CC BY
79
УДК 539.3
Аналитический анализ тепловых напряжений в перфорированном симметричном слоистом композите с квазитреугольным отверстием
M.H. Bayati Chaleshtari1, H. Khoramishad1, and F. Berto2
1 Научно-технологический университет Ирана, Тегеран, 16846, Иран 2 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия
В статье изучено распределение тепловых напряжений вокруг квазитреугольного отверстия в симметричном слоистом композите, состоящем из слоев различных композиционных материалов. На основе решения двумерной задачи термоупругости для стационарного случая в рамках метода комплексных переменных аналитически исследовано поведение композита под воздействием однородного теплового потока. Аналитическое решение получено путем преобразования решения Лехницкого с использованием классической теории слоистых пластин и функции конформного отображения. Изучено влияние ряда параметров на распределение тепловых напряжений вокруг отверстия, включая ориентацию отверстия, степень закругления углов и соотношение сторон отверстия, угол теплового потока, последовательность укладки, а также материал слоев. Результаты приведены для различных последовательностей укладки [45°/-45°]s и [0°/90°]s. Найденное аналитическое решение показывает хорошее соответствие с результатами, полученными методом конечных элементов.
Ключевые слова: метод комплексных переменных, квазитреугольное отверстие, симметричный слоистый композит, тепловое напряжение, последовательность укладки слоев
DOI 10.24411/1683-805X-2020-11008
Analytical thermal stress analysis of perforated symmetric composite laminates containing a quasi-triangular hole
M.H. Bayati Chaleshtari1, H. Khoramishad1, and F. Berto2
1 School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran 2 Department of Mechanical and Industrial Engineering, Norwegian University of Science and Technology, Trondheim, 7491, Norway
In this study, the thermal stress distribution surrounding a quasi-triangular hole within a symmetric composite laminate made of various composite materials was investigated. A composite laminate exposed to uniform heat flux was analyzed using the complex variable method as a thermo-elastic two-dimensional problem under a steady state condition. The analytical solution was achieved with the development of the Lekhnitskii's solution and using the classical laminated plate theory and a conformal mapping function. The influence of important parameters on thermal stress distribution surrounding a hole such as the hole orientation, bluntness and aspect ratio, the heat flux angle and the laminate stacking sequence and material was studied. The results were achieved for different stacking sequences of [45°/-45°]s and [0°/90°]s. The analytical solution in this study was well validated against the finite element results.
Keywords: complex variable method, quasi-triangular hole, symmetric laminated composite, thermal stress, stacking sequence
1. Introduction
Polymeric composites have been adopted as an inevitable class of materials in various industrial sectors such as marine, gas and oil, automotive and aerospace due to their advantages. The advantages of laminated composite materials are well-known such as high corrosion resistance, high impact strength and
low structural weight. Holes are usually made in structures for various reasons, for instance to provide manholes as the points of entry and exit or to reduce the weight of structures. It is very likely that a perforated composite laminate is exposed to thermal loads. The holes and cutouts, as geometrical discontinuities can easily disturb the uniform heat flow in structures
© Chaleshtari M.H.B., Khoramishad H., Berto F., 2020
giving rise to thermal stresses. These thermal stresses accumulate on the mechanical stresses and can result in structural failure.
2. Literature review
Many researches have studied thermal stress distribution around holes within isotropic and anisotropic plates using different methods. The complex variable technique is an efficient approach to address this class of problems. The complex variable technique has been used by many researchers for solving two-dimensional boundary value problems [1-5]. Bhullar et al. [6] evaluated thermal stresses induced in aniso-tropic plates containing an elliptical hole using an analytical approach and by considering isothermal state for the hole edges. Tarn and Wang [7] obtained stresses due to thermal loads in an anisotropic elastic plate with a circular hole by applying the Lekhnitskii complex variable technique and by assuming the plane strain and plane stress conditions. Sharma and Chau-han [8] studied stresses induced surrounding a non-circular hole in an anisotropic finite laminate subjected to mechanical loading using the complex variable technique. The parameters such as the plate size and the hole geometry were investigated. Kalyon and Yilbas [9] determined the temperature and thermal stress distributions in a plate under laser pulse heating using a closed-form solution obtained using the Laplace transformation method.
Su et al. [10] studied stress concentration around the triangular and square holes in composite laminates under in-plane tensile loading. They used analytical solution and conformal mapping function to obtained the optimum hole shape. Ju et al. [11] used an analytical solution for obtaining the deformation and stress fields in multilayer composite laminates under mechanical loading. Alshaya and Lin [12] used an analytical solution to determine the stress distribution around a circular hole located in an isotropic plate using the hybrid complex-variable method. Damghani et al. [13] utilized the Lekhnitskii's complex variable method and obtained the stress resultant distribution at the edge of an elliptical or circular hole in an infinite composite plate under membrane loading. Moreover, they studied the effect of composite laminate stiffness on the stress resultant concentration factor around a circular hole.
Khechai and Mohite [14] obtained the optimum fiber and load orientation angles and the hole size in a perforated lamina as well as the symmetric com-
posite plates having circular hole under different inplane loading conditions using an analytical approach. Jafari et al. [15] employed the Florence and Goodier approach to analyze thermoelastic stresses in an isotropic plate with a polygonal hole under constant heat flux. Furthermore, Jafari and Jafari [16] presented the stress distribution surrounding a noncircular hole within an infinite composite plate under steady state condition by utilizing the two-dimensional thermo-elastic theory. Choi [17] analyzed an infinite ortho-tropic plate containing two parallel cracks subjected to constant heat flux. The presence of insulated cracks disturbed the uniform heat flow. Mahmoudi and Atefi [18] obtained a comprehensive analytical solution for thermal stresses in a hollow cylinder subjected to periodic time-varying thermal loading on the inner surface and by insulating the outer circular surface and maintaining the temperature of the both lateral surfaces constant. They employed the Fourier expansion and general properties of Bessel functions. Chao et al. [19] introduced an analytical solution based on the complex potential technique to assess the stress distribution surrounding multiple circular holes under constant heat flux. Wang et al. [20] analyzed the stress distribution in a perforated plate made of different materials containing a circular cutout using the complex variable technique. Zenkour and Abouel-regal [21] studied the interactions between the effective parameters influencing the thermal stress distribution around a perforated cylinder. They applied thermoelasticity method and Laplace solution to derive the solution of the problem. Ukadgaonker and Rao [22] studied symmetric laminates under in-plane loading containing various holes by applying a complex variable technique. Zha and Zuo [23] utilized analytical and experimental methods to investigate the strength and stress distribution in laminates containing holes. They used finite element method to validate the proposed method. Dehghani et al. [24] investigated thermal stresses surrounding triangular and square holes in a finite isotropic plate using analytical solution. They applied conformal mapping function and complex variable technique to obtain the thermal stress value around the hole.
