Mechanics Of Deformable Bodies Research Articles

Use of the integration by part to derive beam/plate equilibrium equations from elasticity with applications to beams

This article proposes a rigorous mathematical justification of the derivation of the governing equations of any beam and plate theory by the three three-dimensional (3D) elasticity equations. This is done by the application of the integration by part theorem via weight residual-like methods. The Fis polynomials (subscripts i and s denote the i-direction and the generic s-term of the displacement expansion uis, respectively) related to a given beam/plate theory are used as weight functions for each of the displacement unknown. To show how the method works a number of simple problems are discussed. These are related to the equilibrium equations and boundary conditions of bending beams according to shear deformation theory. Three methods for obtaining these equations are compared in some detail. The first two are the well-known methods used in mechanics of deformable bodies: (1) use of the Principle of Virtual Displacements; (2) use of the six cardinal ‘Newton’ equations of statics. The third method coincides with the one proposed and rigorously justified in this paper which is (3) the direct integration of the equations of 3D elasticity. Beam theories based on Lagrange polynomials are used as well. The latter makes it possible to immediately develop theories for layered beams and obtain so-called ‘layer-wise’ models, which are only applied here to the case of sandwich structures. Three appendixes discuss: A – The problem in which the derivative of one of the unknowns appears among the unknowns, as is the case with the Euler–Bernoulli theory; B – The extension to dynamics; C-Simple derivation of constitutive equations and related closed-form solution via Navier method.

Geomechanical Analysis of the Main Roof Deformation in Room-and-Pillar Ore Mining Systems in Relation to Real Induced Seismicity

Rockbursts represent one of the most serious and severe natural hazards emerging in underground copper mines within the Legnica–Glogow Copper District (LGCD) in Poland. The contributing factor determining the scale of this event is mining-induced seismicity of the rock strata. Extensive expertise of the copper mining practitioners clearly indicates that high-energy tremors are the consequence of tectonic disturbances or can be attributed to stress/strain behaviour within the burst-prone roof strata. Apparently, seismic activity is a triggering factor; hence, attempts are made by mine operators to mitigate and control that risk. Underlying the effective rockburst control strategy is a reliable seismicity forecast, taking into account the causes of the registered phenomena. The paper summarises the geomechanics analyses aimed to verify the actual seismic and rockburst hazard levels in one of the panels within the copper mine Rudna (LGCD). Two traverses were designated at the face range and comparative analyses were conducted to establish correlations between the locations of epicentres of registered tremors and anomaly zones obtained via analytical modelling of changes in stress/strain behaviours within the rock strata. The main objective of this study was to evaluate the likelihood of activating carbonate/anhydrite layers within the main roof over the excavation being mined, with an aim to verify the potential causes and conditions which might have triggered the registered high-energy events. Special attention is given to two seismic events giving rise to rockbursts in mine workings. Results seem to confirm the adequacy and effectiveness of solutions provided by mechanics of deformable bodies in the context of forecasting the scale and risk of dynamic phenomena and selecting the appropriate mitigation and control measures in copper mines employing the room-and-pillar mining system.

"SOME REMARKS ON THE PRINCIPLES OF SAINT-VENANT"

"The validity of the generalized Hooke’s law is checked by the relations obtained from the solution of the boundary value problem of the equilibrium of the cylinder, the boundary conditions on the basis of which are simplified on the basis of the narrowed and general principles of Saint-Venant. However, these relations are not a generalized Hooke’s law, but the kind in which these principles are invisibly present. With this in mind, they should be called Hooke–Saint-Venant relations. In this light, the deviation of experimental data from these relations should be discussed not as a deviation from the generalized Hooke’s law, but from its simplified form by the principles of Saint-Venant. The ratios turned out to be inconsistent, the experimental points are not located in the way they envisage. This, it would seem, is the end of the question. But no, the load-strain diagram has gained independence. Based on the assumption, as shown in this paper, it began to be considered as an experimentally determined physical law. The real deformation, as shown, cannot be explained at all by expressions (4), according to which the parts of the body do not interact with each other at all. But the acceptance of this assumption has become fixed – the tense state must necessarily be uniaxial. Keywords: Saint-Venant principle, Hooke–Saint-Venant relation, Hooke’s law, deformation, mechanics of deformable body."

