Three-dimensional Stress-strain State Research Articles

Comparative evaluation of mathematical models of polymer composite material with the implementation of a three-dimensional stress–strain state in the simulation of impact

Comparative evaluation of mathematical models of polymer composite material with the implementation of a three-dimensional stress–strain state in the simulation of impact

Construction of homogeneous solutions of the torsion problem for a radially inhomogeneous transversely isotropic cylinder

The torsion problem for a radially inhomogeneous transversely isotropic cylinder of small thickness was investigated by the method of asymptotic integration of elasticity theory equations. It is assumed that the side part of the cylinder is stress-free, and boundary conditions are set at the ends of the cylinder, leaving the cylinder in equilibrium. The elastic moduli are thought to be arbitrary continuous functions of the variable along the cylinder radius. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thin-walledness of the cylinder. Homogeneous solutions are built, i.e. any solutions of the equilibrium equation satisfying the condition of no stresses on the side surfaces. It is shown that the solution of the torsion problem consists of a penetrating solution and a boundary layer character solution similar to Saint-Venant's edge effect in the theory of inhomogeneous plates. The penetrating solution determines the internal stress-strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the torsional moments of stresses acting in the cross-section perpendicular to the cylinder axis. Solutions having the boundary layer character are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in applied shell theories. Asymptotic formulas for displacement and stresses are built, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous transversely isotropic cylinder of small thickness. Based on the obtained asymptotic expansions, it is possible to assess the applicability of applied theories and build a refined applied theory for radially inhomogeneous cylindrical shells

Three-Dimensional Stress-Strain State Analysis of the Bimetallic Launch Vehicle Propellant Tank Shell

Three-Dimensional Stress-Strain State Analysis of the Bimetallic Launch Vehicle Propellant Tank Shell

Analysis of Stress-Strain State of a Cylinder with Variable Elasticity Moduli Based on Three-Dimensional Equations of Elasticity Theory

Introduction. Functionally graded materials are of great use, because heterogeneity of properties enables to control the strength and rigidity of structures. This has caused great interest in the topic in the world scientific literature. The construction of solutions to such problems depends significantly on the type of boundary conditions. In this paper, we consider the equilibrium of a thin-walled circular cylinder whose mechanical properties change along the radius. Homogeneous boundary conditions were set on cylindrical surfaces that had not been considered before, the effect was on the ends. The mathematical formulation of the problem was carried out in the linear theory of elasticity in the framework of axisymmetric deformation. Expressions were constructed for the components of the stress-strain state of the cylinder, in which some coefficients were found from the solution to the resulting system of linear algebraic equations.Materials and Methods. The material of the cylinder was linearly elastic, the elastic modulus of which depended linearly on the radial coordinate. The basic research method was the asymptotic method, in which half the logarithm of the ratio of the outer and inner radii acted as a small parameter. Iterative processes were used to construct the characteristics of the stress-strain state of the cylinder.Results. Homogeneous solutions to the boundary value problem were obtained for a linearly elastic functionally gradient hollow thin-walled cylinder. An analysis of these solutions made it possible to reveal the nature of the stress-strain state in the cylinder wall. For this purpose, an asymptotic analysis of the solutions was carried out, relations for displacements and stresses were obtained. It was determined that those solutions corresponded to the boundary layer, while their first terms determined Saint-Venant edge effect similar to the plate theory.Discussion and Conclusion. The analytical solution to the equilibrium problem of a thin-walled cylinder inhomogeneous in radius constructed by asymptotic expansion can be used for numerical solution to a specific problem. For this, it is required to solve the obtained systems of linear algebraic equations and determine the corresponding coefficients. The resulting asymptotic representations provide analyzing the three-dimensional stress-strain state. The selection of the number of expansion terms makes it possible to calculate displacements and stresses with a given degree of accuracy. This analysis can be useful in assessing the adequacy of applied calculation methods used in engineering practice.

