Electronic Supplementary Materials
Analytical Solutions of the Displacement and Stress Fields of the Generic
Nanocomposite Structure of Biological Materials
Gang Liu, Baohua Ji, Keh-Chih Hwang, Boo Cheong Khoo
Appendix A: The unknown functions of the initial solutions
0
0
0
0
0
0
0
The unknown functions f i     , gi    , i    ,  i    ,      , p     , q     , and
s  0   in the initial solutions (Eq.(22) and Eq.(25)) can be determined by applying appropriate
m 0
p 0
m 0
p 0
boundary conditions. Firstly, based on the continuity conditions ( vI    v   , u I    u   ,
 xym,0I    xyp 0 ,  yym,0I    yyp 0 ) at the interface between mineral I and the protein layer at   d p  2d m  ,
we can obtain the following relations,
 d 
m  p  f I 0    g I 0   
 2d m 
l 1   p2   d p   0
l 1   p   d p   0
 d p   0
 0


    

 s     p 
 p     q  
EmCE  2d m 
2 EmCE  2d m 
 2d m 
2
(A1)
l  d p   0
 0

     p  
GmCG  2d m 
(A2)
Em  d p   0
 0
 0

 f I    I      
l  2d m 
(A3)
f I 0   

2
2
 d p  0
 d p  0
Em  d p   0
 0
 0

 f I    
 I      I     
      s  
2l  2d m 
 2d m 
 2d m 
(A4)
m 0
p 0
m 0
p 0
m 0
p 0
m 0
p 0
Based on the continuity conditions ( vII    v   , u II    u   ,  xy, II   xy  ,  yy, II   yy  ) at the
interface between mineral II and the protein layer at    d p
1
 2d m  , we have,
 d p   0
 0
 f II     g II   
d
2
 m
m 
l 1   p2   d p   0
l 1   p   d p   0
 d p   0
 0


    

 s     p 
 p     q  
EmCE  2d m 
2 EmCE  2dm 
 2d m 
2
f II
0
   
2
l  d p   0
 0

     p  
Gm CG  2d m 
(A5)
(A6)
Em  d p   0
 0
 0

 f II    II      
l  2d m 
(A7)
2
 d p   0
 d p  0
Em  d p   0
0
 0

 f II    
 II      II    
      s   .
2l  2d m 
 2d m 
 2d m 
(A8)
m 0
0
Then, according to the symmetric boundary conditions on AB (see Fig.2), we set vI    C   and
 xym,0I   0 at    d m  d p   2d m  . That is,
 d m  d p  0
 0
 0
m 
 f I     g I    C
 2d m 
(A9)
Em  d m  d p   0
 0

 f I    I    0.
l  2d m 
(A10)

m 0
0
m 0
For the symmetric boundary conditions on GH (Fig. 2), we set vII    C   and  xy, II  0 at
    d m  d p   2d m  , hence
 d m  d p   0
 0
 0
 f II     g II    C
 2d m 
m 
(A11)
Em  d m  d p   0 
 0

 f II    II    0.
l  2d m 
(A12)
Secondly, we solve the unknown functions with above boundary conditions. Combining Eq. (A7) and
Eq. (A3), we obtain

Em  d p   0
Em  d p   0
 0
 0

 f I   

 f II    I    II    0.
l  2d m 
l  2d m 
2
(A13)
Similarly, subtracting Eq. (A12) from Eq. (A10), we have

Em  d m  d p   0
Em  d m  d p   0
 0
 0

 f I   

 f II    I    II    0.
l  2d m 
l  2d m 
(A14)
By comparing Eq. (A13) with Eq. (A14), we get
0
0
f I      f II    .
(A15)
And substituting Eq. (A15) into Eq. (A10) and Eq. (A12) arrives
I 0    II 0   
Em  d m  d p   0

 f I   .
l  2d m 
(A16)
According to Eq. (A16) and Eq. (A3), we obtain
  0   
Em  0
f I   .
2l
(A17)
Next, summation of Eq. (A2) with Eq. (A6) gives
f I 0    f II 0    2 p  0   .
(A18)
Combining the second order derivatives of Eq. (A18) and Eq.(A15), we have
p  0    0.
(A19)
which yields,
p  0     C1 0   C2 0 .
(A20)
Substituting Eq. (A17) and Eq. (A20) into Eq. (A2), we have
1

