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Abbas I.A., Razavi S.M.J. / Физическая мезомеханика 21 1 (2018) 41-45
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УДК 539.42
A mode I crack problem for a thermoelastic fibre-reinforced anisotropic material using finite element method
I.A. Abbas12, S.M.J. Razavi3
1 Department of Mathematics, Faculty of Science, Sohag University, Sohag, 82524, Egypt
2 Nonlinear Analysis and Applied Mathematics Research Group, Department of Mathematics,
King Abdulaziz University, Jeddah, 21589, Saudi Arabia 3 Department of Mechanical and Industrial Engineering, Norwegian University of Science and Technology, Trondheim, 7491, Norway
In this article, the theory of generalized thermoelasticity with one relaxation time is used to investigate the thermoelastic fiber-reinforced anisotropic material with a finite linear crack. The crack boundary is due to a prescribed temperature and stress distribution. By using the finite element method, the numerical solutions of the components of displacement, temperature and the stress components have been obtained. Comparisons of the results in the absence and presence of reinforcement have been presented.
Keywords: finite element method, fiber-reinforced material, mode I crack
DOI 10.24411/1683-805X-2018-11006
Решение двумерной задачи о трещине отрыва в термоупругом армированном волокном анизотропном материале методом конечных элементов
I.A. Abbas12, S.M.J. Razavi3
1 Сохагский университет, Сохаг, 82524, Египет 2 Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия
3 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия
В рамках теории обобщенной термоупругости с одним временем релаксации рассмотрен термоупругий упрочненный волокном анизотропный материал, в объеме которого присутствует конечная линейная трещина. Границы трещины определяются заданной температурой и распределением напряжений. Методом конечных элементов получены численные решения для компонент смещений, температуры и напряжений. Проведено сравнение результатов для неупрочненного и упрочненного материала.
Ключевые слова: метод конечных элементов, упрочненный волокном материал, трещина отрыва
1. Introduction
Fibers are assumed to be an inherent property of matter, rather than some form of inclusion in such models as Spencer [1]. Fiber-reinforced composite materials are widely used in technical structures. These materials have considerably high strength with respect to their weight, even at high temperatures they keep their stiffness. Continuum models are commonly used to explain the mechanical properties of these materials. There are some researches in the field of thermoelastic behavior of fibrous composites, among which two of them are well-known. Firstly, Lord and Shulman [2] presented the generalized thermoelastic theory with one relaxation time by postulating a new law
of thermal conduction instead of the classical Fourier law. Secondly, Green and Lindsay [3] presented two relaxation times effects on the generalized thermoelastic theory. Dha-liwal and Sherief [4] extended the generalized thermoelastic theories for the anisotropic medium. The material strength in the presence of cracks is an attracting problem of fracture mechanics and the knowledge of the elastic stress fields is potentially useful for strength estimation based on the available theories for brittle fracture [5-7]. Several researches have been published which treated the stress distributions in an unbounded solid due to the application of normal pressure or temperature on the faces of a circular internal flat crack [8, 9].
© Abbas I.A., Razavi S.M.J., 2018
The exact solution of the basic equations of generalized thermoelastic models for linear/nonlinear coupled system exists for initial and boundary issues which are very specific and simple cases. Therefore one can choose the finite element method. Basically there are three steps to apply the finite element method. The first step is to take the overall behavior of the variables so as to satisfy the differential equations given unknown field. The second step is temporal integration. The temporal derivatives of the unknown variables must be determined by the previous results. The last step is to solve the resulting equations from the first and second steps by the algorithm of the finite element method [10].
The present paper investigates a Lord and Sherman model in a two-dimensional thermoelastic medium containing a mode I crack. The nondimensional equations have been solved numerically using the finite element method.
