Plane Problem Of Elasticity Theory Research Articles

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ПЛОСКОГО НАПРЯЖЕННОГО СОСТОЯНИЯ ПРЯМОУГОЛЬНОЙ ПЛАСТИНЫ МЕТОДОМ ГРАНИЧНЫХ УРАВНЕНИЙ

The article presents the results of numerical simulation of a plane problem of elasticity theory by the method of boundary integral equations. The developed algorithm is implemented on the example of a square beam-wall from the action of a horizontal distributed load. The results of numerical simulation are compared with known solutions.

ИССЛЕДОВАНИЕ ВЛИЯНИЯ УКРЕПИТЕЛЬНОЙ ЦЕМЕНТАЦИИ ПОРОД НА НАПРЯЖЁННОЕ СОСТОЯНИЕ ОБДЕЛОК ПАРАЛЛЕЛЬНЫХ ТОННЕЛЕЙ

The results of determining the stress state of the rock mass and the lining of two parallel tunnels constructed under the slope are presented, obtained using the developed analytical calculation method, which is based on the solution of the plane problem of elasticity theory, obtained using the mathematical apparatus of the theory of functions of a complex variable. The obtained results illustrate the influence on the stress state of the elements of the geomechanical system under consideration of the order of tunneling, the lag in the construction of the lining from the face, as well as the appearance of zones of reinforced soil around the tunnel walls. During the study, a computer program was used that implements the developed method, which makes it possible to effectively perform multivariate calculations in the design of underground structures and for research purposes.

Fluctuations of the ground surface at blasting operations on tunnel structures

The paper considers the fluctuations of the ground surface during blasting operations on tunnel structures. To estimate the vibration levels of the soil surface, under the explosive action inside a cylindrical cavity, we will consider the plane problem of elasticity theory as a first approximation. The main purpose of the work is to determine the stress-strain state of pipelines under the influence of seismic waves. As a result of this impact, longitudinal and transverse waves will propagate in the ground. To solve the problem, the reciprocity theorem is applied, for dynamic problems of elasticity theory and the results obtained are compared by wave dynamics methods. The obtained solution makes it possible to determine the vibration levels of the soil surface that occur during explosive impacts inside the cylindrical cavity of the elastic half-space. The results can be used to assess the vibration levels of the soil surface that occur in the structures of shallow tunnels laid by drilling and blasting.

ГЕОМЕХАНИЧЕСКАЯ ОЦЕНКА ВЛИЯНИЯ ЗОНЫ УКРЕПЛЕННЫХ ПОРОД НА НЕСУЩУЮ СПОСОБНОСТЬ ОБДЕЛКИ ПОДВОДНОГО ТОННЕЛЯ

The approach for assessing the stress-strain state and bearing capacity of monolithic linings of underwater tunnels constructed with the use of reinforcing grouting of the surrounding soil massif is proposed. The method makes it possible to determine the necessary parameters of underground structures based on the analysis of the boundaries of their areas of application. In order to implement the developed theoretical provisions, an original analytical solution of the corresponding plane problem of elasticity theory was applied, implemented in the form of a computer program.

Uniaxial Compression Mechanical Analysis of Cylindrical Rock Specimens Based on Space Axisymmetric Problem

The common failure forms in the uniaxial compression test of standard cylindrical rock specimens are symmetric cone failure and splitting failure. In the past, two kinds of failure forms were explained from the plane problems of elasticity theory. However, there was a lack of research based on space problems. In this paper, based on spatial axisymmetric elastic mechanics theory, three-dimensional stress function is constructed for the cylindrical rock specimen under uniform axial compression by adopting the semi-inverse method. According to the stress function, the stress components and the principal stress are deduced when considering the end effect. The failure form is speculated, which can well explain the two common types of damage in uniaxial compression. This work may enrich the basic research of rock mechanics.