2
f I 0    f I 0    C1 0  C2 0 .
(A21)
of which the solution is
f I 0    C1 0  C2 0  C3 0 e  C4 0 e 
where   l 2 E p
 E d d 1    .
m
m
p
p
(A22)
Substituting Eq. (A22) into Eq. (A16) and Eq. (A17),
respectively, we obtain,
3
I 0    II 0   
2GmCG  d m  d p 
ld p
 C  e
0

3
 C4 0 e 

(A23)
and


2GmCG d m  0 
C3 e  C4 0 e .
ld p
  0   
(A24)
Substituting Eq. (A20) and Eq. (A24) into Eq. (A6) yields
f II 0    C1 0   C2 0  C3 0 e  C4 0 e  .
(A25)
Substituting Eq. (A22) into Eq. (A9), we have
g I
0
 d m  d p   0
 0
 f I     C
 2d m 
 d m  d p   0 
 dm  d p
 0  
 m 
 m 
 C3 e  C4 e
 2d m 
 2d m
   m 


  0
0
 C2  C

(A26)
and substituting Eq. (A25) into Eq. (A11), we obtain
 d m  d p   0
0
0
g II     m 
 f II     C
 2d m 
 d m  d p   0 
 d m  d p   0
 0  
 0
 m 
 m 
 C3 e  C4 e
 C2  C .
 2d m 
 2d m 


(A27)
Then, subtracting Eq. (A5) from Eq. (A1), we obtain
s  0   
2 EmCE d m  0 EmCE m d m   p d p   0
C 
C2 .
l 1   p2  d p
l 1   p2  d p
(A28)
and adding Eq. (A1) to Eq. (A5), we get
q
0

   

m
2

1    d
p
8d m
p

  C3 0 e  C4 0 e  .




Based on Eqs. (A4), (A17), (A22), (A23), (A28), we obtain
4
(A29)
2
 d 
 d 
E  d 
 I      p    0    s 0    m  p  f I 0     p  I 0  
2l  2dm 
 2dm 
 2dm 
 0

GmCG d p
2ldm

 C3 0 e  C4 0e

2EmCE dm  0 EmCE m dm   p d p   0

C 
C2 .
ld p 1   p2 
ld p 1   p2 
(A30)
Based on Eqs. (A8), (A17), (A23), (A25), and (A28), we obtain
2
 d 
 d 
E  d 
II     p    0    s 0    m  p  f II 0     p  II 0  
2l  2dm 
 2dm 
 2dm 
 0

GmCG d p
2ldm

 0 
 0
 C3 e  C4 e


2EmCE dm  0 EmCE mdm   p d p   0

C 
C2 .
ld p 1  p2 
ld p 1  p2 
(A31)
Finally, according to the loading conditions (for instance, displacement loading u0 (0) 2  1 on
boundary BD) and considering the equilibrium of the nanostructure and the anti-symmetry of
displacement in mineral I respect to mineral II, we have
uIm 0    uIIm 0   
(A32)
uIm 0
(A33)


 dm  d p 
dp

 u0 0 2
1
4
2 dm
2 dm
14
1 4
 xxm,0I  d  0 where   
 yym,0I  d  0 where  
dm  d p
2d m
1
4
(A34)
.
(A35)
m 0
Substituting the expression for ui   into Eq. (A32) and Eq. (A33), and also substituting the expressions
for  xx,i  and  yy,i  into Eq. (A34) and Eq. (A35), respectively, we can obtain
m 0
C  0  
m 0

m
d m   p d p   sh


4 u  0


d m   sh  4ch 
4
4

(A36)
0
C1 0  0
(A37)
5
C2  
2 sh
0


4
 sh  4ch
4
C3   C4  
0
0
u0
0


(A38)
4
1
 sh  4ch
4

u0  ,
0
(A39)
4
Then we obtain the expressions of all the unknown function by substituting Eqs. (A36) to (A39) into Eqs.
(A22), (A25), (A26), (A27), (A23), (A30), (A31), (A20), (A29), (A24) and (A28)
fi 
gi
0
   A   2 sh