2. Basic equations and formulation of the problem
An infinite space -«> < y < «>, < x < containing a crack on the y axis, |x| < b, x = ±0 was considered for the problem. The crack surface is subjected to a prescribed temperature and normal stress distribution. The preferred direction of the x axis was considered for the fiber direction as a = (1, 0, 0). All the considered functions depend on x and y with the time t. Thus, the components of displacement vector are u(x,y, t) and v(x,y, t). In this case, the governing equations have the following form [11]:
do xx do
dx do
xy
dy do
yy
K
dx dy d 2T
= P
= P. 2
dt2'
d^v dt2
(1) (2)
+ K 9 T
11 ~dx2 K22 ~
dy 2
dt
}2 >
+ Tn
'dt2
id d
pCeT + T0Vu dU + T.Ï22 -f dx dy
du
o xx = (A + 2a + 4^ L - 2\iT + p)— +
dx
dv
+ (A + a)— -Yn(T-T.),
dy
= (A + 2]iT)d!V + (A+a)ÏU-Y22(T- T.), dx
(3)
(4)
(5)
(6)
dy
( dv du
o xy = ^L (ax +ay
where
Y11 = (2A + 3a + 4fi L - 2\x T +P )a11 + (A + a )a 22, Y 22 = (2A + a )a11 + (A+ 2^T )a 22, a11, a 22 are the linear thermal expansion coefficients, T0 is the reference uniform temperature, T is the incremental temperature, K11 and K22 are the thermal conductivity components, p is the mass density, ce is the specific heat at
constant strain, X and |iT are the elastic constants, Txx, Txy and tyy are the stress components, a, P, (|L - |T) are the elastic parameters of a fiber reinforced material. We will use the nondimensional form of the previous equations. The nondimensional parameters are
(o'xx , o'xy, o'w ) = — (oxx , oxy, oyy I
m
xx xy yy
(t', T.) = c"
C (t, T0) T'= T - T.
n
(7)
(x , y , u , v ) = — (x, y, u, v),
n
where
n = Kn/(pCe), c2 = mp, m = X + 2a + 4|L - 2|T + p.
With respect to the nondimensional quantities in Eq. (7), after neglecting the primes for convenience, the previous equations reduced to
d 2u d2 v d 2u dT d 2u
-+ s
dx2 1 dxdy 2 dy2 3 dx dt2
(8)
d2v d2u
d2 v
dT d2 v
s4TT + s1^- + s2TT - s5^- = ^T, (9)
dz2 dxdy dx2 dy dtz
d 2T _
dx2 + S6 dy2
d2T (d d2 ^ dt+To d2
V
o du dv
oxx =^- + (s1 - s2k--s3T,
dx dy
, .du dv
oyy = (S1 - s2) ^ + s4 dy ~ S5T'
^ du dv
T + Sn — + s,
dx dy
o xy = S2
where
dx
du dv — + —
dy dx
a + A + uT s =-—
m
ft22T0
, s6
P11T) m
, s2 = m S3 = s4 =
K22 =Ju. Sc. = YM.
K11 ,s , s7 PCe , s8 PCe
(10) (11) (12) (13)
A + 2p.r
To solve this problem, the boundary and initial conditions must be considered. The initial conditions are the following:
T ( x, y, 0 ) =A T ( x, y, 0 ) = 0,
dt
u(x, y, 0) =— u (x, y, 0) = 0, dt
v (x, y, 0) = — v (x, y, 0) = 0. dt
(14)
At x = 0, the boundary conditions are assumed as (Fig. 1) u = 0, |y| > b, (15)
^ = 0, |y| > b, (16)
dx
T = T1H(t), |y| < b, ox = -PH(t), |y| < b,
(17)
(18)
Fig. 1. Fiber-reinforced plate
G^ = 0, -~<7<~, (19)
where T, and P, are constants and H is the Heaviside unit step function. This means that mechanical and heat loading are applied on the crack surface (Fig. 1).
