A version of endochronic theory of plasticity for describing non-proportional cyclic deformation

002s 7462/93 $6 00 + .oO C 1993 Pergamon Press Ltd

Inr. J. Non-Linew Mrchanics, Vol. 28. No. 2. pp. 267 278. 1993 Pnnred in Great Britain.

A VERSION OF ENDOCHRONIC THEORY OF PLASTICITY FOR DESCRIBING NON-PROPORTIONAL CYCLIC DEFORMATION N. Institute

for Problems

K.

of Strength,

KUCHER

and M. V.

BORODII

Timiryazevskaya street 2, Academy 252014, Kiev, Ukraine

of Sciences of the Ukraine,

Attract-Constitutive equations of the endochronic theory of plasticity for describing nonproportional cyclic deformation of materials are presented. New rules are introduced for characterizing isotropic and kinematic hardening taking into account the non-proportionality parameter of the plastic strain history in a cycle. Basic experiments and calculation methods for specifying constitutive relationships are described. Stress field distribution has been determined in thin-walled tubular specimens under combined action of axial load and torque with reference to various plastic strain paths. A comparison of calculated and experimental data is presented.

1. INTRODUCTION

The necessity of a more accurate prediction of mechanical behaviour of structural elements subject to cyclic loads calls for further development of deformable solid mechanics. This is brought about by both the practical needs and inner Iogic of the continuum mechanics development. Surveys of investigations associated with the formulation of constitutive equations of the present-day theory of plasticity and the analysis of the adequacy of the material real behaviour description under cyclic loading have been presented by Miller Cl], Chaboche [2], Vasin [3], Kadashevich and Mosolov [4]. Appreciable success in predicting complex processes of deformation has been achieved, in particular, within the framework of the endochronic theory of plasticity. According to the theory proposed by Valanis [5], an instantaneous value of the stress tensor depends on the total history of variation of thermodynamic parameters (strain, temperature, etc.). This approach allows one to describe the most general laws of deformation and to predict efficiently the material mechanical behaviour. The endochronic theory made it possible to describe some specific features of material elastoplastic straining under active loading and at unloading, based on simple assumptions and from a unified point of view. These are, for instance, linear and non-linear hardening, a lag in vector and scalar properties of the material, with a kink on the strain path, hysteresis and hysteresis loop stabilization under cyclic loading, inelasticity of unloading, etc. In the initial version of the theory, the length of the total strain path was used as a deformation process measure, However, it was criticized seriously by Sandler [6] and Rivlin [7] due to the important error in the prediction of the stress state for complex loading histories. To eliminate the disadvantages discovered, Valanis [8] proposed to use another measure of intrinsic time although in the latter case, Drucker’s postulates were also violated and unusual (from the standpoint of the classical theory of plasticity) effects of cyclic creep and relaxation occurred. The attempts to avoid those phenomena resulted in the use of Odquist’s parameter as a measure of deformation processes. We should point out one more way in the development of present-day theories of plasticity which do not involve the notion of a yield surface as their basis. It is associated with a more involved representation of both the plasticity functional [9-l 11 and deformation process measure [ 121. In some works by Valanis [ 131, Watanabe and Atluri [ 141, Kucher and Borodii [15], constitutive equations of the endochronic theory of plasticity as applied to the inherent time Contributed

by W. F. Ames. 267

268

N. K.

KM.HFK

and

M. V. BOK(JI)II

measure in the form of Odquist’s parameter were used to analyst cyclic processes of elastoplastic straining. Here, a good agreement of the calculated and experimental data was observed in the investigations of unidimensional cyclic deformation of metals. The possibility of describing tests under non-proportional loading with the loads varying in time was studied, in particular. by Valanis and Jinghong [I 63, Kucher and Borodii [ 171, who pointed out the necessity of modifying the constitutive equations used. The purpose of the present paper is to develop a model of a hardening body for describing non-proportional, cyclic deformation of materials on the basis of one of the versions of the endochronic theory of plasticity. The efficiency of the constitutive equations proposed will be examined with the analysis of straining of thin-walled tubular specimens subject to the action of an axial force and a torque with different paths in two-dimensional plastic strain space. 2. HYPOTHESES

