Elastic Deformation Equation Research Articles

The effect of thickness coordinate to radius ratio on free vibration of moderately thick and deep doubly curved cross-ply laminated shell

This paper presents the free vibration analysis of moderately thick and deep doubly curved laminated shell with \(C^{0}\) finite element model based on higher-order shear deformation theory. The strain–displacement relationships are developed using the accurate equations of elastic deformation of shell structure. An eight-noded isoparametric shell element with nine degrees of freedom per node is used to formulate the present model. The effect of incorporating the ratio of thickness coordinate to radius of curvature (z / R) in the strain components has been taken into account in the present study. The numerical results in terms of natural frequencies obtained by the present formulations are compared with those available in the published literature to validate the proposed model.

An accurate [formula omitted] finite element model of moderately thick and deep laminated doubly curved shell considering cross sectional warping

A C0 finite element model based on the higher order shear deformation theory has been developed for moderately thick and deep doubly curved laminated composite shells of various shapes. The new accurate equations of elastic deformation by incorporating cross sectional warping of the laminated shells have been developed. The present finite element model has been developed by using eight noded isoparametric shell element with nine degrees of freedom per node. The numerical analysis of laminated shells using the present model has been carried out under sinusoidal loading. The present formulation has been validated by comparing the results in terms of deflection and stresses for moderately thick and deep shells with those obtained from the three dimensional elasticity solution.

Non-invasive determination of transcatheter pressure gradient in stenotic aortic valves: An analytical model

Aortic stenosis (AS), in which the opening of the aortic valve is narrowed, is the most common valvular heart disease. Cardiac catheterization is considered the reference standard for definitive evaluation of AS severity, based on instantaneous systolic value of transvalvular pressure gradient (TPG). However, using invasive cardiac catheterization might carry high risks knowing that undergoing multiple cardiac catheterizations for follow-up in patients with AS is common. The objective of this study was to suggest an analytical description of the AS that estimates TPG without a need for high risk invasive data collection. For this purpose, Navier–Stokes equation coupled with the elastic-deformation equation was solved analytically. The estimated TPG resulted from the suggested analytical description was validated against published in vivo and in vitro measurement data. Very good concordances were found between TPG obtained from the analytical formulation and in vivo (maximum root mean square error: 3.8 mmHg) and in vitro (maximum root mean square error: 9.4 mmHg). The analytical description can be integrated to non-invasive imaging modalities to estimate AS severity as an alternative to cardiac catheterization to help preventing its risks in patients with AS.

A unified theoretical structure for modeling interstitial growth and muscle activation in soft tissues

The objective of this paper is to develop a new unified theoretical structure for modeling interstitial growth and muscle activation in soft tissues. The model assumes a simple continuum with a single velocity field. In contrast with many other formulations, evolution equations are proposed directly for a scalar measure of elastic dilatation and a tensorial measure of elastic distortional deformation. The evolution equation for elastic dilatation includes a rate of mass supply or removal that controls volumetric growth and causes the elastic dilatation to evolve towards its homeostatic value. Similarly, the evolution equation for elastic distortional deformation includes a rate of growth that causes the elastic distortional deformation tensor to evolve towards its homeostatic value. Specific forms for these inelastic rates of growth and the associated homeostatic values have been considered for volumetric, area and fiber growth processes, as well as for muscle activation. Since the rate of growth appears in the evolution equations and not a growth tensor it is possible to model the combined effects of multiple growth and muscle activation mechanisms simultaneously. Also, robust, strongly objective, numerical algorithms have been developed to integrate the evolution equations.

