Displacement Equilibrium Equations Research Articles

УТОЧНЁННАЯ ЛИНЕЙНАЯ ТЕОРИЯ УПРУГОСТИ С УЧЁТОМ КВАДРАТИЧНЫХ СЛАГАЕМЫХ

A variant of the linear theory is proposed, based on taking into account in Taylor expansions for stresses and in power series and the expansion of the radical into a binomial series for deformations of not only linear, but also quadratic terms. In this case, static equilibrium equations in stresses are written in the form of second-order partial differential equations. The law of pairing of tangential stresses is observed. The relationship between deformations and displacements is described by relations similar to Cauchy relations, but taking into account the second derivatives of displacements along spatial coordinates. The equations of continuity of deformations for this version of the theory of elasticity are written in the form of two groups of relations. The first group of relations establishes the continuity equations for the linear terms of the deformation components; The second group of relations establishes the continuity equations for the quadratic terms of the deformation components. The physical equations for this variant of the linear theory of elasticity are written in the same way as for the classical linear theory of elasticity. Equilibrium equations in stresses are represented by second-order partial differential equations. The equilibrium equations in displacements are written in the form of fourth-order partial differential equations. The equations of equilibrium, as well as the equations of continuity of deformations, along with other parameters – the physical constants of the medium – contain small parameters dx, dy, dz, the magnitude of which, as shown by numerical modeling, has little effect on the nature of the stress-strain state. To determine them, it is necessary, apparently, to use experimental data. The construction of resolving equations in both stresses and displacements for special cases of the stress-strain state of an elastic continuous medium is considered: uniaxial dressed-up state; uniaxial deformed state; plane strain; generalized plane stress state. The problem of the uniaxial deformed state of a continuous medium (solution in stresses) is numerically solved. The proposed version of the linear theory of elasticity extends the classical linear theory of elasticity and, with appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

Solving equations of 3D elasticity for orthotropic bodies

A mathematical model of the three-dimensional orthotropic body was used. A technique has been developed for integrating equilibrium equations in displacements that describe the stress state of orthotropic materials. We find the solution of equilibrium equations in displacements for orthotropic bodies. The expression of displacements, strains and stresses is obtained through the introduced displacement function, which satisfies the sixth-order equation for partial derivatives. We develop the algorithm of the analytical and numerical solutions of boundary value problem for orthotropic rectangular prism. Solution of boundary value problems is reduced to minimize a generalized quadratic form.

Theoretical analysis and experimental measurement of coupling dynamic characteristics for transversely isotropic rectangular plate based on modified FSDT assumption

The theoretical analysis of thick plate vibration behavior has been investigated in the literature, but most of the studies were focused on flexural dynamic characteristics and lack experimental verification. In this study, the analytical solutions based on the superposition method for both the flexural and extensional vibrations are presented to obtain resonant frequencies and associated mode shapes for a transversely isotropic thick rectangular plate. The displacement equilibrium equations and boundary conditions of the modified first-order shear deformation theory (FSDT) are derived by utilizing Hamilton’s principle and the variation method. To verify the validity of the theoretical model, the finite-element method (FEM) and impact experiment results for thick plates are employed in this work. Excellent agreement of resonant frequencies and associated mode shapes is obtained for FEM calculation and theoretical analysis. To excite vibrations of a thick rectangular plate, a steel ball controlled by an electromagnet is utilized. A steel ball is dropped freely from a height of 231 mm on the top of the plate surface, and a transient wave will be generated after the ball impact on the specimen. The frequency spectrum of the thick rectangular plate is constructed by using the fast Fourier transform of the time-domain transient response. The excited resonant frequencies obtained from experimental measurement are compared with theoretical results. The comparisons show that the modified FSDT provides an excellent prediction of the resonant frequencies and mode shapes for the dynamic characteristics of thick rectangular plates.