OF THE

MODEL

As it was pointed out by Lamba and Sidebottom 1181. after a certain number of load cycles with specified strain paths. many structural materials reveal a steady-state response when stress distribution does not actually change. Stabilization of the deformation process is also observed for constant plastic strain paths 1191. Hardening materials under strain-controlled cyclic loading reveal an increase in the stress vector range. (The stress vector range is the maximum modulus of the difference in stress vectors per loading cycle.) The stress vector range tends asymptotically to a constant value which is related to the plastic strain path amplitude and shape. With a constant plastic strain amplitude, the minimum hardening is characteristic of uniaxial alternating straining, while the maximum is the characteristic of circular strain paths [ 19, 201. The duration of non-steady-state transient processes depends essentially on the accumulated plastic strain in a load cycle. The longer the plastic strain path length, the longer it takes the material to attain a saturated deformation state [l9]. A coefficient of nonproportionality CD,introduced by McDowell 1211 with reference to two-surface yield theory, can be used as a characteristic of plastic strain cycle shape. To describe the effects of the type of isotropic hardening in endochronic theory of plasticity, the so-called hardening function F which depends on the accumulated plastic strain or on the intrinsic time parameter is used. Basing on the above experimental results, it is assumed in the present paper that the hardening function is related to the intrinsic time parameter 2, plastic strain path amplitude and length in a cycle, and coefficient of nonproportionality 0. However, even with those assumptions, the constitutive equations do not take a complete account of the kinematic hardening for complex cyclic loading histories. For this reason, a function of additional kinematic hardening p(App, @) has been introduced into the constitutive equations, which allows the correlation between the calculated and experimental data to be improved appreciably. In addition, when analysing mechanical behaviour of type 316 steel for prescribed plastic strain paths (cruciform cycles of two types, stellate, square, circular) Tanaka ct a/. 1191 showed that the cycle of each shape (i.e. each magnitude of 0) has a corresponding value of the stress vector amplitude in the saturated state. In this case, the magnitude of the maximum value in cyclic straining along the prescribed path is independent of the preceding loading history. Therefore, after stabilization of the deformation process with a transition to another cyclic loading path, the relation between the stress and plastic strain vectors can be defined by an integral relationship of the hereditary type with a new intrinsic time scale. The initial intrinsic time value for the loading stage considered is assumed to be equal to zero. The influence of the preceding straining history upon the stress-state kinetics is taken into account by assuming the equality of the isotropic hardening function at the end of the preceding stage and at the beginning of the current stage. 3. CONSTITIJTIVE

EQUATIONS

We shall confine ourselves to the consideration of mechanical behaviour of plastically incompressible materials and to the case of small strains. Assume also that the bodies considered are in a natural non-deformed state.

Endochronic

theory

for non-proportional

cyclic deformation

Relationships .for the prescribed plastic struin cycles Within the framework of the endochronic theory of plasticity, the relation stress deviator vectors and plastic strain deviator vector ep can be represented

269

3.1.

s=s,-+

dep

between the in the form:

’ o pp(z - z’) $$ dz’

dz

(1)

s

d5 Z = F(z, Be”, lp, @) d<’ = k(deP+de*)

(3)

where sy, and k are the material constants, p(z) is the kernel: Ffz, Aer, P, @) and p(AeP, (b) are the functions of the type of isotropic and additional kinematic hardening, respectively; z is the intrinsic time parameter, AeP = 11eP(zi) - e”(zz) 11is the plastic strain vector range equal to the maximum modulus of the difference between two plastic strain vectors in a cycle, @ is the parameter of the plastic strain path shape which is tailed coefficient of nonproportionality, N = (ep(zr) - eP(zz))/AeP is the unit vector,
p(z) = i

(xi 3 0).

