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Journal articles on the topic 'Half-plane method'

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Published: 4 June 2021

Last updated: 3 February 2022

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1

Li, YingChun, and ZhiHong Liu. "Convolutions of harmonic right half-plane mappings." Open Mathematics 14, no. 1 (2016): 789–800. http://dx.doi.org/10.1515/math-2016-0069.

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AbstractWe first prove that the convolution of a normalized right half-plane mapping with another subclass of normalized right half-plane mappings with the dilatation $ - z(a + z)/(1 + az)$ is CHD (convex in the horizontal direction) provided $a = 1$ or $ - 1 \le a \le 0$. Secondly, we give a simply method to prove the convolution of two special subclasses of harmonic univalent mappings in the right half-plane is CHD which was proved by Kumar et al. [1, Theorem 2.2]. In addition, we derive the convolution of harmonic univalent mappings involving the generalized harmonic right half-plane mappings is CHD. Finally, we present two examples of harmonic mappings to illuminate our main results.

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2

Kontoni, Dionyssia-Pinelopi N., Dimitri E. Beskos, and George D. Manolis. "Uniform half-plane elastodynamic problems by an approximate boundary element method." Soil Dynamics and Earthquake Engineering 6, no. 4 (1987): 227–38. http://dx.doi.org/10.1016/0267-7261(87)90004-2.

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3

Yu-ying, Huang, and Yin Lei-fang. "A direct method for deriving fundamental solution of half-plane problem." Applied Mathematics and Mechanics 8, no. 12 (1987): 1181–90. http://dx.doi.org/10.1007/bf02450912.

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4

Fan, C. W., and Chyanbin Hwu. "Punch Problems for an Anisotropic Elastic Half-Plane." Journal of Applied Mechanics 63, no. 1 (1996): 69–76. http://dx.doi.org/10.1115/1.2787211.

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By combining Stroh’s formalism and the method of analytical continuation, several mixed-typed boundary value problems of an anisotropic elastic half-plane are studied in this paper. First, we consider a set of rigid punches of arbitrary profiles indenting into the surface of an anisotropic elastic half-plane with no slip occurring. Illustrations are presented for the normal and rotary indentation by a flat-ended punch. A sliding punch with or without friction is then considered under the complete or incomplete indentation condition.

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5

Chen, Y. Z. "Image method for Green's function of anisotropic half-plane with inclined boundary." International Journal of Engineering Science 34, no. 14 (1996): 1563–71. http://dx.doi.org/10.1016/s0020-7225(96)00054-7.

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6

Wang, Y., and R. K. N. D. Rajapakse. "An Exact Stiffness Method for Elastodynamics of a Layered Orthotropic Half-Plane." Journal of Applied Mechanics 61, no. 2 (1994): 339–48. http://dx.doi.org/10.1115/1.2901450.

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A method is presented in this paper to compute displacements and stresses of a multilayered orthotropic elastic half-plane under time-harmonic excitations. The half-plane region under consideration consists of a number of layers with different thicknesses and material properties. Exact layer stiffness matrices describing the relationship between Fourier transforms of displacements and tractions at the upper and bottom surface of each layer are established explicitly by using the analytical general solutions for displacements and stresses of a homogeneous orthotropic elastic medium. The global stiff ness matrix which is also symmetric and banded is assembled by considering the traction continuity conditions at the interface between adjacent layers of the multilayered half-plane. The numerical solution of the global stiffness equation results in the solutions for Fourier transform of displacements at layer interfaces. Thereafter displacements and stresses of the multilayered plane can be obtained by the numerical integration of Fourier integrals. Only negative exponential terms of Fourier transform parameter are found to appear in the elements of layer stiffness matrices. This ensures the numerical stability in the solution of the global stiffness equation. In addition, the size of the final equation system is nearly onehalf of that corresponding to the conventional matrix approach for layered media based on the determination of layer arbitrary coefficients. The present method provides accurate solutions for both displacements and stresses over a wide range of frequencies and layer thicknesses. Selected numerical results are presented to portray the influence of layering, material orthotropy, and frequency of excitation on the response of five layered systems. Time-domain solutions are also presented to demonstrate the features of transient surface displacements due to a surface loading pulse applied to layered orthotropic half-planes.

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7

Attiya, Ahmed M. "Transmission of Pulsed Plane Wave Into Dispersive Half-Space: Prony's Method Approximation." IEEE Transactions on Antennas and Propagation 59, no. 1 (2011): 324–27. http://dx.doi.org/10.1109/tap.2010.2090467.

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8

Ike, Charles Chinwuba. "Fourier Sine Transform Method for Solving the Cerrutti Problem of the Elastic Half Plane in Plane Strain." Mathematical Modelling in Civil Engineering 14, no. 1 (2018): 1–11. http://dx.doi.org/10.2478/mmce-2018-0001.