Construction of homogeneous solutions in the torsion problem for a transversally isotropic sphere with variable elastic moduli

The object of research is the problem of torsion for a radially inhomogeneous transversally isotropic sphere and the study based on this three-dimensional stress-strain state. To establish the scope of applicability of existing applied theories and to create more refined applied theories of inhomogeneous shells, it is important to study the stress-strain state of inhomogeneous bodies based on three-dimensional equations of elasticity theory. The problem of torsion of a radially inhomogeneous transversally isotropic non-closed sphere containing none of the poles 0 and π is considered. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the lateral surface of the sphere is free from stresses, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The formulated boundary value problem is reduced to a spectral problem. After fulfilling the homogeneous boundary conditions specified on the side surfaces of the sphere, a characteristic equation is obtained with respect to the spectral parameter. The corresponding solutions are constructed depending on the roots of the characteristic equation. It is shown that the solution corresponding to the first group of roots is penetrating, and the stress state determined by this solution is equivalent to the torques of the stresses acting in an arbitrary section θ=const. The solutions corresponding to the countable set of the second group of roots have the character of a boundary layer localized in conic slices. In the case of significant anisotropy, some boundary layer solutions decay weakly and can cover the entire region occupied by the sphere. On the basis of the performed three-dimensional analysis, new classes of solutions (solutions having the character of a boundary layer) are obtained, which are absent in applied theories. In contrast to an isotropic radially inhomogeneous sphere, for a transversely isotropic radially inhomogeneous sphere, a weakly damped boundary layer solution appears, which can penetrate deep far from the conical sections and change the picture of the stress-strain state.

Analysis of the stress-strain state of a radially inhomogeneous transversely isotropic sphere with a fixed side surface

Introduction. The paper considers an axisymmetric problem of elasticity theory for a radially inhomogeneous transversally isotopic nonclosed sphere containing none of the 0 and 𝜋 poles. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the side surface of the sphere is fixed, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The work objective is an asymptotic analysis of the problem of elasticity theory for a radially inhomogeneous transversally isotropic sphere of small thickness, and a study of a three-dimensional stress-strain state based on this analysis.Materials and Methods. The three-dimensional stress-strain state is investigated on the basis of the equations of elasticity theory by the method of homogeneous solutions and asymptotic analysis.Research Results. After the homogeneous boundary conditions set on the side surfaces of the sphere are met, a characteristic equation is obtained, and its roots are classified with respect to a small parameter characterizing the thickness of the sphere. The corresponding asymptotic solutions depending on the roots of the characteristic equation are constructed. It is shown that the solutions corresponding to a countable set of roots have the character of a boundary layer localized in conic slices. The branching of the roots generates new solutions that are characteristic only for a transversally isotropic radially inhomogeneous sphere. A weakly damping boundary layer solution appears, which can penetrate deep away from the conical sections and change the picture of the stress-strain state.Discussion and Conclusions. Based on the solutions constructed, it is possible to determine the applicability areas of existing applied theories and propose a new more refined applied theory for a radially inhomogeneous transversally isotropic spherical shell.

Прогнозування параметрів вивалоутворення в очисних вибоях глибоких вугільних шахт

Purpose. Process research of the parameters of the cave-in rock formation in roof of the 9th northern longwall of block #10 “Mines Control Pokrovske” Method. Numerical modeling of the volumetric stress-strain state (finite element method), the method of continuous media, the method of contact interaction of surfaces, analytical studies of the process of the formation of rock cave-in of the immediate roof. Results. The article presents the results of numerical modeling of the three-dimensional stress-strain state of a layered rock massif, with boundary conditions of contact interaction between rock layers and with the load of gravitational forces. The results of the grapho-analytical method of modeling the volume formation of fallout formation are presented on the example of working out the 9th northern bench of block No. 10 “PRAT Mine Management Pokrovske”. Calculated collapse parameters of the direct and main roof. The obtained parameters of fallout and their location. Scientific novelty The results of predicting the formation of cave-in rock the immediate roof of cleaning potholes using the volumetric stress-deformed state of a layered massif with the gravitational load of rock layers are presented. The results of predicting the formation of cave-in rock using the rock crack resistance criterion are presented. Practical significance. The practical significance lies in the method of predicting the processes of cave-in rock formation in the immediate roof during the working of coal seams and determining their parameters. The calculation process of numerical modeling of the step-by-step increase of the produced space gives a general picture of the state of rock layers above the produced space and their destruction zones, and determines the following measures to strengthen the roof in the coal face.

