Limit equilibrium analysis and the minimum thickness of circular masonry arches to withstand lateral inertial loading

In this paper, we compute the location of the imminent hinges and the minimum thickness, t, of a circular masonry arch with mid-thickness radius, R, and embracing angle, β, which can just sustain its own weight together with a given level of a horizontal ground acceleration, ε g. Motivated from the recent growing interest in identifying the limit equilibrium states of historic structures in earthquake prone areas, this paper shows that the value of the minimum horizontal acceleration that is needed to convert an arch with slenderness (t/R, β) into a four-hinge mechanism depends on the direction of the rupture at the imminent hinge locations. This result is obtained with a variational formulation and the application of the principle of stationary potential energy, and it is shown that a circular arch becomes a mechanism with vertical ruptures when subjected to a horizontal ground acceleration that is slightly lower than the horizontal acceleration needed to create a mechanism with radial ruptures. The paper explains that the multiplicity on the solution for the minimum uplift acceleration is a direct consequence of the multiple possible ways that a masonry arch with finite thickness may rupture at a given location. The paper further confirms that the results obtained with commercially available distinct element software are in very good agreement with the rigorous solution.     

Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

The role of shape and relative submergence on the structure of wakes of low-aspect-ratio wall-mounted bodies

The effects of shape and relative submergence (the ratio of flow depth to obstacle height, d/H) are investigated on the wakes around four different low-aspect-ratio wall-mounted obstacles at Re _H  = 17,800: semi-ellipsoids with the major axes of the base ellipses aligned in the streamwise and transverse directions, and two cylinders with aspect ratios matching the ellipsoids (H/D = 0.89 and 0.67, where D is the maximum transverse dimension). Particle Image Velocimetry was used to interrogate the flow. Streamwise features observed in the mean wake include counter-rotating distributions of vorticity inducing downwash (tip structures), upwash (base structures), and horseshoe vortices. In particular, the relatively subtle change in geometry produced by the rotation of the ellipsoid from the streamwise to the transverse orientation results in a striking modification of the mean streamwise vorticity distribution in the wake. Tip structures are dominant in the former case, while base structures are dominant in the latter. A vortex skeleton model of the wake is proposed in which arch vortex structures, shed from the obstacle, are deformed by the competing mechanisms of Biot-Savart self-induction and the external shear flow. The selection of tip or base structures in the ellipsoid wakes is caused by tilting of the arch structures either upstream or downstream, respectively, which is governed by ellipsoid curvature. An inverse relationship was observed between the relative submergence and the strength of the base structures for the ellipsoids, with a dominant base structure observed for d/H = 1 in both cases. These results demonstrate a means by which to achieve significant modifications to flow structure and thereby also to transport mechanisms in the flow. Therefore, this work provides insight into the modeling and control of flow over wall-mounted bodies.     

Natural Convection as an Asymmetrical Factor of the Transport Through Porous Membrane

For nonionic substances, which density of solution depends on its concentration, concentration polarization of the membrane in horizontal plane depends not only on diffusion but on the hydrodynamic instabilities at the membrane surfaces also. Such instabilities are the cause of asymmetry of membrane transport in gravitational field. On the basis of results of glucose transport through the Nephrophan membrane in horizontal plane we can state that this asymmetry was observed for the cases with concentration Rayleigh number greater than critical value (R _C )_crit = 1709.3. The mathematical model based on Kedem–Katchalsky equations and Rayleigh number was presented. On the basis of this model and the dependence of volume flux through the Nephrophan membrane as a function of glucose concentration in the upper (configuration B) and lower (configuration A) chamber of the membrane system, the dependencies of thickness of concentration boundary layer, Rayleigh number, and introduced coefficient of asymmetry as a function of glucose concentration were presented for both configurations. These dependencies show that asymmetry of the membrane transport is observed for glucose concentration higher than 0.015 mol l^?1.     

