On extremum properties of plasticity conditions

ON EXTREMUM PROPERTIES OF PLASTICITY CONDITIONS (OB EKSTREMAL'NYKB SVOISTVAKB PLASTICHNOSTI) PMM Vo1.24,

h’o.5, D.

1960,

The theory

of

isotropic

rigid-plastic

of

the mathematical

feature

of

the ideally

ly,

an appearance

ulation

of

In [7 a body ible,

]

these

their fy

the

(f)

normally is

isotropic

1

the

isotropic

that

the

an ideally

plastic

which

correspond

well

flow

plasticity

rigid-plastic

for

[l-6

Tresca to

isotropy);

1:

condition

leads

to

equations

which

for

permits

the shear

behavior

1439

general

case

give

of a given

with

it.

a homogeneous, condition.

a theory

qualitative of

satis-

admissible

properties also that

the Tresca

unique

in the

reverse which

law associated

body with

and

equal.

otherwise,

among all

It is also known that any other plasticity condition, cases when such a condition essentially coincides with tion,

are

isotropy).

mechanical We note

is

for

in tension also

condition body

a theory incompress-

bodies

(anormal

group of condition.

a plastic

a true

behavior

(normal

was advanced

d)

velocities

form-

equations.

and compression

isotropic

bodies

and,consequent-

of

isotropic,

the

the

A characteristic

The mathematical

like

sign,

1.

mechanism differential

tension

sequel

most developed

construction

cl

exhibits

bodies

determines

isotropic,

shear

hyperbolic

for

within a given only one true

the [l-6

surfaces.

of

stress

in the

shown that

known [ 8,9

developed

the

proposition

condition

its

ideal,

which

points

them anormally

It was also

istics, flow.

of call

conditions there exists

to

the problem

yield

normal

is

b)

is

plasticity

and slip

reduces

and f) the

reversing

The following

plasticity

lines

1960)

body

of

flow

homogeneous,

We shall

call

plasticity materials

It

a)

i.e.

condition

ideal,

slip

rigid-plastic

sign.

we shall

of

phenomena

is:

compression, and, with

plastic

was considered

which e)

theory

951-955

1

2 July

(Received

1.

pp.

IVLEV

D.

(Voronezh

chapter

USLOVII

ideally

to

be

characterplastic

except for the the Tresca condiresults

which

do

1440

D.D.

Iuleu

not correspond to the qualitative nature of ideal plastic flow. Among others. the von Mises condition belongs to this group, and it leads in the general case to elliptic differential equations. The class of all family of functions

admissible plasticity conditions convex relative to the origin: f (&,

(&I) =

where 3, x3 are the second and third tensor respectively.

is determined

const

invariants

For torsion and plane-strain problems (and for plasticity conditions are reduced to one, namely

zp,= const

by a

(1.1) of the stress

some other

deviator

cases)

all

(1.2)

For plane-stress and three-dimensional general axisymmetric problems there exists 4 priori a possibility of choice of various plasticity conditions within the class (1.1). In [ 7 1 it was assumed that the value of the yield stress in tension and compression for all plasticity conditions (1.1) is the same. Thus. in each specific case an arbitrary constant was dett!rmined, which appears on the right-hand side of (1.1). and thus a class of all admissible plasticity conditions has been determined (Fig. 1). Obviously, if an additional requirement is set forth that a class of admissible plasticity conditions satisfies some other given stress combination, then the relative position of yield surfaces will be different [ 10 1 (Fig. 2). In the case, when for a class of admissible plasticity conditions a value of the yield stress in shear is fixed, then the relative position of yield surfaces is as shown in Fig. 3. For all considered cases of admissible plasticity conditions only one experimental point is common. This fact is explained as follows. For a given ideally plastic body the initiation of yield for a given stress combination must be completely determined. However, this condition, viz. the specification of one experimental point, does not impose any limitations on a class of admissible plasticity conditions. The selection of an experimental point determines only relative positions of the yield surfaces. In [ 7

1

three

theorems have been formulated.

Local theorem. For all given deformation increments of an element of a body the work of external forces attains its minimum for the true

Eztrerur

plasticity

properties

of

plasticity

1441

conditions

condition.

Fig.

1.

Fig.

2.

Fig.

3.

Integral theorem. (I) For given boundary displacements of a body the work of external forces attains its minimum for the true plasticity condition among all admissible plasticity conditions. (II) The work of given external forces attains its maximum for the true plasticity condition among all admissible plasticity conditions.