Comparison of Stress-Strain States of Rail-Wheel Pair Made of Steel and MoNiCa

Contact interaction in a wheel-rail pair and its corresponding stress-strain state for various combinations of materials of this pair and the weights of passenger and freight cars were considered. The materials under study were steel and high-strength cast iron based structural material MoNiCa with strength and wear-fatigue characteristics close to steel ones.The load on each wheel was: 53.9 and 294.3 kN for an empty and loaded freight car, 146.8 kN and 230.5 kN for an empty and loaded passenger car. The bottom surface of the rail was rigidly fixed.Young's modules were adopted as follows: 200 GPa for steel and 165 for MoNiCa. The results of finite element calculations of the three-dimensional stress-strain state showed that the largest von Mises stresses of 613.08 MPa occur in the wheel (steel)-rail (steel) pair of the loaded freight car. For all types of loading, the lowest stresses (15.8 % – 16.7 % less than in the steel-steel pair) are observed in the wheel (MoNiCa)-rail (MoNiCa) pair. This indicates the appropriateness of using MoNiCa for the manufacturing elements of the wheel-rail system.

Calculation of the improved steel beams of buildings and structures of the mining and metallurgical complex

Load-bearing elements of buildings and structures of the mining and metallurgical complex in recent decades need to develop new more effective design solutions due to the intensification of technological processes, an increase in temperature loads and aggressiveness of the environment. The main direction of increasing the efficiency of such elements is their design from economically alloyed steel, which allows to increase the resource of structures and prevent accidents with a significant increase in temperature. Due to the fact that alloyed steels have higher mechanical characteristics at elevated temperatures, the question arises of creating lightweight beam structures from such steels, reducing their material consumption while maintaining the stability and fatigue strength of beams, the most promising is the use of welded beams with a perforated wall and composite beams.
 The creation of the most effective cross-sectional shape of metal beams with a perforated wall and welded beams, as well as crane beams in transverse bending, considering strength, local stability, flat bending stability and fatigue strength is considered. It is shown that an effective shape of beams with a perforated wall is a box-shaped structure made of perforated channels. A calculation was carried out to select a rational design made of an assortment of hot-rolled channel profiles. It is shown that due to the use of the proposed sectional shape, significant savings in the weight of the structure can be achieved. Considering the three-dimensional stress-strain state, the fatigue strength of welded metal crane girders operating in severe conditions is estimated. The efficiency of using a hot-rolled I-beam as the upper chord of such welded beams is shown. The necessity of using a hot-rolled I-beam and to ensure the fatigue strength of the lower chord is demonstrated.
 The use of the previously proposed combined method for calculating the structures of industrial buildings and structures and the use of economically alloyed steels allows us to create new designs of critical elements that reduce their material consumption and increase their resource. Further research can be carried out for real object designs in order to reduce their cost and increase reliability during operation in the conditions of mining and metallurgical production.

