Imperfection sensitivity of hilltop branching points of systems with dihedral group symmetry

Imperfection sensitivity of a hilltop branching point occurring as a coincidence of a limit point and a double bifurcation point of a finite-dimensional, elastic, conservative system equivariant to the dihedral group is investigated. In the neighborhood of this point, the potential is expanded into a power series of independent state variables, loading parameter and imperfection magnitude. The form of the expansion is determined through exploitation of dihedral-group symmetry. For the perfect system, the hilltop branching point and bifurcated paths are shown to be all unstable. For an imperfect system, equilibrium paths in general break into a series of paths: including fundamental, complementary and aloof paths. The imperfection sensitivity laws for maximum (critical) points of loading on these paths are obtained as a novel finding of this paper. Critical points on the fundamental and complementary paths enjoy a piecewise linear law, which is less severe than a one-half or two-thirds power law for the double bifurcation point. By contrast, maximum points on aloof paths suffer more severe sensitivity. The hilltop branching point thus displays complex system of imperfection sensitivities. As numerical examples, imperfection sensitivity of simple structural models with the hilltop point is investigated to ensure the validity of the present formulation.     

Singularities of homogeneous deformations of constrained hyper-elastic materials

Singularity analysis is performed for homogeneous deformations of any hyper-elastic, constrained anisotropic material, under any type of conservative quasi-static loading. Critical conditions for branching of the equilibrium paths are defined and their post-critical behavior is discussed. Classification of the simple (cuspoids) and compound (umbilics) singularities of the total potential energy function is effected. The theory is implemented into an umbilic elliptical singularity of an isotropic and totally inextensible unit cube under normal loading.     

DIVERGENCE INSTABILITY OF VARIABLE-ARC-LENGTH ELASTICA PIPES TRANSPORTING FLUID

A flexible elastic pipe transporting fluid is held by an elastic rotational spring at one end, while at the other end, a portion of the pipe may slide on a frictional support. Regardless of the gravity loads, when the internal flow velocity is higher than the critical velocity, large displacements of static equilibrium and divergence instability can be induced. This problem is highly nonlinear. Based on the inextensible elastica theory, it is solved herein via the use of elliptic integrals and the shooting method. Unlike buckling with stable branching of a simply supported elastica pipe with constant length, the variable arc-length elastica pipe buckles with unstable branching. The friction at the support has an influence in shifting the critical locus over the branching point. Alteration of the flow history causes jumping between equilibrium paths due to abrupt changes of direction of the support friction. The elastic rotational restraint brings about unsymmetrical bending configurations; consequently, snap-throughs and snap-backs can occur on odd and even buckling modes, respectively. From the theoretical point of view, the equilibrium configurations could be formed like soliton loops due to snapping instability.     

Complete equilibrium paths for Mises trusses

This paper presents determination of equilibrium paths for Mises trusses with different ratio of height to span. Unsymmetrical deformation modes are considered and the structure is treated as a two DOF system. First, a few special equilibrium configurations are resolved from considerations of free body diagrams. Complete equilibrium paths are determined by solving numerically the governing non-linear equilibrium equations. The stability of possible equilibrium configurations is checked using the second partial derivative test for the total potential energy. The positive definiteness of the appropriate Hessian matrices is checked numerically using the Sylvester criterion.     

On the singularities of a constrained (incompressible-like) tensegrity-cytoskeleton model under equitriaxial loading

Singularity theory is applied for the study of the characteristic three-dimensional tensegrity-cytoskeleton model after adopting an incompressibility constraint. The model comprises six elastic bars interconnected with 24 elastic string members. Previous studies have already been performed on non-constrained systems; however, the present one allows for general non-symmetric equilibrium configurations. Critical conditions for branching of the equilibrium are derived and post-critical behaviour is discussed. Classification of the simple and compound singularities of the total potential energy function is effected. The theory is implemented into the cusp catastrophe for the case of one-dimensional branching of the buckling-allowed tensegrity model, and an elliptic umbilic singularity for compound branching of a rigid-bar model. It is pointed out that singularity studies with constraints demand a quite different mathematical approach than those without constraints.     

Perturbation formulation of continuation method including limit and bifurcation points

This paper gives the perturbation formulation of continuation method for nonlinear equations. Emphasis is laid on the discussion of searching for the singular points on the equilibrium path and of tracing the paths over the limit or bifurcation points. The method is applied to buckling analysis of thin shells. The pre-and post-buckling equilibrium paths and deflections can be obtained, which are illustrated in examples of buckling analysis of cylindrical and toroidal shells.     

Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells

We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching and self-intersecting shells. The conditions are derived by performing direct through-the-thickness integration in the global equilibrium conditions of continuum mechanics. At each regular internal and boundary point of the base surface our exact, local equilibrium equations and dynamic boundary conditions are equivalent, as expected, to the ones known in the literature. As the new equilibrium relations we derive the exact, resultant dynamic continuity conditions along the singular surface curve modelling the branching and self-intersection as well as the dynamic conditions at singular points of the surface boundary. All the results do not depend on the size of shell thicknesses, internal through-the-thickness shell structure, material properties, and are valid for an arbitrary deformation of the shell material elements.     

