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УДК 539.37
Коэффициенты интенсивности напряжений множественных наклонных трещин в слоистом композите при нагружении в плоскости
S. Li, J. Wang
Пекинский университет, Пекин, 100871, Китай
В слоистых композиционных материалах и многослойных конструкциях возможно зарождение множественных наклонных трещин. В статье рассмотрена задача распространения трещин в трехслойной конструкции, средний слой которой содержит множественные наклонные трещины в условиях растяжения и поперечного сдвига. Исследованы три типа конфигурации трещин: единичная трещина, периодическая система наклонных трещин одинаковой длины и две параллельные трещины различной длины. Взаимодействие между трещинами в условиях нагружения смешанного типа I—II и типа III изучено с использованием коэффициентов интенсивности напряжений. Полученные решения применены для армированных волокнами слоистых композитов. Результаты показывают, что на коэффициенты интенсивности напряжений для композитов с множественными трещинами существенное влияние оказывают ограничивающее воздействие внешних слоев и эффект экранирования между трещинами. В случае разноразмерных трещин главную роль в концентрации напряжений играет трещина большей длины. Показано влияние особенностей конструкции слоистого материала, распределения и ориентации трещин, а также их размера на концентрацию напряжений в вершинах наклонных трещин в трехслойном слоистом композите. Полученные результаты могут быть использованы при анализе распространения трещин в слоистых материалах.
Ключевые слова: наклонные трещины, взаимодействие, коэффициент интенсивности напряжений, нагружение смешанного типа I—II, нагружение типа III, слоистый композит с различной ориентацией волокон, преобразование Фурье
DOI 10.24411/1683-805X-2019-12003
The stress intensity factors of multiple inclined cracks in a composite laminate
subjected to in-plane loading
S. Li and J. Wang
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871, PR China Inclined multiple cracks may appear in composite laminates and sandwich structures. In this paper, we solve the fracture problem of a three-layer sandwich structure which contains multiple inclined cracks in the central layer under tension and antiplane shear. Three types of crack configurations are considered: an isolated crack, a periodic array of inclined cracks with the same length, and two parallel cracks with different lengths. In these cases, we examine the interaction among the cracks under mixed I—II mode and pure mode III based on the stress intensity factors. Then, we apply the solutions to fibre-reinforced composite laminates. The results show that the stress intensity factors of the multiple cracks are significantly affected by the constraining effect of the outer sublaminates and the shielding effect among cracks. For cracks with significantly different sizes, the long crack dominates the stress concentration. This work reveals the influences of the laminate configuration, crack distribution, crack orientation and crack size on the stress concentration at the tips of inclined cracks in the three-layer composite laminate, and the results may be used to analyze the crack propagation in the laminates.
Keywords: inclined cracks, interaction, stress intensity factor, mixed I—II mode, pure mode III, angle-ply composite laminate, Fourier transform
1. Introduction
Discrete multiple cracks may appear in layered structures under thermal stress or external loading. For example, multiple intralaminar matrix cracks are usually the first damage mode in composite laminates which are composed of
unidirectional fibre-reinforced laminas, and the cracks cause deterioration of integrity and stiffness of the laminates [15]. Generally, when a layered structure is subjected to inplane tensile loading, the emerging multiple cracks are usually perpendicular to the interfaces between the layers, such
© Li S., Wang J., 2019
as in the case of fibre-reinforced composite laminates. Thus, many studies have been devoted to the topic of multiple cracks that are perpendicular to the interfaces in laminated structures [6-13].
On the other hand, experimental results have also shown that inclined intralaminar cracks, which are not normal to the interfaces, may appear when a layered structure is subjected to out-of-plane loading such as transverse static loading, low velocity impact [14, 15], or to shear loading [16]. As a matter of fact, some researchers have solved fracture problems of interacting inclined cracks in layered structures. Zhang et al. [17] calculated the stress field and the energy release rate of multiple edge cracks in a surface coating layer bonded to a substrate in which the interface has waviness such that the otherwise perpendicular cracks become inclined to the interface. They show that interface waviness significantly affects the strain energy release rate, the stress distribution, and the crack propagation patterns. Petrova and Sadowski [18] solved the stress intensity factors for two inclined and interacting edge cracks in a functionally graded layer bonded to a homogeneous substrate, and analyzed the crack propagation, and recently, Petrova and Schmauder [19] investigated the case of multiple inclined and interacting edge cracks in the layered structure.
Previously, the authors have solved the stress intensity factor and examined the crack propagation behavior of a single inclined crack in the central layer of a three-layer composite laminate subjected to in-plane tension, and found that due to the constraining effects of the outer sublaminates, the crack propagation may be different from that of a perpendicular crack. Considering that multiple inclined cracks may emerge in composite laminates [14-16], and that the laminates with existing multiple cracks may be subjected to inplane loading, including in-plane tension and shear. In this paper, we solve the stress intensity factors at the tips of two and multiple periodic inclined cracks embedded in the central layer of a three-layer composite laminate under in-plane tension and in-plane shear. Then, we apply the results to analyze the behavior of multiple inclined intralaminar cracks in the 90° layer of a [±6^/90°^ angle-ply fibre-reinforced composite laminate. The major difference between the current problem and those mentioned above is the combined effect of the constraining effect of the outer sublaminates and the interaction among cracks on the inclined cracks. The problem is studied by the Fourier integral transform method combined with the superposition principle which has been used to solve multiple cracks [20, 21]. It is found that the stress intensity factors are strongly influenced by the crack spacing. The influence of the longer crack on the shorter one is great when two cracks, with significantly different sizes, are close enough. It is also shown that for the crack with tips close to the interface, the most dangerous state of the crack in the problem with multiple cracks may be different from that with an isolated crack due to the shielding effect.
2. Model
A three-layer composite laminate that contains multiple inclined intralaminar cracks in its central layer is shown in Fig. 1, a. We assume that the length of the cracks in the z-direction is much larger than the lengths of the cracks in the xy-plane such that the problem can be approximated by a two-dimensional one in the xy-plane. This assumption is reasonable for fibre-reinforced composite laminates, in which the fibres may be aligned with the z-axis. As mentioned above, multiple inclined cracks may be caused by transverse static loading, or low velocity impact. Then, the composite laminate may be subjected to in-plane (here, the "plane" means the plane of the laminate) loading, and the in-plane loading can be decomposed into in-plane tension/ compression along the x- and z-axes, and in-plane shear in the xz-plane. Therefore, we consider the cases when the laminate is subjected to tensile loading along the x-axis, and in-plane shear in the xz-plane.