Numerical Modelling of Deformation of Soil Foundations

The algorithm for investigation of soil foundations under its own weight and surface loading is considered. A soil is presented as a continuous medium, which has specific physical and mechanical properties. Structurally, there are three groups of soils, the first – sand, the second – clay and the third – rocks. The main difference is a bonding strength between the individual crystal grains and the medium continuum.Kinematics of a medium continuum is described by the spatial gradient of rate tensor, the deformation rate tensor and the rate of rotation tensor. The state of stress is described by the Cauchy stress tensor and objective Jaumann rate of Cauchy stress. An isotropic elastic-plastic material is considered. The linearized constitutive equations of elastic deformation are obtained as a function of objective Jaumann rate of Cauchy stress in the current state. The theory of a flow and an additive representation of the total deformation rate on elastic and plastic parts are used. The Drucker-Prager yield criterion is applied.The research algorithm is based on the incremental method. The principle of virtual work in terms of the virtual velocity is used. After linearization, the system of linear equations is obtained, where an increment of displacement in the current state is unknown. The radial return method with an iterative refinement of the current mode of deformation is applied. This procedure is based on the introduction into constitutive equations a work of the additional stresses in virtual deformation of the rate.The numerical implementation is based on the finite element method. An eight-node isoparametric hexahedron element is used. The calculation of the soil foundation is considered as an example. The fields of displacements and stresses are obtained. The intensity of stresses and plastic deformations are shown.

A unifying model for fluid flow and elastic solid deformation: A novel approach for fluid–structure interaction

Fluid–structure coupling is addressed through a unified equation for compressible Newtonian fluid flow and elastic solid deformation. This is done by introducing thermodynamics within Cauchy׳s equation through the isothermal compressibility coefficient that is experimentally measurable for both fluids and solids. The vectorial resolution of the governing equation, where every component of velocity vectors and displacement variation vectors is calculated simultaneously in the overall multi-phase system, is characteristic of a monolithic resolution involving no iterative coupling. For system equation closure, mass density and pressure are both re-actualized from velocity vector divergence, when the shear stress tensor within the solid phase is re-actualized from the displacement variation vectors. This novel approach is first validated on a two-phase system, involving a plane fluid–solid interface, through the two following test cases: (i) steady-state compression and (ii) longitudinal and transverse elastic wave propagations. Then the 3D study of compressive fluid injection towards an elastic solid is analyzed from initial time to steady-state evolution.

Elastohydrodynamic Lubrication Analysis for Transversely Isotropic Coating Layer

The effects of the transversely isotropic coating layer on the elastohydrodynamic lubrication (EHL) circular contact problems are analyzed and discussed under constant load condition. The equivalent elastic modulus for an equivalent isotropic half-space problem is applied to simplify the present transversely isotropic coating. The finite element method (FEM) is utilized to solve the Reynolds equation, the load balance equation, the rheology equations, and the elastic deformation equation simultaneously. The simulation results of the present equivalent model are compared with those of an anisotropic material elasticity matrix to evaluate the applicable range of coating thickness under a fixed relative error. The pressure distribution tends to gradually escalating and concentrating toward the center with increasing longitudinal Young's modulus. The variations of pressure and film thickness become significant as the coating thickness becomes thinner. The deformations of interface are smaller than the deformations of the surface. The film thickness and pressure characteristics of the lubricant are discussed for various parameters. These characteristics are important for the design of the mechanical element with coating layer.

Mimicking the effect of gravity using an elastic membrane

Comparing astrospace with an elastic membrane is an interesting analogy but it lacks a theoretical basis and experimental support. We develop a theoretical model that brings to light the relationship between the conceptual model of a gravity well and an elastic deformation equation of a membrane supporting a heavy ball, and further derive the ‘gravitational constant’ for such a small ‘elastic space’. The experimental data obtained are consistent with the prediction of our model, in mimicking the revolution of a small planet. Teaching practice shows that using an elastic membrane is a simple, intuitive and reliable method to enhance the quality of learning about the effect of gravity.

