Finite Shear Modulus Research Articles

The role of constitutive properties on the longitudinal compressive strength of composites

The longitudinal compressive strength of fibre-reinforced composites is often a limiting factor for structural design. This paper uses micromechanical finite-element analyses to examine how the constitutive properties of the matrix, fibres, and interface affect the composite’s strength. For a typical carbon/epoxy composite, the matrix shear strength has a larger effect than the modulus; a linear-elastic perfectly-plastic approximation of the matrix shear curve may overestimate the composite’s strength by over 15%. A pressure-dependent and dilatant matrix plasticity response may strengthen the composite by 30%, considering current uncertainties in friction and dilation angles of epoxy matrices. The finite shear modulus of carbon fibres may weaken the composite by more than 10%. Considering a cohesive interface using traction–separation laws significantly affects the predicted response of the composite, even for interfaces stronger than the matrix. Opportunities for improving the characterisation, modelling and performance of the constituents in terms of matrix plasticity, fibre shear modulus, and interface response are discussed.

Correlations of tensor field components in isotropic systems with an application to stress correlations in elastic bodies.

Correlation functions of components of second-order tensor fields in isotropic systems can be reduced to an isotropic fourth-order tensor field characterized by a few invariant correlation functions (ICFs). It is emphasized that components of this field depend in general on the coordinates of the field vector variable and thus on the orientation of the coordinate system. These angular dependencies are distinct from those of ordinary anisotropic systems. As a simple example of the procedure to obtain the ICFs we discuss correlations of time-averaged stresses in isotropic glasses where only one ICF in reciprocal space becomes a finite constant e for large sampling times and small wave vectors. It is shown that e is set by the typical size of the frozen-in stress components normal to the wave vectors, i.e., it is caused by the symmetry breaking of the stress for each independent configuration. Using the presented general mathematical formalism for isotropic tensor fields this finding explains in turn the observed long-range stress correlations in real space. Under additional but rather general assumptions e is shown to be given by a thermodynamic quantity, the equilibrium Young modulus E. We thus relate for certain isotropic amorphous bodies the existence of finite Young or shear moduli to the symmetry breaking of a stress component in reciprocal space.

Structure and Rheology in Vertex Models under Cell-Shape-Dependent Active Stresses.

Biological cells can actively tune their intracellular architecture according to their overall shape. Here we explore the rheological implication of such coupling in a minimal model of a dense cellular material where each cell exerts an active mechanical stress along its axis of elongation. Increasing the active stress amplitude leads to several transitions. An initially hexagonal crystal motif is first destabilized into a solid with anisotropic cells whose shear modulus eventually vanishes at a first critical activity. Increasing activity beyond this first critical value, we find a re-entrant transition to a regime with finite hexatic order and finite shear modulus, in which cells arrange according to a rhombile pattern with periodically arranged rosette structures. The shear modulus vanishes again at a third threshold beyond which spontaneous tissue flows and topological defects of the nematic cell shape field arise. Flow and stress fields around the defects agree with active nematic theory, with either contractile or extensile signs, as also observed in several epithelial tissue experiments.

Explicit Analytical Solution for Random Close Packing in d=2 and d=3.

We present an analytical derivation of the volume fractions for random close packing (RCP) in both d=3 and d=2, based on the same methodology. Using suitably modified nearest neighbor statistics for hard spheres, we obtain ϕ_{RCP}=0.658 96 in d=3 and ϕ_{RCP}=0.886 48 in d=2. These values are well within the interval of values reported in the literature using different methods (experiments and numerical simulations) and protocols. This statistical derivation suggests some considerations related to the nature of RCP: (i)RCP corresponds to the onset of mechanical rigidity where the finite shear modulus emerges, (ii)the onset of mechanical rigidity marks the maximally random jammed state and dictates ϕ_{RCP} via the coordination number z, (iii)disordered packings with ϕ>ϕ_{RCP} are possible at the expense of creating some order, and z=12 at the fcc limit acts as a boundary condition.

