Crack Paths 2006

It should be noted that here we deal only with a scalar model, when only one of two

existing types of waves may propagate in the elastic medium. However, the proposed

model may also be connected with the so-called “anti-plane” (or, shear, SH-) two

dimensional problem Achenbach [8].

S O M SEI M P L EG E O M E T R I E ST: R E N G TA NHA L Y S I S

The equations of equilibrium are described by the following system of partial

differential equations and the components of the stress tensor are defined by the Hook’s

relations, where μ and are classical elastic constants, and denotes the displacement

vector, for the linear isotropic homogeneous elastic medium.

Let us first consider the two-dimensional plane (y,z)-strain,

when the

components of the displacement vector are ={0, uy(y,z), uz(y,z)}, and the non-trivial

yz(y,z). The

components of the stress tensor are

22 =

yy(y,z), 33 =

zz(y,z), 23 =

general equations read in two-dimensional case as follows (1), with positions (2):

1 c

2

2

2

°

w

w

w

u

c

u

u

0

2

2

z

2

2

2

z

2

y

2

y

2

2

y

2

2

z

w w

yu z

(1)

°

w

y

w

z

® °

w

w

w

u

c u

1 c

0

°

w w

y z ,

¯

w

z

w

y

O P

O P

O P

O P

P

u P U

U

W

yz V

P

V

u

u

2 l u

w

w

w

w

y

z

2

2 s l

2 s

2 1 2 c

2 1 2 c

yy

y

z

zz

y

z

2

u

y

u w

z

2 c

y

z

2

w

w

w

w

w

(2)

z

y

,

c

2 c ,

1 c ,

,

c

,

,

w

w

Here is the mass density, and c expresses the ratio of the transverse and

longitudinal wave speeds [8]. The general equations should be completed by the

following boundary conditions (3), if we start from the very simple problem about a

shear of an infinite layer by a constant shear load 0 .

y z 0 . W W f z ;

(3)

y 0 u 0 y h

:

;

:

z

The exact solution to equations (1)--(3) is (4); the solution depends only on the

coordinate y, so the problem is in fact one-dimensional.

W

0 u y

(4)

P

yy V V

yz W W {

y u 0 {

,

,

0

,

.

{

z

zz

0

It should be noted that the well knownclassical result of the theory of elasticity,

which establishes that tangential stress vanishes on the principal elementary areas of

normal stress and vice versa [9]. The basic conclusion, which can be extracted from