The main purpose of this paper was investigating thermal stresses surrounding a quasi-triangular hole within a symmetric composite laminate using an analytical method. Many researches have analyzed stresses around circular and elliptical holes in metallic plates and few researchers have examined noncircular holes
in symmetric composite laminates under mechanical loading using analytical methods. In this paper, an analytical method was developed for analyzing thermal stresses in perforated symmetric laminates containing a quasi-triangular hole subjected to uniform heat flux. An equivalent thermal conductivity coefficient was defined and thermal stresses around the hole were determined using the Lekhnitskii complex variable method, classical laminate plate theory and conformal mapping. Moreover, the effect of the laminate stacking sequence as an important parameter [25] together with other significant parameters such as the hole orientation, aspect ratio, bluntness and the heat flux angle was investigated on thermal stresses. The composite laminates made of graphite/epoxy (AS/ 3501), E-glass/epoxy, carbon/epoxy woven and unidirectional carbon/epoxy with two different stacking sequences of [45°/-45°]s and [0°/90°]s were studied.
3. Problem definition
A symmetric composite laminate containing a quasi-triangular hole under a remote uniform heat flux q under steady state condition (Fig. 1) was considered in this study. The material behavior of the composite laminate was assumed orthotropic and linear elastic governed by the generalized Hooke's law. The edges of the quasi-triangular hole were considered thermally insulated. The size of the hole was sufficiently small compared to the laminate size, so the laminate can be assumed infinite. As can be seen in Fig. 1, the hole rotation angle indicating the hole orientation with respect to the horizontal axis was denoted by p. The uniform heat flux was disturbed due to the presence of a thermally insulated quasi-triangular hole giving rise to thermal stresses surrounding the hole. The thermal stresses on the hole edges were confined to the
tangential component a0 because of the hole boundary conditions. The small deformation and plane stress condition were assumed in the study.
4. Analytical solution
According to classical laminate plate theory, the thermal stress vector in a laminated composite can be obtained by the equation [26]
{°}T = -1 i [q] m'tj, h i=i
(i)
where {o}T is the thermal stress vector, [Q] is the
reduced stiffness matrix corresponding to the lth layer,
T is the temperature change, t, is the thickness of each
_i l
lamina and {a} is the thermal expansion coefficient
of the lth lamina in off-axis coordinate system and its
components can be determined using equations
a x = a11m2 + a 22n2,
— 2 2 a =ann + a22m ,
axy = 2mn(an -a22),
(2)
where a,, and a22 are the coefficients of thermal expansion in on-axis coordinate system. Moreover, m and n are cosine and sine of the fiber angle y, respectively. Equation (1) can be simplified as
{of = {QfT (3)
in which
Q
x
Q
xy
1 N
=—i Hà
Q11
Q12 Q16
Q12
Q22 Q26
Q16
Q26
Q66
I
a x
x
a y
y
a
xy
(zl - zl-1). (4)
Fig. 1. Symmetric composite laminate containing a quasi-triangular hole under a remote uniform heat flux (color online)
According to the Lekhnitskii's complex potential method and by considering the Airy's stress function U(x, y), the stress components are defined as
d 2U
d2U = d 2U
' Txy = '
dy2~' °y dx2 ' Ty dxdy (5)
The compatibility equation for a planar geometry is presented as
d 2< + d =d 2Y
(6)
dy2 dx2 dxdy
Combining Eq. (5), the compatibility equation and the stress-strain relations gives the compatibility equation of an anisotropic material in terms of the stress function [3]:
„ , d4U
-+ (2ai2 + a66)-.
Hi
d 4U 2ai 6 d 4U
dy 4 dxdy3
2a26 d 4U
dx3dy + a22
d x2d y2
+ a
xy
d 2T dxdy
--a
dU
dx4 d2 T
y dx2 '
■=-av
d 2T dy2'
(7)
In Eq. (7), a., are the coefficients of the reduced compliance matrix of the symmetric laminate and ax, ay and axy are the thermal expansion coefficients in the global coordinate system and can be obtained as Eq. (8). For simplicity, the overall thickness of the laminate was considered unity:
a x = aiPx + ai2ß y + ai6&xy > a y = ai2^x + a22&y + a26Qxy > a xy = ai6^x + a26ß y + a66&xy '
(8)
The general solution of Eq. (7) can be considered as the summation of the homogeneous U and particular UP parts:
U = Uh + UP
(9)
The homogeneous form of Eq. (7) can be presented
as
d 4U h aii dy4
- 2a0
- 2a
d 4U h
i6
dxdy
+ (2ai2 + a66>
d 4U h
d 4U h
d 4U h
ai2 T a66> -, 2 dx dy2
-+ «22 "V = 0 (10)
dx dy dx According to the Lekhnitskii's method, by introducing the four linear first-order differential operators Dk, Eq. (10) can be expressed as [3]
AD2D3D4U11 = 0, Dk = ^, (11)
dy dx
in which \ (k = 1, 2, 3, 4) are the roots of the characteristic equation
an\4 -2ai6\3 + (2ai2 + a66)\2 -- 2a26\ + a22 = 0 (12) The roots of Eq. (12) can be presented as \1 = a1 + z'Pp \2 = \1 = a1 - iP1,
_ (13)
\3 =a2 + 'P2, \4 =\3 = a2 -'P2, where a1, a2 and P2 are real numbers. For a symmetric laminate a16 = a26 = 0, so the characteristic equation is reduced to
a11\4 + (2a12 + a66)\2 + a22 = 0. (14)
The function Uh is considered as
Uh = 2 Re S Uk (Zk ), k=i
(15)
where Zk is the mapping function and is defined as Zk = x + \ky, k = 1,2. (16)
By substituting Eq. (15) into Eq. (10) and considering Zk, function U is derived. Then Eq. (9) can be rephrased as
U = U i(Zi) + U 2( Z 2) + U i(Zi) +
+ U2( Z2) + Up. (17)
The new stress function ^ is derived from the stress function U in the form
^ = Vi( Zi) + Z 2) + Vi( Zi) + dz
+ ^2) + Vp. (18)
By substituting Eqs. (17) and (18) into Eq. (5), the stress components are determined as
d 2U p
a x = 2Re{|i12Vi (Zi ) + ^(Z 2)} + -
dy2
a y = 2Re{V (Zx ) + V(Z2)} +
d 2U p
dx2
(19)
Txy = -2 Re {Vi V' (Zi) + V2V2 (Z2)} -
d 2U p dxdy
in which (Z1) and V2( Z 2) are the derivatives of functions V1(Z1) and ty2(Z2) with respect to Z1 and Z2, respectively. The Fourier's law was employed in order to relate the on-axis heat flux q and temperature gradient in a lamina as seen in Eq. (20) [27]:
9i kii 0 0
92 ■ = - 0 k22 0
93. 0 0 k33
dT.