УЧЕТ ВНУТРЕННЕГО ВЯЗКОГО ТРЕНИЯ В УРАВНЕНИЯХ КОЛЕБАНИЙ БАЛКИ КАК ДЛИННОЙ ПРЯМОУГОЛЬНОЙ ПОЛОСЫ

Abstract. On the base of the method of simple iterations generalising methods of semi-inverse one of Saint-Venant, Reissner and Timoshenko the one-dimensional theory is constructed using the example of dynamic equations of a plane problem of elasticity theory for a long elastic strip. The resolving equation of that one-dimensional theory coincides with the equation of beam vibrations. The other problems with unknowns are determined without integration by direct calculations. In the initial equations of the theory of elasticity the terms corresponding to the viscous friction in the Navier-Stokes equations are introduced. The asymptotic characteristics of the unknowns obtained by the method of simple iterations allow to search for a solution in the form of expansions of the unknowns into asymptotic series. The resolving equation contains a term that depends on the coefficient of viscous friction.

Methodological aspects of research into the stress-strain state of panel-frame vertical load-bearing systems

Aim . To analyse the possibilities of various computational methods (primarily numerical) in terms of investigating the stress-strain state of complex multi-dimensional vertical loadbearing systems. Methods . The fundamental essence of the finite-difference method (grids) and the finite element method is revealed, their advantages and disadvantages are described in terms of solving the plane problem of elasticity theory, in particular, in calculating the stress-strain state of panelframe structures united in a single vertical multi-connected system. Results . The obtained results can be used to optimize the methodology of theoretical stress-strain state analysis of complex multiconnected systems, taking into account available computer equipment and licensed software packages for automated calculation of building structures. Conclusion . The conducted analysis shows that, provided there is a sufficiently powerful computer, the finite element method is the most versatile and effective method. This method was the basis of a large number of software packages permitting analysis of the stress-strain state of any designs characterized by the complexity of form, topology, load, etc.

Stresses in Viscoelastic Half Space at Given on the Boundary Non-stationary Normal Displacement

Within the framework of linear model with coinciding volume and shear relaxation kernels non-stationary problem for viscoelastic half space with normal displacements given on its boundary is considered. Solution representation in the form of generalized convolution of corresponding plane elasticity theory problem solution and one-dimensional viscoelasticity theory problem solution is used. These solutions are written down as convolution of boundary conditions with the kernels - surface Green functions. Time Laplace transform is used for kernels constructing. Its inversion is carried out exactly using either analytical representations of the transforms (for plane problem of elasticity theory) or asymptotic method (for one-dimensional problem of viscoelasticity). Explicit formulas for normal and shear stresses on the boundary of viscoelastic half space are obtained using the structure of Green functions. Examples of calculation show that contrary to elastic medium discontinuities of the first kind don’t exist on the wave front.

Combined Bending with Tension of Isotropic Plate with Crack Considering Crack Banks Contact and Plastic Zones at its Tops

Abstract Stress-strain state of isotropic plate with rectilinear through-crack at combined action of bending and tension, realized by applying distributed forces and bending moments at infinity, the vectors of which are parallel and perpendicular to the crack, is investigated. Under the influence of the internal stress the crack faces contacts on area of constant width near the upper base of plate, and plastic zones forms in its tips. Using methods of the theory of complex variables, complex potentials plane problem of elasticity theory and the classical theory of plates bending, solving of the problem is reduced to the set of linear conjugation problems and their analytical solution is built in a class of functions of limited plastic zones in the crack tips. The conditions of existence of the solution of the problem in these terms are determined. Using Treska plasticity conditions in the form of surface layer or the plastic hinge, the length of plastic zone and crack opening displacement are found analytically. Their numerical analysis for various parameters of the problem is conducted.

Constitutive relations in multidimensional isotropic elasticity and their restrictions to subspaces of lower dimensions

The mechanical meaning and the relationships among material constants in an n-dimensional isotropic elastic medium are discussed. The restrictions of the constitutive relations (Hooke’s law) to subspaces of lower dimension caused by the conditions that an m-dimensional strain state or an m-dimensional stress state (1 ≤ m < n) is realized in the medium. Both the terminology and the general idea of the mathematical construction are chosen by analogy with the case n = 3 and m = 2, which is well known in the classical plane problem of elasticity theory. The quintuples of elastic constants of the same medium that enter both the n-dimensional relations and the relations written out for any m-dimensional restriction are expressed in terms of one another. These expressions in terms of the known constants, for example, of a three-dimensional medium, i.e., the classical elastic constants, enable us to judge the material properties of this medium immersed in a space of larger dimension.

