Crack Paths 2012

The displacement field u : Q —> R3 with (smooth enough) componentsu,- can be

transformed to curvilinear coordinates by the defining relation

W6) I ui-(flei I fii-(wg’ty)

for al

x I @(y).

ly'l

<< 1

Vector fields in global C A R T E S I AcNoordinates x I (x1,x2,ac3)T are related to

the standard unit basis of R3: {e1,e2,e3}. While this basis is fixed in R3, the

displacement field can be identified by the vector of its CARTESIANcomponents I (u1(ac),u2(:1c),u3(9c))T at any point 96 € R3. This is no longer true in curvilin

ear coordinates, wherethe functions

represent the covariant componentosf the

displacement vector over the contravariant basis {g1(y), g2(y), g3

which varies

with @(y) I :16 Q T. W e identify the vector a I with the vector of its covariant

components whereas fi :I fligi is the (physical) displacement vector, see e.g. [5] for

moredetails.

W e formulate the equilibrium equations (1) in curvilinear coordinates. The

derivatives of a vector field in curvilinear coordinates are defined by the relations

ajmlxl : (fikflllgklilgllj) (y),

95 I @(y),

I: gk ' 9i.

(always with sum convention), the covariant derivative is defined by

57,-“,-

z:

— FIZZ-6p

with the CHRISTOFFELsymbols

Ffj :: gp - oigj.

Remark,that the choice of the basis as shownin figure l specifies the enumeration

of derivatives as follows:

A 86 AA 517 vo:(5o,5a5o), a o : _ , t:_, t : _ . y 1 2 3 1 g m 2 as 3 a m AA 817

The CHRISTOFFEsLymbols F2- of the second kind can be expressed in terms of the

metric tensor by CHRISTOFFEsLymbols of the first kind,

1 Fijq i: 5 (31911 + aigjq I @1919‘),

by the relation P;- I gpqFiJ-q where (gpq) I (gig-Y1. The (covariant) components of

the strain tensor in curvilinear coordinates are defined by the relation

1 A A — (uilj + uni).

.

.

m I 1.2.3.

31mm) II 6M”;x)lgi(y)lklgj(y)lh

517mm) I 2

and similar the (contravariant) componentsof the stress tensor are

Fill-(fig) I: vkl(u;w)lgi(y)lklgj(y)li7

it]. : 1>2>3‘

548