We study three configurations of the inclined cracks: an isolated crack in Fig. 1, b, a periodic array of multiple inclined cracks in Fig. 1, c, and two parallel interacting cracks in Fig. 1, d. The central layer (region I) of thickness H is an isotropic material, which is characterized by two independent elastic constants, e.g. Young's modulus E and Poisson ratio v. The two outer sublaminates (regions II and III) of thickness h are an orthotropic material in the xy-plane with the elastic symmetry planes along the axes of the Cartesian coordinate system, and the stress-strain relations of the orthotropic material are
O — CiiE,
: + C12Eyy,
O yy — c21£ x
: + C22Eyy,
Txy c66^xy'
where c11, c12, c21(— c12), c66 are the elastic constants. The centers of the cracks are on the midline of the central layer. For the periodic array of cracks in Fig. 1, c, a local coordinate system (n, s) is set to define the orientations and positions of the cracks. Then the cracks occupy (k X cos y, a + k X sin y) to (kX cos y, b + kX sin y) in the local coordinate system (n, s), where k is the crack number from to a and b are the coordinates of the crack tips A and B on the s-axis, respectively. The relationship between the global coordinates (x, y) and the local coordinates (n, s) is
(n\ (a -pV x ^
(1)
s
V y
p
a
y
where a and P represent cosy and siny, respectively. For the two parallel cracks with different lengths shown in Fig. 1, d, two local coordinate systems (n1, s1) and (n2, s2) are introduced. The coordinates of the crack tips Ay and B1 are (0, a1) and (0, b1) in the local coordinate system (n1, s1), and those of A2 and B2 are (0, a2) and (0, b2) in the local coordinate system (n2, s2). The relationship between the global coordinate system (x, y) and the local coordinate system (n1, s1) is in the form of (1), and that between the two local coordinate systems is
H
II lb
I /
ni /7
H
n\
"2
h II l£
H - y /
h III \ X
y.
II [d_ S2/
I WBl yA\ té*2 /A /a2
III x
Fig. 1. 3D model of a composite laminate with inclined cracks (a); a cross section in the xy-plane with an isolated crack (b); an array of periodic cracks (c); two parallel interacting cracks (d)
+ À
(2)
The composite laminate is assumed to be infinitely long in the x- and z-direction, and the cracks are long enough in the z-direction, so that only a section in the xy-plane is considered. For the loading case of a unidirectional tension along the x-direction, the central layer undergoes a uniform tensile stress a in the x-direction such that the inclined cracks are under mixed I-II loading mode. When the composite laminate is subjected to in-plane shear loading, the central layer is under a uniform shear stress Tm along z-direction such that the inclined cracks are under mode III loading. In terms of the superposition principle,
one only needs to solve the corresponding perturbation problems, in which the faces of all cracks are subjected to an equal loading axx = -a and tzx = -tiii, respectively.
3. Solutions of displacement and stress fields under mixed I-II mode
For the above perturbation problem under mixed I-II mode, the solution of displacement and stress fields must satisfy the continuity conditions at the isotropic-orthotropic interfaces
a®} (x, H) = a® (x, H), t®} (x, H) = t® (x, H), u™ (x, H) = u® (x, H), u Jn) (x, H) = u® (x, H), a(JiI)(x, 0) = a® (x, 0), T<f (x, 0) = t® (x, 0),
u™(x, 0) = u® (x, 0), u(x,0) = u® (x, 0), where -^ < x < + and the free surface conditions o®> (x, H + h) = 0, t®> (x, H + h) = 0,
G®I}(x, - h) = 0, t^ (x, - h) = 0
r(ni)(
(4)
in the (x, y) coordinate system. In the above equations, ux and uy denote the displacement components in the x- and y-directions, respectively, and the superscripts (I), (II) and (III) represent the regions of the three-layer laminate. It should also satisfy the boundary conditions of the crack faces. For the problem with periodic cracks (Fig. 1, c), the boundary conditions in the (n, s) coordinate system are
a® (kXa, s) = -P(s), TnS (kXa, s) = -Q(s), (5) where a + kXP < s < b + kXP, k = ..., + and
P(5) = aa2, Q(s) = aaß.
(6)
(7)
For the problem with two cracks (Fig. 1, d), the boundary conditions in the corresponding local coordinates are
a® (0, ) = -P(S), T® (0, s, ) = -0(s),
ak < sk < h, k = 1,2
The solutions to the above boundary-value problem with multiple cracks are obtained from that of an isolated crack given by Li and Wang [22], consistent with the superposition procedure of Nied [21], and Kaw and Besterfield [9]. To this end, we will start with the solution for an isolated crack.
3.1. An isolated crack
The solution to the problem with a single crack [22] will be recapitulated here without details. Considering the isolated crack along the s-axis, the stresses a® (n, s) and a® (n, s) in the central layer I are
a£(n, s) = 1J
2 2 . . 3n + (t - s) ,..
(t- s), 2 , .2l2 f (t) +
+ n
-(t - s)2
[n2 + (t - s)2]2
[ ^ + ( t - s rr g ( t ) dt +
+ ± j [/^(Ç, n, s)eÇ(in-s^ 2Y + n 0
+ h^Ç, n, s)eÇ(in+s)eiï+12y + + h2(Ç, n, s)e~Çin+s'e>1+'2Y + + h4(Ç, n, s)e-Ç(m-s'e"Y_i2Y-- c2(-Ç)eÇ(m-s)e-y -C2(Ç)e"Ç(in+s)e" +
+ C4(-Ç)eÇ(in+s'^ + C4(Ç)e-Ç('n-s)^" ]^dÇ,
1 b T
(n, s) = - j
t®
n
n2 - (t - s)2 ^
n—^-'yt f (t ) -
[n2 + (t - s)2]2
(8)
, n2 - (t - s)2 _
- (t - s) 2 ;. g (t)
[n2 + (t - s)2]
dt +
(9)
1 _• +1 j [h(Ç, n, s)eÇ('n-s^2Y-n 0
- hj(Ç, n, s)eÇ(,n+s'eiï+'2Y-
- h2(Ç, n, s)e"Ç(,n+s)e'ï+'2Y +
+ h4(Ç, n, s'e"^-^2Y]•¿dÇ,
where i2 =-1 and
hi (Ç, n, s) = C (-Ç) + [-1 + Ç(-pn + as)]C (-Ç) h (Ç, n, s) = Ci (Ç) + [-1 + Ç(-pn + as)]C2 (Ç), h (Ç, n, s) = C3 (-Ç) + [1 + Ç(-pn + as)]C4(-Ç), hA (Ç, n, s) = C3 (Ç) + [1 + Ç(-pn + as)]C4 (Ç). The four coefficients Cj (Ç) (j = 1, ..., 4) can be expressed by functions f(t) and g(t) using the continuity conditions (3) and free surface conditions (4) [22].
