Analytic calculation of the stresses in an ensiled granular mediumCalcul analytique des contraintes dans un matériau granulaire ensilé

We present in this Note a new analytic approach, of continuous medium type, which improves the Janssen theory and enables us to calculate the stresses in an ensiled granular medium. This approach is based on the two dimensional equilibrium equations, coupled with the Mohr–Coulomb criterion and a slip condition at the walls of the silo. An analytic resolution is developed to compute the stresses for cohesive and non cohesive materials in the whole silo. To cite this article: O. Millet et al., C. R. Mecanique 334 (2006).     

Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part II: Numerical bounds and assessment of the theoretical model

This second part of the two-part study is devoted to the numerical Limit Analysis of a hollow sphere model with a Mohr–Coulomb matrix and its use for the assessment of theoretical results. Brief background and fundamental of the static and kinematic approaches in the context of numerical limit analysis are first recalled. We then present the hollow sphere model, together with its axisymmetric FEM discretization and its mechanical position. A conic programming adaptation of a previous iterative static approach, based on a piecewise linearization (PWL) of the plasticity criterion, was first realized. Unfortunately, the resulting code, no more than the PWL one, did not allow sufficiently refined meshes for loss of convergence of the conic optimizer. This problem was solved by using the projection algorithm of Ben Tal and Nemriovski (BTN) and the (interior point) linear programming code XA. For the kinematic approach, a first conic adaptation appeared also inefficient. Then, an original mixed (but fully kinematic) approach dedicated to the general Mohr–Coulomb axisymmetric problem was elaborated. The final conic mixed code appears much more robust than the classic one when using the conic code MOSEK, allowing us to take into account refined numerical meshes. After a fine validation in the case of spherical cavities and isotropic loadings (for which the exact solution is known) and comparison to previous (partial) results, numerical lower and upper bounds (a posteriori verified) of the macroscopic strength are provided. These bounds are used to assess and validate the theoretical results of the companion (part I) paper. Effects of the friction angle as well as that of the porosity are illustrated.     

基于破坏概率的岩土试件剪切破坏角分析

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岩土体剪切破坏时剪切破坏角是一个一直在探讨的问题,传统理论解为Mohr-Coulomb解和Roscoe解,基于试件的宏观破坏现象受微观破坏概率的分布制约这一前提,从岩土体试件内部破坏概率分布函数的角度重新研究了这一问题。通过对试件内部微观破坏概率由微观到宏观的统计分析与坐标变换,发现在形式上破坏概率主轴即相当于主应力,并基此给出了破坏概率破坏莫尔圆。还对Mohr-Coulomb破坏准则与微观破坏概率统计之间的关系开展了探索分析,结果表明 Mohr-Coulomb准则是描述试件剪切破坏时大量微破坏行为共同作用的统计结果,Weibull分布仅是岩土试件内部破坏概率分布形式的一种描述,基于数学概率推导的岩土试件破坏模型能更好的与试件的破坏准则联系起来。

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Turbulent shear stress profiles in a bubbly channel flow assessed by particle tracking velocimetry

Particle tracking velocimetry (PTV) is applied to a bubbly two-phase turbulent flow in a horizontal channel at Re = 2 × 10⁴ to investigate the turbulent shear stress profile which had been altered by the presence of bubbles. Streamwise and vertical velocity components of liquid phase are obtained using a shallow focus imaging method under backlight photography. The size of bubbles injected through a porous plate in the channel ranged from 0.3 to 1.5 mm diameter, and the bubbles show a significant backward slip velocity relative to liquid flow. After bubbles and tracer particles are identified by binarizing the image, velocity of each phase and void fraction are profiled in a downstream region. The turbulent shear stress, which consists of three components in the bubbly two-phase flow, is computed by analysis of PTV data. The result shows that the fluctuation correlation between local void fraction and vertical liquid velocity provides a negative shear stress component which promotes frictional drag reduction in the bubbly two-phase layer. The paper also deals with the source of the negative shear stress considering bubble’s relative motion to liquid.     

