Jim Barber's blog
https://imechanica.org/blog/768
enIntermediate Mechanics of Materials - 2nd Edition
https://imechanica.org/node/9342
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Springer has just published the second edition of my book<br /><a href="http://www.springer.com/materials/mechanics/book/978-94-007-0294-3">`Intermediate Mechanics of Materials'</a>. The book covers a selection of topics appropriate to a second course in mechanics of materials. Many books with titles like 'Advanced Mechanics of Materials' are pitched at a much higher level than most introductory courses and this can present a significant barrier to undergraduate students. My intention in this book is to make this transition smoother by discussing simple examples before introducing general principles and by restricting the mathematical level to topics that can be treated using ordinary differential equations rather than PDEs.<br />
See below for the <a href="#toc">Table of Contents</a> and the <a href="#preface">Preface</a>. Parts of the book can be browsed and the book can also be ordered at this <a href="http://www.springer.com/materials/mechanics/book/978-94-007-0294-3">website</a>.</p>
<p>
The ISBN number is 978-94-007-0294-3.</p>
<p>ERRATA<br />
I would like to thank those who reported errors in the first edition and who made other suggestions for improvement. If you find any errors in this edition, please let me know at <a href="mailto:jbarber@umich.edu">jbarber@umich.edu</a>. </p>
<p>SOLUTION MANUAL<br />
A solution manual is available, containing detailed solutions to all the problems, in some cases involving further discussion of the material. Bona fide instructors should contact me at <a href="mailto:jbarber@umich.edu">jbarber@umich.edu</a> if they need the manual and I will send it out as zipped .pdf files.</p>
<p><a name="toc" title="toc" id="toc"></a>TABLE OF CONTENTS</p>
<p>
</p>
<p>CHAPTER 1 Introduction<br />
The Engineering Design Process,<br />
Design optimization,<br />
Relative magnitude of different effects,<br />
Formulating and Solving Problems,<br />
Review of Elementary Mechanics of Materials.</p>
<p>CHAPTER 2 Material Behaviour and Failure<br />
Transformation of Stresses,<br />
Failure Theories for Isotropic Materials,<br />
Cyclic Loading and Fatigue.</p>
<p>CHAPTER 3 Energy Methods<br />
Work Done on Loading and Unloading ,<br />
Strain Energy,<br />
Load-displacement relations,<br />
Potential Energy,<br />
The Principle of Stationary Potential Energy,<br />
The Rayleigh-Ritz Method,<br />
Castigliano's First Theorem,<br />
Linear Elastic Systems,<br />
The Stiffness Matrix,<br />
Castigliano's Second Theorem.</p>
<p>CHAPTER 4 Unsymmetrical Bending<br />
Stress distribution in bending,<br />
Displacements of the beam,<br />
Second moments of area,<br />
Further properties of second moments.</p>
<p>CHAPTER 5 Non-linear and Elastic-Plastic Bending<br />
Kinematics of Bending,<br />
Elastic-Plastic Constitutive Behaviour,<br />
Stress Fields in Non-linear and Inelastic Bending,<br />
Pure Bending about an Axis of Symmetry,<br />
Bending of a Symmetric Section about an Orthogonal Axis,<br />
Unsymmetrical Plastic Bending,<br />
Unloading, Springback and Residual Stress,<br />
Limit Analysis in the Design of Beams.</p>
<p>CHAPTER 6 Shear and Torsion of Thin-walled Beams<br />
Derivation of the shear stress formula,<br />
Shear center,<br />
Unsymmetrical sections,<br />
Closed sections,<br />
Pure torsion of closed thin-walled sections,<br />
Finding the shear center for a closed section,<br />
Torsion of thin-walled open sections.</p>
<p>CHAPTER 7 Beams on Elastic Foundations<br />
The governing equation,<br />
The homogeneous solution,<br />
Localized nature of the solution,<br />
Concentrated force on an infinite beam,<br />
The particular solution,<br />
Finite beams,<br />
Short beams.</p>
<p>CHAPTER 8 Membrane Stresses in Axisymmetric Shells<br />
The meridional stress,<br />
The circumferential stress,<br />
Self-weight,<br />
Relative magnitudes of different loads,<br />
Strains and Displacements.</p>
<p>CHAPTER 9 Axisymmetric Bending of Cylindrical Shells<br />
Bending stresses and moments,<br />
Deformation of the shell,<br />
Equilibrium of the shell element,<br />
The governing equation,<br />
Localized loading of the shell,<br />
Shell transition regions,<br />
Thermal stresses,<br />
The ASME pressure vessel code.</p>
<p>CHAPTER 10 Thick-walled Cylinders and Disks<br />
Solution Method,<br />
The thin circular disk,<br />
Cylindrical pressure vessels,<br />
Composite cylinders, limits and fits,<br />
Plastic deformation of disks and cylinders.</p>
<p>CHAPTER 11 Curved Beams<br />
The governing equation,<br />
Radial stresses,<br />
Distortion of the cross-section,<br />
Range of application of the theory.</p>
<p>CHAPTER 12 Elastic Stability<br />
Uniform Beam in Compression,<br />
Effect of Initial Perturbations,<br />
