Principle Of Virtual Velocities Research Articles

Galileo, Coulomb, the origins of Yield Design Analyses

The Memoir presented to the French Academy of Sciences by Coulomb in 1773 is often considered as the starting point for “yield design” stability analyses in geotechnics and, more generally, in various fields of civil and construction engineering. In this Essai (1773), Coulomb implemented a reasoning based on the necessary condition that the equilibrium of the structure under study and the resistance of its constituent materials should be mathematically compatible with each other. A previous out sketch of this rationale can be identified in Galileo’s solution to the problem of the bearing capacity of a cantilever beam in the Discorsi (1638). In the present contribution, after outlining the philosophy of Galileo’s argument, we will highlight Coulomb’s clarifying insight and essential input before the fundamental concepts of the mechanics of continuous media, such as the notion of stress and the principle of virtual velocities, were established. Various aspects of yield design or “limit equilibrium” stability analyses in geotechnics will be evoked. The theory of Yield design, in line with Coulomb’s reasoning, will be briefly presented, making it possible, in particular, to overcome the difficulties encountered when implementing “classical” methods and clarifying a possible confusion with the mathematical theory of plasticity.

STRENGTH OF CONCRETE UNDER LOCAL COMPRESSION TAKING INTO ACCOUNT THE RELATIONSHIP OF THE ELEMENT HEIGHT TO THE SIZE OF THE LOADING AREA

Formulation of the problem. Reliability of buildings and structures from reinforced concrete largely depends on efficiency of construction solutions of supporting areas and joins of elements of bearing systems. At the same time, an important place is given to the issues of ensuring their bearing capacity. One of the ways to solve this problem is to increase the strength characteristics of concrete, in particular by using basalt fiber. The purpose of the article is to improve the method of calculating the strength of concrete elements under the local compression on a general theoretical basis and to clarify the impact on the strength of determining factors. Conclusion. To calculate the strength of concrete elements under the local compression, the theory of concrete plasticity is promising. The application of the variational method and the principle of virtual velocities is proposed. According to research, the strength of concrete, in addition to the ratio of the area of application of the load and the cross-sectional area, is affected by the ratio of the height of the element to the width of the stamp and both characteristics of concrete strength. The proposed kinematic failure schemes have found experimental confirmation. Analysis of test results of concrete elements and assessment of their strength on the basis of the theory of plasticity allowed making the program of experimental research of fibroconcrete on basalt fibers under local compression.

Rate-related study on mechanical properties and fracture characteristics in aluminium alloy via electromagnetic ring expansion test

The uniform one-dimensional tension can be easily realized by the electromagnetic ring expansion (EMRE) test at high strain rates of 103-104 s−1, which utilizes strong pulsed electromagnetic force to deform materials. However, the accurate and rapid acquisition of the time-dependent stress, strain and strain rate in EMRE test has been a bottleneck that hinders its wide application. Based on the electromagnetism and the principle of virtual velocity in plastic dynamics, a stress analytical model considering both the electromagnetic and inertial components is established. The whole EMRE process of deformation, strain localization and fracture for 5052 aluminium alloy (AA5052) which is widely used in the automotive and aerospace industries is collected by a measurement system with microsecond time-resolved imaging, and then the true stress-strain-strain rate relationship is calculated based on the above analytical model. In addition, the dynamic mechanical properties of AA5052 under EMRE test are contrastively analyzed with those under the split Hopkinson tension bar (SHTB) and quasi-static tests. The ductility and formability of AA5052 under EMRE are improved due to the inertial and eddy current effects which is demonstrated by fracture characteristics observed using Scanning Electron Microscopy (SEM).

