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IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 09, Issue 6 (June. 2019), ||S (III) || PP 24-42
International organization of Scientific Research 24 | Page
Thermo-mechanical Analysis of a Crack in an Infinite
Functionally Graded Elastic Layer
Rajesh Patra1, S. P. Barik
2, P. K. Chaudhuri
3
1(Department of Mathematics, Hooghly Engineering & Technology College, Vivekananda Road,
Hooghly-712103, India) 2(Department of Mathematics, Gobardanga Hindu College, 24-Parganas (N), Pin-743273, India)
3(Retired Professor, Department of Applied Mathematics, University of Calcutta,92, A. P. C. Road,
Kolkata - 700009, India)
Corresponding Author: S. P. Barik
Abstract: This paper aims to develop a steady state thermoelastic solution for an infinite functionally graded layer of finite thickness with a crack in it lying in the middle of the layer and parallel to the faces of the layer.
The faces of the layer are maintained at constant temperature of different magnitude. The layer surfaces are
supposed to be acted on by symmetrically applied concentrated forces of magnitude P
2 with respect to the centre
of the crack. The applied concentrated force may be compressive or tensile in nature. The problem is solved by
using integral transform technique. The solution of the problem has been reduced to the solution of a Cauchy
type singular integral equation, which requires numerical treatment. Both normalized thermo-mechanical stress
intensity factor (TMSIF), thermal stress intensity factor (TSIF) and the normalized crack opening displacement
are determined. Thermal effect and the effects of non-homogeneity parameters of the graded material on various
subjects of physical interest are shown graphically.
Key words and phrases: Fourier integral transform, Singular integral equation, Stress-intensity factor. ----------------------------------------------------------------------------------------------------------------------------- ----------
Date of Submission: 11-06-2019 Date of acceptance: 28-06-2019
----------------------------------------------------------------------------------------------------------------------------- ----------
I. INTRODUCTION Every solid material has its own characteristics in respect of its elastic behavior, density, porosity,
thermal and electrical conductivity, magnetic permeability, and so on. In numerous situations, a particular solid
does not necessarily fulfill all the criteria for a particular purpose. For example, aerospace industry requires light
materials with high strength; the outer part of a space craft body should be a non-conductor of heat such that the
heat generated due to friction does not disturb the interior part made of high strength metal. It is difficult to find
materials with light weight but high strength or high strength metal with zero heat conductivity. These
difficulties are overcome at present using two or more solids at a time to generate a new solid which will fulfill
most of our requirements. Composite materials, fibre-reinforced materials and functionally graded materials
(FGM) are some of newly manufactured materials which are used at present in various areas of applications.
Two solids say A and B with specific properties are used to form a FGM in such a way that the composition
gradually vary in space following a definite designed rule. This means that at a particular point in the medium
the FGM shows (100 - x) % of A's property and x% of B's property, (0 โค x โค 100). The idea was originated in Japan in early part of the eighties in the last century and has been found to be very useful in respect of
applications in various areas like resisting corrosiveness, controlling thermal activity, increasing strength and
toughness in materials etc. FGMs have also several biomedical applications.
Thermal loading on solids has significant effects in their after load behavior and so should be dealt with
utmost care. As such, the study of thermoelastic problems has always been an important branch in solid
mechanics1,2
. In the design of a structure in engineering field , considerable attention on thermal stress is a
natural task, because many structural components are subjected to severe thermal loading which might cause
significant thermal stresses in the components, especially around any defect present in the solid. Thermal
stresses along with the stresses due to mechanical loadings can give rise to stress concentration in an around the
defects and can lead to considerable damage in the structure.
In literature, problems related to defects such as cracks in solids have been studied in detail for various
kinds of solid medium. Cracks in a solid may be generated due to several reasons such as uncertainties in the
loading process, compositional defects in materials, inadequacies in the design, deficiencies in construction or
maintenance of environmental conditions, and several others. Consequently, almost all structures contain cracks,
either due to manufacturing defects or due to inappropriate thermal or mechanical loading. If proper attention to
Thermo-mechanical Analysis Of a Crack In An Infinite Functionally Graded Elastic Layer
International organization of Scientific Research 25 | Page
load condition is not paid, the size of the crack grows, catastrophically leading to a structure failure. A list of
work on crack problems by earlier investigators has been provided 3,4,5,6,7,8,9,10,11,12
etc. Among the recent works
on crack problems in solids of above mentioned characteristics, notable are the works 13,14,15,16,17,18,19,20,21
etc.
For a solid with a crack in it loaded mechanically or thermally, determination of stress intensity factor
(SIF) becomes a very important task in fracture mechanics. The SIF is a parameter that gives a measure of
stress concentration around cracks and defects in a solid. SIF needs to be understood if we are to design fracture
tolerant materials used in bridges, buildings, aircraft, or even bells. Polishing just won't do if we detect crack.
For a thermoelastic crack problem thermal stress intensity factor (TSIF) is a very important subject of physical
interest. Literature survey shows good number of papers dealing with thermal stress intensity factors. Among
them mention may be made of the works 22,23,24,25,26
etc.
The present investigation aims to find the elastostatic solution in an infinite layer with a crack in it and
is under steady state thermal loading as well as mechanical loading. Following the integral transform technique
the problem has been reduced to a problem of Cauchy type singular integral equation , which has been solved
numerically. Finally , the stress-intensity factors and the crack opening displacements are determined for various
thermal and mechanical loading conditions and the associated numerical results have been shown graphically.
II. NOMENCLATURE ฮป , ฮผ : Lame's constants for isotropic elastic material ฮฝ : Poisson's ratio of elastic material ฮฑt : Thermal expansion coefficient of the material ฯx ; ฯxy ; ฯy : Stress components in cartesian co-ordinate system
ฮบ : Thermal diffusivity of the material P : Applied load in isothermal problem
ฮด(:) : Dirac delta function p0 : Internal uniform pressure along crack surface ฮฒ : Non homogeniety parameter for elastic coefficients k(:) : Stress intensity factor in a medium with a crack in it
T : Absolute temperature
T0 : Reference temperature T1 ; T2 : Constant temperature supplied to the material in the lower and upper surfaces respectively
III. FORMULATION OF THE PROBLEM We consider an infinitely long functionally graded layer of thickness 2h weakened by the presence of
an internal crack. Cartesian system co-ordinates will be used in our analysis. We shall take y-axis along the
normal to the layer surface. The layer is infinite in a direction perpendicular to y-axis. A line crack of length
2b is assumed to be present in the middle of the layer. We shall take x-axis along the line of the crack with origin at the center of the crack and investigate the problem as a two dimensional problem in the x-y plane. The
crack faces are supposed to be acted upon by tensile force p0 and the layer surfaces are acted upon by concentrated forces at points on the layer surfaces at distance 2a , symmetrically positioned with respect to the center of the crack. The concentrated forces are either of compressive in nature or tensile in nature. Fig.1
displays the geometry of the problem.
In this figure two sided arrows actually correspond to two distinct problems: the inward drawn arrows
correspond to compressive loading , while outward drawn arrows correspond to tensile loading. The two
different types of loading are (i) a pair of concentrated compressive loads each of magnitude P
2 applied
symmetrically with respect to the center of the crack at a distance 2a apart , (ii) a pair of concentrated tensile
loads each of magnitude P
2 applied symmetrically with respect to the center of the crack at a distance 2a apart.
