Yield criterion and elasto-plastic damage constitutive model for frozen sandy soil

A series of triaxial compression tests was carried out on a frozen sandy soil under confining pressures of 0–18 MPa at −6 °C. The experimental results indicate that, the strength of frozen sandy soil increases versus the increase in the confining pressures when σ₃ ? 3 MPa, but decreases when σ₃ > 3 MPa. This phenomenon is called the strengthening and weakening effects of confining pressures. A yield function, considering both effects, is proposed using the experimental method according to Drucker’s postulate, and the mathematical expression of the hardening parameter, which can describe the softening and hardening phenomenon, is provided. An elasto-plastic constitutive model for frozen sandy soil is developed. Based on the continuum damage theory, the cross anisotropic damage variables are deduced and their change regularities are investigated. Then the elasto-plastic damage constitutive model is proposed by introducing damage variables into elasto-plastic constitutive model. The validity of the model is verified by comparing its modeling results with experimental results obtained from triaxial tests. It is found that, this model can predict the deformation regularity of frozen soil exactly. It can simulate the stress–strain process under high confining pressures when pressure melting phenomena appear especially well.     

Non-coaxiality of strain increment and stress directions in cross-anisotropic sand

An experimental program was carried out in a recently developed torsion shear apparatus to study the non-coaxiality of strain increment and stress directions in cross-anisotropic deposits of Fine Nevada sand. Forty-four drained torsion shear tests were performed at constant mean confining stress, σ_m, constant intermediate principal stress ratios, as indicated by b = (σ₂ − σ₃)/(σ₁ − σ₃), and constant principal stress directions, α. The experiments were performed on large hollow cylinder specimens deposited by dry pluviation and tested in an automated torsion shear apparatus. The specimens had height of 40 cm, and average diameter of 20 cm, and wall thickness of 2 cm. The stress–strain behavior of Fine Nevada sand is presented for discrete combinations of constant principal stress direction, α, and intermediate principal stress. The effects of these two variables on the non-coaxiality are presented. The experiments show that the directions of the strain increments do not in general coincide with the directions of stresses, and there is a switch from one to the other side between the two quantities.     

A micromechanics-based model for sand-silt mixtures

Experimental observations have shown the significant impact of fines or coarse grains on the behavior of sand-silt mixtures. To describe the behavior of sand-silt mixtures under both drained and undrained conditions, this paper presents a mathematical model based on a micromechanical approach. The novelty of this model is the introduction of the equivalent mean size and the evolution of the position of the critical state line with fines content for various sand-silt mixtures. The predictive capability of the model was evaluated by comparing the model simulations with experimental results on undrained triaxial tests of Foundry sand-silt mixtures with fines content, f_c = 0–100% and Ottawa sand-silt mixtures with fines content f_c = 0–50%, and on drained triaxial tests of Hong Kong Completely Decomposed Granite (HK-CDG) mixtures before and after erosion. The predicted local behavior in the contact planes has also been examined. It shows that all local contact planes are mobilized to different degrees in terms of local stress and strain and that a few active contact planes contribute dominantly to the deformation of the assembly, leading to an anisotropic global behavior when the soil is subjected to external loading.     

Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10⁻⁶ is used for Al₂O₃ fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0₄], [90₄], and [0/90]₂ fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90]₂ laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of (σ, τ) = (500, 70) MPa, the center of the yield locus translated to (σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.     

On the particle inertia-free collision with a partially contaminated spherical bubble

The collision between a contaminated spherical bubble and fine particles in suspension is considered for r_p/r_b ? 1 (r_p being the radius of the particles in suspension and r_b the radius of the bubble). The collision probability or efficiency is defined as the number of particles colliding the bubble surface to the number of particles initially present in the volume swept out by the bubble. In this note we show that the collision probability can be expressed as P_c(r_p/r_b,Re) = g(r_p/r_b)f(Re) for both mobile and immobile interfaces. For partially contaminated bubbles a linear or quadratic dependency in r_p/r_b is found depending on the level of contamination and the value of r_p/r_b. These behaviors are given by the flux of particles near the surface which is controlled by the tangential velocity for mobile interfaces and by the velocity gradient for immobile interfaces. The threshold value (r_p/r_b)_th between the r_p/r_b and (r_p/r_b)² evolution is shown to vary as sin^n(Re)(θ_clean/n(Re))sin(3θ_clean/4), θ_clean being the angle describing the front clean part of the bubble and n(Re) varying from n = 2 to n = 1 from small to large Reynolds number.     