Effect of Lateral Load (Beam-Columns),<br />
Indeterminate Problems,<br />
Suppressing Low-order Modes,<br />
Beams on Elastic Foundations,<br />
Energy Methods,<br />
Quick Estimates for the Buckling Force.</p>
<p>APPENDIX A The Finite Element Method<br />
Approximation,<br />
Axial loading,<br />
Solution of differential equations,<br />
Finite element solutions for the bending of beams,<br />
Two and Three-dimensional Problems,<br />
Computational considerations,<br />
Use of the Finite Element Method in Design.</p>
<p><a name="preface" title="preface" id="preface"></a>PREFACE</p>
<p>
Most engineering students first encounter the subject of mechanics of materials in a course covering the concepts of stress and strain and the elementary theories of axial loading, torsion, bending and shear. There is broad agreement as to the content of such courses, there are many excellent textbooks and it is easy to motivate the students by using simple examples with obvious engineering relevance.</p>
<p>The second course in the subject presents considerably more challenge to the instructor. There is a very wide range of possible topics and different selections will be made (for example) by civil engineers and mechanical engineers. The concepts tend to be more subtle and the examples more complex making it harder to motivate the students, to whom the subject may appear merely as an intellectual excercise. Existing second level texts are frequently pitched at too high an intellectual level for students, many of whom will still have a rather imperfect grasp of the fundamental concepts.</p>
<p>Most undergraduate students are looking ahead to a career in industry, where they will use the methods of mechanics of materials in design. Many will get a foretaste of this process in a capstone design project and this provides an excellent vehicle for motivating the subject. In mechanical or aerospace engineering, the second course in mechanics of materials will often be an elective, taken predominantly by students with a design concentration. It is therefore essential to place emphasis on the way the material is used in design.</p>
<p>Mechanical design typically involves an initial conceptual stage during which many options are considered. During this phase, quick approximate analytical methods are crucial in determining which of the initial proposals are feasible. The ideal would be to get within plus or minus 30% with a few lines of calculation. The designer also needs to develop experience as to the kinds of features in the geometry or the loading that are most likely to lead to critical conditions. With this in mind, I try wherever possible to give a physical and even an intuitive interpretation to the problems under investigation. For example, students are encouraged to estimate the location of weak and strong bending axes and the resulting neutral axis of bending by eye and methods are discussed for getting good accuracy with a simple one degree of freedom Rayleigh-Ritz approximation. Students are also encouraged to develop a feeling for the mode of deformation of engineering components by performing simple experiments in their outside environment, for example, estimating the radius to which an initially straight bar can be bent without producing permanent deformation, or convincing themselves of the dramatic difference between torsional and bending stiffness for a thin-walled open beam section by trying to bend and then twist a structural steel beam by hand-applied loads at the ends. </p>
<p>In choosing dimensions for mechanical components, designers will expect to be guided by criteria of minimum weight, which with elementary calculations, often leads to a thin-walled structure as the optimal solution. This demands that students be introduced to the limits imposed by elastic instability. Some emphasis is also placed on the effect of manufacturing errors on such highly-designed structures --- for example, the effect of load misalignment on a beam with a large ratio between principal stiffnesses and the large magnification of initial alignment or loading errors in a column below, but not too far below the buckling load.</p>
<p>No modern text of mechanics on materials would be complete without a discussion of the finite element method. However, students and even some instructors are often confused as to the respective roles played by analytical and numerical methods in engineering practice. Numerical methods provide accurate solutions for complex practical problems, but the results are specific to the geometry and loading modelled and the solution involves a significant amount of programming effort. By contrast, analytical methods may be very idealized and hence approximate, but they are often quick to apply and they provide generality, permitting a whole family of designs to be compared or even optimized. </p>