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Abstract The Fourier sine transform method was implemented in this study to obtain general solutions for stress and displacement fields in homogeneous, isotropic, linear elastic soil of semi-infinite extent subject to a point load applied tangentially at a point considered the origin of the half plane. The study adopted a stress based formulation of the elasticity problem. Fourier transformation of the biharmonic stress compatibility equation was done to obtain bounded stress functions for the elastic half plane problem. Stresses and boundary conditions expressed in terms of the Boussinesq-Papkovich potential functions were transformed using Fourier sine transforms. Boundary conditions were used to obtain the unknown constants of the stress functions for the Cerrutti problem considered; and the complete determination of the stress fields in the Fourier transform space. Inversion of the Fourier sine transforms for the stresses yielded the general expressions for the stresses in the physical domain space variables. The strain fields were obtained from the kinematic relations. The displacement fields were obtained by integration of the strain-displacement relations. The solutions obtained were identical with solutions in literature obtained using Cerrutti stress functions.

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9

Yeh, Chau-Shioung, Tsung-Jen Teng, Wen-Shinn Shyu, and I.-Chang Tsai. "A Hybrid Method for Analyzing the Dynamic Responses of Cavities or Shells Buried in an Elastic Half-Plane." Journal of Mechanics 18, no. 2 (2002): 75–87. http://dx.doi.org/10.1017/s1727719100002045.

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AbstractIn this paper, based on a variational formalism which originally proposed by Mei [1] for infinite elastic medium and extended by Yeh, et al. [2,3] for elastic half-plane, a hybrid method which combines the finite element and series expansion method is implemented to solve the diffraction of plane waves by a cavity buried in an elastic half-plane. The finite domain which encloses all inhomogeneities including the cavity can be easily formulated by finite element methods. The unknown boundary data obtained by subtracting the known free fields from the total fields which include the boundary nodal displacements and tractions at the interface between the finite domain and the surrounding elastic half-plane are not independent of each other and can be correlated through a series representation. Due to the continuity condition at the interface, the same series representation is still valid for the exterior elastic half-plane to represents the scattered wave. The unknown coefficients of this series are treated as generalized coordinates and can be easily formulated by the same variational principle. The expansion function of the series is composed of basis function. Each basis function is constructed from the basis function for an infinite plane by superimposing an additional homogeneous reflective term to satisfy both traction free conditions at ground surface and radiation conditions at infinity. The numerical results are made against those obtained by boundary element methods, and good agreements are found.

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10

Theotokoglou, E. E., and Glaucio H. Paulino. "Interaction between an Embedded Crack and an Interface Crack in Nonhomogeneous Coating System." Materials Science Forum 492-493 (August 2005): 397–402. http://dx.doi.org/10.4028/www.scientific.net/msf.492-493.397.

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A general methodology is constructed for the fundamental solution of a crack in the homogeneous half-plane interacting with a crack at the interface between the homogeneous elastic half-plane and the nonhomogeneous elastic coating in which the shear modulus varies exponentially with one coordinate. The problem is solved under plane strain or generalized plane stress condition using the Fourier integral transform method. The stress field in the homogeneous half plane is evaluated by the superposition of two states of stresses, one is associated with a local coordinate system in the infinite fractured plate, while the other in the infinite half plane defined in a structural coordinate system.

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11

Li, Xin-yuan, and Guo-bin Liu. "A Complex Variable Solution for Rectangle Pipe Jacking in Elastic Half-Plane." Mathematical Problems in Engineering 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/5713063.

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In mechanics, the solution of soil stresses and displacements field caused by shallow rectangular jacking pipe construction can be simplified as half-plane problem. Both the boundary conditions of the surface and the cavity boundary must be taken into account. It is the essential prerequisite for mechanical analysis of the pipe jacking with the complex variable theory that the mechanical boundary must be transformed from the half-plane with a rectangle cavity to the concentric ring. According to Riemann’s existence theorem and basic complex variable theory, a conformal mapping function is established. Both sides of boundary conditions equation are developed into Laurent series, and then the coefficients of complex stress function are solved by power series method. The derived solution is applied to an example and a comparison is made using FEM method to show the accuracy of the methods. The result shows the following: (1) the method presented in this paper is applicable to a shallow-buried rectangular tunnel; (2) the complex function method proposed in this paper is characterized by clear steps, fast convergence, and simple operation.

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12

Onyshkevych, V. M., and G. T. Sulym. "Consideration of wear in plane contact of rectangular punch and elastic half-plane." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1 (2019): 138–41. http://dx.doi.org/10.17721/1812-5409.2019/1.31.

