一般ダイアコプティックス的刻接による, 高次擾乱を含む塑性物質多様体の表現について

　 MEMorRS

　

OF

　

SAGAMI

INSTITUTE 　OF ［VECHNO 夏

」

OGY

　 　 　 Vo1

，

5

．

　No

．

1

，

1971

On

　

a

　

Representation

　

of

　

Plastic

　

Material

　

Manifold

　

With

　

Higher

　

Order

　

Anomalies

　

bV

　

General

セ

ed

　　　　　　

Diakoptical

　

Tea

 

n9

＊

shojiro

　

sAKATA

＊ ＊

一

_般

_{ダ イア} コ プテ ィ ッ ク ス 2｝

_{的 刻 接}

_{に よ る} ，

高 次 擾 乱

を

含

む

　　　　　　　　

塑 性

物 質

多 様

体

の

_表

_現

に つ い て 阪 田 省 二

_郎

＊ ＊

Contents

§1

．

　 Introduction

．

§2

．

　

Teari

ロgs　of　Disturbances 　of　Higher 　

Order．

§

3

． Physical

　and 　

Geometrical

　

Meanings

　of　

Higher

　

Order

　

Tearings．

§4

．

Plastic　Energy 　in　the　Third 　

Grade

　

Materia1．

目 次 §

1．

導 　入 §2

．

高 次 擾 乱の 刻 接 §3

，

高 次 刻 接の物 理 的

，

幾何学 的意 味 §

4．

塑 性エ ネル ギ

ー

　 　 　 　“

一

般ダ イ ア コプティ ッ クス 2｝的刻接 に よ る， 高次 擾乱を含む 塑性物 質 　 　 　 　　 　 　　 　 　 　　 多様体の表 現 につ いて” _の 要 旨 　連 続 体 力学で ば

，

加え ら れた外 力と， それに よる物体の 変形との関 係 を 規 定 するこ と が 重要な 問 題となる。 結 晶 体で あ れ， 非 晶質の固体であれ， 加え られた外 力 が 小さい と きは

，

それを構 成 する原 子 間の 位相的関 係 を 保存する よ うな

，

ホ ロ ノ

ー

ムな変形をする と仮定し，応 力と 歪 との線形関係 を 前 提 と し た 弾性 論が有 効である

。

こ の場 合， 変 形は

，

変 位ベ ク トル Ui （

i

＝

1，2

_，

3

）_によ_っ て記述され_， 歪の状態を表わす歪 テン ソ ル は 　 　 　 　1

　　　　　　　

・壱ゴ＝ ∂〔朔 1ヨ 百（∂izei ＋∂ゴ％ 言）・ で 与 え られ る。 とこ ろ が連 続 的 微 分 可 能 な 変 位tei の場 合で は，原 子 配 列の な す格子構 造 を保存しない ような 微 視 的変形を， 規定するこ とは で きない

。

そ して 物 体の塑性的性質 とか かわ りの ある

，

微 視 的 状 態を規定で きないな らば

，

応 力と歪との 関 係を有 効に 論 ず るこ とは不 可能にな る。

　

さて ，東 京 大 学 教 授工学博士

　

近 藤

一

夫 先 生に よ っ て

，

塑性状 態の

一

_{つ の 表 現 と し}て

，

非 ホ ロ ノ

ー

ムな変形 とい う概 念が導入 さ れ

，

徴 分幾何 学 的 関 係に ょっ て， 塑性状 態を特 徴づける立場が確立 された。 そして

，

こ の見 地から 塑性 的現 象の解明 が 多 くの研 究 者に

＊

　

This　_paper　is　a　revised 　one 　of　the　original 　Note8｝ bythe_{same 　author}

，

with 　sorne

　 　remarks 　not 　stated 　in　the　

Note ．

＊ ＊

講 師　数理工学 科　 　46 年 1 月

30

目受 理

一

　9　

一

Shojiro

　

SAKATA

よっ て行な われた1）。 　

一

方

，

変 位 場 ui の

2

階 以上の_微係 数が関 係 する よ

．

うな現 象に対 し て

Cosserat

連続 体 論を始め とする，

一

般化 連続体力 学とい う 立場か らの ア ブ P

一

チが

，

最 近 多

．

くな され

　　　　　　　

　

　

　

ロ

　

　

　

　

　

　

コ

　

　

　

てい るts［ ，

．

後 者は

，

よ り微視的な高次官由度の_導入

．

を図るもの で あ

．

る が， 結局i 前者の 幾 何学 的立場に帰すべ _ぎもの と考え られる。 　塑 性 多様体 を 表 わ す リ

ー

マ _ンあるい は非 リ

ー

マ _{ン空間の幾何 学的な構造は} ， 連続体を

．

　

　

ロ

　

　

■

　

　

　

ロ

　

　

コ

微 小な素 片に 刻接 するとい う， 思考実 験 的 な 操作に よ り定義される。 著者は刻接の概 念

　　　　　　　■

　

　

　

　

を

一

般 化 して， よ り高 次の塑 性 的不整 を 規 定 するよ うな

，

幾 何学的に 不変な変形 量を導

　　　　　　　ロ

　

　

　

入 し， よ り微 視 的 な 擾 乱 を 包 含 する連 続 体 変 形 論へ の

一

つ の幾何 学的ア プ ロ

ー

チを 展 開 する

。

　なお

，

こ の 研 究に際 して_， 近 藤

一

夫 先生の御 指導と， 相 模工業 大学教授 理学博士 河 口_{商 次 先生}の_{御 助}言を得た こ とを文頭に_述べ ， 感謝の意を表わし たい と思 う。

§

1

。

　

IntroductiOn

　　　　

The

　

innumerable

　_{atoms 　composing 　a　material} 　

form

　_a　much 　or 　

less

regular 　

pattern

　

proper

七〇

the

　specified 　nature 　of 　

the

　material

．

　

