Elastica Solution Research Articles

On the Cauchy Problem for a Dynamical Euler's Elastica

The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order elliptic equation for its tension. A Hasimoto transformation is used to rewrite the equations in convenient coordinates for the space and time derivatives of the tangent vector. A feature of this elastica is that it exhibits time-dependent monodromy. A frame parallel-transported along the elastica is in general only quasi-periodic, resulting in time-dependent boundary conditions for the coordinates. This complication is addressed by a gauge transformation, after which a contraction mapping argument can be applied. Local existence and uniqueness of elastica solutions are established for initial data in suitable Sobolev spaces.

Three-dimensional analysis of alveolar structure in usual interstitial pneumonia

The etiology of usual interstitial pneumonia (UIP), a progressive lung disease, remains unclear. We examined alveolar structure in UIP three-dimensionally. Lung biopsy specimens from five patients with idiopathic pulmonary fibrosis were used. Sections 150-microm thick were stained with elastica solution for elastic fibers, with alpha-smooth muscle actin antibody for myofibroblasts, with anti-Thomsen-Friedenreich antibody for type-II pneumocytes and with anti-CD34 antibody for blood vessels. We examined them three-dimensionally using a laser confocal microscope or light microscope. In the fibrotic lesions, the thick elastic fibers forming the alveolar framework were not particularly dense considering the reduction in alveolar volume. Near the fibrotic lesions, some of the thin elastic fibers in the alveolar wall were slightly sinuous and ended with rounded tips. Type-II pneumocytes had proliferated and were distributed uniformly over the alveolar surface. Smooth muscle actin filaments were detected only around the alveolar orifice. These findings show that in UIP destruction of the elastic fiber framework of the alveoli may lead to irreversible focal alveolar collapse after damage to the alveolar epithelial cells, and proliferation of type-II pneumocytes may be involved with this elastolysis.

Behaviour of the extensible elastica solution

The general form of the virtual work expression for the large strain Euler–Bernoulli beam theory is derived using the nominal strain (Biot's) tensor. From the equilibrium equations, derived from the virtual work expression, it turns out that a linear relation between Biot's stress tensor and the (Biot) nominal strain tensor forms the differential equation used to derive the elastica solution. Moreover, in the differential equation one additional term enters which is related to the extensibility of the beam axis. As a special application, the well-known problem of an axially loaded beam is analysed. Due to the extensibility of the beam axis, it is shown that the buckling load of the extensible elastica solution depends on the slenderness, and it is of interest that for small slenderness the bifurcation point becomes unstable. This means the bifurcation point changes from being supercritical, which always hold for the inextensible case, i.e. the classical elastica solution, to being a subcritical point. In addition, higher order singularities are found as well as nonbifurcating (isolated) branches.

NONLINEAR BUCKLING OF MARINE ELASTICA PIPES TRANSPORTING FLUID

This paper addresses the nonlinear buckling and post-buckling behavior of an extensible marine elastica pipe conveying fluid. The mathematical model employed in the nonlinear buckling analysis is developed based on the extensible elastica theory and the large strain formulation, so that the high extensibility of the pipe due to large axial strains is tackled thoroughly. The boundary value problem of the model is solved by the shooting method, and the numerical elastica solutions are obtained. For stability examination, the method of adjacent nonlinear equilibrium is exploited. It is revealed that the fundamental mode of nonlinear buckling of the pipe is reached when the pipe experiences either the critical top tension or the critical weight. Postbuckling behavior of the pipe is recognized to be unstable. The investigation is extended to studying various parameters that impinge on the limit states of the pipe. These parameters are the dimensionless quantities that relate to density of pipe material, densities of external and internal fluids, applied top tension, Poisson's ratio, slenderness ratio, vessel offset, seawater depth, current-drag coefficients, current velocity, and internal flow velocity.

