Isotropic Body Research Articles

АНАЛИЗ ХАРАКТЕРИСТИК НАПРЯЖЕННО-ДЕФОРМИРОВАННОГО СОСТОЯНИЯ КОМЛЕВОЙ ЧАСТИ ЛОПАСТИ НЕСУЩЕГО ВИНТА ВЕРТОЛЕТА

The butt part of the main rotor blade (RB) of the helicopter is presented. A geometric model of the calculated connection was developed using the CATIA V5 system. With the help of the ANSYS system, a finite element model of the butt part of the main rotor blade was developed, and the calculation and analysis of the characteristics of the results obtained were carried out. To describe the response of materials to an external action, a model of an elastically deformable isotropic body was used with the assignment of the corresponding elastic constants of the material. The case of loading a bolted joint by a centrifugal force arising during the rotation of a blade is considered. To model the contact interaction, a generalized surface-to-surface contact interaction model was adopted. The analysis of the calculation results includes the determination of reactions at the attachment points, the values of the maximum displacements of structural elements and stresses in the stress concentration zones. In the course of the calculation, reactions were determined at the points of attachment of the tip along the holes in the lugs. The nature of the deformation of the bolted joint as a whole and of its individual elements is analyzed. The assessment of total displacements and displacement components is carried out. To assess the static and fatigue strength of the elements of the bolted connection of the tip with the blade spar, the stresses in the load-bearing elements were analyzed for typical concentrators. When assessing the static strength, the distribution of equivalent stresses according to Mises was analyzed. The distribution of stresses in the zone of holes along the rows is analyzed, starting from the root section in the direction of increasing the Y coordinate. The maximum von Mises stresses for typical stress concentration zones were compared with the ultimate strength of the structural element material. To assess the fatigue strength, we analyzed the distribution of the main tensile stresses in the load-bearing elements by typical stress concentrators. The maximum values of these stresses were compared with the fatigue limit of the structural element material.

Elastodynamic transformation cloaking for non-centrosymmetric gradient solids

In this paper we investigate the possibility of elastodynamic transformation cloaking in bodies made of non-centrosymmetric gradient solids. The goal of transformation cloaking is to hide a hole from elastic disturbances in the sense that the mechanical response of a homogeneous and isotropic body with a hole covered by a cloak would be identical to that of the corresponding homogeneous and isotropic body outside the cloak. It is known that in the case of centrosymmetric gradient solids the balance of angular momentum is the obstruction to transformation cloaking. We will show that this is the case for non-centrosymmetric gradient solids as well.

Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i.e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

Solution of the first basic physically nonlinear problem of elasticity theory for anisotropic bodies

The study looks upon the process of the physically non-linear deformation of transversely isotropic homogeneous continuous solid bodies made from composites where the reinforcing elements are far more rigid than the binder. The solution of the physically non-linear problems employs the simplified theory of plasticity as proposed by B.E. Pobedra. The study proposes an approach to the writing out of an explicit solution that builds on the small parameter method. Ilyushin plasticity functions (that fall within the generalized Hooke’s law) are assigned discrete small values, and the resulting equations are then decomposed into power series in small parameters. The values of such small parameters are the measures of the deflection of the non-linear vs. the linear medium. Such decomposition produces an analytical coordinate & small parameters function enabling an immediate solution of the non-linear problem using just one linear elasticity field, which, in turn, is also created exclusively in reliance on the method of boundary states. Below are the results of solutions of test problems featuring a transversely isotropic cylinder with homogeneous boundary conditions. High precision in this case is achieved as early on as the third iteration. In problems with non-trivial boundary conditions, maximal precision is achieved at the first iteration and heavily depends on small parameter values. Each of the problems presented provides a detailed convergence analysis and graphic illustrations of the results.

