Squeeze flow between plane and spherical surfaces

Experiments are described in which a constant force F squeezed a fluid, either between two parallel circular plates, or between a plate and convex spherical lens. Newtonian fluids obeyed the relation of Stefan (1874) for plates, and the relation of Adams et al. (1994) for plate and lens. The non-Newtonian yield stress fluids Brylcreem, Laponite and Sephadex were squeezed between plates of various diameter D to attain a stationary separation h. Only for separations greater than h ^* (which depended on the fluid) did Brylcreem and Laponite obey the relation F/D ³ ∝ h ⁻¹ of Scott (1931) and give a yield stress in agreement with the vane method. For Sephadex the dependence of F/D ³ on h disagreed with Scott's relation, but varied as h ^−5/2 for h > 0.6 mm and h ^−3/2 for h < 0.6 mm. On rotating one plate in its plane the yield stress fluids at a fixed F suffered a marked decrease of h. This, and the existence of h ^*, are discussed in terms of the soft glassy material model of Sollich et al. (1997) and Sollich (1998). Brylcreem and Laponite were squeezed between a plate and lenses of various curvature and their yield stress obtained using the relation of Adams et al. (1994) was compared with measurements by plate-plate squeeze-flow and vane methods. Received: 12 April 2000 Accepted: 26 October 2000     

Deformation of a Convex Hydrogel Shell by Parallel Plate and Central Compression

To characterize the materials parameters and deformation of a convex shell of axial symmetry, a hydrogel contact lens is mechanically deformed by two loading configurations: (a) compression between two parallel plates and (b) central load applied by a shaft with a spherical tip. A universal testing machine with nano-Newton and submicron resolutions is used to measure the applied force, F, as a function of vertical displacement of the plate/shaft, w ₀, while a homemade laser aided topography system records the in-situ deformed shell profile and the contact radius or central dimple, a. A nonlinear shell theory and an iterative finite difference method are used to account for the large elastic deformation, the central buckling for the central load compression, and the interrelationship between the measureable quantities (F, w ₀, a).     

Flow characteristics and flow-induced forces of a stationary and rotating triangular cylinder with different incidence angles at low Reynolds numbers

In this paper, the problem of two-dimensional fluid flow past a stationary and rotationally oscillating equilateral triangular cylinder with a variable incident angle, Reynolds number, oscillating amplitude, and oscillating frequency is numerically investigated. The computations are carried out by using a two-step Taylor-characteristic-based Galerkin (TCBG) algorithm. For the stationary cases, simulations are conducted at various incident angles of α=0.0–60.0° and Reynolds numbers of Re=50–160. For the oscillation cases, the investigations are done at various oscillating amplitudes of θ_max=7.5–30.0° and oscillating frequencies of F_s/F_o=0.5–3.0 considering two different incidence angles (α=0.0°, 60.0°) and three different Reynolds numbers (Re=50, 100, 150). The results show that the influences of key parameters (incidence angle, Reynolds number, oscillating amplitude, and oscillating frequency) are significant on the flow pattern and hydrodynamic forces. For the stationary cases, at smaller angle of incidence (α≤30.0°), Reynolds number has a large impact on the position of the separation points. When α is between 30.0° and 60.0°, it was found that the separation points are located at the rear corners. From a topological point of view, the diagram of flow pattern is summarized, including two distinct patterns, namely, main separation and vortex merging. A deep analysis of the influence of Reynolds number and incidence angles on the mean pressure coefficient along the triangular cylinder surface is presented. Additionally, for the oscillating cases, the lock-on phenomenon is captured. The dominant flow patterns are 2S mode and P+S mode in lock-on region at α=0.0°. It is found at α=60.0°, however, that the flow pattern is predominantly 2S mode. Furthermore, except for the case of F_s/F_o=2.0, the mean drag decreases as the oscillating amplitude increases for each Reynolds number at α=0.0°. At α=60.0°, the minimum mean drag for F_s/F_o=1.5 is lower than that for stationary case, and occurs at θ_max=15.0° (Re=100) and θ_max=22.5° (Re=150), respectively. Finally, the effect of Reynolds number on a rotational oscillation cylinder is elucidated.     

