CH2 The Meaning of the
Constitutive Equation
Prof. M.-S. Ju
Dept. of Mechanical Eng.
National Cheng Kung University
2.1 Introduction
Living system at cellular, tissue, organ and
organism level sufficient to take Newton’s
laws of motion as axiom
The smallest volume considered contains a
very large no. of atoms and molecules. The
materials can be considered as a continuum
Isomorphism between real number and
material particles. Between any two material
particles there is another material particle.
Continuum: in Euclidean space, between any two
material particles there is another material particle, each
material particle has a mass.
Mass density of a continuum at a point P is defined as:
M
 ( p )  lim
v 0 V
V : a sequence of volumes enclosing p
M : mass of particles in V
P
V
M
Modification of definition
Observation of living organisms at various levels of size: e.g.,
naked eye, optical microscope, electrical microscope,
scanning tunnel microscope or atomic force microscope
M
lim
  ( p )  tolerable error
v v0 V
v0 : finite lower bound, v0  0
Blood example
Whole blood – continuum at scale of
heart, large arteries, large veins
Two-phase fluid (plasma & blood cells)
at capillary blood vessels, arterioles and
venules.
At smaller scale: red cell membrane as
continuum and red cell content as
another continuum
2.2 Stress
Stress expresses the interaction of the
material in one part of the body on another.
Unit: 1 Pa = 1 N/m2
1 psi = 6.894 kPa
1 dyn/cm2 = 0.1 Pa
1 atmosphere = 1.013 x 105 N/m2 =
1.013bar
F

F : force exerted by the particles at positive side of surface
S
Ｓon the negative side
Assume that as S tends to zero the ratio F/S tends to a
definite limit, dF/ds, and moment of the force acting on the
surface S tends to zero.
F
Stress 
n
x3
F
N
  unit :1Pa  1 2
S
A
m
N
1MPa  1
mm2
S
B
o
x1
x2
Let vector n be normal of S
Positive side: surface pointed by n
Positive side exerts force F on the negative side
F depends on location and size of S and orientation of n
When S→0


F dF
lim

S 0 S
ds
moment of F exerts on S →0

V
dF
T
: Traction or stress vector
ds
force per unit area acting on the surface
Components of stress

T   11ê1   12ê2   13ê3 2
T   21ê1   22ê2   23ê3
3
T   31ê1   32ê2   33ê3
1
êi～unit vecto r

x3


33
32
n 
 


31
23
13
 11 , 22 , 33～normal stresses
 12 , 23 , 31～shear stresses
12
22
21
11
ds
o
x1
x2
Cauchy' s formula
v  v1ê1  v2 ê2  v3ê3
(1 )
V
Ti  v1 1i  v2 2i  v3 3i  v j ji
V
or　Ti   ji v j
V
Ti  
v
ij j
  ji   ij
Note: knowing components of a stress tensor one can write
down the stress vector acting on any surface with unit outer
normal vector n
(2) For a body in equilibrium we have
  11   12  13


 x1  0  x1  x2  x3
 21  22  23


 x2  0
 x1  x2  x3
 31  32  33


 x3  0
 x1  x2  x3
 ij
or　 x i  0
x j
Xi: components of body force
(per unit volume) along ith axis
(3) ij   ji
Due to equilibrium of moments
(4) change of coordinate system
( x1 , x2 , x3 )  ( x , x , x )
xk   ki xi   k1 x1   k 2 x2   k 3 x3


 ki : directiona l cosine of xk axis w.r.t xi axis
'
1
'
2
'
3
   ki  mj ij
 km
’km components of stress tensor in the new coordinate system
ij components of stress tensor in theold coordinate system
2.3 Strain
Deformation of a solid described by strain
Take one-dimensional problem as an example:
elongation of a string, initial length L0
L  L0
L  L0

, ' 
L0
L
Other definitions:
L2  Lo 2
e
2 L2
L2  Lo 2

2 Lo 2
For example :
3
3
( a )L  2　Lo  1　e   
8
2
( b )L  1.01　Lo  1　e  0.01　  0.01
  0.01　 '  0.01
Note: above strain measures are equal for
infinitesimal elongation
Another is Shear Strain，consider the twist of a circular
cylindrical shaft. tana or tana/2 is defined as the shear strain
a
M
M
Constitutive Equation: a relationship between stress and strain
  Ee　Hooke' s law, E : Young' s　modulus
  G tana　, G : modulus of rigidity
Deformation of living system is more complicated and require a
general method. Let a body occupy a space S，let coordinate of a
particle before deformation be (a1,a2,a3)，coordinate after
deformation(x1,x2,x3)，
a3 ,x3
P’
P”
P
S
a1, x1
Q”
u
Q’
Q
(a1, a2, a3)
a2, x2
o
(x1, x2, x3)


