effective mechanical properties of the composite are governed by the matrix. Laying the groundwork for a generic analysis of inhomogeneously swollen elastomers, we develop a model based on incremental mean-field homogenization of a hyperelastic matrix. The framework allows for the computation of the macroscopic effective stiffness for arbitrary hyperelastic matrix materials. For an in-depth quantification of the local effective stiffness, we extend the concept of elastic stiffness maps to incompressible materials. For strain-stiffening materials, stiffness maps in the swollen state highlight pronounced radial stiffening with a non-monotonic change in stiffness in the hoop direction. Stiffening characteristics are sensitive to the form of constitutive models, which may be exploited in the design of hydrated actuators, soft composites and metamaterials. For validation, we apply this framework to a Yeoh material, and compare to recently published data. Model predictions agree well with experimental data on elastomers with highly swollen embedded microgel particles. We identify three distinct regimes related to an increasing degree of particle swelling: first, an initial decrease in composite stiffness is attributed to particle softening upon liquid intake. Second, dilute particle swelling leads to matrix stiffening dominating over particle softening, resulting in an increase in composite stiffness. Third, for swelling degrees beyond the dilute limit, particle interactions dominate further matrix stiffening.

Effects of in-situ temperature in deep reservoirs on shale fracture properties

The development of deep shale reservoirs is a major strategic demand to meet national energy requirements. However, the high-temperature characteristics of in-situ reservoirs make it difficult to understand the fracture properties of deep shale clearly, which restricts the development of reservoir stimulation technologies. Based on digital image correlation method and fracture mechanics theory, this study investigates the issue of rock fracture parameters testing in high-temperature environments. This study builds an in-situ high-temperature environment rock fracture visualization test system that can dynamically capture crack-tip initiation and propagation. Further, the influence law of temperature on shale fracture behaviour has been analysed. The results have shown that, under the condition of no confining pressure, with increases in the temperature, the macroscopic material stiffness and fracture toughness of shale increase; this shows that high temperatures inhibit crack initiation and propagation. Further, with increases in the temperature, the temperature-dependent CTOD is positively correlated with the fracture toughness, the simultaneous increase of these two factors leads to the increase of local plasticity in the crack tip region. When considering high-temperature conditions alone, the fracture tip still experiences a small-scale yield state; the research conclusions can provide a reference for studies on deep reservoir fractures.

Anomalous temperature dependent stiffness evolution in hydrogels

• A water-filled cavity model simulates the mechanical response of a hydrogel. • The model predicts the temperature-dependent viscoelastic response accurately. • The polymer and water dominant behaviours result in an anomaly in elastic response. We investigate the anomalous stiffness evolution of a hydrogel with an increase in temperature and provide an insight into the underlying mechanism through a finite element model. The continuum model considers the interplay between the water-filled fluid cavity, the surrounding viscoelastic chitosan matrix, and the temperature. It is observed that in the low-temperature regime (30 °C to 70 °C), water cavity volume fractions play a dominant role, while at higher temperatures (70 °C to 200 °C), the polymer viscoelastic properties govern the macroscopic stiffness of the system.

Effects of tow arrangements on the homogenized response of carbon fiber woven composites

A set of micro-mechanical analyses is carried out to investigate the sensitivity of the homogenized response of two-ply carbon fiber woven composites on relative positioning of tow arrangements. In this paper, first, we present a convergence study to validate the applicability of the characteristic unit cell to capture the macroscale mechanical response of the composite. The study showcases the capacity of our models to capture the macroscale mechanical properties at different microscopic resolutions of the macroscale. The effects from phase different tow arrangements are then analyzed to quantify the mechanical response variability at the macroscale. Different tow arrangements considered in this work naturally stems from the relative inter-ply sliding at the manufacturing stage of woven composites. We consider several tow arrangements that can be interpreted as intermediate configurations between the two extremes; fiber in-phase and out-of-phase. Tows are assembled and constrained systematically to resemble different tow arrangements for the subsequent multiscale modeling. Corresponding macroscopic ABD stiffness matrices are extracted using the principle of virtual work, whereby the unit cell deformation is homogenized using periodic boundary conditions. The obtained results not only quantify the variability in A and D sub-matrices of the ABD stiffness matrix, but also explain possible reasons for the observed deviations between experimental and numerical results.

