On Two-Scale Model of Fracture Mesomechanics of Composites with Cracks under Compression

A two-level fracture model of composites with cracks under compression is proposed. Fiber-reinforced laminated panels with a hole are considered. The panels are compressed along the fibers so that two cracks propagate from the hole boundary at a right angle to the loading direction. The fracture mechanism is analyzed at two levels. The first level is the compression-induced stress concentration at the tip of a finite-thickness crack filled with a fractured material. At this level, the theory of elasticity of linear orthotropic body is used. The second level is compression fracture at the crack tip. The analysis performed at this level involves the three-dimensional linearized theory of stability of solids, considering microcracks in the interface between the fractured and intact materials, and the exact solution describing the compression-induced instability of the cracked interface. The second level is where results obtained by the author are used. Thus, the approach proposed employs the theory of elasticity of linear orthotropic body and the three-dimensional linearized theory of stability of solids to analyze a specific mechanism of fracture__________Published in Prikladnaya Mekhanika, Vol. 41, No. 5, pp. 141–144, May 2005Reported at ICF-11 (Turin, Italy, March 20–25, 2005). The ICF-11 proceedings have been published on CD-ROM. The author ‘s other reports presented at ICF-6 (1984) to ICF-10 (2001) are listed in Appendix.     

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 __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 38–46, November 2005.     

Setting up a theory of stability of fibrous and laminated composites

The results obtained in setting up a theory of stability of fibrous and laminated composites in the case where the plane Π is in an arbitrary position are analyzed. The plane Π is formed by the points of a buckling mode that have equal phases relative to the line of compression. This theory follows from the linearized three-dimensional theory of stability of deformable bodies and is used to determine the critical compressive load and the associated position of the plane Π. Numerical examples are presented. A brief historical sketch is given     

On the Ill-Posedness of the Prandtl Equations in Three-Dimensional Space

In this paper, we give an instability criterion for the Prandtl equations in three-dimensional space, which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations, in contrast to the classical well-posedness theory of the two-dimensional Prandtl equations under the Oleinik monotonicity assumption. Both linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three-dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work (A well-posedness theory for the Prandtl equations in three space variables. arXiv:1405.5308, 2014) on the well-posedness theory for the three-dimensional Prandtl equations with a special structure.     

Plane Problem of Three-Dimensional Stability for a Hinged Plate with Two Symmetric End Cracks

The plane stability problem for a rectangular plate with two symmetric end cracks is solved in three-dimensional formulation. The three-dimensional linearized theory of stability and the finite-difference method are used. The effect of the crack parameter on the critical load is examined__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 47–52, April 2005.     

Three-dimensional theory of stability of a carbon nanotube in a matrix

The three-dimensional theory of stability of a carbon nanotube (CNT) in a polymer matrix is presented. The results are obtained on the basis of the three-dimensional linearized theory of stability of deformable bodies. Flexural and helical (torsional) buckling modes are considered. It is proved that the helical (torsional) buckling modes occur in a single CNT (the interaction of neighboring CNTs is neglected) and do not occur in nanocomposites (the interaction of neighboring CNTs is taken into account) __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 23–37, January 2006.     

用三维弹性力学方法研究圆板的稳定问题

用三维弹性力学方法研究任意边界条件圆板的轴对称稳定问题，利用Ｈ变换和Ｓｔｏｃｋｅｓ变换，导出位移函数及其偏导数的一种新型双重极数式，并由数学弹性定理论的基本方程和边界条件建立的特征方程，求得最小临界载荷的精确解，文末以简支圆板为例进行数字计算，结果表明：在弹性失稳范围内，三维弹性力学方法求得的临界载荷略低于经典理论的结果，对于薄板的弹性稳定问题，经典板理论有足够的精度。     

Stability of a Laminate Between Two Homogeneous Half-Spaces under Inelastic Deformation

The surface instability of inelastic laminated coatings is analyzed using a piecewise-homogeneous model and the three-dimensional linearized theory of stability. The general formulation of the problem is given, and the characteristic equations are derived. Specific problems for some viscoelastic and elastoplastic models are solved__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 22–31, May 2005.     

Three-dimensional stability of a rectangular plate under uniaxial tension

The paper studies the three-dimensional stability of an isotropic, linear elastic, rectangular plate under a uniform tensile load applied to its sides. The concept of free strains is used to reduce the three-dimensional problem to a two-dimensional one. It is solved using the three-dimensional linearized theory of stability. An approximate solution of the buckling problem is obtained by the finite-difference method. Numerical results are presented __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 116–123, July 2006.     

Three-dimensional stability theory of deformed bodies. Internal instability

Conclusions In studying internal instability effects for elastic (which is fully obvious) and elastoplastic models of deformable bodies the approximate approach [12, 15] in the three-dimensional stability theory leads to results which disagree quantitatively and qualitatively with the corresponding results of the three-dimensional linearzed stability theory of deformable bodies (the second variant of the theory of small subcritical deformations). In this connection, in studying internal instability effects for various models of deformable bodies, in which elastic or elastoplastic deformations are substantial, the use of this approximate approach is recommended.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 21, No. 11, pp. 3–17, November, 1985.     

