ÓÄÊ 539.3

**Methods of the theory complex functions and singular
integral equations in problems of reinforcement of cracked anisotropic plates**

(Submitted by academician L.A. Agalovyan 29/X 2001)

The fracture problem of the materials and constructions
constitutes of the contemporary topics of the mechanics of deformable bodies.
Although the fracture mechanism cannot be only referred to a phenomenon of crack
propagation, however the examination of the conditions under which the body
starts creating a crack or a system of cracks which later propagate in it,
constitutes one of the main and interesting sides of the fracture problem. This
side of the problem is referred to the study of the stress-strain state of the
body in the region of the imperfections (defects) and the singular points.

_{1} it is put which has as aim the reinforcement
of the plate because the crack starts propagating.

_{1} we have
equal displacements, as well as equal and opposite stresses i.e.

b) On the two lips of the crack l the loadings are generally
given:

ì

ï

í

ï

î

d

dt[ u
_{1}(t) + iv_{1}(t)] =d

dt[ u(t) + iv(t)]

s _{n}^{1 }+ is_{t}^{1 }= -(s_{n }+ is_{t}) = f^{*}(t).(1)

The complex potentials F

s
_{n}^{±} - is_{1}^{±}ê

êt ˙(2) _{0}(z_{1}) and y_{0}(z_{2}), on the basis of which the
stress-strain state of the cracked plate, which is loaded in the infinite and on
which concentrated force X + iY on the point z_{0} acts can be studied,
are given in the form [1-4]:

where j(t

F(z
_{1}) =1

2pi

ó

õ

l_{1}
j(t
_{1})

t_{1} - z_{1}dt
_{1 }+ G +c
_{11}X + c_{12}Y

z_{1} - x_{1}

y(z
_{2}) =1

2pi

ó

õ

l_{2}
j(t
_{2})

t_{2} - z_{2}dt
_{2 }+ G^{¢ }+c
_{21}X - c_{22}Y

z_{2} - x_{2}(3) _{k}), y(t_{k})-densities on the l_{k} (k = 1,2) and

z _{k }= x + m_{k}y, x_{k }= x_{0 }+ m_{k}y_{0}; z = x + iy, z_{0 }= x_{0 }+ iy_{0}

and
m
_{k} (k = 1,2) are the roots of the equation:

with E

m
^{4 }+æ

ç

è
E
_{x}

G_{xy}- 2n
_{x}ö

÷

ø m
^{2 }+E
_{x}

E_{y}= 0 (Im m _{k} > 0)_{x}, E_{y}, n_{x}, n_{y} being elastic constants of the plate in
directions x and y respectively, G_{xy} is the shear modulus. Apart
this, for the constants G, G^{¢} we know that:

Now, we shall proceed to the determination of the complex potentials of
the composite cracked plate (S and S
_{1}).

For the finite elastic isotropic inclusion S_{1} the
complex potential is given in the form:

The stresses
and the displacements on the boundary g of the
inclusion are given by the relationships [1]:

F(z) =
ì

ï

í

ï

î

F
^{+}(z) =1

2pi

ó

õ

g_{1}
G(t)

t - zdt,

z Î S _{1}

0,

z Ï S _{1}(4)

(5)

where k
(6) _{1 }= [(3 - v_{1})/(1 + v_{1})], m_{1 }= [(E_{1})/(2(1 + v_{1}))] and E_{1},
v_{1} being the elastic constants of the indusion S_{1}.

Taking into consideration the Plemelj formulae for the
function F(z) when z ® t, t Î g
[1,5]

and solving Eq. (5) to obtain y
(7) ^{+}(t), the expression of the other complex
potential y(z) takes the following form:

Substituting, next, the limit values of F(z),
y(z) z ® t Î g
(8) _{1}, on the basis of
Plemelj formulae, in Eq. (6) we obtain the following singular integral equation

where l(t) = 2m
(9) _{1}[d/dt](u_{1}(t) + in_{1}(t)), G(t) = [(f^{*}(t))/(1 + k_{1})] + [(l(t))/(1 + k_{1})] - density of Cauchy integral.