Two-phase frictional pressure drop multipliers for SUVA R-134a flowing in a rectangular duct

The adiabatic two-phase frictional multipliers for SUVA, R-134a flowing in a rectangular duct (with D_H = 4.8 mm) have been measured for three nominal system pressures (0.9 MPa, T_sat = 35.5 °C; 1.38 MPa, T_sat = 51.8 °C; and 2.41 MPa, T_sat = 75.9 °C) and three nominal mass fluxes (510, 1020 and 2040 kg/m²/s). The data is compared with several classical correlations to assess their predictive capabilities. The Lockhart–Martinelli model gives reasonable results at the lowest pressure and mass flux, near the operating range of most refrigeration systems, but gives increasingly poor comparisons as the pressure and mass flux are increased. The Chisholm B-coefficient model is found to best predict the data over the entire range of test conditions; however, there is significant disagreement at the highest pressure tested (with the model over predicting the data upwards of 100% for some cases). The data shows an increased tendency toward homogeneous flow as the pressure and flow rate are increased, and in fact the homogeneous model best predicts the bulk of the data at the highest pressure tested.     

The interlayer shear effect on graphene multilayer resonators

Graphene nanostrips with single or few layers can be used as bending resonators with extremely high sensitivity to environmental changes. In this paper we report molecular dynamics (MD) simulation results on the fundamental and secondary resonant frequencies f of cantilever graphene nanostrips with different layer number n and different nanostrip length L. The results deviate significantly from the prediction of not only the Euler-Bernoulli beam theory (f∝nL⁻²), but also the Timoshenko's model. Since graphene nanostrips have extremely high intralayer Young's modulus and ultralow interlayer shear modulus, we propose a multibeam shear model (MBSM) that neglects the intralayer stretch but accounts for the interlayer shear. The MBSM prediction of the fundamental and secondary resonant frequencies f can be well expressed in the form f−f_mono∝[(n-1)/n]^bL^−2(1−b), where f_mono denotes the corresponding resonant frequency as the layer number is 1, with b=0.61 and 0.77 for the fundamental and secondary resonant modes. Without any additional parameters fitting, the prediction from MBSM agrees excellently with the MD simulation results. The model is thus of importance for designing multilayer graphene nanostrips based applications, such as resonators, sensors and actuators, where interlayer shear has apparent impacts on the mechanical deformation, vibration and energy dissipation processes therein.     

Determination of thermal conductivity and interfacial energy of solid Zn solution in the Zn-Al-Bi eutectic system

The equilibrated grain boundary groove shapes for solid Zn solution (Zn-3.0 at.% Al-0.3 at.% Bi) in equilibrium with the Zn-Al-Bi eutectic liquid (Zn-12.7 at.% Al-1.6 at.% Bi) have been observed from quenched sample with a radial heat flow apparatus. Gibbs-Thomson coefficient, solid-liquid interfacial energy and grain boundary energy for solid Zn solution in equilibrium with Al-Bi-Zn eutectic liquid have been determined to be (5.1 ± 0.4) × 10⁻⁸ K m, (80.1 ± 9.6) × 10⁻³ and (158.6 ± 20.6) × 10⁻³ J m⁻² from the observed grain boundary groove shapes, respectively. The thermal conductivity variation with temperature for solid Zn solution has been measure with radial heat flow apparatus and the value of thermal conductivity for solid Zn solution has been determined to be 135.68 W/km at the eutectic melting temperature. The thermal conductivity ratio of equilibrated eutectic liquid to solid Zn solution, R = K_L(Zn)/K_S(Zn) has also been measured to be 0.85 with Bridgman type solidification apparatus.     

Subcooled flow boiling of R-134a in vertical channels of small diameter

Subcooled flow boiling heat transfer for refrigerant R-134a in vertical cylindrical tubes with 0.83, 1.22 and 1.70 mm internal diameter was experimentally investigated. The effects of the heat flux, q″ = 1–26 kW/m², mass flux, G = 300–700 kg/m² s, inlet subcooling, ΔT_sub,i = 5–15 °C, system pressure, P = 7.70–10.17 bar, and channel diameter, D, on the subcooled boiling heat transfer were explored in detail. The results are presented in the form of boiling curves and heat transfer coefficients. The boiling curves evidenced the existence of hysteresis when increasing the heat flux until the onset of nucleate boiling, ONB. The wall superheat at ONB was found to be essentially higher than that predicted with correlations for larger tubes. An increase of the mass flux leads, for early subcooled boiling, to an increase in the heat transfer coefficient. However, for fully developed subcooled boiling, increases of the mass flux only result in a slight improvement of the heat transfer. Higher inlet subcooling, higher system pressure and smaller channel diameter lead to better boiling heat transfer. Experimental heat transfer coefficients are compared to predictions from classical correlations available in the literature. None of them predicts the experimental data for all tested conditions.     