0
i
p
q
0
0

 2ch 
4

(A40)
  dm  d p 
 m d p   p d p 

 m sh  
  sh 
4
dm


  dm 
   A   
i 0    A
 0

Em  d m  d p 
ld m
    A
Em d p2
4ld
2
m
   2 A sh

   A  
m
2
 2 ch
(A42)
 3 sh
(A43)


(A41)
(A44)
4
1    d
p
8d m

E
  0    A m  2 ch
l
p

 2 sh


(A45)
(A46)
s  0    0.
(A47)
6
Appendix B: The first order perturbation correction
In this section, we solve the first order perturbation correction of the perturbation solutions of the
governing equations. According to the perturbation expansions obtained in section 3.2, the governing
equations of the first order perturbation (the second expansion term) for the mineral crystal are,
 m1
 xx 


m1
 yy  

 m1
 xy  

 xym 0 
1
1 u m1
m 1
m 0 
 xx  m yy 
Em
l 


1
1 v m1
 yym 0  m xxm1 
Em
l 


(B1)
Gm  u m1 v m 0 



 
l  
(B2)
1
 xym1
2Gm
  xym1  m1
 xx  0





 m1
m1
 yy 
  xy

 0.
 


(B3)
and those for the protein are,
 p1
1
1 u p1
p 1
p 1







 xx
xx
p yy
EmCE
l 

p 1

 p1  1  p1    p1  1 v
 yy
yy
p xx
EmCE
l 

p
1

1
1  u   v p 0 
p 1
 xyp1 

 xy  

 
2GmCG
2l  

(B4)
  xyp1  p 0
 xx  0


 
 p1
 yyp1
  xy

 0.
 


(B5)




Solving above governing equations we can obtain the displacement and stress fields of the mineral
7
and protein of the first order perturbation as below

 m1
1
ui  f i    Li  , 
 m1
1
1
vi   m f i     gi    M i  , 

Em 1
m1
f i     N i   , 
 xx ,i 
l

Em
 m1
1
1
 xy ,i   l  f i    i    Oi  , 

 m1  Em  2 f 1    1    1   P  ,
  i   i   i 
i
 yy ,i
2l
(B6)
for mineral, where i  I and II, standing for mineral I and mineral II, respectively, and
l
 p 1
1
1
u  G C     p    Q  , 
m G

2

l 1   p2  1
l 1   p  2 1
1
p


v 
     
 s     p p 1    q 1    R  , 

Em C E
2 Em C E

Em CE 1
 p 1
1
1
p      p s       S   , 
 xx   2   p     
l

 xyp 1   1    T  , 

 p 1   1    s 1    U  , 
 yy



(B7)
for protein. The unknown functions Li  ,  , M i  ,  , N i  ,  , Oi  ,  , Pi  ,  , Q  ,  ,
1
1
1
1
1
R  ,  , S  ,  , T  ,  , U  ,  ; and f i     , gi    , i    ,  i    ,      ,
p 1   , s 1   , q 1   can be determined by applying the boundary conditions.
The unknown functions in Eq.(B6) are
Li  ,   
2  m 2  0
l
 fi     gi 0    i 0  
Gm
2
(B8)
l 1  m   0
l 1  m  2  0
1  2m 3  0

M i  ,  
 fi    m  2 gi 0   
 i    
i  
6
2
2Em
Em
2
N i  ,   
2
Em 2  0
E
 fi    m  gi 0     2  m i 0    m  i 0  
l
l
8
(B9)
(B10)
Em 3  0
E
2  m 2  0
 fi
 i    m i 0  
   m  2 gi 0   
3l
2l
2
E
E
2   m 3  0

0
0
 i    m  2  i 0  
Pi  ,    m  4 f i      m  3 gi    
12l
6l
6
2
Oi  ,  
(B11)
(B12)
where i  I and II denoting the mineral I and mineral II, respectively.
And the unknown functions in Eq.(B7) are
Q  ,  
R  ,  
l 1 p  3  p 
6EmCE
l 1  p 
12EmCE
S  ,   
 4  0   
p
6
2 p
2
2  p
6
 0
2
 3  2     
T   ,   
U  ,  

