1. 20.0 1
Milton 2
Berryman-Milton
15.0 1 3
Gibiansky-Torquato
10.0 2
5.0 3
0.0
0.0
5.0
10.0
15.0
20.0
Figure 2-1

2. 1.00 1
Bristow
0.75 2
Milton 3
Gibiansky-Torquato
0.50 3 2
0.25 1
0.00
0.00
0.25
0.50
0.75
1.00
Figure 2-2

3. Equivalent for conductivity
Not equivalent for elasticity
Figure 3-1

4. Reff /a =a/c Figure 4-2 0 = 0.10 0 = 0.25  0 = 0.50 0.00 0.25
0.75
1.00
0 = 0.10
0 = 0.25
0 = 0.50
Reff /a
=a/c
r = a
r = a - c a r c s
Figure 4-2

5. 15 2 10 2 -3 4 5 -6 1 3 4 -9 1 3 -5
-12
0.01
0.1 1
0.01
0.1 1
B1, D1
B3, D3 1 3
B2, D2
B4, D4 2 4
Figure 4-5

6. Figure 4-6 Rigid inclusions 100 crack-like 0.01 1 100 needle-like 0.01
Soft inclusions
Rigid inclusions
Rigid inclusions
100
100
crack-like
0.01 1
100
needle-like
crack-like
0.01 1
100
needle-like
0.01
0.01
Soft inclusions
Soft inclusions
Figure 4-6

7. Figure 4-7 Rigid inclusions 100 crack-like 0.01 1 100 needle-like 0.01
Soft inclusions
Rigid inclusions
Rigid inclusions
100
100
crack-like
0.01 1
100
needle-like
crack-like
0.01 1
100
needle-like
0.01
0.01
Soft inclusions
Soft inclusions
Figure 4-7

8. 10 1 4 5 6 5 2 3 -5
-10 A1 g
-15
0.01
0.1
1.0 10
100 6 6 4 5 1 2 3 2 4 -2 A2 g -4
0.01
0.1
1.0 10
100
Figure 4-9

9.  Reff /a Elasticity problem Conductivity problem Figure 4-10 1.00
0.75
0.50
Conductivity problem
0.25 
0.00
0.00
0.25
0.50
0.75
1.00
Figure 4-10

10. 2
0.5 2 4 1 1 3
-0.5 1 -1 2 4 3 g g -1
-1.5
0.01
0.1 1 10
100
0.01
0.1 1 10
100 2
0.5 2 4 1 1 1 3
-0.5 2 4 g g 3 -1 -1
0.01
0.1 1 10
100
0.01
0.1 1 10
100
Figure 5-1

11. g g g g 3 4 2 1 1 2 4 3 Copper reinforced with diamond particles 4 2 3
0.6
0.1 4 3
0.4
0.0 2
0.2 1
-0.1 1
0.0
-0.2 2
-0.2
-0.3 4 3 g g
-0.4
-0.4
0.01
0.1 1 10
100
0.01
0.1 1 10
100
Copper reinforced with diamond particles
0.6
0.1 4
0.4
0.0 2
0.2 3 1
-0.1
0.0 1 2
-0.2
-0.2 3 g 4 g
-0.4
-0.3
0.01
0.1 1 10
100
0.01
0.1 1 10
100
Poly(phenylene sulfide) reinforced with glass particles
Figure 5-2

12. g g 1 1 2 2 Poly(phenylene sulfide) reinforced with glass particles
0.8 1
0.6
1.5
0.4 1 1 2
0.2 2 g g
0.5
0.01
0.1 1 10
100
0.01
0.1 1 10
100
Poly(phenylene sulfide) reinforced with glass particles
Porous aluminum
Exact connection 1
Coefficient at Young’s modulus
Coefficient at shear modulus
Approximate connection 2
Figure 5-3

