a mathematical modification of two surface model ...

A MATHEMATICAL MODIFICATION OF TWO SURFACE MODEL FORMULATION IN PLASTICITY K. Department

of Agricultural

HASHIGUCHI

Engintyring.

Kyushu

University.

Fukuoka

812. Japan

Abstract-Various constitutive models for the description of the elastoplaslic deformation with an anisotropic hardening and also a transition from the elastic to the normal-yield (fully-plastic) state hate been proposed in the past. Among them the two surface model has bevn widely used for predictions ofmctal or soil deformation behavior. which is regarded as a simplification of the multi surface model (a field of hardening moduli) extended from the kinematic hardening model. The past formulation of this model is not given in a mathematically exact form with generality applicable to materials with not only hardening but also softening behavior. In this paper. a reasonable formulation of the two surface model is given by deriving a mathematical condition which must be satisfied in order that surfaces do not inlerstTt in their relative translation and which will be called ;I “non-intcrrection condition” and by proposing a reasonable measure to represent an approaching dcgrce to the normal-yield stale. This model is extcndcd to the three surface model which enables a smooth elastic pl;l%cticIransition lo bc described. Besides. a loadingcritcrion for hardening,‘softcning marcrials is dcrivcd in a stress formulation. Finally, some comments on past models or f’ormulations arc made comparrd with the prcscnl formulation.

lNTRODUCTlON

An cxtcnsion of the kincmalic hardening model (Edclman 1954; Pragcr. 1956) so as to dcscribc cvcn a deformation thcclastic to the normal-yield (fully-plastic) 1926). has been attempted by Mroz cxtcntlcd mod&

(1967,

from

1969) and lwan (1967) indcpcndently.

Their

arc. however, of complsx form assuming multiple “subyicld (nesting yield)

surfaces” cncirclcd by a “normal-yield

(bounding or limiting)

surface model”. Thcrcaftcr,

a normal-yield

surface and only one “subyield

elastic domain has been formulated Kricg,

1951 ; Ishlinski,

state. which would obey Masing’s rule (Masing.

callctl a “multi

1977:

and Druckcr.

proceeding in the transition

1975 ; Mroz

model employing

(inner yield) surface” enclosing ;L purely

by many workers

PI crl., 1979;

surface”. which has been

based on them, a simplified

Hashiguchi,

(Dafalias

and Popov.

1975, 1976,

198 I) which has been called a “two

surfacc niodcl”. The past constitutivc have: not been formulated l3pccially.

cquations in the framework

of the multi or two surface model

exactly on the basis of mathematical

littlc consideration

has been given to formulating

which regulates a surface so as not to protrude keeps surfaces from intcrscction

Therefore,

even though the

of specialized materials exhibiting

hardening behavior, they give rise to a mathematical contradiction assumed surfaces in the deformation

however.

from an outor larger surcdce. i.e. which

at their rclativc translation.

past equations could analyze the deformation

verifications.

the mathematical condition

only a

on the translation

of

analysis of gcncralizcd material with not only hard-

cning but also softening behavior. In this paper, the two surface model is formulated this formulation translation

the “non-intcrscction

condition”

in mathcmaticully

exact forms. On

of surfaces is dcrivcd, and then the

rule of the subyicld surface is formulated so as to satisfy this condition.

Besides,

assuming a rcnsonahlc mcasurc to dcscribc an approaching dcgrcc to the normal-yield state. the plastic strain rate: is li,rmulatcd so ;LS not to violate the non-intcrscction condition. Further. this model is cxtondcd so as to describe the smooth elastic-plastic transition by incorporating the third surface. named a “subloading surface”, which exists within the subyicld surface and is gcomctrically similar to it. This model is called a “three surface model”. which is regarded as a combination of the two surface model and the subloading surface modal (Hashiguchi. 1980). Besides. the loading criterion for hardcninglsoftening materials is derived within the framework of the stress space formulation. S&S2,:,3-*

987

K.

f(i)

Stress

HASHICLCHI

= F

Strain

Fig. I. The normal-yield

BASIC

CONSTITUTIVE

A typical stress/strain Fig.

I. First,

state illustrated as an envelope curve of reloading state

EQUATIONS

IN THE

curve of elastoplastic

one assumes that the normal-yield

NORMAL-YIELD

STATE

materials is schematically illustrated

in

state shown by the envelope curve of

reloading curves is described by

f(d)

(1)

- F(K) = 0

where one sets

(3

CtZtF--i.

