1) The document discusses the effect of a ceramic coating on the thermal buckling of functionally graded plates under a non-uniform temperature rise. It presents a model to analyze the buckling behavior of plates with an initial geometric imperfection and a ceramic coating. 2) The model uses nonlinear stability equations and considers the plates as thin with simply supported boundary conditions. It derives an expression for the critical temperature of buckling in terms of the plate properties and dimensions. 3) The results show that increasing the thickness of the ceramic coating or decreasing the size of the initial geometric imperfection improves the stability of the functionally graded plate by increasing its critical buckling temperature. The analysis provides useful information for designing functionally graded plates

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