Academic literature on the topic 'Non-linear elastic behaviour'

An experimental study of the fracture of unidirectional carbon fibre-organic matrix composites under compression show that fibre kinking is the principal compressive failure mode. All materials tested have a non linear elastic behaviour during loading. This behaviour can be attributed to the intrinsic non linear elastic behaviour of the fibres.

Purpose The purpose of this paper is to perform an analytical study of non-linear elastic delamination fracture in the multilayered functionally graded split cantilever beam (SCB) configuration. The SCB studied may have an arbitrary number of vertical layers. The material in each layer is functionally graded along the layer thickness. Also, the material properties may be different in each layer. The analytical solution derived was applied for parametric investigations in order to evaluate the effects of material properties and delamination crack location on the non-linear fracture behaviour. Design/methodology/approach The delamination fracture was studied in terms of the strain energy release rate. The SCB mechanical response was described by using a power-law stress-strain relation. A non-linear analytical solution for the strain energy release rate was derived by considering the SCB complementary strain energy. In order to verify the solution, an additional analysis of the strain energy release rate was developed by considering the complementary strain energy in the beam cross-sections ahead and behind the crack front. Findings The effects of material gradient, crack location along the beam width and non-linear material behaviour on the delamination fracture were evaluated. The analytical solution derived is useful for parametric studies of non-linear fracture in multilayered functionally graded beams. Originality/value Delamination fracture in the multilayered functionally graded SCB configuration was analysed with considering the non-linear material behaviour.

Longitudinal tensile tests have been conducted on unidirectional SiC fibre reinforced 6061 aluminium matrix composites in the annealed and as-manufactured conditions. The results are presented in terms of stress-strain curves and tangent modulus-strain relations, which show considerable non-linearity. Corresponding micromechanical finite element modelling is performed including the effects of the manufacturing process on the matrix in-situ properties. The analysis shows that the non-linear behaviour of the composite is caused by the elastic-plastic deformation of the matrix alloy. The matrix fully yields during the cooldown from manufacturing. Residual stress relaxation plays an important role in the stress-strain characteristics of the annealed aluminium matrix composite by introducing some initial elastic deformation. The amount of elastic deformation for the as-manufactured condition is greater because of subsequent age hardening. However, more linear elastic deformation was observed than predicted in the as-manufactured specimens, which is believed to be due to higher precipitation hardening caused by metallurgical effects induced in the manufacturing process.

In this work the elastic behaviour of metallic powder compacts is studied. Cylindrical specimens with different levels of density have been submitted to uniaxial compression tests with loading and unloading cycles. The analysis of the elastic loadings shows a non linear elasticity which can be mathematically represented by means of a potential law. Results are explained by assuming that the total elastic strain is the contribution of two terms one deriving from the hertzian deformation of the contacts among particles and another that takes into account the linear elastic deformation of the powder skeleton. A simple model based in a one pore unit cell is presented to support the mathematical model.

Purpose The purpose of this paper is to develop an analysis of longitudinal fracture behaviour of a functionally graded non-linear-elastic circular shaft loaded in torsion. It is assumed that the material is functionally graded in both radial and longitudinal directions of the shaft (i.e. the material is bi-directional functionally graded). Design/methodology/approach The Ramberg–Osgood stress-strain relation is used to describe the non-linear mechanical behaviour of the functionally graded material. The fracture is studied in terms of the strain energy release rate by analysing the balance of the energy. The strain energy release rate is obtained also by differentiating of the complementary strain energy with respect to the crack area for verification. Findings Parametric studies are carried out in order to evaluate the influence of material gradients in radial and longitudinal directions, the crack location in radial direction and the crack length on the fracture behaviour of the shaft. It is found that by using appropriate gradients in radial and longitudinal directions, one can tailor the variations of material properties in order to improve the fracture performance of the non-linear-elastic circular shafts to the externally applied torsion moments. Originality/value A longitudinal cylindrical crack in a bi-directional functionally graded non-linear-elastic circular shaft loaded in torsion is analysed by using the Ramberg–Osgood stress-strain relation.

