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Mehtiyev Rafail Kerim oghlu, Azerbaijan Technical Universityy Е-mail: rafail60mehtiyev@mail.ru Jafarova Saida Allahverdi kizi, Azerbaijan, Baku
CONTiNiUSiNG SHEAR OF THROUGH CRACKS iN A COMPOSiTE REiNFORCED WiTH UNiDiRECTiONAL ORTHOTROPiC FiBERS
Abstract: The problem of fracture mechanics on the interaction of orthotropic elastic inclusions which surface is equally covered with a material cylindrical slight (layer) is considered. The fiber is weakened by a rectilinear crack of a collinear ordinate axis. The surface is weakened by two doubly periodic systems of rectilinear cracks collinear to the axes ofthe abscissas and ordinates in the orthotropic plane.
Keywords: doubly periodic lattice, coating thickness, coating fibers, coating-binder, average stresses, linear algebraic equations.
Formulation of the problem Pmn = + ( = °> ±!> ±2,- ■ ■);
Suppose the issue there is an isotropic elastic bind- (0i = 2;o2 = ^ • hew; h > 0;Ima>2 > 0 er weakened by a doubly periodic system of circular apertures having radii 1(1< l) and centers at points
Figure 1.
The plane under consideration is subjected to shea Ty = t™, rx = 0 (shear at infinity) (Fig. 1). On the basic of the symmetry of the boundary conditions and the geometry of the place D occupied by the stress medium, they are doubly periodic functions with principal periods a T and <a2.
Expressing voltages and displacement w through an analytic function is like that:
u = u = 0, w = w(x,y); ox = oy = oz = txy = 0; tx =
~ dx
dw (1)
dy
1
T-Ty = f'(z) = F(z), Tt + hn = F(zya, w = -Ref (z), z = x + iy, ^
(where H- is the constant of the medium material; i = V—1, t, n are the natural coordinates), we write the boundary conditions of the problem in the form [2]
The operated form for the problem of the interaction of orthotropic elastic inclusions and through cracks
l + tL »
t
1-
M
ft O-
v
s J
. » J
\
M
fb (T ) = 2 ft (t), (2)
f (t) = 2fs (t), (3)
1-
v M j
f'(t )-f(t) = 0, f'(t 1 )-ffo) = 0, (4) where t =Xe'6 +ma>1 + na2,zl = (- h )e'e +mal +
-n®
'2 >
m,n = 0,± 1,±2,..., h* - coating thickness; 0- polar
angle; t and t1 affixes of points of cracks on the abscissa and ordinate axes, simultaneously. The summers related to the coating, the washer and the plane are hereinafter referred to as t, b and s, respectively: Solving a boundary value problem The solution of the boundary value problem is sought in the form
fs(z) = fi(z) + f2(z), fb(z) = fib(z) + f2b(z), (5)
x z 2k+1
fib (z) = E«2k 2kTT> f (z)=! b2kz2k+1, (6)
k=0 2k + 1 k=-rn
2k+2
k=0
l2k+2p(2k )(z ) (2k +1)!
(7)
f2'(z ) = J g (t )ç(t - z )dt + A,
'g (t )
f2 (z) = — J g^(it 1 -z)dti + B f2b = - f^^-,(8)
m L n - t - z
where the integrals in (8) are taken along the lines
LT ={{-b, - a ] + [a, b ]} ; L2 ={{-d, - r ] + [r ,d ]},
22
g (t), g i (t i) - the de-
P(z ) = |-m j
sin
-z
1 (3 \m
m
sired functions characterized the shift of the banks of the prefracture zones.
g (x ) = [W+ (x ,0) - w " (x ,0)] on L1;
2 dx
gi(y) = W (0, y) - w- (0, y)l on L2.
2 ayL J
Additional conditions are added to the basic concepts (5) - (8), which follow from the physical meaning of the task
-a f. -b r
| g (t )dt = 0; | g (t)dt = 0; j gi(ti)dti = 0; j g i(tx)dtx = 0.(9)
-f a -r b
The unknown functions g (x ) and g 1(y ), and the coefficients a2k, b2k, a2k must be determined from the boundary conditions (2)-(4). To formulate equations for unknown coefficients, we transform the boundary condition (3) to the form
f ^ 1 + 4
V Ms y
f ^ 1-4
v 4s
ft (t) = 2[ft (t ) + if2 (t)].(10)
Regarding the function f2 (t ) s and if2 (t ), we will assume that they decompose on a contour |t| = x into Fourier series.
