Mechanics of plasticity for finite strains : an engineering approach

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Mechanics of plasticity for finite strains : an engineering

approach

Citation for published version (APA):

Veenstra, P. C., & Mot, E. (1967). Mechanics of plasticity for finite strains : an engineering approach. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0173). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1967

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techntsche hogeschool eindhoven

laboratorlum voor mechanl.che technologie en werkploot.techniek

blz.O van 1,blz.

rapport nr.01

7'

rapport van de .ectle:

Werkplaatstechniek

. tltel:

Mechanios of Plaeticit7 tor ttDite straina,

AD Bagilleerug Approaoh

codering: P1,

P 6a

auteur(.):

_Prot.dr.

_p.e.

_Veenetra

Ir. B. Mot

trefwoord:

flaaticiteit••

mechanUa

.ectielelcler:-hoogl.,aar:

Prot

.dr.

p.e.

Veell.tra

In this artiole a model tor tinite plastic

straining of a straiDhardell1ng material i.

pre.ell-ted. Inetead of the constitutive equations for

finite shears· the following h7Pothe.1a

18

usedl

The

principal directiol18 of stress and straill are

.c91Pcident

1B

techllical processes. This h7pothes18

18

a teohnical approxiaation of the exact theore.

stating that the principal directiol1s of stre.s

cotDcid. with tho•• of .trd.nrat•• This procedure

-proves to 71.1d reasonable re.ults which are both

slJl~er to

a ttain

~d

ph781cal17

INch

olearer

and

f

lIOl'e o.bv1oWl thaD the introduotion of t1D1t. shear

ooapo.e.u

. f • •

train t ....

I'.

" . . .

..

datum: 7

april

1967 --aantal bll, 13 g••chikt yoor publicotie in:

!International

~ournal

of .,

r.c.ohaD1oa1

Isoience.

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- 1 ~

MECHANICS OF PLASTICITY FOR FINITE STRAINS, An Engineering Approach •

P.C. Veenstra E. Mot. ••

•

Summary.

In this article a model for finite plastic straining of a -etrainhardening material is presented. Instead of the

consti-tutive equations for finite shears the following hypothesis is used: The principal directions of stress and strain are coincident in technical processes.

This hypothesis is a technical approximation of the exact

_.

theorem stating that the principal directions of stress coin-cide with those of strainrate.

This procedure proves to yield reasonable results which are both simpler to attain and physically much clearer and more obvious than the introduction of finite shear components of a strain tensor •

• ••

Professor of Production Engineering, Technological Univer-sity Eindhoven, Netherlands •

Research Officer, Department of Production Engineering, Technological University Eindhoven, Netherlands.

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a, b a 1, b1 c 1 1 0 m p, q, u, r r' r 0

x,

y, z 1, 2, ₃

I, II

gamma y delta b

-

b delta b. etha _T'I nu v ksi _t si~a a

-

a tau 't 't ij 2 -List of symbols.

initial length in direction X, Y deformed length in direction X, Y

-material constant (stress if b = 1) deformed length of an element of line initial length of an element of line strain hardening exponent

v, w deformation quantities

deformed length of an arbitrary line

. .

partially deformed length of an arbitrary line initial length of an arbitrary line

coordinate directions (indices) principal directions (indices) stages of strain (indices)

angle determining the shear strain natural (= logarithmic) strains effective natural strain

linear strains

director of an initial arbitrary line director of a deformed arbitrary line

director of a partially deformed arbitrary line tensile stresses

effective stress shear stresses stress tensor

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3

-It is generally recognised that finite straining problems entail complicated mathematical procedures, especially when finite shear is involved. Moreover, the physical meaning of finite shear tensor components is not clear.

