Deformation Formula Research Articles

Deformation and leakage mechanisms at hydraulic clearance fit in deep-sea extreme environment

Hydraulic system is widely used as a power source in deep-sea operation equipment. The sealing performance and the relative movement of the parts in hydraulic components are significantly affected by the structural deformation at the clearance fit and the viscosity increase of the hydraulic fluid medium, which are both caused by high seawater pressure. In this paper, the deformation formula at the clearance fit in the deep-sea environment is deduced, which indicates that the deformed height of the fit clearance decreases linearly in the axial direction. A minimum clearance design criterion is proposed, and it is found that the smaller the difference between the bulk modulus of the matching parts and the fit radius are, the smaller the variation of the height of the fit clearance is under various working conditions and the smaller the required minimum initial height is. The leakage flow rate formula at the clearance fit in the deep-sea environment is deduced as well, which introduces modified factors to consider the effects of structural deformation, viscosity increase, and eccentricity. The calculation result shows that under the condition of 11 000 m deep sea, the leakage flow rate calculated by the classic formula is about five times larger than that of the modified formula. Multi-parameter fluid–solid-interaction simulations are carried out to verify the correctness of the deduced deformation and leakage flow rate formulas. The leakage flow rate of the situation with inclination involved is also analyzed through numerical simulations.

Co-seismic internal deformations in a spherical layered earth model

SUMMARYUsing a numerical integral method, we deduced a set of formulae for the co-seismic internal deformation in a spherically symmetric earth model, simultaneously taking self-gravitation, compressibility and realistically stratified structure of the Earth into account. Using these formulae, we can calculate the internal deformation at an arbitrary depth caused by an arbitrary seismic source. To demonstrate the correctness of our formulae, we compared our numerical solutions of radial functions with analytical solutions reported by Dong & Sun based on a homogeneous earth model; we found that two sets of results agree well with each other. Our co-seismic internal Green's functions in the near field agree well with the results calculated by the formulae of Okada, which also verifies our Green's functions. Finally, we calculated the Coulomb stress changes on the Japanese Islands and Northeast China induced by the Tohoku-Oki Mw 9.0 earthquake using the methods described above. We found that the effect of layered structure plays a leading role on the near field, while curvature occupies a dominant position on the deep region of the far field. Through a comparison of the Coulomb stress changes at a depth of 10 km on a layered earth model calculated by our method along with the corresponding results of Okada, we found that the discrepancy between them in near field was ∼31.5 per cent, and that of far field was >100 per cent of the signals.

Experiment and finite element analysis study on the deflection of aluminum extruded 6063-T5 hollow structural beam

In this study, the tensile tests were performed on samples extruded from aluminum beam to confirm the material properties of the various parts. Then, the beam was cut into lengths of 1160 mm and 680 mm to perform the bending test, and the beam loading point deformation was recorded. At the same time, the amount of beam deformation caused by three type beam fixing methods was compared. The Solidwork 3D CAD software used to build the beam model and perform finite element analysis. Among them, the analysis results of different boundary conditions were compared with the experimental results. The experimental load conditions were used to calculate the theoretical deformation formula amount through the beam theory deformation formula of material mechanics. It is to compare beam deformation data of experiments, finite element analysis, and theoretical formulas, and explore possible causes of differences.

The field of tensions of a rotating anisotropic disc of a variable thickness loaded with undistracted forces on the outer contour

The work gives a solution of the plane elasticity problem for rotating polar-orthotropic annular disks of a variable thickness. The disk is loaded with a system of equal focused forces on the outer contour applied evenly along the rim and symmetric concerning the diameter. The disk is seated with an interference fit on the flexible shaft so that a constant contact pressure acts on the interior contour. The stresses and deformations arising in such a rotating anisotropic annular disk will be non-axisymmetric. A conclusion of a fourth-order partial differential equation for the effort function is drawn. Its general solution is searched out in the form of a Fourier series of cosines with even numbers. As a result, an infinite system of ordinary differential equations is solved for the coefficients of the series. These differential equations correspond to the linear Volterra integral equations of the 2 nd kind, which are solved using resolvents. Constants of integration are determined from the border conditions. Expressions for the stress components are written through the effort function by the well-known formulas. We find the components of the displacement vector in the disk by the integration of the Hooke’s law equations for the polar-orthotropic plate. We calculate the deformation components in a ring anisotropic disk by Cauchy differential relations if we know the displacements. The solved formulas for stresses, deformations and displacements completely describe the stress-deformed state in a rotating polar-orthotropic disc of variable thickness with a system of focused forces on the outer contour. The results of the work can be used in the design of working disks of turbomachines and turbo compressors, as well as rotors of centrifugal stands.