Heterogeneous digital stiffness programming

A heterogeneous design of mechanical metamaterial enables digital stiffness programmability. The prototype is composed of an elastomer matrix containing diamond-shaped cavities that are selectively confined along their diagonals by semi-rigid plastic beam inserts. Unit-cell perturbations caused by insert placement or removal reshape the global constitutive relation, whose lower and upper bounds correspond to configurations with all holes empty and all inserts in place, respectively, due to a gap between the elastomer and insert moduli. Bidirectional operation is accomplished by varying the orientation of the inserts, with longitudinal inserts increasing the macroscopic stiffness in compression and transverse inserts increasing that in tension. Arranged digital representations of such local insert states form the explicit encoding of global patterns, allowing both statical and in situ systematic stiffness programming with minimal mass changes. These properties establish a new paradigm for actively tuning vibration isolation systems in response to changes in the base structure’s resonances.

Fatigue property evaluation for fiber reinforced plastics based on mode conversion effect of guided wave

Fiber reinforced plastics (FRPs) are widely used in aerospace engineering industry. When the composite structure is subjected to the cyclic load, the initiation, propagation and accumulation of the fatigue damage in FRPs lead to structural failure. This paper presents a guided wave based fatigue property evaluation method to characterize matrix crack accumulation in FRPs, which provides a potential way to predict the remaining useful life of the structures. In this paper, a finite element model for FRPs is firstly developed to simulate the fatigue induced matrix cracks, which are randomly distributed in the structures. Both the initiation and propagation behaviors of the matrix cracks are modeled by the regulation of the random parameters, such as the number of fatigue elements and the degree of the stiffness reduction. Based on a series of models representing the fatigue evolution process of FRPs, the mode conversion effect of the guided wave is then investigated. A guided wave characteristic MCI (Mode Conversion Index) quantifying the mode conversion effect is used to evaluate the fatigue property of FRPs. Compared with the macroscopic stiffness, MCI shows higher sensitivity to the degree of matrix crack accumulation. Finally, controlled fatigue tests under different stress levels are carried out by using a fatigue testing machine, in which a laser ultrasonic system is employed to obtain the guided wave propagation in FRPs with different numbers of load cycles. The relationship between MCI and macroscopic stiffness is also obtained, providing a basis for using mode conversion feature to predict fatigue life.

Atomistic study on simultaneous achievement of partial crystallization and rejuvenated glassy structure in thermal process of metallic glasses

ABSTRACT The relaxation state and partial crystallization are crucial factors that affect the material properties of metallic glasses. Rejuvenation induces a less relaxed glass state and hence facilitates the control of the relaxation state. The rejuvenation and crystallization during thermal processing are associated closely with the phase changes of metallic glasses upon heating. Most nanocrystalline metallic glasses formed via conventional thermal annealing have a relaxed glassy matrix. In this molecular dynamics study, we investigate the feasibility of thermal processing to simultaneously realise both partial crystallization and rejuvenation using model alloy system. Dynamic mechanical analyses reveal relaxation behaviours and local deformation features of metallic glasses composed of a crystalline phase and a rejuvenated glass matrix. The crystalline phase increases the macroscopic shear stiffness of the composite model over a wide temperature range, whereas it does not significantly affect the internal friction of the rejuvenated glass matrix. In addition, dispersed nanocrystals suppress the development of sharp and concentrated shear localisation in the rejuvenated glass matrix. The simultaneous adjustment of the relaxation state and crystalline phase in the glass matrix is discussed from a viewpoint of the microstructure design of metallic glasses.

A frictional cohesive zone model for characterizing transverse compression responses of unidirectional fiber-reinforced polymer composites

The Coulomb friction law is incorporated into the traction separation law, so a friction cohesive zone model (FCZM) combining sliding friction, interfacial debonding, and the coupling of the debonding and friction is developed. The proposed interface model is used to describe the interfacial mechanical responses of fiber/resin composite. To verify this model, the finite element model of microdroplet is built, then the calculated results of the finite element model are compared with the microdroplet test. To analyze the effect of interface friction on the macroscopic strength and stiffness of unidirectional composites, the FCZM is applied to representative volume element (RVE), and the mechanical behaviors of composites is simulated by finite element analysis of RVE, then the results are compared with the test in reference [1, 2]. It is found that the interfacial debonding is accelerated and the macroscopic strength is reduced due to the friction.