Microscopic Instabilities Research Articles

Twinning in porous elastomers

It has been known for some time that constitutive models for the large-deformation response of porous elastomers can develop ‘macroscopic’ instabilities as a consequence of loss of strong ellipticity. Indeed, constitutive models obtained by homogenization methods for porous elastomers with periodic and random microstructures can lose strong ellipticity under appropriate loading conditions. For periodic microstructures, it has been shown theoretically, and verified experimentally and numerically, that ‘microscopic’ instabilities consisting in solutions that are periodic on ensembles of unit cells tend to occur before the long-wavelength ‘macroscopic’ instabilities that are captured by the loss of strong ellipticity of the homogenized response. But what about the response of porous elastomers with random microstructures? In this case, microscopic instabilities can be excluded, and the question then arises as to what happens after the onset of a macroscopic instability. Building on earlier work for reinforced elastomers, it is shown here that porous elastomers can undergo twinning after the onset of a macroscopic instability. For this purpose, use is made of a generalized Maxwell-type construction arising from the theory of relaxation and applied to linear comparison variational homogenization estimates for a certain class of two-dimensional porous elastomers consisting of aligned cylindrical pores in a rubber matrix subjected to plane strain loadings. It is shown that such porous elastomers recover Legendre-Hadamard stability by twinning after the onset of a macroscopic instability. Interestingly, the porous elastomers behave like elastic fluids in the twinned region, losing their ability to support shear stresses.

Effects of interfacial debonding on the stability of finitely strained periodic microstructured elastic composites.

Predicting failure initiation in nonlinear composite materials, often referred to as metamaterials, is a fundamental challenge in nonlinear solid mechanics. Microstructural failure mechanisms encompass fracture, decohesion, cavitation, compression-induced contact and instabilities, affecting their unconventional static and dynamic performances. To fully take advantage of these materials, especially in extreme applications, it is imperative to predict their nonlinear behaviour using reliable, accurate and computationally efficient numerical methodologies. This study presents an innovative nonlinear homogenization-based theoretical framework for characterizing the failure behaviour of periodic reinforced hyperelastic composites induced by reinforcement/matrix decohesion and interaction between contact mechanisms and microscopic instabilities. Debonding and unilateral contact between different phases are incorporated by employing an enhanced cohesive/contact model, which features a special nonlinear interface constitutive law and an accurate contact formulation within the context of finite strain continuum mechanics. The theoretical formulation is demonstrated using periodically layered composites subjected to macroscopic compressive loading conditions along the lamination direction. Numerical results illustrate the ways in which debonding phenomena, in conjunction with fibre microbuckling, may influence the critical loads of the examined composite solid. The sensitivity of the results obtained through the proposed contact-cohesive model at finite strain with respect to its implementation is also explored. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Microscopic instabilities in single crystal matrix composites

A finite strain micromechanical analysis is presented for the prediction of the loss of microscopic stability of a class of metal matrix composites that are subjected to axial compressive loading and undergoing large deformations. The metallic constituent behavior is modeled by the single crystal anisotropic plasticity theory in which, due to the resolved shear stresses, plastic deformations occur along certain pre-defined slip planes. Thus, this incremental plasticity theory is capable of providing the effect of the applied axial loading on the induced shear stresses which dominate the microbuckling. The composites are assumed to possess slight imperfections at the interfaces, and in order to satisfy the interfacial conditions, a perturbation expansion is employed which yields zero and first order micromechanical analysis problems. The zero order problem corresponds to the micromechanical modeling of the composite with no imperfections, whereas the solution of the first order problem is utilized to obtain the critical stresses and deformations at which bifurcation buckling occurs. Both problems are solved by employing the finite strain high-fidelity generalized method of cells (HFGMC) micromechanics. Applications are given for various types of single crystal matrix composites including layered, particulate, continuous and short fiber composites. Finally, a comparison between the compressive strengths of a standard metal matrix boron/aluminum and SiC/single crystal composites is presented and discussed.