Towards clinical prostate ultrasound elastography using full inversion approach

Various types of cancers including prostate cancer are known to be associated with biological changes that lead to tissue stiffening. Digital rectal examination is based on manually palpating the prostate tissue via the rectum. This test lacks sufficient accuracy required for early diagnosis which is necessary for effective management of prostate cancer. To develop an effective prostate cancer diagnostic technique, the authors propose an imaging technique that maps the distribution of the relative prostate tissue's elasticity modulus. Unlike digital rectal examination, this technique is quantitative, capable of accurately detecting small prostate lesions that cannot be sensed by manual palpation, and its accuracy is independent of the physician's experience. The proposed technique is a quasistatic elastography technique which uses ultrasound imaging to acquire tissue displacements resulting from transrectal ultrasound mechanical stimulation. The system involves a standard ultrasound imaging unit with accessibility to its radiofrequency data. The displacements are used as data for the tissue elasticity reconstruction. This reconstruction does not require tissue segmentation and is based on physics governing tissue mechanics. It is formulated using an inverse problem framework where elastic tissue deformation equations are fully inverted using an iterative scheme where each iteration involves stress calculation followed by elastic modulus updating until convergence is achieved.In silico and tissue mimicking phantom studies were conducted to validate the proposed technique, followed by a clinical pilot study involving two prostate cancer patients with whole-mount histopathology analysis on prostatectomy specimens to confirm a cancer location. The phantom studies demonstrated robustness and reasonably high accuracy of the proposed method. Obtained Young's modulus ratios indicated reconstruction errors of less than 12%. Reconstructed elastic modulus images of the two clinical cases were compared to whole-mount histopathology slides where cancerous areas were identified. This comparison indicated marked tissue stiffening in the cancer area with reasonably accurate consistency observed between cancerous lesions identified by histopathology and high stiffness areas of the elastography images. Results obtained from the phantom and patient studies indicate that the proposed method is reasonably accurate for detecting cancerous lesions. The proposed system does not require any additional hardware attachment for mechanical stimulation or data acquisition while the elasticity reconstruction algorithm can be easily implemented, leading to a low cost system that can be potentially utilized as an effective clinical tool for prostate cancer diagnosis.

Hydrodynamic analysis of multileaf gas foil bearing with backing springs

In this paper, a one-dimensional beam element model on elastic foundations is used to study the static and dynamic performances of a gas-lubricated multileaf foil journal bearing. The two-dimensional Reynolds equation is solved by using the finite element method coupled with the elastic deformation equation simultaneously for the converged steady state. Subsequently, a perturbation method is used to calculate the dynamic characteristics including stiffness and damping coefficients. The effects of structural and operating parameters, such as bump stiffness, leaf foil thickness, attitude angle, bearing number, and eccentricity ratio are discussed in detail. The results show that the parameters greatly influence the bearing performance, which is helpful information for designing and manufacturing an experimental prototype.

Influencing of Fluid-structure Interactions on Static and Dynamic Characteristics of Oil Hydrostatic Guideways

Oil hydrostatic guideways are widely applied in ultra-precision machine tools, the structural parts are often considered as rigid bodies without deformations in the process of design and calculation. In other words, the effects of the structural parts on the oil film thickness and the pressure distribution are ignored in the course of working. The fluid-structure interactions model of oil hydrostatic guideways with annular slit restrictors is established. This model is based on the dynamic Reynolds equation on the land, the elastic deformation equation of the upper slideway and the table dynamic equilibrium equation. The effects of fluid-structure interactions on the deformations, static and dynamic stiffness and settling time under the step load are explored. The results show that the deformations of the upper slideway increase linearly with recess pressure, the static and dynamic stiffness are decreased by the deformations, the settling time is extended by the deformations. Measured values of the static and dynamic stiffness agree well with the theoretical values with the deformations.

Effect of surface roughness on pure squeeze EHL motion of circular contacts

PurposeThe purpose of this paper is to explore the pure squeeze elastohydrodynamic lubrication motion of circular contacts with surface roughness under constant load conditions. The proposed model can reasonably calculate the effects of surface roughness on the transient pressure profiles, film shapes, and normal squeeze velocities during the pure squeeze process.Design/methodology/approachBased on Christensen's stochastic theory, the transient modified Reynolds equation is derived in polar coordinates to consider the effects of surface roughness. The finite difference method and the Gauss‐Seidel iteration method are used to solve the transient modified Reynolds equation, the elasticity deformation equation, load balance equation, and lubricant rheology equations simultaneously.FindingsThe simulation results reveal that the circular type roughness possesses storage oil capacity. Comparatively, the radial type roughness possesses leak oil capacity. Therefore, the film thickness is found with circular type roughness, followed by smooth, and then radial type roughness. In additional, the central dimensionless pressure is found with radial type roughness, followed by smooth, and then circular type roughness.Originality/valueA numerical method for general applications with surface roughness was developed to investigate the pure squeeze action in an isothermal EHL spherical conjunction under constant load conditions, but without asperities contact.