3.2. Periodic cracks
Due to the translational symmetry, the solution to the problem with the periodic cracks can be obtained from that to the single crack. The perturbation of the stress fields for the crack along the s-axis is caused by multiple cracks, and it is obtained from the sum of contributions of the cracks located at kÀa (k = ..., -1,1,..., + in the (n, s) system. By setting n = -kÀa, s = -kÀP in Eqs. (8) and (9), and summing the contributions over k varying from to we get the stress fields for the crack face at k = 0 (a < s < b)
^£1(0, s )=1 j
n a
1
i-s
2
■f (t )+
( -1
2 + 2
k=- 0 k =1 )
V
f (t ) X
(r + k tan Y)(3k2 + (r + k tan y)2)
- g (t)
(k2 + (r + k tan y)2)2 Àa k (k2 - ( r + k tan y)2) (k2 + (r + k tan y)2)2 Àa
dt +
1 - • - • +1 j {[h (Ç, 0, s)e"^Y + h4 (Ç, 0, s)e^iY ]e-2y +
+ [h2(ç,0, s)e"çseiï + h3(ç,0, s)eçseï ]ei 2 Y- ct(-ç)e"^- cT(ç)e"çseiy +
+ c4(-ç)eçse% c4(ç)eçse"iy ]}x
2
1 + 2 2 cos(kÇÀ)
k =1
dÇ,
r®
1 b
(0, s) = - j
1
f-s
g (t )+
( -1
2 + 2
k=- 0 k =1 )
V
(10)
- f (t ) X
k (k 2 - ( r + k tan Y)2) -(k2 + (r + k tan y)2)2 Àa
(r + k tan Y)(k2 - (r + k tan Y)2)
- g (t)
(k2 + (r + k tan y)2)2 Àa
dt +
+ - j {[h, (Ç, 0, s)e"Çs^ + hA (Ç, 0, s)eÇs^Y ] e
- 2Y -
- [h2(Ç,0, s)e-ÇseiY + h3(Ç,0, s)eÇs^ ]é 2y }x
•Ç2
X— 2
1 + 2 2 cos(kÇÀ)
k=1
dÇ,
where
Àa
(11)
The integration and summation in Eqs. (10) turn to be convergent because of the following equation
1 J
k=1
= ( 1)m dm (-À + cn coth (cn/À)
= dC™l 2cÀ
(12)
where c is a complex number except for a purely imaginary number.
If the crack spacing X tends to infinity, Eq. (11) is
lim r = 0, and Eq. (12) tends to be
(13)
lim
à^+K
2 \Çme~cÇ cos(kÇÀ'dÇ
k=1 0
= lim
à^+K
+ lim
(-1)^^ Ml
dcm l 2cÀ
(-1)m
dm ( cn coth(cn/À)
dcm
2cÀ
=(-1) (_i 1+ dcm l 2c 1
+ (-1) "
dc"
— lim | — coth—
2c À^+^l À À
dm (-1 ^ dm ( 1 = (-1)^-^1 — |+ (-1)m—I — 1 = 0,
dcm \2c) dcm [ 2c 1
(14)
where
lim
X^+c
cn . I cn —coth I — X I X
= lim
X^+c
(e
pnix + e-cnA
) cn/X
cnfX _ e_crc/X
= lim
X^+c
[(e'
crc/X + e~cn/X
) cn/X]
(ecn/X _ e"cn/X)'
cn/X (ecn/X _ e"cn/X) + ecn/X + e_ cn/X ,
im —-—--*--—77-= 1.
= lim -n-tt
X^+c ecn'X + ecn'X
= 1. (15)
Based on Eqs. (13) and (14), Eqs. (10) can degenerate into (8), that is, the solution to the periodic cracks can degenerate into that to an isolated crack.
Substituting (10) into the boundary conditions (5), we get two singular integral equations
JM dt+ja (s, t) f (t) dt+jl (s, t) g (t) dt =
. s = -TCP(s),
J ^ dt + J Lg (s, t) f (t) dt + JL4 (s, t)g (t) dt =
(16)
f_ s
= _nQ( s),
where a < s < b. In order to ensure the continuity of the material, we must impose two conditions [20]
Jf(t)dt = 0, Jg(t)dt = 0. (17)
a a
Then, numerical calculations are needed to solve the two singular integral equations (16) and (17) to obtain f(t) and g(t). Once f(t) and g(t) are determined, the stresses can be obtained.
3.3. Two parallel cracks
For the two cracks with different sizes (Fig. 1, d), there is no symmetry, and then the crack surface conditions must be considered separately. The solutions of the two cracks contain two parts: one is the solution of the isolated crack; the other is the effect of the other crack. These two parts can be obtained from Eqs. (8) and (2). Thus the stress fields for the left crack are
(0, s1) = n! f+ r+f & 0, si )x
'1 _ s1
xe^V hlA(l,0, s!)eSs1e_ii ]e"'2Y +
+ [h12 (S, 0, s )e-Sse + /13 (S, 0, s )eSs>eii ]x
xe'2Y _C12(_S)e~Ss1e^ _ C2 (S)e"Ss1e'T +
+ Cu(_S)eSse + q4(S)eSs'^} SfdS +
1 b2 n a
(r2 + tan y)(3 + (r2 + tan y )2)
(1 + (r2 + tan y)2)2 Xa
f2(t2) _
1 _ (r2 + ^ g 2(t2)
dt2 +
(1 + (r2 + tan y)2)2 Xa
1 +c
+ - J {[/21 (S, _ aX, s _ pX)e-'2y _ C^ (_S)]x n 0
x e"Ssle_iY-'SX + [h22(S, _aX, s1 _pX)e' 2y _ _ C22(S)]e"Ss1eiY+,SX + [MS, _aX, s _pX) x xe'2y + C24(_S)]eSsleiY-'SX + [h24(S, _aX, s1 _
_ pX)e"'2 y + C24(S)]eSs1e_'Y+,SX} and
(0, s) =1J Ml dt1 +1 +[ [h 1 (S, 0,3)x
(18)
r(I) «1s1
n a t1 _ s1
xe-Ssle"Y-'2y + h4(S, 0, s)eSsle_Y-,2y _ h,2 (S, 0, s )x
xe"Ss1e'Y+'2y _ h3(S, 0, s )eSs1e'Y+'2y +
1 b2
+ ^J n a
1 _ (r2 +tan y)2 f ( )
x2x2. f 2 (t2 )
(1 + (r2 + tan y)2)2 xa"
dt2 +
1 _ (r2 + tan y) , s
_ 2 + tan Y),, . -g2 (h)
(1 + (r2 + tan y)2)2 Xa
+1 J [h21 (S, _ aX, s1 _ pX) e~Ssle_Y-'SX-'2y _ n 0
_ h22 (S, _ aX, s1 _ pX )e~Ss1e'Y+'SX+'2 Y _ _ h23 (S, _ aX, s1 _ pX )eSsleiY-'SX+' 2y +
+ h24(S, _aX, s _pX)eSsle_Y+'SX-'2Y]'2-dS, where a1 < s1 < b1, and the parameter r2 is defined as
(19)
r2 =
f2 s1
a
The other functions are
hm1 (S, n, s) = Cm1 (_S) + [_1 + S(_pn + as)]Cm2 (_S), hm2 (S, n, s) = Cm1 (S) + [_1 + S(_pn + as)]Cm2 (S), hm3 (S, n, s) = Cm3 (_S) + [_1 + S(_pn + as)]Cm4(_S), hm4 (S, n, s) = Cm3 (S) + [_1 + S(_pn + Os)]^S),
h
C1(S) = B _1(S) J G(S, t1)dt1,
(20)
(21)
C2(S) = B_1(S) J G(S, t2)dt
2
where m = 1,2.