On the Post-buckling Behavior of Elastic Fixed-End Column with Central Brace

This paper investigates the large displacement post-buckling behavior of a column with central brace using the elastica solution. The method is able in predicting the buckling load and in tracing the exact post-buckling paths, both primary and secondary paths. Depending on the brace stiffness, different shapes of symmetrical and anti-symmetrical post-buckling paths can be shown to exist. Whilst previous work reported only one threshold value of brace stiffness, the results also show the existence of infinite number of threshold values, each indicates a change in the post-buckling behavior.

2403 有限要素法による梁の3次元大変形解析(C01-1/解析コンテスト(1))(C01/解析コンテスト)

Finite element method using SO(3), which is named special orthogonal group, is formulated by decomposing deformation gradient tensor into SO(3) and stretch tensor. Then, tangent stiffness' using four types of stress rates, which are Truesdell, Jaumann, Green and Linear stress rates, are obtained. Next, euler-buckling analyses and post-buckling analyses using beam elements of this formulation are made. As a result, the buckling loads of the beam element using Linear stress rate are perfectly the same as Timoshenko's theoretical solutions and the results of post-buckling analyses of the beam element using Linear stress rate are also perfectly the same as the solutions of elastica.

Elastica solution for the hygrothermal buckling of a beam

An elliptic integral solution for the post-buckling response of a linear-elastic and hygrothermal beam fully restrained against axial expansion is presented. Whereas in the classical solutions the extension of the beam can be neglected, a well-posed formulation of the title problem must include the extension. The solution for the limiting case of a string is presented. The present solution shows that the magnitude of the compressive axial load is a maximum at the onset of buckling and decreases as the potential for free expansion is increased; this is in contrast to the approximate solutions found in the literature.

Analytical bending solutions of elastica with one end held while the other end portion slides on a friction support

Treated herein is an elastica under a point load. One end of the elastica is fully restrained against translation, and elastically restrained against rotation, while the other end portion is allowed to slide over a friction support. The considered elastica problem belongs to the class of large-deflection beam problems with variable deformed arc-lengths between the supports. To solve the governing nonlinear differential equation together with the boundary conditions, the elliptic integral method has been used. The method yields closed-form solutions that are expressed in a set of transcendental equations in terms of elliptic integrals. Using an iterative scheme, pertinent bending results are computed for different values of coefficient of friction, elastic rotational spring constant and loading position, so that their effects may be examined. Also, these accurate solutions provide useful reference sources for checking the convergence, accuracy and validity of results obtained from numerical methods and software for large deflection beam analysis. It is interesting to note that this class of elastica problem may have two equilibrium states; a stable one and an unstable one.

Analytical solution of spatial elastica and its application to kinking problem

An analytical solution is presented for the spatially large deformation of a thin elastic rod (spatial elastica) which is naturally straight and uniform with equal principal stiffnesses and is subjected to terminal loads. The elastica can suffer not only flexure and torsion as in the classical Kirchhoff theory, but also extension and shear. The present solution is expressed in integral form and described in terms of only four parameters. This solution clears the difficulty with the polar singularity in the use of Euler angles. Hence, the numerical analysis is possible for various boundary value problems with no limitation.In this paper we study the post-buckling behavior of an elastica under the terminal twist and uniaxial end-shortening, and give a theoretical explanation to commonly observed phenomena such as secondary bifurcation, formation of a kink, snap-through behavior. The contact problem is analyzed in the case where the elastica contacts with itself and forms a kink. These results are available for other analysis, e.g., based on finite element approximations.

Elastic nonlinear behavior of truss system under follower and non-follower forces

This paper presents a nonlinear analysis of elastic plane and space trusses, incorporating buckling of individual member(s) of the trusses. The nonlinear equilibrium equations are expressed as relations between force intensity and positions of free nodes. Using this approach, the nonlinear behavior under both follower and non-follower forces can be formulated in a concise and simple way. When a member buckles, an additional degree of freedom, the (equal) buckling angles of end nodes appear. At this state, a member has two governing differential equations. From the first governing differential equation and using elastica solution, its corresponding nonlinear equilibrium equation is derived. On the other hand, the second governing differential equation leads to a nonlinear relation between buckling angle and positions of end nodes which serves as an additional equation to accommodate the above additional degree of freedom. The versatility of the proposed analysis is first applied to simple two bar plane truss, under both follower and non-follower forces and then to more complex plane and space trusses.