Dynamic analysis programs and empirical formulae are often used to compute the period of vibration of single-storey steel buildings. Recent ambient vibration tests of buildings in Quebec and British Columbia have shown that the predicted period of vibration is typically much longer than that measured. Software and empirical formulae do not usually take into account the stiffening effects of the non-structural components; this could be the source of the discrepancy between the results in the field and the results obtained by computational methods.<br>This research project concentrates on the roof diaphragm system of single-storey steel buildings and the contribution of the non-structural components to diaphragm stiffness. It is believed that the non-structural components, roofing materials such as gypsum board and fibreboard, add to the overall stiffness of the system. A roofing system called AMCQ SBS-34 consisting of gypsum board, ISO insulation board and fibreboard, all hot bitumen adhered, was studied. The full roof system, as well as its individual components and connections, were first studied through laboratory testing. The flexural and shear stiffness of the fibreboard and gypsum panels, as well as the shear stiffness and equivalent flexural stiffness of the complete roof system and shear stiffness of the roofing connections were determined.<br>Linear elastic finite element models, using the SAP2000 software, were developed to replicate the behaviour of bare sheet steel and clad diaphragm test specimens. The test based properties of the roofing components and connections were incorporated into the definition of the elements. The models were then calibrated based on the results of large-scale diaphragm tests by Yang. Once the elastic behaviour of the diaphragms had been matched, a parametric study was performed in order to assess the importance of the contribution of the roofing assembly relative to the roof deck panel thickness.<br>It was shown that as the deck thickness increases, the relative contribution of the non-structural components decreases on a percentage basis, but does not become non-negligible. The increase in shear stiffness of the diaphragm ranges from 58.6% for the 0.76 mm deck panel to 4.7% for the 1.51 mm roof deck panel, dependent on the sidelap and deck-to-frame connection configuration.

The flexural capacity of the new hollow flange steel section known as LiteSteel beam (LSB) is limited by lateral distortional buckling for intermediate spans, which is characterised by simultaneous lateral deflection, twist and web distortion. Recent research based on finite element analysis and testing has developed design rules for the member capacity of LiteSteel beams subject to this unique lateral distortional buckling. These design rules are limited to a uniform bending moment distribution. However, uniform bending moment conditions rarely exist in practice despite being considered as the worst case due to uniform yielding across the span. Loading position or load height is also known to have significant effects on the lateral buckling strength of beams. Therefore it is important to include the effects of these loading conditions in the assessment of LSB member capacities. Many steel design codes have adopted equivalent uniform moment distribution and load height factors for this purpose. But they were derived mostly based on data for conventional hot-rolled, doubly symmetric I-beams subject to lateral torsional buckling. In contrast LSBs are made of high strength steel and have a unique crosssection with specific residual stresses and geometrical imperfections along with a unique lateral distortional buckling mode. The moment distribution and load height effects for LSBs, and the suitability of the current steel design code methods to accommodate these effects for LSBs are not yet known. The research study presented in this thesis was therefore undertaken to investigate the effects of nonuniform moment distribution and load height on the lateral buckling strength of simply supported and cantilever LSBs. Finite element analyses of LSBs subject to lateral buckling formed the main component of this study. As the first step the original finite element model used to develop the current LSB design rules for uniform moment was improved to eliminate some of the modelling inaccuracies. The modified finite element model was validated using the elastic buckling analysis results from well established finite strip analysis programs. It was used to review the current LSB design curve for uniform moment distribution, based on which appropriate recommendations were made. The modified finite element model was further modified to simulate various loading and support configurations and used to investigate the effects of many commonly used moment distributions and load height for both simply supported and cantilever LSBs. The results were compared with the predictions based on the current steel code design rules. Based on these comparisons, appropriate recommendations were made on the suitability of the current steel code design methods. New design recommendations were made for LSBs subjected to non-uniform moment distributions and varying load positions. A number of LSB experiments was also undertaken to confirm the results of finite element analysis study. In summary the research reported in this thesis has developed an improved finite element model that can be used to investigate the buckling behaviour of LSBs for the purpose of developing design rules. It has increased the understanding and knowledge of simply supported and cantilever LSBs subject to non-uniform moment distributions and load height effects. Finally it has proposed suitable design rules for LSBs in the form of equations and factors within the current steel code design provisions. All of these advances have thus further enhanced the economical and safe design of LSBs.