КРАЕВОЕ НАПРЯЖЕННО-ДЕФОРМИРОВАННОЕ СОСТОЯНИЕ КРУГЛОЙ ПЛАСТИНЫ ПЕРЕМЕННОЙ ТОЛЩИНЫ НА ОСНОВЕ НЕКЛАССИЧЕСКОЙ ТЕОРИИ

A variant of the refined theory on calculation of the stress-strain state of circular plates with symmetrically various thicknesses according to an arbitrary law in the radial direction was presented. Equations of the plate state were established by using the three-dimensional elasticity theory. The required displacements were approximately calculated according to upright direction to the middle plane by polynomials with two degrees higher than in the classical Kirchhoff - Love theory. The differential equation at equilibrium in displacements with various coefficients was obtained by using means of the Lagrange variational principle. The direct integration of the equilibrium equations in the three-dimensional elasticity theory was used to determine the transverse normal and shear stresses. Of an isotropic circular plate with changing in thickness by using the analyzing Fourier chain, the obtained differential equilibrium equations in displacements with variable coefficients containing supplement components and taking into account of the effect of thickness on the stress-strain state of the plate. Examples of calculating the stress state of a circular plate with a thickness varying according to linear and parabolic laws under the action of a uniformly distributed load were considered. The limited difference method was employed to solve the boundary value problem. Comparison results of the refined and classical theories were investigated. It is demonstrated that the study on the stress state in the zones of its distortion (compounds, local loading zones, etc.) should use a refined theory, since the additional corresponding stresses of the “boundary layer” type are of the same order with the values of the main (internal) stress state. This is important to increase the reliability of strength calculations of such elements of aircraft-rocket structures as the power housings of aircraft, their various transition zones and connections, as well as objects in various engineering industries.

НАПРЯЖЕННО-ДЕФОРМИРОВАННОЕ СОСТОЯНИЕ СФЕРИЧЕСКОЙ ОБОЛОЧКИ ПОД ДЕЙСТВИЕМ ПРОИЗВОЛЬНОЙ НАГРУЗКИ НА ОСНОВЕ НЕКЛАССИЧЕСКОЙ ТЕОРИИ

Considered is the stress state of an isotropic spherical shell exposed to an arbitrary load based on a non-classical theory. When building a mathematical model of the shell, three-dimensional equations of the theory of elasticity are applied. Displacements are represented in the form of polynomials along the coordinate normal to the middle surface two degrees higher relative to the classical theory of the Kirchhoff-Love type. As a result of minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are obtained. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of displacements and external loads into trigonometric series in the circumferential coordinate. Displacements are represented in form of polynomials along the coordinate normal to the middle surface by two degrees higher relative to the classical theory of the Kirchhoff-Love type. Resulting from minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are received. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of the displacements and the external loads into trigonometric series as per the circumferential coordinate. The formulated boundary problem is solved by the methods of finite differences and matrix sweep. As a result, displacements are obtained in the grid nodes, for approximation of which splines are used. The shell deformations are found using geometric relationship; tangential stresses are received from the correlations of Hooke's law. One of the features of this paper lies in the fact that the transverse stresses are determined by the direct integration of the equilibrium equations of the three-dimensional theory of elasticity. An example of the calculation of a hemispherical shell rigidly restrained along the lower base contour is brought. The shell is exposed to the wind load. Comparison of the results received by the refined theory with the data of the classical theory has shown that in the zone of distortion of the stressed state, the normal tangential stresses are substantially revised and the transverse normal stresses, which are neglected in the classical theory, are of the same magnitude with the maximum values of the main bending stress. Considered is the influence of the relative thickness on the stress state of the shell. It was discovered that the shell thickness significantly increases the error of the classical theory, while determining the stresses and assessing the strength of the elements of the aircraft structures.

Revisit of non-ordinary state-based peridynamics

The force density vector in the Non-Ordinary State-Based (NOSB) PeriDynamics (PD) replaces the internal force vector derived from the divergence of the stress tensor in the classical (local) stress equilibrium equations. It involves only the non-local form of the first order derivatives of stress and displacement components. Inherent in the NOSB PD formulation is the presence of oscillations especially in the regions of steep displacement gradients. This study introduces an alternative form of the force density vector by considering the internal force vector derived directly from the displacement equilibrium equations. It involves only the non-local form of the second-order derivatives of the displacement components. The numerical results from this form of the force density vector do not present any oscillations. Therefore, it is referred to as the Refined NOSB (RNOSB) – PD. The simulations concern the comparisons of NOSB and RNOSB PD predictions for an isotropic plate with or without a notch or a crack under quasi-static and dynamic tensile loading. The RNOSB PD proves to be effective and accurate for cracking and fracture analysis without any numerical instability.

Peridynamic differential operator and its applications

The nonlocal peridynamic theory has been proven extremely robust for predicting damage nucleation and propagation in materials under complex loading conditions. Its equations of motion, originally derived based on the principle of virtual work, do not contain any spatial derivatives of the displacement components. Thus, their solution does not require special treatment in the presence of geometric and material discontinuities. This study presents an alternative approach to derive the peridynamic equations of motion by recasting Navier’s displacement equilibrium equations into their nonlocal form by introducing the peridynamic differential operator. Also, this operator permits the nonlocal form of expressions for the determination of the stress and strain components. The capability of this differential operator is demonstrated by constructing solutions to ordinary, partial differential equations and derivatives of scattered data, as well as image compression and recovery without employing any special filtering and regularization techniques.