3. Finite element solution
The formulation of finite element method for the thermo-elastic problem can easily be obtained using standard procedure. In the finite element method, the temperature {T} and the components of displacement {u, v}T are connected to the corresponding nodal values {Te} and {ue, ve}T by
{u, v}T = [M]{ue, ve}T, {T} = [M']{Te}, (20) where [M] and [ M'] are shape functions which given by
[ M ]:
M
0
M
M
0 M
Mn 0
0 M
(21)
[M'] = {M i M 2 ... Mn}, where n indicates the nodes number. According to the equation linking strain and temperature the following relationship can be written
ei (un + ut, j )> T' = Ti
(22)
i 2 j'1
which yields
{e} = [Fj]{ue}, T = [F2]{Te}. (23)
The variation form of the above equation is 5{e} = [Fj]5{ue}, 8r = [F2]5{Te}. (24)
Considering two-dimensional problem statement, [ Fi ] and [F2] can be written in the following form: [ F,] =
dMj
dx 0
3MJ
0
dM, dy 3M,
dM 2
dx
0 dM 2
dy dx dy dM1 dM2
[ F2
0
dM 2 dy dM 2
dx dM„
dx
0
dM„
0
dM dy
dM„
dy dx
(25)
dx
dM, _
_ dy dy
dx dM 2
dx
dy J
(26)
In the domain V and the boundary A, the principle of virtual displacement can be given by
Fig. 2. Contour plots of the temperature distribution with the crack tip at t = 0.1 (a), 0.5 (b), 1.0 (c), 2.0 (d)
0.0 -0.4 -0.8 -1.2
0.0 -0.2 -0.4 -0.6
\ T .......1 -2
-4
-2
-4
-2
0
d
4 y
V -2
f
4 y
Fig. 3. The distribution of temperature T (a), horizontal u (b) and vertical displacement v (c), axx (d), a^ (e), ayy (f stress components with (1) and without fiber reinforcement (2) for x = 0.2 and t = 0.5
J (5{u e}pu + 5{Te} [pce (T + t0T) + T0 Yij (Ui j +
V
+ x0Ui, j )])dV + J (5{e}T {a j} + d{T'}KiiTi )d V =
V
= J (5{ue}T{T} + S{T e}q )dA, (27)
A
where {x} are the components of the traction vector and q represents the heat flux. Equation (27) can be expressed
by
M d + C d + Kd = Fext, (28)
where d = [u v
F ext
are the vectors of external force, C is the damping matrix, M is the mass matrix and K is the stiffness matrix. Based on an implicit time integration method, the time derivatives of the unknown variables have been determined.
4. Numerical results and discussion
We assume that the plate is fiber-reinforced. The physical parameters are listed below [12]:
p = 2660 kg/m3, |T = 2.46x1010 N/m2 , X = 5.65x1010 N/m2 , | = 5.66x1010 N/m2 ,
a22 = 0.015x 10-4 deg
-1
ce = 0.787x 103 J kg-1 deg-1, T0 = 293 K, L = 0.5, T = 1, P = 1, a = -1.28x 1010N/m: P = 220.90x1010N/m2, a11 = 0.017 x 10-4 deg-1, a = 1, K11 = 0.0921x 103Jm-1 s-1 deg-1, K
= 0.0963x 103Jm-1 s-1
-1
t0 = 0.05.
l22 = 0.0963x 10 j m s deg
To visualize the crack growth under thermal and mechanical loading, a series of contour plots of the temperature distributions at different nondimensional moment of time is presented in Fig. 2. The temperature at the crack tip increases as the time t increases. Figure 3 show two curves predicted by generalized thermoelastic interaction on the fiber-reinforced medium (with fiber) and on the isotropic medium (without fiber), the solid lines refer to the solution obtained for the isotropic medium (i.e. a = 0, P = 0 and |l - |T = 0) while the dotted lines refer to the solution obtained for the fiber-reinforced medium. The reinforcement has a great effect on field quantities as expected.
