Lit StressStrain YCFung1984

8/15/2019 Lit StressStrain YCFung1984

1/11

Structure and Stress-Strain Relationship of Soft TissuesAuthor(s): Y. C. FungSource: American Zoologist, Vol. 24, No. 1 (1984), pp. 13-22Published by: Oxford University PressStable URL: http://www.jstor.org/stable/3882748

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2/11

Amer.

Zool.,

24:13-22

(1984)

Structure

and

Stress-Strain

Relationship

of

Soft

Tissues1

Y.

C.Fung

Department

of

AMES

/Bioengineering, Universityof California,

San

Diego,

La

Jolla,

California

92093

Synopsis.

The mechanical

properties

ofa soft

tissue are related

to

its structure. We shall

illustrate

this

by

the

properties

of

the

arteries

and

the

lung. Viscoelasticity,

strain

rate

effects,

pseudo-elasticity,

and constitutive

equations

are discussed. The

mecahnical

prop?

erties

of an

organ

is, however,

not

only

based on the

tissues

of the

organ,

but

also on its

geometry

and

relationship

to the

neighboring organs.

A

typical

example

is

the

blood

vessel. The

capillary

blood vessels of the

mesentery

are

"rigid";

those

in

the bat's

wing

are

"distensible";

whereas the

capillaries

ofthe

lung

are "sheet" like:

rigid

in one

plane,

and

compliant

in

another. The stress-strain

relationship

of

the

systemic

arteries

is

highly

nonlinear,

stiffening exponentially

with

increasing

strains;

yet

that

of the

pulmonary

arteries

in

the

lung

is linear. The

systemic

veins

are

easily

collapsible;

yet

the

pulmonary

veins

in the

lung

are not:

they

remain

patent

when

the blood

pressure

falls below

the

alveolar

gas pressure.

The

explanation

of

these differences

lies in

the varied

interactions

between the blood vessels and the surrounding tissues in different organs. The implications

of these differences on blood circulation are

pointed

out.

The

role of

ultrastructure

is

discussed.

Introduction

Soft tissues

are

major components

of ani?

mal

body:

the

muscle makes locomotion

possible.

The

skin

protects

the

internal

milieu.

A

variety

of soft

tissues

make

up

the internal

organs.

The

function

of all

organs

is

closely

related to the mechanics

of soft

tissues,

about which this article is

concerned.

Soft tissues are made

of

collagen,

elastin,

muscle and other

cells,

and

ground

sub?

stances.

Their

mechanical

properties

de?

pend

not

only

on

their chemical

composi?

tion,

but

also on

structural details. For

organs,

their mechanical

property

depends

not

only

on

their

own

materials

and

struc?

tures,

but also on the

environment. We

shall

illustrate

this

with

several

examples.

Some

general

features

Some features

of

the mechanical

prop?

erties are common to all

soft tissues.

They

are

pseudo-elastic,

that

is,

they

are

not elas?

tic,

but

under

periodic

loading

and unload-

ing

a

steady-state

stress-strain

relationship

exists

which

is

not

very

sensitive to

strain

rate.

For

example,

Figure

1,

from

Vawter,

Fung

and West

(1978),

shows

the

stress-

1

From

the

Symposium

on

Biomechanics

presented

at the Annual

Meeting

of

the American

Society

of

Zoologists,

27-30 December

1982,

at

Louisville,

Ken?

tucky.

strain

relationship

of

the

lung

tissue

of the

dog

(with

the

airspace

filled

with

saline so

that the

surface

tension between the

alveo-

lar

gas

and the moist alveolar

walls is

replaced

by

the

very

small

liquid-solid

interfacial tension.

The

tissue was

pre-

pared

in the form of a

slab,

and biaxial

loading

was

used,

while the strains were

monitored

in

the

middle

portion

of the

specimen,

away

from the

edges

(in

order

to

avoid the

"edge

effect" as much

as

pos?

sible).

After

a

number

of

cycles

of

loading

and

unloading,

a

repeatable

stress-strain

loop

as shown

in

Figure

1

was

obtained.

The

existence of the

loop

shows that the

tissue

is

viscoelastic,

and not

elastic.

But

since

the

loop

is

repeatable

we can

treat

the

loading

and

unloading

curves

sepa-

rately

and borrow the

method

of

the

the?

ory

of

elasticity

to describe the

mechanical

properties.