The stress fields for the right crack are
tO)
(0, s2) =1J f2~~2—-dt2 +1 +[ {[h21 (S, 0, s2 )x
n a t2 s2
xe_Sie + h,4(S, 0, s2 )eSie '' ]e_2Y + fe (S, 0, s, )x
£s2e-'< ]e-i2y _
xe"^ + Mi 0, ]e2Y - C,2 (-5)e"^ -- C22&e-i2e"' + C24(-Z)ei2e'" + C24(i) e^'} x
x^-di + - f
2S n i
(r - tan у )(3 + (r - tan y) )
(1 + (r - tan y)2)2 Aa
/¡ft) +
1 - (r - tan y) (1 + (r - tan y)2)2 Aa
gj(h)
dt1 +
1
+ - ff {/ (i, aA, S2 + PA) e-2y - (-i)]x п 0
x + [h12 (i, aA, s2 + pA) e' 2y -
- C12 (i)] e-^'-'^ + [/1з (i, aA, S2 + pA) ef 2y + + C14 (-i)] e^-^ + / (i, aA, S2 + PA) e-' 2y +
+ CM©] e^'-'^} Jldi
(22)
and
Л)
(0, S2) = - J g^d?2 +17 / (i,0, S2 )x
П a, f2 - S2
xe-is2e-iY-'2Y + h24(i, 0, S2 )eiie^1-i2Y - /22 (i, 0, s )x
2y - /23 (i, 0, s2 )ei2eiY+'2Y di +
x e 1 b + Ь
П a
1 -(r1 -tan y ) f ) -(1 + (r - tan y)2)2 Aa 1 1
- (r - tan Y )(1 1:(ri"tan)Y))2 A *«)
(1 + (r - tan y)2)2 Aa
dt +
+ — J [h11(i, aA, s2 +pA) e
-is^+iiA-i 2y
- /2 (i aA, s2 + pA)e^'-'^' 2y -
- h13(i, aA, s2 +pA) e^'-'^' 2y +
+ /14 (i, aA, S2 + pA^"'-^-' 2y ] '¿di, (23)
where a2 < s2 < b2, and
_ - s2
aA
(24)
The convergence of Eqs. (18), (19), (22) and (23) are the same as (12). Based on the analysis of Eq. (14), the solution to this problem can also degenerate into that to an isolated crack when A ^ +
Substituting (18), (19), (22) and (23) into the boundary conditions (7) of the crack faces, we obtain the singular integral equations
¡MA at, +1 jLk- (s,, t) f (t )dt +
at tk sk a
+ 2 J L2U (sk , t' )g (t )dt =-nP(Sk ),
bk \ 2 b
J gk-k) dtk + 2 J L3k- (Sk, ') / (t) dt + (25)
ak k Sk ' -1 at
2 b
+ 2 J L4k (Sk , t' )g (t )dt- =-nQ(Sk), '' =1
1 ii
where ak < sk < bk, k = 1, 2. The continuity conditions are
h bk
(26)
J /k(tk)&k = 0, J gk (tk )dtk = 0.
4. Solutions of displacement and stress fields under mode III
When the laminate is subjected to shear loading in its plane where the applied nonzero shear stresses are Txz = Tzx, the cracks are under mode III loading. In order to solve the stress intensity factors at the crack tips, we only need to solve the perturbation problems where the crack surfaces are subjected to shear stresses. For the prob-lemwith an isolated crack (Fig. 1, b) or periodic cracks (Fig. 1, c), the boundary conditions in the (n, s) system are T®(kAa, s) = - R(s), (27)
where R(s) = тш cos ', a + kAP < s < b + kAP, k = -<», ..., For the problem with two cracks (Fig. 1, d), the boundary conditions in the corresponding local coordinates are т®(0, sk) = -R(s), ak < Sk < bk, k = 1,2. (28)
For mode III, the displacement of the material in the z-direction is denoted by w(x, y). The displacement and stress fields must satisfy the continuity conditions at the two interfaces
w(II) (x, H) = w(I) (x, H), т® (x, H) = т®(*,H),
(29)
w(III) (x, 0) = w(I) (x,0), т™ (x,0) = т® (x, 0), where < x < -+», and the free surface conditions
КЦЧx, H + h) = 0,
Kftx, - h) = 0
(30)
in the (x, y) coordinate system.
In region I, the basic solutions of the stress and displacement are constructed from two parts: the solutions for an uncracked finite strip, and that for a cracked infinite region
w(I) (x, y) = w(I) (x, y ) + w2I) (x, y ), (31)
where the subscripts 1 and 2 refer to the solution of the uncracked strip and that of the cracked infinite region, respectively. Due to the isotropy of region I, the shear stresses corresponding to w^ are
T(I) =,, dw[I)( ^ y) t _llдWL)(x, y)
dy '
yx(1)
Lzx (1)
dx
(32)
where , denotes the shear modulus of the isotropic mate rial. The equilibrium equation is
a2 w(I)( x, y) + a2 w(I)(x, y)
'=1
dx
dy 2
= 0.
