DOCSLIB.ORG

Copyrighted Material

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Copyrighted Material 1 Elastic Domains: Yield Conditions 1.1. Introductory remarks Figure 1.1. Henri Tresca (1814–1885)1 honored on the North pillar of the Eiffel Tower One hundred and fifty years ago, Henri Tresca (Figure 1.1), then a Professor of applied mechanics at the Conservatoire des arts et métiers in Paris, submitted a series of memoirs to The French Academy of Sciences (Tresca 1864a, 1867a, 1867c, 1869a, 1870). They were devoted to recording the extensive series of experiments he had carried out investigating punching, rolling, forging, stamping and extruding processes of metals, where he had definitely identified a phenomenon; heCOPYRIGHTED called it the “fluidity” of metals MATERIAL subjected to very high pressures (Figure 1.2). 1 Archives of The French Academy of Sciences, Paris. 2 Elastoplastic Modeling Figure 1.2. “Flow of two iron slices, hot, under the action of a forging-hammer” In addition to the precise description of the experiments that were initially performed on various lead specimens and, afterwards, on other metals (copper, iron, tin and zinc) or materials (ceramics and modeling wax), Tresca had proposed a mechanical theory for the observed phenomena with his “Fundamental assumption as to the resistance to fluidity”, which he stated as follows (Figure 1.3): “Beyond this limit, we propose to consider, for the state of fluidity, that the forces that are developed are absolutely constant and entirely independent of the relative displacements, which amounts to admitting that they can always be evaluated by means of a resistance coefficient K per square meter or per square centimeter, this coefficient K remaining the same for the assessment of any molecular deformation developed during the fluidity period.” Elastic Domains: Yield Conditions 3 Figure 1.3. Excerpt from Tresca (1869a, p. 777) As Tresca was not yet a member of the Academy, his memoirs had to be approved before abstracts, written by himself, could be published in the Comptes rendus (Tresca 1864b, 1867b, 1867d, 1869b, 1870). In 1870, Saint-Venant (1870b, 1871), who was one of the commissioners in charge of reviewing them, published a note whose rather long title may be considered as a perfect definition of what is now called the plastic behavior of materials: “Sur les équations des mouvements intérieurs opérés dans les solides ductiles au delà des limites où l’élasticité pourrait les ramener à leur premier état”. [On the equations of internal movements operated in ductile solids beyond the limits where elasticity could bring them back to their initial state]. In this note, considering the particular case of a two-dimensional problem, he derived five equations that governed what he called “hydrostéréodynamique” or “plasticodynamique”. They consisted of two differential equations of equilibrium, one equation expressing the fluidity condition2, another equation expressing incompressibility of matter and a last one expressing the coincidence of the principal directions of the two-dimensional strain-rate and stress tensors3. 2 Similar equations had been written by Lévy (1867) in his theory of earth pressure. 3 Original French wording: “afin d’obtenir la coïncidence des directions de plus grand glissement et de plus grande résistance au glissement.” 4 Elastoplastic Modeling We may say that, for the first time, both questions that are the cornerstones of plastic modeling were thus evoked4: – When? i.e. the question of the yield condition; – What and How? i.e. the question of the plastic flow rule. These two issues will be addressed successively in this chapter and the one following it. 1.2. An overview of the model 1.2.1. The infinitesimal transformation framework Classical elastic–plastic modeling of ductile materials is expressed within the framework of infinitesimal transformation5. The model is well established by now and, although much work has been devoted to the formulation of finite transformation elastic–plastic theories, infinitesimal transformation is always the first framework – and often the only one – chosen when practical applications to structural analysis are concerned, which often proves sufficient. Moreover, it is clear that finite transformation theories call for the knowledge of infinitesimal models as they must match them under the infinitesimal assumption. Therefore, the presentation in this book focuses on infinitesimal plasticity within the framework of the small perturbation hypothesis (SPH) unless otherwise stated (see Chapter 4, section 4.3.7). 1.2.2. Time variable In the same way as elastic modeling, the classical elastic–plastic model excludes any viscosity effect (no viscoplasticity) and does not take any ageing of the concerned material into account. It follows that the corresponding constitutive equations shall be invariant in any translation or positive rescaling of the time variable. Nevertheless, unlike the elastic case, the introduction of a time variable proves necessary due to the irreversibility of plastic deformation, which calls for an unambiguous chronology of loading and response histories. Such a time variable is just a “time-coding” parameter, monotonously increasing with the physical time. As it will be discussed in Chapter 2 (section 2.2.2), this status implies homogeneity conditions, with respect to time, to be satisfied by the constitutive equations. 