Linear Bifurcation Analysis Research Articles

Instability of Incompatible Bilayered Soft Tissues and the Role of Interface Conditions.

Mechanical stability analysis is instructive in explaining biological processes like morphogenesis, organogenesis, and pathogenesis of soft tissues. Consideration of the layered, residually stressed structure of tissues, requires accounting for the joint effects of interface conditions and layer incompatibility. This paper is concerned with the influence of imposed rate (incremental) interface conditions (RICs) on critical loads in soft tissues, within the context of linear bifurcation analysis. Aiming at simplicity, we analyze a model of bilayered isotropic hyperelastic (neo-Hookean) spherical shells with residual stresses generated by "shrink-fitting" two perfectly bonded layers with radial interfacial incompatibility. This setting allows a comparison between available, seemingly equivalent, interface conditions commonly used in the literature of layered media stability. We analytically determine the circumstances under which the interface conditions are equivalent or not, and numerically demonstrate significant differences between interface conditions with increasing level of layer incompatibility. Differences of more than tenfold in buckling and 30% in inflation instability critical loads are recorded using the different RICs. Contrasting instability characteristics are also revealed using the different RICs in the presence of incompatibility: inflation instability can occur before pressure maximum, and spontaneous instability may be excluded for thin shells. These findings are relevant to the growing body of stability studies of layered and residually stressed tissues. The impact of interface conditions on critical thresholds is significant in studies that use concepts of instability to draw conclusions about the normal development and the pathologies of tissues like arteries, esophagus, airways, and the brain.

The uplift effect of bottom plate of aboveground storage tanks subjected to wind loading

Because aboveground storage tanks are usually empty during construction, inspection and repair periods, they are more vulnerable to buckling due to wind loading than they are in use filled with liquid product. The uplift effect of the bottom plate of empty storage tanks due to wind pressure is investigated in this work. The buckling behavior of two sets of tanks were studied using finite element analysis (FEA): (1) tanks with bottom plate modeled directly in the analysis and (2) tanks without bottom plate but suitable boundary conditions applied in the analysis. Linear bifurcation analysis (LBA) and geometrically nonlinear analysis including imperfections (GNIA) are performed to obtain the deformation shapes and buckling capacities of tanks. The soil supporting the with-bottom tanks were modeled using nonlinear compression only springs connected to each node of the bottom plate. The buckling modes and the results from GNIA show that when a bottom is included into a tank FEA simulation, instead of fixed, the buckling pressure load will be substantially reduced for slender tanks having height to diameter ratio greater than 1/3. However, the broad tanks having height to diameter ratio less than 1/3 experience relatively small reduction in buckling load.

A descrição matemática do desempenho eletroanalítico do compósito poli(alaranjado da acridina)-oxihidróxido de vanádio na detecção eletrorredutiva da entacapona

Por meio de uma análise teórica, foi avaliado o desempenho do compósito do oxihidróxido do vanádio trivalente com o polímero do corante alaranjado da acridina, na detecção eletroquímica do fármaco antiparquinsônico entacapona. O processo eletroanalítico é baseado na eletrorredução do fármaco mencionado. Do desenvolvimento e da análise do modelo matemático correspondente, mediante a teoria de estabilidade linear e anáise de bifurcações, foi possível concluir que o compósito pode ser um modificador eficiente para a determinação da entacapona. Os comportamentos oscilatórios e monotônico, neste sistema, também são passíveis de realizar.

Influence of Internal Inward Pressure on Stability of Open-Top Aboveground Steel Tanks Subjected to Wind Loading

The results of wind tunnel tests indicate that there is an internal inward pressure induced by wind excitation when open-top tanks are examined, but not when close-top tanks are examined. This internal pressure is considered in many design documents outside of the U.S., however, ASCE-7 and API 650 do not explicitly address this factor. This study examined the effect of this internal pressure by conducting finite element analyses. Open-top tanks with height to diameter ratios of 0.11, 0.2, 0.5, 1.0, 2.0, and 4.0 were modeled using a finite element program. A linear bifurcation analysis and a post-buckling analysis were then conducted to verify the tank's stability when subjected to wind loading in accordance with the wind profiles specified in the selected design documents. To ensure the quality of the analyses, a study on mesh convergence and the load increment of Riks analysis was conducted. It was determined that the presence of the additional internal pressure term has a drastic impact on the buckling capacity of all the tanks examined. As a consequence, it can be concluded that the additional internal pressure generated by the wind on an open-top tank should not be neglected.

