Plastic Stress-Strain Relations for Isotropic Metals

Citation preview

PLASTIC STRESS-STRAIN RELATIONS FOR ISOTROPIC METALS

Thesis Submit •'.eci in The Support of The Decree of Doctor of Philosophy to The Victoria University of Manchester by It 1

Cheng Y a n g , B. Eng*

- M a y , 19^0-

ProQuest N um ber: 13841490

All rights reserved INFORMATION TO ALL USERS The q u a lity of this re p ro d u c tio n is d e p e n d e n t u p o n the q u a lity of the co p y su b m itte d . In the unlikely e v e n t that the a u th o r did not send a c o m p le te m a n u scrip t and there are missing p a g e s, these will be n o te d . Also, if m a te ria l had to be re m o v e d , a n o te will in d ic a te the d e le tio n .

uest P roQ uest 13841490 Published by ProQuest LLC(2019). C o p y rig h t of the Dissertation is held by the A uthor. All rights reserved. This work is p ro te cte d a g a in s t u n a u th o rize d co p yin g under Title 17, United States C o d e M icroform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346

UiiTiWSJTfi!> If* ok

*■ ’ ’ "* >iiag^ ■9N; \ I9?« k x .o

pit.fr

LibRARY

The experimental part ox* tlio following investigation was carried out in tho Faculty of Technology of The Victoria University of Manchester* during the oouslon 1947-50*

The

writer wishes to express his thanlco to all members of the staff in the Faculty who helped in various ways in the set­ ting up of the test apparatus and in the carrying out of tho research* D*Sc *

3

Special thanks are duo to Professor H. Yh Bakers

M«I*M*E*, under whose supervision the investigation

was undertaken, to Dr* J* II* Lamble, Moling. , Ph. M.IoK.A.,

in whose Laboratory tho experiments were performs,

and especially to Mr* So Go Gill, iw® So* Tech* his

D. , M* I* IU E* ,

9

A. M® I* M* E*

continual, encouraging help throughout the work*

9

for

CONTENTS Page Chapter I

Introduction

1

Chapter II

The Plastic Behaviour of Isotropic Metals Under Uniaxial Litress

4

Chapter III

The Plastic Behaviour of isotropic Metals Under Compound Utross

III -1

General

III -2

The Criterion of Yield

III -3

The General Pattern of Plastic StressStrain Relations

8 11

IG

III -4

The Strain Hardening Characteristic

83

III -3

Connection between Strain Hardening Characteristic and Tho Pattern of Plastic Stress-fltrain Relations

31

Connection between Yield Criterion and The Pattern of Plastic Stress-Strain Relations

,33

III -6

Chapter IV

Experimental Investigation of Plastic Strcss-Litrain Relations

IV -1

Work of Previous Investigators

36

IV -2

Scope of Present Experimental Work

44

IV -3

Method of Analysis for Experimental Work

49

IV -4

Experimental Details

68

IV -6

Test Results

7y

IV -6

Interpretation of Results

73

Chapter V Appendices Bibliography

Conclusion

03 91

NOTATION S ^ 98|j29&zx

Stress components

0x\,cijy,e;£,exu»eii£*ezx.

Strain components

3,>S,,S^

Principal stresses

c,,e 2 ,e3

Principal strains

2 t ,S 7 , Q r

Tangential 9 a x i a l , and cylinder

radial stresses in

e, ,e7 ,cr

Tangential, axial, and cylinder

radial strains in

Ss,e$

Shear stress and shear

strain in cylinder

e

e xx

*

e * c 22

e

1

p * p 8 e sr 8. P * * c EX8 9Qc t y

if p 5* o'* 21 » Cxy > '• UJ 9 e Zx* G t 9 0 Z 9 C r

vxx

11 0

C>

v3 S o - i (B , *>s2 yij + 3 u ) :5 : * 1 1 (^ yO _ ) ct " •>e i + C-, + e 3 ) ) ( uxx

Q > Q 0 ,Qg

L ia o tic

; pu -

“ ( o

a

a

P la s tic ut * s z * s r ) et

v

c

)

Criterion stresses of yield, or reprer.cntat ive s tr esses

n ,a0 9 dg

H e rr esentat i ve s tr ains

\ Q , VG

Octahedral shear strain and maximum shear r. train

STjC-njO-rr X ix Xu.