(33)
We solve (33) by taking the Fourier transform with respect to x
W(I) (S, y) = F[W1(I) (x, y)] = J wf (x, y)eSdx. (34)
Then the above equilibrium equation (33) can be written as
d2 wf(S, y)
dy2
-S2 wf^S, y ) = 0.
The general solution of (35) is
W1(I)(S, y) = 4(S )e_Sy + A2(S )elSy, (36)
and thus the displacement in the z-direction is 1
w1(I)(x,y) = -L J [A(S)e_|S|y + A(S)eSy]e"''S(37) Now, we consider the solution
for an isotropic infinite region with a crack shown in Fig. 2. We obtain the solution following the procedure of Krenk [20] that was for mode I and II crack problems in a strip of finite size. There is a continuity condition x^) (0+, s) = x^) (0_, s) of the crack face. Combining the continuity condition of the crack face and (36), the displacement
wf in
the local
coordinate system (n, s) after Fourier transform with respect to s is
w2i)(n, n) = sgnn G(nKN|, (38)
(i)
TSP(2)(n, s) is an odd function of n, and xs'j2)(0+ , s) -
Tsz(2)(0 , s) can be written as
^(2)(0+, S)-^(2)(0", S) = M
(i)
aw2'}(0+, s) ds
dw2')(0-, s ) ' ds
= M J -inG(n)e~insdn.
n-L
(39)
Then the so-called dislocation distribution function L(s) is introduced
L( s) = — [ w£>(0+, s ) - w2')(0-, s )], os
r(')
so ts_(2) is reproduced as
sz (2) tsz(2)
= M
(0+, s )-xsiyo-, s ) = 9w2')(0+, s) 9w2')(0-, s )'
ds
ds
= pL(s).
(40)
(41)
Fig. 2. The model of an infinite plane with an isolated crack
By (39) and (41), the undetermined function G(n) is expressed as
G( n) J L(t )eint dt.
2 n a
(42)
Substituting (42) into (38), taking inverse transform wp (n, n) and using the finite part of a integral method
(35) [23], the displacement in the z-direction is obtained
w
(')
I +L
(n, s) = — J w2')(n, n)e-nsdn =
2 n
1 ? r
= — I arctani
2 n{ 1
s -1
L(t )dt.
(43)
Substituting (37) and (43) into Eq. (31) gives the displacement of region I
w
(')
1 +L
( xy )=I
J (-ie
. -|S(y-at)|
sgn ( y -a t )/(2£, )-
+ yô(S)) eiStßL (t)dt + (A (S)e~IS|y +
-iSiy.
+ A2( S)elS| y )
(44)
The stress field in region I is expressed as
^J sgn S
-|S(y-at)|,
1 +L
^ x. y ) = ¿j
x eiStß L(t )dt + M | S | ( - A (S )eHS|y + + A2(S)eIS|y )
x2?( n, s ) = fj
iSx dS,
t - s
(45)
+ {[A (-S)e
2n a (t - sY+n
_S)e_iY-S(-ßn+as) _
-L(t )dt +
2 n
+ A2 (-S)eiY+S(-ßn+as) ] eiS(an+ßs) -- [ A1 (S )eiY-S(-ßn+as ) + A2 (S )e"iY+S(-ßn+as ) ] x
x e
- iS( an+ß s )
}iSdS.
T('') = c .
(46)
In region II, the constitutive relations of the orthotropic material are
dw(II)(x, j) _(II) = dw(II)(x, j) 3 , Tzx c55 3 ,
dj dx
where c44 and c55 are the two shear moduli. The equilibrium equation of the orthotropic material is expressed by displacement
d 2 w(II)( x, j ) d 2 w(II)(x, j )
55
- + c
= o.
(47)
dx2 44 9y2
By taking the Fourier transform with respect to x, Eq. (47) is rewritten as
d2W(II)(S, y) C55 „2-™,
(48)
Ç2w(II'(Ç, j) = 0.
9j c44
The general solution of (48) is
w(11) (Ç, j ) = A3 (Ç )e " j -h ' + A4 (Ç )e^|Ç|( j -h ', (49)
+L
where p = c55/c44. Thus the displacement in the z-direc-tion of region II is
w
,(n)
1
(x, y) =J-J [A3tt)e~p|l|(y-H> + 2n:
+ A4(l)ep|l|( y-H V'^dl. (50)
Substituting (50) into (46), the stress is obtained as
T yz
1
(x y) = — J c44P\ l|x
x[-A3©e"pK|(y-H> + ^)e^|(y-H V^ (51) The corresponding displacement and stress components of orthotropic region III are
1 +<*>
<,(in)
(x, y) = J [A5(l)e"p|ly +
+ A6(l)ep|l|y ]e~ ilxdl,
T(iii)
(x y) = — J c44P\ l|x
(52)
x[-A5 ©e"my + Ag ©epK|y In order to express the above unknown functions Ai (£) (i = 1, ..., 6) by L(i), substituting equations (44), (45), (50)-(52) into Eqs. (29), (30), a system of equations are obtained in the following form:
[M kg,
[ A]«*, =
-[G]
6x1,
(53)
where [A] represents the undetermined functions Ai (l) (i = = 1, ..., 6), and [M], [G] and A are shown in Appendix A in detail. Up to now, all of the stresses and displacements are expressed by the unknown function L(t).
Substituting TZn (n, s) in Eq. (45) into the boundary condition (27), we get one singular integral equation
t - s
- + K (s, t)
L(t )dt = -nR( s),
(54)
where a < s < b. In order to ensure the continuity of the material, one must impose the condition [20]
J L(t )dt = 0.
Then, numerical calculations are needed to solve the above singular integral equations (54) and (55) to obtain L(i). Once L(t) is determined, the stresses can be obtained.
4.1. Periodic cracks
For the model with periodic inclined cracks in the central layer (Fig. 1, c), the corresponding solution is obtained from that of a single crack mentioned above, consistent with the superposition procedure shown in Sect. 3.2. Then the shear stress on the crack face at k = 0 (a < s < b) is obtained
t2(0, s) = fj 2n „
-1
2+2
1
t - (s - kXP)
k=— k=i I (t - (s - kXP))2 + (-kXa)2
xL(t)dt + J [A,(+
2n
+ A2K)eiY+^eiï - A,©^-^ -
- A2©e-iY+^ ]
1 + 2 2 cos (kIX)
k=1
il dl.
