BRIDGED CRACK APPROACH TO MODEL MATERIALS
SELF-HEALING
M. Perelmuter1
1 Institute for Problems in Mechanics of RAS, Pr. Vernadskogo 101-1, 119526, Moscow,
Russia - e-mail: perelm@ipmnet.ru
Keywords: self-healing, crack bridged model, stress intensity factors
ABSTRACT
The subject of this paper is the mathematical modelling of the crack self-healing process. For self-healing modeling of cracked structures the bridged crack approach is used. In the frames of this approach is assumed: 1) there is a crack in a structure after service loading; 2) there are bonds between surfaces of a crack (which are induced by a healing mechanism, the interface layer); 3) this layer is considered as a part of the crack with distributed nonlinear spring-like bonds between surfaces of crack (bridged zone). The bonds properties define the stresses at the crack bridged zone, its size and, hence, the fracture toughness of the material. The main goal of the modeling consists in the computational analysis of the bridging stresses distribution and in the computing of the stress intensity factors which are the main characteristics of self-healing efficiency. The mathematical background of the stress problem solution is based on the singular integral-differential equations method and the boundary elements method. Different self-healing methods (microcapsules filled with a self-healing agent, microvascular fibres, mendable polymers) with various mechanisms of self-healing are analyzed. The thermo-fluctuation kinetic Zurkov’s model is used to evaluate the regeneration and formation of the crack bridged zone. The healing time and efficiency are dependent on the chemical reaction rate of the healing agent, crack size and the external loads. The non-local fracture criterion is used to evaluate the fracture toughness and the critical external loading in the frames of the bridged crack model. The model can be use for the evaluation of composite materials healing and durability.
1. THEORY BACKGROUND
We use the bridged crack approach for self-healing modeling of cracked structures. It is assumed that due to a healing process the crack surfaces interact in some zone starting from the crack tips. As a general case the plane elasticity problem on a crack at the interface of two dissimilar joint half-planes is considered. The crack surface interaction exists in the bridged zones,
d
x
. As a simple mathematical model of the crack surfaces interaction we will assume that the linearly elastic bonds act through out the crack bridged zones.Denote by
( )
x
the stresses arising in the bonds2
( )
x
yy( )
x
i
xy( ) ,
x
i
1
, (1)where
yy( )
x
and
xy( )
x
are the normal and shear components of the bond stresses. The crack opening,u x
at
d
x
is determined as follows [1]1 2
( )
y( )
x( )
( ( )
yy( )
( )
xy( )),
bH
u x
u x
i u x
x
x
i
x
x
E
(2)where
u
y x,( )
x
are the projections of the crack opening on the coordinate axes, H is a linear scale related to the thickness of the intermediate layer adjacent to the interface,E
b is the effective elasticity modulus of the bond and 1,2 aredimensionless functions of the coordinate x.
By incorporating linearity of the problem one can represent the crack opening as follows
( )
o( )
b( ) ,
u x
u x
u x
(3)where
u x
0
,
u x
b
are the crack opening caused by the bond stresses
x
and external loads, respectively.By incorporating formulae (2)-(3) one can obtain a system of integral-differential equations relative to the bond stresses
( )
x
.Introduce the new variable,
s
x
/
, and differentiate the relations in (3) to obtain1
( )
yy( )
2( )
xy( )
b( )
b o( )
b,
H
s
s
s
s
u s E
u s E
s
(4)where the right side is the given function of the coordinate.
Taking into account the results given in [2] one can obtain the following formula for the derivative of the function
u x
b
[1]1 2 2 2 2 1 /
2 (1
) 1
1
( )
(1
) ( )
( )
( )
1
1
i b y x dA
s
t
u s
iA
s
sq t
itq t dt
s
t
s
s
(5) where1
( )
( ) ( ( )
( ))
,
1
i yy xy y xs
s
i
s
q s
iq s
s
(6)and parameters A, and depend on the elastic properties of the materials and bonds.