Mathematical modeling using finite element analysis in forensic medical examination

The purpose of the work is to develop methods of mathematical modeling using finite element analysis in forensic medical examination. The stages of the methodology for solving problems of deformable body mechanics in forensic medicine are considered, which allows to reliably establish the possibility of formation and morphology of damage under specific conditions and circumstances, to focus the researcher's attention on problem points when creating and evaluating the model. The use of simplified models of the human body makes the expert's conclusion more reasonable, which increases the confidence of law enforcement agencies in the activities of the forensic medical expert service and allows for a new look at solving the problems of forensic medicine and forensic medical examination.

Моделювання впливу кільцевого включення на концентрацію напружень навколо видовженого еліптичного отвору у сферичній оболонці

Shell structures are widely used in various branches of technology and industry due to a combination of a high strength and a relatively light weight. In the majority of cases, actual structures have openings for manufacturing or design reasons, thus leading to a sharp increase in local stresses and, as a result, to a decrease in the strength and reliability of the structure as a whole. That is why reducing stress concentration in thin-walled structural elements is an important and topical problem in deformable body mechanics. This paper presents the results of a computer simulation and finite-element analysis of the stress and strain field of a thin-walled spherical shell with an elongated elliptical opening and an annular inclusion that surrounds the opening at a certain distance therefrom. The effect of the geometrical and mechanical parameters of the inclusion and its distance from the opening contour on the concentration of the stress and strain field parameters of the shell is studied. The stress and strain intensity distribution in the local stress concentration zones is obtained. It is shown that a rigid annular inclusion located at a certain distance from an opening allows one to reduce the stress concentration factor by nearly 27 percent with a proportional decrease in strain intensity in the vicinity of the opening. The elliptical opening elongation degree greatly affects the concentration of the stress and strain field parameters. If an opening is reinforced with a rigid annular inclusion immediately along its contour, the stress intensity in its vicinity increases, while the strain intensity decreases. The numerical calculations conducted show that surrounding an opening with a rigid annular inclusion located remotely therefrom reduces both the stress and the strain intensity in the vicinity of the opening. If an opening is reinforced immediately along its contour, a decrease in the maximum strain intensity is somewhat greater in comparison with the case where the rigid annual inclusion surrounding the opening is located at some distance therefrom. The use of specially selected and located reinforcements of elongated elliptical openings in spherical shells allows one to control the stress and strain intensity distribution and magnitude in the zones of local concentration of their stress and strain field parameters.

Preface

This volume presents up-to-date contributions in the field of mechanical engineering. The following topics are covered: Machine Design; Tribology, Materials and Surface Engineering; Mechatronics; Computer Aided Design; Mechanical Vibrations; Theory of Mechanisms and Machinery; Robotics; Mechanics of Deformable Bodies; Automotives; Engine and Transmission; Road Safety; Applied Thermodynamics, Heat Transfer, and Renewable Energy; Thermal Systems; Technologies in Agriculture and Food Processing. The results reported in this volume should interest researchers, scientists, industrial experts, teachers and students in the field of mechanical engineering related to design, control and applications.The authors discussed their results at The 10th International Conference on Advanced Concepts in mechanical Engineering (ACME 2022), held in Iasi, Romania, on June 09-10, 2022. The contributions had rigorously reviewed and selected by independent reviewers, members of the International Scientific Committee of this conference. ACME 2022 was organized by Mechanical Engineering Faculty, at “Gheorghe Asachi” Technical University of Iasi, Romania, under the aegis of Romanian Ministry of National Education and Scientific Research, Romanian Academy of Technical Sciences, Academy of Romanian Scientists, in the partnership with American University of Madaba, Jordan, SIAR - Society of Automotive Engineers of Romania, ARoTMM - Romanian Association for Mechanisms and Machine Science, SROMECA – Romanian Association of Mechatronics, ART – Romanian Tribology Association, SRT – Romanian Society of Thermodynamics, AFCR – Romanian Association for Refrigeration and Cryogenics Engineers.We are grateful to the authors for their valuable contributions and to the reviewers for their timely reviews and recommendations as improvement guidance for the selected papers. We thank also the members of Organizing Committee for the enthusiastic work of organizing this conference and the publisher, IOP Publishing, who paved the way to the timely publication of this volume with their excellent technical and editorial support.List of CONFERENCE CHAIRMEN, HONORARY COMMITTEE, SCIENTIFIC COMMITTEE, ORGANIZING COMMITTEE are available in this pdf.