Hence the term

"pseudoelas-

ticity." Figure

2

shows

the stress-strain

relationship

ofthe

same

lung

tissue in

load?

ing

at different strain rates. Each

cycle

was

done

at

a constant

rate.

The

period

of each

cycle

is

noted

in

the

figure.

It

is seen

that

over

a

360-fold

change

in

strain

rate

there

was

only

a minor

change

in

the

stress-strain

relationship.

The

hysteresis, H,

defined

as

the ratio

of

the area

of

the

hysteresis loop

divided

by

the area under

the

loading

curve,

is also noted

in

Figure

2.

H

is seen

to be

variable,

but its

variation with strain

rate

13


3/11

14

Y.

C. Fung

60

x

40

20

H

?

STRETCH

?

RELEASE

*m^^m&}

J

08 1.0 1.2

1.4

1.6 1.8

X?

(EXTENSION

ATIO

,

/L0I),

DIMENSIONLESS

Fig. 1.

A

typical

stress-strain

curve for uniaxial load?

ing. Every

fourth data

point

is

plotted.

Note that the

unloading

curve is

different

from

the

loading

curve,

showing

the

existence

of

hysteresis.

From

Vawter et

al. (1978), by permission.

is

not

large.

Similar

experience

is

encoun?

tered

with

other tissues.

Records of skel?

etal

and cardiac

muscles, ureter,

taenia

coli,

arteries,

veins,

pericardium,

mesentery,

bile

duct, skin, tendon,

elastin

(lig.

nuchae

with

collagen

denatured),

cartilage,

and

other

tissues

show

the same characteristics.

The

stress-strain

relationships

of some tissues

have been tested in a range of strain rate

covering

a million-fold difference between

the slowest

and

the fastest

cycling,

and

the

stresses at

the same

strain

are

usually

found

to

differ

by

less than a factor

of

2.

The

fastest stress

cycle

can be

imposed

by

ultra-

sound,

and

it

is known that

for

most tissues

the

damping per

cycle

of

oscillation

remains

almost constant

as

frequency

varies. The

slowest

cycling

in

the

laboratory

is often

done

by step-by-step testing

with a

long

period

of

waiting

between

steps.

Hysteresis

loops

do not vanish

in

these "static"

tests;

in

fact,

they

usually

remain

comparable

in

size to

those

obtained

at

moderate fre?

quencies.

The

features

shown

in

Figures

1 and

2

may

be described

by saying

that

living

soft

tissues

are

nonlinearly

pseudo-elastic.

The

stress-strain

relationship

is

nonlinear,

the

viscoelasticity

is

pseudoelastic?hysteresis

may

be

sizable,

but it varies

only mildly

over a wide

range

of strain rates.

Nonlinear

elasticity

Treating

the

loading

and

unloading

curves

separately,

we can

borrow the

85

80

60

CYCLE

IME,

HYSTERESIS

?

18

SEC,

H

=

027

?

60

SEC,

H

=

032

a

220

SEC.H

030

? 900 SEC,H=028

?

6500

SEC.

H

=

035

40

<

tn

5

?

g

20

<

?*?

'?

?"

i-1-1-1-r~

0.8

10

II

12 14

16

X,

(STRETCH

RATIO

LI/L0I)

Fig.

2.

Loading

phase

at different

strain rates.

Vary-

ing

strain rate

over

2.5

decades caused

only

small

changes

in

response.

The

hysteresis,

H,

is

the ratio

of

the area of

hysteresis

loop

(not shown)

to the

area

under the

loading

curve. The

period

of

cycling

and

the values of H are

given

in

the insert. From

Vawter

et al.

(1978),

by

permission.

method of

elasticity

to describe the stress-

strain

relationship.

For a nonlinear mate?

rial

the

simplest

way

is

to introduce a

pseudo-elastic potential

(also

called

a

strain

energy

function),

p0W,

which

is a

function

of

the Green's strain

components

Ey.

The

partial

derivatives

of

p0W

with

respect

to

Ejj

gives

the

corresponding

stresses

Sy

(Kirchoff

stresses).

W

is

defined for a

unit

mass

of

the

tissue,

p0

is the

density

of

the

tissue

in

the initial

state,

hence

p0W

is

the

strain

energy

per

unit initial volume. Thus

Sy"

(i,j=l,2,3).

(1)

If

the material

is

incompressible

(volume

does

not

change)

then k can

take

on

a

pres?

sure

that

is

independent

ofthe

deformation

of

the

body.