(56)
Substituting Eq. (53) into Eq. (56), the stress T® (0, s) at the crack face is rewritten as
1 b M
t®(0, s) =-J 2 n 2
1
t - (s - kXP)
-1
2+2
k=— k=1J (t - (s - kXP))2 + (-kXa)2 1 b..
xL(t)dt + i-J J -M{[(M12 -Mn)e"l(H-(t-s)a) +
n a 0 2A
+ (M13 + M14 )e"l(t+s)a ] cos [l(s -1 )P - Y]+ + [(M22 - M21)e"l(H-(t+s)a) + + (M23 + M24 )e"l(t -s )a ] cos [l(s -1 )P + Y ]} x
1 + 2 2 cos (kIX)
k=1
dl dt.
(57)
The convergence of (57) is the same as (12). Based on the analysis of Eq. (14), the solution to this problem can also degenerate into that to an isolated crack when A ^ -+». Substituting (57) into the boundary condition (27), one singular integral equation is obtained, which is the same type as (54), but has a different expression.
4.2. Two parallel cracks
As in the analysis in Sect. 3.3, the stress field is the superposition of the effects of the two cracks. Thus, the stress for the left crack in the local coordinate system (n(, s() and the stress for the right crack in (n2, s2) are
rWr
(0, s1) L1(t1)dt1 +
b1
2n a t1 - s1
b1
(55) + J J {[(M12 -Mn)e"l(H-(^ +
a1 0
+ (M13 + MM)e"l(t1 +s1)a ]cos(l (s1 -1 )P-Y ) + + [(M22 - M21)e"l(H-t1+s1>a) + + (M23 + M24)e"l(t1 -s1)a ]cos(l (s1 - t1)p + Y )}x
x Mild idt1 +^J
t2 - (s1-pX)
2n a>2 - (s1 -PX))2 + (-aX)2
M b2
xL2(t2)dt2 + ^J J {[(M12 -Mn)e"l(H-(t2-s1)a) +
2n a2 0
+ (M13 + M14 )e"l(t2 +s1)a ]cos (l(s1 -12 )P - Y)e",lX + + [(M22 - M21 )e"l(H-t2 +s1)a) + (M23 + M24 ) x
xe
-l(t2-s1)a]cos(l(s1 -12)P + Y)eilX}L(t2)dldt2. (58)
A
x
r®
(0, s2) = L1(t1)dt1 +
2n a t2 - s2
+ T^j j {[(M12 -M„)e"Ç(H-(t2-s2)a) +
2n a2 0
+ (M13 + M14)e"Ç(tT +s2)a]cos (Ç(s2 - t2)p + Y)+ + [(M22 - M21 )e"Ç(H-(t2+s2)a) + + (M23 + M24)e"Ç(tT]cos(Ç(s2 -12 )p + Y)}x
XLMdÇdt2 + JN. j_* - (s2 + pÀ' 2 X
A 2 2nai(t1 + (s2 -pÀ))2 + (aÀ)2
N b1
XL1(t1)dt1 + j {[(M12 -Mu )e-Ç(H-(t1 -s2)a) + 2n
a1 0
+ (M13 + M14 )e"Ç(t1 +s2)a ]cos (Ç(s2 - Op - Y)eiÇÀ + + [(M22 - M21)e"Ç(H-(t1+s2)a) + + (M23 + M24)e"Ç(t1 -s2'a]cos(Ç(s2 -t1)p +Y)e_iÇÀ}x
(59)
X Ml^d Çdt1. A 1
Substituting (58) and (59) into the boundary conditions (28) of the crack faces, we obtain the singular integral equations
N JL— dtk +N 2)KM ( sk, t ) Li ft )dt =
2 attk - sk 2 ,=1 at
= -nR(sk ), (60)
where ak < sk < bk, k = 1,2. The continuity conditions are
j Lk (tk )dtk = 0.
(61)
5. Stress intensity factors
We apply the quadrature method in the previous work [20, 22] to solve the above integral equations (16), (25), (54) and (60). We introduce variables Tk and Zk to normalize the intervals of integrations in these equations:
lk 1 = bk - ak ( Tk | + bk + ak
(62)
l sk ) 2 l Zk -1 <Tk, Ck <1.
We then define functions fk (tk ), gk (tk ), and Lk (tk ) as follows:
Fk (Tk )
fk (tk ) = ¥fk (Tk ) =
1
-Tk
gk (tk ) = ¥gk (Tk ) =
Gk (Tk)
Lk (tk ) =¥l (Tk ) =
Hk (Tk )
r-
(63)
Tk2
where k is the crack number. k = 0 represents the solution to the single crack or the periodic cracks because of the
same form of the corresponding integral equation, and k = = 1, 2 represent that to the two cracks. Following the quadrature method [20, 22], Eqs. (16), (17), (25), (26), (54), (55), (60), (61) are discretized into the systems of linear equations. Here, we only take the model with periodic cracks as an example, and the corresponding systems of linear equations for cracks in mixed I—II mode are as follows:
N
- NP(Zi ) = £
j=1
1 b - a T .
+ ; LXZi, t j )
t , -Z
j ™ b - a N
X F (t j ) + b-a £ Lt(Z, , T j )G(t j ),
2 j=1
b - a N
- NQ(Zi ) = b-a £ L5(ci- , t j ) F (t j ) +
2 j=1
(64)
N
+ £
j=1
1
b - a
i_tj -z,
L4(Z, , Tj )
G(t , ),
N N
£ F(Tj ) = 0, £ G(tj ) = 0 j=1 j=1 and those in pure mode III are
2 N
- N - R(Zi ) = £
N j=1
N
£ h (t j ) = 0, j=1 where
2 j-1,
1
_Tj -Z,
+ k (z, , T j )
H(T,),
(65)
Tj = cos
2N
n |, j = 1,2,..., N,
(66)
Z' = cos|mnj, i = 1,2,..., N_ 1,
Ty and Zi satisfy the following equations:
TN (Tj) = 0, UN—1 (Zi) = 0, (67)
where TN and UN_1 are the Chebyshev polynomials of the first kind and the second kind, respectively [24]. The numerical solutions of functions Ld (Z t, t j ) (d = 1, ..., 4) are obtained by the Gauss-Laguerre numerical integration formulas for S.
