Strength of material

International Journal of Plasticity, Vol. 12, No. 2, pp. 191-213, 1996 Copyright 0 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0749+X19/96 %lS.oO + .OO

EVOLUTION EQUATIONS FOR DISTORTIONAL PLASTIC HARDENING

T. Kurtyka and M. Zyczkowski

Politechnika Krakowska (Cracow University of Technology)

(Received in final revised form 15 May 1995)

Ash&act-In an earlier paper the authors proposed a geometrical description of distortional plastic hardening. The parameters of this model describe, in turn, all the elements of transformation of the initial yield surface, namely proportional expansion, translation, affine deformation, rotation, and distortion. The present paper is devoted to evolution equations for individual parameters: after a general discussion and analysis of existing experimental data, some effective equations are proposed. In particular, they are adjusted to describe Bui’s experiment on subsequent yield surfaces under combined compression and torsion of a thin-walled tube.

I. INTRODUCTION

During a process of plastic deformation of a hardening (or softening) material, the initial neutral surface (yield surface) is subject to subsequent transformations. Experi- mental results show that these transformations in a stress space consist of the following five elements: (i) Proportional expansion (ii) Translation (iii) Affine deformation (iv) Rotation (v) Distortion, exceeding affine deformation. The simplest case of pure proportional expansion is called “isotropic hardening”, though an, isotropic material may remain isotropic under much more general transformations (Zyczkowski [1981]). Pure translation is called “kinematic hardening”, and is particularly extensively analyzed in the literature, starting with the early papers by Melan [ 19381, Ishlinsky [1954], and Prager [ 19561. The combined case of expansion and translation is called “mixed isotropic-kinematic hardening”, or briefly “mixed hardening” and is often sufficiently accurate for engineering applications.

However, more precise description of plastic hardening requires taking the remaining three of the above-mentioned elements into account. General affine transformations of the Huber-Mises-Hencky (HMH) yield condition, mentioned by Edelman and Drucker [1951], were investigated in detail by Baltov and Sawczuk [1965], Backhaus [1968], Danilov [1971], Tanaka and Miyagawa [1975], Rees [1982]. Extensive discussion and comparison of the relevant equations was given by Rees [1989] and Skrzypek [1993].

191

192 T. Kurtyka and M. iyczkowski

Numerous experimental investigations were gathered and discussed by Ikegami [1976], Michno and Findley [1976], and Rees [1989]. Most of them show remarkable distortion of subsequent yield surfaces, connected with increase of their curvature in the vicinity of the control point (generic stress point), and flattening on the opposite side. This effect was first (partly) described by the slip theory of Batdorf and Budiansky [1949]: their theory predicts even a vertex (corner) at the control point, but no flattening on the opposite side. Further proposals used mostly stress invariants or mixed invariants containing stress components in the third or fourth degree: Freudenthal and Gou [1969], Williams and Svensson [1971], Shrivastava et al. [ 19731, Phillips and Weng [1975], Shiratori et al. [1976, 19791, Axelsson [1978], Rees [1982], Ortiz and Popov [1983], Eisenberg and Yen [1984].

A relatively simple geometric description of distortional plastic hardening, with clear meaning of individual parameters, was given by Kurtyka and Zyczkowski [1985]. The central idea of this description is to have the initial yield condition presented as a hypersphere and to consider various transformations of such a hypersphere. For isotropic materials subject to the HMH yield condition this is obtained in the five- dimensional auxiliary stress space, proposed by Ilyushin [1954, 19631. For other materials this is also possible in modified Ilyushin’s spaces, provided the material is insensitive to the first stress invariant (a “deviatoric” material); otherwise a five-dimensional space is not sufficient, and hence not adequate. Such modifications for relatively broad classes of isotropic and anisotropic materials were proposed by Zyczkowski and Kur- tyka [1984], [1990]. Parameter identification in the description of plastic hardening proposed was discussed by Kurtyka [1988]. Equivalence of vectorial and tensorial description of plastic hardening was investigated by Kurtyka [ 19901.

Recently, some other approaches to the description of distortional hardening were suggested. Mazilu and Meyers [1985], Gupta and Meyers [1986,1992] used common invariants of two stress tensors: actual stress deviator and the stress deviator corresponding to maximal prestrain. Helling and Miller [1987] developed the so-called MATMOD4V model using a variable Hill-type anisotropy tensor. Watanabe [1987] extended the Valanis endochronic theory of plasticity to cover distortion of subsequent yield surfaces up to formation of a vertex; his approach uses, in principle, just two parameters. Voyiadjis and Foroozesh [1990] based their proposal on variations of the fourth-order anisotropy tensor and used three parameters to describe subsequent yield surfaces.