4 “Tresca’s paper on the flow of solids may approximately be regarded as the birth of the mathematical theory of plasticity” (Koiter, 1960, p. 179). 5 As defined in Salençon (2001, p. 57). Elastic Domains: Yield Conditions 5 1.3. One-dimensional approach 1.3.1. Uniaxial tension test 1.3.1.1. Work-hardening material Figure 1.4. Uniaxial tension test performed on a work-hardening specimen6 Figure 1.4 shows a typical test piece for a uniaxial tension–compression test, made from a homogeneous material and with specific geometrical features such that the stress field is assumed to be a simple homogeneous tension (or compression) field, parallel to the axis of the specimen7. In its simplest form, this experiment involves recording the change in the tensile force F or the ratio F S0 , where S0 is the initial value of the cross-section in the central part of the specimen, as a function of the relative extension Δ. The experiment is carried out at a fixed deformation rate, slow enough to be considered as quasi-static, with temperature maintained constant. Figure 1.4a presents a classical diagram recorded in the case of a test piece made from stainless steel (Salençon and Gary 1999). σ It shows first the reversibility of the material response up to a threshold 0 , σ marked by point A on the curve. After loading up to a level below 0 , total or 6 See: https://www.researchgate.net/publication/325757756. 7 Note that in the case of anisotropic materials, the simple off-axis tension experiment requires additional special precautions to be taken. 6 Elastoplastic Modeling partial unloading causes the point (Δ ,)F S0 to run back down the same straight line in the opposite direction. The material behavior is linear elastic along OA . σ When F S0 exceeds this threshold following the loading curve up to a value B along OAB , subsequent unloading shows that, when the load is made to decrease, point (Δ ,)F S0 does not regress along the loading curve BAO but moves down a straight line BC parallel to OA . In particular, when the load has been reduced to zero, the test piece still retains a certain degree of extension represented by OC , which is the plastic deformation of the specimen. Then, a new loading process being carried out, starting from point C or any σ point on CB , it is observed that as long as F S0 remains below B , the material response is reversible and point (Δ ,)F S0 follows the same straight path CB ≤≤σ σ for any loading or unloading process with 0 FS0 B . When F S0 exceeds B , point (Δ ,)F S0 moves along a curve in the prolongation of the first loading σ curve OAB , which means that, for this new tension loading process, B stands as the new threshold. These experimental observations are the basis of the stress–strain diagram shown in Figure 1.4b for modeling the elastic–plastic behavior in tension, where σ stands σ σσ=⊗ ε for the xx component of a uniaxial tension stress tensor xx eexx and ε εε=⊗8 represents the xx component of the linearized strain tensor ijeeij: σ – 0 is the initial elastic limit or initial plastic yield point for this experiment; σ – after loading has been carried out up to B , the current elastic limit or current σ plastic yield point is equal to B , this result sometimes being called “Coulomb’s principle”9; – the permanent strain after complete unloading is the plastic strain ε p . An example of the influence of the deformation rate ε on the experimental results regarding the stress–strain diagram, showing a viscosity effect, is shown in Figure 1.5 which refers to experiments on aluminum test pieces (Bui and Zarka 1972). This phenomenon is not taken into account in classical elastoplastic models10. ⊗ = 8 Symbol “ ” denoting the tensor product. eii ,1,2,3 unit vector. 9 As in Bouasse (1920). 10 See Bingham (1916), Hohenemser and Prager (1932), Oldroyd (1950) and Perzyna (1963, 1966, 1971, 2005). Elastic Domains: Yield Conditions 7 Figure 1.5. Uniaxial tension tests on aluminum specimens No ageing effect being considered either, it follows that the response of the material in any uniaxial tension test is only characterized by the stress–strain diagram in Figure 1.4, which is independent of ε or, equivalently, σ 11. The phenomenon observed in Figure 1.4, where the current plastic yield point actually depends on the plastic strain, characterizes work-hardening or strain- hardening of the material. Finally, one important conclusion to be retained from this first approach to σ elastoplastic behavior is that, in the plastic range (i.e., once threshold 0 has been crossed), there is no longer a one-to-one relationship between stress σ and strain ε .