The method of boundary states with perturbations as applied to the analysis of geometrically non-linear elastostatic bodies

This study makes a case for the application of the numeric and analytic method of boundary states with perturbations (MBSP) to analytic problems focused on the stress-strain states (SSS) of geometrically non-linear isotropic elastic bodies. The defining relations are represented through a weakly non-linear operator equation that contains (on an additive basis) a non-linear operator decomposable into a linear combination of a sequence of linear operators. A solution is proposed for a simple case of a linearly heterogeneous uniaxial loading problem for a long half-pipe with butt ends subjected to loads. Even in this load scenario, geometrical non-linearity affects the way the body alters its shape and causes buckling. This study analyzes the SSS involved, draws conclusions, and addresses application prospects.

Numerical simulation of borehole wall remodel during gas drainage

Gas drainage bearing coal and rock mass can be simplified as isotropic homogeneous body with gas saturation. Numerical method was used to study the influence of borehole wall remodel-size, remodel-length and negative pressure on gas drainage effect which based on gas diffusion and seepage equation of pore fracture system and governing equation of coal and rock mass degeneration. The results showed that borehole wall remodel can maintain the support of coal skeleton in borehole and prevent blocking the gas flow channel, gas pressure drop in remodelled section is more obvious than that in the non-remodelled section, the longer the remodelled length of borehole wall the more smooth the gas flow channel between borehole and coal body around borehole, with the increase of negative pressure, the decrease range of gas pressure at the same depth increases, but the difference is small under different negative pressure. On the premise of preventing borehole creep instability, borehole wall remodel can provide high-quality gas flow channel for coal seam gas drainage, maintain the continuous influence of negative pressure on coal body along borehole depth to extend the life cycle of gas drainage and is not sensitive to borehole wall remodel size and negative pressure.

Debye model for the surface phonons

A quantum description of the surface waves in an isotropic elastic body without the use of the semiclassical quantization is proposed. The problem about the surface waves is formulated in the Lagrangian and Hamiltonian representations. Within the framework of the generalized Debye model, the contribution of the surface phonons (“rayleighons”) to thermodynamic functions is calculated. It is emphasized that the role of the surface phonons can be significant and even decisive in low-dimensional systems, granular and porous media, and that their contribution to the total heat capacity increases with decreasing temperature.

BESLE: Boundary element software for 3D linear elasticity

BESLE is the first available parallel open-source code to analyse the mechanical behaviour of heterogeneous materials using the boundary element method (BEM) in 3D and in both an elastostatic and elastodynamic setting. Unlike all the other codes that are presently available, the software presented here is capable of simulating both isotropic and anisotropic materials comprised of single or multiple domains. Furthermore, the user-interface has been designed to provide a convenient way for configuring simulations involving many complex material constituents. Results are described by the displacement and traction fields, also, the stress and strain tensors are available for post-processing. BESLE is largely implemented in Fortran-MPI, but some of the sub-packages are based on other languages such as C and C++, and as such a the software is best used on a multi-core architecture where the parallelisation can be fully exploited. The main features and functionality of BESLE are presented here, and the User's Guide, available from the repository listed below, gives further details and outlines how users can carry out bespoke simulations. Program summaryProgram Title: BESLECPC Library link to program files:https://doi.org/10.17632/vx4vg47hzg.1Developer's repository link:https://github.com/Afgr1087/BESLE_v1.0.gitLicensing provisions: GPL-v3.0Programming language: Fortran 90, C++/CExternal libraries: LAPACK [1], BLAS [2], SCOTCH [3], ScaLAPACK [4], MUMPS [5], Voro++ [6], Triangle [7]Nature of problem: The solution of 3D elasticity models of heterogeneous materials is often arduous owing to the complexity of the underlying system of partial differential equations. Some analyses that require intensive computation are solids under quasi-static, inertial, and high-rate loading (all of which are treated by BESLE).Solution method: BESLE provides a strategy to configure and solve complex problems of 3D heterogeneous solids using the elastostatic and elastodynamic formulations of the boundary element method (BEM). Moreover, it provides flexible means create surface meshes, to impose both Neumann and Dirichlet boundary conditions, and comes with a material database for fast parameterisation. It allows large scale problems to be treated in a straightforward framework. The mechanical behaviour of isotropic and anisotropic bodies which can include several domains, each comprised of heterogeneous materials with a diverse range of constitutive properties and complex morphologies, is feasibly analysed.Additional comments including restrictions and unusual features: Fortran is limited in the size of arrays that can be allocated. Thus, for very large problems the number of degrees of freedom in the discretised system can exceed that allowed in Fortran, even if the machine has a large amount of RAM. To avoid over-allocation we have added functionality to check whether BESLE, and it's underlying Fortran code, will be able to handle the requisite number of degrees of freedom in advance of a calculation being executed. In the event that this limitation will be approached, BESLE will return an error message advising that the number of entries required will exceed 70% of the limit imposed by Fortran.