Though the model proposed by the authors in [1985] uses, in total, many more parameters, nevertheless the number of parameters responsible for distortions is com- parable: it is equal to the number of dimensions of the auxiliary stress space, and hence it reduces to two in the most important two-dimensional cases. Moreover, all parameters have clear geometrical meaning. Hence, we return to that model and in the present paper we propose the relevant evolution equations: their general form and some particular cases. Basic assumptions are as follows: (i) The material is isotropic, subject to the HMH initial yield condition. However, in

view of possible above-mentioned modifications of Ilyushin’s space, a broader class of isotropic and anisotropic materials may also be considered.

(ii) The material is elastic-plastic, rate independent. However, further generalizations to viscoplastic_materials seem also possible; some preliminary ideas were given by Kurtyka and Zyczkowski, [1991].

(iii) The material is conforming to Ilyushin’s postulate of isotropy.

Considerations are restricted to small strains only. The processes under consideration are isothermal, temperature changes are neglected.

64 Plastic strains are incompressible. (vii) Phenomenological approach to the description of plastic hardening is used.

Distortional plastic hardening 193

II. MODEL OF DISTORTIONAL PLASTIC HARDENING

II. 1. llyushin s auxiliary Jive-dimensional stress and strain spaces

Ilyushin [ 19541, [ 19631 introduced an auxiliary five-dimensional stress space, namely the stress vector u = aiwi (i = 1,2,...5) is defined as follows

3 01 =-s11, 2 02 = +i+s22), c73 = d&2, a4 = x&23, cj = d&3,, (1)

where sij are deviatoric stress components, and wi are versors (unit vectors) of the axes. The Huber-Mises-Hencky yield condition takes here the form

InI = ffo (2)

(where oO. stands for the yield stress in uniaxial tension), and hence it is represented by a hypersphere in the five-dimensional space. In uniaxial tension in the physical direction ‘11’ we have cl = cl i.

In view of the initial yield surface being a hypersphere in Ilyushin’s space, initial yielding does not depend on the direction of the trajectory in that space. This fact makes it possible to formulate and apply Ilyushin’s postulate of isotropy, which, roughly speaking, states that in the five-dimensional space the transformation of the initial. yield surface depends only on the geometrical invariants of the trajectory and does :not depend on the direction.

The postulate of isotropy has a ‘mathematical part’ and a ‘physical part’. The mathematical part is connected with rotation of the physical system of coordinates in the body since then the auxiliary space is subject to a change. This part was proved by Ilyushin: in fact, any rotation of physical coordinates does not affect geometrical invariants in the auxiliary space. On the other hand, the physical part of the postulate refers to plastic hardening under essentially different loading trajectories, e.g. hardening due to normal stresses and due to shearing stresses. This part cannot be proved and remains simply as a hypothesis. It has been verified experimentally for isotropic bodies by many investigators. In the present paper we assume the postulate of isotropy to hold and make use of a purely geometric description of subsequent yield surfaces, proposed by the authors in [1985]. Moreover, Ilyushin introduces an auxiliary, five- dimensional strain space. We define the strain vector 3=3i wi (i = 1,2...5) as follows

31= ell, 32= -$ (?j + e22), 33= -$2, 34= -$23, 35= -$e3*, (3)

where ev denote deviatoric strain components. In this space the initial HMH yield condition takes the form


and also is represented by hypersphere. Total strain vector may be decomposed into its elastic and plastic part, 3 = se + +‘.

Ilyushin’s spaces were employed to describe plastic hardening by many investigators. We mention here Shiratori et al. [ 1976, 19791, Ohashi and Tanaka [ 198 11, Andrusik and Rusinko [ 19931.

11.2. Geometric description of distortional hardening

The hypersphere (2), representing the initial yield condition, during the process of plastic hardening (or softening) is subject to subsequent transformations. Such transformations consist of five elements listed in Section I. They are described in a fairly general way by the geometric proposal of Kurtyka and Zyczkowski [1985]: expansion, translation, affine deformation and rotation are quite general, whereas distortion is confined to a certain class, presented in two variants: broader and narrower. Now we recall the main idea of this description.

It is well known that an ellipse may be obtained from two concentric circles by a projecting procedure where the pole of projecting radii coincides with the centres of both circles, Fig. l(a). Now, a much more general curve may be obtained by similar projecting if we distinguish the centres of circles O(i) and OC2) and the pole A, Fig. l(b). Indeed, such a curve resembles a subsequent yield curve obtained from experiments for a general curvilinear trajectory.

