Mechanical relaxation theory of fibrous structures

Advances in Molecular Relaxation and Interaction Processes, 11 (1977) 165-190 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

MECHANICAL

RELAXATION

D. R. AXELRAD Department

and S.

THEORY

(Received

McGill

STRUCTURES

BASU.

of Mechanical

Laboratory,

OF FIBROUS

165

Engineering,

University,

21 December

Micromechanics

Montreal,

of Solids

Canada.

1976)

ABSTRACT

The present

study is concerned

of fibrous media. analysis

the mechanical

investigated. considers

relaxation

The theory

the significant

stochastic

with a mechanical

On the basis of probabilistic

functions

phenomena

of fibrous

admits the existence field quantities

relaxation

concepts

theory

and functional

structures

are

of the microstructure

as random variables

and

or

of such variables.

INTRODUCTION

In the study of the rheological such as fibrous

structures

ically predicted

response

ably from experimental to the existence random nature various

observed

materials

that the theoret-

force field deviates

consider-

of such materials the application

that are closely

related

which

ascribed

is of

of a force field to to the rheological

of the material. work simple

have been discussed [l-5].

models

for discrete

materials

micromechanics

Thus it has been shown that linear stochastic

to describe

Furthermore

by means

for a specified

the material

relaxation

on the basis of probabilistic

be employed

valid

during

of discrete

This fact can be largely

of a microstructure

and subjected

In previous

solids

it is frequently to an external

observations.

random parameters

behaviour

behaviour

the material

response

of a characteristic intermediate

"material

domain

body and by using distribution

field quantities

the transition

on a "local"

models

can

level.

operator"

called "mesodomain" functions

of

which

is

within

of the involved

from the local to the "macroscopic"

166 description

of the rheological

theory requires

response

functions,

element

the interfacial

Consequently

the rheological

respectively,

of a structural

that characterizes elements.

can be achieved.

from the onset of the analysis

creep or relaxation internal

phenomena

of fibrous

the use of two distinct

one of which

represents

or "microelement"

behaviour

the theory permits

behaviour

The present

between

a unified

structures

the

and another

two overlapping representation

of

formed by ensembles

of

such elements. In the subsequent

study, the "local response"

first and then by the application distribution

functions

al formalism, material

the macroscopic

on the rheological

ular with

the bonding between

quantities, operator

Hence

the definition

them.

the proposed

with

deals with a and in partic-

is devoted

to

field

of an appropriate

material

In the final section

remarks will be made as to the applicability

theory and the experimental

distributions

section

of the required

time.

of the significant

work required

physical

the

for the discrete

The third section functions

involving

and an operation-

of microelements

and establishment

as well as its evolution

this paper concluding

relations

the following

behaviour

of the distribution

concepts

field quantities

rheological

will be formulated.

discussion

considerations

of statistical

of the relevant

will be considered

parameters

to obtain

involved

of of

the

in the

analysis.

It

is evident

mechanical

that the theory

relaxation

characteristics those which to include

of fibrous

are included.

attempted

general, referring

symbols

description

representing

ensemble

index notation vector

to the local behaviour

of structural

theories

in the analysis.

elements

or

However effect

bonding".

direct notation

will be used

will also be employed.

or tensor-valued

of elements

"f" for the fibres and "b" for the bonds.

"m".

the microstructural

have been able so far

for the case of "hydrogen

standard

of the

study is made in that the bonding

the entire presentation

but when necessary,

in which

effect of bonding

in the present

will only be considered Throughout

systems

None of the conventional

a statistical

the important

one restriction

is aimed at the description

will

quantities

and

carry a superscript

Those symbols

will be designated

In

related

to an

by the superscript

167 RHEOLOGICAL

BEHAVIOUR

Before

OF MICROSTRUCTURAL

discussing

description

the rheological

to the formulation

to restate

of a general

Thus the first concept the smallest

of which

an intermediate

of introducing

in the material

a statistical

ensemble

The second operator

fundamental

concept

and physical

parameters

tains in general variables

related

over a specific

processes,

the temperature

subsequent

analysis,

of the structure

behaviour.

micromechanics

can be accounted Finally,

field

as will be seen in the

form of constitutive

relations.

in the analysis

and strains

since it

acting on the elements

discontinuities

in the system of

for in the formulation

the relevant

functions

It also con-

the stress with the deformation

the fourth concept

theory considers

or stochastic

random geometrical

structure.

the most important

so that the existing

elements

of a material

of the relevant

This operator,

of stresses

material mesodomains.

the time t and for non-isothermal

the conventional

is perhaps

aims at a generalization

overlapping

8.

the notion

and

(a = l,....N;

the significant

is used to connect

space and thus replaces The third concept

mesodomain,

scales,

to as a "mesodomain"

of non-intersecting

functions

led

[4,5].

The next scale concerns

to the fibrous

the distribution

deformations

of the macroscopic

introduces

structures,

three measuring

referred

in its argument

in the

concepts which

of microelements

formed by a finite number

that contains

involved

of fibrous

to a microelement.

The third scale is the domain

body itself

variables

phenomena

the four fundamental theory of stochastic

consists refers

region

contains

N large).

response

field quantities

of the creep and relaxation

it may be instructive

which

ELEMENTS

of the

of the probabilistic

field quantities

of such variables

as random

[5].

