Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2019-81-3-324-332
T. Yakovleva, V. Bazhenov, V. Kruzhilin, V. Krysko
A theory of contact interaction of a plate locally supported by a beam, under the influence of external lateral load and external additive color noise (pink, red, white) was constructed. Also described design is in a stationary temperature field. For the plate, the Kirchhoff kinematic model was adopted; for the beam, Euler - Bernoulli, the physical nonlinearity is taken into account according to the theory of small elastic-plastic deformations. The temperature field is taken into account according to the Duhamel - Neumann theory, and there are no restrictions on the temperature distribution over the plate thickness and the height of the beam. The temperature field is determined from the solution of the three-dimensional (plate) and two-dimensional (beam) heat conduction equations. The theory of B.Ya. Cantor. The heat conduction equations are solved by the finite difference method of the second and fourth order of accuracy. The system of differential equations is reduced to the Cauchy problem by the Bubnov - Galerkin methods in higher approximations and finite differences in spatial variables. Next, the Cauchy problem is solved by the fourth-order Runge - Kutta method and the Newmark method. At each time step, the iterative procedure of I. Birger was applied. The results of a numerical experiment are given. To analyze the results, the methods of nonlinear dynamics were used (construction of signals, phase portraits, Poincare sections, Fourier power spectra and Morlet wavelet spectra, analysis of the sign of Lyapunov indices by three methods: Wolf, Kantz, Rosenstein). The effect of color noise on the contact interaction between the plate and the beam has been studied. It has been established that red additive noise has the most significant effect on the oscillation pattern of the lamellar-beam structure compared to pink and white noise.
{"title":"MATHEMATICAL MODELING OF NONLINEAR VIBRATIONS OF A PLATE WITH EXPOSURE TO COLOR NOISE TAKING INTO ACCOUNT OF CONTACT INTERACTION WITH THE BEAM","authors":"T. Yakovleva, V. Bazhenov, V. Kruzhilin, V. Krysko","doi":"10.32326/1814-9146-2019-81-3-324-332","DOIUrl":"https://doi.org/10.32326/1814-9146-2019-81-3-324-332","url":null,"abstract":"A theory of contact interaction of a plate locally supported by a beam, under the influence of external lateral load and external additive color noise (pink, red, white) was constructed. Also described design is in a stationary temperature field. For the plate, the Kirchhoff kinematic model was adopted; for the beam, Euler - Bernoulli, the physical nonlinearity is taken into account according to the theory of small elastic-plastic deformations. The temperature field is taken into account according to the Duhamel - Neumann theory, and there are no restrictions on the temperature distribution over the plate thickness and the height of the beam. The temperature field is determined from the solution of the three-dimensional (plate) and two-dimensional (beam) heat conduction equations. The theory of B.Ya. Cantor. The heat conduction equations are solved by the finite difference method of the second and fourth order of accuracy. The system of differential equations is reduced to the Cauchy problem by the Bubnov - Galerkin methods in higher approximations and finite differences in spatial variables. Next, the Cauchy problem is solved by the fourth-order Runge - Kutta method and the Newmark method. At each time step, the iterative procedure of I. Birger was applied. The results of a numerical experiment are given. To analyze the results, the methods of nonlinear dynamics were used (construction of signals, phase portraits, Poincare sections, Fourier power spectra and Morlet wavelet spectra, analysis of the sign of Lyapunov indices by three methods: Wolf, Kantz, Rosenstein). The effect of color noise on the contact interaction between the plate and the beam has been studied. It has been established that red additive noise has the most significant effect on the oscillation pattern of the lamellar-beam structure compared to pink and white noise.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132219611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2020-82-2-215-224
V. Erofeev, I. Samokhvalov
A numerical study of the survivability of the flange assembly is carried out upon reaching a critical load and in the presence of a defect in one of the design areas, taking into account the calculated values of the aerodynamic coefficients. An experiment is being carried out to determine the values of the wind load acting on the supporting legs of a metal tower. The calculation of the stressstrain state is performed using software system as SCAD Office and IDEA StatiCa 10.0. After calculating the forces in the core model of the structure, a threedimensional plate model of the assembly is formed and prepared for calculation. According to the results of the experiment, a graph was compiled with the values of aerodynamic coefficients, which were used in calculating the stressstrain state of the node. The analysis of the calculation results revealed that in the design (defectfree) state of the structure, the safety factor of the bearing units and elements is 35-40% (equivalent stresses were 165 MPa). If there is a defect in the metal structures of the belt in the region of the flange, the equivalent stresses increase to 247.6 MPa in the region of the cleavage (defective hole), thus, the margin in bearing capacity drops to 0.4%. As a result of the assessment of the survivability of the flange connection, it was revealed that the connection has a high potential survivability, in turn, the flange itself is able to work in the presence of some defects without reducing its bearing capacity to a critical level. The aerodynamic coefficients obtained in this work will determine the wind load on this type of profile and can be used in design calculations of tower structures for wind loads.
