Python Notes for mca i year students osmania university.docx
Fg micropolates
1. latrobe.edu.au CRICOS Provider 00115M
Analyses of functionally graded
microplates
Son Thai
School Engineering and Mathematical Sciences, La Trobe University
2017 Borland Forum and Materials Careers Forum
September 18, 2017
4. 4La Trobe University
Functionally Graded Materials (FGMs)
- Classified as composite materials
(manufactured by two phases of
materials: Ceramic and Metal)
- Smooth and continuous variation of
material compositions and properties
Ceramic:
-High thermal resistance
-Low toughness and brittle
-Cannot be directly used in
engineering applications
Metal:
-Tough and Ductile
-Ideal for engineering
applications
FGMs
7. 7La Trobe University
Size-dependent effects
Strain energy in the modified-strain gradient theory
where
- = classical component
- = Non-classical components account for the size-effects
Constitutive relations
( ) ( )
( )1 11
2
s s
i i ijkij i ijk ijj
V
ijpU dVmγ τ η χσ ε += + +∫
( ) ( )
( ) ( )
2
0
1 1
2
0
2
1
2
2
3
2
2 2
2
i i
ijk ijk
ij ij kk
s s
ij
ij j
j
i
i
p l
l
m l
T T
µ
σ µε λε δ α λ
µ
µ
γ
τ η
µ χ
δ
=
=
+ −
=
= + −
ij ijσ ε
( ) ( )1 1 s s
i i ijk ijk ij ijp mγ τ η χ+ +
µ
where
λand are Lame constants; and are three material length scale parameters( )0 1 2, ,l l l
8. 8La Trobe University
Models of microplates
P(z) is a generic material properties (E, v, ρ)
( ) c c m mP z PV P V= +
Rule of mixture
1
; 1
2
n
c c m
z
V V V
h
= + + = ÷
Volume fraction of material
9. 9La Trobe University
Models of microplates
Kinematic modelling
( ) ( )
( ) ( )
( ) ( )
3
1 2
3
2 2
3
4
, , , , ,
3
4
, , , , ,
3
, , , , ,
x x
y y
z w
u x y z t u x y t z
h x
z w
u x y z t v x y t z
h y
u x y z t w x y t
θ θ
θ θ
∂
= + − + ÷
∂
∂
= + − + ÷∂
=
10. 10La Trobe University
Non-Uniform RationalB-splines (NURBS) basis functions
Governing equationof motion
( )
( ) ( )
( ) ( )
, , ,,
,
ˆ ˆ ˆ ˆˆ ˆ , , ,1 1
, i p j p i jp q
i j n m
i p j p i ji j
N M w
R
N M w
ξ η
ξ η
ξ η= =
=
∑ ∑
( )γε η χ++ + =+K K KK d Md f&&
NURBS element
Lagrange element
Control point
Isogeometric analysis
11. 11La Trobe University
Static bending analysis
Smaller Gradient index n smaller deformation
Smaller scale greater influence of size effect smaller deformation
12. 12La Trobe University
Nonlinear bending analysis
1
; 1
2
n
c c m
z
V V V
h
= + + = ÷
( ) ( )
2
0
1 12
1
2
2
0 1 2
2
2
2
i i
ijk ijk
s s
ij ij
p l
l
m l
l l l l
µ γ
τ µ η
µ χ
=
=
=
= = =
13. 13La Trobe University
Free vibration analysis
Smaller Gradient index n greater natural frequency
greater influence of size effect greater natural frequency.
15. 15La Trobe University
Buckling analysis
Smaller Gradient index n greater buckling load
Smaller size greater buckling load
16. 16La Trobe University
Post-buckling analysis (Mechanical loads)
1
; 1
2
n
c c m
z
V V V
h
= + + = ÷
( ) ( )
2
0
1 12
1
2
2
0 1 2
2
2
2
i i
ijk ijk
s s
ij ij
p l
l
m l
l l l l
µ γ
τ µ η
µ χ
=
=
=
= = =
18. 18La Trobe University
Buckling analysis (Thermal loads)
( ) 0
d dT
z
dz dz
κ
=
1
; 1
2
n
c c m
z
V V V
h
= + + = ÷
( ) ( )
2
0
1 12
1
2
2
0 1 2
2
2
2
i i
ijk ijk
s s
ij ij
p l
l
m l
l l l l
µ γ
τ µ η
µ χ
=
=
=
= = =
19. 19La Trobe University
Conclusions
Numerical models based on IGA have been developed to investigate the
structural behaviour of micro-plates.
