Kinematic Boundary Conditions Research Articles

Verification of a design model and evaluation of a technique of manufacturing aircraft engine composite parts on the basis of dynamic tests

The results of finite element modeling and experimental evaluation of dynamic characteristics (vibration modes and frequencies) of the composite element of aircraft power plant structures are presented. The aim of the work is to develop effective design and technological schemes for thin-walled complex-geometry composite parts, taking into account the peculiarities of mechanical behavior of polymer composite materials to the fullest extent possible. A method for determining the parameters of natural oscillations of composite parts in a free state using a ping test is developed, which allows excluding the influence of kinematic boundary conditions and obtaining frequency characteristics that depend only on the local characteristics of the material which are primarily determined by the manufacturing technology. According to the results of measurements of the amplitude-frequency parameters of the dynamic response, spectrograms of damped oscillations are obtained the peak values of which correspond to the experimental evaluation of the natural frequencies of the composite part. Verification of the design model was carried out according to the results of the ping test and a method for assessing the quality of technological processes of manufacturing thin-walled carbon fiber structures was proposed.

Extended Hill’s lemma for non-Cauchy continua based on a modified couple stress theory

Two extended versions of Hill’s lemma for non-Cauchy continua are provided using a modified couple stress theory, which includes a constraint relation between the displacement gradient and the micro-rotation, unlike the Cosserat (micro-polar) elasticity theory. The first version is used to obtain the classical elasticity tensor that relates the Cauchy (force) stress to the strain, while the second version is employed to determine the couple stress elasticity tensor that links the couple stress to the curvature. The kinematic (displacement- or strain-based) boundary conditions used in each version of the extended Hill’s lemma are then modified to accommodate composites with periodic microstructures by introducing periodic parts to the displacement and micro-rotation fields and reconstructing the Hill–Mandel condition. To illustrate the two newly proposed versions of the extended Hill’s lemma, homogenization analyses are performed for a two-phase composite using a meshfree radial point interpolation method incorporating the extended Hill’s lemma and the modified couple stress theory, which is newly developed in the current study.

Saint-Venant principle for kinematic boundary conditions

A variant of the extended formulation of the Saint-Venant principle is formulated, including its generalization to kinematic parameters. Using the LIRA CAD software package, a numerical experiment is carried out, during which the zones of tangential stress disturbances in the support zone of a rigidly clamped I-beam under various kinematic boundary conditions are determined.

Modelling a torus cloud employing the van Wijngaarden Ansatz and the Gilmore equation

The present work analyses the dynamics of cloud cavitation in a compressible flow. Recent experiments highlight that a common cloud geometry for cavitation clouds is a horseshoe. This paper presents a calculation method describing the dynamics of a cavitation cloud applying analytical methods. With the Helmholtz vortex theorem in mind, the horseshoe is artificially completed to be a generic torus shaped cloud. Following van Wijngaarden, the mixture of cavitation bubbles and liquid inside the cloud is treated as a continuous medium, i.e. a homogenous model with the void fraction α and the single bubble radius R. By doing so, the bubble radii are a function of the radial position inside the cloud and time only. The individual bubble dynamics depend on the given dynamic and kinematic boundary conditions of the model. A pressure history at infinity and a strain rate or circulation represent these two types of excitations. The problem results in a system of non-linear differential equations consisting of the Gilmore equation, the continuity equation and the Euler equation. The Gilmore equation describes the dynamics of the individual bubbles inside the cloud whereas the continuity and Euler equations describe the movement and interaction of the bubbles. The resulting pressure and the bubble radii inside the cloud as well as the cloud radius are highlighted. They depend on the excitation of the cloud, the Mach number M, the Reynolds number Re, the Weber number We and the polytropic index k.

Blade Section Profile Array Lifting Surface Design Method for Marine Screw Propeller Blade

Abstract The lifting surface model is widely used in screw propeller design and analysis applications. It serves as a reliable tool for determination of the propeller blade mean line and pitch distribution. The main idea of this application was to determine the blade shape that would satisfy the kinematic boundary condition on its surface with the prescribed bound circulation distribution over it. In this paper a simplified lifting surface method is presented – in which the 3D task for the entire blade is replaced by a set of 2D tasks for subsequent blade section profiles.