An incremental plate theory for polymer gels in equilibrium

Based on the finite-strain plate theory derived by one of the authors, an incremental plate theory for deformations superimposed on a general base state is formulated in this paper. The unknown functions of the incremental deformation are first expanded into Taylor series in terms of the thickness variable and then expanded around the base state by retaining third-order nonlinearity. From the field equations and the boundary conditions at the top and bottom surfaces, the recursive relations of the expansion coefficients as well as the incremental balance equations are derived. With the constitutive relation for swollen polymer gels in equilibrium, an incremental plate theory is obtained. As an application, this theory is used to study the incremental deformation of a polymer gel layer with a homogeneous base state in a plane strain setting. A linear bifurcation analysis is carried out, which gives the critical values of the external chemical potential and the mode number. The results agree with those obtained directly from a full two-dimensional analysis. Post-bifurcation is also conducted through a perturbation procedure, which reveals that the bifurcation is of supercritical type.

A diffuse interface model for the analysis of propagating bulges in cylindrical balloons

With the aim to characterize the formation and propagation of bulges in cylindrical rubber balloons, we carry out an expansion of the nonlinear axisymmetric membrane model assuming slow axial variations. We obtain a diffuse interface model similar to that introduced by van der Waals in the context of liquid–vapour phase transitions. This provides a quantitative basis to the well-known analogy between propagating bulges and phase transitions. The diffuse interface model is amenable to numerical as well as analytical solutions, including linear and nonlinear bifurcation analyses. Comparisons to the original membrane model reveal that the diffuse interface model captures the bulging phenomenon very accurately, even for well-localized phase boundaries.

On Rate Boundary Conditions for Soft Tissue Bifurcation Analysis.

Mechanical instability of soft tissues can either risk their normal function or alternatively trigger patterning mechanisms during growth and morphogenesis processes. Unlike standard stability analysis of linear elastic bodies, for soft tissues undergoing large deformations it is imperative to account for the nonlinearities induced by the coupling between load and surface changes at onset of instability. The related issue of boundary conditions, in context of soft tissues, has hardly been addressed in the literature, with most of available research employing dead-load conditions. This paper is concerned with the influence of imposed homogeneous rate (incremental) surface data on critical loads and associated modes in soft tissues, within the context of linear bifurcation analysis. Material behavior is modeled by compressible isotropic hyperelastic strain energy functions (SEFs), with experimentally validated material parameters for the Fung-Demiray SEF, over a range of constitutive response (including brain and liver tissues). For simplicity, we examine benchmark problems of basic spherical patterns: full sphere, spherical cavity, and thick spherical shell. Limiting the analysis to primary hydrostatic states we arrive at universal closed-form solutions, thus providing insight on the role of imposed boundary data. Influence of selected rate boundary conditions (RBCs) like dead-load and fluid-pressure (FP), coupled with constitutive parameters, on the existence and levels of bifurcation loads is compared and discussed. It is argued that the selection of the appropriate type of homogeneous RBC can have a critical effect on the level of bifurcation loads and even exclude the emergence of bifurcation instabilities.

THE MATHEMATICAL DESCRIPTION FOR SUCRALOSE ELECTROCHEMICAL DETECTION ON THE NOVEL ACRIDINIC DERIVATIVE

A new acridinic derivative has been synthetized in order to assist the sucralose electrochemical determination. The mathematical model, correspondent to the electroanalytical process, has been developed and analyzed by means of linear stability theory and bifurcation analysis. It was shown that the 9-saccharinoacridine may serve as a modifier for sucralose electrochemical detection in neutral solutions, participating in the electroanalytical process as an immobilization substrate and as a mediator. The electroanalytical process is diffusion-controlled. The possibility of oscillatory and monotonic instabilities has also been verified.