Basic invariants of stress tensor

T , dn ,(^£

Functions of C 1 ,d 1L :r.is. 3 ^

Q-- ,hp.

1 unctions of stress components

X, $,0, B

'scalar functions of Q L o d e 1s parameter of strcs; .-ratio L o d c fs parameters of strain-ratio

^

Shear modulus

p

Internal pressure

M

Torque

w

Equivalent weight on torque arm

d*L,d0 ,dtn

Inside, outside and mean diameters of cylinder

r t »r0 9 r m

Inside, outside and mean radii of cylinder

t

Wall thichness of cylinder

a ,b ,c

Dimensional constants of cylinder

a.

Acute angle between direction of maximum or in­ cisal stress with normal plane to axis of cy­ linder

£

Acute angle between direction of maximum prin­ cipal plastic strain increment with normal piano to axis of cylinder

m

Exponential coefficient

A,B,Iw

‘Abbreviations! notations for functions of strer components

f»F

arbitrary functions

h,n

arbitrary constants

Other symbols arc either hexincd in the text 01* are obvious from their context®

vV;

* In tho thesis the following tex\.*e are used in connection with the plastic strains: Resultant strain — nlusLic strain v .Vlastic strains v.here plastic strain is either tiitt. tota! 'elastic str' in a** given by a total strain law or tho integral of the plastic strain increments as given by an incremental strain la;:; (.Refer to section ill - 6 ) .

CIIAPTEH I* IU THO DUG TION The plastic deformation of metals has become an impor­ tant subject for engineering research during the pact few 1c*cades arising out of its two main applications in the in­

dustrial field, namelysa )The shaping or forming processes ox’ metals by 'elas­ tic deformation— e g * , cold drawing, rolling of structural steels, etc* (gojThe use of metals in engineering design at stresses beyond the elastic range— eg*, the autofrettage of hig.. presy sure cylinders, etc* In the first group of industrial a b d i c a t i o n s , the deformation is usually so large that the elastic part of the deformation becomes negligible in the total.

Because of the

great changes in the dimensions of the deformed body, it is often necessary,

in the analytical study of such problems,

to introduce some definition of the strain which is based o 1 the current dimensions of the body under deformation,

in­

stead of tne original ones. In the second group

of plastic hexoihmitions, the total

changes of dimensions of the deform*, d body are usually small and negligible v.iicn compared '.vita the original dimensions. Thus there is no difficulty in specifying the strain as v eil as the stress in the conventional -

1

-

v n y 9 since the

specifications of these quantities whether referred to the original or to the current dimensions would, give no signifi­ cant difference.

However, it should be borne in mind in stu­

dying this group of problems wherein the metals are only deformed moderately into the plastic region*

that the elas­

tic part of the strains would, not be entirely negligible in the total. Henoo due account of them must be taken in the establishing or applying of any stress-s train relationship. It is with this latter range of deformation that the present investigation is chiefly concerned. Although it has been established by many investigators ;1,2,3;

tnat the stress-strain relation in plastic deforma­

tion depends appreciably on the temperature of the metal, well as on the rate of deformstion*

as

it is to be understood,

for the present investigation, that when the deformation takes place at room temperature and at a rate sufficiently slow to allow for the initial settling-down at each loading stage, there exists for most engineerin ; metals a unicue stress-strain relationship which may be regarded primarily / as independent of strain-rate fyi,oJ The anisotropy resulting from t.r. elastic flow of a metal is also to be ignored in the pin-sent investigation, again owing to the small range of defor.mtion being

* Numbers in brackets refer to the bibliography at the end of the thesis. - 3 -

considered.

The neglecting of anisotropy is of course specu­

lative, since the nature of the plastic deformation is chief ly directional* and so anisotropic.

But this factor is gen­

erally considered to he more of a corrective nature than a dominant one, if the metal behaves as an isotropic material in its original state.