<p>The traditional approach to mechanics is to define the basic concepts, derive a general theory and then illustrate its application in a variety of examples. As a student and later as a practising engineer, I have never felt comfortable with this approach, because it is impossible to understand the nuances of the definitions or the general treatment until after they are seen in examples which are simple enough for the mathematics and physics to be transparent. Over the years, I have therefore developed rather untraditional ways of proving and explaining things, relying heavily on simple examples during the derivation process and using only the bare minimum of specialist terminology. I try to avoid presenting to the student anything which he or she cannot reasonably be expected to understand fully <em> now</em>. </p>
<p>The problems provided at the end of each chapter range from routine applications of standard methods to more challenging problems. Particularly lengthy or challenging problems are identified by an asterisk. The Solution Manual to accompany this book is prepared to the same level of detail as the example problems in the text and in many cases introduces additional discussion. It is available to <em>bona fide </em> instructors on application to the author at <a href="mailto:jbarber@umich.edu">jbarber@umich.edu</a>. Answers to even-numbered problems are provided in Appendix D.</p>
<p>This book evolved out of a set of notes that I wrote for a second-level course at the University of Michigan and the resulting interaction with my students and colleagues has played a crucial role in the development of my thinking about the subject. Special thanks go to Przemislaw Zagrodzki of Warsaw University of Technology and Raytech Composites Inc. for his invaluable help with the appendix on finite element methods. I also wish to thank the many people who have made suggestions for improvements and corrections to the first edition which I have incorporated wherever possible.</p>
</div></div></div>Fri, 19 Nov 2010 15:09:35 +0000Jim Barber9342 at https://imechanica.orghttps://imechanica.org/node/9342#commentshttps://imechanica.org/crss/node/9342Elasticity, 3rd edition, J.R.Barber
https://imechanica.org/node/7261
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Springer has just published the third edition of my book<br /><a href="http://www.springer.com/engineering/book/978-90-481-3808-1">`Elasticity'</a>. </p>
<p>
This new edition contains four additional chapters, including two concerned with the use of complex-variable methods in two-dimensional elasticity. In keeping with the style of the rest of the book, I have endeavoured to present this material in a such a way as to be usable by a reader with minimal previous experience of complex analysis who wishes to solve specific elasticity problems. I have emphasised the relation between the complex and real (Airy and Prandtl) stress functions, including algorithms for obtaining the complex function for a stress field for which the real stress function is already known. The complex variable methods and notation are also used in the development of a hierarchical treatment of three-dimensional problems for prismatic bars of fairly general cross-section in a later chapter. The other major addition is a new chapter on variational methods, including the use of the Rayleigh-Ritz method and Castigliano's second theorem in developing approximate solutions to elasticity problems. The full Table of Contents, a sample chapter, and other information can be accessed <a href="http://www-personal.umich.edu/~jbarber/elasticity/book.html">here</a>. For purchasing information or to request inspection copies, please contact the <a href="http://www.springer.com/engineering/book/978-90-481-3808-1">publisher</a>.
</p>
<p>
As in the second edition, I encourage the reader to become familiar with the use of symbolic mathematical languages such as Maple and Mathematica, since these tools open up the possibility of solving considerably more complex and hence interesting and realistic elasticity problems. They also enable the student to focus on the formulation of the problem (e.g. the appropriate governing equations and boundary conditions) rather than on the algebraic manipulations, with a consequent improvement in insight into the subject and in motivation. Finally, they each posess post-processing graphics facilities that enable the user to explore important features of the resulting stress state. The reader can access numerous files for this purpose at my University of Michigan homepage <a href="http://www-personal.umich.edu/~jbarber/elasticity/book.html">http://www-personal.umich.edu/~jbarber/elasticity/book.html</a>, including the solution of sample problems, electronic versions of the tables in Chapters 21,22, and algorithms for the generation of spherical harmonic potentials. Some hints about the use of this material are contained in Appendix A, and more detailed tips about programming are included at the above website.