Analytical Solution of the Problem of Symmetric Thermally Stressed State of Thick Plates Based on the 3D Elasticity Theory

An important place among thermoelasticity problems is occupied by the plane elasticity problem obtained from the general three-dimensional problem after using plane stress state hypotheses for thin plates. In the two-dimensional formulation, this problem has become widespread in the study of the effect of temperature loads on the stress state of thin thermosensitive plates. The article proposes a general three-dimensional solution of the static problem of thermoelasticity in a form convenient for practical application. To construct it, a particular solution of the inhomogeneous equation, the thermoelastic displacement potential, was added by us to the general solution of Lamé's equations, the latter solution having been previously found by us in terms of three harmonic functions. It is shown that the use of the proposed solution allows one to satisfy the relation between the static three-dimensional theory of thermoelasticity and boundary conditions, and also to construct a closed system of partial differential equations for the introduced two-dimensional functions without using hypotheses about the plane stress state of a plate. The thermoelastic stress state of a thick or thin plate is divided into two parts. The first part takes into account the thermal effects caused by external heating and internal heat sources, while the second one is determined by a symmetrical force load. The thermoelastic stresses are expressed in terms of deformations and known temperature. A three-dimensional thermoelastic stress-strain state representation is used and the zero boundary conditions on the outer flat surfaces of the plate are precisely satisfied. This allows us to show that the introduced two-dimensional functions will be harmonic. After integrating along the thickness of the plate along the normal to the median surface, normal and shear efforts are expressed in terms of three unknown two-dimensional functions. The three-dimensional stress state of a symmetrically loaded thermosensitive plate was simplified to the two-dimensional state. For this purpose, we used only the hypothesis that the normal stresses perpendicular to the median surface are insignificant in comparison with the longitudinal and transverse ones. Displacements and stresses in the plate are expressed in terms of two two-dimensional harmonic functions and a particular solution, which is determined by a given temperature on the surfaces of the plate. The introduced harmonic functions are determined from the boundary conditions on the side surface of the thick plate. The proposed technique allows the solution of three-dimensional boundary value problems for thick thermosensitive plates to be reduced to a two-dimensional case.

Separation of the 3D stress state of a loaded plate into two-dimensional tasks: bending and symmetric compression of the plate

The three-dimensional stress-strain state of an isotropic plate loaded on all its surfaces is considered in the article. The initial problem is divided into two ones: symmetrical bending of the plate and a symmetrical compression of the plate, by specified loads. It is shown that the plane problem of the theory of elasticity is a special case of the second task. To solve the second task, the symmetry of normal stresses is used. Boundary conditions on plane surfaces are satisfied and harmonic conditions are obtained for some functions. Expressions of effort were found after integrating three-dimensional stresses that satisfy three equilibrium equations. For a thin plate, a closed system of equations was obtained to determine the harmonic functions. Displacements and stresses in the plate were expressed in two two-dimensional harmonic functions and a partial solution of the Laplace equation with the right-hand side, which is determined by the end loads. Three-dimensional boundary conditions were reduced to two-dimensional ones. The formula was found for experimental determination of the sum of normal stresses via the displacements of the surface of the plate.

Modeling of Three-Dimensional Stress-Strain State and Strength Analysis of Metal Composite Cylinders under Impulsive Loading Using the Conwep Method

The application package previously developed at the Pisarenko Institute of Problems of Strength of the National Academy of Sciences of Ukraine for the numerical calculation of the dynamic non-axisymmetric three-dimensional stress-strain state and strength of multilayer helically orthotropic cylinders of finite length under impulsive loading was modified by the CONWEP method for load due to the explosion of a spherical explosive charge arbitrarily located in air inside a cylinder. Its accuracy was investigated by the variation of grid size, the number of the terms of Fourier expansions of unknowns, boundary conditions, charge coordinates and other input parameters. The three-dimensional dynamics and strength of composite and metal-composite cylinders loaded by a non-axisymmetric internal explosion were studied. Strength was determined by the generalized Mises criterion. The influence of displacement of the explosive charge relative to the center of symmetry of the cylinder along the radius and axis on stress-strain state and strength was studied. It is shown that at the constant relative mass of the charge, metal-composite cylinders can be made equally strong by increasing the thickness of the metal layer from zero to a quarter of the shell thickness, while increasing the mass of the charge by a factor of almost two. The inefficiency of the use of reinforced composites with low ultimate tensile strength across fibers in such structural elements has been established. It is shown that in some cases adding an outer composite layer to the inner steel shell leads not to an increase but on the contrary to a decrease in the strength of the object as a whole.