Analytical modeling and simulation on lateral mechanical characteristics of high-speed train traction motor hanging leaf spring

In this paper, using the theoretical analysis method, according to the actual structure of the hanging leaf spring of the traction motor mounted on the frame, the lateral force model of the hanging leaf spring of the traction motor was established. Then, through theoretical deduction, the deformation analytical calculation formula and the stress analytical calculation formula of the hanging leaf spring were established. The correctness of the leaf spring’s lateral force model was established and its deformation and stress analytical formulae were verified using ANSYS finite element analysis software. Based on this, according to the deformation analytical formula and the stress analytical formula of the leaf spring established, the influence of the main structural parameters on the mechanical characteristics of the leaf spring was discussed, and the reliability of the analytical analysis method of the lateral mechanical characteristics of the traction motor hanging leaf spring was verified by the loading–unloading test. The results show that the deformation and the load of the leaf spring change linearly. The changes of leaf spring’s stress at different positions can be considered as being composed of three sections: a linear change section in the root, a nonlinear change section in the middle, and a nonlinear change section in the end. In the structural parameters, the end thickness [Formula: see text] has the greatest influence on the stiffness and the stress of the leaf spring, and the maximum thickness of the leaf spring eye [Formula: see text] has the least influence on the stiffness and the stress of the leaf spring. The influence degree of other parameters on the stiffness of the leaf spring is [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] in order, and the influence degree on the stress of the leaf spring is [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] in order. In addition, when the root thickness [Formula: see text] is greater than a certain value, the maximum stress point of the leaf spring appears at the end position [Formula: see text]. This study can provide a useful reference for the intelligent forward design and the rapid analysis of the mechanical characteristics of high-speed train traction motor hanging leaf spring.

Novel Segmented Roadside Plugging-Filling Mining Method and Overlying Rock Mechanical Mechanism Analyses

The no-pillar mining method is widely used in coal mining engineering because of its superiority in resolving mine pressure hazards and protecting natural resources. In view of the geological conditions of stable strata in a coal mine of the Shandong Coal Zone, a novel segmented roadside plugging-filling mining method is proposed by introducing the filling coefficient into roadside filling. The operation process is designed with a new grouting filling forming device. Based on the relationship between the theoretical deflection of a cantilever beam and geometric settlement and parallel settlement models, strength formulas applicable to the segmented roadside plugging-filling (SRPF) method at different migration stages are obtained, and the deformation formulas of a roadway toward a rock slab are solved by an elastic equivalent model. Further, the determination procedure of the filling mode under the mining method is clarified. The SRPF method was implemented on a test stope, and the test results indicated that the theoretical deformation produced by the SRPF method was small and can meet the mining requirements. Through on-site test monitoring, the deformation of surrounding rock was 0–160 mm and the filling body under the SRPF method could maintain its own strength and the stability of the surrounding rock. In addition, entire successful mining been completed in the working stope, which further proves the applicability of this method. The backfilling cost of the gob-side filling was reduced by approximately 50%, and the backfilling efficiency was improved in the mine. The description of the novel mining method may provide theoretical and practical guidance for coal mining in similar geological conditions.

CALCULATION AND ANALYSIS OF THE PHYSICAL STOREY DRIFT FOR HIGH-RISE FRAME-DIAGRID STRUCTURES

The high-rise frame-diagrid structure is a new type of dual system structure. The inner frame part can create a large space, and the external diagrid part can provide a larger lateral stiffness. In this paper, the lateral deformation formula for the high-rise frame-diagrid structures is derived. The bending deformation of the structure is divided into the bending rotational deformation and the floor rigid rotational deformation. The physical storey drift is proposed. The physical storey drift is directly related to the structural damage. When the structure is in the plastic state, the structure maximum storey drift and maximum physical storey drift are in different positions. It is recommended to use both storey drift and physical storey drift as structural deformation limitation criteria. Finally, the proposed method is used to structural parameters analysis for the high-rise frame-diagrid structure. It provides reference for the structural design.