Load more

Recommended publications
  • Fatigue Characterization and Cyclic Plasticity Modeling of Magnesium Spot-Welds
    Fatigue Characterization and Cyclic Plasticity Modeling of Magnesium Spot-Welds by Seyed Behzad Behravesh A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Mechanical Engineering Waterloo, Ontario, Canada, 2013 © Seyed Behzad Behravesh 2013 AUTHOR'S DECLARATION I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. Seyed Behzad Behravesh ii Abstract The automotive industry is adopting lightweight materials to improve emissions and fuel economy. Magnesium (Mg) alloys are the lightest of engineering metals, but work is required to assess their structural strength, especially for spot-welded applications. In the present research, fatigue behavior of magnesium spot-welds was characterized and compared with steel and aluminum spot-welds. A fatigue model was proposed to predict the failure location and crack initiation life in magnesium structures. The material under investigation, AZ31B-H24 Mg alloy, and its spot-welds were characterized from microstructural and mechanical points of view. Microstructure and hardness of the base metal (BM) and different regions in the spot-welds were studied. Monotonic testing of the BM demonstrated asymmetric hardening behavior under tension and compression. Under cyclic loading, the BM had an asymmetric hysteresis loop. Static behavior of spot-welds was studied with different specimen configurations. The effect of nugget size on the static peak load was similar to that of aluminum and less than steel.
    [Show full text]
  • Experimental Characterization and Modeling of High Strength Martensitic Steels Based on a New Distortional Hardening Model
    EXPERIMENTAL CHARACTERIZATION AND MODELING OF HIGH STRENGTH MARTENSITIC STEELS BASED ON A NEW DISTORTIONAL HARDENING MODEL BY ELIZABETH K BARTLETT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2018 © 2018 Elizabeth K Bartlett 2 To my daughter, Gwyneth Webb, who helps me to have fun, laugh, and love through the most difficult times in my life. 3 ACKNOWLEDGEMENTS First, I would like to thank all the members of my supervisory committee for their support. I would especially like to thank my committee chair, Prof. Oana Cazacu, for her support and constructive criticisms. It was an honor and a privilege to study her research. I would also like to thank Dr. Benoit Revil- Baudard for his expertise in numerical methods. I am so grateful for the support of the 96th Test Wing, my sponsoring facility in the Science, Mathematics, and Research for Transformation (SMART) scholarship program. Specifically, I would like to thank my commanders and supervisor, Mr. Ron Lutz, as well as, Mrs. Linda Busch and Dr. Betta Jerome. I also greatly appreciate the support of the Air Force Research Laboratory (AFRL) in the experimental characterization of Eglin steel: Dr. Geremy Kleiser and Dr. Philip Flater for their training in quasi-static testing, Dr. Brad Martin and Dr. Xu Nie for their expertise in dynamic split Hopkinson pressure bar (SHPB) testing, Dr. Rachel Abrahams and Dr. Sean Gibbons for their instruction in microscopy and material science, and Mr. Richard Harris for his expertise in material characterization at the Advanced Weapons Effects Facility (AWEF).
    [Show full text]
  • Mechanical Models in Nonparametric Regression
    IMS Collections From Probability to Statistics and Back: High-Dimensional Models and Processes Vol. 9 (2013) 5–19 c Institute of Mathematical Statistics,2013 DOI:⃝ 10.1214/12-IMSCOLL902 Mechanical models in nonparametric regression 1, 2, Vladim´ır Balek ∗ and Ivan Mizera † Comenius University and University of Alberta This note is devoted to an extrapolation of a solid-mechanic motif from nonparamet- ric regression (or, as it is often referred to, smoothing) and numerical interpolation, athemeoriginatingintheexpositionofspline-basedmethods.Theattentionis turned to the fact that a mechanical analogy, akin to that from elasticity theory for quadratic spline penalties, can be elucidated in plasticity theory for certain penal- ties of different type. It is this—in the (slightly altered) words of a referee, “what would happen if the metaphor underlying splines were replaced by a metaphor of plastic energy”—what is the objective of what follows; everything else, in particular discussion of the origins of (certain types of) splines, or an exhaustive bibliograph- ical account, is outside our scope. To give an idea what mechanical analogies we have in mind, let us start by the quote from the Oxford English Dictionary, referencing (univariate) “draftsman spline” as “a flexible strip of wood or hard rubber used by draftsmen in laying out broad sweeping curves”.Bivariatecasebringsmoresophistication:thedesignation “thin-plate spline” is regularly explained by a story about the deformation of an elastic flat thin plate—see page 139 of Green and Silverman [12]orpage108ofSmall [31]: if the plate is deformed to the shape of the function ϵf,andϵ is small, then the bending energy is (up to the first order) proportional to the smoothing penalty.