3
3
0
 
 2   

p
2
2
 0
p
1 2   p  
l p 1 p 
 2 s 0   q 0  
p
1 2 p   2s0   EmCE q0 
EmCE 2  0
 p   
 
 
l
l
2
(B15)
3
0
3EmCE
 3s 0   
(B13)
(B14)
6
  
2EmCE
2
 2q 0  
p
 
  
l 1 p 
 2  0   
EmCE
 p  0     p s  0  
l
 3  0   

EmCE 2  0
 p    p  2 s  0   .
2l
2
2
(B16)
(B17)
Substituting Eqs. (A40) to (A47) into above equations, we have
Li  ,   

2  m 2  0
l
 fi    gi 0    i 0  
Gm
2
Em  dm  d p  2
d d 
2  m 2
l
 A  2 2ch   A  m p   2mch   A
 ch
Gm
ldm
2
 dm 
 2  m   d m  d p  2
2  m 2

 A  2 2ch    A
 ch
dm
2

 d  dp   2
 A  2  m    2   m
   ch
 dm  

9
(B18)
2
l 1m2   0
l 1m  2  0
1 2m 3  0
m 2  0

Mi  ,  
 fi     gi   
 i   
i  
Em
6
2
2Em

 d d 

1 2m 3
 A 2 3sh   m  2 A m p   3msh
6
2
 dm 
l 1m2   Emd p2 3

l 1m  2 Em  dm  d p  3
A
 sh 
   A 2  sh 

2Em
4ldm
ldm
Em


2
(B19)
2
2
 1 2
dm  d p 1 2m 2 1m  d p  3
m 3
 
   sh
 
 A 


4dm2
3
2
dm


Em 2  0
E
 fi    m  gi 0     2 m i 0   m i 0  
l
l
 d d 
E
E
  m  2 A 2 3sh   m  A m p   3m sh
l
l
 dm 
Ni  ,   
Em  dm  d p 
 Emd p2 3

  2  m  A
 sh m   A 2  sh 
4ldm
ldm


 2E
E  d2 
2E  dm  d p 
 A  m  2  m
  m m2 p   3sh
 l
4ldm 
l
dm


(B20)
3
Em 3  0
E
2  m 2  0
 fi    m  2 gi 0   
 i   mi 0  
3l
2l
2
 d  dp  4
E
E
 m  3 A  2 4ch   m  2 A  m
  mch
3l
2l
 dm 
Oi  ,  


Emd p2 4
2  m 2 Em  dm  d p  4

 A
 ch m   A 2  ch 
2
4ldm
ldm


2
 2E
E  dm  d p  2 Emm d p  4
 A  m  3  m
 
   ch
 3l

4ldm2
l
dm


10
(B21)
Em 4  0
E
2 

 fi
   m 3 gi0    m 3i0    m  2 i0  
12l
6l
6
2
 dm  d p  5
E
E
  m  4 A 2 5sh   m  3 A
  m sh
12l
6l
 dm 
Pi  ,   

 E d2

2 m 3 Em  dm  d p  5

A
 sh  m  2   A m 2p  5sh 
ldm
6
2 
4ldm

(B22)
2
 E
Em  dm  d p  3 Emmd p 2  5
m 4
 
 A   
   sh
 6l

dm
3l
8ldm2


Q,  
l 1p  3 p 

6EmCE

l 1p  3 p 
6EmCE
 2    p    l 1   s  

2
3
 0
3 A
 
p
2
2
0
 
p
2EmCE
2 0
  q0  

1p  dp  2ch
Em 4
 ch  Am 

l
4dm 


(B23)
 1p  3 p 

1p  dp   2ch
3 2  m 
 A 


6CE
4dm  

 

R  ,  

l 1 p 
2
 4  0   
12EmCE
l 1 p 
12EmCE
2
4 A
1 2   p  
p
6
3
0
 
l p 1 p 
3EmCE
3s 0   
p
2
 2q 0  


1p  d p  3sh
Em 5
 sh  p  2 Am 

l
2
4dm 


(B24)
 1 2
 
1p  d p 2  3sh
p
 A
 4 2  p m 
 12CE
2
4dm  

 