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The plane contact problem on wear of elastic half-plane by a rigid punch has been considered. The punch moves with constant velocity. Arising thermal effects are neglected because the problem is investigated in stationary statement. In this case the crumpling of the nonhomogeneities of the surfaces and abrasion of half-plane take place. Out of the punch the surface of half-plane is free of load. The solution for problem of theory of elasticity is constructed by means of Fourier integral transformation. Contact stresses are found in Fourier series which coefficients satisfy the dual integral equations. It leads to the system of nonlinear algebraical equations for unknown coefficients by a method of collocations. This system is reduced to linear system in the partial most interesting cases for computing of maximum and minimum wear. The iterative scheme is considered for investigation of other nonlinear cases, for initial approximation the mean value of boundary cases is used. The evolutions of contact stresses, wear and abrasion in the time are given. For both last cases increase or invariable of vertical displacement correspondently is obtained. In the boundary cases coincidence of results with known is obtained.

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13

YASUDA, Y., and T. SAKUMA. "A TECHNIQUE FOR PLANE-SYMMETRIC SOUND FIELD ANALYSIS IN THE FAST MULTIPOLE BOUNDARY ELEMENT METHOD." Journal of Computational Acoustics 13, no. 01 (2005): 71–85. http://dx.doi.org/10.1142/s0218396x05002591.

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The fast multipole boundary element method (FMBEM) is an advanced BEM, with which both the operation count and the memory requirements are O(Na log b N) for large-scale problems, where N is the degree of freedom (DOF), a ≥ 1 and b ≥ 0. In this paper, an efficient technique for analyses of plane-symmetric sound fields in the acoustic FMBEM is proposed. Half-space sound fields where an infinite rigid plane exists are typical cases of these fields. When one plane of symmetry is assumed, the number of elements and cells required for the FMBEM with this technique are half of those for the FMBEM used in a naive manner. In consequence, this technique reduces both the computational complexity and the memory requirements for the FMBEM almost by half. The technique is validated with respect to accuracy and efficiency through numerical study.

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14

Richter, Christoph, and G�nther Schmid. "A Green's function time-domain boundary element method for the elastodynamic half-plane." International Journal for Numerical Methods in Engineering 46, no. 5 (1999): 627–48. http://dx.doi.org/10.1002/(sici)1097-0207(19991020)46:5<627::aid-nme691>3.0.co;2-q.

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15

Jiang, Aimin, and Haojiang Ding. "Green's functions and boundary element method for a magneto-electro-elastic half-plane." Structural Engineering and Mechanics 20, no. 2 (2005): 259–64. http://dx.doi.org/10.12989/sem.2005.20.2.259.

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16

Legros, Benoı̂t, Sofia G. Mogilevskaya, and Steven L. Crouch. "A boundary integral method for multiple circular inclusions in an elastic half-plane." Engineering Analysis with Boundary Elements 28, no. 9 (2004): 1083–98. http://dx.doi.org/10.1016/j.enganabound.2004.02.010.

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17

Dejoie, Alexandre, Sofia G. Mogilevskaya, and Steven L. Crouch. "A boundary integral method for multiple circular holes in an elastic half-plane." Engineering Analysis with Boundary Elements 30, no. 6 (2006): 450–64. http://dx.doi.org/10.1016/j.enganabound.2005.12.005.

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18

Yee, Hooi Min, and Abdul Malek Nurul Afiqah. "Membrane Structures with Half-Costa Surface in YZ-Plane." Materials Science Forum 995 (June 2020): 222–28. http://dx.doi.org/10.4028/www.scientific.net/msf.995.222.

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This paper presents the computational form-finding analysis of half-Costa tensioned membrane structure model in YZ-plane with different boundaries. The computational form-finding analysis is carried out based on nonlinear analysis method. The tensioned membrane structure in the form of half-Costa models in YZ-plane with different geometry have been found to converge with least square error of total warp and fill stress deviation. The outcome of this paper can serve as a reference in selecting satisfactory parameters to allow the performance increase of tensioned membrane structure in the form of half-Costa in YZ-plane respected to their boundary condition. These models will be selective forms of tensioned membrane structure for design engineers and architect to ponder on as it is a resource efficient structure hence preventing further damage to the environment.

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19

Yamazaki, Tadashi. "Rankin-Selberg method for Siegel cusp forms." Nagoya Mathematical Journal 120 (December 1990): 35–49. http://dx.doi.org/10.1017/s0027763000003226.

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Let Gn (resp. Γn) be the real symplectic (resp. Siegel modular) group of degree n. The Siegel cusp form is a holomorphic function on the Siegel upper half plane which satisfies functional equations relative to Γn and vanishes at the cusps.

