Numerical simulation of vortex-induced vibration of a circular cylinder at low mass-damping using RANS code

Fundamental research on vortex-induced vibration (VIV) of a circular cylinder is still needed to build more rational VIV analysis tools for slender marine structures. Numerical results are presented for the response of an elastically mounted rigid cylinder at low mass damping constrained to oscillate transversely to a free stream. A two-dimensional Reynolds-averaged Navier–Stokes (RANS) code equipped with the SST k−ω turbulence model is applied for the numerical calculations. The numerical results are compared in detail with recent experimental and computational work. The Reynolds-averaging procedure erases the random disturbances in the vortex shedding process, so that the comparison between experimental data and the numerical results obtained by RANS codes may reveal some random characteristics of the VIV response. How random disturbance affects the observation in the experiments is discussed in this paper and the issues influencing the appearance of the upper branch in experiments are especially investigated. The absence of the upper branch in RANS simulations is explained in depth on account of discrepancies, which exist between experiments and RANS simulations. In addition, the formation of the 2P vortex shedding mode and its transition through the lock-in region are well reproduced in this investigation.     

On the dynamic mechanical properties of open-cell metal foams – A re-assessment of the ‘simple-shock theory’

Metal foams are increasingly used for energy absorption especially in lightweight structures and to resist blast and impact loads. This requires an understanding of the dynamic response of these materials for modelling purposes. As a supplement to Tan et al., 2005a, Tan et al., 2005b, hereinafter referred to as T–L for brevity, this paper provides experimental data for the dynamic mechanical properties of open-cell Duocel® foams having a three-dimensional (3D) distribution of cells. These confirm significant enhancement of the foam’s compressive strength, accompanied by changes in their deformation pattern in certain loading régimes, particularly what has come to be described as the ‘shock’ régime by Zheng et al. (2012). This paper examines experimentally, in a similar fashion as T–L, how the structural response of the individual cell walls is affected by cell-shape anisotropy at the cell (meso)-scale and how this, in turn, alters the pattern of cell crushing and the dynamic, mechanical properties. The distinctive role of cell microinertia and ‘shock’ formation are discussed in relation to the mechanical properties measured for these 3D cylindrical specimens. For consistency the same procedures described in T–L are used. The features identified are shown to be consistent with those observed in finite-element simulations of two-dimensional (2D) honeycombs as estimated by the one-dimensional (1D) steady-shock theory summarised in T–L. The different deformation patterns that develop in the various loading régimes are categorised according to the compression rate/impact speed. Critical values of impact velocity, corresponding to the transition from one pattern to the other, are quantified and predictive formulae for the compressive uniaxial strengths in the directions of two of the principal axes of the material in each loading régime are derived and discussed. The accuracy of the predictive formula in T–L is shown to critically depend on the ‘densification strain’ of the foam specimens. This parameter and the discussion that follows could assist the formulation and validation of alternative theoretical/computational models on the dynamic deformation of such materials.     

On optimizing crash energy and load-bearing capacity in cellular structures

The energy absorption and load-bearing capacity under axial compression of some model cellular structures are studied with an eye toward optimization based on structural mass or volume available for deformation. Three configurations are considered: multilayer, multi-cell and multi-tube, all of a rectangular-cell topology. Loading is applied either parallel or normal to the cell axis. The cell’s aspect ratio and the relative density of the material ρ are systematically varied. The specimens are laterally confined by rigid walls to stabilize the deformation, but the effect of confinement diminishes for sufficiently large number of cells. A square-cell topology seems to be optimal. Together with an appropriate value for ρ, this provides an optimal constraint on the wavelength of the characteristic buckle and consequently extensive energy dissipation throughout the material body. When considering mean stress, crush energy and stroke or densification strain on the basis of minimum mass and volume simultaneously, ρ ≈ 0.5 seem to be a viable compromise among conflicting trends. The mechanical performance in this case is considerably improved over common cellular structures, for which ρ is typically <0.1.     

Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers

Shape memory polymers (SMPs) are polymers that can demonstrate programmable shape memory effects. Typically, an SMP is pre-deformed from an initial shape to a deformed shape by applying a mechanical load at the temperature T_H>T_g. It will maintain this deformed shape after subsequently lowering the temperature to T_L<T_g and removing the externally mechanical load. The shape memory effect is activated by increasing the temperature to T_D>T_g, where the initial shape is recovered. In this paper, the finite deformation thermo-mechanical behaviors of amorphous SMPs are experimentally investigated. Based on the experimental observations and an understanding of the underlying physical mechanism of the shape memory behavior, a three-dimensional (3D) constitutive model is developed to describe the finite deformation thermo-mechanical response of SMPs. The model in this paper has been implemented into an ABAQUS user material subroutine (UMAT) for finite element analysis, and numerical simulations of the thermo-mechanical experiments verify the efficiency of the model. This model will serve as a modeling tool for the design of more complicated SMP-based structures and devices.     

Computational and experimental studies of crushing of metallic hemispherical shells

Axial compression of aluminium spherical shells of R/t values ranging from 25 to 43 was performed under central loading. Quasi-static tests were conducted on an INSTRON machine (model 1197) of 50 T capacity. Spherical shells were tested to identify their modes of collapse and to study the associated energy absorption capacity. In experiments all the spherical shells were found to collapse due to formation of an axisymmetric inward dimple associated with a rolling plastic hinge. A Finite Element computational model of development of the axisymmetric mode of collapse is also presented. Experimental and computed results of the deformed shapes and their corresponding load–compression and energy–compression curves were presented and compared to validate the computational model. The computed variations of the different strains and stresses were also studied. On the basis of the computational results mechanics of the development of the axisymmetric inward dimple mode of collapse has been presented, analysed and discussed.     

Transfert de champs plastiquement admissibles

This paper presents a new approach to interpolate the mechanical fields associated to a given mesh of the computational domain which satisfy the equilibrium equations together with the mechanical criteria which are quadratical in terms of these fields. The method is based on the diffuse approximation techniques. These allow us to construct a field of globally arbitrary order of continuity which approximates accurately the initial discrete mechanical fields. Indeed, the construction is based locally on the resolution of a quadratical optimisation problem under degenerate quadratical constraints for which we propose an analytical solution. The method is applied, in particular, to an equilibrium problem of elastoplastic solid with non linear hardening. To cite this article: P. Villon et al., C. R. Mecanique 330 (2002) 313–318.     

Spark-generated bubble near an elastic sphere

The interaction between a spark-generated bubble and an elastic sphere is investigated. A spark-generated bubble is created at various distances horizontally away from a suspended elastic sphere made of silicone rubber or super absorbent polymer (of shear modulus of elasticity G of between 5 and 312 kPa), using a low-voltage spark discharge method. We observe pronounced deformation and elongation of the elastic sphere when the spark-bubble is generated very close to a sphere. This happens when the elastic sphere has a small modulus of elasticity and a small size ratio R’ between the bubble and the elastic sphere (i.e. the bubble and the sphere have similar radii). Numerical simulations are also conducted using a Boundary Element Method (BEM) model coupled with a Finite Element Method (FEM) solver. The simulation results compare well with the experimental data. The numerical model is then extended to study the effects of elasticity and experimental parameters, such as the dimensionless stand-off distance H’, and size ratio R’, on the degree of deformation of the elastic cell and the dynamics of the bubble.     

Local Minimizers and the Schmid Law in Corner-Shaped Domains

This paper focuses on the mathematical analysis of biaxial loading experiments in martensite, more particularly on how hysteresis relates to metastability. These experiments were carried out by Chu and James and their mathematical treatment was initiated by Ball, Chu and James. Experimentally it is observed that a homogeneous deformation y ₁ is the stable state for “small” loads while y ₂ is stable for “large” loads. A model was proposed by Ball, Chu and James which, for a certain intermediate range of loads, predicts crucially that y ₁ remains metastable (that is, a local—as opposed to global—minimiser of the energy). This result explains convincingly the hysteresis that is observed experimentally. It is easy to get an upper bound on the load at which metastability finishes. However, it was also noticed that this bound (the Schmid Law) may not be sharp, though this required some geometric conditions on the sample. In this research, we rigorously justify the Ball–Chu–James model by means of De Giorgi’s Γ-convergence, establish some properties of local minimisers of the (limiting) energy and prove the metastability result mentioned above. An important part of the paper is then devoted to establishing which geometric conditions are necessary and sufficient for the counter-example to the Schmid Law to apply, namely, the presence of sharp corners in the sample.     