The transverse crack problem for elastic bodies stiffened by thin elastic coating

For two types of large‐scale bodies, a half‐plane and a strip, each of which is weakened by a straight transverse crack, the static problem of elasticity theory is considered. The upper boundary of each body is reinforced by a thin flexible coating. The coating is modeled by special boundary conditions on the upper faces of considered bodies. Three different cases of boundary conditions on the lower face of the strip were studied.By application of generalized integral transforms to the equilibrium equations in displacements the problems were reduced to the solutions of singular integral equations of first kind with Cauchy kernel to the respect of derivative of the crack opening function. In all considered cases the integral equations consists of a singular term, corresponding to crack behavior in an infinite plate, and a regular term, reflecting the influence of various geometric and physical parameters.For various sets of model parameters the solutions of the integral equations were built by small parameter and collocation methods; their structure was analyzed. The values of stress intensity factor in the vicinity of the tips of the crack were obtained and analyzed for different coating materials and geometric parameters of the crack.From the analysis of the numerical results of the problem, it can be concluded that thin flexible coatings significantly reduce stress intensity at a crack tip and therewith significantly increase a reliability of considered elastic bodies.

Computational Investigation of Elastoplastic Deformation of a Sandwich Beam on the Elastic Foundation

Elastoplastic bending of a sandwich beam with a rigid compressible filler on an elastic base has been studied. To describe kinematics of asymmetrical across thickness pack in the bearing layers the Bernoulli hypotheses have been accepted. Displacements in the filler vary linearly over thickness. The distributed superficial loading simulates the hydrostatic effect of the ambient liquid. The reaction of the base is described by Winkler’s model. A system of equilibrium equations for displacements has been obtained and solved. The numerical results for a sandwich metal-polymer-metal beam are cited.

Elastoplastic bending of a sandwich bar on an elastic foundation

The elastoplastic bending of a sandwich bar with a stiff compressible core on an elastic foundation is studied. The kinematics of the bar, which is asymmetric across the thickness, is described adopting Bernoulli’s hypotheses for the face layers. The displacements of the core are assumed to vary linearly across the thickness. The foundation is described by the Winkler model. A system of equilibrium equations for displacements is derived and solved. Numerical results for a metal-polymer sandwich bar are presented

Stability and Load-Carrying Capacity of Elastic Reinforced Cylindrical Shells

By using the equilibrium equations in displacements, we develop an analytic method for the evaluation of upper critical loads in elastic cylindrical shells with transverse reinforcement. As a result, we deduce analytic expressions for the evaluation of critical stresses in momentless shells and propose a procedure for the evaluation of the lower limit of the load-carrying capacity of shells based on the application of the method of reduced stiffness. The numerical results are compared with the available theoretical and experimental data.

Stress state of a transversally isotropic medium with a parabolic crack under a linearly varying shear load

In this paper we examine an unbounded transversally isotropic medium containing a parabolic crack in the plane of isotropy. Let varying shear forces of the type be applied to the surface of the crack. In solving the problem, we can use the expression of the general solution of the equilibrium equations in displacements in terms of the potential functions.

Stress state of a transversely isotropic medium with a parabolic crack when linearly changing pressure is applied to its surface

We are studying the stress-strain state of an infinite, transversely isotropic medium containing a parabolic crack in the plane of isotropy. We assume that the surface crack is loaded by linearly changing pressure, and we will solve the problem with the use of an expression for the general solution of the equilibrium equations in displacements that employ potential functions.

Buckling of edge-damaged composite cylindrical shells subjected to radial pressure

The stability of filament-wound composite cylindrical shells subjected to radial pressure is examined, for the case in which edge-damage is present. The displacement equilibrium equations, based on Flügge's quasi-linear theory, are solved using a finite complex Fourier transform together with the introduction of a displacement function. The zero of a determinantal equation, arising from the nontrivial fulfillment of the boundary conditions, furnishes the value of the critical radial pressure. The edge-damage is modeled by nonuniform boundary conditions. From computed results it is concluded that isotropic shells are capable of withstanding edge-damage up to 50% of their circumference before a reduction in the critical pressure occurs. Anisotropy tends to weaken this sturdiness with all single and bilayered shells considered suffering a fairly sharp drop in the critical pressure sustainable after only up to 20% of the edge is “loosened”. This represents a reversal of the roles of isotropy and anisotropy found by the authors for the case when the shells were subjected to axial compression.

