Reduced equilibrium equations for perfectly elastic materials

Sufficient conditions are given on the coordinate systems which enable reduced equilibrium equations to be derived for perfectly elastic materials involving deformations which depend in an essential way only on two of the three coordinates. Reduced equilibrium equations given previously for plane and axially symmetric deformations are special cases of the equations given here. These equations considerably reduce the calculations involved in investigating possible solutions of finite elasticity, either exact semi-inverse solutions or approximate perturbation solutions. Moreover a formula for the pressure function appearing in the reduced equilibrium equations is given which relates to the corresponding pressure function associated with the inverse deformation. This formula is similar to one given previously for fully three dimensional deformations.     

Tensile stability of cylindrical membranes

The tensile stability of rotationally symmetric thin membranes composed of isotropic, incompressible and elastic materials is considered by investigating under what conditions the initial equilibrium configuration can bifurcate to another rotationally symmetric equilibrium mode.The general equilibrium equations of a rotationally symmetric membrane are first derived in cylindrical coordinates. The initially cylindrical membrane is studied. The classic solution, which assumes homogeneous deformations, is shown to be a special case of the general equations. Perturbation theory is employed to find the bifurcation points from the homogeneous mode.This study shows that, for the chosen boundary conditions, no rotationally symmetric equilibrium mode exists near the cylindrical mode except the cylindrical mode itself. This corresponds to all experimental data that the author is aware of. The initially cylindrical membrane either remains cylindrical or goes into a non-rotationally symmetric mode.     

Stability analysis of helical rod based on exact Cosserat model

Helical equilibrium of a thin elastic rod has practical backgrounds, such as DNA, fiber, sub-ocean cable, and oil-well drill string. Kirchhoff's kinetic analogy is an effective approach to the stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff's theory without the extension of the centerline and the shear deformation of the cross section are not adoptable to real soft materials of biological fibers. In this paper, the dynamic equations of a rod with a circular cross section are established on the basis of the exact Cosserat model by considering the tension and the shear deformations. Euler's angles are applied as the attitude representation of the cross section. The deviation of the normal axis of the cross section from the tangent of the centerline is considered as the result of the shear deformation. Lyapunov's stability of the helical equilibrium is discussed in static category. Euler's critical values of axial force and torque are obtained. Lyapunov's and Euler's stability conditions in the space domain are the necessary conditions of Lyapunov's stability of the helical rod in the time domain.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

Anti-plane Shear Deformations in Compressible Transversely Isotropic Materials

Anti-plane shear deformations in a compressible, transversely isotropic hyperelastic material are under investigation. The displacement is assumed to be along the direction of the symmetry axis and is independent of the axial position. The resulting equations of equilibrium form an overdetermined system of partial differential equations for which solutions do not exist in general. Necessary and sufficient conditions are derived for such materials to sustain anti-plane shear deformations in the sense that every solution to the axial equation automatically satisfies the other two in-plane equations. Comparison is made with results for isotropic materials. A weaker version of the conditions specialized to axisymmetric anti-plane shear deformations is also obtained.     

Stabilization of beam parametric vibrations with shear deformations and rotary inertia effects

The purpose of this theoretical work is to present a stabilization problem of beam with shear deformations and rotary inertia effects. A velocity feedback and particular polarization profiles of piezoelectric sensors and actuators are introduced. The structure is described by partial differential equations with time-dependent coefficient including transverse and rotary inertia terms, general deformation state with interlaminar shear strains. The first order deformation theory is utilized to investigate beam vibrations. The beam motion is described by the transverse displacement and the slope. The almost sure stochastic stability criteria of the beam equilibrium are derived using the Liapunov direct method. If the axial force is described by the stationary and continuous with probability one process the classic differentiation rule can be applied to calculate the time-derivative of functional. The particular problem of beam stabilization due to the Gaussian and harmonic forces is analyzed in details. The influence of the shear deformations, rotary inertia effects and the gain factors on dynamic stability regions is shown.     