Localization,delocalization and compression fracture in externally pressurized thick cross-ply (very) long cylindrical shells with material non-linearity: A multi-scale and multi-physics analysis

Compression fracture in carbon fiber reinforced plastics (CFRP) involves multiple physical mechanisms operating at multiple scales ranging from angströms to cms and beyond. First, at the macro/meso-scale, combined effects of modal imperfections, transverse shear/normal deformation along with the non-linear hypo-elastic transverse shear (G_TT) material property on the emergence of interlaminar shear crippling type instability modes, related to the localization (onset of deformation softening), delocalization (onset of deformation hardening) and propagation of mode II compression fracture/damage, in thick imperfect cross-ply very long cylindrical shells under applied hydrostatic pressure, are investigated. The primary accomplishment is the (hitherto unavailable) computation of the layer-wise mode II stress intensity factor, energy release rate and kink crack band-width, under hydrostatic compression, from a non-linear finite element analysis (FEA), using Maxwell’s construction and Griffith׳s energy balance approach. Numerical results include the effects of hypoelastic (G_TT only) material property, on localization and delocalization leading to compression fracture.At the micro-scale, a novel three-dimensional eigenfunction expansion technique, based in part on separation of the cylinder length-variable and partly utilizing a modified Frobenius type series expansion in conjunction with an affine transformation to compute the local stress singularity, in the vicinity of a kinked-fiber/matrix trimaterial junction front. Such computed stress singularities represent a measure of the degree of inherent flaw sensitivity of unidirectional CFRP under compression. Finally, dislocation glide in graphite crystallites plays a dominant role in kink band nucleation and propagation at the nano-meter scale.     

Segmental orientation in plastically deformed glassy PMMA

In this study, spatial orientational distribution functions of labeled chain segments of cross-linked and linear PMMA were obtained by solid-state NMR as a function of finite deformation (far) below and (far) above the glass transition temperature T_g. The applied data analysis allows comparison of theoretical predictions and experimental data, both in terms of the orientational probability distributions as a function of two polar angles, as well as in terms of moments of the distribution. Orientation-strain relationships of chain segments agreed above and below T_g with predictions from the rubber-elastic affine network model, but suggests a much denser network below T_g than given by the cross-link density or the entanglement density in the melt. This suggested network structure is believed to be the generator of segmental orientation during plastic deformation in the glassy state, independent of the range of applied cross-link densities and deformation rates used in this study.     

Intrinsic nonlinearity from LAOStrain—experiments on various strain- and stress-controlled rheometers: a quantitative comparison

Strain-controlled large amplitude oscillatory shear (LAOStrain) experiments on a polyisoprene melt and a polyisobutylene solution were conducted on four different rheometers. The results are compared using nonlinear quantities such as the normalized intensity of the third harmonic (I _3/1) and the intrinsic nonlinearity in order to assess the reproducibility of the experiments. Two of the investigated instruments were strain-controlled rheometers, another two, were advanced stress-controlled rheometers. Since the stress-controlled rheometers are able to conduct strain-controlled tests when employing an active deformation control loop, the two different rheometer types could be compared. Experimental details like the gain of the deformation control loop, and the method of temperature control have been shown to play crucial roles in achieving reasonable reproducibility across the different instruments. Furthermore, deviations from the quadratic scaling of I _3/1 with the strain amplitude and the influence of instrument inertia on nonlinear quantities were observed for one of the stress-controlled instruments. The standard deviation of the intrinsic nonlinearity Q ₀(ω ₀) at a specific angular frequency as determined by measurements on the same instrument was found to be 8 % or lower. The relative deviations of Q ₀ across different instruments were instead up to 12 % in the investigated frequency range with an exception for a specific instrument and one of the samples, where the deviation was considerably larger.     