On the anisotropic (orthotropic) cracked plate where along
g the effect of the inclusion is transmitted, it is created such a
stress-strain field that can be described by the complex potentials
F_{0}(z_{1}) and
y_{0}(z_{1})

(10)

where F(z
(11) _{1}) = [1/(2pi)]/ (t_{1} - z_{1})]dt_{1},
y(z_{2}) = [1/(2pi)]/ (t_{2} - z_{2})]dt_{2}, g(t_{1}),
y(t_{2})-unknown densities along L_{1}, L_{2} and
t_{k}^{* }= x + m_{k}y, with x + iy = z (k = 1,2).

Now, we shall describe the boundary conditions (1) and (2)
by the aid of the boundary values of complex potentials (10) and (11) on the
basis of Plemelj formulae.

The stress components s_{xx}, s_{yy},
t_{xy} and the displacements for any point of
the anisotropic plate are given by and of complex potentials by the following
expressions [3]:

and

ì

ï

í

ï

î

s _{xx }= 2Re[ m_{1}^{2}F_{0}(z_{1}) + m_{2}^{2}y_{0}(z_{2})]

s _{yy }= 2Re[ F_{0}(z_{1}) + y_{0}(z_{2})]

t _{xy }= -2Re{ m_{1}F_{0}(z_{1}) + m_{2}y_{0}(z_{2})}(12)

where p

ì

í

î

u(z) = 2Re[ p _{1}j_{0}(z_{1}) + p_{2}y_{0}(z_{2})]

n(z) = 2Re[ q _{1}j_{0}(z_{1}) + q_{2}y_{0}(z_{2})](13) _{j }= [(m_{j}^{2})/(E_{x})] - [(v_{y})/(E_{x})], q_{j }= [1/(E_{y}m_{j})] - [(v_{x})/(E_{y})]m_{j} (j = 1,2).

By derivating Eq. (13) and taking into account the
relationship

we obtain the following
relationships on the boundaries considered:

s
_{n }+ is_{i }=1

2(s
_{xx }+ s_{yy}) -e
^{-2iq}

2(s _{xx} - s_{yy }+ 2is_{xy})(14)

(15)

Solving for y
(16) ^{+}(t_{2}) - y_{-}(t_{2}) after
some algebra and taking into account the Plemelj formulae we obtain:

(17)

where

2. By substraction of the Eq. (16) and taking into account
the Plemelj formulae for the complex potentials F
_{0}(z_{1}) and y_{0}(z_{2}), when z_{k} ® t_{k} Î l_{k}
k = 1, 2 after some algebra the following singular integral equation results:

which describes the conditions
(2) and where q
(18) ^{*}(t) = q^{+}(t) + q^{-}(t).

Finally, substituting in Eq. (15) the limit values of the
complex potentials by the Plemelj formulae we have the following singular
integral equation on the g of the anisotropic plate.

Eqs. (9), (18) and (19) are completed with the single
valuedness condition for the displacements on the crack l
(19)

and the equilibrium condition
of the stresses on g

ó

õ

l
d

dt[(u ^{+ }- u^{-}) + i(n^{+} - n^{-})]dt = 0(20)

where Eq. (20), after
substitution of the relationships for the displacements through the complex
functions on the basis of Eq. (15) and using Plemelj formulae, will take the
following form

ó

õ

g f ^{*}(t)ds = 0(21)

Eqs. (9), (18), (19), (21) and (22) give the possibility to
describe the stress-strain state field of the plane cracked orthotropic plate
which is reinforced with an elastic isotropic inclusion.
(22)

The work is fulfilled in the framework of agreement on
scientific collaboration between Athen National Technical University and the
Institute of Mechanics of NAS RA

National Technical University of Athens

1. *Muskhelishvili N.I.*
Some basic problems of the mathematical theory of
elasticity. P. Noordhoff. Groningen. 1965.

2.
*Lekhnitski S.G.* Anisotropic plates. M-L. 1947.
(in Russian).

3. *Bardzokas D.,
Parton V.Z., Theocaris P.S.* - DAN. SSSR. 1989. T. 309.
N 5. P. 1072-1077. (in Russian).

4. *Bardzokas D., Parton V.Z.* - DAN. Rus. 1992. T. 325. N
2. P. 255-259. (in Russian).

5. *Gakhov F.D.* Boundary value problems. Pergamon.
Oxford. 1966.