{"title":"EVALUATION OF THE VITALITY OF A FLANGED CONNECTION WITH A STEEL TOWER STRUCTURE WITH ACCOUNT OF EXPERIMENTAL DETERMINATION OF AERODYNAMIC COEFFICIENTS","authors":"V. Erofeev, I. Samokhvalov","doi":"10.32326/1814-9146-2020-82-2-215-224","DOIUrl":"https://doi.org/10.32326/1814-9146-2020-82-2-215-224","url":null,"abstract":"A numerical study of the survivability of the flange assembly is carried out upon reaching a critical load and in the presence of a defect in one of the design areas, taking into account the calculated values of the aerodynamic coefficients. An experiment is being carried out to determine the values of the wind load acting on the supporting legs of a metal tower. The calculation of the stressstrain state is performed using software system as SCAD Office and IDEA StatiCa 10.0. After calculating the forces in the core model of the structure, a threedimensional plate model of the assembly is formed and prepared for calculation.\u0000\u0000According to the results of the experiment, a graph was compiled with the values of aerodynamic coefficients, which were used in calculating the stressstrain state of the node. The analysis of the calculation results revealed that in the design (defectfree) state of the structure, the safety factor of the bearing units and elements is 35-40% (equivalent stresses were 165 MPa). If there is a defect in the metal structures of the belt in the region of the flange, the equivalent stresses increase to 247.6 MPa in the region of the cleavage (defective hole), thus, the margin in bearing capacity drops to 0.4%. As a result of the assessment of the survivability of the flange connection, it was revealed that the connection has a high potential survivability, in turn, the flange itself is able to work in the presence of some defects without reducing its bearing capacity to a critical level.\u0000\u0000The aerodynamic coefficients obtained in this work will determine the wind load on this type of profile and can be used in design calculations of tower structures for wind loads.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132458461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2020-82-4-507-523
A. Petrov, M. V. Grigoryev
Computer modeling based on the boundary element method is performed for the problem of loading in terms of the Heaviside step function inside a cubic cavity located in a partially saturated poroelastic half-space. A poroelastic medium is represented by a heterogeneous material-based model consisting of an elastic matrix phase and two phases of fillers – liquid and gas filling the pore system. The material model corresponds to a three-component medium. The constitutive relations of poroelastic medium written in terms skeleton displacements and pore pressures of fillers are considered. The original initial-boundary value problem is reduced to a boundary value problem by using the formal application of the Laplace transform. The research technique is based on the direct approach boundary integral equations of 3D isotropic linear theory of poroelasticity. Boundary integral equations corresponding to the boundary value problem are solved by the boundary element method in combination with the collocation method. In this study 8-noded elements have been adopted to discretize the boundary of poroelastic half-space. It is assumed that the element is linear with respect to displacements and pore pressures, while only one central node is used to represent tractions and fluxes. Algorithms for eliminating singularities, decreasing the order and subdividing elements are employed to compute the integral coefficients of a discrete analogue of the boundary integral equation. Regular integrals are calculated using the Gauss quadrature formula. The solution in time is obtained by numerical inversion of the Laplace transform. The numerical inversion method relies on quadrature formulas for computing the convolution integral. The time dependences of unknown displacement functions and pore pressures at points on the surface of the half-space and the cavity are plotted. The corresponding graphs are given. The influence of the cavity depth and degree of saturation on dynamic responses is investigated. The solution obtained by using the model of a fully saturated poroelastic material is compared to that of partially saturated poroelastic material. It is noted that the model used for solving this problem leads to an underestimation of displacement and overestimation of pore pressure estimates.