Uncovering Pattern-Transformable Soft Granular Crystals Induced by Microscopic Instability

Abstract Upon compression, some soft granular crystals undergo pattern transformation. Recent studies have unveiled that the underlying mechanism of this transformation is closely tied to microscopic instability, resulting in symmetry breaking. This intriguing phenomenon gives rise to unconventional mechanical properties in the granular crystals, paving the way for potential metamaterial application. However, no consistent approach has been reported for studying other unexplored transformable granular crystals. In this study, we present a systematic approach to identify a new set of pattern-transformable diatomic granular crystals induced by microscopic instability. After identifying the kinematic constraints for diatomic soft granular crystals, we have generated a list of feasible particle arrangements for instability-induced pattern transformation under compression. Instead of computationally intensive finite element models (FEMs) with continuum elements, we adopt a simplified mass-spring model derived from granular contact networks to efficiently evaluate these feasible particle arrangements for pattern transformation. Our numerical analysis encompasses quasi-static analysis and microscopic/macroscopic instability analyses within the framework of linear perturbation. Subsequently, the pattern transformation of the identified particle arrangements is confirmed through quasi-static analyses employing detailed finite element (FE) simulations with continuum elements. Additional numerical simulations with continuum elements reveal that the pattern transformations of particle arrangements are significantly influenced by the initial void volume and some transformed granular crystals may exhibit strong low-frequency directional phononic band-gaps, which were not observed in the initial granular crystals.

A variational model for finger-driven cell diffusion in the extracellular matrix

We present a simple chemo-mechanical variational model for a fibrous material that describes (i) the emergence of the anisotropy due to microscopic buckling instabilities (ii) a diffusion in the substrate of the cell phase driven by the new created macroscopic bands characterized by intense compressive deformation. The model is applicable for simulating the spreading of cells within tissues and their interaction with tissue remodeling during mesenchymal migration.

Microscopic deformation and fragmentation energy consumption characteristics of soils with various moisture contents using discrete element method

Increased soil moisture content commonly leads to higher resistance and energy consumption during agricultural operations. However, the impact of moisture content on soil microscopic deformation and fragmentation energy consumption characteristics remains unclear. In this study, discrete element models of sandy clay loam with different gravimetric water contents (15.0%, 18.8%, 21.6%, 24.1%, and 27.3%) were constructed for simulation experiments using coupled contact method (Hertz-Mindlin with bonding and Hertz-Mindlin with JKR). The degree of soil fragmentation and the rate of crack propagation were investigated. The results indicated that at fixed gravimetric water content, fissure rate exhibited a pattern of slow increase, rapid increase, and eventual stabilization with increasing compression displacement, while the fissure transfer coefficient showed a pattern of first increase and then decrease. At a fixed compression displacement, the increase in moisture content led to a decrease in both fissure rate and fissure transfer coefficient. The analysis of soil stress-strain characteristics and energy dissipation patterns found that soil with a gravimetric water content of 27.3% (close to soil liquid limit) exhibited no compression yield point; at a soil gravimetric water content of 15.0%, the minimum crushing energy (Udmin) was achieved (1.32 J), with a maximum elastic strain energy (Uemax) of 1.83 J. Similarly, at a soil gravimetric water content of 24.1%, the minimum crushing energy (Udmin) was reached (2.17 J), with a maximum elastic strain energy (Uemax) of 2.25 J. The comparison between actual and simulated results revealed low relative errors (<6%) in Udmin for all the above soil gravimetric water contents, suggesting the feasibility of our coupled discrete element model for assessing soil crushing energy consumption. This study revealed microscopic deformation and instability fragmentation mechanisms of soils with various moisture contents under vertical axial loads. Our findings provide novel method and strategy for developing the high-efficiency low-consumption soil fragmentation model.

