Saint-Venant Principle Research Articles

Lateral Size-Dependence in UHF Mode-Coupled ZnO FBARs to Suppress Undesirable Eigen-Modes and Weaken Mounting Effect.

Mode-coupled vibrations in an ultra-high frequency (UHF) ZnO thin film bulk acoustic resonator (FBAR) operating at thickness-extensional (TE) mode are studied by employing weak boundary conditions (WBCs), constructed based on Saint-Venant's principle and mixed variational principle in the piezoelectric theory. The frequency spectra, describing the lateral size-dependence of mode couplings between the main mode (TE) and undesirable eigen-modes, for clamped lateral edges are compared with the existing frequency spectra for free lateral edges to illustrate the boundary influence. The displacement and stress variations in FBAR volume are also presented to intuitionally understand and distinguish the difference of frequency spectra between these two different lateral edges, and then we discuss how to select outstanding lateral sizes to weaken the mounting effect. The frequency spectra predicted from our approximate WBCs are also compared with and agree well with those predicted by the finite element method (FEM) using COMSOL, which proves the correctness and accuracy of our theoretical method. These results indicate that the WBCs could have potentials in the valid predictions of lateral size-dependence of mode couplings in piezoelectric acoustic wave devices.

Two-dimensional beams in rectangular coordinates using the radial point interpolation method

Abstract The three-dimensional Theory of Elasticity equations lead to a complex solution for most problems in engineering. Therefore, the solutions are typically developed for reduced systems, usually symmetrical or two-dimensional. In this context, computational resources allow the reduction of these simplifications. The most recognized methods of algebraic approximation of the differential equations are the Finite Differences Method and the Finite Element Method (FEM). However, they have limitations in mesh generation and/or adaptation. As follows, Meshless Methods appear as an alternative to these options. The present work uses the Radial Point Interpolation Method (RPIM) to evaluate the stress in two-dimensional beams in regions close to loading (Saint Venant's Principle). Formulations based on the Fourier Series Theory and the RPIM are presented. Multiquadrics Radial Basis Functions were used to obtain the stiffness matrix. Two numerical examples demonstrate the validity of the RPIM for the proposed theme. The results were obtained from the formulations cited and the Finite Element Method for comparison.

Minimum Required Distance of Strain Gauge from Specimen for Measuring Transmitted Signal in Split Hopkinson Pressure Bar Test

The minimum required distance of the strain gauge on the transmitted bar of the split Hopkinson bar has been determined from the position of a metallic specimen via an explicit finite element analysis. The minimum required distance was determined when the strain-time profiles at r = 0, 0.5Ro and 1.0Ro, were coincident (r is the radial position and Ro is the radius of the bar.). The determined minimum required distance, f(x), is presented as a function of the relative specimen diameter to that of the bar (x = D/D0): j(x) = - 0.9385.x3 + 0.6624.x2 - 0.7459.x + 1.4478 (0.1 ≤ x ≤ 0.9). This result demonstrates the Saint-Venant's principle of rapid dissipation of localized stress in transient loading. The result will be useful for the design/modification of the pseudo-one-dimensional impact instruments that utilise a stress pulse transmitted through the specimen. The result will also allow one to avoid unnecessarily remote strain gage position from the specimen.

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.

Numerical Study of the Shear Stress Distribution in an I-Beam in the Load Application Zone

A study on the Saint-Venant principle implementation for a rigidly clamped I-beam loaded with various loads at the free end was carried out. When using the software package LIRA SAPR, the tangential stresses perturbations zones are determined in order to compare their distribution with the materials resistance solution.

A generalization of the Saint-Venant’s principle for an elastic body with dipolar structure

This study is concerned with the linear elasticity theory for bodies with a dipolar structure. In this context, we approach transient elastic processes and the steady state in a cylinder consisting of such kind of body which is only subjected to some boundary restrictions at a plane end. We will show that at a certain distance $$d=d(t)$$, which can be calculated, from the loaded plan, the deformation of the body vanishes. For the points of the cylinder located at a distance less than d, we will use an appropriate measure to assess the decreasing of the deformation relative to the distance from the loaded plane end. The fact that the measure, that assess the deformation, decays with respect to the distance at the loaded end is the essence of the principle of Saint-Venant.

