Large Deflection Research Articles

The non-plane initiation and propagation mechanism of multiple hydraulic fractures in tight reservoirs considering stress shadow effects

Multi-cluster fracturing in horizontal wells is an important stimulation technology for unconventional tight reservoirs. The interference among multiple simultaneously growing hydraulic fractures results in a non-uniform fluid distribution into fractures and non-planar fracture geometries. In this paper, the competition mechanism between perforations under the stress shadow, the formation mechanism of complex non-plane fractures near the wellbore, and the initiation and propagation mechanism of multiple hydraulic fractures are investigated by using the 3D lattice method. Also, the effects of fracture geometry, fracture spacing and rock mechanics parameters on the stress interaction among fractures are analyzed by the dislocation theory. The results showed the complexity of fracture initiation near the wellbore and the low efficiency of the perforation initiation in the case of spiral perforation, due to a strong interaction between perforations. However, these initial micro-cracks will eventually coalesce to be a macroscopical fracture which perpendicular to the direction of minimum principal stress. The study indicated that stress interaction and fracture propagation is significantly affected by inclined well orientation. The hydraulic fracture is more likely to deflect and propagate non-uniformly with the increasing width of hydraulic fracture, the decreasing spacing between perforation clusters and the increasing rock modulus. When the horizontal principal stress difference is small, it is obvious to cause a large deflection angle and sometimes furcation of fractures. This study can provide a theoretical basis and technical guidance for hydraulic fracturing design in tight reservoirs.

Improved Power Series Solution of Transversely Loaded Hollow Annular Membranes: Simultaneous Modification of Out-of-Plane Equilibrium Equation and Radial Geometric Equation

The ability to accurately predict the shape of a transversely loaded hollow annular membrane is essential to the design of bending-free hollow annular shells of revolution, which requires a further improvement in the hollow annular membrane solution to meet the needs of this accurate prediction. In this paper, the large deflection problem of a transversely loaded hollow annular membrane is reformulated by simultaneously modifying the out-of-plane equilibrium equation and radial geometric equation, and a newer and more refined power series solution is derived. The reason why the classical radial geometry equation induces errors is revealed. The convergence and asymptotic behavior of the power series solution obtained is analyzed numerically. The newly derived solution is compared with the two previously derived solutions graphically, showing that the newly derived solution performs basically as well as expected. In addition, the anticipated use of the hollow and not-hollow annular membrane solutions for the design application of bending-free annular shells of revolution is discussed.

Functionally graded nanobeams subjected to large deflection by considering surface effects

Functionally graded nanobeams subjected to large deflection by considering surface effects

Accurate identification of cross-sectional bending stiffness in large-scale wind turbine blades through static loading test

To ensure the safe and stable operation of wind turbines, it is essential that the blades have sufficient bending stiffness to prevent excessive deflection and tower impact. A method combining static loading test and numerical analysis is proposed to solve the problem of identifying the cross-sectional bending stiffness of actual blades. Two working conditions, namely multi-point static test and single-point static calibration, are considered. A mathematical model of a cantilever beam with an initial small curvature and variable cross section is established. By fitting the deflection measurement data obtained from the static loading test, the deflection differential equation is solved by the finite difference method to obtain expressions for the cross-sectional bending stiffness under large and small deflection. The analysis results of several blade models show that the bending stiffness identification error for most blade sections is less than 10%, confirming the effectiveness of this method. This provides a practical and theoretical basis for the design, analysis, and refinement of test parameters for large-scale blades.

Strengthening of Precast Segmental Bridge built in 1960s using External Tendons

AbstractRecently more attention has been paid to ageing infrastructure under increased traffic on the roads in Slovakia. As a result, selected prestressed concrete bridges have been inspected. Evidence of their poor technical condition has been discovered including serious faults such as large permanent deflection, wide cracks, extensive prestressing degradation, and others. The original conceptual design and poor quality of prestressing steel protection are often the decisive factor. This paper provides an example of bridge rehabilitation design of the concrete precast segmental bridge in an emergency condition. The bridge in question which is located over the water reservoir Ruzin was built in 1967 as one of the first‐generation precast segmental bridges in Slovakia. Its service life was successfully extended using structural system modification and strengthened using external prestressing. The initial assumptions for the rehabilitation have been based on the detailed analysis of the defects, and assessment of existing conditions including the analytical approach of residual prestressing determination. This concept represents the potential solution for achieving enhanced service life of similar bridges.