Stability of an elastic cube under dead loading: Two equal forces

A cube of incompressible neo-Hookean material undergoes a pure homogeneous deformation and is held in equilibrium by three specified pairs of equal and opposite forces, two of which are the same, applied normally to its faces and uniformly distributed over them. The possible equilibrium states are determined and the stability of each is studied with respect to arbitrary superposed infinitesimal deformations. The stability limits are found to be different from those obtained when only infinitesimal deformations having the same principal directions as those of the basic equilibrium state are considered. The differences arise from rotational and shearing types of instabilities that may occur in the general case. A critical inference is drawn concerning the nature of the dead loading conditions employed.     

One-dimensional deformations of nonlinearly elastic micropolar bodies

We find families of finite deformations of a Cosserat elastic continuum on which the system of equilibrium equations is reduced to a system of ordinary differential equations. These families can be used to describe the expansion, tension, and torsion of a hollow circular cylinder, cylindrical bending of a rectangular slab, straightening of a circular arch, reversing of a cylindrical tube, formation of screw and wedge dislocations in a hollow cylinder, and other types of deformations. In the case of a physically nonlinear material model, the above-listed families of deformations can be used to construct exact solutions of several problems of strong bending of micropolar bodies.     

Superposition of Generalized Plane Strain on Anti-Plane Shear Deformations in Isotropic Incompressible Hyperelastic Materials

The purpose of this research is to investigate the basic issues that arise when generalized plane strain deformations are superimposed on anti-plane shear deformations in isotropic incompressible hyperelastic materials. Attention is confined to a subclass of such materials for which the strain-energy density depends only on the first invariant of the strain tensor. The governing equations of equilibrium are a coupled system of three nonlinear partial differential equations for three displacement fields. It is shown that, for general plane domains, this system decouples the plane and anti-plane displacements only for the case of a neo-Hookean material. Even in this case, the stress field involves coupling of both deformations. For generalized neo-Hookean materials, universal relations may be used in some situations to uncouple the governing equations. It is shown that some of the results are also valid for inhomogeneous materials and for elastodynamics.     

Nonlinear analysis of members curved in space with warping and Wagner effects

The centroidal axis of a member that is curved in space is generally a space curve. The curvature of the space curve is not necessarily in the direction of either of the principal axes of the cross-section, but can be resolved into components in the directions of both of these principal axes. Hence, a member curved in space is primarily subjected to combined compressive, biaxial bending and torsional actions under vertical (or gravity) loading. In addition, warping actions in particular may occur in curved members with an open thin-walled cross-section, and as the deformations increase, significant interactions of the compressive, biaxial bending and torsional actions occur and profoundly nonlinear deformations are developed in the nonlinear range of structural response. This makes the nonlinear behaviour of a member curved in space very complicated, making it difficult to obtain a consistent differential equation of equilibrium for the nonlinear analysis of members curved in space. In addition, because torsion is one of the primary actions in these members, when the torsional deformations become large, the Wagner effects including both Wagner moment and the conjugate Wagner strain terms are increasingly significant and need to be included in the nonlinear analysis. This paper takes advantage of the merits of so-called “geometrically exact beam theory” and the weak form formulation of the differential equations of equilibrium in beam theory, and it develops consistent differential equations of equilibrium for the nonlinear elastic analysis of members curved in space with warping and Wagner effects. The application of the nonlinear differential equations of equilibrium to various problems is illustrated.     

On the one-dimensional modeling of camber bending deformations in active anisotropic slender structures

The paper presents a one-dimensional model for anisotropic active slender structures that captures arbitrary cross-sectional deformations. The 1-D geometrically-nonlinear static problem is derived by an asymptotic reduction process from the equations of 3-D electroelasticity. In addition to the conventional (bending–extension–shear–twist) beam strain measures, it includes a Ritz approximation to account for arbitrary deformation shapes of the finite-size cross-sections. As a particular case, closed-form analytical expressions are derived for the linear static equilibrium of a composite thin strip with surface-mounted piezoelectric actuators. This solution is based on a boundary-layer approximation to the static equilibrium equations in regions where Saint-Venant’s principle for elastic bodies cannot be applied and includes camber bending deformations to account for the local bimoments induced by the distributed actuation in a finite-width strip.     