In

　real materials ，　

the

　

perfect

　regUlarity 　are 　

disturbed

　

by

　several 　sources 　 of

imperfections

，　 which 　can 　 never 　

be

　reliesed 　only 　

by

　removing

．

the

　ex

−

ternal

　

force

．

　

The

　so

−

called 　

plastic

　nature 　of　matter 　originates 　

in

　such

microscQpic



disturbances

．



P

】映stic



structures



of



continua



_can



appro

曽

priately

　

be

　

described

　

by

　

the

　

geometrical

　

termillology

　of　

Riemannian

　alld

non

−

Riemannian

　space

_（

K

．

　

Kondoi

｝

_）

．

　　　　

The

　

geometrical

　 structures 　 are 　

determined

　

from

　

the

　 concept 　 of

tearing

七

he

　material 　

body

　

into

　small 　

pieces3

）4〕5）

．

　

The

　simplest 　

interpre

−

tation

　

of

　

it

　

is

　

that

　

an

　

infinitesimal

　

line

−element

　

dxi

_〈

i

・：

1

，

2

，

3

、　or　a

small 　

piece

　_{of　material} 　

is

　_{cut 　off　and 　naturalized} 　

being

　_relaxed 　

from

the

　constraints 　of　

the

　surroundings

．

　

One

　may 　

prescribe

　

its

　naturalized

con 丘

_guration

　

by

　

the

　

linear

　

formula

（

d

¢

）

a　・＝」

A

：

・

　

（

x）

dx

‘ ， 〈

1

．

1

）

（

where 　

the

　

quantities

．

4

窒 are 　

the

　

functions

　 of　

position

　coordina 七es　 x

‘

，

de

且ned 　

in

　

the

　

limit

　when 　

dxi

−

→

0

，　while 　

the

　

differentiable

　

structures

of　material 　manifold 　

is

　overall 　assumed

，　wherein 　

the

　atomic 　structures

combined 　_with 　

the

　 _{continuous 且eld　are 　 averaged 　 and 　 smoothed 　out}

．

Owing

　

to

　

the

　

plastic

　

deformation

_，

　

this

　

transformation

。

4

_牙

is

　

in

　

general

non

−

holonomic

，　or　

in

　other 　words ，　

the

　rotatioll 　of ∠

4

砦

does

　not 　vanish ，

∂_匚か

4

_窒

］≠

0

，

（

1

．

2

）

in

　_{cQntrast 　 with} 　

the

　_elastic 　

holonomic

　

deformation

_{，　of　which} 　

the

　

A

_兄

are

　reduced 　

to

　

the

　

partial

　

derivatives

∂ltxa

（

　＝ _∂xa1_∂xh

）

of　_one

−

_valued 　

func

−

tions

　xa

_〈

  h

_）

．

　 The

　

point

　

functions

　

AS

　

of

　what 　

we

　call　

deformation

且

eld

On

a

Representation

of

Ptastic

_Mdteriat

_Mdnifbld

with

Higher

Order

Anomattes

of

the

first

order should

be

defined

as regards minute

but

finite

material

pieces,

for

eaeh

torn

element cannot

be

so small'as

to

reach a

lattice

unit.

If

only sueh a

deformation

A:･

of

Iine-element

dx`

is

taken

into

consideration,

the

_macroscopic

but

finite

torn

_piece

is

deprived

of any

other characteristics

_possibly

relating

to

_plasticity.

We

refer

to

this

situation as

the

_point

tearing

or zeTo-dimensional

tearing.

The

more

higher

degrees

of

freedom

of

imperfeetions

in

material

must

be

disclosed

when we

tear

off

the

more microscopieally.

If

a

torn

small

_piece

under consideration

is

finite,

we must

introduee

a

defor-mation

_quantity

correlated

with

its

finiteness,

even

though･it

is

mathe-matically

brough

to

the

infinitesimal

limit.

The

point

tearing

may

be

considered

in

comparison

with

the

as-sumption of elasticity.

In

the

latter

context

the

_point

deformation

functions

Af･

are

particularlized

to

the

first

order

derivatives

Oixa.

The

configurations

of a eontinuous medium are not sufieiently expressed

without referring

to

the

sueeesive

higher

_gradients

OhAr･,OhaiA:,･･･

at

eaeh

_point

x`, whereas

the

classieal elasticity

theory

assumes

that

only

the

first

order

_point

gradients

Ag,,

whieh are

Oix"

in

_particular,

influenee

the

material energy

density

in

the

elastie material.

In

some

case

the

higher

order

_gradients

_prove

to

be

indispensable

for

describing

the

configuratienii'

so

that

something more

than

the

_point

tearing

<rep-resented

by

A:･

_alone) must

be

introduced.

Furthermore

we remark

that

the

so-called

_plasticity

is

a certain summary of many complicated microscopic

features

with certain meehanical

_processes

including

tearing.

There

is

missed something, whieh

is

somewhat

_global,

by

this

way of'

recognition.

In

the

last

few

years

there

have

been

several approaches

to

the

formulation

ef

_generalized

mechanics of eontinuai3}.

Our

investigation

is

direeted

to

an

invariant

formulation

of

higher

order effects and

its

non-holonomy

in

the

terminology

of

_generalized

tearings.

g2.

Tearings

of

Disturbances

of

Higher

Order

As

the

material eonnot

be

torn

beyond

the

lattice

unit, a cut off

piece

eannot

be

_perfectly

naturalized,

involving

several microseopie

anomalies whieh are not

torn

off.

The

_point

tearing

is

an

idealized

eoncept.