{"title":"NUMERICAL MODELLING OF DYNAMIC RESPONSE OF A PARTIALLY SATURATED POROELASTIC HALF-SPACE IN CASE OF A LOAD ACTING INSIDE A CUBIC CAVITY","authors":"A. Petrov, M. V. Grigoryev","doi":"10.32326/1814-9146-2020-82-4-507-523","DOIUrl":"https://doi.org/10.32326/1814-9146-2020-82-4-507-523","url":null,"abstract":"Computer modeling based on the boundary element method is performed for the problem of loading in terms of the Heaviside step function inside a cubic cavity located in a partially saturated poroelastic half-space. A poroelastic medium is represented by a heterogeneous material-based model consisting of an elastic matrix phase and two phases of fillers – liquid and gas filling the pore system. The material model corresponds to a three-component medium. The constitutive relations of poroelastic medium written in terms skeleton displacements and pore pressures of fillers are considered. The original initial-boundary value problem is reduced to a boundary value problem by using the formal application of the Laplace transform. The research technique is based on the direct approach boundary integral equations of 3D isotropic linear theory of poroelasticity. Boundary integral equations corresponding to the boundary value problem are solved by the boundary element method in combination with the collocation method. In this study 8-noded elements have been adopted to discretize the boundary of poroelastic half-space. It is assumed that the element is linear with respect to displacements and pore pressures, while only one central node is used to represent tractions and fluxes. Algorithms for eliminating singularities, decreasing the order and subdividing elements are employed to compute the integral coefficients of a discrete analogue of the boundary integral equation. Regular integrals are calculated using the Gauss quadrature formula. The solution in time is obtained by numerical inversion of the Laplace transform. The numerical inversion method relies on quadrature formulas for computing the convolution integral. The time dependences of unknown displacement functions and pore pressures at points on the surface of the half-space and the cavity are plotted. The corresponding graphs are given. The influence of the cavity depth and degree of saturation on dynamic responses is investigated. The solution obtained by using the model of a fully saturated poroelastic material is compared to that of partially saturated poroelastic material. It is noted that the model used for solving this problem leads to an underestimation of displacement and overestimation of pore pressure estimates.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117287390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2020-82-4-428-441
M. V. Bezhentseva, L. I. Vutsin, A. I. Kibets, L. Kruszka
The 3D problem of wood deformation under shock loading is considered. The governing system of equations is formulated in Lagrange variables. A defining system of equations in a three-dimensional formulation is presented. The equation of motion is derived from the balance of the virtual powers of work. Wood is modeled as a unidirectionally reinforced material with a description of the descending branch of the deformation diagram. Deformations and stresses are determined in a local basis, the position of which in space is related to the direction of the wood grain. Wood material is represented as a combination of reinforcing fibers and a matrix, the elastoplastic deformation of which is described by the relations of the theory of flow with combined kinematic and isotropic strengthening. The deformation characteristics of the matrix and fibers are determined on the basis of a computational and experimental study of the mechanical properties of wood along and across the fibers. In numerical simulation, the moment scheme of the finite element method and an explicit time integration scheme of the “cross” type are used. Discretization of the computational domain is based on an eight-node isoparametric finite element adapted to the specifics of the problem under consideration. Software realization of the developed mathematical model and numerical methodology is implemented within the computing complex “Dynamics-3”. Computer simulation of compression of an experimental specimen of spruce along and across the fibers has been performed. The reliability of the calculation results is confirmed by good agreement with the experimental data.