13. 1.5
1.5 4 4
1.0
1.0
(a)
(b)
0.5
0.5 1 1
0.0
0.0 2 3 2 3  
-0.5
-0.5 1 2 3 4 1 2 3 4
1.5
1.0
(d) 4
1.0
(c)
0.5
0.5 1
0.0 2 3  
-0.5
0.0 1 2 3 4
0.0
2.0
4.0
Figure 5-4

14. 1.5
1.5
1.0
1.0 1 1 2 2
(a)
(b)
0.5
0.5 4 4
0.0
0.0 3  3 
-0.5
-0.5 1 2 3 4 1 2 3 4
1.5
1.0 1 2
(c)
0.5 4
0.0  3
-0.5 1 2 3 4
Figure 5-5

15. 1.00
1.00
E1/E2
E1/E2
0.80
0.90
k3 / k0=0.6
k3 / k0=0.8
0.60
0.80
k1/k2
k1/k2
0.70
0.40
0.80
0.85
0.90
0.95
1.00
0.60
0.70
0.80
0.90
1.00
1.00
1.00
E1/E3
E1/E3
0.80
0.80
0.60
k3 / k0=0.8
k3 / k0=0.6
0.60
0.40
k1/k3
k1/k3
0.20
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.60
0.70
0.80
0.90
1.00
g = 0 (cracks)
g = 3 (prolate pores)
g = 1/3 (oblate pores)
g (cylinders)
Figure 5-6

16. E1 / E3
E1 / E3
3.0
1.5
k3 / k0=0.5
k3 / k0=0.7
2.0
1.0
1.0
0.5
(a)
(b)
0.0
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
k1 / k3
k1 / k3
E1 / E3
1.5
g = 0 (cracks)
k3 / k0=0.9
1.0
g = 1/3 (oblate pores)
g = 3 (prolate pores)
0.5
g (cylinders)
(c)
0.0
0.0
0.5
1.0
1.5
k1 / k3
Figure 5-7

17. 1.0
0.0
-1.0
-1.0
0.0
1.0
Figure 6-6

18. Figure 7-1

19. 1.0
0.9
0.8
0.7
0.6
0.5 p
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 7-2

20. E/E0 p Figure 7-3 0.2 Direct measurements Prediction via
cross-property connection
0.1 p
0.0
0.6
0.7
0.8
0.9
1.0
Figure 7-3

21. 30 60
1.5
1.6
0.5
1.0
2.0
2.0
2.2
0.5
1.0
1.5 20 40 10 20 γ γ
0.01
0.1 1 10
0.01
0.1 1 10
Figure 7-4

22. p Figure 7-5 4 Non-interaction approximation Mori-Tanaka method 3
Levin-Kanaun method
Differential scheme 2
Formula (6.4)
Experiment 1 p
0.5
0.6
0.7
0.8
0.9
Figure 7-5

23. k, x106S/m
E, GPa 25 40 20 30 15 20 10 10 5 p p
0.5
0.6
0.7
0.8
0.9
1.0
0.5
0.6
0.7
0.8
0.9
1.0
Non-interaction approximation
Differential scheme
Mori-Tanaka method
Experiment
Levin-Kanaun method
Figure 7-6

24. E, GPa 70 65 60 55 10 100 1000 Number of cycles Figure 7-7 1 2 3 15%
20%
25% 65 1 2 60 3 55 10
100
1000
Number of cycles
Figure 7-7

25. (a)
(b)
(c)
(d)
Figure 7-8

26. 1.0
2.5 1 3
(a)
(b)
0.9
2.0 2 2 1 4
0.8
1.5 3
0.7
1.0 5 10 15 20 25 5 10 15 20 25
Aging time (days)
Aging time (days) 1 1 2 2 3 3 4
Figure 7-9

27. 0.0
0.0 1 1
-0.1
-0.2 3
-0.2
-0.4 3 2 2
-0.3
-0.6
-0.4
(a)
(b)
-0.8
-0.5 5 10 15 20 25 5 10 15 20 25
Aging time (days)
Aging time (days)
0.0 1
Measured 2
Alternative cross-property connections (6.6b)
-0.05 2 1
-0.10 3
Cross-property connection (5.9b) 3
(c)
-0.15 5 10 15 20 25
Figure 7-10
Aging time (days)