Dynamics of a new generalized fractional one-dimensional map: quasiperiodic to chaotic

Discovering new chaotic maps is always essential for secure communication, cryptography, image encryption and decryption when pseudo-number generation is mandatory; however, it is still very fascinating to come across new complex dynamics of very simple maps exhibiting chaotic behavior. Despite the various forms already presented in the literature, we deal with the fractional forms of one-dimensional chaotic map with one system parameter; yet while generalization, two parameters were inserted to the map as the multiplier and the power. Therefore, in this paper, we present a novel and generalized version of a map exhibiting a strange behavior in discrete time and real number space, while detailed analyses regarding the new map with intervals of various parameters are also included. We mainly focus on a simple one-dimensional chaotic map and propose various instances with linear stability, bifurcation and Lyapunov analyses for each instance, to enhance the understanding of unstable fractional chaotic maps. It is found that the fractional map exhibits quasiperiodicity as well as periodic behavior for the smallest power parameter; while the chaotic states emerge for larger values.

The Theoretical Evaluation For The Use of Vanadium (Iii) Oxyhydroxide For The Electrochemical Determination of Benzodiazepines

The theoretical evaluation for the use of vanadium (III) oxyhydroxide as an electrode modifier for benzodiazepines’ electrochemical determination in acid solutions has been made. The correspondent mathematical model has been developed and analyzed by means of the linear stability theory and bifurcation analysis. It was shown that VO(OH) may be an efficient electrode modifier for the benzodiazepines electrochemical determination in lightly and moderately acid solution. Its use in concentrated acidic solution may be compromised by modifier destruction. The oscillatory behavior in this system may be caused not only by the DEL influences of the electrochemical stage, but also by surface instabilities, caused by VO(OH) dissolution.

Evaluación teórica del desempeño del electrodo, modificado por el oxihidróxido de cobalto (III) en la detección del ácido úrico

La posibilidad del empleo de oxihidróxido de cobalto como modificador del electrodo en la detección del ácido úrico se investigó desde el punto de vista teórico. Se sugiere el mecanismo del desempeño del compuesto inorgánico en el proceso, y el modelo, correspondiente al proceso, se analizó mediante la teoría de estabilidad lineal y análisis de bifurcaciones. Se concluye que el oxihidróxido de cobalto puede servir como un modificador eficiente, facilitando la oxidación del analito en medio neutro en el modo galvanostático. La inestabilidad monotónica para este caso no puede realizarse, y la oscilatoria es causada por el único factor.

A descrição matemática da análise eletroquímica do fármaco hidroxizina em soluções ácidas sobre um ácido sulfônico, baseado num polímero condutor

EnglishAn interesting case of the application of the functionalized conducting polymer in the analysis of hydroxyzine drug in acid solutions is described from the theoreticalpoint of view. A mathematical model, correspondent to the case, is developed and analyzed by means of linear stability theory and bifurcation analysis. It was detected that the use of an acid conducting polymer may permit the use of less acid solutionwithout mischief of the electroanalytical efficiency. Moreover, the influence of the aminogroups´ behavior to the process was also evaluated. The possibility of the oscillatory and monotonic instabilities has also been verified. portuguesUm caso interessante da aplicacao dos polimeros condutores funcionalizados na analise do farmaco hidroxizina em solucoes acidas vem sendo descrito do ponto de vista teorico. Um modelo matematico, correspondente ao caso, e desenvolvidoe analisado mediante a teoria de estabilidade linear e da analise de bifurcacoes. Foi detectado que o uso de um polimero condutor de natureza acida no processo podedeixar o uso das solucoes menos acidas do farmaco sem prejuizo para a eficiencia eletroanalitica. Outrossim, foi avaliada a influencia para o processo do comportamentodos grupos amina. A possibilidade das instabilidades oscilatoria e monotonica tambem foi verificada.

The Mathematical Description for CoO(OH)-Assisted Hydroxylamine Electrochemical Determination in Neutral Media

The possibility of hydroxylamine electrochemical determination over a CoO(OH)-modified electrode, accompanied by complex-formation, has been studied from the theoretical point of view. The correspondent mathematical model has been developed and analyzed by means of linear stability theory and bifurcation analysis. It was shown that CoO(OH) may be applied as an efficient electrode modifier in the detection of hydroxylamine in neutral and slightly alkaline media. The electrochemical response has to be clear and easy to interpret. The possibility of the oscillatory and monotonic instabilities has also been verified.

