Finite Shear Research Articles

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.

The twist-fit problem: finite torsional and shear eigenstrains in nonlinear elastic solids

Eigenstrains in nonlinear elastic solids are created through defects, growth or other anelastic effects. These eigenstrains are known to be important as they can generate residual stresses and alter the overall response of the solid. Here, we study the residual stress fields generated by finite torsional or shear eigenstrains. This problem is addressed by considering a cylindrical bar made of an incompressible isotropic solid with an axisymmetric distribution of shear eigenstrains. As particular examples, we consider a cylindrical inhomogeneity and a double inhomogeneity with finite shear eigenstrains and study the effect of torsional shear eigenstrains on the axial and torsional stiffnesses of the circular cylindrical bar.

Offset‐variable density improves acoustic full‐waveform inversion: a shallow marine case study

ABSTRACTWe have previously applied three‐dimensional acoustic, anisotropic, full‐waveform inversion to a shallow‐water, wide‐angle, ocean‐bottom‐cable dataset to obtain a high‐resolution velocity model. This velocity model produced an improved match between synthetic and field data, better flattening of common‐image gathers, a closer fit to well logs, and an improvement in the pre‐stack depth‐migrated image. Nevertheless, close examination reveals that there is a systematic mismatch between the observed and predicted data from this full‐waveform inversion model, with the predicted data being consistently delayed in time. We demonstrate that this mismatch cannot be produced by systematic errors in the starting model, by errors in the assumed source wavelet, by incomplete convergence, or by the use of an insufficiently fine finite‐difference mesh. Throughout these tests, the mismatch is remarkably robust with the significant exception that we do not see an analogous mismatch when inverting synthetic acoustic data. We suspect therefore that the mismatch arises because of inadequacies in the physics that are used during inversion. For ocean‐bottom‐cable data in shallow water at low frequency, apparent observed arrival times, in wide‐angle turning‐ray data, result from the characteristics of the detailed interference pattern between primary refractions, surface ghosts, and a large suite of wide‐angle multiple reflected and/or multiple refracted arrivals. In these circumstances, the dynamics of individual arrivals can strongly influence the apparent arrival times of the resultant compound waveforms. In acoustic full‐waveform inversion, we do not normally know the density of the seabed, and we do not properly account for finite shear velocity, finite attenuation, and fine‐scale anisotropy variation, all of which can influence the relative amplitudes of different interfering arrivals, which in their turn influence the apparent kinematics. Here, we demonstrate that the introduction of a non‐physical offset‐variable water density during acoustic full‐waveform inversion of this ocean‐bottom‐cable field dataset can compensate efficiently and heuristically for these inaccuracies. This approach improves the travel‐time match and consequently increases both the accuracy and resolution of the final velocity model that is obtained using purely acoustic full‐waveform inversion at minimal additional cost.

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.

Elastoconductivity as a probe of broken mirror symmetries

We propose the possible detection of broken mirror symmetries in correlated two-dimensional materials by elastotransport measurements. Using linear response theory we calculate the``shear conductivity'' ${\mathrm{\ensuremath{\Gamma}}}_{xx,xy}$, defined as the linear change of the longitudinal conductivity ${\ensuremath{\sigma}}_{xx}$ due to a shear strain ${\ensuremath{\epsilon}}_{xy}$. This quantity can only be nonvanishing when in-plane mirror symmetries are broken and we discuss how candidate states in the cuprate pseudogap regime (e.g., various loop current or charge orders) may exhibit a finite shear conductivity. We also provide a realistic experimental protocol for detecting such a response.

Shearrate diffusion and constitutive relations during transients in simple shear

Granular matter, consisting of hard, frictional, cohesionless spheres, sheared in a simple shear geometry with smooth walls undergoes a velocity driven transition from a jammed or creeping state (low wall velocity) to a flow state with a finite shear rate in the bulk (high wall velocity). In the flow state, the state variables volume fraction \(\nu \), inertial number I and the macroscopic friction \(\mu \) of the bulk follow an exponential transient. The characteristic time of this progression grows with the wall velocity and the system size and is typically large compared to the inverse shear rate. It is shown that I, first being stationary in the shear zones, spreads diffusively into the bulk. The other state variables follow according to the constitutive laws, well known from the steady state.

