Polar Cases of Elasticity and Constant Elasticity

Draft DRAFT Lecture Notes Introduction to CONTINUUM MECHANICS and Elements of Elasticity/Structural Mechanics c  VICTOR E. SAOUMA Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO 80309-0428 Draft 0–2 Victor Saouma Introduction to Continuum Mechanics Draft 0–3 PREFACE Une des questions fondamentales que l’ing´enieur des Mat´eriaux se pose est de conna ˆ itre le comporte- ment d’un materiel sous l’effet de contraintes et la cause de sa rupture. En d´efinitive, c’est pr´ecis´ement la r´eponse `a c/mat es deux questions qui vont guider le d´eveloppement de nouveaux mat´eriaux, et d´eterminer leur survie sous diff´erentes conditions physiques et environnementales. L’ing´enieur en Mat´eriaux devra donc poss´eder une connaissance fondamentale de la M´ecanique sur le plan qualitatif, et ˆetre capable d’effectuer des simulations num´eriques (le plus souvent avec les El´ements Finis) et d’en extraire les r´esultats quantitatifs pour un probl`eme bien pos´e. Selon l’humble opinion de l’auteur, ces nobles buts sont id´ealement atteints en trois ´etapes. Pour commencer, l’´el`eve devra ˆetre confront´eauxprincipesdebasedelaM´ecanique des Milieux Continus. Une pr´esentation d´etaill´ee des contraintes, d´eformations, et principes fondamentaux est essentiel. Par la suite une briefe introduction `a l’Elasticit´e(ainsiqu’`alath´eorie des poutres) convaincra l’´el`eve qu’un probl`eme g´en´eral bien pos´e peut avoir une solution analytique. Par contre, ceci n’est vrai (`a quelques exceptions prˆets) que pour des cas avec de nombreuses hypoth`eses qui simplifient le probl`eme (´elasticit´e lin´eaire, petites d´eformations, contraintes/d´eformations planes, ou axisymmetrie). Ainsi, la troisi`eme et derni`ere ´etape consiste en une briefe introduction `alaM´ecanique des Solides, et plus pr´ecis´ement au Calcul Variationel. A travers la m´ethode des Puissances Virtuelles, et celle de Rayleigh-Ritz, l’´el`eve sera enfin prˆet `a un autre cours d’´el´ements finis. Enfin, un sujet d’int´erˆet particulier aux ´etudiants en Mat´eriaux a ´et´eajout´e, `a savoir la R´esistance Th´eorique des Mat´eriaux cristallins. Ce sujet est capital pour une bonne compr´ehension de la rupture et servira de lien `aun´eventuel cours sur la M´ecanique de la Rupture. Ce polycopi´ea´et´eenti`erement pr´epar´e par l’auteur durant son ann´ee sabbatique `a l’Ecole Poly- technique F´ed´erale de Lausanne, D´epartement des Mat´eriaux. Le cours ´etait donn´eaux´etudiants en deuxi`eme ann´ee en Fran¸cais. Ce polycopi´ea´et´e´ecrit avec les objectifs suivants. Avant tout il doit ˆetre complet et rigoureux. A tout moment, l’´el`eve doit ˆetre `amˆeme de retrouver toutes les ´etapes suivies dans la d´erivation d’une ´equation. Ensuite, en allant `a travers toutes les d´erivations, l’´el`eve sera `amˆeme de bien conna ˆ itre les limitations et hypoth`eses derri`ere chaque model. Enfin, la rigueur scientifique adopt´ee, pourra servir d’exemple `a la solution d’autres probl`emes scientifiques que l’´etudiant pourrait ˆetre emmen´e`ar´esoudre dans le futur. Ce dernier point est souvent n´eglig´e. Le polycopi´e est subdivis´edefa¸con tr`es hi´erarchique. Chaque concept est d´evelopp´e dans un para- graphe s´epar´e. Ceci devrait faciliter non seulement la compr´ehension, mais aussi le dialogue entres ´elev´es eux-mˆemes ainsi qu’avec le Professeur. Quand il a ´et´ejug´en´ecessaire, un bref rappel math´ematique est introduit. De nombreux exemples sont pr´esent´es, et enfin des exercices solutionn´es avec Mathematica sont pr´esent´es dans l’annexe. L’auteur ne se fait point d’illusions quand au complet et `a l’exactitude de tout le polycopi´e. Il a ´et´e enti`erement d´evelopp´e durant une seule ann´ee acad´emique, et pourrait donc b´en´eficier d’une r´evision extensive. A ce titre, corrections et critiques seront les bienvenues. Enfin, l’auteur voudrait remercier ses ´elev´es qui ont diligemment suivis son cours sur Polar Cases of Elasticity and Constant Elasticity Polar Cases of Elasticity and Constant Elasticity By: OpenStaxCollege There are two extreme cases of elasticity: when elasticity equals zero and when it is infinite A third case is that of constant unitary elasticity We will describe each case Infinite elasticity or perfect elasticity refers to the extreme case where either the quantity demanded (Qd) or supplied (Qs) changes by an infinite amount in response to any change in price at all In both cases, the supply and the demand curve are horizontal as shown in [link] While perfectly elastic supply curves are unrealistic, goods with readily available inputs and whose production can be easily expanded will feature highly elastic supply curves Examples include pizza, bread, books