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Journal articles on the topic 'Elasticity - Cauchy's Problem'

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Published: 4 June 2021
Last updated: 6 September 2023

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1

Makhmudov, O. I., and I. E. Niyozov. "Cauchy Problem for Dynamic Elasticity Equations." Differential Equations 56, no. 9 (September 2020): 1130–39. http://dx.doi.org/10.1134/s0012266120090037.

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2

Ang, Dang Dinh, and Nguyen Dung. "A Cauchy like problem in plane elasticity." Vietnam Journal of Mechanics 29, no. 3 (September 22, 2007): 245–48. http://dx.doi.org/10.15625/0866-7136/29/3/5536.

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Let \(\Omega \) be a bounded domain in the plane, representing an elastic body. Let \(\Gamma_0 \) be a portion of the boundary \(\Gamma\) of \(\Omega \), \(\Gamma_0 \) being assumed to be paralled to the \(x\)-axis. It is proposed to determine the stress field in \(\Omega \) from the displacements and surface stresses given on \(\Gamma_0 \). Under the assumption of plane stress, it is shown that \(u_x + u_y\) is a harmonic function. An Airy stress function is introduced, from which the stress field is computed.
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3

Koenemann, Falk H. "Unorthodox Thoughts about Deformation, Elasticity, and Stress." Zeitschrift für Naturforschung A 56, no. 12 (December 1, 2001): 794–808. http://dx.doi.org/10.1515/zna-2001-1202.

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Abstract The nature of elastic deformation is examined in the light of the potential theory. The concepts and mathematical treatment of elasticity and the choice of equilibrium conditions are adopted from the mechanics of discrete bodies, e. g., celestial mechanics; they are not applicable to a change of state. By nature, elastic deformation is energetically a Poisson problem since the buildup of an elastic potential implies a change of the energetic state in the sense of thermodynamics. In the Euler-Cauchy theory, elasticity is treated as a Laplace problem, implying that no change of state occurs, and there is no clue in the Euler-Cauchy approach that it was ever considered as one. The Euler-Cauchy theory of stress is incompatible with the potential theory and with the nature of the problem; it is therefore wrong. The key point in the understanding of elasticity is the elastic potential.
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4

Nezam, Ali. "Numerical Results of Sub-Cauchy Problem for Linear Elasticity." International Journal for Research in Applied Science and Engineering Technology 7, no. 6 (June 30, 2019): 1693–99. http://dx.doi.org/10.22214/ijraset.2019.6284.

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5

Makhmudov, O. I., and I. E. Niezov. "A cauchy problem for the system of elasticity equations." Differential Equations 36, no. 5 (May 2000): 749–54. http://dx.doi.org/10.1007/bf02754234.

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6

Marin, L., L. Elliott, D. B. Ingham, and D. Lesnic. "Boundary element method for the Cauchy problem in linear elasticity." Engineering Analysis with Boundary Elements 25, no. 9 (October 2001): 783–93. http://dx.doi.org/10.1016/s0955-7997(01)00062-5.

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7

Baranger, Thouraya N., and Stéphane Andrieux. "An optimization approach for the Cauchy problem in linear elasticity." Structural and Multidisciplinary Optimization 35, no. 2 (May 3, 2007): 141–52. http://dx.doi.org/10.1007/s00158-007-0123-5.

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8

Cheng, Jin, Victor Isakov, Masahiro Yamamoto, and Qi Zhou. "Lipschitz stability in the lateral Cauchy problem for elasticity system." Journal of Mathematics of Kyoto University 43, no. 3 (2003): 475–501. http://dx.doi.org/10.1215/kjm/1250283691.

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9

Joumaa, Hady, and Martin Ostoja-Starzewski. "Stress and couple-stress invariance in non-centrosymmetric micropolar planar elasticity." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2134 (May 11, 2011): 2896–911. http://dx.doi.org/10.1098/rspa.2010.0660.

