Plane Elasticity Problems Research Articles

The field of tensions of a rotating anisotropic disc of a variable thickness loaded with undistracted forces on the outer contour

The work gives a solution of the plane elasticity problem for rotating polar-orthotropic annular disks of a variable thickness. The disk is loaded with a system of equal focused forces on the outer contour applied evenly along the rim and symmetric concerning the diameter. The disk is seated with an interference fit on the flexible shaft so that a constant contact pressure acts on the interior contour. The stresses and deformations arising in such a rotating anisotropic annular disk will be non-axisymmetric. A conclusion of a fourth-order partial differential equation for the effort function is drawn. Its general solution is searched out in the form of a Fourier series of cosines with even numbers. As a result, an infinite system of ordinary differential equations is solved for the coefficients of the series. These differential equations correspond to the linear Volterra integral equations of the 2 nd kind, which are solved using resolvents. Constants of integration are determined from the border conditions. Expressions for the stress components are written through the effort function by the well-known formulas. We find the components of the displacement vector in the disk by the integration of the Hooke’s law equations for the polar-orthotropic plate. We calculate the deformation components in a ring anisotropic disk by Cauchy differential relations if we know the displacements. The solved formulas for stresses, deformations and displacements completely describe the stress-deformed state in a rotating polar-orthotropic disc of variable thickness with a system of focused forces on the outer contour. The results of the work can be used in the design of working disks of turbomachines and turbo compressors, as well as rotors of centrifugal stands.

Correspondence principle in plane and axisymmetric mixed boundary-value problems of elasticity

This paper advances the relations by A. Ja. Aleksandrov between axisymmetric and plane strain states to the case of mixed boundary-value problems. We revise two models of a strip-shaped and a circular stamp indented into a half-plane and a half-space, respectively, when the normal and tangential traction components are unknown a priori. Also, two models of a strip-shaped and a penny-shaped interfacial crack are considered. By using the theory of Abelian integral operators and the Riemann-Hilbert problem on a segment we derive the solutions to the model problems and establish the relations between the governing systems of integral equations associated with the four models and their solutions. These relations can be interpreted as mappings between (i) plane and axisymmetric contact problems, (ii) plane and axisymmetric fracture models, (iii) plane contact and fracture problems, and (iv) axisymmetric contact and fracture problems. The mappings enable us to write down the governing systems of integral equations and the solutions to any three models by making use of the governing system and the solution to the fourth problem. The transformations are specified in the scalar cases when there is no friction in the contact zone and when a crack is in a homogeneous elastic medium. By considering the contact frictionless problem of an annulus stamp it is shown that, although an exact solution to the plane strain frictionless contact problem of two stamps is available, a transformation of the plane to the axisymmetric solution in this case is not possible to obtain, and derivation of a closed-form solution to the annulus stamp model is still an open mathematical problem.

Application of evolutionary structural optimization based on the boundary element method using initial stress fields

This paper presents the application of the topological optimization in plane elasticity problems applying the boundary element method (BEM) coupled with the evolutionary structural optimization (ESO) procedure by using initial stress fields as the new strategy to create cavities in the topological optimization domain. In this approach, in the domain region, where must be removed in the heuristic process, a stress field is added that is equal to and of opposite sign in the initial elastic stress formulation resulting in null domain stresses. Thus, a virtual cavity is created, avoiding the introduction of elements in the domain of the structure or other cumbersome strategy integrated in the formulation for considering the void region in BEM. The advantages of the present approach, as compared to other existing methods, are that, firstly, during the iterative process, there is no need to remesh and recalculate the boundary equations at each iteration once the introduction of the initial stress - to simulate each void region - is computed in the right-hand side vector of the final algebraic equation and, secondly, there is a significant reduction in the computational cost. Numerical examples are presented and compared to others available in literature, showing the robustness and efficiency of the proposed algorithm.

Analysis of an orthotropic elastic plane problem of a half plane with an edge crack using a mapping function

It is a challenging problem to derive a solution for an orthotropic elastic plane problem using a mapping function. The complex variable method is employed to introduce a mapping function. Then, comparatively arbitrary configurations can be analyzed. The boundary condition equation is represented by two complex variables. This fact makes solving the present problem difficult. This difficulty is conquered by applying Cauchy integrals. Also, mapping functions represented by an infinite terms of fractional expressions are introduced to conquer the mathematical difficulties of the Cauchy integral in the process of the derivation of the stress function. One of two stress functions is derived by analytical continuation. The final exact stress functions are represented by an irrational mapping function in a closed form. Stress components are represented by one complex variable. Therefore, the calculation of the stress components is easier than that of an isotropic problem. Half planes with a notch or a mound expressed by an irrational mapping function can be solved using their mapping functions. Such a polygonal mapping function can be derived by Schwarz–Christoffel’s transformation. As a demonstration, a half plane with a vertical edge crack subjected to uniform tension is analyzed and the stress distributions are shown for two examples of two characteristic roots of the characteristic equation for the orthotropic elastic plane.

