Sec. 1.8
HISTORICAL R EMARKS
25
then develop classical and computational methods to cover it. Then biology
becomes a new source of fresh water to irrigate our field of mechanics.
The present book will not deal with biomechanics any further. Some
reference
24
I NTRODUCTION
Chap. 1
atherosclerosis studies, a characteristic dimension for hemodynamics in the
vessel is the diameter of the coronary arteries, that for the shear stress
on the vessel wall is the thickness of the endothelial cell, in the micrometers
Sec. 1.7
BIOMECHANICS
23
Biomechanics has the following salient features:
( 1) Material Constitution
Every living organism has a solid structure that gives it a unique shape
and size, and an internal fluid flow that transports materials and keep the
organ
22
I NTRODUCTION
Chap. 1
first reflection, when the stress was 2pcV0, but at the third reflection, Le.,
the second reflection at the top of the wire, when the tensile stress reached
2.15 pCV0.
If the mass of the wire were negligible, then the wave velocit
Sec. 1.6
P ROTOTYPE O F WAVE DYNAMICS
21
application to an experiment initiated by John Hopkinson (1872) who hang
a steel wire vertically from a ceiling, attached a stopper at the lower end as
shown in Fig. 1.6:1, then dropped a massive weight down the wi
20
Chap. 1
I NTRODUCTION
force acting at the lower end is u A , where A is the cross-sectional area of
the wire. The force acting a t the upper end is A ( a d a). For a continuous
function a(.) t he differential da is equal to ( da/dx)dx. T he acceleratio
Sec. 1.6
P ROTOTYPE O F WAVE DYNAMICS
19
Fig. 1.6:l.
Fig. 1.6:2.
t he length of the wire, with an origin 0 chosen at the lower end. When
the wire is loaded, each particle in the wire will be displaced longitudinally
from its original position by an amount
18
I NTRODUCTION
Chap. 1
a nd its dependence on the parameters w , , and the frequency spectrum of
E
the forcing function s (t).
T he simple solution given by Eqs. ( 13) a nd (14) h as important applications to the problems of isolation of delicate instru
Sec. 1.5
VIBRATIONS
17
body at a circular frequency of w rad/sec. It can exist without external
load. Hence w is called the natural frequency of a free vibration. If the
damping constant c were zero, and the forcing function i is periodic with
the same fr
16
Chap. 1
I NTRODUCTION
Fig. 1.5:l.
represent the displacement of t he body relative to the ground, then Eq. (1)
may be written as
( 3)
My+cy+ky=-Ms.
T he solution of Eq. ( 3) represents a forced vibration of a d amped system.
If t he forcing function MS
Sec. 1.5
VIBRATIONS
.i.
.l
Deflection
Deflection
(a) Ideal plasticity
15
(b) Linear elastic
- ideal
plastic material
Fig. 1 4 l
.:.
Structural steel behaves pretty much like an ideal plastic material, except that when the load (measured in terms of the ma
14
Chap. 1
I NTRODUCTION
F requency
Fig. 1.3:2. A typical relaxation spectrum. (After C. M . Zener, Elasticity and A nelasticity of Metals, T he University of Chicago P e s 1948.)
rs,
oscillations of metal wires a t various temperatures, reduced to room t
Sec. 1.3
SINUSOIDAL OSCILLATIONS I N A VISCOELASTIC
.
13
Similarly, if uo = B ei$, t hen a simple harmonic oscillatory displacement
u (t)= uOeiWt is either the real part or t he imaginary part of
(2)
u ( t )= uOeiWt = B cos(wt + $1
+ i~
sin(wt
+ $) .
O n
12
Chap. 1
INTRODUCTION
a
l
e
l.2
E
L
0
.
c
0)
Q
0
Time
I
Time
Fig. 1 2 3 Relaxation function, of ( a) Maxwell, (b) Voigt, ( c) Kelvin solid.
.:.
For t he Kelvin solid, a similar interpretation is applicable. The constant rEis the time of relaxation of lo
Sec. 1.2
LINEAR SOLIDS WITH MEMORY: MODELS O F
.