Thermoelastic Memory-dependent Responses to an Infinite Medium with a Cylindrical Hole and Temperature-dependent Properties

The present research discusses a generalized thermoelastic model with variable thermal material properties and derivatives based on memory. Based on this new model, an infinitely long homogeneous, isotropic elastic body with a cylindrical hole is analyzed for thermal behavior analysis. The governing equations are deduced by the application of the principle of memory-dependent derivatives and the generalized law on heat conduction. In a numerical form, the governing differential equations are solved utilizing the Laplace transform technique. Numerical calculations are shown in graphs to explain the effects of the thermal variable material properties and memory dependent derivatives. In addition, the response of the cylindrical hole is studied through the effects of many parameters such as time delay, the kernel function and boundary conditions. The results obtained with those from previous literature are finally verified.

Explicit Analytic Solution for the Plane Elastostatic Problem with a Rigid Inclusion of Arbitrary Shape Subject to Arbitrary Far-Field Loadings

We provide an analytical solution for the elastic fields in a two-dimensional unbounded isotropic body with a rigid inclusion. Our analysis is based on the boundary integral formulation of the elastostatic problem and geometric function theory. Specifically, we use the coordinate system provided by the exterior conformal mapping of the inclusion to define a density basis functions on the boundary of the inclusion, and we use the Faber polynomials associated with the inclusion for a basis inside the inclusion. The latter, which constitutes the main novelty of our approach, allows us to obtain an explicit series solution for the plane elastostatic problem for an inclusion of arbitrary shape in terms of the given arbitrary far-field loading.

A CA-XFEM for mixed-mode variable-amplitude fatigue crack growth

In this research, the extended finite element method (XFEM) in conjunction with the combined approximation (CA) is utilized for estimating the fatigue life of two-dimensional isotropic bodies under variable-amplitude loading. In the proposed method, called CA-XFEM, in addition to the fact that no re-meshing process is required, the crack growth path is determined without the need to solve the whole system of equations which these features significantly reduce computational costs. The Willenborg model is employed for modeling the retardation effect due to the overloads in the load history. For validation of the method, the numerical results are compared with the existing experimental data for a compact tension specimen made of Al 7075-T6 and a compact tension shear specimen made of Al 5083-H111. It is observed that the developed CA-XFEM Matlab code has excellent capability in modeling variable-amplitude fatigue crack propagation considering the retardation effect. Also, the effect of ratio and sequence of overload and mixed-mode overloading on the fatigue crack growth are studied.