Field Quantities

In accordance probabilistic the analysis corresponding

position

the above outlined

microdeformation stimulus

vector

concepts

a&

These quantities

will be functions

of

of prime interest

UE(t) acting on an element

given to the microelement

Both quantities

with respect

fundamental

the field quantities

are the microstress

as a stochastic response.

with

micromechanics,

may be regarded

and its corresponding

in a kinematic

sense of the

%J(t) at time t to the centre of mass of the element

to a fixed Lagrangian

frame.

in

a and the

On the assumption

that the

168 relaxation

phenomena

deformations write

entails

its evolution

and that the additivity

the stress

to a microelement

u!(t) = Llfg(t)

property

bonding

from a system

in general Heaviside

acting

in more detail,

of the description

for the purpose

geneous medium stress.

complex.

kinematics

for example,

of which

analysis,

which may be operators" and

It is evident

effects,

of its response

the inclusion behaviour

In this sense and for the simplicity

so that the microstress

is considered

of the

as a homo-

fl(t) can be taken as a Cauchy

to the above form for-microstress,

one can write

total deformation of a fibrous element, in terms of the deformation b f the fibre u(t) and in the bonding zone u(t) so that:

yl(t) = Llfu(t) zone between

overlapping

"generalized

force" that is derivable

potential

discussed

acting

fibres must be introduced

in

in the interaction

in terms of a

from the corresponding

later in accordance

with the general

bonding

form due to

[9]. In view of the probabilistic

it is perhaps

instructive

and topological

response vectors

uv(t):uv;(t);

state vectors the material ation.

the mechanical

of the fibrous

space".

the structure

briefly

the probabilistic

states

that represent

structure Thus,

the rheological

i = 1, . .. . . . r; u = 1, . .. . . . N whereby as stochastic

and response

during

that these state vectors analysis

Hence

analysis

of the above field quantities.

can also be regarded

in the present

used in the present

system in terms of a set of r-dimensional

characteristics

It is evident

function

concepts

to discuss

characteristics

one can consider

which

the

+ L,b&4t)

It should be noted that the above microstress

Yvon

and

of discrete media

in [5-81.

impurity

the fibre or fibre segment

Analogous

operators

in the fibre element

The deformation

itself exhibits

present

form as follows: (1)

that the fibrous element

would be extremely

of

one can in general

theory point of view as "filtering

zone, respectively.

has been studied

holds,

in an operational

ff(t), b c(t) are the stresses r

where

small increments

+ LzbS(t)

in which L 1, L2 represent regarded

during

is identified

it follows

is discussed

functions

expressing

the mechanical

belong

state

these

relax-

to a state space

with a "probabilistic

that the rheological

in terms of an "abstract

behaviour

dynamical

of

system

169 [X, F,P 1" whereby

F is the u-algebra

open sets or events where

is the appropriate

P

in considering

However,

due to the application "transition

intensity

For simplification subspaces

vector

u"(t) can be regarded

quantities

theory of such materials of

in order to obtain

by the quadruplet

a

[X, F,

Q] in which

P,

matrix.

of the probabilistic

in an approximate

the concept

In this case the abstract

of the subsequent

employ

space X and

on these open sets.

force field,

formulation.

is represented

Q is the transition

measure

has to be introduced

mathematical

system

of the function

relaxation

of an external

probabilities"

more rigorous dynamical

probability

a general

to

of Bore1 sets that correspond

that are the elements

analysis

function

to be composed manner

it is convenient

to

space X so that the state

of the rheological

field

as follows:

(3)

in which each of these field quantities

belong

to the following

sub-

spaces:

EfE;

fE(t) I:

be(t)

,bg ; fE Ubg= :, c

b

EbU ;

z

x (4)

f u(t) efU where

fibrous

< uv. L

ations.

I

space and U the deformation

that represent

then to the

a mechanical

state,

manner:

< 3v. + A3v i ; i=l, .. . . . r1 1 3~3

+.

these state vectors

It may be noted

rndicates

can be assessed

from experimental

that the choice of the above subspaces

imposed by the present

permit observations

of microdeformations

distribution

(5)

the range and A3vi the accuracy

due to restrictions

the relevant

space as sub-

correspond

together

as an open set in the following

the the superscript

with which

ucx

fE and bE, respectively,

and bonds

can be expressed i

f UUbU=

the stress

The events

elements

E = f3" where

g(t)

E designates

spaces of X.

which

;

functions

experimental

only.

Hence

and transition

observis mainly

techniques it is obvious

probabilities

of

which that

170 measurable

microdeformations

"material

operator"

a material

must be related

This can be achieved

microstresses.

that links the two subspaces.

operator

actual physical

as discussed

characteristics

form can only be obtained In studying

to the distribution

by the application

subsequently of the fibrous

by an appropriate

in general

material

the rheological

consider

the mechanical

u,(X) which

define

Alternatively

: U + U ; ftcR

in which

the subspace

and bonding

the subspace

relation

[lo-131.

mechanical

in the present

creep or relaxation expressed

to the fibre

It has been shown,

for instance,

phenomenon

of discrete

Chapman-Kolmogorov

relation

[14].

Considering

[t,s] in which

of the fibrous

connects

relations

functional

the stochastic

analytic

approach

taken

this relation with respect

the incremental

structure

in

involv-

deformations

may occur,

to

during

it can be

by:

PIt,s}

= IU PIt,r]dPIr,s]

(7)

in which P is the transition

probability

representing

of a microelement

the probability

go a transition process,

systems

lead to response

theory with the functional

analysis

space

corresponding

by the well-known

a closed time interval

of the function

Tt such that:

probabilities

This functional

relaxation

[U, FU, P"]

the spaces

respectively.

of transition

that can be expressed

funct-

e.g. u(X, t) or _in the deformation space.

(6)

[4,5] that the creep or relaxation ing the concept

systems,

; ~,cgEU

U includes

behaviour,

of fibrous

In this sense, one can

process

family of mappings +

model.

behaviour

deformations,

a random deformation

considering

such

its explicit

theory in terms of measurable

of the occurring

X and a one parameter Tt

relaxation

whilst

in its arguments

structures,

one can invoke the theory of Markov processes.

ions characteristic

However,

contains

of

of a distinct

a variable

time.