ВЛИЯНИЕ ЛЕДЯНОГО ПОКРОВА НА ВОЛНЫ КЕЛЬВИНА И ПУАНКАРЕ

This work presents theoretical foundations of Kelvin and Poincare waves in the homogeneous ocean under an ice cover. The ice is considered as thin elastic plate of uniform thickness, with constant values of Young’s modulus, Poisson’s ratio, density, and compressive stress. The boundary conditions are such that the normal velocity at the bottom is zero, and at the undersurface of the ice the linearized kinematic and dynamic boundary conditions are satisfied. We present and analyze explicit solutions for the Kelvin and Poincare waves and the dispersion equations. The problem is examined in the context of a unified theory and without the hydrostatic assumption.

Free-surface laminar flow of a Herschel–Bulkley fluid over an inclined porous bed

The current work presents a mathematical solution for the flow of a Herschel--Bulkley fluid over a porous bed. To obtain the velocity profile, a dimensional analysis is firstly conducted for the Darcian-like flow, showing that the ratio between pore and flow scales impose the domain of validity for the Beaver-like kinematic boundary condition. A friction velocity dependent only on fluid and porous medium properties was found to identify the scale of the Darcian velocity. After applying a modified kinematic boundary condition between free flow and Darcian-like flow, free-surface flow velocity profile was obtained, which can effectively predicts non-porous solutions for less rheological complex fluids. Dimensional analysis was then performed for the velocity profile solution, which allowed to identify the effect of the porous bed on flow properties. Experimental results from the literature were employed to identify the necessary conditions for yield stress fluids to have Darcian-like flow in natural open-channel flows. Sensitivity analysis pointed out that the porous medium permeability and non-Newtonian fluid parameters have greater influence on the velocity profile for pseudoplastic fluids than for dilatant ones.

Existence of a weak solution to the fluid-structure interaction problem in 3D

We study a nonlinear fluid-structure interaction problem in which the fluid is described by the three-dimensional incompressible Navier-Stokes equations, and the elastic structure is modeled by the nonlinear plate equation which includes a generalization of Kirchhoff, von Kármán and Berger plate models. The fluid and the structure are fully coupled via kinematic and dynamic boundary conditions. The existence of a weak solution is obtained by designing a hybrid approximation scheme that successfully deals with the nonlinearities of the system. We combine time-discretization and operator splitting to create two sub-problems, one piece-wise stationary for the fluid and one in the Galerkin basis for the plate. To guarantee the convergence of approximate solutions to a weak solution, a sufficient condition is given on the number of time discretization sub-intervals in every step in a form of dependence with number of the Galerkin basis functions and nonlinearity order of the plate equation.

Surface parametric instability of star-shaped oscillating liquid drops

The star-shaped oscillation of water drops has been observed in various physical situations under vertical excitation with different sources. In the past, the motion of such drops was simplified to two dimensions and only the azimuthal oscillation modes were considered. The parametric instability mechanism explained with the 2D (two-dimensional) model and the corresponding dispersion relation are not satisfactory. In this paper, we show that the external excitation induces a Faraday-wave-like parametric instability on the upper surface of the drop, and the surface patterns and azimuthal oscillations are coupled to produce star-shaped oscillations, which induces a significant softening to oscillation frequencies. We build a 3D (three-dimensional) theoretical model, in which the surface patterns and azimuthal oscillations are connected via kinematical boundary conditions and vary at the same frequency. Given the surface and azimuthal mode numbers, we propose a quasi-3D dispersion relation, which shows better consistency with the experimental data compared with the previous quasi-2D dispersion relation. Our theoretical model provides a more accurate description of the dynamics of liquid drops and will motivate a wide range of applications.