A minimization problem related to the estimation of a lower bound for the temperature in a free boundary value problem related to the combustion of a solid

This paper investigates the least time τ_* of the first zero of the bounded solution to an initial boundary value problem for the heat equation. The heat equation is considered in the domain $$\left\{ {(x,t)| - \infty< x< s(t),0< t \leqslant T} \right\}$$ . The initial conditionu(x, 0)=φ(x) and the boundary conditionu _x (s(t),t)=?R are specified. Let τ=τ(φ,R, s) denote the first zero ofu onx=s(t), that is,u(s(τ), τ)=0. Let τ_*=min τ, where the minimum is taken over a class of functionss=s(t). The existence of τ_* is demonstrated, and a generalization of the problem is discussed.     

Self-similar collapse of 2D and 3D vortex filament models

In this article, a very simple toy model for a candidate blow-up solution of the Euler equation by Boratav and Pelz (vortex dodecapole) is investigated. In this model, vortex tubes are replaced with straight vortex filaments of infinitesimal thickness, and the entire motion is monitored by tracing the motion of a representative point on one vortex filament. It is demonstrated that this model permits a self-similar collapse solution which provides the time dependence of the length scale as (t _c ? t)^1/2, (t < t _c), where the collapse time t _c depends on the initial configuration. From the conservation of circulation, this time dependence implies that vorticity ω scales as (t _c ? t) ^?1, which agrees with the one observed in the direct numerical (pseudo spectral) simulations of the vortex dodecapole. Finally, possible modification of the model is considered.     

On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System

In this paper we extend and complement the results in Chiodaroli et al. (Global ill-posedness of the isentropic system of gas dynamics, 2014) on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law p(ρ) = ρ ^γ , γ ≥ 1. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi (Ann Math 170:1417–1436, 2009), (Arch Ration Mech Anal 195:225–260, 2010). Moreover we prove that for some of these Riemann problems and for 1 ≤ γ < 3 such solutions have a greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in (J Diff Equ 14:202–212, 1973) does not favour the classical self-similar solutions.     

Influence of aspect ratio on the flow above the free end of a surface-mounted finite cylinder

The flow above the free end of a surface-mounted finite-height cylinder was studied in a low-speed wind tunnel using particle image velocimetry (PIV). Velocity measurements were made in vertical and horizontal measurement planes above the free end of finite cylinders of aspect ratios AR = 9, 7, 5 and 3, at a Reynolds number of Re = 4.2 × 10⁴. The relative thickness of the boundary layer on the ground plane was δ/D = 1.7. Flow separating from the leading edge formed a prominent recirculation zone on the free-end surface. The legs of the mean arch vortex contained within the recirculation zone terminate on the free-end surface on either side of the centreline. Separated flow from the leading edge attaches onto the upper surface of the cylinder along a prominent attachment line. Local separation downstream of the leading edge is also induced by the reverse flow and arch vortex circulation within the recirculation zone. As the cylinder aspect ratio is lowered from AR = 9 to AR = 3, the thickness of the recirculation zone increases, the arch vortex centre moves downstream and higher above the free-end surface, the attachment position moves downstream, and the termination points of the arch vortex move upstream. A lowering of the aspect ratio therefore results in accentuated curvature of the arch vortex line. Changes in aspect ratio also influence the vorticity generation in the near-wake region and the shape of the attachment line.     

Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous mediaNon unicité de solutions des équations différentielles pour un problème de couches limites en milieux poreux

The free convection, along a vertical flat plate embedded in a porous medium, can be described in terms of solutions to $\text{f?+}\text{α+1}\text{2}\text{ff″?αf′}{}^2\text{=0,}$ for all t∈(0,+∞). The purpose of this Note is to study the nonuniqueness of solutions to this problem, with the initial conditions, $\text{f(0)=a∈}\text{R}$ and f′(0)∈{0,1}, where $\text{α∈(?}\text{1}\text{3}\text{,0).}$ No assumption at infinity is imposed. We show that this problem has an infinite number of unbounded global solutions. Moreover, we prove that the first and the second derivative of solutions tend to 0 as t approaches infinity. To cite this article: M. Guedda, C. R. Mecanique 330 (2002) 279–283.     