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The stress-invariance problem for a chiral (non-centrosymmetric) micropolar material model is explored in two different planar problems: the in-plane and the anti-plane problems. This material model grasps direct coupling between the Cauchy-type and Cosserat-type (or micropolar) effects in Hooke's law. An identical strategy of invariance is set for both problems, leading to a remarkable similarity in their results. For both problems, the planar components of stress and couple-stress undergo strong invariance, while their out-of-plane counterparts can only attain weak invariance, which restricts all compliance moduli transformations to a linear type. As an application, when heterogeneous (composite) materials are subjected to weak invariance, their effective (volume-averaged) compliance moduli undergo the same linear shift as that of the moduli of the local phases forming the material, independently of the microstructure, geometry and phase distribution. These analytical results constitute a valuable means to validate computational procedures that handle this particular type of material model.
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10

Niyozov, I. E., and O. I. Makhmudov. "The Cauchy problem of the moment elasticity theory in R m." Russian Mathematics 58, no. 2 (February 2014): 24–30. http://dx.doi.org/10.3103/s1066369x14020042.

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11

Marin, L. "Conjugate Gradient-Boundary Element Method for the Cauchy Problem in Elasticity." Quarterly Journal of Mechanics and Applied Mathematics 55, no. 2 (May 1, 2002): 227–47. http://dx.doi.org/10.1093/qjmam/55.2.227.

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12

Comino, Lucia, Liviu Marin, and Rafael Gallego. "An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity." Engineering Analysis with Boundary Elements 31, no. 8 (August 2007): 667–82. http://dx.doi.org/10.1016/j.enganabound.2006.12.009.

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13

Marin, L. "Boundary element-Landweber method for the Cauchy problem in linear elasticity." IMA Journal of Applied Mathematics 70, no. 2 (December 16, 2004): 323–40. http://dx.doi.org/10.1093/imamat/hxh034.

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14

Lazar, Markus. "Incompatible strain gradient elasticity of Mindlin type: screw and edge dislocations." Acta Mechanica 232, no. 9 (June 28, 2021): 3471–94. http://dx.doi.org/10.1007/s00707-021-02999-2.

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AbstractThe fundamental problem of dislocations in incompatible isotropic strain gradient elasticity theory of Mindlin type, unsolved for more than half a century, is solved in this work. Incompatible strain gradient elasticity of Mindlin type is the generalization of Mindlin’s compatible strain gradient elasticity including plastic fields providing in this way a proper eigenstrain framework for the study of defects like dislocations. Exact analytical solutions for the displacement fields, elastic distortions, Cauchy stresses, plastic distortions and dislocation densities of screw and edge dislocations are derived. For the numerical analysis of the dislocation fields, elastic constants and gradient elastic constants have been used taken from ab initio DFT calculations. The displacement, elastic distortion, plastic distortion and Cauchy stress fields of screw and edge dislocations are non-singular, finite, and smooth. The dislocation fields of a screw dislocation depend on one characteristic length, whereas the dislocation fields of an edge dislocation depend on up to three characteristic lengths. For a screw dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type agree with the corresponding ones in simplified incompatible strain gradient elasticity. In the case of an edge dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type are depicted more realistic than the corresponding ones in simplified incompatible strain gradient elasticity. Among others, the Cauchy stress of an edge dislocation obtained in incompatible isotropic strain gradient elasticity of Mindlin type looks more physical in the dislocation core region than the Cauchy stress obtained in simplified incompatible strain gradient elasticity and is in good agreement with the stress fields of an edge dislocation computed in atomistic simulations. Moreover, it is shown that the shape of the dislocation core of an edge dislocation has a more realistic asymmetric form due to its inherent asymmetry in incompatible isotropic strain gradient elasticity of Mindlin type than the dislocation core possessing a cylindrical symmetry in simplified incompatible strain gradient elasticity. It is revealed that the considered theory with the incorporation of three characteristic lengths offers a more realistic description of an edge dislocation than the simplified incompatible strain gradient elasticity with only one characteristic length.
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15

Marin, L., L. Elliott, D. B. Ingham, and D. Lesnic. "Boundary Element Regularisation Methods for Solving the Cauchy Problem in Linear Elasticity." Inverse Problems in Engineering 10, no. 4 (January 2002): 335–57. http://dx.doi.org/10.1080/1068276021000004698.