Vibration of axially moving hyperelastic beam with finite deformation

In this paper, we study the vibration of an axially moving hyperelastic beam under simply supported condition. The kinematic of the axially moving beam have been described by Eulerian-Lagrangian formulation. In continuum mechanics frame, the finite deformation formula and a higher order shear deformation beam theory are applied to describe the deformation of the axially moving hyperelastic beam. In these formulas the material parameter, shear deformation and the geometric non-linearity have been taken into account. Through the Hamilton principle, the governing equations of nonlinear vibration are obtained, where the transverse vibration is coupled with the longitudinal vibration. When the velocity is a constant, the critical speed and natural frequencies are determined by solving the corresponding linear equations. Meantime, effects of the geometrical and material parameters on the critical speed and natural frequencies have been investigated. Comparisons among the critical velocities of the hyperelastic and Euler linear beam are also made. The results show that the critical velocity of hyperelastic beam is larger than that of linear Euler–Bernoulli beam. For the natural frequencies, we have the same conclusions. Lastly, by the multiple scales method, the leading order analytical solutions of the equilibrium state of axially moving hyperelastic beam in the supercritical regime are obtained. Furthermore the amplitudes of analytical solutions of the hyperelastic beam have been compared with that of linear Euler–Bernoulli beam. The effects of the material and geometrical parameters on the asymptotic solutions and the amplitude has been analyzed.

Possibility of the right-opened Mn-oxo intermediate (R-oxo(4444)) among all nine intermediates in the S3 state of the oxygen-evolving complex of photosystem II revealed by large-scale QM/MM calculations

Large-scale quantum mechanics/molecular mechanics (QM/MM) calculations were performed to elucidate the possible intermediates in the S3 state of the oxygen-evolving complex of photosystem II. We have investigated nine S3 candidates in the R-/L-opened Mn-hydroxide, Mn-oxo, Mn-peroxide and Mn-superoxide intermediate states, and their optimized CaMn4O6 structures and the Mn valence states were compared to the experimental results revealed by recent serial femtosecond crystallography (SFX) and X-ray emission spectroscopy (XES). By using the Jahn-Teller (JT) deformation formulae, the Mn4-O(5) and O(5)-O(6) distances were estimated from the Mn4-Mn3 distances. Among all the calculated S3 intermediates, not only the right-opened hydroxo intermediate (R-hydroxo), but also the right-opened Mn-oxo intermediate (R-oxo(4444)) were found to be plausible based on the calculated CaMn4O6 structure and the Mn valence state (4444).

On the relevance of generalized disclinations in defect mechanics

The utility of the notion of generalized disclinations in materials science is discussed within the physical context of modeling interfacial and bulk line defects like defected grain and phase boundaries, dislocations and disclinations. The Burgers vector of a disclination dipole in linear elasticity is derived, clearly demonstrating the equivalence of its stress field to that of an edge dislocation. We also prove that the inverse deformation/displacement jump of a defect line is independent of the cut-surface when its g.disclination strength vanishes. An explicit formula for the inverse deformation jump of a single localized composite defect line in terms of given g.disclination and dislocation strengths is deduced based on the Weingarten theorem for g.disclination theory (Weingarten-gd theorem) at finite deformation. The Burgers vector of a g.disclination dipole at finite deformation is also derived.

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

In this survey paper, we give an overview of our recent works on the study of the $W$-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the $W$-entropy formula for the heat equation associated with the Witten Laplacian on $n$-dimensional complete Riemannian manifolds with the $CD(K, m)$-condition, and the $W$-entropy formula for the heat equation associated with the time dependent Witten Laplacian on $n$-dimensional compact manifolds equipped with a $(K, m)$-super Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$. Furthermore, we prove an analogue of the $W$-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two $W$-entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the $W$-entropy formula for the Langevin deformation. Finally, we make a discussion on the $W$-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory.

Energy Method of Static Analysis of Stiffened Plates on Elastic Foundation

This paper is devoted to derive the calculating formula of the deformation and the internal force of stiffened plate on elastic foundation, which is beneficial for the engineering design. Nowadays, we generally use the finite element method to solve the deformations and internal forces of stiffened plates on elastic foundation under loads, which the calculate setting is complex and the amount of calculation is large. In this paper the calculating deformation formula of this construction on elastic foundation under loads is derived by means of energy method. Calculating the potential energy of the foundation plate, that of the foundation beams, that of the loads on the stiffened plate and that of the foundation reactions forces, separately, and substituting them into the minimum condition of total potential energy while the structure is equilibrium, the calculating deformation formulas of this construction is obtained. The internal forces of foundation plate can be obtained according to formulas of mechanics of elasticity. The internal forces of foundation beams can be obtained according to the differential relationships among the deformation, bending moment and shearing force of the beam. Using the formulas in this paper, the deformations and the internal forces of stiffened plate on elastic foundation can be calculated out easily. Comparing with the finite element method, the amount of computation can be reduced greatly by using these formulas in this paper.