Postcritical Imperfection Sensitivity of Functionally Graded Piezoelectric Cylindrical Nanoshells Using Boundary Layer Solution

The nonlocal (NL) and the strain gradient (SG) based equivalent continuum theories were proposed for nanomechanics by accommodating the long-range molecular interactions and to bypass the computationally expensive atomistic simulations. Nanostructures are often made of smart materials (e.g., piezoelectric, piezoceramic, and flexoelectric) for multifunctional properties. Applications of nanostructures in critical systems deserve accurate analysis. The buckling and postcritical behavior of an axially loaded, piezoelectric, nonlocal strain gradient (NLSG) thin cylindrical shells with functionally graded elastic properties were presented in earlier studies following Donnell’s approach, leading to a set of stiff, nonlinear, partial differential equations, accommodating the prebuckling nonlinearity and large postcritical deflection. This study extends the previous work by accommodating geometric imperfections in the formulation. Furthermore, a consistent thickness-wise distribution for the electric potential is also adopted following the Maxwell equation. The boundary layer (BL) concept is employed to solve the resulting nonlinear equations via asymptotic expansions of the regular and the BL fields. The solution is numerically illustrated on moderately short shells, illustrating the influence of the geometric imperfections. The critical load remains imperfection sensitive, yet the imperfections change an unstable postcritical path into a stable one. The influence of the functional gradation and the external electric fields on the imperfection sensitivity behavior are illustrated.

Model-based Design of the COAST Guidewire Robot for Large Deflection.

Minimally invasive endovascular procedures involve the manual placement of a guidewire, which is made difficult by vascular tortuosity and the lack of precise tip control. Steerable guidewire systems have been developed with tendon-driven, magnetic, and concentric tube actuation strategies to enable precise tip control, however, selecting machining parameters for such robots does not have a strict procedure. In this paper, we develop a systematic design procedure for selecting the tube pairs of the COaxially Aligned STeerable (COAST) guidewire robot. This includes the introduction of a mechanical model that accounts for micromachining-induced pre-curvatures with the goal of determining design parameters that reduce combined distal tip pre-curvature and minimize abrupt changes in actuated tip position for the COAST guidewire robot through selection of the best flexural rigidity between the tube pairs. We present adjustments in the kinematics modeling of COAST robot tip bending motion, and use these to characterize the bending behavior of the COAST robot for varying geometries of the micromachined tubes, with an average RMSE value for the tip position error of 0.816 mm in the validation study.

Design Guidelines for Thin Diaphragm-Based Microsystems through Comprehensive Numerical and Analytical Studies.

This paper presents comprehensive guidelines for the design and analysis of a thin diaphragm that is used in a variety of microsystems, including microphones and pressure sensors. It highlights the empirical relations that can be utilized for the design of thin diaphragm-based microsystems (TDMS). Design guidelines developed through a Finite Element Analysis (FEA) limit the iterative efforts to fabricate TDMS. These design guidelines are validated analytically, with the assumption that the material properties are isotropic, and the deviation from anisotropic material is calculated. In the FEA simulations, a large deflection theory is taken into account to incorporate nonlinearity, such that a critical dimensional ratio of a/h or 2r/h can be decided to have the linear response of a thin diaphragm. The observed differences of 12% in the deflection and 13% in the induced stresses from the analytical calculations are attributed to the anisotropic material consideration in the FEA model. It suggests that, up to a critical ratio (a/h or 2r/h), the thin diaphragm shows a linear relationship with a high sensitivity. The study also presents a few empirical relations to finalize the geometrical parameters of the thin diaphragm in terms of its edge length or radius and thickness. Utilizing the critical ratio calculated in the static FEA analysis, the basic conventional geometries are considered for harmonic analyses to understand the frequency response of the thin diaphragms, which is a primary sensing element for microphone applications and many more. This work provides a solution to microelectromechanical system (MEMS) developers for reducing cost and time while conceptualizing TDMS designs.

