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УДК 539.42
A Green-Naghdi model in a 2D problem of a mode I crack in an isotropic thermoelastic plate
I.A. Abbas12, F.S. Alzahrani2
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
In this article, the generalized thermoelastic theory under Green and Naghdi models are used to study the thermoelastic interaction in an isotropic material containing a finite crack inside the material. The crack boundary is due to a prescribed temperature and stress distribution. Based on the Green-Naghdi type II and type III models, the formulation is applied to generalized thermoelasticity with an appropriate choice of parameters. Numerical solutions of the displacement components, temperature, and stress components are obtained using the finite element method. The results have been verified numerically and are represented graphically. Comparisons were made with expected results from Green and Naghdi model of type III and Green and Naghdi model of type II.
Keywords: Green and Naghdi models, finite element method, mode I crack, thermoelasticity
DOI 10.24411/1683-805X-2018-11001
Модель Грина-Нагди в двумерной задаче для трещины отрыва в изотропной термоупругой пластине
I.A. Abbas12, F.S. Alzahrani2
1 Сохагский университет, Сохаг, 82524, Египет 2 Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия
В статье в рамках теории обобщенной термоупругости на основе моделей Грина-Нагди рассмотрено термоупругое взаимодействие в изотропном материале, в объеме которого присутствует трещина конечной длины. Границы трещины определяются заданной температурой и распределением напряжений. Постановка задачи выполнена в рамках теории обобщенной термоупругости на основе моделей Грина-Нагди типа II и типа III с соответствующим выбором параметров. С помощью метода конечных элементов найдены значения компонент смещения, температуры и компонент напряжений. Выполнена численная верификация результатов. Проведено сравнение с результатами на основе моделей Грина-Нагди типа III и типа II.
Ключевые слова: модели Грина-Нагди, метод конечных элементов, трещина отрыва, термоупругость
1. Introduction
Two generalized thermoelasticity theories well-investigated and well-established. Replacing the classical Fourier law by postulating a new thermal conduction law, the theory of generalized thermoelasticity containing one relaxation time has been proposed by Lord and Shulman [1]. Green and Lindsay [2] established the generalized of thermoelastic theory containing two relaxation times. For the anisotropic medium, Dhaliwal and Sherief [3] extended the generalized thermoelastic theories. Entropy based on equality rather than inequality usually entropy, Green and Naghdi [4-6]
established three new theories of thermoelasticity. The constitutive hypotheses for the heat flux vector in each theory are different. So they got three theories of thermoelasticity called types I, II, and III. We get the classical thermoelasti-city system when the type I model is linearized. Type II model (a limiting case of type III) does not admit energy dissipation.
The strength of a material with cracks is an attracting problem in fracture and the knowledge of elastic stress fields is potentially useful for strength estimation based on the theory of brittle fracture. Several articles have appeared
© Abbas I.A., Alzahrani F.S., 2018
which treat the stress distributions in an unbounded solid due to the application of normal pressure or temperature on the faces of a circular internal fiat crack. Mathematically, the basic equations for cracking problems in piezo-elasticity and magnetoelectroelasticity are identical to their analogues in pure elasticity as in Ref. [7]. Sherief and El-Maghraby [8, 9] studied mode I crack problems using the method of regularization. Prasad et al. [10] applied the method of regularization in a two dimensional thermoelastic problem of a mode I crack under Green and Naghdi type III model. Lotfy and Othman [11] studied the effect of magnetic field for a mode I crack on a two-dimensional problem under generalized thermoelastic theory. Abdel-Halim and Elfalaky [12] studied an unbounded thermoelastic solid with internal penny-shaped crack. Elfalaky and Abdel-Halim [13] investigated an unbounded thermoelastic space containing a mode I crack.
The analytical solution of the basic equations of the generalized thermoelastic theory for a coupled and linear/ nonlinear system exists only for very special and simple initial and boundary issues. Therefore one can chose the finite element method. Three steps have been involved 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. In the third step, the solutions of equations resulting from the first and second steps will obtained by the finite element algorithm as in Ref. [14].
The present paper investigates a GN model in a two dimensional problem of a mode I crack in a thermoelastic medium using the finite element method. The results have been verified numerically and represented graphically.
2. Basic equation
For a homogenous, isotropic, and linear thermoelasticity, the basic equations can be written in the form [10]
Mu, jj + (X + - yTt = pu, i, j = 1, 2, 3, (1) where X, ^ are elastic parameters, p is the mass density, U are the components of displacement, T is the change in temperature of a particle of material, y = (3X + 2^)at, at is the linear thermal expansion coefficient and t is the time.