Grain size–inclusion size interaction in metal matrix composites using mechanism-based gradient crystal plasticity

Metal matrix composites (MMCs) comprising nano/microcrystalline matrices and reinforcements exhibit impressive mechanical behaviors derived by exploiting the size effects due to development of geometrically necessary dislocations. In such nanostructured MMCs intricate interactions between the grain size d_g and inclusion size d_i may exist in their overall response, but are difficult to isolate in experiments and are also not accounted for in the size-dependent homogenized models. In this paper, we computationally investigate the grain size–inclusion size interaction in model MMCs architectures wherein the grains and inclusions are explicitly resolved. A mechanism-based slip-gradient crystal plasticity formulation (Han et al., 2005a) is implemented in a finite element framework to model polycrystalline mass as an aggregate of randomly oriented single crystals that host elastic inclusions. The slip gradients that develop across grain boundaries and at inclusion–grain interfaces during deformation result in length-scale dependent responses that depend on both d_g and d_i, for a fixed inclusion volume fraction f. For a given d_i and f, the overall hardening exhibits a nonlinear dependence on grain size for d_g ? d_i indicating that interaction effects become important at those length-scales. Systematic computational simulations on bare polycrystalline and MMC architectures are performed in order to isolate the contributions due to grain size, inclusion size and the interaction thereof. Based on these results, an analytical model developed for the interaction hardening exhibits a Hall–Petch type dependence on these microstructural sizes that can be incorporated into homogenized approaches.     

Numerical study of the effects of constitutive models on plastic buckling of plate elements

Compared with experiments, the J₂ deformation theory of plasticity is known to predict plastic buckling with better accuracy than the more accepted incremental J₂ flow theory. This paradox is commonly known as the ‘plastic buckling paradox’. In an attempt to analyse this discrepancy, the two mentioned constitutive models were implemented in a non-linear finite element code, along with a third non-associative J₂ flow theory. The latter model incorporates a vertex-type plastic flow rule. Using these three constitutive models, the buckling behaviour of plate outstand elements was investigated. Comparisons between the buckling strengths derived are presented. The non-linear static buckling simulations show that the instability introduced by the alternative flow rule of the non-associative model has substantial influence on the buckling behaviour. The acceptance of only small departures from normality was shown to reduce the predicted ultimate capacity of the plates. Furthermore, for plates with small plate slendernesses it was found that the imperfection sensitivity was significantly reduced when using the non-associative flow rule.     

Indentation of an elastic layer by a rigid cylinder

The Green’s functions for the indentation of an elastic layer resting on or bonded to a rigid base by a line load are found efficiently and accurately by a combination of contour integration with a series expansion for small arguments. From the form of the equations it is clear that the function is oscillatory when the layer is free to slip over the base, but for the bonded layer, the function simply decays to zero after a single overshoot.The deformation due to pressure distributions of the form of the product of a polynomial with an elliptical (“Hertzian”) term is calculated and the coefficients chosen to match the indentation shape to that of a cylindrical indenter. The resulting pressure distributions behave much as in Johnson’s approximate theory, becoming parabolic instead of elliptical as the ratio b/d of contact width to layer thickness increases, or, for the bonded incompressible (ν = 1/2) layer, becoming bell-shaped for very large b/d.The relation between the approach δ and the contact width b curves has been investigated, and some anomalies in published asymptotic equations noted and, perhaps, resolved.A noticeable feature of our method is that, unlike previous solutions in which the full mixed boundary value problem (given indenter shape / stress-free boundary) has been solved, the bonded incompressible solid causes no problems and is handled just as for lower values of Poisson’s ratio.     