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16

Bilotta, Antonio, and Emilio Turco. "A numerical study on the solution of the Cauchy problem in elasticity." International Journal of Solids and Structures 46, no. 25-26 (December 2009): 4451–77. http://dx.doi.org/10.1016/j.ijsolstr.2009.09.006.

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17

Yarmukhamedov, Sh Ya, T. I. Ishankulov, and O. I. Makhmudov. "Cauchy problem for a system of equations of three-dimensional elasticity theory." Siberian Mathematical Journal 33, no. 1 (1992): 154–58. http://dx.doi.org/10.1007/bf00972948.

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18

Yin, Jingjing, Huaijian Li, Lingdi Kong, Junwei Cheng, Tonghui Niu, Xiaonan Wang, and Peizhi Zhuang. "Analytical Solution for Elastic Analysis around an Ellipse with Displacement-Controlled Boundary." Mathematical Problems in Engineering 2022 (June 8, 2022): 1–11. http://dx.doi.org/10.1155/2022/2148947.

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This paper presents novel analytical solutions for the analysis of an elliptical cavity within an infinite plane under plane strain conditions, considering typical displacement-controlled boundaries at the inner cavity and biaxial stresses at infinity. The problem is investigated by the plane theory of elasticity using Muskhelishvili’s complex variable method. The complex displacement boundary conditions are represented using the conformal mapping technique and Fourier series, and stress functions are evaluated using Cauchy’s integral formula. The proposed solutions are validated at first by comparing them with other existing solutions and then used to show the influences of displacement vectors on the distributions of induced stresses and displacements. The new solutions may provide useful analytical tools for stress and displacement analysis of an elliptical hole/opening in linear elastic materials which are common in many engineering problems.
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19

Durand, Bastien, Franck Delvare, and Patrice Bailly. "Numerical solution of Cauchy problems in linear elasticity in axisymmetric situations." International Journal of Solids and Structures 48, no. 21 (October 2011): 3041–53. http://dx.doi.org/10.1016/j.ijsolstr.2011.06.017.

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20

Valerii, Mirenkov. "Hydraulic fracture in the nonlinear stress field." Izvestiya vysshikh uchebnykh zavedenii Gornyi zhurnal, no. 4 (June 25, 2020): 12–20. http://dx.doi.org/10.21440/0536-1028-2020-4-12-20.

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Introduction. The article considers a variant of a straight finite fracture modeled by a mathematical cut in the elastic plane. Aim. The new model proposed differs from the existing models by the damage zone bounded by the elastic material at the fracture tip up to the moment of the fracture growth. The process of fracturing is essentially nonlinear. Methodology. The model is based on the full-scale tension experiments with a reference sample of rocks enclosing a fracture and having the characteristic stress points, namely, proportionality limit, elasticity limit, plasticity domain and the domain in the vicinity of destructive stresses. Results. The problem with fracture is considered as an experiment to determine deformation with growing pressure in the fracture. The problem has no correct analytical solution. The problem on hydrofracture 20 "Izvestiya vysshikh uchebnykh zavedenii. Gornyi zhurnal". No. 4. 2020 ISSN 0536-1028 assumes the presence of the initial stress field in rock mass, which is essentially used in formulation of boundary conditions. Conclusions. All such problems belong to the class of Cauchy’s problems with an infinitely distant point in the computational domain. This article proposes the correct formulation of the fracture theory problem in the static, kinematic and dynamic framework.
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21

Cheng, Pan, and Ling Zhang. "Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity." Mathematical Problems in Engineering 2018 (May 31, 2018): 1–8. http://dx.doi.org/10.1155/2018/6932164.

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This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods (MQMs) are presented to solve the nonlinear equations where the accuracy of the solutions is of three orders. According to the asymptotical compact convergence theory, the errors with odd powers asymptotic expansion are obtained. Following the asymptotic expansion, the accuracy of the solutions can be improved to five orders with the Richardson extrapolation. Some results are shown regarding these approximations for problems by the numerical example.
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22

Yeih, Weichung, Tatsuhito Koya, and Toshio Mura. "An Inverse Problem in Elasticity With Partially Overprescribed Boundary Conditions, Part I: Theoretical Approach." Journal of Applied Mechanics 60, no. 3 (September 1, 1993): 595–600. http://dx.doi.org/10.1115/1.2900845.