Elastic instabilities of soft laminates with stiffening behavior

This paper investigates the elastic instability behavior in soft periodic laminates subjected to finite strains, with a focus on both macroscopic and microscopic instabilities. Considering the deformation-induced phase stiffening, the Gent model with a high bulk-to-shear modulus ratio describes the behavior of incompressible phases. This non-Gaussian statistics-based model captures the non-linear constitutive results from the limited extensibility of polymeric molecular chains. This paper derives an analytical prediction for the onset of macroscopic (or longwave) instability and microscopic instability as functions of material parameters. Moreover, a numerical Bloch-Floquet analysis is imposed on identifying the instability behavior under compression. We consider a wide range of phase combinations and find that the relatively rapid stiffening of the matrix compared to the stiff layer increases the stability of laminates by decreasing the critical stretch ratio. Essentially, properly manipulating the stiffening parameters can produce an absolutely stable region without observed instability. This paper also systematically illustrates the changes in instability and the transition between macro and micro instability in fully Gent laminates, which show higher stability than fully neo-Hookean laminates with larger critical stretch ratios. The critical characteristics of instabilities, such as critical stretch ratios and critical wavenumbers, can be controlled by the choice of stiffening parameters and other material properties, enlarging the tuning of soft laminates for desired buckling patterns in practical applications.

Influence of the Phase Fractions on the Formation of Eutectic Colonies: A Large‐Scale Phase‐Field Study

The growth of two‐phase eutectic colonies is a frequently observed phenomenon in the microstructure of directionally solidified ternary alloys. The formation of these macroscopic structures is driven by microscopic instabilities, caused by minor component impurities, diffusing into the liquid from the two solidifying phases. Due to an accumulation of this impurity component, a morphological instability at the eutectic front leads to the formation of eutectic colonies. In this work, the phase‐field method is used to investigate the influence of different melt composition and hence of the adjusting phase fractions on the growth of eutectic colonies. For this purpose, specially designed model systems, N‐xA‐yC, with defined phase fractions of the solids are generated on the basis of the high performance material NiAl‐34Cr. Based on these models, the evolution of eutectic colonies is investigated in two‐ and three‐dimensional (3D) large‐scale simulations with up to 3⋅109 cells. To perform these highly computationally intense large‐scale 3D simulations, the computational framework used is optimized in several layers. The results obtained show the influence of the melt composition on the formation and characteristics of the evolved eutectic colonies and provide new insights into the formation of these macroscopic structures.

Elastic Wave Propagation Control in Porous and Finitely Deformed Locally Resonant Nacre-like Metamaterials.

Recent studies have shown that the mechanical properties of bioinspired periodic composite materials can be strongly influenced by finite deformation effects, leading to highly nonlinear static and dynamic behaviors at multiple length scales. For instance, in porous periodic nacre-like microstructures, microscopic and macroscopic instabilities may occur for a given uniaxial loading process and, as a consequence, wave attenuation properties may evolve as a function of the microstructural evolution, designating it as metamaterials. The numerical outcomes provide new opportunities to design bioinspired, soft composite metamaterials characterized by high deformability and enhanced elastic wave attenuation capabilities given by the insertion of voids and lead cores.

Gyrokinetic Particle-in-Cell simulation of the effect of compressional magnetic perturbations on the microscopic instabilities in Tokamak

The compressional magnetic component δB|| has been implemented in the gyrokinetic Particle-in-Cell (PIC) code GEM for the fully electromagnetic simulation of microscopic instabilities in high-β tokamak plasmas. The simulation model is formulated for the split-weight scheme that is based on the canonical momentum formulation of the gyrokinetic equation. A pseudo-spectral method is used to solve the coupled gyrokinetic field equations while preserving all finite-Larmor-radius (FLR) effects. The algorithm is verified by the method of manufactured solutions. Simulations of the ion-temperature-gradient-driven (ITG) mode, the kinetic balloon mode (KBM) and the trapped electron mode (TEM) are presented.