13th International Conference on Films and Coatings

13th International Conference on Films and Coatings

Influence of Pipe Ramming on Stress State of Surrounding Soil and Nearby Tunnel Lining

ANNOTATIONThis article proposes an approach to the prediction of stress state and assessing the strength of a circular tunnel lining and the surrounding rock mass under construction near the production used by the micro-tunneling technology of pipe ramming. The basis of this method is an analytical solution of the corresponding plane problem of elasticity theory.

Distribution of Stresses Near V-Notches in an Orthotropic Plane Under Symmetric Loads

The solution of a plane problem of elasticity theory for an orthotropic plane with semiinfinite rounded V-notch under the action of symmetric loads is obtained by the method of singular integral equations. By using this solution, we deduce the relations between the stress intensity factor at the sharp tip of the V-notch and the normal stresses at the tip of the corresponding rounded notch. For bounded bodies with V-notches, the indicated solution is obtained as an asymptotic relation for small radii of rounding of their tips. The presented relation can be used in the limit transitions in order to find the stress intensity factor at the tips of sharp V-notches according to the solutions for the corresponding rounded stress concentrators.

A two-level method for calculation of microstress on reinforced plates with circular hole in case of extension normal to principal direction

The stress concentration must often be examined at two levels while analyzing the stress condition of composite materials. The macroconcentration depends on the presence of holes, notches and other local areas of a construction. Typical dimensions of macroconcentration distribution areas are of the order of 0,01-0,1 m. Macroconcentration analysis is performed using the models of homogeneous material. Microstress concentration occurs in structurally inhomogeneous composites due to the structural heterogeneity of the composite. The sizes of concentration areas in regular structures are defined by the sizes of periodically recurring areas. In fibrous composites, such areas have the size of approximately 0,0001 m or less. This makes it necessary to use a two-level approach for the analysis of the stress concentration in the construction of composite materials. The aim of the present study was to compute the stress concentration in unidirectional reinforced composite plate with circular hole with respect to the volume ratio of the component materials in composite. The contour of the circular hole and its dependency on the structure of plates was calculated in order to study the behaviors of macro- and microstresses. The boundary conditions at a large distance from the hole are pressure, uniformly distributed on the plate. Also this problem is analyzed with the finite element method by package ANSYS. Macroconcentration is defined based on the solution of the plane problem of elasticity theory of the orthotropic material by the virtue of functions of a complex variable. The finite element method was used to investigate the stress distribution at microlevel. Boundary conditions that model the state of the specified two-dimensional representative cell in the composite structure were established. The results demonstrated the macro- and microstresses and behavior of the orthotropic plate with a circular hole calculated for two different structures.

On the compatibility equations in terms of stresses in many-dimensional elastic medium

By equating to zero all components of the Kroner incompatibility tensor of rank 2n − 4 or of the Riemann tensor dual to the Kroner tensor, n2(n2 − 1)/12 independent consistency equations for the stresses in an n-dimensional isotropic elastic medium are derived. The problem concerning the equivalence of the system of these equations to systems following from equating to zero either all n(n + 1)/2 components of the Ricci tensor or only one curvature invariant is investigated. It is shown that the answer to this question depends on the dimension of the space. Three cases are singled out: n = 2 (plane problem of elasticity theory), n = 3 (spatial problem of elasticity theory), and n ⩾ 4.

Plane problem of elasticity theory for anisotropic bodies with thin elastic inclusions

Based on the coupling principle for continua of different dimensions, we constructed integral equations for determining the plane stress of an anisotropic elastic body with thin inhomogeneities of the material’s structure. For the numerical solution of these equations, we used the boundary element method and proposed new basic functions for the description of the near-end area of an inhomogeneity. The interpolation quadratures and the polynomial transformations are adopted for the efficient numerical evaluation of the corresponding singular and hypersingular integrals. Numerical examples show a high efficiency of the approach developed for the determination of the stressed state of anisotropic bodies that contain cracks and thin elastic and rigid inclusions.