Mechanical Modeling the Processes of Additive Manufacturing Bearing Pillars of Viscoelastic Aging Materials by Thickening over the Lateral Surface under Axial Loading

A nonclassical mechanical model for the process of additive manufacturing a bearing circular conical pillar by means of its layer-by-layer thickening over the lateral surface using a viscoelastic aging material under simultaneous action of an axial variable central force is developed. The exact in the sense of Saint-Venant principle analytical solution of the stated corresponding quasi-static boundary value problem is constructed. The obtained solution allows to retrace the evolution of the stress-strain state of the manufactured pillar after the start and on completing the piecewise continuous additive process under consideration.

Repairing force for deformed casing shaping with spinning casing swage and damage behaviour of cement sheath

In oil and gas fields, the repairing force (bit weight) is generally determined by experience when the deformed casing is repaired using the spinning casing swage. However, unreasonable repairing force easily damages the cement sheath around the deformed casing and causes pipe sticking, which results in failure of the wellbore integrity. Hence, based on the Hertz contact theory, the present study established a mechanical model to calculate the repairing force required to repair the deformed casing without a cement sheath by spinning casing swage, and the reshaping force was determined by combining the structural features of the spinning casing swage with the method of mechanics and kinetics analysis regarding axial loading and circumferential deformation of the deformed casing. Finally, a mechanical model was presented that could calculate the repairing force of the deformation casing with cement sheath using the inversion method. Repairing experiments involving three types of deformed casings (casing without cement sheath, casing with undamaged cement sheath and casing with damaged cement sheath) were performed, from which the accuracy and reliability of the mechanical model was validated. The damage behaviour of the cement sheath after casing repair was investigated based on the experimental results and the damage mechanism was analysed based on Saint-Venant's deformation compatibility principle. Analysis results showed that three types of damage, including micro-annulus, transverse fracture and longitudinal fracture, were caused by high contact pressure between the steel ball on the spinning casing swage and the internal wall of the deformed casing and pressure fluctuation during repairing. The research results provide important guidance and decision making for practical repairing measures.

Damage detection algorithm based on using surface mounted fiber-optic sensors on Bragg gratings

In the paper the problem associated with the use of fiber-optic strain sensors based on Bragg gratings glued to the surface of the material for the defects registration and their development was studied. The first part of the work is devoted to the numerical and experimental estimation of the error of strain values calculated on the basis of information on physical quantities recorded by sensors. In the second part of the work, a method for defects registration for a given external loads is presented. A background of the technique is the Saint-Venant principle, according to which the change in the strain values with the appearance of a defect occurs in the area defined by 3 – 5 characteristic defect sizes. Based on this, in order to register a defect without preliminary information about the strain distribution, it is necessary to provide in the observed object a strains measurement by the sensors that form a spatial grid with a step of 3 to 5 sizes of the expected defect. Last can be practically unrealistic. In the framework of the considered methodology, based on the results of numerical simulation, the sensors are located in stress concentration zones and areas with moderate stress concentration. Further, in the object without defects on the basis of numerical or experimental data, all variants of the relationships of the strain values obtained by two different sensors are recorded. The change in the monitoring process at least of one of these relationships allows to make a conclusion about the defect appearance as well as its location and evolution. The effectiveness of the technique is illustrated on a sample of a polymer composite material, in which holes of different diameters are made, imitating the appearance and development of a defect.

Method of calculating the parameters of the mountain pressure epure

Elevated bearing load pressure is formed near the exposed part of the coal seam, compared with static stresses normal to the reservoir. The loading of the near-bottom part of the coal seam is formed by linearly damped, according to the principle of Saint-Venant, from the bottom of the face to the massif of the tangential stresses from contact friction between the formation and lateral enclosing rocks in the form of a reference rock pressure, the epure of which is described by a convex quadratic function whose initial value is normal stress at the top of the bottom hole fracture, and the final stress is to the rock pressure in the zone of the intact massif. In connection with the above scientific position, a method has been developed for determining the vertical normal stress at the top of the bottom hole fracture, the length of the epure, and the distance from the bottom to the maximum of the reference pressure.