{"title":"FINITE ELEMENT METHOD FOR NUMERICAL MODELING OF ELASTIC-PLASTIC DEFORMATION OF WOOD UNDER SHOCK LOADING","authors":"M. V. Bezhentseva, L. I. Vutsin, A. I. Kibets, L. Kruszka","doi":"10.32326/1814-9146-2020-82-4-428-441","DOIUrl":"https://doi.org/10.32326/1814-9146-2020-82-4-428-441","url":null,"abstract":"The 3D problem of wood deformation under shock loading is considered. The governing system of equations is formulated in Lagrange variables. A defining system of equations in a three-dimensional formulation is presented. The equation of motion is derived from the balance of the virtual powers of work. Wood is modeled as a unidirectionally reinforced material with a description of the descending branch of the deformation diagram. Deformations and stresses are determined in a local basis, the position of which in space is related to the direction of the wood grain. Wood material is represented as a combination of reinforcing fibers and a matrix, the elastoplastic deformation of which is described by the relations of the theory of flow with combined kinematic and isotropic strengthening. The deformation characteristics of the matrix and fibers are determined on the basis of a computational and experimental study of the mechanical properties of wood along and across the fibers. In numerical simulation, the moment scheme of the finite element method and an explicit time integration scheme of the “cross” type are used. Discretization of the computational domain is based on an eight-node isoparametric finite element adapted to the specifics of the problem under consideration. Software realization of the developed mathematical model and numerical methodology is implemented within the computing complex “Dynamics-3”. Computer simulation of compression of an experimental specimen of spruce along and across the fibers has been performed. The reliability of the calculation results is confirmed by good agreement with the experimental data.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123192594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2020-82-4-483-492
V. Firsanov, L. Nguyen
Based on the refined theory in this paper presents the stress-strain state of cylindrical shells taking into account the piezoelectric effect. The mechanical displacements and electrical potentials of the shell are approximated by polynomials in the normal coordinate two degrees higher in relation to the classical theory of the Kirchhoff-Love type. The equations of the theory of elasticity and the laws of electrostatics are used to obtain model of electroelasticity behavior. By using Lagrange variational principle a system of differential equations of equilibrium in displacements and potentials with boundary conditions is derived. Trigonometric Fourier series in the circumferential coordinate is used to reduce partial differential equations system to ordinary differential equations. The formulated boundary value problem of the electroelastic state of the shell is solved by an operator method based on the Laplace transform. Transverse normal and tangential stresses of the linear equilibrium equation of the three-dimensional theory of elasticity. Examples of calculating the stress state of a cylindrical piezoelectric shell with clamped support are provided. Two cases are analyzed: the shell is under the influence of mechanical loads and electrical potentials. A comparison of the results obtained according to the proposed theory and the classical theory is carried out. It has been established that there is an additional stress state of the "boundary layer" type. It allows to confirm the practical value of the developed mathematical model and a significant contribution to the general stress-strain state with the strength and durability of cylindrical shells modeled elements of mechanical engineering structures taking into account the piezoelectric effect.
{"title":"STRESSED STATE IN EDGE ZONE OF CYLINDRICAL SHELLS BASED ON A NON-CLASSIC THEORY WITH THE PIEZOELECTRIC EFFECT","authors":"V. Firsanov, L. Nguyen","doi":"10.32326/1814-9146-2020-82-4-483-492","DOIUrl":"https://doi.org/10.32326/1814-9146-2020-82-4-483-492","url":null,"abstract":"Based on the refined theory in this paper presents the stress-strain state of cylindrical shells taking into account the piezoelectric effect. The mechanical displacements and electrical potentials of the shell are approximated by polynomials in the normal coordinate two degrees higher in relation to the classical theory of the Kirchhoff-Love type. The equations of the theory of elasticity and the laws of electrostatics are used to obtain model of electroelasticity behavior. By using Lagrange variational principle a system of differential equations of equilibrium in displacements and potentials with boundary conditions is derived. Trigonometric Fourier series in the circumferential coordinate is used to reduce partial differential equations system to ordinary differential equations. The formulated boundary value problem of the electroelastic state of the shell is solved by an operator method based on the Laplace transform. Transverse normal and tangential stresses of the linear equilibrium equation of the three-dimensional theory of elasticity.\u0000\u0000Examples of calculating the stress state of a cylindrical piezoelectric shell with clamped support are provided. Two cases are analyzed: the shell is under the influence of mechanical loads and electrical potentials. A comparison of the results obtained according to the proposed theory and the classical theory is carried out. It has been established that there is an additional stress state of the \"boundary layer\" type. It allows to confirm the practical value of the developed mathematical model and a significant contribution to the general stress-strain state with the strength and durability of cylindrical shells modeled elements of mechanical engineering structures taking into account the piezoelectric effect.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123421771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2019-81-1-5-18
A. Nasedkin
The present paper considers the homogenization problems for mixed piezoelectric composite materials with stochastic distributions of inclusions or pores and with taking into account the mechanical imperfect interphase boundaries. The accepted interface statements correspond to the Gurtin - Murdoch model and give a significant contribution only for nanostructured composites. To determine the effective properties, an integrated approach was used, based on the theory of effective moduli, on the modelling of representative element volumes and on the finite element method. An aggregate of boundary value problems was described, which allow one to find a complete set of effective stiffness moduli, piezomoduli, and dielectric constants for a piezocomposite of arbitrary anisotropy class. The numerical solution of homogenization problems was carried out in the ANSYS finite-element package, which was used both for modelling of representative element volumes and for computation of the effective properties of composite material. The representative volume consisted of a regular cubic array of piezoelectric finite elements with the material properties of the two phases. The contact boundaries between materials of different phases were covered with elastic membrane elements that simulated interface surface stresses. Specific implementation was performed for nanoporous piezoceramic composites, for which both the initial phases and the homogeneous material were materials of the hexagonal symmetry class, and the pores were considered as piezoelectric material with negligibly small stiffness moduli and piezomoduli. For this composite the membrane elements inherited the anisotropy structure of volume elements on their common edges. As an example, the results of calculations of effective moduli for porous ferroelectric soft piezoceramics PZT-5H were presented. It was noted that the surface stresses on the boundaries of the pores can significantly increase the values of the effective stiffness moduli. However, they had a weak influence on the values of the effective piezomoduli and dielectric constants.
