Compressive Instabilities Research Articles

Compressive instabilities enable cell-induced extreme densification patterns in the fibrous extracellular matrix: Discrete model predictions.

We present a new model and extensive computations that explain the dramatic remodelling undergone by a fibrous collagen extracellular matrix (ECM), when subjected to contractile mechanical forces from embedded cells or cell clusters. This remodelling creates complex patterns, comprising multiple narrow localised bands of severe densification and fiber alignment, extending far into the ECM, often joining distant cells or cell clusters (such as tumours). Most previous models cannot capture this behaviour, as they assume stable mechanical fiber response with stress an increasing function of fiber stretch, and a restriction to small displacements. Our fully nonlinear network model distinguishes between two types of single-fiber nonlinearity: fibers that undergo stable (supercritical) buckling (as in previous work) versus fibers that suffer unstable (subcritical) buckling collapse. The model allows unrestricted, arbitrarily large displacements (geometric nonlinearity). Our assumptions on single-fiber instability are supported by recent simulations and experiments on buckling of individual beams with a hierarchical microstructure, such as collagen fibers. We use simple scenarios to illustrate, for the first time, two distinct compressive-instability mechanisms at work in our model: unstable buckling collapse of single fibers, and snap-through of multiple-fiber groups. The latter is possible even when single fibers are stable. Through simulations of large fiber networks, we show how these instabilities lead to spatially extended patterns of densification, fiber alignment and ECM remodelling induced by cell contraction. Our model is simple, but describes a very complex, multi-stable energy landscape, using sophisticated numerical optimisation methods that overcome the difficulties caused by instabilities in large systems. Our work opens up new ways of understanding the unique biomechanics of fibrous-network ECM, by fully accounting for nonlinearity and associated loss of stability in fiber networks. Our results provide new insights on tumour invasion and metastasis.

Compressive Flow Behaviour and Deformation Instabilities of Fe-Mn-Al-Ni-C Lightweight Duplex Steel

In this work, uniaxial hot compression tests of Fe-Mn-Al-Ni-C steel are conducted in the domain of strain rates (0.001–10 s−1) and temperatures (1223–1423 K). The apparent activation energy of the present steel is estimated as ~400 kJ mol−1. A constitutive equation is formulated by using the experimental flow stress data to depict the flow characteristics of the steel. An excellent correspondence (R = 0.99 and AARE = 5.2%) between the experimental and predicted flow stress data has been obtained. Further, we have predicted the deformation instabilities in the present steel based on the various existing criteria (viz. Prasad, Gegel, Alexander, Jonas, and Semiatin). The predicted unstable domains are further validated by the microstructural characterization. The instability features like boundary decohesion, micro-cracks, and flow localization are observed in the regions 1223 K/0.1–10 s−1 and 1273–1373 K/1–10 s−1 while the wedge-cracks at the triple junctions of the boundaries are observed in the region 1350–1423 K/0.001 s−1. The Prasad’s instability criterion could precisely identify the flow instabilities in the present steel. However, the Gegel’s and Alexandar’s criteria over-estimated the unstable regions (<1300 K/0.003–0.1 s−1), (1300–1373 K/0.003–1 s−1) and (1373–1423 K/0.002–5.62 s−1) in the material. The Jonas’ and Semiatin’s criteria, on the other hand, underrated the extent of flow instabilities in the lightweight duplex steel.

Modeling hybrid polymer–nanometal lightweight structures

Increasing the environmental sustainability of aviation is a key design goal for commercial aircraft for the foreseeable future. From the perspective of structural engineering, this is accomplished through reducing the mass of aircraft components and structures. Advanced manufacturing techniques offer new avenues for design, enabling more complex structures which can have highly tailored properties. One advanced manufacturing concept is the use of 3D printed polymer preforms that are coated with nanocrystalline metal through electrodeposition. This enables the use of high-performance materials in virtually any geometry. To exploit this manufacturing approach, it is incumbent to have well-established mechanical models of the behavior of such hybrid structures. In particular, hybrid polymer–nanometal structures tend to fail due to compressive instabilities. This paper describes a model of local shell buckling, a typical compressive instability, as it applies to hybrid polymer–nanometal structures. The analysis depends upon the Southwell stress function model for radially loaded solids of rotation, and couples this with the Timoshenko analysis of local shell buckling. This combination is applicable to a range of practical configurations for truss-like hybrid structures.

Size scale dependence of compressive instabilities in layered composites in the presence of stress gradients

The compressive strength of unidirectionally or layer-wise reinforced composite materials in direction parallel to their reinforcement is limited by micro-buckling instabilities. Although the inherent compressive strength of a given material micro-structure can easily be determined by assessing its stability under a uniform compressive load, this is often not sufficient for predicting failure initiation within a larger structure. In cases, where the composite material micro-structure is locally subjected to strongly non-uniform loadings, compressive instabilities depend not only on the maximum compressive stress but also on spatial stress or strain gradients, rendering failure initiation size scale dependent. The present work demonstrates and investigates the aforementioned effect through numerical simulations of periodically layered structures with notches and holes under bending and compressive loads, respectively. The presented results emphasize the importance of the reinforcing layer thickness on the load carrying capacity of the investigated structures, at a constant volumetric fraction of the reinforcement. The observed strengthening at higher values of the relative layer thickness is attributed to the bending stiffness of the reinforcing layers.