Grain size,strain rate,and temperature dependence of flow stress in ultra-fine grained and nanocrystalline Cu and Al: Synthesis,experiment, and constitutive modeling

For the first time, high quality bulk nanocrystalline (nc) fcc metals, with least amounts of imperfections, exhibiting high strength and ductility at room and different temperatures, under quasi-static and dynamic types of loading, were prepared and a comprehensive study on their post-yield mechanical properties was performed. This investigation included study of the effect of temperature on stress–strain responses of mechanically milled bulk nc Cu and Al. The samples after preparation through mechanical milling and consolidation processes were subjected to uniaxial compressive loading at quasi-static and dynamic strain rates of 10⁻² s⁻¹ and 1840–3105 s⁻¹, respectively, at temperatures ranging from 223 to 523 K. In both materials strong dependency of flow stress to temperature was observed; this dependency was rather more pronounced when the materials were tested at the quasi-static strain rate. Further, a new grain size and temperature dependent viscoplastic phenomenological constitutive equation, Khan–Liang–Farrokh (KLF) model was developed based on the Khan–Huang–Liang (KHL) constitutive equation. The model was featured to correlate different characteristic behaviors of polycrystalline materials in the plastic regime, as the result of grain refinement. In addition, the viscoplastic responses of bulk Cu and Al of different grain sizes (from sub-micron to nanometer range), and those from bulk nc Cu and Al at different strain rates (quasi-static to dynamic), recently published (21 and 22), were simulated using the newly developed equation. The results confirmed reasonable capability of the developed model to correlate a wide spectrum of the viscoplastic responses of these fcc metals.     

Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

The mechanical and thermal characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing power-law velocities U _w∝x ^m , m<?1 are considered. Comparing to the well studied cases of the increasing stretching velocities (m>0) several new features of basic significance have been found. Thus: (i) for m<?1 the boundary layer equations admit self-similar solutions only if a lateral suction is applied; (ii) the dimensionless suction velocity f _w<0 must be strong enough, i.e. f _w<f _w,max(m) where f _w,max(m) depends on m so that its absolute maximum max (f _w,max(m))=?2.279 is reached for m→?∞, while for m→?1, f _w,max(m)→?∞; (iii) the case {m→?∞, f _w,max(m)=?2.279} of the flow boundary value problem is isomorphic to the stretching problems with exponentially decreasing velocities U _w∝e ^ax with arbitrary a<0; (iv) for any fixed m<?1 and f _w<f _w,max(m) the flow problem admits a non-denumerable infinity of multiple solutions corresponding to the values of the dimensionless skin friction f ^″(0)≡s belonging to a finite interval s∈ [s _min(f _w,m), s _max(f _w,m)]; (v) the solution is only unique for f _w=f _w,max(m) where s=s _min(f _w,m)= s _max(f _w,m) holds; (vi) to every one of the multiple solutions of the flow problem there corresponds a unique solution of the heat transfer problem with a wall temperature distribution T _w?T _∞∝x ⁿ and a well defined and distinct value of the dimensionless wall temperature gradient ?^′(0), except for the cases n=(|m|?1)/2 where ?^′(0) has the same value ?^′(0)=Pr·f _w for the whole class of flow solutions with s∈[s _min(f _w,m), s _max(f _w,m)]; (vii) for f _w→?∞ one obtains the `asymptotic suction profiles' corresponding to s=s _min(f _w,m)?f _w and ?^′(0)?Pr·f _w in an explicit analytic form. The paper includes several examples which illustrate the dependence of the heat and fluid flows induced by surfaces stretching with rapidly decreasing velocities on the physical parameters f _w, m, n and Pr.     

