CH2 The Meaning of the Constitutive Equation Prof. M.-S. Ju Dept. of Mechanical Eng. National Cheng Kung Unive rsity
Dec 20, 2015
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:
VM
pVV
Mlim)p(v
in particles of mass :
enclosing volumesof sequence a :
0
P
M
V
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
0 bound,lower finite 00
0
v:v
errortolerable)p(V
Mlim
vv
Blood example
Whole blood – continuum at scale of heart, large arteries, large veinsTwo-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
side negative on the
surface of side positiveat particles by the exerted force :
S
FS
F
Assume that as S tends to zero the ratio F/S tends to a defi
nite limit, dF/ds, and moment of the force acting on the surfac
e S tends to zero.Stress
2
2
11
11 :unit
mm
NMPa
m
NPa
A
F
x1
x3
x2o
S
B
S
F
Let vector benormal of SPositive side: surface pointed by Positive side exerts force F on the negative sideF depends on location and size of S and orientation of When S→0
ds
Fd
S
FS
0lim
moment of F exerts on S →0
vectorstressor Traction : ds
FdTV
force per unit area acting on the surface
Components of stress
stressesshear
stresses normal
runit vecto
312312
332211
333232131
3
323222121
2
313212111
1
~~
~
,,
,,
eeeT
eeeT
eeeeT i
x1
x3
x2o
11
13 12
33
23
21
22
31
32
ds
jji
V
i
jijiii
V
i
vTor
vvvvT
)(evevevv
332211
332211 1
formula sCauchy'
ijjijij
V
viT
Note: knowing components of a stress tensor one can write
down the stress vector acting on any surface with unit outer
normal vector
(2) For a body in equilibrium we have
0x
0x
0x
0x
33
33
2
32
1
31
23
23
2
22
1
21
13
13
2
12
1
11
ij
ij
xor
xxx
xxx
xxx
jiij )3(
Xi: components of body force (per unit volume) along ith axis
Due to equilibrium of moments
(4) change of coordinate system
ijmjkikm
ikki
kkkikik
'''
xx:
xxxxx
)x,x,x()x,x,x(
axis w.r.t axis of cosine ldirectiona332211
321321
’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
LL',
L
LL 0
0
0
010010
01001010112
3
8
312
:exampleFor 2
2
2
22
2
22
.'.
..eLo.L)b(
eLoL)a(
Lo
LoL
L
LoLe
Other definitions:
Note: above strain measures are equal for infinitesimal elongation
Another is Shear Strain , consider the twist of a circular cylindrical shaft. tan or tan/2 is defined as the shear strain
M M
Constitutive Equation: a relationship between stress and strain
rigidity of modulus ,
modulus sYoung' law, sHooke'
:GtanG
:EEe
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) ,
333222111 ,,
are of components The . ofnt displaceme:or
axuaxuaxu
upuPQ
a1, x1
a3 ,x3
a2, x2o
SP
P’P”
Q
Q’Q”u
(a1, a2, a3)(x1, x2, x3)
field)ent (displacem321321 ),,i()a,a,a(xx ii )1(3,2,1),,,( 321 ixxxaa ii
Q: stretching and distortion of the body?
)(dadadadS
dSpp)daa,daa,daa(pppassume
o
o
2
distance toclose 23
22
21
2
332211
,,
After the deformation p and p’ displaced to Q and Q’ , QQ’ distance dS
)3(23
22
21
2 dxdxdxdS
mlm
j
l
iijjiij
mlm
j
l
iijjiij
ijijij
jj
iij
j
ii
dadaa
x
a
xdxdxdS
dxdxx
a
x
adadadS
ji,ji,
dxx
adada
a
xdx
2
20
01delta Kronecker using
jiji
ij
mlm
j
l
iijjiij
jiijji
jiijmlm
j
l
iij
dxdx]x
a
x
a[
dxdxx
a
x
adxdxdSdS
dada]a
x
a
x[
dadadadaa
x
a
xdSdS
dSdS
20
2
20
2
20
2
or
nsorstrain te define
toused becan and between difference
Definition of strain tensors:
iji
ijij
iijji
ij
x)Eulerian(tensorStrains Almansi)x
a
x
a(e
a)a
x
a
x(E
on based 2
1
on based rangian)tensor(Lag train ssGreen 2
1
body rigid implies 0or 0
0 and 0 implies 0 :note
2
2
20
2
20
2
20
2
ijij
ijij
jiij
jiij
eE
eEdsds
dxdxedsds
dadaEdsds
Show that Eij and eij are symmetric
When ui is small
eij reduces to Cauchy’s infinitesimal strain tensor:
negligible are and 2 )x
u
x
u()
x
u(
e
k
j
i
j
i
zxwu
yxvu
xxuu
,,,j,i)x
u
x
u(
j
i
i
jij
33
22
11
3212
1
Cauchy strain tensor
zyyzzz
zxxzyy
yxxyxx
)y
w
z
v(
z
w
)z
u
x
w(
y
v
)y
u
x
v(
x
u
2
1
2
1
2
1
In the infinitesimal deformation, no distinction between Lagrang
ian and Eulerian strain tensors.