Dionysian Hard Sphere Packings Are Mechanically Stable at Vanishingly Low Densities.

High strength-to-weight ratio materials can be constructed by either maximizing strength or minimizing weight. Tensegrity structures and aerogels take very different paths to achieving high strength-to-weight ratios but both rely on internal tensile forces. In the absence of tensile forces, removing material eventually destabilizes a structure. Attempts to maximize the strength-to-weight ratio with purely repulsive spheres have proceeded by removing spheres from already stable crystalline structures. This results in a modestly low density and a strength-to-weight ratio much worse than can be achieved with tensile materials. Here, we demonstrate the existence of a packing of hard spheres that has asymptotically zero density and yet maintains finite strength, thus achieving an unbounded strength-to-weight ratio. This construction, which we term Dionysian, is the diametric opposite to the Apollonian sphere packing which completely and stably fills space. We create tools to evaluate the stability and strength of compressive sphere packings. Using these we find that our structures have asymptotically finite bulk and shear moduli and are linearly resistant to every applied deformation, both internal and external. By demonstrating that there is no lower bound on the density of stable structures, this work allows for the construction of arbitrarily lightweight high-strength materials.

Universal G' ∼ L-3 Law for the Low-Frequency Shear Modulus of Confined Liquids.

Liquids confined to sub-millimeter scales have remained poorly understood. One of the most striking effects is the large elasticity revealed using good wetting conditions, which grows upon further decreasing the confinement length, L. These systems display a low-frequency shear modulus in the order of 1-103 Pa, contrary to our everyday experience of liquids as bodies with a zero low-frequency shear modulus. While early experimental evidence of this effect was met with skepticism and abandoned, further experimental results and, most recently, a new atomistic theoretical framework have confirmed that liquids indeed possess a finite low-frequency shear modulus G', which scales with the inverse cubic power of confinement length L. We show that this law is universal and valid for a wide range of materials (liquid water, glycerol, ionic liquids, non-entangled polymer liquids, isotropic liquids crystals). Open questions and potential applications in microfluidics mechanochemistry, energy, and other fields are highlighted.

Finite element simulation of multi‐component vesicle morphologies undergoing phase separation

AbstractMorphological changes in lipid bilayer vesicles are due to phase transitions and surface deformations occurring in unison. We present a dynamic chemo‐mechanical finite element model to study such changes. This is achieved by coupling two fourth order partial differential equations (PDEs): The Cahn‐Hilliard [2] mass balance equation based on irreversible thermodynamics, and the Kirchhoff‐Love [1] rotation free thin shell equation. The Helmholtz free energy consists of Helfrich energy to model elastic bending, and also includes in‐plane elastic energy with a finite shear modulus for the purpose of regularization. The geometry is discretized by C1‐continuous NURBS shape functions. An implicit second order accurate generalized‐α scheme is used for time integration. Newton‐Raphson iterations are utilized to solve the resulting linearized weak form monolithically. The proposed formulation is illustrated by a numerical example.

Rheology of hard glassy materials

Glassy solids may undergo a fluidization (yielding) transition upon deformation whereby the material starts to flow plastically. It has been a matter of debate whether this process is controlled by a specific time scale, from among different competing relaxation/kinetic processes. Here, two constitutive models of cage relaxation are examined within the microscopic model of nonaffine elasto-plasticity. One (widely used) constitutive model implies that the overall relaxation rate is dominated by the fastest between the structural (α) relaxation rate and the shear-induced relaxation rate. A different model is formulated here which, instead, assumes that the slowest (global) relaxation process controls the overall relaxation. We show that the first model is not compatible with the existence of finite elastic shear modulus for quasistatic (low-frequency) deformation, while the second model is able to describe all key features of deformation of ‘hard’ glassy solids, including the yielding transition, the nonaffine-to-affine plateau crossover, and the rate-stiffening of the modulus. The proposed framework provides an operational way to distinguish between ‘soft’ glasses and ‘hard’ glasses based on the shear-rate dependence of the structural relaxation time.

First-Order Phase Transition in Liquid Ag to the Heterogeneous G-Phase.

