Finite Shear Research Articles

Apparent elongational yield stress of soft matter

Apparent elongational yield stresses of soft matter including polymer gels, highly concentrated emulsions, and aggregated suspensions have been determined from step stretch experiments. Materials display apparent shear yield stresses in the range 1–100 Pa and large but finite shear relaxation times tR. For all investigated fluids, the Laplace pressure within the stretched filaments is essentially constant during an initial period of time after the step strain. Then, it increases rapidly and finally the filaments break. Filament lifetime tf strongly increases with decreasing stretching ratio e. The apparent elongational yield stress is identified as the initial value of the Laplace pressure obtained at a critical stretching ratio ec corresponding to a Deborah number De = tR /tf = 1. For all fluids, the ratio of this elongational yield stress to shear yield stress is √3 in agreement with the von Mises plasticity criterion, irrespective of the physical nature of structural breakdown. Elongational experiments performed at different e or tf covering Deborah numbers between 0.1 and 100 reveal a universal relationship between the initial plateau value of the Laplace pressure normalized to the shear yield stress and De. This stress ratio varies between 0.5 and 5, and equals √3 only for De ≈ 1.

Mechanical instability and ideal strengths of layered M2SC (M = Ti, Zr, and Hf) compounds

The behavior of M2SC (M = Ti, Zr, Hf) under tensile and shear deformations was investigated using the plane-wave pseudopotential density functional method. The microscopic mechanism that determines the structural stability is studied using the results of electronic structure calculations. It is shown that the structural failure of M2SC can be ascribed to the breakage of weak covalent M – S bond under uniaxial tensile tension, and the plasticity of these ternary materials can be triggered by electronic instabilities at finite shear deformations. Layered structural stability of M2SC is determined by the strength M – S bond under tensile tension, which is less resistive to shear deformation than to tensile strain. The stress-strain relationships of these ternary materials are presented and compared with corresponding binary carbides. Our study demonstrates that the ideal shear strength for M2SC is limited by the electronic instabilities which in turn lead to elastic instability. M2SC ceramics are predicted to be intrinsically brittleness in nature based on low bulk-modulus-to-shear-modulus ratios and negative Cauchy pressure (C12−C44). We propose that the brittleness of M2SC originates from larger movement of a dislocation in a glide plane and this can be quantized by larger Peierls stress, which is in turn initiated by novel layered structure for M2SC ceramics.

The jamming perspective on wet foams

Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can {\em jam} into a rigid, disordered state where they withstand finite shear stresses before yielding. The jamming transition has been studied extensively, in particular in computer simulations of frictionless, soft, purely repulsive spheres. Foams and emulsions are the closest realizations of this model, and in foams, the (un)jamming point corresponds to the wet limit, where the bubbles become spherical and just form contacts. Here we sketch the relevance of the jamming perspective for the geometry and flow of foams --- and also discuss the impact that foams studies may have on theoretical studies on jamming. We first briefly review insights into the crucial role of disorder in these systems, culminating in the breakdown of the affine assumption that underlies the rich mechanics near jamming. Second, we discuss how crucial theoretical predictions, such as the square root scaling of contact number with packing fraction, and the nontrivial role of disorder and fluctuations for flow have been observed in experiments on 2D foams. Third, we discuss a scaling model for the rheology of disordered media that appears to capture the key features of the flow of foams, emulsions and soft colloidal suspensions. Finally, we discuss how best to confront predictions of this model with experimental data.

Glass Elasticity from Particle Trajectories

Using positional data from video microscopy of a two-dimensional colloidal system and from simulations of hard disks, we determine the wave-vector-dependent elastic dispersion relations in glass. The emergence of rigidity based on the existence of a well defined displacement field in amorphous solids is demonstrated. Continuum elastic theory is recovered in the limit of long wavelengths which provides the glass elastic shear and bulk modulus as a function of temperature. The onset of a finite static shear modulus upon cooling marks the fluid-glass transition in an intuitive and unique way.