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20

CHEN, Y. Z., and Z. X. WANG. "MULTIPLE AND PERIODIC NOTCH PROBLEMS OF ELASTIC HALF-PLANE BY USING BIE BASED ON GREEN'S FUNCTION METHOD." International Journal of Computational Methods 07, no. 04 (2010): 539–57. http://dx.doi.org/10.1142/s0219876210002428.

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This paper provides a modified complex potential for the fundamental solution, which is composed of a principal part and a complementary part. The suggested complex potential satisfies the traction free condition along the boundary of half-plane in advance. After using the Somigliana identity or the Betti's reciprocal theorem between the physical field and the field of the fundamental solution, a complex variable boundary integral equation (BIE) for the notch problem in elastic half-plane is obtained. A compact derivation for the BIE is presented. By using the BIE, multiple notch problems of elastic half-plane can be solved numerically. In the method, there is no limitation for the configuration of notches. Many problems with the elliptic notch or the square notch are solved successfully. For the periodic notch problem, the remainder estimation technique is suggested. This technique provides an effective way for the solution of periodic notch problem.

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21

Mikayelyan, G. V., and F. V. Hayrapetyan. "ON INTEGRAL LOGARITHMIC MEANS OF BLASCHKE PRODUCTS FOR A HALF-PLANE." Proceedings of the YSU A: Physical and Mathematical Sciences 52, no. 3 (247) (2018): 166–71. http://dx.doi.org/10.46991/pysu:a/2018.52.3.166.

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22

Bich, Nguyen Dang. "Doing the plane dynamic elasto-plastic problem by the method of elastic root." Vietnam Journal of Mechanics 7, no. 4 (1985): 9–14. http://dx.doi.org/10.15625/0866-7136/10395.

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In this paper, the method of clastic root used to answer the plane dynamic elasto plastic problem, for the half of plane made by material according to the theory of small elasto-plastic deformation, on the boundary supporte1 degree loading of motion with over sound speed, root in the first -step given analytical form.

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23

Malyutin, K. G., and A. A. Revenko. "Extreme problems in the space of meromorphic functions of finite order in the half plane. II." Matematychni Studii 54, no. 2 (2020): 154–61. http://dx.doi.org/10.30970/ms.54.2.154-161.

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The extremal problems in the space of meromorphic functions of order $\rho>0$ in upper half-plane are studed.The method for studying is based on the theory of Fourier coefficients of meromorphic functions. The concept of just meromorphic function of order $\rho>0$ in upper half-plane is introduced. Using Lemma on the P\'olya peaks and the Parseval equality, sharp estimate from below of the upper limits of relations Nevanlinna characteristics of meromorphic functions in the upper half plane are obtained.

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24

Bergeron, Hervé, and Jean-Pierre Gazeau. "Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane." Entropy 20, no. 10 (2018): 787. http://dx.doi.org/10.3390/e20100787.

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Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.

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25

Lin, Yuan, and Timothy C. Ovaert. "Thermal Distortion of an Anisotropic Elastic Half-Plane and its Application in Contact Problems Including Frictional Heating." Journal of Tribology 128, no. 1 (2005): 32–39. http://dx.doi.org/10.1115/1.2125907.

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The thermal surface distortion of an anisotropic elastic half-plane is studied using the extended version of Stroh’s formalism. In general, the curvature of the surface depends both on the local heat flux into the half-plane and the local temperature variation along the surface. However, if the material is orthotropic, the curvature of the surface depends only on the local heat flux into the half-plane. As a direct application, the two-dimensional thermoelastic contact problem of an indenter sliding against an orthotropic half-plane is considered. Two cases, where the indenter has either a flat or a parabolic profile, are studied in detail. Comparisons with other available results in the literature show that the present method is correct and accurate.

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26

Wang, Y. J., C. F. Gao, H. P. Song, and S. C. Xing. "The Generalized Two Dimensional Thermal-Electro-Elastic Solution for the Cracked-Half-Elliptical-Hole Problem in a Half Plane." Journal of Theoretical and Applied Mechanics 45, no. 2 (2015): 21–44. http://dx.doi.org/10.1515/jtam-2015-0009.

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AbstractThe half elliptical hole with an edge crack in a thermopiezoelectric material is studied by using the complex variable method. First, the mapping function which maps the outside of the elliptical hole and the crack in the right half plane into the outside of a circular hole in a full plane is given by the method of conformal mapping. Then, the complex potential functions and the field intensity factors (FIF) are presented according to the boundary conditions, respectively. Some useful results can be found by numerical analysis: 1) The influence of the heat flux on FIF depends on the model of the crack; 2) The shape and the size of the hole possess a significant effect on the field distribution at the crack tip.