Oscillations of offshore drilling platforms on layerd anisotropic base

The article presents the results of the studies, including the elements of a theoretical calculation, factors of internal strength and deformation field during volumetric vibrations of offshore drilling platforms on a layered basis. The influence of factors such as seismic dynamic impacts at the regional scale and dynamics of the geomorphology of the sea floor have been studied. Structures with spatial resilient links connected rigidly, hingedly or otherwise are subjected to dynamic force action. Depending on the size of the external forces and the forces of inertia, the flexible elements of the structure of the offshore drilling platform are in a complex three-dimensional stress-strain state. In dynamic calculation of the characteristics of the systems, the finite element method was used by replacing the platform with a system with a finite number of nodal points interconnected by inertia less rods. The masses are concentrated at the nodal points.

Stresses in a Lumber Pile in the Process of Drying in a Convective Drier

Numerical calculation of the three-dimensional stress–strain state of a lumber pile in the process of drying in a convective drier is presented. The solution is obtained with the aid of the program complex ANSYS allowing one to develop and visualize three-dimensional geometric constructions. The results of the work can be used for carrying out computational experiments on drying materials in chambers and in developing the regimes of drying lumber piles by analyzing their stress–strain state and for designing drying facilities.

The cross-disciplinary approach to analysis and forecast of operational damage tolerance of the oil pipeline system – part 2

The paper presents the cross-disciplinary approach to the analysis of the oil pipeline system based on the methodologies of tribo-fatigue and mechano-thermodynamics. The pipeline section is analyzed as a complex system pipe-soil- flow of liquid subject to the set of mechanical, thermal and friction loads. It is shown that theses loads are mainly repeatedlyalternated, and the pipe metal works in the multi-cycle fatigue conditions. The procedure of resonance accelerated fatigue tests is proposed, and their results are presented. Also, the unorthodox method of integrated wear-fatigue tests of the pipeline steel was proposed with the model of simultaneous pressure and wall friction actions. The presented field test results of pipes subject to the long-term operation showed that their fracture may occur not only in the near-weld zone, but also in the vicinity of internal corrosion damages. New models of three-dimensional stress-strain state and volumetric damage tolerance for the system pipe-soil-liquid flow were developed. These models were applied with regard to the pipe internal corrosion damages, defined using the inline inspection technique. A new efficient method to describe static and cyclic elastic-plastic fracture of the pipe steel with crack using the transverse strain is proposed and tested. Results of the computer-simulated propagation of the crack-like damage are based upon the model of deformed solid with dangerous volume. The new model is proposed for risk and safety assessment with regard to the ultrasonic inspection data. The algorithm of the ‘oil line pipe’ problem solution is presented for drafting a short-term plan of particular R&D actions.

The cross-disciplinary approach to analysis and forecast of operational damage tolerance of the oil pipeline system – part 1

The paper presents the cross-disciplinary approach to the analysis of the oil pipeline system based on the methodologies of tribo-fatigue and mechanothermodynamics. The pipeline section is analyzed as a complex system pipe-soil- flow of liquid subject to the set of mechanical, thermal and friction loads. It is shown that theses loads are mainly repeatedly-alternated, and the pipe metal works in the multi-cycle fatigue conditions. The procedure of resonance accelerated fatigue tests is proposed, and their results are presented. Also, the unorthodox method of integrated wear-fatigue tests of the pipeline steel was proposed with the model of simultaneous pressure and wall friction actions. The presented field test results of pipes subject to the long-term operation showed that their fracture may occur not only in the near-weld zone, but also in the vicinity of internal corrosion damages. New models of three-dimensional stressstrain state and volumetric damage tolerance for the system pipe-soil-liquid flow were developed. These models were applied with regard to the pipe internal corrosion damages, defined using the inline inspection technique. A new efficient method to describe static and cyclic elastic-plastic fracture of the pipe steel with crack using the transverse strain is proposed and tested. Results of the computer-simulated propagation of the crack-like damage are based upon the model of deformed solid with dangerous volume. The new model is proposed for risk and safety assessment with regard to the ultrasonic inspection data. The algorithm of the ‘oil line pipe’ problem solution is presented for drafting a short-term plan of particular R&D actions.