La mousse de polyuréthane est un matériau cellulaire caractérisé par un spectre de propriétés mécaniques intéressant : une faible densité, une capacité à absorber l'énergie de déformation et une faible raideur.Elle présente également des propriétés telles qu'une excellente isolation thermique et acoustique, une forte absorption des liquides et une diffusion complexe de la lumière. Ce spectre de propriétés fait de la mousse de polyuréthane un des matériaux couramment utilisés dans de nombreuses applications phoniques, thermiques et de confort. Pour contrôler la vibration transmise aux occupants des sièges, plusieurs dispositifs automatiques de régulation et de contrôle sont actuellement en cours de développement tels que les amortisseurs actifs et semi-actifs. La performance de ces derniers dépend bien évidemment de la prédiction des comportements de tous les composants du siège et en particulier la mousse. D'une façon générale, il est indispensable de modéliser le comportement mécanique complexe de la mousse de polyuréthane et d'identifier ses propriétés quasi-statique et dynamiques afin d'optimiser la conception des systèmes incluant la mousse en particulier l'optimisation de l'aspect confort. Dans cette optique, l'objectif principal de cette thèse consiste à implémenter des modèles mécaniques de la mousse de polyuréthane fiables et capables de prévoir sa réponse sous différentes conditions d'essais. Dans la littérature, on retrouve les divers modèles développés tels que les modèles de mémoire entier et fractionnaire. L'inconvénient majeur de ces modèles est lié à la dépendance de leurs paramètres vis-à-vis des conditions d'essais, chose qui affecte le caractère général de leur représentativité des comportements quasi-statique et dynamique de la mousse polyuréthane. Pour pallier à cet inconvénient, nous avons développé des modèles qui, grâce à des choix judicieux de méthodes d'identification, assurent une représentativité plus générale des comportements quasi-statique et dynamique de la mousse polyuréthane. En effet, nous avons démontré qu'on peut exprimer les paramètres dimensionnels des modèles développés par le produit de deux parties indépendantes ; une regroupant les conditions d'essais et une autre définissant les paramètres adimensionnels et invariants qui caractérisent le matériau. Ces résultats ont été obtenus à partir de plusieurs études expérimentales qui ont permis l'appréhension du comportement quasi-statique (à travers des essais de compression unidirectionnelle) et dynamique (à travers des tests en vibration entretenue). La mousse, sous des grandes déformations, présente à la fois un comportement élastique non linéaire et un comportement viscoélastique. En outre, une discrimination entre les modèles développés particulièrement en quasi-statique a été effectuée. Les avantages et les limites de chacun y ont été discutés.

Le présent travail est consacré à l'étude du comportement thermo-hydro-mécanique linéaire et non linéaire des roches poreuses de type argilites par approche de changement d'échelle. A partir des observations microstructurales de ces matériaux, un modèle conceptuel a été proposé. Dans ce modèle, le volume élémentaire représentatif du milieu hétérogène est composé d'une phase matricielle argileuse contenant des inclusions sphériques de minéraux de quartz et de calcite et des inclusions ellipsoïdales aplaties représentant l'espace poreux. Dans un premier temps, le procédé de la modélisation a été exploité par la détermination des propriétés effectives isotropes et isotrope transverses des argilites : la conductivité thermique et les propriétés thermo-hydro-mécanique croisées. En outre, de nombreuses études numériques ont mis en évidence l'influence de la morphologie de l'espace poreux, de la minéralogie et des schémas d'estimation sur les résultats prédictifs. Dans un deuxième temps, nous avons modélisé le comportement mécanique non linéaire (élasto-plastique, élasto-viscoplastique) des roches argileuses. La comparaison entre les simulations numériques et les résultats expérimentaux disponibles (essai de compression triaxiale, essai de fluage) a confirmé la validation du modèle développé<br>The present work deals with the linear and non-linear thermo-hydro-mechanical behaviour of porous rocks such as the argillite by the multiscale modelling approach. Based on microstructure observations, a conceptual model was proposed. In this model, the representative elementary volume of a heterogeneous medium is composed of an argillaceous matrix containing spherical inclusions of minerals quartz and calcite and ellipsoidal inclusions representing the pore space. In a first step, the process of modelling has been exploited by determining the isotropic and transversely isotropic effective properties of the argillite: thermal conductivity and thermo-hydro-mechanical properties. Furthermore, many numerical studies have highlighted the influence of the morphology of the pore space, of the mineralogy and of the estimate schemes to the predictive results. In a second step, we modelled the non linear mechanical behaviour (elasto-plastic, elasto-viscoplastic) of argillaceous rocks. The comparison between numerical simulations and available experimental results (triaxial compression test, creep test) confirmed the validation of the model developed