In

order

to

reveal non-loeal or macroseopic effects of averaged

microscopie

disturbances,

we assume

that

a

_part

of

the

still

finer

defor-mation

is

invariantly

retained

in

a

torn

infinitesimal

material

pieee

having

the

_property

of

the

line-segment,

area- or volume-element.

Let

us use

the

terminologies

of

the

one-,

two-

or

three-dimensional

tearing,

-!1-j

Shojiro

SAKATA

or alternatively segment, area or volume

tearing

in

this

sense.

In

general

a

eut

off material'element

_possibly

has

a

very

compli-cated

form

resultingfrom naturalization.

The

deformation

represented

by

an arbitrary

A:･

is

_generally

non-holonomic and

is

the

average

feature

of

the

deformation

over

the

material

_pieee

by

a zero-dimensional

tearing.

On

considering a

further

approximation reaehing

the

first

order we need

to

handle

a

distribution

of

deformations

or

the

strain-rotation

gradient

averaged on

the

line-element

along

which

the

material

a

is

torn.

Let

this

quantity

be

denoted

by

Bp.

It

also represents

the

curvature

and

the

torsion

averaged

along

the

axis of

the

slender

material

a

pieee

dx`.

From

this

strueture,

the

quantity

Bp,

which represents a

mieroscopic

local

_{configuration around}

the

end

_point

x`+elx" of

the

torn

pieee

starting at

the

point

xi, must

be

_given

as

follows

a

B,=6g+B;is"(x)axr,

<2.1)

where

aff

is

the

Kronecker

delta.

This

is

to

tear

_perfeetly

at mi

but

a

not

at

x`+dx`.

Hence

we

must remark

that

the

value of

Bp

at

the

starting

_point

x'(elx`=O) of

the

torn

pieee

is

equal

to

ap"

and

the

seeond

term

of

the

above

formula

is

not necessarily equal

to

the

_gradient

O,Bka

of any

function

Bfo".

The

_integral

iB;badx'

eannot

be

determined

as a

_point

function

without

depending

on a

_path

of

integration.

We

refer

to

this

situation as

the

dependenee

on

the

one-dimensional

ct

eharacterirtics

of

the

more microscopie

distributions

of

deformation

Bp.

In

order

to

compare

the

configurations

of

the

neighbouring

_points

x` and

x`+dx`, we must adopt

the

naturalized state

(a),

whieh

is

introdueed

at

eaeh

_point

by

the

zero-dimensional

tearing,

as

the

reference

system

(a)

in

the

above considerations.

Thus

the

difference

of neighbouring

cen-figurations,

which we

denote

by

AS(aA):･

symbolically,

is

_given

in

a

Eulerian

coordinates system

<i)

as

follows

a

AZ(6A)

:･

=

Ag(B,A}･

(x

+

dx)

-

OtAS

(x))

=AS(O,A:･

+Bi;･a)dxn,

(2.2)

where

Bvia=BaqAe･Al,

A:･=A:<x),

etc. , a

and we must remark

that

the

upper

index

a of

Bb

is

different

in

nature

from

the

lower

index

b,

the

latter

indicating

the

microscopie measuring

apparatus

(frame)

in

the

torn

small

piece

and

the

former

the

maeros-copic coordinates common

to

all

the

_points

on

the

torn

piece,

and

there-t

On

a

RepTesentation

of

Ptastic

Mictteriat

Mtznfold

with

Higher

Order

Anomaldes

fore

the

upper

index

must

be

contraeted with

the

a of

Ag

at

x'.

Hence-forth,

measuring

the

difference

of

the

microscopic eonfigurations along

a

one-dirnensional

element

dx`

by

the

above mentioned manner will

be

referred

to

as

segment

tearing

of

one-demensional

tearing.

If

we

identify

the

_geometrical

eoncept of eonnection along a curve with

the

integration

of one-dimensional

tearings

of

line-elements

eornposing

it,

the

coeMcients of connection are

_given

from

the

formula

(eompare

with

S,

Amari3)).

I'{,dxh

==

Al(OA):

=A2(6hA:-

+BL;･a)dxh

,

(2.3)

and

hence

it

is

specified

by

the

first

order

deformation

P;h

defined

by

P;h-(A:.-6:.)6:,

(2.4)

and

the

second order

B3L"

(we

use

this

terminology

throughout

this

_paper)

*

rL,=Tl,+Bx,j,

(2.s)

where

the

quantity

*

r,t･=AZO,Ag･,

(2.6>

is

derived

from

the

point

tearing

A:･.

Now

we consider

the

eonneetion along an

infinitesimal

loop

encircl-ing

an

infinitesimal

area

df`i

(in

other words,

the

segment

tearing

of

a

torus

which encireles

the

area

clfij

and

has

an

infinitesimally

small

eross section,

preserving

only

its

one-dimensional characteristies).

At

the

cut

off

point

we

have

a

discrepancy

of

location

of

the

cross

section

of

the

torus

dxh

=:

-Sshdf"J,

(2.7)

and a ehange of

the

material veetor vk attached

to

the

cross sections

1

Avh

==--l;R;y,hvk(byC'id,

(2.8)

where

the

quantity

given

as

follows

in

any

holonomic

coordinates

(x`),

S;l.h=1're,,.,,

(2.9)

CE)

is

the

torsion

tensor

and

the

_quantity

R;}:,h=20ur,-h],+2I'[,ft.11-tjM],

(2.10)

(h)

-13-t

Shojiro

SAKATA

is

the

curvature

tensor,

Which

is

elimifiated

by

the

zero-dimensional

,

_*

'

tearing,

_pr

in

other

words,

which vanishes

if

the

re･j

are

taken

in

_plaee

of

the

r.h･i

in

the

forrpula

(2.10).

From

the

formula

(2.5)

of

the

eoeMcients ef eonnection,

iove

have

,

R;}:,fi=2fi[,B;･,ihi2B,;i-lhB;,;.m.