These theorems were formulated for completely determined configurations of admissible yield surfaces (Fig. 11. If an experimental point in shear is given (Fig. 3). then the word nminimumll has to be replaced by “maximum’, and vice versa. Let us consider in greater detail the case when one experimental point is given. Assume that on the basis of an experiment it was established that plastic flow of an ideal, normally isotropic, incompressible, rigidB plastic body takes place after some combination of the stresses achieves the value represented by point A in Fig. 4. Figure 4 represents one sixth of a deviatoric plane; this is sufficient for further considerations. Assume that OA = 1. All admissible convex Fig. 4. yield polygons are bounded by broken lines BCD. . . and BIC,D,... . Assume now that an angle 8 will be measured angle AOB is fixed and is equal to y. A variable counter-clockwise from OB. Note that angle BOD= l/3 II and angle BOC= l/6 w. We shall assume that the displacement increment vector dr is given and for simplicity we set ( dr 1 = 1. To begin with, let us consider the Tresca plasticity condition represented by the segment BD. The direction of flow for the interior points of BD is 8 = l/6 o, and the other remaining flow directions correspond to the points B and D.

1442

D.D.

It

is

given have

easy

to see

deformation the

that

the

increments

Ivleu

magnitude for

the

of

the work of

Tresca

the

plasticity

stresses

condition

along will

form (1.3)

dA T To the segment ing

segment

corresponds

BICl

corresponds

C,D,

directions

the

correspond

the work for

this

the

flow

to

flow

direction

the vertex

plasticity

8 = l/3

Cl.

condition

direction

8 = 0, n.

Clearly,

and to the

The other

the

remain-

magnitude

of

is (1.4)

The von Uses of

the

plasticity

condition

is

characterized

by the magnitude

work dA,=ude=C

Hencky [ll case

of

Figure

1 established

the von Mises 5 represents

Fig.

for

of the

dA*

is

the

the

stationary

the variation

variation

of

of

dAT

The horizontal

plasticity plasticity

represented

character

of

the work for

the

condition.

Fig.

by df e.

von Mises

Obviously, quirement

plasticity

5.

Furthermore, that

the

(1.5)

the

work with

angle

Fig.

6.

is

line

8.

represented

7.

by a curve

mn corresponds

to the

abc

and

work dAy

condition. condition by the

satisfying

broken

line

the

BIACAID.

minimum work reIn this

case

the

work is

dAl = cos (0 - r) dA1 = cos (l/a x -

A corresponding

curve

in Fig.

(0 < 0 -

5 is

y)

8 < ‘,‘a 4

(l/6 x < dkble.

(1.6)

0 < l/3 n)

If y = 0,

i.e.

if

the

Extremum

properties

of

plasticity

1443

conditions

original experiment is simple tension or compression, then the work diagram is as shown in Fig, 6. In this case the Tresca plasticity condition satisfies the minimum work requirement 17 I. if the original experiment is pure shear (aI = -u2, If y = l/6 R, i.e. 2 CT3- olu* = 0). then the Tresca plasticity condition satisfies the maximumwork requirement among all other admissible plasticity conditions (Fig. 7). The von Uses plasticity condition does not satisfy the extremum condition of work along given displacement increments among all admissible plasticity conditions. The von Mises circle is bounded below by a broken line B,ACA,D,. . . , and from above by a broken line EFG.. . (Fig. 4). In other words, for an arbitrary original experimental point and for arbitrary deformation increments it is possible to find within a class of admissible plasticity conditions two such plasticity conditions for which (1.7) where dAI and dA, are the increments of work of the stresses specially selected plasticity conditions.

for

some

We shall now formulate the above point of view. In the theory of a homogeneous, ideal, incompressible, normally isotropic, rigid-Plastic body there exists only one true plasticity condition (a basic PrOPOsition). There are evidences (such as qualitative behavior of observed flows of metals being close to the ideally plastic behavior; the experiments which demonstrate that the metals with the more pronounced plateau at the yield point behave in closer conformity with the Tresca plasticity condition) which permit the assertion that this true plasticity condition is the Tresca condition. There exist energy criteria which sin@le condition among the class of all admissible as the criterion of the class of admissible initiation of yield in tension, compression

out the Tresca plasticity conditions, if one accepts conditions a known value of or pure shear.

The development of such concepts may permit us to formulate a criterion for the determination of a true relationship between the stresses and deformations within a given group of the mechanical Properties of materials. We would like of this work one determination of tensile stress is

to dwell longer on the work [lo 1. In the first section finds the following: “close attention was given to the a plasticity condition for the case when a limiting given and when the two extremal principles are utilized’.

1444

D.D. Ivlev

For a body along

given

vector

dr

element

of

was given.

simplicity,

1

in [‘I

increments

an extremum of

deformations

(The modulus

was taken

of

this

vector,

Then the

as unity).

the work of

was determined; for

following

the

stresses

in other

words,

the

of

sake

expression

a

was con-

sidered: CL4= ode = 6 cos (u, de) In [lo]

two extremal

body.