For the inclined cracks (y ^ 0), the tips are under mixed I-II loading mode when the laminate is subjected to tensile loading along the x-direction, and under pure mode III when the laminate is subjected to shear loading in the xz-plane. The mode I, II and III stress intensity factors at crack tips (0, a) and (0, b) in the corresponding local coordinate systems are
Ki(a) = F(_1)a.-n, Ki(b) = _F(1)a.n,
Kn(a) = G(-1)aJb-an, K11 (b) = -G(1)aJ^n, (68)
K m (a) = H (-1)Tiii where
b - a
n, K m (b) = H (1)Tn
b -a
(69)
F (1) =—x N
x% sin [(2 N - 1)/(4N )(2j - 1)n ] % sin [(2N -1)/(4N)n] jh
F (-1) = — x N
x% sin [(2 N - 1)/(4N )(2j - 1)n ] % sin [(2 N -1)/(4 N)n] F (TN+1-j)'
G (1) = — x N
x% sin [(2 N - 1)/(4N )(2j - 1)n ] (T ) % sin [(2N -1)/(4N)n] j
G (-1) = — x N
% sin [(2 N -1)/(4 N )(2j - 1)n] G X%1 sin [(2N -1)/(4N)n] G(TN+1-j)'
H (1) = *-1 x 2N
x% sin [(2 N - 1)/(4N )(2j - 1)n ] % sin [(2N -1)/(4N)n] ih
H (-1) = H 1 x 2N
x% sin [(2 N - 1)/(4N )(2j - 1)n ]
x%1 sin [(2 N -1)/(4 N)n] H(TN+>-j)-
6. Application to a fibre-reinforced composite laminate
In this section, we apply the above analysis of the stress intensity factors to a [±0n2/9O°n]s angle-ply composite laminate. The stacking sequence of this laminate is the same as that in the papers of Wang and Karihaloo [11], and Li and Wang [22]. It is composed of continuous fibre-reinforced unidirectional laminas, and the material properties of unidirectional laminas are listed in Table 1. The AS4/ 3501-6 graphite/epoxy (fibre/matrix) composite is used in all the following computations. From these material properties, the elastic constants of the central layer and the outer sublaminates can be calculated [11]. In the following numerical analysis, we introduce a length parameter l in the corresponding local coordinate system (n, s) to represent the crack size
l = b - a. (70)
6.1. Results of the cracks in mixed I—II mode
The analysis of the isolated crack in mixed I-II mode has been shown in the paper of Li and Wang [22], and only the results for the periodic cracks and two parallel cracks with different lengths are analyzed here.
Firstly, we examine the effect of the geometrical parameters and crack spacing on the stress intensity factors
Table 1
Material properties [25]
Material AS4/3501-6
Longitudinal modulus EL, GPa 126
Transverse modulus ET, GPa 11
In-plane shear modulus GLT, GPa 6.6
Out-of-plane shear modulus GTT, GPa 3.9
In-plane Poisson ratio v LT 0.28
Out-of-plane Poisson ratio v TT 0.4
for the case of multiple periodic cracks. The influences of the crack spacing X and orientation y on the normalized stress intensity factors k/(a^/rcl/2) are shown in Fig. 3, when the outer sublaminates have the strongest constraining effect (0 = 0°). In Fig. 3, K J (aV^) decreases mono-tonically with increasing y, and K„/(a/rc^I) increases first and then decreases with the increase of y. Due to the shielding effect between cracks, the value of y increases when Kn/ (a^/rcl/ 2) reaches the maximum value.
Secondly, the interaction between two parallel cracks with different sizes is analyzed, when the ply angle is 0 = = 0°; the crack orientation is y = 45°; the thickness of the outer constraining sublaminates is h = H; and the lengths of the two cracks are lx = 0.8H and l2 = 0.1H. The effect of the crack spacing on the normalized stress intensity factors is shown in Fig. 4. For both of the two cracks, the stress intensity factors of the two modes, Kj and Kn, tend to be stable with the increase of X; and their values tend to those of the isolated crack when X is larger than three times the thickness H of the central layer. For the shorter crack of the two, Kj and Kjj decrease generally with the decreasing X because of the stronger shielding effect between the two cracks. However, for the longer crack, Kj and Kn decrease with the decrease of X first due to the strong shield-
Fig. 3. Influence of crack orientation y and crack spacing X on the normalized stress intensity factor l = 0.8H and 0 = 0° (color online)
Kl/(a(nlk/2)1/2)
1.0-
* Point A1 Point B1 Point A2 Point B2
n
0 2 K„/(ö(TC/k/2)1/2)
6 yj H
0.6-
0.4
0.2
Point A1 Point B1 Point A2 Point B2
n
0 2 4 6 À/ H
Fig. 4. Influence of crack spacing À on normalized stress intensity factors when 0 = 0°, Y = 45°, h = H, l1 = 0.8H, l2 = 0.1H, KJ(^nlkl2) (a), KJfrJnlJ2) (b) (color online)
ing effect, and then increase with further decreasing À because the longer crack dominates the stress concentration.
6.2. Results of the cracks in mode III
Firstly, the results for one inclined crack are shown in Figs. 5, 6. The normalized stress intensity factors, as functions of h and Y, are presented in Fig. 5. 0 = 0° is used here. For a fixed crack size l, the normalized stress intensity factors decrease monotonically with h, and tend to be stable when h is larger than twice of the thickness of the central layer. For different Y in Fig. 5, the values of the normalized
Fig. 6. Influence of crack orientation y and ply angle 0 on the normalized stress intensity factor Km/ (Tm = 1, l = 0.8H, y = 0° (1), 45° (2), 60° (3)
stress intensity factors vary more for y = 0° than that for other value of Y within the same interval of h, which means that the constraining effect of the outer sublaminates on the mode III crack is more significant for the one with the larger size.
The influences of crack orientation y and ply angle 0 on the normalized stress intensity factor KIII/(TIII when h/H = 1 are shown in Fig. 6. K In / ( t InVnV2) decreases with the increase of Y, which means that the constraining effect of the outer sublaminates on the crack becomes weak as the crack tips are away from interfaces. It also decreases with the increase of 0 first until 0 = 45°, and then increases with further increase of 0. Because the effective shear modulus Gxz reaches the maximum value at 0 = 45°, the
minimum value of Km/(Tm appears
at 0 = 45°. It means that this state has the strongest constraining effect on the mode III crack propagation.