In the general five-dimensional case we introduce five hyperspheres with various radii RC,) and various centres O,+ a pole A, and a system of mutually perpendicular projecting directions. The hyperspheres must have a common domain and the pole must be located in this domain. Then the current radius intersects all the hyperspheres and the

( /

Fig. 1. Proposed description of subsequent yield surfaces (b) as a generalization of quadratic surfaces (a).

Subsequent yieldsurFace


projecting procedure means simply taking one coordinate (in the rotated system of projecting directions) from each hypersphere.

The simplest analytical description is obtained in a moving system of coordinates, Si, translated and rotated with respect to the original system rri. The directions of Si coin- cide with the projecting directions. The versors of 6i will be denoted by Ci, they are related to the versors wi of the original system oi by the orthogonal transformation (rotation) formulae

Gi = Qijwj, i,j = 1,2..5, (5)

where QU denotes an orthogonal tensor (five-dimensional, in general). Now we define in the moving system of coordinates the vector “active stress”, B = ci+i, related to the stress vector CT by the formula

Si = Qij(gj - aj), (6)

where the vector a = aiwi describes the translation of the centre of the moving coordinate system and may be interpreted as a vector of residual microstresses.

The position of the centres of hyperspheres 0~~~ is defined in the moving coordinate system by five vectors di = dcl>jtij. These vectors are responsible for nonelliptic (non- affine) distortion of the yield surface, hence the notation d. Here a bracketed index denotes a label (number of individual hypersphere) and is not subject to summation.

11.3. Equations for the general (broader) case

Kurtyka and iyczkowski [1985] derived the following parametric equations of the distorted yield surface in Ilyushin’s space

aj = Q,‘{dci)ktk + [(dci)ktk)2 - d(i)dG)k + RtiJ]1’2}ti + aj, (7)

where the summation goes over i (without brackets), and over k; ij,k = 1,2,...5. The quantities ti denote current parameters, namely Cartesian coordinates of the unit vector of the radius-vector in spherical rotated coordinate system. The conditions of the pole to be located within all the hyperspheres have here the form

dci)kd(i)k < R?~J, i, k = 1,2,. . .5. 63)

Equation (7) contains in the general, five-dimensional case 45 fixed parameters (counted as scalars): five of them are responsible for expansion and affine deformation of the surface (radii Rc,)), 5 - for translation (aJ, 10 - for rotation (Qii> and 25 - for distortion (dCzIk). The first 20 of them appeared in the proposals of Edelman-Drucker and of Baltov-Sawczuk, whereas the remaining 25 are new. The last number seems too big for applications, and in the above-mentioned paper a simplified, narrower version was also proposed.

11.4. Equations for the simpl$ed (narrower) case

The description (7) is much simpler, but for most applications sufficiently general, if we assume

dcib = 0 for j # i (9)


Then only five non-zero parameters responsible for distortion remain, namely d(+ denoted briefly by di. The total number of fixed parameters is reduced here to 25. Equations (7) are simplified to

9 = &’ [& + (dftf - di + Rf)1/2]ti + aj, (10) ij, = 1,2,..5, summation over i, no summation over 1. Using “active stresses” (6) we may also write

c?i = [diti + (dftf - d!’ + R;)‘I*]& (11)

In this case the parameters ti may easily be eliminated and one obtains implicit equation

5

c c$ i=l Rf + 2di6i - 4 = ’ ’ (12)

It is seen that in the case di = 0 eqn (11) corresponds to a hyperellipsoid, hence, in the general case, (! 1) describes a distorted hyperellipsoid. Convexity of (11) was proved by Kurtyka and Zyczkowki [1985] for the two-dimensional case. On the other hand, general surfaces (7) may also exhibit concavities.

The conditions (8) are here reduced to

]dil < Riy i= 1,2...5. (13)

III. EVOLUTION EQUATIONS FOR SIMPLE LOADING

Consider first the case of proportional change of all deviatoric stress components, as the control process responsible for the formation of subsequent yield surfaces:

Sij = Sijdi (t) 7 (14)

or, using (l),

gi = ~ifzfl (1) (15)

where t is a time-like parameter; the derivatives with respect to t will be marked by dots. The function f,(t), may be increasing or decreasing, positive or negative (as it happens in cyclic loading). Loading of this type is called ‘simple loading”, since then a multidimensional case is reduced to a one-dimensional case (though the state of stress is, in general, multiaxial).