A New Approach to Equations with Memory

We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function ${G : \mathbb{R}^+ \to \mathbb{R}^+}We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G : \mathbbR⁺ ? \mathbbR⁺{G : \mathbb{R}^+ \to \mathbb{R}^+} such that

$G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0      11. The Minimal Thickness of a Semicircular Arch with a Tension-Free Crown∗      Piero Villaggio 《基于设计的结构力学与机械力学》2013,41(4):515-527 ABSTRACT When a semicircular elastic arch is loaded with a single point force at the vertex, the stress at interior points depends on the thickness of the arch and on the way the abutments react. Applying the theory of plane elasticity, three typical load-bearing abutment constraints are evaluated to find the minimal thickness at which no tensile stress occurs at a given interior point. Such thickness is found to be quite large.      12. Flow visualization of supersonic laminar flow over a backward-facing step via NPLS      Z. Chen  S.H. Yi  L.F. Tian  L. He  Y.Z. Zhu 《Shock Waves》2013,23(4):299-306 An experimental study on a supersonic laminar flow over a backward-facing step of 5 mm height was undertaken in a low-noise indraft wind tunnel. To investigate the fine structures of Ma = 3.0 and 3.8 laminar flow over a backward-facing step, nanotracer planar laser scattering was adopted for flow visualization. Flow structures, including supersonic laminar boundary layer, separation, reattachment, redeveloping turbulent boundary layer, expansion wave fan and reattachment shock, were revealed in the transient flow fields. In the Ma = 3.0 BFS (backward-facing step) flow, by measuring four typical regions, it could be found that the emergence of weak shock waves was related to the K–H (Kelvin–Helmholtz) vortex which appeared in the free shear layer and that the convergence of these waves into a reattachment shock was distinct. Based on large numbers of measurements, the structure of time-averaging flow field could be gained. Reattachment occurred at the location downstream from the step, about 7–7.5 h distance. After reattachment, the recovery boundary layer developed into turbulence quickly and its thickness increased at an angle of 4.6°. At the location of X = 14h, the redeveloping boundary layer was about ten times thicker than its original thickness, but it still had not changed into fully developed turbulence. However, in the Ma = 3.8 flow, the emergence of weak shock waves could be seen seldom, due to the decrease of expansion. The reattachment point was thought to be near X = 15h according to the averaging result. The reattachment shock was not legible, which meant the expansion and compression effects were not intensive.      13. The Lavrentiev Phenomenon in Nonlinear Elasticity      M. Foss  W. Hrusa  V.J. Mizel 《Journal of Elasticity》2003,72(1-3):173-181 In 1985 J.M. Ball and V.J. Mizel raised the question of whether there exist nonlinearly elastic materials possessing a physically natural stored energy density, i.e., one which is independent of an observer's coordinate frame (objective) and is invariant under the group of orthogonal linear transformations of space (isotropic), as well as physically reasonable boundary value problems for such materials such that the infimum of the total stored energy for those continuous deformations of the material meeting the boundary condition (admissible deformations) which belong to a Sobolev space W 1 p 2 for some p 2>1 is strictly greater than its infimum for those admissible continuous deformations belonging to some Sobolev space W 1 p 1, p 1<p 2, despite the density of W 1 p 2 in W 1 p 1. The question was motivated by M. Lavrentiev's demonstration in 1926 of the presence of such a gap for a 1-dimensional variational boundary value problem on a bounded interval whose smooth integrand satisfied the conditions of Tonelli's existence theorem (as well as the development of improved versions in the 1980's). The present article describes a positive response to the question raised in 1985. Namely, we provide examples of nonlinearly elastic materials in 2-dimensions and physically reasonable boundary value problems for these materials in which a positive gap exists between the infimum of the total stored energy over admissible continuous deformations belonging to a Sobolev space W 1 p 2 and its infimum over admissible continuous deformations belonging to a Sobolev space W 1 p 1, with p 1<p 2. The physical and computational significance of such results is also discussed.      14. Oscillating radial flow between parallel plates      A. F. Elkouh 《Flow, Turbulence and Combustion》1975,30(6):401-417 An analysis is presented for laminar radial flow due to an oscillating source between parallel plates. The source strength varies according to Q=Q 0 cos ωt, and the solution is in the form of an infinite series in terms of a reduced Reynolds number, R a * =Q 0/4πνa/(r/a)2. (Q 0 = amplitude of source strength, ω = frequency, a = half distance between plates, r = radial coordinate, t = time, and ν = kinematic viscosity.) The results are valid for small values of R a * and all values of the frequency Reynolds number, α=ωa 2/ν. The effects of the parameters R a * and α are discussed.      