The gravitational force has not been taken into consideration. In deriving analytical solution in the present study
the elastic parameters ฮป and ฮผ have been assumed to vary exponentially in the direction perpendicular to the plane of the crack , following the law
ฮป = ฮป0eฮฒ y , ฮผ = ฮผ0e
ฮฒ y , โh โค y โค h, (1) where ฮป0 and ฮผ0 are the elastic parameters in the homogeneous medium and ฮฒ is the non-homogeneity parameter controlling the variation of the elastic parameters in the graded medium.
Thermo-mechanical Analysis Of a Crack In An Infinite Functionally Graded Elastic Layer
International organization of Scientific Research 26 | Page
Fig.1: Geometry of the problem.
The strain displacement relations , linear stress-strain relations and equations of equilibrium are , respectively ,
given by
ฮตx =โu
โx , ฮตy =
โv
โy , ฮณ
xy=
1
2
โu
โy+
โv
โx , (2)
๐๐ฅ =๐
๐ โ1 1 + ๐ ๐๐ฅ + 3 โ ๐ ๐๐ฆ ,
๐๐ฆ =๐
๐ โ1 3 โ ๐ ๐๐ฅ + 1 + ๐ ๐๐ฆ ,
๐๐ฅ๐ฆ = 2๐๐พ๐ฅ๐ฆ . (3)
๐๐๐ฅ
๐๐ฅ+
๐๐๐ฅ๐ฆ
๐๐ฆ= 0, (4)
๐๐๐ฅ๐ฆ
๐๐ฅ+
๐๐๐ฆ
๐๐ฆ= 0, (5)
where ๐ = 3 โ 4๐ and ๐ is Poisson's ratio. Before further proceeding it will be convenient to adopt non-dimensional variables by rescaling all length variables by the problem's length scale ๐ and the temperature variable by the reference temperature scale ๐0 :
๐ขโฒ =๐ข
๐ , ๐ฃ โฒ =
๐ฃ
๐ , ๐ฅ โฒ =
๐ฅ
๐ , ๐ฆ โฒ =
๐ฆ
๐ , โฒ =
๐,
๐ โฒ =๐
๐0 , ๐1
โฒ =๐1
๐0 , ๐2
โฒ =๐2
๐0 , ๐ผ๐ก
โฒ = ๐ผ๐ก๐0. (6)
In the analysis below, for notational convenience , we shall use only dimensionless variables and shall
ignore the dashes on the transformed non-dimensional variables. Mathematically, the problem under
consideration is reduced to the solution of thermoelasticity equations with thermal expansion coefficient ๐ผ๐ก and the quantity H , the dimensionless thermal conductivity of the crack surface defined
by Carslaw and Jaeger
27.
(๐) Equilibrium equations:
2 1 โ ๐ ๐2๐ข
๐๐ฅ2+ 1 โ 2๐
๐2๐ข
โ๐ฆ2+
๐2๐ฃ
๐๐ฅ๐๐ฆ+ ๐ฝ 1 โ 2๐
๐๐ข
๐๐ฆ+
๐๐ฃ
๐๐ฅ = 2 1 + ๐ ๐ผ๐ก
๐๐
๐๐ฅ , (7)
1 โ 2๐ ๐2๐ฃ
๐๐ฅ2+ 2 1 โ ๐
๐2๐ฃ
๐๐ฆ2+
๐2๐ข
๐๐ฅ๐๐ฆ+ ๐ฝ 2 1 โ ๐
๐๐ฃ
๐๐ฆ+ 2๐
๐๐ข
๐๐ฅ = 2 1 + ๐ ๐ผ๐ก
๐๐
๐๐ฆ , (8)
(๐๐) Steady state heat conduction equation: ๐2๐
๐๐ฅ2+
๐2๐
๐๐ฆ2= 0, 0 < ๐ฅ < โ , (9)
Thermo-mechanical Analysis Of a Crack In An Infinite Functionally Graded Elastic Layer
International organization of Scientific Research 27 | Page
and
(๐๐๐) The boundary conditions: (๐) Thermal boundary conditions:
๐ ๐ฅ , โ = ๐1 , 0 < ๐ฅ < โ , (10)
๐ ๐ฅ , = ๐2 , 0 < ๐ฅ < โ , (11)
๐
๐๐ฆ๐ ๐ฅ, 0+ =
๐
๐๐ฆ๐ ๐ฅ, 0โ = ๐ป ๐ ๐ฅ, 0+ โ ๐ ๐ฅ, 0โ , 0 < ๐ฅ < 1 , (12)
๐ ๐ฅ, 0+ = ๐ ๐ฅ, 0โ , ๐ฅ โฅ 1 , (13)
๐
๐๐ฆ๐ ๐ฅ, 0+ =
๐
๐๐ฆ๐ ๐ฅ, 0โ , ๐ฅ โฅ 1 , (14)
(๐) Elastic boundary conditions:
๐๐ฅ๐ฆ ๐ฅ , 0 = 0, 0 < ๐ฅ < โ , (15)
๐๐ฅ๐ฆ ๐ฅ , = 0, 0 < ๐ฅ < โ , (16)
๐๐ฆ ๐ฅ , = โ๐
2๐ฟ ๐ฅ โ ๐ , 0 < ๐ฅ < โ , (17)
๐
๐๐ฅ ๐ฃ ๐ฅ, 0 =
๐ ๐ฅ , 0 < ๐ฅ < 10 , ๐ฅ > 1
, (18)
๐๐ฆ ๐ฅ, 0 = โ๐0 , 0 โค ๐ฅ โค 1 , (19)
where u and v are the x and y components of the displacement vector ; ๐๐ฅ , ๐๐ฆ , ๐๐ฅ๐ฆ are the normal and
shearing stress components ; ๐(๐ฅ) is an unknown function and ๐ฟ ๐ฅ is the Dirac delta function. In Eq. (17) positive sign indicates tensile force while negative sign corresponds to compressive force.
IV. METHOD OF SOLUTION (๐) Thermal part: To determine temperature field ๐(๐ฅ, ๐ฆ) from Eq. (9) and boundary conditions (10)-(14) we assume
๐ ๐ฅ , ๐ฆ = ๐ ๐ฅ, ๐ฆ + ๐ ๐ฆ , (20) where ๐(๐ฅ, ๐ฆ) and ๐(๐ฆ) are two unknown functions satisfying the conditions
๐ ๐ฅ , โ = ๐ ๐ฅ, = 0, (21) and
๐ โ = ๐1 , ๐ = ๐2 , (22) Under these considerations we get
๐ ๐ฆ = ๐2โ๐1
2๐ฆ +
๐1+๐2
2 , (23)
and
๐ ๐ฅ, ๐ฆ = ๐๐๐ฆ โ ๐๐ 2โ๐ฆ ๐ด1 , ๐ฆ โฅ 0;
๐๐๐ฆ โ ๐โ๐ 2+y 1+๐2๐
1+๐โ2๐๐ด1 , ๐ฆ โค 0.