PQ or u : displaceme nt of p. The components of u are
u1  x1  a1
,
u2  x2  a2
,
u3  x3  a3
xi  xi ( a1 , a2 , a3 ) ( i  1,2,3 ) (displacem ent field)
ai  ai ( x1 , x2 , x3 ), i  1, 2, 3 (1)
Q: stretching and distortion of the body?
assume p close to p，p( a1  da1 , a2  da2 , a3  da3 ，
) pp distance dSo
dSo  da12  da22  da32
2
　(2)
After the deformation p and p’ displaced to Q and Q’，QQ’ distance dS
dS  dx  dx  dx (3)
2
2
1
2
2
2
3
xi
ai
dxi 
da j dai 
dx j
a j
x j
using Kronecker delta  ij ,  ij  1　i  j ,  ij  0　i  j
ai a j
dS   ij dai da j   ij
dxl dxm
xl xm
xi x j
2
dS   ij dxi dx j   ij
dal dam
al am
2
0
difference between dS 2 and dS 02 can be used to
define strain te nsor
xi x j
dS  dS   ij
dal dam   ij dai da j
al am
xa x
 [  a
  ij ] dai da j
ai a j
2
2
0
or
ai a j
dS  dS   ij dxi dx j   ij
dxl dxm
xl xm
aa a
 [  ij   a
] dxi dx j
xi x j
2
2
0
Definition of strain tensors:
xa x
1
Eij  (  a
  ij )　Green s　s train tensor(Lag rangian) based on ai
2
ai a j
aa a
1
eij  (  ij   a
)　Almansi s　Strain　tensor( Eulerian ) based on xi
2
xi x j
ds 2  ds02  2 Eij dai da j
 2
2
ds  ds0  2eij dxi dx j
note : ds 2  ds02  0 implies Eij  0 and eij  0
Eij  0 or eij  0 implies rigid body
Show that Eij and eij are symmetric
When ui is small
(
ui 2
u u
) and ( i k ) are negligible
x j
x j xe
eij reduces to Cauchy’s infinitesimal strain tensor:
1  u j  ui
 ij  (

)　i , j  1,2,3,
2  xi  x j
u1  u　x1  x
u2  v　x2  y
u3  w　x3  z
Cauchy strain tensor
or
u
1 v u
 xx 
 xy  (

)   yx
x
2 x y
v
1 w u
 yy 
 xz  (

)   zx
y
2 x z
w
1 v w
 zz 
 yz  (

)   zy
z
2 z y
In the infinitesimal deformation, no distinction between
Lagrangian and Eulerian strain tensors.
Geometric meaning
u
u
y
u
dx
x
u
u
 0,v  0
x
u
u
dx
x
u
 0,v  0
x
x
x
y
y
x
x
u
v
 0,  0
y
x
u
v
u
 0,  0,  0
y
x
x
y
x
u
v
 0,  0
y
x
2.4 Strain rate
For fluid motion, consider velocity field and rate of strain.
At point (x,y,z) velocity vector


V ( x, y, z )  u ( x, y, z )iˆ  v( x, y, z ) j  w( x, y, z )kˆ
or　Vi ( x, y, z ) i  1, 2, 3
For continuous flow, Vi : continuous and differentiable
 Vi
d Vi 
d x j i , j  1,2,3
 xj
 Vi 1  Vi  V j
1  V j  Vi
where
 (

) (

)
 x j 2  x j  xi
2  xi  x j
Define strain rate tensor Vij
1  Vi  V j
Vij  (

)
2  x j  xi
, Vij  V ji symmetric
Define vorticity tensor ij
1  V j  Vi
 ij  (

)
2  xi  x j
,  ij   ji anti  symmetric
dVi  Vij d x j  ij d x j
2.5 Constitutive equations
Properties of materials are specified by
constitutive equations
Non-viscous fluid, Newtonian viscous fluid and
Hookean elastic solid are most widely models for
engineering materials
Most biological materials can not be described
by above equations
Constitutive equations are independent of any
particular set of coordinates. A constitutive
equation must be a tensor equation: every term
in it be a tensor of same rank.
2.6 The Nonviscous Fluid
 ij   p  ij
Stress tensor:
D Kronecker delta, p: pressure (scalar)
For ideal gas: equation of state
p
 RT