15. On the hyperbolicity of 3D plasticity equations in isostatic coordinates      Yu. N. Radaev 《Mechanics of Solids》2008,43(5):756-764 We consider the problem of classifying partial differential equations of the three-dimensional problem of ideal plasticity (for stress states corresponding to an edge of the Tresca prism) and the problem of finding a change of independent variables reducing these equations to the simplest normal Cauchy form. The original system of equations is represented in an isostatic coordinate system and is substantially nonlinear. We state a criterion for the simplest normal Cauchy form and find a coordinate system reducing the original system to the simplest normal Cauchy form. We show that the condition obtained in the present paper for a system to take the simplest normal form is stronger than the Petrovskii t-hyperbolicity condition if t is understood as the canonical isostatic coordinate whose level surfaces in space form fibers normal to the principal direction field corresponding to the maximum (minimum) principal stress.      16. Numerical simulation of secondary vortex chamber effect on the cooling capacity enhancement of vortex tube      Nader Pourmahmoud  Farid Sepehrian Azar  Amir Hassanzadeh 《Heat and Mass Transfer》2014,50(9):1225-1236 A vortex tube with additional chamber is investigated by computational fluid mechanics techniques to realize the effects of additional chamber in Ranque–Hilsch vortex tube and to understand optimal length for placing the second chamber in order to have maximum cooling effect. Results show that by increasing the distance between two chambers, both minimum cold and maximum hot temperatures increase and maximum cooling effect occurs at Z/L = 0.047 (dimensionless distance).      17. PIV measurement of separated flow over a blunt plate with different chord-to-thickness ratios      Liu Liu Shi  Ying Zheng Liu  Jun Yu 《Journal of Fluids and Structures》2010,26(4):644-657 The influence of the chord-to-thickness ratio (c/t) on the spatial characteristics of the separated shear layer over a blunt plate and the leading-edge vortices embedded in the separated shear layer was studied extensively using planar particle image velocimetry (PIV). Three systems corresponding to different shedding modes were chosen for the comparative study: c/t=3, 6 and 9. The Reynolds number based on the plate's thickness (t) was Ret=1×103. A gigapixel CCD camera was used to acquire images with a spatial resolution of 0.06t×0.06t in the measurement range of 9.5t×4.5t. Distributions of statistical quantities, such as the streamline pattern, streamwise velocity fluctuation intensity, shear stress and reverse flow intermittency, showed that the separated shear layer in the system with c/t=3 did not reattach to the plate's surface, while the near‐wake behind the trailing edge was highly unstable because the energetic leading-edge vortices were shed into the wake. The separated shear layer of the system with c/t=6 periodically reattached to the plate's surface, which resulted in intensified fluctuations of the near wake behind the trailing edge. In the longest system (c/t=9), the separated shear layer always reattached to the plate's surface far upstream from the trailing edge, which did not induce large fluctuations of the near wake. Furthermore, the proper orthogonal decomposition (POD) was extensively employed to filter the original velocity fields spatially to identify the large-scale vortices immersed in the separated shear layer easily. The distribution of the v-v correlation coefficients of the spatially filtered flow fields reflected the organized large-scale vortices in the three systems. The number of alternations of the positive and negative correlation coefficients across the flow field were determined to be 1, 2 and 3 for the systems with c/t=3, 6 and 9, respectively; this is in agreement with the shedding mode of each system. The distribution of the swirling strength of the separated shear layer accurately determined the positions and structures of the large-scale vortices formed above the plate surface.      18. Some improvements on the energy absorbed in axial plastic collapse of hollow cylinders      《International Journal of Solids and Structures》2006,43(6):1543-1560 The control of the plastic flow mechanism during axial collapse of metallic hollow cylinders is of particular interest in the present work for the absorbed energy. Hence, an experimental methodology is developed during which some different tubular structures are loaded under compressive quasi-static strain rate. These structures of various geometrical parameters η = Rm/t and λ = Rm/L (Rm: mean radius, L: initial length and t: thickness of tube) are made either from copper or aluminum considered as an energy dissipating system. At this point, the effects of both parameters on the mean collapse load and absorbed energy are appropriately studied. The role of η ratio, which has been largely investigated previously, is studied again. Moreover, it is found that the λ ratio has a non-negligible influence on the deformation mode for a given η. It is well known that the absorbed energy is influenced by the deformation mechanism, i.e., for the axisymmetric mode, the related absorbed energy becomes more important than that of the diamond fold mechanism for a given cylinder. Accordingly, to maximize the absorbed energy, two different structural solutions, namely fixed-ends and subdivided structure, are developed for encouraging the axisymmetric mode. It is convenient to consider the classical axial collapse situation (noted as free-ends) as a comparison reference. In this work, it is recognized that the subdivided solution is relatively the best solution. As a result, the absorbed energy increases up to 21% in comparison with the free-ends situation for copper tubes.      