Critical Curvature Research Articles

On a general constitutive description for the inelastic and failure behavior of fibrous laminates––Part I: Lamina theory

It is well known that a structural design with isotropic materials can be accomplished only based on a stress failure criterion. This is, however, generally not true with laminated composites. Only when the laminate is subjected to an in-plane load, the ultimate failure of the laminate can correspond to its last-ply failure, and hence a stress failure criterion may be sufficient to detect the maximum load that can be sustained by the laminate. Even in such case, the load shared by each lamina in the laminate cannot be correctly determined if the lamina instantaneous stiffness matrix is inaccurately provided, since the lamina is always statically indeterminate in the laminate. If, however, the laminate is subjected to a lateral load, its ultimate failure occurs before the last-ply failure and the only use of the stress failure criterion is no longer sufficient; an additional critical deflection or curvature condition must be employed as well. This necessitates development of an efficient constitutive relationship for laminated composites in order that the laminate strains/deflections until the ultimate failure can be accurately calculated. A general constitutive description for the thermo-mechanical response of a fibrous laminate up to the ultimate failure with applications to various fibrous laminates is presented in these two parts of papers. The constitutive relationship is obtained by combining the classical lamination theory with a recently developed bridging micromechanics model, through a layer-by-layer analysis. The present paper focuses on the lamina analysis. Attention has been given to the applicability of the constitutive theory to the fibrous laminates stacked with a wide variety of composite laminae, including multidirectional tape laminae, woven and braided fabric composites, and knitted fabric reinforced composites, which have different constituent behavior such as elasto-plasticity and elastic-visco-plasticity. The laminate analysis and the application examples will be presented in the subsequent paper.

Area-minimizing cones over the canonical imbedding of symmetric R-spaces

In this paper, using a criterion of Lawlor's (curvature criterion), we prove that the cones over the canonical embedding of symmetric R-spaces are area-minimizing.

Using Polyhedral Models to Automatically Sketch Idealized Geometry for Structural Analysis

Simplification of polyhedral models, which may incorporate large numbers of faces and nodes, is often required to reduce their amount of data, to allow their efficient manipulation, and to speed up computation. Such a simplification process must be adapted to the use of the resulting polyhedral model. Several applications require simplified shapes which have the same topology as the original model (e.g. reverse engineering, medical applications, etc.). Nevertheless, in the fields of structural analysis and computer visualization, for example, several adaptations and idealizations of the initial geometry are often necessary. To this end, within this paper a new approach is proposed to simplify an initial manifold or non-manifold polyhedral model with respect to bounded errors specified by the user, or set up, for example, from a preliminary F.E. analysis. The topological changes which may occur during a simplification because of the bounded error (or tolerance) values specified are performed using specific curvature and topological criteria and operators. Moreover, topological changes, whether or not they kept the manifold of the object, are managed simultaneously with the geometric operations of the simplification process.

Large deformation of thin films and layered flat panels: effects of gravity

A general theory is presented for the large deformation of thin films and layered flat panels in which gravitational forces have a marked influence on the evolution of curvature, shape and instability. Isotropic, linear elastic deformation is considered with small strains and moderate rotations. The thermomechanical properties of the layered material are allowed to vary through the panel thickness so as to derive a general result for multilayers and graded materials. Explicit analytical expressions are derived for the critical curvature and the critical “effective load” at which curvature bifurcation occurs. The analysis considers square, circular and rectangular panels that are simply supported at three points, with the thin film on the panel facing either up or down. A boundary layer analysis is presented for rectangular panels specifically to examine the effect of panel shape on curvature evolution and geometric stability. Computational simulations involving full three-dimensional hyperelastic formulations with large rotations and two-dimensional hyper elastic formulations with moderate rotations were used to assess the validity of the analytical results. Systematic experiments on the large deformation characteristics of flat glass panels with and without silicon nitride thin film deposits were carried out to check the predictive capabilities of the theory. The trends predicted by the theory and its quantitative predictions of bifurcation with and without thin film deposits on the panels were found to be in reasonable agreements with experiments. The limits of the solutions of the present theory for the special case of thin films on substrates with only mismatch strains are shown to converge to prior analytical results. Furthermore, the theory is shown to capture the experimental trends observed during large deformation in thin-film/substrate systems in the absence of gravitational effects.