Understanding Two Different Structures in the Dark Stable State of the Oxygen-Evolving Complex of Photosystem II: Applicability of the Jahn-Teller Deformation Formula.

Tanaka et al. (J. Am. Chem. Soc., 2017, 139, 1718) recently reported the three‐dimensional (3D) structure of the oxygen evolving complex (OEC) of photosystem II (PSII) by X‐ray diffraction (XRD) using extremely low X‐ray doses of 0.03 and 0.12 MGy. They observed two different 3D structures of the CaMn4O5 cluster with different hydrogen‐bonding interactions in the S1 state of OEC keeping the surrounding polypeptide frameworks of PSII the same. Our Jahn–Teller (JT) deformation formula based on large‐scale quantum mechanics/molecular mechanics (QM/MM) was applied for these low‐dose XRD structures, elucidating important roles of JT effects of the MnIII ion for subtle geometric distortions of the CaMn4O5 cluster in OEC of PSII. The JT deformation formula revealed the similarity between the low‐dose XRD and damage‐free serial femtosecond X‐ray diffraction (SFX) structures of the CaMn4O5 cluster in the dark stable state. The extremely low‐dose XRD structures were not damaged by X‐ray irradiation. Implications of the present results are discussed in relation to recent SFX results and a blue print for the design of artificial photocatalysts for water oxidation.

A universal construction of universal deformation formulas, Drinfeld twists and their positivity

In this paper we provide an explicit construction of star products on U(g)-module algebras by using the Fedosov approach. This construction allows us to give a constructive proof to Drinfel'd theorem and to obtain a concrete formula for Drinfel'd twist. We prove that the equivalence classes of twists are in one-to-one correspondence with the second Chevalley-Eilenberg cohomology of the Lie algebra g. Finally, we show that for Lie algebras with K\"ahler structure we obtain a strongly positive universal deformation of *-algebras by using a Wick-type deformation. This results in a positive Drinfel'd twist.

Research on machining characteristic of double-layer elastomer in pneumatic wheel method

For improvement of finishing effect to the laser hardening mold’s free-form surface with high hardness, double-layer elastic mechanics theory of pneumatic wheel based with softness consolidation abrasives (SCA) is analyzed. Under the annular stress around, ratio coefficient m of modulus of elasticity of abrasive layer to modulus of rubber layer and ratio coefficient n of thickness of abrasive layer to radius of stress have been bought in for establishing machining force model and deformation formula of double-layer elastic wheel. After that, a double-layer elastic model of the wheel has been established by ANSYS and the machining process has been simulated. Stress distribution and deformation rule are given by simulation. In the experiments, we use fiber to reinforce inner rubber layer and take binder to hold on abrasive particles. Microscopic analysis demonstrates that pneumatic wheel accords with the situation of double-layer elastic mechanic theory. Moreover, machining platform is established and empirical results show that pneumatic wheel can finish laser hardening mold’s free-form surface efficiently. And conclusion shows pneumatic wheel with lower factor n can help to decrease R a when it faces with concave surface and pneumatic wheel with higher factor n can improve machining efficiency to convex surface.

Radial deformation computation and test of axial symmetry rubber bushing for vehicle suspension

To facilitate the initial design and stiffness matching of rubber bushing, according to the elastic mechanics theory, based on the analysis of the measured curve of radial deformation, by imposing virtual constraint stress on both ends of the bushing, the deformation analytic formula of rubber bushing on a radial concentrated force was established. With a practical example, the radial static deformation formula was computed and verified by simulation using ANSYS and test, and the analytical value is close to the simulated and the tested. The results show that the method of radial deformation analytic computation of rubber bushing under small deformation assumption is reliable and this method has important reference value for suspension design of vehicles.