Strain energy spectral density and information content of materials deformation

A broader range of analytical tools can enhance understanding of the unusual mechanical properties of metamaterials and other advanced material systems. Here, we discuss a mechanics analogue of the Parseval's energy theorem that leads to a density of the strain energy in the reciprocal space. It reflects the ways for an elastic medium to translate static deformation patterns between two points in space. A normalized spectral density also provides an information entropy of deformation at those points. Both differential and discrete (numerical) entropies of the Shannon's type are discussed. Spectral entropy is a basic measure of information available in the material interior about surface loads, or a measure of disorder introduced into elastic medium by the deformation. An exact analytical entropy function is derived for an isotropic plane solid under Gauss-distributed and point loads. Approaches to numerical calculation of spectral entropy in computational solid mechanics are also discussed. Energy spectral density and spectral entropy of an elastic continuum is shown to translate, logically, in agreement with the Saint-Venant's principle. However, it also becomes clear that microstructured media may demonstrate anomalous pathways of evolution of the strain energy spectrum, enabling interesting transformation mechanics studies of engineered material systems.

Scalloping and Stress Concentration in DRIE-Manufactured Comb-Drives

In the last decades, microelectromechanical systems have been increasing their number of degrees of freedom and their structural complexity. Hence, most recently designed MEMSs have required higher mobility than in the past and higher structural strength and stability. In some applications, device thickness increased up to the order of tens (or hundred) of microns, which nowadays can be easily obtained by means of DRIE Bosch process. Unfortunately, scalloping introduces stress concentration regions in some parts of the structure. Stress concentration is a dangerous source of strength loss for the whole structure and for comb-drives actuators which may suffer from side pull-in. This paper presents an analytical approach to characterize stress concentrations in DRIE micro-machined MEMS. The method is based on the linear elasticity equations, the de Saint-Venant Principle, and the boundary value problem for the case of a torsional state of the beam. The results obtained by means of this theoretical method are then compared with those obtained by using two other methods: one based on finite difference discretization of the equations, and one based on finite element analysis (FEA). Finally, the new theoretical approach yields results which are in accordance with the known value of the stress concentration factor for asymptotically null radius notches.

The study of attenuation of the shear stresses in a straight rod near the point of load application

For engineering calculations must be used the principle of Saint-Venant. Usually assume that the "perturbations" of the stress-strain state decay approximately at a distance of one transverse dimension of the rod from the point of application of load [1]. In a previous article [2] a study was conducted of the normal stresses in the cross sections rigidly clamped rod for various loads applied to its free end. At the present work is the study of shearing stresses in the zone of application of the loads and the near kinematic heterogeneities related to the conditions of attachment of a rod. The limits of the applicability of the Saint-Venant principle are investigated while using the LIRA-CAD app.

A door to model reduction in high-dimensional parameter space

Model reduction techniques such as Proper Generalized Decomposition (PGD) are decision-making tools that are about to revolutionize many domains. Unfortunately, their computation is still problematic for problems involving many parameters, for which one has to face the “curse of dimensionality”. An answer to this challenge is given in solid mechanics by the so-called “parameter-multiscale PGD”, which is based on Saint-Venant's principle. In this article, a model problem composed of up to a thousand parameters is presented, showing that the method is able to overcome the “curse of dimensionality”.

Position-based dynamic of a particle system: a configurable algorithm to describe complex behaviour of continuum material starting from swarm robotics

In a previous work, we considered a two-dimensional lattice of particles and calculated its time evolution by using an interaction law based on the spatial position of the particles themselves. The model reproduced the behaviour of deformable bodies both according to the standard Cauchy model and second gradient theory; this success led us to use this method in more complex cases. This work is intended as the natural evolution of the previous one in which we shall consider both energy aspects, coherence with the principle of Saint Venant and we start to manage a more general tool that can be adapted to different physical phenomena, supporting complex effects like lateral contraction, anisotropy or elastoplasticity.