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A Cauchy problem in linear elasticity is considered. This problem is governed by a Fredholm integral equation of the first kind and cannot be solved directly. The regularization method, which has been originally employed by Gao and Mura (1989), is formulated from a different perspective in order to address some of the difficulties experienced in their formulation. The theoretical details are discussed in this paper. Numerical examples are treated to Part II.
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23

MICHELITSCH, THOMAS M., GÉRARD A. MAUGIN, MUJIBUR RAHMAN, SHAHRAM DEROGAR, ANDRZEJ F. NOWAKOWSKI, and FRANCK C. G. A. NICOLLEAU. "A continuum theory for one-dimensional self-similar elasticity and applications to wave propagation and diffusion." European Journal of Applied Mathematics 23, no. 6 (August 16, 2012): 709–35. http://dx.doi.org/10.1017/s095679251200023x.

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We analyse some fundamental problems of linear elasticity in one-dimensional (1D) continua where the material points of the medium interact in a self-similar manner. This continuum with ‘self-similar’ elastic properties is obtained as the continuum limit of a linear chain with self-similar harmonic interactions (harmonic springs) which was introduced in [19] and (Michelitsch T.M. (2011) The self-similar field and its application to a diffusion problem. J. Phys. A Math. Theor.44, 465206). We deduce a continuous field approach where the self-similar elasticity is reflected by self-similar Laplacian-generating equations of motion which are spatially non-local convolutions with power-function kernels (fractional integrals). We obtain closed-form expressions for the static displacement Green's function due to a unit δ-force. In the dynamic framework we derive the solution of the Cauchy problem and the retarded Green's function. We deduce the distributions of a self-similar variant of diffusion problem with Lévi-stable distributions as solutions with infinite mean fluctuations. In both dynamic cases we obtain a hierarchy of solutions for the self-similar Poisson's equation, which we call ‘self-similar potentials’. These non-local singular potentials are in a sense self-similar analogues to Newtonian potentials and to the 1D Dirac's δ-function. The approach can be a point of departure for a theory of self-similar elasticity in 2D and 3D and for other field theories (e.g. in electrodynamics) of systems with scale invariant interactions.
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24

Froiio, Francesco, and Antonis Zervos. "Second-grade elasticity revisited." Mathematics and Mechanics of Solids 24, no. 3 (April 24, 2018): 748–77. http://dx.doi.org/10.1177/1081286518754616.

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We present a compact, linearized theory for the quasi-static deformation of elastic materials whose stored energy depends on the first two gradients of the displacement (second-grade elastic materials). The theory targets two main issues: (1) the mechanical interpretation of the boundary conditions and (2) the analytical form and physical interpretation of the relevant stress fields in the sense of Cauchy. Since the pioneering works of Toupin and Mindlin et al. in the 1960’s, a major difficulty has been the lack of a convincing mechanical interpretation of the boundary conditions, causing second-grade theories to be viewed as ‘perturbations’ of constitutive laws for simple (first-grade) materials. The first main contribution of this work is the provision of such an interpretation based on the concept of ortho-fiber. This approach enables us to circumvent some difficulties of a well-known ‘reduction’ of second-grade materials to continua with microstructure (in the sense of Mindlin) with internal constraints. A second main contribution is the deduction of the form of the linear and angular-momentum balance laws, and related stress fields in the sense of Cauchy, as they should appear in a consistent Newtonian formulation. The viewpoint expressed in this work is substantially different from the one in a well known and influential paper by Mindlin and Eshel in 1968, while affinities can be found with recent studies by dell’Isola et al. The merits of the new formulation and the associated numerical approach are demonstrated by stating and solving three example boundary value problems in isotropic elasticity. A general finite element discretization of the governing equations is presented, using C1-continuous interpolation, while the numerical results show excellent convergence even for relatively coarse meshes.
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25

Marin, L., L. Elliott, D. B. Ingham, and D. Lesnic. "An iterative boundary element algorithm for a singular Cauchy problem in linear elasticity." Computational Mechanics 28, no. 6 (June 1, 2002): 479–88. http://dx.doi.org/10.1007/s00466-002-0313-3.