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27

Wang, He Hui, Meng Xi Hu, Yi Fan Chen, Dong Liang Wang, and Ke Di Xie. "Analysis of Slant Surface Cracks with Step Notches Subject to Contact Loadings by a Variational Boundary Integral Method." Key Engineering Materials 353-358 (September 2007): 1125–28. http://dx.doi.org/10.4028/www.scientific.net/kem.353-358.1125.

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Modes I and II stress intensity factors are analyzed by means of a variational boundary integral method (VBIM) for slant surface-breaking cracks in a half-plane with surface steps subject to contact loadings. This method represents the crack as a continuous distribution of dislocation loops. The crack opening displacements, which are related to the geometry of loops and their Burgers vectors, can be determined by minimizing the elastic potential energy, obtained from the known expressions of the interaction energy of a pair of dislocation loops, of the solid. In contrast to other methods, this approach finally reduces to a symmetric system of equations with milder singularities of the type 1/R, which facilitate the numerical treatments. By modeling the surface boundary of the half-plane as half part of an infinite crack breaking through an infinite solid, this paper demonstrates that the VBIM can be well extended to solve the fracture problems of inclined surface-breaking cracks in a half-plane with curve or step notches subject to combined contact loadings, and presents results of stress intensity factors for a variety of loadings, cracks and step surface configurations. Numerical results of test examples are in good agreement with the existing results in the literature.

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28

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 (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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29

Lee, Jung-Ki, and Duck-Young Ku. "Elastic Analysis of a Half-Plane Containing Multiple Inclusions Using Volume Integral Equation Method." Transactions of the Korean Society of Mechanical Engineers A 32, no. 2 (2008): 148–61. http://dx.doi.org/10.3795/ksme-a.2008.32.2.148.

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30

Tan, Eng Leong, and Ding Yu Heh. "POLE-ZERO ANALYSIS OF MICROWAVE FILTERS USING CONTOUR INTEGRATION METHOD EXPLOITING RIGHT-HALF PLANE." Progress In Electromagnetics Research M 78 (2019): 59–68. http://dx.doi.org/10.2528/pierm18102301.

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31

Guo, Min, Juan chen, and Yawei Peng. "The Control Method of Multivariable Time-delay Square System Containing Right Half Plane Zeros." Procedia Engineering 15 (2011): 1004–9. http://dx.doi.org/10.1016/j.proeng.2011.08.186.

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CHEN, Y. Z., Z. X. WANG, and X. Y. LIN. "SINGULAR INTEGRAL EQUATION METHOD FOR MULTIPLE FLAT PUNCH PROBLEM FOR AN ELASTIC HALF-PLANE." International Journal of Computational Methods 06, no. 04 (2009): 605–14. http://dx.doi.org/10.1142/s0219876209002029.

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When a flat punch is indented on elastic half-plane, the singular stress distribution at the vicinity of the punch corners is studied. The angle distribution for the stress components is also achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor is defined. The multiple punch problem can be considered as a superposition of many single punch problems. Taking the stress distribution under the punch base as the unknown function and the deformation under punch as the right-hand term, a set of the singular integral equations for the multiple punch problem can be achieved. After the singular integral equations are solved, the stress distributions under punches can be obtained. In addition, the exerting locations of the resultant forces under punches can also be determined. Two numerical examples with the calculated results are presented.

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Exadaktylos, G., and G. Xiroudakis. "The G2 constant displacement discontinuity method – Part II: Solution of half-plane crack problems." International Journal of Solids and Structures 47, no. 18-19 (2010): 2578–90. http://dx.doi.org/10.1016/j.ijsolstr.2010.05.014.

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34

Weilin, Zang, and Peter Gudmundson. "An integral equation method for piece-wise smooth cracks in an elastic half-plane." Engineering Fracture Mechanics 32, no. 6 (1989): 889–97. http://dx.doi.org/10.1016/0013-7944(89)90005-2.

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35

Lee, Jungki, Duckyoung Ku, and Ajit Mal. "Elastic analysis of a half-plane with multiple inclusions using volume integral equation method." Engineering Analysis with Boundary Elements 35, no. 3 (2011): 564–74. http://dx.doi.org/10.1016/j.enganabound.2010.08.012.

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36

Mkhitaryan, S. M. "On the Application of the Method of Hypersingular Integral Equations to Solving Problems for an Elastic Plane with a Collinear System of Cracks." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 2 (206) (June 18, 2020): 72–83. http://dx.doi.org/10.18522/1026-2237-2020-2-72-83.