IMPROVED JOINTS OF REINFORCED CONCRETE ELEMENTS IN PRECAST-AND-CAST-IN-PLACE CONSTRUCTION BASED ON DESTRUCTION MECHANICS

Systematic theoretical and experimental studies of the keyed joints of concrete and reinforced concrete elements have been carried out at PoltNTU over the recent thirty years that made it possible to formulate recommendations for their design and to improve the existing design solutions of joints. The keyed joints have higher bearing capacity. The method for calculating the strength of the joints is proposed. This method is based on the theory of concrete plasticity, the variational method and the principle of virtual velocities. It considers the stage of destruction and takes into account the full set of influence factors. The research takes into account the influence of the following factors on the joints strength: the ratio of the key depth to its height; compression level; the intensity of reinforcement and the nature of the reinforcement location. The keys of a rectangular profile of circular cross section are considered, which are formed in the voids of the slabs at their ends surface when girders concreting. For their reinforcement, a spatial cage in the form of a hollow cylinder is proposed. The results of experimental verification of the quantitative and qualitative influence on the joints strength of these factors are presented. The boundaries of the efficiency of reinforcement intensity, the degree of compression are determined, the optimal ratio of the depth of the key to its height is accepted. The design features of the joints of multi-hollow floor slabs (roofs) with a monolithic girder and between themselves in a precast-cast-in-place frame structural system, with wall panels in large-panel houses, as well as joints of wall panels on flexible loops in large-panel buildings, are taken into account. Examples of improved structural solutions for connecting elements of modern load-bearing systems based on the specifics of their work are given.

A Development of the Principle of Virtual Laws and its Conceptual Framework in Mechanics as Fundamental Relationship between Physics and Mathematics

Generally speaking, virtual displacement or work concerns to a timely idea according to which a motion of a certain body is not the unique possible motion. The process of reducing this motion to a particular magnitude and concept, eventually minimizing as a hypothesis, can be traced back to the Aristotelian school. In the history and philosophy of science one finds various enunciations of the Principle of Virtual Laws and its virtual displacement or work applications, i.e., from Aristotle to Leibniz’s vis viva, from Maupertuis’ least action to Euler and Lagrange with calculus of variations (statics and dynamics) to Lazare Carnot’s mechanics. In this case study, I will demonstrate that a particular approach used by Lazare Carnot is original by explaining within the historical context of rival approaches such as the development of the Principle of virtual Laws (also known as the Principle of virtual velocity or of virtual work). I will also discuss Carnot’s geometric motion as one of the possible but invertible movements applied to virtual displacement as employed in his theories of machines and collisions. I will then go on to explore how the originality of an invertible motion within his mechanical and, in general, mathematical research program permitted Carnot to introduce a new way of structuring a scientific theory and making mechanics, with respect to the Newtonian paradigm, to scholars and his students of the École polytechnique de Paris.

Estimation of the underwater implosion pulse from cylindrical metal shells

Underwater implosion, the rapid and catastrophic collapse of a structure caused by hydrostatic pressure, generates a short-duration, high-pressure pulse in the surrounding water that is potentially damaging to adjacent structures or personnel. This paper presents a method to estimate the energy in the pressure pulse as the difference between the known total potential energy and energy lost by plastic deformation. The implosion pulse energy is proportional to the maximum system kinetic energy developed during collapse. The pulse was approximated by considering only the first phase of collapse (up to the instant of first contact), because the maximum kinetic energy occurs very near that instant. The plastic energy dissipated by the structure during the initial phase of implosion was evaluated analytically, using the principle of virtual velocities. The solution for energy dissipation agreed with numerical simulation within 1% at the end of the initial collapse phase. It was found that for a representative aluminum cylinder, the implosion pulse represents only 15% of the total available energy.

Production of ultrafine grained aluminum by cyclic severe plastic deformation at ambient temperature