In

that case a

pressure

term

should be added

to the

right

hand

side

of

Equation

(1).

The value

ofthe

pressure

(as

in

water)

can

vary

from

point

to

point,

and

it

can be determined from

the

equations

of

motion

and

continuity,

and

boundary

conditions.

An

example

of

pseudo-elastic

potential

for

arteries and

veins is

the

following

(Fung

etal,

1979):


4/11

Soft Tissue

Mechanics

15

p0W&

=

C

exp[a,E,2

+

a2E22

+

2a4E,E2]

(2)

Here

the

superscript

(2)

over

p0W(2)

signi-

fies that this

is a two-dimensional

approx-

imation, which can

yield only

a

relationship

between

the

average

circumferential

and

axial stresses

(S1? S2)

and strains

(E1? E2).

Differentiation

of

p0W(2)

with

respect

to

Et

yields

Slf

that

with

respect

to

E2

yields

S2.

Figure

3 shows

a

comparison

of

the

fitting

of

Equation

(1)

to

experimental

data

on

rabbit arteries

subject

to

increasing

inter?

nal

pressure

and

longitudinal stretching.

The

constants

C,

a1?

a2,

a4

are

the

material

constants that

characterize the

artery.

Other forms of strain

energy

function

such as

polynomials

can be used

which

can

also

yield good

fitting

with

experimental

results. Most soft tissues

can be described

by

a

strain

energy

function

similar to

Equa?

tion

(1).

For a

body

subjected

to

small

changes

in

strains,

the

corresponding changes

in

stresses are

also small and the relation

between the

incremental

stresses

and

strains

can be

linearized if the strains are

sufficiently

small.

The linearized

relation?

ship

is

the

Hooke's

law,

for which

the

familiar material constants

are

the

incre?

mental

Young's

modulus and

incremental

shear modulus.

For soft

tissues

a

general

feature

implied

by

Equation

(2)

is

that the

incremental moduli

increase

with

increas?

ing

stresses.

Viscoelasticity

We have shown in Figures 1 and 2 that

soft tissues

are viscoelastic

in

a

special

way:

the

hysteresis

loop

is

relatively

insensitive

to

strain

rate.

If

relaxation

under

constant

strain

is measured it

will

be found that

elas-

tin relaxes

very

little, tendon,

mesentery,

skin,

blood

vessels,

lung,

and smooth mus?

cles relax

more and more

in

the order listed.

Creep

under constant load exists

for all

these tissues.

A

mathematical

model of

viscoelasticity

of a tissue must cover all features of

hys?

teresis,

relaxation,

and

creep.

One

of the

most

popular

models

of

viscoelasticity

is

the

Maxwell model

of

a

spring

in

series

with

a

dashpot.

The

other

is

the

Voigt

model with a

spring

and a

dashpot

in

par?

allel. A

third

is

the

Kelvin

model

which

is

a

combination

of a

spring

in

parallel

with

a Maxwell

body

(see

Fig.

4a, b,

c).

None of

these

can

represent

a soft

tissue,

because

when a

material

represented by any

one

of

these

models is

subjected

to

a

cyclic

strain,

the

hysteresis

will

not

be

insensitive to strain

rate:

as

frequency

increases the

dashpot

in

the Maxwell

body

will

move

less and

less

at same

load so

the

hysteresis

decreases

with

frequency.

On the

other

hand,

the

Voigt

body

will

let

the

dashpot

take

up

more and

more of

of

the load so that the

hysteresis

increases

with

frequency

(see

Fig.

4d,

e).

For

the

Kelvin

body

there exists a char?

acteristic

frequency

at

which

the

hysteresis

is a

maximum

(see

Fig.

4f).

None of

these

has

the feature

of

nearly

constant

hyster?

esis as

soft tissues

do.

A

model

suitable

for

the

soft tissue is

shown in

Figure 4g,

which has an

infinite

number of

springs

and

dashpots.

In

the

corresponding

hysteresis diagram

shown

in

Figure

4h

there

are an

infinite

number

of

bell-shaped

curves

which add

up

to

a

continuous curve of nearly constant height

over a

very

wide

range

of

frequencies.

In

this

situation,

we

say

that the soft tissue has

a

continuous relaxation

spectrum.

