Introduction to Continuum Mechanics - Mechanical Engineering

203 Pages·2008·45.53 MB·English

UNIVERSITY OF CALIFORNIA AT BERKELEY
DEPARTMENT OF MECHANICAL ENGINEERING

ME 185

P.M. NAGHDI’S NOTES ON CONTINUUM MECHANICS

2001
Preface

‘These classnoes were developed by Professor Paul M. Naghd (1924-1994) fora fst course on
Continuum mechanes, which has been offered in the Mechanical Enginering Department con-
tinuously singe 1960, Tt vs orginally Fisted as ME289A, "Foundations of the Theory of Can-
tinuous Media 1,” bot aftr few years became MEISS, “Intoduction to Continuum Mechan-
ics "Tho course offers concise, Figoross, elegant, and genera reament of the mechanics of
Gformable contin, The enoliment hav traditionally been incoming graduate stents plas
Sone advanced ndengrasex

‘These notes first appeared in hand-written mimeographed form and were subsequently typewrit~
ten, and later digitized, A few minor corrections have been made fram time tn time.

1. Casey
Berkeley
January 2001
UNIVERSITY OF CALIFORNIA
Department of Mechanical Engineering,

MEIS5

A.List of Generel References

Cartesian Tensor, by Sir Harold Jefteys, Cambridge University
ress, 1952 [Ist ed, 1931, 7th impression 1969].

Schaum's Outline Series - Theory and Problems of Continuurn
Mechanics, by G. E. Mase, MeGraw-Hill Book Co,, 1970,

Chapters 1-7 of Mechanics, by E. A. Fox, Harper & Row, 1967,
Continuum Mechanics, by Y.C Fung, Prentice-Hall Ine., 1969,
Continuum Mechanies, by P. G. Hodge, J, MeGraw-Hill, 1970

Mechanics of Continua, by A. C. Fringen, John Wiley & Sons
Ine, 1967,

Continuum Mechanics, by P. Chadwick, G. Allen & Unwin
Eid, Landon, 1976, (Dover, 1999).

‘Continuum Mechanics, by A.J.M. Spencer, Longman Group
Ted, London, 1980.

An latroduction to the Theory of Plasticity, by RJ. Atkin and
'N-Fox, Longman Ine, New York, 1980.
Table of Contents

1. Kinematies
1. Bodies, configurations, motions, mass, and mass decsity.
2. Deformation gradient. Measure of strain. Rovation and stretch.
3. Further developments of kinematical results
4, Further developments of kinematical results

5, Velocity gradiont. Rate of deformation. Vortcty
5, Superposed rigid body motions.
7. Infintesimal deformation and infinitesimal strain mensures
8, The transport theorem,
1. Conservation Laws and Some Related Results
1, Conservation of mass
Appendix to See. { of Part TL

2, Forees and couples. Euler's laws.
3. Further consideration af the ste vector. Existence of the sess tensor and
its relationship to the stress veer. 3
4. Derivation of spatial (or Evlerian) form of the equations of mation, ine * 58
8, Derivation ofthe equations of motion in referential (or Lagrangian forr), 61
6, Invariance under superposc¢ rigid body motions 4
7. The principle of balance of energy... nl
TT, Examples of Consitive Equations und Applicaions, Invisid and Viscous Fluids
‘and Nenlinear and Linear Elastic Solids m4
Te Inviseid Bid. 4
2, Viscous fui. %
3. lastc solids: Nonlinear constitutive equations 81
4, Flastc solids: Linear constitutive equations 85
5. Same steady flows of viscous fluids °0
6. Some unsteady flow problems. 95
7, Further constiuive results in Tinea elasticity and ilusirative examples. 7
8, Saint-Venant torsion of a cireular cylinder, : 101
Appendix to Part TT (Sees. 5-8). 103
Supplements tothe Main Text 109

Appendix L: Some results ftom linear algebra 150
MESS

Partl: Kinematics
1. odie, configurations, motions, mass, and mass densi

A body isa set of particles, Sometimes inthe literature a body ais referred to as a mani-
fol of panicles. Particles of a body 2 will be designated as X (ee Fig. 1.1) and may be
‘dened by any convenient system of labels, such 35 for example a set of colors In mechanies
the body is assumed to be smoot and, by assumption, can be put into correspondence with &
domain of Euclideen space, Thas, by assumption, a panicle X ean be put ilo a one-to-one
coarspondonce with the triples of rel numbers X,%.X3 in region of Bucidean Sspace ©
‘The mapping from the body manifold oato the domain of 2s assumed t be one-to-one iaver-
te, and differentiable as many times as desiod and for mos: purposes two or the times
soffce

Each part (or subse) Sof the body at each instant of time is assumed to be endowed with

positive measure 44(9, i. areal number > O, called the mass ofthe part Sand the whole body
is assumed to be endowed with « nonnegative measure (a) called the mass of the body. We
sretum t9 a further consideration of mass later in this section,

Tiodies are seen only in their configurations, ic. the regions of they oocupy at each
instant of time ¢ (© <t<4H0}, These configurations should not be confused with the bodies
themselves.