Large Deflection Effects on the Energy Release Rate and Mode Partitioning of the Single Cantilever Beam Sandwich Debond Configuration

Abstract This paper investigates the effects of large deflections on the energy release rate and mode partitioning of face/core debonds for the single cantilever beam sandwich composite testing configuration, which is loaded with an applied shear force and/or a bending moment. Studies in this topic have been done by employing geometrically linear theories (either Euler–Bernoulli or Timoshenko beam theory). This assumes that the deflection at the tip of the loaded debonded part is small, which is not always the case. To address this effect, we employ the elastica theory, which is a nonlinear theory, for the debonded part. An elastic foundation analysis and the linear Euler Bernoulli theory are employed for the “joined” section where a series of springs exists between the face and the substrate (core and bottom face). A closed form expression for the energy release rate is derived by the use of J-integral. Another closed-form expression for the energy release rate is derived from the energy released by a differential spring as the debond propagates. Furthermore, a mode partitioning angle is defined based on the displacement field solution. Results for a range of core materials are in very good agreement with the corresponding ones from a finite element analysis. The results show that large deflection effects reduce the energy release rate but do not have a noteworthy effect on the mode partitioning. A small deflection assumption can significantly overestimate the energy release rate for relatively large applied loads and/or relatively long debonds.

Incorporating Geometric Nonlinearity in Theoretical Modeling of Muscle-Powered Soft Robotic Bio-Actuators

Abstract Biohybrid actuators aim to leverage the various advantages of biological cells over artificial components to build novel compliant machines with high performance and autonomy. Significant advances have been made in bio-fabrication technologies, enabling the realization of muscle-powered bio-actuators. However, the mechanics of muscle-scaffold coupling has been relatively understudied, limiting the development of bio-actuators to intuitive or biomimetic designs. Here, we consider the case of implementing muscle-based actuation for soft robotic swimmers operating at low Reynolds numbers. We develop an analytical model to describe the elasto-hydrodynamic problem and identify key design parameters. Muscle contraction dynamics is characterized experimentally and the implications of nonlinear amplitude-frequency relationship of muscle-based actuation are discussed. We show that a novel bio-actuator with high performance can be developed by introducing compliant flexural mechanisms undergoing large deflection. Geometric nonlinearities are accounted for in the analysis of the force-deflection relationship for the flexural mechanism. Our results show that for expected muscle contraction forces, this novel bio-actuator can outperform previous muscle-powered swimmers by up to two orders of magnitude in swimming speed.

Large deflection deformation behavior of a Zr-based bulk metallic glass for compliant spinal fixation application

A novel compliant spinal fixation designed based on the concept of compliant mechanisms can reduce the stress-shielding effect and adjacent segment degeneration (ASD) effectively, but propose higher requirements for the properties of the used materials. Bulk metallic glasses (BMGs), as a kind of young biomaterials, exhibiting excellent comprehensive properties, which are attractive for compliant spinal fixation. Here, according to the practical service condition of the basic elements in compliant spinal fixation, large deflection deformation behaviors of Zr61Ti2Cu25Al12 (at.%, ZT1) BMG beam, including elastic, yielding and plastic were investigated systematically. It was shown that the theoretical nonlinear analytical solution curve as the benchmark not only with the capacity to predict the nonlinear load-deflection relation within the elastic deformation regime, but also assists to capture the yielding event roughly, which can be used as a powerful design tool for engineers. To capture the beginning of the yielding event exactly, bending proof strength (σp,0.05%) accompanied with tiny permanent strain of 0.05% was proposed and determined for BMGs in biomedical implant applications, which is of significance for setting the allowable operating limits of the basic flexible elements. By approach of interrupted loading-unloading cycles, plastic deformation driven by the bending moment can be classified into two typical stages: the initial stage which mainly characterized by the nucleation and intense interaction of abundant shear bands when the plastic strain below the critical value, and the second stage which dominated by the progressive propagation of shear bands and coupled with the emergence of shear offsets on tensile side. The plasticity of BMG beam structures depends on the BMG's inherent plastic zone size (rp). When the half beam thickness less than that of the rp, the plastic deformation of BMGs will behave in a stable manner, which can be acted as the margin of safety effectively.