Acoustic radiation force on a sphere in tissue

A theory is presented for the acoustic radiation force on a sphere embedded in a soft elastic medium that possesses a shear modulus μ several orders of magnitude smaller than its bulk modulus. Scattering of both compressional and shear waves is taken into account. There is no restriction on the size of the sphere or, apart from axisymmetry, the form of the incident compressional wave. The analysis employs the Piola-Kirchhoff pseudostress tensor and Lagrangian coordinates. In the linear approximation an analytical solution is obtained for the scattered waves. The nonlinear stress and full radiation force are calculated at the next order of approximation. For a small sphere and μ≈0 the classical result for a particle in liquid is recovered. For small but finite shear modulus the radiation force is evaluated for a gas bubble driven at a frequency below resonance. The predicted magnitude of the radiation force on the bubble is found to be less than that in liquid by the factor [1+(4/3)μ/γ P 0]-1, where P 0 is the ambient pressure and γ the ratio of specific heats of the gas. Influence of the scattered shear wave in this limit is negligible. [Work supported by NIH DK070618.]

Identification of a low-frequency elastic behaviour in liquid water

This article deals with the identification of solid-like properties measured at room temperature at a sub-millimetre length scale in liquid water. At a macroscopic scale, normal liquids (i.e. above melting temperature), and in particular water, are typically and empirically defined by the absence of shear elasticity, in contrast to solids or plastic fluids that require a stress threshold for flowing. A novel method optimizing the transmission of the shear stress to the sample enables a more complete probe of the mechanical response of liquids. It reveals that glass formers and viscous alkanes actually exhibit finite macroscopic shear elasticity away from any phase transition. This protocol is here applied for the first time to liquid water at room temperature, revealing, at the sub-millimetre scale, a low-frequency solid-like property.

Extended stress field analysis of surface strip footing on sand

Maksimovic's non-linear failure envelope and the consistent model of footing roughness provide the backdrop to the development of a comprehensive extension of the stress field method, which is applied to the analysis of surface footings on sand in plane strain. The note derives governing non-dimensional equations and discusses issues pertinent to the extended stress field method: modelling of the shear strength in the ultra-low stress range; internal lengths that produce scale-dependent bearing capacity factors; and the relation between slip-lines and characteristics. For constant friction angles the note produces scale-independent solutions that perfectly match already published results. For friction angles degrading with pressure, the note numerically quantifies and theoretically explains the scale dependence of the bearing capacity factor Nγ. The note shows why friction models with finite shear strength at ultra-low stresses produce Nγs ranging from Nγ(φB + ∆φ) for very small to Nγ(φB) for very large surface footings, and identifies the internal length parameter, γB/pN, as the controlling factor. Theoretical bearing capacity factors are shown to match well with experimentally obtained factors.

Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded cylindrical shells subjected to mechanical loadings. Eight types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded cylindrical shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.

Non-linear oscillatory rheological properties of a generic continuum foam model: Comparison with experiments and shear-banding predictions

The occurrence of shear bands in a complex fluid is generally understood as resulting from a structural evolution of the material under shear, which leads (from a theoretical perspective) to a non-monotonic stationary flow curve related to the coexistence of different states of the material under shear. In this paper we present a scenario for shear-banding in a particular class of complex fluids, namely foams and concentrated emulsions, which differs from other scenarios in two important ways. First, the appearance of shear bands is shown to be possible both without any intrinsic physical evolution of the material (e.g. via a parameter coupled to the flow such as concentration or entanglements) and without any finite critical shear rate below which the flow does not remain stationary and homogeneous. Secondly, the appearance of shear bands depends on the initial conditions, i.e. the preparation of the material. In other words, it is history dependent. This behaviour relies on the tensorial character of the underlying model (2D or 3D) and is triggered by an initially inhomogeneous strain distribution in the material. The shear rate displays a discontinuity at the band boundary whose amplitude is history dependent and thus depends on the sample preparation.