One does

there,

not

principles

find,

and moreover

are

however,

in both

cases

(2.1)

mentioned

for

a given

any formulation

of

an extremum of

the

element

these

of

a

principles

same quantity

(2.1)

was considered. When an element and compare

of

deformation

increments

impossible,

however,

and thus

a body

the magnitude

it

is

is

of

for

considered

all

the

admissible

to assign

meaningless

one can assign

the work of

speak

about

case

a vector

(2.1)

plasticity

in a general

to

stresses

given

conditions. the

two local

dc

along

It

stress-vector

is 0.

variational

Prin-

ciples. In the face

of

circle, for

first

section

an admissible then

the work (2.1)

the von Mises

circle

admissible

plasticity

work (2.1)

is

But.this

is

deviatoric is

(2.1)

Note that

lies

to

case

the of

yield

surface

(they

former

Expressions

because

A simple

section

review

The problem

(2.1)

he does of

is

is

the

larger are

of [ 10

1

take

into

not

inaccuracy

of

of [ 7 I would not

that

same expression

if

yield

of

the

the von Mises

circle.

*

1u.N. [ 7

1

experiment

Rabotnov

will

drew the

was in progress.

situation.

is

a given

said:

the

vector

‘D.D.

consideration

of

in the condition

points

other

out

in

experimental

was made. An experiment

however, is dealing only a model. Thus,

to this

the

latter.

procedures”.

experiments.

show more drastically

attention

then the

no such statement the

the

dr.

Ivlev the

experimental

show that

author’s

If

l

an then

plasticity

to the work for

may be executed perfectly. A given experiment, real materials and the ideally plastic body is more accurate

of

circle

may have common points), for

accuracy

surface

the von Uses

or equal

it

sur-

to the

some admissible

equal

the yield

or equal

a more general

another

if

the von Mises

same for

for

that

outside

inside

of

the

out

lies

conversely,

surface

work that

points

and,

greater

a yield

In the second his

is

or equal

pointed

condition

condition

a special

plane

outside

work

less

1 it is

of [lo plasticity

the

deviations

fact

in

with a in the

1958 while

Extreaun

behavior

of real

properties

materials

of

plasticity

1445

conditions

and models.

The use of one experimental point, as has been demonstrated, does not impose any limitations on the class of admissible plasticity conditions, and permits the consideration of the totality of all admissible plasticity conditions. Next in [ 10 1 it was said: “Since from the point of view of the simplest isotropic,ideally plastic theory the selection of a fixed experimental point should not influence the results of the investigations, it can be asserted that in the most general case of specification of an experimental point only the von Mises plasticity condition remains a physical invariant characteristic. n To date,the ideally plastic theory has dealt with the concept of an isotropic body but not with the concept of “an isotropic theory” (this terminology appears twice in [lo I). This terminology should be clarified. If the von Mises plasticity condition were only a physical invariant characteristic. then the question would automatically be resolved in favor of the author of [ 10 1 . It is not so, however, and this has been condition formknown since the time of Tresca. Moreover, any plasticity ulated in terms of the invariants is an invariant characteristic. In the framework of the considered energy criteria there does not exist a single case where the von Yises plasticity condition appears to be preferable to any other condition.

BIBLIOGRAPHY 1.

Sokolovskii, V.V., Teoriia plastichnosti Gostekhteoretizdat, 1950.

2.

Hill,

R.,

Theory

3.

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Nadai, A., and

plasticity). Plastiehnost’

Fracture

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(The

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plastichnosti

(The

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razrsshenie

IIL,

of

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tvezdykh

tel

(Plastic

Flow

1955.

4.

plastichnosti Kachanov, L. Id., Osnooy teorii of Plasticity). Gostekhteoretizdat. 1957.

5.

Prager, V., Teoriia plasticbnosti (The theory of plasticity). book by Prager. V. and Hodge. F.G., The Theory of Ideally Body. IIL, 1956.

(Foundation

of

the

Theory

In the Plastic

1446

6.

D.D.

Ivlev

Khandi, B., Ploskaia zadacha teorii plasticity). Moshinostroenie, Sb. No. 3. 1955.

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(Plane problem of

perev.

in.

i

obz.

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lit.

7.

teorii ideal’noi plastichnosti (On the Ivlev, D. D. , K postroeniiu determination of an ideal theory of plasticity). PMM Vol. 22, No.6, 1958.

8.

plastichnosti i Ivlev. D.D., Ob obshchikh uravneniiakh ideal’noi statiki sypuchei sredy (On general equations of ideal plasticity and statics of pulverulent media). PMM Vol. 22, No. 1. 1958.

9.

opredliaiushchikh plasticheskoe Ivlev, D. D. , 0 sootnosheniakh, techenie pri uslovii plastichnosti Treska i ego obobshcheniiakh (On the relationships determining plastic flow with the Tresca plasticity condition and its generalization). Dokl. Akad. Nauk .SSSR 124, No. 3. 1959.

teorii idealno plasticheskogo tela 10. Shesterikov, S. A., K postroeniiu (On the construction of the theory of an ideally plastic body). PMM Vol. 24, No. 3, 1960. Deformationen und der hierdurch 11. Hencky. H.. Zur Theorie plastischer in Material hervorgerufenen Nachspannungen. 2. angem. Math. und Mech. Vol. 4, No. 4, 1924.

Translated

by

R.M.

E.-I.