Compression response, strength and post-peak response of an axial fiber reinforced tow

This paper discusses the results of a finite element (FE)-based study of the compressive instabilities of axial fiber tows in 2D triaxial braided carbon fiber composites (2DTBC). A 3D FE model of an axial tow is used in conjunction with the ABAQUS commercial FE code. To accommodate unstable snap-back post-peak behavior, an arc-length solution technique is adopted to solve the FE equations. Explicit account of the braid microstructure (geometry and packing) and the measured inelastic properties of the matrix (the in-situ properties) are accounted for via the use of the FE method. This type of detailed modeling is necessary for developing a mechanism-based failure prediction capability. The computational model provides a means to assess the compressive strength of the axial fiber tow and its dependence on various microstructural parameters. It also serves as a tool to assess the most significant parameters that affect compressive strength of a 2DTBC.

On Fermi acceleration and magnetohydrodynamic instabilities at ultra-relativistic magnetized shock waves

Fermi acceleration can take place at ultra-relativistic shock waves if the upstream or downstream magnetic field has been remodelled so that most of the magnetic power lies on short spatial scales. The relevant conditions under which Fermi acceleration becomes efficient in the presence of both coherent and short-scale turbulent magnetic fields are addressed. Within the magnetohydrodynamic (MHD) approximation, this paper then studies the amplification of a pre-existing magnetic field through the streaming of cosmic rays upstream of a relativistic shock wave. The magnetic field is assumed to be perpendicular in the shock-front frame, as generally expected in the limit of large shock Lorentz factor. In the MHD regime, compressive instabilities seeded by the net cosmic-ray charge in the shock precursor (as seen in the shock-front frame) develop on the shortest spatial scales but saturate at a moderate level δB/B ∼ 1, which is not sufficient for Fermi acceleration. As we argue, it is possible that other instabilities outside the MHD range provide enough amplification to allow successful Fermi acceleration.

A Mathematical Model for Determination of Limiting Blank Holding Force and Cavity Pressure in Hydromechanical Deep Drawing

A mathematical model for the prediction of a safe zone for die cavity pressure and blank holding force/pressure, to avoid tensile and compressive instabilities in hydromechanical deep drawing is presented in this paper. Tensile instability is predicted using the load maximization principle whereas a solution for the stability equations provided the critical condition for compressive instability. Predictions of the present work are compared with results available in the literature and are in good agreement. A comprehensive parametric study is carried out to study the effect of process variables on the safe zone.

Micromechanical Analyses of Instabilities in Braided Glass Textile Composites

The results of a finite element (FE)-based study of the multiaxial compressive instabilities in two-dimensional triaxial braided glass fiber composites are discussed. The micromechanics-based study was carried out on a two unit-cell size three-dimensional FE model. Computational simulations were carried out first to determine the orthotropic elastic engineering constants. The results were validated by comparison against available experimental data. Subsequently, the uniaxial compressive response and the biaxial tension/compression response of the micromodel were established using an arc-length method in conjunction with the ABAQUS commercial FE code. The approach is different from those reported in the literature where classical methods based on the technique of homogenization are used to model the elastic and inelastic response of braided composites. Instead, explicit account of the braid microstructure (geometry and packing) and the measured inelastic properties of the matrix (both the virgin properties and the in situ properties) are accounted for via the use of the FE method. This detail is necessary for developing a mechanism-based failure prediction capability. The computational model provides a means to assess the compressive and biaxial strength of the braided composites and its dependence on various microstructural parameters. It also serves as a tool to assess the most significant parameters that affect compressive strength.

Effect of reinforcement connectivity on the elasto-plastic behavior of aluminum composites containing sub-micron alumina particles

The mechanical properties of composites consisting of an aluminum matrix with 34 and 37 vol.% sub-micron Al 2O 3 particles were studied in compression for two reinforcement architectures: interconnected and discontinuous. Both the elastic and plastic behaviors of these composites are successfully modeled using a self-consistent approach: the classical self-consistent and the three-phase self-consistent models for the interconnected and discontinuous architectures, respectively. At ambient temperature, an interconnected architecture offers only a modest increase in stiffness and strength over a discontinuous architecture of equal volume fraction. At elevated temperatures (250, 500 and 600 °C), the interconnected reinforcement becomes increasingly more effective at strengthening the composites. However, the relative increase in strength due to interconnectivity can only be exploited at small strains (1–5%) due to the early development of compressive flow instabilities in the interconnected composites. While microstructural damage controls the instability strain of the interconnected composites at ambient temperature, their low strain-hardening coefficient is the main contribution to flow instabilities at elevated temperature.

Do faults trigger folding in the lithosphere?