(6)

i-l

Here Ri and cli are the material characteristics. The average pressure is considered to be a linear function volume: @kk

=

of the relative

change

(7)

3KOCkk

(3K, is the bulk modulus), and the total strain elastic and plastic component increments:

in the

vector increment

is presented

as a sum of

de = deP + 2 2Po where 2~~ is the shear modulus. One can see that the i-value for polygon-type paths is the ratio of the doubled strain range 2Aep to the total length of the considered cycle path lp, i.e.

plastic

2A,oP ~=---. 1p . From relationships (9) and (5), it follows that for uniaxial alternating straining, i = 1 and (I, = 0; while for a circular path, L = 2/7r and Cp= 1. Let p(AeP, 0) = 1. In this case, equation (1) for one-dimensional deformation processes will differ from the known constitutive equations (8) only by the representation of function F. For hardening materials which after a certain number of loading cycles attain a saturated deformation state, isotropic hardening function F can be taken in the form F = C -(C

- l)eeP’

(10)

where C = C(AeP, CD),

p = @(AeP, P).

(11)

Iim F(z, AeP, t*, Cp)= C(AeP, @) -_- I#

(12)

Since

function C(AeP, Q} should be sought for various cyclic loading paths.

by analysing

the steady-state

stress field djstribution

270

N. K. KW‘HER and M. V. BOROIIII

WC shall represent the parameter deformation process, as

/i, which characterizes

the rate of the non-steady-state

/I = A (A$‘) + R(AeP) ez(r -- ip AC’“),

(13)

Then at constant A@‘. parameter /I’will have the maximum value if In = ZAP. It means that the duration of transient processes will be the shortest for uniaxial alternating deformation and the longest, if Ip 4 AC”. An example of the latter case is the &late-type path in a circle of radius Aci’:‘2. with a great number of beams. Note that for polygon-type paths, in accordance with equation (9) the exponent in equation (13) is equal to 2n -- 4,E:. For such load cycles. the hardening function I;‘ depends on three parameters only: Z, AeP and @.

3.2. ~~~r~stiti~ti~~~ cyuutioas fir iwtnpirr cyclic ~~~f~~r~~l~it~o~l iaistorics The object of inv~stigat~~~ in this section is constitLitive equations for a hardening body for different sequences of plastic strain paths under the condition of process stabilization at each loading stage. Before starting the presentation of constitutive relationships, we would like to state again the main conclusions which follow from the work of Tanaka ut ul. [19]: (I) At a prescribed plastic strain anlplitude (AcJ’j2 = constant), a certain magnitude of the maximum stress vector range in the saturated state corresponds to each shape of the cycle. (2) The magnitude of the stabilized maximum range in a subsequent loading block is independent of the preceding strain history. (3) If after alternating tension-compression up to the deformation process stabilization, the material is subjected to alternatmg torsion, a further increase in the stress vector range is observed. In this case, the resulting hardening is equal to the sum of hardenings from the types of loading considered. The relation between the stress deviator vector s, and the plastic strain et: for the tlth loading block can be defined by the expression

deviator

vector

(1.5) d;’

= ~(de~*de~).

The inffuence of plastic strain history on the relationship into account by the assumption

(16) for the nth loading

lim F‘,,(:, Acle,“,I:, Cp,) = lim F,,_ 1(z, Al>,“_.,. Ifi’_, . CD,. , ). :m+i) p-i

block is taken

(17)

From equation (17) it follows that isotropic hardening function E‘,, is ~o~ltinuous in transition from one block of loading to another. If, in the course of loading, there is a change in the strain amplitude or the cycle shape, a discontinuity in the :-value occurs. Its subsequent value may both increase and decrease, a fact which changes appreciably the initial definition of the intrinsic time. For hardening materials with respect to block loading, the isotropic hardening function F, can be taken in the form F,, = C,, - (C,, ~~ C,, _ , ) e-. ‘I,*‘. Co = 1, II = 1, 2, 3,