Deformations of helical spring and cuboid into hollow cylinders

We study finite inhomogeneous deformations of a helical spring with a rectangular cross-section and a long cuboid. Two surfaces of the spring or the cuboid are joined to obtain a hollow cylinder. When body forces are absent the equilibrium equations reduce to ordinary differential equations. The stress-strain states are the same in each cross-section. The proposed deformations correspond to an inflation, an extension and a torsion of the obtained hollow cylinders. If the obtained cylinders are free of external applied loads, then they have residual stresses.     

In-plane stability of arches

Classical buckling theory is mostly used to investigate the in-plane stability of arches, which assumes that the pre-buckling behaviour is linear and that the effects of pre-buckling deformations on buckling can be ignored. However, the behaviour of shallow arches becomes non-linear and the deformations are substantial prior to buckling, so that their effects on the buckling of shallow arches need to be considered. Classical buckling theory which does not consider these effects cannot correctly predict the in-plane buckling load of shallow arches. This paper investigates the in-plane buckling of circular arches with an arbitrary cross-section and subjected to a radial load uniformly distributed around the arch axis. An energy method is used to establish both non-linear equilibrium equations and buckling equilibrium equations for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained. Approximations to the symmetric buckling of shallow arches and formulae for the in-plane anti-symmetric bifurcation buckling load of non-shallow arches are proposed, and criteria that define shallow and non-shallow arches are also stated. Comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate, and that classical buckling theory can correctly predict the in-plane anti-symmetric bifurcation buckling load of non-shallow arches, but overestimates the in-plane anti-symmetric bifurcation buckling load of shallow arches significantly.     

On the influence of prebuckling deformations on the buckling of thin elastic shells

A buckling theory valid for finite prebuckling deformations is presented for thin homogeneous, isotropic and elastic shells. It is subject to the restriction of the Kirchhoff hypothesis. A set of stability equations is derived by decomposing strain and stress components into four classes according to their characteristics.The influence of the prebuckling deformations on the buckling of thin circular cylindrical shells under lateral pressure is investigated with the aid of the basic equations derived above and the results are compared with the solutions of the Flügge equations and those obtained by Yamaki.     

Incremental Magnetoelastic Deformations,with Application to Surface Instability

In this paper the equations governing the deformations of infinitesimal (incremental) disturbances superimposed on finite static deformation fields involving magnetic and elastic interactions are presented. The coupling between the equations of mechanical equilibrium and Maxwell’s equations complicates the incremental formulation and particular attention is therefore paid to the derivation of the incremental equations, of the tensors of magnetoelastic moduli and of the incremental boundary conditions at a magnetoelastic/vacuum interface. The problem of surface stability for a solid half-space under plane strain with a magnetic field normal to its surface is used to illustrate the general results. The analysis involved leads to the simultaneous resolution of a bicubic and vanishing of a 7×7 determinant. In order to provide specific demonstration of the effect of the magnetic field, the material model is specialized to that of a “magnetoelastic Mooney–Rivlin solid”. Depending on the magnitudes of the magnetic field and the magnetoelastic coupling parameters, this shows that the half-space may become either more stable or less stable than in the absence of a magnetic field.      

Stability of nonlinearly elastic rods

This paper describes previously unknown stabilities and instabilities of planar equilibrium configurations of a nonlinearly elastic rod that is buckled under the action of a dead-load. The governing equations are derived from variational principles, including ones of isoperimetric type. Properties of stability are accordingly determined by study of the second variation. Stabilities to deformations both in the plane and out of the plane are considered.Among the newly discovered properties are: secondary bifurcation from the first buckled mode, marked differences between stability to two-dimensional and to three-dimensional variations, and the stabilizing influence of resistance to twist. In the isoperimetric examples, the analysis makes crucial use of a novel device to account for the dependence of the second variation on constraints.This research was supported by the U.S. National Science Foundation, the Army Research Office, the Air Force Office of Scientific Research, and the Office of Naval Research.     

Axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading

Summary Axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading systems are investigated by means of a so-called deformation map. The deformation map was further used for stability considerations of geometrically nonlinear shells, see Shilkrut [1, 2]. The map reveals the complete picture of the axially symmetric deformations and the stability of the investigated structure. The equilibrium differential equations for the above mentioned circular plate were derived by Timoshenko [3]. The boundary value problem of the investigated structure is transformed to an initial value problem (Cauchy's problem). Then the Runge-Kutta (R. K.) method can be used to solve numerically the equilibrium equations. The geometrically nonlinear, simply supported circular plate subjected to uniform radial force and uniform radial bending moment acting along the supported edge is investigated as example, and some new qualitative and quantitative results are obtained. This approach can be used without essential difficulties for the investigation of axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading systems in elastic and non-elastic fields.

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Axialsymmetrische Verformung und Stabilität geometrisch nichtlinearer Kreisplatten unter mehrparametrischer statischer Belastung

Übersicht Zur Untersuchung axialsymmetrischer, geometrisch nichtlinearer Verformung von Kreisplatten und ihrer Stabilität bei mehrparametrischer Belastung wird eine sog. Deformationskarte benutzt. Sie wurde auch für Stabilitätsbetrachtungen geometrisch nichtlinearer Schalen benutzt, s. Shilkrut [1,2]. Die Karte zeigt das vollständige Bild der axialsymmetrischen Verformung und die Stabilität der untersuchten Struktur. Das Randwertproblem zu den differentiellen Gleichgewichtsbedingungen, die für die betrachtete Platte von Timoshenko [3] hergeleitet wurden, wird in ein Anfangswertproblem (Caudy-Problem) überführt, welches numerisch nach der Methode von Runge-Kutta gelöst wird. Als Beispiel wird die nichtlineare Kreisplatte unter radialer Zug-und Biegemomentenbelastung am einfach gestützten Umfang untersucht, und man erhält einige neue qualitative und quantitative Ergebnisse. Die Methode läßt sich ohne wesentliche Schwierigkeiten auch auf axialsymmetrische, nichtlineare Verformungen und die Stabilität von Kreisplatten unter anderen mehrparametrischen statischen Belastungen im elastischen und nichtelastischen Bereich anwenden.

     

On stability and non-uniqueness of the dilatation states of a compressible isotropic unit cube subjected to dead loading

This paper considers a unit elastic cube, made of compressible isotropic material, with its faces subjected to certain dead-load tractions that produce a possible equilibrium state of non-uniform dilatation. It is seen that, at the considered equilibrium state, the cube material acquires properties of pseudo-transverse isotropy. Conditions are obtained for the stability of such an equilibrium state with respect to superimposed pure homogeneous deformations having principal directions parallel to the cube edges. The problem of non-uniqueness of the cube dilatation states is also addressed, and non-uniqueness is illustrated in an example application dealing with an isotropic cube made of the Blatz-Ko material. The nature and the stability features of these equilibrium states are studied in depth.     

Stability for manifolds of equilibrium state of generalized Hamiltonian system with additional terms

For a generalized Hamiltonian system with additional terms, stability for the manifolds of the equilibrium state is presented. Equilibrium equations, disturbance equations and the first approximate equations of the system are given. A theorem for the stability of the manifolds of the equilibrium state of a general autonomous system is used for the generalized Hamiltonian systems with additional terms, and three propositions on the stability of the manifolds of the equilibrium state of the system are obtained. An example is given to illustrate the application of the method and results. At last, we study the stability for manifolds of the equilibrium state of the Euler equations of a rigid body subjected to external moments of force, by using of the method in this paper.