Asymmetric Collapse Modes of Pipes under Combined Bending and External Pressure

The installation and operation of pipelines in high ambient pressure environments, such as deep water, requires design against the potential of collapse. The subject of pipe buckling and collapse under combined external pressure and bending is revisited in order to investigate the causes of angled buckles observed in recent experiments and the associated scatter in the critical loads. Stainless steel 304 tubes with diameter-to-thickness ratio of 18.3 are bent to collapse under various values of external pressure. Tubes bent at pressures higher than 0.72 exhibited angled buckles which were oriented at 20 deg to 45 deg to the axis of bending. In this pressure regime, the scatter in the results was larger than usual. At lower pressures, the tubes buckled in the expected mode with the flattening being along the axis of bending. A previously developed formulation is extended so that it can handle asymmetric imperfections and buckling modes. The analysis is first used to reproduce the experimental results and subsequently to study the problem parametrically. The orientation of the initial ovality is found to play a role in the final buckling mode and in the value of the critical curvature for bending under high values of pressure. In addition, residual stresses can interact with the initial ovality affecting the critical curvature both positively and negatively. For lower pressures these effects are small.

Inference of Mechanical Properties from Instrumented Depth Sensing Indentation at Tiny Loads and Indentation Depths

Different hardness measures as Vickers, Brinell and Meyer hardness are discussed with respect to their physical interpretation. Meyer hardness is found to be best suited as a measure of plastic properties. The outline of a depth sensing nanoindentation experiment is described, and particular emphasis is given to the correct deduction of the contact area from indenter penetration data. Experimental complications, as phase transformations, finite machine compliance, thermal and pieco drift, and sample creep are detailed with, and their impact on the calculation of hardness and elastic modulus is shown. Furthermore the onset of yielding in dependence on critical load, indenter curvature and yield strength is discussed.

A new approach for nonlinear distortion correction in endoscopic images based on least squares estimation.

Images captured with a typical endoscope show spatial distortion, which necessitates distortion correction for subsequent analysis. In this paper, a new methodology based on least squares estimation is proposed to correct the nonlinear distortion in the endoscopic images. A mathematical model based on polynomial mapping is used to map the images from distorted image space onto the corrected image space. The model parameters include the polynomial coefficients, distortion center, and corrected center. The proposed method utilizes a line search approach of global convergence for the iterative procedure to obtain the optimum expansion coefficients. A new technique to find the distortion center of the image based on curvature criterion is presented. A dual-step approach comprising token matching and integrated neighborhood search is also proposed for accurate extraction of the centers of the dots contained in a rectangular grid, used for the model parameter estimation. The model parameters were verified with different grid patterns. The distortion-correction model is applied to several gastrointestinal images and the results are presented. The proposed technique provides high-speed response and forms a key step toward online camera calibration, which is required for accurate quantitative analysis of the images.

An efficient indexing scheme for image storage and recognition

This paper presents a model-based vision system to achieve robust recognition of planar contours that are scale invariant of known models. Planar contours are partitioned into segments by using constant curvature criterion. A set of descriptors that are invariant with respect to scale, rotation, and translation are extracted from the geometric features of the segments. The descriptors are used to carry out an efficient indexed search over the models so as to reduce the search space. Fragments of contours extracted from partially occluded scenes can be individually matched by using the local shape descriptors. Pruning of large portions of the models is carried out by keeping only some matched classes which received the highest vote. This significantly reduces the search and enables the use of finer matching operators, such as comparing the positioning of segments in the scene to positioning of matched segments in the model. More sophisticated matching is applied in later stages over a much restricted number of hypotheses. Therefore, the dependency of the recognition time over the size of the models is significantly reduced. Evaluation shows the ability of our approach to recognize scenes with real partially occluded objects. Entirely visible objects are recognized with a reasonably high efficiency (80%), even with a change in viewpoint of up to 25/spl deg/. The efficiency smoothly decreases, but remains above 60% when the percentage of visible segments drops to 50% and the change in viewpoint is as above.

Technique of distortion correction in endoscopic images using a polynomial expansion.