Flow patterns of n-hexadecane–CO2 liquid–liquid two-phase flow in vertical pipes under high pressure

To promote a better understanding of liquid–liquid two-phase flow behavior, particularly under high pressure, flow patterns of n-hexadecane–CO₂ liquid–liquid two-phase upward flow in vertical stainless steel pipes were experimentally investigated. Observations were made in two 0.0015 m I.D. pipes of different lengths (0.068 m and 0.5 m) under high pressure varying from 10.3 to 29.6 MPa using a high pressure visualization system. The total flow rate was fixed at 2.0 × 10⁻⁶ m³/min, while the flow rate ratio (φ) varied from 0.05 to 19. Bubbly flow, plug flow, slug flow, annular flow, and near-one-phase flow regions were found in both pipes, while stratified flow was observed only in the 0.068 m pipe. Flow pattern maps were constructed in the flow rate ratio versus pressure graph, which demonstrates significant impacts of flow rate ratio, pipe length, and pressure on flow patterns. These impacts are discussed in detail. To the authors’ best knowledge, this work is the first attempt to observe complex liquid–liquid two-phase flow behavior with flow pattern transitions under high pressure, and contributes to a better understanding of liquid–liquid two-phase flow behavior.     

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n < ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.     

Three phase liquid–liquid–gas flows in 5.6 mm and 7 mm inner diameter pipes

Three phase liquid–liquid–gas flow maps in pipes of medium inner diameters (5.6 mm and 7 mm), are presented. A low viscosity paraffin oil (4.5 × 10⁻³ Pa s viscosity and 818.5 kg m⁻³ density at 20 °C), deionised water and air are flowing concurrently in Schott Duran^® glass pipes. A decreasing pipe diameter changes the flow pattern maps and also the behavior of the transition boundaries. Flow patterns are determined by high speed photography. To illuminate the pipe, laser induced fluorescence (LIF) is applied. The laser sheet is cutting through the axial vertical plane of the pipe. The laser light excites a fluorescent dye (uranine) in the water phase to separate the phases optically. The resulting flow maps are compared with literature data and a theoretical model.     

Regularity of the Inverse of a Planar Sobolev Homeomorphism

Let be a domain. Suppose that f ∈ W^1,1_loc(Ω,R²) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J_f. We show that f^-1 ∈ W^1,1_loc(f(Ω),R²) and that Df⁻¹(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f⁻¹ ∈ W^1,q(f(Ω),R²) for some 1<q≤2 are also given.     

The compressive failure of aluminum nitride considered as a model advanced ceramic

Uniaxial quasi-static compression, uniaxial dynamic compression and confined dynamic compression experiments were performed to characterize the failure of Aluminum Nitride (AlN) using a servo hydraulic machine and a modified Kolsky bar set-up respectively. High-speed digital cameras are used to visualize the failure processes. A summary of the available experimental results, including that in the literature, shows that the compressive strength of the AlN is sensitive to strain rate in the range from 10⁻³ to 10³ s⁻¹, and that the deviatoric strength of AlN is linearly dependent on pressure at low pressures and nearly independent of pressure above a transitional pressure (about 2 GPa). TEM characterization of fragments obtained after dynamic loading is used to characterize the deformation mechanisms in the AlN for varying confinement. The transition in the pressure dependent behavior is shown to be the result of a change of deformation mechanism. Classical wing crack micromechanics is used to predict the transition in the deformation mechanism, and to explain the observed behavior at low pressure.     

Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres

Gas–solid momentum transfer is a fundamental problem that is characterized by the dependence of normalized average fluid–particle force F on solid volume fraction ? and the Reynolds number based on the mean slip velocity Re_m. In this work we report particle-resolved direct numerical simulation (DNS) results of interphase momentum transfer in flow past fixed random assemblies of monodisperse spheres with finite fluid inertia using a continuum Navier–Stokes solver. This solver is based on a new formulation we refer to as the Particle-resolved Uncontaminated-fluid Reconcilable Immersed Boundary Method (PUReIBM). The principal advantage of this formulation is that the fluid stress at the particle surface is calculated directly from the flow solution (velocity and pressure fields), which when integrated over the surfaces of all particles yields the average fluid–particle force. We demonstrate that PUReIBM is a consistent numerical method to study gas–solid flow because it results in a force density on particle surfaces that is reconcilable with the averaged two-fluid theory. The numerical convergence and accuracy of PUReIBM are established through a comprehensive suite of validation tests. The normalized average fluid–particle force F is obtained as a function of solid volume fraction ? (0.1 ? ? ? 0.5) and mean flow Reynolds number Re_m (0.01 ? Re_m ? 300) for random assemblies of monodisperse spheres. These results extend previously reported results of  and  to a wider range of ?, Re_m, and are more accurate than those reported by Beetstra et al. (2007). Differences between the drag values obtained from PUReIBM and the drag correlation of Beetstra et al. (2007) are as high as 30% for Re_m in the range 100–300. We take advantage of PUReIBM’s ability to directly calculate the relative contributions of pressure and viscous stress to the total fluid–particle force, which is useful in developing drag correlations. Using a scaling argument, Hill et al. (2001b) proposed that the viscous contribution is independent of Re_m but the pressure contribution is linear in Re_m (for Re_m > 50). However, from PUReIBM simulations we find that the viscous contribution is not independent of the mean flow Reynolds number, although the pressure contribution does indeed vary linearly with Re_m in accord with the analysis of Hill et al. (2001b). An improved correlation for F in terms of ? and Re_m is proposed that corrects the existing correlations in Re_m range 100–300. Since this drag correlation has been inferred from simulations of fixed particle assemblies, it does not include the effect of mobility of the particles. However, the fixed-bed simulation approach is a good approximation for high Stokes number particles, which are encountered in most gas–solid flows. This improved drag correlation can be used in CFD simulations of fluidized beds that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior.     

Falling film flow along steep two-dimensional topography: The effect of inertia

Steady film flow along a vertical wall with isolated step changes is studied numerically for Reynolds numbers Re ∼ O(10⁻³–10²) and capillary numbers Ca ∼ O(10⁻²–10¹). The lengthscale of free surface capillary features upstream of a step-in or step-out decreases uniformly with Re and switches from a −1/3 to a −1/2 power-law dependence on Ca. The height of the capillary features first grows with Re, but eventually diminishes when inertia forces overpower capillary forces. Simultaneously, the key dynamics move from upstream to downstream of the step, and switch from capillary arrest to inertial re-directioning of the falling liquid. The latter mechanism involves a low-pressure region originating from the edge of the step. At a step-out, a new free surface feature appears with increasing Re, which is caused by liquid overshoot in the horizontal direction and is restrained initially by capillary and subsequently by inertial forces. Simple scaling arguments are shown to predict many of the above characteristics.     

The comparison model method: A new arithmetic approach to the discrete inverse problem of groundwater hydrology

Let the steady-state pressure z(·) of a fluid in a one-dimensional domain be governed by the equation d _x (a d _x z) = f subject to Dirichlet boundary conditions. We consider the identification of the transmissivity a (·), given f(·), and measured pressure z(·) by the comparison model method, a direct method which has been known and applied for some time but lacked theoretical background. With reference to a domain in one spatial dimension, we examine both the infinite-(‘continuous’) and finite-(discrete) dimensional cases. In the former, the method is based on the solution p(·) of an auxiliary flow equation, where f(·) and the two-point boundary conditions are unchanged with respect to the original or z(·) equation, whereas a tentative constant value b is assigned to the auxiliary transmissivity. The ratio of the first derivatives of p(·) and z(·) multiplied by b yields a solution ã(·) to the inverse problem. We examine in detail the nonuniqueness of ã(·) as a function of b, some cases where existence implies uniqueness, the role of positivity constraints, and a special feature: self-identifiability. We then translate all available results into the discrete case, where the good unknowns for the inverse problem are the internode coefficients. Several algebraic and numerical examples are presented.     

A new unified constitutive model with short- and long-range back stress for lead-free solders of Sn–3Ag–0.5Cu and Sn–0.7Cu

A series of tensile tests of Sn–3Ag–0.5Cu and Sn–0.7Cu lead-free solders were investigated at various strain rates from 1 × 10⁻⁴ s⁻¹ to 1 × 10⁻² s⁻¹ and over a wide temperature range from 25 ^oC to 150 ^oC. Two-step strain rate jump tests, three-step short term creep tests with stress jump, and uniaxial ratcheting tests were also conducted. Based on the test data, a new constitutive model was proposed with a simple formulation and only eight material constants which can be easily obtained. The model employs two carefully defined back stress components to simulate the loading/unloading asymmetry phenomenon in uniaxial ratcheting tests. Different evolution rules of short-range back stress were given for loading and unloading stage, which provides the model ability to simulate the asymmetry in hysteresis loops. The proposed model presents good simulation of uniaxial tensile tests, strain rate jump tests, short term creep tests with stress jump, and uniaxial ratcheting tests.