For real gas:
f (p, , T) = 0
Incompressible fluid:
 = constant
2.7 Newtonian Viscous Fluid
Shear stress is proportional to strain rate
Stress-strain rate relationship
 ij   p ij  DijklVkl
ij : stress tensor, Vkl: strain rate tensor, p: static pressure
• p= p(, T) equation of state
• Elements of Dijkl depend on temperature but stress or strain
rate
• Isotropic materials: a tensor has same array of components
when frame of reference is rotated or reflected (isotropic
tensor)
Dijkl   ij kl   ( ik  jl   il jk )
 ij   p ij  Vkk  ij  2Vij isotropic Newtonian　fluid
 kk  3 p  ( 3  2 )V kk
2
 kk  3 p ( 3  2  )  0     
3
2
 ij   p  ij  2 Vij   Vkk  ij Stokes fluid
3
for incompress ible fluid Vkk  0
 ij   p ij  2Vij incompress ible viscous
let   0  ij   p ij nonviscous
Coefficient of viscosity 
Newton proposed
du
 
dy
Units:
1 poise= dyne. s /cm2 = 0.1 Ns/m2
Viscosity of air: 1.8 x 10-4 poise
Water: 0.01 poise at 1 Atm. 20 deg C
Glycerin: 8.7 poise
2.8 Hookean Elastic Solid
Hooke’s law: stress tensor linearly proportional
to strain tensor
 ij  Cijkl ekl
(1)
 ij : stress tensor,
e kl : strain ten sor
Cijkl : tensor of elastic constants or moduli
Note: elastic moduli are independent of stress or strain
Isotropic materials
 ij   eaa  ij  2 eij (2)
 ,  are Lame constants,   G : Shear modulus
 xx  ( exx  e yy  ezz )  2Gexx
 yy  ( exx  e yy  ezz )  2Geyy
 zz  ( exx  e yy  ezz )  2Gezz
 xy  2G exy
 yz  2G e yz
 zx  2G ezx
Inverted form
1 n
n
eij 
 ij   kk  ij
E
E
1
exx  [ xx  v( yy   zz )] ,
E
1
e yy  [ yy  v( xx   zz )] ,
E
1
ezz  [ zz  v( xx   yy )]，
E
1 v
1
exy 
 xy 
 xy
E
2G
1 v
1
e yz 
 yz 
 yz
E
2G
1 v
1
ezx 
 zx 
 zx
E
2G
Ｅv
E

G 
(1  v)(1  2v)
2(1  v)
E
 (1  v)(1  2v)
v
 1　E 
 2G (1  v)
2G
v
E: Young’s modulus, G: shear modulus, n: Poisson’s ratio
2.9 Effect of Temperature
The constitutive equations are stated at a given
temperature T0
Dijkl, Cijkl,  depend on temperature
If temperature is variable: Duhamel-Neumann
form
 ij  Cijkl ekl   ij ( T  To )
Cijkl  Cijkl ( T0 )
For isotropic material
 ij    ij
 ij  Cijkl ekl   ( T  To ) ij
  ekk  ij  2 G eij   ( T  T0 )  ij
or
1 n
n
eij 
 ij   kk  ij  a ( T  To ) ij
E
E
a: linear expansion coefficient
2.10 Materials with more complex
mechanical behavior
In limited ranges of temperature, stress and strain,
some real materials may follow above constitutive
equations
Real materials have more complex behavior:

Non-Newtonian fluids: blood, paints and varnish, wet
clay and mud, colloidal solutions

Hookean elastic solid: structural material within elastic
limit, disobey Hooke’s law for yielding & fracture

Few biological tissues obey Hooke’s law
2.11 Viscoelasticity
Features: hysteresis, relaxation, creeping
Stress Relaxation

Body is suddenly strained and maintained constant, the
corresponding stress decreases with time
Creep

Body is suddenly stressed and maintained constant, the
body continues to deform
Hysteresis

Body subjected to cyclic loading, the stress-strain
relationship is different between loading cycle and
unloading cycle.
Mechanical Models of
Viscoelastic Materials
Maxwell model (series)
Voigt model (parallel)
Kelvin model (standard linear solid) (series + parallel)
Lumped mass model consisted of linear springs and
dashpots
spring constant: 
viscous coefficient of dashpot: h
Maxwell model
h

F
u  u1  u 2
u  u1  u 2
u1
u
F
u2
F  hu1  u2
 u1 
F
h
u 2 
F

 u 
F


F
h
( 1 )
when F is suddenly applied at t  0, initial condition
F( 0 )
u( 0 ) 

