Pub Date : 2025-02-09DOI: 10.1134/S0025654424603471
L. V. Fomin, A. A. Dalinkevich, Yu. G. Basalov
The stress-strain state is considered and the time to fracture of a composite tensile rod during creep in an active medium is determined. The influence of the active medium is determined by a non-classical diffusion process, with the active substance penetrating into the material in two states: free and bound. The process of such diffusion is described by a modified diffusion equation that takes into account the two-phase state of the active substance in the material. A system of equations has been obtained that models the creep of a composite rod, in which its parts are rigidly connected to each other without slipping, and also includes kinetic equations for the accumulation of damage in parts of the rod. The influence of the active medium is taken into account by introducing into the indicated kinetic equations the function of the influence of the active medium - a function of the integral average concentration. Stress distributions and damage accumulation processes over time in various parts of the composite rod are analyzed. Calculations were carried out in two cases, namely, classical and non-classical diffusion processes are considered. The setting of these differences is determined by the choice of appropriate parameters in the diffusion model under consideration. Dependences of damage accumulation and stress distribution in parts of the rod over time were obtained.
As a result, it was determined that the destruction of a composite rod in the classical case occurs earlier than in the case of the considered non-classical diffusion process.
{"title":"On The Influence of a Non-Classical Diffusion Process on the Long-Term Fracture of a Composite Tensile Rod During Creep","authors":"L. V. Fomin, A. A. Dalinkevich, Yu. G. Basalov","doi":"10.1134/S0025654424603471","DOIUrl":"10.1134/S0025654424603471","url":null,"abstract":"<p>The stress-strain state is considered and the time to fracture of a composite tensile rod during creep in an active medium is determined. The influence of the active medium is determined by a non-classical diffusion process, with the active substance penetrating into the material in two states: free and bound. The process of such diffusion is described by a modified diffusion equation that takes into account the two-phase state of the active substance in the material. A system of equations has been obtained that models the creep of a composite rod, in which its parts are rigidly connected to each other without slipping, and also includes kinetic equations for the accumulation of damage in parts of the rod. The influence of the active medium is taken into account by introducing into the indicated kinetic equations the function of the influence of the active medium - a function of the integral average concentration. Stress distributions and damage accumulation processes over time in various parts of the composite rod are analyzed. Calculations were carried out in two cases, namely, classical and non-classical diffusion processes are considered. The setting of these differences is determined by the choice of appropriate parameters in the diffusion model under consideration. Dependences of damage accumulation and stress distribution in parts of the rod over time were obtained.</p><p>As a result, it was determined that the destruction of a composite rod in the classical case occurs earlier than in the case of the considered non-classical diffusion process.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 5","pages":"2735 - 2745"},"PeriodicalIF":0.6,"publicationDate":"2025-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143373237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-09DOI: 10.1134/S0025654424605184
Guangjie Li, Jiawen Xu, Lijun Chang, Taiwei Chen, Zhihua Cai
To examine the physical properties of loofah sponge structures, the article employed air-dried loofah samples subjected to quasi-static compression experiments at varying heights and strain rates. In quasi-static compression, the results indicate that the natural structure of the loofah sponge has a lower initial peak stress and greater specific energy absorption. Meanwhile, to determine the durability of the woven structure of the loofah sponge, cyclic compression experiments were conducted on samples of the loofah sponge at strain rates of 0.1 and 0.01 s–1 after the compression test. Analysis of experimental data collected after ten cyclic compression cycles revealed that the natural loofah sponge structure exhibited remarkable recovery and durability, with an energy loss efficiency of merely 32 and 29%, respectively. The research demonstrates that natural loofah sponge possesses remarkable energy absorption capabilities and durability. It can be considered an ideal example for the application of tubular, lightweight load-bearing components in the field of impact.