{"title":"ANALYSIS OF SURFACE STRESS INFLUENCE ON THE EFFECTIVE PROPERTIES OF NANOPOROUS PIEZOCOMPOSITES","authors":"A. Nasedkin","doi":"10.32326/1814-9146-2019-81-1-5-18","DOIUrl":"https://doi.org/10.32326/1814-9146-2019-81-1-5-18","url":null,"abstract":"The present paper considers the homogenization problems for mixed piezoelectric composite materials with stochastic distributions of inclusions or pores and with taking into account the mechanical imperfect interphase boundaries. The accepted interface statements correspond to the Gurtin - Murdoch model and give a significant contribution only for nanostructured composites. To determine the effective properties, an integrated approach was used, based on the theory of effective moduli, on the modelling of representative element volumes and on the finite element method. An aggregate of boundary value problems was described, which allow one to find a complete set of effective stiffness moduli, piezomoduli, and dielectric constants for a piezocomposite of arbitrary anisotropy class. The numerical solution of homogenization problems was carried out in the ANSYS finite-element package, which was used both for modelling of representative element volumes and for computation of the effective properties of composite material. The representative volume consisted of a regular cubic array of piezoelectric finite elements with the material properties of the two phases. The contact boundaries between materials of different phases were covered with elastic membrane elements that simulated interface surface stresses.\u0000\u0000Specific implementation was performed for nanoporous piezoceramic composites, for which both the initial phases and the homogeneous material were materials of the hexagonal symmetry class, and the pores were considered as piezoelectric material with negligibly small stiffness moduli and piezomoduli. For this composite the membrane elements inherited the anisotropy structure of volume elements on their common edges. As an example, the results of calculations of effective moduli for porous ferroelectric soft piezoceramics PZT-5H were presented. It was noted that the surface stresses on the boundaries of the pores can significantly increase the values of the effective stiffness moduli. However, they had a weak influence on the values of the effective piezomoduli and dielectric constants.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128643319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2020-82-1-100-106
A. Dubinsky
The field of application of Functionally Graded Materialsis steadily expanding, which stimulates research in the relevant areas. In relation to penetration mechanics, these are primarily experimental studies of multilayer barriers consisting of plates “in contact” with various mechanical properties. Despite intensive research, explicit formulas for integral penetration characteristics (penetration depth and ballistic limit) cannot be obtained, except for the case when sequential penetration of layers (barriers with large gaps between layers). In this article, explicit formulas for the depth of penetration into an semi-infinite shield and for the ballistic limit velocity applying penetration into a shield of a finite thickness are derived assuming that the hardness of the barrier material varies continuously depending on barrier depth. The theoretical analysis is based on a model that represents the normal stress at points on the surface of the penetrating body that are in contact with the barrier as a quadratic function of the normal component of local impactor velocity with a zero linear term (the Vitman - Stepanov model). Difference of the dynamic hardness in different points of impactor-barrier contact is taken into account. It is also assumed that the nose of the striker has the form of a straight circular cone and the initial stage of penetration when the striker is not completely immersed in the barrier is ignored.