For these reasons, the present authors would suggest a diffe-rent approach, which has already proved its value in several cases, e.g. for a model of metal cutting [1J

Though mathematically not quite correct,this approach seems to describe reality pretty well. It is known that in plastici-ty the principal directions of stress and strainrate are coin-cident. Now, in technical processes, a material is deformed efficiently, that is, the ultimate shape is attained from the initial one without unnecessary deformations. As a consequence of this technical fact, the straining will take place approxi-mately in a constant direction during the entire course of the process. In that case, the principal directions of strain and strainrate will be coincident, hence, so will be the principal directions of stress and strain.

Under these assumptions, we can derive a relatively simple closed set of equations without tne introduction of finite shear components.

In order to solve a problem, the following equations are available (a) The equations of equilibrium, in Cartesian coordinates:

~ij,j

=

0 (1)

(b) Transformation equations for principal directions of . stresses, giving values of 01' 02 and 03 and the angles between the principal directions and the coordinate system originally chosen.

(c) The yield condition according to Von Misesl

- 2 2 2

i

°

=

(°1+°2+°3-°1°2-°2°,-°,°1)

(2)

(d) The L'vy-Von Mises' equations for principal directionSI

db

°2+°3

db 1 =

=-

(01- 2 ) ' cyclic

°

in which b 1

=

ln

!

1 0 (~)

(6)

- 4 - .

(e) The incremental effective strain db •

(jl

(d6

1

)2+(~62)2+(d6,)2}

)

t

(t) A deformation equation, giving the amount of strainhardening

-_{CJ •} ₀₆":'tIl ₍₆₎

(g) Two equations stating that the principal directions of stress and strain are coinoiden t (The third equation·. of this kind will be dependent).

Summarising, we have:

number

number of

equations new of new

variables _variables equations

(a) equilibrium eq. a a a 6

,

x y Z "rXY'ryz'rzx

(b) transformation eq. for

princi-°102Q, 3 3

pal directions

(0 ) yield condition

-

a 1 1

(d) L&vy-Von Mises' eq. for princi- 6

16263 4- 3

pal directions

'6

(e) incremental effective strain

-

1

(f) deformation eq.

-

1

(g) principal direction of strain =

2

-

-principal direction of stress =

TOTAL 14-

14-In order to obtain the principal directions of strain an arbi-trary element of line is considered both in its initial and de-formed states and its (natural) strain is calculated. Then dif-ferentiation with respect to its direction yields the directions of extreme elongation, which have been defined as principal di-rections.

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.. 5

-Example.

For a general state of strain in which two shear components are zero, we have the following situation (Fig. 1):

(Fig.1)

Fig. 1. Deformation of ABC D to A~CDo

o 0 0 0 0

A finite, rectangular element ABC D is uniformly deformed

o 0 0 0

to

A

BCD. Tensile strain may occur in directions X, Yand

Z,

o

shear strain only in the XY-plane. Hence, Z is a principal direction. In Fig. 2 we have the same straining process, split up into two stadia:

(I) ABC D ----+ o 0 0 0 (II) A_oB'C'D' ~ AB'C'D' o ABCD o

First we consider the straining of an arbitrary element of line

A E

to

A E'.

In the next stage, A

E'

is strained to

0 0 0 0

A E. The directions of the line element considered are given

o

by the angles 11, t and " , respectively. Its length is rot r' and r respectively.

For the first stage we have

6

=

ln r'/r

r

I 0

and for the second stage

6

=

ln r/r'

r

II

(8)

(10)

6

-. (II)

Fig. 2. Strain of Fig. 1, split up into two stadia.

Hence, the total strain follows from

6 = ln r/r = 6 + 6

r 0 r

I rII

Apparently - even in this non linear case - we are allowed to superimpose logarithmic strains.

We also introduce linear strains:

a₁-a b₁-b

.1_a1

=

a

L\1

= ~

From Fig. 21, we find that

r2 = x2 + y2

0 0 0

(9)

7

-Introducing

v

=

u • (1+6 )2

xI

we find from Fig. 21

(r'/ro, 2 . u sin2" + v cos2"

2 v 2

Sinc e tan "

=-

tan ~ , we find from (13), using

u

sin2"

=

tan2 " (1+tan2,,).1 and cos2"

=

(1+tan2,,).1

1 1 2 1 2

0r I

-

2

In uv +

2

InC1+tan t) -

2

In(u+vtan t)

For the second stage of strain we find

(12) . that

o

_r

•

In r/r' • In (cos~/cosv) II (15) (18) (16) (17) D'D + D'E', 1 2

o

_r

•

-In cosy • -2 In (1+tan Q

II

Hence, using

(9)

and (14) we find

1 1 2

Or

=

2

In uv - In cos v.