Les matériaux composites à matrice organique sont largement utilisés dans l'industrie des transports et notamment dans le domaine aéronautique. Pour permettre un dimensionnement optimal des structures, il est nécessaire d'étudier le comportement des matériaux CMO sur une large gamme de vitesses et de températures.L'objectif de cette thèse est de proposer un modèle de comportement et de rupture permettant de prédire la réponse des CMO sur une large gamme de vitesses de sollicitation et de températures. Les recherches se sont intéressées dans un premier temps à la caractérisation de la transition entre les régimes de comportement linéaire et non linéaire du matériau unidirectionnel T700GC/M21 (renforts de fibres de carbone, résine époxy), ainsi qu'à la dépendance de cette transition à la vitesse de sollicitation et à la température. Les travaux se sont ensuite focalisés sur l'étude expérimentale du régime de comportement non linéaire endommageable du T700GC/M21. Enfin, au terme de ces deux étapes, une version améliorée du modèle disponible à l'ONERA pour les composites stratifiés (OPFM) a été proposée, version intégrant un critère de transition linéaire/non linéaire de comportement, et une prise en compte de l'influence de la vitesse de sollicitation et de la température sur la réponse du matériau<br>Nowadays, organic matrix composite materials are widely used in the transportation industry, and particularly in the aeronautical industry. To provide an optimal dimensioning of the structures, it is necessary to study the mechanical behavior of OMC on a large range of strain rates and temperatures. The aim of this PhD thesis is to propose a behavior and a rupture model to predict the mechanical response of OMC for a large range of strain rates and temperatures. The research was initially focused on the characterization of the transition between the linear and nonlinear behavior of the material T700GC/M21, a carbon / epoxy unidirectional laminate as well as the strain rate and temperature dependencies of this transition. The work was then focused on the experimental study of the nonlinear damaged behavior of the T700GC/M21. Finally, completing these first two steps, an updated version of the behavior model available at ONERA (OPFM) was proposed which includes the transition between linear and nonlinear behavior and the influence of strain rate and temperature on the mechanical response of the material

Concrete, which was hitherto considered as a brittle material, has shown much better softening behavior after the post peak load than anticipated. This behavior of concrete did put the researchers in a quandary, whether to categorize concrete under brittle materials or not. Consequently concrete has been called a quasi-brittle material. Fracture mechanics concepts like Linear elastic fracture mechanics (LEFM) and Plastic limit analysis applicable to both brittle and ductile materials have been applied to concrete to characterize the fracture behavior. Because of quasi-brittle nature of concrete, which lies between ductile and brittle response and due to the presence of process zone ahead of crack/notch tip instead of a plastic zone, it is found that non-linear fracture mechanics (NLFM) principles are more suitable than linear elastic fracture mechanics (LEFM) principles to characterize fracture behavior. Fracture energy, fracture process zone (FPZ) size and the behavior of concrete during fracture process are the fracture characteristics, which are at the forefront of research on concrete fracture. Another important output from the research on concrete fracture has been the size effect. Numerous investigations, through mathematical modeling and experiments, have been carried out and reported in literature on the effect of size on the strength of concrete and fracture energy. Identification of the sources of size effect is of prime importance to arrive at a clear analytical model, which gives a comprehensive insight into the size effect. With the support of an unambiguous theory, it is possible to incorporate the size effects into codes of practices of concrete design. However, the theories put forth to describe the size effect do not seem to follow acceptable regression. After introduction in Chapter-1 and literature survey in Chapter-2, Chapter-3 details the study on size effect through three point bend (TPB) tests on 3D geometrically similar specimens. Fracture behavior of beams with smaller process zone size in relation to ligament dimension approaches LEFM. The fracture energy obtained from such beams is said to be size independent. In the current work Size effect law (Bazant et al. 1987) is used on beams geometrically similar in three dimensions with the depth of the largest beam being equal to 750mm, and size independent fracture energy G Bf is obtained. In literature very few results are available on the results obtained from testing geometrically similar beams in three dimensions and with such large depth. In the current thesis the results from size effect tests yielded average fracture energy of 232 N/m. Generally the fracture energies obtained from 2D-geometrically similar specimens are in the range of 60-70 N/m as could be seen in literature. From 3D-geometrically similar specimens, the fracture energies are higher. The reason is increased peak load, could be due to increased width. The RILEM fracture energy Gf , determined from TPB tests, is said to be size dependent. The assumption made in the work of fracture is that the total strain energy is utilized for the fracture of the specimen. The fracture energy is proportional to the size of the FPZ, it also implies that FPZ size increases with increase in (W−a) of beam. This also means that FPZ is proportional to the depth W for a given notch to depth ratio, because for a given notch/depth, (W−a) which is also W(1 − a ) is proportional to W`because (1 − a ) is a constant. WWThis corroborates the fact that fracture energy increases with size. Interestingly, the same conclusion has been drawn by Abdalla & Karihaloo (2006). They have plotted a curve relating fracture process zone length and overall depth the beam. In the present study a new method namely Fracture energy release rate method is suggested. In the new method the plot of Gf / (W−a) versus (W−a) is obtained from a set of experimental results. The plot is found to follow power law and showed almost constant value of Gf / (W−a) at larger ligament lengths. This means that fracture energy reaches a constant value at large ligament lengths reaffirming that the fracture energy from very large specimen is size independent. The new method is verified for the data from literature and is found to give consistent results. In a quasi-brittle material such as concrete, a fracture process zone forms ahead of a pre-existing crack (notch) tip before the crack propagates from the tip. The process zone contains a scatter of micro-cracks, which coalesce into one or more macro-cracks, which eventually lead to fracture. These micro-cracks and macro-cracks release stresses in the form of acoustic waves having different amplitudes. Each micro or macro crack formation is called an acoustic emission (AE) event. Through AE technique it is possible to locate the positions of AE events. The zone containing these AE events is termed the fracture process zone (FPZ). In Chapter-4, a study on the evolution of fracture process zone is made using AE technique. In the AE study, the fracture process zone is seen as a region with a lot of acoustic emission event locations. Instead of the amplitudes of the events, the absolute AE energy is used to quantify the size of the process zone at various loading stages. It has been shown that the continuous activities during the evolution of fracture process zone correspond to the formation of FPZ, the size of which is quantified based on the density of AE events and AE energy. The total AE energy released in the zone is found to be about 78% of the total AE energy released and this is viewed as possible FPZ. The result reasonably supports the conclusion, from Otsuka and Date (2000) who tested compact tension specimens, that zone over which AE energy is released is about 95% can be regarded as the fracture process zone. As pointed out earlier, among the fracture characteristics, the determination of fracture energy, which is size independent, is the main concern of research fraternity. Kai Duan et al. (2003) have assumed a bi-linear variation of local fracture energy in the boundary effect model (BEM) to showcase the size effect due to proximity of FPZ to the specimen back boundary. In fact the local fracture energy is shown to be constant away from boundary and reducing while approaching the specimen back boundary. The constant local fracture energy is quantified as size independent fracture energy. A relationship between Gf , size independent fracture energy GF , un-cracked ligament length and transition ligament length was developed in the form of equations. In the proposed method the transition ligament length al is taken from the plot of histograms of energy of AE events plotted over the un-cracked ligament. The value of GF is calculated by solving these over-determined equations using the RILEM fracture energies obtained from TPB tests. In chapter-5 a new method involving BEM and AE techniques is presented. The histogram of energy of AE events along the un-cracked ligament, which incidentally matches in pattern with the local fracture energy distribution, assumed by Kai Duan et al. (2003), along the un-cracked ligament, is used to obtain the value of GF , of course using the same equations from BEM developed by Kai Duan et al. (2003). A critical observation of the histogram of energy of AE events, described in the previous chapter, showed a declining trend of AE event pattern towards the notch tip also in addition to the one towards the specimen back boundary. The pattern of AE energy distribution suggests a tri-linear rather than bi-linear local fracture energy distribution over un-cracked ligament as given in BEM. Accordingly in Chapter-6, GF is obtained from a tri-linear model, which is an improved bi-linear hybrid model, after developing expressions relating Gf , GF , (W−a) with two transition ligament lengths al and blon both sides. The values of Gf , and GF from both bi-linear hybrid method and tri-linear method are tabulated and compared. In addition to GF , the length of FPZ is estimated from the tri-linear model and compared with the values obtained from softening beam model (SBM) by Ananthan et al. (1990). There seems to be a good agreement between the results. A comparative study of size independent fracture energies obtained from the methods described in the previous chapters is made. The fracture process in concrete is another interesting topic for research. Due to heterogeneity, the fracture process is a blend of complex activities. AE technique serves as an effective tool to qualitatively describe the fracture process through a damage parameter called b-value. In the Gutenberg-Richter empirical relationship log 10N=a−bM, the constant ‘b’ is called the b-value and is the log linear slope of frequency-magnitude distribution. Fault rupture inside earth’s crust and failure process in concrete are analogous. The b-value, is calculated conventionally till now, based on amplitude of AE data from concrete specimens, and is used to describe the damage process. Further, sampling size of event group is found to influence the calculated b-value from the conventional method, as pointed out by Colombo et al. (2003). Hence standardization of event group size, used in the statistical analysis while calculating b-value, should be based on some logical assumption, to bring consistency into analytical study on b-value. In Chapter-7, a methodology has been suggested to determine the b-value from AE energy and its utilization to quantify fracture process zone length. The event group is chosen based on clusters of energy or quanta as named in the thesis. Quanta conform to the damage stages and justify well their use in the determination of the b-value, apparently a damage parameter and also FPZ length. The results obtained on the basis of quanta agree well with the earlier results.

In this chapter, through some illustrative examples, the applicability of the Discrete Finite Element Method (DFEM) to analysis of unreinforced masonry structures such as rock pillars, open rock slopes, underground openings, tunnels, fault propagations, and fault-structure interactions is examined and discussed. In the numerical study, the behavior of contacts and blocks is assumed to be elasto-plastic or elastic. The Mohr-Coulomb yield criterion, representing material behavior of contacts, is implemented in the developed codes for DFEM used in the analysis. The secant stiffness method with the updated Lagrangian scheme is employed to deal with non-linear behavior. The constant strain triangular element with two degrees of freedoms at each node, formed by properly joining the corners and contact nodes of an individual block, is adopted for finite element meshing of the blocks. The DFEM provides an efficient and promising tool for designing, analyzing, and studying the behavior of unreinforced masonry structures.

The AA 5083 aluminium alloy is widely used in high speed marine vehicles. The aim of this scientific work was to develop and validate a procedure, starting only from hardness measurements, to predict the elastic-plastic behavior of AA 5083 welded joints under static loading using non-linear FEA analyses. The hardness measurements allowed identifying the different zones and to assess their different mechanical properties, which were considered in the finite element model. Finally, the finite element model results were validated experimentally, comparing the results with the measurements obtained by means of a full-field technique such as the Digital Image Correlation technique.

The rolling of a single biological cell is analysed using modelling of the local kinetics of successive attachment and detachment of bonds occurring at the interface between a single cell and the wall of an ECM (extracellular matrix). Those kinetics correspond to a succession of creations and ruptures of ligand-receptor molecular connections under the combined effects of mechanical, physical (both specific and non-specific), and chemical external interactions. A three-dimensional model of the interfacial molecular rupture and adhesion kinetic events is developed in the present contribution. From a mechanical point of view, this chapter works under the assumption that the cell-wall interface is composed of two elastic shells, namely the wall and the cell membrane, linked by rheological elements representing the molecular bonds. Both the time and space fluctuations of several parameters related to the mutual affinity of ligands and receptors are described by stochastic field theory; especially, the individual rupture limits of the bonds are modelled in Fourier space from the spectral distribution of power. The bonds are modelled as macromolecular chains undergoing a nonlinear elastic deformation according to the commonly used freely joined chains model, while the cell membrane facing the ECM wall is modelled as a linear elastic plate. The cell itself is represented by an equivalent constant rigidity. Numerical simulations predict the sequence of broken bonds, as well as the newly established connections on the ‘adhesive part’ of the interface. The interplay between adhesion and rupture entails a rolling phenomenon. In the last part of this chapter, a model of the deformation induced by the random fluctuation of the protrusion force resulting from the variation of affinity with chemiotactic sources is calculated, using stochastic finite element methods in combination with the theory of Gaussian random variables.

Non-linear elastic-plastic techniques have been used to study the level A operating conditions of a relief valve subject to a small number of severe thermal shock transients during operation. The analysis uses a detailed FE mesh verified to capture the elastic-plastic behaviour of the valve. In addition, short duration transient heat loading on the valve was calculated using computational fluid dynamics using a conjugate heat transfer approach. The non-linear plasticity behaviour of the model was simulated using a kinematic hardening model incorporating non-linear hardening. Due to the very localised plasticity around small radius fillets in the model, a highly refined mesh strategy was needed. An innovative meshing strategy was therefore incorporated utilising a similar methodology to that used for fluid dynamics meshing; the fluid facing surfaces and nozzles were finely meshed with hexahedral elements, while parts of the internal bulk material were meshed using high order tetrahedral elements. Primary strength was analysed using traditional elastic methods. The progressive distortion check was based on the elasto-plastic through-wall strain distribution and the fatigue analysis based on the equivalent strain range.

Abstract The use of perturbation methods for fourth order PDE’s has not yet been examined extensively. Usually approximating power series are applied, which are truncated to one or two modes. Very little — or nothing — is said about the relation between this approximation and the exact solution. In this paper initial boundary value problems for the following equation will be discussed: w t t + w x x x x + ϵ ( u ( π , t ) − u ( 0 , t ) + ∫ 0 π w x 2 d x ) w x x = ϵ g ( x , t , w , w t ) . This equation can be regarded as a model describing wind-induced oscillations of flexible structures like elastic beams, where the small term on the right hand side of the equation represents the windforce acting on the structure. Existence and uniqueness for solutions of these problems will be discussed, as well as finding approximations using a multiple time-scale method. Finally the asymptotic validity of these approximations will be considered.

Nexans Norway is, together with Marintek, currently developing a software for detailed analysis of complex umbilical cross-section designs. The software development project combines numerical methods with small-scale testing of involved materials, as well as full-scale testing of a wide variety of umbilical designs, essential for calibration and verification purposes. Each umbilical design is modelled and comparisons are made with respect to global behaviour in terms of: • Axial strain versus axial force; • Axial strain versus torsion; • Torsion versus torsion moment for various axial force levels; • Moment versus curvature for different tension levels. The applied theory is based on curved beam and curved axisymmetric thin shell theories. The problem is formulated in terms of finite elements applying the Principle of Virtual Displacements. Each body of the cross-section interacts with the other bodies by contact elements which are formulated by a penalty formulation. The contact elements operate in the local surface coordinate system and include eccentricity, surface stiffness and friction effects. The software is designed to include the following functionality: • Arbitrary geometry modelling including helical elements wound into arbitrary order; • The helical elements may include both tubes and filled bodies; • Elastic, hyper-elastic, and elastic-plastic material models; • Initial strain; • Contact elements, including friction; • Tension, torsion, internal pressure, external pressure, bending and external contact loading (caterpillars, tensioners, etc.). The paper focuses on the motivation behind the development program including a description of the different activities. The theory is described in terms of kinematics, material models and finite element formulation. A test example is further presented comparing predicted behaviour with respect to full-scale test results.

Internal fluid flow parameters in conjunction with elastomechanical properties of conveyance systems have significantly modulated flow induced vibrations in pipeline and riser systems. Recent advances on the mechanics of sandwich elastic systems as effective vibration and noise reduction mechanisms have simulated the possibility of replacing traditional steel pipes with sandwich pipes in deepwater environment. The dynamic behaviour and stability of sandwich elastic pipes conveying a non-Newtonian fluid are investigated in this paper. For this problem, a set of generalised non-linear equations governing the vibration of sandwich pipes held together in pressurised environment and conveying a non-Newtonian fluid is presented. By linearizing the governing partial differential equation matching the problem physics, under slight perturbation of the internal fluid velocity and other flow variables closed form analytical results for the system dual natural frequencies and stability under external excitation are computed for field designs and applications. Results show that for a given length of pipe, beyond the critical velocity, instability increases with the velocity of conveyance.

Bounding techniques for calculating shakedown loads are of great importance in design since this eliminates the need for performing full elasto-plastic cyclic loading analyses. The classical Melan’s lower bound theorem is widely used for calculating shakedown loads of pressure vessel components under proportional loading. Polizzotto extended the Melan’s theorem to the case of non-proportional loading acting on a structure. This paper presents a finite element method, based on Polizzotto’s theorem, to estimate the elastic shakedown load for a structure subjected to a combination of steady and cyclic mechanical loads. This method, called non-linear superposition, is then applied to investigate the shakedown behaviour of a pressure vessel component — a nozzle/cylinder intersection and that of a biaxially loaded square plate with a central hole. Results obtained for both problems are compared with those available in the literature and are verified by means of cyclic elasto-plastic finite element analysis.

This paper is concerned with the mechanics of woven fabrics under tensile loading. The yarns are treated as elastica. The yarns bent into shape for both warp and weft are assumed to be elastic, homogenous, and weightless. During deformation the yarns are subjected to bending, extension and transverse compression. The initial geometry of the yarns in the fabric, under no external loading, is first obtained using a force-equilibrium method based on Love’s ordinary approximate theory, a generalisation of the Bernoulli-Euler theory of elastic rods. A non-linear boundary-value problem with a system of five differential equations has been formulated and solved. Application of load will further change the shape of the bent yarns due to bending and stretching. For a yarn with given initial geometry, as obtained by the force-equilibrium method, the solution of the deformed configuration is obtained from the solution of two nonlinear differential equations using appropriate boundary conditions. The formulation of the latter problem is based on the energy method. The sum of the energy terms due to bending, stretching together with the potential energy due to the applied load provides an expression for the total energy of the system. The variation of the total energy in terms of the variations of two parameters is then obtained, using the techniques from calculus of variations. One parameter described the deviation of the bent yarn from a straight line while the other is the length as measured along the yarn axis. This leads to a set of differential equations that fully describe the deformed yarns. The models, initially developed for plain weave, are being currently extended to non-plain weaves and 3D woven fabrics.

The non-linear dynamic response interaction of a pipeline on a non-stationary ocean-bed is herein investigated where the pipeline is idealized as a beam on a non-linear elastic foundation. By employing regular perturbation and integral transform approach, this paper presents generalized closed form expressions for the natural frequency of vibration and the dynamic interaction response profile. In particular, the modulation of the natural frequency by flow parameters such as, the internal fluid transmission velocity, pipeline sediment cover and the geo-mechanical behaviour of the ocean bed are highlighted while the design implications for such transmission pipelines are analyzed.

Abstract In the last years modeling, simulation, and control of flexibel structures have made an essential progress, especially stimulated by the requirements of space operations. For this working field, flexible lightweight robots can enhance the work space of space robots very well, if the length of the robot arm is variable. Contributions in the corresponding literature mostly consider models of special configurations and perform simple approximations or even assume linear models to describe the elastic motions of the robot. However, especially in the case of lightweight robots undergoing large reference motion, its nonlinear dynamic behaviour should be modeled as exact as possible. The mechanical system should be controlled to improve its dynamical behaviour. Unfortunately, it is not possible to measure the position of the endeffector. The only measurements that can be done are that of the strains of the beam. The control input should be the velocity (translational and rotational) transmitted to the base of the beam. This includes a correction of the actual path planning values as a kinematical control. Here the specified task is solved in two parts: Firstly, the variable length, the geometric nonlinear beam behaviour and the large reference motion of the driven joint are taken into consideration with a systematic modeling approach in (Söffker, 1995b; Söffker, 1995c). Nonlinear control approaches can often be used only for special classes of problems assumptions have to be made, which cannot be generally fulfilled in practice. Here a new controller is developed which is realizable for on-line applications with the mentioned restrictions. Therefore, in a second part the mechanical plant will be controlled by a new observer-based dynamic compensation scheme. Based on a linear time-invariant model, nonlinear effects and unmodeled dynamics are estimated in a first step by an Proportinal-Integral (PI) observer scheme, whereby the nonlinearities and the effects of the non-considered time-varying parameters etc. are assumed as external disturbances. The used PI-observer (Söffker, 1995a) estimates the states of the system and the external disturbances under some weak assumptions. Using this informations about the external disturbances to the nominal time-invariant system, the task of an extended regulator scheme is to compensate these effects in a second step. Because of the structure of the given mechanical system, usual static disturbance compensation schemes are not useful. Therefore a new dynamic approach is developed, which uses the estimations of the extended observer. The developed method does not depend on the structure of the physical problem and can be also used more generally. This is mainly due to the robustness of the observer-based dynamic compensation scheme. Two examples of a very flexible spatial telescopic robot arm demonstrate the effects of disturbances compensation and control of the elastic vibrations induced by the initial conditions and the reference motion.

This paper is devoted to the study of the dynamic behaviour of a tower structure with an attached damper. Two dampers are attached to the upper end of the rod in two mutually perpendicular planes. The oscillation damper is a mass attached to the rod by elastic coupling and a viscous friction. The mathematical model of the object being under consideration includes a system of two non-linear differential equations. The numerical simulation was used to study the dynamic behavior of a dissipative system with limited excitation, containing a pendulum vibration damper. The characteristics of the system amplitude − the speed of the wind flow is obtained. It is shown that the pendulum vibration damper can significantly reduce the amplitudes of resonant oscillations of the cantilever rod. It is also shown that the choice of system parameters makes it possible to avoid the establishing of resonant oscillations of the system. at the first resonance zone where the amplitudes of oscillations achieved the greatest values.

The purpose of this paper is to demonstrate the application of the Linear Matching Method (LMM) to cracked bodies subjected to cyclic histories of loads and temperatures. The method [1,2,3], related to the methods of Elastic Compensation and Gloss r-node, used in design calculations for a number of years, involves matching the behaviour of a non-linear material to that of a linear material. The significance of the developed programming methods, in engineering design, is then discussed for two applications. The first is in the identification of ratchet limits for cracked structures subjected to variable loads and temperatures. Solutions are presented for both perfect plasticity and complete cyclic hardening conditions, for differing values of crack lengths, for the classical axisymmetric Bree problem. The other is in the investigation of the relationship between the near crack-tip fields and the cyclic loading histories. The analysis reveals the strong influence of the elastic stresses, immediately outside the stress singularities, allowing an understanding of the behaviour of mechanically and thermally induced crack-tip fields to be developed.