Small Subcritical Strains Research Articles

Establishing the Foundations of the Mechanics of Fracture of Materials Compressed Along Cracks (Review)

The basic results of establishing the foundations of the mechanics of fracture of homogeneous materials compressed along cracks and inhomogeneous (composite) materials compressed along interface cracks are analyzed. These results were obtained using elastic, plastic, and viscoelastic material models. This review consists of three parts. The first part discusses the basic concept that the start (onset) of fracture is the mechanism of local instability near the cracks located in a single plane or parallel planes. The fracture criterion and the basic problems arising in this division of fracture mechanics are also formulated. Two basic approaches to establishing the foundations of the mechanics of fracture of materials compressed along cracks are outlined. One approach, so-called beam approximation, is based on various applied theories of stability of thin-walled systems (including the Bernoulli, Kirchhoff–Love, Timoshenko-type hypotheses, etc.). This approach is essentially approximate and introduces an irreducible error into the calculated stresses. The other approach is based on the basic equations and methods of the three-dimensional linearized theory of stability of deformable bodies for finite and small subcritical strains. This approach does not introduce major errors typical for the former approach and allows obtaining results with accuracy acceptable for mechanics. The second part offers a brief analysis of the basic results obtained with the first approach and a more detailed analysis of the basic results obtained with the second approach, including the consideration of the exact solutions for interacting cracks in a single plane and in parallel planes and results for some structural materials. The third part reports new results for interacting cracks in very closely spaced (or coinciding, as an asymptotic case) planes. These results may be considered a transition from the second approach (three-dimensional linearized theory of elastic stability) to the first approach (beam approximation). This is how the accuracy of results produced by the first approach is evaluated and the boundary conditions near the crack tip are established in the second approach.

Stability of layered coatings with outer layer made of ceramics

Introduction. The factors responsible for the performance of coatings in various elements of modern metallurgical equipment [6, 7] (such as the combustion chamber and air tuyere of a blast furnace) include: external conditions (temperature and stress in structural elements, composition and intensity of corrosive components in the gas flow); internal conditions (composition of the coating, material of the substrate, coating technology). The reactions to external and internal factors include thermal fatigue, physical and chemical development and formation of interlayers and, as a consequence, local bulging of coatings; erosive damage and cracking; change in the behavior of the interaction between the components of the gas flow and the materials of the layered coating (hydrogenation) and the oxidation processes, which give rise to new surface layers. One of the possible causes of bulging may be the surface microbuckling of the coating. The condensed metal coatings currently used for protection purposes are capable of resisting the erosive action of the gas flow. Note that the more functions the coating performs, the more layers it consists of and the higher the probability that it will lose stability. This, in turn, may lead to failure of such elements of metallurgical equipment, with all that ensues. It is of interest to study the surface instability of layered coatings in tuyeres and the effect of the basic mechanical, physical, and chemical processes in the blast furnace on this process. The three-dimensional stability of layered coatings with variable structure was addressed in [4, 8–13] where a general problem-solving approach was developed and some specific results were discussed. In [3, 14], the effect of high temperature is incorporated in buckling problems by correcting the values of the mechanical characteristics of the layers at set temperature of the coating. Following [3, 14], we will study (using a plane strain assumption) the effect of the outer ceramic layer on the stability of a layered coating in the operating conditions of the air tuyere of a blast furnace. We will use a piecewise-homogeneous medium model for linear elastic bodies. We will give a problem statement, outline a problem-solving method, and solve specific surface-buckling problems for layered coatings with outer layer made of ceramics. Small subcritical strains in the composite under biaxial thermomechanical loading will be assumed. 1. Problem Formulation. Governing Equations. The air tuyere of a blast furnace can be modeled by a layered

Surface instability of orthotropic laminated materials under compression

The piecewise-homogeneous material model and the three-dimensional linearized theory of stability with the assumption of small subcritical strains are used to study the surface buckling of orthotropic and transtropic laminates. A plane problem is formulated, and characteristic equations are derived. A solution is found for a specific transtropic material with different orientations of the isotropy axis

On the determination of loads upon the ground support in underground mine workings

The piecewise-homogeneous model and the three-dimensional linearized theory of stability for small subcritical strains are used to study the surface instability of a regularly layered rock mass under biaxial loading. A plane problem is formulated. Basic characteristic equations are derived. A specific problem is solved as an example to demonstrate the selection of loads and the interaction of support elements with the wall rock

On the possibility of local buckling of the surface of an elastic half-space under compression

The possibility of local buckling of the free surface of the lower half-plane under compression is studied in a static formulation within the framework of plane deformation. It is shown that in some media small subcritical strains can lead to local buckling of the half-plane surface. It is found that two forms of local surface buckling correspond to one critical compression load.