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26

Isakov, Victor. "A nonhyperbolic cauchy problem for □b□c and its applications to elasticity theory." Communications on Pure and Applied Mathematics 39, no. 6 (November 1986): 747–67. http://dx.doi.org/10.1002/cpa.3160390603.

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27

Sagindykov, Bimurat. "Analytical functions of generalized complex variables and some appllications." INTERNATIONAL JOURNAL OF RESEARCH IN EDUCATION METHODOLOGY 5, no. 1 (April 30, 2014): 569–75. http://dx.doi.org/10.24297/ijrem.v5i1.3922.

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The object of this work is to use functions of a generalized complex variable to solve the problems of fluid dynamics and elasticity theory. In this paper, we obtain Cauchy-Riemann conditions, generalized Laplace equation and the generalized Poisson formula for such functions.
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28

Mousavi, S. Mahmoud, Juha Paavola, and Djebar Baroudi. "Distributed non-singular dislocation technique for cracks in strain gradient elasticity." Journal of the Mechanical Behavior of Materials 23, no. 3-4 (August 1, 2014): 47–58. http://dx.doi.org/10.1515/jmbm-2014-0007.

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AbstractThe mode III fracture analysis of a cracked graded plane in the framework of classical, first strain gradient, and second strain gradient elasticity is presented in this paper. Solutions to the problem of screw dislocation in graded materials are available in the literature. These solutions include various frameworks such as classical elasticity, and the first strain and second strain gradient elasticity theories. One of the applications of dislocations is the analysis of a cracked medium through distributed dislocation technique. In this article, this technique is used for the mode III fracture analysis of a graded medium in classical elasticity, which results in a system of Cauchy singular integral equations for multiple interacting cracks. Furthermore, the technique is modified for gradient elasticity. Owing to the regularization of the classical singularity, a system of non-singular integral equations is obtained in gradient elasticity. A plane with one crack is studied, and the stress distribution in classical elasticity is compared with those in gradient elasticity theories. The effects of the internal lengths, introduced in gradient elasticity theories, are investigated. Additionally, a plane with two cracks is studied to elaborate the interactions of multiple cracks in both the classical and gradient theories.
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29

Uskov, Vladimir, and Arina Panteleeva. "PROPERTIES OF A CERTAIN LOADED INTEGRAL OPERATOR WITH WEIGHT." Voronezh Scientific-Technical Bulletin 4, no. 4 (January 25, 2021): 4–10. http://dx.doi.org/10.34220/2311-8873-2021-4-4-4-10.

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In this paper, we consider a certain integral operator with a weight, loaded with operator term. It acts in the space of continuous functions. The conditions under which this operator is limited are determined, the form of its semigroup is es-tablished. The Cauchy problem for an integro-differential equation is considered as an application. Such equations arise in the theory of elasticity and models of biological processes: Proctor's problem on the equilibrium of an elastic beam, Volterra's problem on torsional vibrations, Prandtl's problem for calculating an airplane wing, in analysis of economic models, etc.
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30

Marin, Liviu. "The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity." International Journal of Solids and Structures 46, no. 5 (March 2009): 957–74. http://dx.doi.org/10.1016/j.ijsolstr.2008.10.004.

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31

Ellabib, Abdellatif, Abdeljalil Nachaoui, and Abdessamad Ousaadane. "Mathematical analysis and simulation of fixed point formulation of Cauchy problem in linear elasticity." Mathematics and Computers in Simulation 187 (September 2021): 231–47. http://dx.doi.org/10.1016/j.matcom.2021.02.020.

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32

Faverjon, B., B. Puig, and T. N. Baranger. "Identification of boundary conditions by solving Cauchy problem in linear elasticity with material uncertainties." Computers & Mathematics with Applications 73, no. 3 (February 2017): 494–504. http://dx.doi.org/10.1016/j.camwa.2016.12.011.