</p>
<p>
A solution manual is available, containing detailed solutions to all the problems, in some cases involving further discussion of the material and contour plots of the stresses etc. Bona fide instructors should contact me at <a href="mailto:jbarber@umich.edu">jbarber@umich.edu</a> if they need the manual and I will send it out as zipped .pdf files. Please let me know whether you have the second edition or the third edition.
</p>
</div></div></div>Thu, 17 Dec 2009 16:36:44 +0000Jim Barber7261 at https://imechanica.orghttps://imechanica.org/node/7261#commentshttps://imechanica.org/crss/node/7261Optimal structural design against elastic instability
https://imechanica.org/node/5251
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<br />
In the classical Euler buckling problem, the critical buckling load can be increased by a factor of four if the first mode is suppressed by placing an additional simple support at the mid-point.</p>
<p>If we solve the more general problem where the additional support is placed at some different point z=a in 0<z<L, the critical load will be found to increase above that for the unspported first mode, but the maximum increase is achieved when the support is at the mid-point and buckling then occurs of course in what would have been the second mode of the unsupported beam.</p>
<p>It seems likely that this behaviour would apply to all elastic stability problems, but I have been unable to find a rigorous proof or even a formal statement of the result in classical stability texts. To promote some discussion, I propose the following theorem:-
</p>
<p>
<br />
"Suppose the stability problem for an elastic structure under a load P is formulated as a small-displacement linear eigenvalue problem with critical loads P1,P2,... such that P1<P2<P3,... etc with corresponding eigenmodes u1,u2,u3,.... If a single support is now placed at the (presumably unique) node of the eigenmode u2, so as to constrain the displacement there to zero, the critical load of the new system will be larger than that obtained by placing a single support at any other point in the structure.''
</p>
<p>
<br />
The new critical load will of course be P2 and the support will actually not be required to transmit any force. By contrast, if the support is placed at any other point, it will transmit a force when the structure buckles.</p>
<p>If anyone can give me a reference to a statement and/or a proof of this theorem or something equivalent or related, I would be very grateful. If not I propose it as a challenge, either to prove the result or disprove it by finding a counter-example. It is potentially a useful result for structural design, since efficient (low weight) structures tend to be thin-walled and hence limited by stability considerations. Also, increasing the stability threshold by adding supports in this way is an efficient solution, since the added supports theoretically carry no load and hence are not required to be particularly strong. They do however need to have some critical elastic stiffness.</p>
<p>These results are easily established for the classical Euler buckling problem of a simply-supported beam of length L loaded by an axial force. A more interesting case concerns the cantilever beam of length L1 loaded by an axial force P a distance b from the built in end, for which the optimal support is a distance (2b/3Pi) <em>behind</em> the force.