The

two

ends

of the

spectrum,

marked

by

frequen?

cies

Tj-1

and

r2_1

in

Figure

4h,

define

two

characteristic times

r{

and

r2

which can be

determined

from

experimental

data

(see

Fung

[1972;

1981,

pp.

232

et.

seq.]

for

mathematical

details).

Tanaka and

Fung

(1974)

have found

rx

and

r2

for various

arteries of the dog: rY lies in the range of

several hundred

to thousands

of

seconds,

r2

lies

in

the

range

of

0.05

to 0.36

see.

Chen

and

Fung

(1973)

showed

that

for

the mes?

entery

Tj,

t2

are

1.869

x

104

see and

1.735

x

10"5 see.

Woo

et al.

(1979)

have

found

that

for

the

cartilage

r2

=

0.006

see,

rx

=

8.38

see.

Why different

blood

vessels

behave so

differently

One of the beauties and

puzzles

of the

biological

world

is

that

it has a

great

variety

of

things,

and

sometimes,

the same

thing

has

quite

different

properties

in

different

circumstances.

Take

blood vessels

as

an


5/11

16

Y. C.Fung

0.2

0.4

Green's

Strain,

Eqq

Fig. 3.

Comparison

of

the

stress-strain

relationships

obtained

from

Eqs.

(2)

and

(1)

with

experimental

data

on four

normal rabbit

arteries.

The

symbols

are

defined

in the

inset.

The curves

joining

experimental

data

points

are not

always

smooth

because stresses

and strains

in

two

dimensions are

coupled

and

any

disturbance

in

EZ2

esults

in a kink

in

the

S?

vs.

Ege

urve,

and vice versa.

From

Fung

et al.

(1979),

by

permission.

example.

The aorta

and

thoracic

arteries

have

nonlinear stress-strain

curves as shown

in

Figure

3.

The

pulmonary

arteries

and

veins,

in

contrast,

have

linear

pressure-

diameter

relationships

as shown

in

Figure

5,

though

this

does not

imply

a linear stress-

strain

curve.

The

capillary

blood vessels

of

the

mesentery

appears

to

be

rigid?with?

out

measurable

change

in

diameter

when

blood

pressure

changes

over a

range

of 100

mm

Hg

(Baez,

1960;

Fung

et

al,

1966).

But the

capillary

blood vessels

in

the

lung

are

very

distensible

(see

Fig.

6),

in which

the

variation

of

the thickness

of

the

cap-

illary

sheet

with

the transmural

pressure

AP

(blood

pressure

minus alveolar

gas pres?

sure)

is shown. The

pulmonary

capillaries

are

closely

knit

into

a dense

network

which

occupies

about

90%

of

the total

space

in

the interalveolar

septa

(the

alveolar

walls).

It

is

best to

describe

this network as

a

sheet.

Each sheet is

exposed

to

gas

on

both sides.

The sheet thickness varies with the

trans?

mural

pressure. Figure

6 shows that the

thickness ofthe

pulmonary

capillary

sheets

ofthe

eat

increases

linearly

with

increasing

transmural

pressure

at

a

rate of

0.22

nm

per

cm

H20

when AP is

positive.

But

when


6/11

Soft

Tissue Mechanics

17

0.5

0.7

0.9

Green's

Strain,

En

Fig.

3. Continued.

1.1

AP is negative the thickness quickly drops

to zero.

There is a sudden

change

of

thick?

ness

at

AP

=

0.

When AP

tends to 0

from

the

positive

side the

limiting

value

of

the

sheet thickness is

4.28

jum

for the

eat. When

AP

<

?

1

cm

H20

the

capillaries

are

all

collapsed.

Why

do the

capillaries

of

the

lung

behave

so

differently

from

those

of

the

peripheral

circulation?

There

is

another

important

property

of

the

blood vessels

that

has an

important

physiological

effect: the

stability

ofthe ves?

sel when

the external

pressure

exceeds the

internal

pressure.

We have seen

in

Figure

6

that

the

pulmonary

capillaries collapse

whenAP

<

0. But we

know

that

peripheral

capillaries do not collapse when blood pres?

sure falls below tissue

pressure

(see

Baez,

1964;

Fung

etal,

1966).

On the other

hand,

it

is

common

knowledge

that

peripheral

veins

and vena cava

collapse

when

the

blood

pressure

falls below the

pressure

in

the

sur-

rounding

media.