{"title":"Study on the Mechanical Properties of Loofah Sponge in Cyclic Compression","authors":"Guangjie Li, Jiawen Xu, Lijun Chang, Taiwei Chen, Zhihua Cai","doi":"10.1134/S0025654424605184","DOIUrl":"10.1134/S0025654424605184","url":null,"abstract":"<p>To examine the physical properties of loofah sponge structures, the article employed air-dried loofah samples subjected to quasi-static compression experiments at varying heights and strain rates. In quasi-static compression, the results indicate that the natural structure of the loofah sponge has a lower initial peak stress and greater specific energy absorption. Meanwhile, to determine the durability of the woven structure of the loofah sponge, cyclic compression experiments were conducted on samples of the loofah sponge at strain rates of 0.1 and 0.01 s<sup>–1</sup> after the compression test. Analysis of experimental data collected after ten cyclic compression cycles revealed that the natural loofah sponge structure exhibited remarkable recovery and durability, with an energy loss efficiency of merely 32 and 29%, respectively. The research demonstrates that natural loofah sponge possesses remarkable energy absorption capabilities and durability. It can be considered an ideal example for the application of tubular, lightweight load-bearing components in the field of impact.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 5","pages":"3097 - 3107"},"PeriodicalIF":0.6,"publicationDate":"2025-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143373384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1134/S0025654424604592
A. Fomekong Tanechue, F. P. Ewolo Ngak, G. E. Ntamack, B. Djeumako
In this study, a new analytical model based on an nth-order shear deformation theory formulation is used to analyse the post buckling of porous FGM plates resting on an elastic Winkler–Pasternak type foundation. The model presented contains a smaller number of variables than other higher-order theories in the literature. In addition, with this model, the effective properties of the structure are calculated as a function of the even and odd distributions of the porosity, and these distributions follow the power and sigmoid laws. The behaviour of the elastic foundation is governed by the constant Winkler parameter, which represents the reaction of the elastic springs, and the Pasternak parameter in the form of a shear layer of the foundation. The non-linear equilibrium equations are based on Von Karman’s theorem, the principle of virtual work and the equilibrium criterion. To solve these equations, approximate solutions and boundary conditions are considered. The accuracy of the nth-order HSDT model used takes into account the uniform, linear and non-linear variation of temperature across the thickness. We obtained several results for the evolution of the critical temperature: as a function of the amplitude/height ratio, as a function of the porosity and as a function of the foundations. The relative error between our results and those in the literature is generally less than 5%.
{"title":"Thermal Post-Buckling of Porous Thin Plates Following a Power and Sigmoid Distribution Law Based on an Elastic Foundation","authors":"A. Fomekong Tanechue, F. P. Ewolo Ngak, G. E. Ntamack, B. Djeumako","doi":"10.1134/S0025654424604592","DOIUrl":"10.1134/S0025654424604592","url":null,"abstract":"<p>In this study, a new analytical model based on an nth-order shear deformation theory formulation is used to analyse the post buckling of porous FGM plates resting on an elastic Winkler–Pasternak type foundation. The model presented contains a smaller number of variables than other higher-order theories in the literature. In addition, with this model, the effective properties of the structure are calculated as a function of the even and odd distributions of the porosity, and these distributions follow the power and sigmoid laws. The behaviour of the elastic foundation is governed by the constant Winkler parameter, which represents the reaction of the elastic springs, and the Pasternak parameter in the form of a shear layer of the foundation. The non-linear equilibrium equations are based on Von Karman’s theorem, the principle of virtual work and the equilibrium criterion. To solve these equations, approximate solutions and boundary conditions are considered. The accuracy of the nth-order HSDT model used takes into account the uniform, linear and non-linear variation of temperature across the thickness. We obtained several results for the evolution of the critical temperature: as a function of the amplitude/height ratio, as a function of the porosity and as a function of the foundations. The relative error between our results and those in the literature is generally less than 5%.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 4","pages":"2440 - 2469"},"PeriodicalIF":0.6,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1134/S0025654424603318
V. I. Karev, Yu. F. Kovalenko
One of the main problems when conducting laboratory tests of rock specimens aimed at determining their mechanical and strength properties is to transfer the test results of relatively small specimens to sufficiently large areas of a rock massif, often with a complex structure. This is due to the fact that generalized numerical indicators characterizing the degree of influence of structural heterogeneities of various sizes on the deformation and destruction of rocks and massifs are not yet available. In addition to heterogeneity, other factors also affect the processes under study, such as the stress state of the massif, the presence of geological disturbances, macrofractures, etc. These issues are studied in this paper based on a comparison of the results of experiments performed on the Triaxial Independent Loading Test System of the Institute of Problems in Vechanics of the Russian Academy of Sciences using the “hollow cylinder” scheme on specimens with a central hole of 10 and 20 mm in diameter and physical modeling of deformation processes in the vicinity of wells with a decrease in pressure at their bottomhole for reservoir rocks of the Prirazlomnoye oil field.