P ij(t)

of the structure

from a state "i" to a state "j" during

T designates

in matrix

matrix with elements

Relation

to under-

the deformation

(7) can also be written

form:

P(t+s)

= P(t)P(s)

(8)

from which

it may be seen that in the case of steady-state

relaxation

as well as for the time independent

structure,

the abstract

group property

[15].

dynamical

system

elastic

[U, FU,pu]

For these two stages,

creep or

response

exhibits

the occurring

of the

the semi-

deformations

171 can be regarded difference

as a time-homogeneous

of time

(t-s) only

Markov

(Doob

process Hence

[16]).

it can be shown that the elements

for example,

P. (t) will satisfy ij

the following

limiting

that depends

following

on the

Kendall

of the transition

[17],

matrix

conditions:

(9)

for a u-measurable qlj(t)

function

E Q(t) is a measure

Q(t) is the transition description Basu

of the relative

intensity

of the abstract

property

that this property

range of the rheological during

the transient intensity

that makes

the process

following

[7,8].

immediately

associated

with

process.

certain

domain

throughout

A detailed

equation

includes

"breakdown

to replace

transition

probabilities.

has then to deal with are either

the first-order

related

onset of the steady-state to exist.

The mathematical

phenomena"

which

probability

transition

behaviour

formulation

con-

point

by higher of view one

probabilities

that

before

the steady-state

of such Markov

at a

it becomes

of the material

after

in the

are initiated

deformation,

transition

or the breakdown

certain

the

Since such a process

From a mathematical

to the transient

and in

of the Markov

will not satisfy

sense unless

the time-dependent

process

in creep, must be

microstructure

time or at the end of the steady-state

necessary

factor

deformation

to deal with a function evidently

the

of such a case is given

deformation

in the usual

Thus,

for example,

that a general

of the existing

necessary

the entire

medium.

so that the semi-group

discussion

the steady-state

and

it cannot be

and is a decisive

non-Markovian

for such a case can be specified.

physical

However,

of the structured

It is evident

creep or

the above relations

(8).

time dependent

distinctly

in the

for the steady-state

of relation

In that case the process

Chapman-Kolmogorov ditions

becomes

and accordingly

earlier

It has been shown by

will be satisfied

the breakdown

that case it becomes

system.

that satisfy

behaviour

does not hold.

in references

transition

stage of the creep behaviour,

transition

property

In the above expressions

introduced

probability

leads to a formulation

that of the semi-group expected

matrix

dynamical

[7] that the transition

relaxation

order

and for all tsR+.

processes

the

ceases can be

172 achieved

on the basis

of the work by Bharucha-Reid

[lOI and Dynkin

[13]

for example. Without

going into detail of an analysis

the transition

probabilities

of the creep behaviour summarized

which

of this kind,

the role of

cover the above outlined

four stages

of a fibrous

structure,

for example,

can be

as follows:

(i)

; P(t) P(s) = P(t+s)

Pij (t) = Qj

(ii)

gCt)

;

>

(elastic)

P(t) P(s) # P(t+s)

Pij(t)

=

PLj(t)

= f(t) ; P(t) P(s) = P(t+s)

(transient) >

(iii)

Pij (t) = h(t)

(iv)

; P(t) P(s) # P(t+s)

(final state) /

j = i+l, i+2, .,. It is seen from the above brief ation of an abstract

dynamical

creep or the relaxation that it becomes

experimental

with

of these quantities be substituted

In this context,

at present

so that solutions

ential

representative

equation

in themselves

of accuracy,

as well

permit

of

the measurement

analytical

models must

f(t), g(t) and h(t) in relations

of the Chapman-Kolmogorov

of a particular

the rheological

can only be considered

the structure,

it is convenient

which actually

is located between

Thus following

one can consider

differ-

stage during

the stresses

due to bonding.

then possible

to superpose

Thus, in general

due to the application fI+(t)

: f!+)

behaviour

the de-

of single fibres

as a part of an element

to refer to a so-called two bonding

the concept

that arising

effects.

probabilities

of Single Fibres

In order to formulate

fibres.

of the

comain of the material,

can be achieved.

Creep and Relaxation

which

the applic-

due to the absence

that would

the functions

(10) explicitly

formation

in the physical

with any degree

to express

concerning

for the representation

to study the transition

time.

possibilities

discussion

system

phenomenon

important

as their evolution

(10)

(steady-state)

areas of overlapping

expressed

acting

in references

in the single

By an appropriate

the deformation

= fy(t) - f'f

[5,8,18,19]

fibre sepately

model

analysis

for the case of small deformations

of an external

of

"fibre segment"

occurring

it is the two

in a single

load can be written

from

fibre

as follows: (11)

173 In which

the subscript

"I" refers

to the coordinate

centre of mass of the corresponding

bonding

refer to the undeformed

configuration

As mentioned

2.1 the single

which

in section

is only an approximation,

depends

on other parameters

significant

aspect

forming

from the internal

obvious

that

f u$t) where

Again

the fibre behaviour

For example,

in natural

is the fibrilar

angle.

fixed reference

fibres a

By transframe it is

(12) matrix

One can introduce

f

:

=

+

Gik

‘j

+

the fibre.

In general,

The gradient

operation

6’ J“

fibre as mentioned the conventional

Aihk

(13)

complex

previously

inhomogeneities

strain measure vector

to the attached is regarded

description

for

only, will be as follows:

is due to the inherent

a more

and

strain

in the fibre segment

deformations

on the deformation

has to be taken with respect

at an appropriate

strain measure

the above approximation

of both the external

the base vectors

and for the elastic

e.. 1J

in the usual sense of deformation

in order to arrive

Thus, an approximate

small strains

I

point.

fibres are looked upon as continua

to an external

body frame, respectively,

e

as well.

is a transformation

kinematics.

f

symbols

= fnIi fuIW

fAIi

measure.

to the

in the fibre at any arbitrary

since generally

in this context

frame attached

zone and the majuscule

of

would be required.

in the above relation

body frame.