To derive the resolving equations, we substitute the boundary conditions (2) (3) instead of the functions fb (z), ft (z), fi (z) their expansions into Laurent series in the nearly of the zero point, and instead f1 (z ) and if2 (t ) of the - Fourier series on the contour |t| = 1 and equating the coefficients with the same powers exp(id) in both parts of the boundary conditions, we get after some transformations, the set of infinite systems of linear algebraic equations [6]:
b2k =
1 + ^
. 4t
I
2k+1
*2k
2 (2k +1)
2K+1
(11)
_ (Я-h)4k+2Я_
b_ Я3Й0
4Я
■2k _2
1 +^ Ht
_2k _1
2k
_2k _2
V « y
2 (2k + 1)Я
да
(g 1 + f 2h:H;+ c0 +^«2k+2^2k+2rc
la.
0,k>
{K f 2 + g 2
k=1 = - a2,
1 2 k+1 a
A3 a2k
4Я
2k+1
да
( + f4k+Х) = а2ХХ1+^а2Р+1Р+\
p=1
+ 2k p, ^2k '
4^2k+1 (g2 + f h2 ) = -a2k+2.
Here
gx =
h: =
1 + H
V Ht
1 -ËL
V H j
л Ht
1 + —
V H y
\
1 -Hl
. H.
л f
g 2 =
, h2 =
1 + H
V Ht, r >
, ft
1 + —
v ft y
1-H
V Hs y
1_Hb_
V ft.
= (2 p + 2k +1)! g p+k+1 =A-h = , „
rp,k (2p)!(2k +1)!22p+2k+1 ' f Я ' C2k C'k + °2k' С 'k = " — Î g (t ) f2 k (t d,
Oi i
•1
С2'k = " — f g 1(t 1)ф2k (it 1)dt 1>
L-2
X2k+2
3 2k
fk(t > W'" 1 (t )-(2k +1)!
«>(t), r(t) = ctgnt,
W
y 2k y 2k+2
Ф ( 1 ) = ' (" 1 )-(Ь)) +21 1 )•
Y1 (t ) = ctgПit1, (k = 0, ± 1, ±2,...),
ю
* _ 1 _ ^44 _ H2
gp+k+1 _ ^___2(p+k+1) , g _ n H
55 ft
_1 m
Я3 = —,
2
2g
v1 + g y
2
v1 + g y
» r0,0 =
Now accounting that the functions (5) - (8) satisfy the boundary condition on the banks of the pre-fracture zones, we obtain a system of two singular integral equations with calculation to g (x) and g 1 (y):
- J g (t )g(t - x )dt - Im [ A + f((x)] = 0 on L, (12)
- J g1 (t1 )ç(-У)t1-ImГА + f/(x)] = 0 on L2, (13)
ni L J
1 { - Im f (x )] = 0. (14)
Infinite algebraic systems (11) together with singular integral equations (12) - (14) are the main resolving equations of the problem, allowing to determine g 1 (y),g(x) the coefficients «2*,b2k ,a2*. Algebraic systems (11) and integral equations (12) -(14) turned out to be connected and should be solved together. After determining the complex potentials fs (z), fs (z) and ft (z) it is possible to find the stressstrain-state of a piecewise-homogeneous medium Algebrization of basic solving equations Taking advantage of the expansion of the function g(z) in the main period band, and also taking into account that, g(x) = -g(-x), gi(y) = -gi(-y) and applying the change of variables, the integral equations (12) - (14) are reduced to the standard form. The use of quadrature formulas [4, 5, 6] allows replacing the basic resolving equations (12) - (14) with two finite systems of algebraic equations for the approximate values p°, R° of the desired functions at the nodal points
t"m,kpk0 - 1Im[А + f' (nm )] = 0,
k=1 2
(m = 1, 2, ..., М - 1),
R - 1Im[/1b (nm )] = 0.
v=1 2
(14)
Здесь
2M
1 вт +(-1)'m"k| 0k
sinO
"ctg
2M
1 вт +(-1)'m"V| ву
sinO
ctg
B (m ,Tk ) B*(m ,Tv )
To systems (14) it is necessary to add additional conditions (9), which in discrete form take the form
I
k=1
M
I
0 p°
^1/2 (1-Я12 )(k + 1) + Л2
= 0,
R0
(15)
= 0.