1  2 p   2 s0   EmCE q0 
EmCE 2  0
 p   
 
 
6
2
l
l
 3  2 p   3 A Em  5sh  EmCE  A   1   p  d p   3sh

 m
6
4d m 
l
l


S  ,   
 3  2     
p
3
0
  
  3  2 p  Em
1   p  d p    3sh
E C 
 A －
 3 2－ m E m 

6l
4dm  
l 

 

11
(B25)
T   ,   
2 p
2
2  p
 2  0   
Em C E
 p  0     p s  0  
l
Em 4
 ch
l
2
 2   p  Em  2 4ch
 A
2l

U   ,  
2  p
6
2 p
2A
 3  0   
(B26)

Em CE 2  0
 p    p  2 s  0  
2l
2
Em 5
 sh
l
6
 2   p  Em  3 5 sh .
A
6l

3A
(B27)
It can be found that
LI  ,     LII  , 
(B28)
M I  ,    M II  , 
(B29)
N I  ,     N II  , 
(B30)
OI  ,    OII  , 
(B31)
PI  ,     PII  , 
(B32)
Q  ,    Q  , 
(B33)
R  ,    R  , 
(B34)
S  ,     S  , 
(B35)
T  ,    T  , 
(B36)
U  ,    U  ,  .
(B37)
In the following, we will determine the coefficients in above unknown functions. Firstly, according to
m1
p1
m1
p1
m1
p1
m1
p1
the continuity conditions ( u I    u   , vI    v   ,  xy, I   xy  ,  yy, I   yy  ) on the interface of the
mineral I and the protein at   d p
 2d m  , we have
12
 d p  1
 dp 
 dp 
1
m 
 f I     g I    M I   ,
  R  ,

 2d m 
 2d m 
 2d m 
(B38)
l 1   p2   d p  1
l 1   p   d p  1
 d p  1
1


    

 s     p 
 p     q  
2 EmCE  2d m 
EmCE  2d m 
 2d m 
2
2
 dp 
l  dp
1
f I     LI   ,


 2d m  GmCG  2d m

 1
 dp 
1
     p    Q   ,


 2d m 
(B39)
 dp 
 dp 
Em  d p  1
1
1

 f I    I    OI   ,
      T   ,

l  2d m 
 2d m 
 2d m 
(B40)
2
 d p  1
 dp 
Em  d p  1
1

 f I    
 I      I    PI   ,

2l  2d m 
 2d m 
 2d m 
(B41)
 d 
 d 
   p   1    s 1    U   , p  .
 2d m 
 2d m 
m 1
And according to the continuity condition ( u II
u
p 1
the interface of the mineral II and the protein at    d p
m1
, vII
v
p 1
,  xy, II   xy  ,  yy, II   yy  ) on
m1
p1
m1
p1
 2d m  , we obtain
dp 
dp 
 d p  1


1
 f II     g II    M II   , 
  R  , 

2d m 
2dm 
 2d m 


m 
l 1   p2   d p  1
l 1   p   d p  1
 d p  1
1


    

 s     p 
 p     q  
2EmCE  2dm 
2
EmCE  2dm 
d
 m
2
2
dp 

l
1
f II     LII   , 

2d m  GmCG

dp 
 d p  1

1

     p    Q   , 

2d m 
 2d m 

dp 
dp 


Em  d p  1
1
1

 f II    II    OII   , 
      T   , 

2d m 
2d m 
l  2d m 


(B42)
(B43)
(B44)
2
dp 
 d p  1

Em  d p  1
1

 f II    
 II      II    PII   , 

2l  2d m 
2d m 
 2d m 

d 
 d 

  p   1    s 1    U   ,  p  .
2d m 
 2d m 

(B45)
m1
1
m1
And from the symmetric boundary conditions on AB (see Fig. 2), we set vI    C   and  xy, I  0 at
13
   d m  d p   2d m  , then we have
 d  d p  1
 dm  d p
1
m  m
 f I     g I    M I   ,
2d m
 2d m 


