Constitutive equations and physical reliability in the modern theory of plasticity

1~. 1. Engng Sci. Vol. 32, No. 5. pp. 743-753, IY94 Copyright @ lW4 Elscvicr ScicnccLtd

Pergamon

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CONSTITUTIVE EQUATIONS AND PHYSICAL RELIABILITY IN THE MODERN THEORY OF PLASTICITY V. S. LENSKY Department of Elasticity, Moscow State University, Moscow 117234, Russia

E. V. LENSKY Department of Material Science, Fluminenase Federal University, Volta Redonda, RJ, CEP 27260-740,Brazil (Communicated by G. A. MAUGIN)

Abstract-Two of the most widespread theories of plasticity are analyzed regarding their domain of validity. A trinomial stress-strain relation of the local theory elastoplastic processes conformability to an arbitrary complex loading, being well-grounded experimentally, provides the physical reliability of a solution of the corresponding boundary value problem. The latter is correctly posed, and a convergent algorithm for resolution is proposed and used in some problems. On the other hand, it is shown that the postulates of the linear differential theory of plasticity (modern flow theory) and ones of the Prandtl-Reuss theory contradict considerably the variety of experimental data at complex straining, and their applicability is estimated as the strain process of small curvature.

1.

INTRODUCTION

equations in the mechanics of solids are the basis of the mathematical theory of design of a construction which must ensure the operative fitness and safety of a unit under both normal and extraordinary situations. That means that the solution of the corresponding boundary value problem has to be physically reliable, i.e. the predicted distributions of displacements, stresses and other physical parameters have to be in accord with real ones. The problem of physical reliability is especially manifested in the theory of plasticity [l] where there are several stress-strain relations differing in their basic principles and mathematical structure rather than in details. It is strange that some experts ignore essential discordance of a constitutive equation with experimental data supposing that some mathematical conveniences are in position to compensate the discrepancy with reality. Moreover, there are reasons to suspect that some serious craches could be connected with the utilization of an inadequate theory of design. At least three essential claims related to constitutive equations should be mentioned here: (i) working concepts and subjects have to be defined precisely, to permit their one-valued interpretation and unique experimental reproduction; (ii) basic postulates and/or their direct consequences provide the theory of experiment and are able to be corroborated experimentally except when they are founded by previously established physical laws; (iii) these postulates must in fact be corroborated experimentally with an appropriate accuracy or be grounded theoretically. From this point of view, two the most widespread theories of plasticity are analyzed here, namely the theory of elastoplastic processes and the lineardifferential theory of plasticity which is also termed the modern flow theory. It is shown that in the first theory the basic postulates are in concord with various experimental data and the universal stress-strain relation for arbitrary complex loading is deduced and corroborated experimentally, while the second one is in evident contradiction with experiments at combined loading, and it cannot be recommended for the computation of construction in complex conditions of exploitation. Constitutive

743

V. S. LENSKY

744

2. THEORY

OF

and

E. V. LENSKY

ELASTOPLASTIC

PROCESSES

In order to represent a process of strain deviator eii(t) and stress deviator s;j(‘), let US consider the five-dimensional (5-D) space with orthonormal coordinate frame e,, e2, e3, e4, es where the above mentioned deviators are defined as the strain vector 2 = e,,F” and stresses vector a = r,t?,, with components e, =eli. tl =

3s,,/2,

e3 = 2e,Jti,

e2 = (ez2 - e&tip r2 = ti

(Q2 - Rn)/2,

The stress vector can be referred strain trajectory p(t) in the form

e4 = 2e,,lti,

%=lh,

7J4 =

ti

e5 = 2eSIlti s23,

t5

=

;

l&4,.

to FrCnCt’s frame p,, (n = 1, 2, . . . , 5) at any point of the lY= &a” cos 8”

(2. I)

where a, = Ial, cos 8, = @,/au. The so-called image of process is defined as the aggregate of the strain trajectory together with stress vectors at all points of the trajectory and function T(s) of the arc length s defined by the element ds = (de, dei)“‘. Here T is the set of parameters such as pressure, temperature, dose of irradiation, and other parameters of physical nature which do not depend on the geometry of strain trajectory. The postulate of isotropy [2] assumed that the image of process is invariant with respect to the transformation of reflection in a plane containing the coordinate origin and/or rotation around a ray issued from the origin. The postulate leads to the conclusion that the representative stress a,, and angles of orientation 0, in (2.1) the functionals of a process with generating functions as K,(s)-curvatures of the strain trajectory and T(s), i.e.