3D theory of microscopic instabilities driven by space-charge forces

Microscopic, or short-wavelength, instabilities are known for drastic reduction of the beam quality and strong amplification of the noise in a beam. Space charge and coherent synchrotron radiation are known to be the leading causes for such instabilities. In this paper we present rigorous 3D theory of such instabilities driven by the space-charge forces. We define the condition when our theory is applicable for an arbitrary accelerator system with 3D coupling. Finally, we derive a linear integral equation describing such instability and identify conditions when it can be reduced to an ordinary second order differential equation.

Workability and dynamic restoration mechanisms of Al-6.2Zn-2.1Mg-0.15Zr alloy during medium temperature compression

Medium temperature compressions were implemented in the range of 433–593 K and 0.001–5 s−1 to reveal the workability and dynamic restoration mechanisms of quenched Al-6.2Zn-2.1Mg-0.15Zr alloy. The activation energy evolution with processing parameters was inspected through the modified kinetic equation that derived by integrally considering the strain and deformation parameters, and the results suggested activation energies decreased with the increasing strain rate and deformation temperature while less affected by the increase of strain at a specific strain rate condition. The 3D processing map was constructed to optimize the processing regions, in which two unstable domains yielded by distinct microscopic instabilities were marked whilst the safe working regions could be identified at 533–593 K, 0.001–0.1 s−1 with dissipation efficiency >20%. The microstructural evolution of the compressed specimens was examined, and the results confirmed different types of dynamic recovery (DRV) involving dislocation migration and rearrangement were dominate restoration process during the compression. Also, the microstructures of the specimens deformed at unstable domains were characterized, the results revealed fluctuating flow localization, micro cracking and particles debonding as well as dynamic strain aging were responsible for compression failure.

Enhancing elastocaloric effect of NiTi alloy by concentration-gradient engineering

By formulating the martensite transformation start temperature (Ms) of NiTi shape memory alloys (SMAs) as a function of Ni concentration, a Ni-concentration dependent phase field model is newly proposed in this work. Then, based on correspondent phase field simulations, the elastocaloric effects (eCEs) of the NiTi SMA samples with different Ni-concentration gradients and various aspect ratios are investigated and discussed. The simulated results show that the martensite transformation start stress of NiTi SMAs in the sample with a Ni-concentration gradient along the tensile direction progressively increases during the tensile loading, leading to a step-by-step martensite transformation; hence the suppression of microscopic instability results in a small stress-strain hysteresis loop in the process of tensile-unloading and a tremendous improvement in the coefficient of performance of material (COPmat) if addressing the eCE of the sample. When the aspect ratio of the sample is relatively small (e.g., 1:1), the microscopic instability can be suppressed due to the strong boundary constraint, indicating that reducing the aspect ratio is an efficient way to obtain a small stress-strain hysteresis loop and improve the COPmat of NiTi SMA samples. Moreover, for the sample with a Ni-concentration gradient, when its aspect ratio is relatively large, its eCE is mainly determined by the Ni-concentration gradient; while, when the aspect ratio becomes relatively small, the eCE of the sample is strongly dependent on both the Ni-concentration gradient and the aspect ratio. It implies that the NiTi SMA sample with both a small Ni-concentration gradient and a small aspect ratio and that with both a large Ni-concentration gradient and a large aspect ratio demonstrate the optimal eCE. In other words, a pioneering approach, i.e., concentration-gradient engineering, can be put forward to enhance the eCE of NiTi SMAs based on the detailed phase field simulations. The new findings in this work provide available guidance for improving the eCE of NiTi SMAs and open a novel way for developing advanced SMA-based elastocaloric materials.

