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Contributed by W. F. Ames.*Corresponding author. Tel.: #886 2 2366 1931; fax: #886

2 2362 2975; e-mail: msvlab@msvlab.ce.ntu.edu.tw.

International Journal of Non-Linear Mechanics 34 (1999) 279288

Internal symmetry in bilinear elastoplasticity

Hong-Ki Hong* and Chein-Shan LiuDepartment of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan

Received 1 February 1998

Abstract

Internal symmetry of a constitutive model of bilinear elastoplasticity (i.e. linear elasticity combined with linear kinematichardeningsoftening plasticity) is investigated. First the model is analyzed and synthesized so that a two-phase two-stagelinear representation of the constitutive model is obtained. The underlying structure of the representation is found to beMinkowski spacetime, in which the augmented active states admit of a Lorentz group of transformations in the on phase.The kinematic rule of the model renders the transformation group inhomogeneous, resulting in a larger grouptheproper orthochronous Poincare group. ( 1998 Elsevier Science Ltd. All rights reserved.

Keywords: bilinear elastoplasticity, internal symmetry, Minkowski spacetime, Poincare& group

1. Introduction

A constitutive law is said to possess internal sym-metry if it retains the form of a certain expressionfor constitutive phenomena after some changes inthe point of view from which the phenomena areobserved. The changes or transformations made tothe constitutive law which leave the form un-changed in the effects are naturally linked with theinvariance of conserved quantities. A procedure forparameter estimation (and model identification)with effective utilization of the invariance propertyalong the experimental path will be more capable ofcapturing key features during elastoplastic defor-

mation, and a numerical algorithm which preservessymmetry in time marching will have long-termstability and much improved efficiency and accu-racy. So the issue of internal symmetries in consti-tutive laws of plasticity is not only important in itsown right, but will also find applications to com-putational plasticity and to experimental researchand industrial testing practice.

Recently, Hong and Liu [1] considered a consti-tutive model of perfect elastoplasticity, revealingthat the model possesses two kinds of internal sym-metries, characterized by the translation group(n) in the off (or elastic) phase and by the pro-jective proper orthochronous Lorentz groupPSO

o(n, 1) in the on (or elastoplastic) phase, and

has symmetry switching between the two groupsdictated by the control input. The perfect elastop-lasticity model is the simplest to use, but beinglinearly elastic and perfectly plastic it cannot

0020-7462/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 6 2 ( 9 8 ) 0 0 0 2 9 - 8

Fig. 1. The restoring forcerelative displacement curves ofa unit-directional seismic isolator, showing elastic deformationtaking place at infinitely many two-way line segments whileelastoplastic flow occurring at two one-way lines.

predict the Bauschinger effect which is observedexperimentally in most metals under reversed cyclictesting. The Bauschinger effect refers to a particulartype of directional anisotropy in stress space in-duced by plastic deformationan initial plasticdeformation of one direction reduces the sub-sequent yield strength in the opposite direction. Tomodel the Bauschinger effect, Prager [2] suggesteda linear kinematic hardening rule. Kinematic hard-ening is the hypothesis that the yield hypersurfacetranslates as a rigid body in the stress space duringthe plastic deformation, maintaining the size, shapeand orientation of the yield hypersurface in thesubsequent plastic deforming [3,4].

In this paper let us limit our scope to linearkinematic hardeningsoftening and extend thesymmetry study to a constitutive model of bilinearelastoplasticity, investigating the influence of linearkinematic hardeningsoftening on the structureof the underlying vector space and on the trans-formation group of symmetries. Here bilinear elas-toplasticity is used to abbreviate the combinationof linear elasticity with linear kinematic harden-ingsoftening plasticity.

It may be interesting to name a few applicationareas of the bilinear elastoplasticity model [ofequations (1)(8) below]. It is well known that themodel (of dimension n"5) has been used to de-scribe the deviatoric part of the stress versus strain-rate relation of a three-dimensional elastoplasticmaterial. Less known is that the model (of dimen-sion n"1 or 2 or more) has been used in theanalyses of isolation systems of buildings andequipment in recent years (see, e.g. [5] or [6]),where the model describing the relation betweenthe restoring force and the relative velocity of thetwo end-plates of a seismic isolator was combinedwith the equation of motion to simulate the hys-teretic motion and dissipation capacity of theisolator. Fig. 1 depicts the restoring force-relativedisplacement curves for a uni-directional isolator(the model of dimension n"1).