A continuum-kinematics-inspired peridynamic model of anisotropic continua: Elasticity, damage, and fracture

In this work, linearized elastic isotropic continuum-kinematics-inspired-peridynamics (CPD) is further generalized to study the elastic, damage, and fracture behavior of anisotropic continua, focusing on 3D transversely isotropic and 2D orthotropic body. The anisotropy is generated by changing the material properties, including the micromodulus, critical stretch, and critical micropotential energy of basic finite kinematic elements with their direction in the principal material axes in terms of eight-order double Fourier expansion. It is critically proven that the volumetric micromodulus of finite volume elements in the 3D case is independent of their direction, which is similarly applied to the areal micromodulus in the 2D case. For this reason, the 2D orthotropic CPD formulation characterized by three independent elastic moduli and the 3D transversely isotropic CPD formulation characterized by four independent elastic moduli are distinguished. The accuracy of the computational models applied to elastic analysis is assessed by comparing the predicted displacement fields with the results from finite element analysis. Quantitative simulation of an orthotropic lamina with a central rectangular hole under a tensile load, eccentric three-point bending test for orthotropic lamina, and compact tension test for cortical bone are performed to verify the ability of the proposed model to describe the damage and fracture behavior of anisotropic materials.

Fracture analysis of functionally graded material by hybrid meshless displacement discontinuity method

The recent progress in the development and research of functionally graded materials (FGMs) has prompted the development of numerical methods in engineering analysis for continuously non-homogeneous media. This paper presents the hybrid Meshless Displacement Discontinuity Method (MDDM) for a cracked structure with functionally graded materials. The fundamental solutions of displacement discontinuity for an isotropic homogenous body form a part of the general solutions for non-homogenous materials, in order to create the gap on the crack surface. The governing equations are satisfied by using the Meshless Local Petrov-Galerkin method (MLPG) at the scattered nodes in the domain. The stress intensity factors (SIFs) are evaluated by either crack opening displacement (COD) method or equivalent stress technique. The crack growth paths with different FGMs gradient parameters are investigated and compared with the solutions given by the finite element method (FEM). The efficiency and accuracy of hybrid MDDM are demonstrated by four examples and comparisons have been e with other analytical and numerical methods.

A class of transversely isotropic non-linear elastic bodies that is not Green elastic

An implicit constitutive relation is proposed to study transversely isotropic bodies. The relation is obtained assuming the existence of a Gibbs potential that depends on the second Piola–Kirchhoff stress tensor, from which the Green Saint-Venant strain tensor is obtained as the derivative with respect to the stress. The responses of unconstrained as well as inextensible bodies are studied, and some boundary value problems are analysed. An inextensible body, where the constraint of inextensibility appears only in tension is also considered.

Micromechanical modeling of a cracked elliptically orthotropic medium

In this paper, we derive the second-order crack opening displacement tensor for an arbitrarily oriented elliptical crack in an elliptically orthotropic (EO) matrix. This result is obtained in explicit closed form. The approach is based on the Saint-Venant’s idea of linear transformation between boundary value problems for elliptically orthotropic and isotropic bodies. The solution utilizes the classical representation of an ellipsoid crack where the smallest aspect ratio approaches zero and the transformation of the Taylor expansion of the corresponding Hill tensor. It is shown, in particular, that transformed cracks have neither the same in-plane aspect ratio nor the same vanishing aspect ratio. It requires a correction factor in the crack opening displacement tensor. Some specific relative orientations of the crack with respect to the symmetry planes of the EO matrix are considered in detail and effective properties are calculated in the case of randomly distributed cracks. The result is also extended to the case of a cylindrical (plane strain) crack.

Identification of characteristics of a functionally graded isotropic cylinder

On the basis of the general problem formulation for an inhomogeneous isotropic body, the problems of steady-state longitudinal-radial and torsional vibrations of a cylinder are considered. The Lame parameters and the density are variable in radial coordinate. The inverse coefficient problem on determining the laws of variation of cylinder’s properties from the data on the amplitude–frequency characteristics specified at different frequency ranges is studied. The numerical solution is constructed by using a three-stage iterative process based on the derived systems of the Fredholm integral equations of the first kind with smooth kernels with respect to the unknown corrections; the A.N. Tikhonov regularization method is employed to solve these equations. The reconstruction examples for monotonic laws are presented.