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33

Marin, Liviu, and Daniel Lesnic. "The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity." International Journal of Solids and Structures 41, no. 13 (June 2004): 3425–38. http://dx.doi.org/10.1016/j.ijsolstr.2004.02.009.

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34

Marin, L., and D. Lesnic. "Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition." Computer Methods in Applied Mechanics and Engineering 191, no. 29-30 (May 2002): 3257–70. http://dx.doi.org/10.1016/s0045-7825(02)00262-1.

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35

Delvare, Franck, Alain Cimetière, Jean-Luc Hanus, and Patrice Bailly. "An iterative method for the Cauchy problem in linear elasticity with fading regularization effect." Computer Methods in Applied Mechanics and Engineering 199, no. 49-52 (December 2010): 3336–44. http://dx.doi.org/10.1016/j.cma.2010.07.004.

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36

Ahmad, B., A. Alsaedi, and M. Kirane. "Nonexistence results for the Cauchy problem of time fractional nonlinear systems of thermo-elasticity." Mathematical Methods in the Applied Sciences 40, no. 12 (January 16, 2017): 4272–79. http://dx.doi.org/10.1002/mma.4303.

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37

Zemlyanukhin, Aleksandr, and Andrey Bochkarev. "Exact Solutions and Numerical Simulation of the Discrete Sawada–Kotera Equation." Symmetry 12, no. 1 (January 9, 2020): 131. http://dx.doi.org/10.3390/sym12010131.

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We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi elliptic function, a class of new exact periodic solutions is constructed, in particular stationary ones. Using an exponential generating function for Catalan numbers, Cauchy’s problem with the initial condition in the form of a step is solved. As a result of numerical simulation, the elasticity of the interaction of exact localized solutions is established.
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38

Amel’kin, V. V., M. N. Vasilevich, and L. A. Khvostchinskaya. "On one approach to the solution of miscellaneous problems of the theory of elasticity." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 57, no. 3 (October 7, 2021): 263–73. http://dx.doi.org/10.29235/1561-2430-2021-57-3-263-273.

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Herein, a miscellaneous contact problem of the theory of elasticity in the upper half-plane is considered. The boundary is a real semi-axis separated into four parts, on each of which the boundary conditions are set for the real or imaginary part of two desired analytical functions. Using new unknown functions, the problem is reduced to an inhomogeneous Riemann boundary value problem with a piecewise constant 2 × 2 matrix and four singular points. A differential equation of the Fuchs class with four singular points is constructed, the residue matrices of which are found by the logarithm method of the product of matrices. The single solution of the problem is represented in terms of Cauchy-type integrals when the solvability condition is met.
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39

Méjri, Bochra. "Shape sensitivity analysis for identification of voids under Navier’s boundary conditions in linear elasticity." Journal of Inverse and Ill-posed Problems 27, no. 3 (June 1, 2019): 385–400. http://dx.doi.org/10.1515/jiip-2018-0029.

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Abstract This work is devoted to the study of the void identification problem from partially overdetermined boundary data in the 2D-elastostatic case. In a first part, a shape identifiability result from a Cauchy data is presented, i.e. with traction field and boundary displacement as measurements. Then this geometric inverse problem is tackled by the minimization of two cost functionals, an energy gap functional and an {L^{2}} -gap functional, which enable the reconstruction of voids under Navier’s boundary conditions. The shape derivatives of these cost functionals are computed for the purpose of sensitivity analysis.
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40

Galybin, AN, and GA Rogerson. "An ill-posed Cauchy type problem for an elastic strip." Mathematics and Mechanics of Solids 24, no. 9 (February 2019): 2986–98. http://dx.doi.org/10.1177/1081286519826395.

