Differential equations of the thermoelastic state of shells under thermal impact on the surface

Section: Introductionmentioning

confidence: 99%

Stress state around a circular hole in a prestressed transversely isotropic spherical shell

Kondratenko

2008

“…The efficiency (rapid convergence and accuracy) of this approach is demonstrated by solving test problems for thick plates that can also be solved exactly or approximately by other methods. A numerical solution is obtained to the bending problem for orthotropic nonthin plates of constant and varying thickness Keywords: nonthin plate, constant and varying thickness, bending, numerical solutions, comparison of resultsIn searching for numerical solutions describing the elastic deformation and stability of thin plates and shells, finite-difference schemes are used to approximate the governing equations and relations of the theory of thin shells [3][4]. The discrete-orthogonalization method is successfully used to solve spatial problems for thick shells [8][9][10][11].…”

mentioning

confidence: 99%

“…In searching for numerical solutions describing the elastic deformation and stability of thin plates and shells, finite-difference schemes are used to approximate the governing equations and relations of the theory of thin shells [3][4]. The discrete-orthogonalization method is successfully used to solve spatial problems for thick shells [8][9][10][11].…”

mentioning

confidence: 99%

Numerical stress-strain analysis of a nonthin elastic plate

Chernopiskii

2006

A numerical solution to elastic-equilibrium problems for nonthin plates is proposed. The solution is obtained by using the curvilinear-mesh method in combination with Vekua's method. The efficiency (rapid convergence and accuracy) of this approach is demonstrated by solving test problems for thick plates that can also be solved exactly or approximately by other methods. A numerical solution is obtained to the bending problem for orthotropic nonthin plates of constant and varying thickness Keywords: nonthin plate, constant and varying thickness, bending, numerical solutions, comparison of resultsIn searching for numerical solutions describing the elastic deformation and stability of thin plates and shells, finite-difference schemes are used to approximate the governing equations and relations of the theory of thin shells [3][4]. The discrete-orthogonalization method is successfully used to solve spatial problems for thick shells [8][9][10][11]. Note that to determine the stress-strain state (SSS) in stress concentration zones (holes or abrupt changes in the geometry of the boundary surfaces) with adequate accuracy, it is necessary to use algebraic systems of equations or stiffness matrices with a great number of variables. Such systems as a rule converge weakly when solved by iteration. The reason is that finite-difference or finite-element schemes of approximating the differential relations of the theory of shells disregard rigid-body displacements of the displacement vector.The approach proposed in [4] made it possible to efficiently solve a number of problems for thin shells in view of rigid-body displacements. It was also used to find a numerical solution to the stress-strain problem for shells and to compare it with the exact solutions of test problems.Here we set forth an efficient numerical algorithm, based on the approach from [4], for solving the elastic problem for nonthin plates.

“…There are a number of methods to reduce three-dimensional problems to two-dimensional. Most popular of them are the method of hypotheses [2,14], the asymptotic method [10,15], and the method of series expansion in positive powers of the thickness coordinate [3,17,20] or in orthogonal functions such as Legendre polynomials [6,8,16,18,19].The present paper describes a method of reducing the three-dimensional problem of thermopiezoelectricity for nonthin anisotropic ceramic shells to a two-dimensional one by expanding the unknown functions into a Fourier-Legendre series in thickness coordinate. …”

mentioning

confidence: 99%

Thermopiezoelectric Equations for Nonthin Ceramic Shells

Khoma

2005

The paper presents a method for deriving the differential equations of thermopiezoelectricity for nonthin anisotropic ceramic shells of constant thickness. The method is based on expanding the unknown functions into Fourier-Legendre series in thickness coordinate Keywords: thermopiezoelectricity theory, anisotropy, nonthin ceramic shells of constant thickness Complete and precise information on the influence of temperature, electric, and other fields on the stress distribution in an elastic body can be obtained by solving the corresponding three-dimensional problems [1,5,7,9,12,[21][22][23]. For thin-walled structural members, however, such problems are very difficult to solve. Therefore, two-dimensional models of the theory of shells and plates become of necessity. There are a number of methods to reduce three-dimensional problems to two-dimensional. Most popular of them are the method of hypotheses [2,14], the asymptotic method [10,15], and the method of series expansion in positive powers of the thickness coordinate [3,17,20] or in orthogonal functions such as Legendre polynomials [6,8,16,18,19].The present paper describes a method of reducing the three-dimensional problem of thermopiezoelectricity for nonthin anisotropic ceramic shells to a two-dimensional one by expanding the unknown functions into a Fourier-Legendre series in thickness coordinate.