19. Open and closed-loop experiments to identify the separated flow dynamics of a thick turbulent boundary layer      T. Shaqarin  C. Braud  S. Coudert  M. Stanislas 《Experiments in fluids》2013,54(2):1-22 Open and closed-loop flow control experiments were performed on the transient attachment and separation mechanisms of a thick turbulent boundary layer (TBL). Without actuation, the TBL is subjected to an adverse pressure gradient and separates downstream of a sharp variation in the wall geometry. Departing from a given geometry and steady operations of vortex generator actuators, the control objective was to attach the flow in the separated region with a minimum of injected fluid using adaptation of the closed-loop control. The large scale of the facility (i.e., δ = 20 cm upstream of separation) induces large time scales and large Reynolds numbers of the flow to be controlled. It is found to consequently induce large time scales of the separation/attachment mechanisms, making the dynamic closed-loop implementation easier. Open-loop tests were first performed to extract the adequate input/output variables for closed-loop implementations. The chosen input variable was the Duty Cycle, DC, which enables sending of a control action at least 10 times faster than the time scales of the attachment/separation process. The chosen output variable was the voltage signal from a hot-film probe located on the flap which characterizes the degree of separation. In open loop, both the large scale (i.e., large time scales) of the present facility (Carlier and Stanislas in J Fluid Mech 535(36):143–188, 2005) and the well-defined excitation (Braud and Dyment in Phys Fluids 24:047102, 2012) help to extract the different time scales involved and to identify the whole system (actuators, baseline flow and sensor). Three Reynolds numbers based on the momentum thickness of the boundary layer near the actuators and upstream of separation were investigated (Re θ  = 7,500, 10,500 and 12,600) through variation of the free-stream velocity (U ∞ = 5, 8, 10 m/s). These three systems were found to behave like first-order linear systems, with coefficients that need to be adapted depending on the Reynolds number. From Re θ  = 7,500 to Re θ  = 12, 600, the time scale and static gain of the linear system needed to be almost doubled. A simple controller (Proportional-Integral) was implemented in closed-loop configuration, improving the reactivity of the system. Robustness was tested by varying the free-stream velocity. Closed-loop control based on a fixed reference was unsuccessful as it failed to account for the effect of the Reynolds number. This was successfully overcome by tracking a given state of the flow using a simple P controller to adapt the reference according to variations of Re. The P controller, acting on the DC variable, compensates the corresponding variations of VR (ratio between the free-stream and the jet exit velocity).      20. A study of different modes of collapse in metallic hemispherical shells resting on flat platen and compressed with hemispherical nosed indenter      P.K. Gupta  N.K. Gupta 《International Journal of Solids and Structures》2014 The paper presents experimental and analytical studies on axial compression of aluminium spherical shells having Radius/wall thickness (R/t) ratios between 23 and 135. Quasi-static compressive load was applied centrally and with offset through a indenter having diameter of 22 mm. Testing was carried out on an INSTRON machine having 250 T capacity. Shells having different radius and wall thickness were tested, to classify their modes of collapse and their corresponding energy absorption mechanism. In experiments shells of lower R/t values were found to collapse due to formation of an inward dimple associated with a rolling plastic hinge in central as well as in offset loading. On the other hand, shells of higher R/t values were collapsed initially with formation of an axisymmetric inward dimple, but in later stage of compression showed buckling of non-symmetric shape consisting of integral number of lobes and stationary plastic hinges. The stationary hinges were formed between consecutive lobes. Experimental observations are used to propose an analytical model for prediction of load–compression and energy–compression curves. The results obtained from analytical model compared with the experimental results and found match fairly well.