Consider a configuration of the body at time {in which configuration ® occupies a region
‘Rin 2 bounded by a closed surface OR (Fig. 1.2). Let x be the position vector of the place

‘occupied by a typical partiele X al time |. Since the body can be mapped smoothly onto a

domain of £3, we write

= 1X0, rab)

In (1.1), X refers to the particle, tis the time, x [the value of the function 7] is the place occu
pied by the particle X at time t and the mapping function x is assumed to be differentiable as
many times as desired both with respect to X and t, Also, for each t, (1.1) is assumed to be

invertible so taat
x=Heo,

2»

‘where the symbol 27? designates the inverse mapping. The mapping (1.1) is called 8 motion of
the body The description of motion in (I.1) is similar to that in particle mechanics and is
traditionally referred to as the material description,

During @ motion of the body 4 2 particle X (by occupying successive points in 75) moves
‘rough space and describes a path C; the equation of this path is parametially represented by
(1.1). The tate of change of place with time as X traverses Cis called the velocity and is tangent
to the curve C._ Similarly, the second rate of change of place with time as X traverses Cis the
acceleration. Thus, in the material descristion of the motion, the velocity v and acceleration a
are defined by
Pun,

re a3)

In the above formulae, the particle velocity ¥ is the partial derivative of the function x with
respoct to t holding X fixed; and, similerly, the particle acceleration a is the second partial
derivative of x with respect to t holding X fixed. Also, the superposed dot, which designates the
_matcral time differentiation, can be utilized in conjunetion with any funetion Fund signifies dif-

ferentiation with cespect tot holding X fixed.

Reference configuration, Often itis convenient to select one particular conigurnsion and
refer everything concerning the body and iis mation to this configuration, The reference
configuration need not necessarily be a configuration tat sacral occupied by the body in any
ofits motions. In panicular, the reference configuration need not be the initia! configuration oF
the body,

Let nx bea reference configuration of Sin which X is the position veetor ofthe place oveu-
pied by the purtiole X (Fig. 1.3), Then, the mapping from X tothe place X in the configuration

‘kg may be writen as
aR R
Body 8

Fig. 1.1 Present configuration < at tine t.

& Fig. 1.2

Fixed reference configuration <q

Fig. 3
X= KR (X) aay

‘which specifics the position vector X occupied by the particle X in the reference configuration
sp. ‘The mapping (1.4) is assumed fo be differentiable as many times as desired and invertible

The inverse mapping is

X= as)

‘Then, the mapping (1.1) al time {from the place x the particle X may be expressed in terms of |
(5) ie,

X= EPG OO) = Hag Kt) (18)

“The second of (1.6), wich involves the fanetion fg (with a subserpe wy emphasizes tha he
‘motion described. in this manner represents a. sequence of mappings of the reference
configuration In the fare, however, for simplicity’s sake we omit the subscript xg and

‘write (1.6)g and its inverse as

x, X= Hey an

with the understanding that X in the argument (1.6) is the position vector of X in the reference
configuration x, Returning 1 (1.6), we observe tat for each vy. diferent function Fy
reli and he chose of refeencecootguation, snarl tthe chive of coordinate, sabi
trary ands intoduced fr conveisnes.

A necessary an sucient condon forthe invert of the mappings of (1.7). is ha
the determinant Jf th rnsirmation frm X ox be onze,

1 = aac » 00 as
Description of the motion, There ate several methods of desribing the motion af a body:
Here, we describe three but now chat, because of our smoothness assumptions, they are all
ceauivalem.

te was noted carlor that the deseription utilized in (1.1) i the material deveription. In such
a desription one deals with abstract panicles X which gether with ime tare the independent
variables.

[A desertion such as (7) in which the position X of X at some time, e t= 0, is used a8
4 label for the particle X is called the referential description. In the referential descrpon,
‘which is also known as Lagrangian, the independent variables are X and t, For some pusposes,
itis convenient display and to utlize (1.7) in toms ofits rectangular Cartesian coordinates.
Thus, let Fx be constant ontbonontal basis vectors associated withthe reference configuration
and similarly denote by e, constant orthonormal basis vector’ associated with the present

configuration al time {, Then, the positions X and x referred to Ex and e, respectively, are

X=XcRk xe me a9

where Xx are the rectangular Cartesian coordinates of the position X and similarly xj are the
rectangular Cartesian coondinates of the position x. Referred to the basis e,, the component
form of (1.7), can he displayed as

X= 0%) + 19)

‘where without loss in generality (since Ex are constant orthonormal basis) we have also replaced

the argument X by its components Xx. In the future we often display the various formulae both

in their "coordinate-free" forms, as well as their component forms.
‘The particle velocity and acceleration in a material description of motion are given, respec-
tively, by (1.3); The comesponding expressions for velocity and acceleration in the referential

scription (1.7) wil have the same forms a5 (1.3), but with the argument X replaced by X:

vei= Bay, ante Saxy, ay