Em
l

1
C

(B46)
 d m  d p  1
 dm  d p 
1

 f I      I    OI   ,
  0.
2d m 
 2d m 

(B47)
And for the symmetric boundary conditions on GH (Fig. 2), we set vII    C   and  xy, II  0 , at
m1
1
m1
    d m  d p   2d m  , then we have
dm  d p
 d m  d p  1

1
 f II     g II    M II   , 
2d m
 2d m 

m 

1
  C

(B48)
dm  d p 

Em  d m  d p  1
1

 f II    II    OII   , 
  0.
2d m 
l  2d m 

(B49)
Secondly, we use above relations obtained from the boundary conditions and continuity conditions to
solve
the
unknown
functions.
Subtracting
Eq.
(B40)
from
Eq.
(B44),
and
applying
OI  ,    OII  ,  , T  ,    T  ,  , we obtain

Em
l
 d p  1
Em  d p  1
1
1

 f I    

 f II    I    II    0.
l  2d m 
 2d m 
(B50)
And subtracting Eq. (B47) from Eq. (B49) and applying OI  ,    OII  ,  , we obtain

Em  d m  d p  1
Em  d m  d p  1
1
1

 f I   

 f II    I    II    0.
l  2d m 
l  2d m 
(B51)
Combining Eq. (B50) and (B51) leads to
f I     f II     0.
1
1
(B52)
Then substituting Eq. (B52) into Eq. (B47) and Eq. (B49), and considering OI  ,    OII  ,  , we
obtain
I1    II1   
 dm  d p
Em  d m  d p  1

 f I    OI   ,
2d m
l  2d m 

14

.

(B53)
And substituting Eq. (B53) into Eq. (B40), we obtain
 1   
 d 
 d  dp
Em 1
f I    OI   , p   OI   , m
2l
2d m
 2d m 


 dp 
  T  ,
.

 2d m 
(B54)
Next, by adding Eq. (B39) to Eq. (B43), and considering the relationship LI  ,     LII  ,  ,
we obtain
f I1    f II1    2 p 1   .
(B55)
According to Eq. (B52) and (B55), we have
2 p      0.
1
(B56)
of which the general solution is
p 1    C11  C21 .
(B57)
Substituting Eq. (B54) and Eq. (B57) into Eq. (B39), we obtain
 d 
 d 
f I1    f I1    C11  C21  LI   , p   Q   , p 

 2d m 
 2d m 
1
2
 dm  d p 
 dp 
2l   d p 





O
O
T
,
,





 ,
  .
I
I
Em 2   2d m 
2d m 

 2d m  
(B58)
of which the general solution is
f I     C1   C2   C3  e  C4  e
1
1

1
1
AB 2  
ch2


e  ch sh  ch2     e  ch sh 
   
3 
48dm 
2


0
where A  u0
1
(B59)



2
2
2
3
  sh  4ch  , B  12m d p d m  3d m d p   p d m d p  4d m .
4
4

From Eq. (B55) and Eq. (B57), we obtain
f II1    C11  C21  C31 e  C41 e 

AB 2  
ch2


e  ch sh  ch 2     e  ch sh 
    .
3 
48d m 
2


15
(B60)
From Eqs. (B46), (B48), (B53), (B54), (B57), (B41) and (B45), we obtain
 d  d p  1
 dm  d p
g I1    m  m
 f I     M I   ,
2d m
 2d m 


1
C

(B61)
dm  d p 
 d m  d p  1

1
1
g II     m 
 f II     M II   , 
C
2d m 
 2d m 

(B62)
 dm  d p 
Em  d m  d p  1

 f I    OI   ,

l  2d m 
2d m 

(B63)
I1    II1   
 1   
 d 
 d  dp
Em 1
f I    OI   , p   OI   , m
2l
2d m
 2d m 


 dp 
  T  ,


 2d m 
p 1    C11  C21
(B64)
(B65)
2
 d 
 d 
E  d 
 I     m  p  f I1     p   1     p  I1  
2l  2d m 
 2d m 
 2d m 
1
 d 
 d 
 s 1    U   , p   PI   , p 
 2d m 
 2d m 
(B66)
2
 d 
 d 
E  d 
 II     m  p  f II1     p   1     p  II1  
2l  2d m 
 2d m 
 2d m 
1
d 
d 


 s    U   ,  p   PII   ,  p  .
2d m 
2d m 


(B67)
1
Then subtracting Eq. (B38) from Eq. (B42), and considering the relationship M I  ,    M II  , 
and R  ,    R  ,  , we obtain
s     
1
EmCE d m
l 1   p2  d p
 m d m   p d p  1