Experimental investigation of material effects on free vibration of a splitter plate

Fluid-structure interaction resulting from free vibration is a complex phenomenon, not fully understood today. In the present study the flow separation from the trailing edge of a splitter plate in a convergent channel involves Vortex-Induced Vibration (VIV) modifying the fundamental instability related to vortex shedding. Under certain conditions, the VIV produces cellular vortex shedding at the trailing edge. In this paper, we attempt experimentally to further investigate the important parameters affecting VIV phenomenon. We present results on measurements on the effect of plate material. Experimental techniques include Laser Telemetry (LT), which is a laser displacement sensor used to measure the vibrational response of the plate and Particle Image Velocimetry (PIV), which is used to measure the corresponding effect on the vortex shedding. Combining data from these techniques the variation in the response of the plate due to material effects can be addressed together with the imprint of the excited vibration mode on the flow. Measurements were performed with five different plate materials over a range of Reynolds numbers. The results show that the vibrational response of the combined fluid-structure system is modified by the VIV instability. A characteristic vibrational behaviour with a stepwise increase of the frequency of the dominant vibration mode is formed as the vortex shedding frequency (f _s) synchronizes to the plate vibration frequency (f _o). The synchronization takes place over a range of Re numbers. After certain Re number threshold is exceeded the frequency jumps to a new synchronization region. The dimensionless vibration frequency (St _o) of the plate, being a Strouhal number characterized by f _o forms a saw tooth profile centered to reduced velocity value inside the range of highest amplitude response. This behavior is explained by the natural frequencies of the combined fluid-structure system. The results further show that the vibration frequency and amplitude are modified due to material properties. As the mass ratio (M*) is increased the vibration frequency increases and the dimensionless amplitude (A/d) decreases. The number of synchronization regions decreases and the ranges extend wider in terms of Re number with increasing M*.     

Three-dimensional streamlined finite elements: Design of extrusion dies

A method to determine three-dimensional die shapes from extrudate swell and vice versa is presented using a three-dimensional Galerkin finite element method based on a streamlined formulation with the fluid velocities and pressures represented by triquadratic and trilinear basis functions respectively. The three-dimensional streamlined method, an extension of the two-dimensional formulation, uses successive streamsurfaces to form a boundary-conforming co-ordinate system. This produces a fixd, computational domain leaving the spatial location of the elements as unknowns to be determined with the standard primary variables (u, v, w, p). The extrudate produced by a die of a given shape is considered for moderate Reynolds numbers. Finally, the method is extended to address the problem of die design, where a die profile is sought to produce a target extrudate shape.     

Thermal residual stresses in thick graphite/epoxy composite laminates—Uniaxial approach

Thermal residual stresses have been known to be very large in laminates of continuous-fiber-reinforced polymer composites. When the thickness of the laminate is large, however, the measurement of the residual stresses raises questions on the accuracy of the conventional methods. A novel concept of layer separation is developed to measure quantitatively and precisely the tensile residual stresses in thick plates with layered distribution of residual stresses. It is applied to thick [O₂/9O₄]_13s, AS/3501-6 graphite/epoxy laminates. The test specimens were mechanically modeled into the thin strips for the application of the new concept of layer separation. The tensile residual stresses measured in the 90-deg layers of these laminates are nonuniform throughout the specimen, and vary from 55.6 MPa to 71.4 MPa. It is very interesting to compare these values with the transverse strengthF ₂ ^tu of AS/3501-6 unidirectional composites, which is 65.4 MPa.     