Stability of an Elastic Layer Stack Between Two Half-Spaces Under Compressive Loads

A stability problem is solved for a multicomponent medium consisting of a layer stack and two structurally homogeneous half-spaces. The analysis is made within the framework of the piecewise-homogeneous model and the three-dimensional linearized theory of stability for small subcritical strains. The properties of isolated elements of the medium are determined within the framework of the orthotropic linearly elastic model. Specific numerical results are obtained for the case of plane strain

Combined effect of temperature and compressive surface loads on the stability of elastic multilayer coatings with small subcritical strains

Combined effect of temperature and compressive surface loads on the stability of elastic multilayer coatings with small subcritical strains

Stability of multilayer antifriction coatings for small subcritical strains

Stability of multilayer antifriction coatings for small subcritical strains

Stability of two noncircular cylinders in an elastic matrix with small subcritical strains

Stability of two noncircular cylinders in an elastic matrix with small subcritical strains

The spatial problem of the compression of a material along a periodic system of parallel circular cracks

The non-axisymmetric problem of the biaxial uniform compression of a material along a periodic system of parallel circular cracks is considered. A facture criterion is used /1, 2/ within the framework of linearized stability theory according to which the beginning of fracture of the material under compression along the cracks is characterized by local buckling near the cracks. Within the framework of this approach, axisymmetric and plane problems were considered earlier for different material models (highly-elastic, composite and plastic) for one or two internal cracks, near-surface cracks and a periodic system of cracks /1–13/★★ Proceeding of the Eleventh Scientic Conf. of Young Scientists. Inst. Mechanics, Ukraine Academy of Sciences, Kiev, 1986. 154–161, Dep. VINITI 5507-86, July 28, 1986 Nazarenko, V.M. and Starodubtsev, I.P., On material fracture under compression along two parallel cracks in the case of plane strain. Non-classical and Mixed Problems of the Mechanics of a Deformable Body: Materials of a Seminar of Young Scientists, Kiev, 1985, 142–145, Dep. 5531-85 in VINITI, July 29, 1985.) The investigation is performed in general form for an arbitrary kind of elastic potential for compressible and incompressible materials, the theory of large and modifications of small sub-critical strains, and can be extended to other models of a deformable body (composites, plastic bodies, etc.).

Fracture mechanics of solids in compression along cracks

The following results were obtained in regard to compression of solids along planes containing cracks. 1. For the case of loading examined, we proposed a criterion for the beginning of fracture corresponding to the stability criterion in the three-dimensional linearized theory of stability of deformable bodies. 2. An exact solution of a two-dimensional problem of local loss of stability was obtained for a finite number of plane cracks of constant width, when the cracks are infinite in one direction and located in one plane. This solution was obtained in general form for elastic compressible and incompressible bodies with Tinite and small subcritical strains and elastic-plastic compressible and incompressible bodies with small subcritical strains. 3. It was shown for specific models of isotropic elastic and elastic-plastic bodies that, within the framework of plane strain, loss of stability next to a finite number of plane cracks occurs in the form of surface instability of a half-space and half-plane. It was shown that flexural and symmetrical modes of instability have the same critical load.

Stability of elastic bodies under omnilateral compression

We have analyzed here the stability of the equilibrium of a simply connected isotropic compressible body with the elastic potential of arbitrary form and under uniform omnilateral deformation. A survey has been given here of earlier results obtained by other authors. The basic celations have been stated in a general form covering the theory of finite subcritical strains and two variants of the theory of small subcritical strains. For the latter theory new relations have been rigorously derived from which perturbations of “tracking” surface loads can be calculated, on the basis of corresponding expressions in the theory of finite subcritical strains. It has been proven that the sufficient conditions for the applicability of the static method of analysis are satisfied when the same boundary conditions are given over the entire body surface, as well as in several cases of different boundary conditions given at different segments of the boundary surface. It has been shown in a general form, for the theory of finite subcritical strains and for two variants of the theory of small subcritical strains, that the equilibrium of an elastic body under omnilateral deformation is stable, if a “tracking” load, is given over the entire boundary surface. As an example of problems with different boundary conditions at different segments of the boundary surface, we have considered the conventional problem concerning the stability of a bar on hinge supports and under uniform omnilateral deformation. It has been rigorously proven that in this case the equilibrium is stable when “tracking” loads are given at the lateral surfaces and is unstable when “dead” loads are given at the lateral surfaces. These conclusions apply to the theory of finite subcritical strains as well as to the theory of small subcritical strains, and they represent the complete version pertaining to compressible bodies.

Stability of a transversely isotropic cylindrical shell in axial compression

The problem of the stability of a transversely isotropic cylindrical shell subjected to axial compression is considered with the aid of three-dimensional linearized equations for small subcritical strains. Transcendental equations are obtained for the critical loads corresponding to axisymmetric and nonaxisymmetric deformation. It is shown that in the case of the axisymmetric buckling mode the Kirchhoff-Love hypothesis is asymptotically exact irrespective of the properties of the material. The dependence of the critical load on the properties of the shell material is investigated numerically. Graphs of the variation of the critical load with relative shear modulus are presented.

Some problems concerning the stability of nonlinear elastic bodies under finite and small subcritical strains

Some problems concerning the stability of nonlinear elastic bodies under finite and small subcritical strains