,

'(2.11>

*

1 *

where

7[i...]

is

the

Cartan

operator

for

the

eovariant

differentiation

7i

* *

with respect

to

the

eonnection coeMcients

l'ij･j.

The

torsion

tensor

is

divided

into

two

constituents

as

follows

*

S;].h

..

S;;n+B[;;X,

_,

(2.12>

where * *

S;sh=l-'[,hi]=A2a,,Ag･]

.

'

(2.13)

Because

of

the

second

term,

S;;･h

is

not

in

_general

tearable

in

the

sense

that

S;･;lt

is

not

direetly

associated

with

a non-holonomie object

9:b

of

a

non-holonomic

coordinates

system

(a).

If

it

were so

tearable,

we would

have

SI･)lt

=:

Af･,b･,nn2,

==

A:O[,A;]

_,

(2.14>

where

A2;,h=A:･Ae･Ae

is

an

abridged

notatidn.

The

differential

equations

O[,A3]=S;;hA:,

(2.15>

or

1

7[,Ag.]

==O,

(2.16>

are not always

integrable,'

while a suMcient condition

for

its

integra-bility

is

that

the

alternating components

R[xi,･lr

of

the

curvature

tensor

vanish

identically,

((e,,e,A:-,>A:

=)

-ll-

R,i;-1-S

=e,,S};k, =o _.

(2.17>

Because

this

is

entailed

by

the

condition

that

the

jr'e-j

'are

reduced

to

* *

the

ri･i,

we shall call

the

torsion

tensor

S;･3k

to

be

simply or

zero-*

dimensionally

tearable.

The

compQnent

S;;k,

which

is

zero-dimensionally

tearable,

is

conventionally

identified

with

the

dislocation

density

in

the

continuous

theory

of

disloeations.

From

the

foregoing

consideration,

"

On

a

Representation

of

Ptastic

imteriat

Mdndfbld

with

lligher

OrdeT

Anomaldes

however,

we

find

the

_possibility

of eonstructing a more

_general

torsion

tensor

field

not only

from

the

dislocational

imperfections

but

also

from

more microscopie non-tearable struetures of

the

material

_pieee.

The

more

_general

torsion

tensor

of

the

conne ¢

tion

is

not zero-dimensionally

but

at

Ieast

one-dimensionally

torn.

This

has

not

been

excluded

from

the

_general

diakoptical

tearing

eoncept suggested

by

K.

Kondo2'.

The

formula

_(2.11)

suggests

the

Proposition.

Non-teleparallelism

_(dvklO

or

Ri;･}klO)

is

entailed

by

the

existence of

the

microscopic anomalous

features

which ean

be

regarded

as a microseopic souree of stresses owing

to

which

the

second order

deformation

of

the

material

_piece

involves

a

discrepancy

or anomaly of

directienal

(one-dimensional>

characteristics.

This

proposition

allows us

to

use

the

rather mechanical

terminology

of "segment

tearing"

in

the

meaning of a

tearing

of a material seg-ment, er

its

immediate

neighbourhood

into

different

orientations, where

the

failure

of

the

seeond

integrability

(non-teleparallelism)

signifies

the

uncertainty

of

the

orientation

of a material

line-elemente'.

In

a similar

way, zero-dimensional

tearing

is

more suitably

defined

as a

tearing

of

a

_point,

where

the

failure

of

the

first

integrability

(1.2)

signifies

the

uncertainty of

the

_position

of a material

_point.

So

far,

we

have

not required metrieity,

the

displaeement

by

means

of

rli

from

a

_point

x`

to

the

neighbouring

_point

x`+dx`

is

also

treated

without referring

to

invariant

metrie eommon

to

the

eonfigurations

(i>

and

(a).

Therefore

the

tensor

defined

by

g,,･(xh)=a.bAg-A,b･(xh)

(2.18>

is

not

_yet

aseertained

to

be

metrie

tensor.

In

fact,

by

straightforward

ealculation,

we

have

def.

Qil･',･

==

7hgij=-2Bh(ij}.

(2.19>

If,

and

only

if,

it

vanishes,

i.e.,

Qi;,,

:=O,

(2.20>

the

well-defined

metric connection

having

the

well-known structure

re-i

--

{,k,}+

Z;le,

(2.21>

is

entailed, whereas

the

_general

non-metric eonnection obtained as above

is

-15-g

Shojiro

SAKATA

rij･,

--

{,k,-}+

Ti;k-

-ll-

_(Q;e,

-Qkd+

_Q;;k)

,

(2.22)

where

{`ki}

is

the

Christoffel

symbol and

Ti;k

==

S;}k-S)k.,+SS,,d

_.

(2.23)

The

tensor

T};k,

whieh

is

sometimes called

the

strueture curvature,

is

decomposed

into

*

tTli;k=Ti;k+r;)k,'

'

(2.24)

*

where

71i3k

is

the

structure curvature

due

to

the

teleparallelism

con.nee-*

tion

I'Si

and

_r;･;'k

is

the

antisymmetric

_part

Bi[jk]

of

Bijk

with respeet

to

the

last

two

indices

i

k.

We

call ridicrotational strain

tensor,

with which

the

relative rotation of

the

neighbourhood of a

_point

xh along

dxh

is

eorrelated.

The

zero-dimensionally

torn

material

piece

is

assumed

to

involve

no

imperfeetions

in

it.

In

_a similar manner any

_point

defects

_{such as}

interstitial

atoms and vacaneies ete.,

by

whieh

the

non-metricity

is

eaused, may

be

excluded

from

the

one-dimensional material

_piece,

while

it

may not

be

free

from

the

surface

defects

like

staeking

faults,

twin

boundaries

etc.