Dynamic stiffness formulation and free vibration analysis of specially orthotropic Mindlin plates with arbitrary boundary conditions

A novel dynamic stiffness formulation which includes the effects of shear deformation and rotatory inertia is proposed to carry out the free vibration analysis of thick rectangular orthotropic plates i.e. the formulation is based on an extension of the Mindlin theory to orthotropic plates in a dynamic stiffness context. The modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in series form, permitting the derivation of an infinite system of linear algebraic equations which connect the Fourier coefficients of the kinematic boundary conditions of its four edges. The boundary conditions are essentially the deflections and angles of rotations constituting the displacement vector and shear forces and bending moments constituting the force vector. The force-displacement relationship is then obtained by relating the two vectors via the dynamic stiffness matrix. Essentially the ensuing infinite system forms the fundamental basis of the paper, which defines the dynamic stiffness matrix for an orthotropic Mindlin plate in an exact sense. Numerical results are obtained by using the Wittrick-Williams algorithm as solution technique. The theory is first validated using published results and then further illustrated by a series of numerical examples. The paper concludes with significant conclusions.

Modelling and simulation of mitral valve for transapical repair applications

Development and application of the numerical model for the simulation of human heart mitral valve (MV) transapical repair is presented. Transapical repair with neochordae implantation is a novel surgical technique allowing beating-heart correction of mitral regurgitation caused by chordae tendineae rupture through a minimally-invasive approach. In the present study, the structural finite element model of the MV decoupled from the blood flow is considered. It comprises two leaflets and chordae tendineae described by nonlinear material model. Geometry of the model and kinematic boundary conditions for fixed points of MV annulus, papillary muscles, and left ventricle apex are defined by patient-specific data. Decoupled behavior of blood is specified by the time-dependent physiologic transvalvular pressure. Personalized computational modelling strategy is applied to perform virtual transapical MV repair by positioning neochordae following the real-life surgery procedure executed by surgeons. A transient analysis in time frame between end-diastole and peak systole is conducted to evaluate post-repair MV function. Computational MV simulation and modelling results provide quantitative information about the neochordae contribution to the MV function improvement and present practical value for the surgical planning of transapical MV repair.

Variational nonlocal gradient elasticity for nano-beams

In the paper (Zaera, Serrano, & Fernández-Sáez, 2019) it is claimed that the nonlocal strain gradient theory (NSGT) leads to ill-posed structural problems. This conclusion was motivated by observing that Constitutive Boundary Conditions (CBC) conflict with non-standard Kinematic and Static Higher-Order Boundary Conditions (KHOBC) - (SHOBC). In the present study, it is shown that no ill-posedness holds if NSGT is established by an adequate variational formulation, with appropriate test fields. KHOBC and SHOBC have nothing to do with the proper mathematical formulation and thus they have not to be prescribed, while standard kinematic and static boundary conditions and CBC have to be imposed to close the relevant nonlocal gradient problem. This conclusion follows from a well-posed abstract variational scheme conceived for nonlocal gradient inflected beams. The treatment provides as special cases most of the size-dependent models adopted in Engineering Science to assess size-effects in nanostructures, such as NSGT, strain-driven and stress-driven local-nonlocal elasticity approaches. Additionally, a well-posed Nonlocal Stress Gradient (NStressG) model is presented, coupling the stress-driven nonlocal strategy (Romano and Barretta, 2017) with the stress gradient elasticity. The presented methodology is elucidated and validated by investigating the structural behavior of a variety of inflected nano-beams of nanotechnological interest, such as sensors and actuators. NStressG is able to predict both softening and hardening nonlocal responses and, unlike the special stress gradient elasticity model, leads to well-posed structural problems in Nanomechanics.

Geometric regularity criteria for incompressible Navier–Stokes equations with Navier boundary conditions

We study the regularity criteria for weak solutions to the 3D incompressible Navier–Stokes equations in terms of the direction of vorticity, taking into account the boundary conditions. A boundary regularity theorem is proved on regular curvilinear domains with a family of oblique derivative boundary conditions, provided that the directions of vorticity are coherently aligned up to the boundary. As an application, we establish the boundary regularity for weak solutions to Navier–Stokes equations in round balls, half-spaces and right circular cylindrical ducts, subject to the classical Navier and kinematic boundary conditions.