Relation between extrudate swelling of polypropylene and molecular weight

Six samples of polypropylene produced by Montepolimeri were carefully characterized in dilute solution and studied, with a capillary extrusion rheometer, in large ranges of shear and temperature. The extrudate swelling behaviour is found to be regular as far as its shear stress and temperature dependence is concerned, but the influence of molecular weight is abnormal. With increasing molecular weight the amount of elastic recovery (as measured by the extrudate swell) decreases markedly and the shear modulusG (obtained from the data by making use of simplifying assumptions) appears to be an increasing function ofM _w at all the shear stress investigated. No similar results appear to be reported in the literature.Presented in part at the VIIIth Intern. Congress on Rheology, Naples, September 1–5, 1980.     

Variational principles of perforated thin plates and finite element method for buckling and post-buckling analysis

On the basis of the general theory of perforated thin plates under large deflections^{[1, 2]}, variational principles with deflectionw and stress functionF as variables are stated in detail. Based on these principles, finite element method is established for analysing the buckling and post-buckling of perforated thin plates. It is found that the property of element is very complicated, owing to the multiple connexity of the region. Project supported by National Natural Science Foundation of China.     

Viscous and Inviscid Instabilities of Flow Along a Streamwise Corner

Here we consider the stability of flow along a streamwise corner formed by the intersection of two large flat plates held perpendicular to each other. Self-similar solutions for the steady laminar mean flow in the corner region have been obtained by solving the boundary layer equations for zero and nonzero streamwise pressure gradients. The stability of the mean flow is investigated using linear stability analysis. An eigensolver has been developed to solve the resulting linear eigenvalue problem either in a global mode to obtain an approximation to all the dominant eigenmodes or in a local mode to refine a particular eigenmode. The stability results indicate that the entire spectrum of two-dimensional and oblique viscous modes of a two-dimensional Blasius boundary layer is active in the case of a corner layer as well, but away from the cornerline. In a corner region of finite spanwise extent, the continuous spectrum of oblique modes degenerates to a discrete spectrum of modes of increasing spanwise wave number. The effect of the corner on the two-dimensional viscous instability is small and decreases the growth rate. The growth rate of outgoing oblique disturbances is observed to decrease, while the growth rate of incoming oblique disturbances is enhanced by the corner. This asymmetry between the outgoing and incoming viscous modes increases with increasing obliqueness of the disturbance. The instability of a zero pressure gradient corner layer is dominated by the viscous modes; however, an inviscid corner mode is also observed. The critical Reynolds number of the inviscid mode rapidly decreases with even a small adverse streamwise pressure gradient and the inviscid mode becomes the dominant one. Received 17 March 1998 and accepted 28 April 1999     

Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes

Results are presented for the numerical simulation of vortex-induced vibrations (VIVs) of a cylinder at low Reynolds numbers (Re). A stabilized space–time finite-element formulation is utilized to solve the incompressible flow equations in primitive variables. The cylinder, of low nondimensional mass (m^*=10), is free to vibrate in, both, the transverse and in-line directions. To investigate the effect of Re and reduced natural frequency, F_n, two sets of computations are carried out. In the first set of computations the Reynolds number is fixed (=100) and the reduced velocity (U^*=1/F_n) is varied. Hysteresis, in the response of the cylinder, is observed at the low- as well as high-end of the range of reduced velocity for synchronization/lock-in. In the second set of computations, the effect of Reynolds number (50Re500) is investigated for a fixed reduced velocity (U^*=4.92). The effect of the Reynolds number is found to be very significant for VIVs. While the vortex-shedding mode at low Re is 2S (two single vortices shed per cycle), at Re300 and larger, the P+S mode of vortex shedding (a single vortex and one pair of counter-rotating vortices are released in each cycle of shedding) is observed. This is the first time that the P+S mode has been observed for a cylinder undergoing free vibrations. This change of vortex-shedding mode is hysteretic in nature and results in a very large increase in the amplitude of in-line oscillations. Since the flow ceases to remain two-dimensional beyond Re200, it remains to be seen whether the P+S mode of shedding can actually be observed in reality for free vibrations.     