The second-order tensor o is a stress. and the scalar K and the second-order tensor ai arc paramctcrs to dcscribc. rcspcctivcly, the expansion/contraction

and the translation

of the

surface. For simplicity, one assumes that the surface dcscribcd by cqn (I), called a “normalyield surface”. expands/contracts retaining iI gcomctrical similarity fore. function /’ is to bc a homogcncous function

in a stress space. Thcrc-

of arguments. Then,

Ict the dcgrcc of /

be drnotcd by II. Let k whcrc a superposed dot designates a nmtcrial time dcrivativc bc a function of some plastic internal

state variables and the plastic strain

rate C in dcgrcc one by the

dimensional invariance of time. Further,

let k besgivcn as

in accordance with the previous paper (Hashiguchi.

it

G

198 I). where (4)

tr ip

A (20) and B (20) arc scalar functions of K and 5, and the notation rcprcsent the magnitude. By differentiating eqn (I) and substituting the relation

(‘J-

nl;

_

= ---- -- n Tca tr (riti)

11 11is used to

(5)

(6) one has the consistency condition

A mathematical modification of two surface model formulation in plasticity

989

(7) which is given as

tr[s{&Aw?tr (P&y_:6}]=0

(8)

by incorporating eqn (3). Here, one assumes that the elastic deformation only can proceed when the stress exists within the normal-yield surface and that the elastic property is not affected by the plastic deformation. Then. in accordance with Drucker’s interpretation (Drucker. 1951). one incorporates the associated flow rule ip = i/j where j? is a positive proportionality strain rate is given as

(9)

factor. By substituting eqn (9) into eqn (8) the plastic

dP=--_--

tr (rib) n

whore

(11)

3 is a scalar function of stress, plastic internal state variables and ri in degree one, which is given by

f(&, Fig. 2. Norn~al-yield

= F and subyield surfaces.

K. HASHlGCCHl .

r;-= k;,E:.

(13)

similar to the One introduces the subyield surface (Fig. 2) which is geometrically normal-yield surface and translates within the normal-yield sat-faze. Here. assume that the current stress exists on or within the subyield surfslce and that the elastoplastic deformation can proceed when it exists on the subyield surface, but only the elrtstic deformation can proceed when it exists within that surface. Then, fet the subyield surface be described by

f’(a)-SF(K)

= 0

(14)

where one sets

r (0 -=cr < I) is :I material constant and the second-order tensor 32is a parameter to describe a translation of the surrttcc. Now. let tho conjugate point on the norltt~~-yield surface having the same outer-norrn~~~ direction as that of the subyield surface at thu current stress Q be denoted by C, (Fig. 2). Namely, it holds that

a

Ify

_

=

n

t171

whcrc

kfcrc, assume that translation rule 13) of the normal-yield surfxe holds ako in the subyield state, provided that the stress in eyn (3) is regarded as the conjugate stress 6,. Hence, noting relation ( 16), one has

Now, one considers the translation rule of the subyield surface, i.e. &. Since the subyicld surf&e must exist within thu normal-yield surface, it must hold that

where

A ma~emat~cat modification of two surface mode1Formulation in plasticity

991

ri.,

tctting a, denote the intcrscctin~ point of the subyicld surfircc and the half line starting the point d and passing through the point a’ in the stress sprtcc (Fig. 2). Expression (33) cun bc written 3s



(50)

0

one has

;i =

Hi

(51)

which is nonncgativc. and thcrcforc cxprcssion (39) or (41) is :dways satisfied. Eventually,

the plastic strain rate is given from cqns (3X), (44) and (50) as follows :

ip=

tr (G) _ _ D+ff n.

Here, note that in the hardening state satisfying

(52)

tr (id) > 0 and 0 > 0, the smaller the

parameter h the larger the mngnitudc of plastic strain rirtc is. From this one knows that the function

I/ for the general material with a hardening behavior should bc ;I monotonically

increasing function of h. The above modification

based on the non-intersection

the multi surface model formulation

(Mroz.

TllKEE

condition

is rcquircd also for

1967).