(24)
for certain constant ๐ด1. The appropriate temperature field satisfying the boundary conditions and regularity condition can be expressed
as:
๐ ๐ฅ , ๐ฆ = ๐๐๐ฆ โ ๐๐ 2โ๐ฆ โ
โโ๐ท ๐ ๐๐๐ฅ๐๐๐ + ๐ ๐ฆ , ๐ฆ โฅ 0 , (25)
๐ ๐ฅ , ๐ฆ = ๐๐๐ฆ โ ๐โ๐ 2+๐ฆ โ
โโ๐๐๐ ๐ท ๐ ๐
โ๐๐ฅ๐๐๐ + ๐ ๐ฆ , ๐ฆ โค 0 , (26)
where ๐ท ๐ is an unknown function to be determined and
Thermo-mechanical Analysis Of a Crack In An Infinite Functionally Graded Elastic Layer
International organization of Scientific Research 28 | Page
๐๐๐ =1+๐2๐
1+๐โ2๐ , (27)
Let us introduce the density function ๐ฉ ๐ฅ , as
๐ฉ ๐ฅ =๐
๐๐ฅ๐ ๐ฅ, 0+ โ
๐
๐๐ฅ๐ ๐ฅ, 0โ , (28)
It is clear from the boundary conditions (13) and (14) that
๐ฉ ๐ ๐๐ 1
โ1= 0 , (29)
and
๐ฉ ๐ฅ = 0 , ๐ฅ โฅ 1 , (30)
Substituting Eqs. (25) and (26) into Eq. (28) and using Fourier inverse transform , we have
๐ท ๐ =๐ 1+๐โ2๐
4๐๐ ๐โ2๐โ๐2๐ ๐ฉ ๐ ๐๐๐ ๐ ๐๐
1
โ1, (31)
Substituting Eqs. (25) and (26) into Eq. (12) and applying the relation (31) , we get the singular integral
equation for ๐ฉ ๐ฅ as follows
1
๐
1
๐ โ๐ฅ+ ๐1 ๐ฅ, ๐ ๐ฉ ๐ ๐๐
1
โ1=
๐1โ๐2
, (32)
where
๐1 ๐ฅ, ๐ = 1 โ2๐ป
๐+
2+๐2๐ +๐โ2๐
๐โ2๐โ๐2๐ ๐ ๐๐ ๐ ๐ฅ โ ๐ ๐๐
โ
0, (33)
After determining ๐ฉ ๐ from the singular integral equation (32) we have the temperature field along the axes as
๐ ๐ฅ, 0 =
1
4 ๐ ๐๐๐ ๐ฅ โ ๐ ๐ฉ ๐ ๐๐ +
๐1+๐2
2
1
โ1 , ๐ฆ โฅ 0;
โ 1
4 ๐ ๐๐๐ ๐ฅ โ ๐ ๐ฉ ๐ ๐๐ +
๐1+๐2
2
1
โ1 , ๐ฆ โค 0.
(34)
๐ 0, ๐ฆ = โ
1
4๐
1+๐โ2๐
๐โ4๐โ1 ๐โ๐ 2โ๐ฆ โ ๐โ๐๐ฆ
๐ ๐๐ ๐ ๐
๐๐๐ ๐ฉ ๐ ๐๐
1
โ1+
๐2โ๐1
2+
๐1+๐2
2
โ
0 , ๐ฆ โฅ 0;
โ1
4๐
1+๐โ2๐
๐โ4๐โ1 ๐๐๐ฆ โ ๐โ๐ 2+๐ฆ
๐ ๐๐ ๐ ๐
๐๐๐ ๐ฉ ๐ ๐๐
1
โ1+
๐2โ๐1
2+
๐1+๐2
2
โ
0 , ๐ฆ โค 0.
(35)
(๐) Elastic part: First of all we observe that due to symmetry of the crack location with respect to the layer and of the applied
load with respect to the crack , it is sufficient to consider the solution of the problem in the regions 0 โค ๐ฅ < โ and 0 โค ๐ฆ โค . To solve the partial differential equations (7) and (8), Fourier transform is applied to the equations with respect to the variable x.
Utilizing the symmetric condition the displacement components ๐ข , ๐ฃ may be written as
๐ข ๐ฅ , ๐ฆ =2
๐ ๐ท ๐, ๐ฆ ๐ ๐๐ ๐๐ฅ ๐๐
โ
0
, (36)
๐ฃ ๐ฅ , ๐ฆ =2
๐ ๐น ๐, ๐ฆ ๐๐๐ ๐๐ฅ ๐๐
โ
0
, (37)
where ๐ท ๐, ๐ฆ and ๐น ๐, ๐ฆ are Fourier transforms of ๐ข(๐ฅ, ๐ฆ) and ๐ฃ ๐ฅ, ๐ฆ , respectively with respect to the coordinate ๐ฅ , and ๐ is the transformed parameter.
Thermo-mechanical Analysis Of a Crack In An Infinite Functionally Graded Elastic Layer
International organization of Scientific Research 29 | Page
Substituting ๐ข(๐ฅ, ๐ฆ) and ๐ฃ(๐ฅ, ๐ฆ) from Eqs. (36) and (37) into the equations of equilibrium (7) and (8) we obtain the differential equation for the determination of ๐ท ๐, ๐ฆ ,
๐น ๐ท1 ๐ท = ๐ผ๐ก๐ ๐ โ 1 7 โ ๐
๐ + 1 ๐2๐๐
๐๐ฆ2+
1
2๐ผ๐ก๐ 7 โ ๐ 3 โ ๐ + ๐ฝ
7 โ ๐ ๐ โ 1 2
๐ + 1 ๐๐๐
๐๐ฆ
+7 โ ๐
4 ๐ + 1 ๐ผ๐ก๐ ๐ฝ 3 โ ๐ ๐ + 1 ๐ โ 1 + ๐ฝ + 4๐
2 ๐๐ , (38)
where
๐น ๐ท1 = ๐ท14 + 2๐ฝ๐ท1
3 โ 2๐2 โ ๐ฝ2 ๐ท12 โ 2๐2๐ฝ๐ท1 + ๐
2 ๐2 โ ๐ โ 3
