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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 Keywords: nanocomposites, CNT fiber, polymer matrix, compressive failure, microbuckling, three-dimensional linearized theory of stability of deformable bodiesIntroduction. Today's literature on the fracture mechanics of composites considers the paper [17] to be the first to describe, in 1960, fiber microbuckling as a compressive failure mechanism for unidirectional fibrous composites. In the years that followed, several authors set forth different approximate models for the quantitative and qualitative description of this mechanism. These models are based on a number of assumptions and hypotheses, of which the following are worth mentioning: no (neglected) subcritical stresses in the matrix, use of applied one-and two-dimensional theories of stability of rods and plates to study microbuckling, modeling of the matrix by a one-dimensional elastic object, etc.The paper [14] was apparently the first to propose, in 1965, a highly approximate model for the quantitative description of microbuckling in composites within the framework of a plane problem (in fact, the fibrous composite was modeled by a laminated composite), using the above-mentioned assumptions and hypotheses (the same results were reported in [14]). Despite the highly approximate model, the results from [14] were used in many publications, including the seven-volume collective monograph [13], and are generally recognized and widely cited. In the literature on the fracture mechanics of composites (see, e.g., [15]), these results are named the Dow-Gruntfest-Rosen-Schuerch theory, after the authors of the first publications [14,17,45,46].Thus, to describe the failure mechanism in question [17], we need a stability theory for unidirectional fibrous composites (Fig. 1) or laminated composites (Fig. 2) subjected to axial compression. The paper [17] addressed a fibrous composite; thus, it is expedient to construct a stability theory for the material represented in Fig. 1. In this connection, a stability theory for laminated composites (Fig. 2) is to be developed to attain two ends (two different cases). In the first case, a stability theory is needed to describe the f...

The paper proposes a basic approach to study the mechanical properties of nanocomposite materials with polymer matrix and the deformation of nanocomposites and structural members made of them. The notions of homogenization and continualization are discussed with reference to nanocomposite materials. Four main tasks for nanocomposite mechanics are defined: (i) description of the properties of nanoformations, (ii) description of the properties of the matrix (binder), (iii) description of phenomena at the matrix-nanoformations interfaces, and (iv) determination of the effective properties of nanocomposites to change over to the mechanics of structural members. Particular attention is given to the interface conditions between the matrix and reinforcement. The role of lower and upper bound estimates is pointed out. The basic models of linear or nonlinear micro-and nanocomposites are considered. These models are used in a numerical analysis. The analysis makes it possible to observe and describe the peculiarities of the processes of fracture, deformation, and wave propagation in nanocomposite materials with polymer matrix. The numerical results are presented in the form of plots Keywords: basic approaches in the mechanics of nanomaterials, relationship between models of micro-and nanomechanics, fiber-reinforced polymer-matrix composites, continuum models, mechanical properties, plane wave propagation 1. On the Evolution of the Mechanics of Polymer-Matrix Nanocomposites. The changeover from the microlevel to the nanolevel in the science of materials broke new ground in the mechanics of materials and structures. This was preceded by important events in the physics and chemistry of nanoformations, which were partially described in [28] and will be briefly outlined below.The scanning atomic-force microscope and scanning tunneling microscope, which allow seeing atoms, were created relatively recently [44]. They have changed nanotechnology considerably: it went from futurological sciences [21] to modern natural sciences (primarily physics and chemistry) [16,20,34,44]. Over the past twenty years, more and more researchers in various fields of natural science have addressed themselves to the research of nanoscale objects. The interest in such objects is practical (unfortunately, practicality is becoming the prevailing philosophy of the scientific community): a new knowledge of nanoscale objects can quickly be implemented in a variety of technologies, which is why such a research is said to pertain to nanotechnologies. The world's leading countries are prioritizing the development of nanotechnologies and have lavishly financed special nanotechnology programs. Most scientific results regarding nanoscale objects have been obtained by physicists, chemists, and materials scientists.As a separate division partially concerned with continuum physics and materials science, the mechanics of materials and structures has quite recently become involved in the analysis of nanoformations with the anticipated properties of deformable material...