If the single

as a homogeneous

in terms of hereditary

medium

integrals

then

[2,20] can

be used so that: f f -1 f tf f (t) = g s(t) + 1 -m

Y(t-rC) f<(r) dr

(a) (14)

f

t(t) = fE fe(t) - I

t. f

g(t-T)

fe(T)

dr

-m in which

f E = is the elastic

pliance,

fg(t),

in the fibre segment.

of the fibrous rheological structures heredity

I I fE--1 the corresponding =

com-

fg(t) the tensorial creep and relaxation functions, f f and e(t), C(t), the time dependent microstrain and z r

respectively, microstress

tensor modulus,

(b)

structures

behaviour. [21,22].

effects

It is well-known,

of practical

importance

however,

exhibit

that most

non-linear

This has been shown to be true for cellulosic In that case, considerations

lead to a formulation

of non-linear

of the following

type:

174

fe(t) = glLfS(t),

El' 12'

. ..I

t

+ J

f

[ S(T),

-m

CL,

dr

. ..].(t-T)

(15)

's(t) = g21fe(t),

kl,

k2, . ..I

- 1

t

f

. ..]l$(t-r)

C E(T), bl,

dr

-m where

the functions

g,(s) and g,(s) represent

of the response whilst linear hereditary dimensional

by a method

structures

in [22].

approximation

the present relaxation

of data from three-

due to Distefano

analysis

is concerned

of fibrous

structures

behaviour

in [2,20].

of the single

form for the hereditary

rheo-

flj*(t)f$(t) . _

=

It-m fIi(t-T)

f$(~)dr

fs*(t)fe(t) :

=

tf I g(t-T) -ca

f

case

and for cellulosic

with a more general and hence

theory

an operational

Thus considering

fibre, one can write

part of the relation

part

for the non-

of the non-linear

The latter case has also been considered

will be used as outlined

viscoelastic

in absence

the above formulation

of differential

of mechanical

elastic

has only been used so far for the one-dimensional

[21].

However,

the non-linear

hl(*) and h2(') account

In general,

effects.

Todeschini

formalism

the functions

experiments,

logical behaviour

I

the linear

the following

(14) as follows:

(16)

so that (14) becomes f:(t) = Lfg-1 +

e(r) dr

now:

fy*(t)lfp (17)

f

c(t) = lfE_- fl$*(t)]fe(t> z z

It is evident operator

1

from the above relation

for the fibre can be written

that the single element

material

for the case of creep or relaxation

as follows: fi1Ct) = fg-l +

f* y (t)

(18) fJ2(t) = fE, -

fg*w

It is to be noted however which

that due to present

in most cases only permit

above general

formulation

uniaxial

experimental

tests on single

can only be stated

formally

restrictions fibres,

the

for the three-

175 In order

to evaluate

dimensional

case.

dimensional

case a minimization

account

of such a procedure

Fibre-Fibre

interaction

indicated

occurring

in a network

various

of hydrogen models

of hydrogen

the hydrogen of interest,

whereby

Y

bonding

=Y

1

+Y+Y 2

the bonding

3

bonding.

+P

will be essen-

In this context.general

have been suggested [29,30].

of Lippincott

in the following

functions

A detail

the fibre-fibre

can be found in references

in references

bond model

sections,

of fibrous elements

to hydrogen

bonding

past and were reviewed

for the one-

can be found in [21,22].

in the foregoing

tially one with reference theories

functions

has to be adopted.

Bond Behaviour

Interaction:

As already

the kernel

procedure

Thus

[26-281 in the

In a quantitative

and Schroeder

is expressed

[23-251.

manner

[27] for example

is

in terms of potential

form: (19)

4

in which

Iyl =

Y1011-exp[-v*(r*-r~)2/2r*]]

= Y20{l-exp[-v(r-ro)2/2r]]

p2

The interaction quantities

potentials

accounting

ent dissociation distances which

Iy3 where

3

energies.

and Y

(21)

Y1 and Y2 in equation

The quantities

in the potential

Y

- Y20

for the bond behaviour,

(19) are the significant

whilst

YlO and Y20 repres-

r and r* represent

curve and v, v* are spectroscopic

are to be determined

potentials

(20)

by experimental

observations.

the atomic constants,

The two other

are of the form:

4

; Y4 =-B

=Aexp(-bR)

(22)

R"

Y

represents a Van der Waal's repulsion force and Y an electro4 3 static attraction force. The above model suggested in [27] corresponds to one that can be derived

[281.

also from quantum mechanical

Thus one can express Y = w(r*)

in which w(r*)

-2cr w e-b(r-ro) 0

designates

and in which

bonding, between

+ ale

it is assumed

-ae

the potential

the other

the H--B atoms.

concerned

an interaction

potential --cr

(23)

2 energy

terms account

involved

in the A-H

for the bonding

So far as the mechanics in the present

considerations

in the form of:

analysis

relations

of bond deformation

is

that the A-H bonding

is

176 much stronger an external fibrous

than the H--B bonding

structure

the potentials Hence

are rather negligible. potential

and that due to the application

force field during the process

in the bonding

of mechanical

Y3 and Y4 in the empirical

the following

of

relaxation

of a

relation

(22)

model of an interaction

area will be adopted:

Y(r) = Yl + Y2 in which

(24)

Y(r) can be expressed

approximate

Y

corresponds

0

distance general

the interaction

order terms in ((r-rol)

It may be noted

The spectroscopic

constant

that for a cellulosic

approaches

fibres of the structure

the "ultimate

strength"

concerning

in the bonding

when the structure

to an external

work concerning

is subjected

the bonding behaviour

of matching

points

in the form of a "relative

that the number

of bonds actually

area does not change during the probable

considers

occurrance

and its evolution formulation ification process"

existing

with

Following

time are investigated.

the strongly

a specific

A more recent paper

to a representation

In these con-

bonding

force field.

or total bond breakage

leads then, under the assumption

situation

that a

of the analysis,

of an external

bonds and for which

and is a well-known

In previous

can be taken into

vector".

within

models of the mechanical

number N(t) of hydrogen

has been

group" of hydrogen

for simplicity

of a partial

in the formulation.

stochastic

force field.

analytically

the application

of

that occurs

[3,5] it has been assumed

deformation

it has also been assumed,

in over-

observations

of the bonding

in the "hydroxyl

may occur, which

for example,

not included

structure,

area only, no account

deformation

Hence

in

(20,21)

the bonding

based on experimental

given so far as to the relative

siderations

potential

in relations

1311 that r. = 1.72, lo = 4.5 Kcal/mole; o-1 v in general, is of the order 2A .