9xj dT
dx2 dT
(20)
dx3
In Eq. (20), [k] is the on-axis anisotropic thermal conductivity matrix and T is the temperature change. It was assumed that the unidirectional laminae were transversely isotropic (i.e. isotropic in a plane normal to the fibers). Using the transformation matrix, Eq. (20) can be rewritten based on the off-axis coordinate as follows:
[T(Y)Moff =-¿on T(Y)]Vroff, (21)
where [T(y)] is the inverse of the transformation matrix ([T(-y)]) defined in Eq. (22): cos y - sin y 0 [t(-y)]= siny cos y 0 . (22)
0 0 1 Therefore, the Fourier's law of thermal conduction in the global coordinate system can be presented as
{q}off = -off VToff. (23)
As a result, the off-axis thermal conductivity matrix can be obtained using the equation
koff = [T (-Y) ]kon [T (Y) ]. (24)
By considering n = sin y and m = cos Y, the components of the off-axis thermal conductivity matrix koff = [k ] are defined as
k11 = m2kjj + n2k22, k33 = k33,
k22 = n2kll + m^ k13 = k23 = 0
(25)
k12 =-mn(k11 -k22>
The resultant of thermal conductivity coefficients for the laminate can be determined by integrating the off-axis thermal conductivity coefficients across the laminate thickness using the equation
1 H/2
[K] = — J [k]ldt. (26)
H
- H/ 2
Equation (26) can be converted into the summation form:
[K] = ^X [k]ltl. (27)
H 1=1
In Eq. (27), l is the layer number and [X] is the thermal conductivity resultant matrix of the laminate. Due
to the stationary heat flow in the laminate, Eq. (28) should be considered:
V.q = 0 (i = x, y, z). (28)
By substituting Eq. (23) into Eq. (28), the governing equation for temperature is expressed as d 2T d 2T d 2T
K'd7+2K"dXdTy+K>d7 -0 (29)
A harmonic function of T(x, y) satisfying Eq. (29) can be obtained as the temperature distribution in laminate. The solution of Eq. (29) can be considered as T = U(x + |y) in which | are the roots of the characteristic equation
KyV2 + 2Kxyvt + Kx = 0. (30)
As the thermal conductivity matrix is invertible and positive definite (KxKy > K^y), the characteristic equation (Eq. (30)) contains two complex conjugate roots. Hence, the general solution of Eq. (29) can be considered as [7]
T = Ut (Zt) + UXZ) = 2Re(Ut (Zt)), (31)
Zt = x + ltУ,
in which Ut is a complex function. By substituting Eq. (31) into Eq. (7), the particular solution for the stress function U can be obtained as
U p = 2 Re(n U t (Zt)), (32)
where
n = (-ay +«xy^t -«xh2) X
X (an\lAt - 2a16^3 + (2aj2 + a66)^2 -
- 2a26lt + a22 )-1' (33) Moreover, by substituting Eq. (32) into Eq. (19), the stress components and displacement field can be attained in the form of stress functions as
a x = 2Re{liiV1 (Zi) + ^(Z 2)} +
+ 2Re(i1|i2V),
ay = 2Re {v'i (Zi) + (Z2)} + 2 Re (nV t Txy = -2 Re{l V'l (Zi) + 12V2 (Z2 )} -
- 2Re(nit V X
2
Ux = 2 Re X PkVk + 2Re (PtVtX
k=1
2
Uy = 2Re X qk v k + 2Re q VtX
k=1
where
Pk = a11lk2 + a12 - a16lk, k = 1,2, Pt =n(a11lt2 + a12 -a16lt) + ax,
(34)
P = 1
4
£,-plane
Fig. 2. Conformal mapping (color online)
qk = ai2Mk +—- a26' k = 1,2 Vk
K L a
qt =n
22
a12Vt + "" a26 Vt
Vt
4.1. Conformal mapping
The infinite area outside the quasi-triangular hole was transfer into the outside region of a unit circle to develop the analytical solution of a circular hole into a quasi-triangular hole and to facilitate the application of Cauchy integral equation, as can be seen in Fig. 2.
Equation (36) was used in order to define the mapping function:
: w(x) = x + ¡vy, k = 1, 2, t,
(36)
such that x and y represent the Cartesian coordinates within a laminate with a quasi-triangular hole and can be represented in terms of 9 as follows [28]: x = X(cos 9 + w cos (29)),
(37)
y = X(c sin 9-w sin (29)). v ;
Moreover, in Eq. (36), \k are the characteristic equation roots of an anisotropic material as defined in Eq. (13) and i, within the mapped plane, is defined in the equation
(38)
i = pe = p(cos 9 + i sin 9). For a unit circle p = 1. In Eq. (37), c controls the hole elongation in y-direction and w is the bluntness parameter that determines the hole corners curvature. Figure 3 illustrates the influence of the parameter w on the hole shape.
Based on Eqs. (37) and (38), the conformal mapping function for a quasi-triangular hole is obtained as
(39)
Zk = w© = X/ 2(Aik ^ + A2k +
+ A3k + A4 k iT2), in which Aik, i = 1, 2, 3, 4 are as below: X
Aik = y[(i - ic\k )cos P - (ic + \k )sin P], X
A2k = ^[(i + ic\k )cos P + (ic - \k )sin P], Xw
A3k = Xw[(i + i\k)cos P + (i -\k) sin P],
Xw
A4k = xw[(i - i\k)cos P- (i + \k ) sin P].
Equation (41) can be applied to model the hole rotation
(40)
cos ß sin ß - sin ß cos ß
(41)
in which P is the hole rotation angle, and x and y were defined in Eq. (37).
Fig. 3. Effect of the parameter w on the hole shape
4.2. Determination of the complex analytical functions v and vt
Based on Eq. (34), to achieve the stress components, the complex analytical functions v and vt need to be determined. For convenience in formulations, some matrices are defined in equations
(42)
v(S) = f (S) - L-1Lf OT1) - L~llGt (S) -
r-1;
L = 1 M*2 , l =
1 1 _ n _
v =
v1 v2
, a = Pt P1 P2
_ qt _ _ q1 q2 _
The triangular hole boundary is not subjected to external loading, thus Eq. (42) is used to rewrite the mechanical boundary conditions as
Lv + Lv + lvt +lvt =0. (43)
Moreover, as the boundary of the quasi-triangular hole is insulated, the Neumann boundary condition is also considered as
v'(S)-v© = 0. (44)
The function v' © can be presented by two functions f(S) and gt(S) (see Eq. (45)) that are holomorphic in the inner and outer areas of the unit circle, respectively [7]:
v © = ft © + gt (S). (45)
By considering the points on the hole boundary (S = a) and substituting Eq. (45) into Eq. (44) and multiplying it by da/(2ni(a - S)) and applying the Cauchy integral, v' © can be rephrased as
v (S) = ft © + f (iT1). (46)
The function v't © can be expressed as the Laurent series in which the term of S-1 exists. Subsequently, by integrating v' (S), the function of vt© contains log S:
vt (S) = Ft (S) + Gt (S) + r log S, (47)
in which Ft(S) and Gt(S) are the holomorphic functions inside and outside the unit circle, respectively. Moreover, the function of v(S) can be considered as [29]
v(S) = fS) + g(S) + hlogS, (48)
where f(S) and g(S) are holomorphic inside and outside the unit circle, respectively. By substituting Eqs. (47) and (48) into Eq. (43) and multiplying it by da/(2ni(a-S)) and applying the Cauchy integral, v(S) can be rewrited as
- r'lFt (S-1) + h log S
-h
where
h = L_1(B - B)-1(ra -ra) +
+ Ap~\B _1 - B _1)-1(ri-ri), (50)
B = ApL~l,
in which L and A^ were defined in Eq. (42), and r should be calculated by the application of boundary conditions. When the heat flux is applied to an anisotropic laminate without hole, the thermal stress function can be obtained as [7]
v'~ = KZt, (51)
where
q (cos 8 + |t sin 8)
(52)
k = - _
Kt (lt -lt)
Kt = i(KxKy - KXy )12.