TG)l~;=,,, 0, = F,[G,(& TWl&=w

o, = (P[G(E),

(2.2)

On account of (2.2) the relation (2.1) represents the general form of constitutive equation in plasticity under complex loading [3]. To complete the theory, the mathematical structure of functionals (2.2) is to be established. To corroborate this postulate and investigate functionals of plasticity (2.2), a new type of testing CL (complex loading) machine was created in the 1950s in Russia. Now such machines are being produced in several countries. By means of these machines, dozens of tests were carried out embracing several metals and a variety of programming strain trajectories firstly by V. S. Lensky [4-61 and later on by many authors [7-10 and others], in which a thin-walled tubular specimen was loaded by axial tension, The maximum deviation from the internal pressure and torsion in their combination. predictions of postulate observed in these experiments did not exceed 6%. As an example, in Fig. 1 the values of the angle of pursuit 8, vs As = s - so are shown for four stainless steel tubular specimens tested according to two symmetrical strain trajectories: (i) tension up to e, =s,,=O.5% and subsequent torsion at e, = const. (marked 1.3); (ii) torsion up to e3 =s,) = 0.5% and subsequent tension at e? = const. (marked 3.1). A very small scatter of experimental points is to the credit of the postulate. Figure 2 shows the corresponding experimental results for a, vs As which are in good agreement with the postulate toot. The principle of delay [2] asserts that the values of 13, at a point of strain trajectory depend on the intrinsic geometry of some limited part A of the forerunning strain trajectory rather than on the whole preceding strain history. The trace of delay A is one of the mechanical characteristics of the material. Figure 1 shows that for stainless steel the trace of delay can be estimated (with tolerance of 5”) as 0.9%. Figure 2 shows that the same takes place for a,, which tends to the curve of strain hardening at proportional loading. The agreement of the principle of delay with experimental data was established in a variety of tests with a high accuracy. tThese experiments were carried out by Ohashi [7] following author’s were sent to us with the kind consent to publish with this footnote.

recommendations,

and their numerical

results

Modern

theory of plasticity

745

9o A

L Stainless

steel

Y.Ohashi

60

I) (I .3)

Tomion-tension

(3.

Tension-toryton

0

-3.1

A

-3.1

0

-

0

-1.3

1.3

~oAQoAl&OC, “p

L

0

50

100

0

,,

,

150

200

ds*D4 Fig. 1. Dependence

3. LOCAL

of pursuit angle on arc length of strain trajectory

THEORY

OF ELASTOPLASTIC

after corner point.

PROCESSES

So long as it would be unreal to try to establish the mathematical construction of the functionals of plasticity theoretically or directly on the basis of experiments, the hypothesis of focal determinability was proposed in [5] which assumed that the velocity of the change of stress vector was a function (not functional) of instant values of parameters, i.e. the functionals of plasticity were representable in a differential form: d&,l~ =L(&r a,, K,), du,,/ds = T+!J(&,a,, K,),

(n, k = 1, 2, . . . 5, r = 1, 2, 3, 4). 0

350

(3-l)

A

ga OAo OA OA. Torsion-tension

(3. I)

q

0 0

o Stainless

steel

Y.Ohashi

;; $

300

d 0

-3.1

A

-3.1

0

-1.3

0

-1.3

250 0

100

200

ds*D4 Fig. 2. Dependence

of representative

stress on arc length of strain trajectory

after corner point.

V. S. LENSKY and E. V. LENSKY

746

Heref,

and r&are universal functions of the material. In particular d0,ld.s = L(8,, UJ - K~cos &/sin 8,.