Finally, one can obtain the system of two singular integral-differential equations relative to the bond stresses
yy( )
x
and
xy( )
x
in the following form [1]1 1 /
( )
( , )
j( , ) ( )
( , ) ( )
( ), ,
1,2
ij ij j ij j i ddf s
T s
W s
f s
G s t f t dt
Z s
i j
ds
(7) ICSHM2013_________________________________________________________________________________ 139where
f s
j
are unknown function depending on bond stresses,T W G Z
ij,
ij,
ij,
i are the equation kernels and the parameter
depend on the materials and bond properties. For numerical solution of these equations we use a collocation scheme with piecewise quadratic approximation unknown of the bond stresses. See details in [1].Having the distribution of the bond stresses
yy( )
x
and
xy( )
x
over the bridged zone one can calculate the stress intensity factors (SIF)K , K
I II following to [3]
0lim 2
( )
( )
i,
yy xyK
iK
i
(8)where
yy( )
and
xy( )
are the stresses ahead the crack tip caused by the external loads and by the bonds stresses, represents the small distance to the crack tip.On the other hand, the SIF can be written as follows
int int
(
ext)
(
ext) ,
K
iK
K
K
i K
K
(9) where ext,I II
K
andK
I IIint, are the SIF caused by the external loads and the bond stresses.By incorporating the formula for the stress distribution ahead the interface crack tip under arbitrary loads [2] and using (8), (9) we obtain for the external tension load
0 1 0 2 1 /( ( )
( ))
2cosh(
)
(1 2 )
(2 )
1
y x i dq t
itq t
K
iK
i
dt
t
(10)The computation results for bonds stresses at the interface crack bridged zone and the crack opening over the crack for different combination of the materials parameters are presented. The dependencies of the SIF on bonds properties during the self-healing process were analyzed.
2. RESULTS of COMPUTATIONS
The above proposed approach was used for several problems analysis. A crack on the interface between different materials under the external tension
0 was considered. It was assumed that at the initial time instant (when the surfaces of a crack are free of constraints) some healing process is activated inside of a crack and bridges between the crack surfaces are built.The numerical calculations were performed for plane strain conditions and the following elastic constants of the joint materials and bonds (Cu-epoxy polymer):
1
25
E
GPa
,E
2
135
GPa
;E
b
E
2,
1
2
0.35
. The purpose of calculations is the dependence analysis of the self-healing process efficiency (the measure ofefficiency is the level of SIF at the crack tip) on the bridged zone length (the crack filling with bonds) and on the bonds stiffness.
In Fig. 1 for different values of the relative bonds stiffness (see eq.(4))
H
the dependencies of the SIF module (it can be obtained from eq. (10)) vs the relative bridged zone length are shown. For bonds with relative stiffness more than 10 the healing efficiency reaches the saturation if the crack has filled with bonds more than on the half of its length.The evolution of the healing process as the dependence of SIF module vs relative bond stiffness (in logarithmic scale) is shown in Fig. 2. Saturation of the healing effect is observed for bonds with rather big stiffness.
Figure 1: SIF module vs relative bridged zone length, K0 0
Figure 2: SIF module vs relative bond stiffness
,
K0 0Analysis of the healing process in finite size structures can be performed by boundary elements method [4]. Growth prediction of healed cracks can be performed on the basis of the bridged cracks growth criterion [5].
AKNOWLEDGEMENTS
This study was partially supported by Russian Foundation for Basic Research, research project No. 11-08-01243a
REFERENCES
[1] R.V. Goldstein, M.N. Perelmuter, Modeling of bonding at the interface crack, Int. J. Fracture (1999) 99 (1-2) 53-79.
[2] L.I.Slepjan, Crack Mechanics, Publ. “Sudostroenie”, Leningrad, 1981 (in Russian). [3] J.R. Rice, Elastic fracture mechanics concepts for interface cracks, Trans. ASME. J. Applied Mech. (1988) 55 98-103.
[4] M. Perelmuter, Boundary element analysis of structures with bridged interface cracks, Computational Mechanics (2012), P.1-12, DOI: 10.1007/s00466-012-0817-4 (online first paper).
[5] M. Perelmuter, A criterion for the growth of cracks with bonds in the end zone, Journal of Applied Mathematics and Mechanics (PMM) (2007) 71 137-153 (in English).
