FGM_Iso_Ortho Elastostatic analysis of isotropic and orthotropic FGMs using the radial point interpolation method

Elastostatic analysis of isotropic and orthotropic FGMs
using the radial point interpolation method

1. Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
1. Introduction
2. Apply RPIM formulation to FGMs model
3. Numerical examples
4. Conclusions

2. Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
1. Introduction

3. 1. Introduction
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
1.1. Functionally graded materials (FGMs)
Functionally graded materials (FGMs)
- Functionally graded materials are special types of composite
- The material properties vary in one or more directions with predefined profiles
- Application of FGMs
+ thermal barrier coating for space applications
+ energy conversion system
+ thermal-electric and piezoelectric devices
+ dental and medical implants, and so on

4. 1. Introduction
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
1.2. Meshless RPIM for FGMs elastostatic problems
-In this study, an meshfree radial point interpolation method (RPIM)
with the thin plate spline (TPS) radial basis functions is applied to
simulate 2D linear elasto-static problems with isotropic and orthotropic
FGMs
-The finite Element Method (FEM) has been widely used for computing the
behavior of FGMs
-The obtained results are compared with either analytical solutions or
numerical solution form FEM that presented by other authors.
-To improve the FEM for isotropic and orthotropic FGMs, fully
isoparametric graded elements were developed by using the same shape
functions to interpolate the material properties

5. Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
2. Apply RPIM formulation
to FGMs model

6. 2. Apply RPIM formulation to FGMs model
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
2.1. RPIM formulation for 2D elasto-static problem
Consider a 2D dynamic problem of solid mechanics that has a domain and a
boundary . The weak form for this problem is expressed as  0
t
T T T
d d d
  
−−=
εσubut
The Galerkin weak form for this elastic problem x y   ( , )E E x y= ( , )xy= t x y   11
( , )E E x y= t 1
E 2
E 22
( , )E E x y= 12 12
( , )xy= 12 12
( , )G G x y=
where::u
vector of displacements:σ
stress matrix:ε
stress matrix:b
body force:t
surface load

7. 2. Apply RPIM formulation to FGMs model
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
2.1. RPIM formulation for 2D elasto-static problem
Using the meshless procedure, the discretized form can be written as a linear system
of equation=KuF
Mass matrix:T
IJ I J
d

=
M ΦΦ
Stiffness matrix:T
IJ I J
d

=
KBDB
Force vector:t
TT
I II II
dd

=+
FΦbΦt
vector of RPIM shape
functions:
I
Φ ,
,
,,
0
0
Ix
I I y
I y I x


=



B
Displacement
gradient matrix:
where:

8. 2. Apply RPIM formulation to FGMs model
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
2.2. Isotropic and orthotropic FGMs material properties
For isotropic FGMs, the matrix D is the function of global coordinate x()( )
2
1()0
()
()10
1()
00(1())/2
1()()0
()
1()0
112
00(12())/2
E
E








 
 
 
−
  −
  
=
−

−
+−
 − 
x
x
x
x
x
D
xx
x
x
x
for plane stress
for plane strain
For orthotropic FGMs, the matrix D has the form1 2 12
2 12 2
22
12
1
()()()0
1
()()()0
()
1() 0 0 ()
()
ExE
EE
E
G
E





=

− 

xx
D xxx
x
x x
x
for plane stress

9. Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3. Numerical examples

10. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
Consider a square isotropic FGM plate with material variation in the Cartesian x-
direction
3.1. Isotropic FGM plate
• Exponential variation0
();
x
ExEeconst

== 12
(0)1;()8EEEWE==== log(8)/9= 0.3=
• Linear variation0
();ExExconst=+= 12
(0)1;()8EEEWE==== 7/9= 0.3=
The analytical solutions are given by Kim H. J et al in [2].

11. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.1. Isotropic FGM plate
Displacement and stress component distributions along y = 0
for tension load applied to the exponential material gradationx
u yy


12. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.1. Isotropic FGM plate
Stress component distributions along y = 0
for bending load applied to the exponential material gradationyy
 yy


13. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.1. Isotropic FGM plate
Displacement and stress component distributions along y = 0
for tension load applied to the linear material gradationx
u yy


14. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.2. Orthotropic FGM plate
Rectangular plate with an edge slant crack
A rectangular orthotropic FGM plate is considered, the material variation in the Cartesian
x-direction11
22
12
0
11 11
0
22 22
0
12 12
12
();
();
();
x
x
x
ExEe
ExEe
GxGe
const




=
=
=
= 0 0 0
11 22 12
1;0.1;0.5;0.3EEG ==== 11 22 12
log(8)/9==== 12
0.3=
The analytical solutions are given by Kim H. J et al in [2].

15. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.2. Orthotropic FGM plate
Tension load
Displacement and stress component distributions along y = 0 and y = 9
for tension load applied to the exponential material gradationx
u yy


16. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.2. Orthotropic FGM plate
Bending load
Displacement and stress component distributions along y = 0 and y = 9
for bending load applied to the exponential material gradationx
u yy


17. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.3. Isotropic FGM link bar
Consider the isotropic FGM link bar, the material variation in the Cartesian y-
direction
Material properties: 107,0.34
375,0.14
Ti Ti
TiB TiB
EGPa
EGPa


==
== (),()
E
yy
Ti Ti
EyEeye


== 11
log;log
TiB TiB
E
Ti Ti
E
HEH




 
==
 
 

18. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.3. Isotropic FGM link bar
-The model of this example is the same with the one in the publication of Kim, H.
J. et al (2002) in which the results are obtained by FEM using graded elements
and by MLPG given by Gilhooley D.F (2008)
-According to the results given by Kim, using a FGM bar can reduce stress
concentration and the location of the maximum axial stress also changes to
another place in the bar.
-The similar solutions obtained from RPIM in this study are compared with FEM
results given by Kim et al [2] and MLPG results given by Gilhooley, D. F. et al
[23].

19. 3. Numerical examples
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
3.3. Isotropic FGM link bar
homogenous link bar
Method Material propertyLocation ILocation II
FEM (Graded element) [2]
Homogeneous 2.908 2.137
FGM 2.369 2.601
MLS-MLPG1 [21]
Homogeneous 2.918 2.140
FGM 2.360 2.594
RBF-MLPG5 [23]
Homogeneous 2.901 2.131
FGM 2.403 2.607
RPIM-TPS
Homogeneous 2.937 2.153
FGM 2.358 2.584
FGM link bar

20. Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
4. Conclusions

21. 4. Conclusions
Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
-The meshless RPIM with TPS radial basis functions is used to perform several
elasto-static problems with either isotropic or orthotropic functionally
graded materials.
-The shape functions constructed using RBFs interpolation satisfy the
Kronecker’s delta property, so the procedure for enforcing essential boundary
conditions is quite simple like FEM.
-Results for several example problems with different loading conditions and
gradation laws are taken into account show the agreement of the proposed
method using for FGM problems.
-In the comparison with homogenous material, to apply meshless method for
nonhomogeneous material like FGMs, an additional task is defining the
functions for material variation and this is a very simple task. Thus the method
is so flexible to apply for various types of FGMs model.

22. Department of Engineering Mechanics – Faculty of Applied Science02/10/2025
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