


Vol 51, No 5 (2016)
- Year: 2016
- Articles: 16
- URL: https://ogarev-online.ru/0025-6544/issue/view/9877
Article
On the stress monitoring problem for parallel gallery regions
Abstract
We consider the problem of estimating the stress–strain state in underground hard mineral mines where long parallel underground galleries are formed. There are a number of papers dealing with local causes of gallery accidents due to variations in the medium stress–strain state occurring as the minerals are withdrawn. At the same time, the authors’ theory of hidden effects shows that gallery fracture can be caused both by local actions on the defect region and by some long-range factors that can affect these defects remotely by localizing the stress–strain state in the defect region. It should be noted that the stress in the gallery region is redistributed each time a new gallery is formed or the dimensions of the existing galleries are changed. In this paper, a theory for estimating the stress-strain state in underground mines with arbitrarily many parallel galleries of various dimensions is developed.
It is shown that the stresses due to remote factors can be monitored in all gallery regions. The study is based on factorization methods, the block element method, and the topological approach.



Fracture of sedimentary rocks under a complex triaxial stress state
Abstract
Most sedimentary rocks have layered structure, and their strength properties are therefore anisotropic; as a consequence, the rock strength depends on the direction of the applied stresses. In this case, various fracture mechanisms are possible. The following two possible fracture mechanisms are considered: actions along the bedding planes, which are weakening surfaces, and along the planes where stresses exceeding the total rock strength are attained. A triaxial independent loading test bench was used to study the fracture conditions for layered rocks composed of productive oil-and-gas strata in complex true triaxial loading tests. The study shows a good qualitative agreement between experimental results and theoretical estimates.



Generalized Timoshenko–Reissner model for a multilayer plate
Abstract
A multilayer plate with isotropic (or transversally isotropic) layers strongly differing in rigidity is considered. This plate is reduced to an equivalent homogeneous transversally isotropic Timoshenko–Reissner plate whose deflections and free transverse vibration frequencies are close to those of the multilayer plate. By comparison with the exact solution of test three-dimensional problems of elasticity, the error of the proposed method is estimated both for the static problem and for free vibrations. This comparison can readily be carried out for the hinged edges of the plate, and explicit approximate formulas are obtained for the vibration frequencies. The scope of the proposed model turned out to be rather wide (the Young moduli of soft and rigid layers can differ by a factor of 1000). In the case of boundary conditions other than hinged support, a closed-form solution cannot be constructed in general. For rigidly fixed edges, the asymptotic method proposed by V. V. Bolotin is generalized to the case of a Timoshenko–Reissner plate.



Ill-posed problems in mechanics
Abstract
The notion of ill-posed initial and boundary value problems for partial differential equations was introduced by Hadamard, who also presented the first example of an ill-posed problem for a specific partial differential equation. At the same time, there are numerous examples of ill-posed problems in any field of mechanics.
Hadamard and some of his successors believed that any ill-posed problem has no physical meaning and such problems should not be posed.
The present paper contains several examples of ill-posed problems. We show that if one deals with an applied problem, then overcoming the ill-posedness mathematically can help one to improve the structure in practice, which justifies the study of ill-posed problems.



Measurement of inhomogeneous strain fields by fiber optic sensors embedded in a polymer composite material
Abstract
Experimental results of strain field measurement in polymer composite specimens by Bragg grating fiber optic strain sensors embedded in the material are considered. A rectangular plate and a rectangular plate with “butterfly” shaped cuts are used as specimens. The results of uniaxial strain experiments with rectangular plates show that fiber optic strain sensors can be used to measure the strains, and these results can be used to calculate the calibration coefficients for fiber optic strain sensors. A gradient strain field is attained in a plate with cuts, and the possibility of measuring this field by fiber optic strain sensors is the main goal of this paper. The results of measurements of gradient strain fields in the plate with cuts are compared with the results obtained by using the three-dimensional digital optic system Vix-3D and with the results of numerical computations based on finite element methods. It is shown that the difference between the strain values obtained by these three methods does not exceed 5%.



Modeling the influence of the coating deposition technology on the contact interaction characteristics
Abstract
For a multilayer elastic half-space, we consider an axisymmetric loading model taking into account damage on the interface between the layers. The influence of intermediate layers arising in various coating technologies on the contact and internal stresses occurring in the coating and the substrate under elastic indentation conditions is studied for relatively rigid and nonrigid coatings.



New solution of the plane problem for an equilibrium crack
Abstract
We consider the classical plane problem of elasticity about a crack in an isotropic elastic unbounded plane resulting in a singular solution for the stresses near the crack edge. Relations of generalized elasticity with a small parameter characterizing the medium microstructure are derived, and the higher order of these relations permits eliminating the singularity of the classical solution. An experimental method for determining the medium parameter is proposed, and the corresponding experimental results are given.



