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Study on a fractional model of viscoelasticity of human cranial bone Jiaguo Liu, Mingyu Xu School of Mathematics, Shandong University, Jinan, 250100, P.R. China.
Study on a fractional model of viscoelasticity of human cranial bone
Jiaguo Liu, Mingyu XuSchool of Mathematics,
Shandong University, Jinan, 250100, P.R. China.
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1. IntroductionBone is anisotropic and viscoelasticStudy 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.
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Cranial Bones: eight bones
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• Zhu et al’ study on the behavior of cranial bone by classical St.Venant model• Classical Maxwell and Zener model’s fractional order generalizations
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2. Fractional generalization of classical St.Venant model2.1 The classical (integer order) St.Venant
model is shown as follows
Its constitutive equation is
(1))()()()( 1
11 tEtEtt rr
6
)(t where and denote the stress and strain, is the elastic coefficients, and is the viscosity.
(2) Obviously, (3)
)(t
,21
21
EEEEE
,21 EEr
2Ed
21, EE
.1 dr EE
)()()()( 1 tEtEtt rr
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2.2 Fractional generalization of St.Venant model
Riemann-Liouville fractional operators:
(4)
(5)
,0Re,)(:)(
0
1
0
dfttfDt
t
.0,0,)(:)( 00 qnqtfDdtdtfD nq
tn
nqt
8
Let ,
(6) .
(7) is equivalent with Eq. (1).Integrals from to give
, (8)
, (9) where and are initial values of and
respectively.
)()()]([ 1 ttt r
)()()]([ 11 tEtEt r
01
011
0 )()( ttDD trt
))(()( 011
011
0 tEtDED trt
)]([)]([ tt
0 0 )(t )(t
t0
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Substituting by in (8), and
by in (9), we obtain
results the fractional St.Venant model:
(10)
,)()(~00
ttD qt
qr
,))(()(~010
tEtDE tr
)(0 tD qt
qr )(1
01 tDtr
)(10
1 tDtr )(0 tD qt
qr
.)1,0( q
~~
))(()()()( 01000 tEtDEttD tr
qt
qr
10
3. Solutions of fractional St.Venant model3.1 Relaxation and creep function of fractional
St.Venant model
Laplace transform of (10) gives
. (11)
Let , where is the Heaviside unit step function, from (11) we obtain
(12)
1011
10 )(ˆ)(ˆ)(ˆ)(ˆ pEpEppEpppp r
qqr
)()( 0 tt
.1
)(ˆ11
0qq
r
r
ppEpp
)(t
11
The discrete inverse Laplace transform of (12) give the relaxation modulus of fractional St.Venant model
, (13)
where is the H-Fox function.
0 0
11
0
01 )1()1()(k k
qkqkr
kqkqkr
k pEpLtG
qk
rk k
kqk
r
k tqk
Etqk
E0 0
1 )1()1(
)1()1(
q
qrr
q
qr
tHtqEtH
qE
1,0
1,;)1,0(
1,12,1
1,0
1,0;)1,0(
1,12,1
1
)(1,12,1 xH
12
The deduction uses the following properties of the H-Fox function:
(14)
where is also called Maitland’s generalized hypergeometric function.
(15)
,)(;),(),(
)1();1,0(),1(
)(!
)()(
,
,11,
0
1
1
z
BbAa
BbAa
zHnBbn
nAaz
ppqp
pppqp
nq
jjj
p
jjj
n
.),(),(
),(),(1 ,
,,,
ppKnmqp
ppnmqp Kb
KazH
ba
zHK
)( zqp
13
In a similar way, the creep compliance of fractional St.Venant model can be obtained
(16)
where
,11)(
1,0
1,;)1,0(
1,12,1
1
1,0
1,0;)1,0(
1,12,1
1
qr
q
rr
tWHtE
tWHE
tJ
.1
1
EEW
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When , (13) and (16) reduce to the relaxation modulus and creep compliance of classical (integer order) St.Venant model
(17)
(18)
1q
1,01,1;)1,0(
1,12,1
1,01,0;)1,0(
1,12,11)(
rrr
tHtEtHEtG
r
tEEE exp1
1,01,1;)1,0(
1
1,12,1
1
1,01,0;)1,0(
1
1,12,1
1
11)(rrr E
EtHtEE
EtHE
tJ
d
tEE exp111
21
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3.2 The fractional relaxation and creep functions under quasi-static loading
The loading processes of relaxation and creep tests are, respectively,
(19)
(20)
where and are constant strain and stress rates, respectively, and and are constants.
,
0
1
1
1 ttttAt
t
,
0
1
1
1 ttttBt
t
A B
11
16
By Boltzmann superposition principle
from (13) and (16) we obtain the relaxation and creep response functions
,0
dtGtt
,0
dtJtt
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1
1,0
1,1;)1,0(
11,12,1
111,0
1,1;)1,0(
1,12,1
1,0
1,1;)1,0(
11,12,1
111,0
1,1;)1,0(
1,12,1
1
1
1,0
1,1;)1,0(
1,12,1
1,0
1,1;)1,0(
1,12,1
1
0
,
tt
ttHttqttAEtHt
qAEt
ttHqttAEtH
qtAE
tt
tHtqAEttH
qtAE
t
q
qrr
q
qrr
q
qr
q
qr
q
qrr
q
qr
(21)
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1
1,0
1,1;)1,0(
11,12,1
1
1
11,0
1,1;)1,0(
1,12,1
1
1,0
1,1;)1,0(
11,12,1
1
11,0
1,1;)1,0(
1,12,1
1
1
1,0
1,1;)1,0(
1,12,1
1
1,0
1,1;)1,0(
1,12,1
1
0
,
tt
ttWHttEttBWtHt
EBt
ttWHEttBWtH
EBt
tt
WtHtEBtWtH
EBt
t
qr
q
rq
r
q
r
rr
qr
q
rr
(22)
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When , the relaxation and creep functions of classical St.Venant model are
(23)
(24)
1q
1
111
11
,exp1exp
0,exp1
ttttEEAAEt
tttEEAAEtt
rtr
rr
1
1
2
1
12
,exp1exp
0,exp1
ttttEB
EBt
tttEB
EBt
t
dd
d
d
d
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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.
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(***): 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.
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(***): 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.
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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.
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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.