Slip and wear at a corner with Coulomb friction and an interfacial strength

Traditional analyses of slip at corners of contacts, based on linear elasticity and a Coulomb friction law, are very sensitive to the details of local geometries, owing to the effects of elastic singularities. Following the use of cohesive-zone models to address such issues in mode-II fracture, we present analyses of slip and wear at corners of contacts when a finite interfacial shear strength is incorporated with a Coulomb friction law. We show that the concept of an instantaneous cohesive-length scale, borrowed from the field of fracture mechanics, can be used to describe the nature of stress fields around corners, and defines when linear-elasticity and Coulomb friction can provide an accurate description of the interfacial behavior. We also show that the sensitivity of slip analyses to geometrical details decreases when the cohesive-length scale increases. We also show that the cohesive strength of an interface plays a crucial role in the propagation of a wear scar across an interface. If only Coulomb slip is assumed to occur, a wear scar may not progress beyond the original stick-slip boundary. If a finite interfacial shear strength is introduced into the analysis, the wear scar can propagate along the interface.

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.

A Nano-Cantilever Bending Method for Interface Cracking in Nanoscale Materials

A nanocantilever bending method is developed to investigate the interface cracking in multilayered nanoscale materials basing on the technology of the focused ion beam (FIB) and the transmission electron microscopy (TEM). With FIB, a nanocantilever specimen consisting of 20-nm-thick copper (Cu) layer and 500-nm-thick silicon nitride (SiN) layer on a silicon (Si) substrate is fabricated from a macroscale multi-layered material (Si/Cu/SiN) with the proposed method. By using a minute loading apparatus, the loading experiment is conducted in TEM, and the crack initiation at the edge of Cu/Si interface in different specimens is in situ observed. The critical stress fields at crack initiation are analyzed with the finite element method, and both normal and shear stresses concentrate at the region of 100 nm from the interface edge in all specimens. In addition, the normal stress is much larger than the shear one. A close observation on stress fields shows that the normal stress field at the area 20 nm–30 nm away from interface edge produces the local criterion for crack initiation at the edge of Cu/Si interface in nanoscale components.

Nonlinear interaction of finite longitudinal shear with finite torsion of a rubber-like bushing

In the framework of nonlinear elasticity, an example of longitudinal shear and torsion of a cylindrical elastomer bushing pressed between two rigid holders is used to study the dependence of torsional rigidity on the longitudinal shear and the dependence of the longitudinal shear rigidity on the angle of torsion for two potentials of the strain energy. An analytic model of interaction between the longitudinal shear and the transverse torsion is proposed in the asymptotic approximation.

Topology of pair-sphere trajectories in finite inertia suspension shear flow and its effects on microstructure and rheology

The relative motion of pairs of spheres in finite inertia shear flow is examined. The pair relative trajectory is studied, using lattice-Boltzmann simulation, both for pairs which are isolated and for pairs in suspension of solid volume fraction of ϕ ≤ 0.3. For the suspension, the average trajectory and aspects of its dispersion are considered. For neutrally buoyant particles in simple-shear flow, particle-scale inertia is represented by a Reynolds number Re=ργ̇a2μ, where γ̇ is the shear rate, a is the particle radius, and ρ and μ are the density and viscosity of the Newtonian suspending fluid. The pair trajectories in a dilute inertial suspension have the same basic features as the streamlines around an isolated particle at similar Re: reversing, in-plane and off-plane spiraling, and open but fore-aft asymmetric trajectories are observed. The origin of the off-plane spirals is examined in detail in this work. The average pair trajectory space in a suspension of finite volume fraction is found to be qualitatively similar to the dilute suspension pair trajectories, as the spiraling and reversing zones are retained; the influence of ϕ and Re on the extension of different zones is described. The role of the average pair trajectory in setting the microstructure of the suspension is analyzed through a convection equation description of the pair distribution function, showing extreme accumulation at contact consistent with sampling from simulation. The influence of the microstructure on the bulk rheology is considered, with a focus on the combined effects of ϕ and Re on the near contact structural effects on rheology.