(18)

P,, = P,bX,

(19)

Ci,

where C,, = C,(Aef, at,) except when @ = 0 at tr 2 2. With a sequence of uniaxial alternating loading in the direction of basis vrectors of plastic strain space, the material cyclic hardening laws will be described by different relationships. Considering conclusion 3, one can, probably, say that each subsequent straining in the direction coinciding with that of one of the basis vectors leads to an increase in the maximum value of the hardening

Endochronic

theory

for non-proportional

271

cyclic deformation

function. The rate of the transient processes progress does not change. Then in order to use equation (18), we assume c. =jC,(Ae,P,O) -j

- 1

(20)

where j is the integer which is equal to the quantity of loadings in the direction of the basis vectors and which does not exceed the dimensionality of the basis. Other more accurate formulations of the hardening laws for these types of loading require additional experimental investigations. Note that at n = 1 relationships (18) and (19) coincide completely with the analogous expressions (10) and (11) for a non-prestrained material. 4. BASIC OF

EXPERIMENTS

THE

AND

MATERIAL

COMPUTATION

CHARACTERISTICS

In order to determine the values of k, sy and functions F, p(z), p in expressions (l)-(5), two test series will be sufficient: alternating torsion and straining along circular plastic strain paths at various constant amplitudes of the range dep. From the alternating torsion tests (p(AeP, 0) = l), function F(z, AeP, I;, 0) of equation (10) is defined on the basis of the stress decrease analysis during inelastic unloading, by the method of Valanis [13]. In this case, the shape of the elastoplastic hysteresis loops yields the information for defining C(AeP,O), while the rate of non-steady-state processes progress enables obtaining B(AeP,1;). Considering the condition p(AeP, 0) = 1 and setting the final number of terms in series (6), the characteristics of the kernel p(z) can be readily calculated approximating the slz-eyz dependence by a corresponding expression. One can find a detailed description in [17]. C(AeP, 1) and p(Ae”, I’;) are determined in a similar way from the test for circular plastic strain paths. The obtained relationships define the parameter b(Aep, I’) according to equation (13). For the computation of the function C(AeP, Q), we used a linear approximation: C(AeP, a’) = [C(AeP, 1) - C(AeP, 0)] @ + C(AeP, 0)

(21)

which yielded good results when making particular calculations. To obtain p, we use McDowell’s finding about the linear dependence of the stress amplitude A5/2 on the non-proportionality parameter 0 during stabilization of the deformation process. For type 316 steel at AeP/2 = 0.2%, this dependence can be obtained by processing the experimental data of Tanaka et al. [19] (see Fig. 1). Assume that at z = z, and z = z, + AZ, the maximum difference 11 s(z, + AZ) - s(z,) 11 subsists. Then, according to equation (l), we can write deP(z, + AZ) + ‘- + *’ s(z, + AZ) = sY pp(z, dz s0

+ AZ - z’) g

dz’

deP(z,) + ‘m aeP s(z,) = SY___ PP(Z, - z’) Z dz’. dz 0 s

(23)

. P ;J” 200 A Data by Tanaka

et al [ 191

t 100 t 0

0.2

0.4

0.6

0.8

1.0

0 Fig. 1. Stress amplitude

in saturated

state versus non-proportionality

(22)

parameter

for type 316 steel.

277

N. K. KI c

HFK

M. V. k)KOI)II

and

If in loading the plastic strain path shape does not change. function p can be ;t factor outside the integral sign. Subtracting equation (23) from equation (22), we obtain -t A:) -- ~(2, ) = a + ph

s(z,

(24)

where + A:)

de”(z,

a ==,\

(25)

dz

(26) The expression for vector b can lx simplitied I)(:, + AZ) ; /I(:, ) =I const. Then

considering

R. b = p(z , )[eP(z ,

Lp(:,

+ AZ) ~ @‘I_-, )] +

large I, .

that at appreciably

+ A: - I’) -. /,(I,

,7e1’ ~ r’)] x ;;; d:‘.

