Integral equation methods in plane asymmetric elasticity

The boundary integral equation method is used to solve the interior and exterior Dirichlet, Neumann and mixed problems of plane micropolar elasticity. In the exterior case, a specific far-field pattern for the displacements and microrotation is introduced without which the classical scheme fails to work. Finally, we discuss the direct method and establish a connection with results obtained previously.     

Eshelby integral formulas in gradient elasticity

Eshelby integral formulas play a fundamental role in mechanics of composite materials, because they provide an efficient tool for determining the average properties of dispersion-filled materials. For example, their use in the framework of the self-consistent averaging method actually gives a final and quite precise solution to the problem of determining effective physical and mechanical properties of filled composites up to large relative contents of inclusions and almost all relations between the phase characteristics of the composite. In the present paper, we generalize the Eshelby integral formulas to the gradient theory of elasticity. This provides the possibility for using efficient methods for estimating the average characteristics of micro and nano-structured materials in the framework of gradient theories, which permit taking the scale effects into account correctly, and hence find wider and wider applications in describing the mechanical and physical processes.     

Radiation conditions in asymmetric elasticity

In the present paper the radiation conditions of the Sommerfield type for a linear homogeneous and isotropic micropolar elasticity are discussed. A regular solution (u, ) of the fundamental system of field equations in an infinite domain has been defined using the radiation conditions for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% WbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGWbGaaiykaaqabaGccaGGPaaaaa!4834!\[(u_{(p)} , + \varphi _{(p)} )\] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% ZbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGPaaaaa!483A!\[(u_{(s)} , + \varphi _{(s)} )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaadwhacqGH9aqpcaWG1bWaaSbaaSqaaiaa% cIcacaWGWbGaaiykaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGSaGaaeiiaiabgEGirlabgEna0kaa% dwhadaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9iaaic% dacaGGSaGaaeiiaiabgEGirhrbmv3yPrwyGm0BUn3BSvgaiyGacaWF% 1bWaaSbaaSqaaGqaciaa+HcacaGFZbGaa4xkaaqabaGccqGH9aqpca% aIWaGaaiilaiaabccacqaHvpGAcqGH9aqpcqaHvpGAdaWgaaWcbaGa% aiikaiaadchacaGGPaaabeaakiabgUcaRiabew9aQnaaBaaaleaaca% GGOaGaam4CaiaacMcaaeqaaOGaaiilaiaabccacqGHhis0cqGHxdaT% cqaHvpGAdaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9i% aaicdacaGGSaGaaeiiaiabgEGirlabew9aQnaaBaaaleaacaqGOaGa% ae4CaiaabMcaaeqaaOGaeyypa0JaaGimaaaa!809B!\[u = u_{(p)} + u_{(s)} ,{\text{ }}\nabla \times u_{(p)} = 0,{\text{ }}\nabla u_{(s)} = 0,{\text{ }}\varphi = \varphi _{(p)} + \varphi _{(s)} ,{\text{ }}\nabla \times \varphi _{(p)} = 0,{\text{ }}\nabla \varphi _{{\text{(s)}}} = 0\], and formulae of the Betti type for an infinite domain with a cavity have been derived.

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Zusammenfassung Betrachtet werden die Ausstrahlungsbedingungen der Sommerfeldschen Art für lineare homogene mikropolare Elastizitätstheorie. Die reguläre Lösung (u, ) der grundlegenden Gleichungen für einen unendlichen Raum wird mit Hilfe der Bedingungen für Paare % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% WbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGWbGaaiykaaqabaGccaGGPaaaaa!4834!\[(u_{(p)} , + \varphi _{(p)} )\] und % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% ZbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGPaaaaa!483A!\[(u_{(s)} , + \varphi _{(s)} )\] definiert, wobei % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaadwhacqGH9aqpcaWG1bWaaSbaaSqaaiaa% cIcacaWGWbGaaiykaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGSaGaaeiiaiabgEGirlabgEna0kaa% dwhadaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9iaaic% dacaGGSaGaaeiiaiabgEGirhrbmv3yPrwyGm0BUn3BSvgaiyGacaWF% 1bWaaSbaaSqaaGqaciaa+HcacaGFZbGaa4xkaaqabaGccqGH9aqpca% aIWaGaaiilaiaabccacqaHvpGAcqGH9aqpcqaHvpGAdaWgaaWcbaGa% aiikaiaadchacaGGPaaabeaakiabgUcaRiabew9aQnaaBaaaleaaca% GGOaGaam4CaiaacMcaaeqaaOGaaiilaiaabccacqGHhis0cqGHxdaT% cqaHvpGAdaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9i% aaicdacaGGSaGaaeiiaiabgEGirlabew9aQnaaBaaaleaacaqGOaGa% ae4CaiaabMcaaeqaaOGaeyypa0JaaGimaaaa!809B!\[u = u_{(p)} + u_{(s)} ,{\text{ }}\nabla \times u_{(p)} = 0,{\text{ }}\nabla u_{(s)} = 0,{\text{ }}\varphi = \varphi _{(p)} + \varphi _{(s)} ,{\text{ }}\nabla \times \varphi _{(p)} = 0,{\text{ }}\nabla \varphi _{{\text{(s)}}} = 0\] ist. Es wird gezeigt, dass derartige reguläre Lösung mit Hilfe der Formel Bettischer Art für einem unendlichen Raum mit einem Hohlraum dargestellt werden kann.