</p>
</div></div></div>Fri, 10 Apr 2009 18:15:34 +0000Jim Barber5251 at https://imechanica.orghttps://imechanica.org/node/5251#commentshttps://imechanica.org/crss/node/5251Three-dimensional anisotropic elasticity - an extended Stroh formalism
https://imechanica.org/node/957
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/669">Stroh formalism</a></div><div class="field-item odd"><a href="/taxonomy/term/670">anisotropic elasticity</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Tom Ting and I have recently developed a method of extending Stroh's anisotropic formalism to problems in three dimensions. The unproofed paper can be accessed at <a href="http://www-personal.umich.edu/%7Ejbarber/Stroh.pdf">http://www-personal.umich.edu/~jbarber/Stroh.pdf </a>. It is particularly convenient for problems where boundary conditions are imposed on the plane, such as contact problems, dislocations and crack problems.</p>
</div></div></div>Fri, 02 Mar 2007 14:23:12 +0000Jim Barber957 at https://imechanica.orghttps://imechanica.org/node/957#commentshttps://imechanica.org/crss/node/957IINTERMEDIATE MECHANICS OF MATERIALS
https://imechanica.org/node/805
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/179">solid mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/597">mechanics of materials</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>J.R.BARBER: INTERMEDIATE MECHANICS OF MATERIALS
</p><p>Many of you may know my book on <a href="http://www-personal.umich.edu/~jbarber/elasticity/book.html">Elasticity</a>, but may not be aware that I also wrote an undergraduate book on <a href="http://www.mhhe.com/engcs/mech/barber/">Intermediate Mechanics of Materials</a> (Published by McGraw-Hill - ISBN 0-07-232519-4). This picks up from the typical elementary Mechanics of Materials course and deals with the next range of topics such as energy methods, elastic-plastic bending, bending of axisymmetric cylindrical shells and axisymmetric thick-walled cylinders. A full Table of Contents and the Preface are given below. </p>
<p>Table of Contents</p>
<p>Chapter 1: Introduction </p>
<p>Chapter 2: Material Behavior and Failure </p>
<p>Chapter 3: Energy Methods </p>
<p>Chapter 4: Unsymmetrical Bending </p>
<p>Chapter 5: Elastic-Plastic Bending </p>
<p>Chapter 6: Shear and Torsion of Thin-Walled Beams </p>
<p>Chapter 7: Membrane Stresses in Axisymmetric Shells </p>
<p>Chapter 8: Beams on Elastic Foundations </p>
<p>Chapter 9: Axisymmetric Bending of Cylindrical Shells </p>
<p>Chapter 10: Thick-walled Cylinders and Disks </p>
<p>Chapter 11: Curved Beams </p>
<p>Chapter 12: Elastic Stability </p>
<p>Appendix A: The Finite Element Method </p>
<p>Preface</p>
<p>Most engineering students first encounter the subject of Mechanics of Materials in a course covering the concepts of stress and strain and the elementary theories of axial loading, torsion, bending and shear. There is broad agreement as to the content of such courses, there are many excellent textbooks and it is easy to motivate the students by using simple examples with obvious engineering relevance. </p>
<p>The second course in the subject presents considerably more challenge to the instructor. There is a very wide range of possible topics and different selections will be made (for example) by Civil Engineers and Mechanical Engineers. The concepts tend to be more subtle and the examples more complex making it harder to motivate the students, to whom the subject may appear merely as an intellectual excercise. Existing second level texts are frequently pitched at too high an intellectual level for students, many of whom will still have a rather imperfect grasp of the fundamental concepts. </p>
<p>Most undergraduate students are looking ahead to a career in industry, where they will use the methods of Mechanics of Materials in design. Many will get a foretaste of this process in a capstone design project and this provides an excellent vehicle for motivating the subject. In Mechanical or Aerospace Engineering, the second course in Mechanics of Materials will often be an elective, taken predominantly by students with a design concentration. It is therefore essential to place emphasis on the way the material is used in design. </p>
<p>Mechanical Design typically involves an initial conceptual stage during which many options are considered. During this phase, quick approximate analytical methods are crucial in determining which of the initial proposals are feasible. The ideal would be to get within plus or minus 30 percent with a few lines of calculation. The designer also needs to develop experience as to the kinds of features in the geometry or the loading that are most likely to lead to critical conditions. With this in mind, I try wherever possible to give a physical and even an intuitive interpretation to the problems under investigation. For example, students are encouraged to estimate the location of weak and strong bending axes and the resulting neutral axis of bending by eye and methods are discussed for getting good accuracy with a simple one degree of freedom Rayleigh-Ritz approximation. Students are also encouraged to develop a feeling for the mode of deformation of engineering components by performing simple experiments in their outside environment, for example, estimating the radius to which an initially straight bar can be bent without producing permanent deformation, or convincing themselves of the dramatic difference between torsional and bending stiffness for a thin-walled open beam section by trying to bend and then twist a structural steel beam by hand-applied loads at one end. </p>