Estimated delamination of an elastic-plastic liner in a high-pressure metal composite vessel

To carry out delamination analysis in high-pressure metal composite vessels, a software approach is proposed in which the dynamic calculation algorithm is implemented in software systems. The algorithm is built on three points: the introduction of technological deviations (at the level of permissible defects), the determination of a three-dimensional stress-strain state, and a real-time solution. As a calculation program, LS-DYNA was selected in a dynamic formulation with the structure divided into 3D solid elements and unilateral constraint on the contact surface of the metal (liner) and composite shells was introduced. A geometrically and physically nonlinear problem is solved. The moment of liner delamination from the composite shell is determined by the change in the liner stress-strain state and by the liner deformable surface in time presented visually.

Asymptotic analysis of three-dimensional problem of elasticity theory for radially inhomogeneous transversally-isotropic thin hollow spheres

In this study, three-dimensional problem of the theory of elasticity (3DPTE) for radially inhomogeneous (INH) transversally-isotropic thin hollow spheres is investigated using the asymptotic integration method. The basic relations and equilibrium equations for radially inhomogeneous transversally-isotropic thin hollow spheres are formed and inhomogeneous solutions (INHSs) and homogeneous solutions (HSs) are constructed. The built solutions completely reveal the qualitative structure of a three-dimensional stress-strain state of radially inhomogeneous transversely-isotropic spheres of small thickness and serve as an effective apparatus for solving boundary value problems, the basis for evaluating existing applied theories and for creating new, more refined applied theories. Asymptotic formulas are obtained that allow the calculation of the three-dimensional stress-strain state of spheres. New solution groups (boundary layer solutions) have been found that are absent in applied theories. The behavior of homogeneous solutions in the inner parts of the sphere and in the vicinity of conical sections has been studied, when the thin-walled parameter of the sphere tends to zero. The nature of the constructed homogeneous solutions is clarified. On the basis of the theoretical analysis, three types of the stress-strain state (SSS) in the radially inhomogeneous transversally-isotropic hollow spheres (RINHTIHSs) are considered: a penetrating stress state, a simple edge effect, and a boundary layer. Finally numerical calculations are made and the influences of inhomogeneity on the stress distributions are investigated.

Determination of Plane Stress-Strain States of the Plates on the Basis of the Three-Dimensional Theory of Elasticity

We construct a theory of plates loaded only on their sides by loads parallel and symmetric to the median surface. We use the general representation of three-dimensional stress-strain state and exactly satisfy the trivial boundary conditions on the plane surfaces of the plate. The numerical analysis of the three-dimensional stressed state of the plate is reduced to the determination of its two-dimensional state under the assumption that the normal stresses perpendicular to the median surface are negligible. The displacements and stresses are expressed via two two-dimensional harmonic functions. We obtain the homogeneous solutions and develop a numerical-analytic algorithm for the solution of the boundary-value problems posed for rectangular plates.

Reduction of a Three-Dimensional Problem of the Theory of Bending of Thick Plates to the Solution of Two Two-Dimensional Problems

We propose a new theory of bending of thick plates in the case where its stressed state is not described by the Kirchhoff–Love or Timoshenko hypotheses. It is assumed that the three-dimensional stress-strain state of the plate can be split into symmetric bending and symmetric compression. To describe symmetric bending, we use three harmonic functions. Integrating over the thickness of the plate, we express the bending and torsional moments and shear forces via two two-dimensional functions. The relations of the three-dimensional theory of elasticity are satisfied and a closed system of sixth-order partial differential equations for the introduced functions is constructed without using any hypotheses about the geometric character of deformation of the plate. The analytic and numerical methods for their solution are proposed.