A comparative analysis is made of the infinite-fiber and finite-fiber models in the three-dimensional theory of stability of composites. The results analyzed have been obtained using the three-dimensional linearized theory of stability of deformable bodies. A historical sketch is given of the theory of stability for and approaches used in the mechanics of laminated and fibrous composite materials Keywords: laminated and fibrous composite materials, three-dimensional theory of stability, infinite-fiber model, finite-fiber model, historical sketch Introduction. For nearly half a century now, since the publication [18], researchers have been studying the microbuckling of composites and have been developing the fundamentals of the fracture mechanics of compressed composites assuming that failure begins with microbuckling. Depending on the relative dimensions of reinforcement (mean cross section and length), two essentially different models may be used: an infinite fiber and a finite fiber. The overwhelming majority of recent results have been obtained using the infinite-fiber model and analyzing periodic (along fibers) buckling modes. These results are reported in numerous papers in periodicals and in a number of reviews. Mathematically, the finite-fiber model is certainly more rigorous and general than the infinite-fiber model. In this connection, the finite-fiber model is of importance for the cases where fibers cannot be considered infinitely long.However, the use of the finite-fiber model involves mathematical and computational difficulties because the dimension of the problem cannot be reduced by separating periodic factors dependent on one coordinate reckoned along fibers.For consistency between the infinite-fiber and finite-fiber models, the following obvious conditions should be satisfied. The results predicted by the finite-fiber model should asymptotically tend to the results predicted by the infinite-fiber model as the fiber length tends to infinity.It is only recently that the finite-fiber model has been employed to obtain the results outlined in [7-14, 17, 34-37, etc.]. This makes it possible to compare the two models, which is done in the present paper.It what follows, we will use the terms "infinite-fiber model" and "finite-fiber model." When it comes to laminated composites, one should speak about infinitely long layers and finite-size layers. In this connection, it would be more logical to unite fibrous and laminated composites by using the terms "infinitely long reinforcing elements" and "finite-size reinforcing elements." It however appears that "infinite fiber" and "finite fiber" are more informative terms for experts in composite materials, which is why these terms will be applied below to both fibrous and laminated composites.1. Preliminary Discussion. Here we will discuss the issues in question, including a historical sketch on the studies being conducted and information on modeling fibrous materials in microbuckling studies.1.1. Historical Sketch. It is generally agreed in the modern scientifi...

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...

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