2. Bilinear elastoplasticity

Since our aim is to reveal symmetry in a constitut-ive model, we need an appropriate setting to make

the presentation clearer and simpler; therefore, wechoose to postulate the constitutive model directlyin Euclidean vector form as in the following:

q5 "q5 e#q5 p, (1)

Q"Qa#Q

b, (2)

Q0 "keq5 e, (3)

QaqR a0"Q0

aq5 p, (4)

Q0b"k

pq5 p, (5)

EQaE)Q0

a, (6)

qR a0*0, (7)

EQaEqR a

0"Q0

aqR a0, (8)

in which the generalized elastic modulus ke, the

generalized kinematic modulus kp, and the general-

ized yield stress Q0a

are the only three propertyconstants needed in the model, with the limitationsthat

ke0, k

p!k

e, Q0

a0.

280 H.-K. Hong, C.-S. Liu / International Journal of Non-Linear Mechanics 34 (1999) 279288

Table 1Generalized kinematic modulus k

pand hardeningsoftening factor b

Property Range Hardening Perfect Softeningconstants plasticity plasticity plasticity

kp

Rkp!k

eRk

p0 k

p"0 0k

p!k

eb 0(b(R 0(b(1 b"1 1(b(R

Fig. 2. Schematic drawing showing the connection of mechan-ical elementstwo springs and one slide-damperfor themodel of bilinear elastoplasticity.

Depending upon the value of kp, the model is ca-

pable of treating hardening, perfect plasticity, andsoftening, as shown in Table 1.

The bold faced symbols q, qe, qp, Q, Qa

andQ

bare the (Euclidean) (column) vectors of general-

ized strain, generalized elastic strain, generalizedplastic strain, generalized stress, generalized activestress and generalized back stress, respectively, allwith n components, whereas qa

0is a scalar called the

equivalent generalized plastic strain, with Q0aqR a0

be-ing the (specific) power of dissipation. The general-ized strain vector q and the generalized strain ratevector q5 are assumed to be so small that no accountis taken of the geometric non-linearity effect (in-cluding spinning) and the rate effect as well as theinertia effect.

Here the norm of a vector, sayQ"col(Q1, Q2, . . . , Qn)"col(Q

1, Q

2, . . . , Q

n), in

n-dimensional Euclidean space endowed with theEuclidean metric I

nis denoted by EQE:"JQ5Q"

J+ni/1

QiQi, where a superscript t indicates the

transpose and Inis the identity tensor of order n. All

q, qe, qp, Q, Qa, Q

band qa

0are functions of one and

the same independent variable, which in most casesis taken either as the ordinary time or as the arclength of a control path; however, for convenience,the independent variable no matter what it is willbe simply called (the external) time and given thesymbol t. A superimposed dot denotes differenti-ation with respect to time, that is d/dt.

The mechanical-element model displayed in Fig.2 may help explain the meanings of Eqs. (1)(8).It is easy to comprehend and appreciate Eqs.(1)(7): Eq. (1) decomposes the generalized strainrate q5 into an elastic part and a plastic part; Eq. (2)decomposes the generalized stress Q into an activepart and a back (or kinematic or translational) part;Eq. (3) gives a linear law for the elastic part; Eq. (4)is an associated plastic-flow rule; Eq. (5) is a

linear kinematic hardeningsoftening rule (oftenknown as Pragers rule); Eq. (6) specifies an admiss-ible range of generalized active stresses (thus,EQ

aE"Q0

ais the yield hypersphere with center

Qb); and Eq. (7) or Q0

aqR a0*0 requires the

non-negativity of the (specific) power of dissipation.But Eq. (8) may need more explanations: With theaid of the two inequalities (6) and (7), it simplyrequires qR a

0be frozen if EQ

aE(Q0

a, so that qR a

0"0

drastically reduces Eqs. (1), (3) and (4) to Q0 "keq5 ,

and Eq. (5) to Q0b"0. The complementary trios

H.-K. Hong, C.-S. Liu / International Journal of Non-Linear Mechanics 34 (1999) 279288 281

(6)(8) furnish the model with two phases, as toe analyzed in the next section.