Further contributions on the flow past a stationary and confined cylinder: Creeping and slowly moving flow of Power law fluids

The steady flow of generalized Newtonian fluid around a stationary cylinder placed between two parallel plates was studied numerically. Finite volume method was applied to solve the momentum equations along with the continuity equation and the Power law rheological model within the laminar flow regime for a range of the Reynolds number Re and the Power law index n values. The values of the Reynolds number, based on physical and rheological properties, cylinder radius and bulk velocity, were varied between 0.0001≤Re≤10, while the Power law index values mapped the 0.50≤n≤1.50 range, allowing for the investigation of both shear-thinning and shear-thickening effects at the creeping as well as slowly moving fluid flow conditions. We report accurate results of a systematic study with a focus on the most important characteristics of fluid flow past circular cylinder. It is shown that for the creeping flow regime there exist finite sized redevelopment length, drag and loss coefficient. Last but not least, the present numerical results indicate that the shear-thinning viscous behaviour decreases the onset of flow separation.     

超高强度平头圆柱形弹体对低碳合金钢板的高速撞击实验

为分析不同组分低碳合金钢板抗超高强度低碳合金钢弹体的高速撞击性能及破坏模式,以两种典型防弹特种钢SS、AS以及常见的Q235A钢为研究对象,通过静态拉伸、静态压缩及动态压缩测试,获得静态拉伸和压缩性能参数以及1 000~6 000 s-1应变率范围内的力学行为,分析了材料组分与力学性能的相关性。采用弹道枪加载撞击方法,获得了两种超高强度合金钢平头圆柱形弹体对3种钢板(14.5~15.9 mm厚)的弹道极限速度,通过分析获得了不同工况下的极限比吸收能,讨论了合金钢板在弹体高速撞击下破坏模式的差异,分析了材料力学性能与破坏模式的相关性。研究表明:3种合金钢板抗弹体撞击性能与材料屈服强度正相关,但其性能间的差异远小于屈服强度间的差异;在超高强度合金钢平头圆柱形弹体的高速撞击下,3种钢板的失效机制与其力学性能密切相关,Si和Mn含量高的AS钢呈硬脆性特征,其断裂失效主要取决于材料的剪切强度,而Si和Mn含量较低的SS钢和Q235A钢具有良好的塑性,其断裂失效主要取决于材料的压缩强度和剪切强度。     

The general solution of one-dimensional hexagonal quasicrystal

A general solution is given for three-dimensional dynamic problems in 1D hexagonal quasicrystalline material with Laue class 6/m_hmm. In the general solution, the phonon and phason displacements and stresses are controlled by two displacement functions Ψ and F, which satisfy a second-order partial differential equation, and a sixth-order partial differential equation, respectively. The general solution for static equilibrium problems is treated as a special case.     

Yield stress of structured fluids measured by squeeze flow

Various structured fluids were placed between the parallel circular plates of a squeeze-flow rheometer and squeezed by a force F until the fluid thickness h was stationary. Fluid thickness down to a few microns could be measured. Most fluids showed two kinds of dependence of f on h according to an experimentally-determined thickness h ^*. If h > h ^* then F varied in proportion to h ⁻¹ as predicted by Scott (1931) for a fluid with a shear yield stress τ₀. The magnitude of τ₀ from squeeze-flow data in this region was compared with the yield stress measured by the vane method. For some fluids τ₀ measured by squeeze flow was less than the vane yield stress, suggesting that the yield stress of fluid in contact with the plates was less than the bulk yield stress. If h < h ^* then F varied approximately as h ^−5/2 and the squeeze-flow data in this region analysed with Scott's relationship gave a yield stress which increased as the fluid thickness decreased. This previously unreported effect may result from unconnected regions of large yield stress in the fluid of size similar to h ^* which are not sensed by the vane and which become effective in squeeze flow only when h < h ^*. Received: 13 December 1999/Accepted: 4 January 2000     

Squeeze flow of a power-law fluid between non-parallel plates

Squeeze flow in the gap between non-parallel circular plates of radius R is discussed. The test material is assumed to be a power-law fluid, with a no-slip boundary condition at the plates. If the mean separation between the plates is h, and the angle of inclination between the plates is ? ? h/R, the force on the plates is perturbed only at O(?²) and is increased by less than 10% if ? < 0.35h/R. A torque O(?) tends to return the plates to a parallel configuration.     