{"title":"EXPLICIT FORMULA FOR DEPTH OF PENETRATION OF CONE-NOSED IMPACTOR INTO ANISOTROPIC SHIELDS","authors":"A. Dubinsky","doi":"10.32326/1814-9146-2020-82-1-100-106","DOIUrl":"https://doi.org/10.32326/1814-9146-2020-82-1-100-106","url":null,"abstract":"The field of application of Functionally Graded Materialsis steadily expanding, which stimulates research in the relevant areas. In relation to penetration mechanics, these are primarily experimental studies of multilayer barriers consisting of plates “in contact” with various mechanical properties. Despite intensive research, explicit formulas for integral penetration characteristics (penetration depth and ballistic limit) cannot be obtained, except for the case when sequential penetration of layers (barriers with large gaps between layers).\u0000\u0000In this article, explicit formulas for the depth of penetration into an semi-infinite shield and for the ballistic limit velocity applying penetration into a shield of a finite thickness are derived assuming that the hardness of the barrier material varies continuously depending on barrier depth. The theoretical analysis is based on a model that represents the normal stress at points on the surface of the penetrating body that are in contact with the barrier as a quadratic function of the normal component of local impactor velocity with a zero linear term (the Vitman - Stepanov model). Difference of the dynamic hardness in different points of impactor-barrier contact is taken into account. It is also assumed that the nose of the striker has the form of a straight circular cone and the initial stage of penetration when the striker is not completely immersed in the barrier is ignored.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121292784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2019-81-4-449-460
V. Saurin
Issues related to eigen-vibrations of elastic beams of variable cross-section are discussed. It is noted that one of the common features characteristic of boundary-value problems of mathematical physics is certain ambiguity of their formulations. A boundary-value problem of determining eigen-frequencies of a variable cross-section beam in displacements is formulated. By introducing new variables characterizing the behavior of the system, the boundary-value problem is reduced to three ordinary differential equations with variable coefficients. The new variables have a distinct physical meaning. One of the functions is linear density of the pulse and the other is bending moment in the cross-section of the beam. Such a formulation of the problem of free vibrations of a variable cross-section beam makes it possible to reduce the system of differential equations to a single fourth-order equation written in terms of pulse functions. This equation is equivalent to the initial one, formulated in displacements, but has a different form. A method of integral-differential relations, alternative to classical numerical approaches, is described. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions resulting from the method of integral-differential relations is studied. The projection approach to analyzing spectral problems of nonlinear beam theory is considered. The efficiency of the method of integral-differential equations is demonstrated, using the problem of free vibrations of a rectangular beam with a constructional depth quadratically varying along its length. Energy-based evaluations of the accuracy of the approximate solutions constructed using polynomial approximations of the sought functions are presented. It is shown that applying standard Bubnov-Galerkin's method to the problem of free vibrations leads to the appearance of complex eigen-frequencies. At the same time, the ratio of the imaginary component to the real one of the eigen-value is a relative inaccuracy of the solution of the boundary-value problem. The introduced numerical algorithm makes it possible to evaluate unambiguously the local and integral quality of numerical solutions obtained.
{"title":"INTEGRAL-DIFFERENTIAL RELATIONS IN THE PROBLEM OF FREE BENDING VIBRATIONS OF VARIABLE CROSS-SECTION BEAMS","authors":"V. Saurin","doi":"10.32326/1814-9146-2019-81-4-449-460","DOIUrl":"https://doi.org/10.32326/1814-9146-2019-81-4-449-460","url":null,"abstract":"Issues related to eigen-vibrations of elastic beams of variable cross-section are discussed. It is noted that one of the common features characteristic of boundary-value problems of mathematical physics is certain ambiguity of their formulations. A boundary-value problem of determining eigen-frequencies of a variable cross-section beam in displacements is formulated. By introducing new variables characterizing the behavior of the system, the boundary-value problem is reduced to three ordinary differential equations with variable coefficients. The new variables have a distinct physical meaning. One of the functions is linear density of the pulse and the other is bending moment in the cross-section of the beam. Such a formulation of the problem of free vibrations of a variable cross-section beam makes it possible to reduce the system of differential equations to a single fourth-order equation written in terms of pulse functions. This equation is equivalent to the initial one, formulated in displacements, but has a different form. A method of integral-differential relations, alternative to classical numerical approaches, is described. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions resulting from the method of integral-differential relations is studied. The projection approach to analyzing spectral problems of nonlinear beam theory is considered. The efficiency of the method of integral-differential equations is demonstrated, using the problem of free vibrations of a rectangular beam with a constructional depth quadratically varying along its length. Energy-based evaluations of the accuracy of the approximate solutions constructed using polynomial approximations of the sought functions are presented. It is shown that applying standard Bubnov-Galerkin's method to the problem of free vibrations leads to the appearance of complex eigen-frequencies. At the same time, the ratio of the imaginary component to the real one of the eigen-value is a relative inaccuracy of the solution of the boundary-value problem. The introduced numerical algorithm makes it possible to evaluate unambiguously the local and integral quality of numerical solutions obtained.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"23 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120990296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2021-83-3-324-334
D. Derendyaev, N. Derendyaev
Earlier, one of the authors proposed and developed (together with coworkers) an original method to study the stability of stationary rotation of rotary systems containing a viscous liquid and having a drive that maintains the angular velocity of rotation constant. It was assumed that the rotor has axial symmetry, the anchors of its axis are isotropic. The method is based on two theorems, according to which a change in the degree of instability is associated with the possibility of a perturbed motion of the circular precession type. This motion has a remarkable property: the velocity field and the shape of the liquid surface do not depend on time in a specially selected non-inertial reference frame associated with the line of centers. Finding the conditions for the feasibility of circular precession makes it possible to effectively construct the boundaries of the stability regions of the stationary rotation regime in the space of problem parameters. In addition, the study of the occurrence of circular precession allows us to find the conditions under which a subcritical (supercritical) Andronov-Hopf bifurcation takes place in the rotor system and to identify "dangerous" (“safe”) sections of the boundaries of the stability regions. In this paper, the previously proposed method of stability research applies to systems in which the rotor axis is located in anisotropic Laval type anchors. In the study of rotary systems of this type, it is possible to link the change in the degree of instability with the feasibility of perturbed movements of the elliptical precession type. It can be shown that the imaginary characteristic numbers of the equations in deviations from the stationary rotation mode are possible only in the case when there is a perturbed motion in the form of an elliptical precession. An example of a study of the stability of stationary rotation of a typical rotary system is given. Mechanical effects caused by the fact that gyroscopic stabilization becomes impossible with anisotropic fixing of the rotor axis are noted.
{"title":"THE RESEARCH OF STABILITY OF STATIONARY ROTATION A ROTOR SYSTEM WITH A LIQUID, THE AXLE OF WHICH IS LOCATED IN ANISOTROPIC FIXINGS","authors":"D. Derendyaev, N. Derendyaev","doi":"10.32326/1814-9146-2021-83-3-324-334","DOIUrl":"https://doi.org/10.32326/1814-9146-2021-83-3-324-334","url":null,"abstract":"Earlier, one of the authors proposed and developed (together with coworkers) an original method to study the stability of stationary rotation of rotary systems containing a viscous liquid and having a drive that maintains the angular velocity of rotation constant. It was assumed that the rotor has axial symmetry, the anchors of its axis are isotropic. The method is based on two theorems, according to which a change in the degree of instability is associated with the possibility of a perturbed motion of the circular precession type. This motion has a remarkable property: the velocity field and the shape of the liquid surface do not depend on time in a specially selected non-inertial reference frame associated with the line of centers. Finding the conditions for the feasibility of circular precession makes it possible to effectively construct the boundaries of the stability regions of the stationary rotation regime in the space of problem parameters. In addition, the study of the occurrence of circular precession allows us to find the conditions under which a subcritical (supercritical) Andronov-Hopf bifurcation takes place in the rotor system and to identify \"dangerous\" (“safe”) sections of the boundaries of the stability regions. In this paper, the previously proposed method of stability research applies to systems in which the rotor axis is located in anisotropic Laval type anchors. In the study of rotary systems of this type, it is possible to link the change in the degree of instability with the feasibility of perturbed movements of the elliptical precession type. It can be shown that the imaginary characteristic numbers of the equations in deviations from the stationary rotation mode are possible only in the case when there is a perturbed motion in the form of an elliptical precession. An example of a study of the stability of stationary rotation of a typical rotary system is given. Mechanical effects caused by the fact that gyroscopic stabilization becomes impossible with anisotropic fixing of the rotor axis are noted.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127314888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.32326/1814-9146-2019-81-4-488-499