A molten metal is an atomic liquid that lacks directional bonding and is free from chemical ordering effects. Experimentally, liquid metals can be undercooled by up to ∼20% of their melting temperature but crystallize rapidly in subnanosecond time scales at deeper undercooling. To address this limited metastability with respect to crystallization, we employed molecular dynamics simulations to study the thermodynamics and kinetics of the glass transition and crystallization in deeply undercooled liquid Ag. We present direct evidence that undercooled liquid Ag undergoes a first-order configurational freezing transition from the high-temperature homogeneous disordered liquid phase (L) to a metastable, heterogeneous, configurationally ordered state that displays elastic rigidity with a persistent and finite shear modulus, μ. We designate this ordered state as the G-phase and conclude it is a metastable non-crystalline phase. We show that the L-G transition occurs by nucleation of the G-phase from the L-phase. Both the L- and G-phases are metastable because both ultimately crystallize. The observed first-order transition is reversible: the G-phase displays a first-order melting transition to the L-phase at a coexistence temperature, TG,M. We develop a thermodynamic description of the two phases and their coexistence boundary.

Elastic and Flow Properties of Densely Packed Binary Microgel Mixtures with Size and Stiffness Disparities

Densely packed soft microgels (pastes) behave as a yield-stress fluid. They act as soft elastic solids with finite equilibrium shear modulus (G0) below a critical stress (σc) whereas they flow like liquids above σc. The effects of size and stiffness heterogeneities in the constituent microgels on the rheological properties of the pastes are revealed using the binary microgel mixtures by an oscillatory rheometer and diffusive wave spectroscopy (DWS). The binary blends with various degrees of size and stiffness disparities are made by mixing the pastes with the same apparent particle-volume fraction (ϕeff) at various values of relative weight fraction of soft microgels (fsoft). The G0–fsoft relations for the soft/hard microgel mixtures are significantly influenced by size disparity: The relations for small size disparities well obey the logarithmic mixing rule, while those for large size disparities have a wide fsoft regime in which G0 are almost equal to those of the single small-microgel pastes (G0,small)...

Elastic moduli of a Brownian colloidal glass former

The static, dynamic and flow-dependent shear moduli of a binary mixture of Brownian hard disks are studied by an event-driven molecular dynamics simulation. Thereby, the emergence of rigidity close to the glass transition encoded in the static shear modulus is accessed by three methods. Results from shear stress auto-correlation functions, elastic dispersion relations, and the elastic response to strain deformations upon the start-up of shear flow are compared. This enables one to sample the time-dependent shear modulus consistently over several decades in time. By that a very precise specification of the glass transition point and of is feasible. Predictions by mode coupling theory of a finite shear modulus at the glass transition, of α-scaling in fluid states close to the transition, and of shear induced decay in yielding glass states are tested and broadly verified.

Geometry and the onset of rigidity in a disordered network.

Disordered spring networks that are undercoordinated may abruptly rigidify when sufficient strain is applied. Since the deformation in response to applied strain does not change the generic quantifiers of network architecture, the number of nodes and the number of bonds between them, this rigidity transition must have a geometric origin. Naive, degree-of-freedom-based mechanical analyses such as the Maxwell-Calladine count or the pebble game algorithm overlook such geometric rigidity transitions and offer no means of predicting or characterizing them. We apply tools that were developed for the topological analysis of zero modes and states of self-stress on regular lattices to two-dimensional random spring networks and demonstrate that the onset of rigidity, at a finite simple shear strain γ^{★}, coincides with the appearance of a single state of self-stress, accompanied by a single floppy mode. The process conserves the topologically invariant difference between the number of zero modes and the number of states of self-stress but imparts a finite shear modulus to the spring network. Beyond the critical shear, the network acquires a highly anisotropic elastic modulus, resisting further deformation most strongly in the direction of the rigidifying shear. We confirm previously reported critical scaling of the corresponding differential shear modulus. In the subcritical regime, a singular value decomposition of the network's compatibility matrix foreshadows the onset of rigidity by way of a continuously vanishing singular value corresponding to the nascent state of self-stress.