The mechanical problems on additive manufacturing of viscoelastic solids with integral conditions on a surface increasing in the growth process

Quasistatic mechanical problems on additive manufacturing aging viscoelastic solids are investigated. The processes of piecewise-continuous accretion of such solids are considered. The consideration is carried out in the framework of linear mechanics of growing solids. A theorem about commutativity of the integration over an arbitrary surface increasing in the solid growing process and the time-derived integral operator of viscoelasticity with a limit depending on the solid point is proved. This theorem provides an efficient way to construct on the basis of Saint-Venant principle solutions of nonclassical boundary-value problems for describing the mechanical behaviour of additively formed solids with integral satisfaction of boundary conditions on the surfaces expanding due to the additional material influx to the formed solid. The constructed solutions will retrace the evolution of the stress-strain state of the solids under consideration during and after the processes of their additive formation. An example of applying the proved theorem is given.

The development of estimated methodology for interfacial adhesion of semiconductor coatings having an enormous mismatch extent

The long-term reliability of multi-stacked coatings suffering the bending or rolling load was a severe challenge to extend the lifespan of foregoing structure. In addition, the adhesive strength of dissimilar materials was regarded as the major mechanical reliability concerns among multi-stacked films. However, the significant scale-mismatch from several nano-meter to micro-meter among the multi-stacked coatings causing the numerical accuracy and converged capability issues on fracture-based simulation approach. For those reasons, this study proposed the FEA-based multi-level submodeling and multi-point constraint (MPC) technique to conquer the foregoing scale-mismatch issue. The results indicated that the decent region of first and second-order submodeling can achieve the small error of 1.27% compared with the experimental result and significantly reduced the mesh density and computing time. Moreover, the MPC method adopted in FEA simulation also shown only 0.54% error when the boundary of selected local region was away the concerned critical region following the Saint-Venant principle. In this investigation, two FEA-based approaches were used to conquer the evidently scale mismatch issue when the adhesive strengths of micro and nano-scale multi-stacked coating were taken into account.

The study of the perturbation of normal stresses in the rod of the I-section in the zone of application of the load

The study of the implementation of the Saint-Venant principle for restrained rods of the I-section exposed to various loads at its free end is carried out. When using the program complex LIRA SAPR are defined zones of disturbance of normal stresses.

Quasistatic problems for piecewise-continuously growing solids with integral force conditions on surfaces expanding due to additional material influx

The piecewise continuous processes of additive forming of solids are studied. The being formed solids exhibit properties of deformation heredity and aging. The approaches of linear mechanics of growing solids in the framework of the theory of viscoelasticity of the homogeneously aging isotropic media are applied. Nonclassical boundary-value problems for describing the mentioned processes with the integral satisfaction of force conditions on some expanding due to the influx of additional material parts of the formed solid surface are investigated. A proposition about the commutativity of the time-derived integral operator of viscoelasticity with a limit depending on the solid point with the integration over an arbitrary, expanding due to the growth, surface inside or on the boundary of the growing solid is given. This proposition provides a way to construct the solution of the corresponding growing solids mechanics problem on the basis of Saint-Venant principle. The solution will retrace the evolution of the stress-strain state of the solid under consideration during and after the process of its additive formation. An example of applying the announced technic to modelling the processes of additive forming solids of conical shape under simultaneous action of end loads that are statically equivalent to an axial time-varying force is demonstrated.

Finite bending of a dielectric elastomer actuator and pre-stretch effects

A sandwiched dielectric elastomer actuator consisting of a hyperelastic material layer and two pre-stretched dielectric elastomer layers is introduced in this work, which is bi-directionally bendable when subject to a direct current voltage on the top or bottom layer. By using the pure bending deformation assumption and the nonlinear electroelasticity theory, the voltage-induced finite bending deformation of the actuator is studied theoretically with a focus on the pre-stretch effects. Although pure bending deformation may be not exactly ensured at the free end of the considered structure, according to the well known Saint Venant's principle, the boundary effect will not affect the deformation and stress distribution outside the boundary layers at the free end. Through theoretical modeling and numerical analyses, it is revealed that pre-stretch helps to reduce the drive voltage required to induce a desired bending deformation, and when the pre-stretch is comparatively large, there exists a pre-stretch dependent optimum thickness ratio so that the drive voltage is minimized. This work is expected to provide guide for the design and fabrication of high-performance soft actuators and soft robotics.