Voigt model
h
F1
F1  h u
F2   u

F2
F
u
F  F1  F2  h u   u  F  h u   u ( 2 )
when F is suddenly applied at t  0,
initial condition
u( 0 )  0
Kelvin model
h1
F  F0  F1
F0   0u
u  u1  u 2
u  u1  u 2
F1  h1u1  1u 2  u1 
u2 
F1
F
F1
1
u1
F1
u2
0
F0
F
u
h1
 u  u1
1

h
F  1 F   0u  h1( 1  0 )u　1
1
or
F    F  E R ( u    u )　 (3)
h1 relaxation time for
 
1 constant strain
 0 relaxation time for
h1
 
(1  )
0
1 constant stress
  F (0)  E R  u(0)
Creep function
When F(t) is unit-step function, solutions of (1)(2)(3)
Maxwell　c(t )  (
Voigt c(t ) 
1

1


(1  e
1
h

h
 t
t )1(t )
)1(t）
   
1
Kelvin　c(t ) 
[1  (1  )e ]1(t )
ER

t
1　ｔ 0
1
1(t )   t  0
2

0 t  0
unit-step function
Creep function
u
Maxwell
u
Voigt
1/ER
1/
1/h
Kelvin
u
1
t
F
1
t
t
F
F
1
1
t
t
t
Relaxation function
When u(t) is a unit-step function, F(t)=k(t)
Maxwell　k (t )  e

h
 t
1(t )
Voigt k (t )  h (t )  1(t )

Kelvin　k (t )  E R [1  (1  )e




f (t ) (t )dt  f (0) (  0)

t

]1(t )

1
Relaxation function

Maxwell
F
h(t-t0) Voigt

Kelvin
 F

ER
t
u
u
u
1
1
1
t0
t0
t
General linear viscoelastic model
by Boltzmann
Lumped mass  continuum (Boltzmann model)
F(t)
F(t)
u(t)
u
t
t
t
F


t
simple bar model
Assumptions
F() continuous & differentiable
In d, increment of F() = (dF/d)d
Increment of u(t) due to F(): du(t), t > 
Relationship between du(t) and F’(t)dt
d F ( )
d u (t )  c(t   )
d
d
creep function
d F ( )
u (t )   c(t   )
d
0
d
t
convolution integral
Similarly, we can define the relaxation function
du ( )
F (t )   k (t   )
d
0
d
t
relaxation function
Notes:
1) Maxwell, Voigt & Kelvin models are special case of
Boltzmann model
2) Relaxation function can be approximated by
N
k (t )   a n e
n 0
n n t
Fourier series
N
k (t )   a n e
n n t
a
n 0
Spectrum of relaxation func.
an: coefficient
vn: characteristic frequency
an (vn ) : discrete spectrum
n1
n2
n3
n4
n5
Note: in living tissue such as mesentery continuous spectrum
is required
n
Generalization to viscoelastic
materials
Assumptions: small deformation, infinitesimal
displacements, strains and velocities
F  , u  , c, k  tensor
t
 ekl 


 ij ( x , t )   Gijkl ( x , t   )
( x , ) d


or
t
  kl 


eij ( x , t )   J ijkl ( x , t   )
( x , ) d


Gijkl～tensorialrelaxation function
J ijkl～tensorial creeping function
Initial condition
Assume ij=eij=0, t<0
t
 ekl 

 

 ij ( x , t )  Gijkl ( x, t )ekl ( x , o )   Gijkl ( x , t   )
( x , t )d
0

 
eij ( x , o )  lim eij ( x, t )
t 
  0　  0
Note: the 1st term is due to the initial disturbance (condition)
2.12 Response of a viscoelastic
body to harmonic excitation
Most biological tissues ~ viscoelastic, periodic
oscillation is a simple method
Simple harmonic motion: x
x  A cos( t   )
A: amplitude, : phase angle
x = projection of a rotating vector (phasor) on real axis
ei (t  )  cos(t   )  i sin( t   )
x  iy  Aei (t  ) ;
Aei (t  )  Aei eit
 Be it  B  Aei
x  A cos(t   ) ;
y  A sin( t   )
y
Aei(t+)
A
t+
x
x
Response of Maxwell body to
harmonic excitation
Maxwell material
u 
F


F
h
it

F (t )  i F e  i F (t )
it
F (t )  F e
d
u
let
then
it
it

u
 iU e  i u (t )
u  Ue
dt
Maxwell material
Substitute into above equation
iu 
iF (t )

 i  U e i t 
 i U 
U  [
1


h
i F e i t

iF


F (t )

1
ih

F e i t
h
F
h
]F
F
1
i
h  i
G (i )  