    [Show full text]
  • A Force Is Whatever That Can Cause an Object with Mass to Accelerate
    1 www.onlineeducation.bharatsevaksamaj.net www.bssskillmission.in “Foundations of mechanical engineering”. In Section 1 of this course you will cover these topics: Statically Determinate Force Systems Statically Determinate Stress Systems. Stress-Strain Relations. Torsion Topic : Statically Determinate Force Systems Topic Objective: After studying this topic the student should be able to: Define Newtons Law of Motion Definition/Overview: Force: In physics,WWW.BSSVE.IN a force is whatever that can cause an object with mass to accelerate. Force has both magnitude and direction, making it a vector quantity. According to Newton's second law, an object with constant mass will accelerate in proportion to the net force acting upon it and in inverse proportion to its mass. An equivalent formulation is that the net force on an object is equal to the rate of change of momentum it experiences. Forces acting on three-dimensional objects may also cause them to rotate or deform, or result in a change in pressure. The tendency of a force to cause angular acceleration about an axis is termed torque. Deformation and pressure are the result of stress forces within an object. www.bsscommunitycollege.in www.bssnewgeneration.in www.bsslifeskillscollege.in 2 www.onlineeducation.bharatsevaksamaj.net www.bssskillmission.in Key Points: 1. Newtonian mechanics Newton's second law of motion relates the concept of force with the time-derivative of momentum: . In 1687, Newton went on to publish his Philosophiae Naturalis Principia Mathematica, which used concepts of inertia, force, and conservation to describe the motion of all objects. In this work, Newton set out three laws of motion that to this day are the way forces are described in physics.
    [Show full text]
  • Doctorat Paristech T H È S E L'école Nationale Supérieure Des Mines De
    N°: 2009 ENAM XXXX École doctorale n° 364 : Sciences fondamentales et appliquées Doctorat ParisTech T H È S E pour obtenir le grade de docteur délivré par l’École nationale supérieure des mines de Paris Spécialité “ Mécanique Numérique ” présentée et soutenue publiquement par Zhigang LIU le 23 Avril 2012 Numerical and experimental study of AZ31-O magnesium alloy warm sheet forming Directeur de thèse : Michel BELLET Co-Directeur de thèse : Elisabeth MASSONI Jury Mme Salima BOUVIER , Professeur, Robertval , Université technologique de Compiégne Rapporteur T M.Tudor BALAN , Maître de conférences – HDR, LEM3, Arts et Métiers ParisTech Rapporteur M. Jean-Claude GELIN, Professeur, ENSMM Examinateur H M. Patrice LASNE, Ingénieur, Transvalor Examinateur Mme Elisabeth MASSONI, Maître de recherche, CEMEF, Mines ParisTech Co-Directeur de thèse È M. Michel BELLET, Professeur, CEMEF, Mines ParisTech Directeur de thèse S E MINES ParisTech Centre de Mise en Forme des Matériaux Rue Claude Daunesse B.P.207, 06904 Sophia Antipolis Cedex, France Abstract Lightweight materials have been studied widely at present. Because weight reduction while maintaining functional requirements is one of the major goals in industries in order to save materials, energy and costs, etc. As the lightest structural alloys, magnesium alloys offer great potential to displace the most commonly used materials, because its density is about 2/3 of aluminum and 1/4 of steel. However, due to HCP (Hexagonal Close-Packed) crystal structure, magnesium provides only limited ductility for cold forming operations. But its formability improved obviously at elevated temperature. In this project, the material is AZ31- O magnesium alloy sheet, including 3% aluminum and 1% zinc.
    [Show full text]
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Lagrangian and Eulerian Forms of Finite Plasticity Permalink https://escholarship.org/uc/item/55v4x159 Author de Vera, Giorgio Tantuan Publication Date 2016 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Lagrangian and Eulerian Forms of Finite Plasticity By Giorgio Tantuan De Vera A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering – Mechanical Engineering in the Graduate Division of the University of California, Berkeley Committee in charge: Professor James Casey, Co-chair Professor George C. Johnson, Co-chair Professor Oliver M. O’Reilly Professor Filippos Filippou Fall 2016 Lagrangian and Eulerian Forms of Finite Plasticity Copyright 2016 by Giorgio Tantuan De Vera Abstract Lagrangian and Eulerian Forms of Finite Plasticity by Giorgio Tantuan De Vera Doctor of Philosophy in Mechanical Engineering University of California, Berkeley Professor James Casey, Co-chair Over the past half century, much work has been published on the theory of elastic-plastic materials undergoing large deformations. However, there is still disagreement about certain basic issues. In this dissertation, the strain-based Lagrangian theory developed by Green, Naghdi and co-workers is re-assessed. Its basic structure is found to be satisfactory. For the purpose of applications, the theory is recast in Eulerian form. Additionally, a novel three-factor multiplicative decomposition of the deformation gradient is employed to define a unique intermediate configuration. The resulting theory of finite plasticity contains an elastic strain tensor measured from the intermediate stress-free configuration. The consti- tutive equations involve the objective stress rate of the rotated Cauchy stress, which can be expressed in terms of the rate of deformation tensor.
    [Show full text]