Abbas I.A., Razavi S.M.J. / Физическая мезомеханика 21 1 (2018) 41-45
45
Figure 3, a shows the temperature variation along the y direction and it indicates that temperature field has its maximum value along the crack line (-1 < y < 1), and it starts to decrease just near the crack edges (y = ± 1) where it smoothly decreases and finally reaches zero. Figure 3, b displays the variation of horizontal displacement with respect to y and it indicates that the magnitude of the displacement has maximum value along the crack line (-1 < y < 1), and it starts to decrease just near the crack edges (y = ±1), and then decreases to zero to obey the boundary conditions. Figure 3, c displays the variation of vertical displacement along the y axis. The vertical displacement starts decreasing at both ends of the crack, and has a minimum value at the middle of the crack, after which it starts increasing and reaches a maximum value just near the crack edges (y = ±1) and then it decreases to become zero. The Gxx, Gxy and gyy stress components are shown in Figs. 3, d-f, respectively.
5. Conclusion
In this work, the solution of a two-dimensional problem on fiber-reinforced thermoelastic plate with a finite linear crack was studied. The differences of the predicted field quantities were remarkable in the presence and absence of fiber reinforcement. The properties of the fiber-reinforced material tend to increase the temperature variation and reduce the magnitudes of the other considered variables, which may be significant in some practical applications.
References
1. Spencer A.J.M. Deformations of Fibre-Reinforced Materials. - Oxford: Clarendon Press, 1972.
2. LordH. W., Shulman Y. A generalized dynamical theory of thermoelas-ticity // J. Mech. Phys. Solid. - 1967. - V. 15. - No. 5. - P. 299-309.
3. Green A.E., Lindsay K.A. Thermoelasticity // J. Elasticity. - 1972. -V. 2. - No. 1. - P. 1-7.
4. Dhaliwal R.S., Sherief H.H. Generalized thermoelasticity for anisotropic media // Q. Appl. Math. - 1980. - V. 38. - No. 1. - P. 1-8.
5. Ayatollahi M.R., Razavi S.M.J., Rashidi Moghaddam M, Berto F. Mode I fracture analysis of polymethylmetacrylate using modified energy-based models // Phys. Mesomech. - 2015. - V. 18. - No. 4. -P. 326-336.
6. Ayatollahi M.R., Rashidi Moghaddam M., Razavi S.M.J., Berto F. Geometry effects on fracture trajectory of PMMA samples under pure mode-I loading // Eng. Fract. Mech. - 2016. - V. 163. - P. 449-461.
7. Rashidi Moghaddam M, Ayatollahi M.R., Razavi S.M.J., Berto F. Mode II brittle fracture assessment using an energy based criterion // Phys. Mesomech. - 2017. - V. 20. - No. 2. - P. 142-148.
8. Abbas I.A. A GN model for thermoelastic interaction in an unbounded
fiber-reinforced anisotropic medium with a circular hole // Appl. Math. Lett. - 2013. - V. 26. - No. 2. - P. 232-239.
9. Abbas I.A., Zenkour A.M. The effect of rotation and initial stress on thermal shock problem for a fiber-reinforced anisotropic half-space using Green-Naghdi theory // J. Comput. Theor. Nanos. - 2014. -V. 11. - No. 2. - P. 331-338.
10. Wriggers P. Nonlinear Finite Element Methods. - Berlin: Springer, 2008.
11. Abbas I.A., Abd-alla A.N., Othman M.I.A. Generalized magneto-thermoelasticity in a fiber-reinforced anisotropic half-space // Int. J. Thermophys. - 2011. - V. 32. - No. 5. - P. 1071-1085.
12. Abbas I.A. A two-dimensional problem for a fibre-reinforced aniso-tropic thermoelastic half-space with energy dissipation // Sadhana Acad. P. Eng. Sci. - 2011. - V. 36. - No. 3. - P. 411-423.
Поступила в редакцию 09.03.2017 г.
Сведения об авторах
Ibrahim A. Abbas, Prof. Dr., Prof., Sohag University, Sohag, Egypt, ibrabbas7@yahoo.com
Seyed M. Javad Razavi, PhD student, Norwegian University of Science and Technology, Norway, mj_razavi@yahoo.com, javad.razavi@ntnu.no