Effects of Surface Roughness and Flow Rheology on the EHL of Circular Contacts with Power-law Fluid

The coupled effects of surface roughness and flow rheology on the Elastohydrodynamic lubrication (EHL) circular contact problems are analyzed and discussed. The average flow model is adapted for the interaction of the flow rheology of lubricant and surface roughness. The averaged Reynolds type equation, the rheology equations, the elastic deformation equation, and the force balance equation are solved simultaneously. The multilevel multi-integration (MLMI) algorithm and Gauss-Seidel iteration method are utilized to calculate the film thickness and pressure distributions. The effects of flow index, Peklenik number, and standard deviation of composite surface roughness on the film thickness and pressure distributions are discussed. The results show that the transverse type roughness and standard deviation of composite roughness enhance the pressure and film thickness in the central contact region. Moreover, the greater the flow index is, the greater the pressure distribution is in the central contact region, and the greater the film thickness is in all regions.

속도의 영향에 따른 3차원 거친 표면의 혼합윤활해석

Dept. of Mechanical Engineering, Pusan National University(Received September 5, 2012 ; Revised November 11, 2012 ; Accepted November 12, 2012)Abstract − This study examined the effects of surface roughness in the mixed lubrication regime of smooth andrough surfaces for roller bearings. The average flow model was adopted for interaction between the flow rhe-ology of the lubricant and the surface roughness. The average Reynolds equation and related flow factor thatdescribes the coupled effects of surface roughness and flow rheology, the viscosity–pressure and density–pressureequations, the elastic deformation equation, and the force balance equation were solved simultaneously. Theresults showed that the effects of surface roughness on the film thickness and pressure distribution should be con-sidered, especially in elastohydrodynamic lubrication contact problems. Keywords − average Reynolds equation (평균레이놀즈방정식), flow factor (유동계수), surface roughness(표면거칠기), roller bearing (롤러베어링), dimensionless film parameter (유막계수)

Solving elastic deformation of some parallel manipulators with linear active legs using computer-aided design variation geometry

It has been a significant and challenging issue to determine the elastic deformation of parallel manipulators for their precision analysis and control. A new method is proposed and studied for solving the elastic deformation of some parallel manipulators with linear active legs using computer-aided design variation geometry. First, an original simulation mechanism of a parallel manipulator is constructed; each of the vectors in the force transformation matrix of the parallel manipulators is constructed by this simulation mechanism. The active/constrained wrench and their pose are determined based on the Newton–Euler formulation. Second, the elastic deformed dimensions of the active legs are determined based on the elastic deformation equation and the active/constrained wrench. Third, a new simulation mechanism of this parallel manipulator is constructed by replacing the original dimensions of active legs with the deformed dimensions of active legs and the elastic deformations of parallel manipulators are solved using the pose difference between the original and new simulation mechanisms. Finally, two parallel manipulators are illustrated and their elastic deformations are solved and verified by both analytic approach and finite element method.

Short-range and long-range order of phyllomanganate nanoparticles determined using high-energy X-ray scattering