Deformation of olivine in torsion under hydrous conditions

We performed torsional deformation experiments on pre-hydrated fine-grained olivine aggregates using an innovative experimental assembly to investigate water weakening in mantle rocks at high shear strains. San Carlos olivine powder was cold-pressed and then hot-pressed under hydrous conditions, producing aggregates with average grain sizes of 7 or 15μm. Deformation experiments were performed in a high-resolution gas-medium apparatus equipped with a torsional actuator, under a confining pressure of 300MPa, a temperature of 1200°C, and constant shear strain rates ranging from 8×10−5 to 1.4×10−4s−1. Maximum shear stresses range from 150 to 195MPa. These values are 30% lower relative to those determined in previous torsion experiments on dry, fined-grained dunites under similar conditions. Textures and microstructures of the starting and deformed specimens were characterized by scanning and transmission electron microscopy. All deformed aggregates exhibit a shape-preferred orientation marking a foliation and lineation, as well as a reduction in mean grain size from 15μm down to 3–4μm due to dynamic recrystallization. Olivinecrystallographic fabrics developed rapidly (γ<0.1), but their strength, characterized by the J-index, is low compared to naturally deformed peridotites or to polycrystalline olivine deformed at similar finite shear strains under dry conditions. The crystallographic fabrics are consistent with deformation by a dislocation accommodated creep mechanism with activation of multiple {0kl}[100] systems, among which the (010)[100] slip system is dominant, and minor participation of the (010)[001] slip system. Transmission electron microscopy confirmed the occurrence of dislocations with [100] and [001] Burgers vectors in most grains. Analysis of unpolarized infrared spectra indicates that hydrogen concentration in the olivine lattice is below the saturation level of 18ppm wt H2O, which is similar to those typically observed in spinel-bearing peridotite xenoliths, and also provide evidence for water-rich inter-granular material trapped in pores and grain boundaries. Seismic properties computed from the CPO observations correspond to those most commonly observed in naturally deformed mantle peridotites with fast P-wave propagation and S-wave polarization subparallel to the shear direction. These torsion experiments on fine-grained olivine polycrystals under hydrous conditions indicate that water weakening under lithospheric conditions is linked to various defects with hydrogen in the olivine structure, as well as with water-derived species in grain boundaries or pores.

Crustal-scale transcurrent shearing in the Paleoproterozoic Sefwi-Sunyani-Comoé region, West Africa

The Paleoproterozoic Sefwi-Sunyani-Comoé region that straddles the Ghana-Ivory Coast border in West Africa has been characterised as resulting from a combination of compression and simple shear with leucogranite intrusion either being prior to the compression or synchronous with late shearing. The analysis of regional magnetic datasets combined with field observations allows us to better define the geometry of the major lithostratigraphic packages and their structural contacts in this region. This analysis reveals a series of elongate rounded leucogranite intrusions enveloped by deformed metasediments.Recent finite element modelling of two-phase aggregates has shown that we can analyse the geometry of these systems both in terms of their finite deformation and their mechanical contrast. We interpret the geometries we see in the Sefwi-Sunyani-Comoé region as reflecting the activity of a major crustal deformation zone which was dominated by simple shear. The comparison with our modelling suggests a finite shear strain of approximately 5 gamma, which in turn implies a lateral displacement of 400km parallel the Sefwi Greenstone Belt, which places Southern Ghana (EoGhana?) near eastern Burkina Faso prior to deformation. Our analysis also suggests that the leucogranites were already acting as more rigid bodies during the shearing, suggesting that their emplacement was predominantly pre-kinematic.

Mechanics of kink-bands during torsion deformation of muscovite aggregate

The growth, geometry and bulk mechanical behavior of grain-scale kink-bands were investigated during layer parallel non-coaxial deformation (torsion) of synthetic polycrystalline aggregates of muscovite at constant displacement rate (γ˙=3.10−4s−1) and various confining pressures (CP=50–300MPa), temperatures (T=300–800°C) and finite shear strains (γ=0.2–5.0). The kink-instabilities were observed during elastic to plastic transition (yield) at finite shear strain of ca. 0.2. The shear stress required to form the kinks is a function of pressure and temperature and range from ca. 20 to 90MPa within the experimental conditions. The kink bands formed in non-coaxial torsion tests are asymmetric, involve strain partitioning, show complex microstructures with increased deformation and have a limited lifetime as the structure is destroyed at γ=ca. 5. These results suggest a domain of pressure and temperature conditions where kink-band structures should be common. Results obtained from torsion experiments are compared to those of layer parallel axial compression conducted with the same displacement rate at 600°C and 300MPa for 5, 10 and 17% bulk shortening. Combining microstructures viewed on surfaces parallel and perpendicular to the torsion plane, a 3D geometrical model is constructed to discuss the kink mechanism, the microstructural evolution and the mechanical behavior.