Coupling of the interfacial and bulk flow in knife-edge viscometers

The operation of the knife-edge viscometer requires knowledge of the interfacial velocity profile in order to determine the viscous traction between the surface film and the knife edge and hence measure the surface shear viscosity of the film. The interfacial velocity profile can be obtained analytically in two limiting regimes. One is the limit of the surface shear viscosity going to infinity, in which case the interfacial velocity profile is independent of the bulk flow and a simple analytic expression is available. The other limit corresponds to vanishing bulk flow inertia, allowing one to reduce the Navier–Stokes equations to the Stokes equation, and the resulting linear system can be solved analytically. For finite inertia and finite surface shear viscosity, the knife-edge viscometer hydrodynamics is governed by the coupled nonlinear set of equations. Here, we study these numerically, explore the coupling between the interfacial and bulk flow, and delineate the ranges of surface shear viscosity and knife-edge rotation rates where the analytic approximations are appropriate. We also examine a variant of the knife-edge viscometer, known as the double-wall ring viscometer, which is essentially the same geometry but with the addition of a stationary inner cylinder so that the bulk fluid is contained in an annular channel rather than a cylinder.

Application of Damp Newton Algorithm on FEPG Platform

Damp Newton algorithm based on FEPG system is developed for calculating strain softening problem. The algorithm can solve displacement and yield surface equation simultaneously. Soften modulus is introduced into D-P yield function. The example result shows that, the algorithm with damp factor can be used to solve softening problems. The plastic theory described by finite element weak form has no requirement of continuity, so appropriate outcome can be obtained by first-order element. Descent segment of stress-strain curve and cloud picture of finite shear band are presented.

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.

Three dimensional graded finite element elasticity shear buckling analysis of FGM annular sector plates

In the present paper, shear buckling analysis of functionally graded annular sector plates is investigated for the first time. A three-dimensional elasticity approach is employed instead of the approximate plate theories. In-plane shear loads have been applied to radial, circumferential, or all edges of annular sector plates. Moreover, buckling of annular sector plates subjected to in-plane shear load at upper/lower surfaces is investigated for the first time. Three different boundary conditions have been examined: (1) Movable simply supported edges. (2) Immovable simply supported radial edges and free circumferential edges. (3) Immovable simply supported circumferential edges and free radial edges. The material properties are assumed to have transverse heterogeneity according to a power law distribution while Poisson's ratio is assumed to be constant. Results are extracted based on the principle of minimum total potential energy and a graded finite element method. Buckling loads are obtained based on a generalized geometric stiffness concept. In addition to different loading and boundary conditions, the effects of material gradient exponent, sector angles, aspect ratio and thickness ratio on the shear buckling loads and mode shapes of FGM annular sector plates have been investigated in detail.

Finite Stretching and Shearing of an Internally Balanced Elastic Solid

When seeking to separately describe elastic and inelastic effects in solids undergoing finite strain, it is typical to factor the deformation gradient \(\bf F\) into parts, say \(\mathbf{F}={\hat {\mathbf{F}}}{\mathbf{F}}^{*}\). Such a factorization can also be used to describe a notion of internal elastic balance. For equilibrium deformations this balance emerges naturally from an appropriately formulated energy minimization by considering the variation with respect to the multiplicative decomposition itself. The internal balance requirement is then a tensor relation between F, \({\hat {\mathbf{F}}}\) and F∗. Such theories holds promise for describing various substructural reconfigurations in solids. This includes the development of surfaces that localize slip once a threshold value of load is obtained. We consider an incompressible internally balanced elastic material with a constitutive law that is motivated by neo-Hookean behavior in the standard hyperelastic theory. The role of the internal balance relation is examined in uniaxial loading and also in simple shearing. In uniaxial loading the principle directions remain fixed. In simple shearing the principle directions change as the deformation proceeds. To address the latter, we establish a specific sense in which the internal balance relation can be solved analytically for general isochoric \(\bf F\).

Shear-stress relaxation and ensemble transformation of shear-stress autocorrelation functions.