1
1
i 1
ih
U


1
 ih
 h

Maxwell material
F (
1


1
) 1U
ih
1
1 1 it
F (t )  ( 
) U e  G (i )U eit
 ih
 F eit
F
1 1 1 1
1 1 i 1 1
 G (i )   ( 
) ( ) (  )
U
 i h
i
 h
i 1 1
 i (  )
~ complex modulus of elasticity

h
Kelvin materials
F    F  ER (u    u )
u  U eit u  iu
F  F e it F  iF
 F  i  F  ER (u  i  u )
(1  i  ) F e
it
 ER (1  i  )U e
it
F ER (1  i  )
 
 G (i )  G (i ) ei ( )
U
1  i 
Kelvin materials
G (i )  ER
1  
2
1   2 
2
2
～Complex modulus of elasticity
  tan 1    tan 1  
      (     )
 tan  

2
1       1   2   
dB
40
20
0
log10
90
 >
Lead compensator
1/
1/
log10
黏彈模型應用
由實驗得到鬆弛及潛變曲線，或由弦波得到頻率
響應函數G(i)
曲線崁合至模型
若特定模型之鬆弛函數，潛變函數，遲滯，複彈
性係數皆與實驗數據吻合，則該生物材料力學行
為 可由此模型表示
求出材料常數如h, , , , ER
構成方程式可用以分析其他問題
2.14 Methods of Testing
Different aspects of biological materials

Lack large samples of biological materials

Keep samples viable & close to in vivo condition

nonhomogeneous
Viscometry (biofluids)

Ostwald viscometer

Couette viscometer

Cone-plate viscometer

Weissenberg’s rheogoniometer (general instrument)

Other types
Ostwald viscometer
Ostwald viscometer
For Newtonian fluid in a laminar flow
R
4
dp
h 
8Q d L
Unit: dyn.s/cm2 or poise, (CGS)
N.s/m2 (MKS)
1 poise = 0.1 N.s/m2
Couette Viscometer
R2
h
R1
Couette Viscometer
Flow between two coaxial cylinders
Rotating outer cylinder (R2), angular velocity ,
inner cylinder (R1) torque M, height of liquid h
M (R  R )
h
4  h R R
2
2
2
1
2 2
1 2
Cone-plate Viscometer
Cone-plate Viscometer
Higher accuracy, cone gap yield
constant shear rate
operated in steady rotation or oscillatory
or step change modes
Complex modulus of viscoelastic
materials
Weissenberg’s rheogoniometerr
Weissenberg’s effect – uptrust or
normal force occurs in some nonNewtonian fluids; in Couette flow the
fluid will climb up inner cylinder
Measure force at all angles
Others
Vertical rod oscillated sinusoidally within a hollow
cylinder containing the material
Placed between a spherical bob and a concentric
hemisphere cup
Tuning fork to drive an oscillatory rod in fluid either
in axial or lateral motion
Falling or rising small sphere, a metal or plastic
sphere falls or rises through a known distance and
measure the time. For Newtonian flow, Reynold no.
<<1
2
2 r  g
v
9h
Biosolids
Material testing machines
Specimen is clamped and stretched or
shortened at a specific rate while both the
force and displacement are recorded.
For biological materials, small size & need
to keeping specimen viable!
Example of noncontact method
3D analysis of blood vessels in vivo and in vitro
2.15 Mathematical Development
of Constitutive Equations
Nonviscous ideal fluid, Newtonian viscous
fluid, Hookean elastic solid, linear viscoelastic
(Maxwell, Voigt, Kelvin), Boltzmann
Finite deformation, nonlinear between strain
and deformation gradients: Cauchy, Green, St.
Venant, Almansi & Hamel -> nonlinear
constitutive equations for elastic, viscoelastic,
viscoplastic materials [Rivlin, Trusdell 50s60s]
Nonlinear viscoelastic material, Green & Rivlin
(’57) & Green, Rivlin & Spencer (’59), multiple
integral constitutive equation, series solutions.
Non-Newtonian fluid mechanics developed for
polymer plastics industry (Bird et. al. ’77)
Constitutive laws of most biological tissues
were not known, can not formulate boundaryvalue problems, nor analysis, prediction.
To give an account of mechanical properties of
living tissues, consolidated in constitutive
equations
結 論
連體力學包含固體力學與流體力學，本章討
論非黏性流體，牛頓黏性流體及虎克固體，
生物材料多半具黏彈性
黏彈性材料可用Maxwell, Voigt, Kelvin或
Blotzmann模型來描述
研究器官運動，人體內外流體流動，人體內
應力，有賴於組織構成方程式。