The Minimal Thickness of a Semicircular Arch with a Tension-Free Crown∗

ABSTRACT

When a semicircular elastic arch is loaded with a single point force at the vertex, the stress at interior points depends on the thickness of the arch and on the way the abutments react. Applying the theory of plane elasticity, three typical load-bearing abutment constraints are evaluated to find the minimal thickness at which no tensile stress occurs at a given interior point. Such thickness is found to be quite large.     

Flow visualization of supersonic laminar flow over a backward-facing step via NPLS

An experimental study on a supersonic laminar flow over a backward-facing step of 5 mm height was undertaken in a low-noise indraft wind tunnel. To investigate the fine structures of Ma = 3.0 and 3.8 laminar flow over a backward-facing step, nanotracer planar laser scattering was adopted for flow visualization. Flow structures, including supersonic laminar boundary layer, separation, reattachment, redeveloping turbulent boundary layer, expansion wave fan and reattachment shock, were revealed in the transient flow fields. In the Ma = 3.0 BFS (backward-facing step) flow, by measuring four typical regions, it could be found that the emergence of weak shock waves was related to the K–H (Kelvin–Helmholtz) vortex which appeared in the free shear layer and that the convergence of these waves into a reattachment shock was distinct. Based on large numbers of measurements, the structure of time-averaging flow field could be gained. Reattachment occurred at the location downstream from the step, about 7–7.5 h distance. After reattachment, the recovery boundary layer developed into turbulence quickly and its thickness increased at an angle of 4.6°. At the location of X = 14h, the redeveloping boundary layer was about ten times thicker than its original thickness, but it still had not changed into fully developed turbulence. However, in the Ma = 3.8 flow, the emergence of weak shock waves could be seen seldom, due to the decrease of expansion. The reattachment point was thought to be near X = 15h according to the averaging result. The reattachment shock was not legible, which meant the expansion and compression effects were not intensive.     

The Lavrentiev Phenomenon in Nonlinear Elasticity

In 1985 J.M. Ball and V.J. Mizel raised the question of whether there exist nonlinearly elastic materials possessing a physically natural stored energy density, i.e., one which is independent of an observer's coordinate frame (objective) and is invariant under the group of orthogonal linear transformations of space (isotropic), as well as physically reasonable boundary value problems for such materials such that the infimum of the total stored energy for those continuous deformations of the material meeting the boundary condition (admissible deformations) which belong to a Sobolev space W ^{1 p} 2 for some p ₂>1 is strictly greater than its infimum for those admissible continuous deformations belonging to some Sobolev space W ¹ p ₁, p ₁<p ₂, despite the density of W ^{1 p} 2 in W ¹ p ₁. The question was motivated by M. Lavrentiev's demonstration in 1926 of the presence of such a gap for a 1-dimensional variational boundary value problem on a bounded interval whose smooth integrand satisfied the conditions of Tonelli's existence theorem (as well as the development of improved versions in the 1980's). The present article describes a positive response to the question raised in 1985. Namely, we provide examples of nonlinearly elastic materials in 2-dimensions and physically reasonable boundary value problems for these materials in which a positive gap exists between the infimum of the total stored energy over admissible continuous deformations belonging to a Sobolev space W ^{1 p} 2 and its infimum over admissible continuous deformations belonging to a Sobolev space W ^{1 p} 1, with p ₁<p ₂. The physical and computational significance of such results is also discussed.     