3D elasticity solutions for stress field analysis of FGM circular plates subject to concentrated edge forces and couples

In this paper, the complex variable method is applied to study three-dimensional problems of transversely isotropic functionally graded circular plates subjected to concentrated edge forces and couples. To that end, the extended England–Spencer plate theory is adopted to obtain the general solutions of the governing equations in which four analytic functions $$\alpha (\zeta )$$ , $$\beta (\zeta )$$ , $$\phi (\zeta )$$ and $$\psi (\zeta )$$ are involved. The material properties can vary along the thickness direction in an arbitrary fashion. The cylindrical boundary of the plate is considered to be free which is well known as the first kind basic problem in plane elasticity. Four analytic functions can be determined by the Cauchy’s integral formula and conformal mapping technology. As a result, the 3D stress field is investigated for a transversely isotropic FGM circular plate whose cylindrical boundary is subjected to concentrated forces and couples. The proposed elasticity solutions can be used as benchmarks to validate solutions obtained based on various simplified plate theories or numerical methods.

Microcrack interaction with circular inclusion and interfacial zone

A geometrically simplified plane elasticity problem of a finite small crack emanating from a thin interfacial zone surrounding the circular inclusion situated in the finite bounded domain is investigated. The crack is arbitrarily oriented and modelled using the distribution dislocation technique. This model represents the inner solution of the studied problem. The corresponding fundamental solution is based on the application of Muskhelishvili complex potentials in the form of the Laurent series. The coefficients of the series are evaluated from the compatibility conditions along the interfaces of the inclusion, the interfacial zone and the enclosing matrix. The fundamental solution is also used in the solution of the boundary integral method approximating the stress and strain relations of the so-called outer solution. The asymptotic analysis at the point of the crack initiation combines the inner and the outer solution and results in the evaluation of the stress intensity factors of the crack tip, which lies in the matrix. The topological derivative is subsequently used to approximate the energy release rate field associated with the perturbing crack in the matrix. The extreme values of the energy release rate allow one to assess the crack path direction of the initiated microcrack.

Edge Effect in a Two-layer Orthotropic Strip

The article discusses the edge effect (boundary layer, Saint-Venant effect) of a two-layer strip of orthotropic materials. Two methods are used to solve the problem of plane elasticity (solution in stresses and displacements). Both methods are constructed in such a way that they can describe the edge effect. This allows us to construct a theory for calculating the edge effects of single-layer strips based on which we manage to solve the problem of calculating the edge effect of two-layer strips analytically accurately. In fact, the problem can be summarized by saying that it is calculation of a single-layer strip. The obtained numerical results show that both methods of solution lead to the completely coinciding results.

带裂纹十次对称二维准晶平面弹性的无摩擦接触问题

With the classical complex function method, a frictionless contact problem of 2D decagonal quasicrystal semiplane elasticity with arbitraryform cracks was addressed under the action of a rigid convex basal punch. Based on complex expressions of stresses and displacements of 2D decagonal quasicrystals, the problem was converted into solvable boundary value problems with analytic functions, and then reduced to a class of Riemann boundary problems. Solutions to the Riemann boundary problems give the stress functions in closed form, the explicit expressions of the stress intensity factors at crack tips and the contact stress distribution under the punch. The expression of the contact stress shows that, it has singularity at the edge of the contact zone and the crack tips. Without the effect of the phason field, the obtained results match well with those classical conclusions for elastic materials. Numerical examples illustrated the solutions to the frictionless contact problem in 2D decagonal quasicrystal semiplane elasticity with a vertical crack and a horizontal straight crack under a rigid punch. The work provides a theoretical reference for the application of quasicrystalline materials.

Architectonic logic of construction on the example the plane elasticity problems solving by the finite element method

The development of architecture, within the process of architectural form making, is based on architectonic logic. From the constructive-physical side, the architectonic logic can be studied by certain methods of elasticity theory, in particular, by the approximate numerical method of finite elements. The article deals with the algorithm of revealing the architectonic logic of eccentricities on the examples of the plane problem solving of elasticity theory. The formal characteristics of the eccentricity logic in connection with the process of form making are considered. The article describes the action of logic within the historical context of the process of architectural form making development, on the example of the constructivists and deconstructionists schools.