SURI:ACE

MODEL

The two surface model brings about ;rn abrupt change of the strain ratc,!strcss rate relation when loaded from the stress state within the subyiold surface since the interior of this surface is assumed to be ir purely elastic domain. so thitt it cannot dcscribc ir smooth elastic-plastic transition. This trend is rcmarkablc especially when a current stress passes through the contact point of the normal-yield and the subyicld surfaces. As a more fundamental problem, the loading criterion must include the judgcmcnt whether a current stress lies on the subyicld surfitcc or not. i.e. yield condition (14) which Icads to a disitdvantagc in a stress/strain calculation, since it assumes an chrstic domain. Now, one extends the foregoing two surface model so as to express the smooth elasticplastic transition in accordance with the three surface theory (Hashiguchi. 1981) which incorporates a subloading surface (Hirshiguchi. 1980) within the subyield surface. The subloadingsurfacc is the surface that is geometrically similar to the subyield surfitcc with

A mathematical modification of two surface model formulation in plasticity Normal-yield I

995

surface

, Subyteld surface ng surface

f(i)

=

F

Fig. 4. Sarmrtl-yictd. suhyield and suhloadinp surfaces.

respect to i (Fig. 3) and pitsscs always through 3 current stress point not only in 3 loading st;ltc but

illSO

in an unloading state, and thus the loading criterion

ncod not include ;1

judgcmcnt whother a stress lies on a subyicld surface or not. In the subloading surfiicc concept t hc f~~ll~)~~ill~~~IlXillCtCr

is introduced :

R

=

! _/x4 _I .-_ “% r

I I

(53)

F

which designates the ratio of the size of the loading surface to that of the subyicid surface. H~r~in~lft~r

let the conjugate point on the subyieid surface bc denoted by u, and then it

holds that - =(iJR 6s

(54)

t?, = ii

(5%

whcrc (56)

c7, = a,-1

(57)

Diliixcntiation

of cqn (53) &ads to

tr f&)

From eqn (36) and rewriting

= tr [P{~-~i

f+

3~}].

(58)

CYas u,. noting eqn (55), one has

(59)

Substituting

oqn (58) into eqn (59). one obtains

9%

K.

finSHIC;CCHZ

(60)

Here. one assumes

that d, R is nonnegative

in the loading

state and is given by

d,‘R = UI/ip/I where c’ (20)

is a monotonic~Ily

decreasing Lj=O

Further, assume that the associated eqn (61) is rewritten as

function

when

(61) of R, satisfying

the condition

R= I.

(69,)

flow rule holds also for the loading

surface. Then,

k/R = iJ tr (EC’)

(63)

d/R = t/i.

(64)

or

Substituting cqns (60) and (61) into eqn (34) with cqn (35) and eqn (52) with eqn (43) with regarding (7 in equations of the two surface model to bc 6,. one has

(66)

$1’

=

.__.

Df

tr (ki) .._. ._. --_/j f/I+ U tr (r?cs)

(67)

c in fi given by eqn (42) being a current stress. d and h are given by the same equations as eyns (2 I ) and (45). respectively. The two surface model predicts a discontinuous stress rate:‘strain rate relation and is incapabk ofdcscribing II plastic deformation due to the stress change within the subyield surface the interior of which is assumed to bc ;I purely elastic domain. These shortcomings are excluded in the three surface model. On tho other hand, the three surface model predicts an excessive mechanical ratchctting effect, since the center of similarity of the subyield and the norm~ll-yield surfaces is fixed in the center of the subyield surface while an unloading (elastic deformation) proceeds until the stress returns the center of similarity. This shortcoming would bc modified by Ictting the center of similarity translutc within the subyicld surface.

A LOADING

The plastic strain

rate obeying

CRITERION

the associated

ir = hl where

flow rule is generally

given by

(68)

A mathematical

moddication

of

two surface

model formulation

in plasticity

tr (nri) : AE L

997

(69)

as seen in eqns (IO). (52) and (67). In eqns (68) and (69). i is a positive proportionality factor. n a unit outer normal of the loading surface and L a function of stress and some plastic internal state variables. By substituting & = Ed’ d’

=

(70)

C_CQ

(71)

and eqn (68) into eqn (69). one has tr {nE(C-in)]

j’.- -

(77)

L

where C is a strain rate. C’ an elastic strain rate and the fourth-order

tensor E an elastic

modulus. While from eqn (72) one obtains the expression of L by the strain rate instead of the stress rate, Ict it bc dcnotcd by h

(73) It holds that tr (nci) < 0

(74)

in the unloading state. while cxprcssion (74) holds also in the loading state with a softening. Since it holds that 15 = f:ii in the unloading, cxprcssion (74) is rcwrittcn as (75) in the unloading state.