and pencils Similarly, perfectly elastic demand is an extreme example But luxury goods, goods that take a large share of individuals’ income, and goods with many substitutes are likely to have highly elastic demand curves Examples of such goods are Caribbean cruises and sports vehicles Infinite Elasticity The horizontal lines show that an infinite quantity will be demanded or supplied at a specific price This illustrates the cases of a perfectly (or infinitely) elastic demand curve and supply curve The quantity supplied or demanded is extremely responsive to price changes, moving from zero for prices close to P to infinite when price reach P 1/5 Polar Cases of Elasticity and Constant Elasticity Zero elasticity or perfect inelasticity, as depicted in [link] refers to the extreme case in which a percentage change in price, no matter how large, results in zero change in quantity While a perfectly inelastic supply is an extreme example, goods with limited supply of inputs are likely to feature highly inelastic supply curves Examples include diamond rings or housing in prime locations such as apartments facing Central Park in New York City Similarly, while perfectly inelastic demand is an extreme case, necessities with no close substitutes are likely to have highly inelastic demand curves This is the case of life-saving drugs and gasoline Zero Elasticity The vertical supply curve and vertical demand curve show that there will be zero percentage change in quantity (a) supplied or (b) demanded, regardless of the price This illustrates the case of zero elasticity (or perfect inelasticity) The quantity supplied or demanded is not responsive to price changes Constant unitary elasticity, in either a supply or demand curve, occurs when a price change of one percent results in a quantity change of one percent [link] shows a demand curve with constant unit elasticity As we move down the demand curve from A to B, the price falls by 33% and quantity demanded rises by 33%; as you move from B to C, the price falls by 25% and the quantity demanded rises by 25%; as you move from C to D, the price falls by 16% and the quantity rises by 16% Notice that in absolute value, the declines in price, as you step down the demand curve, are not identical Instead, the price falls by $3 from A to B, by a smaller amount of $1.50 from B to C, and by a still smaller amount of $0.75 from C to D As a result, a demand curve with constant unitary elasticity moves from a steeper slope on the left and a flatter slope on the right—and a curved shape overall 2/5 Polar Cases of Elasticity and Constant Elasticity A Constant Unitary Elasticity Demand Curve A demand curve with constant unitary elasticity will be a curved line Notice how price and quantity demanded change by an identical amount in each step down the demand curve Unlike the demand curve with unitary elasticity, the supply curve with unitary elasticity is represented by a straight line In moving up the supply curve from left to right, each increase in quantity of 30, from 90 to 120 to 150 to 180, is equal in absolute value However, in percentage value, the steps are decreasing, from 33.3% to 25% to 16.7%, because the original quantity points in each percentage calculation are getting larger and larger, which expands the denominator in the elasticity calculation Consider the price changes moving up the supply curve in [link] From points D to E to F and to G on the supply curve, each step of $1.50 is the same in absolute value However, if the price changes are measured in percentage change terms, they are also decreasing, from 33.3% to 25% to 16.7%, because the original price points in each percentage calculation are getting larger and larger in value Along the constant unitary elasticity supply curve, the percentage quantity increases on the horizontal axis exactly match the percentage price increases on the vertical axis—so this supply curve has a constant unitary elasticity at all points 3/5 Polar Cases of Elasticity and Constant Elasticity A Constant Unitary Elasticity Supply Curve A constant unitary elasticity supply curve is a straight line reaching up from the origin Between each point, the percentage increase in ... GAO United States Government Accountabilit y Office Testimon y Before the Committee on Education and Labor, House of Representatives SECLUSIONS AND RESTRAINTS Selected Cases of Death and Abuse at Public and Private Schools and Treatment Centers Statement of Gregory D. Kutz, Managing Director Forensic Audits and Special Investigations For Release on Delivery Expected at 10:00 a.m. EDT Tuesday, May 19, 2009 GAO-09-719T What GAO Found United States Government Accountability Office Why GAO Did This Study Highlights Accountability Integrity