However, in the presence of a quasi-triangular hole, in addition to v'~, the Unction of v'M, which is holomorphic outside the unit circle, should be added to the thermal stress function. Therefore, v' (S) is derived as
v (S) = K(A1t S + A 2t S-1 + A3t S2 + + A4tS-2) + v'M (S). (53)
By comparing Eq. (53) and Eq. (46), the function f(Z) can be obtained as Eq. (54), so v' (S) can be rewritten as Eq. (55):
ft (S) = KA1t S+KA 3t S 2,
(54)
v't (S) = kAjtS + KAtS-1 + KA3tS2 + KA3tS-2 (55) By integration Eq. (55) and comparing with Eq. (47), r can be obtained as
r = -A1t(kA2t - KA^1t) - 2A3t (KA4t - KA3t). (56) As mentioned, vT expresses the thermal stress function for a laminate with no hole, as a result, it only causes the laminate deformation and does not generate stress. Thus, to achieve the thermal stresses, vtM should be determined using Eqs. (51), (53) and (55) as presented in the equation
v'M (S) = -(kA2t -KA1t) S-1 -
- (KAtt -KA3t)S-2. (57)
Therefore, the temperature stress function of vM(Z) can be obtained as Eq. (58) by integrating Eq. (57):
V M (i) = [-An (KA 2,-KA;) - 2A3, x
4t
x (kA4, - kA3t)] log i + [Ait (kA
-ka^-1 - 2[ a21 (KA2t -ka^-
■ 3[ A2t(KA41 - KA3t) + 2A41(KA2,
KA;t )]i-3-[2A3t (KA2t-KA;t )]■ 1
[2A 4, (KA 4,-KA 3, )]i4.
(58)
Finally, by comparing Eqs. (58) and (47), the functions Ft(£) and Gt(i) are obtained and by inserting them into Eq. (49), the complex analytical function of can be achieved. Then, the stress components are obtained using Eq. (34).
5. Validation of the analytical solution
In order to validate the analytical results, ABAQUS/ standard finite element code was used. Three dimensional finite element models were developed for the perforated composite laminates. The perforated lami-
nates were considered considerably larger than the holes. Four-noded quadrilateral elements (S4R) were used for modeling composite laminates. A mesh sensitivity study was carried out and an acceptable number of elements were considered. According to Fig. 4, the region around the hole was modeled with fine mesh in order to capture the stress concentration. In this region, the number of elements was increased from 40 to 360 and it was observed that further increasing the number of elements did not change the results. Figure 5 compares the analytical and numerical circumferential stress g9 distributions around a quasi-triangular hole in the composite laminates made of different materials including graphite/epoxy (AS/ 3501) and E-glass/epoxy with different stacking sequences containing the holes with different specifications. In Fig. 5, angle of 9 specifies the angular position on the border of the hole with respect to the horizontal axis. It can be seen from Fig. 5 that the
g0, MPa
Fig. 4. Mesh refinement for composite laminate containing quasi-triangular hole. w = 0.10 (a), 0.05 (b), P = 0° (a), 60° (b)
Fig. 5. Comparison between the finite element method (FEM) and the analytical results: graphite/epoxy (AS/3501) with [45°/ -45°]s lay-up (a), E-glass/epoxy with [0°/90°]s lay-up (b); w = = 0.15 (a), 0.05 (b), P = 0° (a), 60° (b), c = 1, 5 = 270°
The material properties of the composite laminates
Material En, GPa E11, GPa G12, GPa v12 a„, K-1 a22, K-1 Kn, Wm-1K-1 K22, Wm-1K-1
Graphite/epoxy (AS/3501) 144.80 9.70 4.1 0.30 -3.0-10-6 2.8-10-5 4.62 0.72
E-glass/epoxy 35.00 9.00 4.7 0.28 5.5-10-6 2.5-10-5 2.20 1.10
Carbon/epoxy woven 91.82 91.82 19.5 0.05 2.5-10-6 2.5-10-6 3.50 2.60
Carbon/epoxy unidirectional 200.00 9.45 5.5 0.27 0.4-10-6 30-10-6 0.70 1.21
G12 G13 G23, v12 v13 v23, a33 a22, K33 K22, K12 0-
analytical results correlated reasonably well with the finite element method results.
6. Results and discussion
In this study, thermal stresses in symmetric laminated composites containing quasi-triangular hole were investigated. The distribution of thermal stress surrounding the hole was examined in terms of vari-
ous parameters including the hole corner bluntness w, the hole aspect ratio c, the hole rotation angle P, the heat flux angle 8 and the stacking sequence of laminated composite made of different materials. The mechanical properties of the laminates are listed in Table 1. In order to represent the effect of different parameters on the thermal stress distribution around the hole, the normalized maximum thermal stress was defined as
Table 2
Normalized thermal stress of each lamina in [45°/-45°]s laminate of different composite material in w = 0.1
Graphite/epoxy (AS/3501) E-glass/epoxy Carbon/epoxy woven Unidirectional carbon/epoxy
ö First layer Second layer First layer Second layer First layer Second layer First layer Second layer
0° 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
5° -0.0497 -0.0848 -0.1259 -0.1806 -0.1583 -0.1583 -0.0492 -0.0859
10° -0.0799 -0.2318 -0.2096 -0.4190 -0.3257 -0.3257 -0.0773 -0.2360
20° -0.1151 -0.8930 -0.2991 -0.9673 -0.6619 -0.6619 -0.1005 -0.9154
30° -0.1391 -1.9756 -0.3350 -1.3146 -0.8669 -0.8669 -0.1041 -2.0372
40° -0.1453 -2.0465 -0.3506 -1.3727 -0.9056 -0.9056 -0.1091 -2.1102
50° -0.1440 -1.4948 -0.3644 -1.3055 -0.8751 -0.8751 -0.1184 -1.5361
60° -0.1483 -1.0796 -0.3871 -1.2211 -0.8400 -0.8400 -0.1309 -1.1062
70° -0.1618 -0.8138 -0.4276 -1.1474 -0.8198 -0.8198 -0.1498 -0.8316
80° -0.1892 -0.6297 -0.4984 -1.0783 -0.8177 -0.8177 -0.1814 -0.6416
85° -0.2114 -0.5554 -0.5517 -1.0399 -0.8240 -0.8240 -0.2055 -0.5648
90° -0.2427 -0.4892 -0.6229 -0.9957 -0.8367 -0.8367 -0.2388 -0.4962
95° -0.2879 -0.4296 -0.7191 -0.9436 -0.8584 -0.8584 -0.2863 -0.4342
100° -0.3558 -0.3758 -0.8503 -0.8827 -0.8942 -0.8942 -0.3569 -0.3777
110° -0.6451 -0.2853 -1.2689 -0.7381 -1.0381 -1.0381 -0.6551 -0.2794
120° -1.6195 -0.2225 -1.8317 -0.5807 -1.2600 -1.2600 -1.6593 -0.1963
130° -2.6372 -0.1776 -1.6827 -0.4240 -1.1075 -1.1075 -2.7212 -0.1307
140° -0.7456 -0.1092 -0.8793 -0.2861 -0.6084 -0.6084 -0.7636 -0.0974
150° -0.2297 -0.0702 -0.3961 -0.1849 -0.3013 -0.3013 -0.2340 -0.0674
160° -0.0866 -0.0436 -0.1770 -0.1119 -0.1494 -0.1494 -0.0878 -0.0430
170° -0.0301 -0.0220 -0.0675 -0.0545 -0.0629 -0.0629 -0.0304 -0.0219
180° 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
G0, max kx
A(l, l)q0 X 1 ax
in which g0 is the maximum circumferential stress
o,max
induced around the hole. The effect of various geometrical and material parameters of the perforated composite laminate on the pick thermal stress value induced around the quasi-triangular hole was studied analytically. Among the studied cases in each section, the minimum value of the pick normalized thermal stress around the hole achieved by changing the parameters is called the desirable thermal stress and the maximum value of the pick normalized thermal stress is called undesirable thermal stress.