In the present study the possibilities of grain refinement was investigated by applying large-scale of cyclic plastic deformation to aluminum at ambient temperature. The specimens are processed by multiaxial forging, which is one of the severe plastic deformation techniques. The aim of the experiments with the aluminum alloy 6082M was the determination of the equivalent stress and strain by multiaxial forging and the investigation of evolution of mechanical properties in relation with the accumulated deformation in the specimen. The mechanical properties of raw material was determined by plane strain compression test as well as by hardness measurements. The forming experiments were carried out on Gleeble 3800 physical simulator with MaxStrain System. The mechanical properties of the forged specimens were investigated by micro hardness measurements and tensile tests. A mechanical model, based on the principle of virtual velocities was developed to calculate the flow curves using the measured dimensional changes of the specimen and the measured force. With respect to the evolution of these curves, the cyclic growth of the flow stress can be observed at every characteristic points of the calculated flow curves. In accordance with this tendency, the evolution of the hardness along the middle cross section of the deformed volume has also a nonmonotonous characteristic and the magnitudes of these values are much smaller than by the specimen after plane strain compression test. This difference between the flow stresses respect to the monotonic and non-monotonic deformation can be also observed. The formed microstructure, after a 10-passes multiaxial forging process, consists of mainly equiaxial grains in the submicron grain scale.

Determination of flow curve at large cyclic plastic strain by multiaxial forging on MaxStrain System

Abstract In this present work, the determination of flow curve and equivalent strain were aimed by multiaxial forging (cyclic compression test), which was carried out on the MaxStrain System of Gleeble 3800 thermo-mechanical simulator. Therefore, the dimensions of the modifying deformed volume were measured and the surfaces, which are contacted with the tools, were tracked by optical means continuously during the physical simulation. These parameters and the measured force data were applied in the mechanical model, based on the principle of virtual velocities. In the examined range of 0–3 of the equivalent strain, the flow curve of the monotonic deformation is significantly different from the cyclic flow curve. The experiments were performed with AlSi1MgMn heat treatable aluminium alloy at room temperature.

History of virtual work laws: a history of mechanics prospective

Logic status of virtual work laws.- Greek origins.- Arabic and Latin science of weights.- Italian Renaissance statics.- Torricelli's principle.- European statics during the XVI and XVII centuries.- The principle of virtual velocities.- The Jesuit school of the XVIII century.- Lagrange's contribution.- Lazare Carnot's mechanics of collision.- The debate in Italy.- The debate at the Ecole polytechnique.- Poinsot's criticism.- The treatises of mechanics.- Virtual work laws and continuum mechanics.- Thermodynamical approach.

Localized indentation of sandwich beam with metallic foam core

The localized behavior of sandwich structures with foam core subjected to indentation loading is investigated in this article. Based on the principle of virtual velocities, concisely explicit solutions for the indentation forces and shape functions of deformation zones of sandwich beams are derived. Both flat and cylindrical indenters are considered. The indentation force varies linearly with the square root of indenter displacement and thesurface deformation profiles are proportional to the square of the distance from the contact center along the beam direction. Finite element models have been established using the ABAQUS/Explicit code to verify the validity and applicability of the analytical solutions. The theoretical predictions of the profiles of deformation zones and the denting loads of indenters are in good agreement with those by numerical simulation.

Can Science Education Research Give an Answer to Questions posed by History of Science and Technology? The Case of Steam Engine’s Measurement

According to the principle of virtual velocities, if on a simple machine in equilibrium we suppose a slight virtual movement, then the ratio of weights or forces equals the inverse ratio of velocities or displacements. The product of the weight raised or force applied multiplied by the height or displacement plays a central role there. British engineers used the same product in the eighteenth century in order to measure steam engines’ effectiveness. The question is whether this measure was obviously empirical or had its origin in theory of mechanics and particularly in the principle of virtual velocities. According to science education research, this measure is not likely to have arisen intuitively and most probably could not have been formulated without any acquaintance with theory of mechanics.

Estimation of load-carrying capacity of multi-layered plated structures

The load-carrying capacity of plated structures consisted of three-layered plates with metallic isotropic face layers and metallic or non-metallic orthotropic core is determined by means of four groups of methods, namely analytical–numerical methods, the effective width approach, numerical methods—finite element and finite strip methods as well as kinematical method based on principle of virtual velocities. Theoretical backgrounds of all applied methods are presented. Results of case studies concerning separated simply supported plate under compression and thin-walled girder under pure bending are discussed. Exemplary results of calculations are presented in diagrams. Some remarks derived from the comparative analysis of obtained results based on the methods under consideration are derived.