ITERATIVE ALGORITHMS FOR SOLVING A PLANE PROBLEM FOR AN ENVIRONMENT WITH SIGNIFICANT MANIFESTATION OF INTERNAL FRICTION

The purpose of research is to develop an algorithm for solving of a flat nonlinear problem of mechanics of discrete environment, which laws of deformation of take into account the effect of the internal friction. Described the explanation of the iterative algorithm for the numerical solution of the plane physically nonlinear boundary value problems of mechanics of discrete environment. Feature of the problem is the consideration of the effect of internal Coulomb friction on the deformation of the environment. When solving the problem in displacements for finite simplex elements, the continuity conditions are always satisfied at the nodes and at the faces of the element. Solving the system of canonical linear equations of the displacement method ensures the fulfillment of the equilibrium conditions in each node of the discrete computational domain. Therefore, the calculation procedure is organized in such a way that the obtained solutions also comply with the laws of deformation of the material. Known iterative methods of variable stiffness, initial stresses or initial deformations, which differ only in the method of obtaining solutions, can be used for this purpose. The analysis showed that the most effective method for solving the formulated physically nonlinear problem is the variable stiffness, on the basis of which the iterative algorithm was developed. Described in the article iterative algorithm for solving of a flat boundary value problems of mechanics of a discrete environment allows to take into account the influence on the process of deformation internal friction. It can also be used to solve problems in the mechanics of solid deformable body with a most influence of internal friction, thermo-elasticity problems, etc.

MATHEMATICAL MODELS OF HUMAN RESPIRATORY AND BLOOD CIRCULATORY SYSTEMS

Aim. To analyze modern approaches to mathematical modeling of human respiratory and blood circulatory systems. Methods. Comprehensive review of scientific literature sources extracted from domestic and international resources databases. Results. Historical information and modern data concerning mathematical modeling of human functional respiratory and blood circulatory systems were summarized and analyzed in present ¬review; current trends in approaches to the construction of these models were revealed. Conclusions. Currently, two main approaches to the mathematical modeling of respiratory and blood circulatory systems exist. One of them is the construction of models of the mechanics of respiration and blood circulation. They are based on the models of mechanics of solid deformable body, thermomechanics, hydromechanics, and continuum mechanics. This approach uses complex mathematical apparatus, including Navier-Stokes equation, which makes it possible to obtain a number of theoretical results, but it is hardly usable for real problems solutions at present time. The second approach is based on the model of F. Grodins, who represented the process of breathing as a controlled dynamic system, described by ordinary differential equations, in which the process control is carried out according to the feedback principle. There is a significant number of modifications of this model, which made it possible to simulate various disturbing influences, such as physical activity, hypoxia and hyperemia, and to predict parameters characterizing functional respiratory system under these disturbing influences.

Задача устойчивости криволинейной арки малого подъема из пористого материала при случайном ее нагружении

Classical mechanics of deformable bodies is based on the continuity hypothesis. However, many structural elements are made of porous materials. Porous natural materials (soil, rocks) have invariable porosity. Porous synthetic materials (ceramics, concrete, graphite, and pressed powder metals) have controlled porosity. To calculate the strength and hardness of the structure, the material is assumed to be conditionally continuous with the adjusted porosity. Nowadays, there are many available works presenting mechanical characteristics of materials with different porosities. This paper proposes a new class of problems in mechanics of deformable solids. Considering a low arch stability problem, which is important in construction practice, the problem of optimal arch design is solved by controlling the properties of the material. The solution to the problem of stability of the low arch made of porous material is presented. The flat arch with a rectangular crosssection is exposed to equally distributed loading. The near-rational law of the porosity distribution over the cross-section is used. The load is considered as a random variable. The solution to the problem is obtained using the theory of stationary random processes. A comparative analysis of the reliability and material consumption is carried out for the arch with continuous and porous sections. The calculation shows that the porous structure of the arch reduces the material consumption by 13.3% without stability and reliability losses.