The thermal stresses induced in different layers of composite laminates made of graphite/epoxy (AS/ 3501), E-glass/epoxy, carbon/epoxy woven and unidirectional carbon/epoxy were determined. The stacking sequences considered for the laminates were [45°/ -45°]s and [0°/90°]s. Table 2 presents the normalized thermal stress values of different layers of the laminate with a stacking sequence of [45°/-45°]s and made of different materials. According to Table 2 and Fig. 6, the maximum thermal stress for the 45° layer of the graphite/epoxy (AS/3501) composite laminate was obtained at 0 = 128° and 325° with the values of -2.8583 and 2.1802, respectively. Moreover, for the -45° layer, the maximum thermal stress values were obtained as -0.1446 and 0.1909 at 0 = 35° and 232°, respectively. It is obvious that by changing the geometrical and material parameters, the thermal stress values were changed.
Table 3 presents the normalized thermal stress values induced in different layers of the composite
laminate with a stacking sequence of [0°/90°]s and made of different materials. According to Table 3 and Fig. 6, the maximum thermal stresses for the 0° and 90° layers of the E-glass/epoxy composite laminate were obtained at 0 = 257° and 233° with the values of 1.76004 and 0.5216, respectively. Figure 6 demonstrates the normalized thermal stress distributions in different layers of the composite laminates made of graphite/epoxy (AS/3501) and E-glass/epoxy and with the stacking sequences of [45°/-45°]s and [0°/ 90°]s around a quasi-triangular hole.
6.1. The effect of hole aspect ratio
In this section, for studying the effect of hole aspect ratio c on the maximum thermal stress around the hole, the flux angle, the bluntness and the rotation angle of the hole were cosidered 8 = 270°, w = = 0.05 and P = 0°, respectively. Figure 7 illustrates the effect of c parameter on the shape of the quasi-triangular hole. The effect of hole aspect ratio on the normalized maximum thermal stress value around the quasi-triangular hole in the perforated composite laminate is shown in Table 4. It is observed for the stacking sequence of [45°/-45°]s that the desirable and undesirable thermal stresses were achieved at c = 1 and 2.5, respectively. It can be seen in Table 4 that the thermal stress value around the hole is considerably dependent on the hole aspect ratio. It can be seen that the thermal stress around the hole, initially decreased from c = 0.5 to 1.0 and further increase of c resulted in increase of thermal stress. However, for the stacking sequence of [0°/90°]s, the thermal stress value continuously increased by increasing the parameter c.
Fig. 6. Thermal stress distribution for the first and second layers of composite laminates: graphite/epoxy (AS/3501) laminate of [45°/ -45°]sstacking (a), E-glass/epoxy laminate of [0°/90°]s stacking (b)
Table 3
Normalized thermal stress of each lamina in [0o/90o]s laminate of different composite material in w = 0.1
e Graphite/epoxy (AS/3501) E-glass/epoxy Carbon/epoxy woven Unidirectional carbon/epoxy
First layer Second layer First layer Second layer First layer Second layer First layer Second layer
00 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
50 -0.0222 -0.2487 -0.0747 -0.2740 -0.1758 -0.1758 -0.0177 -0.2526
100 -0.0411 -0.2614 -0.1451 -0.4326 -0.2896 -0.2896 -0.0366 -0.2642
200 -0.0767 -0.1851 -0.2673 -0.4785 -0.3709 -0.3709 -0.0740 -0.1858
300 -0.1115 -0.1398 -0.3660 -0.4270 -0.3933 -0.3933 -0.1100 -0.1391
400 -0.1475 -0.1158 -0.4493 -0.3810 -0.4119 -0.4119 -0.1469 -0.1142
500 -0.1899 -0.1032 -0.5315 -0.3527 -0.4391 -0.4391 -0.1900 -0.1006
600 -0.2490 -0.0973 -0.6312 -0.3403 -0.4834 -0.4834 -0.2501 -0.0936
700 -0.3488 -0.0967 -0.7743 -0.3423 -0.5570 -0.5570 -0.3512 -0.0910
800 -0.5538 -0.1012 -0.9998 -0.3581 -0.6799 -0.6799 -0.5593 -0.0916
850 -0.7490 -0.1060 -1.1583 -0.3716 -0.7678 -0.7678 -0.7578 -0.0928
900 -1.0696 -0.1133 -1.3493 -0.3887 -0.8747 -0.8747 -1.0846 -0.0941
950 -1.5647 -0.1240 -1.5551 -0.4092 -0.9912 -0.9912 -1.5913 -0.0954
1000 -2.0345 -0.1353 -1.7199 -0.4320 -1.0875 -1.0875 -2.0737 -0.0978
1100 -1.1446 -0.1374 -1.5823 -0.4763 -1.0349 -1.0349 -1.1595 -0.1169
1200 -0.3736 -0.1460 -0.9469 -0.5105 -0.7251 -0.7251 -0.3751 -0.1404
1300 -0.1485 -0.1668 -0.4804 -0.5200 -0.4962 -0.4962 -0.1469 -0.1658
1400 -0.0706 -0.1915 -0.2474 -0.4755 -0.3598 -0.3598 -0.0676 -0.1923
1500 -0.0373 -0.2084 -0.1322 -0.3723 -0.2527 -0.2527 -0.0338 -0.2105
1600 -0.0203 -0.1950 -0.0695 -0.2427 -0.1572 -0.1572 -0.0168 -0.1978
1700 -0.0092 -0.1244 -0.0301 -0.1167 -0.0741 -0.0741 -0.0069 -0.1267
1800 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
woven material. Figure 8 shows the effect of hole aspect ratio c on the variation of normalized thermal stress and composite materials behavior in the laminates containing quasi-triangular hole for two stacking sequence of [450/-450]s and [0o/90O]s. According to Fig. 8, for the stacking sequence of [450/-450]s, the desirable stresses achieved for the graphite/ep-oxy (AS/3501), E-glass/epoxy, carbon/epoxy woven and unidirectional carbon/epoxy materials were in the range of 0.5 < c < 1.