Analysis of Large Deflection of Nanobeams with Circular Cross Section under Distributed Load Taking into Surface Effects

Analysis of Large Deflection of Nanobeams with Circular Cross Section under Distributed Load Taking into Surface Effects

Mechanism of characteristic change of panel flutter caused by oblique shock impingement

Compared to the shock-free condition, the weak shock impingement stabilizes the flexible panel, while the strong shock impingement leads to the early onset of panel flutter with a significant increase in flutter amplitude and frequency. However, the reason for this change by shock impingement remains unclear. The current research examines the mechanism of this change by an in-house code where the von Kármán’s large deflection plate theory is coupled with two-dimensional Euler equations. Compared to the shock-free condition, the oblique shock impingement leads to the change of local dynamic pressure on the panel as well as the static pressure differential across the panel. The analysis on the influence of these changes indicates that, on the one hand, the average dynamic pressure on the panel becomes larger than the shock-free condition, accelerating the onset of panel flutter. On the other hand, the change of the static pressure differential across the panel alters the coupling characteristic between different natural frequencies (modes) of the panel structure. The dynamic response of panel flutter under shock impingements is dominated by the coupling between the second and third modes instead of the first two modes for panel flutter under the shock-free condition. The combined effect of these two changes leads to the change of flutter characteristics of the panel under shock impingement. These findings provide valuable insights into the mechanism of shock-induced panel flutter.

Analytical Modeling and Application for Semi-Circular Notch Flexure Hinges

Flexure-based compliant mechanisms can be used to achieve bio-imitability and adaptability in the applications of biomedical engineering. However, a nonlinear load-displacement profile increases the design complexity of this type of compliant mechanism, especially when the cross-section of the flexure hinge is not constant. This paper proposes two general analytical models by analyzing the compliance and stress characteristics of the semi-circular notch flexure hinge undergoing large deflections, which is a typical variable cross-section of a flexure hinge, based on the Castigliano’s second theorem and the finite elements analysis method. As a case study for verification, three compliant four-bar linkage mechanisms are designed based on the proposed design approach, the design method proposed by Howell, and the equations proposed by Lobontiu, respectively. The results show that the design accuracy is improved 36% in comparison with designs from Howell and Lobontiu. Finally, a flexure-based artificial finger is designed and manufactured based on the proposed optimization approach. The performance test of the prototype shows that the artificial finger has good bio-imitability and adaptability with respect to joint movements.

Numerical studies on the aeroelasticity of a shipboard helicopter rotor during engagement and disengagement with ship motions using CFD/CSD coupling approach

When the shipboard helicopter rotor engages and disengages on a ship deck, due to the influence of wind-over-deck environment and ship airwake, the heterogeneity of rotor airload is large. This large heterogeneity leads to the fluctuation of blade deflection which changes the aerodynamic forces and causes danger of over deflection. To investigate the aeroelasticity of a rotor blade and find the rule of deflection and the deformed blade aerodynamic force in the rotor/ship coupling flow field, we establish a loose computational fluid dynamics/computational structural dynamics (CFD/CSD) coupling model. This coupling model simulates the aeroelastic characteristics of the shipboard helicopter rotor during engagement and disengagement with ship motions. In the simulation, the CFD solver is based on Reynolds-averaged Navier-Stokes (RANS) equations and the CSD solver is based on the moderate deflection beam model. We compare the results with the experimental data and verify the effectiveness of the simulation. Through the analysis of the flow field, blade deflection, and aerodynamic forces under various wind-over-deck conditions at different positions, we find in certain cases, that the elastic deflection has a significant impaction the blade aerodynamic force. As the wind direction anglesincrease, the rotor blade is more likely to have a large negative elastic twist deflection, causing drop of aerodynamic forces and danger of over deflection.

An Improved Mathematical Theory for Designing Membrane Deflection-Based Rain Gauges

This paper is devoted to developing a more refined mathematical theory for designing the previously proposed membrane deflection-based rain gauges. The differential-integral equations governing the large deflection behavior of the membrane are improved by modifying the geometric equations, and more accurate power-series solutions of the large deflection problem are provided, resulting in a new and more refined mathematical theory for designing such rain gauges. Examples are presented to illustrate how to analyze the convergence of the power-series solutions and how to numerically calibrate membrane deflection-based linear rain gauges. In addition, some important issues are demonstrated, analyzed, and discussed, such as the superiority of the new mathematical theory over the old one, the reason why the classical geometric equations cause errors, and the influence of changing design parameters on the input–output relationships of rain gauges.