In most phenomenological

account

of the Y(r) curve at a

it has been found

for example,

bonding,

(25)

potential

r. from the origin representing and where higher

translation

in an

- 2 exp(-vlr-ro/)l

to the equilibrium

have been neglected.

lapping

type potential"

form as follows:

Y(r) = YoIexp(-2v/r-ro/) where

in terms of a "Morse

failure

was

[32], however, of a certain

a probability

distribution

The resulting of a controlled

that corresponds

analytical humid-

to a "pure birth

form in the theory of Markov processes. idealized

model,

e.g. that the bond during

177 the whole perfect

process

with

that corresponds

to an increased

to the hydroxyl

area, a certain two surface figuration

that between

layers of the overlapping

will change

response and hence

a

above will be

matching

points

a "unit cell" of the bonding

fibres

the distance

between

in the undeformed

By the application

distance

to a b6(t)

specific

vector bA will designate

of the structure.

field, this primitive

vector mentioned

groups within

distance

to remain nearly

of the structure

instability

deformation

Thus, it is assumed

introduced.

can be assumed

of the last stage of the rheological

the relative

bond breakage,

belonging

of creep or relaxation

the exception

characteristic

con-

of an external

of a certain

force

type of bond

so that this change per unit cell can be expressed

by: bdI(t)

With reference co-ordinate

bdi(t) in which

bfj(t)

= b$(t) =

- b;

to the external

transformation

the subscript

matrix

"1" refers

the corresponding

then:

bAIi

solids

fibre interface

behaviour

the above deformation librium

position

written

as:

given in [2,5]. involving

where bPo is the equilibrium spectroscopic

material

an individual

bond.

ized interaction

fibres,

equi-

can be

(28)

v the previously

and Ibd(t)I the relative

observations. analysis

function

mentioned deformation

small magnitude

that one can only determine

of

of

its expected

Consequently,

to consider

it

this parameter

J'ibd(t)].

to note that in reference

potential"

in which

with a specific

due to the relative

in the subsequent

It is of interest

the fibre-

type potential,

of two bonded

potential,

from experimental

in terms of its distribution

to that for

- 2 exp(-vIbd(t)/)]

constant

However,

it is obvious

rate-dependent

the Morse

the deformation

bY = bYo{exp(-Zv~bd(t)j)

will be necessary

is analogous

As a consequence

vector will be associated

during

and variance

to the body frame and "i" to the exter-

The above model

frame.

polycrystalline

this vector

fixed frame and using

= bd(t) = bAIi bdI(t)

nal fixed reference

value

(26)

has been proposed

force can be established.

[4], a "generalized

according

to which

It is then possible

a generalto use a

178 differential

law of deformation

strain can be evaluated. in the interaction "generalized

zone between

potential"

case.

deformation

defined

distinct

whilst

such a

for the mathematical

difficult

above several

the bond behaviour.

in equation

able approximation phenomena.

However

to be valid

to assess

formulation

it in the

This is due to the fact that apart from the significant

vector

simultaneously potential

it is extremely

stress or

can be regarded

two fibres.

may be introduced

of the bond behaviour, present

so that the time-dependent

These quantities

(28) is regarded

in the formulation

Following

points

reference

in the present

may effect form of the

case as an accept-

of the mechanical

relaxation

force bi(t) acting at

layers of the overlapping

can be expressed

bA ably(t) F(t) = a(lbd(t)l)

the suggested

[4] the discrete

on the surface

of the above potential

other mechanisms

Hence

fibres in terms

as follows:

(29)

g -

in which

g denotes the base vector of the co-ordinate frame attached to _ the surface layer of the fibre. This discrete force may be generalized

in the sense of Gel'fand I

= ~6(~d - bd),

-

_

and Vilenkin

[33] to read:

b;_ >

(30)

n in which 'd is the discrete matching

points

generalized

of the hydroxyl

b5(t) in which

b

the inner product.

eij(t)

in sign.

microstrains

area between

in the outward

Correspondingly

fibre is opposite

to the expression b

bond, r(bd) the _ I function and

Dirac-delta

The microstress

in the

as:

n is the unit normal

the corresponding bonding

at specific

= r(bd) bnI_

layer of the fibre. lapping

indicates

zone may be defined

vector

group in the hydrogen

force, 15(a) the three-dimensional

the angular bracket bonding

value of the deformation

the surface normal Following

Tonti

which would be valid

overlapping

concerning

direction

of the surface of the over-

[34], for example, for the total

fibres can be formulated

the single fibres

= $(SkiVj + AkjVi)'dk(t)

analogously

(eqn. 13) so that:

(32)

179 The above relation values

can only be considered

of the relative

deformation

directly

but only its distribution

Finally,

by rewriting

as a formal expression

vector

function

the expressions

since

are not obtainable

bd(t)

can be established

directly.

in terms of the potential

for

(28) it is seen that: b c(t) = 2vbYo{exp(-2vlbd(t)I) I in which

the base vector

form, in a similar manner strain

relations

(33)

- exp(-v/bd(t)l)}.g5bn

g is identical

In operational

to bd(t)/lbd(t)l.

as shown for the fibre segments

the bond stress

become:

b

b

b3 bC(t) _

E(t) =

(34)

> bE(t) z

= bP4 b+ I

where b?3 and b"4 are corresponding to the bonding considered

behaviour

operators

for all practical

that are related

purposes

may be

to be linear operators.