Xn other words* if the crystal grains

in the metal are sufficiently small in nice and random in orientation,

then no appreciable directional property^shown

by the outward behaviour of the metallic body of finite dimensions, so long as the randomness of grain orientations is not seriously affected by the deformation. alius, in the present investigation,

the metal under

consideration is to some extent idealised.

It is isotrooic

in the original state, and would retain its isotropy durinn tno plastic deformation, ponse

ihe change of deformation in. res­

oO a cnange oi external lours would be independent of

the rate of l o a d i n *, and would reach an equilibrium state after ihe initial, settling— down,

unis idealisation, as all

ideuli^aoions, c ; •*og be strictly t' ue 1 or any actual metal, but is certainly justified by the comparative simplicity of mathematical analysis which it permits, and by the fact that ouch analyses are in usual cases engineering design.

- 5 -

sufficiently accurate for

CHAPTER II THE PLASTIC BEHAVIOUR OF ISOTROPIC METALS UNDER UNIAXIAL STRESS

when an isotropic metal is subjected to the action of a uniaxial stress* the stress-strain relation can be most con­ veniently described by a stresa-strain c u r v e y such as typi­ cally shown in Fig*

1* in which there is usually no well-

defined elastic limit* Within the range of small

s. i

cleforiuation, there are some well-established facts which may be regarded as a primary description of the behaviour o

p"

of such a system*

p' F\G I

These ex­

perimental facts are enumera-

t eC. a g the folio w iiig :(a)The elastic modulus of the deformed metal as repre­ sented by the slope of the unloading line P R U (fig. 1 ) iro.._

ew i.i c

practically

on the stress— a c m in curve remains

one same as that of the- original^ (Y,baJ

{ z ) »hen the deformed metal ij reloaded after being un­

loaded from f, it will behave elastically with tae srune modulus until the saint ? is approximately reached again*

After this, it will deform plastically along

the some s t r o s s—s train curve of u

original metal as

if under an uninterrupted test*(9910a) (c)For moot engineering iaetalo9 the stress-otrain curve of a continuous uniaxial loading will show £, as a mo­ notonously increasing function of e ^ l Q b ) *

then this

is viewed together with (b ) 9 it would be seen that the elastic limit (or the criterion stress of yield; of a metal in the deformed state v/ili in general increase with the amount of plastic deformation it had under­ gone before it was unloaded*

This phenomenon is common­

ly known as strain hardening* (a)The permanent change of volume is approximately zero (1 1 )— This fact does not seem to have been very extensively investigated by workers in this field, but has often been accepted by various author!ties( 9 ,1 2 a; as an a priori postulate, from perhaps the experience of the. incompressibility of most liquids as usually accepted in classical fluid dynamics.

Although some

investigators have found an appreciable deviation from this postulate in one plastic deformation of some so­ lids (1:5,1'-;, tiie results obtained so far have not seemed, to be sufficiently conclusive for a reneral con­ tradiction to this postulate for engineering metals. Hence in tno present investiy at ion,

oiie volume change

of metals under deformation is s till regarded as pri­ marily

elastic, and will tisus vanisi. vdieii tiie load is

r£m«jv«:d. (e)For most isotropic inetals* the stress-otrain curve for pure tension is approximately identical with that for pure compression. (81)* 15) It ought to be mentioned* however* that in the enumera­ tion above, some secondary behaviour of plastically deformed metals, such as the hysteresis loop* the Bauschinger effect, etc* have been ignored.

These secondary effects have general­

ly oeen considered as play’ing much less important roles in static cases thn.11 in dynamic ones.

In tiie former* reversal V

of stresses seldom takes place* and even when it does, the difficulty could often be tuckhed by treating the problem in reversed stresses as if on a new metal. In accordance with the enumerated facts, tia.i i.ole to split., die resultant strain two p a r t s : the elastic part plastic part e " = oP".

it is thus t ius-

at any stage P into

e,f- 3, / K = P fP"

(Fig.

1) , and the

These definitions are consistent with

the conception tnuo the elastic strain is recoverable

while

the plastic strain is a permanent set, as can, oe visualised from Fir.

1.

