. University of California, BerkeleyCIV ENG 130CE130N-book

Engineering Mechanics of Deformable Solids:
A Presentation with Exercises
Sanjay Govindjee
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c by Sanjay Govindjee, 2009, all rights rese
Preface
This text was developed for a Strength of Materials course I taught at the University of California, Berkeley for 15 years. The students in this course are typically
second semester Sophomores and first semester Juniors. They have already studied 1 semester of statics and dynamics in the Physics Department, had a separate
two unit engineering course in statics, and most have also completed or are concurrently completing a 4 semester mathematics sequence in calculus, linear algebra,
and ordinary and partial differential equations. Additionally they have already
completed a laboratory course on materials. With regard to this background, the
essential prerequisites for this text are the basic physics course in mechanics and
the mathematics background (basic one- and multi-dimensional integration, linear
ordinary differential equations with constant coefficients, introduction to partial
differentiation, and concepts of matricies and eigen-problems). The additional
background is helpful but not required. While there is a wealth of texts appropriate for such a course, they uniformly leave much to be desired by focusing heavily
on special techniques of analysis as opposed to basic principles of mechanics. The
outlook of such books is perfectly valid but does not put students in a good position
for higher studies.
The goal of this text is to provide a self-contained concise description of the
main material of this type of course in a modern way. The emphasis is upon kinematic relations and assumptions, equilibrium relations, and constitutive relations.
The emphasis is upon the construction of appropriate sets of equations in a manner
in which the underlying assumptions are clearly exposed. The preparation given
puts weight upon model development as opposed to solution technique. This is
not to say that problem solving is not a large part of the material presented. But
it does mean that “solving a problem” involves two key items: the formulation of
the governing equations of a model and then their solution. A central motivation
for placing emphasis upon the formulation of governing equations is that many
problems, and especially many interesting problems, first require modeling before
solution. Often such problems are not amenable to hand solution and thus they
are solved numerically. In well posed numerical computations one needs a clear
definition of a complete set of equations with boundary conditions. For effective
further studies in mechanics this viewpoint is essential and thus the presentation,
in this regard, is strongly influenced by the need to adequately prepare students
for further study in modern methods.
Sanjay Govindjee
Berkeley and Z¨urich
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Mechanics is the paradise of mathematical science, because here we come to
the fruits of mathematics. – Leonardo da Vinci
Theory is the captain, practice the soldier. – Leonardo da Vinci
Mechanics is not a spectator sport. – The Author
Contents
1 Introduction 1
1.1 Force systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Characterization of force systems . . . . . . . . . . . . . . . . 3
1.2.1 Distributed forces . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Equivalent forces systems . . . . . . . . . . . . . . . . 6
1.3 Work and power . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Conservative Forces . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Conservative systems . . . . . . . . . . . . . . . . . . . 10
1.4 Static equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Equilibrium of a body . . . . . . . . . . . . . . . . . . 10
1.4.2 Virtual work and virtual power . . . . . . . . . . . . . 11
1.5 Equilibrium of subsets: Free body diagrams . . . . . . . . . . 11
1.5.1 Internal force diagram . . . . . . . . . . . . . . . . . . 12
1.6 Dimensional homogeneity . . . . . . . . . . . . . . . . . . . . 15
1.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 The One Dimensional Case 19
2.1 Displacement field and strain . . . . . . . . . . . . . . . . . . 19
2.1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 Strain at a point . . . . . . . . . . . . . . . . . . . . . 22
2.2 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Pointwise equilibrium . . . . . . . . . . . . . . . . . . . 26
2.3 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 One dimensional Hooke’s law . . . . . . . . . . . . . . 29
2.3.2 More constitutive behaviors . . . . . . . . . . . . . . . 29
2.4 One dimensional response . . . . . . . . . . . . . . . . . . . . 30
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