3. Two phases

Since once dt is factored out the differentialequations (1), (3)(5), (7) and (8) become incrementalequations independent of time, the flow modelrepresented by axioms (1)(8) restrict itself to time-independent elastoplasticity; therefore, we are most-ly concerned with paths rather than histories in thepresent paper. We require a path as a continuouscurve with piecewise continuous tangent vectors.From Eqs. (4), (7) and (8) and Q0

a0, it is not

difficult to prove that

qR a0"Eq5 pE, (9)

which indicates that qa0

is the arc length of a path inthe generalized plastic strain space.

From Eqs. (1)(3) and (5) it follows that

Q0a#k

pq5 p"k

e(q5 !q5 p). (10)

Inserting Eq. (4) into Eq. (10) we have

Q0a#ke#kp

Q0a

qR a0Q

a"k

eq5 , (11)

or

d

dt(X0

aQ

a)"k

eX0

aq5 , (12)

where

X0a:"expA

qa0

bqyB (13)

with the generalized yield strain

qy:"Q0a

ke

(14)

and the hardeningsoftening factor

b :" keke#k

p

. (15)

The ranges of values of b as well as kpare shown in

Table 1.

The inner product of Qa

with Eq. (11) is

(Qa)5Q0

a#ke#kp

Q0a

qR a0(Q

a)5Q

a"k

e(Q

a)5q5 , (16)

which, due to the constancy of Q0a, asserts the state-

ment

EQaE"Q0

aNb (Q

a)5q5 "Q0

aqR a0. (17)

Since Q0a0 and b0, the statement

EQaE"Q0

aNM(Q

a)5q5 08q5 a

00N (18)

is true. Thus

MEQaE"Q0

aand (Q

a)5q5 0NNqR a

00. (19)

On the other hand, if qR a00, axiom (8) assures

EQaE"Q0

a, which together with Eq. (18) asserts

that

qR a00NMEQ

aE"Q0

aand (Q

a)5q5 0N. (20)

Statements (19) and (20) tell us that the yield condi-tion EQ

aE"Q0

aand the straining condition

(Qa)5q5 0 are sufficient and necessary for plastic

irreversibility qR a00.

In view of Eqs. (6) and (7) and of Eqs. (9) and (17),statements (19) and (20) are logically equivalent tothe following onoff switching criteria for themechanism of plasticity:

qR a0"Eq5 pE"

GbQ0

a

(Qa)5q5 0 if EQ

aE"Q0

aand (Q

a)5q5 0,

0 if EQaE(Q0

aor (Q

a)5q5 )0.

(21)

Based on the criteria and the complementarytrios (6)(8), the model of elastoplasticity has twophases (and just two phases): the ON phase inwhich qR a

00 and EQ

aE"Q0

aand the OFF phase in

which qR a0"0 and EQ

aE)Q0

a. In the on phase the

mechanism of plasticity is on so that the modelexhibits elastoplastic behavior, which is irrevers-ible, while in the off phase the mechanism of plastic-ity is off so that the model responds elastically andreversibly.

282 H.-K. Hong, C.-S. Liu / International Journal of Non-Linear Mechanics 34 (1999) 279288

1Recall that Eq. (12) is obtained by combining Eqs. (1)(5).

4. Homogeneous coordinates

Let us normalize the generalized active stress Qa,

the generalized back stress Qb, and the generalized

stress Q with respect to the generalized yield stressQ0

a, and then consider their homogeneous coordi-

nates:

Xa"C

Xsa

X0aD"

X1a

X2a

F

Xna

X0a

:"expA

qa0

bqyB

Q0a

Q1a

Q2a

F

Qna

Q0a

"expA

qa0

bqyB

Q0a

CQ

aQ0

aD, (22)

Xb"col(X1

b, X2

b, . . . , Xn

b, X0

b) :"C

Qb

Q0a

0 D , (23)X"col(X1, X2, . . . , Xn, X0) :"X

a#X

b

"CX0

a

Qa

Q0a

#QbQ0

a

X0a D . (24)

Here and henceforth the index s refers to the total-ity of the internal space coordinates, while theindex 0 refers to the internal time coordinate.Together the homogeneous coordinates will be util-ized to describe the intrinsic properties of the con-stitutive model in its internal spacetime.