The form of heat equation can be given by
K % + nKT u = pcj + YTU. i, i, j = 1, 2, 3, (2)
where K is the thermal conductivity, ce is the specific heat at constant strain, To is the reference uniform temperature, K * is the material constant characteristic of the theory, n = 1 refers to the theory of Green and Naghdi of type III (with energy dissipation) while n = 0 refers to the theory of Green and Naghdi of type II (without energy dissipation). The constitutive equations have the form
j (Xufi -Y(T-T,))8j +^(Uj+ Uj), (3)
i, j = 1, 2, 3,
where 8j is the Kronecker symbol, and Gy are the stress components.
3. Formulation of the problem
An infinite space -x < x < x, -x < y < x was considered in this problem with a crack on the x axis, |x| < a, y = 0. The surface of the cracks is subjected to a prescribed temperature and to the normal stresses. The displacement components ui are (u(x, y, t), v(x, y, t), 0). In this case, the governing equations have the following form [10]: d 2u „ N d2 v
(X + 2^)- 2
dx
dxdy
dy2
dT
d 2u
"YaX" = pat 2
(4)
d2 v
dy2
d 2u dxdy
d2 v dx2
K + nK— dt
dT d2 v
-Y—= p-r>
dy dt2
a y a2t a2t
(5)
ax2'+"3-.2
52 f
dt2
pCeT + YTo
dy
fdu + dv Y
dx dy
axx = (X + 2^)^ + -Y(T - To),
ox dy
a y = (X + 2^i) ^ + -Y(T - T0),
f du dv ^
a xy =M>
dy dx
(6)
(7)
(8) (9)
It should take the nondimensional form for the previous equations. Thus, the nondimensional parameters are given by
T - T , c
T' =-0, (v, u', x , y) = — (v, u, x, y),
To X
1
2 C t
(10)
(a'xx, a'xy, a'yy ) =- (axx, axy, ayy X S = ■
where % = K/ (pce) and c2 = (X + 2^)/ p.
In terms of the dimensionless quantities (10), after neglecting the primes, the above equations can be reduce to
a2u a2u ,„, Ja2u a2v')
di+(P2 -1)
a x a y
dT „7 a2u
= aix,
^+(p2 -1)f dy2 dx2
dT „2 d2v dy dt2
dx2 dxdy
d2u d2v
(11)
dxdy d y
d
n—he
V 2T d
dt
il "dt2
dx
- \
T + e 2
dy2
rdv du^
dy dx
J J
axx = 2dU + (P2 - 2)
dx
a yy = 21 ^
a = du + dv
xy dy d x '
r du + 9v ^
dx 9 y
V J J
du dv — + —
dx 9 y
V J J
where p
2 À + 2^ _yT0
-a T,
-aT, tf *
a = ^-iL, ej =-2, and e2 =
H H pcec2 pc,
(13)
(14)
(15)
(16) Y
4. Application
To solve the problem, boundary and initial conditions should be taken into account. The initial conditions of the problem are
u (x, y, t) = — u (x, y, t ) = 0,
dt
9
T (x, y, t ) = — T ( x, y, t ) = 0, dt
dv
v(x, y, t) = — (x, y, t) = 0, t = 0. dt
(17)
At y = 0, the boundary conditions will described by (see Fig. 1)
r\T
— = 0, | x | > a, (18) dy
v = 0, |x| > a, (19)
T = TiH(t)H(a- |x|), |x|<a, (20)
Gyy =-pH(t)H(a- |x|), |x|<a, (21)
axy = 0, x<~, (22)
where Ti and Pi are constants and H is the step function of the Heaviside unit. This means that mechanical and thermal loading are applied on the surface of crack as in Fig. 1.