Brittle fracture dynamics with arbitrary paths III. The branching instability under general loading

The dynamic propagation of a bifurcated crack under arbitrary loading is studied. Under plane loading configurations, it is shown that the model problem of the determination of the dynamic stress intensity factors after branching is similar to the anti-plane crack branching problem. By analogy with the exact results of the mode III case, the energy release rate immediately after branching under plane situations is expected to be maximized when the branches start to propagate quasi-statically. Therefore, the branching of a single propagating crack under mode I loading should be energetically possible when its speed exceeds a threshold value. The critical velocity for branching of the initial single crack depends only weakly on the criterion applied for selecting the paths followed by the branches. However, the principle of local symmetry imposes a branching angle which is larger than the one given by the maximum energy release rate criterion. Finally, it is shown that an increasing fracture energy with the velocity results in a decrease in the critical velocity at which branching is energetically possible.     

Effect of thermal radiation exchange between boundaries on convection in a gas heated from below

Natural convection problems offer many examples of branching of the solutions [1]. Usually, such branching (from the standpoint of catastrophe theory) can be described by a Whitney fold or cusp. A characteristic feature of nontrivial branching is the presence of some small but finite disturbance of the convective equilibrium conditions. In this study the perturbation disturbing the convective equilibrium of a fluid heated from below is Stefan-law thermal radiation exchange between the boundaries of the enclosure. Natural convection with lateral heating and allowance for radiative heat transfer was previously investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 47–51, September–October, 1992.     

Numerical Investigation of the Plane Problem of Convection of a Multicomponent Fluid in a Porous Medium

Scenarios of the development of continuous families of steady-state regimes branching off from mechanical equilibrium are investigated for the plane problem of filtrational convection of a multicomponent fluid saturating a porous block of rectangular cross-section. Convection of two- and three-component fluids is considered and unidirectional and differently directed vertical temperature and concentration gradients are analyzed. A new scenario of the formation of a continuous family of steady-state solutions realized in the case of oscillatory instability of mechanical equilibrium is studied.     

Numerical investigation of the first transition in the three-dimensional problem of convective flow in a porous medium

The branching off of steady-state regimes from mechanical equilibrium is studied for the problem of filtration convection in a parallelepiped. The conditions for the geometric parameters under which stable continuous families of steady-state regimes develop are found. The stability of equilibria of the family with respect to three-dimensional perturbations is analyzed in a numerical experiment using a finite-difference method.     

Bifurcational instability of an atomic lattice

In a recent article N.H. Macmillan and A. Kelly (1972) have confirmed on the basis of a linear eigenvalue analysis that a mechanically stressed perfect crystal can exhibit a bifurcational instability at stresses ranging to 20 per cent below that of the limiting maximum of the primary stress-strain curve. The question thus arises as to whether the branching point is in a non-linear sense either stable or unstable. In the former case, perfect and slightly imperfect crystals would be capable of sustaining stresses over and above the eigenvalue critical stress. In the unstable case, however, this eigenvalue stress would represent the ultimate strength of a perfect solid, while an imperfect crystal would fail at a limiting stress substantially below the eigenvalue.At 20 per cent below the limit point such a branching point is essentially distinct, and the non-linear stability analysis needed to answer this question is provided by a recently established general branching theory for discrete conservative systems. Often, however, the two critical equilibrium states are much nearer than this, and the branching theory is here suitably extended to cover the case of near-compound instabilities.An illustrative study of a close-packed crystal under uniaxial tension is next presented. A kinematically-admissible displacement field is employed and a bifurcation point is located on the primary equilibrium path just before the limiting maximum, the eigenvector being associated with a transverse shearing strain. Under these conditions a corresponding small transverse shearing stress would represent an ‘imperfection’, and the non-linear branching problem is next studied using the new general theory. This shows (in excellent quantitative agreement with an ad hoc numerical solution) that the branching point is non-linearly unstable with a quite severe imperfection-sensitivity which manifests itself as a sharp cusp on the failure-stress locus.     

Post-buckling of a hinged-fixed beam under uniformly distributed follower forces

Based on geometrically non-linear theory for extensible elastic beams, governing equations of statically post-buckling of a beam with one end hinged and the other fixed, subjected to a uniformly distributed, tangentially compressing follower forces are established. They consist of a boundary-value problem of ordinary differential equations with a strong non-linearity, in which seven unknown functions are contained and the arc length of the deformed axis is considered as one of the basic unknown functions. By using shooting method and in conjunction with analytical continuation, the non-linear governing equations are solved numerically and the equilibrium paths as well as the post-buckled configurations of the deformed beam are presented. A comparison between the results of conservative system and that of the non-conservative systems are given. The results show that the features of the equilibrium paths of the beams under follower loads are evidently different from that under conservative ones.     