Therefore

the

seeond order strain components

Bwk]

vanish without

the

other

types

of

disturbanees,

and

the

material

mani-fold

with

the

connection

coeMeients

Te-j

and

the

metric

_g,i

specified

by

the

point

tearing

and segment

tearing

is

a metric

(in

general,

non-teleparallelism)

space.

(Compare

with.the

foregoing

_paperS').

It

does

not suffiee

for

representing all sorts of anomalies

to

tear

the

material zero- or one-dimensionally at every

_point

separately

from

one another.

Some

other anomalies

than

such

tearings

are of necessity

involved.

They

are related

to

some aspects of connecting neighbouring

material elements

in

a more

_general

manner.

When

we consider only

the

one-dimensional

tearing,

two

kinds

of

diserepancies

have

been

reeognized and

have

been

_given

by

the

formulae

_(2.7)

and

(2.8)

with respect

to

the

torsion

tensor

S;;k

and

the

eurvature

tensor

Rvjk

in

the

ordinary sense.

The

areal element

of'i

involving

areally

distributed

imperfections,

the

diserepancies

are

exhibited which are not characterized only

by

the

I"f･j.

Let

us cut a

dise-shaped

material

_piece

into

separate sectors with

infinitesimal

angle and naturalize

them.

ff

they

are connected without any compelling

forces

in

them

and

form

a eontinuous aggregate, whieh

is

a

disk

with

a

break,

the

diserepancies

at

the

edge are caused

by

the

two-dimensional

anomalies.

<

On

a

Representation

of

Plastic

Mttterial

Mttnifbld

with

thgher

Oraer

Anomalies

For

the

skew-symmetric

tensor

of rank

three,

_geometry

is

preserved

if

the

B:･;k

is

arbitrarily chosen.

However,

no similar

_generality

is

guaranteed

for

an arbitrary

tensor

of rank

four.

If,

_{and only}

if,

Xi,1･k

satisfies

the

Bianchi

identity

2

7[,Xi;]Lva=O,

(2.25)

for

arbitrary assumed eoeficients of eonneetion

rij･j,

it

agrees with

ordinary

Riemann-Christoffel

eurvature

tensor,

_provided

that

it

satisfies

also

the

other

identities.

A

four-rank

tensor

Xil･)･k

which

does

not

satisfy

the

Bianchi

identity

may

be

assumed as a representaton of

the

anomaly arising

first

at

the

area

tearing.

Thus

we refer

the

ctrea

tearing

to

speeifying

the

following

_quantities

S';;k,R'A;1･k

by

measuring

the

diserepancies

exhibited on

the

disc-shaped

material element with a

break,

so

that

they

are

_given

by

dxk=-S';･;icdfij,

<2.26)

dv"=-

-li-

_R'i,

_1-}-kdfhivj

,

(2.27)

where

the

_above

_quantity

R'i}}-k

is

not always

the

curvature

tensor

Ri;lh

in

regard

to

the

connection with

the

eoeMeients

re･j

defined

by

the

segment

tearing,

_while

the

rank-three

tensor

S';3ic

is

not

intrinsieally

new and

it

can

be

equated with

the

tensor

S;]".

Let

us call

R'is･k

de-formation

of

the

third

order.

Only

the

symmetric

components

R'h,i,/ik)

are essentially new, and

the

antisymmetrie components

Rhwm

are as--summed

to

be

equal with

the

Rhijk

of

the

metrie conneetion

r;･j.

Lastly

we

introduce

the

volume

tearing,

which

is

eorrelated with

the

eutting off ef an

infinitesirnal

piece

without eliminating

the

three-dimensional

eharaeteristics of

imperfections

distributed

in

it.

As

the

area

tearing,

by

our

definition,

is

to

prescribe

the

quantity

R'L;-vk,

whieh

does

not always satisfy

(2.25),

by

measuring

the

discrepancy

dvk

ex-hibited

on

the

_periphery

of areal element, so

the

volume

tearing

is

cutting off along

the

_periphery

and a

generating

line

of _an

infinitesimal

conical material

_piece

df'"j

and measuring

the

discrepaneies

dxk

and

Avk,

whieh are shown as

the

differenee

of

the

_positions

and

eonfigura-tions

of

both

sides of

the

edge, and which are

given

by

lixk

=:

-M'A;yledfh`j

,

(2.28)

zdvM=-L'A;･1:krnctf-h'ijvk.

(2.29)

We

need not consider

the

tearing

of more

than

three

dimensions

-17-i

Shojiro

SAKATA

in

the

ordinary

three-dimensional

material space, except

in

the

case of

the

four-dimensional

material space-time,

for

_{example, eneountered}

in

the

relativistie

theory.

g3.

Physical

and

Geometrical

Meanings

of

Higher

Order

Tearings

Now

let

us

discuss

the

area and

volume

tearing

first,

leaving

the･

meanings of

the

non-holonomic

transformation

coeM ¢

ients

A:

and

the

connection

_parameters

l'#･j

till

later

on.

Given

the

quantity

R'x'ijk

intro-duced

by

the

area

tearing,

we

do

not always

have

a

quantity

r:･f･

such

that

the

formula

is

valid

R'nt.･,k

==

20,,r',S,+21"'[,?.lr',T,

,

(3.1>

that

is,

the

above

differential

_equations

in

_general

have

_{no solution, of}

whieh

the

only

integrability

condition

is

what

is

called

the

Bianehi

identity

2

J7E,R';;,kM=O,

(3.2>

2

where

7Eh...]

is

the

Cartan

operator with respect

to

the

l",k･i.

If

(3.2)

eannot

be

satisfied

with

respect

to

arbitrary

assumed

coeMcients of

conneetion

r':j,

we

have

the

non-vanishing

quantity

L,`

,:,':km=

-ll-

fi

[,

R';b]km

,

(3

.3>･

2

where

7[h.,.]

is

the

Cartan

operator corresponding

to

the

r,k-j

of

the

segment

tearing.