<p>In choosing dimensions for mechanical components, designers will expect to be guided by criteria of minimum weight, which with elementary calculations, often leads to a thin-walled structure as the optimal solution. This demands that students be introduced to the limits imposed by elastic instability. Some emphasis is also placed on the effect of manufacturing errors on such highly-designed structures --- for example, the effect of load misalignment on a beam with a large ratio between principal stiffnesses and the large magnification of initial alignment or loading errors in a column below, but not too far below the buckling load. </p>
<p>No modern text of Mechanics of Materials would be complete without a discussion of the finite element method. However, students and even some instructors are often confused as to the respective roles played by analytical and numerical methods in engineering practice. Numerical methods provide accurate solutions for complex practical problems, but the results are specific to the geometry and loading modelled and the solution involves a significant amount of programming effort. By contrast, analytical methods may be very idealized and hence approximate, but they are often quick to apply and they provide generality, permitting a whole family of designs to be compared or even optimized. </p>
<p>The traditional approach to mechanics is to define the basic concepts, derive a general theory and then illustrate its application in a variety of examples. As a student and later as a practising engineer, I have never felt comfortable with this approach, because it is impossible to understand the nuances of the definitions or the general treatment until after they are seen in examples which are simple enough for the mathematics and physics to be transparent. Over the years, I have therefore developed rather untraditional ways of proving and explaining things, relying heavily on simple examples during the derivation process and using only the bare minimum of specialist terminology. I try to avoid presenting to the student anything which he or she cannot reasonably be expected to understand fully NOW. </p>
<p>The problems provided at the end of each chapter range from routine applications of standard methods to more challenging problems. Particularly lengthy or difficult problems are identified by an asterix. The Solution Manual to accompany this book is prepared to the same level of detail as the example problems in the text and in many cases introduces additional discussion. </p>
</div></div></div>Tue, 06 Feb 2007 21:56:00 +0000Jim Barber805 at https://imechanica.orghttps://imechanica.org/node/805#commentshttps://imechanica.org/crss/node/805ASYMPTOTIC ELASTIC STRESS FIELDS AT SINGULAR POINTS
https://imechanica.org/node/676
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/347">elasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/505">asymptotic methods</a></div><div class="field-item even"><a href="/taxonomy/term/506">singular points</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Singular elastic stress fields are generally developed at sharp re-entrant corners and at the end of bonded interfaces between dissimilar elastic materials. This behaviour can present difficulties in both analytical and numerical solution of such problems. For example, excessive mesh refinement might be needed in a finite element solution.
</p><p> Williams (1952) pioneered a method for determining the strength of the dominant singularity by expressing the local field as an asymptotic expansion. The same method has since been used for a variety of situations leading to singular points, including bonded dissimilar wedges and frictionless or frictional contact between bodies with sharp corners. </p>
<p> Information about the strength of the singularity can be used in analytical solutions to choose an appropriate representation for the fields (for example in the choice of quadrature to use in an integral equation formulation of the problem). It can also be used in numerical solutions to suggest the most appropriate form of graded mesh refinement into the corner, or (better) to develop special corner elements with the analytically determined form. However, asymptotic analysis is seldom the primary purpose of such research and it is tempting just to use a large number of elements in the corner and hope for the best. </p>