Stress singularity evaluation for three-dimensional cracks using characteristic tensor method

The Characteristic tensor method (CTM) is a new method to evaluate the singular stress field with a characteristic tensor (CT) which is calculated by averaging the stress in an infinitesimal volume at a crack tip. The CT has a linear relation with the stress intensity factor (SIF), which characterizes the singular stress field around a crack tip in an isotropic homogeneous elastic body. Even under mixed-mode loading conditions, the SIFs of each mode can be easily and efficiently estimated using CT. For the practical use in industries, the evaluation of the CTM for a three-dimensional crack under general three-dimensional stress state is necessary. In this study, the reliability and accuracy of CTM is investigated by estimating SIFs in several representative types of three-dimensional cracks where stress states are calculated by the finite element method. The results show that the SIFs estimated by the CTM have high accuracy even for mixed-mode conditions. It was, thus, demonstrated that CTM provides a practical and reliable approach for the efficient estimation of SIFs of three-dimensional cracks.

МЕТОД ГРАНИЧНЫХ СОСТОЯНИЙ ПРИ РЕШЕНИИ СМЕШАННОЙ ЗАДАЧИ ТЕОРИИ АНИЗОТРОПНОЙ УПРУГОСТИ С МАССОВЫМИ СИЛАМИ

The paper presents a methodology for determining a stress-strain state of transversely isotropic bodies of revolution under conditions of a mixed problem of the elasticity theory, i.e. displacements of the boundary points are specified on the one part of the surface, and forces are assigned on the other part. At the same time, the body is exposed to mass forces. The problem solving involves the development of the boundary state method. A theory is created to construct the bases of spaces for internal and boundary states. The basis of the internal states includes displacements, strains, and stresses. The basis of the boundary states includes forces at the boundary, displacements of the boundary points, and mass forces. Spaces are conjugated by an isomorphism. It allows one to reduce the determination of the internal state to a study of the boundary state. Characteristics of the stress-strain state are presented in terms of the Fourier series. Finally, the determination of the elastic state is reduced to the solving of an infinite system of algebraic equations. A result of the study is presented as a solution to the main mixed problem for a hemisphere clamped on a plane surface and exposed to a concentrated compressive force and mass forces.

Stress–temperature equations of motion of Ignaczak and Beltrami–Michell types in arbitrary curve coordinate system

يبيًّن البحث بالنموذج الرياضي لجسم مرن متماثل المناجي Isotropic)) ومتجانس Homogeneous))، ومهمل البنية الجزيئية ويعاني من انفعالات مرنة لامتناهية في الصغر، والذي يُرمز له اختصاراً بـ (H)، وسيتم مناقشة الشكل التنسوري الناطق في نظام إحداثي منحني كيفي، لكلٍ من النموذج الرياضي التقليدي ونموذج لامي الرياضي لجسم هوك المرن (H)، المًعتبر، في حالته التحريكية، بوجود حرارة. سيتم مناقشة الشكل التنسوري الناطق في نظام إحداثي منحني كيفي على المتنوعة التقليدية ثلاثية الأبعاد، لكل من معادلات بيلترامي- ميشيل المعممة ومعادلات إغناتشاك بالإجهادات والحرارة، ومسألة الاتمام المتعلقة بها، وذلك لأجل الجسم الثرموديناميكي المرن (H) الخاضع للحرارة.

Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics of Objects

The paper presents an approach developed on the basis of the functional voxel method to the geometric representation of the thermal expansion of objects and temperature stresses in a material when exposed to a surface of a heat source. A discrete geometric law of a single temperature stress in an isotropic heat-conducting body is derived, applicable in the concept of functional voxel modeling. Based on this law, functional-voxel models of thermal stress are developed for a single and distributed application of a heat source. Algorithms of functional-voxel modeling of temperature stress and expansion in the case of distributed thermal loading are presented, which make it possible to construct a loading region of a complex configuration, uniformly form a contour (surface) after material expansion and obtain information about changes in the length (volume) of products. The advantages of the proposed functional-voxel approach to modeling thermal expansion and stress over approaches based on the FEM are substantiated.