Présentation

1. 1
Ceramic Coating effect on thermal buckling of functionally
graded plates under a non-uniform temperature rise
El Ibrahimi Mohamed*1, Lahcen Azrar2, Abderrahim Samaouali1
1 Team of Thermodynamic-energy research center, Faculty of Science Rabat, Mohamed V University, Morocco
2 M2CS, CRSTIS, Department of Applied Mathematics and Informatics, ENSET Rabat, Mohamed V University, Morocco
*mohamed.elibrahimi@um5s.net.ma
2nd International Materials Science
and Engineering for Green Energy Conference
April 25-27, 2018
Rabat Morocco

2. 2
1
MODEL
INTRODUCTION
2
2
3
Concept of FGM
Role of the Coating
CONCLUSION
PURPOSE AND HYPOTHESES
RESULTS
4
6
RESOLUTION METHOD
5
contents

3. 3
Introduction
100% Ceramic
100% Metal
FGM plate
The functionally graded material Concept
- Composite material
- Gradual variation
- Continuity of properties
- Eliminating interface
problems
- Mitigating thermal stress
concentrations
Fig2: Configuration of functionally graded plate
Fig1: Schematic description of LZ7C3/8YSZ functionally
graded plate [1]
[1] Sumei Zhao and al, «http://dx.doi.org/10.1016/j.jallcom.2014.01.001»

4. 4
Role of the Coating
4
The coating must be ceramic with a high
corrosion resistance to protect the
functionally graded plate.
Bottom face Temperature =Tm
z
x
y h
hc
Top face Temperature = Tc
Ceramic coating
FGM plate
a
b
Where: V z =
z
h
+
1
2
k
; −
ℎ
2
≤ 𝑧 ≤
ℎ
2
− ℎ𝑐
1 ; +
ℎ
2
− ℎ𝑐 ≤ 𝑧 ≤
ℎ
2
P z = Pm + Pc − Pm V z
EFFECTIVE PROPERTIES
(YOUNG MODULUS, THERMAL CONDUCTIVITY,…)
Fig3: Configuration of functionally graded plate with
ceramic coating
Why the coating?

5. 5
5
Purpose & Hypotheses
• Thin plates (Kirchhoff- Love)
• Simply supported plate
Hypotheses
Purpose
• How the FG plate response change in presence of thermal load, ceramic coating and
geometrical imperfections?
Ultra high temperature environment
Geometrical imperfections
Ceramic Coating
Functionally graded plate
Fig 4: The buckling of FG plate simply supported
• Initial Geometrical imperfections
• Large Deformations

6. 66
Where : D =
E2E0−E1
2
(1−ν2)E0
and
h and hc: Thickness of the plate and ceramic coating respectively.
w and f : The lateral displacement component and the Airy stress function respectively.
w∗
and E : The initial small imperfection of the rectangular plate and Young Modulus respectively.
Non linear Stability Equation
D𝛻4
w −
𝜕2
f
𝜕y2
𝜕2
w
𝜕x2
+
𝜕2
w∗
𝜕x2
+
𝜕2
f
𝜕x2
𝜕2
w
𝜕y2
+
𝜕2
w∗
𝜕y2
− 2
𝜕2
f
𝜕x𝜕y
𝜕2
w
𝜕x𝜕y
+
𝜕2
w∗
𝜕x𝜕y
= 0 (1)
E0
𝜕2
w
𝜕x𝜕y
2
−
𝜕2
w
𝜕x2
𝜕2
w
𝜕y2
+ 2
𝜕2
w
𝜕x𝜕y
𝜕2
w∗
𝜕x𝜕y
−
𝜕2
w
𝜕x2
𝜕2
w∗
𝜕y2
−
𝜕2
w
𝜕y2
𝜕2
w∗
𝜕x2
−
𝜕4
f
𝜕x4
+ 2
𝜕4
f
𝜕x2 𝜕y2
+
𝜕4
f
𝜕y4
= 0
Compatibility Equation
(2)


2/
2/
)(
h
h
i
i dzzzEE
Model

7. 7
(Non uniform Temperature rise)
One can obtain the Critical temperature of buckling :
∆Tcr =
pi2
D n2 b
a
2
+ m2
1 − ν
b2
Ph
Ph + μh
+ E1
pi2
b2
3 − ν2
n4 b
a
4
+ m4
+ 4ν
b
a
2
n2
m2
16 1 + ν b4(
n
a
2
+
m
b
2
)
Ph Ph + 2μh
+ 4E2
pi2
b2
n4 b
a
4
+ m4
+ 2νn2
m2 b
a
2
mnb4 1 + ν (
b
a
2
n2 + m2)
Ph
S
(h − 𝐡 𝐜)(Q+R)
and Q= 𝑖=0
𝑖𝑛𝑓
h−hc
h
𝑖𝑘+1
−
𝐾 𝑐𝑚
𝐾 𝑚
𝑖
𝑖𝑘+1
α 𝑚E 𝑚
𝑖𝑘+2
h−hc
h
𝑖𝑘+1
+
α 𝑐𝑚E 𝑚+α 𝑚E 𝑐𝑚
(𝑖+1)𝑘+2
h−hc
h
(𝑖+1)𝑘+2
+
α 𝑐𝑚E 𝑐𝑚
(𝑖+2)𝑘+2
h−hc
h
(𝑖+2)𝑘+2
Where: 𝑃ℎ=P/h ; 𝐸1=
𝐸1
ℎ
; 𝐸2=
𝐸2
ℎ2 ; S= 𝑖=0
𝑖𝑛𝑓
−1 𝑖
𝐾 𝑐𝑚
𝐾 𝑚
𝑖
𝑖𝑘+1
; R=α 𝑐E 𝑐 𝑖=0
𝑖𝑛𝑓
h−hc
h
𝑖𝑘+1
−
𝐾 𝑐𝑚
𝐾 𝑚
𝑖
(𝑖𝑘+1)(𝑖𝑘+2)
1 −
h−hc
h
𝑖𝑘+2
Critical Temperature of Buckling
Results

8. 8
Increasing μ improves the stability of the plate.Increasing hc improves the stability of the plate
Geometrical Imperfection effectCoating thickness effect
𝑃ℎ𝑃ℎ
Fig 5: Effect of size of imperfection μ on thermal buckling of FG plates with coatingFig 6: Effect of coating thickness hc on thermal buckling of imperfect plates

9. 9
Conclusion
• Thickness of coatings and size of initial small imperfections has a significant effect on
the buckling behavior of the rectangular plate.
• The results provide the useful and important information for the design problems of
FG plates with coatings.

10. 10