A new technique to correct non-linear distortion of endoscopic images based on L2-norm approximation is proposed. A mathematical model is defined that maps the endoscopic images from the distorted image plane onto an undistorted image plane. A set of model parameters is defined, consisting of the image distortion centre, corrected centre and expansion polynomial coefficients representing radial distortion correction. A new method to find the image distortion centre based on a curvature criterion is also developed. The expansion coefficients are estimated on the basis of the degree of straightness of the grid lines in an image of a test grid consisting of dots in a rectangular pattern. The corrected image centre is computed by ensuring that the distances between neighbouring test dots of different grid lines in the corrected image are the same. The quantitative data providing the results and errors of the expansion characteristics are presented. The performance of the proposed distortion correction algorithm is validated with grid patterns of different orientations. The algorithm is also applied to typical endoscopic images, and the results are presented. The high-speed response of the proposed technique is a key step towards on-line camera calibration.

Relative chaos in gravitating systems with massive centre

A geometric method, namely the Ricci curvature criterion, is used to study the relative instability of evolving N-body gravitating systems with massive centres. The performed numerical experiments show that the Ricci curvature decreases with the increase in central concentration, thus indicating an increasing instability for systems with massive centre — either in the form of a point mass or a core formed during the evolution of single and multimass systems. Simulations of such systems do indeed suggest that they have faster gravothermal evolution than less concentrated ones.

In-situobserved deformation of carbon nanotubes

Abstract Carbon nanotubes have a whisker-like structure so an extremely high strength and lack of plasticity are generally predicted. In-situ observations made by transmission electron microscopy, however, prove that carbon nanotubes bend plastically at room temperature. The bending process at the atomic level is suggested by a molecular mechanics calculation: a nanotube deforms elastically until certain critical curvature is attained; then the atomic bonding in the stressed side changes from a graphite-like bonding state to a diamond-like state.

The Normal Detonation Shock Velocity–Curvature Relationship for Materials with Nonideal Equation of State and Multiple Turning Points

We present a model and a simple-to-implement numerical procedure that obtains the normal detonation shock velocity (Dn)–curvature (κ) relationships for an explosive material with nonideal equation of state and an arbitrary reaction rate law. In addition we illustrate numerically (for a nonideal equation of state) and analytically (for an ideal equation of state with a large activation energy rate law) that for sufficient rate-state–sensitive explosives, the corresponding Dn, κ-response curve can have two turning points, at Dn, κ,-pairs [(Dn)1, κ1], [(Dn)2, κ2], such that κ1 > κ2 > 0 for DCJ ≥ (Dn)1 ≥ (Dn)2, and such that the curve has a Z-shape. The top branch of the Z response curve has been previously associated with detonation extinction at a critical curvature. The bottom branch can be possibly associated with low-velocity detonation and rapid change from low-order detonation to high-order.

Electrical turbulence as a result of the critical curvature for propagation in cardiac tissue.

In cardiac tissue, the propagation of electrical excitation waves is dependent on the active properties of the cell membrane (ionic channels) and the passive electrical properties of cardiac tissue (passive membrane properties, distribution of gap junctions, and cell shapes). Initiation of cardiac arrhythmias is usually associated with heterogeneities in the active and/or passive properties of cardiac tissue. However, as a result of the effect of wave front geometry (curvature) on propagation of cardiac waves, inexcitable anatomical obstacles, like veins and arteries, may cause the formation of self-sustained vortices and uncontrolled high-frequency excitation in normal homogeneous myocardium. (c) 1998 American Institute of Physics.

Immobilization of smooth convex figures: Some extensions

In “Immobilization of Smooth Convex Figures” J. Bracho et al [2] prove a conjecture by Kuperberg, namely that any plane convex figure with twice continuously differentiable boundary, different from a circular disk, can be immobilized with three points. Here we extend the theorem to say that the figure can be immobilized firmly (meaning that the sufficient secondorder curvature criterion for immobilization holds), give a proof of their main lemma, show that fixing points are subsets of level sets of certain functions, and prove that for triples firmly fixing a figure, at least one point can be shifted uniquely to adjust for small shifts in the other two and maintain the figure firmly fixed.

A Note on the Toda Criterion for Interacting Dipole–Quadrupole Vibrations

The Toda criterion of the Gaussian curvature is applied to calculate analytically the transition energy from regular to chaotic motion in a schematic model describing the interaction between collective dipole and quadrupole modes in atomic nuclei.