Oscillating radial flow between parallel plates

An analysis is presented for laminar radial flow due to an oscillating source between parallel plates. The source strength varies according to Q=Q ₀ cos ωt, and the solution is in the form of an infinite series in terms of a reduced Reynolds number, R _a ^* =Q ₀/4πνa/(r/a)². (Q ₀ = amplitude of source strength, ω = frequency, a = half distance between plates, r = radial coordinate, t = time, and ν = kinematic viscosity.) The results are valid for small values of R _a ^* and all values of the frequency Reynolds number, α=ωa ²/ν. The effects of the parameters R _a ^* and α are discussed.     

On the hyperbolicity of 3D plasticity equations in isostatic coordinates

We consider the problem of classifying partial differential equations of the three-dimensional problem of ideal plasticity (for stress states corresponding to an edge of the Tresca prism) and the problem of finding a change of independent variables reducing these equations to the simplest normal Cauchy form. The original system of equations is represented in an isostatic coordinate system and is substantially nonlinear. We state a criterion for the simplest normal Cauchy form and find a coordinate system reducing the original system to the simplest normal Cauchy form. We show that the condition obtained in the present paper for a system to take the simplest normal form is stronger than the Petrovskii t-hyperbolicity condition if t is understood as the canonical isostatic coordinate whose level surfaces in space form fibers normal to the principal direction field corresponding to the maximum (minimum) principal stress.     

Numerical simulation of secondary vortex chamber effect on the cooling capacity enhancement of vortex tube

A vortex tube with additional chamber is investigated by computational fluid mechanics techniques to realize the effects of additional chamber in Ranque–Hilsch vortex tube and to understand optimal length for placing the second chamber in order to have maximum cooling effect. Results show that by increasing the distance between two chambers, both minimum cold and maximum hot temperatures increase and maximum cooling effect occurs at Z/L = 0.047 (dimensionless distance).     

PIV measurement of separated flow over a blunt plate with different chord-to-thickness ratios

The influence of the chord-to-thickness ratio (c/t) on the spatial characteristics of the separated shear layer over a blunt plate and the leading-edge vortices embedded in the separated shear layer was studied extensively using planar particle image velocimetry (PIV). Three systems corresponding to different shedding modes were chosen for the comparative study: c/t=3, 6 and 9. The Reynolds number based on the plate's thickness (t) was Re_t=1×10³. A gigapixel CCD camera was used to acquire images with a spatial resolution of 0.06t×0.06t in the measurement range of 9.5t×4.5t. Distributions of statistical quantities, such as the streamline pattern, streamwise velocity fluctuation intensity, shear stress and reverse flow intermittency, showed that the separated shear layer in the system with c/t=3 did not reattach to the plate's surface, while the near‐wake behind the trailing edge was highly unstable because the energetic leading-edge vortices were shed into the wake. The separated shear layer of the system with c/t=6 periodically reattached to the plate's surface, which resulted in intensified fluctuations of the near wake behind the trailing edge. In the longest system (c/t=9), the separated shear layer always reattached to the plate's surface far upstream from the trailing edge, which did not induce large fluctuations of the near wake. Furthermore, the proper orthogonal decomposition (POD) was extensively employed to filter the original velocity fields spatially to identify the large-scale vortices immersed in the separated shear layer easily. The distribution of the v-v correlation coefficients of the spatially filtered flow fields reflected the organized large-scale vortices in the three systems. The number of alternations of the positive and negative correlation coefficients across the flow field were determined to be 1, 2 and 3 for the systems with c/t=3, 6 and 9, respectively; this is in agreement with the shedding mode of each system. The distribution of the swirling strength of the separated shear layer accurately determined the positions and structures of the large-scale vortices formed above the plate surface.     