11
(10) Kelvin solid:
Here we have used the symbol s ( t ) t o indicate the unit-impulse function,
or Dimc-delta function, which is defined as a function with a singularity at
the origin:
qt)=0
( f (t)b(t)d
Chap. 1
10
INTRODUCTION
( 1)
Maxwell model:
( 2)
Voigt model:
(3)
Kelvin model:
F
u=-+-
F
Prl
+ q U,
F + T ~ = E R(U+ T , I ~ ) ,
F
F
= PU
u (0)= 0 ,
T,F(O) = ERT,u(O),
where T ~T~ a re two constants. When these equations are t o be integrated,
,
the init
Sec. 1.2
LINEAR SOLIDS WITH MEMORY: MODELS O F
.
9
1.2. LINEAR SOLIDS WITH MEMORY: MODELS OF
VISCOELASTICITY
Most structural metals are nearly linear elastic under small strain, as
measurements of load-displacement relationship reveal. The existence of
no
8
Chap. 1
I NTRODUCTION
strain energy, we have, from ( 3) and ( 4),
If we differentiate Eq. ( 6) with respect to P we obtain
i,
i = 1 , 2 , . * . , 2.
7
But, the right-hand side is precisely ui; ence, we obtain
h
( F) C astiglianos theorem
( 7)
dU
dPa
-=u
S ec. 1.1
H OOKE'SLAW
7
f orces is equal to the work d one b y t he second set of f orces acting through
the corresponding displacements produced b y t he f irst s et o f f orces.
A straightforward proof is furnished by writing out the u( and u: in
terms
6
Chap, 1
I NTRODUCTION
If we multiply the first equation by P I , t he second by P2, e tc., and add, we
obtain
(3)
9 .1
+ P2u2 + . . . + Pnu, = CllP? + C12PlP2 + . . + ClnPlP,
+ C21PlP2 + c22Pz+ . . . + c,P,2.
*
T he quantity above is independent of the
Sec. 1.1
H OOKESLAW
of P I). Similarly, ~ 3 - cL2 can only be a function of
2
as
6
4 . If we write Eq. (d)
then the left-hand side is a function of P alone, and the right-hand side
2
is a function of PI alone. Since PI and Pz are arbitrary numbers, the on
4
Chap. 1
I NTRODUCTION
( H3) There exists a unique unstressed state of the b ody, t o which the
body returns whenever a ll the external forces are removed.
A body satisfying these three hypothesis is called a linear elastic s olid.
A number of deductions
Sec. 1.1
H OOKESLAW
3
Under this hypothesis the atomistic structure of the body is ignored
and the body is idealized into a geometrical copy in Euclidean space whose
points are identified with the material
particles of the body. Continuity is defined in m
2
I NTRODUCTION
Chap. 1
often, they are limited by insufficient scientific knowledge. Thus they study
mathematics, physics, chemistry, biology and mechanics. Often they have
to add to the sciences relevant to their profession. Thus engineering sciences
ar
1
INTRODUCTION
Mechanics is the science of force and motion of matter. Solid mechanics is
the science of force and motion of matter in the solid state. Physicists are
of course interested in mechanics. The greatest advances in physics in the
twentieth cen
Sec. 2.16
PHYSICAL COMPONENTS O F A V ECTOR
65
Riemann-Christoffel curvature tensor are all equal to a c onstant, which may be
written as 1. For a flat plate, they are zero. For c ertain hyperboloidal surface all
t he nonvanishing components of curvature
64
C hap. 2
TENSOR ANALYSIS
A ns.
= -xl,
2.26.
Show that
= l / z l , all other components = 0.
r&, is symmetric in m and n ; i.e., rLn= rKm.
2.27.
Show that the necessary and sufficient condition that a given curvilinear coordinate system be orthogonal is
Sec. 2.16
2.22.
P HYSICAL C OMPONENTS O F A V ECTOR
63
Prove that the Laplacian of Prob. 2.18 c an be written
H n : Use the results of Prob. 2.21.
it
2.23. Show that the covariant differentiation of sums and products follows
the usual rules for partial di
62
C hap. 2
T ENSOR ANALYSIS
( b) In rectangular Cartesian coordinates g ap = J ap, t he scalar g ap$Iap reduces
to the form (writing z' = x ,x 2 = y, z3= z )
a2* d
- +-+-.2*
dy2
8 x2
d2*
dz2
( c) Hence, the Laplace equation in curvilinear coordinates wit
Sec. 2.16
PHYSICAL COMPONENTS O F A V ECTOR
61
not have the same physical dimensions. This difficulty (and it is also a g reat
convenience!) arises because we would like to keep o ur freedom in choosing
arbitrary curvilinear coordinates. Thus, in spherica