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This study deals with an incorrectly posed, plane elasticity, boundary value problem for a strip. The strip is loaded by a concentrated load of known intensity applied to one side and the displacements on this side are also known. The problem is therefore over-determined on one side of the boundary; in contrast no boundary conditions are specified on the other side of the strip. Therefore, the problem is ill-posed with the specified boundary conditions. The problem can be reduced to a system of integral equations derived from basic properties of holomorphic functions, which are used to prove uniqueness of the considered boundary value problem. An analytical solution of the problem is obtained by applying Fourier transforms. The inversion of the Fourier transform is performed with the use of the Stieltjes integral. This is a non-stable operation, which necessitates the application of a regularisation technique in order to build stable solutions. For numerical implementation we discuss the regularisation procedure based on the singular value decomposition truncation method.
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41

Demirkan, Erol, Murat Çelik, and Reha Artan. "Slope Deflection Method in Nonlocal Axially Functionally Graded Tapered Beams." Applied Sciences 13, no. 8 (April 11, 2023): 4814. http://dx.doi.org/10.3390/app13084814.

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In this study, the slope deflection method was presented for structures made of small-scaled axially functionally graded beams with a variable cross section within the scope of nonlocal elasticity theory. The small-scale effect between individual atoms cannot be neglected when the structures are small in size. Therefore, the theory of nonlocal elasticity is used throughout. The stiffness coefficients and fixed-end moments are calculated using the method of initial values. With this method, the solution of the differential equation system is reduced to the solution of the linear equation system. The given transfer matrix is unique and the problem can be easily solved for any end condition and loading. In this problem, double integrals occur in terms of the transfer matrix. However, this form is not suitable for numerical calculations. With the help of Cauchy’s repeated integration formula, the transfer matrix is given in terms of single integrals. The analytical or numerical calculation of single integrals is easier than the numerical or analytical calculation of double integrals. It is demonstrated that the nonlocal effect plays an important role in the fixed-end moments of small-scaled beams.
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42

Tavakoli, Mohamad, and Ali Reza Fotuhi. "Anti-plane stress analysis of a half-plane with multiple cracks by distributed dislocation technique in nonlocal elasticity." Mathematics and Mechanics of Solids 24, no. 5 (October 13, 2018): 1567–77. http://dx.doi.org/10.1177/1081286518802330.

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The effective role of a distributed dislocation technique accompanied by a nonlocal elasticity model has been demonstrated for the crack problem in a half-plane. The dislocation solution is employed to model and analyze the anti-plane crack problem for nonlocal elasticity using the distributed dislocation technique. The solution of dislocation in the half-plane has been extracted through the solution of dislocation in an infinite plane by the image method. The dislocation solution has been utilized to formulate integral equations for dislocation density functions on the surface of a smooth crack embedded in the half-plane under anti-plane loads. The integral equations are of the Cauchy singular type, and have been solved numerically. Multiple cracks with different configurations have been solved; results demonstrate that the nonlocal theory predicts a certain stress value in the crack tip.
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43

Chraptovič, Ela, and Juozas Atkočiūnas. "ROLE OF KUHN-TUCKER CONDITIONS IN ELASTICITY EQUATIONS IN TERMS OF STRESSES." JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT 6, no. 2 (April 30, 2000): 104–12. http://dx.doi.org/10.3846/13921525.2000.10531573.

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Solution of the elasticity problem in terms of stresses leads to the stress vector six components, satisfying the Beltrami compatibility eqns and boundary conditions, evaluation. A direct integration of the nine differential eqns system in respect of the six stress components is difficult to realise practically. This is the reason why often the Casigliano variation principle to solve the boundary elasticity problem in terms of stresses is applied. An application of the above-mentioned principle ensures the satisfaction of all the six Saint-Venant strain compatibility eqns (see the works of Southwell, Kliushnikov, a.o.). Castigliano variation principle does not define the number of independent strain compatibility eqns. Thus, it is not clear whether the elasticity problem eqns system in terms of stresses is over-defined or not. The strain compatibility eqns for an ideal elastic body is investigated in the article by means of the mathematical programming theory. A mathematical model to evaluate the statically admissible stresses is formulated on the basis of complementary energy minimum principle. It is proved that the strain compatibility eqns mean the Kuhn-Tucker optimality conditions of the mathematical programming problem. The method to formulate the strain compatibility eqns in respect of the statically admissible stresses defining eqns formulation technique is revealed. The proposed method is illustrated to achieve the six component stresses vector in functional space for the three-dimension problem: usually the solution of the elasticity problem in terms of the stresses is realised via the nine eqns system integration. The Kuhn-Tucker conditions allowed to confirm an original but not usually applied Washizu conclusion about Cauchy geometrical compatibility eqns.
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44

Liu, Chein-Shan. "To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator." Inverse Problems in Science and Engineering 22, no. 4 (October 24, 2013): 641–71. http://dx.doi.org/10.1080/17415977.2013.848434.