(27)

?0 When going over from vector equation

(24) to a scalar relation.

we get

AS’ = 11a in’ + p’ 11b’i’ + 2(a - b)p

(28)

A.i’ = 11 s(; , + A:) ~~s(:, ) ~‘.

(29)

where Solving

the quadratic

equation

(28), we find an expression

for 11 (30)

For hardening materials. condition p 2 1.

the sign before the radical

in equation

(30) is defined

by the

Let us use the above constitutivc equations to describe the mechanical behaviour of tubular specimens of type 316 steel subjected to combined action of axial force and torque along various paths in the plastic strain space. For those loading conditions. stress r~ and plastic strain I:~ vectors will have the form

<‘onstituti\c

equations

( I ) are dclincd

by the cxpreasion (33)

where (T, = js,. In the expansion

for kernel

b.‘(z) = Q3).

E(r). uc shall confine

oursclvcs

(34) to two tcrmx of the series

I<(z) = E, 1’ Xz+ kl. To determine the material characteristics. UC use two series of experiments: alternating torsion and straining along circular plastic strain paths at three constant amplitudes AP2 equal to 0. I %. 0.2’:/0, 0.411, [19, 23. 231. The dependence of the stress vector amplitude AC(Ar:P. @):3 in the steady-state regime of deformation upon the non-proportionality coefficient @ is written in the form (35)

Air 2 = X,(Ai:“)@ + h,(AC). To determine

functions

k, (AC”) and h, (AC”), use a quadratic

approximation.

X, (AC”) = ~~2.916.666(AP)’

+ 55.OOOAi:” + 26.666

h, (AX”) = ~~ 1.666.666(Ar:“)’

+ ?O.OOOA>:”+ 196.6.

Then

Endochronic

theory

for non-proportional

cyclic deformation

213

5oofi

400 300 200 cd 9

100 -

tVI I,

O-100 -200 -300 -400 I -500 ’ -500-400

-300 -200 -100

0

100 200

I 300 400

500

100 200

300 400

500

(T MPa

(a> 500 400 300 200 cd F?

100

1

-500-400 (b)

-300 -200 -100

0

c MPa Fig. 2. (a) and (b).

NLM 28:2-J

N. K. Kr:c‘ttrR

and M. V HOKOI)II

500 400 300 200 (d ;

100

t‘,m

0 -100 -200 -300 -400

I-

I -SO? ,-_, OC) -400

I -300

!

-200

I - 100

I

I

0

100

I 200

I 300

I

400

500

I 400

500

a MPa w 500 400 300 200 (d Liz

100

t‘,m

0 -100 -200 -300 -400

1

I -500 -500 -400 W Fig. 2. Stress response

I -300

I I -200 -100

I 0

I

I

100

200

I

300

CJ MPa for plastic strain paths (a) cruciform cycles, (b) stellate cycles, (d) circular cycles at A@‘/2 = 0.2%.

cycles, (c) square

Endochronic

theory

for non-proportional

cyclic deformation

275

Similarly, from the analysis of stabilized elastoplastic hysteresis loops for alternating and straining along circular paths of the steel studied, we find C(Asp, Q) = k2(AeP)@ + h2(AeP)

torsion

(36)

where kz(A~:p) = -- l4,583(A~~)* + 275Acp + 0.1333 &(AE~) = - 6250(Acp)’ + 87.5A~~ + 1. For the computation

of the parameter

p, we get an equation

p = 3 + 4e?(~ ~~/“Ad, In addition, from the approximation sion, it follows that o,~ = 100 MPa;

(37)

of the t--yP dependence

E, = 182,025 MPa;

by the corresponding

E2 = 7,754 MPa,

expres-

cx= 1,301.