TRANSIENT THERMAL STRESSES IN A FINITE CYLINDER OF NONUNIFORM CROSS SECTION

Boundary fitted coordinate variables are used to map the region of interest into a square. The governing transient heat conduction and displacement equilibrium equations are rewritten in terms of the new coordinate variables and are solved by a direct power series approach through the application of the Lanczos-Chebyshev and the discrete least squares methods. A finite cylindrical cone section is used to demonstrate how numerical solutions can be obtained

Buckling of edge-damaged filament-wound composite cylindrical shells under combined torsional/axial loads

The stability of filament-wound composite cylindrical shells under combined torsional/axial loading is considered, for the case in which edge-damage is present at one end of the cylinders. The displacement equilibrium equations, based on Flügge's quasi-linear theory, are solved using a finite complex Fourier transform together with the introduction of a displacement function. The zero of a determinantal equation, arising from nontrivial fulfilment of the boundary conditions, furnishes the value of the critical torque/axial load. The edge-damage is modelled by nonuniform boundary conditions. Results indicated that for isotropic shells the degree of sensitivity to edge-damage decreased the lower the applied axial load, irrespective of the direction of the applied torque. However, for anisotropic shells certain torques can actually enhance the critical axial load sustainable. This effect persists in the presence of edge-damage, and, in addition, can desensitize the shell to edge-damage over a larger section of its circumference.

Stress analysis for end effects in cylindrical rods by the lanczos-chebyshev method

The Lanczos-Chebyshev method is used to reduce the two-dimensional displacement equilibrium equations in cylindrical co-ordinates, to a set of coupled ordinary differential equations. The eigenvalues and eigenvectors defining the axial decay of self-equilibrating stress systems in a solid cylinder are obtained, and used to solve a particular problem of specified self-equilibrating end stresses applied to a semi-infinite cylinder, for which an accurate reference solution is known. The method is compared with a Galerkin and a finite element method of achieving the same result, and is shown to have definite advantages in simplicity, speed and accuracy, over the two approximation methods.

On the theory of elasticity of inhomogeneous media: PMM, vol. 35, n≗ 5, 1971, pp. 853–860

The general solution is obtained of the equilibrium equations in displacements for inhomogeneous isotropie media, whose elastic characteristics are differentiable functions of the Cartesian coordinates. It is shown that the components of the displacement vector in the three-dimensional problem of elasticity theory can always be expressed in terms of two functions which satisfy second and fourth order linear partial differential equations, respectively. Of the earlier research devoted to analogous problems, the paper [1] should first be noted in which an equation is derived for the Airy stress function in the two-dimensional problem of the theory of elasticity of an inhomogeneous medium. A general solution of the equilibrium equations in displacements is obtained in [2, 3] for the case of axisymmetric deformation of bodies of revolution whose elastic moduli vary exponentially as a function of the coordinate z. A powerlaw change in the elastic modulus was investigated in [4] with primary attention paid to the plane problem. General solutions of the equilibrium equations in displacements for the three-dimensional case and an arbitrary law of variation of the elastic characteristics of an inhomogeneous medium has apparently not been examined.

Difference schemes for the solution of some static problems in the theory of elasticity

IN this paper we investigate difference schemes for the solution of basic plane problems in the theory of elasticity in a rectangle, which are described by a set of equilibrium equations in displacements and boundary conditions of one of the three following types: 1. 1) the second boundary value problem (stresses are given on the boundary); 2. 2) the problem of hard contact or the third boundary value problem (the normal component of the displacement vector and the tangential component of the stress vector are given on the boundary); 3. 3) a mixed problem (conditions of various types are given at different parts of the boundary). The existence and uniqueness of the solution of the difference problems is proved and an apriori estimation of the solutions in the network norm [1] is obtained, which is homogeneous in h, and from which the convergence of the constructed difference scheme with speed O ( h 1 2 + h 2 2) follows for a sufficiently smooth solution, where h 1 and h 2 are network steps in the direction of x 1 and x 2. Difference schemes for the solution of a number of problems in the theory of elasticity are discussed in [1–7]. In [1] a decomposition scheme was developed for the solution of a two-dimensional dynamical problem in a rectangle with given displacements on the boundary and its convergence with speed O ( τ 2) was proved for ƒ h i = const . In [2] an iteration process was developed for the solution of a two-dimensional static problem in a rectangle with the same boundary conditions as in [1] and an estimate of the speed of convergence was obtained. In [3] both dynamic and static problems of the theory of elasticity in a p-dimensional parallelopiped are considered. Displacements on the boundary are also given. For the solution of the dynamic problem difference schemes are developed which are absolutely stable and converge with speed O ( τ + h 2) and O ( τ 2 + h 2). For the solution of the static problem an iteration process is developed and an estimate of the rate of convergence obtained. In [4] a problem is considered which converges to a plane problem of the theory of elasticity in a rectangle with boundary conditions which contain derivatives of the displacements. For this by a variational method a difference scheme of the second order of approximation is developed and an estimate obtained of the accuracy of O ( h) in the integral metric of W 2 (1) (the solution of the difference problem was used semilinearly). Some problems in the theory of elasticity in a rectangle (in particular a mixed one) are considered in [5,6]. In [7] a difference scheme is developed for the second boundary value problem of the theory of elasticity in a parallelopiped by a variational method; the boundary conditions are approximated with order O ( h). For its solution an iteration process is developed.