High-energy X-ray scattering (HEXS) is used to explore the pH-dependent structure of randomly stacked manganese oxide nanosheets of nominal formula δ-MnO2. Data are simulated in real space by pair distribution function (PDF) analysis and in reciprocal space by both the Bragg-rod method and the Debye equation in order to maximize the information gained from the total scattering measurements. The essential new features of this triple-analysis approach are (1) the use of a two-dimensional supercell in PDF modeling to describe local distortions around Mn layer vacancies, (2) the implementation in Bragg-rod calculations of a lognormal crystal size distribution in the layer plane and an empirical function for the effect of strain, and (3) the incorporation into the model used with the Debye equation of an explicit elastic deformation of the two-dimensional nanocrystals. The PDF analysis reveals steady migration at acidic pH of the Mn atoms from layer to interlayer sites, either above or below the Mn layer vacancies, and important displacement of the remaining in-layer Mn atoms toward vacancies. The increased density of the vacancy–interlayer Mn pairs at low pH causes their mutual repulsion and results in short-range ordering. The layer microstructure, responsible for the long-range lateral disorder, is modeled with spherically and cylindrically bent crystallites having volume-averaged radii of 20–40 Å. Thebunit-cell parameter from the hexagonal layer has different values in PDF, Bragg-rod and Debye equation modeling, because of the use of different weighting contributions from long-range and short-range distances in each method. The PDFbparameter is in effect a measure of the average inlayer Mn...Mn distance and consistently deviates from the average structure value determined by the Bragg-rod method by 0.02 Å at low pH, as a result of the local relaxation induced by vacancies. The layer curvature increases the Bragg-rod value by 0.01–0.02 Å with the cylindrical model and as much as 0.04–0.05 Å with the spherical model. Therefore, in principle, the diffraction alone can unambiguously determine with good accuracy only a volume-averaged apparent layer dimension of the manganese oxide nanosheets. Thebparameter is model dependent and has no single straightforward interpretation, so comparison ofbbetween different samples only makes sense if done in the context of a single specified model.

Numerical analysis of a nonlinear elastic Rayleigh wave

The equations of two-dimensional quadratic nonlinear elastic deformation described by the Murnaghan model are derived and used as the basis for the analysis of a nonlinear Rayleigh wave. In the case of geometrical nonlinearity alone, a preliminary theoretical analysis of the nonlinear equations is carried out, and the first and second approximations are found and commented on. Some numerical results on the approximate representation of Rayleigh wave are presented. The dependence of displacements on the distance traveled by the wave is plotted for different material properties and different initial amplitude and frequency of the wave

Tribodynamic modeling of piston compression ring and cylinder liner conjunction in high-pressure zone of engine cycle

Piston compression ring and cylinder liner contact contributes a significant part of friction loss in an engine. Most of this loss occurs during compression and power stroke transition (i.e., between 300° to 400° crank position). It is because of the combustion gas pressure is higher in this region to enhance ring–liner contact friction. In this paper, we developed a tribodynamic model to study the transient thermoelastohydrodynamics of ring–liner contact. It takes into account the combined solution of Reynolds equation, energy equation, and elastic deformation equation considering ring–liner conformability and rheology change. We estimate the minimum film profile, friction force, and friction power loss within a high-pressure zone of a high-performance engine. Roughness of the liner is characterized using Rk parameter for better surface representation.

Effects of Surface Roughness and Surface Force on the Thin Film Elastohydrodynamic Lubrication of Circular Contacts

The effects of surface roughness and surface force on thin film elastohydrodynamic lubrication (TFEHL) circular contact problems are analyzed and discussed under constant load condition. The multi-level multi-integration (MLMI) algorithm and the Gauss-Seidel iterative method are used to simultaneously solve the average Reynolds type equation, surface force equations, the load balance equation, the rheology equations, and the elastic deformation equation. The simulation results reveal that the difference between the TFEHL model and the traditional EHL model increase with decreasing film thickness. The effects of surface forces become significant as the film thickness becomes thinner. The surface forces have obvious effects in the Hertzian contact region. The oscillation phenomena in pressure and film thickness come mainly from the action of solvation forces

The Analysis of Lubrication for HPD Diesel Engine Piston Pin Bearing

Based on the unified Reynolds equation model and fast Fourier transform (FFT) method, the lubrication performance of the piston pin bearing for high power density diesel engine was studied by numerical simulation. First of all, through the coupled solving of a unified Reynolds equation and elastic deformation equation, the orbit of journal center for piston pin bearing is investigated. The eccentricity ratio of the piston pin bearing in vertical direction of the piston stroke is smaller, however it is much larger in the downward direction of the piston stroke, which indicate that the below area of the piston pin bearing bears greater load and occurs larger deformation. This is consistent with the reality that the below area of the piston pin bearing is prone to damage and wear. Secondly, the influence of the different bearing clearances and width on the minimum oil film thickness is discussed, The results show that the minimum oil film thickness is increased, while the width of piston pin bearing is increased or the clearance of piston pin bearing is decreased.