Stress-Strain State in a Buckled Thermal Barrier Coating on an Elastic Substrate

The paper analyzes the stress-strain state of a thermal barrier coating buckled on an elastic substrate under thermal shock. For the analysis, the stability loss of an orthotropic plate on an elastic substrate (Winkler foundation) is modeled using the Euler method on the assumption that compressive stresses arise in the plate as it expands on heating while the substrate remains cold and dimensionally stable. The deflection amplitude and stresses in the heated plate are determined, and the stress-strain state in the substrate is estimated by solving a plane elastic problem with displacement boundary conditions. The data suggest that both compressive and tensile stresses arise at the coating-substrate interface since the coating becomes unstable and that extrusion and intrusion zones appear in the substrate, showing their chessboard distribution in the plane strain case.

Application of the internal and external Williams functions to plane elasticity problem for A Mode I crack

The main idea of this research is in applying Williams functions to calculate stress intensity factors (SIF) in case of 2D elastic bodies with cracks. Firstly, we employ the concept of Williams functions converging on infinity which can be used along withtraditional functions to analyse infinite bodies and to study partially loaded crack boundaries. We show the convergence of SIF for the case of a strip and for the bodies with circular boundary depending on the number of Williams functions and on the number ofintervals on the boundary in case of numerical integration. Secondly, we complement the standard set of equations with the global equilibrium conditions of the elastic body to improve the accuracy of the calculation. We compared the computed stress with the onedefined on the boundary.

An extended family of nonconforming quasi-Wilson elements for solving elasticity problem

This contribution (part two) focuses on numerical implementation and efficiency aspects of an extended family of nonconforming quasi-Wilson elements type. Such a class of nonconforming elements has been introduced recently in Achchab et al. (2015). Here, based on a rectangular mesh, it is used for the approximate solution of a planar elasticity problem. It is shown that this family passes the patch-test, and that by suitably adding some orthogonality conditions, on a general class of enrichment functions, we can derive higher order consistency error estimates. Our general theoretical results, see Theorems 4.1 and 4.2, unify, simplify and extend a number of existing works on the improvement of the order of consistency error. Numerical experiments are carried out to demonstrate that our method is optimal for various Lamé parameter μ, shear modulus λ and locking free when the Poisson parameter ν approaches close to 0.5.

An asymptotic analysis of crack initiation from an interfacial zone surrounding the circular inclusion

A geometrically simplified plane elasticity problem of a finite small crack emanating from a thin interfacial zone surrounding the circular inclusion situated in the finite bounded domain is investigated. The inclusion can model a particle in a composite material. The crack is modelled using the distribution dislocation technique, which represents the so called inner solution or boundary layer of the studied problem. Its application is conditioned by the knowledge of the fundamental solution of the dislocation interacting with the inclusion and its interfacial zone. The fundamental solution is based on the application of Muskhelishvili complex potentials in the form of the Laurent series. The coefficients of the series are evaluated from the compatibility conditions along the interfaces of the inclusion, the interfacial zone and the matrix. Another supplement of the fundamental solution is its utilization for the so-called outer solution, which is the solution of the boundary integral method approximating the stress and strain relations along the external boundary of the domain containing the inclusion. The asymptotic analysis is introduced at the point of crack initiation to control the mismatch between the outer and the inner solutions along the external boundary of the studied domain. The asymptotic analysis results in the evaluation of the stress intensity factors as the leading terms of the asymptotic series expressed from the stress distribution near the crack tip, which lies in the matrix. The topological derivative is used to approximate the energy release rate field associated with the perturbing of the matrix by the finite small crack emanating from the interfacial zone. The assessed values of the energy release rate allow one to study the influence of the interfacial zone dimension and elastic properties on crack initiation under the conditions of finite fracture mechanics.

A transform method for the biharmonic equation in multiply connected circular domains

AbstractA new transform approach for solving mixed boundary value problems for the biharmonic equation in simply and multiply connected circular domains is presented. This work is a sequel to Crowdy (2015, IMA J. Appl. Math., 80, 1902–1931) where new transform techniques were developed for boundary value problems for Laplace’s equation in circular domains. A circular domain is defined to be a domain, which can be simply or multiply connected, having boundaries that are a union of circular arc segments. The method provides a flexible approach to finding quasi-analytical solutions to a wide range of problems in fluid dynamics and plane elasticity. Three example problems involving slow viscous flows are solved in detail to illustrate how to apply the method; these concern flow towards a semicircular ridge, a translating and rotating cylinder near a wall as well as in a channel geometry.

A family of virtual element methods for plane elasticity problems based on the Hellinger–Reissner principle

In the framework of 2D elasticity problems, a family of Virtual Element schemes based on the Hellinger–Reissner variational principle is presented. A convergence and stability analysis is rigorously developed. Numerical tests confirming the theoretical predictions are performed.