Test Methods for the Mechanical Characterization of Flexure Hinges

BackgroundThe design of compliant mechanisms requires detailed knowledge about the stiffness properties of their flexible segments. However, there are no standardized test methods for flexure hinges, and therefore the influence of manufacturing-specific effects, such as anisotropy, on the stiffness properties cannot be quantified.ObjectiveThis paper presents novel test methods for variable cross-section flexure hinges subjected to large deformations and pure bending loading, which determine the bending stiffness of flexure hinges over their entire deflection range using a universal testing machine.MethodsThe novel test methods for flexure hinges are based on the tensile test, the four-point bending test (FPBT), and the column bending test (CBT). These test methods were initially formulated for constant cross-section specimens, but are adapted in this study to examine variable cross-section specimens. The derived test methods are validated by using isotropic materials with well-known properties and by comparing the calculated deflections with deflections measured by means of image processing.ResultsThe deflection validation shows that the adapted CBT (aCBT) is accurate over the entire deflection range, achieving curvature of up to kappa =0.40;text {mm}^{-1}, whereas the maximum curvature in the adapted FPBT (aFPBT) is limited by the test methodology to about kappa =0.15;text {mm}^{-1}. At small strains, the flexural modulus determined in the aCBT and aFPBT agrees well with the Young’s modulus determined in the tensile test, as would be expected for isotropic materials.ConclusionThe aCBT proves to be a suitable test method for flexure hinges at large deflections, whereas the stiffness characterization at small deflections can be performed with both the aCBT and the aFPBT. The presented test methods validated on isotropic materials form the basis for characterizing anisotropic flexure hinges with geometry-dependent stiffness properties.

A SDoF Response Model for Elastic–Plastic Beams under Impact at Any Point on the Span

This study aims to investigate the elastic–plastic response of a clamped-clamped beam struck by a rigid heavy mass with a low velocity at any point on the span. The impact system is simplified as a single-degree-of-freedom (SDoF) mass-spring model to formulate the beam’s equations of motion during loading and unloading. With the consideration of material elasticity and large deflection, elastic–plastic analytical solutions are derived to predict the global deformation behavior of the beam. Validation and comparison are conducted against numerical simulations performed using ABAQUS, and satisfactory agreement is achieved for the predictions of the structural dynamic behavior. Meanwhile, a parametric study is presented to assess the influence of the impact location on the characteristic response parameters, which suggests that the structural stiffness increases as the impact location approaches the beam’s support. The findings drawn from analytical and numerical studies can be useful in the anti-impact design of engineering structures.

Modeling Method for Static Large Deflection Problem of Curved Planar Beams in Compliant Mechanisms Based on a Novel Governing Equation

Abstract Slender beams serve as typical flexible components to transfer motion, force, and energy in compliant mechanisms. Therefore, the accurate and efficient kinetostatic modeling for the slender beams is highly needed in the synthesis and analysis of compliant mechanisms. Several impressive and great modeling methods have been completed by the pioneering researchers based on the governing equation of slender beams, which can solve the deflection of the beam while the beam-tip loads are known. However, parts of the beam-tip loads are unknown in some scenarios and the traditional governing equation becomes inefficient in these scenarios. The aim of this paper is to propose a novel modeling method for the compliant mechanism, which can solve the large deflection problem without iterative algorithm. In this research, a novel governing equation of slender beams has been established based on Castigliano’s principle and simplified by the differential geometry. In the proposed governing equation, the statically force equilibrium equations and geometric constraint equations between different slender beams can be established in the boundary conditions, therefore the model of compliant mechanisms can be solved directly even if parts of the beam’s tip loads and displacements are unknown. The proposed governing equation provides the deformed shapes and cross-sectional loads of the slender beams, which help the designer to check the mechanical interference and strength in the design process. The numerical examples are presented to demonstrate the feasibility of proposed method.

Geometric nonlinear analysis of slender layered non-prismatic beams with interlayer slip

This paper presents a novel model that, for the first time, simultaneously captures the effects of variable cross-section and interlayer slip on the geometrically nonlinear response of slender symmetrically layered beams within the framework of a beam theory. While the central layer has a constant cross-section along its length, the height of the two outer layers can vary arbitrarily. A nonlinear axial strain–displacement relationship captures the influence of moderately large deflections for immovably supported beams. The kinematic hypothesis of the Euler–Bernoulli theory is applied layerwise. The relationship between the interlaminar shear tractions and the interlayer slips is assumed to be linear. The Galerkin method is used to solve the derived initial–boundary value problem. Several application examples demonstrate the influence of the interlayer slip, the variable cross-section, and the geometrically nonlinear formulation of the equations of motion on the static and dynamic response. The results of comparative analyses with a numerically much more computational expensive finite element model discretized by plane stress continuum elements are in excellent agreement with the outcomes of the presented theory.