Secondly, the constraining effect of the outer sublaminates and the shielding effect between the cracks for the periodic cracks are shown in Figs. 7, 8. For the fixed 0 = 0° and Y = 45°, the influences of the thickness of the outer
Fig. 5. Influence of crack orientation y and thickness of the outer sublaminates h on the normalized stress intensity factor Km/(x rnVrcV^) when 0 = 0°, l = 0.8H, y = 0° (1), 45° (2), 60° (3)
Fig. 7. Influence of thickness h and crack spacing X on the normalized stress intensity factor K m/(xm sjn^l) when 0 = 0° and Y = 45°, hlH = 0.5 (1), 1.0 (2), 2.0 (3)
Km/(Tin(rc/*/2)1/2)
Fig. 8. Influence of ply angle 0 and crack spacing X on the normalized stress intensity factor Km/(tm and Y = 45°, 0 = 0° (1), 45° (2), 90° (3)
0.6-
0.4
0.2
Point A1 Point B1 Point A2 Point B2
1 2 X/H
Fig. 10. Influence of crack spacing X on normalized stress intensity factors Km/() when 0 = 0°, Y = 45°, h = H, l1 = = 0.8H, l2 = 0.1H (color online)
sublaminates and the shielding effect between the cracks are shown in Fig. 7. Due to the joint effects, K m/ ( t m increases first and then decreases as the crack spacing decreases. The weaker the constraining effect of the outer sublaminates is, the smaller the value of X becomes, when Khi /(ThiVnl/ 2) reaches the maximum value. For the fixed h = 0.5H and Y = 45°, the influences of the ply angle 0 of the outer sublaminates and shielding effect between cracks are shown in Fig. 8. The values of X when K m/ ( tiii reaches the maximum value are almost the same for different 0. The influences of the crack orientation and shielding effect of the cracks are shown in Fig. 9. h = H, l = 0.8H and 0 = 0° are used here. For a fixed crack size, Km/ ( tihV nl/ 2) decrease with the increase of crack orientation y.
Thirdly, the effect of the crack spacing on Km/(tiii x Xyjnl/ 2) for the two parallel cracks is shown in Fig. 10. For the shorter crack of the two, Km/(tmVnl/2) at the two tips A2 and B2 decrease with decreasing X, especially when X is smaller than the thickness H of the central layer. However Km/(tiii
for the longer crack do not change significantly as X decreases.
Fig. 9. Influence of crack orientation y and crack spacing X on the normalized stress intensity factor Km/ ( tiii when h = H,
l= 0.8H and 0 = 0°
7. Conclusions
In summary, we solved the fracture problem of a three-layer composite laminate with multiple inclined cracks embedded in the central layer when the laminate undergoes a unidirectional tensile stress and in-plane shear stress. Three kinds of crack configurations, namely, an isolated crack, an infinite array of periodic cracks and two parallel cracks with different sizes, are examined. For all cases, we calculated the stress intensity factors at the crack tips. The stress intensity factors are all strongly affected by the orientation, interaction between the cracks, and the constraining effect of the outer sublaminates. For two cracks with significantly different sizes, the influence of crack spacing on the shorter one is more significant than that on the longer one. Therefore, this work reveals how the laminate configuration, crack distribution, crack orientation and crack size influence the stress concentration at the inclined crack tips in the three-layer composite laminate, and the results may be used to analyze the crack propagation in composite laminates.
Acknowledgments
The authors thank the support of the National Natural Science Foundation of China (grants Nos. 11232001 and 11521202), and Professor Bhushan L. Karihaloo of Cardiff University for helpful discussion.
Appendix A. The expressions of coefficients A(£) (ii = 1, 6) in Sect. 4
Substituting Eqs. (45)-(52) into Eqs. (29), (30) and then taking Fourier transform of the system of equations with respect to x, we obtain the expressions of Ai(£) (i = 1, ..., 6) in the following form:
[IT*]
t[G]6x, =
[ALX1 = m-u G =[F ]6x6
|[F ]
I 6x6
[M ]
16x6
A
[G ]
I 6x1 '
(A1)
where
G(I) =
2_ e-|( -at) + y8(|) J )dt
a 21
a | 2_ «-I«+y8(^) ) e^^ )dt
a 2|
(A2)
F (I) =
-e
-e1
№
-elH c44p c44 p 0 0
M M
-1 -1 0 0 1 1
1 -1 0 0 c44p c44 p
M M
0 0 -e" pj I jh ep|||h 0 0
0 0 0 0 -eplI|h e-p II jh
(A3)
A is the determinant of [F], M^ are the elements of adjoint matrix [F ], and the used elements are as follows:
A=f M+c44p (el||( H+2 ph) -e-\!\(H+2 ph)) +
| M
+ 2 ^ - cIP - ¿VP) +
M2
+ | M*— c44p ]" (elI!( H-2 ph) - e-|||( H-2 ph)),
M„ =
c44p | c44p
M
M
e II|2 ph +
M
c44p ] + c44p
M
J|2ph - 2 ( c44p
M
M12 =| C-Mp -1 I e12 ph -I C-f- +1 I e"lIj2 ph - 2,
M13 =-
c44p ] + c44p
J|j(H+2 ph) .
c44p ) c44p
M
M
eI( ff-2 ph) + 21 c44p | ,
M
M14 +1| eI( +2 ph)-
c44p -1 I eI(H-2 ph) + 2e|ff
M 21 =-
c44p | + c44p M
JI2ph .
c44p | c44p
M
M
-|2ph + 21 c44p
M
M22 = -| ^ + 1 Ie12ph +I ^-1 IeHIj2ph-2,
M 23 =
c44p | c44p
M
M
el|j( -H+2 ph) +
M
c44p | + c44p
M
eI( -H-2 ph) - 21 c44p | ,
M J
M
= c44p -1 I eI( - H+2 ph) +
24
+ | c44p + 1 )eI(-H-2ph) + 2e-||jff
(A4)
M31 =
c44p M
--1 I (e1
+1 I (e1
l|(H+2ph) -A!H
- ) +
M32 =-1
c44p
+ 1 I (e1
ei( h+2 ph )+
+ ) +
+ | S44L -1 | (e|(~H+2ph) + ),
M33 = (e12ph -1),
M34 = 2(elIj2 ph +1),
M41 =-
c44p
M42 =-
c44p
+ 1 I (e1
£j( - H-2 ph)
) -
-1 I (e1
ei( h-2 ph) - ),
c44p M
-1 I (e1
+ 1 | (eI( ~H-2 ph) +elH) + + e~lH),
M43 = (eH|2ph -1), m44 = 2(e"12ph +1).
M
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Received December 24, 2018, revised December 24, 2018, accepted January 10, 2019
CeedeHua 06 aemopax
Shu Li, PhD student, Peking University, China, lishu_2013@pku.edu.cn Jianxiang Wang, Prof., Peking University, China, jxwang@pku.edu.cn