A number of observations reveal large periodic undulations within the oceanic and continental lithospheres. The question if these observations are the result of large‐scale compressive instabilities, i.e. buckling, remains open. In this study, we support the buckling hypothesis by direct numerical modeling. We compare our results with the data on three most proeminent cases of the oceanic and continental folding‐like deformation (Indian Ocean, Western Gobi (Central Asia) and Central Australia). We demonstrate that under reasonable tectonic stresses, folds can develop from brittle faults cutting through the brittle parts of a lithosphere. The predicted wavelengths and finite growth rates are in agreement with observations. We also show that within a continental lithosphere with thermal age greater than 400 My, either a bi‐harmonic mode (two superimposed wavelengths, crustal and mantle one) or a coupled mode (mono‐layer deformation) of inelastic folding can develop, depending on the strength and thickness of the lower crust.

Asymptotic analysis of stability for prismatic solids under axial loads

This work addresses the stability of axially loaded prismatic beams with any simply connected crosssection. The solids obey a general class of rate-independent constitutive laws, and can sustain finite strains in either compression or tension. The proposed method is based on multiple scale asymptotic analysis, and starts with the full Lagrangian formulation for the three-dimensional stability problem, where the boundary conditions are chosen to avoid the formation of boundary layers. The calculations proceed by taking the limit of the beam's slenderness parameter, ε (ε 2 ≡ area/length 2), going to zero, thus resulting in asymptotic expressions for the critical loads and modes. The analysis presents a consistent and unified treatment for both compressive (buckling) and tensile (necking) instabilities, and is carried out explicitly up to o( ε 4) in each case. The present method circumvents the standard structural mechanics approach for the stability problem of beams which requires the choice of displacement and stress field approximations in order to construct a nonlinear beam theory. Moreover, this work provides a consistent way to calculate the effect of the beam's slenderness on the critical load and mode to any order of accuracy required. In contrast, engineering theories give accurately the lowest order terms ( O( ε 2)—Euler load—in compression or O(1)—maximum load—in tension) but give only approximately the next higher order terms, with the exception of simple section geometries where exact stability results are available. The proposed method is used to calculate the critical loads and eigenmodes for bars of several different cross-sections (circular, square, cruciform and L-shaped). Elastic beams are considered in compression and elastoplastic beams are considered in tension. The O( ε 2) and O( ε 4) asymptotic results are compared to the exact finite element calculations for the corresponding three-dimensional prismatic solids. The O( ε 4) results give significant improvement over the O( ε 2) results, even for extremely stubby beams, and in particular for the case of cross-sections with commensurate dimensions.

A cosmic-ray-driven plasma instability

The stability of the MHD equations describing the mutual interaction of cosmic rays, thermal plasma, magnetic field and Alfvén waves used in cosmic-ray-shock acceleration theory (e.g. McKenzie &amp; Völk 1982) is analysed for linear compressive instabilities. It is found that the inclusion of wave effects implies that the forward propagating sub-Alfvénic mode is unstable on wavelength scales greater than 1 parsec. The role of the instability in astrophysical models is considered.

Short-wavelength compressive instabilities in cosmic ray shocks and heat conduction flows

In this paper we discuss the stability of three genetically similar non-uniform flows to compressive disturbances whose wavelengths are much shorter than the length scales characterizing the background flow. The results are relevant to theoretical models of cosmic ray shocks and solar wind type flows involving heat conduction. A JWKB expansion solution yields an equation which determines how the amplitudes of the perturbations may grow (or decay) as they propagate within such structures. It is shown that, in all three of the models considered, the perturbations exhibit spatial growth if the background flow is sufficiently supersonic and decelerating. The associated equations describing the evolution of the wave action are also studied with a view to deciding whether or not the behaviour of this attractive variable can provide an unambiguous answer to the question of stability. In the case of a shock transition dominated by heat conduction, it is shown that the effects of dissipative heating within the transition more than offset those of wave growth, with the result that wave amplification is accompanied by wave action decay. Therefore in general it would appear that the wave action equation alone cannot unambiguously settle stability questions.

Magnetohydrodynamic plasma instability driven by Alfvén waves excited by cosmic rays

Hydrodynamical equations describing the mutual interaction of cosmic rays, thermal plasma, magnetic field and Alfvén waves scattering the cosmic rays used in cosmic ray shock acceleration theory (e.g. McKenzie &amp; Völk 1982; Drury 1983; Webb 1983) are analysed for long-wavelength linear compressive instabilities. The Alfvén wave field may contain a pre-existing component as well as a component excited by the cosmic ray streaming instability. In the case of no Alfvén wave damping, adiabatic wave growth and Alfvén wave generation by the cosmic ray streaming instability, it is found that the backward propagating slow magneto-acoustic mode is driven convectively unstable by the pressure of the self-excited Alfvén waves, provided the thermal plasmaβis sufficiently large. The equations are also analysed for the case where the Alfvén wave growth is balanced by some nonlinear damping mechanisms. In the latter case both the forward and backward propagating slow magneto-acoustic modes may be driven unstable if the plasmaβis sufficiently small. The conditions under which the instabilities occur are delineated, and sample calculations of growth rates given. Possible applications of the instabilities to astrophysical situations are briefly discussed.