Reliability Ma y 19, 2009 SECLUSIONS AND RESTRAINTS Selected Cases of Death and Abuse at Public and Private Schools and Treatment Centers Highlights of GAO-09-719T, a testimony before the Committee on Education and Labor, House of Representatives T GAO recently testified before the Committee regarding allegations of death and abuse at residential programs for troubled teens. Recent reports indicate that vulnerable children are being abused in other settings. For example, one report on the use of restraints and seclusions in schools documented cases where students were pinned to the floor for hours at a time, handcuffed, locked in closets, and subjected to other acts of violence. In some of these cases, this type of abuse resulted in death. Given these reports, the Committee asked GAO to (1) provide an overview of seclusions and restraint laws applicable to children in public and private schools, (2) verify whether allegations of student death and abuse from the use of these methods are widespread, and (3) examine the facts and circumstances surrounding cases where a student died or suffered abuse as a result of being secluded or restrained. GAO reviewed federal and state laws and abuse allegations from advocacy groups, parents, and the media from the past two decades. GAO did not evaluate whether using restraints and seclusions can be beneficial. GAO examined documents related to closed cases, including police and autopsy reports and school policies. GAO also interviewed parents, attorneys, and school officials and conducted searches to determine the current employment status of staff involved in the cases. GAO found no federal laws restricting the use of seclusion and restraints in public and private schools and widely divergent laws at the state level. Although GAO could not determine whether allegations were widespread, GAO did find hundreds of cases of alleged abuse and death related to the use of these methods on school children during the past two decades. Examples of these cases include a 7 year old purportedly dying after being held face down for hours by school staff, 5 year olds allegedly being tied to chairs with bungee cords and duct tape by their teacher and suffering broken arms and bloody noses, and a 13 year old reportedly hanging himself in a seclusion room after prolonged confinement. Although GAO continues to receive new allegations from parents and advocacy groups, GAO could not find a single Web site, federal agency, or other entity that collects information on the use of these methods or the extent of their alleged abuse. GAO also examined the details of 10 restraint and seclusion cases in which there was a criminal conviction, a finding Original article Genetic control of stiffness of standing Douglas fir; from the standing stem to the standardised wood sample, relationships between modulus of elasticity and wood density parameters. Part I Cécile Mamdy, Philippe Rozenberg Alain Franc, Jean Launay, Nicolas Schermann Jean-Charles Bastien Inra Orléans, 45160 Ardon, France (Received 18 December 1997; accepted 1 October 1998) Abstract - The Institut national de la recherche agronomique (Inra) developed a tree-bending machine, similar to the device elabo- rated by Koizumi and Ueda, and used it to measure the stiffness of standing tree trunks (modulus of elasticity, MOE). There are mod- erate or good relationships between trunk MOE and MOE based on destructive samples successively sawn in the study stems: the modulomètre is able to rank genetic units for a trait related to the MOE of the wood of the stem. Our study showed that there exists a strong genetic effect on trunk MOE. This trait and the MOE measured on destructive samples are moderately related (best r 2 from 0.37 to 0.42) with ring density parameters (based on trimming the ring in two parts: earlywood and latewood), and closely related (best r 2 from 0.58 to 0.73) with parameters describing the shape of a mean density profile segment, mostly located in the latewood part of the ring. (© Inra/Elsevier, Paris.) genetics / modulus of elasticity / stem mechanics / wood density / Douglas fir Résumé - Modélisation du module d’élasticité à l’aide de données microdensitométriques : méthodes et effets génétiques. 