The buckling analysis of a cold-formed steel C-sectional column resting on an elastic foundation

The paper is focused on stability analysis of different models of steel cold-formed C-section column. The analysed element may be applied as a primary load-bearing member in steel trusses or silo columns. The column can be used as a support for a wall of roof plates made of corrugated sheets. The wall sheeting that restrains the column against buckling may be considered as an elastic foundation. Due to the design solutions the foundation may be modelled as a series of lateral translational or rotational elastic supports. Such model is used for example in silo column analysis. In the paper linear bifurcation analysis (LBA) of axially compressed columns with different type of elastic foundation is carried by means of ABAQUS software. The analysed thin-walled column (C 250×100×33×4) is regarded as a load-bearing element in a steel hall. The column was modelled by shell elements, the foundation modelled by means of elastic supports. The column length was 5000 mm. The authors present a 3D image of an influence of lateral translational and rotational elastic supports on the buckling resistance of the column.

Stability of Open-Topped Storage Tanks With Top Stiffener and One Intermediate Stiffener Subject to Wind Loading

Aboveground vertical steel storage tanks use stiffener rings to prevent their shell wall from buckling under wind loading. The existing stiffener rings design rules from API 650 standard is known to be overly conservative. This study investigates the possibility of modifying the design rules by reducing the required size of the top stiffener ring to the same size as the intermediate stiffener ring. In this study, we used finite element analysis (FEA) to perform linear bifurcation analysis (LBA) and geometrically nonlinear analysis including imperfections (GNIA) to obtain failure load of modeled tanks. The buckling pressure load was obtained to ensure it is larger than the design pressure. Moreover, the effects of higher strength materials, different buckling modes, and various wind profiles were also studied to ensure the design suggested by this study is practical and universal to different situations. The results show that for cylindrical storage tanks, which only needs one intermediate stiffener ring, the size of the top stiffener ring can be set to the same size as the intermediate stiffener ring.

Investigations of local/global buckling of cylindrical metal silos with corrugated sheets and open-sectional column profiles

The paper deals with buckling of cylindrical metal silos with corrugated sheets and columns. Attention was paid to local buckling of cold-formed open-sectional thin-walled columns in metal cylindrical silos with corrugated sheets. The effect of the column profile dimensions, silo height, silo diameter, corrugated wall thickness, column number/column distance, corrugation height and bulk solid's type was comprehensively investigated. Three-dimensional linear bifurcation analyses were performed for silos using the finite element method and compared with analytical formulae of Eurocode 3. Based on calculation outcomes some recommendations for silo dimensioning were elaborated.

Elastic buckling of externally pressurized Cassini oval shells with various shape indices

This study examines the effect of shape index on the elastic buckling of Cassini oval shells under uniform external pressure. Shells are evaluated under a uniform wall thickness (2mm) and capacity (3.63×106mm3), with the shape index, kc=c/a, ranging from 0 to 0.99. Several numerical computations, involving linear elastic bifurcation analysis and geometrically nonlinear elastic analysis including eigenmode imperfections, are conducted for these shells. Some of them are investigated experimentally. The results show that a Cassini oval shell with a stable character seems to be an unfavorable shape due to its low load-carrying capacity, which is at variance with previous findings regarding this problem. Notably, the kc=0.1 Cassini oval geometry, exhibiting a high load-carrying capacity, appears to be a favorable shape in various fields such as underwater tanks, pressure hulls, and artificial capsules.

04.10: Analysis of stiffened curved panels in compression: A preliminary numerical study for experimental tests

ABSTRACTThe use of different kinds of curved steel panels in bridge design has been increasing in the past few years, but the design recommendations and the knowledge in this field remain scarce. Taking in account the fact that very few experimental tests on life size curved panels have been carried out so far, an experimental study on stiffened and unstiffened curved panels in compression is planned in the near future. For this purpose, a preliminary numerical study on curved panels was conducted, in order to establish an appropriate experimental test setup. This paper deals with a parametric study on stiffened and unstiffened transversally curved steel panels subjected to pure in‐plane compression performing linear buckling analysis (LBA) and fully nonlinear analysis with geometric imperfections (GMNIA). Considering that the shell‐like structures are highly sensitive to initial imperfections, which significantly lowers their ultimate strength, a linear bifurcation analysis is performed first in order to get the buckling modes of the analysed panels. The buckling shapes are later used as equivalent geometric imperfections in the nonlinear analysis, based on the rules given by EN1993‐1‐5. Furthermore, a finite element analysis, considering geometric and material nonlinearity with initial imperfections, is conducted in ABAQUS software. Simply supported, transversally curved and longitudinally stiffened panels are analysed in order to determine their resistance to pure in‐plane compression. A large parametric study is performed to evaluate the effects of different parameters on the ultimate resistance of the panel. Panels that differ in curvature, aspect ratio and stiffness of the stiffeners are studied. Finally, the numerical results of the parametric study on all, the unstiffened, longitudinally stiffened, both curved and flat panels with the same dimensions are compared. Effects of panel curvature on the critical elastic stress and the ultimate resistance are presented.

04.04: Buckling of elastic cylindrical shells under symmetric but non‐uniform bending moment distributions

ABSTRACTThe behaviour of cylindrical shells and circular hollow sections under uniform bending has long received significant research attention. This is mainly due to the phenomenon of Brazier cross‐sectional ovalisation which is known to lead to an almost 50% reduction in the buckling moment relative to the classical analytical prediction. However, very little research appears to have been carried out to investigate the ovalisation behaviour under non‐uniform bending moment distributions (BMDs). This appears to be because until only relatively recently, most studies on the topic were analytical in nature which made the buckling analysis of anything but a uniform BMD difficult. With the rise of modern computing it is now possible to undertake extensive parametric studies to investigate any desired load condition.This paper employs the finite element method to explore the elastic stability of perfect simply‐supported cylinders or tubes under load cases that represent symmetric parabolic and triangular BMDs, and to compare them with the behaviour under the hitherto most‐commonly studied uniform BMD. Each non‐uniform BMD is investigated by four different three‐dimensional shell loading arrangements. While these all share the same resultant BMD, they exhibit very different buckling behaviours illustrating the ambiguity inherent in realistically modelling such systems. This is a consequence of the loss of uniqueness when projecting a 1D construct onto a 3D space.In addition to linear bifurcation analyses, nonlinear elastic buckling analyses are presented to explore the effect of ovalisation at long lengths under the proposed loading arrangements. It is found that ovalisation becomes fully developed at approximately the same dimensionless length and leads to a similar ∼50% reduction in the buckling strength of asymptotically long cylinders for almost any representation of the three considered symmetric BMDs.

Bifurcation analysis of density wave oscillations in natural circulation loop

The present study explores the linear stability and bifurcation analysis of two-phase flow in the single heated channel for the natural circulation loop. The bifurcation analysis of this natural circulation system is limited to the detection of Hopf bifurcation, generalized Hopf bifurcation and turning point or limit point bifurcation of limit cycles. These natural circulation loops have several engineering applications and prone to thermal-hydraulic instability, which may lead to an operationally inefficient system. Stability boundary (linearly) obtained on a Npch- Nsub plane predicts the existence of limit cycles (a nonlinear phenomenon) for these loops and gives motivation for the further study on the detection of nonlinear phenomena. The subsequent results show a novel case of stability boundary (listed as Type A in the present work) emerging for higher subcooling numbers on Npch- Nsub parameter plane, which has not been noted in literature earlier and lies near to the Ledinegg stability boundary, which is a static instability. The characteristics of points nearby Type A stability boundary predicts that this boundary is different from the Ledinegg stability boundary. This Type A stability boundary divided by subcritical and supercritical Hopf bifurcation, which is separated by a generalized Hopf bifurcation (GH) point. Sustained oscillations in flow velocity can be observed near to the Hopf points on this boundary. These subcritical (hard and dangerous) and supercritical (soft and safe) Hopf bifurcations have been examined with time series graphs for all types (Type A-C) of stability boundaries present for the loops and occurring on the Npch- Nsub parameter plane. The two other type of boundaries known earlier is characterized here as Type B and C, respectively, which is basically Type I and Type II density wave oscillations. The transition of stability boundaries is discussed on the Npch- Nsub parameter plane, which shows that Type A stability boundary contains GH bifurcation coming from Type B stability boundary. The bifurcation of limit cycles originating from the Hopf points in the system has been examined and it is found that limit point bifurcation of limit cycles is occurring in the system, originating from the generalized Hopf points.