We revisit the relation between the shear-stress relaxation modulus G(t), computed at finite shear strain 0<γ≪1, and the shear-stress autocorrelation functions C(t)|(γ) and C(t)|(τ) computed, respectively, at imposed strain γ and mean stress τ. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general G(t)=C(t)|(τ)=C(t)|(γ)+G(eq) for t>0 with G(eq) being the static equilibrium shear modulus. G(t) and C(t)|(γ) thus must become different for solids and it is impossible to obtain G(eq) alone from C(t)|(γ) as often assumed. We comment briefly on self-assembled transient networks where G(eq)(f) must vanish for a finite scission-recombination frequency f. We argue that G(t)=C(t)|(τ)=C(t)|(γ) should reveal an intermediate plateau set by the shear modulus G(eq)(f=0) of the quenched network.

Prediction of Fatigue Failure at Asphalt Concrete Layer Interface from Monotonic Testing

This study investigated the characterization of interface shear behavior in asphalt concrete through the estimation of the stress ratio (SR). This parameter, originally identified as the ratio between predicted interface stress from a finite element model (FEM) and interface shear strength at the corresponding normal stress, was assumed to be dynamic. As part of the experimental plan, monotonic tests on double-layered asphalt specimens were performed. Dynamic evaluations of the number of repetitions to failure under several stress conditions, equal to or higher than stresses computed from an FEM of the pavement structure, were also performed. The failure curves of the two testing modalities show similar patterns on the Mohr plane. The Hoek–Brown shear strength failure criterion and the three-dimensional surface that best fits the dynamic outcomes were considered. In this scenario, the SR referred to the proportion between the applied shear stress conditions in the dynamic modality and the maximum stress from monotonic tests. For the same predicted failure repetitions, SR assumed a constant value. Correlating monotonic and dynamic results could be an important approach both in furthering knowledge of interface shear strength and in predicting information about failure under repetitive loading applications based on simple monotonic tests.

The mechanical and microstructural behaviour of calcite-dolomite composites: An experimental investigation

The styles and mechanisms of deformation associated with many variably dolomitized limestone shear systems are strongly controlled by strain partitioning between dolomite and calcite. Here, we present experimental results from the deformation of four composite materials designed to address the role of dolomite on the strength of limestone. Composites were synthesized by hot isostatic pressing mixtures of dolomite (Dm) and calcite powders (% Dm: 25%-Dm, 35%-Dm, 51%-Dm, and 75%-Dm). In all composites, calcite is finer grained than dolomite. The synthesized materials were deformed in torsion at constant strain rate (3 × 10−4 and 1 × 10−4 s−1), high effective pressure (262 MPa), and high temperature (750 °C) to variable finite shear strains. Mechanical data show an increase in yield strength with increasing dolomite content. Composites with <75% dolomite (the remaining being calcite), accommodate significant shear strain at much lower shear stresses than pure dolomite but have significantly higher yield strengths than anticipated for 100% calcite. The microstructure of the fine-grained calcite suggests grain boundary sliding, accommodated by diffusion creep and dislocation glide. At low dolomite concentrations (i.e. 25%), the presence of coarse-grained dolomite in a micritic calcite matrix has a profound effect on the strength of composite materials as dolomite grains inhibit the superplastic flow of calcite aggregates. In high (>50%) dolomite content samples, the addition of 25% fine-grained calcite significantly weakens dolomite, such that strain can be partially localized along narrow ribbons of fine-grained calcite. Deformation of dolomite grains by shear fracture is observed; there is no intracrystalline deformation in dolomite irrespective of its relative abundance and finite shear strain.

Effect of rotation on the growth rate of magnetogravitational instability of a viscoelastic medium

In the present paper, we investigate the effect of rotation on the onset of gravitational collapse and the growth rate of magnetogravitational instability of a finitely electrically conducting viscoelastic medium under both strongly and weakly coupled plasma limits for transverse and longitudinal modes of wave propagation. A general dispersion relation, which is uniformly valid for both the transverse and longitudinal mode of wave propagation, is obtained using the normal mode analysis method. It is observed that the rotation has no effect on the instability criterion which governs the onset of gravitational collapse of the viscoelastic medium in the presence of a magnetic field for either the longitudinal or transverse mode of wave propagation in strongly coupled plasma (SCP) or weakly coupled plasma (WCP). Furthermore, the effects of rotation (Coriolis force), finite electrical resistivity and shear viscosity on the growth rate of Jeans instability in both SCP and WCP for transverse modes of propagation have been numerically calculated and the results obtained are depicted graphically. From these results, we concluded that the rotation reduces the growth rate of Jeans instability.