The superdeep penetration of a particle into an elastic–plastic medium

Mathematical simulation of impact-induced deep penetration of a rigid spherical particle into an elastic–plastic medium is performed; the law of particle motion and the distance covered by the particle are determined. In solving the boundary-value problem, the principle of virtual velocities and stresses and the phenomenological theory of fracture are used. Realization of the decision is carried out with the use of the MATLAB system of computer mathematics.

Load-capacity estimation and collapse analysis of thin-walled beams and columns—recent advances

The present paper is devoted to the recent results of research in the area of load-capacity and post-failure behaviour of thin-walled beams and columns (among them thin-walled cold-formed profiles). It deals with ultimate load and collapse of box-section girders (tubes) of different cross-sections under bending, as well as of lipped and plain channel-section beam-columns. The paper contains the presentation of theoretical analysis and experimental investigation of plastic mechanisms of failure and collapse behaviour of these thin-walled sections. The short review of results obtained in recent years in general precedes those obtained by the Department of Strength of Materials and Structures, TUL. The problem of post-failure behaviour is solved using the rigid-plastic theory adopted and modified for the purposes of the solution taking into consideration strain-hardening of the member’s material. On the basis of experimental investigations theoretical models of plastic mechanisms of failure are produced for different sections. Theoretical analysis is based on the principle of virtual velocities. The problem is solved in an analytical–numerical way. The particular attention has been paid to the influence of the strain-hardening of the material after yielding upon the collapse structural behaviour and also to the influence of cross-section shape and dimensions on the character of collapse. The upper-bound estimation of the load-carrying capacity of analysed thin-walled sections by combining results of non-linear, elastic post-buckling analysis with the results of plastic mechanism analysis is carried out. Results are presented in diagrams showing post-failure curves as well as curves representing structural behaviour in the whole range of loading up to and beyond the ultimate load. Some results are compared with experimental results and those obtained from FE analysis. A comparison of lower- and upper-bound estimation of the load-carrying capacity is discussed and illustrated in diagrams. Conclusions dealing with the influence of strain-hardening phenomenon displayed by the material upon the load-carrying capacity and collapse behaviour of examined sections are derived. Also conclusions concerning different upper- and lower-bound estimations of the load-carrying capacity of analysed sections are presented.

Numerical study of the flange earring of deep-drawing sheets with stronger anisotropy

A Barlat–Lian anisotropy yield function is introduced into a quasi-flow corner theory of elastic–plastic finite deformation and the elastic–plastic large deformation finite element formulation based on the principle of virtual velocity and the discrete Kirchhoff triangle plate shell element model. The focus of the present researches is on the numerical simulation of the flange earring of deep-drawing process of circular sheets with stronger anisotropy, based on which, the schemes for controlling the flange earring are proposed.

有限要素法による等方引張膜の有限変位応力解析

Shape-finding analysis of isotonic membrane structures and the large deformation analysis under loads including follower force like pressure are carried out with the same algorithm as problem where the corresponding equilibrium equations are solved. Exact formulation based on the principle of virtual velocity is possible to extend large strain problem of membrane structures. Modulus of tension in plane, which is equivalent with bulk modulus, is applied in this paper into the constitutive equation to simplify the property of membrane materials. Two kinds of examples are illustratively demonstrated in order to prove the validity of the formulation.

A method for calculating the stress–strain state in the general boundary-value problem of metal forming—part 1