6.2. The effect of hole rotation angle
The effect of rotation angle parameter P on the normalized thermal stress distribution around the quasi-triangular hole for the stacking sequences of [45o/-450]s and [00/900]s was studied. The effect of P on the perforated composite laminates with different materials and stacking sequences are presented in Table 5. In this section, for studying the effect of
For the stacking sequence of [00/900]s, the lowest value of the maximum thermal stress was obtained for the graphite/epoxy (AS/3501) material and the highest value was obtained for the carbon/epoxy
---c = 0.5
c~ 1.0 -1.5
......c= 1.5
•0.5----n\
r 0.0 ]
0 J v
s*-0.5____yJi L*-1.5 • *
Fig. 7. The effect of c on the shape of quasi-triangular hole
Stacking sequence Graphite/epoxy (AS/3501) E-glass/epoxy Carbon/epoxy woven Unidirectional carbon/epoxy
c anorm c anorm c anorm c anorm
[45°/-45°]s 0.5 1.6606 0.5 1.2262 0.5 1.2894 0.5 1.6827
1.0 1.3693 1.0 1.0896 1.0 1.1421 1.0 1.3873
1.5 1.4814 1.5 1.2238 1.5 1.2809 1.5 1.5009
2.0 1.6493 2.0 1.3982 2.0 1.4616 2.0 1.6710
2.5 1.8441 2.5 1.5881 2.5 1.6581 2.5 1.8676
[0°/90°]s 0.5 0.7965 0.5 0.8130 0.5 0.8204 0.5 0.7970
1.0 1.0247 1.0 1.0180 1.0 1.0287 1.0 1.0255
1.5 1.2682 1.5 1.2555 1.5 1.2689 1.5 1.2693
2.0 1.5150 2.0 1.4979 2.0 1.5140 2.0 1.5163
2.5 1.7559 2.5 1.7416 2.5 1.7601 2.5 1.7573
l symmetric laminates for different hole aspect ratios
[457-45°]
[0790°
0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
1.6606
1.3693
1.4814
1.6493
1.8441
0.7965
1.0247
1.2682
1.5150
1.7559
E-glass/epoxy
0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
an
1.2262
1.0896
1.2238
1.3982
1.5881
0.8130
1.0180
1.2555
1.4979
1.7416
Carbon/epoxy woven
0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
an
1.2894
1.1421
1.2809
1.4616
1.6581
0.8204
1.0287
1.2689
1.5140
1.7601
Unidirectional carbon/epoxy
0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
an
1.6827
1.3873
1.5009
1.6710
1.8676
0.7970
1.0255
1.2693
1.5163
1.7573
the rotation angle, it was assumed that 8 = 270°, w = = 0.05 and c = 1. It is observed that for the stacking sequence of [45°/-45°]s the desirable thermal stress
was achieved at a rotation angle of P = 30°. The desirable thermal stresses obtained at P = 30° for different materials of graphite/epoxy (AS/3501), E-glass/
Fig. 8. Effect of the hole aspect ratio on anorm for different composite materials and stacking sequences: graphite/epoxy (AS/3501) (a), E-glass/epoxy (b), carbon/epoxy woven (c), unidirectional carbon/epoxy (d)
Normalized thermal stress values for different rotation angles of hole
Stacking sequence Graphite/epoxy (AS/3501) E-glass/epoxy Carbon/epoxy woven Unidirectional carbon/epoxy
P ^norm P ^norm P ^norm P ^norm
[450/-450]s 00 1.3693 00 1.0896 00 1.1421 00 1.3873
300 1.1986 300 0.9897 300 1.0351 300 1.2143
450 1.2944 450 1.0553 450 1.1047 450 1.3113
600 1.3693 600 1.0896 600 1.1421 600 1.3873
900 1.3548 900 1.0399 900 1.0918 900 1.3726
[00/900]s 00 1.0247 00 1.0180 00 1.0287 00 1.0255
300 1.1095 300 1.0965 300 1.1083 300 1.1105
450 1.086 450 1.0754 450 1.0869 450 1.0869
600 1.0247 600 1.0180 600 1.0287 600 1.0255
900 0.9078 900 0.8972 900 0.9068 900 0.9086
epoxy, carbon/epoxy woven and unidirectional car- tively. However, for the stacking sequence of [0o/90o]s bon/epoxy having the stacking sequence of [450/ the desirable thermal stress was obtained at the rota--450]s were 1.1986, 0.9897, 1.0351 and 1.2143, respec- tion angle of P = 90°. Moreover, for the stacking se-
Fig. 9. Effect of the hole rotation angle on the normalized thermal stress for different values of w: 0.050 (1), 0.075 (2), 0.100 (3), 0.125 (4); [450/-450]s (a, c), [00/900]s (b, d)
Normalized thermal stress values for different heat flux angles
Stacking sequence Graphite/epoxy (AS/3501) E-glass/epoxy Carbon/epoxy woven Unidirectional carbon/epoxy
5 ^norm 5 ^norm 5 ^norm 5 ^norm
[450/-450]s 0o 1.1986 0o 0.9897 0o 1.0351 0o 1.2143
30o 1.6061 30o 1.2106 30o 1.2716 30o 1.6274
45o 1.6575 45o 1.2287 45o 1.2917 45o 1.6796
60o 1.5989 60o 1.1786 60o 1.2394 60o 1.6202
90o 1.3693 90o 1.0896 90o 1.1421 90o 1.3873
[00/900]s 0o 1.1095 0o 1.0965 0o 1.1083 0o 1.1105
30o 0.9646 30o 0.9731 30o 0.9825 30o 0.9653
45o 0.7952 45o 0.8312 45o 0.8383 45o 0.7958
60o 0.8873 60o 0.8817 60o 0.8910 60o 0.8880
90o 1.0247 90o 1.0180 90o 1.0287 90o 1.0255
quence of [0o/90o]s, the minimum normalized thermal stress was obtained for the E-glass/epoxy material with the value of 0.8972 and the maximum normalized thermal stress was obtained for the unidirectional carbon/epoxy material with the value of 1.1105. The variations of the normalized thermal stress with the hole rotation angle for different values of the hole bluntness and the stacking sequences of [450/-450]s and [0o/900]s for the carbon/epoxy woven and the unidirectional carbon/epoxy materials are shown in Fig. 9.