Macroscopic Analysis of Vector Approximate Message Passing in a Model-Mismatched Setting

In this study, macroscopic properties of the vector approximate message passing (VAMP) algorithm for inference of generalized linear models are investigated using a non-rigorous heuristic method of statistical mechanics when the true posterior cannot be used and the measurement matrix is a sample from rotation-invariant random matrix ensembles. The focus is on the correspondence between the non-rigorous replica analysis of statistical mechanics and the performance assessment of VAMP in the model-mismatched setting. The correspondence of this kind is well-known when the measurement matrix has independent and identically distributed entries. However, when the measurement matrix follows a general rotation-invariant matrix ensemble, the correspondence has been validated only under limited cases, such as the Bayes optimal inference or the convex empirical risk minimization. The result presented in this paper is to extend the scope of such correspondence. Herein, we heuristically derive the explicit formula of state-evolution equations, which macroscopically describe VAMP dynamics for the current model-mismatched case, and show that their fixed point is generally consistent with the replica symmetric solution obtained by the replica method of statistical mechanics. We also show that the fixed point of VAMP can exhibit a microscopic instability, which indicates that message variables continue to move by VAMP while their macroscopically summarized quantities converge to fixed values. The critical condition the for microscopic instability agrees with that for breaking the replica symmetry that is derived within the non-rigorous replica analysis. The results of the numerical experiments cross-check our findings.

Wave Emission of Nonthermal Electron Beams Generated by Magnetic Reconnection

Magnetic reconnection in solar flares can efficiently generate nonthermal electron beams. The energetic electrons can, in turn, cause radio waves through microscopic plasma instabilities as they propagate through the ambient plasma along the magnetic field lines. We aim at investigating the wave emission caused by fast-moving electron beams with characteristic nonthermal electron velocity distribution functions (EVDFs) generated by kinetic magnetic reconnection: two-stream EVDFs along the separatrices and in the diffusion region, and perpendicular crescent-shaped EVDFs closer to the diffusion region. For this purpose, we utilized 2.5D fully kinetic Particle-In-Cell code simulations in this study. We found the following: (1) the two-stream EVDFs plus the background ions are unstable to electron/ion (streaming) instabilities, which cause ion-acoustic waves and Langmuir waves due to the net current. This can lead to multiple-harmonic plasma emission in the diffusion region and the separatrices of reconnection. (2) The perpendicular crescent-shaped EVDFs can cause multiple-harmonic electromagnetic electron cyclotron waves through the electron cyclotron maser instabilities in the diffusion region of reconnection. Our results are applicable to diagnose the plasma parameters, which are associated to magnetic reconnection in solar flares by means of radio wave observations.

Detailed characterization of a laboratory magnetized supercritical collisionless shock and of the associated proton energization

Collisionless shocks are ubiquitous in the Universe and are held responsible for the production of nonthermal particles and high-energy radiation. In the absence of particle collisions in the system, theory shows that the interaction of an expanding plasma with a pre-existing electromagnetic structure (as in our case) is able to induce energy dissipation and allow shock formation. Shock formation can alternatively take place when two plasmas interact, through microscopic instabilities inducing electromagnetic fields that are able in turn to mediate energy dissipation and shock formation. Using our platform in which we couple a rapidly expanding plasma induced by high-power lasers (JLF/Titan at LLNL and LULI2000) with high-strength magnetic fields, we have investigated the generation of a magnetized collisionless shock and the associated particle energization. We have characterized the shock as being collisionless and supercritical. We report here on measurements of the plasma density and temperature, the electromagnetic field structures, and the particle energization in the experiments, under various conditions of ambient plasma and magnetic field. We have also modeled the formation of the shocks using macroscopic hydrodynamic simulations and the associated particle acceleration using kinetic particle-in-cell simulations. As a companion paper to Yao et al. [Nat. Phys. 17, 1177–1182 (2021)], here we show additional results of the experiments and simulations, providing more information to allow their reproduction and to demonstrate the robustness of our interpretation of the proton energization mechanism as being shock surfing acceleration.

Homogenization of fully nonlinear rod lattice structures: on the size of the RVE and micro structural instabilities

This work investigates the possibility of applying two-scale computational homogenization to rod lattice structures emerging, for instance, from additive manufacturing. The influence of the number of unit cells within the representative volume element (RVE), thus, the RVE’s size on the homogenized mechanical response is studied for occurring microscopic structural instabilities. Therein, the macro-scale, described in terms of three-dimensional continuum mechanics, is coupled to the micro-scale described by geometrically exact rods, enabling arbitrary large deformations and rotations. A special feature of the presented framework is that the rods building the lattice structures are not restricted to deform purely elastically but may deform inelastically. The mechanical response of lattice structures is investigated by applying the developed homogenization method to an exemplary lattice. Under special loads the structure reaches an instable state and may buckle. The appearance of instabilities depends on the geometric properties of the lattice’s underlying rods and the RVE’s size.