๐ + 1 ๐ฝ2 , (39)
and ๐๐ ๐, ๐ฆ is the Fourier Cosine transform of ๐(๐ฅ, ๐ฆ) defined by
๐๐ ๐ , ๐ฆ = ๐ ๐ฅ, ๐ฆ ๐๐๐ ๐๐ฅ ๐๐ฅ
โ
0
, (40)
The solution of the Eq. (38) is of the form
๐ท ๐, ๐ฆ = ๐ด๐ ๐ ๐๐ ๐๐ฆ + ๐๐
๐ ๐
4
๐=1
, ๐ = 1,2 (41)
where ๐ด๐ ๐ , ๐ = 1, โฆ . ,4 are constants to be determined from the boundary conditions , ๐๐ ๐ = 1, โฆ . ,4 are the four complex roots of two biquadratic equation
๐4 + 2๐ฝ๐3 โ 2๐2 โ ๐ฝ2 ๐2 โ 2๐2๐ฝ๐ + ๐2 ๐2 โ๐ โ 3
๐ + 1๐ฝ2 = 0, (42)
and
๐1 ๐ ๐ =
1
๐น ๐ท1
7โ๐ ๐ผ๐ก๐
4 ๐ +1 ๐ฝ 3 โ ๐ ๐ + 1 ๐ โ 1 + ๐ฝ + 4 ๐2
โ
โโ
+๐
2 7 โ ๐ 3 โ ๐ ๐ผ๐ก๐ +
๐ผ๐ก๐ฝ ๐ โ1 2 7โ๐ ๐
๐ +1
+ ๐2 ๐ โ1 7โ๐ ๐ผ๐ก๐
๐ +1
๐๐
๐2โ๐2๐ท ๐ ๐๐๐
2โ๐ ๐๐๐ฆ๐๐ , ๐ = 1,2 (43)
๐2 ๐ ๐ =
1
๐น ๐ท1 โ
7โ๐ ๐ผ๐ก๐
4 ๐ +1 ๐ฝ 3 โ ๐ ๐ + 1 ๐ โ 1 + ๐ฝ + 4 ๐2
โ
โโ
+๐
2 7 โ ๐ 3 โ ๐ ๐ผ๐ก๐ +
๐ผ๐ก๐ฝ ๐ โ1 2 7โ๐ ๐
๐ +1
โ ๐2 ๐ โ1 7โ๐ ๐ผ๐ก๐
๐ +1
๐๐
๐2โ๐2๐ท ๐ ๐๐๐
2โ๐ ๐ โ1 ๐ .๐ 2โ๐ฆ ๐๐ , ๐ = 1,2 (44)
๐3 ๐ ๐ =
1
๐น ๐ท1
7โ๐ ๐ผ๐ก๐
4 ๐ +1 ๐ฝ 3 โ ๐ ๐ + 1 ๐ โ 1 + ๐ฝ + 4๐2
๐2โ๐1
๐๐ฟ ๐ ๐ฆ , ๐ = 1,2 (45)
๐4 ๐ ๐ =
1
๐น ๐ท1
1
2 7 โ ๐ 3 โ ๐ ๐ผ๐ก๐ +
๐ผ๐ก๐ฝ ๐ โ1 2 7โ๐ ๐
๐ +1
Thermo-mechanical Analysis Of a Crack In An Infinite Functionally Graded Elastic Layer
International organization of Scientific Research 30 | Page
โ 7โ๐ ๐ผ๐ก๐
4 ๐ +1 ๐ฝ 3 โ ๐ ๐ + 1 ๐ โ 1 + ๐ฝ + 4๐2
๐2โ๐1
๐๐ฟ ๐ , ๐ = 1,2 (46)
where ๐ = 1, 2 are for the lower and upper region respectively. The function ๐น ๐, ๐ฆ can then be determined as
๐น ๐, ๐ฆ = ๐๐ ๐ ๐ด๐ ๐ ๐๐ ๐๐ฆ + ๐๐๐๐
๐ ๐
4
๐=1
, ๐ = 1,2 (47)
where
๐๐ ๐ = ๐ โ1 ๐ ๐
2+๐ฝ ๐ โ1 ๐ ๐โ๐2 ๐ +1
๐ 2๐ ๐+๐ฝ ๐ โ1 , ๐ = 1, โฆ . ,4 , (48)
๐๐ = ๐ โ1 ๐ท1
2+๐ฝ ๐ โ1 ๐ท1โ๐2 ๐ +1
๐ 2๐ ๐+๐ฝ ๐ โ1 , ๐ = 1, โฆ . ,4 , (49)
It follows from Eq. (42) that ๐3 = ๐ 1 and ๐4 = ๐ 2 where
๐1 = โ๐ฝ
2+ ๐2 +
๐ฝ2
4+ ๐๐๐ฝ
3โ๐
๐ +1 , (50)
๐2 = โ๐ฝ
2โ ๐2 +
๐ฝ2
4+ ๐๐๐ฝ
3โ๐
๐ +1 , (51)
Substituting Eqs. (36) and (37) into Eqs. (2) and (3) and utilizing Eqs. (41) and (47) we obtain
1
2๐๐๐ฅ ๐ฅ, ๐ฆ =
2
๐
1
2 1 โ ๐ โ 1 + ๐ ๐ โ 3 โ ๐ ๐๐๐๐ ๐ด๐๐
๐ ๐๐ฆ 4
๐=1
โ
0
+ โ๐ 1 + ๐ ๐๐ ๐
โ 3 โ ๐ ๐ท1 ๐๐๐๐ ๐
๐๐๐ ๐๐ฅ ๐๐ , ๐ = 1,2 (52)
1
2๐๐๐ฆ ๐ฅ, ๐ฆ =
2
๐
1
2 1 โ ๐ โ 1 + ๐ ๐๐๐๐ โ 3 โ ๐ ๐ ๐ด๐๐
๐ ๐๐ฆ 4
๐=1
โ
0
+ โ 1 + ๐ ๐ท1 ๐๐๐๐ ๐
โ 3 โ ๐ ๐ ๐๐ ๐
๐๐๐ ๐๐ฅ ๐๐ , ๐ = 1,2 (53)
1
2๐๐๐ฅ๐ฆ ๐ฅ, ๐ฆ =
2
๐ โ
๐
2๐๐ +
๐๐2
๐ด๐๐๐ ๐๐ฆ
4
๐=1
โ
0
+ โ๐
2๐๐๐๐
๐ +
1
2๐ท1 ๐๐
๐ ๐ ๐๐ ๐๐ฅ ๐๐ , ๐ = 1,2 (54)
From the boundary conditions (15)-(18),the unknown constants ๐ด๐ ๐ = 1, โฆ ,4 can be found out from the following linear algebraic system of equations expressed in matrix form:
๐ฟ๐ด = ๐ต , (55)
where the matrices ๐ฟ , ๐ด , ๐ต are
๐ฟ =
๐1๐
๐1 ๐2๐๐2 ๐3๐
๐3
๐บ1๐๐1 ๐บ2๐
๐2 ๐บ3๐๐3
๐1 โ ๐บ1 ๐2 โ ๐บ2 ๐3 โ ๐บ3
๐4๐๐4
๐บ4๐๐4
๐4 โ ๐บ4๐บ1 ๐บ2 ๐บ3 ๐บ4
, ๐ด =
๐ด1๐ด2๐ด3๐ด4
, ๐ต =
โ๐ 1 โ ๐1๐2
โ๐ 2 โ ๐3๐4
and
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๐ 1 ๐ = ยฑ ๐ โ 1
2๐๐ ๐๐๐ ๐๐ + ๐๐๐ โ๐๐ , ๐1 = ๐ท4๐
โฒ ๐๐ ๐
,
4
๐=1
๐ 2 ๐ = ๐ ๐ฅ ๐ ๐๐ ๐๐ฅ ๐๐ฅ , ๐2 = ๐ท4๐๐๐ ๐
,
4
๐=1
โ
0
๐3 = ๐๐๐๐0 ๐
, ๐4 = ๐ท4๐๐๐0 ๐
, ๐ = 1,2,
4
๐=1
4
๐=1
(56)
๐๐ = 1 + ๐ ๐๐๐๐ + 3 โ ๐ ๐ ,
๐บ๐ = โ๐๐๐ + ๐๐ , ๐ = 1, โฆ . . ,4 .