6.3. The effect of heat flux angle
Table 6 presents the influence of the heat flux angle 8 on the normalized thermal stress value around the quasi-triangular hole in a four-layer symmetric composite laminate made of different materials with two stacking sequences. It was assumed that the heat flux
angle was varied between 0o and 90o and its effect on the normalized thermal stress was studied. It was found out that the effect of 8 on the normalized thermal stress value for the two stacking sequences of [45o/-450]s and [0o/900]s were different. According to Table 6, the minimum and maximum normalized thermal stress values were obtained for 8 = 0° and 45° for the stacking sequence of [450/-450]s, respectively. While, for the stacking sequence of [0o/90o]s the minimum and maximum thermal stress values were obtained for 8 = 450 and 0o, respectively.
For the stacking sequence of [00/900]s, the desirable normalized thermal stress values obtained for the graphite/epoxy (AS/3501), E-glass/epoxy, carbon/ epoxy woven and unidirectional carbon/epoxy materials were equal to 0.7952, 0.8312, 0.8383 and 0.7958, respectively. In order to better observe the heat flux angle effect on the normalized thermal stress for dif-
Fig. 10. Effect of heat flux angle on anorm for the laminate made of graphite/epoxy (AS/3501) for different values of w: 0.050 (1), 0.075 (2), 0.100 (3), 0.125 (4); [450/-450]s (a), [00/900]s (b)
Fig. 11. Effect of heat flux angle on anorm for the E-glass/epoxy and different values ofw: 0.050 (1), 0.075 (2), 0.100 (3), 0.125 (4); [450/ -450]s(a), [00/900]s (b)
ferent values of the hole bluntness, Figs. 10 and 11 are presented. As observed in Fig. 10, by increasing the value of w and the sharpening the hole corners, both of the desirable and undesirable thermal stresses were increased. For the graphite/epoxy (AS/3501) material with the stacking sequence of [450/-450]s, for all values of w, the desirable normalized thermal stress occurred at the angles of approximately 70° and 280°. Whereas, for the stacking sequence of [00/ 900]s, the desirable normalized thermal stress occurred at the angles of approximately 145° and 215°.
Moreover, it is clear in Fig. 11 that for the glass/ epoxy material with the stacking sequence of [450/
-450]s for all values of w, the desirable normalized thermal stress occurred at the angles of approximately 80° and 280°. Wherase, for the stacking sequence of [00/900]s, the desirable normalized thermal stress occurred at the angles of approximately 145° and 215°. It should be noted that the results of Table 6 are presented with the assumptions of P = 00, w = 0.05 and c = 1.
6.4. The effect of bluntness
Table 7 represents the effect of the hole corner bluntness w on the normalized thermal stress. It can be seen in Table 7 that the thermal stress value in-
Table 7
Normalized thermal stress for different bluntness values
Stacking sequence Graphite/epoxy (AS/3501) E-glass/epoxy Carbon/epoxy woven Unidirectional carbon/epoxy
w ^norm w ^norm w ^norm w ^norm
[450/-450]s 0 1.252 0 0. 9831 0 1.031 0 1.2685
0.05 1.3693 0.05 1.0896 0.05 1.1421 0.05 1.3873
0.10 1.5247 0.10 1.2243 0.10 1.2833 0.10 1.5449
0.15 1.7385 0.15 1.3990 0.15 1.4956 0.15 1.7615
0.20 2.0137 0.20 1.6292 0.20 1.7080 0.20 2.0394
0.25 2.4158 0.25 1.9579 0.25 2.0508 0.25 2.4467
0.30 2.9758 0.30 2.4457 0.30 2.5639 0.30 3.0128
[00/900]s 0.00 0.9986 0.00 0.9869 0.00 0.9975 0.00 0.9994
0.05 1.0247 0.05 1.0180 0.05 1.0287 0.05 1.0255
0.10 1.1006 0.10 1.1047 0.10 1.1159 0.10 1.1015
0.15 1.2293 0.15 1.2423 0.15 1.2546 0.15 1.2302
0.20 1.4225 0.20 1.4399 0.20 1.4540 0.20 1.4236
0.25 1.7020 0.25 1.7238 0.25 1.7411 0.25 1.7032
0.30 2.1305 0.30 2.1581 0.30 2.1796 0.30 2.1322
creased with increasing the hole bluntness. The minimum thermal stress value occurred when the bluntness parameter was zero (w = 0). As shown in Fig. 3, w = 0 is equivalent to a circular hole. Therefore, the thermal stress value in a perforated symmetric laminate with a circular hole was always lower compared to the quasi-triangular holes. For the stacking sequence of [450/-450]s, the desirable thermal stress obtained at w = 0 for the E-glass/epoxy composite laminate with the value of 0.9831 and the undesirable thermal stress value obtained at w = 0.3 for the unidirectional carbon/epoxy composite laminate with the value of 3.0128. However, in the stacking sequence of [00/900]s, the desirable thermal stress value obtained at w = 0 for the E-glass/epoxy composite laminate with the value of 0.9869 and the undesirable thermal stress value obtained at w = 0.3 for the carbon/epoxy woven composite laminate with the value of 2.1796. Figure 12 illustrates the variations of normalized thermal stress in the laminates con-
taining quasi-triangular hole in terms of the bluntness parameter for two stacking sequences of [450/-450]s and [00/900]s. It can be observed that the normalized thermal stress increased by increasing the bluntness parameter and yet the rate of this increase was steeper for the larger bluntness values. Furthermore, it is evident from Fig. 12 that the normalized stress in the stacking sequence of [450/-450]s was always higher than that of the stacking sequence of [00/900]s in all composite materials. Figure 13 shows the normalized thermal stress distribution in different perforated composite materials around the quasi-triangular hole with different values of w and P parameters. The results presented in this section were obtained by considering P = 0o, 8 = 270o and c = 1.
7. Conclusions
The thermal stress distribution around the quasi-triangular hole in a symmetric laminated plate with various composite materials under a uniform heat flux
Fig. 12. Effect of bluntness on anorm for different composite materials and stacking sequences: graphite/epoxy (AS/3501) (a), E-glass/ epoxy (b), carbon/epoxy woven (c), unidirectional carbon/epoxy (d)
Fig. 13. Thermal stress distribution around the quasi-triangular hole with P = 30° (a, e), 0° (b, d, f h), 45° (c, g); w = 0.2 (a, e), 0.1 (b, f ), 0.15 (c, g), 0.05 (d, h); anorm = 1.465 (a), 1.5247 (b), 1.3895 (c), 1.0255 (d, h), 1.6643 (e), 1.1047 f), 1.3807 (g). Graphite/epoxy (AS/ 3501) (a, e), E-glass/epoxy (b,f), carbon/epoxy woven (c, g), unidirectional carbon/epoxy (d, h). Stacking sequence of [450/-450]s(a-d), [00/900]s (e-h)
was studied analytically using the Lekhnitskii's complex variable technique and a conformal transferring function. It was assumed that the edges of the hole were thermally insulated and had Neumann boundary condition. The analytical method was validated with finite element results. The thermal stress distribution in each lamina for different composite materials was analyzed. Furthermore, the effects of the laminate stacking sequence, the heat flux angle and the hole bluntness, aspect ratio and rotation angle as important parameters and behavior of composite materials were investigated on the thermal stress distribution. It was found out that for the composite materials and the two stacking sequences of [450/-450]s and [00/900]s studied in this research, the minimum normalized thermal stress value was obtained for the circular hole (i.e. w = 0). Moreover, increasing the hole aspect ratio increased the thermal stress surrounding the quasi-triangular hole. The results showed that the hole orientation can influence the thermal stress value considerably. The hole rotation angles of 300 and 900 resulted minimum thermal stress values in the perfo-
rated laminates with stacking sequences of [450/-450]s and [00/900]s, respectively. Moreover, it was found that applying the heat flux angles of 00 and 450 on the perforated laminates with the stacking sequences of [450/-450]s and [00/900]s, respectively, caused lower thermal stress values around the hole.