But the

pulmonary

veins

do

not

collapse

when the

airway

pressure

exceeds

the

blood

pressure,

as the data

shown

in

Figure

5 demonstrates.

We

have

shown further

(Fung

et

al.,

1982)

that

pul?

monary

venules do not

collapse

under the

same condition.

Why

do

these vessels

behave

so

differently

while their

compo?

sition and anatomical

and

histological

structures are

very

similar?


7/11


8/11

Soft Tissue

Mechanics

19

?

100

3

cc

UJ

Z

<

o

2

<

^

=

-10cmH20(*98P$}

?

100"200m*200400m

?

400000

tm

^

8001200

m

+

MEAfttSD

-15

200

1

150

-15 -10

?

100-200

m

a 200 00 m

o

400

00

m

a

8001200nm

-

0:

h

=

h0

+

aAp.

(6)

a is the

compliance

constant ofthe alveolar

capillaries.

Other

symbols

in

Equation

(5)

are:

Area

=

area

of

the

alveolar

wall, #

is

viscosity

of

blood,

k and f are numerical

factors

which

depend

on

the

sheet

geom-

etry,

L"~2 is the mean

path length

of blood

between the

entry

and exit sections.

In the

lung,

the

entry

section

is

located at

the

pulmonary arterioles, the exit is located at

the

pulmonary

venules.

Equation

(5),

together

with

the

knowl?

edge

that

pulmonary

veins

and venules do

not

collapse

when

the

pulmonary

alveolar

gas pressure

exceeds the

blood

pressure,

explains

an

important phenomenon

which

is

known

by

the name

of

"waterfall"

or

"sluicing." Figure

7

illustrates

the

phe?

nomenon:

Let

a

lung

be

perfused

with fixed

airway

pressure (pA)

and

arterial

pressure

(pa),

while the left atrium

pressure pv

is

varied. Let the

flow be

measured

when

pv

is

gradually

reduced. When

pv

>

pa

there

is

no flow. When

pv

is

decreased

below

pa

flow

starts,

and it increases with

decreasing


9/11

20

Y.

C. Fung

-f?i?i?r?i?|

i

i?r?i?]?i

i

i

i?|?i?r?i?i?j?r-1?i

i

|

i?i

i

i

|

i

x-l^r

h

=

4.28+0.219

Ap

(cat)

?MEAN??STD

DEV

Approx.

used

in

this

po

per

h-h0

+

aAp

when

Ap>0

~h=0 whenAp^O

i

i

i

1 I_L_l_L_l_I

I

I

I_L_l_I

I I

-5

5 10 15

20

25

30

35

Ap,CAPILLARY-ALVEOLAR

PRESSURE,

cm

H20

40

Fig.

6.

The variation ofthe

thickness

ofthe

pulmonary capillary

sheet ofthe eat with the

transmural

pressure

Ap

=

capillary

blood

pressure

minus

the alveolar

gas pressure.

From

Fung

and

Sobin

(1972),

by

permission.

pv.

But

an

upper

limit

is

reached when

pv

becomes equal to pA. From there on fur?

ther decrease of

pv

does not

increase the

flow: a

flow

limitation

is

reached. This is

analogous

to a

waterfall

whose

volume flow

rate does

not

depend

on

the

height

of the

fall. The

explanation

lies

in

Equation

(5):

when

pv

<

pa,

Ah

vanishes and the flow

Fig. 7. The variation of the blood flow in the

lung,

Q,

with

decreasing

left

atrium

pressure (pvp)

at three

fixed values

of

pulmonary

arterial

pressures.

As

pvp

decreases,

the

flow reaches a

plateau, resulting

in

a

phenomenon

called

"vascular waterfall." From

Per-

mutt

et

al.

(1962),

by permission.

depends entirely

on

Ap

at

the

pulmonary

arteriole. For further decrease in pv the

pulmonary

veins and

venules do not

col?

lapse,

only

the

capillaries

can

collapse.

The

site

of

flow

limitation,

or

sluicing gate,

must

be located at

the

junctions

of

capillaries

and

venules. The last term

in

Equation

(5)

is

either

negligible

compared

with

rest,

or

is zero

(see

Fig.

6),

and

Q

remains constant.

The

similarity

of

Equations

(4)

and

(5)

suggests

that other

flow

limitation

phe?

nomena can be

similarly

explained.