Solving a plane problem of elasticity theory with the use of a hybrid FEM formulation

In the framework of a plane problem of elasticity theory in strength analysis based on the finite element method in the hybrid formulation, we suggest that the approximation of displacements and stresses be used as magnitudes of vector and tensor fields, respectively. The quadrilateral finite element, the displacements and stresses of which are accepted as nodal unknowns is proposed. The mixed Reissner functional is applied to form a finite element deformation matrix.

On Kolosov’s relations in the plane problem of elasticity theory

On Kolosov’s relations in the plane problem of elasticity theory

Stress state of compound polygonal plate

The constructions made of bars and plates with holes, openings and bulges of various forms are widely used in modern industry. By loading these structural elements with different efforts, there appears concentration (accumulation) of stress whose values sometimes exceeds the admissible one. The durability of the given element is defined according to the quantity of these stresses. Since the failure of details and construction itself begins from the place where the stress concentration has the greatest value. Therefore the exact determination of stress distribution in details (bars, plates, beams) is of great scientific and practical interest and is one of the important problems of the solid fracture. Compound details (when the nucleus of different material is soldered to the hole) are often used to decrease the stress concentration. In the present paper, we study a stress–strain state of polygonal plate weakened by a central elliptic hole with two linear cracks info which a rigid nucleus (elliptic cylinder with two linear bulges) of different material was put in (soldered) without preload. The problem is solved by a complex variable functions theory stated in papers [Theory of Elasticity, Higher School, Moscow, 1976, p. 276; Plane Problem of Elasticity Theory of Plates with Holes, Cuts and Inclusions, Publishing House Highest School, Kiev, 1975, p. 228; Bidimensional Problem of Elasticity Theory, Stroyizdat, Moscow, 1991, p. 352; Science, Moscow (1996) 708; MSB AH USSR OTH 9 (1948) 1371]. Kolosov–Mushkelishvili complex potential ϕ( z) and ψ( z) satisfying the definite boundary conditions are sought in the form of sums of functional series. After making several strict mathematical transformations, the problem is reduced to the solution of a system of linear algebraic equations with respect to the coefficients of expansions of functions ϕ( z) and ψ( z). Determining the values of ϕ( z) and ψ( z), we can find the stress components σ r , σ θ and τ rθ at any point of cross-section of the plate and nucleus on the basis of the known formulae. The obtained solution is illustrated by numerical example. Changing the parameters A 1, m 1, e, A 2, and m 2 we can get the various contour plates. For example, if we assume m 1=0, A 1= r, then the internal contour of L 1 becomes the circle of radius r with two rectilinear cracks (for the nucleus––a rectilinear bulges). Further, if we assume a small semi-axis of the ellipse b 1 to be equal to zero ( b 1=0), then a linear crack becomes the internal contour of L 1 (and the nucleus becomes the linear rigid inclusion made of other material). For m 2=0; A 2= R, the external contour L 2 turns into the circle of radius R. The obtained method of solution may be applied and in other similar problems of elasticity theory; tension of compound polygonal plate, torsion and bending of compound prismatic beams, etc.

Stability of a Circular Sandwich Ring under an Axially Symmetric Temperature Field Inhomogeneous across the Thickness

The linearized problems on the stability of a circular sandwich ring of symmetric structure under an axially symmetric temperature field inhomogeneous across the core thickness are stated and analytical solutions to them are given. The first problem deals with the mixed flexural buckling form (BF) of the ring as a whole, realized as a result of buckling in one of the load-carrying layers due to formation of precritical pressure stresses in the layer. The second problem considers purely shear BFs when one load-carrying layer is rotated relative to the other. The deformation processes for the load-carrying layers are described by the Kirchhoff-Love model, and for the core of arbitrary thickness - by two models, namely by the equations of the plane problem of elasticity theory and by the model of a transversely soft layer of arbitrary thickness (the same equations simplified by the assumption of zero circumferential normal stresses). Within the frames of the first model adopted for the core, the shear BF is theoretically possible but practically unrealizable, since the mixed flexural BF arises earlier than the shear BF.