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

A solution is found to the two-dimensional buckling problem for a composite material reinforced with a periodic row of collinear short fibers and compressed along the fibers. The problem formulation is based on the piecewise-homogeneous model and the three-dimensional theory of stability of deformable bodies. The dependence of the critical strain and buckling mode on the fiber spacing is studied for various material and geometrical characteristics of the composite components Keywords: composite material, short fibers, three-dimensional theory of stability of deformable bodies, inhomogeneous stress-strain state, finite-difference method, critical strain, buckling modeIntroduction. Microbuckling is the basic mechanism of failure of fibrous and laminated composites under compression. It is obvious that this very complex phenomenon cannot be adequately described under two-dimensional applied theories of stability of thin-walled members (beams, plates, and shells)-it is expedient to apply the three-dimensional theory of stability of deformable bodies.It should be noted that the results to be discussed below have been obtained using the piecewise-homogeneous model and the three-dimensional linearized theory of stability of deformable bodies [1,6,7,9]. This formulation for composite micromechanics problems is currently the most rigorous and exact in solid mechanics. Solving such problems under the three-dimensional linearized theory of stability of deformable bodies is very difficult because of the inhomogeneity of the resulting (two-or three-dimensional) subcritical state and is only possible with up-to-date numerical methods.Of considerable interest is the question of whether the results presented here can be applied to study nanomaterials and nanocomposites. It should be noted that the term "nanocomposite" in nanomechanics seems to refer to two different types of nanomaterials. One type is nanocomposites consisting of a matrix and nanoparticles as reinforcement; such terminology is used in the micromechanics of polymer-or metal-matrix composites. Approaches and methods of composite micromechanics are applicable to composites for which the applicability conditions for continuum solid mechanics formulated in [13] are satisfied. The other type is nanocomposites in which each nanoparticle or nanoformation is a rather complex structure that may be considered a nanocomposite with its own microstructure. There is no material medium (analog of matrix in composite micromechanics) between particles (as in the case of nanotubes); and particles interact via interatomic forces. This interpretation of the term "nanocomposite" is ruled out in the micromechanics of polymer-or metal-matrix composites because it treats each element of the reinforcement as a solid body (solid mechanics is a division of continuum mechanics) and requires multilevel models to be developed under fracture mesomechanics, as shown in [8]. However, it seems to be preferred to describe a particle under continuum mechanics if it is an element of the microstructu...

Section: Solution Techniquementioning

confidence: 99%

Two-dimensional buckling problem for a composite reinforced with a periodic row of collinear short fibers

Декрет

2006

“…Let us now address, following [4,7,8], the questions of constructing base factors, setting up global discrete problems, and solving finite-difference equations.…”

Section: Finite-difference Equationsmentioning

confidence: 99%

Three-dimensional stability of a rectangular plate under uniaxial tension

Kokhanenko

2006

Self Cite

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

“…To design arbitrarily outlined plates, use is commonly made of complex-variable theory [10], the balancing-load method [3][4][5][6], the potential method [7,9,17], or various numerical methods [1,7,[13][14][15][16]. If Green's function is known, then a solution can be obtained by integrating over the mid-surface of the plate.…”

Section: The Bending Problem For An Arbitrarily Outlined Thin Plane Wmentioning

confidence: 99%

“…A technique based on the methods of potentials and balancing loads is proposed for constructing Green's function for the Germain-Lagrange equation. This technique ensures high accuracy of approximate solutions, which is checked against Levi's solution for rectangular platesTo design arbitrarily outlined plates, use is commonly made of complex-variable theory [10], the balancing-load method [3][4][5][6], the potential method [7,9,17], or various numerical methods [1,7,[13][14][15][16]. If Green's function is known, then a solution can be obtained by integrating over the mid-surface of the plate.…”

mentioning

confidence: 99%

Solving the elastic bending problem for a plate with mixed boundary conditions

Boborykin¹

2006

The bending problem for an arbitrarily outlined thin plane with mixed boundary conditions is solved. A technique based on the methods of potentials and balancing loads is proposed for constructing Green's function for the Germain-Lagrange equation. This technique ensures high accuracy of approximate solutions, which is checked against Levi's solution for rectangular platesTo design arbitrarily outlined plates, use is commonly made of complex-variable theory [10], the balancing-load method [3][4][5][6], the potential method [7,9,17], or various numerical methods [1,7,[13][14][15][16]. If Green's function is known, then a solution can be obtained by integrating over the mid-surface of the plate. Green's functions for arbitrarily outlined plates can conveniently be found using the potential method [12,18] with, as a rule, uniform boundary conditions throughout the entire boundary.The present paper is concerned with the bending of an arbitrarily outlined thin plate with mixed boundary conditions. The Kirchhoff and small-deflection hypotheses are acepted. We will outline a technique for constructing Green's function based on the methods of potentials and balancing loads. Auxiliary boundary conditions with unknown shear forces and bending moments will be formulated. The kernel of the potential representation of the solution of the biharmonic equation will be constructed as linear combinations of the fundamental solution and a special harmonic function. The boundary integral equations are of the first and second kinds and have logarithmic singularities. If the boundary has hinged and/or clamped sections, then additional boundary integral equations will appear. When numerically implemented, the method ensures high accuracy of approximate solutions, which is checked against Levi's solutions for rectangular plates. Being a variation of the method of boundary integral equations, the proposed technique reduces the dimension of the problem by one and, hence, the amount of computation compared to the finite-element method, which has currently been widely used to design plates with mixed boundary conditions. The present paper continues the study [12].