A mathematical model is proposed to describe the static deformation of a unidirectional fiber-reinforced composite under plane-strain conditions in a plane perpendicular to the fiber direction. The model is based on the stochastic static equations for an elastic homogeneous two-component medium with nonzero body forces. Applying the method of conditional moments and double Fourier transform yields a nonlocal model in the form of integro-differential equations. Expanding the integral kernels into series in expansion coefficients yields differential equations whose order depends on the number of terms in the series. The zero-order approximation leads to the theory of effective moduli, the first-order approximation to the refined theory of effective moduli, and the second-order approximation to fourth-order equilibrium equations for mean displacements (mathematical expectations) and formulas for displacements, strains, and stresses averaged over the composite and its components. All the coefficients in theses formulas are expressed in terms of the elastic constants of the components and the geometric parameters of the structure. The model is valid for heavy stress gradients. In a particular case, the static theory of two-component elastic mixtures follows from the model Introduction. The mechanics of composite materials has intensively been developed in recent decades. This is because of the increasingly widespread use of composites in various industries and creation of new materials and necessity of developing reliable methods for their design and strength prediction. Thus, fundamental research responded to practical needs. The basic model of the mechanics of composites is the equations governing the physicomechanical processes in structural components and their interaction. It should be emphasized that studies on the mechanics of composites were conducted at different structural levels, which is typical of fracture mechanics [1,[12][13][14]. Both macromechanics and micromechanics of composites employ various mathematical models.One of the best-known and well-developed theories in the mechanics of composites is the theory of effective moduli (continuum theory of the first order) [9]. It suggests modeling an inhomogeneous material by a homogeneous medium with effective characteristics. According to this theory, the macroscopic characteristics are stresses, strains, and displacements averaged over macrovolumes and macroareas and satisfying the classical equations of solid mechanics. The macrovolumes and macroareas must be much greater than the structural components. The microstructure of the material manifests itself as effective elastic moduli appearing in the macrostress-macrostrain relations and depending on the elastic moduli of the components and the geometry of the structure. In this case, it is necessary to determine the effective elastic moduli for a composite of given structure. The corresponding problem is formulated for an infinite body that is in a macroscopically homogeneous stress-strain state. As evide...
The paper presents the fundamental solutions established for the equilibrium equations written in terms of displacements in the gradient model describing the deformation of two‐component elastic composites with stochastic structure. In the paper 2D and 3D problems are constructed on the basis of the theory of generalized functions.
The gradient model of stochastically inhomogeneous media is used to study the stress concentration around a circular hole in a two-component elastic composite. The study is based on a general solution of the system of equilibrium equations expressed in terms of harmonic functions and functions that satisfy the Helmholtz equation. This solution is used to solve problems for an infinite plane with a circular hole under uniform and uniaxial tension. The results obtained are compared with the solutions found using the theory of effective moduli, which is simpler Introduction. Wide application of composite materials in various fields of modern engineering and construction calls for comprehensive analysis of problems associated with the selection of components and structure of materials, modeling and prediction of their mechanical behavior and properties, and calculation of the load-bearing capacity of structures. Such studies are based on certain mathematical models that allow for specific features of materials and describe the behavior of loads at the macro, micro, and nano-levels [1,3,5,10,11,16,17,19,20]. One of the first and the most developed and widely used continuum theories in the mechanics of composites is the theory of effective moduli, which deals with dynamic and kinematic parameters averaged over elementary macrovolumes and macroareas. Effective constants in this theory are determined in necessarily simple and feasible experimental programs or from some mathematical models. Either deterministic [2,12,13] or stochastic [15,18] approaches can be used. A shortage of this theory is that it disregards gradients of external loads at distances comparable with the dimensions of structural members.Mixture theory [6, 8] is more exact and free from this shortage. It deals with dynamic and kinematic parameters averaged over components of elementary macrovolumes and macroareas. However, here the number of constants relating the dynamic and kinematic parameters is much greater than in the theory of effective moduli, and there is a need to substantiate the equations of state and to correctly determine (experimentally and theoretically) the corresponding constants for composites. It should be noted that the concept of a mechanical two-component mixture applied to describe the mechanical behavior of coupled charges in a dielectric [9] makes it possible to set up coupled equations of electromagnetomechanics of dielectrics invariant to the Galilean transformations from which the Maxwell equations follow as a special case.One of the few constructive theories free from the shortcomings of the theory of effective moduli and mixture theory is the static gradient model [4,7] for two-component particulate and unidirectional fibrous composites of stochastic structure under static loads. The zero approximation of this model is the second-order equations of the theory of effective moduli, the first approximation is the refined equations of the theory of effective moduli, and the second approximation is the fourth-order equations for...
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