Macro to Micro Stress and Strain Conversion in Porous Ceramics

In this work, we propose an analytical model, capable of distinguishing the contribution of porosity, pore morphology and solid domain properties to the macroscopic elasticity of a porous ceramic material. A practical method is shown for the evaluation of the dense material elastic properties in porous (and microcracked) polycrystalline materials, making use of in-situ neutron diffraction experiments. By this method, axial and transverse microstrains measurements can reveal the average values for Young’s modulus and Poisson’s ratio of the dense material, as well as the value of pore morphology at known porosity. The approach has been validated on porous SiC. Finite Element Modeling is shown to allow calculating the three-dimensional strain and stress state under applied uniaxial stress, highlighting that small but finite shear stresses arise. Stress-strain curves of porous and microcracked materials have been generated, which correlate qualitatively well with the measured properties and can be used for quantitative numerical simulation of the materials strength. Predictions of FEM coincide very well with analytical calculations, thus corroborating the validity of the analytical method proposed.

Quark polarization in a viscous quark-gluon plasma

Quarks produced in the early stage of noncentral heavy-ion collisions could develop a global spin polarization along the opposite direction of the reaction plane due to the spin-orbital coupling via parton interaction in a medium that has finite longitudinal flow shear along the direction of the impact parameter. We study how such polarization evolves via multiple scattering in a viscous quark-gluon plasma with an initial laminar flow. The final polarization is found to be sensitive to the viscosity and the initial shear of local longitudinal flow.

Transport bifurcation induced by sheared toroidal flow in tokamak plasmas

First-principles numerical simulations are used to describe a transport bifurcation in a differentially rotating tokamak plasma. Such a bifurcation is more probable in a region of zero magnetic shear than one of finite magnetic shear, because in the former case the component of the sheared toroidal flow that is perpendicular to the magnetic field has the strongest suppressing effect on the turbulence. In the zero-magnetic-shear regime, there are no growing linear eigenmodes at any finite value of flow shear. However, subcritical turbulence can be sustained, owing to the existence of modes, driven by the ion temperature gradient and the parallel velocity gradient, which grow transiently. Nonetheless, in a parameter space containing a wide range of temperature gradients and velocity shears, there is a sizeable window where all turbulence is suppressed. Combined with the relatively low transport of momentum by collisional (neoclassical) mechanisms, this produces the conditions for a bifurcation from low to high temperature and velocity gradients. A parametric model is constructed which accurately describes the combined effect of the temperature gradient and the flow gradient over a wide range of their values. Using this parametric model, it is shown that in the reduced-transport state, heat is transported almost neoclassically, while momentum transport is dominated by subcritical parallel-velocity-gradient-driven turbulence. It is further shown that for any given input of torque, there is an optimum input of heat which maximises the temperature gradient. The parametric model describes both the behaviour of the subcritical turbulence (which cannot be modelled by the quasi-linear methods used in current transport codes) and the complicated effect of the flow shear on the transport stiffness. It may prove useful for transport modelling of tokamaks with sheared flows.

Semi-brittle deformation of granitoid gouges in shear experiments at elevated pressures and temperatures

The physical and chemical processes acting in the ‘brittle-to-plastic’ transition are of great interest for a better understanding of fault rheology. We performed a series of experiments on granitoid gouge material under high confining pressures (Pc = 500–1500 MPa), temperatures (T = 300 °C and 500 °C) and fast shear strain rates (∼1.8 × 10−4 s−1) where the material deforms by semi-brittle flow. Samples deformed at 500 °C are systematically weaker than samples deformed at 300 °C over the whole examined confining pressure range indicating a non-frictional component of the deformation. All samples develop an S–C′ fabrics and deformation localizes in slip zones containing ‘amorphous’ feldspar material with an intermediate composition (Na, Ca and K-rich). Further, we observe changes in composition of feldspars (enrichment in the albite component) in the highly fragmented – but crystalline – regions with increasing finite shear strain. Our results indicate that mass-transfer processes keep pace with frictional deformation even at high strain rates and together with viscous flow of the ‘amorphous’ material are responsible for the observed strength difference.