Factorization method in the geometric inverse problem of static elasticity
Abstract
The factorization method, which has previously been used to solve inverse scattering problems, is generalized to geometric inverse problems of static elasticity. We prove that finitely many defects (cavities, cracks, and inclusions) in an isotropic linearly elastic body can be determined uniquely if the operator that takes the forces applied to the body outer boundary to the outer boundary displacements due to these forces is known.



Limit velocities of lamb waves: Analytic and numerical studies
Abstract
The Lamb wave propagation in elastic isotropic and orthotropic layers is studied by numerical and analytic methods. An analytic solution is obtained by using the Cauchy formalism for the entire frequency range. Numerical solutions are obtained in a neighborhood of the second limit velocity corresponding to very small frequencies. The influence of variations in the layer geometry on the dispersion curves is studied.



On the dispersion relations for an inhomogeneous waveguide with attenuation
Abstract
Some general laws concerning the structure of dispersion relations for solid inhomogeneous waveguides with attenuation are studied. An approach based on the analysis of a first-order matrix differential equation is presented in the framework of the concept of complex moduli. Some laws concerning the structure of components of the dispersion set for a viscoelastic inhomogeneous cylindrical waveguide are studied analytically and numerically, and the asymptotics of components of the dispersion set are constructed for arbitrary inhomogeneity laws in the low-frequency region.



Localized waves in a string of infinite length lying on a damaged elastic base under finitely many impacts
Abstract
Asymptotic solutions of the problem of dynamics of an infinitely long string lying on an elastic base with prescribed damage under the action of finitely many periodic impacts are constructed in the two cases of small and large damage of the elastic base. The condition of resonance origination in the string is obtained in the case of small damage when the standing wave is localized in the region of damage. At the final stage of the damage growth in the elastic base, when its value is close to the critical one, the localized mode and the resonance are absent, and only a traveling wave exists in the string.



Dynamic equations of a prestressed magnetoelectroelastic medium
Abstract
The constitutive relations of nonlinear mechanics of a magnetoelectroelastic medium subjected to initial mechanical stresses are linearized in the framework of material (Lagrangian) coordinates. The final expressions are constructed independently of the choice of curvilinear coordinates and are represented in a form convenient for theoretical and applied studies. The constitutive relations for the motion of a prestressed magnetoelectroelastic medium are given in rectangular Cartesian coordinates. The influence of the initial mechanical stresses on piezomagnetoelectric materials of the class 6mm is studied.



Mathematical models of carbon-carbon composite deformation
Abstract
Mathematical models of carbon-carbon composites (CCC) intended for describing the processes of deformation of structures produced by using CCC under high-temperature loading are considered. A phenomenological theory of CCC inelastic deformation is proposed, where such materials are considered as homogeneous ones with effective characteristics and where their high anisotropy of mechanical characteristics and different ways of resistance to extension and compression are taken into account. Micromechanical models are proposed for spatially reinforced CCC, where the difference between mechanical characteristics of components and the reinforcement scheme are taken into account. Themodel parameters are determined from the results of experiments of composite macrospecimens in the directions typical of the material. A version of endochronictype theory with several internal times “launched” for each composite component and related to some damage accumulation mechanisms is proposed for describing the inelastic deformation. Some practical examples are considered.



Estimating the plastic strain with the use of acoustic anisotropy
Abstract
Experimental verification is used to show that reference specimens and structure unloading do not permit obtaining an adequate estimate of plastic strain by measuring the acoustic anisotropy. Analytic estimates of the speed of propagation of a plane acoustic wave of various polarizations in an elastoplastic material in the direction orthogonal to the action of preliminary uniaxial stress are obtained. An analysis of the obtained relations reveala an advantage of using absolute values of the velocity of longitudinal and transverse waves for the plastic strain identification. In contrast to acoustic anisotropy, the velocities vary monotonically in a wider range of plastic strains. At the same time, the elastic strain does not affect the longitude wave velocity, which allows one to use the measurement results to estimate the character of strains.



Development of inhomogeneous fields under postcritical deformation of steel specimens in extension
Abstract
The paper deals with experimental studies of inhomogeneous strain fields in the “neck” region in extension of plane and cylindrical specimens at the postcritical stage, which directly precedes the fracture, by using a video system and a digital image correlation method. We consider the problems of interpretation of extension diagrams obtained for specimens of various length with strain inhomogeneity in the working section of the specimen with a “neck.”We obtain experimental data about the distribution of longitudinal, transverse, and shear strain fields and the displacement fields in the region of the plane specimen localization by using the digital image correlation method.



Potentiality of isotropic nonlinear tensor functions relating two deviators
Abstract
In the theory of constitutive relations, isotropic quadratic nonlinear tensor functions modeling media with second-order effects, in particular, with misalignment of the force and kinematic tensors, are considered. It is very interesting to consider tensor functions with a scalar potential relating two symmetric deviators of rank two. In this case, the potentiality conditions are integrated, and it is shown that the first integral contains two arbitrary functions of the quadratic invariant of the tensor argument and one arbitrary function of the cubic invariant. A tensorially nonlinear generalization of the rigid-viscoplasticmodel (a two-contact Binghamsolid) is carried out.