If L+ tr (n/.n) c 0, ;I plastic dcformntion cannot occur for tr (nEti) condition

h > 0 for i” # 0. Further.

only for tr (nf%) = 0. These Fxts deformation

and cxprcssion

cannot occur for tr (nE)

of any direction

can occur that any

ditTerent from a stress rate). On the other

> 0. a plastic deformation

deformation proceeds for tr (n&) direction, it must hold that

(75) lead to the shortcoming

> 0. while real materials can undergo a strain rate

(this point is fundamentally

hand. if L + tr (nEn)

> 0 by the collateral

if L+ tr (nEn) = 0, a plastic deformation

proceeds for tr (nEd)

6 0. Eventually,

f, + tr (nEn)

> 0, while an elastic

in order to describe a strain rate of any

> 0

(76)

while L can take both positive and negative values but tr (nEn) > 0 since tr (nEn) is of the quadratic form and E is a positive dclinitc as known from tr (icEi’) A loading criterion

while constitutivc

= tr (G’) > 0 (Cc# 0).

should bc given as

equations

ip # 0

for

tr (nl%) > 0

dP = 0

for

tr (nE)

with

< 0

a purely elastic domain

(77) require a further

judgemcnt

whcthcr a yield condition (cqn (I) for the classical theory and eqn (14) for the two surface model) is satisfied or not. Equations (77) wcrc shown by Hill (1958) presupposing L > 0, i.e. a hardening material and assumed apriori by Mroz and Zienkiewicz (1984) in a different approach. i.e. a strain space formulation of plastic constitutive equations.

I 0.

the unloading tr (na) < 0 in the state L < 0 also leads to i > 0. Thus.

necessary condition

but not a sufficient condition

for a loading state. Eventually.

be adopted as a quantity to define a loading criterion hardening but also a softening behavior.

COMMENTS

ON PAST MODELS

for materials exhibiting

AND

;’ > 0 is a i cannot not only a

FORMULATIONS

Various models assuming plural surfaces have been proposed in the past : the two, the multi and the infinite surface models. In what follows, some comments on these past models and formulations

are made comparing with the two and the three surface models formulated

in this paper.

Translrrtion rule of .sub_vidrl .wrfim In the multi surface model. so-called a “model of a field of hardening moduli” proposed by Mroz

(1967). the translation

rule of the active subyield surface was assumed N priori as

follows :

h = ri,fi which was used by Mroz

(1969).

Prcvost

(1977.

(1985) for the muiti surface model formulation (I 978). (1985a.

Provost

(IX!),

197% 1982). Faruquc (19X3).

Bruhns

(1984),

Mcdowell

(1985). Chabochc (1986) and Ohno and Kachi (1986) for

the two surface model formulation. normal-yield

(1985) and Gilat

and by Kricg (197.5). Lamba and Sidcbottom

Chabochc and Roussclicr

b). Shaw and Kyriakidcs

the non-hardening

(78)

Equation

(78) coincides with cqn (34) in the cast of

surface. i.e. i; = 0 and IJ!= 0.

Also for the multi surface model formulation,

Mroz

(‘I ul. (1978) adopted, howovcr,

the ditrercnt equation

which was used by Mroz formulation. Further,

(11trl. (1979) and Hashiguchi

Dafalias and Popov (1975, 1976, 1977) incorporated the equation

for the two surface model formulation. proportionality

(198 I) for the two surface model

In cqns (78)-(80).

factors which can bc cxprcsscd explicitly

Ii,, Ii, and ii3 ;irc positive

by substituting

these equations

into the consistency condition (36). whereas they do not examine the positivity of Ii,, Ii2 and /iI in their formulations of plastic strain rate equations. Equations (6X) -(70) do not satisfy gcncrally the non-intersection condition (30).

Mcuwrc

CI/’cqymxdliti~y tiiyrcv la Ihe rlorr~lrrl-~,~ic~ltl .sIuIc

In the formulation of the two surface model. Kricg (1975). Dafalias and Popov (1975) and Mroz ct (11.(1979) assumed the parameter

as a measure to describe the approaching dcgrec to the normal-yield

state.