Likewise

we

have

the

_quantity

Ml;}･k=fi,,S;･

_IS-

-ll-

_{R'[M･]･,k}

.

(3.4)･

In

order

to

inquire

what

is

meant

in

such a ease, we must review"

the

interpretation

of

Bianchi

identity

_given

by

the

_geometeri`'.

This

is

to

consider a

three-dimensional

submanifold

Xk

through

a

point

xi

in

o

the

non-Riemannian manifold and a elosed

X>

surrounding xi

in

the

XL..

o

It

is

shown

that,

if

the

_point

with radius vector v` at x` undergoes a

o o

Cartan

displacement

from

x`

to

a

point

x` _of

IYh,

_{and along}

the

border'

e

of

a

surface element

df`i

of

X

through

x`,

in

the

sense of

df`i

baek

to

xi and

finally

from

x`

to

x`,

then

the

change of v`

is

given

by

o o

On

a

Representation

of

Ptastic

imteriat

Ml

nifbld with

Higher

Order

Anomatdes

zt?m =

(

-

fl;･

}･m

-

(x

in

-

`yre)o,Sg-]m

-

-;-{i

;y,m?k

+

-li'

{ir;}'hpt

(Xh

-eeh)

+

-li-

{ir;･YkMe"{iit.

_(xh

-?h)

--}

_(xh-?h)

O,R;;･,M

?kldfij

,

(3.5)

where

rt･i,

S;-;ic

_,

R;ykM

are

field

values at x`, and

the

higher

order

terms

oe o o

with respeet

to

clx`(==x'-xi)

are neglected.

Aeeording

to

the

theorem

o

ef

Stokes

the

integral

of

this

difference

vector over

Xi

is

equal

to

!.,I

-{l;[;;'"ChT"

-

ll

_R[;)'tLft?kl

'hTn

-O[hS;;･T

+

-ll-

{P,tum,

+

-ll-

i:e[;･

}･

i

cr

]:"hT.e"

-

-S-

?ko[h

R;;,Lm

l

df,dh

=

I.,I

-l

_[hS;;',"

+

-ll-

R[A;･1･r-

-ll-?k

er[,R;;]im

l.cW,hii

,

(3.6)

o

and

this

integral

vanishes on account of

the

Bianehi

identities

as can

be

seen easily.

If,

diseriminating

the

segment

tearing

along

the

vector

like

material element

elx`

and

the

area

tearing

of

the

areal element

df`d,

whieh

does

not

include

the

direction

clx`,

we substitute

the

R'vijk

for

the

Rvi'jk

in

the

above

formula

(3.6),

we

have

a

discrepancy

as

follews

{-Mm-vkLLm}dV,

(3.7)

where

dV=(113!)ehij(if"'j=(116)ehijdx[hdf`j]

is

a volume element, and

the

quantities

Lh'i;'kM,

Mi;;nt

are

_given

by

(3.3)

and

(3.4)

and

Mm=(113!)EhijMi;;M,

(3.8)

LLm=(1!3!)ek`s'Li,:;:,".

(3.9)

This

diserepancy

results as a special case of

the

three-dimensional

tear-ing.

Thus

the

one-dimensional

distribution

of

the

_area

tearing

R'vak

brings

about

a

three-dimensional

discrepancy

as above.

Next

it

must

be

argued whether

there

is

a

_quantity

R'i;･1-k

such

that

L'A;･'i',M=-ll-J;[,R'i,:]iM,

_(3.10)

for

an arbitrary assumed conneetion

r.k･d,

_given

_a

_quantity

L's;･)･k"

pre-seribed, otherwise arbitrarily,

by

the

volume

tearing.

But

this

_problem

,

Shojiro

SAKATA

ean

be

solved

immediately

by

notieing

that

our material spaee

is

three-dimensional.

The

integrability

eondition of

the

above

differential

equa-tions

is

3 32

27[pL'h';;]kpt

==

"[p7hR'i`;]int

=

+ll'

R[iAihTR'is:]

ek

-

-illL

R[i'hdLrrR';}']Tn

_,

(3･11)

whieh

is

an

identity,

for

both

sides _vanish

because

_of

the

four

alter-nating

indiees.

Therefore

the

volume

tearing

reveal no

higher

order anomalies

than

represented

by

the

two-dimensional

_quantity

R'E;･}･k.

Frorn

the

preceding

considerations,

it

follows

that

material

imper-fections

are

classified

into

the

three

types,

which are represented

by

the

quantities

A:,

I"ts･i

and

R',"i}･k

respeetively.

This

may suggest

the

fact

that

the

lattiee

imperfections

are

_grouped

into

the

three

classes

of

point,

line

and

_plane

defects

in

the

crystallography,

where

vacaneies,

interstitial

atoms,

foreign

atoms and

_possibly

solute atoms

in

solid

solu-tion

etc. are

_point

defects,

and

dislocations

are

line

defects

and

stack-ing

faults,

twin

boundaries

and

habit

_planes

of martensitic

trans-formation

etc. are

_plane

defects,

Concerning

the

structural similarity

in

the

two

systems of classifieations of material

defects,

we

inquire

some

link

between

them.

A

dislbeation

is

intrinsically

different

from

a

train

of vacancies, and a

_grain

boundary

with a small angle

is

enly

a

two-dimensional

distribution

of

disloeations,

but

not

a

_plane

defect.

On

the

other

hand,

the

two-dimensional

distribution

of segment

tearings

should

be

discriminated

from

the

area

tearing.

The

internal

energy

density

of

_plasticity

material should

be

a

func-tion

of

the

deformation

quantities

A:,

1-tj

and'

R'Avjk,

u=u(A:･,

I'S,

R'LX).