{"title":"Scale Effect in Modeling of Mechanical Processes in the Vicinity of a Borhole on a True Triaxial Loading Setup","authors":"V. I. Karev, Yu. F. Kovalenko","doi":"10.1134/S0025654424603318","DOIUrl":"10.1134/S0025654424603318","url":null,"abstract":"<p>One of the main problems when conducting laboratory tests of rock specimens aimed at determining their mechanical and strength properties is to transfer the test results of relatively small specimens to sufficiently large areas of a rock massif, often with a complex structure. This is due to the fact that generalized numerical indicators characterizing the degree of influence of structural heterogeneities of various sizes on the deformation and destruction of rocks and massifs are not yet available. In addition to heterogeneity, other factors also affect the processes under study, such as the stress state of the massif, the presence of geological disturbances, macrofractures, etc. These issues are studied in this paper based on a comparison of the results of experiments performed on the Triaxial Independent Loading Test System of the Institute of Problems in Vechanics of the Russian Academy of Sciences using the “hollow cylinder” scheme on specimens with a central hole of 10 and 20 mm in diameter and physical modeling of deformation processes in the vicinity of wells with a decrease in pressure at their bottomhole for reservoir rocks of the Prirazlomnoye oil field.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 4","pages":"1940 - 1948"},"PeriodicalIF":0.6,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1134/S0025654424603240
K. B. Ustinov
Expressions for T-stresses are obtained based on the exact analytical solution of the two-dimensional problem of a strip of orthotropic material with the principal axes of the elasticity tensor directed parallel and perpendicular to its boundaries and a central semi-infinite crack. A balanced system of loads in the form of four independent active loading modes is assumed to be applied sufficiently far from the crack tip. It is shown that for two (antisymmetric) loading modes the T-stresses are equal to zero, and for the other two (symmetric) modes they are determined by one or two parameters composed of the elasticity tensor components. The T-stress dependencies for symmetric loading modes are obtained in the form of double integrals from combinations of elementary functions depending on one of the dimensionless parameters, the second of the dimensionless parameters is included in the expression for the T-stresses of only one of the modes in the form of a multiplicative coefficient.
{"title":"T-Stress in an Orthotropic Strip with a Central Semi-Infinite Crack Loaded Far from the Crack Tip","authors":"K. B. Ustinov","doi":"10.1134/S0025654424603240","DOIUrl":"10.1134/S0025654424603240","url":null,"abstract":"<p>Expressions for T-stresses are obtained based on the exact analytical solution of the two-dimensional problem of a strip of orthotropic material with the principal axes of the elasticity tensor directed parallel and perpendicular to its boundaries and a central semi-infinite crack. A balanced system of loads in the form of four independent active loading modes is assumed to be applied sufficiently far from the crack tip. It is shown that for two (antisymmetric) loading modes the T-stresses are equal to zero, and for the other two (symmetric) modes they are determined by one or two parameters composed of the elasticity tensor components. The T-stress dependencies for symmetric loading modes are obtained in the form of double integrals from combinations of elementary functions depending on one of the dimensionless parameters, the second of the dimensionless parameters is included in the expression for the T-stresses of only one of the modes in the form of a multiplicative coefficient.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 4","pages":"1930 - 1939"},"PeriodicalIF":0.6,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1134/S0025654424602891
E. A. Nikonova
The article analyzes an approach that goes back to Maxwell to the representation of a potential, in particular, the potential of the Newtonian field of gravity as a sum of potentials of multipoles of different orders. Critical cases of the algorithm for finding the parameters of a multipole, namely, its axes and moment, are indicated. The cases take place when the body has certain symmetries in the mass distribution. Recommendations for overcoming the identified difficulties are formulated. For a body with a triaxial ellipsoid of inertia, explicit expressions for the axes and moment of a second-order multipole that are expressed via second-order inertia integrals are given. It is shown that the axes of the multipole are orthogonal to the circular cross-sections of the ellipsoid of inertia of the body. Critical cases of calculating a third-order multipole are considered using the example of a model body with constant density, that has the shape of an equihedral tetrahedron. A method for calculating the axes and moment of a third-order multipole for such a body is given.