Finite deformation and stability behaviour of cylindrical membranes subjected to symmetric line loads2

An ‘exact’ analysis of the complete non-linear load-deflection and stability behaviour of cylindrical membranes without end ‘shear walls’ subjected to longitudinal Symmetric line loads is presented. The analysis includes low as well as high profile structures. In order to determine the lateral stability behaviour, infinitesimal lateral displacements are superimposed on the symmetric finite deflection field.Results indicate that such structures may experience vertical and/or lateral instability depending on their initial geometry. Membranes with initial central angles, θ₀ ? 90° are stable, both vertically and laterally, for all load values. For 42.23° < θ₀ < 90° the structure is laterally stable but becomes vertically unstable at a certain ‘limit load’ W?_V while for θ₀< 42.23°, the structure becomes laterally unstable at a load value w?_l<W?_V.The analysis admits contact between vertical segments of the membrane to either side of the line load as well as between the membrane and the horizontal surface next to the supports.     

Hypoplasticity theory for granular materials—II: Three-dimensional axially symmetric exact solutions

In part I of this paper, we consider the governing equations of hypoplasticity theory for two-dimensional steady quasi-static plane strain compressible gravity flow and determine some exact analytical solutions applying for certain special cases. Similarly, for the three-dimensional situation considered here in part II, we undertake a similar mathematical investigation to determine some simple solutions of the governing equations for three-dimensional steady quasi-static axially symmetric compressible gravity flow for hypoplastic granular materials. We again find that for certain special cases, we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solutions is not a trivial matter, and that the solutions determined here might be exploited as benchmarks for full numerical schemes.     

A new analytical theory for earing generated from anisotropic plasticity

Commercial canmaking processes include drawing, redrawing and several ironing operations. It is experimentally observed that during the drawing and redrawing processes earing develops, but during the ironing processes earing is reduced. It is essential to understand the earing mechanism during drawing and ironing for an advanced material modeling. A new analytical approach that relates the earing profile to r-value and yield stress directionalities is presented in this work. The analytical formula is based on the exact integration of the logarithmic strain. The derivation is for a cylindrical cup under the plane stress condition based on rigid perfect plasticity while force equilibrium is not considered. The earing profile is obtained solely from anisotropic plastic properties in simple tension. The earing mechanism is explained from the present theory with explicit formulae. It has been proved that earing is the combination of the contributions from r-value and yield stress directionalities. From a directionality (y-axis) vs. angle from the rolling (x-axis) plot, the earing profile is generated to be a scaled mirror image of the r-value directionality with respect to 90° (x = 90) and also a scaled mirror image of the yield stress directionality with respect to the reference yield stress (y = 1). Three different materials (Al-5% Mg alloy, AA 2090-T3 and AA 3104 RPDT control coil) are considered for verification purposes. This approach provides a fundamental basis for understanding the earing mechanism. In practice, the present theory is also very useful for the prediction of the earing profile of a drawn and iron cup and its related convolute cut-edge design for an earless cup.     

Delaminations induced by constrained transverse cracking in symmetric composite laminates

Transverse ply cracking and its induced delaminations at the φ/90° interfaces in [. . . /φ_i/φ_m/90_n] _s laminates are theoretically investigated. Three cracked and delaminated model laminates, one five-layer model (FLM) laminate [S^L/φ_m/90_2n/φ_m/S^R] _T and two three-layer model (TLM) laminates I and II, [φ_m/90_2n/φ_m] _T and [S′^L/90_2n/S′^R] _T, are designed to examine constraining mechanisms of the constraining plies of the center 90°-ply group on transverse crack induced delaminations, where S^L, S^R, S′^L and S′^R are sublaminates [. . ./φ_i] _T, [φ_i/. . .] _T, [. . ./φ_i/φ_m] _T and [φ_m/φ_i/. . .] _T, respectively. A sublaminate-wise first-order shear laminate theory is used to analyze stress and strain fields in the three cracked and delaminated laminates loaded in tension. The extension stiffness reduction of the constrained 90°-plies and the strain energy release rate for a local delamination normalized by the square of the laminate strain are calculated as a function of delamination length and transverse crack spacing. The constraining effects of the immediate neighboring plies and the remote plies are identified by conducting comparisons between the three model laminates. It is seen for the examined laminates that the nearest neighboring ply group of the 90°-plies primarily affects the stiffness reduction and also the normalized strain energy release rate, whereas the influences of the remote constraining layers are negligible.     

