scholarly journals Boundary Value Problems of Potential Functions in Decagonal Quasicrystals

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wu Li ◽  
Hao Xin ◽  
Tianyou Fan
Keyword(s):  
Boundary Value Problems ◽  
Edge Crack ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Potential Functions ◽  
Crack Model ◽  
Finite Width ◽  
Single Edge ◽  
Decagonal Quasicrystals ◽  
Crack Problems

A unified form of potential functions in decagonal quasicrystals (QCs) and conformal mappings are applied in a novel way to solve the boundary value problems emanating from the generalized theory of elasticity for decagonal QCs. By executing the reduction of boundary value problem to function equations, two crack problems are investigated. In the first one, an approximate analysis for bending specimen with a crack is given. In the other, a finite width strip with single edge crack of decagonal QCs is analytically estimated. Using the basic idea underlying Dugdale’s crack model, the extent of cohesive force zone in each of the two cases is analytically derived.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

scholarly journals Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies

2019 ◽  
Vol 24 (1) ◽  
pp. 33 ◽  
Author(s):  
Mikhail Nikabadze ◽  
Armine Ulukhanyan
Keyword(s):  
Boundary Value Problems ◽  
Anisotropic Medium ◽  
Eigenvalue Problems ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Theory Of Elasticity ◽  
Initial Boundary Value Problems ◽  
Special Cases ◽  
Initial Boundary Value

The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any orderand of any even rank is formulated, and also some of its special cases are considered. In particular,using the canonical presentation of the TBM of the tensor of elastic modules of the micropolartheory, in the canonical form the specific deformation energy and the constitutive relations arewritten. With the help of the introduced TBM operator, the equations of motion of a micropolararbitrarily anisotropic medium are written, and also the boundary conditions are written down bymeans of the introduced TBM operator of the stress and the couple stress vectors. The formulationsof initial-boundary value problems in these terms for an arbitrary anisotropic medium are given.The questions on the decomposition of initial-boundary value problems of elasticity and thin bodytheory for some anisotropic media are considered. In particular, the initial-boundary problems of themicropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators(tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transverselyisotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators(tensors–tensors) of the initial-boundary value problems are constructed that allow decomposinginitial-boundary value problems. We also find the determinant and the tensor of cofactors to the sumof six tensors used for decomposition of initial-boundary value problems. From three-dimensionaldecomposed initial-boundary value problems, the corresponding decomposed initial-boundary valueproblems for the theories of thin bodies are obtained.

Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials

2004 ◽  
Vol 57 (5) ◽  
pp. 325-343 ◽  
Author(s):  
Tian-You Fan ◽  
Yiu-Wing Mai
Keyword(s):  
Thermal Properties ◽  
Fracture Mechanics ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Higher Order ◽  
Classical Elasticity ◽  
Partial Differential ◽  
Crystalline Materials ◽  
Quasi Crystals

A review is given on the basic concepts and fundamental framework of the theory of elasticity for quasi-crystalline materials, including some 1D, 2D, and 3D quasi-crystals. The elasticity of quasi-crystals embodies some new concepts, field variables, and equations. It is much more complicated and beyond the scope of classical elasticity which holds only for conventional structural materials, including crystalline materials. Hence, some well-developed methods in classical elasticity cannot be directly applied to solve the problems of elasticity of quasi-crystalline materials. But the ideas of the classical theory of elasticity provide beneficial insight to treat this new subject. A decomposition and superposition procedure is suggested to simplify the elasticity problems of 1D and 2D quasi-crystals. Application of displacement and stress potentials further simplifies the problems. The large number of complicated equations involving elasticity is reduced to a single or a few partial differential equations of higher order by this technique. Also, efforts have been made to simplify the equations for 3D cubic quasi-crystals to a single partial differential equation of higher order. Simplification of the basic equations provides the possibility to solve boundary value or initial-boundary value problems of elasticity. For this purpose, some direct and systematic methods of mathematical physics and function theory are developed, and a series of analytic (classical) solutions, mainly for dislocations and cracks in materials, are derived. In addition, attention is drawn to those variational problems and generalized solutions (weak solutions) of boundary value problems and numerical implementation by the finite element method. The above may be seen as a development of the theory and methodology akin to those of classical elasticity. Based on the exact solutions of crack problems with different configurations under different motion states for different quasi-crystal systems, we put forward a framework of fracture mechanics of quasi-crystalline materials. This may be seen as an extension of the development of fracture mechanics for conventional structural materials. Also, some elastodynamic problems for some 1D and 2D quasi-crystals are studied, related results for dislocation and crack dynamics are found, and possible connections with certain thermal properties of quasi-crystalline materials, eg, specific heat and other thermo-dynamic functions, are discussed. There are 75 references cited in this review article.

scholarly journals Stability of mathematical models of the main problems of the anisotropic theory of elasticity

2020 ◽  
Vol 30 (1) ◽  
pp. 112-124
Author(s):  
A.V. Yudenkov ◽  
A.M. Volodchenkov
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Research Result ◽  
Theory Of Elasticity ◽  
Boundary Problems ◽  
Main Research ◽  
Contour Shape ◽  
Variable Function ◽  
Anisotropic Bodies

The boundary problems of the complex-variable function theory are effectively used while investigating equilibrium of homogeneous elastic mediums. The most complicated systems of the boundary value problems correspond to the case when an elastic body exhibits anisotropic properties. Anisotropy of the medium results in the drift of boundary conditions of the function that in general disrupts analyticity of the functions of interest. The paper studies systems of the boundary value problems with drift for analytic vectors corresponding to the primal elastic problems (first, second and mixed problems). Systems of analytic vectors with drift are reduced to equivalent systems of Hilbert boundary value problems for analytic functions with weak singularity integrators. The obtained general solution of the primal boundary value problems for the anisotropic theory of elasticity allows us to check the above problems for stability with respect to perturbations of boundary value conditions and contour shape. The research is relevant as there is necessity to apply approximate numerical methods to the boundary value problems with drift. The main research result comes to be a proof of stability of the systems of the vector boundary value problems with drift for analytic functions on the Hölder space corresponding to the primal problems of the elastic theory for anisotropic bodies in the case of change in the boundary value conditions and contour shape.

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