(3.2)

This hypothesis was examined experimentally with special care [7-IO and others], and quite satisfactory agreement with experimental data was observed in all cases. Let us consider some results. In the test of a tubular specimen of stainless steel along the complex 3-D strain trajectory the last part was the segment of the straight line parallel to the axis e, , and the angle of pursuit 8, ar the origin of this segment was 85”. In Fig. 1 the subsequent values of this angle are shown by a solid line. This line was drawn close to the points obtained in different tests with the same initial values of 8,. Figure 3 shows several types of bigonal and ~lygonal strain trajectories used in the tests of tubular specimens of annealed Steel-45 under combined tension and torsion. Personal specification is shown under Fig. 4, where the results of these tests are displayed. Comparing experimental points for orthogonal trajectories one can conclude that the values of a, have a rather small influence on 8, vs As relation, and that 8, vs As relations with diffemt initial values of 8, coincide with those for orthogonal trajectories when being shifted to the right by a corresponding distant. On the basis of bigonal tests, the analytical approximations of function L in (3.2) were proposed, and then the solutions of (3.2) were obtained for some prescribed strain trajectories, which were found in quite satisfactory agreement with the corresponding experimental data. Some instructive results can also be found in the paper [lo]. The most valuable result referred to the local approach has been obtained on the basis of the hypothesis of coplanarity [ll], which asserts that at an arbitrary loading the relation t?=Aii+B8

(3.3)

takes piace, where the point means d/ds and

e,/(0,L).

B = -sin

A = (cos @$+ o; sin ~,f(cr,L))/q,,

(3.4)

200

150

B

s aJ

100

50 Steel

45

V.Len\ky

0

I

I

SO

100

I

I

I

lxl

200

250

el*D4 Fig. 3. Tested strain trajectories values of PSI plotted near test numbers.

Modern theory of plasticity

Steel 45 V.Lensky 0

60

4

10.1 - 10.2

-

A - 12.1 Cl - 14.1 *

- 14.2

I

100

ds+D4 Fig. 4. Experimental verification of hypothesis of local determinability. (-) Displaced corresponding to oblique segments in Fig. 3 {their numbers set after point).

Fig. S. 3-D strain trajectories.

curves

V.

748

S. LENSKY

and E.

V.

LENSKY

Table I. Average values of (Y” (numerator) and standard deviations from the average (denominator) for 225 measurements in 15 tests

60”

1.0%

-0.9

+

90°

120”

I

2.2

2.4

2.

1.5

2.2

0.9

2.7

0.6

3.4

G

-0.3

1.0

150

0.6

For corroboration of the hypothesis the results of experiments were used in which three series of tubular specimens of annealed steel SEC were tested along 3-D trigonal strain trajectories (Fig. 5)t at sg = e3 = 2%, three values of s, = e2 = 0.25, 0.5 and 1.0% and the values of 30, 60, 90, 120 and 150” for the location angle y of the third segment, placed in the plane P orthogonal to the e,-axis. If the hypothesis is true then the orientation of the plane (3, p) will be constant @ is the unit vector of the third segment). Let a be the angle of discordance, i.e. the angle between normal to a current plane (a, p) and to the initial one (ii,,, p), where a,, is the stress vector at the origin of the third segment. In Table 1 average values of (Y” and standard deviations from these for 270 points of measurements in 15 tests are shown, and that confirms cogently the reliability of the hypothesis. In the experiments [12] with annealed Steel-45 along the 3-D strain trajectories in the form of a circular helix found that (Ydid not exceed 6”. That reaffirms the validity of the hypothesis. Therefore, relation (3.3) together with (3.4) represents the stress-strain relation of the local theory of processes corroborated experimentally. It might be added here that the corresponding boundary value problem is posed and analyzed, and convergent methods of solution are developed, i.e. this theory is in a position to solve a practical problem for a construction or a massive body under arbitrary complex loading with a guarantee of the physical reliability of the solution.

4.

MODERN

FLOW

THEORY

OF

PLASTICITY

In the classical version of this theory the following assumptions are used: (i) as the basic working concept the plastic deformation can be uniquely defined; (ii) the concept of instant yield surface can be introduced, which is reproducible experimentally. Moreover, this surface contains the point of loading; (iii) the incremental vector of plastic strain is collinear to the normal vector at the point of loading (normality rule). In the alternative version, the vector of plastic strain-rate is directed along the stress vector. It should be noted that Drucker’s postulate [13] as such is not connected with these assumptions though its application to the theory of plasticity implies ail three. The vector of plastic strain can be determined with quite satisfactory accuracy when representing the residual strain. However, due to the nonlinearity and irreversibility of the unloading-reloading process, the increment of this vector as determined by the relation A?’ = AZ - AaJ(3G) can differ significantly from the real value. It is well known that at any moment of the loading process an infinite variety of yield surfaces can be constructed, and only one of them contains the point of loading. Moreover, at the next instant this point would belong to another surface constructed on the basis of another tolerance. The most incomprehensivle situation arises when a very precisely determined yield tSee footnote

on

p. 744.