Инвариантные показатели напряженного состояния при кузнечной осадке магния в оболочке

Introduction. For pressure treatment of low-plastic metals, it is necessary to develop special techniques for increasing plasticity. In the cold state, an increase in plastic properties is possible due to an increase in the level of compressive stresses during deformation. In the processes of forging precipitation, this is achieved by using shells or clips of various types. At the same time, the configuration of the precipitation tool also matters. To create additional compressive stresses and increase the ductility of the metal, the working surface of the tool can be configured differently than with a normal free draft, where it is obviously larger than the contact surface area of the workpiece, so that metal broadening can occur. The stress state has a great influence on the plasticity of the processed material. This state is described by methods of tensor representation, but to assess the situation, it is customary to use invariants of tensors in one form or another, which eliminates the influence of coordinates on the results of the analysis. In the sections of deformable body mechanics dealing with the influence of the stress state on plasticity, the first, but sometimes other invariants of the stress tensor are used, the invariants themselves are transformed into the stress state indicator and the lode coefficient. The aim of the work: mathematical evaluation of invariant parameters of the stress state of the magnesium precipitation process at room temperature, according to the results of which it is possible to obtain a positive result in real experiments. Research methods: finite element simulation using the DEFORM software module. Results and discussion. The theoretical justification of increasing the plasticity of the magnesium billet in the process of precipitation in the cage without its compression is carried out. An increase in the stress state index modulo 2...5 times is revealed, which contributes to an increase in the plasticity of the metal. At the same time, a zone with a lode coefficient close to zero is identified. It is adjacent to the middle of the height of the workpiece at the point of contact with the cage and can be a dangerous cross-section from the position of crack formation.

Modern interpretation of Saint-Venant’s principle and semi-inverse method

Relevance. The progressive development of views on the Saint-Venant formulated principles and methods underlying the deformable body mechanics, the growth of the mathematical analysis branch, which is used for calculation and accumulation of rules of thumb obtained by the mathematical results interpretation, lead to the fact that the existing principles are being replaced with new, more general ones, their number is decreasing, and this field is brought into an increasingly closer relationship with other branches of science and technology. Most differential equations of mechanics have solutions where there are gaps, quick transitions, inhomogeneities or other irregularities arising out of an approximate description. On the other hand, it is necessary to construct equation solutions with preservation of the order of the differential equation in conjunction with satisfying all the boundary conditions. Thus, the following aims of the work were determined: 1) to complete the familiar Saint-Venants principle for the case of displacements specified on a small area; 2) to generalize the semi-inverse Saint-Venants method by finding the complement to the classical local rapidly decaying solutions; 3) to construct on the basis of the semi-inverse method a modernized method, which completes the solutions obtained by the classical semi-inverse method by rapidly varying decaying solutions, and to rationalize asymptotic convergence of the solutions and clarify the classical theory for a better understanding of the classic theory itself. To achieve these goals, we used such methods , as: 1) strict mathematical separation of decaying and non-decaying components of the solution out of the plane elasticity equations by the methods of complex variable theory function; 2) construction of the asymptotic solution without any hypotheses and satisfaction of all boundary conditions; 3) evaluation of convergence. Results. A generalized formulation of the Saint-Venants principle is proposed for the displacements specified on a small area of a body. A method of constructing asymptotic analytical solutions of the elasticity theory equations is found, which allows to satisfy all boundary conditions.