Rheological

Behaviour

of Microelements

The microelements that exhibit

forming

rheological

stochastic

process

stochastic

process

density

functions

However

before

when

the probability

Individual

(ii)

Fibres

(iii)

Due to the existence

fibre elements

or fibre elements

other due to the partial

buckling

and the corresponding

behaviour

of "microelements" that fibrous

force 'field will in general

will behave may become or cpmplete

of bonds between

could be expected

viscoelastically.

detached

from each

bond failure. elements

in individual

a possible

fibres,

and

finally, in natural

fibrous

fibre arrangement motion

of any

characteristics:

(i)

(iv)

to a

enter into the formulation.

it should be recognized

to an external

system and

in general

In the description

distribution quantities

the rheological

below,

subjected

of the fibrous

will be subjected

their deformations.

of the relevant

discussing

show the following

the structure

properties

during

which will be defined structures

and which

material

structures

a reorientation

of the

can also occur due to the relative

in the bonding

area.

180 In view of the above possibilities element

above possibilities

(18,221 it is suggested

to define

analysis

a microelement

Other researchers

relaxation

suggested

different

cerned with

single fibres mainly

mechanistic

point of view due to the inherent

impossible

to use definitions

the complexities al model

rheological

behaviour

functional f

involved

, t)

of the bonds

dimension

Hence

the

can be expressed

in

(14).

to a

On the other hand,

the quantities in equations

a is concerned

the

simply as:

; bS = bC(be) z I:

in which

(36)

bE and b E conform to the generalized quantities (31) and (32). Thus so far as the defined micro(a = 1, .. . . . . N; N large) the corresponding

tensor fields for the stresses

babilistic

1

and strains pertaining

with relations

can be written

indicated element

In view of

(35)

in acczrdance

e = bf'bg z

random

From a

it becomes

a correct

is adopted.

for relaxation

fC(t) and fe(t) are the stresses

fibre segmznt

b

[35-371).

for creep

fz(t) = fC(fe , t) I z

response

of fibrous

that are con-

form as follows:

e(t) = fe(fC z z z

in which

models

of finding

of the single fibre segments

structure

will be used in the behaviour

scale factor,

of a microelement

the

work

of the

in these references.

and difficulties

the above definition

two halves

(see for instance,

suggested

previous

of the fibrous

with

This definition

for the mechanical

of a structural

take into account

Thus following

fibre segment"

bonds at each end of the segment.

structure.

that would

is rather difficult.

to consist of an "unsupported

subsequent

the definition

to as a "microelement"

referred

micromechanics

and strains

on the basis of pro-

are given by the following

stochastic

functions: "e(t) = ae(fe

2:z

z

ap)

=

Whilst necessary fibrous

,

b

I: ; t)

e

a.5(fc, b5; cc:

-_

1

1’

t)

the above forms express

for the consideration

structure

to derive

only implicit

of the rheological

explicit

(37)

relations, behaviour

it is of a

forms of these relations.

An

181 attempt

of this kind has been made

not overcome manner.

It

the difficulty is therefore

the microelement

type potential.

fibre segment

f

s=

in which

a

s=p

On the assumption

energy

potential

i=l

bY.= 1

where pl expresses does not exceed

could

the response

of

In particular,

energies.

can be characterized that the internal

by a

energy

of the

can be considered

form for an element

can be written:

f per unit area, E a b to the fibre segment and E a general

in the microelement

pertaining

written

1t

however

E

E is the total energy

bond potential b

to investigate

as shown earlier

the following b

E+

viscoelastic

[22] which

scale factor in a rigorous

and the energy due to the bond potential

to be additive, a

more profitable

in terms of the involved

since the bond behaviour Morse

in reference

of the involved

in a probabilistic

form as follows:

(39)

plnbY

the probability

a certain

that the number

fixed value j. and where

of hydrogen bonds "j" b 'Pi considered to be

the previously discussed Morse potential within the ith bond so that ,,,b,,, designates the resultant bonding potential for n bonds contained in the bonding

area.

It is obvious

bonds would be a uniform

that the simplest

one, in which

distribution

of

case:

j pl(j d jo) = o n However,

(40)

it is more likely

materials. procedure

some remarks

to find the actual distribution

made in the last section Returning rheological

that other distributions

In this context,

se(t) :

= Allah

az(t)

=

az2(t)ae(t) z

an experimental

in fibrous elements

formalism

of a structural

of creep and relaxation,

will occur in actual

will be

of this paper.

to the operational

response

concerning

respectively,

in order to represent

element,

one can write

the following

the

for the case

relations:

(41)

where

(42)

182 It is to be noted operators tions

that in the above relations,

the arguments

'El(t) and u$((t) are the operators

previously

(18) and (34) and that for the simplicity

given in equa-

of the analysis

assumed

these operators

are linear ones.

In general

linear.

Considerations

of the non-linear

type of operators

outside

the scope of the present

Whilst

it may be difficult

a$(t)

operators

'l&(t),

relations

in general,

will be given below

it is

they may be nonare, however,

analysis. to establish

explicit

forms for the above

in order to find the appropriate an explicit

form valid

for the purpose

introducing

for the first quantity

dissipative

type as indicated,

tP, f~ = I[/

of the

for the case of creep only

of illustrating in equation

for example,

stress-strain

the analysis.

(38) a potential

in references

Thus

of the

[38,39] then:

f Sd e]dt I- z

00

(43)

so that

(44)

The microstrain pressed

in the unsupported

fibre segment

can therefore

be ex-

by:

(45)

(eqn. (38)) it is possible

In terms of the total potential microstrain

in a structural

due

aft

a+)=a=T.__l-+T.a a5 3-S In general

element

in the following

afl

abE

abi

aa5

a$

ag

a formal relation

to define

the

manner:

(46)

for the relaxation

case will be of the

form:

ap

sac

afE

=_

=-

aua : which,

however,

.

afe 2

afe 2

+

aae r

abe z

.