The developed model is capable of capturing the (1) size effect, (2)
geometric nonlinearity, (3) shear deformation effect, (4) thermal effect and
(5) nonhomogeneous behaviour of FG materials
The inclusion of the size effect results in an increase in plate stiffness, and
consequently, leads to a reduction of deflection and an increase in natural
frequency as well as buckling load and buckling temperature.
The small scale effect becomes significant when the plate thickness is small,
but it is negligible when the plate thickness becomes larger.
Based on the present models, optimal values of gradient index n could be
obtained for proper design of micro-structures.
Good evening ladies and gentlemen.
I am Son Thai, I come from Latrobe University. It is my pleasure to be here today to present the research I have been working on so far.
My topic today is “Analysis of FG microplates”
This is the layout of my presentation. I am going to have a introduction about the topic, then I will go to the theoretical formulation and numerical results and finally I will draw some conclusions.
In the first section, let’s me briefly introduce about functionally graded materials or FGMs in abbreviation.
First of all, FGMs are categories as composite materials, which means that they are made from two or more constituent materials with significantly different physical and chemical properties.
The main difference between FGMs and traditional laminated composites is the structures of components that are used to frabricate the materials.
In FGM, the material constituents vary smoothly in a prescribed direction. Whereas the constituents in laminated composites are mixed up in a defined layer and the structures are normally made from a set of a number of different layers. This kind of structure can cause stress concentration effect when the structure is applied a severe loading, and a delamination failure could happen consequently.
Recently, The FGMs are considered as advanced composite materials thanks to their favorable characteristic. Not only do they have the ability to eliminate the stress concentration effect that is seen in traditional laminated composite, structures made from FGMs could make the best use of their constituents.
For example, the FGMs are normally fabricated from Ceramic and Metal components. In general, the ceramic is good at thermal resistance, however, it is low toughness and easy to produce the crack. In contrast, the metal is tough and ductile but has higher thermal expansion coefficient. Therefore, by combining those material with their properties vary smoothly from ceramic to metal phrase, we have a new composite material that can exploit the advantages of both constituents.
The concept of FGMs was initiated in japan 30 years ago during a spcae project. In this project, the FGMs was used to make a thermal barrier which can a surface temperature of 2000 K and a temperature gradient of 1000 k across a 10 mm section.
Since then, FGMs has penetrated various field in civil engineering, mechanical engineering and electronic engineering, thanks to their advanced features. For examples in nucluear reactors (plasma wall), space applications (rocket component, space vehicels), mediacal applications (artificial bone, denstistry) and so on…
In addition, the adoption of FGMs to the micro-structures in high-tech divives is increasingly investigated in the last few years. As we can see in these figure, micro structures, indlung micro-beams. And micro plates are the fundamental components in biosensor, atomic force microscopes micro- and nano-electromechanical systems. So the mechanical behaviour of those small-scale structures should be comprehensive investigated for a proper design of those expensive applications.
However, the experimental studies in the literature indicate that the behaviour those small-scale structure are significantly affected by the so-called size-dependent effect. Therefore, the objective of our study so far is to figure out how the size-effect and the charactoristics of FGMs affect to the mechanical respsone of the miro structures.
As I mentioned earlier, the mechanical responses of micro structure are considerably influenced by the size size-effect.
Technically speaking, the classical elasticity theory that we use to predict the behaviour of common structures fail to accurately describe the response of micro structure.
Therefore, in our work, we employed the modified strain gradient theory to capture the size effect phenomenon.
The reason for choosing the MST for developing the numerical models is that the MST is considered as a completed and simple form of general strain gradient theory, it can be converted into other models by setting the values of length scale parameter.
For example, when we prescribed all three length scale parameters all equal to 0, we have the original form of classical elasticity theory.
In general, the essential information that we need to difined in an analysis of FG structures is the distribution of material properties through the structrual dimensions.
In our study, we assumed that plates made from FGMs has 2 constituents; ceramic and metal. The ceramic component is prescribed at the top surface, the metal component is prescibed at the bottom surface, and the materials inside the plate volume are varied continously according to the mixture rule.
Here, we could also use the values of volume fractions to quantify the amount of ceramic and metal at an arbitrary point in the palte volume, the n here denontes the gradient index of ceramic phrase.
When n increases, we see that the volume fraction of cemaic componet in the plate volume reduces.
In structrual mechanics, there are various approaches to model a plate structure. However, the apporach based on the kinematic assumptions are predominant.
In our study, we use the third-order shear deformation theory to describe the displacement field of a plate. The benefits of using this model is that it can eliminate the shear correction factor and shear locking effect, which are cumbersome in comutational mechanics.