Shear behaviour of adhesive layers

An experimental method to determine the complete stress versus deformation relation for a thin adhesive layer loaded in shear is presented. The method is based on a classic specimen geometry; the end-notch flexure specimen. The experiments are evaluated using an inverse method. First, the variation of the energy release rate with respect to the shear deformation at the crack tip is measured during an experiment. Then the traction–deformation relation is derived using an inverse method. The theory is based on the path-independence of the J-integral and considers the effects of a flexible adhesive layer.Quasi-static experiments on three different specimen geometries are performed using a servo-hydraulic testing machine. The experiments give consistent results. This shows that the traction–deformation relation can be taken as independent of the dimensions of the adherends. Thus, the constitutive relation can be considered as a property of the adhesive layer. The deformation process at the crack tip is also monitored during the experiments by the use of a digital camera attached to a microscope.     

Transient elastodynamic analysis of two-dimensional structures using Coons-patch macroelements

Recently, Coons’ interpolation was used for the construction of large finite elements with degrees of freedom appearing mostly along the boundaries of a structure. So far, these so-called Coons-patch macroelements were successfully applied to the analysis of two-dimensional and axisymmetric elastic structures as well as potential problems including Poisson equation and acoustics. Now, this paper continues the research by investigating their applicability and performance in calculating the propagation of elastic waves within continua due to sudden loads. Explicit (central difference) and implicit (θ-Wilson) time-integration schemes have been successfully applied to four typical model problems in conjunction with the proposed Coons-patch macroelements—without and with substructuring—and the results are successfully compared with conventional finite elements having the same number of nodes along the boundary. Finally, theoretical issues between the proposed global technique and well-established computational methods are discussed.     

On the Navier–Stokes equations simulation of the head-on collision between two surface solitary waves

The scope of this Note is to show the results obtained for simulating the two-dimensional head-on collision of two solitary waves by solving the Navier–Stokes equations in air and water. The work is dedicated to the numerical investigation of the hydrodynamics associated to this highly nonlinear flow configuration, the first numerical results being analyzed. The original numerical model is proved to be efficient and accurate in predicting the main features described in experiments found in the literature. This Note also outlines the interest of this configuration to be considered as a test-case for numerical models dedicated to computational fluid mechanics. To cite this article: P. Lubin et al., C. R. Mecanique 333 (2005).     

Experimental Investigation on Mechanical Behavior of Coarse Marble Under Six Different Loading Paths

A series of triaxial compression experiments were preformed for the coarse marble samples under different loading paths by the rock mechanics servo-controlled testing system. Based on the experimental results of complete stress-strain curves, the influence of loading path on the strength and deformation failure behavior of coarse marble is made a detailed analysis. Three loading paths (Paths I–III) are put forward to confirm the strength parameters (cohesion and internal friction angle) of coarse marble in accordance with linear Mohr-Coulomb criterion. Compared among the strength parameters, two loading paths (i.e. Path II by stepping up the confining pressure and Path III by reducing the confining pressure after peak strength) are suggested to confirm the triaxial strengths of rock under different confining pressures by only one sample, which is very applicable for a kind of rock that has obvious plastic and ductile deformation behavior (e.g. marble, chalk, mudstone, etc.). In order to investigate re-fracture mechanical behavior of rock material, three loading paths (Paths IV–VI) are also put forward for flawed coarse marble. The peak strength and deformation failure mode of flawed coarse marble are found depending on the loading paths (Paths IV–VI). Under lower confining pressures, the peak strength and Young’s modulus of damage sample (compressed until post-peak stress under higher confining pressure) are all lower compared with that of flawed sample; moreover mechanical parameter of damage sample is lower for the larger compressed post-peak plastic deformation of coarse marble. However under higher confining pressures (e.g. σ ₃?=?30 MPa), the axial supporting capacity and elastic modulus of damage coarse marble (compressed until post-peak stress under lower confining pressure) is not related to the loading path, while the deformation modulus and peak strain of damage sample depend on the difference of initial confining pressure and post-peak plastic deformation. The friction among crystal grains determines the strength behavior of flawed coarse marble under various loading paths. In the end, the effect of loading path on failure mode of intact and flawed coarse marble is also investigated. The present research provides increased understanding of the fundamental nature of rock failure under different loading paths.     

A new trigonometric shear deformation theory for isotropic,laminated composite and sandwich plates

A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.