Model Of Piecewise-homogeneous Medium Research Articles

Method for Calculating the Near-Surface Effect in Piecewise Homogeneous Bodies at Large Deformations Based on a Two-Level Approach

A method is proposed for calculating the near-surface effect in piecewise homogeneous bodies under large deformations based on a combined use of one-level applied and two-level carcass theories. The applied theory is used for a macromechanical continuation of the solution of the problem, but the carcass theory is used in the final part of the loading path or directly at the final loading of the body. The implementation of the problem by the carcass theory for the body as a whole ends with solution of extreme problems using a highly gradient scheme for assemblies of structural blocks near boundary surfaces of the body. The rotation-caused development of configurations of cylinders reinforced with ring fibers is studied using this method and the model of a piecewise homogeneous medium. The results obtained by the carcass theory and the model of a piecewise homogeneous medium differed only slightly, confirming the high accuracy of the analysis using the two-level approach. A higher stability of the numerical implementation of the carcass theory, in comparison with the model of a piecewise homogeneous medium, was revealed when the macromechanical continuation was carried out within the framework of this theory itself. An even higher stability of the approach was reached by implementation of the macromechanical continuation of the applied theory. Performing calculations only for nodal blocks of the medium at the micromechanical level of the carcass theory, together with application of the applied theory for the macromechanical continuation, led to an extremely high efficiency of the two-level approach. In contrast to the piecewise homogeneous medium model, this approach made it possible to study the behavior of piecewise homogeneous cylinders considering the near-surface effect up to the configurationdetermined ultimate rotation speed.

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

Near-surface instability of composites weakly reinforced with short fibers

The paper is concerned with loss of stability near the free flat surface of a weakly short-fiber-reinforced composite material under compression along the fibers. Emphasis is on the effect of the mechanical and geometrical parameters of the composite and the distance between fibers and to the free surface on the critical strains. The piecewise-homogeneous medium model and the rigorous three-dimensional linearized theory of stability of deformable bodies are used

Plane instability problem for a composite reinforced with a periodic row of short parallel fibers

The instability of a composite material reinforced with a periodic row of parallel short fibers is studied considering the interaction of neighboring fibers. Emphasis is on the mutual influence of short fibers in the matrix during loss of stability, depending on the distance between them. A piecewise-homogeneous medium model and the three-dimensional linearized theory of stability of deformable bodies are used

The effect of cracks’ interaction for transversely isotropic layered material under compressive loading

Laminar composites due to their internal structure and manufacturing methods contain a number of inter- and intra-component defects whose size, dispersion and mutual interaction alter significantly the critical compression strain level. The current paper investigates crack interaction in transversely isotropic materials compressed in a static manner along inter-laminar defects. For laminated composites compressed along layers and, therefore, along the mentioned interfacial defects, the classical Griffith–Irvin criterion of fracture or its generalisation is inapplicable since all stresses intensity factors and crack opening displacements are equal to zero. The statement of the problem is based on the most accurate approach, the model of piecewise-homogeneous medium. The moment of stability loss in the microstructure of material is treated as the onset of the fracture process. The behaviour of each constituent is described by the three-dimensional equations of solid mechanics, provided certain boundary conditions are satisfied at the interfaces. The complex non-classical fracture mechanics problem is solved by finite elements analysis, using a linear buckling model. The results are obtained for fibre volume fraction V f = 0.1 – 0.73 for the typical dispositions of cracks. The model of classical cracks is utilised. It is found that the fibre volume fraction, the crack length and the mutual position of cracks influence the critical strain of the composite.

The effect of the interaction of cracks in orthotropic layered materials under compressive loading

The non-classical problem of fracture mechanics of composites compressed along the layers with interfacial cracks is analysed. The statement of the problem is based on the model of piecewise homogeneous medium, the most accurate within the framework of the mechanics of deformable bodies as applied to composites. The condition of plane strain state is examined. The layers are modelled by a transversally isotropic material (a matrix reinforced by continuous parallel fibres). The frictionless Hertzian contact of the crack faces is considered. The complex fracture mechanics problem is solved using the finite-element analysis. The shear mode of stability loss is studied. The results are obtained for the typical dispositions of cracks. It was found that the interacting crack faces, the crack length and the mutual position of cracks influence the critical strain in the composite.