Dynamic Aspects of Wetting Balance Tests

The relationships between the force measured during wetting balance tests and the observed changes of contact angle and meniscus shape are studied. Experiments using silicone oil at 25, 50, and 100°C on glass plates as well as Pb-Sn eutectic solder on Au-coated glass plates are reported. Discrepancies between the measured force and height and those expected for a static meniscus are detailed. Equilibrium meniscus shapes are computed for wide plates using the elastica solution and for narrow plates using the public-domain software package, “Surface Evolver.” For room temperature experiments with oil, the measured force discrepancy disappears when the meniscus rise is complete. Thus, the force discrepancy may be due to shear stress exerted on the sample by fluid rising up the sample. For static menisci with heated liquids, force and meniscus height discrepancies do not disappear when the meniscus rise is complete. These discrepancies can be explained by Marangoni flow due to temperature gradients in the fluid for the oil experiments but not for the solder experiments.

On the critical and postcritical behaviour of plane frames

Abstract The aim of the present paper is to show an application of some results of the classical elastica solution for exact critical and postcritical analysis of plane frames. To this effect equilibrium and compatibility equations of perturbed frame configurations are derived. They are solved numerically to provide postcritical equilibrium paths and critical points for both perfect and imperfect structures. It turns out that all the postcritical frame parameters can be expressed by a single properly chosen independent variable, say displacement or angle of rotation of a characteristic cross-section. Finally, the total potential energy function at the critical branching point of a simple frame is derived to show that it exhibits a degenerate type of double cusp catastrophe.

Three‐Dimensional Simulations of Initially Straight Elasticas

The governing nonlinear vector differential equations for large deflections of an end‐loaded, three‐dimensionally deformed, initially straight, axisymmetric elastica are developed and simulated numerically without decomposition. All simulations are based on repeated applications of truncated Taylor's expansions to advance along a deformed elastica. Both initial value problems, and two‐point boundary value problems with a corresponding shooting method are discussed. A number of examples are solved and compared with corresponding theoretical solutions, and with a finite element solution. An elastica solution with at least six independent equilibrium configurations satisfying a single set of boundary conditions and not heretofore treated in the literature is also presented.

Elastica Solution of Braced Struts

An exact formulation is derived for the buckling of structs supported elastically at the midspan. The formulation is based on the large deflection theory of the beam column. Two coupled nonlinear equations involving definite integrals govern the postbuckling solution. The equations are solved numerically with an iterative scheme. The postbuckling solutions are found for the range of support stiffness of interest. The load‐deflection history of the postbuckling path, as well as the postbuckling shape, is traced. As the support stiffness increases from zero, the postbuckling path changes from stable to unstable. This finding confirms the results of a previous study using a static perturbation method. The current results, however, also reveal the rate of change of the load‐carrying capacity along the postbuckling path away from the buckling load. For the range of support stiffness that renders the postbuckling path unstable, the rate of reduction of the load‐carrying capacity along the path is found to be small.

Elliptic integral solutions of plane elastica with axial and shear deformations

Closed-form solutions are derived for elastica with axial and shear deformations, using elliptic integrals. The theories used here are the Timoshenko beam theory of finite displacements with finite strains and that with small strains. Elliptic integral solutions are further transformed to normal forms to obtain accurate solutions. As a result, both of the closed-form solutions are expressed in normal forms from the first to the third kind of elliptic integrals. With these solutions, two kinds of structures are analyzed to examine the effect of shear deformations along with the accuracy of the approximate theory of finite displacements with small strains.