<p> We have recently developed a Matlab tool for determining the nature of the stress and displacement fields near a fairly general singular point in linear elasticity, using Williams' method. The basic mathematics for this procedure is given in Section 11.2 of J.R.Barber, <a href="http://www-personal.umich.edu/~jbarber/elasticity/book.html"> <em>Elasticity,</em></a> Kluwer, Dordrecht 2nd edn. (2002), 410pp. However, it is not necessary to have any detailed knowledge of the method in order to use the tool. </p>
<p> A more detailed description of the procedure, including detailed instructions for using the analytical tool has been published in the Journal of Strain Analysis for Engineering Design and can be downloaded <a href="http://www-personal.umich.edu/~jbarber/donghee.pdf"> here.</a> </p>
<p> The user is prompted to input the local geometry of the system, the material properties and the boundary conditions (and interface conditions in the case of composite bodies or problems involving contact between two or more bodies). The tool then computes the dominant eigenvalue and provides as output the equations defining the singular stress and displacement fields and contour plots of these fields. No knowledge of the asymptotic analysis procedure is required of the user. </p>
<p> The tool is written in the software code MATLAB v7.0 with the MATLAB GUI development environment (GUIDE) v2.5 and the MATLAB Symbolic Toolbox v3.1. It provides a graphic interface in which users can define their problem, determine the order of the corresponding singularity and generate the distribution of stress and displacement. Final results are provided in both text and graphic format. </p>
<p> To download the source code, click on <a href="http://www-personal.umich.edu/~donghl/ws/ws.zip">this link </a> and unzip the downloaded file. If and only if you have difficulty opening this link, try <a href="http://www-personal.umich.edu/~jbarber/asymptotics/ws.zip">this one </a>. After downloading and unzipping the file, open MATLAB and start the program with the command `ws'. </p>
<p> Two example programs are included: `williams.wat' and `bogy.wat', which solve the problems of the single wedge and the bi-material wedge respectively. </p>
<p> Please report any problems with the software or any suggestions for additional features or improvements to <a href="mailto:donghl@umich.edu,%20jbarber@umich.edu">Donghee Lee and J.R.Barber</a> </p>
<p> </p>
</div></div></div>Thu, 11 Jan 2007 19:35:50 +0000Jim Barber676 at https://imechanica.orghttps://imechanica.org/node/676#commentshttps://imechanica.org/crss/node/676Surface Roughness and Electrical Contact Resistance
https://imechanica.org/node/671
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/295">contact stiffness</a></div><div class="field-item odd"><a href="/taxonomy/term/501">Roughness</a></div><div class="field-item even"><a href="/taxonomy/term/502">fractal surfaces</a></div><div class="field-item odd"><a href="/taxonomy/term/503">contact resistance</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><a href="/afs/umich.edu/user/j/b/jbarber/Public/html/index.html">J.R.Barber </a> The contact of rough surfaces Surfaces are rough on the microscopic scale, so contact is restricted to a few `actual contact areas'. If a current flows between two contacting bodies, it has to pass through these areas, causing an electrical contact resistance. The problem can be seen as analogous to a large number of people trying to get out of a hall through a small number of doors.
</p><p> Classical treatments of the problem are mostly based on the approximation of the surfaces as a set of `asperities' of idealized shape. The real surfaces are represented as a statistical distribution of such asperities with height above some datum surface. However, modern measurement techniques have shown surfaces have multiscale, quasi-fractal characteristics over a wide range of length scales. This makes it difficult to decide on what scale to define the asperities. </p>
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<p>A theorem bounding electrical contact resistance <a href="/afs/umich.edu/user/j/b/jbarber/Public/html/bounds.pdf"> A recent theorem </a>places upper and lower bounds on the electrical contact resistance due to surface roughness, using only properties of the 'determinate' shape of the contacting bodies and the maximum peak-to-valley roughness. First an analogy is established between the mathematical descriptions of the electrical conduction problem and the elastic contact problem. The electrical potential is governed by Laplace's equation and the interfacial plane is an equipotential surface. The elastic problem can also be cast in terms of a harmonic potential function and the analogous problem is that in which the normal displacement in the contact region is uniform, whilst the normal traction is zero outside the contact region. These boundary conditions define the <em> incremental </em> contact problem - i.e. the response of the system to a small increment of normal load. It follows that there is a strict linear relation between the contact conductance (reciprocal of the resistance) and the incremental normal contact stiffness that depends only on the electrical and elastic properties of the contacting materials.