Viscoelastic nematodynamics.

Nematic liquid crystals exhibit both crystallike and fluidlike features. In particular, the propagation of an acoustic wave shows an interesting occurrence of some of the solidlike features at the hydrodynamic level, namely, the frequency-dependent anisotropy of sound velocity and acoustic attenuation. The non-Newtonian behavior of nematics also emerges from the frequency-dependent viscosity coefficients. To account for these phenomena, we put forward a viscoelastic model of nematic liquid crystals, and we extend our previous theory to fully include the combined effects of compressibility, anisotropic elasticity, and dynamic relaxation, at any shear rate. The low-frequency limit agrees with the compressible Ericksen-Leslie theory, while at intermediate frequencies the model correctly captures the relaxation mechanisms underlying finite shear and bulk elastic moduli. We show that there are only four relaxation times allowed by the uniaxial symmetry.

Polymer glass transition occurs at the marginal rigidity point with connectivity z* = 4.

We re-examine the physical origin of the polymer glass transition from the point of view of marginal rigidity, which is achieved at a certain average number of mechanically active intermolecular contacts per monomer. In the case of polymer chains in a melt/poor solvent, each monomer has two neighbors bound by covalent bonds and also a number of central-force contacts modelled by the Lennard-Jones (LJ) potential. We find that when the average number of contacts per monomer (covalent and non-covalent) exceeds the critical value z* ≈ 4, the system becomes solid and the dynamics arrested - a state that we declare the glass. Coarse-grained Brownian dynamics simulations show that at sufficient strength of LJ attraction (which effectively represents the depth of quenching, or the quality of solvent) the polymer globule indeed crosses the threshold of z*, and becomes a glass with a finite zero-frequency shear modulus, G∝ (z-z*). We verify this by showing the distinction between the 'liquid' polymer droplet at z < z*, which changes shape and adopts the spherical conformation in equilibrium, and the glassy 'solid' droplet at z > z*, which retains its shape frozen at the moment of z* crossover. These results provide a robust microscopic criterion to tell the liquid apart from the glass for the linear polymers.

Slow dynamics of a tagged particle in a supercooled liquid.

The ergodicity-nonergodicity (ENE) transition of the self-consistent mode-coupling theory (MCT) is marked by the point at which the time correlation of collective density fluctuations is not zero in the long-time limit. The nonergodic state, reaching beyond the ENE transition of simple MCT, is characterized by a finite shear modulus. The MCT, formulated in the current set of papers, predicts that the single-particle density correlation, unlike the collective density correlation, decays to zero at long times on either side of the ENE transition. The self-diffusion coefficient remains finite. This differs from the existing MCT results in which both collective and single-particle correlations are simultaniously frozen at the ENE transition. We discuss in this paper mechanisms by which a sharp fall in self-diffusion coefficient may occur within the present model. This overdamping or the so-called adiabatic approximation for the supercooled state does not maintain microscopic momentum conservation. Within this approximation, the self-diffusion constant approaches zero at the ENE transition point. This approximate result, which is similar to the prediction of the existing MCT models, further illustrates the process of cage formation with increase of density. At a qualitative level, our analysis shows that the self-diffusion process depends on the structure as well as short-time transport properties of the supercooled liquid. We solve the integral equations for the nonergodicity parameters to analyze the full implications of the adiabatic approximation.

Discontinuous Shear Modulus Determines the Glass Transition Temperature

A solid - amorphous or crystalline - is defined by a finite shear modulus while a fluid lacks such. We thus experimentally investigate the elastic properties of a colloidal glass former near the glass transition: spectroscopy of vibrational excitations yields the dispersion relations of longitudinal and transverse phonons in the glassy state. From the long wavelength limit of the dispersion relation we extract the bulk and the shear modulus. As expected, the latter disappear in a fluid and we measure a clearly resolved discontinuous behaviour of the elastic moduli at the glass transition. This not only determines the transition temperature T_G of the system but also directly addresses recent discussions about elasticity during vitrification. We show that low frequency excitations in our system are plane waves such that continuum elasticity theory can be used to describe the macroscopic behaviour.