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45

Makhmudov, O. I., and I. E. Niezov. "Regularization of solutions of the Cauchy problem for systems of elasticity theory in infinite domains." Mathematical Notes 68, no. 4 (October 2000): 471–75. http://dx.doi.org/10.1007/bf02676726.

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46

Boutefnouchet, M., H. Erjaee, M. Kirane, and M. Qafsaoui. "Nonexistence results for the Cauchy problem for some fractional nonlinear systems of thermo-elasticity type." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 96, no. 9 (February 18, 2016): 1119–28. http://dx.doi.org/10.1002/zamm.201500134.

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47

Wang, Xu, and Hui Fan. "Interaction between a nanocrack with surface elasticity and a screw dislocation." Mathematics and Mechanics of Solids 22, no. 2 (August 6, 2016): 131–43. http://dx.doi.org/10.1177/1081286515574147.

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In the present analytical study, we consider the problem of a nanocrack with surface elasticity interacting with a screw dislocation. The surface elasticity is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. By considering both distributed screw dislocations and line forces on the crack, we reduce the interaction problem to two decoupled first-order Cauchy singular integro-differential equations which can be numerically solved by the collocation method. The analysis indicates that if the dislocation is on the real axis where the crack is located, the stresses at the crack tips only exhibit the weak logarithmic singularity; if the dislocation is not on the real axis, however, the stresses exhibit both the weak logarithmic and the strong square-root singularities. Our result suggests that the surface effects of the crack will make the fracture more ductile. The criterion for the spontaneous generation of dislocations at the crack tip is proposed.
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48

Yudenkov, Aleksey V., Aleksandr M. Volodchenkov, and Liliya P. Rimskaya. "STABILITY OF SYSTEMS OF SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNEL." T-Comm 14, no. 9 (2020): 48–55. http://dx.doi.org/10.36724/2072-8735-2020-14-9-48-55.

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Singular Cauchy integral equations have been widely used for mathematical simulation of the actual physical and technical systems. They are considered universal at every level of simulation beginning with quantum field theory and up to strength analysis of the underground constructions. Therefore investigating system stability of such models under perturbation of their absolute terms and coefficients appears an urgent scientific task. The aim of the study is to show various aspects of stability of singular Cauchy integral sets of equations which are generalizing simulation models of the primal problems of the elasticity theory for homogeneous isotropic bodies. The methods of study are based on the properties of the Cauchy singular integral, on the general theory of Fredholm operators. When in use, systems of the singular integral equations are reduced to a set of Fredholm integral equations of the second kind and a set of the boundary value problems for analytic functions. The key results of the study are the following: development of the general determination method of the system index for singular integral equations, proof of the system stability against perturbations of the absolute terms of the set. Against perturbations of the boundary coefficients, the singular integral system is unstable. Demonstration of the stability of the singular integral Cauchy sets generalizing primal problems of the elasticity theory appears a significantly new result. The research of singular integral equations sets has been performed conducted on the space of functions satisfying the Holder condition. However the main research results prove to be true if we operate random functions converting in mean square. Stability of singular integral equations sets against perturbations of the absolute terms lays a foundation for calculus of approximations in real world tasks of defining the built-in stress of an elastic complex body.
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49

Sun, Yao, Fuming Ma, and Xu Zhou. "An Invariant Method of Fundamental Solutions for the Cauchy Problem in Two-Dimensional Isotropic Linear Elasticity." Journal of Scientific Computing 64, no. 1 (October 4, 2014): 197–215. http://dx.doi.org/10.1007/s10915-014-9929-7.

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50

Andrieux, Stéphane, and Thouraya N. Baranger. "An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity." Computer Methods in Applied Mechanics and Engineering 197, no. 9-12 (February 2008): 902–20. http://dx.doi.org/10.1016/j.cma.2007.08.022.

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