(3.12)

In

specifying

the

internal

energy

in

a

line,

a

surface and a

body,

the

one-,

two-

and

three-dimensional

distributions

of

them

must

be

inquirediO).

The

two-dimensional

distribution

of

the

first

order

defor-mation

Al-

has

an effect on

the

internal

energy u

in

a surface

in

the

*

form

of

O[ltA:･],

or

invariantly

Sh;-'=AZO[hA:-],

which

is

comparable with

the

density

of

disloeations

passing

through

the

surfaee.

The

surface

distribution

O[hLij

of

the

second order

deformation

r.k-j,

or

its

invariant

eounterpart

Rxiik

ealled

incompatibility,

is

exemplified

by

the

distribution

of untearable

dislocation

pairs,

and

the

trains

of vacaneies

or

intersti-tial

atoms".

The

three-dimensional

distribution

of

A:,

I'e･i

does

not work

in

the

form

O[hOiA:](=O)

and

O[hOiri]ic(==O),

or

in

other words,

the

sur-

-20-,

On a Representation ef Ptastic Mlzteriat 1;dilznifbldwith thgher Oreler Anomalies *

faee

distribution

Svij

of

_point

tearings

A;

is

self-equilibrated

in

a volume

2*

7[,S;･

]･S

==O,

<3.13)

*

and

the

surface

distribution

RAvik

of segment'

tearings

rSi

is

self-equilibrated

in

a

body

2

ff[,R;･1･]LM=O,

(3.14)

while

the

three-dirnensional

distribution

R[nl･B]k

and

Lh',:;'icM

gives

the

divergence

of

the

surface

distribution

Sxij

and

R'si;k,

respectively.

2

R[LZ･1･k]

--

27[,,S;･

}･:,

<3.15)

L,-,'

,-

LM=

-;-

l;'[

,,R'i,']iM .

(3.16)

In

defining

the

area

tearing

as a

tearing

of a material areal

ele-ment

in

the

similar way as

the

eases of

the

tearings

of a

_point

and a segment, we may

be

allowed

to

assume

that

we _can naturalize

the

interior

of eut

off

areal

element exeept

its

_periphery,

whereupon some eonstraints are needed

in

order

to

concentrate all

the

imperfections

on

it.

Thus

tearing

off of

_plane

defeets

except

line

defects

is

considered

to

be

a

kind

of

area

tearing.

The

symmetric eomponents

R'Xi,/;'k,

are

related with

the

metrieal

imperfeetions,

with whieh

point

defects

have

not a slight

]ink,

while

the

antisymmetric

_part

R'kl[;･in

are redueed

to

the

Rh-,:;`k

due

to

the

rotational strain

r,{d.

Therefore

the

area

tearing

has

some relevance with

the

non-metrieity of mieroscopic structure

in

materia].

g4.

Plastic

Energy

in

the

Third

Grade

Material

The

distinction

of

plasticity

from

elasticity

is,

in

essential, nothing

but

the

non-holonomic nature of

the

deformation

of

the

first

order

re-presented

by

A:,

whieh cannot

be

reduced

to

the

partial

differentiations

Otx"

of

functions

x"(xi).

R.A.

Toupin

and othersii'i2'

define

the

elastic

material of

the

seeond

grade

sueh

that

the

second order

position

gradient

OhOix"

as well

as

the

first

order

Oixa

infiuence

the

elastie energy

density.

Tentatively

we

define

the

plastic

material manifold of

the

seeond

grade

having

the

connection

rlr･j

given

by

l'f･,･=o,As+B;-}",

(4.1>

Shojiro

SAKATA

where

B;-;-a

are not reducible

to

_partial

derivatives

O`B'g･

of

funetions

B'S,

that

is,

the

differential

equations

I-t,･=A'kO,A'S,

(4.2)

are not

integrable

because

of

Ri;･;ksO.

While

the

area

tearing

Rl;;k

is

redueed

to

the

two-dimensional

distribution

of _segment

tearings

r,k･i

in

the

second

_grade

material,

the

third

_grade

material

is

totally

prescribed

by

the

point

tearing

A:･,

the

line

tearing

r.k･i

and

the

area

tearing

R's;]･le,

or

equivalently

by

the

invariant

quantities

g,j,

S};k,

Ri;･]･k

and

L'i;･;:,nt.

We

can neglect

the

seeond and

higher

order

terms

of

Pii,

_rhii,

R'ltidk,

provided

that

the

disturbanees

are small.

Then

we

have

the

following

linear

relations,

A:=Bg･

+B,,Bna,

(4.3)

gid=6ii+2eij,

(4.4)

1-'{j=1-,i,

==

O,Pj,+r,d,

,

(4.5)

Sv,i=O[,P,]i+r[,,]i,

(4.6)

RSi;･k=20[hri],･k,

(4.7)

L'iM=(113!)L'i;YkMenij=(1112)shiiO[itR';･)]i.'

,

(4･8)

where

Pij,

eii--P(ij),

_rnii,

R'hisu

are

distortion,

strain, rotational _strain,

deformation

of

the

third

order,

respectively.

By

eontraetion

with

the

Eddington's

symbol ehii, we

introduce

the

following

quantities

of valence

two

which are equivalent with

the

above

quantities

with

three

indices,

with respect

to

the

two

of

whieh

they

are antisymmetrie.

Thus

we

have

1

rhi=-li-Sdimnrh..,

(4.9)

Shd=

-IL

_eh..Srmi=:

-!-

_{(eh..a.i?.i-rdh-6hir)}

,

(4.10)

2

2

,

ILi=

_{(114)slt..siikRik..}

==

Rhi

-

-ll-

_ghiR=

e"kOirkh _,

(4.11)

R'h"i}=-ll'ehmnR'ptptcdiso

,

(4.12)

where

IL`

is

called

Einstein

tensor,

and

_r=

_r;i.