Some improvements on the energy absorbed in axial plastic collapse of hollow cylinders

The control of the plastic flow mechanism during axial collapse of metallic hollow cylinders is of particular interest in the present work for the absorbed energy. Hence, an experimental methodology is developed during which some different tubular structures are loaded under compressive quasi-static strain rate. These structures of various geometrical parameters η = R_m/t and λ = R_m/L (R_m: mean radius, L: initial length and t: thickness of tube) are made either from copper or aluminum considered as an energy dissipating system. At this point, the effects of both parameters on the mean collapse load and absorbed energy are appropriately studied. The role of η ratio, which has been largely investigated previously, is studied again. Moreover, it is found that the λ ratio has a non-negligible influence on the deformation mode for a given η. It is well known that the absorbed energy is influenced by the deformation mechanism, i.e., for the axisymmetric mode, the related absorbed energy becomes more important than that of the diamond fold mechanism for a given cylinder. Accordingly, to maximize the absorbed energy, two different structural solutions, namely fixed-ends and subdivided structure, are developed for encouraging the axisymmetric mode. It is convenient to consider the classical axial collapse situation (noted as free-ends) as a comparison reference. In this work, it is recognized that the subdivided solution is relatively the best solution. As a result, the absorbed energy increases up to 21% in comparison with the free-ends situation for copper tubes.     

Open and closed-loop experiments to identify the separated flow dynamics of a thick turbulent boundary layer

Open and closed-loop flow control experiments were performed on the transient attachment and separation mechanisms of a thick turbulent boundary layer (TBL). Without actuation, the TBL is subjected to an adverse pressure gradient and separates downstream of a sharp variation in the wall geometry. Departing from a given geometry and steady operations of vortex generator actuators, the control objective was to attach the flow in the separated region with a minimum of injected fluid using adaptation of the closed-loop control. The large scale of the facility (i.e., δ = 20 cm upstream of separation) induces large time scales and large Reynolds numbers of the flow to be controlled. It is found to consequently induce large time scales of the separation/attachment mechanisms, making the dynamic closed-loop implementation easier. Open-loop tests were first performed to extract the adequate input/output variables for closed-loop implementations. The chosen input variable was the Duty Cycle, DC, which enables sending of a control action at least 10 times faster than the time scales of the attachment/separation process. The chosen output variable was the voltage signal from a hot-film probe located on the flap which characterizes the degree of separation. In open loop, both the large scale (i.e., large time scales) of the present facility (Carlier and Stanislas in J Fluid Mech 535(36):143–188, 2005) and the well-defined excitation (Braud and Dyment in Phys Fluids 24:047102, 2012) help to extract the different time scales involved and to identify the whole system (actuators, baseline flow and sensor). Three Reynolds numbers based on the momentum thickness of the boundary layer near the actuators and upstream of separation were investigated (Re _θ  = 7,500, 10,500 and 12,600) through variation of the free-stream velocity (U _∞ = 5, 8, 10 m/s). These three systems were found to behave like first-order linear systems, with coefficients that need to be adapted depending on the Reynolds number. From Re _θ  = 7,500 to Re _θ  = 12, 600, the time scale and static gain of the linear system needed to be almost doubled. A simple controller (Proportional-Integral) was implemented in closed-loop configuration, improving the reactivity of the system. Robustness was tested by varying the free-stream velocity. Closed-loop control based on a fixed reference was unsuccessful as it failed to account for the effect of the Reynolds number. This was successfully overcome by tracking a given state of the flow using a simple P controller to adapt the reference according to variations of Re. The P controller, acting on the DC variable, compensates the corresponding variations of VR (ratio between the free-stream and the jet exit velocity).     

A study of different modes of collapse in metallic hemispherical shells resting on flat platen and compressed with hemispherical nosed indenter

The paper presents experimental and analytical studies on axial compression of aluminium spherical shells having Radius/wall thickness (R/t) ratios between 23 and 135. Quasi-static compressive load was applied centrally and with offset through a indenter having diameter of 22 mm. Testing was carried out on an INSTRON machine having 250 T capacity. Shells having different radius and wall thickness were tested, to classify their modes of collapse and their corresponding energy absorption mechanism. In experiments shells of lower R/t values were found to collapse due to formation of an inward dimple associated with a rolling plastic hinge in central as well as in offset loading. On the other hand, shells of higher R/t values were collapsed initially with formation of an axisymmetric inward dimple, but in later stage of compression showed buckling of non-symmetric shape consisting of integral number of lobes and stationary plastic hinges. The stationary hinges were formed between consecutive lobes. Experimental observations are used to propose an analytical model for prediction of load–compression and energy–compression curves. The results obtained from analytical model compared with the experimental results and found match fairly well.