1 re partie. L’Inra a fabriqué une machine servant à mesurer la rigidité du tronc des arbres sur pied (Module d’Elasticité du tronc de l’arbre sur Pied, MEP), inspirée de celle imaginée et construite par Koizumi et Ueda au Japon. Des mesures de module de Young en flexion statique réalisées sur des échantillons de taille variable débités dans les troncs des arbres sur lesquels on a mesuré le MEP sont assez bien ou bien liées avec les mesures sur pied : le modulomètre semble donc capable de classer des unités génétiques pour le module de Young du bois. À partir de la mesure du MEP de cinq clones de douglas x quatre arbres non sélectionnés sur les proprié- tés de leur bois, on a mis en évidence l’existence d’un très fort contrôle génétique du MEP. Ce caractère et le module d’élasticité des échantillons destructifs découpés dans les troncs sont modérément liés (les meilleurs R2 vont de 0,37 à 0,42) aux paramètres micro- densitométriques basés sur la découpe du cerne en bois initial et final, et bien liés (les meilleurs R2 vont de 0,58 à 0,73) à des para- mètres de polynômes décrivant la forme d’un segment de profil situé plutôt vers la fin (bois final) du cerne. (© Inra/Elsevier, Paris.) génétiques / module d’élasticité / mécanique de la tige / densité du bois / douglas * Correspondence and reprints rozenberg@orleans.inra.fr 1. INTRODUCTION Forest resources in temperate regions of the earth are being converted from rather slow-growing naturally regenerated stands to relatively fast-growing planted stands [20, 37]. This evolution will cause a notable decrease of softwood wood quality [18, 20, 21, 33, 36-38, 45, 51]. Most tree geneticists think that this decrease in wood quality could be restrained or impeded if wood quality traits were taken into account in breeding programmes (e.g. [1, 3, 28, 34, 48, 50, 52]. Among the wood quality traits of interest, modulus of elasticity (MOE) is one of the Original article Genetic control of stiffness of standing Douglas fir; from the standing stem to the standardised wood sample, relationships between modulus of elasticity and wood density parameters. Part II Philippe Rozenberg* Alain Franc, Cécile Mamdy Jean Launay, Nicolas Schermann Jean Charles Bastien Inra Orléans, 45160 Ardon, France (Received 18 December 1997; accepted 1 October 1998) Abstract - Fairly strong positive relationships between stiffness and density have often been reported. No stronger relationships have been found when using parameters of density profiles based on an earlywood-latewood boundary. In this study, we attempt to model the relationships among the stiffness of different samples and simple parameters derived from microdensity profiles, not established according to an earlywood-latewood boundary. Furthermore, we try to determine if there is a genetic variation for the relationship between stiffness and density. From the results, we find that the strongest relationship between a single density parameter and stiff- ness is r2 = 0.78, whereas it is r2 = 0.37 when involving a classical within-ring density parameter. At clone level, r 2 ranges from 0.88 to 0.95, while it is 0.51 for the bulked samples. The mathematical form of the models differ from one clone to another: there is a genetic effect on the models. This could mean that different clones different build their stiffness in different ways. (© Inra/Elsevier, Paris.) genetics / modulus of elasticity / wood density / X-ray microdensitometry / Douglas fir Résumé - Modélisation du module d’élasticité à l’aide de données microdensitométriques : méthodes et effets génétiques. 2e partie. On a souvent mis en évidence d’assez fortes relations entre la rigidité et la densité du bois. Ces relations n’étaient pas plus fortes quand on a essayé d’expliquer la rigidité à l’aide de paramètres microdensitométriques intra-cerne basés sur une limite bois ini- tial-bois final. Dans cette étude, nous tentons de modéliser la rigidité d’un échantillon de bois à l’aide de paramètres simples calculés à partir de profils microdensitométriques, mais non basés sur la limite classique bois initial-bois final. De plus, nous cherchons si les modèles décrivant cette relation sont différents d’une unité génétique à l’autre. Les résultats montrent que les modèles bâtis à l’aide de nos nouveaux paramètres sont plus précis que ceux construits à l’aide des paramètres intra-cernes classiques (par exemple, pour les mêmes échantillons, r2 passe de 0,37 à 0,78 quand la rigidité est expliquée à l’aide d’un de ces nouveaux paramètres, plutôt qu’à l’aide de la densité du bois final). Au niveau clonal, le r2 varie de 0,88 à 0,94, alors que tous échantillons confondus, il est seulement de 0,51. De plus, la forme mathématique des modèles est différente d’un clone à l’autre. Donc il existe un effet génétique sur la rela- tion rigidité-densité. Si ces résultats sont confirmés, cela signifie que différents clones ont différentes manières de construire leur rigi- dité. (© Inra/Elsevier, Paris.) génétiques / module d’élasticité / densité du bois / microdensité aux rayons X / douglas * Correspondence and reprints rozenberg@orleans.inra.fr 1. INTRODUCTION Since the end of the nineteenth century, density has been acknowledged as the best single predictor