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In the present paper, using the method of hypersingular integral equations, based on the formulas of the inversion of the corresponding singular integral equations, the exact quadrature solution of the classical problems of the mechanics of an elastic plane with a collinear system of cracks is constructed. The elastic plane is in a state of antiplane deformation or plane deformation; in case of antiplane deformation, crack edges are symmetrically loaded by tangential forces, while in case of plane deformation, they are again loaded symmetrically but by normal forces. Mixed boundary-value problems for an elastic half-plane equivalent to these problems are formulated. Under plane deformation, the mixed boundary-value problem for an elastic half-plane is discussed as well when the plane boundary is reinforced by two similar and symmetrically located semi-infinite stringers between which a system of an arbitrarily final number of stringers is situated. It is considered that the stringers are absolutely rigid for expansion and compression and absolutely flexible for bending. A particular case of two similar symmetrically located cracks is considered more in detail. In this case, the exact solution to the problem is also constructed by the method of Chebyshev orthogonal polynomials.

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37

Lee, Vincent W., and Heather P. Brandow. "Weighted Residual Method for Diffraction of Plane P-Waves in a 2D Elastic Half-Space Revisited: On an Almost Circular Arbitrary-Shaped Canyon." Journal of Earthquakes 2015 (September 30, 2015): 1–21. http://dx.doi.org/10.1155/2015/543128.

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Scattering and diffraction of elastic in-plane P- and SV-waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong-motion seismologists for over forty years. The case of out-of-plane SH-waves on the same elastic canyon that is semicircular in shape on the half-space surface is the first such problem that was solved by analytic closed-form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because the in-plane P- and SV-scattered-waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by the author in the work of Lee and Liu. This paper uses the technique of Lee and Liu of defining these stress-free scattered waves to solve the problem of the scattering and diffraction of these in-plane waves on an on an almost-circular surface canyon that is arbitrary in shape.

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Zhang, Li Xiong, and Rong Gang Gao. "Study of Method for Determination of Thermal Conductivity of Automotive Interior Materials." Applied Mechanics and Materials 526 (February 2014): 46–51. http://dx.doi.org/10.4028/www.scientific.net/amm.526.46.

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Based on the traditional theory of transient plane source for thermal conductivity measurement, this paper designed and developed a new pattern of heating and temperature sensing probe, presented the study of transient heat conduction of half-infinite plane while being heated, established a modified mathematical model of transient plane source method, and achieved the measurement of thermal conductivity of automotive interior material sample by the data processing method of mathematical iteration and liner regression using the modified transient plane source probe. According to the data of experiments, the instrument which this paper designed has a high precision of 5% and a wide range of 0.003-1W/(mK).This paper provides a practicable way for heat capacity determination of automotive interior materials.

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Marmysh, D. E., and U. I. Babaed. "Monte Carlo method for determination and analysis damage to the power system." Doklady BGUIR 19, no. 1 (2021): 21–29. http://dx.doi.org/10.35596/1729-7648-2021-19-1-21-29.

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The purpose of the work, the results of which are presented within the framework of the article, was to develop algorithms for calculating the damage to a solid or a system of solids based on the Monte Carlo method and the analytical boundary element method. The analytical boundary element method was used to calculate and analyze the stress-strain state of a solid under the distributed surface load. Based on indicators of the stress state, the algorithms for numerically assessing the dangerous volume and integral damage using the Monte Carlo methods, have been developed. Based on the pattern of distribution of stress fields, the technique of determining the area for randomly generating integration nodes is described. General recommendations have been developed for determining the boundaries of a subdomain containing a dangerous volume. Based on the features of the Monte Carlo methods, a numerical assessment of the indicators of damage of continuous media for a different number of integration nodes was carried out. Methods and algorithms were used to calculate the dangerous volume and integral damage in the plane and spatial cases for the two most common laws of the distribution of surface forces in the contact mechanics of solids: in case of contact interaction of two non-conformal bodies (Hertz problem) and when a non deformable rigid stamp is pressed into elastic half-plane or half-space. The scientific novelty of the work is to combine analytical and numerical approaches for the quantitative assessment of damage indicators of the power system. As a result the quantitative indicators of the dangerous volume (in the flat case - the dangerous area) and the integral damage of the half-plane and half-space related to the value of the applied load are obtained.

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Rodnikov, A. O., and B. A. Samokish. "Finite difference method in the problem of diffraction of a plane acoustic wave in a half-plane with a cut." Computational Mathematics and Mathematical Physics 49, no. 12 (2009): 2117–34. http://dx.doi.org/10.1134/s0965542509120112.