The February 2010 large landslide at Maierato, Vibo Valentia, Southern Italy

A large landslide formed at Maierato (Vibo Valencia District), Southern Italy, on 15 February 2010, at 1430 hours local time, when rapid failure occurred after several days of preliminary movements. The landslide has an area of 0.3 km2, a runout distance of 1.2 km and an estimated volume of about 10 Mm3. The landslide caused nearly 2,300 inhabitants to be evacuated, with high economic losses. The most probable trigger of the landslide was the cumulative precipitation over the preceding 20 days (having a return period of more than 100 years), which followed a long period of 4–5 months of heavy rainfall (of about 150% of the average rainfall of the period). This report presents a summary of our findings pertinent to the landslide’s activities based on our field investigations. In particular, this report covers (1) details of land deformation caused by the landslide, (2) geology pertinent to landslide development, (3) identification of the landslide mechanism and its triggering factors based on the analysis of the boring core specimens and landform features, as well as the available video of the event, and (4) preliminary evaluation of the stability of the original slope before the landslide using the finite element-based shear strength reduction method. The aim of the paper was to describe the landslide and explain its mechanism of occurrence.

Development of a recrystallized grain size piezometer for hematite based on high-temperature torsion experiments

A series of hematite samples (>98 % Fe 2 O 3, Sishen Mine, South Africa) with different initial grain sizes of −5 s −1 and 4.7 × 10 −5 s −1 , resulting in finite shear strains between 0.4 and 4.7 and finite shear strengths in the range of 32–326 MPa. The data obtained at high shear strain (>3) fit to a power law creep equation with a pre-exponential constant of lnA = −5.03 ± 0.61 MPa −2.5 s −1 , an activation energy of Q = 249 ± 48 kJ mol −1 , and a stress exponent of n = 2.5 ± 0.5, indicating dislocation creep partially assisted by grain boundary sliding as the main deformation mechanism. Irrespective of the initial grain size, the final grain size ( D ) of fully recrystallized samples decreases continuously with increasing (steady state) equivalent stress (σ), yielding a piezometric relationship of the form: D = 1055 × σ 1.03 ± 0.10 . The piezometer is applied to itabiritic hematite ores of the Iron Quadrangle, Minas Gerais, Brazil, consisting mainly of layered hematite and quartz. Compared to hematite the measured recrystallized grain size of quartz is always larger, which is in agreement with the grain size range predicted by recrystallized grain size-based piezometers for hematite and quartz if deformed at similar stress below about 100 MPa.

Numerical homogenisation of an elasto-plastic model-material with large elastic strains: macroscopic yield surfaces and the Eulerian normality rule

This article presents the details of a numerical technique for computing the macroscopic response of a material with a given micro-structure to arbitrary prescribed loading histories. The method uses classical concepts of homogenisation theory in combination with the finite element method and focusses on the computation of macroscopic yield surfaces and inelastic strain rates. It places no restrictions on the magnitude of deformation and allows arbitrary combinations of stress- or strain control including the prescription of histories of the Cauchy stress. The method is illustrated by analysing a model material, consisting of a non-linearly elastic matrix with stiff elasto-plastic inclusions, which exhibits macroscopically associative elasto-plastic material behaviour with finite elastic strains. Yield surfaces and the directions of plastic flow after a prior finite simple shear deformation are computed for this material and are shown to be consistent with an additive decomposition of the Eulerian strain rate into elastic and plastic parts and a suitable formulation of the normality rule in Cauchy stress space; a novel version of the latter is derived which is valid for an arbitrary reference configuration.

Influence of Shear Stress on Current Carrying Capabilities of High Temperature Superconductor Tapes

To achieve high currents single HTSC tapes or Roeble strands must be combined into a cable. Many concepts like Rutherford cable, coated cable in conduit or full size cable model of Roebel strands require twisting of the HTSC tapes. Twisting or torsion exerts longitudinal and shear stress on the tapes which degrade the current carrying capabilities. Finite element analysis was done to explain this behavior. However, the influence of longitudinal stress in HTSC tapes on critical current is well documented, but there is no experimental data for pure shear stress available. To verify the finite element analysis shear stress experiments are of significant interest. With a three point bending method shear stress can be applied to a single HTSC tape. Critical current measurements with varying shear stress are conducted for several HTSC tapes. With the experimental results and the longitudinal stress dependence a correlation between torsion experiments and finite element analysis is possible.