of wood mechanical properties [1, 15, 20-22, 33]. Modulus of elasticity (MOE), or stiffness, is a basic mechanical property for softwoods, especially when they VIETNAM ACADEMY OF SOCIAL SCIENCES GRADUATE ACADEMY OF SOCIAL SCIENCES PHAM HONG LINH MUSIC IN THE BUDDHIST CEREMONY OF GIVING ALMS TO THE DEAD OF THE VIET PEOPLE COMPARISON OF CASES OF HUE AND HO CHI MINH CITY MAJOR : FOLK CULTURE MAJOR CODE : 62 22 01 30 SUMMARY OF DOCTORAL THESIS IN CULTURAL STUDIES HANOI - 2016 The work is completed in: Department of Cultural Studies Graduate Academy of Social Sciences Scientific Supervisor Associate Prof Dr Nguyen Thuy Loan Reviewer 1: Associate Prof Dr Bui Huyen Nga Reviewer 2: Associate Prof Dr Pham Trong Toan Reviewer 3: Associate Prof Dr Nguyen Quang Hung The thesis will be uphold under the criticism of Graduate Academy’s Thesis Censor Council at Graduate Academy of Social Sciences, No 477 Nguyen Trai, Thanh Xuan District, Ha Noi at date .month year 2016 The thesis is available at: - Library of Graduate Academy of Social Sciences - Vietnam National Library INTRODUCTION Reason of choosing the thesis topic Buddhism is a kind of religious culture that has extensive influence in the spiritual life of some ethnic groups in Vietnam Buddhist music is an indispensable component in the rituals of Buddhism That is one of the objects of cultural studies in general and of Buddhist culture in particular In my process of researching into Hue Buddhist music, I noticed that there had been a mutual relation between Buddhism in Hue and that in Ho Chi Minh City (HCMC) through exchange monks and practice documents to give alms to the death in two regions That relationship suggested me some questions related to Buddhism in general and Buddhist music in particular in two regions that were far from each other They urged me to learn about the similarities and differences in Buddhist music between these two regions This was one of aspects performing regional culture- an area that had been attracting the attention of some researchers We just focused on researching the music played in the Buddhish ceremony of giving alms to the dead (TDCT) in Hue and HCMC for some reasons: Firstly, in the ritual system of Mahayana Buddhism, TDCT was one of ceremonies that had the largest scale, frequent usage and extensive influence on cultural and spiritual life of the Viet community throughout the country Secondly, in comparison to other types of ceremonial music of Mahayana Buddhism, the music in TDCT ceremony could make the most use of Buddhist musical elements and was considered as a typical musical phenomenon of Vietnamese Buddhism Thirdly, Hue and HCMC were the cultural centers of both the Central and the South, where demonstrated the most cultural characteristics of each region Therefore, “Music in the Buddhist Ceremony of Giving Alms to the Death of the Viet people - Comparison of cases of Hue and Ho Chi Minh city” was chosen to the thesis topic Purposes and tasks of the thesis 2.1 Reseach purposes - In-depth research on concrete expressions of the similarities, differences and characteristics of the music played in the TDCT ceremony of the Viet people in Hue and HCMC - Research contribution to clarify some aspects related to domestic cultural regions through a specific phenomenon, that was the music played in TDCT ceremony of the Viet people in the two cities - Research and conclusion on some theoretical issues of culture and religious culture that were related to the similarities, differences and characteristics of the music played in TDCT ceremony of the Viet people in Hue and HCMC 2.2 Research tasks Comprehensive surveys and in-depth interviews were conducted on people who had profound knowledge about TDCT ceremony of the Viet people and its cultural aspects, especially its music, in Hue and HCMC Collected data from the field, including audio materials, photographs, interviews and written documents related to the research subject, was handled and analyzed The characteristics of the natural environment, society and history, as well as interior factors in each region were analyzed, compared and considered their ... supply curve has a constant unitary elasticity at all points 3/5 Polar Cases of Elasticity and Constant Elasticity A Constant Unitary Elasticity Supply Curve A constant unitary elasticity supply... with constant unitary elasticity moves from a steeper slope on the left and a flatter slope on the right and a curved shape overall 2/5 Polar Cases of Elasticity and Constant Elasticity A Constant. .. line? The constant unitary elasticity is a straight line because the curve slopes upward and both price and quantity are increasing proportionally 4/5 Polar Cases of Elasticity and Constant Elasticity