Investigation of elasticity problem for the radially inhomogeneous transversely isotropic sphere

Akhmedov, N.K.; Yusubova, Sevinj М.

doi:10.1002/mma.8360

Cited by 6 publications

(9 citation statements)

References 43 publications

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“…Polymer composite materials with diverse compositions are extensively employed in solving a wide range of practical problems, particularly in applications involving unique conditions such as high-temperature environments. These applications encompass various structures like high-temperature pipes, components within gas turbines, nuclear power plant elements and aviation components, among others (Bashirov & Ismailov, 2022;Piriev, 2023;Akhmedov & Yusubova, 2022;Hasanov et al, 2023;Akhmedov, 2021). Regrettably, many of these structures often lack proper consideration of their strength conditions in their design reports, which can lead to unexpected and premature failures during operation (Akhmedov & Yusubova, 2022;Hasanov et al, 2023;Akhmedov, 2021;Boiprav et al, 2023;Fionov et al, 2023;Vavilov, 2020).…”

Section: Introductionmentioning

confidence: 99%

“…These applications encompass various structures like high-temperature pipes, components within gas turbines, nuclear power plant elements and aviation components, among others (Bashirov & Ismailov, 2022;Piriev, 2023;Akhmedov & Yusubova, 2022;Hasanov et al, 2023;Akhmedov, 2021). Regrettably, many of these structures often lack proper consideration of their strength conditions in their design reports, which can lead to unexpected and premature failures during operation (Akhmedov & Yusubova, 2022;Hasanov et al, 2023;Akhmedov, 2021;Boiprav et al, 2023;Fionov et al, 2023;Vavilov, 2020). Hence, the operation of structures under unique conditions and utilizing novel polymer composite materials underscore the significance of assessing their long-term strength.…”

Section: Introductionmentioning

confidence: 99%

“…In the context of material defects, one type of deterioration is associated with random defects that accumulate over time, leading to cumulative damage. This form of deterioration is termed "scattered scattering" (Akhmedov & Yusubova, 2022;Hasanov et al, 2023;Akhmedov, 2021;Boiprav et al, 2023;Fionov et al, 2023). When stress is uniformly distributed, such as in tensile materials, damage increases consistently with volume.…”

Section: Introductionmentioning

confidence: 99%

“…The utilization of this approach has led to the development of numerous theories regarding the long-term degradation of materials, supported by experiments conducted by various researchers (Bashirov & Ismailov, 2022;Fionov et al, 2023;Chertishchev, 2020;Slavin et al, 2022). However, alongside their advantages, these theories also exhibit limitations, which have been examined in (Akhmedov & Yusubova, 2022;Fionov et al, 2023;Vavilov, 2020;Chulkov, 2023;Chertishchev, 2020;Slavin et al, 2022;Aamir et al, 2019;Ge et al, 2018;Shukla, 2019).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Development of a Piezoelectric Defect Detection Device Through Mathematical Modeling Applied to Polymer Composite Materials

Hasanov

2024

NMCA

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In this research article, the focus is on examining the strength characteristics of polymer composite materials under various stress states. The investigation into the material dissolution process is conducted in two distinct stages. Initially, the article employs a criterion approach to derive explicit formulas for the latent decay time (incubation period) of materials, considering singular, exponential and Abelian kernels of the damage operator. In the subsequent stage, the classical strength condition is applied to establish a Volterra type integral equation, which characterizes the damage process for hereditary materials. These resulting differential equations are then solved with appropriate initial conditions. The study utilizes the obtained results to create time-dependent graphs of the damage parameter and performs comparisons. A piezoelectric defect detection device for polymer composite materials has been proposed through the application of mathematical modeling.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Development of a Piezoelectric Defect Detection Device Through Mathematical Modeling Applied to Polymer Composite Materials

Hasanov

2024

NMCA

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“…Asymptotic formulas for displacements and stresses are obtained, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous sphere. In [16], an axisymmetric problem of elasticity theory for a radially inhomogeneous transversally isotropic sphere of small thickness was considered by the method of homogeneous solutions. Based on the asymptotic analysis carried out, three groups of solutions are obtained: a penetrating solution, a solution having the nature of an edge effect, and a solution having the nature of a boundary layer.…”

Section: Introductionmentioning

confidence: 99%

Construction of homogeneous solutions in the torsion problem for a transversally isotropic sphere with variable elastic moduli

Yusubova¹

2022

TAPR

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The object of research is the problem of torsion for a radially inhomogeneous transversally isotropic sphere and the study based on this three-dimensional stress-strain state. To establish the scope of applicability of existing applied theories and to create more refined applied theories of inhomogeneous shells, it is important to study the stress-strain state of inhomogeneous bodies based on three-dimensional equations of elasticity theory. The problem of torsion of a radially inhomogeneous transversally isotropic non-closed sphere containing none of the poles 0 and π is considered. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the lateral surface of the sphere is free from stresses, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The formulated boundary value problem is reduced to a spectral problem. After fulfilling the homogeneous boundary conditions specified on the side surfaces of the sphere, a characteristic equation is obtained with respect to the spectral parameter. The corresponding solutions are constructed depending on the roots of the characteristic equation. It is shown that the solution corresponding to the first group of roots is penetrating, and the stress state determined by this solution is equivalent to the torques of the stresses acting in an arbitrary section θ=const. The solutions corresponding to the countable set of the second group of roots have the character of a boundary layer localized in conic slices. In the case of significant anisotropy, some boundary layer solutions decay weakly and can cover the entire region occupied by the sphere. On the basis of the performed three-dimensional analysis, new classes of solutions (solutions having the character of a boundary layer) are obtained, which are absent in applied theories. In contrast to an isotropic radially inhomogeneous sphere, for a transversely isotropic radially inhomogeneous sphere, a weakly damped boundary layer solution appears, which can penetrate deep far from the conical sections and change the picture of the stress-strain state.

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Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

Akhmedov

2024

Z Angew Math Mech

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In this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

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Investigation of elasticity problem for the radially inhomogeneous transversely isotropic sphere

Cited by 6 publications

References 43 publications

The Development of a Piezoelectric Defect Detection Device Through Mathematical Modeling Applied to Polymer Composite Materials

The Development of a Piezoelectric Defect Detection Device Through Mathematical Modeling Applied to Polymer Composite Materials

Construction of homogeneous solutions in the torsion problem for a transversally isotropic sphere with variable elastic moduli

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

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