Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 407

@ T rr

@

1

r @ @y þ @

T r y T rz

@

T rr T yy

r þ z þ

þ r d r ¼ r u € x r ;

r

@ T r y

@

1

r @ @y þ @

T yy T y z

@

T r r þ r

r þ z þ

d y ¼ r€ x y ;

@ T rz

@

1

r @ @y þ @

T y z T zz

@

T rz
r þ r r þ z þ

d z ¼ r€ x z : (A.205)

Problem

A.15.1 Calculate the components of the rate of deformation-velocity tensor D ,
(3.33) in cylindrical coordinates.

A.16 Laplace Transform Refresher

The solutions to linear differential equations in time are often obtained by the use of
Laplace transforms and the Laplace transforms of discontinuous functions. Laplace
transforms provide a method for representing and analyzing linear systems using
algebraic methods. The Laplace transform variable 's' can directly replace the d /d t
operator in differential equations involving functions whose value at time zero is
zero. Most of the readers of this text will have been introduced to Laplace
transforms at some time in their past and find it convenient to have the salient
points about these transforms refreshed in their minds before solving the differen-
tial equations of this type.
The Laplace transform of a function f ( t ), 0 t 1 , is defined by

1

Þg ¼ f

e st d t

L f f ð t ð s Þ¼ f ð t Þ : (A.206)

0

This integral is absolutely convergent if the function f ( t ) is of exponential order

c e at where c and

a , that is to say if f ( t ) is continuous for 0 t 1 and j f ð t Þj< a

are constants. The notation for the inverse Laplace transform of the function f

ð s Þ is

ff

L 1 ð s Þg ¼ f ð t Þ: (A.207)

The Laplace transforms of derivatives and integrals are some of the most useful
properties of the transform. If the function f ( t ) is of exponential order and the
derivative of f ( t ) is continuous, then for s

> a

,

sf

f 0 ð

L f t Þg ¼ sL f f ð t Þg f ð 0 Þ¼ ð s Þ f ð 0 Þ; (A.208)