As for the two lateral strains

o, and

the same conception of splitting cun >>.• -onlied,

thu:

.o

e p

e, = c»li + e2 e^W= ■)

=3 uonsx»«

^ ^ ^ 3T J

And, according to the Guest-Mohr theory, vt — «»G — Const,

v

— 3^)

( G, > Sz > 3^ )

= Const, t* ^ WhereT is the greatest root of the equation: T 3 - oOi't3 + S o ^ T - U in v.hich,

2X^ “ ^»j Q

ic« would cienend on the

particular Get of suresc-ratios eharaeterictic of each simple loading. Hence* the incremental law (eq. &e -” = k

■becomes*

J

and therefore,

J

ey

Xt^) Q = k.j ^ U X'(^)£q

3.12>

Q

While the total law (eq. Cij “ =

(3.15a)

3.12) becomeo* ky 4 T7(Q)

.............. (3.12a)

If then the predictions by both laws were based on the same data from lues of

another simple loading, i.e., if the va­

e-j "as given by ecu.

(3.15a) and (3.12a) were

made identical to one another for a certain set of then the functions

%'(Q)

and[\^*)

k ;j ,

would be related by

the identity, h,j V'd Z(Q) oti 5 kjj Q T(Qj or,

1 mi 1(Q) ^ '-'o

^ F( h

.............. (3.15) ......(3.17) '

Vvith identity ^5.1 7 ), which is independent of the par­ ticular values of loadings,

k- , it is seen that for all simple

the predictions by the incremental law will those by a suitably constructed total

be identical law, v.ith X U ;

and I (**)

derived from a some simole

te s t.

I:- ehouln oe noted that, in the above discussion for both types of ctress-strain laws, (i)Ihe resultant strain at an,’ stage of deformation is the sum o.i tue plastic strain plastic

is ^iven by the

tress-strain laws mentioned aoove and the

- 19 -

elastic strain as given by the generalised Hooke's law. The complete forma of stress-strafn laws inc­ luding the elastic strains would then be: For the total strain types e X X 12 ^

e XLj = g

^

““

^ I ( Q)

+ F('s») O x^

XI » • • •

)

•••

etc.

(.5.18)

For the incremental strain type: ^ ex.x^ exy

"*

~ ^ ^ £ 2.

+ ^ xeP

...(3 . 3 4 )

.. •. (3.35)

As in the case for whe principal axes of strain® the value of V as defined by eq® (3*35) from the total plastic strains would again be dependent on the history of loading® if the incremental strees-strain lav/ is to be considered*

In order to isolate this effect of load­

ing history, it is again desirable to define a strainratio parameter

from the strain increments®

P =lef'-lef " 1

i.e.,

>S q ">S g \ " ) ..... (3. 3G)

The distributional relationship is then specified by a function

V = f(p)„

This relationship can be

reasonably expected to exist, independent of tho history of deformation# For the Cases of tensile,compressive or torsion tests, isotropy alone would require that V - P . It has also been noticed that for isotropic viscous fluids, the sLrs.i-u—ratio parameter I'' is equal to

p

under any

general coxiipound stress^ nen.ee tneorotical investiga­ tors on the plastic deformation of isotropic metals often

assume that V =s p.

can bo taken as the general distri­

butional relationship.

The relation V = p.

has been

proposed in various f o r m s , many of which imply at the same time the coincidence of principal directions of stress and of incremental strain.

Some of the

impor­

tant form3 ares ( «) _ § Z L =j

=,

£0 zx

=

3z?, .

£ * £ ?■ f e ' J

)

..... (,5.42)

The constants in the above definitions v/ere chosen to .ve £q 0 the possible physical interpretation as tfe maximum lear strain increment in the octahedral plane referred to le current principal axes of strain increment

and SqG as

le maximum shear strain increment at the locality.

These

jpresentative strain increments, o q 0 and (TqG $, as given by is* (3.41), (3.43) 9 would be uniquely defined at any given late of stress, independent of the path through which this late has been attained. Moreover, they are both invariant I dictions of the plastic strain increments with respect to ► rotation of coordinate axes. This can be shown in a manir similar to that discussed on p. 15. Hence they are suitable >r the use as a scalar representation of the amount of chani of deformation in the isotropic metal at any stage of loadig. The representative strains, q c and qG , as obtained by degrating eqe.