A mathrmatul

modification

of two surface model formulation

ftii)

Fig. 5. A mcwwc

of approaching

= F

dcprcc to the norm;&yield

As the current stress appronchcs the normal-yield

state.

state. h becomes smaller. On the

other hand, there exists the c;Ise that h’ bccomcs larger inversely illustrated

999

in plasticity

in that approach as

in Fig. 5 in which b = tr (C/l)ti = hF’ “fi.

While h would he it re:lsonithk

mc;tsurc

of

itppro:lch

to the normal-yield

stats,

h’

is

~In~l~cept~Ibt~its that measure.

I>nf:llias

and Popov (1077).

Mroz

ct (11. (1979,

1981) and Mroz

vi~lidity of the idea of i1 vanishing elastic region as the limiting model. in which the subyield surface diminishes

(IWO)

insisted the

case of the two surface

to it point. Further,

they assumed that a

conjugate point on the norm~ll-yi~td surface ties on the extension of the stress rate vector stemming from the current stress point or the center of the norm&yield

surface. This

idea

may be expected to be capable of describing the dependence of the direction of the plastic strain rate on the stress rate, that is. the “pseudo-corner efTcct” and also the smooth clasticplastic transition. For example, consider the softening

state its shown

in Fig. 6. There

Fig. 6. An cnamplc of disproof for the idca of the vanishing elastic domain.

occurs a fatal

K. HASHIGLC-HI

loo0

contradiction that the conjugate point lies on the opposite side of the normal-yield surface the outer-normal direction of which differs entirely from the actual direction of the plastic strain rate 2. predicting the plastic strain rate dP with an extremely large value of b (or 6’). It would be adequate to assume a small subyield surface in order to express some dependence of the direction of plastic strain rate on the stress rate. It would be. however, physically unreasonable that the direction of the plastic strain rate depends mainly on the stress rate but not on the stress state as is predicted by the idea of a vanishing elastic region. This idea describes the stress rate dependence stronger than the corner effect of the yield surface. since it makes the yield surface shrink to a point. Eventually. this idea is not acceptable physically and mathematically at present. Espcinsion rirlt~ The author

of lootlit~g sirrjim~ (Hashiguchi.

1980) assumed

the equation

ii/R =

r/p

(82)

in the previous three surface model. According to this equation. the current stress cannot approach to the subyicld surface but recedes from it in the non-hardening state t < 0. On the other hand. eqn (61) would be applicable to the gcncralized material with not only hardening but also softening behavior.

In the multi surfacc indopcndcntly, the current model needs to mcmori/c rcvcrsc loading products stress passes through the

model which was proposed by Mroz

(1967) and lwan (1967) stress transfers steadily to larger active subyicld surfaces. This aI1 of the assunicd surfaces. Further, a reloading after il partial an abrupt change of the stress ratc,‘strain rate relation when the stress rcvcrscd point bccausc the stress transfers abruptly to the Inrgc subyicld surfacc which was an active suhyicld surface just before the reverse loading took place. Mroz VI rrl. (IOSI). Mroz and Norris (1083) and Mroz and I’ictruszczuk (1983) rclincd the multi surface model mathematically by imaging a licld of an inlinitc number of subyield surfaces the s&s of which range from a point to the normal-yield surface. A salient fcaturc of this formulation is the hypothesis of “stress reversal surface”. This is the surface which was an active subyicld surface just bcforc a reverse loading took place. Then, a new active subyicld surface expands contacting with the stress reversal surface at the stress reversed point. This model, named an “inlinitc surface model”, may be regarded also as a reasonable formalization of the loading surface concept introduced by Greenstreet and Phillips (1973) for artificial graphite. It is a notable improvement of the multi surface model because it needs to mcmorizc only three kinds of surface : ‘in * active subyield surface on which a current stress exists, a normal-yield (bounding or limiting) surface and stress reversal surfaces. In the cast of the cyclic loading with dccrcasing amplitudes, however. many stress reversal surfaces should bc memorized. on which stress rcvcrsal events took place before then. Further, the abrupt change of the hardening modulus in a reloading after a partial revcrsc loading cannot bc avoided also by that model.

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in plasticity

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