-22-`

On

a

Rerpesentation

of Plastic Mdterial Mtznifbtd with

thgher

Order

Anomalies

The

plastie

energy stored

in

the

third

_grade

material

is

_given

by

a

function

of

the

Shi,

Ihi,

L'{ii).

Therefore

we

assume

the

following

energy

function

U=

I

Ud

V:::;

-ll-

!.

(ILiRih

+2Sh

;rhi+L'

uj]

eid)dV

,

(4.13)

where

the

resistance _against

the

increase

of

these

_quantities

_are

denoted

by

Rih,

2ihi,

edj.

Substituting

the

expressions

(4.10),

(4.11),

(4.8>

into

(4.13),

we

have,

by

integration

by

parts,

U=

-}!.(B.L,+2r,,S,,+Rt,,j,R,,i,)dv

+

-ll-

S

`

.

(rhiRiieikh

+Phi?.ie.kh +

(1!6)R',(,i]

lid)aSk

,

(4.14)

where

iihi=

-!

_{(s,i,O,･B-",,-i,,+6,3)}

,

(4.ls)

2

Zhi=

-ll-

_eijkOi"r-kb,

(4.16)

R',j,=-(116)aig.,,,,.

(4.17)

These

newly

introduced

quantities

satisfy

the

follewing

relations

o,L,=o,

(4.ls)

20,S,,=e,,,L,,

(4.lg)

E,,jO,Rl{..,=O.

(4,2o)

The

equality

(4.18)

is

the

Bianehi

identity

and

the

(4.19)

is

the

seeond

identity,

both

in

the

linearized

form.

The

above

form

(4.13)

except

the

term

R'ijk

of

the

energy

function

and

the

above

equalities suggest

that

ai,

Shi

are stress and couple stress

in

the

Cosserat

eontinua,

pro-vided

that

the

latter

relations

(4.18),

(4.19)

are regarded as

the

equilibrium condition of stresses.

Therein

the

quantities

khi,

'rhi

are ealled stress

functions,

by

which

the

non-Riemannian

_geometries

of

the

stress

funetion

spaee are constructed

by

many authors').

In

our stress

function

space another

_quantity

R;･jk,

rotation of which vanishes

(4.20),

is

introduced.

In

this

regard

the

third

_grade

material

is

different

from

the

non-metric stress

function

space

introduced

in

the

foregoing

_paper

by

the

author"'.

Shojiro

SAKATA

Aclenowledgement

The

author wishes

to

express

deepest

appreeiation

to

Professor

Dr.

A.

Kawaguchi

for

his

kind

advice

in

the

_preparation

of

the

manu-script.

The

author also

is

indebted

to

Professor

Dr.

K.

Kondo

for

his

coordial

_guidanee

and criticisms

in

the

_preparation

of

the

original

Note8'.

References

1)

K.

Kondo,

et al,:

Non-Helonomic

Geometry

of

Plasticity

and

Yielding.

RAAG

Memoirs,

1,

2,

3,

4,Division D, 1955, 1958, 1962, 1968.

2) K. Kondo:

On

"Generalized _{Diakoptics". RAAG Memoirs, 2, F-V}

(1958),

_409-422,

3)

S,

Amari:

On

SomePrimaryStructuresofNon-RiemannianPlasticityTheory, RAAG

Memoirs,

3,

D-IX

(1962),

99-108,

4)

K.

Kondo: The

Non:Holonomic

Structures

of Plastic

Manifolds

and

Metailurgic

tant Parallelism

by

Perfect

Tearing

and Refurbishment

Thereof.

RAAG

Memoirs, 2,

D-X

(1962),

109-133.

5)

K.

Kondo,

M,

Shimbo

and S.Amari: On the Standpoint of Non-Riernannian Plasticity

Theory. RAAG Memoirs, 4, D-XXII

(1968),

205-224.

6) K. Kondo: Monographs en "Plasticity

Theory"

in

_manuscript.

7)

S.

Amari:

A Dualistic Treatment of

Non-Riemannian

Material

Spaces

with a

ment

by

K.

Kondo.

RAAG

Research Notes, No. 125, March 1968.

8)

S,

Sakata;

A

Constructive

Approach

to

Non-Teleparallelism

and Non-Metric

tions of Plastic Material Manifold by Generalized Diakoptical Tearlng, Part

I.

Riemannian Tearing.

RAAG

Research

Notes,

No,

146,

_January

1970.

9)

S.Sakata: A

Constructive

Approach

to

Non-Teleparallelism

and

Non-Metric

tions of Plastic Material Manifold by

Generalized

Diakoptical

Tearing,

Part II.

lerian

Tearing.

RAAG

Research

Notes, No. 149,

January

1970,

10)

M.

Satake;

On

Mechanical

Quantities

in Generalized Continua. Technology Reports,

Tohoku

Univ.,

Vol.

35,

No.

1

(1970),

15-37.

11)

R.A.

Toupin:

Theories of Elasticity with

Couple-Stress,

Arch.

Ration.

Mech.

Anal.,

17

(1964),

85-112.

12)

A.C.

Eringen and

E.S,

Suhubi:

Nonlinear

Theory

of

Simple

Micro-Elastic

Solids-I.

Int.

J.

Engng.

Sci.,

2,

(1964),

189-203.

13)

E.

Krtiner

(ed,):

Mechanics of

Generalized

Continua.

Proceedings of the

Symposium

on the Generalized Cosserat Continuum Theory of Dislocations with

plications,Freudenstadt und

Stuttgart

(Germany),

1967.

Springer

Verlag,

1968,

14)

J.A,

Schouten:

Ricci-Calculus, 2nd ed.,

Springer

Verlag,

1959,

,

{