Preface

This volume presents up-to-date contributions in the field of mechanical engineering. The following topics are covered: Machine Design; Tribology, Materials and Surface Engineering; Mechatronics; Computer Aided Design; Mechanical Vibrations; Theory of Mechanisms and Machinery; Robotics; Mechanics of Deformable Bodies; Automotives; Engine and Transmission; Road Safety; Applied Thermodynamics, Heat Transfer, and Renewable Energy; Thermal Systems; Technologies in Agriculture and Food Processing. The results reported in this volume should interest researchers, scientists, industrial experts, teachers and students in the field of mechanical engineering related to design, control and applications.The authors discussed their results at The 9th International Conference on Advanced Concepts in mechanical Engineering (ACME 2020), held in Iasi, Romania, on June 04-05, 2020. The contributions had rigorously reviewed and selected by independent reviewers, members of the International Scientific Committee of this conference. ACME 2020 was organized by Mechanical Engineering Faculty, at “Gheorghe Asachi” Technical University of Iasi, Romania, under the aegis of Romanian Ministry of National Education and Scientific Research, Romanian Academy of Technical Sciences, Academy of Romanian Scientists, in the partnership with American University of Madaba, Jordan, SIAR - Society of Automotive Engineers of Romania, ARoTMM - Romanian Association for Mechanisms and Machine Science, SROMECA – Romanian Association of Mechatronics, ART – Romanian Tribology Association, SRT – Romanian Society of Thermodynamics, AFCR – Romanian Association for Refrigeration and Cryogenics Engineers.List of Organizer, Conference Chairman, Honorary Committee, Scientific Committee, Organizing Committee, Topics and History And Aim.

SOLUTIONS OF THE PROBLEM OF RANDOM VIBRATIONS IN HEREDITARY-DEFORMABLE SYSTEMS USING IMPULSIVE FUNCTIONS

In recent years, probabilistic-statistical methods for studying various problems of mechanics of solid deformable bodies are being applied more than ever. The principal part of its field of application is the development of a general theory of strength and a hereditarily deformed solid. As known, the theory of random oscillations is increasingly being used in technology. Current research is aimed to present a numerical-analytical approach for studying the dynamic response of a hereditarily deformable system to unsteady input influences. It is established that the dynamic reaction of hereditarily deformable systems to an arbitrary form of random perturbations can also be represented as the Duhamel integral if the impulse transition function satisfies special Cauchy problems for the integro-differential equation (IMU). A study proposes an accurate analytical solution to the IMU of an impulsive transition function in existing weakly singular Rzhanitsyn-Koltunov nucleuses.

Modelling of the Effect of Piece-Wise Potential Volume Forces in the Photo-Elasticity Method

The authors note a special feature complicating the direct modelling process in the photo-elasticity method (using the models made from widely-used epoxy materials) with volume forces even of the simplest kind (mass forces). This problem is more complicated than the problems with constant parameters ν, E, γ, α in the whole body. The authors give some additional relationships between the similarity factors, the use of which may create some restrictions and difficulties in the process of modelling non-uniform problems through traditional methods. The authors use the theoretical principles of mechanics of deformable rigid bodies and show that the solution of the problem in case, when the volume forces in different domains of the body have different potential values, may be reduced to the solutions of the problems with potential volume forces and to the solution of the problem with given stress values. The authors have developed both methodology and engineering questions of modelling piece-wise potential volume forces using the equivalence of effects. They note that the methods of solution of the problems with the loadings substituting the volume forces have been developed in full and give efficient results.

Determination of the Dynamic Vehicle Model Parameters by Means of Computer Vision

This study is devoted to determining the geometric, kinematic and dynamic characteristics of a vehicle. To this purpose, it is proposed to use a complex approach applying the models of deformable body mechanics for describing the oscillatory movements of a vehicle and the computer vision algorithms for processing a series of object images to determine the state parameters of a vehicle on the road. The model of the vehicle vertical oscillations is produced by means of the viscoelastic elements and the dry friction element that fully enough represent the behavior of the sprung masses. The introduced algorithms and models can be used as a part of a complex system for monitoring and controlling the road traffic. In addition, they can determine both the speed of the car and its dynamic parameters and the driving behavior of the individual drivers.