The direction determined by (15) in the five-dimensional stress space will be denoted by “i”, and will be considered as the axis after rotation (6), whereas the remaining four directions cannot be distinguished for simple loading (subsequent yield surfaces remain rotationally symmetric). Denote by ge the effective HMH stress for (14) or (15), then

The rotation tensor in (6) looks as follows

Distortional plastic hardening

-3u & SJO+hQ iod 2 o4 IQ x6% x6%

Qij = I arbitrary, but subject

to orthogonality conditions .

and does not change during the process.

197

(17)

Hence, the number of free parameters in the case of simple loading is equal to five in the general case (RI, Rj, cil, dcl)l, dop,j = 2, 3, 4, 5), and is equal to four in the simplified case, since then don = 0 (Fig. 2). The symbol cir denotes the coordinate of the translation vector in the rotated system coordinates and is particularly convenient when considering a simple loading process.

The radius RI is responsible for isotropic hardening and first we go back to classical descriptions of this type of hardening. Odqvist [1933] proposed to consider it as a

a

Fig. 2. Subsequent yield surfaces corresponding to simple loading: (a) general case, (b) simplified case.


function of the length of the trajectory in the nine-dimensional space of plastic strains (strain-hardening)

RI = a,[1 + $(Z,)] (18)

where Zep denotes this length and is called Odqvist’s parameter

(19)

and T denotes the variable of integration. In Ilyushin’s space Odqvist’s parameter is proportional to the length of the trajectory with the multiplier m. For simple loading (14), assuming similarity of deviators at least for this process,

where the dependence between f2 and fi is determined by the uniaxial stress-strain diagram. Even earlier Taylor and Quinney [1931] proposed to consider isotropic hardening as dependent on the plastic work (work-hardening)

RI = ~4 + 1cI(?+‘91 (21) where

(22)

For simple loading

Wp = Uij&ijo I

*.fl(&(+ = 010 310 J

Lfl(+2(M. (23) 0 0

So, for simple loading, we propose to retain (18) or (21) (in differential form) also for distortional plastic hardening under consideration.

The radius Rj, j = 2,3,4,5, denoted here briefly by Rz, is responsible for cross effects, it means for transversal dimensions of subsequent yield surfaces. Experimental investigations show either an increase of these dimensions (e.g. Theocaris & Hazel1 [1965]), or no cross effects (e.g. Phillips and Tang [1972]), or a decrease of transversal dimensions (e.g. Miastkowski & Szczepinski [1965]). So, denoting $‘(I,) by cp(Z,) we propose a rule similar to (18),

Z& = cA1 + A,)cp(Z,)Z,, R2(0) = 00, (24)

with the coefficient PC1 responsible for cross effects. This coefficient is variable in general, but in the simplest approach may be regarded as constant.

If cp(Z,) > 0, then PC1 > 0 denotes increase of transversal dimensions, -1 < PC1 < 0 denotes also absolute increase, but relative decrease comparing to R,, finally ,BCi < -1 denotes absolute and relative decrease. If cp(Z,) < 0 (isotropic softening), then the conclusions as regards ,B,i are opposite to those given above. In any case, for purely isotropic hardening or softening ,L?=i = 0.


The coordinate of rigid translation, 21, is responsible for kinematic hardening. In the case of simple loading this type of hardening is usually connected directly with the corresponding plastic strain rate,

where

denotes effective plastic strain (intensity of plastic strains). This strain may increase or decre.ase, whereas Odqvist’s parameter Z,, (19), is a monotonically non-decreasing quantity.

Finally, evolution equations for the parameters of distortion are quite new. Analysis of experimental results shows that a decrease of < results in a decrease of di too. So, we propose an equation of the type (25), namely

where the multiplier (R: - 4) ensures the condition (13) to be satisfied. A similar relati’on may be proposed for d*(i) in the broader description (7); reasonable results are obtained if ldc2)i1 < Id(i,il and this fact should be accounted for when constructing particular functions of the type (27).

Change of the rotation tensor QU will be presented in the form

Q = hQ Qij = h&Qkj, i,j,k = 1,2 . . . . . 5, (28)

where the tensor & has the interpretation of a “tensor of angular velocities” and is antisymmetric (hik = -&), with 10 independent components in the general case. In the case of simple loading there is no rate of rotation, and hence fiik = 0.

It ishould be noted that assuming cp(Z,) = const. = c, fa = const., we obtain linear strain hardening with the hardening modulus

(2%

or

since for simple loading 81 = RI + 81. On the other hand, assuming $J = cWP in (21) we do not obtain linear work hardening.