Stress intensity factor for an elastic half plane weakened by multiple curved cracks

• The multiple curved cracks problem in an elastic half plane is formulated by using the complex potential methods. • The integral equations are solved numerically using quadrature rule for the stress intensity factor. • The closer the cracks to the boundary the higher the value of SIF. Modified complex potential with free traction boundary condition is used to formulate the curved crack problem in a half plane elasticity into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function. Numerical examples exhibit the stress intensity factor increases as the cracks getting close to each other, and close to the boundary of the half plane.

ANALYSIS OF POSSIBLE CAUSES AND MECHANISMS OF DESTRUCTION OF BUILDING STRUCTURES

In order to better understand the wave properties of the Timoshenko equation, the derivation of the refined equation from the equations of the plane elasticity problem for a long band is carried out. The simple iterations method is used for the derivation. It includes known methods: the semi-inverse method of Saint-Venant and Picard operator. In accordance with the semi-inverse method, a part of the unknowns is defined, which are interpreted as the values of the initial (zero) approximation. Proceeding from them, a sequential computation is carried out using a sequence of the four Picard operators in such a way that the outputs of the one operator are the inputs for the next. Calculating in this way all the unknowns in the zeroth approximation by the direct integration over the transverse coordinate, the values of the initial approximation are calculated in the first approximation. These quantities are small of the second order with respect to the dimensionless thickness. Expressions for the unknowns are obtained as power functions of the transverse coordinate and as a function of the deriva-tives along the longitudinal coordinate. By the Banach fixed point theorem, the computation process is asymptot-ically convergent one. After this, boundary conditions on the long sides are satisfied by means of the derivatives of the arbitrariness, depending only on the longitudinal coordinate. This gives us the ordinary differential equations for the determination of these arbitrary functions. In turn, the integration constants of the last equations can be found from the conditions on the short sides of the strip. The ordinary differential equations are split into equations for slowly varying and quickly varying quantities. The slowly changing values give the classical solu-tion of the beam oscillations. The quickly varying solutions give the perturbed solutions describing high-frequency oscillations and singularly perturbed wave solutions for time-concentrated effects. Some of these soluions are absent in the Timoshenko equation. It is assumed that the selected shear waves provoke in the buildings subjected to the rapid impacts (shock by airplane, explosions, and seismic movements of the base) the interruptions of interlayers between the floors and subsequent progressive collapse.

Determination of load paths in plates and shells

Load paths represent the load flow in structures and are used to identify how the load is transferred throughout the structure from points of application to support. The load function method is used in this paper for identification and visualization of load paths in plate and shell structures. Authors previously proposed a load function method to calculate the load paths in plane elasticity problems. This work extends the previous formulation to plates and shells. The significant difference from previous work is that the plates and shells equilibrium equations are not divergence free. Therefore, Helmholtz decomposition is used to decompose the stress field to divergence and curl free components to calculate the load function and find load paths. Mathematical formulations are presented to support the proposed method. Numerical and analytical examples show the application of this method.

Fourier Sine Transform Method for Solving the Cerrutti Problem of the Elastic Half Plane in Plane Strain

Abstract The Fourier sine transform method was implemented in this study to obtain general solutions for stress and displacement fields in homogeneous, isotropic, linear elastic soil of semi-infinite extent subject to a point load applied tangentially at a point considered the origin of the half plane. The study adopted a stress based formulation of the elasticity problem. Fourier transformation of the biharmonic stress compatibility equation was done to obtain bounded stress functions for the elastic half plane problem. Stresses and boundary conditions expressed in terms of the Boussinesq-Papkovich potential functions were transformed using Fourier sine transforms. Boundary conditions were used to obtain the unknown constants of the stress functions for the Cerrutti problem considered; and the complete determination of the stress fields in the Fourier transform space. Inversion of the Fourier sine transforms for the stresses yielded the general expressions for the stresses in the physical domain space variables. The strain fields were obtained from the kinematic relations. The displacement fields were obtained by integration of the strain-displacement relations. The solutions obtained were identical with solutions in literature obtained using Cerrutti stress functions.

Influence of mutual position of a noncircular tunnel and a dynamic impact source on stress distribution in lining

Based on the analytical solution of the corresponding dynamic plane elasticity problem, general regularities of the formation of maximum stresses in lining of a noncircular cross-section tunnel are found and studied at different locations of a dynamic impact source. The paper shows that the closely spaced position of the source has a significant effect on the stress state of the underground structure, which in turn depends on the frequencies of the waves radiated by this source. It is revealed that when the dynamic impact source is removed deep in rock mass for a distance of the order of seven average radii of the tunnel cross section, the estimated stresses in the tunnel lining coincide with the stresses found from the solution of the problem on the wave propagation in the corresponding direction from infinity.