{"title":"On the Maxwell Representation of the Gravitational Potential for a Symmetric Body","authors":"E. A. Nikonova","doi":"10.1134/S0025654424602891","DOIUrl":"10.1134/S0025654424602891","url":null,"abstract":"<p>The article analyzes an approach that goes back to Maxwell to the representation of a potential, in particular, the potential of the Newtonian field of gravity as a sum of potentials of multipoles of different orders. Critical cases of the algorithm for finding the parameters of a multipole, namely, its axes and moment, are indicated. The cases take place when the body has certain symmetries in the mass distribution. Recommendations for overcoming the identified difficulties are formulated. For a body with a triaxial ellipsoid of inertia, explicit expressions for the axes and moment of a second-order multipole that are expressed via second-order inertia integrals are given. It is shown that the axes of the multipole are orthogonal to the circular cross-sections of the ellipsoid of inertia of the body. Critical cases of calculating a third-order multipole are considered using the example of a model body with constant density, that has the shape of an equihedral tetrahedron. A method for calculating the axes and moment of a third-order multipole for such a body is given.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 4","pages":"1881 - 1889"},"PeriodicalIF":0.6,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1134/S0025654424604269
Feng Kai, Wang Ling
Based on Euler beam theory and Hamilton’s principle, the free vibration control equation for the variable cross-section beam was established. Galerkin discretization and eigenvalue methods were used to solve the natural frequencies of beams under simply supported-simply supported (SS) and simply-clamped supported (SC) boundary conditions. The vibration characteristics of beams with annular, rectangular frame and rectangular cross-section at different cross-sectional parameters (taper, thickness, and aspect ratio) are explored. The calculation example shows that the natural frequency of the beam is closely related to changes in taper. Especially for rectangular cross-section, the change in taper in the height direction of the beam has a large effect on the frequency, while the change in taper in the width direction has a smaller effect. Increasing the outer thickness of the annular cross-section beam has a different effect on the beam’s natural frequency than the inner thickness, while increasing the thickness of the rectangular frame cross-section beams will result in a decrease in the natural frequency of the beams. Increasing the aspect ratio of a rectangular variable cross-section section beam increases the natural frequency of the beam. Under the same cross-sectional area, the rectangular frame cross-section beam frequencies are higher than other cross-section beams.
