1. 1
Numerical prediction of Effective Young Modulus
(Eeff)
(a) 103 μm3 domain (b) 123 μm3 domain (c) 163 μm3 domain

2. 2
Results
• The (a) Eeff, (b) νeff, (c) Keff and (d) αeff have been compared with the Hashin-
Shtrikman (HS) upper and lower bounds.
0
50
100
150
200
250
0 0.1 0.2 0.3 0.4 0.5
EffectiveYoung's
modulus,Eeff(GPa)
Porosity, P
FV solution
HS upper bound
HS lower bound
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.1 0.2 0.3 0.4 0.5
EffectivePoisson's
ratio,νeff
Porosity, P
FV solution
HS upper bound
HS lower bound
50
70
90
110
130
150
170
190
210
230
250
0 0.1 0.2 0.3 0.4 0.5
EffectiveBulk
modulus,Keff(GPa)
Porosity, P
FV solution
HS upper bound
HS lower bound
30
40
50
60
70
80
90
0 0.1 0.2 0.3 0.4 0.5
EffectiveShear
modulus,Geff(GPa)
Porosity, P
FV solution
HS upper bound
HS lower bound
(a) (b)
(c) (d)

3. 3
Differences between ideal solution vs. realistic solution
(a) (b) (c)
(a) ideal alignment, (b) non-ideal alignment, (c) line of flux
Stress distribution showing in Figures (d), (e) and (f) in the maxZ plane
σzz
σzy
(e) (f)
σzx
(d)

4. 4
Results for the prediction of Effective Thermal
Expansion Coefficient (αeff)
• Predicted displacements in x, y and z directions.
-6.343e-08 < ux < 6.343e-08
x-direction
-5.0145e-11< uy < 1.2506e-07 -6.43e-08 < ux < 6.43e-08
y-direction z-direction

5. 5
(a) Relation between αeff and P (b) αeff vs. Schapery Upper and lower bounds
Comparison of predicted αeff results with Schapery
upper and lower bounds.
9.84E-06
9.86E-06
9.88E-06
9.9E-06
9.92E-06
9.94E-06
9.96E-06
9.98E-06
0.25 0.3 0.35 0.4 0.45
EffectiveThermalExpansion
Coefficient,αeff(/K)
Porosity, P
0.0E+00
5.0E-06
1.0E-05
1.5E-05
2.0E-05
2.5E-05
3.0E-05
0.25 0.27 0.29 0.31 0.33 0.35 0.37 0.39 0.41
EffectiveThermalExpansion
Coefficient,αeff(/K) Porosity, P
FV solution
Schapery upper bound
Schapery lower bound