A Multi-Directional Magnetic Treatment Approach to Improve the Dimensional Stability of Bearing Rings

Comparing with one directional magnetic treatment, a multidirectional magnetic treatment is applied to reduce residual stress in thin-walled bearing rings in order to improve the dimensional stability of bearing rings. The magnetic field was applied in both the axial and the radial directions of the thin walled bearing rings respectively. Six months of repeated testing results demonstrated that the dimension of bearing rings after magnetic treatment had no significant change. Experimental results shows both the major axis(the longer axis of an ellipse)and the minor axis(the shorter axis of an ellipse)appear shorten tendency and the dimension of bearing rings tends to be stable. Thin-walled uniform radial pressure deformation formulas are applied to calculate the reduced stress, and the maximum reduction is up to 45.97MPa. Research shows the multi-directional magnetic treatment can improve the dimensional stability of bearing rings and used in practical engineering.

Flexural performance of U-shape fiber-reinforced polymers profile-concrete composite beams under static and cyclic loads

This paper mainly focuses on flexural behavior of U-shape HFRP profile-concrete composite beams under static and cyclic loading. Four-point-bending experiments of eight pieces of HFRP-RC composite T-shaped beams with shear keys combined with wet-bond interface under both static and cyclic loads were conducted. The fiber-reinforced polymers profiles were produced by using vacuum infusion molding process. Experiment phenomena were detailed, and flexural performance such as failure mode, capacity, stiffness and load–displacement curves was also analyzed. Result shows that the shear keys combined with wet-bond interface is effective. Meanwhile, the flexural performance of the fiber-reinforced polymers-reinforced concrete beams under cyclic load was simulated based on the OpenSees software, in which special constitutive models of different materials were chosen to consider the cyclic loading. The simulation results and the experimental results such as the load–displacement curves, unloading stiffness, residual displacements, etc. were compared. The results show that OpenSees system can simulate the flexural performance of the fiber-reinforced polymers-reinforced concrete beams under both monotonic loads and cyclic loads very well. And then parameters study was conducted by using simulation method, and the influence factors such as concrete strength, steel bar ratio, fiber-reinforced polymers types, tensile stiffness of the longitudinal fiber-reinforced polymers, etc. were analyzed. Meanwhile, the recoverability of the fiber-reinforced polymers-reinforced concrete composite beams was evaluated, and a modified formula for the residual deformation was put forward. Finally, an idealized type of fiber-reinforced polymers-reinforced concrete beams with a medium reinforcement ratio and high post-yield stiffness was proposed, and reasonable values of the secondary stiffness ratio (defined as post-yield tangent stiffness divided by post-crack tangent stiffness) are recommended.

Isogonal deformation of discrete plane curves and discrete Burgers hierarchy

We study deformations of plane curves in the similarity geometry. It is known that continuous deformations of smooth curves are described by the Burgers hierarchy. In this paper, we formulate the discrete deformation of discrete plane curves described by the discrete Burgers hierarchy as isogonal deformations. We also construct explicit formulas for the curve deformations by using the solution of linear diffusion differential/difference equations.

Experimental Study of the Bending Properties and Deformation Analysis of Web-Reinforced Composite Sandwich Floor Slabs with Four Simply Supported Edges.

Web-reinforced composite sandwich panels exhibit good mechanical properties in one-way bending, but few studies have investigated their flexural behavior and deformation calculation methods under conditions of four simply supported edges. This paper studies the bending performance of and deformation calculation methods for two-way web-reinforced composite sandwich panels with different web spacing and heights. Polyurethane foam, two-way orthogonal glass-fiber woven cloth and unsaturated resin were used as raw materials in this study. Vacuum infusion molding was used to prepare an ordinary composite sandwich panel and 5 web-reinforced composite sandwich panels with different spacing and web heights. The panels were subjected to two-way panel bending tests with simple support for all four edges. The mechanical properties of these sandwich panels during the elastic stage were determined by applying uniformly distributed loads. The non-linear mechanical characteristics and failure modes were obtained under centrally concentrated loading. Finally, simulations of the sandwich panels, which used the mechanical model established herein, were used to deduce the formulae for the deflection deformation for this type of sandwich panel. The experimental results show that webs can significantly improve the limit bearing capacity and flexural rigidity of sandwich panels, with smaller web spacing producing a stronger effect. When the web spacing is 75 mm, the limit bearing capacity is 4.63 times that of an ordinary sandwich panel. The deduced deflection calculation formulae provide values that agree well with the measurements (maximum error <15%). The results that are obtained herein can provide a foundation for the structural design of this type of panel.