'^/2 (l-I2 )( +1) + 12
Systems (14), (1.119) are connected with infinite algebraic systems (11), in which a quadrature relation is substituted for the coefficients.
4
1
a , =
m,k
2
1
*
a
m ,v
2
Since stresses are limited in a composite piecewise homogeneous body, the solution of singular integral equations should be sought in the class of everywhere bounded functions (stresses). Consequently, it is necessary to add the constrained stress conditions at the ends of the pre-fracture zones to the resulting systems. Accounting these conditions, we can get:
M , n M n
X(-1 )kp°kctg| = o, X(-1)+Mp0tg| = 0,
k=1 2 k=1 2
MM
X(-1)+MR0tg| = 0, X(-l)vR0ctg| = 0.
v=1 2 v=1 2
Since the dimensions of the pre-fracture zones are unknown, the combined algebraic system of equations turned out to be nonlinear even with linear-elastic constraints. To solve it, the method of successive approximations was used. In the case of the nonlinear law of deformation of bonds in determining the forces in the bonds, an iterative algorithm similar to the method of elastic solutions was also used [1, 4]. To determine the limiting state at which the growth of cracks occurs, the deformation fracture criterion was used. w+ (x, 0 ) - w" (x, 0 ) = 8m, w+(0 , y) - w- ( 0 , y) = 8m.
Here 5mr - is the characteristic of material resis-
Figure 2. Dependence of the critical load on the distance for both ends of the crack
Based on the results obtained in (Fig. 2) in the case of a hard turn on vs = 0 ,3, when the graphs of the critical load rt = Ty*!®!KIIc as a function of the distance for both ends of the crack are plotted, along the abscissa axis (curve 1 corresponds to the left end) at. In (fig. 3-4) show the dependence of the ultimate load on the crack length. In the calculations it was taken.
1
<
3 > \
l*
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
Figure 3. Dependence of maximum load on crack length
vl
2
3
h
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
Figure 4. Dependence of maximum load on crack length
Analysis of the critical equilibrium part in a composite with a periodic structure, in which cracks appear, reduces to a parametric laws of the combined algebraic system and the criterion for the appearance of cracks under different laws of bond deformation, elastic constant materials and geometric characteristics of the composite.
The solution of the combined algebraic system allows us to determine the critical value of the external load, the size of the pre-fracture zones and the
tangential stresses in the bonds in the state of ultimate equilibrium, during which cracks are formed in the bonding composite.
Straightly from the solution of the obtained algebraic systems, the tangential stresses in the bonds and the shift of the coasts of the pre-fracture zones are determined. The obtained relations allow us to investigate the cracking in the composite body during longitudinal shear.
References:
1. Bolotin V. V. Mechanics of nucleation and initial development of fatigue cracks // Fiz.-chemical chem. Mat. 1986.- Vol. 22.- No. 1.- P. 18-23.
2. Van Fo Fe G. A. Theory of reinforced materials.- Kiev: Sciences. Dumka, 1971.- 230 p.
3. Mustafayev A. B. Crack origin in a uniformly rotating thick-walled cylinder // Mechanics. Mechanical Engineering, 2009.- No. 1.- P. 13-15.
4. Mirsalimov V. M. Destruction of elastic and elastoplastic bodies with cracks.- Bak: Elm, 1984.- 124 p.
5. Mehtiyev R. K. The longitudinal shift of bodies with a complex structure weakened by straight-line cracks // Construction mechanics and the calculation of structures issn 0039-2383 No. 52017.- P. 69-72.
6. Mamedov A. T., Mehtiyev R. K. Simulation of a fibrous composite reinforced with unidirectional orthotropic fibers, weakened by rectilinear cracks during longitudinal shear // Mechanics of composite materials and structures. October-December, 2017.- Vol. 23.- No. 4.- P. 579-591.