For convenience Xa, X

b, and X are thus deemed

as (n#1)-dimensional vectors, Xabeing called the

augmented active state vector, Xb

the augmentedback state vector, X the augmented state vector.Then the constitutive model postulated in the statespace of (Q1, Q2, . . . , Qn) may be translated intoa model in the augmented state space of(X1, X2, . . . , Xn, X0), thus Eqs. (2), (12),1 (8),

(6), (7) and (5) become successively

X"Xa#X

b, (25)

CIn

0n]1

01]n

(Xa)5gX

aD X0 a"

1

qyC0n]n

q5

01]n

0DXa, (26)(X

a)5gX

a)0, (27)

XQ 0a*0, (28)

X0b"(1!b) XQ

0a

(X0a)2 C

Xsa

0 D, (29)in terms of the Minkowski metric

g"Cgss

gs0

g0s

g00D"C

In

0n]1

01]n

!1D . (30)The (n#1)-dimensional vector space of augmentedstate X endowed with the Minkowski metric g isreferred to as Minkowski spacetime Mn`1.

Regarding Eqs. (6) and (27), we may further dis-tinguish two correspondences:

EQaE"Q0

a8(X

a)5gX

a"0, (31)

EQaE(Q0

a8(X

a)5gX

a(0. (32)

That is, a generalized stress state Q on the yieldhypersphere EQ!Q

bE"Q0

awith center at Q

bin

the state space of (Q1, Q2, . . . , Qn) corresponds toan augmented state X on the right circular coneMX D (X!X

b)5g (X!X

b)"0N emanating from the

event point Xb

in Minkowski spacetime of(X1, X2, . . . , Xn, X0), henceforth abbreviated tothe cone, whereas a Q within the yield hyperspherecorresponds to an X in the interior MX D (X!X

b)5

g(X!Xb)(0N of the cone. The exterior

MX D (X!Xb)5g (X!X

b)0N of the cone is unin-

habitable since EQ!QbEQ0

ais forbidden ac-

cording to axiom (6). Even though it admits aninfinite number of Riemannian metrics, the yieldhypersphere Sn~1 in the state space of(Q1, Q2, . . . , Qn) does not admit a Minkowskianmetric, nor does the cylinder in the enlarged spaceof (Q1, Q2, . . . , Qn, X0). It is the cone in the aug-mented state space of (X1, X2, . . . , Xn, X0) whichadmits the Minkowski metric.

Taking the Euclidean inner product of Eq.(12) with Q

aX0

a/(Q0

a)2, substituting Eq. (22), and

H.-K. Hong, C.-S. Liu / International Journal of Non-Linear Mechanics 34 (1999) 279288 283

considering the positivity of Q0a, k

eand X0

a,

we have

d

dt[(Xs)5g

ssXs]"(X0a )2ke

(Q0a)2

(Qa)5q5 ,

and, therefore,

(Qa)5q5 08 d

dt[(Xs)5g

ssXs]0, (33)

(Qa)5q5 )08 d

dt[(Xs)5g

ssXs])0. (34)

Hence in the augmented state space what corres-ponds to the yield condition EQ

aE"Q0

ais the cone

condition (Xa)5gX

a"0 and what corresponds to

the straining condition (Qa)5q5 0 is the growing

internal space radial coordinate condition (d/dt)[(Xs)5g

ssXs]0.

5. Two-phase two-stage linear representation

In view of Eqs. (22) and (31)(34), the onoffswitching criteria (21) become

XQ 0a"

G(Xs

a)5

q5qy

0

if (Xa)5gX

a"0 and d

dt[(Xs)5g

ssXs]0,

0

if (Xa)5gX

a(0 or d

dt[(Xs)5g

ssXs])0,

(35)

in the augmented state space. Expressing Eq. (12)in terms of the homogeneous coordinates (22)and arranging them and Eq. (35) together inmatrix form, we obtain a linear system for aug-mented active states

X0a"AX

a(36)

with the control tensor

A :"1qyC0n]nq5 5

q50D

if (Xa)5gX

a"0 and...