Jiaguo Liu, Mingyu Xu
School of Mathematics,
Shandong University,
Jinan, 250100, P.R. China.
1. Introduction
 Bone is anisotropic and viscoelastic
 Study on mechanical behavior of Cranial bone is the
basic work of research on craniocerebral injury.
 The researches on dynamic behavior of bones are
important in guiding orthopaedics diseases, cure of
bone injure, substitutive materials and healing study.
2
Cranial Bones: eight bones
3
• Zhu et al’ study on the behavior of
cranial bone by classical St.Venant model
• Classical Maxwell and Zener model’s
fractional order generalizations
4
2. Fractional generalization of classical
St.Venant model
2.1 The classical (integer order) St.Venant model is
shown as follows
Its constitutive equation is
 r1 (t )   (t )  E r1 (t )  E1(t )
(1)
5
 (t )   r (t )  E (t )  E1 r(t )
where  (t ) and  (t ) denote the stress and strain, E1 , E2 is
the elastic coefficients, and  is the viscosity.
E1E2
E
,
E1  E2

r 
,
E1  E2
d 

E2
(2)
Obviously,
E1 r  E d .
(3)
6
2.2 Fractional generalization of St.Venant model
Riemann-Liouville fractional operators:
 1


t



D
f (t ) : 
f ( )d ,
0 t
0
 
t
n
d
q
D
0 t f (t ) :
dt n
D
0
q n
t

f (t ) ,
Re   0 ,
q  0, n  q  0 .
(4)
(5)
7
Let
[ (t )]   r 1 (t )   (t ) ,
(6)
(7)
 [ (t )]  E r 1 (t )  E1(t ) .
[ (t )]   [ (t )] is equivalent with Eq. (1).
Integrals from 0 to t give
0 D  r
1
t
0 D   E r
1
t
1
1
1
D
,
0 t  (t )   (t )   0
(8)
1
D
0 t  (t )  E1 ( (t )   0 ) ,
where  0 and  0 are initial values of  (t ) and
respectively.
(9)
 (t )
8
1
1
q
q
Substituting  r 0 Dt  (t ) by  r 0 Dt  (t ) in (8), and
 r1 0 Dt1 (t ) by  rq 0 Dtq (t ) in (9), we obtain
~
 r
~  E

r
q
q
D
0 t  (t )   (t )   0 ,

(0  q,   1) .
D
0 t  (t )  E1 ( (t )   0 ) ,
~
  ~ results the fractional St.Venant model:

q
r 0
D  (t )   (t )   0  E r
q
t


D
0 t  (t )  E1 ( (t )   0 )
(10)
9
3. Solutions of fractional St.Venant model
3.1 Relaxation and creep function of fractional St.Venant
model
Laplace transform of (10) gives
 
q q
1
 r p ˆ ( p)  ˆ ( p)   0 p  E r p ˆ( p)  E1ˆ( p)  E1 0 p 1 . (11)
Let  (t )   0 (t ) , where  (t ) is the Heaviside unit step
function, from (11) we obtain
    1
1
 0 p  E r p
ˆ ( p ) 
.
q q
(12)
1 r p
10
The discrete inverse Laplace transform of (12) give the
relaxation modulus of fractional St.Venant model

 0 
k  qk  qk 1
k  qk    qk   1 
G(t )  L   (1)  r p
 E  (1)  r p


k 0
 0 k 0

1

(1)  t 
 
 E1 
(qk  1)   r 
k 0
E1 1,1  t
 H1, 2
q
 r

k
 1
 0 , 
 q
1
( 0 , ) ; 0 ,1
q
qk

 t 
(1)
 
E
(qk    1)   r 
k 0
 E  t 
t
1
,
1
    H1, 2 
 q   r 
 r


k
 1
 0 , 
 q
1
( 0 , ) ;    ,1
q


,

qk  
(13)
where H11,,21 (x) is the H-Fox function.
11
The deduction uses the following properties of the HFox function:


n 0
( z )
p
n
 ( a
j
 A j n)
j 1
q
n! (b j  B j n)
H
1, p
p , q 1
 (1  a p , Ap ) 
(a p , Ap )

; ( z ) ,
z
 p  q 
(
0
,
1
);
(
1

b
B
)
(
b
,
B
)
q, q 

 q q


(14)
j 1
where p q ( z) is also called Maitland’s generalized
hypergeometric function.
 K (a p , K p )
1 m,n  (a p ,  p )
m,n
H p ,q  z
  H p ,q  z
.
(
b
,

)
(
b
,
K

)
K
q
q 
q
q 




(15)
12
In a similar way, the creep compliance of fractional
St.Venant model can be obtained
 t
1
1,1
J (t ) 
H1, 2 W
E1
 r