Aifantis versus Lam strain gradient models of Bishop elastic rods

In this paper, the size-dependent static behavior of Bishop rods is investigated by Lam and Aifantis strain gradient formulations of elasticity. Appropriate constitutive boundary conditions are established for both the theories by making recourse to a variational approach. Unlike contributions of literature, no higher-order kinematic and static boundary conditions, which have not a clear physical meaning, are required to close the relevant gradient problems. The proposed methodology leads to mathematically well-posed elastostatic problems and is illustrated by examining size effects in selected thick rods of nanotechnological interest. Exact solutions of Bishop nano-rods are detected for a variety of loading systems and kinematic boundary conditions. Peculiar properties, merits, and implications of both the strain gradient formulations, equipped with the proper boundary conditions, are illustrated and commented. The outcomes can be useful for the design and optimization of rod-like thick components of nanoelectromechanical systems.

Stress concentration analysis of nanosized thin-film coating with rough interface

The boundary perturbation method combined with the superposition principle is used to calculate the stress concentration along the arbitrary curved interface of an isotropic thin film coherently bonded to a substrate. In the case of plane strain conditions, the boundary value problem is formulated for a four-phase system involving two-dimensional constitutive equations for bulk materials and one-dimensional equations of Gurtin–Murdoch model for surface and interface. Static boundary conditions are formulated in the form of generalized Young–Laplace equations. Kinematic boundary conditions describe the continuous of displacements across the surface and interphase regions. Using Goursat–Kolosov complex potentials, the system of boundary equations is reduced to a system of the integral equations via first-order boundary perturbation method. Finally, the solution of boundary value problem is obtained in terms of Fourier series. The numerical analysis is then carried out using the practically important properties of ultra-thin-film materials.

Numerical study of the landslide tsunami in the South China Sea using Herschel-Bulkley rheological theory

The Herschel-Bulkley rheological theory is used to describe the viscoplastic debris landslide flow. The shallow water equations considering the time-dependent deformation of the seafloors are adopted to simulate the generation, propagation, and run-up of the landslide induced tsunami. The one-way coupled method of the landslide induced tsunami is implemented through satisfying the kinematic bottom boundary condition. The 1998 Papua New Guinea landslide tsunami is simulated to validate the numerical model by comparing with measurements. We found that the mechanism of the 1992 Hainan Island tsunami in the South China Sea is due to a submarine landslide by comparing the numerical results between earthquake and landslide. With respect of the Baiyun slide, the effects of remolding rate, initial, and residual yield strength on landslide and tsunami are studied numerically. To distinguish the potential landslide tsunami hazard in the South China Sea, the scenarios of the landslides with the volume of 10, 50, 100, and 200 km3 in the Baiyun slide and 1200 km3 in the Brunei slide are presented. Comparison with the nondeformation model in the near-field illustrates the crucial role of rheological property in the landslide tsunami modeling. Furthermore, the characteristics of the propagation of the landslide tsunami in the South China Sea and coastal hazards are analyzed.

Electromagnetic induction in a conducting D″ layer --- I: Effects of boundary charge accumulation and the boundary conditions

In response to the primary electric field originated by the fluid-core-dynamo, electric charges accumulate at the boundaries of a solid conducting D″ layer due to the poor circuitry conditions until the irrotational primary electric field is neutralized. The neutralization undoes the internal induction excited by the primary electric field and, as a result, eliminates the secondary toroidal magnetic field inside the D″. The primary magnetic field generated by the fluid-core-dynamo gives rise to a sustainable secondary poloidal magnetic field inside the conducting D″ layer. We introduce the general kinematic boundary conditions that are valid with different induction processes and parameter jumps across the core mantle boundary. We find by dimensional analysis that the strength of the secondary poloidal magnetic field is much weaker, by a factor of ∼10−2, than the irrotational primary. These findings indicate that the D″ is in effect an insulator for the geodynamo regardless of its material composition. Any mechanism for the core-mantle magnetic coupling should take this fact into consideration.