2

necessary

(47)

a"e 2

due to the inaccessibility

of fE(t) and bS cannot be pursued z I It becomes

abe

abc -

of determining

the derivatives

at this stage.

in the description

of the mechanical

relaxaticm

183 of fibrous reflects during

structures

of the single

a creep deformation

ers [40] have considered

a probability

actual number measurable

It is to be noted

a

the transmission

0’

of microstresses

reference

measure

[8] but with

21 and A =3' the partial

afs= fe(t) aft

a somewhat

the following

RBEOLOGICAL

2b !!1(0 + P2

[(l-p21

BEHAVIOUR

variables. ial operator specific

in eqn.

(46) become

now:

for an element

of the fib-

OF FIBROUS

~31aSw 1:

description

of the field quantities

that it is necessary

Furthermore, acting

mesodomain

it has been stressed

of the macroscopic relations

following

involved

two of this paper.

for a stochastic

distribution

as a link between

of Field Quantities

In general,

(50)

STRUCTURES

to use the corresponding

lieu of constitutive

Distribution

function

theory has been given in section

also been mentioned ation theory

mean-

(49)

microstrain-time

The probabilistic the present

different

and by the use of

can be written: 2f

2

derivatives

model

been

3

rous structure

"e(t) =

p2 has already

= f$l(t) f$(t)

*

z

in the

(48)

that the above probability

in an earlier

-

that the

or equal to a

as follows:

On the basis of the above force transmission

the operators

Hence

the meaning

from the

by intro-

; b$ = P2c;

2 = (l-p2Y)

introduced

k = p2(a < ao) with

theory,

areas "a" per fibre is smaller

can be simply expressed

f

ing.

function

other research-

model entirely

In the present

mechanics.

model which

as that of the bonds

In this context,

such a force transmission

of bonded

fixed value

microelement

transmission"

fibre as well

for instance.

point of view of continuum ducing

to a "force

to resort

the contribution

that the motion

stresses

relax-

of these of a mater-

and strains within

material

It has

mechanic

functions

in

a

body must be applied

for the material.

and Material

Gel'fand

Operators

and Vilenkin

[33] one can define a

in

184 distribution

value of the stochastic ribution

function

function

can be written

x t) in terms of this dist$(_x,t) or 5(_, z as follows:

1

mE(t) = m =Ze(x,t)AP(;,t] z

s

or expected

Thus the average

of any random vector pEx,t).

(51) m;(t)

= m<~(_X,t)>=~~(_x,t)AP{x,t} 1

in which

the quantity

age at a particular

in the angular bracket

time instant of the corresponding

is to be noted that the above representation and for a specific the distribution

fibrous

etc., as indicated

their arguments

in equation

of the material

P{gpl

it would

other geometrical

be necessary

= flP(~lW],

operators

Their distribution

It

form

to include

in

parameters

%1(t)

the operators

therefore

aver-

field quantity.

and/or physical

contain

(42).

body will

the spatial

is only of a general

So far as the microelement

are concerned,

mesodomain

structure,

functions

in their arguments. %2(t)

designates

and

&l(t),

82(t),

over a specific

be of the form:

PI~,H (52)

pC~2w 1 = f2PI~2(t)), so that the mesoscopic

PIe( f

pIA,)

stress-strain

= Phi,(t)1

relations

become:

PIE.(t)3 : (53)

PCS(t)1 = PQ12W

PC$(t)l

P

It is of importance the distribution is inaccessible this reason resorted

to note, however,

functions at present,

that previous

to simplified

that the formal representation

of material

operators

due to experimental

investigations

models,

ators in the representation

given in equation restrictions.

(see, for instance,

e.g. by adopting

of the mechanical

constant

relaxation

of (52)

It is for [22]) have

material

oper-

of structured

media.

General

Theory

and Field Equations

In order to develop fibrous

a general

theory of the mechanical

structures

involving

by necessity

the above material

operators

but also their evolution

convenient

to recognise

relaxation

not only the distribution

from a probabilistic

with

of

time, it is

point of view the duality

of

185 between

the stress and deformation

of the probabilistic

structure

considering

the general

relaxation

of a fibrous

the Markovian

character

al relaxation

process

equation

deformation structure

will be represented

matrix

relaxation,

arded as some product of the material can write

corresponding

to the creep or type and of

space,

by the Kolmogorov

Q(t) with

the mechanicdifferential

derived

of microstresses

from the probability

can be regdistribution Thus, one

and that of the microdeformations.

Au(t)1

= P(M (t)) P(L.(t)}

(a)

P(c(t)j z

= P{M (t)} P{!(t)} 54

(b)

f3

q ij(t) has been defined

With this interpretation

matrix.

the distribution

measure

operators

the elements

for the case of creep and relaxation

in which

Thus

way:

(9) and I is the identity

the mechanical

(eqn. (4)).

; P(0) = I

the transition

in equation

process

in the brief review

to be of the homogeneous

in the stress and deformation

in the following

dP(t) = Q(t)P(t) 77 where

space as indicated

of the field quantities

z

the following

forms:

(55)

M (t) and M (t) in general will be tensors of the -4 $3 :: third rank and are derivable from the operators I&(t) and g2(t) by performing

the operators

the gradient

operations

indicated

in equations

(13) and (32),

respectively. Hence, with reference babilistic

function

ter of the deformation the microstresses dP'(t) dt

process,

in which

the transition

bability

of a Markov

creep,

dPu(t) =QU(t)

The concept

matrix

of the pro-

of the Markovian differential

charac-

equation

for

= I

Q' is identified

process which

is regarded

On the other hand,

form to equation

PU(t)

of statistical

together with

the Kolmogorov

; P'(O)

behaviour.

the analogous

t

space as a subspace

takes the form of:

= Q'(t) p'(t)

the relaxation

to the stress

space and on the assumption

with the transition to represent

considering

in general

the case of

(56) becomes:

; P"(o) = I independence

the assumption

pro-

(57) and of the product

that the stochastic

process

measure

during mechan-

of

186 ical relaxation

in the deformation

type results

in the present

characterize

the evolution

Hence considering separately

of the rheological

the probability

PI”(t)}

equations

processes

distribution

in terms of the initial

it follows

tities,

and stress space are of the Markov

theory in differential

that

in these spaces.

in each of these spaces

distributions

of the relevant

quan-

that:

= P"(t) PI"(O)}

(a)

PIE.(t)} = P"(t) PIE(o)} -I r

(b)

(58)

In the case of the creep behaviour into (55a) will result

p”(t)

P{“(O)}

= P{Mz3 (t)) P'(t)

Pc~3(o)l

It is evident

P{S(o)l I

z

(55a) be satisfied

for all instants

=

m13w

is P{$(o)}

=

pN3(t)}

(59) thus gives:

PC(t)

PIS(0) z

microstress

distribution

at the

(62)

P<(t)

that the distribution

related

to that at t = o by means of the transition In general,

(61)

1

one obtains:

showing

functions

of time

will hold: (60)

that for a non-vanishing

p"(t) PO13(0)}

(58)

(59)

P{<(O)}

together with equation

loading which

and P'(t).

substituting

= P{M (0)) P{S(o)) 23

The above relation

initial

structure,

seen that for t = o the following

it is readily

p"(t)

of fibrous

in:

Also in order that equation

P{"(o))

i

of the material

these transition

operator

probabilities

probabilities

of time so that the above relation

at any time t is P"(t)

will be continuous

can be differentiated

to

read: P{M (0)} 53 However,

-d P';3(t)) dt

E”(t)

dt

considering

PC(t)

the creep response

(63)

+ P{M (t)} Es(') dt f3 only,

the transition

probability

P'(t) is equal to unity and the last term in the above relation zero-valued. material

operator

dp{M (t)) x

53

Thus the following

=

with

PW3(0)

relation

time during 1 g”(t)

expresses

will be

the evolution

the creep of a fibrous

of the

structure:

(64)

187 It is obvious

time during

by using

that analogously,

the evolution

the same calculations,

a relaxation

process

dP{M (t)} = W4(o)I .t :4

(55b) and (58) and performing

of the material

(65)

(64) and (65) are the result of the approach

taken here from the point of view of Markov equations

of a fibrous

in terms of the defined material equations

in the development

is evident. discussed

The more

previously

Macroscopic

in references

(51)) within

in which,

mp)

=

in accordance

%1(t)

m;(t)

strains,

second

including

V(e(t))

=

G

v(a(t>)

rheological

for instance,

the occurring

the previously

mesodomain

macro-

given definition

(eqn.

(66) stresses with reference

to a relax-

by:

(67)

the superscript However,

or higher

"m" refers

for a complete order moments

to a specific

description

of stresses

must be included.

second moments

only which

7$(t)

1

mesodomain and

Thus,

for most applications

then:

mgl(t) V(u(t)) z

= %p)

It should be noted

v(p)

?&t)

that in order to establish

as, for example,

outlined

(68)

explicit

forms given in (66) and (67), it will be necessary model

it is

relations

are given by:

z

material

theory

m$t)

for simplicity,

are sufficient

paragraphs,

the macroscopic

the macroscopic

= %,(t)

. . . . . . p.

m = 1,

relaxation

has already been

[5,7,8].

with

of the structure

In both relations

relaxation of these

mpw

and correspondingly ation behaviour

mechanical

made in the foregoing

structure

a particular

The significance

the

Behaviour

formally

strains

during mechanical

form of these relations

to express

for a fibrous

structure"

of a stochastic

general

In view of the statements

scopic

theory and represent

operators.

Creep and Relaxation

now possible

M (t) with ,4

by:

$)

The above set of equations

"governing

operator

will be characterized

before

relations

of the

to use an appropriate

by the "force

trans-

188 mission model"

in dealing

with the creep behaviour

of a fibrous

struc-

ture. In this context mechanical

it may be of interest

relaxation

theory following

[5,7,8], one could use stress-deformation strain relations

for the macroscopic

ponding material

operators

from the operators deformation present

relations

However,

relations

rather

than stress-

In this case the corres-

response.

is due to the fact that experimental

lead only to the determination

it is possible

obtain experimental

given in references

will be M (t) and M (t) that are derivable ;3 ,4 and l++(t). The reason for establishing stress-

$(t)

available

to note that in a more general

the discussion

by means of scanning

measurements

strain fields and hence

techniques

at

of microdeformations.

electron

microscopy

to

that can be used to find compatible

to derive

the more conventional

stress-strain

relations.

CONCLUDING

REMARKS

From the above presentation fibrous (i)

structures,

By using

of the mechanical

the following

conclusions

the four fundamental it is possible

mechanics,

theory for fibrous of such materials

concepts

to formulate

structures

of probabilistic a stochastic

that includes

and in particular

relaxation

theory of

may be drawn: micro-

relaxation

the characteristics

the bonding

effect between

fibres. (ii) A generalization structure

deformation structure, bonding

which

mentally

mine

However,

the breakdown

makes

for a particular

For this purpose double-sided

fibrous materials

holographic

(cellulosic)

the mechanical to extend

[8].

which

are at present interferometry

can

are experi-

of the macroscopic

are required

1411 and scanning

or complete

field quantities

mater-

not only to deter-

but also those of the relevant experiments

of the micro-

it possible

functions,

mesodomain

experiments

of the

of that state of

the point of analyzing

in terms of their distribution

In this context,

that include

behaviour

also to this stage of deformation

these distributions

ators.

with

it has been shown that the relevant

obtainable

ial body.

is associated

relaxation

the exclusion

in terms of Markov process

the analysis

be expressed

with

i.e. the tearing of fibres and partial

failure.

relaxation

Finally,

of the mechanical

has been attempted

material

being carried

operout

on thin strips of

electron

microscopy

in

189 conjunction

with cathodoluminescence

deformation

vector

or rather

to find the important

its distribution

between

relative

overlapping

bond

fibres.

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