1

C2  2C    .


dm


(B68)
By adding Eq. (B38) to Eq. (B42), and considering the relationship M I  ,    M II  ,  and
R  ,    R  ,  , we obtain
16
q 1   
f
4
m
1
I
   f II1     M I   ,


dp 
 dm  d p 
  M I  ,

2d m 
2d m 

(B69)
 d  l 1   p   d p  1
 R  , p  

     .
2
2
2
d
E
C
d
m 
m E 
m 

2
2
Finally, the unknown coefficients C   , C1  , C2  , C3  and C4  in above equations can be
1
1
1
1
1
obtained according to the loading conditions (For instance, displacement loading u0  2  0 on
1
boundary BD) and considering the equilibrium of the nanostructure and the anti-symmetry of
displacement in mineral I respect to mineral II, which are
uIm1  ,   uIIm1   ,  
(B70)
 1 d  dm 
1
uIm1  , p
  u0 2
 4 2d m 

 dm  d p 
 xxm,1I d  0 where   1 4
(B72)
 yym,1I d  0 where    d m  d p   2d m  .
(B73)
2 dm
d p 2 dm

14
1 4
(B71)
which can be further expressed as below,
 f I1    LI  ,   f II1     LII   ,  
(B74)
 1 d p  d m  u01
1 1
f I     LI  ,

4
 4 2d m  2

 dm  d p 
d p 2 dm
2 dm
(B75)
 Em 1  1 
 1 
 l f I    4   N I   4 ,   d  0





(B76)
 E  d p  d m 2 1
 d p  d m  1
 d p  dm

1
1 4  2ml  2d m  f I      2d m  I      I    PI   , 2d m

14

  d   0.

(B77)
According to Eq. (B74) we have C1   0 , C3   C4  , therefore the original five unknown coefficients
1
1
1
can be reduced into three unknown coefficients C   , C2  , C3   C4  . Then we have simplified
1
expression of Eq. (B59) as
17
1
1
1
f I1    C21  2C31 ch 
AB 2  3
1

ch  sh  2 sh  .
3 
48dm  2
2

(B78)
According to Eq. (B75), Eq. (B76) and Eq. (B78), together with Eqs. (B18) and (B20), we can obtain the
unknown coefficients C2  and C3  as below,
1
1
 




2
 B  2sh 2  ch 2     16dm  dm  3d p  sh 2 

4 u1 

 
C21 

 0

 

3 
4ch   sh
24dm  4ch   sh   12  2  m  dm  dm2  2dm d p  sh 

4
4
4
4

 
2

2 sh

A 3
(B79)




 

24  2m dmdp2sh  B 2ch Bch 5B sh 12Bch 

u0
A
4
4
4
4
4 .
C31 C41 





2

 


3
4ch  sh 192dm 4ch  sh  4Bsh 32dm2  dm 3dp  sh 96 2m dm  dm dp  ch 
4
4
4 
4
4
4
 4
1
2
(B80)
The unknown coefficients C   can be obtained through Eq. (B77). By substituting Eq. (B78), Eq. (B63),
1
Eq. (B66) and Eq. (B22) into Eq. (B77), we can get the analytical solution of coefficient C   as below
1
AB 2  d m  2d p 
 m d m   p d p  1
C2
 1   p  sh 
3 
96d m  2d m 
4
2d m
C 1 
(B81)
Finally, basing on the initial solution and the first order perturbation correction of the perturbation
solution of governing equations, we can obtain the analytical expressions of the improved approximate
perturbation solution of the displacement and stress fields in mineral and protein as below,
m 0 
uim  ui
m 1
 2d ui

vim d vim 0  2d vim1
(B82)