2'

In(u+vtan

0

But, since in Fig. 211 we have D'D

=

E'E, or D'E

=

we see that tan

t

=

tan v - tan y.

Introducing tan y

=

w it follows that

o

=- l l n uv - 1n cos v - -21 1n

I

u+v(tanv-w)2} (19)

r 2

We have now expressed b in (u, v, w) which determine the

r

strain of the element and in v which determines the direction of an arbitrary line in this element. Hence, the principal strains are found by requiring that

db

-..!:. = 0

d\v \

from which we derive

(20) .

(21) Introduce

u 1

(10)

8

-The principal directions are then given by tan v 1 • p + (1+p2)t tan v 2 • p - (1+p2)t Substitution in (19) yields

1

₁₀

~+

{

E.+(1+p2)t}~

61 -= 2 .u+v

I

p..w+(1+p2 ) t }2 (24)

And from Fig. 21 it can be seen that

1

5, • -

2 In

uv (24a)

Next, the stresses that caused the strain as given in Fig. 1 are calculated. We will assume that the stresses are built up in such a way that their ratios remain constant during the entire process. Thomsen [2] proved that in this Case eq. (3) may be integrated to the Hencky equations for principal direc-tions, viz.

in which

-

6 • (26)

Numerical example.

We will assume that 'Y

=

60°, !J.

=

0.2

XI We find w

=

1.73; u = 1.44; v = 0.49; p and 51

=

₀₀₅₈ _} • -0.75 .

-6 2 5

=

0.79 6, • 0.17 and!J.

=

-0.3 •. Y1

=

_{1.42; tanv 1}

=

3. 16,

(11)

- 9 ...

(Notice that 6₁+6₂+6,

=

0, but 6 +6 +0_, _x _1. _{z r}~ 0)0_{_}

If c

=

1500 N/mm2 and m

=

0.2 then

(6)

gives ° = 1490 N/mm20 Solving the equations _{(25), we find, using ° 1+°2+°,}

=

0 as an additional condition (since hydrostatic pressure does not in-fluence the plastic straining).

2

01 = 730 N/mm

2 02 =-940 N/mm

0,

=+210 N/mm2

According to Fig. " in which we apply our hypothesis of coin-cidence of principal stress and strain, we find

Fig. 3. Mohr circle and state of strain according to Fig. 1.

tan v 1 = or_xy °1-ox 2 + '(_xy 01-ox = 0y-oz·

These equations yield, either by calculation or construction:

°

=

580 N/mm2

x 2

° ...

_y 790 N/mm

'( =

485 N/mm2

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- 10

-Finally we would emphasise that the reliability of the results produced largely depends both on the accuracy of the experi-mental determination of the materials constants c and m, and on the degree of uniformity of the deformation process consi-dered.

If the deformation process is not uniform, the continuum should be divided into elements which are so small that their deforma-tion is approximately uniform.

In that case the preceeding theory yields an approximate stress distribution.

Literature:

[1] Contribution to the mechanics of machining, P.C. Veenstra, Paper to the CIRP Conference, Liege

1965.

[2J Thomsen, Yang, Kobayashi: Plastic deformation in metal processing.

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Do

y

Co

D

C

---,.---0+---...

..c ..c

8

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y

Do

Eo

Co

•

Of

A

o

_a

a1

E

f

C'

Sf

(I)

x·

y

D'

D

E'

C' E

C

•

_... -- - - --po----...,..---..,...,..---....

x

Ao

8=8'

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• y

.:..---

.

---'