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...
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