(B83)
 xxm ,i   xxm,i0  2d  xxm,1i 


(B84)



 yym ,i 2d  yym,0i   2d  yym,1i
 xym ,i d
m 0 
xy ,i
(B85)
 2d  xym ,1i
(B86)
u p  u p 0  d2 u p1

v p d v p 0  2d v p1
(B87)

(B88)
18








 xxp 2d  xxp 0  2d  xxp1
 yyp 2d
 xyp d
p0
yy
 2d  yyp 1
p 0
xy
 2d  xyp 1
(B89)
(B90)
(B91)
19
1.0
(a)
=40 Analytical
=40 FEM
=30 Analytical
=30 FEM
=20 Analytical
=20 FEM
 u0 Em 
0.6
l xxm ,I
0.8
0.4
0.2
0.0
-0.2
-0.1
0.0
0.1
0.2
-0.2
-0.1
0.0
0.1
0.2
-0.2
-0.1
0.0
0.1
0.2

-2
2.010
(b)
1.6
1.2
l xym , I  u0 Em 
0.8
0.4
0.0

0.50
(c)
uIm u0
0.45
0.40
0.35
0.30

Figure S1
20
Figure S1: The comparison between our theoretical predictions and the FEM simulation results. (a) The
dimensionless normal stress in the mineral crystal; (b) the dimensionless shear stress along the
mineral-protein interface; (c) the dimensionless displacement in the mineral along the longitudinal
direction. Em G p  2400 , and   50%.
(Discussion: One reason for the discrepancy between our theoretical predictions and the FEM results
at the mineral tips is that we assumed a traction free boundary condition at the mineral tips by neglecting
the traction force acted by protein at the mineral tips during the model simplification from Fig. 1(b) to Fig.
2. The other reason is that we neglected the boundary condition at the end of protein layer. We note that
there is non-zero shear stress at the end of protein layer, which violates the Newton’s third law for the
mutual forces of action and reaction between the protein layer and its neighbors at lines CE and DF. This
problem can be solved by constructing a boundary-layer type solution to satisfy the Newton’s third law.
However, according to theSaint-Venant's principle, these simplifications will only affect the tip zone of


the mineral of size about 2 d m  d p . If the aspect ratio of the mineral is large, the ratio of this zone size
to the length of mineral will be small. The consistency between the theoretical predictions and the FEM
results just validated our assumptions)
21
 b    45%
0.4
Eˆ Em
0.3
Our Analytical Solution
TSC Model
FEM Calculation
0.2
0.1
0.0
0
50
100
150
200
250
  Aspect Ratio 
300
350
Figure S2
Figure S2: The equivalent Young’s modulus of the nanocomposite structure versus the aspect ratio of
mineral crystal predicted by our analytical solution, the TSC model and the FEM simulation
( Em G p  2400). The volume fraction of mineral   45% (as the case of bone).
22
1.0
(a)
=30,Em/Gp=3000
=30,Em/Gp=2400
=30,Em/Gp=1800
=30,Em/Gp=1200
 u0 Em 
0.6
l xxm ,I
0.8
0.4
0.2
0.0
-0.2
0.025
-0.1
0.0
0.1
0.2
-0.1
0.0
0.1
0.2
-0.1
0.0
0.1
0.2

(b)
 u0 Em 
0.015
l xym ,I
0.020
0.010
0.005
0.000
-0.2
0.50

(c)
uIm u0
0.45
0.40
0.35
0.30
-0.2

Figure S3
23
Figure S3: The influence of the elastic modulus ratio of mineral to protein on the distribution of the
dimensionless normal stress in mineral, the dimensionless shear stress at the mineral-protein interface and
the dimensionless displacement in mineral.   50%.
7
=2.0
=4.0
=6.0
=8.0
=10.0
=20.0
6
l xxm , I
u0 Em 
5
4
3
2
1
0
-0.2
-0.1
0.0

0.1
0.2
Figure S4
Figure S4: The evolution of the distribution of the normal stress in the mineral along the longitudinal
direction with the variation of the  value.
24