Modern theory of plasticity

149

surface contains neither the origin of stress space nor the point of the loading (see, for instance, Fig. 17 in [14]). The first fact contradicts the definition of plastic strain as a residual one. The second fact nonpluses an investigator who does not know what point of this surface has to be taken to construct the normal. Both these uncertainties cannot be avoided by the choice of tolerance. Some hypotheses of workhardening are just artificial rules to construct the subsequent yield surfaces which are not associated with reality. Moreover, it is shown that, for instance, the use of the hypothesis of combined workhardening rule by Khadashevich and Novozhilov in real computations gives satisfactory results if the process is close to the proportional loading only. The above-stated notions mean that the yield surface is a useful image which provides the qualitative understanding of the region of elastic behavior under a partial unloading. However, it gives an unreliable basis for the construction of a mathematical theory. To elucidate a question concerning the validity of normality rule we use two types of systematical tests of tubular specimens under complex loading. In the first type the bigonal strain trajectories OA B were realized at which OA is tension and A B is torsion at invariable e, . The values of 0.4, 0.5, 0.8, 1.0, 1.25, 1.5, 2.0, 3.0, 5.0% for OA were used in several tests.? It is known from many experiments that the yield surface is regular at the point of loading. Hence, the normal to the yield surface at point A coincides with the e,-axis, and therefore, according to the normality rule, there must be Ae3 = 0 in the vicinity of the corner A. At each test for three points of the segment AB nearest to the point A the values tan (Y= Ae
Steel Sl5C Y.Ohashi

0

0.2

0.4

0.h

0.8

I.0

I.2

I .‘I

a

Fig. 6. Distribution

tSee footnote

on p. 744.

of 30 values of deviation

of plastic strain rate vector from the normal.

750

V.

S. LENSKY

and E.

V.

LENSKY

Brass BsP1 Y.Ohashi

0

Fig. 7. Distribution

0.2

0.4

0.6

of 30 values of deviation

0.8

1.o

of plastic strain-rate

1.2

I ..I

vector from the normal.

values (Y (in radians) among 30 ones obtained for 10 specimens of steel S15C. The quadratic mean is equal to 0.86 rad. A similar histogram is shown in Fig. 7 for specimens of brass BsPl. The quadratic mean is equal here to 0.76 rad. Moreover, there is no tendency to take the value (Y= 0 predicted by the normality rule. Still more convincing results were obtained in 18 identical tests of tubular specimens of steel S15C by torsion up to e3 = 2% and subsequent internal pressure at e = const. For each test again the values of tan LY= Ae$lAez for the three nearest points were calculated, where “distance” Aei did not exceed the value of 5.10p4. The histogram in Fig. 8 represents repetition of values among means that, on the average,

the the

54 ones. Here the quadratic mean is equal to LY= 1.12 rad. That in the vicinity of the corner point the values of Ae, are comparable

with the value of Ae, whereas, according to the normality rule, Ae, must be negligible compared to Ae3. These results show that the normality rule and, therefore, the hneardifferential theory of plasticity (the modern flow theory) is in discrepancy with experimental data under complex loading. The solution of the corresponding boundary value problem would be physically irreliable, and it would be risky to use this theory in design of an engineering unit under complex conditions of exploitation. As far as it is possible to conclude on the basis of the present experimental data the modern flow theory would provide quite acceptable results in the cases of processes with small curvature or in proportional loading. How to coordinate this conclusion with the plasticity postulates by Drucker [13] and Ilyushin [15] which are, to all appearance, trustworthy though they lead to the irreliable normality rule? Ilyushin [15] deduced the corrected rule taken into account the rate of elastic anisotropy due to plastic deformation. It is possible that this correction would improve the current situation. But as a result a new, quite different theory would appear. In the theory of plasticity originated from the Prandtl-Reuss theory it is assumed that the vector of plastic strain-rate f?“’ is collinear to the stress vector 0. To verify the validity of this assumption we shall use again the data of identical tests of tubular specimens of steel S15C at torsion up to e3 = 2% and subsequent internal pressure. At the set of points at the second segment of the trajectory, at which experimental data were fixed, the angle p between the vectors AC/‘/As and ii was calculated. If 2”’ is collinear to 5 then p = 0. Experimental relation

Modem

theory of plasticity

751

26

Steel SIX Y.Ohashi

13

IO

0.8

I .o

0.9

I.1

1.2

1.3

a

Fig. 8. Distribution

of 54 values of deviation

of plastic strain-rate

vector from the normal.