     

On radially symmetric simple waves in elasticity

In this paper I look for simple wave solutions to the equations of nonlinear elastodynamics. The main result shows that, under mild constitutive restrictions, every nontrivial radially symmetric simple wave must be of the form An explicit solution is then obtained in the special case of a linear material.     

THE MIXED BOUNDARY-VALUE PROBLEM FOR NON-LOCAL ASYMMETRIC ELASTICITY

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅｒｅｅｘｉｓｔｓｔｈｅａｒｇｕｍｅｎｔｂｅｔｗｅｅｎＡｔｋｉｎｓｏｎ（ｓｅｅ［１～４］）ａｎｄＥｒｉｎｇｅｎａｎｄｃｏ＿ｗｏｒｋｅｒｓ（ｓｅｅ［５～７］）ｏｖｅｒｔｈｅｎｏｎ＿ｌｏｃａ...     

On the general expression of Fredholm Integral Equations Method in elasticity

In this paper, the concept of covering domain is introduced to develop a general expression for the Fredholm Integral Equations Method, by which elasticity problems of arbitrarily shaped bodies loaded by external forces can be solved. Some special expressions are given for a body with non-zero remote stresses, or subjected to some concentrated forces on its boundary. The relationship between the loading forces and solutions are also discussed. Some analytical solutions can be obtained for simple cases. When numerical computations are needed for the solution, the method proves to have high precision and fast convergency.     

Variational principles of asymmetric elasticity theory of finite deformation

Ｔｈｅｒｅａｒｅｍａｎｙｆｏｒｍｓｏｆｄｅｆｉｎｉｔｉｏｎａｂｏｕｔｔｈｅｆｉｎｉｔｅｓｔｒａｉｎａｎｄｒｏｔａｔｉｏｎｉｎｔｈｅｎｏｎｌｉｎｅａｒｃｏｎｔｉｎｕｕｍｍｅｃｈａｎｉｃｓｔｈｅｏｒｙａｔｐｒｅｓｅｎｔ．ＴｈｅｃｌａｓｓｉｃａｌｎｏｎｌｉｎｅａｒｔｈｅｏｒｙｂａｓｅｄｏｎＧｒｅｅｎ’ｓｓｔｒａｉｎｔｅｎｓｏｒｌａｃｋｓｔｈｅｄｅｆｉｎｉｔｉｏｎｏｆｆｉｎｉｔｅｒｏｔａｔｉｏｎｃｏｍｐａｔｉｂｌｅｗｉｔｈｔｈｅｓｔｒａｉｎ．Ｔｈｅｐｏｌａｒｄｅｃｏｍｐｏｓｉｔｉｏｎｔｈｅｏｒｅｍｌｏｓ…     

Integral representations for a nonhomogeneous region in couple stress theory of elasticity

In an infinite space a Cosserat medium (with forced rotations) is subjected to a singular concentrated force. The medium is nonhomogeneous and consist of two regular regions where the elastic constants differ. Formulas are derived for the displacements in both the internal and the external region. With the help of this Green's function matrix integral representations are derived for the displacements.     

Flow fields in a simulated axisymmetric terminal aneurysm with symmetric and asymmetric outflows

Laser-Doppler velocimetric measurements and flow visualization were performed in a glass axisymmetric aneurysm model with symmetric and asymmetric outflows through the branches. The bifurcation angle was fixed at 140°, and the Reynolds number based on the steady bulk average velocity and diameter of the affarent conduit was 500. The flow characteristics such as flow separation in the afferent conduit and flow activity inside the aneurysm for the symmetric and asymmetric outflow cases were compared in detail, and the case that is susceptible to thrombosis was identified. In addition, the onset of transition from laminar to turbulent flow inside the aneurysm was evidenced by the presence of vortex breakdown and the steep increase in the fluctuation level. Finally, the effect of pulsation on the flow pattern in the aneurysm was examined.     

Solution of three-dimensional problems of elasticity theory using new formulas for harmonic polynomials

Approximate solutions of three-dimensional problems of elasticity theory are sought in the form of linear combinations of vector functions each of which satisfies a differential equation. The linear-combination coefficients are found by energy minimization of the difference between exact and approximate solutions. This can be realized in the first and second basic problems. Simple recursion relations and differentiation formulas for similar harmonic polynomials are obtained. The above-mentioned vector functions are constructed using these formulas and the Trefftz representation. The problem of a truncated pyramid is considered. Odessa University, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 4, pp. 11–18, April, 1999.