{"title":"The Free Vibration Characteristics of Variable Cross-Section Beams under Different Profiles","authors":"Feng Kai, Wang Ling","doi":"10.1134/S0025654424604269","DOIUrl":"10.1134/S0025654424604269","url":null,"abstract":"<p>Based on Euler beam theory and Hamilton’s principle, the free vibration control equation for the variable cross-section beam was established. Galerkin discretization and eigenvalue methods were used to solve the natural frequencies of beams under simply supported-simply supported (SS) and simply-clamped supported (SC) boundary conditions. The vibration characteristics of beams with annular, rectangular frame and rectangular cross-section at different cross-sectional parameters (taper, thickness, and aspect ratio) are explored. The calculation example shows that the natural frequency of the beam is closely related to changes in taper. Especially for rectangular cross-section, the change in taper in the height direction of the beam has a large effect on the frequency, while the change in taper in the width direction has a smaller effect. Increasing the outer thickness of the annular cross-section beam has a different effect on the beam’s natural frequency than the inner thickness, while increasing the thickness of the rectangular frame cross-section beams will result in a decrease in the natural frequency of the beams. Increasing the aspect ratio of a rectangular variable cross-section section beam increases the natural frequency of the beam. Under the same cross-sectional area, the rectangular frame cross-section beam frequencies are higher than other cross-section beams.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 4","pages":"2617 - 2627"},"PeriodicalIF":0.6,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1134/S0025654424604361
Xiulong Chen, Chengsi Ning, Yonghao Jia
Joint clearances and uncertain parameters of a parallel mechanism will cause the error between the actual dynamic response and the desired dynamic response, leading to a dynamic reliability problem. Currently, the reliability analysis of mechanisms mainly focuses on mechanism with clearances or with uncertain parameters, very few involve considering both clearances and uncertainty parameters simultaneously. The main purpose of this article is to propose a dynamic reliability analysis method of a parallel mechanism with revolute clearance and uncertainty parameters. The dynamic reliability analysis of a parallel mechanism with lubrication revolute clearance joints and uncertain parameters is studied in this paper. The 3-RRPaR spatial parallel mechanism is taken as an example, lubrication revolute clearance joints model is established, and the dynamic model of mechanism with lubrication revolute clearance joints is derived. The dynamic reliability model of mechanism with lubrication revolute clearance joints and uncertain parameters is developed. The effects of various random uncertain parameters on the dynamic reliability of mechanism with lubrication revolute clearance joints are analyzed. This can not only provide a new method for dynamic response reliability analysis of mechanism with lubrication clearances and uncertain parameters, but also lay the foundation for dynamic response and reliability optimization design.
{"title":"Dynamic Responses and Reliability Analysis of a Parallel Mechanism with Lubrication Revolute Clearance Joints and Uncertain Parameters","authors":"Xiulong Chen, Chengsi Ning, Yonghao Jia","doi":"10.1134/S0025654424604361","DOIUrl":"10.1134/S0025654424604361","url":null,"abstract":"<p>Joint clearances and uncertain parameters of a parallel mechanism will cause the error between the actual dynamic response and the desired dynamic response, leading to a dynamic reliability problem. Currently, the reliability analysis of mechanisms mainly focuses on mechanism with clearances or with uncertain parameters, very few involve considering both clearances and uncertainty parameters simultaneously. The main purpose of this article is to propose a dynamic reliability analysis method of a parallel mechanism with revolute clearance and uncertainty parameters. The dynamic reliability analysis of a parallel mechanism with lubrication revolute clearance joints and uncertain parameters is studied in this paper. The 3-<u>R</u>RPaR spatial parallel mechanism is taken as an example, lubrication revolute clearance joints model is established, and the dynamic model of mechanism with lubrication revolute clearance joints is derived. The dynamic reliability model of mechanism with lubrication revolute clearance joints and uncertain parameters is developed. The effects of various random uncertain parameters on the dynamic reliability of mechanism with lubrication revolute clearance joints are analyzed. This can not only provide a new method for dynamic response reliability analysis of mechanism with lubrication clearances and uncertain parameters, but also lay the foundation for dynamic response and reliability optimization design.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 4","pages":"2535 - 2555"},"PeriodicalIF":0.6,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1134/S0025654424603288
A. G. Petrov
For conservative mechanical systems, the method of normal coordinates is known, which uses the theorem on the reduction of two quadratic forms to the sum of squares. In this case, the system of differential equations is split into a system of independent oscillators. A linear dissipative mechanical system with a finite number of freedom degrees is defined by three quadratic forms: the kinetic energy of the system and potential energy of the system, and the dissipative Rayleigh function. We study the linear problem of forced oscillations of a double pendulum when the friction coefficients are proportional to the masses. Then all three quadratic forms are reduced to the sum of squares by a single transformation. In normal coordinates the system splits into two independent systems of second order. An analytical solution is constructed in the most general form for arbitrary rod lengths and point masses. A complete analysis of the oscillations in the non-resonant case and in the case of resonances is given. Formulas for the error of the analytical formulas if the proportionality of the friction coefficients and masses is violated are also obtained.