Approximate estimation of hardening-softening behaviour of circular pipes subjected to pure bending

An approximate method for describing the plastic hardeningsoftening behaviour of circular pipes subjected to pure bending is presented. Theoretical estimation based on the uniform ovalization model and local collapse model proposed in the paper is incorporated to give several simple formulations with reasonable accuracy for determining the relationship between bending moment (M) and curvature (k) in the purely bended pipes. Attention is focused on the critical curvature associated with maximum resistant moment and the maximum change in the original diameter before the end of uniform ovalization stage as well as the local collapse behaviour. Some comparisons between analytical results and experimental results are made in order to examine the theory.

Link between the effect of an electric field on wave propagation and the curvature-velocity relation

The link between the effect of an electric field in a 1D medium and the curvature-velocity relation is studied theoretically by means of a two-variable reaction-diffusion system and numerically in the framework of a two-variable Oregonator model. Theoretical results predict the existence of a relationship between critical electric field ( E cr) and critical curvature ( K cr), which is highlighted by means of the so-called V-shaped patterns.

Self-organization of trajectory formation. II. Theoretical model.

Most studies of movement coordination deal with temporal patterns of synchronization between components, often without regard to the actual amplitudes the components make. When such a system is required to produce a composite action that is spatially constrained, coordination persists, but its stability is modulated by spatial requirements effected, we hypothesize, through the component amplitudes. As shown experimentally in part I, when a redundant three-joint system (wrist, elbow, and shoulder) is required to trace a specified arc in space, the joint angles may be frequency- and phased-locked even as the curvature of the trajectory is manipulated. Transitions between joint coordination patterns occur at a critical curvature, accompanied by a significant reduction in wrist amplitude. Such amplitude reduction is viewed as destabilizing the existing coordinative pattern under current task constraints, thereby forcing the joints into a more stable phase relationship. This paper presents a theoretical analysis of these multijoint patterns and proposes an amplitude mechanism for the transition process. Our model uses three linearly coupled, nonlinear oscillators for the joint angles and reproduces both the observed interjoint coordination and component amplitude effects as well as the resulting trajectories of the end effector.

Self-organization of trajectory formation. I. Experimental evidence.

Most studies examining the stability and change of patterns in biological coordination have focused on identifying generic bifurcation mechanisms in an already active set of components (see Kelso 1994). A less well understood phenomenon is the process by which previously quiescent degrees of freedom (df) are spontaneously recruited and active df suppressed. To examine such behavior, in part I we study a single limb system composed of three joints (wrist, elbow, and shoulder) performing the kinematically redundant task of tracing a sequence of two-dimensional arcs of monotonically varying curvature, kappa. Arcs were displayed on a computer screen in a decreasing and increasing kappa sequence, and subjects rhythmically traced the arcs with the right hand in the sagittal plane at a fixed frequency (1.0 Hz), with motion restricted to flexion-extension of the wrist, elbow, and shoulder. Only a few coordinative patterns among the three joints were stably produced, e.g., in-phase (flexion-extension of one joint coordinated with flexion-extension of another joint) and antiphase (flexion-extension coordinated with extension-flexion). As kappa was systematically increased and decreased, switching between relative phase patterns was observed around critical curvature values, kappa c. A serendipitous finding was a strong 2:1 frequency ratio between the shoulder and elbow that occurred across all curvature values for some subjects, regardless of the wrist-elbow relative phase pattern. Transitions from 1:1 to 2:1 frequency entrainment and vice versa were also observed. The results indicate that both amplitude modulation and relative phase change are utilized to stabilize the end-effector trajectory. In part II, a theoretical model is derived from three coupled nonlinear oscillators, in which the relative phases (phi) between the components and the relative joint amplitudes (rho) are treated as collective variables with arc curvature as a control parameter.

Steady-state autowave patterns in a two-dimensional excitable medium with a band of different excitability

A kinematic approach is used to construct steady-state traveling wave patterns in a two-dimensional excitable medium with a band of different excitability. New stationary autowave structures of bell-like shape are shown to exist when the excitability in a band is decreased compared to surrounding areas. The front profiles of the patterns and their propagation velocities are evaluated. Total internal reflection of waves from a band of higher excitability is predicted. The failure of steady-state propagation because of the critical curvature effect is discussed.