W. Cheng, Yang Tonghui, Li Wan, Tao Li, M. Abuziarov, A. V. Kochetkov
The spatial problem of internal explosive loading of an elastoplastic cylindrical container filled with water in Eulerian - Lagrangian variables using multigrid algorithms is considered. A defining system of three-dimensional equations of the dynamics of gas, fluid, and elastoplastic medium is presented. For numerical modeling, a modification of S.K. Godunov scheme of the increased accuracy for both detonation products and liquids, and elastoplastic container is used. At the moving contact boundaries “detonation products - liquid”, “liquid - deformable body”, the exact solution of the Riemann's problem is used. A time dependent model is used to describe the propagation of steady-state detonation wave through an explosive from an initiation region. In both cases, the initiation of detonation occurs at the center of the charge. Two problems have been solved: the first task for the aisymmetric position of the charge, the second for the charge shifted relative to the axis of symmetry. In the first task, the processes are two-dimensional axisymmetric in nature, in the second task, the processes are essentially three-dimensional. A comparison is made of the results of calculations of the first problem using a three-dimensional method with a solution using a previously developed two-dimensional axisymmetric method and experimental data. Good agreement is observed between the numerical results for the maximum velocities and circumferential strains obtained by various methods and experimental data. There is good agreement between the numerical results obtained by various methods and the known experimental data. Comparison of the results of solving the first and second problems shows a significant effect of the position of the charge on the wave processes in the liquid, the processes of loading the container and its elastoplastic deformation. The dynamic behavior of a gas bubble with detonation products is analyzed. A significant deviation of the bubble shape from the spherical one, caused by the action of shock waves reflected from the structure, is shown. Comparison of the results of solving the first and second problems showed a significant effect of the charge position on wave processes in a liquid, the processes of loading a container and its elastoplastic deformation. In particular, in the second problem, shock waves of higher amplitude are observed in the liquid when reflected from the walls of the container.
{"title":"MODELING OF ELASTIC-PLASTIC DEFORMATION OF ELEMENTS OF SPATIAL STRUCTURES DURING PULSE INTERACTION WITH FLUID BASED ON THE GODUNOV'S METHOD OF INCREASED ACCURACY","authors":"W. Cheng, Yang Tonghui, Li Wan, Tao Li, M. Abuziarov, A. V. Kochetkov","doi":"10.32326/1814-9146-2019-81-4-488-499","DOIUrl":"https://doi.org/10.32326/1814-9146-2019-81-4-488-499","url":null,"abstract":"The spatial problem of internal explosive loading of an elastoplastic cylindrical container filled with water in Eulerian - Lagrangian variables using multigrid algorithms is considered. A defining system of three-dimensional equations of the dynamics of gas, fluid, and elastoplastic medium is presented. For numerical modeling, a modification of S.K. Godunov scheme of the increased accuracy for both detonation products and liquids, and elastoplastic container is used. At the moving contact boundaries “detonation products - liquid”, “liquid - deformable body”, the exact solution of the Riemann's problem is used. A time dependent model is used to describe the propagation of steady-state detonation wave through an explosive from an initiation region. In both cases, the initiation of detonation occurs at the center of the charge. Two problems have been solved: the first task for the aisymmetric position of the charge, the second for the charge shifted relative to the axis of symmetry. In the first task, the processes are two-dimensional axisymmetric in nature, in the second task, the processes are essentially three-dimensional. A comparison is made of the results of calculations of the first problem using a three-dimensional method with a solution using a previously developed two-dimensional axisymmetric method and experimental data. Good agreement is observed between the numerical results for the maximum velocities and circumferential strains obtained by various methods and experimental data. There is good agreement between the numerical results obtained by various methods and the known experimental data. Comparison of the results of solving the first and second problems shows a significant effect of the position of the charge on the wave processes in the liquid, the processes of loading the container and its elastoplastic deformation. The dynamic behavior of a gas bubble with detonation products is analyzed. A significant deviation of the bubble shape from the spherical one, caused by the action of shock waves reflected from the structure, is shown. Comparison of the results of solving the first and second problems showed a significant effect of the charge position on wave processes in a liquid, the processes of loading a container and its elastoplastic deformation. In particular, in the second problem, shock waves of higher amplitude are observed in the liquid when reflected from the walls of the container.","PeriodicalId":340995,"journal":{"name":"Problems of strenght and plasticity","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125591310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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