Instability of Dual Eigenvalue Fourth-Order Systems

Investigations of the postbuckling behavior of an elastic structure at a twofold branching point generally involve a potential of the cubic type. However, when the cubic part vanishes identically (for example, when there is symmetry in both active coordinates), the potential becomes of the quartic type. Here, quartic potentials in normalized coordinates are considered and formulas for limit points and bifurcations on imperfect paths are given in terms of trigonometric polynomials. These are used in certain structural examples to show that a theorem of Ho is invalid unless extra conditions are placed on the potential. They are also used in a proof of the modified Ho theorem.     

Homogeneous Equilibrium Configurations of a Hyperelastic Compressible Cube under Equitriaxial Dead-Load Tractions

Non-uniqueness, bifurcation and stability of homogeneous solutions to the equilibrium problem of a hyperelastic cube subject to equitriaxial dead-load tractions are investigated. Besides the basic and theoretical questions raised by the analysis, the study is motivated by the somewhat surprising feature of this nonlinear problem for which the symmetric load may give rise to asymmetric stable deformations. In reality, the equilibrium problem, formulated for general homogeneous compressible isotropic materials with polyconvex energy function, may exhibit primary and secondary bifurcations. A primary bifurcation occurs when there exist paths of equilibrium states that bifurcate from the primary path of three equal principal stretches. These bifurcation branches have two coinciding stretches and along them, through secondary bifurcations, other completely asymmetric bifurcation branches, which are characterized by all three stretches different, may risen. In this case, the cube transforms into an oblique parallelepiped. With increasing loads, they are also possible discontinuous paths of equilibria which evince prompt jumps in the deformation process. Of course, the set of asymmetric solutions admitted by the equilibrium problem depends on the specific form of the stored energy function adopted. In this paper, expressions governing the global development of asymmetric equilibrium branches are derived. In particular, conditions to have bifurcation points are individualized. For compressible neo-Hookean and Mooney-Rivlin materials a wide parametric analysis is carried out showing by means of graphs the most interesting branches. Finally, using the energy criterion, a detailed study is performed to assess the stability of the computed solutions.      

受均布载荷压杆后屈曲形态的数值计算

对受均布载荷压杆的屈曲及后屈曲行为进行了分析.基于杆的大变形理论,考虑杆的轴向伸长,建立了受均布载荷作用下细长压杆的几何非线性平衡方程.采用打靶法和解析延拓法数值求解非线性两点边值问题,得到了杆的后屈曲平衡路径和平衡构形.     

扰动法在结构分枝失稳分析中的应用

对结构进行平衡路径的跟踪分析，是全面了解该结构的受力性能所必须进行的一项工作。目前的结构非线性稳定分析技术一般仅对极值点失稳型问题较为有效，而对分枝点失稳型问题则困难较多。对于具有缺陷敏感性的结构，如拱结构、壳体等，在普通荷载作用下，其失稳路径常包含分枝点。文献[1]提出并认为位移扰动法和力扰动法在分析结构分枝失稳时具有很好的效果。本文采用多个不同类型的算例，对扰动法在结构分枝失稳问题中的应用进行了分析比较，表明该方法具有较强的跟踪能力。最后，就扰动法在结构分枝失稳问题中的应用提出几点建议。     

On the geometric conditions for multiple stable equilibria in clamped arches

Curved structures, such as beams, arches, and panels are capable of exhibiting snap-through buckling behavior when loaded laterally, that is they can exhibit multiple stable equilibria, sometimes after any external loading is removed. This is a consequence of highly nonlinear force-deflection relations with perhaps multiple crossings of the zero-force axis for typical equilibrium paths. However, the propensity to maintain a stable snapped-through equilibrium position (in addition to the nominally unloaded equilibrium configuration) after the load is removed depends on certain geometric properties. A number of clamped arches are used to illustrate the relation between geometry (essentially the shape) and corresponding equilibrium configuration(s), and especially those conditions for which the initial equilibrium configuration is the only stable shape possible. Furthermore, related results are obtained when a change in the thermal environment may cause a system to exhibit a stable snapped-through equilibrium even when the system at ambient thermal conditions does not. Some representative examples are produced using a 3D printer for verification purposes.     

Formal solution of quasi-static problems

When the quasi-static problem is defined by a set of differential equations complemented by initial and boundary conditions, the resulting quasi-static solutions may exhibit a limited reach over the time domain. On the other hand, the infinity of equilibrium paths that can be obtained in a general non-linear problem also indicates that a proper definition of the quasi-static solution must be provided. In inelasticity problems, this infinite number of equilibrium paths occur even when no dissipative bifurcations are present. In the present paper, a general solution for quasi-static problems in Solid Mechanics is defined and explored. Special attention is addressed to material non-linearities though geometric non-linearities are also covered by the definition. Earlier concepts of path and state stability are recovered in order to reduce the number of solutions to those that are physically acceptable. The important link with the original dynamic problem is accounted for by enforcing a preferential load direction. The resulting definition relies on a time-objective criterion with straightforward applicability to the most common numerical models. In the final part of the paper, simple 1D problems are used to illustrate some of the concepts introduced in the present developments.