To simulate metal-forming processes, one has to calculate the stress–strain state of the metal, i.e. to solve the relevant boundary-value problems. Progress in the theory of plasticity in that respect is well known, for example, via the slip-line method, the finite element method, etc.) , yet many unsolved problems remain. It is well known that the slip-line method is scanty. In our opinion the finite element method has an essential drawback. (No one is against the idea of the discretization of the body being deformed and the approximation of the fields of mechanical variables.) The results of calculation of the stress state by the FEM do not satisfy Newtonian mechanics equations (these equations are said to be softened, i.e, satisfied approximately) and stress fields can be considered poor for solution of the subsequent fracture problem. We believe that it is preferable to construct an approximate solution by the FEM and soften the constitutive relations (not Newtonian mechanics equations) , especially as, in any event, they describe the rheology of actual deformable materials only approximately. We seem to have succeeded in finding the solution technique. Here we present some new results for solving rather general boundary-value problems which can be characterized by the following: the anisotropy of the materials handled; the heredity of their properties and compressibility; finite deformations; non-isothermal flow; rapid flow, with inertial forces; a non-stationary state; movable boundaries; alternating and non-classical boundary conditions, etc. Solution by the method proposed can be made in two stages: (1) integration in space with fixed time, with an accuracy in respect of some parameters; (2) integration in time of certain ordinary differential equations for these parameters. In the first stage the method is based on the principle of virtual velocities and stresses. It is proved that a solution does exist and that it is the only possible one. The approximate solution softens (approximately satisfies) the constitutive relations, all the rest of the equations of mechanics being satisfied precisely. The method is illustrated by some test examples.

Porohyperelastic–transport–swelling theory, material properties and finite element models for large arteries

A porohyperelastic–transport–swelling (PHETS) model is presented in which a soft hydrated tissue material is viewed as a continuum composed of an incompressible porous solid ( fibrous matrix) that is saturated by an incompressible fluid (water) in which a mobile species (solute) is dissolved. This PHETS theoretical model is implemented using a finite element model (FEM) including inherent nonlinearity, coupled transport processes, and complicated geometry and boundary conditions associated with soft tissue structures. The PHETS material properties are clearly identified with a physical basis describing and quantifying elasticity, permeability, diffusion, convection, and osmotic properties. The equivalence between the PHETS and the triphasic (TRI) model (Lai et al., 1991) is established using the phenomenological equations, and mathematical expressions are given to relate the PHETS and TRI material properties. A principle of virtual velocities (PVV ) links Eulerian and Lagrangian PHETS formulations and provides correspondence rules between the Eulerian and the Lagrangian field variables and material properties. The PVV is also the basis for a mixed Lagrangian PHETS FEM (Kaufmann, 1996) , which was developed for the analysis of soft hydrated tissues. Selected PHETS FEM results are presented in order to demonstrate the capability of the PHETS model to simulate coupled deformation, stress, mobile water flux, and albumin flux in the arterial wall undergoing finite straining associated with pressurization, axial stretch, and changes in albumin concentration in bath solutions surrounding a segment of rabbit thoracic aorta. Values for isotropic material parameters and specific details of the experiments and data-reduction methods were obtained from Simon et al. (1997 ; 1998) .

Lagrangian Expressions of the Hydrodynamic Forces Acting on a Rigid Body in the Presence of a Free Surface

The formalism of classical analytical dynamics is used in conjunction with the principle of virtual velocity to derive Lagrangian expressions for the hydrodynamic forces acting on a rigid body moving through an in-viscid and incompressible liquid with a free surface. Simultaneously, a corresponding Lagrangian expression is derived for the hydrodynamic pressure acting on the free surface itself. The expressions for the hydrodynamic forces degenerate to the classical ones if the free surface is not present, and the expression for the pressure is reduced to that obtained by Milder if the rigid boundaries are all kept fixed. The derived Lagrangian expressions for the hydrodynamic reactions are used to obtain a complete set of motion equations for the examined hydromechanical system, and to discuss another Lagrangian approach to the ship-motions problem, presented by Wang.

A variational formulation for the finite element analysis of the vibratory and acoustical response of high speed machinery

The principle of virtual velocities is employed herein to develop a variational theorem for determining the vibratory and acoustical response of high speed mechanisms immersed in a perfect fluid (air). Both the solid and fluid media are modeled as continua and the basic functional expression is generalized by using Lagrange multipliers to incorporate field equations, boundary conditions and a kinematic constraint on the intetface region between the two types of continua. The resulting mixed variational equation of motion then provides the basis for finite element analyses of these acousto-mechanical systems by mixed and displacement (velocity) formulations.