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Zheng, Chang-Jun, Wen-Yu Liu, Yong-Bin Zhang, Chuan-Xing Bi, Hai-Feng Gao, and Hai-Bo Chen. "Simulation of Sound Propagation Over an Infinite Impedance Plane by Using a Fast Multipole BEM." Journal of Theoretical and Computational Acoustics 28, no. 02 (2020): 2050020. http://dx.doi.org/10.1142/s2591728520500206.

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In this paper, a half-space fast multipole BEM is developed for the simulation of three-dimensional acoustic problems above an infinite impedance plane. The half-space impedance Green’s function involving a complex line source is used, so that both mass-like and spring-like impedance boundary conditions on the infinite plane can be explicitly satisfied and the infinite plane is not required to be discretized. The Burton–Miller method is employed to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method. Image relations of the multipole expansion coefficients are used and the half-space impedance Green’s function is modified to apply such relations to avoid calculating, translating and saving the multipole/local expansion coefficients in the image domain. An automatic integrator with adaptive interval subdivision is further adopted to calculate the line integral contained in the M2L translation formula accurately and efficiently. Numerical examples are given to show the validity and potential of the method.

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Dancer, E. N. "Some notes on the method of moving planes." Bulletin of the Australian Mathematical Society 46, no. 3 (1992): 425–34. http://dx.doi.org/10.1017/s0004972700012089.

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In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.

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Musayev, Vyacheslav K. "Mathematical modeling of stress waves under concentrated vertical action in the form of a triangular pulse: Lamb’s problem." Structural Mechanics of Engineering Constructions and Buildings 17, no. 2 (2021): 112–20. http://dx.doi.org/10.22363/1815-5235-2021-17-2-112-120.

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The aim of the work. The problem of numerical simulation of longitudinal, transverse and surface waves on the free surface of an elastic half-plane is considered. Methods. To solve the non-stationary dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements was used. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasiregular approach to solving a system of second-order linear ordinary differential equations in displacements with initial conditions and to approximating the area under study is proposed. The method is based on the schemes: point, line and plane. The study area is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the study area is divided into linear end elements with two nodal points. The Fortran-90 algorithmic language was used in the development of the software package. Results. Some information is given about numerical modeling of elastic stress waves in an elastic half-plane with a concentrated wave action in the form of a Delta function. The estimated area under study has 12 008 001 nodal points. A system of equations consisting of 48 032 004 unknowns is solved. The change of elastic contour stress on the free surface of the half-plane at different points is shown. The amplitude of Rayleigh surface waves is significantly greater than the amplitudes of longitudinal, transverse, and other waves with a concentrated vertical action in the form of a triangular pulse on the surface of an elastic half-plane. After surface Rayleigh waves, a dynamic process is observed in the form of standing waves.

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Ziegler, Franz, and Piotr Borejko. "The Method of Generalized Ray-Revisited." Journal of Mechanics 16, no. 1 (2000): 37–44. http://dx.doi.org/10.1017/s1727719100001751.

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ABSTRACTBased on a landmark paper by Pao and Gajewski, some novel developments of the method of generalized ray integrals are discussed. The expansion of the dynamic Green's function of the infinite space into plane waves allows benchmark 3-D solutions in the layered half-space and even enters the background formulation of elastic-viscoplastic wave propagation. New developments of software of combined symbolic-numerical manipulation and parallel computing make the method a competitive solution technique.

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Pozharskii, Dmitrii A. "Periodic Contact and Mixed Problems of the Elasticity Theory (Review)." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 2 (210) (June 28, 2021): 22–33. http://dx.doi.org/10.18522/1026-2237-2021-2-22-33.