28. r0 a) b)
rint
rext
Figure 7-12

29. 1.0
0.8
0.6
0.4
0.2
(a)
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.0
0.8
0.8
0.6
0.4
0.4
0.2
(c)
(b)
0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 8-1

30. H R x3 x2 x1
Figure 8-2

31. (b)
(a)
Figure 8-3

32. C b a
Figure 8-4

33. A B
Figure 8-5

34. Porous aluminum (numerical simulation)
, MPa
, MPa 50
1500
( a )
( b )
4.8 %
11.3 %
15.0 % 40
1000 30
dense
7.6 % 20
500 10
x 10-3
0.0
0.4
0.8
1.2
1.6
0.00
0.02
0.04
0.06
0.08
, MPa
800
( c )
Porous aluminum (numerical simulation)
(b), (c) Dense and porous Ti-6Al-4V (experiment)
600
400
dense
7.6 %
200
0.00
0.05
0.10
0.15
Figure 9-1

35. Slight shift to the right
Increasing pore
volume fraction
Slight shift to the right
Figure 9-2

36. (a) (b) k1/k0 k3/k0 Isotropy p k1/k0 Figure 9-3 1.0 1.0 0.9 0.8 0.8
0.6
0.7
Isotropy
0.4
0.6
(a)
(b) p
k1/k0
0.2
0.5
0.6
0.7
0.8
0.9
1.0
0.00
0.15
0.30
Figure 9-3

37. (a) (b) k3/k0=0.7 k3/k0=0.8 k1/k0 k1/k0 (c) A1 A4 A2 A5 k3/k0=0.9 A3-1 -2 2 1 -1 -2
(a)
(b)
k3/k0=0.7
k3/k0=0.8
1.0
0.9
0.8
0.7
0.6
k1/k0
1.0
0.9
0.8
0.7
0.6
k1/k0 2 1 -1 -2
(c) A1 A4 A2 A5
k3/k0=0.9 A3
1.0
0.9
0.8
0.7
0.6
k1/k0
Figure 9-5

38. (a) (b) k3/k0=0.7 k3/k0=0.8 k1/k0 k1/k0 (c) A1 A4 k3/k0=0.9 A2 A5 A3-2 -1 1 2 -2 -1 1 2
(a)
(b)
k3/k0=0.7
k3/k0=0.8
1.0
0.9
0.8
0.7
0.6
k1/k0
1.0
0.9
0.8
0.7
0.6
k1/k0 -2 -1 1 2
(c) A1 A4
k3/k0=0.9 A2 A5 A3
1.0
0.9
0.8
0.7
0.6
k1/k0
Figure 9-6

39. 4.0 40
 = 0.3
 = 1.0
 = 4.0
 = 0.3
 = 1.0
 = 4.0
3.0 30 A1 A2 20
2.0
1.0 10
0.0
0.7
0.8
0.9
1.0
0.6
0.5
0.7
0.8
0.9
1.0
0.6
0.5
Figure 9-7

40. 1.5 20
 = 0.3
 = 1.0
 = 4.0
 = 0.3
 = 1.0
 = 4.0 15
1.0 A1 A2 10
0.5 5
0.0
0.7
0.8
0.9
1.0
0.6
0.5
0.7
0.8
0.9
1.0
0.6
0.5
Figure 9-8

41. (a) (b) A1 A1 Figure 9-9 Normal distribution of shapes with
1.5 6
(a)
(b) 5
1.0 A1 A1 4 3
0.5 2
0.0 1
0.6
0.8
1.0
0.5
0.7
0.9
0.6
0.8
1.0
0.5
0.7
0.9
Normal distribution of shapes with
Spherical pores
Figure 9-9