0 0

00

08

0””

0

s, o”

00 p”

P

30

Steel

0 0

s I5C

Y.Ohashi

Solid

line-leau

(standard

0

quares

deviation

fit

0.009)

30

60

90

W’

Fig. 9. Deviation

of plastic strain-rate vector from stresses vector (angle j3) vs angle of pursuit (0) 21 lests torsion-pressure after 2% strain corner point.

in

V. S. LENSKY

752

and E. V. LENSKY

Brass BsPl Y.Ohashi 0

30

0

_

-0.5%

0

-1.0%

A

-1.5%

0

-2.0%

x *

-3.0% -4.0%

+

-5.0%

30

60

90

8”

Fig. 10. Least squares fits of experimental data after corner points of plotted strain values in 7 torsion-tension tests.

pvs6, where 8, is the pursuit angle (cos 0, = @,/a,) is represented in Fig. 9 by a great number of points. In spite of the significant scattering it is seen that there is no collinearity. Lines of Fig. 10 correspond to averaging relations /3 vs 0, in the series of bigonal experiments with brass BsPl mentioned above. These results reaffirm the above-stated conclusion that this theory is valid in the case of processes of small curvature or in proportional loading only. 5. CONCLUSION The local theory of elastoplastic processes provides the physically reliable solutions of boundary-value problems since the basis hypotheses of local determinability and coplanarity are corroborated in experiments at essentially complex loading, and the corresponding mathematical theory is developed. On the other hand, the modern flow theory and the Prandtl-Reuss theory of plasticity are in discordance with experimental data at complex loading, and their applicability is limited to processes of proportional loading or to processes of small curvature.

REFERENCES 111 . ~ A. A. ILYUSHfN and V. S. LENSKY. Prikl. Probl. Prochnosti Plastichnosti N.I. Moscow State Universitv. Moscow (1975). [2] A. A. ILYUSHIN, J. Appl. Malh. Mech. (PMM) 18, N.6 (1954) (translated from Russian by Pergamon Press). 131 A. A. ILYUSHIN, Plustichnost. AN SSSR Press, Moscow (1963). i4j V. S. LENSKY, Proc. Znd Symp. on Naval Strucr. Mech., brown University (5-7 April 1960). [5] V. S. LENSKY, Vop. Teor. Pfasrichnosti. Moscow State University, Moscow (1961). 161 1. F. Bell, Encyclopedia of Physics. Vol. VI a/l, Mechanics of Solids. Berlin (1973). [7] Y. OHASHI, Mem. Fat. Engng. Nagoya Univ. 34, No.1 (1982). 181 V. S. LENSKY. Izv. Akad. Nauk. SSSR, OTN, Mechanica Mashinostroenie No.5 (1962).

Modern theory of plasticity

753

[9] DA0 HUI BIG, Vest. Mosk. Univ. No.1 (1966) (translation from Russian by Allerton Press). [IO] Y. OHASHI, M. TOKUDA, Y. KURITA and T. SUZUKI, Mechanics of Sofia% No.6 (1981) (translation from Russian by Allerton Press). [ll] V. S. LENSKY and E. V. LENSKY, Mechanics of Solids No.4 (1985) (translation from Russian by Allerton Press). [ 121 R. A. VASIN, Prochnost, Plastinchnost i Viuzkouprugost Mater. i Cons&. , Sverdlovsk (1986). [13] D. C. DRUCKER, Proc. First US Nutn. Congr. Appl. Mech. (1952). [ 141 A. PHILLIPS, Problem of Plasficity. Int. Symp. Foundafiom of Plasticity, Warsaw, Vol. 2 (1960). [IS] A. A. ILYUSHIN. Voprosy Teorii Plastichnosti. Moscow State Univ., Moscow (1961). (Received

and accepted

18 May

1993)