In addition, we also use the Isogeomatric analysis as a numerical tool to investigate the structural responses of FG structures.
Nowadays, IGA is wildly considered as an advanced computational approach since it has improved the deficiencies of traditional FEM.
Now, let’s me begin the numerical investigation with the linear static bending of micropolates.
The plot in the screen denotes the influences of size effect via the ratio of h/l, the thickness ratio a/h and the gradient index n on the bending of microplates.
In general, we can see that the change of gradient index n, which is the parameter used to defined the variation of material properties through the thickness of the plate, significant affect the bending of the plate.
By increasing n, we have smaller displacement values. This phenomenon is explained by the rise of metal constituent in the plate’s volume. Which makes the plate weaker.
Another noting point is the influence of the size effect via the ratio of h/l. In here, the reduce of this ratio denotes the decrease of the size of the plate.
We can see that the size effect becomes pronounced when the size of the plate become smaller and comparable to the length scale parameter l. In addition, we see that size effect tends to increase the stiffness of the plate, consequently, the displacement reduces with the increase of the size effect.
In this slide, we have the investigation on the nonlinear bending of microplates under the effect of material parameter n and size effects.
Similar findings as the linear bending analysis are also obtained. However, we can see that the microplate almost exhibits linear response when the size effect is dominant or when the ratio of h/l equal to 1
Another important investigation in every mechanical problem is the free-vibration analysis. The results in here show us that the natural frequency of microplate goes up with the increase of the size effect or the decrease of the of the gradient index as these changes make the plate become stiffer.
Besides the free-vibration analysis, the dynamic investigation should be carried out also, especially when the structure is subjected to severe loading cases, For example blast load.
Here, we also figured out the influence of linear and nonlinear analyses together with the effect of material factor n and the size effect.
We can easily see the difference between linear and nonlinear results when the size effect is not considerable or the stiffness of the plate is small as most of the plate volume is metal constituent.
However, when the stiffness of the plate is increased as the results of the size effect or more ceramic constituent, the difference between linear and nonlinear behaviour can be neglectable.
This is the important finding in this example.
Another type of analysis that I would like to present today is the buckling analysis. The buckling is the phenomenon at which the structure, for example the plate in this example, lose it in-plane load-bearing capacity and we are to define the load at which the failure occurs.
Similar to the previous analysis, we also found out that the buckling load increase with the rise of the size effect and the ceramic constituent.
Although the plate loses it load-bearing capacity after the buckling happens. The behaviour of it after buckling should be investigated for the big picture of its mechanical response and how the influence of material parameter or size-effect is.
Therefore, we conducted the so-called post-buckling analysis. Actually, this type of analysis is based on the geometrical analysis with the loadings are the in-plane load applied to the boundary of the plates.
Here, we can see that the bifurcation phenomenon dose happen when a homogenous model is considered, for example when n = 0 or n = inf.
The bifurcation buckling is the case when the plate still remain its initial configuration under the compressive load until a certain value as we calculated as buckling loads. When the load exceed this values, the plate will deform suddenly.
For FGMs plates, as the material is graded through the plate thickness and there is a difference between the middle surface of the plate and the neutral surface. As a result, the plate deform immediately when the compressive load is applied as we can see here.
For the influence of the size effect, similar to previous examples, we also conclude that the size effect become significant when the plate is small enough that its thickness is compared to the length scale parameter.
In this slide, we present some post buckling configuration of the plate with different shapes. It is seen that the buckling shapes are not identical and should be figured out carefully.
The last analysis I would like to introduce today is the buckling analysis of FG plate under thermal load.
For this type of analysis, different temperature are assigned to the top and bottom of the plate and the temperature inside the plate is calculated based on the 1D heat transfer equation.
One thing that should be noted in this example is that the influence of material index value on the response of the plate is not a monotonic function. It is seen that the deformation with n = 1 or 2 lager than that of n = 0. However, when n is greater than 5, the deformations are smaller than that of a full ceramic plate or n = 0.
Finally, some specific conclusions could be drawn from our study as follows
In our study, we have developed a versatile numerical model, which is able to capture various mechanical responses of a micro-structures, e.g. size effect, the geometrical nonlinear, shear deformation …
Secondly, we found that the size effect tends to increase the stiffness of the structure, which lead to a reduction of deflection and the increase of natural free-frequencies and buckling load
Lastly, based on this numerical, we can find the optimal value of the distribution of materials in the plate volume via the gradient index values for a proper design of micro structures.