The function of additional kinematic hardening p(Q) is calculated with equation (31) where it is necessary to substitute A.7 by A8 and ep, p(z) by cp and E(z). For a numerical analysis of complex cyclic deformation processes, a programme has been developed and the necessary computations carried out. Figure 2 shows the stress-state kinetics for cruciform, stellate, square and circular plastic strain paths. Comparison with the experimental data [19] reveals a fair agreement for both the first load cycles and the stage of stabilization. Similar investigations were performed for plastic strain amplitudes equal to 0.1% and 0.4%. In this case, a good correlation between the computed and experimental data was also observed. Variation of equivalent stress depending on the accumulated plastic strain for the above loading conditions together with the experimental results 1191 is shown in Fig. 3. Figure 4 presents variation of equivalent stress obtained by calculation and experimentally for a sequence of different plastic strain paths at constant ArP/2 = 0.2%. In this case also, one can see a sufficiently good correlation between the calculation and the experiment. Transition from one loading block to another induces a change in the maximum stress amplitude. The maximum hardening within each block is practically independent of the preceding strain history with the exception of the processes of uniaxial deformation. The magnitude of equivalent stress in the steady-state regime is higher, the nearer the parameter I. is to 2/n. For the model proposed, the maximum hardenings within a loading block will be equal for plastic strain paths with similar inner geometry. The distribution of equivalent stress depending upon the accumulated plastic strain 5 for strain paths of the same type at stepwise variation of the amplitude AeP/2 is shown in Fig. 5. Similar agreement of the calculated and experimental data is observed for the stress-state kinetics. In this case, the largest disagreement occur for circular paths with a stepwise

-

Theory Data by Tanaka et al [19] A Torrion

+ Cruciform * steuate 0 Square 0 Circular

Fig. 3. Effects of plastic strain

path shapes

on cyclic hardening.

216

N. K. KUCHEKand M. V. BORODII

Endochronic

theory

for non-proportional

271

cyclic deformation

Torsion

0.1% Theory Data by Tanaka et al [22. 231

--_

0

I

Calculation Experiment

I

I

I

I

I

25

50

75

100

125

(a) 600

500 ----400 -

0.1% 200 -

100 -

I 50

0

I 100

I 150

I 200

I 250

3 0

F;%

(b

Circular 600 500 400 -

0.1%

0.4% 200

100 I 0

Fig. 5. Evolution

I

I

50

of the equivalent

I 100

I 200

I 150

I 250

stress with the accumulated plastic strain at stepwise variation the plastic strain amplitude.

in the plastic strain amplitude. However, the results obtained those obtained with the use of two-surface yield theory.

decrease

of

are much better than

6. CONCLUSIONS

In accordance

with the above considerations

the following

conclusions

can be made:

(i) The proposed constitutive equations of the endochronic theory of plasticity with the Odquist parameter as the intrinsic time measure can give a sufficiently efficient prediction of

27X

N. K

KI

(‘III-R

and M

V. BOKOIIII

non-proportional deformation of materials under cyclic loading. The assumption of the dependence of the isotropic hardening function upon the non-proportionality coefficient (1,, plastic strain amplitude and path length within a load cycle agrees with the avaiiablc experimental data. The introduction of the kinematic hardening function p into constitutive equations allows one to take into account the material additional hardening which occurs due to non-proportionality effects under cyclic elasg~plastic deformation. (ii) The hypothesis of the initial count of the intrinsic time I at each loading stage after stabilization of the deformation process makes it possible to obtain simple cnnstitutive equations for complex cyclic loading histories. The restrictions imposed on the isotropic hardening function in going over one block of loading to another allow one to take into account the influence of the plastic strain history on the kinetics of the stress strain state.

I

2. 2. 4. 5 i:

7.

x. V. IO.

11. 12. I?. 1-k. 15 Ih. 17. IX. IY 20. 111. 2:.

73.