Band gap tuning through microscopic instabilities of compressively loaded lightened nacre-like composite metamaterials

Large deformation effects give the opportunity for tuning and manipulating the dynamic properties of periodic microstructured materials, including those related to vibration control and band gap phenomena. We investigate the wave propagation properties in bioinspired nacre-like composite metamaterials subjected to uniaxial loading processes, also considering the onset of instabilities at the microscopic and macroscopic scales. The dynamic response of a new 2D lightened brick-and-mortar microstructure containing periodically arranged cavities is investigated, by analyzing the propagation of elastic waves superimposed on a deformed state by means of a Bloch-wave analysis. The band gap properties of the proposed advanced metamaterial were investigated in terms of spectral quantities as a function of the main geometrical parameters and of the level of applied deformations. Numerical results have shown that the elastic wave propagation is strongly influenced by the prestress state inducing microscopic instability, thus providing new opportunities to design periodic bioinspired composite metamaterials characterized by wave absorption capabilities as a function of microstructural evolution.

Microscopic and macroscopic instabilities in elastomeric foams

The present work is a thorough study on the elastic instabilities of flexible polymeric foams under compression. Tessellation-based topologies are employed to generate micromechanical models of open-cell random foams with controlled morphologies. Solid distribution on the resulting topologies is assigned based on a closed-form expression for the relative density that accounts for material overlap at the nodes. Simulations of foams with different microstructures are used to first reproduce experimentally obtained results, and subsequently to elucidate the connection between microstructure and the corresponding buckling mode. Compression of an anisotropic foam along the rise and transverse directions displays two distinct buckling modes that have previously been reported experimentally: a long-wavelength instability corresponding to a limit-load type response, and a microscopic instability that produces a monotonically increasing nonlinear response. The presence of large cell size distributions, i.e. polydispersity, is shown to also lead to a monotonic response, which is associated with the increased compaction of the largest cells that are present within the domain. The agreement between experimental and numerical responses is very favorable. The critical stresses are subsequently calculated as a function of key microstructural characteristics including foam density, anisotropy, cell-size distributions and ligament uniformity. The results indicate that a strong amount of anisotropy is needed for the transition from a microscopic instability to the macroscopic one while polydisperse foams produce a stable monotonic response, even in the presence of cell anisotropy.

Magnetoelastic instabilities in soft laminates with ferromagnetic hyperelastic phases

We investigate the microscopic and macroscopic instabilities developing in magnetoactive elastomer (MAE) composites undergoing large deformations in the presence of an external magnetic field. In particular, we consider the MAEs with bi-phasic layered microstructure, with phases exhibiting ferromagnetic behavior. We derive an explicit expression for the magnetic field-induced deformation of MAEs with hyperelastic phases. To perform the magnetoelastic instability analysis, we employ the small-amplitude perturbations superimposed on finite deformations in the presence of the magnetic field. We examine the interplay between the macroscopic and microscopic instabilities. We find that the layered MAEs can develop microscopic instability with antisymmetric buckling modes, in addition to the classical symmetric mode. Notably, the antisymmetric microscopic instability mode does not appear in a purely mechanical scenario (when a magnetic field is absent). Furthermore, our analysis reveals that the wavelength of buckling patterns is highly tunable by the applied magnetic field, and by the properties and volume fractions of the phases. Our findings provide the information for designing materials with reconfigurable microstructures. This material ability can be used to actively tune the behavior of materials by a remotely applied magnetic field. The results can be utilized in designing tunable acoustic metamaterials, soft actuators, sensors, and shape morphing devices.