</p><p> The reciprocal theorem is then used to bound the incremental stiffness of the elastic contact between the solutions of two smooth contact problems. Combination of these two results yields the required bounds. </p>
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<p> Figure 1: Two bounding contact problems
</p><p> <img src="/afs/umich.edu/user/j/b/jbarber/Public/html/fig13.jpg" border="0" alt="Two bounding contact problems" width="480" height="330" align="left" /> Suppose a rigid smooth indenter is pressed to a prescribed depth into a half space with a rough surface, identified as <em>(a)</em> in the figure. We compare this state with two limiting smooth contact problems in which the rough half space is replaced by the smooth half-spaces <em>(b)</em> and <em>(c)</em> respectively. Surface <em>(b)</em> passes through the highest point of the surface and <em>(c)</em> through the lowest point, so that the real rough surface <em>(a)</em> is completely contained between <em>(b)</em> and <em>(c)</em>. The vertical distance between these planes <em>s</em> is also the maximum peak-to-valley roughness of the surface Rt. </p>
<p> Common sense suggests that <em>F(b)>F(a)>F(c)</em> on the grounds that adding material to the half space above plane <em>(c)</em> can only make it harder to press in the indenter to the specified depth. The proof of this result is given by <a href="/afs/umich.edu/user/j/b/jbarber/Public/html/bounds.pdf"> Barber (2003)</a>. However, notice that the roughness of surface <em>(a)</em> will generally cause the contact area <em>A(a)</em> not to be completely enclosed within <em>A(b)</em> or <em>A(c)</em>. </p>
<p> The load-displacement relation for the smooth bounding surfaces <em>(b)</em> and <em>(c)</em> are similar but separated by a distance <em>s</em>. The curve for the rough surface <em>A(a)</em> must lie between these extremes. </p>
<p> It can be shown that the contact area <em>A</em> and the incremental stiffness must be non-decreasing functions of the indentation ζ, so the load displacement curve for both smooth and rough surfaces must be concave upwards. The limiting slope (incremental stiffness) at a given force <em>F</em> can therefore be bounded between two tangent lines. For more information on this procedure, please download the file <a href="/afs/umich.edu/user/j/b/jbarber/Public/html/bounds.pdf"> Bounds on the electrical resistance between contacting elastic rough bodies. </a> </p>
<p> In Figure 1, we show the bounding surfaces as planes because this simplifies the solution of the bounding contact problems, which then become Hertzian. However, the argument does not depend on the bounding surfaces being planes. For example, if we have a solution for the rough surface contact problem at some finite level of resolution, the effect of features below the resolution truncation can be assessed by defining two parallel surfaces separated by the maximum peak-to-valley variance of the scales neglected. The finite roughness problem might be solved by direct numerical methods such as finite element (Hyun <em> et al.</em> 2004), or by an asperity model theory with the asperities defined at some finite scale. This provides a means to tighten the bounds in practical cases. It also demonstrates that the 'infinite tail' (e.g. the terms beyond some fairly large number in a <a href="/afs/umich.edu/user/j/b/jbarber/Public/html/Weierstrass.pdf">Weierstrass series</a>) of a theoretical fractal distribution have negigible effect on the electrical resistance. Put another way, the resistance is largely determined by the coarse scale properties of the rough surface, which supports more recent arguments about asperity models due to Greenwood & Wu (2001). </p>
<p>References
</p><ul><li>J.R.Barber,<a href="/afs/umich.edu/user/j/b/jbarber/Public/html/bounds.pdf"> Bounds on the electrical resistance between contacting elastic rough bodies, </a> <em> Proc.Roy.Soc. (London)</em>, Vol. A 459 (2003), pp. 53-66. </li>
<li>M.Ciavarella, G.Demelio, J.R.Barber and Yong Hoon Jang, <a href="/afs/umich.edu/user/j/b/jbarber/Public/html/Weierstrass.pdf"> Linear elastic contact of the Weierstrass profile,</a> <em> Proc.Roy.Soc. (London)</em>, Vol. A 456 (2000), pp. 387-405. </li>
<li> J.A.Greenwood and J.J.Wu, Surface roughness and contact: an apology, <em> Meccanica</em> Vol.36 (2001), pp.617-630. </li>
<li> S. Hyun, L. Pei, J.-F. Molinari, and M. O. Robbins, Finite-element analysis of contact between elastic self-affine surfaces, <em> Physical Review E</em>, Vol. 70, (2004), art. no. 026117. </li>
</ul><p> <a href="/afs/umich.edu/user/j/b/jbarber/Public/html/index.html"> Back to J.R.Barber's homepage</a> </p>
</div></div></div>Wed, 10 Jan 2007 22:02:35 +0000Jim Barber671 at https://imechanica.orghttps://imechanica.org/node/671#commentshttps://imechanica.org/crss/node/671