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

Creep function

u

Ft

t

Maxwell

1

11/

u

F

1

1/

t

t

Voigtu

t

t

1

1/ER

Kelvin

F

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

Relaxation function

1

Maxwell

u

1

Voigt

u

F (t-t0)

t0

t

t

1

Kelvin

u

t0

F

ER

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

x

y

A

t+x

Aei(t+)

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

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 Viscometer

h

R2

R1

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

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!

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 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

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。