or
Geometric meaning
x
u dxx
uu
00
v,x
u
x
u dxx
uu
00
v,x
u
y
x
y
000
x
u,
x
v,
y
u00
x
v,
y
u
y
x
00
x
v,
y
u
y
x
For fluid motion, consider velocity field and rate of strain. At point (x,y,z) velocity vector
3,2,1),,(
ˆ),,(),,(ˆ),,(),,(
izyxVor
kzyxwjzyxvizyxuzyxV
i
For continuous flow, Vi : continuous and differentiable
)x
V
x
V()
x
V
x
V(
x
V
,,j,ixdx
VVd
j
i
i
j
i
j
j
i
j
i
jj
ii
2
1
2
1 where
321
2.4 Strain rate
Define strain rate tensor Vij
symmetric2
1 jiij
i
j
j
iij VV,)
x
V
x
V(V
symmetricanti2
1 jiij
j
i
i
jij ,)
x
V
x
V(
jijjiji xdxdVVd
Define vorticity tensor ij
2.5 Constitutive equations
Properties of materials are specified by constitutive equationsNon-viscous fluid, Newtonian viscous fluid and Hookean elastic solid are most widely models for engineering materialsMost biological materials can not be described by above equationsConstitutive 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
Stress tensor:
D Kronecker delta, p: pressure (scalar)For ideal gas: equation of state
For real gas: f (p, , T) = 0
Incompressible fluid: = constant
ijij p
TRp
2.7 Newtonian Viscous FluidShear stress is proportional to strain rate
Stress-strain rate relationship
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 r
ate• Isotropic materials: a tensor has same array of components w
hen frame of reference is rotated or reflected (isotropic tensor)
klijklijij VDp
)( jkiljlikklijijklD
nonviscous0let
viscousibleincompress2
0 fluid ibleincompressfor
fluid Stokes 3
22
3
20233
233
2
ijij
ijijij
kk
ijkkijijij
kk
kkkk
ijijkkijij
p
Vp
V
VVp
)(p
V)(p
fluidNewtonianisotropicVVp
Coefficient of viscosity
Newton proposed
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
yd
ud
2.8 Hookean Elastic Solid
Hooke’s law: stress tensor linearly proportional to strain tensor
modulior constants elastic of tensor :
sorstrain ten :e tensor,stress:
(1)
ijkl
klij
klijklij
C
eC
Note: elastic moduli are independent of stress or strain
Isotropic materials
modulusShear : constants, Lame are
(2)2
G,
ee ijijij
zxzx
yzyz
xyxy
zzzzyyxxzz
yyzzyyxxyy
xxzzyyxxxx
eG
eG
eG
Ge)eee(
Ge)eee(
Ge)eee(
2
2
2
2
2
2
Inverted form
xyxyxy
yyxxzzzz
zxzxzxzzxxyyyy
yzyzyzzzyyxxxx
GE
ve
vE
e
GE
vev
Ee
GE
vev
Ee
2
11
)]([1
2
11,)]([
12
11,)]([
1
,
ijkkijij EEe
1
)1(2)21)(1(
12
)1(2)21)(1(
vGv
vvE
G
Ev
v
EG
vv
v
E
E: Young’s modulus, G: shear modulus, : 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
)T(CC
)TT(eC
ijklijkl
oijklijklij
0
For isotropic material
ijoijkkijij
ijijijkk
ijoklijklij
)TT(EE
e
)TT(eGe
)TT(eC
1
or
2 0
linear expansion coefficient
ijij
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 cl
ay and mud, colloidal solutions Hookean elastic solid: structural material within elastic l
imit, disobey Hooke’s law for yielding & fracture Few biological tissues obey Hooke’s law
2.11 ViscoelasticityFeatures: 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:
Maxwell model
F F
uu1
u2
)(F)(u
)(FF
uF
uF
u
uuF
uuu
uuu
00
condition initial 0,at t appliedsuddenly is Fwhen
121
21
21
21
Voigt model
F
u
F1
F2
00
condition initial
0,at t appliedsuddenly is Fwhen
221
2
1
)(u
)(uuFuuFFF
uF
uF
Kelvin model
F F
u
u1 u2
F1
F0
)()uu(EFF
u)(uFF
uuF
u
FuuuF
uuu
uuu
uF
FFF
R 3
or
11
010
1
1
11
12
1
1121111
21
21
00
10
)1(1
0
0
1
1
1
relaxation time for constant strain
relaxation time forconstant stress
)0()0( uEF R
Creep functionWhen F(t) is unit-step function, solutions of (1)(2)(3)
00
021
01
)(
)(])1(1[1
)(
()1(1
)(
)()11
()(
t
tt
teE
tcKelvin
tetcVoigt
tttcMaxwell
t
R
t
t
)
1
1
1
1
unit-step function