Exact integration of the von Mises elastoplasticity model with combined linear isotropic-kinematic hardening

The main purpose of this work is to present two semi-analytical solutions for the von Mises elastoplasticity model governed by combined linear isotropic-kinematic hardening. The first solution (SOLε) corresponds to strain-driven problems with constant strain rate assumption, whereas the second one (SOLσ) is proposed for stress-driven problems using constant stress rate assumption. The formulas are derived within the small strain theory Besides the new analytical solutions, a new discretized integration scheme (AMε) based on the time-continuous SOLε is also presented and the corresponding algorithmically consistent tangent tensor is provided. A main advantage of the discretized stress updating algorithm is its accuracy; it renders the exact solution if constant strain rate is assumed during the strain increment, which is a commonly adopted assumption in the standard finite element calculations. The improved accuracy of the new method (AMε) compared with the well-known radial return method (RRM) is demonstrated by evaluating two simple examples characterized by generic nonlinear strain paths.     

Crack separation energy-rates for the DBCS model under biaxial modes of loading

A Modified version of the Dugdale-Bilby-Cottrell-Swinden (DBCS) model simulating the effect of plasticity at the tip of a crack in an infinite region was used by kfouri and rice (1978) to calculate the crack separation energy-rate G^Δ corresponding to a finite crack growth step Δa during plane strain mode I crack extension. The loading consisted of a remote uniaxial tension σp applied normally to the plane of the crack. Using Rice's path-independent integral J to characterize the applied load in the crack tip region, and assuming the length R of the crack tip plastic zone to be small compared with the length of the crack, an analytical expression was derived relating the ratios (J/G^Δ) and (2a/R) for small values of (2a/R). The analytical solution was incomplete in itself in that the value assumed in the plastic region of the DBCS model for the normal stress Y acting on the extending crack surfaces was the product of the yield stress in uniaxial tension σ_Y and an unknown parameter C, the value of which depends on the effect of the local hydrostatic stresses in the case of plane strain conditions. The analytical solution was compared with a numerical solution obtained from a plane strain elastic-plastic finite element analysis on a centre-cracked plate (CCP) of material obeying the von Mises yield criterion. The value used for the yield stress was 310 MN/m² and moderate isotropic linear hardening was applied with a tangent modulus of 4830 MN/m². A uniaxial tension σp was applied on the two appropriate sides of the plate. The comparisons showed that the analytical and finite element solutions were mutually consistent and they enabled the value of C to be established at 1.91. In the present paper similar comparisons are made between the analytical solution and the finite element solutions for the CCP of the same material under different biaxial modes of loading. By assuming further that the form of the analytical solution does not depend on the details of the geometry and of the loading at remote boundaries, a comparison has also been made with the results of a finite element analysis on a compact tension specimen (CTS) made of the same material as the CCP. The different values of C obtained in each case are consistent with investigations by other authors on the effect of load biaxiality on crack tip plasticity.     

A note on the Mohr-Coulomb and Drucker-Prager strength criteria

This paper presents different expressions of the Mohr-Coulomb (M-C) criterion as well as the interrelationships between them, which lays a foundation for the definition of the equivalent M-C friction angle φ_mc. The characteristics of four types of Drucker-Prager cones matched with the M-C surface are compared as the friction angle φ varies from 0° to 90°. The minimum and maximum value of φ for them is given and the influence of the intermediate principal stress σ₂ to the major principal stress σ₁ is demonstrated using their φ_mc.     

Entrance flow of a bingham fluid in a tube

The numerical solution of the entrance flow in a tube has been obtained for a Bingham fluid. The numerical procedure used is that of Patankar and Spalding [1]. The accuracy of the numerical results is demonstrated by comparing the fully-developed velocity profiles with analytical exact solutions. The results of the entrance flow in a tube for the case of a zero yield stress are compared with the entrance flow solution for a Newtonian fluid. Detailed results are presented for a wide range of yield numbers (=τ _y D/ūμ).     

Analytical and numerical study of double diffusive convection in a vertical enclosure

This paper presents an analytical and numerical study of natural convection of a double-diffusive fluid contained in a rectangular slot subject to uniform heat and mass fluxes along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number, Ra _T ; buoyancy ratio, N; Lewis number, Le; Prandtl number, Pr and aspect ratio of the cavity, A. In the first part of the analytical study a scale analysis is applied to the two extreme cases of heat-transfer and mass-transfer-driven flows. In the second part, an analytical solution, based on the parallel flow approximation, is reported for tall enclosures (A?1). Solutions for the flow fields, temperature and concentration distributions and Nusselt and Sherwood numbers are obtained in terms of the governing parameters of the problem. In the limits of heat-driven and solute-driven flows a good agreement is obtained between the prediction of the scale analysis and those of the analytical solution. The numerical solutions are based on the complete governing equations for two-dimensional flows, and cover the range 1≤Ra _T ≤10⁷, 0≤N≤10⁵, 10^-3≤Le≤10³, 1≤A≤20 and Pr=7. A good agreement is found between the analytical predictions and the numerical simulation.     