5. Finite element solution
The finite element formulation of a thermoelastic problem can easily be obtained by the standard procedure. Based on the finite element method, the temperature {T} and the displacement components {u, v} are connected to the cor-
[ M ] =
responding nodal values {Te} and {ue, ve}T as
{u,v}T = [M]{ue,ve}T, {T} = [M']{Te}, (23)
where [M] and [M' are shape functions which given by mi 0 m2 0 .... mn 0 _ 0 mi 0 m2 .... 0 mn j (24)
[ M'] = {mi m2 ... mn }, where n is indicates the node number. The geometric equation can be written as
eij = 2(uji + ui,j ^ T' = Ti. (25)
It yields
{e} = [Fi]{ue}, T' = [F2]{Te}. (26)
The variation forms of above equations are
5{e} = [Fi]5{ue}, 5T' = [F2]5{Te}. (27)
In the considering coordinates x and y, the forms of [ Fi ] and [ F2 ] can be written by
[ Fj] =
[ F2] =
dmj dx
0
dmj
dmj dx dmj
~dy
0
drm
dmj dx dm2 dx dm2
dm2 dx
0
dm2
dy
0
dm2
17
dm2 dx
dmn
dx
dr^n
dy
dmn
dx
0
dm,.
0 dmn
dy dm„
dy dx
(28)
(29)
Fig. 1. Displacement of an external mode I crack
In the domain V and the boundary A, the principle of virtual displacement of can be given in the following form:
V (5{ue}p2u&i + 8{T e}[T + T0 Y iii4 ])d V+
+ jV (5{e}T {Oj-} + 5{T 'KeT + nT; ))d V=
= JA (5{ue}T {t} + 5{Te}q)dA, (30)
where q represents heat flux, and {f} are the traction vector components. Symbolically, Eq. (30) can be expressed as
M d + C d + S d = Fext, (31)
where d = [u v T]T, S is the stiffness matrix, C is the damping matrix, M is the mass matrix, and Fext are the external force vectors. The temporal derivatives of the unknown variables can be determined by an implicit temporal integration method.
6. Numerical results and discussion
Let us assume that the plate is made of isotropic. The material constants are given as following [8]: p = 8954 kg/m3, X = 7.76x 1010 N/m2, ¡1 = 3.86x1010 N/m2, K = 386 J m"1 s"1 deg1, at = 1.78x 10"5 deg-1, ce = 383.1 kg"1 deg"1, To = 293 K, T1 = 1, P1 = 1, a = 1.
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 2. Contour plots of the temperature distribution with the crack tip at t = 0.1 (a), 0.5 (b), 1.0 (c), 1.5 (d)
Before going to the analysis grid independence audit was carried out. The quadrilateral, eight-node isoparametric element is used for temperature and displacement components. The grid size has been refined and consequently the
value of different parameters get stabilized. Further refinement of mesh size (500x500 elements) does not change greatly values, and it is accepted as the grid size for calculation purposes.
u X 10"3_
Fig. 3. The distribution of temperature T (a), horizontal u (b) and vertical displacement v (c), stress component axx (d), axy (e), ayy (f) for GNII (1, 2) and GNIII (3, 4) models at t = 0.2 with y = 0.15 (1, 3) and 0.20 (2, 4)
To visualize the mechanics of crack growth under thermal and mechanical loading, in the context of Green and Naghdi model of type III (GNIII), the series of contour plots of the temperature distributions at different nondimen-sional moment of time is presented in Fig. 2. It can be seen from the figures that temperature at the crack tip increases as the time t increases. Figure 3 demonstrates four curves predicted by generalized thermoelastic interaction with energy dissipation (GNIII) and without energy dissipation (GNII) on isotropic medium for different values of y.
Figure 3, a displays the temperature behavior along to x. The temperature field has an elevated value at the length of the crack (-a < x < a), and near the crack edges (x = ±1) temperature smoothly decreases and finally closes to zero values. Figure 3, b displays the variation of the horizontal displacement along to x axis. We find that the vertical displacement has a minimum value in the crack middle and reaches a maximum in the immediate vicinity of the edges of the crack (x = ±1), and then it decreases to become zero (Fig. 3, c). It is easy to see that the displacement magnitude has a maximum value at the crack length (-1 < x < 1), and it decreases to zero values to obey the boundary conditions. The stress components axx, a and ayy are shown in Figs. 3, d-f respectively. It is evident that both components of stress display different behaviors in the vicinity of crack.
7. Conclusion
In the present work, the solution of two dimensional problem for an isotropic thermoelastic half-space with a finite linear crack, under Green and Naghdi models of types III and II, has been studied by the finite element method. The variations in the behavior of stress components are
presented. The crack size is significant to elucidate the mechanical structure of the solid.
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Ibrahim A. Abbas, Prof. Dr., Prof., Sohag University, Egypt, [email protected]
Faris S. Alzahrani, Assoc. Prof., King Abdulaziz University, Saudi Arabia, [email protected]