 1
 0 , 
 
1
( 0 , ) ; 0 ,1

 1  t q
 t
1,1

  H1, 2 W
 E1   r 
 r


 1
 0 , 
 
1
( 0 , ) ;   q ,1


,


(16)
where
1
E 
W    .
 E1 
13
When q    1, (13) and (16) reduce to the relaxation
modulus and creep compliance of classical (integer
order) St.Venant model
t
G(t )  E1 H 11,,21 
 r

 t


E
( 0,1) ; 0,1 

 r
0,1
 1,1  t
 H 1, 2 

 r

( 0,1) ;  1,1 

0,1
 E  E1  E  exp  t 
r 

J (t ) 

1 1,1  Et
H 1, 2 
E1
 E1 r
 1  t  1,1  Et
  H 1, 2 
( 0,1) ; 0,1  
 E1   r 
 E1 r
0,1
1
1 

 1  exp  t  
 d 

E1 E2 
(17)

( 0,1) ;  1,1 

0,1
(18)
14
3.2 The fractional relaxation and creep functions under
quasi-static loading
The loading processes of relaxation and creep tests are,
respectively,
 At
 t   
 1
 Bt
 t   
 1
0  t  t1 
,
t  t1 
(19)
0  t  t1 
,
t  t1 
(20)
where A and B are constant strain and stress rates,
respectively, and  1 and  1 are constants.
15
By Boltzmann superposition principle
 t    Gt     d ,
t
0
 t    J t      d ,
t
0
from (13) and (16) we obtain the relaxation and creep
response functions
16

 1
 1





 0 , 
 0 , 


AE
t
t
AEt
t
t
1


,
  H11,,21   1q 
H11,,21   1q 

q
 r ( 0, q ) ; 1,1  q   r 
 r ( 0, q ) ; 1  ,1 






0  t  t1 


 1
 1





 0 , 
 0, 


AE1t 1,1  t  q 
AE
t

t
 t   
1,1 t  t1  q 
1
1


H1, 2
H1, 2 
1
1

q
q
 r ( 0, q ) ; 1,1 
  r ( 0, q ) ; 1,1 







 1
 1





 0 , 
 0 , 






AEt
t
t
AE
t

t
t

t
t

t
q
q
1
,
1
1
,
1
1



  H1, 2   1 
 1  H1, 2  1  1 
 q  r 
q
r 
 r ( 0, q ) ; 1  ,1 
  r ( 0, q ) ; 1  ,1 







t  t1 
(21)
17

Wt  0, 1 
 Bt  t  q
Wt  0, 1 

Bt


1
,
1
1
,
1



  H1, 2   1 
H1, 2   1 

E1
  r ( 0,  ) ; 1,1  E1   r 
  r ( 0,  ) ; 1q ,1 ,







0  t  t1 

 1

 Bt  t 
W t  t   0, 1 

Bt 1,1 Wt  0,  
 t   

1,1
1
1



H1, 2 
H1, 2 
1
1

(
0
,
)
;


1
,
1

(
0
,
)
;


1
,
1

E1
E1
r 

 r







q

Wt  0, 1 
 Bt  t   t  t  q
W t  t   0, 1 



Bt
t


1
,
1
1
,
1
1
1 




  H1, 2   1 
 1  H1, 2 
1
( 0 , ) ;  1 q ,1 
 E1   r 
E1   r 
  r ( 0,  ) ; 1q ,1 
 r







t  t1 
(22)
18
When q    1 , the relaxation and creep functions of
classical St.Venant model are


  t  ,


0  t  t1 
AEt

A

E

E
1

exp



r
1


r




 t   
 AEt1  A r E1  E  exp t1   1 exp  t  ,
t  t1 



t
r






Bt B d 

  t  ,
0  t  t1 

1

exp





d 
E
E2 

 t   
B
 Bt1  d  exp t1   1 exp  t  ,
t  t1 
d 
 E
E2 
 d  

(23)
(24)
19
4. Data fitting and comparison
The relaxation and creep functions (21) and (22) are
fitted with the experimental data from Zhu et al’s, and
we take parameters A, B, E1，E2，t1 , τr, τd the same
values as Zhu et al’s.
20
(***): Relaxation experimental data from [5]; (---): the
relaxation function (23) of the standard St.Venant
model; (—):the relaxation function (21) of the
fractional St.Venant model. Here, q=0.965, μ=0.96.
21
(***): Creep experimental data from [5]; (---): the creep
function (24) of the standard St.Venant model; (—):the
creep function (22) of the fractional St.Venant model.
Here, q=0.5, μ=0.47.
22
It is shown that, the fractional St.Venant model is
more efficient than the standard St.Venant model
with integer order in describing the stress-strain
constitutive relations for the viscoelasticity of human
cranial bone.
23
5. Conclusion
 The fractional St.Venant model is more efficient than
the classical model in describing the stress-strain
constitutive relations for the viscoelasticity of human
cranial bone.
 It is efficient that applying fractional calculus method
to describe constitutive relations of biological
viscoelastic materials.
24
Thank you very much!
Jiaguo Liu
liujiaguo@sdu.edu.cn
25