Multivalued solutions of the spatial problems of nonlinear deformation of thin curvilinear rods

We have developed a finite-element model to study the spatial deformation of elastic rods at large displacements. A numerical algorithm for constructing multivalued nonlinear solutions in the presence of many bifurcation and limit points is formulated. Results of a study of the stability supercritical equilibrium forms of elastic rods, which were supported experimentally, are reported. Chaplygin Siberian Aviation Institute, Novosibirsk 630051. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 141–149, March–April, 1998.     

Geometrical nonlinear waves in finite deformation elastic rods

IntroductionSomenewphenomenaofnonlinearwavesinthesolidmediumsuchasshockwave ,solitarywaveetc.arepaidmoreattentiontoincreasinglybyresearchersbecausetheytakeonalotofimportantproperties.ItistheoreticallyanalyzedinRefs.[1 -6]thattheformationmechanismsofshockwaveandsolitarywaveintheelasticthinrodsaswellastheirpropagationproperties.TheexistenceofsolitarywaveintheelasticmediumsuchasarodandaplatehasbeenverifiedinRef.[7]byexperiments.Shockwaveandsolitarywavearesteadilypropagatingtraveling_wavesgenerat…     

Three kinds of nonlinear dispersive waves in elastic rods with finite deformation

On the basis of classical linear theory on longitudinal, torsional and flexural waves in thin elastic rods, and taking finite deformation and dispersive effects into consideration, three kinds of nonlinear evolution equations are derived. Qualitative analysis of three kinds of nonlinear equations are presented. It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, corresponding to solitary wave or shock wave solutions, respectively. Based on the principle of homogeneous balance, these equations are solved with the Jacobi elliptic function expansion method. Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions. These conclusions are consistent with qualitative analysis.     

Analysis of nonlinear large deformation problems by boundary element method

In this paper, the author deduces an approximate solution of nonlinear influence function in rate form for two-dimensional elastic problems on current configuration by the method of comoving coordinate system.Here BEM formulation of large deformation based on Chen’s theory[1] is given. The computational processes of nonlinear boundary integral equation is discussed. The author also compiles a nonlinear computing program NBEM. Numerical examples show that the results presented here is available to the solution of engineering problems.     

On the geometric phase in the spatial equilibria of nonlinear rods

Geometric phases have natural manifestations in large deformations of geometrically exact rods. The primary concerns of this article are the physical implications and observable consequences of geometric phases arising from the deformed patterns exhibited by a rod subjected to end moments. This mechanical problem is classical and has a long tradition dating back to Kirchhoff. However, the perspective from geometric phases seems to go more deeply into relations between local strain states and global geometry of shapes, and infuses genuinely new insights and better understand-ing, which enable one to describe this kind of deformation in a neat and elegant way. On the other hand, visual represen-tations of these deformations provide beautiful illustrations of geometric phases and render the meaning of the abstract concept of holonomy more direct and transparent.     

Elementary solutions of plane nonlinear filtration problems

In the solution of plane problems of filtration theory it is important to study the behavior of the solution near the singular points of the boundary of the flow region (corner points, points of boundary-condition change, and so on) and at infinity (see, for example, [1]). In the present study, this analysis is made for nonlinear filtration problems.Just as in the analogous problems of gasdynamics [2, 3] and nonlinear elasticity theory [4], to find the singular solutions we make the transformation to the filtration velocity hodograph plane. Examples relating basically to filtration with the limiting gradient are presented.The authors wish to thank I. I. Eremlna, T. N. Ericheva, and T. N. Ivanova for assistance in the calculations.     

EXISTENCE OF POSITIVE SOLUTIONS FOR A CLASS OF SINGULAR TWO POINT BOUNDARY VALUE PROBLEMS OF SECOND ORDER NONLINEAR EQUATION

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Numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity

An algorithm for the numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity is considered. The algorithm is based on separation of a power-law dependence from the finite-element solution in a neighborhood of singular points in the domain under study, where singular solutions are possible. The obtained power-law dependencies allow one to conclude whether the stresses have singularities and what the character of these singularities is. The algorithm was tested for problems of classical elasticity by comparing the stress singularity exponents obtained by the proposed method and from known analytic solutions. Problems with various cases of singular points, namely, body surface points at which either the smoothness of the surface is violated, or the type of boundary conditions is changed, or distinct materials are in contact, are considered as applications. The stress singularity exponents obtained by using the models of classical and asymmetric elasticity are compared. It is shown that, in the case of cracks, the stress singularity exponents are the same for the elasticity models under study, but for other cases of singular points, the stress singularity exponents obtained on the basis of asymmetric elasticity have insignificant quantitative distinctions from the solutions of the classical elasticity.