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Results are reviewed collected in the investigations of periodic contact and mixed problems of the plane, axially symmetric and spatial elasticity theory. Among mixed problems, cut (crack) problems are focused integral equations of which are connected with those for contact problems. The periodic contact problems stimulate research of the discrete contact of rough (wavy) surfaces. Together with classical elastic domains (half-plane, half-space, plane and full space), we consider periodic problems for cylinder, layer, cone and spatial wedge. Most publications including fun-damental ones by Westergaard and Shtaerman deals with plane periodic problems of the elasticity theory. Here, one can mention approaches based on complex variable functions, Fourier series, Green’s functions and potential func-tions. A fracture mechanics approach to the plane periodic contact problem was developed. Methods and approaches are considered which allow us to take friction forces, adhesion and wear into account in the periodic contact. For spatial periodic and doubly periodic contact and properly mixed problems, we describe such methods as the localiza-tion method, the asymptotic methods, the method of nonlinear boundary integral equations, the fast Fourier trans-form. The half-space is the simplest model for elastic solids. But for the simplest straight-line periodic punch system, some three-dimensional contact problems (normal contact or tangential contact for shifted cohesive coatings) turn out to be incorrect because their integral equations contain divergent series. Considering three-dimensional periodic problems, I.G. Goryacheva disposes circular punches in special way (circular orbits, polar coordinated are used for centers of the punches), in this case one can prove convergence of the series in the integral equation (it is important that the punches are circular). For the periodic problems for an elastic layer, V.M. Aleksandrov has shown that the series in integral equations converge but the kernels become more complicated. In the present paper, we demonstrate that for the straight-line periodic punch system of arbitrary form the contact problem for a half-space turns out to be correct in case of more complicated boundary conditions. Namely, it can be sliding support or rigid fixation of a half-plane on the half-space boundary, the half-plane boundary should be parallel to the straight-line (the punch system axis) for arbitrary finite distance between the parallel lines. On this way, for sliding support, the kernel of the period-ic problem integral equation kernel is free of integrals, it consists of single convergent series (normal contact, the kernel is given in two equivalent forms). We consider classical percolation (how neighboring contact domains pene-trate one to another, investigated by K.L. Johnson, V.A. Yastrebov with co-authors) for the three-dimensional periodic contact amplification as well as percolation for the straight-line punch system. A similar approach is suggested for the case of periodic tangential contact (coatings system cohesive with a half-space boundary shifted along its axis or perpendicular to it). Here, one can separate out unique solutions of auxiliary problems because the line of changing boundary conditions on the half-space boundary can provoke non-uniqueness. The method proposed opens possibility to consider more complicated three-dimensional periodic contact problems for straight-line punch systems with changing boundary conditions inside the period.

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Oda, Kazuhiro, and Nao-Aki Noda. "Accurate stress intensity factors for kinked interface crack in bonded dissimilar half-plane." International Journal of Modern Physics B 35, no. 14n16 (2021): 2140030. http://dx.doi.org/10.1142/s0217979221400300.

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In this study, the stress intensity factor (SIF) of an interface kinked crack is analyzed by the singular integral equation of the body force method. The problem can be expressed by distributing the body force doublets of the tension and shear types along all the boundaries of the kinked and interface crack parts. The SIFs can be obtained directly from the densities of the body force doublets at the crack tips. Although the problem has already been calculated using the crack connection model, the accuracy of the analysis has not been clarified. From the analysis results in this study, it can be seen that the SIFs calculated by the crack connection model have a nonnegligible error, and the present method gives more accurate results. The advantage of the present method is that the SIFs of the kinked and the interface crack tips can be obtained at the same time with high accuracy.

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Oner, Mete, and Stanley B. Dong. "Analysis of in-plane waves in layered half-space by global-local finite element method." Soil Dynamics and Earthquake Engineering 7, no. 1 (1988): 2–8. http://dx.doi.org/10.1016/s0267-7261(88)80009-5.

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Liu, Jingbo, and Yan Wang. "A 1D time-domain method for in-plane wave motions in a layered half-space." Acta Mechanica Sinica 23, no. 6 (2007): 673–80. http://dx.doi.org/10.1007/s10409-007-0114-1.

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Zheng, Chang-Jun, Hai-Bo Chen, and Lei-Lei Chen. "A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems." Acta Mechanica Sinica 29, no. 2 (2013): 219–32. http://dx.doi.org/10.1007/s10409-013-0023-4.

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CHEN, Y. Z., and X. Y. LIN. "SINGULAR INTEGRAL EQUATION METHOD FOR MULTIPLE CURVED EDGE CRACKS EMANATING FROM BOUNDARY OF HALF-PLANE." International Journal of Computational Methods 03, no. 02 (2006): 205–17. http://dx.doi.org/10.1142/s021987620600076x.

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This paper provides, an elastic solution for multiple curved edge cracks emanating from the boundary of the half-plane. After placing the distributed dislocations at the prospective sites of cracks in an infinite plate, the principal part of the complex potentials is obtained. By using the concept of the modified complex potentials, the complementary part of the complex potentials can be derived. The whole complex potentials satisfy the traction free condition along the boundary of half-plane automatically. This is a particular advantage of the suggested method. This concept or method of the modified complex potentials is a counterpart of the Green's function method, which is universal in mathematical physics. The direct usage of this method cannot provide a solution in detail. Comparing with the line edge crack case, the following points are significant in the presented study. The relevant kernels in the integral equation are more complicated than in the line edge crack case and the relevant integrations in the problem should be completed on curves. This paper solves a rather complicated problem, the multiple curved edge crack problem, and gives the final solution. A singular integral equation is formulated with the dislocation distribution being unknown function and the traction being the right hand term. The singular integral equation is solved by using the curve length method in conjunction with the semiopening quadrature rule. Periodic curved edge crack problem is also addressed. Finally, several numerical examples are given to illustrate the efficiency of the method presented.

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