This

includes

the

phenomena

of maximum flow

limitation

in

the

airway

in

forced

expira-

tion,

and flow

limitation

in

micturition

due

to muscle

sphincter

action

in

male

(with

one

sluicing

section)

and

female

(with

two

sluicing

sections)

urethra.

Ultrastructure

All

the

properties

of

the

organs

and

tis?

sues have an ultrastructural basis. For

soft

tissues,

the

mechanical

properties

can be

analyzed

ultimately

in

terms of

the

net-

works

of

collagen

and

elastin

fibers,

the

muscle and other

cells,

and the

ground

sub?

stances and fluids.

If

the

ultrastructure

is

known,

we

should

be able

to

theoretically


10/11

Soft

Tissue Mechanics 21

deduce

the

mechanical

properties

of

the

tissue. We

have shown

above,

however,

that

at

a

higher

level,

from

tissues to

organs,

we

must

consider

the

geometric

structure

and

interaction

of various

components. By

analogy,

at

a

lower

level,

from

fibers,

cells

and

ground

substances

to

tissues,

we also

have

to

consider

the

geometric

structure

and

interaction

of

these

components.

Studies

of

collagen

and elastin networks

in

arteries were

initiated

by

Roach

and

Burton

(1957),

with

a

method

of

differ?

ential

digestion

with

enzymes.

Later,

the

contribution

of

vascular

smooth muscles

was

evaluated

by

means

of various

vaso-

active drugs. But progress has been slow

because

structural data

about the

collagen

and

elastin

fibers

in

the tissue are

difficult

to obtain. These fibers are

densely

packed.

Looking

at

an

optical

or

electron

micro?

scopic

picture

of

a

selectively digested

artery

is like

looking

at

a

dense

forest

in a

landscape.

It

is

difficult to make

meaning-

ful

measurements.

For

this

reason,

our

more

recent

work

has been

concentrated

on

those tissues in which

the

collagen

and

elastin networks are less dense. Sobin et al.

(1982)

and Wall

et al.

(1981)

have

system-

atically

photographed

these fibers in

the

alveolar walls

of

human

lung

and

obtained

statistical

data on fiber

width

and

curva?

ture. These data

can then form

the

basis

for a

theory connecting

the

fine

structure

with

mechanical

properties.

Perspectives

The

most

crucial

step

in

the

develop?

ment of biomechanics is the identification

of

the

constitutive

equations

of

the tissues

involved,

that

is,

a

concise

mathematical

description

of

the

mechanical

properties.

If

the

constitutive

equations

are

known,

then

biomechanics

problems

can be

for-

mulated as

mathematical

problems

and

solutions can be

definitive.

Without

con?

stitutive

equations,

biomechanics

will

remain

qualitative

in

character. After

the

form

of

the

constitutive

equation

is

deter?

mined, the next

step

is to

systematically

collect

data on

the

material

constants of

various

times.

Until we

have

a

complete

set

of

data on

material

constants,

the

power

of

biomechanics

to

predict

the

function of

an

animal will be limited.

Only

when the

full

power

of

biomechanics to

predict

the

behavior of

an

animal when

certain

param?

eters

are

changed

is

developed

can bio?

mechanics be

of real

service

to

medicine,

surgery,

health

preservation

and

improve-

ment,

sports,

welfare,

and

quality

of

life of

man and animals.

In

this

article we have

attempted

to

show

how

soft tissues behave and

how

constitu?

tive

equations

can be arrived at. We have

used the blood vessels

to show

that

even

though

all

vessels are

fundamentally

simi?

lar,

the

constitutive

equations

and

material

constants

of

one

vessel

can

be

very

differ?

ent from that of another because of struc?

tural

differences and because

of

interac?

tion

with

neighboring

tissues

or

organs.

In

the

immediate future

we

should com?

plete

the

program

of

identifying

constitu?

tive

equations

and material

constants,

and

develop

biomechanics

to

the

point

of

becoming

a

practical

tool for

the service

of

man.

ACKNOWLEDGMENTS

Support

of NIH

through

Grants HL-

26647

and

HL-07089 and

NSF

through

Grant

CME

79-10560

is

gratefully

acknowledffed.

References

Baez, S.,

H.

Lamport,

and A.

Baez. 1960. Pressure

effects

in

living

microscopic

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A.

L.

Copley

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G.

Stainsby

(eds.),

Flow

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122-136.

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Y. L.

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Fung.

1973.

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Y.

C.

1972.

Stress-strain-history

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