{"title":"On Forced Oscillations of a Double Mathematical Pendulum","authors":"A. G. Petrov","doi":"10.1134/S0025654424603288","DOIUrl":"10.1134/S0025654424603288","url":null,"abstract":"<p>For conservative mechanical systems, the method of normal coordinates is known, which uses the theorem on the reduction of two quadratic forms to the sum of squares. In this case, the system of differential equations is split into a system of independent oscillators. A linear dissipative mechanical system with a finite number of freedom degrees is defined by three quadratic forms: the kinetic energy of the system and potential energy of the system, and the dissipative Rayleigh function. We study the linear problem of forced oscillations of a double pendulum when the friction coefficients are proportional to the masses. Then all three quadratic forms are reduced to the sum of squares by a single transformation. In normal coordinates the system splits into two independent systems of second order. An analytical solution is constructed in the most general form for arbitrary rod lengths and point masses. A complete analysis of the oscillations in the non-resonant case and in the case of resonances is given. Formulas for the error of the analytical formulas if the proportionality of the friction coefficients and masses is violated are also obtained.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 4","pages":"1898 - 1908"},"PeriodicalIF":0.6,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1134/S0025654424604622
A. Boulenouar, M. A. Bouchelarm, M. Chafi, N. Benseddiq
This study examines the mixed-mode fracture criterion for Alumina/Zirconia functionally graded materials (FGMs), based on the concept of the equivalent stress intensity factors Keq. For this purpose, a computational algorithm is developed and incorporated into a finite element software, using a combination of five methods (FE method, Crack box technique CBT, Displacement extrapolation technique DET, Crack propagation criteria and Tanaka’s approach), in order to then determine the critical loading necessary to control the risk of crack propagation, as well as the determination of the different parameters (Stress intensity factors SIFs, bifurcation angle and T-stress). The mechanical properties of the Alumina/Zirconia FGM are supposed to change gradually through the cracked plate width, according to an exponential law (E-FGM). The continuous variation in material properties for Alumina/Zirconia FGMs is addressed by defining these properties at the centroid of each finite element. The proposed fracture criterion was identified according to the geometry of the specimen, the loading conditions and the mechanical properties of the FGM material. The frontier of crack propagation given by the proposed criterion is well defined and excellent results are obtained under pure mode-I and mixed-mode loadings.
{"title":"Mixed-Mode Fracture Criterion in Alumina/Zirconia FGMs Using Finite Element Analysis","authors":"A. Boulenouar, M. A. Bouchelarm, M. Chafi, N. Benseddiq","doi":"10.1134/S0025654424604622","DOIUrl":"10.1134/S0025654424604622","url":null,"abstract":"<p>This study examines the mixed-mode fracture criterion for Alumina/Zirconia functionally graded materials (FGMs), based on the concept of the equivalent stress intensity factors <i>K</i><sub>eq</sub>. For this purpose, a computational algorithm is developed and incorporated into a finite element software, using a combination of five methods (FE method, Crack box technique CBT, Displacement extrapolation technique DET, Crack propagation criteria and Tanaka’s approach), in order to then determine the critical loading necessary to control the risk of crack propagation, as well as the determination of the different parameters (Stress intensity factors SIFs, bifurcation angle and T-stress). The mechanical properties of the Alumina/Zirconia FGM are supposed to change gradually through the cracked plate width, according to an exponential law (E-FGM). The continuous variation in material properties for Alumina/Zirconia FGMs is addressed by defining these properties at the centroid of each finite element. The proposed fracture criterion was identified according to the geometry of the specimen, the loading conditions and the mechanical properties of the FGM material. The frontier of crack propagation given by the proposed criterion is well defined and excellent results are obtained under pure mode-I and mixed-mode loadings.</p>","PeriodicalId":697,"journal":{"name":"Mechanics of Solids","volume":"59 4","pages":"2407 - 2423"},"PeriodicalIF":0.6,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}