A microstructural elastoplastic model for unsaturated granular materials

The homogenization technique is used to obtain an elastoplastic stress–strain relationship for dry, saturated and unsaturated granular materials. Deformation of a representative volume of material is generated by mobilizing particle contacts in all orientations. In this way, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. For the non-saturated state, capillary forces at the grain contacts are added to the contact forces created by an external load. They are calculated as a function of the degree of saturation, depending on the grain size distribution and on the void ratio of the granular assembly. Numerical simulations show that the model is capable of reproducing the major trends of a partially saturated granular assembly under various stress and water content conditions. The model predictions are compared to experimental results on saturated and unsaturated samples of silty sands under undrained triaxial loading condition. This comparison shows that the model is able to account for the influence of capillary forces on the stress–strain response of the granular materials and therefore, to reproduce the overall mechanical behavior of unsaturated granular materials.     

Effect of porosity on the properties of strain localization in porous media under undrained conditions

Within the general framework of mixture theory and by introducing the fictitious “fluid phase” as a mixture of a liquid and a gas, the conditions for localization of deformation into a shear band in the incremental response of partially saturated and fully saturated elastic–plastic porous media under undrained conditions are derived. The effect of porosity is included in the derivation. The explicit analytical expressions of the direction of shear band initiation and the corresponding hardening modulus of the porous media for the plane strain case are deduced, and a parametric analysis is made of the influence of the porosity on the properties of strain localization based on Mohr–Coulomb yield criterion. It is found that the dependence of the shear banding properties of partially saturated porous media on the porosity is related to the stress states and Poisson's ratio. However, the properties of the strain localization for the fully saturated porous media are almost independent of Poisson's ratio. Finally, on the basis of Mohr–Coulomb yield criterion, some solutions of the shear banding orientation for water-saturated granular materials are obtained, which are proved to be in good agreement with the experimental results reported by other researchers.     

Strong Uniqueness Theorems and the Phragmèn–Lindelöf Principle in Nonhomogeneous Elastostatics

We find sharp uniqueness classes for the contact problem of nonhomogeneous linear elastostatics in exterior domains. Moreover, we prove that a solution of the displacement problem in a cone of aperture φ enjoys the asymptotic properties expressed by the classical Phragmèn–Lindelöf principle; in particular, uniqueness holds in the class of all functions that are bounded at large distance by r ^{1/c φ} for some positive c. Then, by means of a counter–example we observe that the same properties are no longer satisfied by the solutions of the contact problem.     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

On the characteristic equations of elasto-plastic flow

Three dimensional characteristic surfaces (slip surfaces) of elasto-plastic Navier's equations and the criteria for their existence are discussed, and the solutions are also applied to two dimensional cases. By making use of isotropic yield function, the following results are proved. If, and only if the plastic/elastic moduli ratio is zero and Det$\text{(φ}{}_{\mathrm{ij}}\text{)=0,(φ}{}_{\mathrm{ij}}\text{=}\text{?φ}\text{?σ}{}_{\mathrm{ij}}$, φ: yield function,σ_ij: stress tensor), characteristic surfaces exist. There are two and only two characteristic surface elements at each point, and they are identical with the surfaces of maximum shearing stress.     

Crack nucleation near stress concentrators in quasi-brittle materials

The results of combined tension (compression) and shear experiments with plexiglass specimens are used to construct a Coulomb-Mohr-type strength curve. Fracture experiments were performed with plexiglass square plates with internal through cuts. The specimens were subjected to compression on a Zwick/Roell testing machine until cracks appeared. In the process of testing, nucleation of symmetric cracks was observed, which propagated as the load was further increased. The fracture character (tensile or shear) could not be decided until a subsequent numerical analysis of the plate stress-strain state was performed by the finite elementmethod. The stress concentration loci were found to coincide with the crack nucleation loci. In the plane (σ, τ ), Mohr’s circles were constructed for the stress states at the stress concentration points. If the point at which Mohr’s curve touches the limit curve is known, then one can determine the plane on which the normal and tangential stresses attain critical values, and hence one can determine the crack propagation direction. The experimental results and numerical solutions were found to be in good agreement.