(5.41) and (3.43), would then be scalar quali­

ties representing the total amount of deformation the metal s undergone.

They would not be single-valued functions of - 29 -

i

\

e final state of strain, "but v/ould depend 011 the history of raining started from the "beginning of plasticity.

In general

ses of deformation, especially those wherein the principal as of plastic strain are rotating in the metal, the quantis­ es

qo and

a o test were found to be related by the equation

O where

(4.11) X 0 = value oi

i ,0 w hich is kept const

For a constant -

test* the load components are

related by the equations:

( d - c )ap° * 4aaM2 =s 4K&

9

for

p4-=r*Ll

and,

/^c)KG p=4KGa 9

where

K G = value of

for

u

-

J

(4o 12)

Q G which is kept constant in the

test* Thus, for any given value of ing value of (2)

p

M , the correspond­

was calculated®

E v a l u a t i o n of s t r a i n v a r i a b l e s

The total strains

y e2 and es were directly

m e a s u r e d b y instruments described in the next section.

The procedure for the evaluation of the strain variab­ les for tests (A), (B)p (g)

v^ere similar.

It will hence

oe described by taking test group (A), i.e., the cons­ tant i n t e r n a l pressure tests 9 as an example: ( a ) The slopes

, --1^ , £&£ were first evaluated, M dw Om from tne measured strains using the formula of numeri­ cal diff ere vitiation (61): • % ; , * i f c ( 6s“" -e*-‘ • » « - * «

>

....... (4.13)

where

ax

(b)lhe

slopes of the p l a s t i c s t r ai ns w i t h r e s p e c t to

U

= x k+, - x k

w er e then evaluated by:

-

bo

-

(Since and e 2 9 were constant in tills case) de s'* dM

3.0s dM

dM

(4.14)

des dM

__

d9s dM

dosf da

dM

a 0

(c)The principal plastic straixi increments v/ere cal­ culated forms .te ll

=1

dM

( t e t“

£ v dM

de 2.jU __1 { de-^ dM dtr^ - “ de 3,?

I

dM

v

.

did

jdez^\

/T T ^

. doz \ * drr)

^

dM

1 [/ det * 2 Iras—

dot'* _t_ do W1 \

cm

f/do_t^

dM

du

'

'

_ v dM

de*V+

' J 1

d o * 11 /des\al2 (4.15) dir )vCatrJ J

(Since

e/^ei^e’ JsO)

3

(d)By virtue of axial symmetry of the system* it is obvious ’ V d i i au .... (4o FOa)

fr =

, .... ‘4-aob'

These derivatives were directly calculated by

- 55 -

I

substitution from eqs* (4# 15). (Ehe first few un­ measured values were obtained by extrapolation from later values*) The numerical integration in eq* (4.13) was carried out by the traposoidal rule* The value of q(0> was calculated directly from the measured total plastic strains (ioCop total strains less the elastic strains) at that point, since the loading before that point was a simple one* For test group (B), the derivatives were with res­ pect to

po

For test group (C), either the derivatives with respect to

M

or to

p could be used for the evalua­

tion of the strain variables* From these calculations, the relationships p-P, &o“ Qo and

a-3,

Q 0-U g were found*

( b) Prediction of strains by three types of stres:?strain laws ..: The choice between presentative stress

Q

Q c and

Qc

for the re­

to be used in the scalar flow

functions (or the strain hardening functions) involved in all three types of lav/ under consideration was made by inspecting the strain hardening characteristics — q0 and

Qc

Q g -Q g * Since the former curves did appear

for this particular material to agree better than the -

56-

latter in following a conenon trend*

Q0

v/ao therefore

chosen for use in the strain predictions by all types of law.

The choice ooem3 aloe justified by the results 9 as v/ill he explained

of the neutral loading tests in section H - 6

9 p. 63

The predictions wore based on the data found from a tensile test. Thus* if

e 4 - S, bo the tensile stress-

strain curve obtained* the data

o' o{

be obtained by deducting from

- a, could easily the elastic strains

S — - • The flow functions v/ere deduced from S the following formulae: »i