Relaxation functionWhen u(t) is a unit-step function, F(t)=k(t)
)(])1(1[)(
)()()(
)()(
teEtkKelvin
tttkVoigt
tetkMaxwell
t
R
t
1
1
1
)0()0()()(
fdtttf
1
General linear viscoelastic model by Boltzmann
Lumped mass continuum (Boltzmann model)
u(t)
F(t)F(t)
u
F
t
tt
t
simple bar model
Assumptions
F() continuous & differentiable
In dincrement of F() = (dF/d)dIncrement of u(t) due to F(): du(t), t >
Relationship between du(t) and F’(t)dt
d
d
Fdtctud
)()()(
creep function
d
d
Fdtctu
t
0
)()()(
convolution integral
Similarly, we can define the relaxation function
d
d
dutktF
t
0
)()()(
relaxation function
Notes: 1) Maxwell, Voigt & Kelvin models are special case of
Boltzmann model2) Relaxation function can be approximated by
N
n
tn
netk0
)( Fourier series
N
n
tn
netk0
)(
n: coefficientvn: characteristic frequency
n (vn ) : discrete spectrum
1 52 3 4
Spectrum of relaxation func.
Note: in living tissue such as mesentery continuous spectrum is required
Generalization to viscoelastic materialsAssumptions: small deformation, infinitesimal displacements, strains and velocities
F , u , c, k tensor
function creeping tensorial
function relaxation tensorial~
),(),(),(
),(),(),(
~ijkl
ijkl
tkl
ijklij
tkl
ijklij
J
G
dxtxJtxe
or
dxe
txGtx
Assume ij=eij=0, t<0
00
),(lim),(
),(),(),(),(),(0
txeoxe
dtxe
txGoxetxGtx
ijt
ij
tkl
ijklklijklij
Note: the 1st term is due to the initial disturbance (condition)
Initial 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
)cos( tAx
x = projection of a rotating vector (phasor) on real axis
iti
tiiti
ti
ti
AeBBe
eAeAe
tAytAxAeiyx
tite
)(
)(
)(
)sin(;)cos(;
)sin()cos(
A: amplitude, : phase angle
Response of Maxwell body to harmonic excitation
Maxwell material
)(
)()(then
)(let
tuieUitd
udu
tFieFitF
eUu
eFtF
FFu
ti
ti
ti
ti
Maxwell material
ii
ii
iU
FiG
Fi
U
FFiUi
eFeFieUi
tFtFiui
tititi
1111
1)(
]11
[
)()(
Substitute into above equation
Maxwell material
ti
titi
eF
eUiGeUi
tF
Ui
F
)()11
()(
)11
(
1
1
elasticity of moduluscomplex ~)1
(
)1
()1
()111
()(
1
111
ii
i
iiU
FiG
Kelvin materials
)()()(1
)1(
)1()1(
)(
)(
iR
tiR
ti
R
ti
ti
R
eiGiGi
iE
U
F
eUiEeFi
uiuEFiF
FiFeFF
uiueUu
uuEFF
Kelvin materials
22
11
22
22
1
)(
1tan
tantan
elasticity of modulusComplex 1
1)( ~REiG
0
20
40dB
log10
90
log10
1/ 1/
>
Lead compensator
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 , , , , ER
構成方程式可用以分析其他問題
2.14 Methods of TestingDifferent 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
For Newtonian fluid in a laminar flow
Ld
pd
Q
R8
4
Unit: dyn.s/cm2 or poise, (CGS) N.s/m2 (MKS)1 poise = 0.1 N.s/m2
Couette ViscometerFlow between two coaxial cylinders
Rotating outer cylinder (R2), angular velocity inner cylinder (R1) torque M, height of liquid h
22
21
21
22
4
)(
RRh
RRM
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 non-Newtonian fluids; in Couette flow the fluid will climb up inner cylinder
Measure force at all angles
OthersVertical rod oscillated sinusoidally within a hollow cylinder containing the materialPlaced between a spherical bob and a concentric hemisphere cupTuning fork to drive an oscillatory rod in fluid either in axial or lateral motionFalling 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
9
2 2 grv
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!
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 50s-60s]
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 boundary-value problems, nor analysis, prediction.
To give an account of mechanical properties of living tissues, consolidated in constitutive equations