UWGeodynamics: A teaching and research tool for numerical geodynamic modelling

The UWGeodynamics module facilitates development of 2D and 3D thermo-mechanical geodynamic models (Subduction, Rift, Passive Margins, Orogenic systems etc.). It is designed to be used for research and teaching, and combined the flexibility of the Under- world Application Programming Interface, (Moresi, Dufour, & Muhlhaus, 2002, Moresi, Dufour, & Muhlhaus (2003), Moresi et al. (2007)) with a structured workflow. Designing geodynamic numerical models can be a daunting task which often requires good understanding of the numerical code. UWGeodynamics provides a simple interface with examples to get you started with development of numerical models. Users can start designing their models without any pre-existing knowledge of programming. Expert users can easily modify the framework and adapt it to more specific needs. The code can be run in parallel on multiple CPUs on personal computers and/or High Performance Computing systems. Although UWGeodynamics has been primarily designed to address geodynamic problems, it can also be used to teach fluid dynamics and material mechanics. UWGeodynamics uses the flexibility of the Python language and the Jupyter Notebook environment, which allows leveraging the wide range of scientific libraries available from the Python community. It also facilitates the coupling with existing scientific Python modules such as Badlands (Salles, Ding, & Brocard, 2018). The functionalities include: • Dimensional input values, using user’s choice of physical units. • Automated and transparent scaling of dimensional values. • Sets of predefined geometries that can be combined to define the initial geometry of a model. • Handles Newtonian and non-Newtonian rheologies (Viscous, Visco-plastic and Visco-elasto-plastic). • Database of common rheologies used in geodynamics, which can be personalised / extended by users. • Simple definition of kinematic, stress, and thermal boundary conditions. • Lithostatic pressure calculation • Thermal equilibrium (steady-state) calculation. • Pseudo Isostasy using a range of kinematic or stress boundary conditions. • Partial melt calculation and associated change in viscosity / heat production. • Simple definition of passive tracers and grid of tracers. • Simple Phase changes • 2-way coupling with the surface processes model pyBadlands (Salles et al., 2018). UWGeo comes with a series of examples, benchmarks and tutorial setups that can be used as cookbook recipes. They provide a wide range of teaching materials useful to introduce numerical geodynamic modeling to students. New functionalities are constantly being added to the code and contributions are more than welcomed. You can access the full documentation online at https://uwgeodynamics. readthedocs.io

Analysis of a castable refractory using the wedge splitting test and cohesive zone model

A cohesive zone approach is applied to the wedge splitting test (WST) using the finite element code Abaqus to obtain the tensile strength, the fracture energy and insight about the crack wake region. A finite element model updating (FEMU) method, with a cost function based on the measured load (FEMU-F), is used to calibrate the sought parameters. Digital image correlation (DIC) provided the kinematic boundary conditions, and the images were also used to define the geometry for the finite element analysis. Besides the fracture energy analysis and the experimental load, gray level images and displacement fields are analyzed in order to validate the results. The cohesive region is active in the whole analyzed test as confirmed by estimates using the cohesive length.

Isogeometric configuration design sensitivity analysis of geometrically exact shear-deformable beam structures

In this paper, using an isogeometric approach, a continuum-based adjoint configuration design sensitivity analysis (DSA) method is presented for three-dimensional finite deformation shear-deformable beam structures. A geometrically exact beam model together with a multiplicative update of finite rotation by an exponential map of a skew-symmetric matrix is utilized. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration, which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. We also present a procedure of explicit parameterization of initial orthonormal frame using the smallest rotation (SR) method within the isogeometric analysis framework. Furthermore, it is shown that for curve entities embedded to a smooth surface, the convected basis of the surface can be effectively utilized for reference orthonormal frames in the SR method. Various numerical examples including pressure loads and nonhomogeneous kinematic boundary conditions in built-up structures demonstrate the effectiveness of the developed DSA method.