A complex approach to the solution of problems in mechanics of deformable rigid bodies

There exists an opinion that the modern numerical methods allow to solve practically any problem in mechanics. But it should be noted that both analytical and experimental methods, as before, are urgent, and exactly a complex of methods develops the mechanics of deformable rigid bodies. The statement of a problem in displacements for some possible cases of equivalent substitution of loads allows to formulate necessary and sufficient conditions of existence of an analogy presenting the effect of a forced deformation in the form of the sum of surface and volume forces, the effect of volume forces in the form of the sum of surface forces and forced deformations, the effect of surface forces in the form of the sum of forced deformations and volume forces. The substitution of volume forces for surface loads and forced deformations allows to extend the use of experimental methods and often to solve through an experimental-and theoretical approach the problems, which cannot be solved through other methods. The obtained results are a considerable step in the development of one of the approaches combining experimental, analytical and numerical methods of solution of linear problems in mechanics of deformable rigid bodies.

The informativeness of roof tremor hazard analysis based on seismic measurements and analytical modelling

Rock bursts are a common hazard in Polish hard-coal mines. Their magnitude depends on the state of stress in the mining area, and on underground mining-induced seismicity. Both long-term and ongoing as-sessments of the capacity of the rock mass to generate tremors are based on geological and mining surveys, rock property measurements (in laborato-ries and in-situ) and observations involving signs of rock mass pressure, supported by available concepts applicable to geomechanical and geophys-ical measurements. These include seismic measurements, including profil-ing, scanning, passive/active geotomography and analytical techniques. This paper presents a comparative analysis involving exemplary results provided by both seismic and analytical methods for a selected panel in one of Upper Silesian Coal Basin (USCB) mines. For the purposes of our seismic measurements, we employed seismic tomography for the ge-otomographic reconstruction of velocity fields. Subsequently, we com-pared the results with the results of analytical modelling of rock stresses that relied on classical engineering solutions applicable to problems of me-chanics of deformable bodies. The tremor hazard analysis was based on observations of changes in P-wave velocity distributions in the coal seam environment relative to the evolution of the vertical stress concentration factor at the level of a potentially seismogenic roof layer.

Residual resource of power resistance during building structures deformation

Modern scientific research and design developments, based on the fundamental principles of physics, mechanics and thermodynamics, are developing in a phenomenological direction. This implementation is found both in traditional integrated models of reinforced concrete using the advantages of computer technology, and in discrete models following the grid methods of solid deformable body mechanics. Discrete models in content and chronology over time are hereditary with respect to integral models. The theoretical basis for calculating the residual life of the force resistance to deformation, determines the stiffness of the cross sections of reinforced concrete elements with a maximum bending moment and with zero bending moment are presented in this article.

General theory of cross-sections kinetics in calculating rod elements of structures. Cartesian coordinates

The applied mechanics of deformable bodies evolves to its more rigorously substantiated predominant empirical approaches, which significantly limit calculation possibilities. The most "severe" limitation is applying exclusively the hypotheses of non-warping cross-sections in calculating the structural rod elements. The research was carried out to expand the possibilities and to systematically generalize and complete a traditional approach within the theory of strength of materials, which is based on isolating a subsystem of points in a cross-sectional form. The presented theory is a synthesis of the theory of strength of materials and the applied theory of elasticity. The theory of strength of materials contributed with the paradigm of monitoring the behaviour of cross-sections as a subsystem of points of a deformable body. The theory of elasticity provided differential equations of internal constraints, modifiable according to the traditional assumptions of the theory of material strength, except for those relating to the behavior of cross-sections. The study touches upon the analytical issue of optimizing the forms of various rod junctions, such as threaded, toothed, and strained junctions, and also on the issue of stress concentration in the regions of curvilinear transitions.

The 8th International Conference on Advanced Concepts in Mechanical Engineering

This volume presents up-to-date contributions in the field of mechanical engineering. The following topics are covered: Machine Design; Tribology, Materials and Surface Engineering; Mechatronics; Computer Aided Design; Mechanical Vibrations; Theory of Mechanisms and Machinery; Robotics; Mechanics of Deformable Bodies; Automotives; Engine and Transmission; Road Safety; Applied Thermodynamics, Heat Transfer, and Renewable Energy; Thermal Systems; Technologies in Agriculture and Food Processing. The results reported in this volume should interest researchers, scientists, industrial experts, teachers and students in the field of mechanical engineering related to design, control and applications.