Establishing the Fundamentals of the Theory of Stability of Mine Workings

The fundamentals of the theory of stability of mine workings are analyzed. The theory is based on the linearized three-dimensional theory of stability of deformable bodies. Some results on horizontal and vertical mine workings and on underground closed cavities are analyzed     

Stability of Hartmann flows

The stability of Hartmann flows for arbitrary magnetic Reynolds numbers is investigated in the framework of linear theory. The initial three-dimensional problem reduces to the equivalent two-dimensional problem. Perturbation theory is used to find asymptotic expressions for the eigenvalues. Distinguishing two types of disturbances — magnetic and hydrodynamic — is shown to be advantageous in a number of cases. Simple features of the stability are considered for particular cases. The well-know Lundquist result is generalized. An energy approach is applied to the problem of stability. The results of simulations involving the solution of the linear stability problem are described. A distinctive picture of stability is developed. There are several types of instability and they can develop simultaneously. The hydrodynamic and magnetic phenomena interact with each other in a very complex fashion. The magnetic field can either enhance flow stability or reduce it.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–31, November–December, 1972.     

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     

On the formation of linear folds in regularly layered rock masses under biaxial loading

The piecewise-homogeneous model and the three-dimensional theory of stability for small and finite subcritical strains are used to study the formation of folds in layered rock masses of the Earth’s crust under biaxial loading. The general statement of the problem is given, and the governing characteristic equations are derived. Numerical results are presented for a layered rock mass composed of two alternating layers __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 26–34, December 2005.     

Three-dimensional stability theory of deformable bodies. Stability of construction elements

Conclusion In [8, 9] and in the present paper we analyzed the possibilities of using the approximate approach [15, 18] in the three-dimensional stability theory of deformable bodies as applied to effects of internal and surface instability and to stability of thinwalled structural elements. The analysis mentioned has been performed by comparing for standard problems the results obtained by the approximate approach [15, 18] with the results for the similar problems, obtained within the three-dimensional linearized stability theory of deformable bodies (for example [2–5, 7, 10, 19]), constructed with the accuracy usually adopted in mechanics. The following conclusions are drawn as a result of the analysis.Applied to effects of internal and surface instability, the approximate approach leads to result in disagreement with the corresponding results of the three-dimensional linearized stability theory of deformable bodies.As applied to the study of stability of thin-walled structural elements, the use of the approximate approach is justified if we restrict ourselves to a calculational accuracy of critical loads corresponding to that of the Kirchhoff-Love hypothesis.In connection with the discussion above, numerous publications carried out on the basis of the approximate approach require further study to clarify the validity limits of the results obtained.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 22, No. 2, pp. 3–17, February, 1986.     

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 __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 64–72, July 2006.     

On One Two-Level Model in the Mesomechanics of Compression Fracture of Cracked Composites

The article is devoted to a two-level model in the mesomechanics of compression fracture of composites with cracks. Consideration is given to fiber-laminated panels with an opening compressed along the fibers so that two cracks propagate from the opening edge in a direction perpendicular to the loading axis. The previous results obtained by the author in the three-dimensional theory of stability of deformable bodies and nonclassical fracture mechanics are used to create the above-mentioned two-level model     

Mechanism of three-dimensional boundary-layer receptivity

Boundary-layer receptivity is always a hot issue in laminar-turbulent transition. Most actual laminar-turbulent transitions belong to three-dimensional flows. An infinite back-swept flat-plate boundary layer is a typical three-dimensional flow. Study of its receptivity is important both in theory and applications. In this paper, a freestream turbulence model is established. A modified fourth-order Runge-Kutta scheme is used for time marching, and compact finite difference schemes are used for space discretization. On these bases, whether unsteady cross-flow vortices can be excited in the three-dimensional boundary layer(the infinite back-swept flat-plate boundary layer) by free-stream turbulence is studied numerically. If so, effects of the level and the direction of free-stream turbulence on the three-dimensional boundary-layer receptivity are further studied. Differences of the three-dimensional boundary-layer receptivity are then discussed by considering the non-parallel effect, influence of the leading-edge stagnation point of the flat plate, and variation of the back-swept angle separately. Intensive studies on the three-dimensional boundary-layer receptivity will benefit the development of the hydrodynamic stability theory, and provide a theoretical basis for prediction and control of laminar-turbulent transition.     

Developing a Compressive Failure Theory for Nanocomposites

The paper addresses a compressive-failure theory for polymer-matrix nanocomposites in the case where failure onset is due to microbuckling. Two approaches based on the three-dimensional linearized theory of stability of deformable bodies are applied to laminated and fibrous nanocomposites. According to the first approach (continuum compressive-failure theory), nanocomposites are modeled by a homogeneous anisotropic medium with effective constants, including microstructural parameters. The second approach uses the piecewise-homogeneous model, three-dimensional relations for fibers (CNT) and matrix, and continuity conditions at the fiber-matrix interface. The compressive-failure theory is used to solve specific problems for laminated and fibrous nanocomposites. Some approximate failure theories based on the one- and two-dimensional applied theories of stability of rods, plates, and shells are analyzed__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 3–37, March 2005.     

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 __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 2, pp. 3–44, February 2007.