Studying edge effects near rectangular mine workings weakened by a crack

Edge effects in a rock mass near a system of periodic mine workings crossed by a crack (fold fault) are considered. The stress state near the mine workings is determined by solving an elastic problem using the piecewise-homogeneous medium model. The original differential problem is reduced to a discrete problem based on the concept of a base scheme. Methods are proposed to solve the discrete problem, and the issue of optimizing the numerical analysis is discussed. The stress distribution around mine workings is analyzed. The stress isolines that represent the boundaries of edge effect zones and indicate their maximum decay length are plotted

The effect of fibre volume fraction on the onset of fracture in laminar materials with an array of coplanar interface cracks

The onset of fracture process in a layered composite with coplanar interface cracks under compression along layers is studied within the scope of 3-D theory of stability using the model of piecewise-homogeneous medium. Each layer consists of a matrix reinforced by continuous parallel fibres and is modelled by a compressible, linear elastic, transversely isotropic material. The eigenvalue problem with respect to the controlled parameters, namely, the crack length, the crack spacing and the 0°-plies layer volume fraction is solved for the shear and the extensional modes utilizing the finite element analysis by commercially available FE code. The results are obtained for 0°-plies volume fraction V 0° = 0.1 … 0.9 for the typical dispositions of cracks. The effect of the crack length, the crack spacing and the 0°-plies volume fraction on the critical microbuckling strain in the composite was determined.

Problems of folding in the earth’s stratified crust

The problem of folding in the Earth’s stratified crust is addressed within the framework of solid mechanics. The history and formulation of the problem and approaches to its solution are briefly outlined. The initial stage of folding in strata with different mechanical properties is studied in detail using the piecewise-homogeneous medium model and the three-dimensional theory of stability. Conclusions of applied and fundamental importance are drawn from an analysis of solutions to a wide range of specific problems

The effect of the multi-axiality of compressive loading on the accuracy of a continuum model for layered materials

Two methods of analysis of the internal instability of layered materials are discussed: the continuum approach and the piecewise-homogeneous medium model. Based on the results obtained within the scope of the model of a piecewise-homogeneous medium and the 3-D stability theory, the accuracy of a continuum theory is examined for incompressible non-linear materials undergoing large deformations. Two different loading conditions are compared: biaxial and uniaxial compression. The effect of the multi-axiality of loading on the accuracy of the continuum theory is determined for the particular model of hyperelastic layers described by the Treloar’s potential (i.e. by a neo-Hookean type potential), which is a simplified version of the Mooney’s elastic potential.

Predicting fracture of layered composites caused by internal instability

The present paper estimates the critical strain at which ply instability occurs in compressible layered composites under uniaxial or biaxial compression. This is achieved by using two methods: the continuum approach and the model of piecewise-homogeneous medium in conjunction with the three-dimensional stability theory. The accuracy of the continuum theory is examined by taking into account the influence of layer thickness, stiffness properties and biaxiality of loads. Also, upper and lower bounds for the critical buckling strain of laminates with interfacial defects (cracks with connected edges) are determined using the results for perfectly bonded and sliding layers.

Compressive fracture of non-linear composites undergoing large deformations

Based on the results obtained within the scope of the model of piecewise-homogeneous medium and three-dimensional stability theory, the accuracy of the continuum theory is examined for laminated incompressible materials undergoing large deformations. Estimation of the accuracy of the continuum theory is illustrated by numerical results for the particular models of composites when the layers are hyperelastic materials with the elastic potential of the neo-Hookean type (Treloar’s potential). Based on this the influence of the layers’ thickness and their stiffness on the accuracy of the continuum theory is determined.

A 3-D stability theory applied to layered rocks undergoing finite deformations in biaxial compression

Based on the results obtained within the scope of the model of piecewise-homogeneous medium and three-dimensional stability theory, the asymptotic accuracy of the continuum theory is examined for layered compressible rocks undergoing finite deformations in biaxial compression. The particular mode of stability loss, that corresponds to the continuum approximation is determined. The investigation is carried out for the cases of uniaxial and biaxial compression and is illustrated by several numerical examples for the particular models of rocks. At that the influence of the layers' thickness and their stiffness, as well as the biaxiality of loading, on the accuracy of the continuum theory is determined.

Calculation of stability of laminated composite coatings in tribotechnics

The stability of laminated coatings in tribotechnics under compression loads is investigated on the basis of the three-dimensional linearized theory of stability within the framework of a model of piecewise-homogeneous medium. The general formulation of the problem is given, and the solution of a particular example is considered.