๐ท4๐ = ๐ ๐๐ โ1
๐๐ท1 , ๐ = 1, โฆ โฆ ,4 , (57)
๐ท4๐โฒ = 1 + ๐ ๐ท1๐๐ + 3 โ ฮบ ๐ , ๐ = 1, โฆ โฆ ,4 . (58)
Eq. (55) yields
๐ด๐ ๐ = ๐ท1๐ ๐ ๐ 1 ๐ + ๐ท2๐ ๐ ๐ 2 ๐ + ๐ท3๐ ๐ , ๐ = 1, โฆ โฆ . . ,4 , (59)
where ๐ท๐๐ ๐ = 1,2,3 ๐๐๐ ๐ = 1, โฆ โฆ . . ,4 are shown in Appendix. Substitution of these values into the Eq. (19) will lead to the following singular integral equation:
1
๐ ๐ ๐ก
1
๐ก โ ๐ฅ+ ๐2 ๐ฅ, ๐ก ๐๐ก =
๐ โ 1
๐๐ โ๐0 ยฑ
๐
2๐๐3 ๐ฅ + ๐0๐4 ๐ฅ , โ๐ < ๐ฅ < ๐ (60)
๐
โ๐
where
๐2 ๐ฅ, ๐ก =1
๐ ๐ท2๐๐๐ โ ๐ ๐ ๐๐ ๐ ๐ก โ ๐ฅ ๐ ๐ , ๐ = ๐๐๐
๐โโ ๐ท2๐๐๐ = 4๐ , (61)
4
๐=1
4
๐=1
โ
0
๐3 ๐ฅ = โ2 ๐ท1๐๐๐ ๐๐๐ ๐ ๐ + ๐ฅ ๐๐ , (62)
4
๐=1
โ
0
๐4 ๐ฅ = โ2๐
๐๐0 ๐ โ 1 ๐ท3๐๐๐ โ ๐ท4๐
โฒ ๐๐0 ๐
๐๐๐ ๐๐ฅ ๐๐ , ๐ = 1,2 63
4
๐=1
โ
0
The kernels ๐2(๐ฅ , ๐ก) and ๐3 ๐ฅ , ๐4(๐ฅ) are bounded and continuous in the closed interval โ๐ โค ๐ฅ โค ๐. The integral equation must be solved under the following single-valuedness condition
๐ ๐ก ๐๐ก = 0 , (64)
๐
โ๐
Before further proceeding it will be convenient to introduce non-dimensional variables ๐ and ๐ by rescaling all lengths in the problem by length scale ๐:
๐ฅ = ๐๐ , ๐ก = ๐๐ , (65)
๐ ๐ก = ๐ ๐๐ = ๐0 ๐ โ 1
๐๐๐ ๐ , ๐ = ๐๐ , (66)
In terms of non-dimensional variables the integral equation (60) and single valuedness condition (64) become
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1
๐
1
๐ โ ๐+ ๐2
โ ๐, ๐ ๐ ๐ ๐๐ = โ1 ยฑ๐
๐๐3
โ ๐ + ๐4โ ๐ , โ1 < ๐ < 1 , (67)
1
โ1
where
๐2โ ๐, ๐ =
1
๐ ๐ท2๐๐๐ โ ๐
4
๐=1
๐ ๐๐ ๐ ๐ โ ๐ ๐๐ , (68)
โ
0
๐3โ ๐ = โ2 ๐ท1๐๐๐ ๐๐๐ ๐ ๐ + ๐
โ ๐๐ , (69)
4
๐=1
โ
0
๐4โ ๐ = โ
2๐0๐๐0๐ ๐ โ 1
๐ท3๐๐๐ โ ๐ท4๐โฒ ๐๐0
๐ ๐๐๐ ๐๐ ๐๐ , (70)
4
๐=1
โ
0
๐โ =๐
๐ and ๐ is the load ratio defined as:
๐ =๐
2๐๐0 , (71)
๐๐0 ๐
โก ๐๐ ๐
๐ฆ=0 ,
๐๐ ๐
โก ๐๐ ๐
๐ฆ= ,
๐ = 1, 2,3 ; ๐ = 1, 2 , ๐ท1 โก๐
๐๐ฆ . (72)
V. SOLUTION OF THE INTEGRAL EQUATIONS (๐) Thermal part: The singular integral equation (32) is a Cauchy-type singular integral equation for an unknown function ๐ฉ ๐ . For the evaluation of thermal stress it is necessary to solve the integral equation (32). For this purpose we write
๐ฉ ๐ =๐ถ ๐
1 โ ๐ 2 , โ1 < ๐ < 1 , (73)
where ๐ถ ๐ is a regular and bounded unknown function. Substituting Eq. (73) into Eq. (32) and using Gauss-Chebyshev formula
28 , we obtain
1
๐
1
๐ ๐ โ ๐ฅ๐+ ๐1 ๐ฅ๐ , ๐ ๐ ๐ถ ๐ ๐
๐
๐=1
=๐1 โ ๐2
, ๐ = 1,2, โฆ โฆ โฆ . , ๐ โ 1 , (74)
and
๐
๐ ๐ถ ๐ ๐ = 0 , (75)
๐
๐=1
where ๐ ๐ and ๐ฅ๐ are given by
๐ ๐ = ๐๐๐ 2๐ โ 1
2๐๐ , ๐ = 1,2,3, โฆ โฆ , ๐ (76)
๐ฅ๐ = ๐๐๐ ๐๐
๐ , ๐ = 1,2,3, โฆ โฆ , ๐ โ 1 (77)
We observe that corresponding to (๐ โ 1) collocation points ๐ฅ๐ = ๐๐๐ ๐๐
2 ๐+1 , ๐ = 1,2, โฆ โฆ โฆ , ๐ โ 1 we
have a set of ๐ linear equations in ๐ unknowns ๐ถ ๐ 1 , ๐ถ ๐ 2 , โฆ โฆ โฆ โฆ โฆ โฆ . , ๐ถ ๐ ๐ . This linear algebraic system of equations is solved numerically by utilizing Gaussian elimination method.
(๐) Elastic part: The singular integral equation (67) is a Cauchy-type singular integral equation for an unknown function ๐ ๐ . Expressing now the solution of Eq. (67) in the form
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๐ ๐ =๐ ๐
1 โ ๐ 2 , โ1 < ๐ < 1 , (78)
where ๐(๐ ) is a regular and bounded unknown function and using Gauss-Chebyshev formula28 to evaluate the integral equation (67) , we obtain ,
1
๐
1
๐ ๐ โ ๐๐+ ๐2
โ ๐๐ , ๐ ๐ ๐ ๐ ๐
๐
๐=1
= โ1 ยฑ๐
๐๐3
โ ๐๐ + ๐4โ ๐๐ , ๐ = 1,2, โฆ โฆ . . , ๐ โ 1 (79)
๐
๐ ๐ ๐ ๐ = 0 , (80)
๐
๐=1
where ๐๐ is given by
๐๐ = ๐๐๐ ๐๐
๐ , ๐ = 1,2,3, โฆ โฆ . , ๐ โ 1 (81)
We observe that corresponding to (๐ โ 1) collocation points ๐ฅ๐ = ๐๐๐ ๐๐
2 ๐+1 , ๐ = 1,2, โฆ โฆ โฆ , ๐ โ 1 the
Eqs. (79) and (80) represent a set of ๐ linear equations in ๐ unknowns ๐ ๐ 1 , ๐ ๐ 2 , โฆ โฆ โฆ โฆ โฆ โฆ . , ๐ ๐ ๐ . This linear algebraic system of equations are solved numerically by utilizing Gaussian elimination method.