Stochastic nonlinear bending response of laminated composite plates with system randomness under lateral pressure and thermal loading

In this paper, the effect of sensitivity of randomness in system parameters on the nonlinear transverse central deflection response of laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modelled as basic random variables. A higher order shear deformation theory in the von-Karman sense is used to model the system behavior of the laminated plate. A direct iterative-based C ⁰ nonlinear finite element method in conjunction with the first-order perturbation technique developed by the authors is extended for thermal problem to obtain the second-order response statistics, i.e., mean and variance of the nonlinear transverse deflection of the plate. Typical numerical results of composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and transverse temperature are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The results have been compared with those available in the literature and an independent Monte Carlo simulation.     

Analysis and active control of nonlinear vibration of composite lattice sandwich plates

This paper is devoted to investigate the nonlinear vibration characteristics and active control of composite lattice sandwich plates using piezoelectric actuator and sensor. Three types of the sandwich plates with pyramidal, tetrahedral and Kagome cores are considered. In the structural modeling, the von Kármán large deflection theory is applied to establish the strain–displacement relations. The nonlinear equations of motion of the structures are derived by Hamilton’s principle with the assumed mode method. The nonlinear free and forced vibration responses of the lattice sandwich plates are calculated. The velocity feedback control (VFC) and H_∞ control methods are applied to design the controller. The nonlinear vibration responses of the sandwich plates with pyramidal, tetrahedral and Kagome cores are compared. The influences of the ply angle of the laminated face sheets, the thicknesses of the lattice core and face sheets and the excitation amplitude on the nonlinear vibration behaviors of the sandwich plates are investigated. The correctness of the H_∞ control algorithm is verified by comparing with the experiment results reported in the literature. The controlled nonlinear vibration response of the sandwich plate is computed and compared with that of the uncontrolled structural system. Numerical results indicate that the VFC and H_∞ control methods can effectively suppress the large amplitude vibration of the composite lattice sandwich plates.

     

Elastic-plastic analysis of biaxially loaded center-cracked plates

Center-cracked panels loaded in biaxial tension are examined in this paper. Calibration relations for the J integral and the Q constraint factor are presented for a Ramberg–Osgood power law hardening material under plane stress and plane strain loadings. Two cases are examined: an isolated crack and a periodic array of cracks both under biaxial loading conditions. The latter has previously been studied for plane stress conditions. A number of different J estimation schemes are proposed based on the remote load and displacement and their dependence on geometry, biaxiality, and material properties is discussed. The variation of constraint, as characterised by Q, is also presented for plane stress and plane strain conditions. Simple slip line field solutions are derived for perfectly plastic conditions and the resulting limit load solutions are compared with numerically determined values. Implications for failure of cracked plates under biaxial loading are discussed.     

Evaluation of crack initiation angle under mixed mode loading at diverse strain rates

Crack initiation angle, under mixed mode loading at several strain rates, is analysed using an experimental–numerical approach. The physical phenomenon for the problem at hand is influenced by the local and global conditions. One of such factors is the strain rate at the crack tip. For this purpose, PMMA plates with centred angled cracks under mixed mode loading were tested. The strain rate at the neighbourhood of the crack tip before crack propagation was evaluated. Considering that this material is strain rate sensitive, the numerical models were calibrated with the modulus of elasticity measured in tension tests at the observed strain rates. Numerical evaluations were performed with the finite element method in conjunction with the volume energy density criterion. An improvement in the evaluation of the crack propagation angle was observed. In order to complete the analysis, the crack initiation angle was also evaluated with the strain energy density factor S, considering the mechanical properties of PMMA, as evaluated at the observed strain rates, and the stress intensity factors k₁ and k₂. Results are in agreement with those observed experimentally.     

Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method

Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown significant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations.