Academic literature on the topic 'Strain energy function'

Abstract The Ogden strain-energy function for rubber appears to be gaining popularity among users of finite element analysis. This paper discusses some of the special features of this material model. It explains why nonlinear regression analysis of stress-strain data obtained from just one mode of deformation may yield an Ogden strain-energy function that is unsatisfactory for predicting behavior in other deformation modes. It suggests the regression analysis be constrained such that some of the coefficients are chosen based upon the known behavior of rubbery materials.

The strain energy for the air-filled lung is calculated from a model of the parenchymal microstructure. The energy is the sum of the surface energy and the elastic energies of two tissue components. The first of these is the peripheral tissue system that provides the recoil pressure of the saline-filled lung, and the second is the system of line elements that form the free edges of the alveolar walls bordering the alveolar ducts. The computed strain energy is consistent with the observed linear elastic behavior of parenchyma and the data on large deformations around blood vessels.

To describe elastic material behavior the starting point is the isochoric-volumetric decoupling of the strain energy function. The volumetric part is the central subject of this contribution. First, some volumetric functions given in the literature are discussed with respect to physical conditions, then three new volumetric functions are developed which fulfill all imposed conditions. One proposed function which contains two material parameters in addition to the compressibility parameter is treated in detail. Some parameter fits are carried out on the basis of well-known volumetric strain energy functions and experimental data. A generalization of the proposed function permits an unlimited number of additional material parameters. Dedicated to Professor Franz Ziegler on the occasion of his 60th birthday. [S0021-8936(00)00901-6]

The main purpose of this study was to determine a suitable strain energy function for a specific elastomer. A survey of various strain energy functions proposed in the past was made. For natural rubber, there were some specific strain energy functions which could accurately fit the experimental data for various types of deformations. The process of determining a strain energy function for the specific elastomer was then described. The second-order invariant polynomial strain energy function (James et al., 1975) was found to give a good fit to the experimental data of uniaxial tension, uniaxial compression, equi-biaxial extension, and pure shear. A new form of strain energy function was proposed; it yielded improved results. The equi-biaxial extension experiment was done in a novel way in which the moire techniques (Pendleton, 1989) were used. The obtained strain energy functions were then utilized in a finite element program to calculate the load-deflection relation of an electrometric spring used in an electrical connector.

Kyoto University (京都大学)
0048
新制・課程博士
博士(工学)
甲第14836号
工博第3133号
新制||工||1469(附属図書館)
27242
UT51-2009-F478
京都大学大学院工学研究科マイクロエンジニアリング専攻
(主査)教授 小寺 秀俊, 教授 北條 正樹, 教授 田畑 修
学位規則第4条第1項該当

The thesis presents a wide range of studies on carbon black and silica particulate reinforced rubbers. These include stress-strain, strain energy function, static and cyclic stress relaxation (stress softening), trouser test piece tearing and cyclic crack growth studies. The novel features of the work include the development of a simple strain energy function which is shown to represent the stress-strain behaviour of carbon black and silica filled rubbers up to strains of 100%. The numerical values of the constants in this function are shown to vary in a meaningful and systematic manner with the fraction of reinforcing filler and with the crosslink density. The cyclic stress relaxation studies are the first of their kind and demonstrate a significantly increased relaxation rate resulting from cycling in filled rubbers. The trouser tearing studies give some insight as to the materials and experimental variables that determine the type of tear growth and regime of tearing. The process of stress whitening around the tear tip during steady tearing in silica filled compounds provide the first opportunity to quantitatively relate the tearing energy to the hysteresis energy loss in a known volume of rubber at the tear tip. The cyclic crack growth studies show for the first time a systematic decrease in crack growth per cycle (dc/dn) at a given tearing energy as the carbon black filler content is systematically increased and as the crosslink density is decreased. A novel feature of the work is the demonstration of the effect of pre-strain in one direction on the cyclic growth rate of a crack in this direction when cyclically strained in a direction at right angles. The very large increase in dc/dn with increasing pre-strain is discussed in terms of pre-orientation of the rubber/carbon black structure.

In the classic Hill model for muscle contraction, the split between the muscle and tendon is arbitrary and the problem lacks a unique solution. Instead of reformulating the problem to a differential-algebraic equation and solving for a set of initial conditions, a constant tendon length is commonly assumed in musculoskeletal simulation tools. This assumption has not been thoroughly tested and introduces errors of unknown magnitude to the simulations. In this thesis, the contractile element of the Hill model is modelled as a friction clutch in parallel to a viscous damper. This provides an evolution law for the muscle length by which the muscle speed is numerically calculated taking into account a non-zero tendon speed. A simple biceps curl is simulated with the friction clutch model and compared to corresponding commercial musculoskeletal simulations. Overall, the results are similar, in particular for the muscle lengths which are almost identical in every simulation (0.00-0.42% difference). The difference in tendon speed is 0.00-3.26%, with upwards tendencies. In general, the error percentage of the tendon speed appears to decrease by the same amount that the contraction speed is reduced. Conclusively, it can be said that the introduced friction clutch model delivers comparative outcomes to a commercial musculoskeletal simulation software, while not assuming a constant tendon length. However, while presenting a relatively simple solution, an increased computation time is to be expected due to the need of a differential equation solver. Further investigation regarding implementation and computing times in more complex simulations may provide an alternative approach to conventional musculoskeletal simulations.

This PhD thesis employed high energy synchrotron x-ray radiation to reveal atomic scale structural features occurring in plastically deformed Zr52.5Ti5Cu18Ni14.5Al10 (Vit105) bulk metallic glass (BMG). The study is divided into three parts: Strain evolution during in-situ compression, strain distribution maps in mechanically-imprinted BMG, and residual strain around a single shear band. 1. Strain evolution during in-situ compression The structural rearrangements occurring during compressive deformation of a plastically deformable BMG showed that the elastic and plastic deformation of the BMG is correlated to the structural changes at short- (SRO) and medium range order (MRO). In the elastic regime, the atomic distances at SRO vary linearly with macroscopic stress. Analysis of the area under radial distribution function indicates that a small fraction of bonds in the first shell is broken in the loading direction whereas some new bonds are formed in the transverse direction. Atomic bonds at SRO appeared significantly stiffer than the MRO shells. Compared to the macroscopic values of the elastic strain, Young’s modulus and Poisson's ratio, both SRO and MRO appeared significantly stiffer, implying that the elastic behavior of the BMG is not only ruled by simple compression of the atoms/clusters but also is aided by rearrangement of atoms/clusters. The deviation of MRO atomic strain-stress correlation from linearity at the onset of plastic deformation was attributed to the activation of irreversible shear transformation zones. It was demonstrated by a strong shear strain value at the onset of yielding. This value is in good agreement with the reported value of the critical shear strain needed for activation of an irreversible STZ. The length scale of 12.5 Å indicated the largest shear strain and is probably the most effective length scale in the formation of STZs. The atomic pairs at SRO with smallest shear strain have the least contribution to the STZs. It was also indicated that the typical fracture angle of this BMG can be explained by the orientation of maximum shear strain at the onset of catastrophic shear band formation. 2. Strain distribution map in mechanically-imprinted BMG In mechanical imprinting, the BMG plate is loaded between two tools with a regular array of linear teeth and, as a result, a regular pattern of linear imprints is created on the surface of the plate. Mechanically imprinting results in considerable tensile plasticity of brittle Vit105 BMG plate. The distribution of hardness and Young’s modulus values at the transverse cross section of imprinted plate probed via nanoindentation revealed oscillating soft and hard regions beneath the surface. Spatially-resolved strain maps obtained via high-energy nano-size beam X-ray diffraction exhibited that the plastic deformation during imprinting creates a spatially heterogeneous atomic arrangement, consisting of strong compressive and tensile strain fields as well as significant shear strain fields in the cross section. It was shown that the heat treatment diminishes the heterogeneous structure resulting in brittle behavior in tension. The analysis of strain tensor components based on changes in the first diffraction maximum of the structure function, q1, revealed that Ɛx, the strain perpendicular to the loading direction, changes from the compressive at near to the surface to the tensile mode at the center of the imprinted plate. In contrast, the strain component along the loading direction, Ɛy, changes from tensile near the surface to the compressive at the center. Beneath the surface, Ɛx reaches to values about 1.5% under the imprints where there is a negligible Ɛy. The distribution map of principal strains, Ɛ1 and Ɛ2, indicated that large regions with compressive Ɛ1 and Ɛ2 exist under the imprints which can result in blocking of the propagating shear bands in agreement with microstructural observations of shear banding after uniaxial tension. Moreover, the region beneath the border of the imprinted and un-imprinted parts has the highest residual shear strain. Microstructural observations indicated that such regions can nucleate new shear bands upon tensile loading of imprinted BMG plate. 3. Residual strain around a single shear band In order to probe structural changes in the shear-induced zone around a single shear band, the distribution of residual strains at short- and medium-range order around a single shear band was determined in cold-rolled BMG plate using the nano-focused high energy x-ray diffraction. Plastic deformation results in significant residual normal and shear strains at distances of more than 15 μm around the shear band. The residual normal strains exhibit an asymmetric distribution whereas the residual shear strain is distributed symmetrically around the shear band. The large amount of residual atomic shear strain magnitude at the vicinity of the shear band triggers the nucleation of the new shear bands. The coincidence of the direction of the nucleating secondary shear bands from the main shear band with the orientation of the residual shear strain at the vicinity of the mature shear band highlight the dominant role of the shear strain in determining further plastic deformation at regions near the shear band
Im Rahmen dieser Arbeit wird hochenergetische Synchrotron Röntgenstrahlung zum Aufzeigen der strukturellen Veränderungen in plastisch verformtem Zr52.5Ti5Cu18Ni14.5Al10 metallischen Glas verwendet. Die Arbeit gliedert sich in drei Teile: Dehnungsentwicklung während in-situ Druckversuch, Dehnungsverteilung eines mechanisch geprägten massiven metallischen Glases, und Restdehnungen in der Umgebung eines einzenen Scherbandes. 1. Dehnungsentwicklung während in-situ Druckversuch Die während der Verformung auftretende strukturelle Neuordnung eines plastisch verformbaren metallischen Glases zeigt die Korrelation der elastischen und plastischen Verformung mit den strukturellen Änderungen in den Größenordnungen der Nah- (SRO) und mittelreichweitigen Ordnung (MRO). Im elastischen Bereich verändern sich die Atomabstände in der SRO linear mit der makroskopisch anliegenden Spannung. Die Untersuchung der Fläche unter der Radialen Verteilungsfunktion (RDF) deutet auf ein Aufbrechen eines geringen Anteils der Bindungen der ersten Schale in Druckspannungsrichtung und deren Neubildung quer dazu. Die atomaren Bindungen in der SRO erscheinen wesentlich steifer als in den MRO Schalen. Vergleicht man die Werte von elastischer Dehnung, E-Modul und Querkontraktionszahl mit ihren makroskopischen Gegenstücken erscheinen beide, SRO und MRO, wesentlich steifer. Dies zeigt, dass die elastische Verformung von metallischen Gläsern nicht nur von der einfachen Stauchung der Atome bzw. Atomgruppen bestimmt, sondern auch durch deren Neuanordnung unterstützt wird. Das Abweichen der Dehnungs-Spannungs-Korrelation vom linearen Verhalten in der MRO am Beginn der plastischen Verformung wird der irreversiblen Bildung von Schertransformations-zonen (STZ) zugeschrieben. Dies zeigt sich zudem in den erhöhten Scherdehnungswerten am Beginn der Dehngrenze, welche mit den in der Literatur berichteten Werten für die kritische Scherdehnung zum Bilden einer STZ übereinstimmen. Bei einem Atomabstand von 12,5 Å tritt der höchste Wert der Scherdehnung auf und markiert den effektivsten Längenbereich der STZ Bildung. Andererseits haben die atomaren Paare in der SRO mit der geringsten Scherdehnung den geringsten Beitrag an der STZ. Es zeigt sich außerdem, dass der typische Bruchwinkel dieses metallischen Glases über die Orientierung der maximalen Scherdehnung am Beginn der kritischen Scherbandbildung erklärt werden kann. 2. Dehnungsverteilung eines mechanisch geprägten massiven metallischen Glases Eine Prägung besteht darin, eine Platte metallischen Glases mit zwei Stempel, auf denen eine regelmäßige Anordnung von geradlinigen Kerben angebracht ist, zu belasten. Dadurch wird eine ebenso regelmäßige Anordnung von geradlinigen Kerben auf der Oberfläche des metallischen Glases erzeugt. Die plastische Verformbarkeit der Vit105 Platte im Zugversuch wird durch Prägung im Vergleich zur gegossenen Probe eindeutig verbessert. Die Untersuchung der Härte und des E-Moduls über den Querschnitt der geprägten Probe zeigt die Einbringung von Abwechselnd weichen und harten Regionen an der Oberfläche. Es wurden räumlich aufgelöste Dehnungskarten des geprägten metallischen Glases durch Beugung eines hochenergetischen nanometergroßen Röntgenstrahles erzeugt. Die Ergebnisse offenbaren, dass die durch Prägung eingebrachte plastische Verformung eine räumlich heterogene Atomanordnung erzeugt, welche aus starken Druck- und Zugdehnungsfeldern besteht. Zusätzlich wird eine signifikante Scherdehnung in die Probe eingebracht. Die Wärmebehandlung beseitigt diese heterogene Struktur und führt sie fast auf den Ausgangszustand zurück. Die Analyse der Dehnungstensorkomponenten basierend auf Änderungen im erstem Maximum des Strukturfaktors, q1, zeigt, dass sich Ɛx von der Oberfläche zur Mitte der Platte hin von einer Stauchung in eine Dehnung umwandelt. Im Gegensatz dazu wandelt sich die Komponente Ɛy von der Oberfläche zur Mitte der Platte hin von einer Dehnung in eine Stauchung um. An der Oberfläche unter den Eindrücken, wo Ɛy vernachlässigbar ist, erreicht Ɛx Werte von ca. 1.5 %. Die Verteilungskarten der Hauptdehnungen zeigt, dass beide e1 und e2 unterhalb der Kerben als Stauchungen vorhanden sind. Daraus resultiert das Blockieren und Ablenken der sich ausbreitenden Scherbänder, was an Zugproben im REM beobachtet werden kann. Weiterhin hat der Bereich an der Grenze der geprägten und nicht geprägten Regionen die höchste Restscherdehnung. Mikrostrukturelle Beobachtungen deuten darauf hin, dass solche Bereiche unter Zuglast Keimstellen für neue Scherbänder sind. 3. Restdehnungen in der Umgebung eines einzenen Scherbandes Es wurde ein einzelnes Scherband einer kaltgewalzte Platte mittels Beugung eines hochenergetischen nanometergroßen Röntgenstrahles untersucht. Die strukturellen Unterschiede in der scherinduzierten Zone um ein einzelnes Scherband werden durch die Verteilung der Restdehnungen in SRO und MRO bestimmt. Plastische Verformung führt zu signifikanten Restnormal- und Restscherdehnungen in Entfernungen von mehr als 15 µm um das Scherband. Die Restnormaldehnungen zeigen eine asymmetrische Verteilung, wohingegen die Restscherdehnungen auf beiden Seiten des Scherbandes symmetrisch verteilt sind. Der große Betrag der atomaren Restscherdehnung in der Nähe des Scherbandes führt zur Bildung von neuen Scherbändern. Das Zusammenfallen der Richtung des sich bildenden sekundären Scherbandes und der Orientierung der Restscherdehnung, in der Nähe des primären Scherbandes, demonstriert die dominierende Rolle der Scherdehnung bei weiterer plastischer Verformung in der Nähe des Scherbandes

Cette thèse a porté sur la construction de densités d'énergie de déformation permettant de décrire le comportement non linéaire de matériaux anisotropes tels que les tissus biologiques souples (ligaments, tendons, parois artérielles etc.) ou les caoutchoucs renforcés par des fibres. Les densités que nous avons proposées ont été élaborées en se basant sur la théorie mathématique des polynômes invariants et notamment sur le théorème de Noether et l'opérateur de Reynolds. Notre travail a concerné deux types de matériaux anisotropes, le premier avec une seule famille de fibre et le second avec quatre familles. Le concept de polyconvexité a également été étudié car il est notoire qu'il joue un rôle important pour s'assurer de l'existence de solutions. Dans le cas d'un matériau comportant une seule famille de fibre, nous avons démontré qu'il était impossible qu'une densité polynomiale de degré quelconque puisse prédire des essais de cisaillement avec un chargement parallèle puis perpendiculaire à la direction des fibres. Une densité polynomiale linéaire combinée avec une fonction puissance a permis de contourner cet obstacle. Dans le cas d'un matériau comportant quatre familles de fibre, une densité polynomiale a permis de prédire correctement des résultats d'essai en traction bi-axiale extraits de la littérature. Les deux densités proposées ont été implémentées avec la méthode des éléments finis et en langage C++ dans le code de calcul universitaire FER. Pour se faire, une formulation lagrangienne totale a été adoptée. L'implémentation a été validée par des comparaisons avec des solutions analytiques de référence que nous avons exhibée dans le cas de chargements simples conduisant à des déformations homogènes. Des exemples tridimensionnels plus complexes, impliquant des déformations non-homogènes, ont également été étudiés
This thesis has focused on the construction of strain energy densities for describing the non-linear behavior of anisotropic materials such as biological soft tissues (ligaments, tendons, arterial walls, etc.) or fiber-reinforced rubbers. The densities we have proposed have been developed with the mathematical theory of invariant polynomials, particularly the Noether theorem and the Reynolds operator. Our work involved two types of anisotropic materials, the first with a single fiber family and the second with a four-fiber family. The concept of polyconvexity has also been studied because it is well known that it plays an important role for ensuring the existence of solutions. In the case of a single fiber family, we have demonstrated that it is impossible for a polynomial density of any degree to predict shear tests with a loading parallel and then perpendicular to the direction of the fibers. A linear polynomial density combined with a power-law function allowed to overcome this problem. In the case of a material made of a four-fiber family, a polynomial density allowed to correctly predict bi-axial tensile test data extracted from the literature. The two proposed densities were implemented in C++ language in the university finite element software FER by adopting a total Lagrangian formulation. This implementation has been validated by comparisons with reference analytical solutions exhibited in the case of simple loads leading to homogeneous deformations. More complex three-dimensional examples, involving non-homogeneous deformations, have also been studied

Part 1: Theoretic background information - Review of Hooke’s law for linear elastic materials - The strain energy density of linear elastic materials - Hyperelastic material - Material laws for hyperelastic materials - About selecting the material model and performing tests - Implementation of hyperelastic material laws in Mechanica - Defining hyperelastic material parameters in Mechanica - Test set-ups and specimen shapes of the supported material tests - The uniaxial compression test - Stress and strain definitions in the Mechanica LDA analysis Part 2: Application examples - A test specimen subjected to uniaxial loading - A volumetric compression test - A planar test - Influence of the material law Appendix - PTC Simulation Services Introduction - Dictionary Technical English-German
Teil 1: Theoretische Hintergrundinformation - Das Hookesche Gesetz für linear-elastische Werkstoffe - Die Dehnungsenergiedichte für linear-elastische Materialien - Hyperelastisches Material - Materialgesetze für Hyperelastizität - Auswählen des Materialgesetzes und Testdurchführung - Implementierung der hyperelastischen Materialgesetze in Mechanica - Definieren der hyperelastischen Materialparameter in Mechanica - Testaufbauten und Prüfkörper der unterstützten Materialtests - Der einachsige Druckversuch - Spannungs- und Dehnungsdefinition in der Mechanica-Analyse mit großen Verformungen Teil 2: Anwendungsbeispiele - Ein einachsig beanspruchter Prüfkörper - Ein volumetrischer Drucktest - Ein planarer Test - Einfluss des Materialgesetzes Anhang: - Kurzvorstellung der PTC Simulationsdienstleistungen - Wörterbuch technisches Englisch-Deutsch

The fractal-like finite element method (FFEM) is developed to compute stress intensity factors (SIFs) for isotropic homogeneous and bi-material V-notched plates. The method is semi-analytical, because analytical expressions of the displacement fields are used as global interpolation functions (GIFs) to carry out a transformation of the nodal displacements within a singular region to a small set of generalised coordinates. The concept of the GIFs in reducing the number of unknowns is similar to the concept of the local interpolation functions of a finite element. Therefore, the singularity at a notch-tip is modelled accurately in the FFEM using a few unknowns, leading to reduction of the computational cost.The analytical expressions of displacements and stresses around a notch tip are derived for different cases of notch problems: in-plane (modes I and II) conditions and out-of-plane (mode III) conditions for isotropic and bi-material notches. These expressions, which are eigenfunction series expansions, are then incorporated into the FFEM to carry out the transformation of the displacements of the singular nodes and to compute the notch SIFs directly without the need for post-processing. Different numerical examples of notch problems are presented and results are compared to available published results and solutions obtained by using other numerical methods.A strain energy approach (SEA) is also developed to extract the notch SIFs from finite element (FE) solutions. The approach is based on the strain energy of a control volume around the notch-tip. The strain energy may be computed using commercial FE packages, which are only capable of computing SIFs for crack problems and not for notch problems. Therefore, this approach is a strong tool for enabling analysts to compute notch SIFs using current commercial FE packages. This approach is developed for comparison of the FFEM results for notch problems where available published results are scarce especially for the bi-material notch cases.A very good agreement between the SEA results and the FFEM results is illustrated. In addition, the accuracy of the results of both procedures is shown to be very good compared to the available results in the literature. Therefore, the FFEM as a stand-alone procedure and the SEA as a post-processing technique, developed in this research, are proved to be very accurate and reliable numerical tools for computing the SIFs of a general notch in isotropic homogeneous and bi-material plates.

This thesis addresses a number of challenges designers face when designing deployable origami-based arrays, specifically joint selection, design, and placement within an array. In deployable systems, the selection and arrangement of joint types is key to how the system functions. The kinematics and performance of an array is directly affected by joint performance. This work develops joint metrics which are then used to compare joint performances, constructing a tool designers can use when selecting joints for an origami array. While often a single type of joint is used throughout an array, this work shows how using multiple types of joints within the same array can offer benefits for motion deployment, and array stiffening. Origami arrays are often used for their unique solutions for stowing and deploying large planar shapes. Folds, enabled through joints, within these patterns allow the arrays to fold compactly. However, it can be difficult to fully deploy arrays, particularly array designs with a high number of joints. In addition, it is a challenge to stabilize a fully deployed array from undesired re-folding. This work introduces a strain-energy storing joint that is used to deploy and stiffen foldable origami arrays, the Lenticular Lock (LentLock). Geometry of the LentLock is introduced and the deploying and stiffening performance of the joint is shown. Folds within an origami array create the constraints that link motion between panels, and can be used to create kinematic benefits, such as creating mechanisms with a single degree-of-freedom. While many fold-constraints are required to define motion, this work shows that origami-based system contain many redundant constraints. The removal of redundant joints does not affect the motion of the array nor the observed mobility, but may decrease the likelihood of binding, simplify the overall system and decrease actuation force. This work introduces a visual and iterative approach designers can use to identify redundant constraints in origami patterns, and techniques that can be used to remove the identified redundant constraints. The presented techniques are demonstrated by removing redundant constraints from prototyped origami mechanisms. As a result of this work, designers will be better able to approach and design deployable origami-based mechanisms.

Thesis (M. S.)--Mechanical Engineering, Georgia Institute of Technology, 2005.
David N. Ku, Committee Chair ; Alexander Rachev, Committee Co-Chair ; Elliot L. Chaikof, Committee Member. Includes bibliographical references.

This thesis report aims to help the client developing their new product. The new product to be developed is a safety bracket for a safety system. The safety bracket connects different parts which create the safety system and it should be able to withstand impacts from moving objects. The client has a set of requirements that needs to be addressed during the product development process. One of the most important requirements that must be fulfilled is the given impact energy that the safety bracket must withstand. The methodology used during this thesis work is the product development processes (PDP). The product development process is used to find concepts that have the potential to answer the research questions and to fulfil the requirements. Some methods used in the product development process are brainstorming, brainwriting and combining working principles. The concepts were evaluated with a combination of Pugh´s matrix and weighting matrix. The three best concepts were selected for further development and tested with FEA simulation with Abaqus CAE. The impact simulation gave indications if the concepts could handle the impact energy and if they could fulfil the requirements. All three concepts could withstand the impact energy based on the simulations and most of the requirements could be fulfilled. The concepts with thinner profile walls had a reduction in stress and an increase in impact duration, where the kinetic energy is distributed throughout the impact. A protective shell helps with the reduction of stress and the energy absorption during the impact simulation.

This chapter covers the notion of hyperelasticity—the concept that stress is derived from a strain—energy function–by invoking an analogy between elastic materials and springs. Alternatively, it can be derived by invoking a work inequality; the notion that work is required to effect a cyclic motion of the material.

This chapter details analytical solutions for unconstrained materials using tractable strain–energy functions. Three-dimensional and plane–strain deformations are illustrated. These include a formulation of the cavtitation problem in compressible materials and a discussion of so-called Harmonic materials.

Children with cerebral palsy have major motor impairments that lead to reduced fitness and physical activity levels. Increased energy cost of walking in combination with reduced aerobic fitness induces high levels of physical strain that can lead to fatigue complaints and limited physical activity. To assess fitness, adapted laboratory and field exercise tests with good reliability and validity are available for both ambulant and wheelchair-using children with cerebral palsy. Children with cerebral palsy show increased sedentary time and low physical activity levels which relate to increased risks for health issues and loss of functional abilities at older age. Reducing sedentary time and increasing physical activity are important to counteract the increased health risks and loss of functional abilities. Fitness training of sufficient frequency, intensity, and duration is required to maintain and optimize long-term health and functional ability in children with cerebral palsy.

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

There is indeed a "miracle" in the 1970 film Wanda. This film has survived, despite decades of neglect, to emerge into the fuliginous light of an era that may just be ready to strain at grasping its harsh and brutal truths -- truths that reveal the imbrication of the psychic in the social and the experiential in political structures. Barbara Loden's film dares to suggest that the social and ethical functions of art should not necessarily be redemptive – that salvation is a cheap and spurious form of consolation that few can afford in this world. This film, made by a woman who knew all too well what it means to be defined through and by her material circumstances (and her relationships to men), and that is so relentlessly ferocious in its refusal to assuage and comfort the viewer, has always been a form of future feminism. Wanda does not brook the comforts of positivity, of aspiration, or even the luxury of selfhood. This film, Still Life contends, is so radical in its feminist-anti-capitalist politics of refusal that we are still struggling to keep up with it. It delineates precisely how the personal is political and why this matters now more than ever. Wanda, a film about a woman who refuses to be saved or to save herself, who lacks the means and energy to alter anything in her life, who lives in a permanent state of blockage, impasse and failure is, as this publication suggests, the film of our contemporary moment.

The vessel wall exhibits relatively strong nonlinear properties and undergoes a wide range of deformations. Identification of a strain energy function (SEF) is the preferred method to describe the complex nonlinear elastic properties of the vascular tissue. Once the strain energy function is known, constitutive equations can be obtained.

This paper proposes a new Patient-Specific Aneurysm CFD Model (PSAM) which is based on the energy strain function combined with dilated vessel wall stress-strain relationship to predict aneurysm rupture. The PSAM relies on the available mechanical properties and parameters obtained from a personalized model. A personalized model is developed based on instantaneous arterial deformations obtained from Doppler Ultrasound (US) images at 6–9 MHz. It is shown that PSAM has the ability to correlate the deformation wall energy based on continuous patient-specifics in predicting rupture.

Models to represent the mechanical behavior of the annulus fibrosus are important tools to understand the biomechanics of the spine. Many hyperelastic constitutive equations have been proposed to simulate the mechanical behavior of the annulus that incorporate the anisotropic nature of the tissue. Recent approaches [1,2] have included terms into the energy function which take into account fiber-fiber and fiber-matrix interactions, leading to complex functions that cannot be readily implemented into commercial finite element codes for an efficient simulation of nonlinear realistic models of the spine (which are generally composed of 100,000+ degrees of freedom). An effort is undertaken here to test the capability of a relatively simple strain energy function [3] for the description of the annulus fibrosus. This function has already been shown to successfully represent the mechanical behavior of the arterial tissue and can be readily implemented into existing finite element codes.

The Modal Strain Energy Method (MSEM) is widely used in practice for the prediction of damping levels in structures. MSEM is based on a fundamental assumption that the damped and the undamped mode shapes of a structure are identical. Therefore, when MSEM is to be used, it is essential to ensure that this assumption is an acceptable assumption. However, detailed information on the accuracy of the method as a function of the system parameters including modal (or mode shape) complexity is quite limited. In this paper, the performance of MSEM is assessed in terms of the damping levels of the structure, proportionality of damping distribution and/or the modal complexity. To do so, an effective finite element based MSE approach is proposed first. Then, a proportionally damped structure with different damping levels is modeled and the performance of MSEM is assessed as a function of the structural damping level. After that, a non-proportionally damped structure is studied in order to examine the performance of the method with respect to mode shape complexity. In all cases, a more accurate reference method, based on complex eigenvalue approach, is used for comparison purposes. Furthermore, a few definitions of mode shape complexity are utilized in order to quantify the mode shape complexity. The results show that as long as the mode shapes are real or close to being real, MSEM can predict the damping levels as well as the natural frequencies of a damped structure with good accuracy. However, the accuracy that can be achieved with MSEM decreases as mode shape complexity increases.

The vessel wall exhibits relatively strong nonlinear properties and undergoes wide range of deformations. These characteristics make the identification of a strain energy function (SEF), the preferred method to describe the complex nonlinear elastic properties of the vascular tissue. None of the currently proposed structural models succeeded in describing accurately and simultaneously both the pressure-radius (Pro) and pressure-longitudinal force (P-Fz) curves. We hypothesized that the shortcomings of current models are partly due to unaccounted anisotropic properties of elastin.

Pneumatic actuators used in devices that, by their function require light weight and small size such as orthotics, can benefit from the inclusion of accumulators to harness and recycle energy normally lost in exhausted gases. In order for an accumulator to provide benefits to these small systems, they must possess relatively high gravimetric and volumetric energy densities with adequately high efficiencies as compared to conventional accumulators, like traditional spring piston accumulators. Constructing accumulators that primarily use strain as the primary energy storage method can provide the energy storage capacities and efficiencies needed, as well as a better pressure-volume relationship than a fixed-volume accumulator. This paper outlines the behavior of an elastomeric strain accumulator constructed using natural rubber tubes as the material for an accumulator. Tube shaped accumulators fill to a preset maximum diameter, constrained by a rigid shroud so that the material’s yield strength is not approached and to control the manner in which the accumulator expands. Controlling the manner of expansion for the accumulator allows a relatively constant pressure to be used through the majority of the fill cycle. Natural rubber accumulators were experimentally evaluated and characterized for their energy storage efficiencies over a range of different parameters allowing basic design criteria to be created for use in building accumulators tailored to specific system requirements.

Abstract A penalty function approach is used in conjunction with a multi-criteria optimization method for topology synthesis of compliant mechanisms. This method can help facilitate convergence to physically meaningful solutions for problems with a large number of design variables. The second part of the paper is an investigation of the element strain energy density of the optimal solution, where a second stage size optimization routine is developed. The solution from the topology optimization is used as a starting point for the resizing algorithm, which uses an optimality criteria method based on the overall average strain energy density. This second stage optimization more uniformly distributes the element strain energy densities in order to avoid localized areas of high stress or strain.

In this paper, a new visco-hyperelastic constitutive law for describing the rate dependent behavior of foams is proposed. The proposed model was based on a phenomenological Zener model: a hyperelastic equilibrium spring, which describes the steady-state, long-term response, parallel to a Maxwell element, which captures the ratedependency. A nonlinear viscous damper connected in series to a hyperelastic intermediate spring, controls the ratedependency of the Maxwell element. Therefore, the stress is the sum of equilibrium stress on the equilibrium spring and overstress on the intermediate spring. In hyperelastic theory stress is not calculated directly as in the case of small-strain, linear elastic materials. Instead, stresses are derived from the principle of virtual work using the stored strain energy potential function. In addition, foams are compressible, therefore classic strain energy functions such as the Ogden strain energy function or the Mooney-Rivlin strain energy function are not suitable to describe hyperelastic behavior of foams. So, strain energy functions must include the effect of compressibility. That means the third principal invariant of the deformation gradient tensor F should enter in strain energy functions. For rate-dependent behavior of foams, history integral constitutive law is used. For the equilibrium spring and the intermediate spring, the same strain energy function is employed. In order to use this stain energy function in history integral equation, the kernel function of it is calculated. The effect of compressibility is considered in rate-dependent behavior of foams too. All material constants were obtained from the results of uniaxial tensile tests. Nonlinear regulation was used to find these constants. In these calculations, Average strain rate was employed to find material constants.