VI. DETERMINATION OF STRESS-INTENSITY FACTOR Presence of a crack in a solid significantly affects the stress distribution compared to the state when
there is no crack. While the stress distribution in a solid with a crack in the region far away from the crack is not
much disturbed , the stresses in the neighbourhood of the crack tip assumes a very high magnitude. In order to
predict whether the crack has a tendency to expand further , the stress intensity factor (SIF) , a quantity of
physical interest , has been defined in fracture mechanics. In our present problem the solid under consideration
is acted upon by two types of loading: (๐) Thermal loading (๐) Mechanical loading (concentrated forces of
magnitude ๐
2 applied symmetrically on the crack faces). The stress components ๐๐ฅ , ๐๐ฆ , ๐๐ฅ๐ฆ given by Eqs.
(52)-(54) do not have closed form expressions. As they are expressed in taking of infinite integrals , they are to
be evaluated numerically. In our present discussion we shall be interested to determine the SIF when both the
thermal and mechanical loadings are present (thermomechanical stress intensity factor) and also the stress
intensity factor when only thermal loading is present (thermal stress intensity factor). The stress intensity factor
is defined as
๐ ๐ = ๐๐๐๐โ1
2๐ ๐ โ 1 ๐๐ฆโ ๐, 0 .
The non-dimensional stress intensity factor when both mechanical and thermal load are present (TMSIF) , can
be obtained interms of the solution of the integral equation (67) as
๐๐๐๐ผ๐น = ๐โฒ ๐ =1
๐0 ๐ ๐๐๐
๐โ1 2๐ ๐ โ 1 ๐๐ฆ
โ ๐, 0 = โ๐ 1 , (82)
When there is only thermal loading , the TSIF can be extracted following the same procedure as discussed
above. In this case the Eq.(67) will be modified to
1
๐
1
๐ โ ๐+ ๐2
โ ๐, ๐ ๐ ๐ ๐๐ = โ1 + ๐4โ ๐ , โ1 < ๐ < 1 , (83)
1
โ1
The TSIF at the crack tip can be expressed in terms of the solution of the integral equation (83) as
๐๐๐ผ๐น =1
๐0 ๐ ๐๐๐
๐โ1 2๐ ๐ โ 1 ๐๐ฆ
โ ๐, 0 = โ๐บ 1 , (84)
in this case we assume
๐ ๐ =๐บ ๐
1 โ ๐ 2 , โ1 < ๐ < 1, (85)
where ๐บ(๐ ) is regular and bounded unknown function and ๐ 1 , ๐บ(1) can be found out from ๐ ๐ ๐ and ๐บ ๐ ๐ ๐ = 1,2,3, โฆ โฆ โฆ . . , ๐ using the interpolation formulas given by Krenk
29 .
Following the method as in Gupta and Erdogan30
we obtain the crack surface displacement in the form
๐ฃโฒ ๐, 0 = ๐ ๐
1 โ ๐ 2๐๐ก , โ1 < ๐ < 1 , (86)
๐
โ1
where
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๐ฃโฒ ๐, 0 =๐ฃ ๐, 0
๐
4๐๐
๐0 ๐ โ 1 , (87)
which can be obtained numerically , using say , Simpson's 1
3 integration formula and appropriate interpolation
formula.
VII. NUMERICAL RESULTS AND DISCUSSIONS The present study is related to the study of an internal crack problem in an infinite functionally graded
layer under thermal effect. The main objective of the present discussion is to study the effects of temperature ,
graded parameter as well as of applied loads on stress intensity factor and crack opening displacement.
Following the standard numerical method described , the normal displacement component and the stress
intensity factor are computed and shown graphically.
Before analyzing our numerical results we denote the regions โ โค ๐ฆ โค 0 and 0 โค ๐ฆ โค below and above the line of crack in the layer by ๐ 1 and ๐ 2 respectively. Fig. 2(a) shows the temperature distribution on
the crack faces for various values of ๐
. As expected , the result shows that the temperature increases with the
decrease of layer thickness. Fig. 2(b) shows the temperature distribution along ๐ฅ = 0 ,taking ๐1 > ๐2 . Temperature decreases linearly from the region ๐ 1 to the region ๐ 2 . There is one point to note here that the variations of temperature at a particular point on ๐ฅ = 0 below and above the line of crack are opposite in
nature in respect of the values of ๐
.
Fig.2(a) Temperature distribution on crack face and extension line for different ๐
when ๐ = 1.8 , ๐ป = 1.0, ๐ผ๐ก =
1.5, ๐1 = 2.5, ๐2 = 1.0. (b) Temperature distribution on the line ๐ฅ = 0 for different ๐
when ๐ = 1.8, ๐ป = 1.0,
๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0.
The variation of normalized thermo-mechanical stress intensity factor (TMSIF) ๐โฒ(๐) with crack length ๐
are shown in Fig. 3 for both the cases of two symmetric transverse pair of compressive and tensile
concentrated forces. It is observed from Figs. 3(a , b) that for compressive concentrated forces the TMSIF
decreases with the increase of the load ratio ๐, and the increase of ๐โฒ(๐) is quite significant for smaller values of ๐. It is also observed from Figs. 3(a,b) that the load ratio ๐ does not have much effect on ๐โฒ(๐) when the crack length is sufficiently small. Contrary to this , where the force is of tensile nature , ๐โฒ(๐) increases with ๐. For small crack length , the behavior of ๐โฒ(๐) is similar to the case of compressive concentrated load. In Fig. 4 , TMSIF experiences the effect of graded parameter ๐ฝ for fixed load ratio ๐ for both the regions ๐ 1 and ๐ 2. It is observed that in both compressive and tensile load conditions ๐โฒ(๐) increases slightly with graded parameter ๐ฝ upto a certain distance from the center of the crack , while the effect is reversed and significant afterward. Fig. 5 displays the variation of ๐โฒ(๐) for different position of loading. It is noted that in the case of
compressive concentrated forces , ๐โฒ(๐) increases with increasing ๐
๐ , but it decreases in the case of tensile
concentrated forces.
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Fig.3(a) Variation of TMSIF, ๐โฒ ๐ with ๐
for different loads ๐ in both cases in the region ๐ 2 when (
๐
๐=
0.0, ๐ = 1.8, ๐ฝ = 0.1, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0). (b) Variation of TMSIF, ๐โฒ ๐ with ๐
for different
loads ๐ in both cases in the region ๐ 1 when (๐
๐= 0.0, ๐ = 1.8, ๐ฝ = 0.1, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0).
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Fig.4(a) Effect of graded parameter ๐ฝ on TMSIF, ๐โฒ ๐ for both cases in the region ๐ 2 (
๐
๐= 0.0, ๐ = 12.0, ๐ =
1.8, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0). (b) Effect of graded parameter ๐ฝ on TMSIF,๐โฒ ๐ for both cases in
the region ๐ 1(๐
๐= 0.0, ๐ = 1.0 (Compressive),๐ = 2.0 (Tensile),๐ = 1.8, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 =
1.0).