References
1. Muskhelishvili N. Some Basic Problems of the Mathematical Theory of Elasticity. - Dordrecht: Springer Netherlands, 1954.
2. Savin G.N. Stress Distribution around Holes. - New York: 1961.
3. Lekhnitskiy S.G. Anisotropic Plates. - New York: Epub ahead of print, 1969. - doi 0677206704.
4. Hasebe N.Wang X. Complex variable method for thermal stress problem // J. Therm. Stress. - 2005. - V. 28. -P. 595-648.
5. Florence A.L., Goodier J.N. Thermal stresses due to disturbance of uniform heat flow by an insulated ovaloid hole // J. Appl. Mech. - 1960. - V. 27. - P. 635-639.
6. Bhullar S.K., Wegner J.L. Thermal stresses in a plate with hyperelliptical hole // J. Eng. Technol. Res. - 2009. -V. 1. - P. 152-170.
7. Tarn J.Q., Wang Y.M. Thermal stresses in anisotropic bodies with a hole or a rigid inclusion // J. Therm. Stress.- 1993. - V. 16. - P. 455-471.
8. Chauhan M.M., Sharma D.S. Stress concentration at the corners of polygonal hole in finite plate // Aerosp. Sci. Technol. - 2016. - V. 58. - P. 197-206.
9. Kalyon M., Yilbas B.S. Analytical solution for thermal stresses during the laser pulse heating process // Proc. Inst. Mech. Eng. Part C. J. Mech. Eng. Sci. - 2001. -V. 215. - P. 1429-1445.
10. Su Z., Xie C., Tang Y. Stress distribution analysis and optimization for composite laminate containing hole of different shapes // Aerosp. Sci. Technol. - 2018. - V. 76. -P. 466-470.
11. Ju S.H., Liang W.Y., Hsu H.H., Tarn J.Q. Analytic solution of angle-ply laminated plates under extension, bending and torsion // J. Compos. Mater. - 2019. - doi 10.1177/0021998319873025.
12. Alshaya A., Lin S. Hybrid stress analysis of a near-surface circular hole in finite structures // Proc. Inst. Mech. Eng. C. J. Mech. Eng. Sci. - 2019. - P. 1-16.
13. DamghaniM., Harrison C., Kennedy D. The effects of composite laminate stiffness and loading on stress resultant concentration factor around a hole // Proc. Inst. Mech. Eng. Part C. J. Mech. Eng. Sci. - 2018. - V. 232. -P. 1033-1049.
14. Khechai A., MohiteP.M. Optimum design of perforated symmetric laminates using evolutionary algorithm // J. Compos. Mater. - 2019. - V. 53. - P. 3281-3305.
15. Jafari M., Nazari M.B., Taherinasab A. Thermal stress analysis in metallic plates with a noncircular hole subjected to uniform heat flux // Eur. J. Mech. A. Solids. -2016. - V. 59. - P. 356-363.
16. Jafari M., Jafari M. Effect of hole geometry on the thermal stress analysis of perforated composite plate under uniform heat flux // J. Compos. Mater. - 2019. -V. 53. - P. 1079-1095.
17. Choi H.J. Thermal stresses due to a uniform heat flow disturbed by a pair of offset parallel cracks in an infinite plane with orthotropy // Eur. J. Mech. A. Solids. - 2017.-V. 63. - P. 1-13.
18. Mahmoudi H., Atefi G. Analytical solution for thermal stresses in a hollow cylinder under periodic thermal
loading // Proc. Inst. Mech. Eng. Part C. J. Mech. Eng. Sci. - 2012. - V. 226. - P. 1705-1724.
19. Chao C.K., Chen F.M., Lin T.H. Thermal stresses induced by a remote uniform heat flow interacting with two circular inclusions // J. Therm. Stress. - 2017. -V. 40. - P. 564-582.
20. Wang P., Wang B.L. Thermoelectric fields and associated thermal stresses for an inclined elliptic hole in thermoelectric materials // Int. J. Eng. Sci. - 2017. -V. 119. - P. 93-108.
21. Zenkour A.M., Abouelregal A.E. Thermoelastic interactions in an infinite orthotropic continuum of a variable thermal conductivity with a cylindrical hole // Iran. J. Sci. Technol. Trans. Mech. Eng. - 2019. - V. 43. - P. 281290.
22. Ukadgaonker V.G., Rao D.K.N. A general solution for stresses around holes in symmetric laminates under inplane loading // Compos. Struct. - 2000. - V. 49. -P. 339-354.
23. Zha X., Zuo Y. Theoretical and experimental studies on in-plane stiffness of integrated container structure // Adv. Mech. Eng. - 2016. - V. 8. - P. 1-20.
24. Dehghani M., Fotuhi A.R., Shafiei A.R. Thermal stress analysis in a finite perforated plane // Iran. J. Sci. Technol. Trans. Mech. Eng. - 2019. - V. 43. - P. 705-721.
25. KhoramishadH., Bagheri TofighiM., KhodaeiM. Effect of Stacking Sequence on Low-Velocity Impact Behavior of Metal Laminates // Phys. Mesomech. - 2018. - V. 21.-No. 2. - P. 140-149. - doi 10.1134/S102995991802 0078.
26. Herakovich C.T. Mechanics of Fibrous Composites. -Virginia: Johan Wiley and Sons, 1998.
27. OzisikN.M. Heat Conduction. - New York: John Wiley and Sons, 1993.
28. Jafari M., Moussavian H., Bayati M.H.C. Optimum design of perforated orthotopic and laminated composite plates under in-plane loading by genetic algorithm // Struct. Multidiscip. Optim. - 2018. - V. 57. - P. 341357.
29. Chao C.K., Shen M.H. Thermal stresses in a generally anisotropic body with an elliptic inclusion subject to uniform heat flow // J. Appl. Mech. - 1998. - P. 51-65.
Received 22.01.2020, revised 22.01.2020, accepted 29.01.2020
CeedeHua 06 aemopax
Mohammad Hossein Bayati Chaleshtari, PhD Candidate, School of Mechanical Engineering, Iran University of Science and Technology, Iran, [email protected]
Hadi Khoramishad, PhD, Assoc. Prof., School of Mechanical Engineering, Iran University of Science and Technology, Iran, [email protected] Filippo Berto, Prof., Norwegian University of Science and Technology, Norway, [email protected]