Stability and anomalous entropic elasticity of subisostatic random-bond networks.

We study the elasticity of thermalized spring networks under an applied bulk strain. The networks considered are subisostatic random-bond networks that, in the athermal limit, are known to have vanishing bulk and linear shear moduli at zero bulk strain. Above a bulk strain threshold, however, these networks become rigid, although surprisingly the shear modulus remains zero until a second, higher, strain threshold. We find that thermal fluctuations stabilize all networks below the rigidity transition, resulting in systems with both finite bulk and shear moduli. Our results show a T(0.66) temperature dependence of the moduli in the region below the bulk strain threshold, resulting in networks with anomalously high rigidity as compared to ordinary entropic elasticity. Furthermore, we find a second regime of anomalous temperature scaling for the shear modulus at its zero-temperature rigidity point, where it scales as T(0.5), behavior that is absent for the bulk modulus since its athermal rigidity transition is discontinuous.

Dynamic modeling of Kevlar KM2 single fiber subjected to transverse impact

In this paper, the transverse impact onto a Kevlar KM2 single fiber is studied using analytical and numerical models. The impact response of fibers with reduced longitudinal shear moduli (‘string’ model) is studied with a 3D finite element model and the wave propagation results converge to the classic 1D analytic solution that supports only axial loads. A dispersive flexural wave mode is predicted numerically for the single fiber during transverse impact due to its finite longitudinal shear modulus. The numerical results are confirmed with an analytical solution derived for the response of an infinitely long Euler–Bernoulli beam subjected to a constant velocity impact. Fiber bounce is predicted for impact with a cylindrical projectile with fiber velocity exceeding impact velocity by 60–80%. At short time scales significant transverse compressive stresses and strains develop in the fiber and the magnitude depends on the impact velocity. The breaking speed predicted by the 3D model based on maximum principal stress criterion for a single fiber is between 26% (membrane failure) and 76% (combined membrane and flexure failure) less than the 1D analytical solution. The flexural wave and projectile fiber interactions induce curvatures in the fiber significant enough to induce compressive fiber failure and fibrillation.

Effects of fluid-shear resistance and squirt flow on velocity dispersion in rocks

Laboratory measurements of rocks saturated with high-viscosity fluids (such as heavy-oil, bitumen, magma, kerogen, etc.) often exhibit considerable seismic velocity dispersion, which is usually underestimated by the Biot theory. Over the years, grain-scale dispersion mechanisms such as squirt (local-flow) and shear relaxation (nonzero shear stress in the pore fluid) have been more successful in explaining the measured dispersion. We developed a new method to quantify the combined high-frequency effects of squirt and shear dispersion on the effective moduli of rocks saturated with viscous fluids. Viscous fluid at high frequencies was idealized as an elastic solid of finite shear modulus, hydraulically locked in stiff and soft pores. This method entailed performing solid substitution in stiff pores of a dry rock frame, which itself was unrelaxed due to solid-filled soft pores. The unrelaxed frame stiffness solutions required information on the pressure dependency of the rock stiffness and porosity. This method did not have any adjustable parameters, and all required inputs can be directly measured. With various laboratory and numerical examples, we noted that accounting for combined effects of squirt and shear relaxation was necessary to explain laboratory-measured velocities of rocks saturated with fluids of high viscosity. Predictions of the new method were in good agreement with the laboratory data.

Cavitation in drying droplets of soft matter solutions.

When a droplet of a soft matter solution is dried, cavities are often formed in the droplet, giving a hollow sphere in the end. A theoretical model is given for this phenomenon. It is shown that the formation of a gel-like layer (skin layer), which has a finite shear modulus, is essential for the phenomenon to take place. The condition for cavity formation (how it depends on the shear modulus and thickness of the skin layer), and the variation of the droplet volume and cavity volume after the cavity formation are examined.