Fig.5(a)Variation of TMSIF,๐โฒ ๐ for different values of
๐
๐ for both cases in the region ๐ 2 (๐ = 16.0
(Compressive), ๐ = 12.0(Tensile),๐ผ = 1.8, ๐ฝ = 0.1, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0). (b) Variation of
TMSIF, ๐โฒ ๐ for different values of ๐
๐ for both cases in the region ๐ 1(๐ = 2.0, ๐ = 1.8, ๐ฝ = 0.1, ๐ป =
1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0).
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Fig. 6 depicts the variation of normalized crack surface displacement ๐ฃโฒ ๐, 0 with ๐ for diffeerent values of load ratio ๐. It is clear from Figs. 6(a , b) that for compressive nature of forces ๐ฃโฒ ๐, 0 decreases as load ratio ๐ increases, but for tensile nature of loading ๐ฃโฒ ๐, 0 decreases as load ratio ๐ decreases. For both the cases of compressive and tensile concentrated forces the graphs show that the normalized crack surface
displacement is symmetrical with respect to origin. The effect of graded parameter ๐ฝ on ๐ฃโฒ ๐, 0 is observed in Fig. 7 for both the cases of compressive and tensile concentrated forces for the both the regions ๐ 1 and ๐ 2.
Fig.6(a) Normalized crack surface displacement ๐ฃโฒ ๐, 0 for various values of ๐ for both cases in the region
๐ 2 (๐
๐= 0.0,
๐
= 1.0 , ๐ = 1.8, ๐ฝ = 0.1, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0). (b) Normalized crack surface
displacement ๐ฃโฒ ๐, 0 for various values of ๐ for both cases in the region ๐ 1(๐
๐= 0.0,
๐
= 1.0, ๐ = 1.8, ๐ฝ =
0.1, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0).
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Fig.7(a) Effect of graded parameter ๐ฝ on normalized crack surface displacement ๐ฃโฒ ๐, 0 for both cases in the
region ๐ 2 ( ๐
๐= 0.0,
๐
= 1.0, ๐ = 6 (Compressive), ๐ = 16 (Tensile), ๐ = 1.8, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 =
2.5, ๐2 = 1.0). (b) Effect of graded parameter ๐ฝ on normalized crack surface displacement ๐ฃโฒ ๐, 0 for both
cases in the region ๐ 1 ( ๐
๐= 0.0,
๐
= 1.0, ๐ = 1, ๐ = 1.8, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0).
Fig. 8 illustrates the role of the point of application of loading on the normalized crack surface
displacement for a particular load ratio ๐ = 6.0 for the compressive forces while ๐ = 12.0 for the tensile forces for the region ๐ 2 and the load ratio ๐ = 1.0 for both types of compressive and tensile forces for the
region ๐ 1 with ๐
= 1.0. It is observed in Figs. 6.8 (a , b) , that for compressive concentrated loading the
normalized crack surface displacement increases with the increased values of ๐
๐ but behavior is just opposite for
tensile concentrated loading.
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Fig.8(a)Normalized crack surface displacement ๐ฃโฒ ๐, 0 for various values of
๐
๐ for both cases in the region
๐ 2 (๐ = 6.0 (Compressive),๐ = 12(Tensile), ๐
= 1.0, ๐ = 1.8, ๐ฝ = 0.1, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5,
๐2 = 1.0). (b) Normalized crack surface displacement ๐ฃโฒ ๐, 0 for various values of ๐
๐ for both cases in the
region ๐ 1 (๐ = 1.0,๐
= 1.0, ๐ = 1.8, ๐ฝ = 0.1, ๐ป = 1.0, ๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0).
Fig. 9 displays the comparison of the variation of thermo-mechanical stress intensity factor (TMSIF) and
thermal stress intensity factor (TSIF) with ๐
for both the regions ๐ 1 and ๐ 2 .
Fig.9 Comparison of TMSIF and TSIF for both the regions ๐ 1 and ๐ 2 (
๐
๐= 0.0 , ๐ = 1.8, ๐ฝ = 0.1, ๐ป = 1.0,
๐ผ๐ก = 1.5, ๐1 = 2.5, ๐2 = 1.0).
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VIII. CONCLUSION The present discussion relating thermo-mechanical loading on a functionally graded layer with a crack in it
yielded the following observations:
(a) The SIF and the crack surface displacement ๐ฃ(๐ฅ, 0) are very much affected by thermal loading. (b) The position and the magnitude of mechanical loading affects the SIF and ๐ฃ(๐ฅ, 0). (c) Grading of medium has significant effects on SIF and ๐ฃ(๐ฅ, 0). Both ๐ฃ(๐ฅ, 0) and SIF increase with increase of rigidity.
(d) The crack length to layer thickness ratio also plays important roles on the behavior of SIF and ๐ฃ(๐ฅ, 0).
IX. APPENDIX We set ๐ฅ = ๐ฟ โ1 , then for ๐ท๐๐ (๐ = 1,2,3 and ๐ = 1, โฆ โฆ ,4) we have the following expressions:
๐ท11 ๐ = ๐ฅ ๐4๐บ2๐บ3 โ ๐3๐บ2๐บ4 ๐๐2 + ๐2๐บ3๐บ4 โ ๐4๐บ2๐บ3 ๐
๐3 + ๐3๐บ2๐บ4 โ ๐2๐บ3๐บ4 ๐๐4 ,
๐ท12 ๐ = ๐ฅ ๐3๐บ1๐บ4 โ ๐4๐บ1๐บ3 ๐๐1 + ๐4๐บ1๐บ3 โ ๐1๐บ3๐บ4 ๐
๐3 + ๐1๐บ3๐บ4 โ ๐3๐บ1๐บ4 ๐๐4 ,
๐ท13 ๐ = ๐ฅ ๐4๐บ1๐บ2 โ ๐2๐บ1๐บ4 ๐๐1 + ๐1๐บ2๐บ4 โ ๐4๐บ1๐บ2 ๐
๐2 + ๐2๐บ1๐บ4 โ ๐1๐บ2๐บ4 ๐๐4 ,
๐ท14 ๐ = ๐ฅ ๐2๐บ1๐บ3 โ ๐3๐บ1๐บ2 ๐๐1 + ๐3๐บ1๐บ2 โ ๐1๐บ2๐บ3 ๐
๐2 + ๐1๐บ2๐บ3 โ ๐2๐บ1๐บ3 ๐๐3 ,
๐ท21 ๐ = ๐ฅ ๐บ3๐บ4๐2 โ ๐บ2๐บ3๐4 ๐ ๐2+๐4 + ๐บ2๐บ3๐4 โ ๐บ2๐บ4๐3 ๐
๐3+๐4 + ๐บ2๐บ4๐3 โ ๐บ3๐บ4๐2 ๐
๐2+๐3 ,
๐ท22 ๐ = ๐ฅ ๐บ1๐บ3๐4 โ ๐บ3๐บ4๐1 ๐ ๐1+๐4 + ๐บ1๐บ4๐3 โ ๐บ1๐บ3๐4 ๐
๐3+๐4 + ๐บ3๐บ4๐1 โ ๐บ1๐บ4๐3 ๐
๐1+๐3 ,
๐ท23 ๐ = ๐ฅ ๐บ2๐บ4๐1 โ ๐บ1๐บ2๐4 ๐ ๐1+๐4 + ๐บ1๐บ2๐4 โ ๐บ1๐บ4๐2 ๐
๐2+๐4 + ๐บ1๐บ4๐2 โ ๐บ2๐บ4๐1 ๐
๐1+๐2 ,
๐ท24 ๐ = ๐ฅ ๐บ1๐บ2๐3 โ ๐บ2๐บ3๐1 ๐ ๐1+๐3 + ๐บ1๐บ3๐2 โ ๐บ1๐บ2๐3 ๐
๐2+๐3 + ๐บ2๐บ3๐1 โ ๐บ1๐บ3๐2 ๐
๐1+๐2 ,
๐ท31 ๐ = ๐ฅ ๐4๐2๐2๐บ3 + ๐4๐1๐บ2๐บ3 โ ๐3๐2๐2๐บ4 โ ๐3๐1๐บ2๐บ4 ๐๐2
+ ๐2๐2๐3๐บ4 + ๐2๐1๐บ3๐บ4 โ ๐4๐2๐3๐บ2 โ ๐4๐1๐บ2๐บ3 ๐๐3
+ ๐3๐2๐4๐บ2 + ๐3๐1๐บ2๐บ4 โ ๐2๐2๐4๐บ3 โ ๐2๐1๐บ3๐บ4 ๐๐4
+ ๐4๐4๐3๐บ2 โ ๐4๐4๐2๐บ3 + ๐3๐3๐บ2๐บ4 โ ๐4๐3๐บ2๐บ4 โ ๐3๐2๐บ3๐บ4 + ๐4๐2๐บ3๐บ4 ๐ ๐2+๐3
+ โ๐3๐4๐4๐บ2 โ ๐3๐4๐บ2๐บ3 + ๐4๐4๐บ2๐บ3 + ๐3๐4๐2๐บ4 + ๐3๐2๐บ3๐บ4 โ ๐4๐2๐บ3๐บ4 ๐ ๐2+๐3
+ ๐2๐4๐4๐บ3 + ๐3๐4๐บ2๐บ3 โ ๐4๐4๐บ2๐บ3 โ ๐2๐4๐3๐บ4 โ ๐3๐3๐บ2๐บ4 + ๐4๐3๐บ2๐บ4 ๐ ๐3+๐4 ,
๐ท32 ๐ = ๐ฅ โ๐4๐2S1๐บ3 โ ๐4๐1๐บ1๐บ3 + ๐3๐2๐1๐บ4 + ๐3๐1๐บ1๐บ4 ๐๐1
+ ๐4๐2๐3๐บ1 + ๐4๐1๐บ1๐บ3 โ ๐1๐2๐3๐บ4 โ ๐1๐1๐บ3๐บ4 ๐๐3
+ โ๐3๐2๐4๐บ1 + ๐1๐2๐4๐บ3 โ ๐3๐1๐บ1๐บ4 + ๐1๐1๐บ3๐บ4 ๐๐4
+ โ๐4๐4๐3๐บ1 + ๐4๐4๐1๐บ3 โ ๐3๐3๐บ1๐บ4 + ๐4๐3๐บ1๐บ4 + ๐3๐1๐บ3๐บ4 โ ๐4๐1๐บ3๐บ4 ๐ ๐1+๐3
+ ๐3๐4๐4๐บ1 + ๐3๐4๐บ1๐บ3 โ ๐4๐4๐บ1๐บ3 โ ๐3๐4๐1๐บ4 โ ๐3๐1๐บ3๐บ4 + ๐4๐1๐บ3๐บ4 ๐ ๐1+๐4
+ โ๐4๐4๐4๐บ3 โ ๐3๐4๐บ1๐บ3 + ๐4๐4๐บ1๐บ3 + ๐1๐4๐3๐บ4 + ๐3๐3๐บ1๐บ4 โ ๐4๐3๐บ1๐บ4 ๐ ๐3+๐4 ,
๐ท33 ๐ = ๐ฅ ๐4๐2๐1๐บ2 + ๐4๐1๐บ1๐บ2 โ ๐2๐2๐1๐บ4 โ ๐2๐1๐บ1๐บ4 ๐๐1
+ โ๐4๐2๐2๐บ1 โ ๐4๐1๐บ1๐บ2 + ๐1๐2๐2๐บ4 + ๐1๐1๐บ2๐บ4 ๐๐2
+ ๐2๐2๐4๐บ1 โ ๐1๐2๐4๐บ2 + ๐2๐1๐บ1๐บ4 โ ๐1๐1๐บ2๐บ4 ๐๐4
+ ๐4๐4๐2๐บ1 โ ๐4๐4๐1๐บ2 + ๐3๐2๐บ1๐บ4 โ ๐4๐2๐บ1๐บ4 โ ๐3๐1๐บ2๐บ4 + ๐4๐1๐บ2๐บ4 ๐ ๐1+๐2
+ โ๐2๐4๐4๐บ1 โ ๐3๐4๐บ1๐บ2 + ๐4๐4๐บ1๐บ2 + ๐2๐4๐1๐บ4 + ๐3๐1๐บ2๐บ4 โ ๐4๐1๐บ2๐บ4 ๐ ๐1+๐4
+ ๐1๐4๐4๐บ2 + ๐3๐4๐บ1๐บ2 โ ๐4๐4๐บ1๐บ2 โ ๐1๐4๐2๐บ4 โ ๐3๐2๐บ1๐บ4 + ๐4๐2๐บ1๐บ4 ๐ ๐2+๐4 ,
๐ท34 ๐ = ๐ฅ โ๐3๐2๐1๐บ2 โ ๐3๐1๐บ1๐บ2 + ๐2๐2๐1๐บ3 + ๐2๐1๐บ1๐บ3 ๐๐1
+ ๐3๐2๐2๐บ1 + ๐3๐1๐บ1๐บ2 โ ๐1๐2๐2๐บ3 โ ๐1๐1๐บ2๐บ3 ๐๐2
Thermo-mechanical Analysis Of a Crack In An Infinite Functionally Graded Elastic Layer
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+ โ๐2๐2๐3๐บ1 + ๐1๐2๐3๐บ2 โ ๐2๐1G1G3 + m1P1G2G3 em3h
+ โm3P4S2G1 + m3P4S1G2 โ P3S2G1G3 + P4S2G1G3 + P3S1G2G3 โ P4S1G2G3 e m1+m2 h
+ m2P4S3G1 + P3S3G1G2 โ P4S3G1G2 โ m2P4S1G3 โ P3S1G2G3 + P4S1G2G3 e m1+m3 h
+ โm1P4S3G2 โ P3S3G1G2 + P4S3G1G2 + m1P4S2G3 + P3S2G1G3 โ P4S2G1G3 e m2+m3 h .
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Rajesh Patra" Thermo-mechanical Analysis of a Crack in an Infinite Functionally Graded
Elastic Layer" IOSR Journal of Engineering (IOSRJEN), vol. 09, no. 06, 2019, pp. 24-42.