IV. GENERAL FORM OF EVOLUTION EQUATIONS

Consider now just the simplified (narrower) case of distortional hardening, described by the implicit equation (12). It may be written in the matrix form


F = (a - a)rQrDQ(n - a) - 1 = 0 (31)

where the diagonal functional matrix D = D(a, Ri, a, diy Q), has the diagonal elements

DE = [Rt + 2diQg(ai - Uj) - d;?]-’ (32)

with summation overi. So, F depends on the stress vector and on the parameters of the yield surface, regarded as internal state variables

F = F(cii; Ri, ai, di, Qij), i,j = 172, ..5. (33)

Denote by p a generalized vector of appropriately ordered state variables (with 25 components)

{P} = {Ri, ai, di, Qij}y (34)

then (33) may be written briefly

F= F(c,p). (35)

Subsequent yield surfaces F for a stable material should change continuously. This is ensured by differential form of evolution equations for p: the right-hand side may be discontinuous, but must not be of Dirac’s type, since such a type would result in a discontinuity of p. These evolution equations should determine P in such a way as to have it rate-independent. The simplest law conforming to this requirement has the form

P = By, hi = B.+‘l ‘J J’ (36)

where B = B(p, CT, S-,...) is a matrix (25 x 5), depending on the arguments discussed in detail in Section VI. Dependence on vectors or tensors denotes here, of course, just a dependence of invariants of these quantities (except considering the vector a and the tensor Q, when this is not necessary). The simple rule (36), proposed for other types of plastic hardening, e.g. by Mroz [1974], Lubliner [1974], Nguyen and Bui [1974]. Dafa- lias and Popov [1976], is more general than one may expect. Assume the constitutive equations in the form

ip = AnG, 3; = /i&j, (37)

where i is a scalar multiplier, and nc is the unit external normal to a certain function G(ai) (plastic potential),

nG = l~~$~~i, IlGi = dG/dui

J(aG/d~~)(dG/d~j)

not necessarily equal to F, (33). Considering the lengths of the vectors in (37) we find

(39)

hence


Further, introduce the notation

2 J

?BnG = b,

where: b is a matrix 1 x 25, we can write briefly

p = bi EPP’

(41)

(42)

The notation (42) is called the standard formulation of evolution equations, Bergander [198Ojl, Bergander et al. [1992] (for finite strains). All the equations discussed in Section III may be presented in the form (36) or (42). The generalized vector p consists in our case of 25 coordinates: 10 scalars Ri and Dip 5 coordinates of the translation vector a and 10 independent coordinates of the rotation tensor Q.

During a plastically active process (loading) the stress point should remain at the yield #surface (Prager’s consistency condition), hence

dF dF -sj+-fij=o dOj dPj

Substituting (36) and (37) we obtain

(43)

(44)

and final general constitutive equations including evolution equations take the form

Plastically active process takes place, if (8F/l3t~)~& > 0, otherwise $’ = 0.

V. ASSOCIATED FLOW RULE

For the sake of generality we admitted any plastic potential G, but for most materials th’e plastic properties of which do not depend on the first stress invariant, the associated, flow rule holds, G = F. Then in eqns (37), (38) (40), (41) and (45) nG should be replaced by nF, with F given by (12) or (31). Introduction of the associated flow rule brings essential effects, since distortion of yield surfaces results in much larger differ- ences in the values of derivatives than of the functions themselves.

The derivatives appearing now in the above-mentioned equations are equal

dF ’ aai- j=, -c ZQj/cQji(gk - ak)[Rj - 4 •t diQjm(am - am)]

[RT - 4 + 24Qjn(an - an)]* (46)

(47) dF dF aai aOi


dF -24IQ&7ic - wc)l* dRi = [Rf - df + 2diQ&n - an)]*

aF 2k4 - @k(flk - Uk)][&,&& - u,)]2 -_= l3di [Rf - df + 2diQin(nn - d,J]* ’

(48)

(49)

(50)

These formulae need additional summation over k,m,n, but no summation over i. They look complicated, but, for example, in a two-dimensional subspace, particularly important in applications, they are fairly simple.

VI. MORE EFFECTIVE EVOLUTION EQUATIONS FOR ARBITRARY LOADING

VI. 1. General remarks

A deformation process may be prescribed in stresses, strains, or in a mixed way. Choosing any six of the twelve quantities au, eii as independent, we obtain 12!/6!6! = 924 combinations (iyczkowski [1981]). Here, considering just deviatoric materials (insensitive to the first stress invariant), we may have 10!/5!5! = 252 combinations. In fact, the number of really independent combinations is much smaller. It would be difficult, for example, to assume that the stress o1 and the relevant strain s1 are both independent. So, if we eliminate the combinations with stresses and strains corresponding to each other, we obtain just 2 5 = 32 combinations. Independent quantities, which may serve as control variables of the process were discussed by iyczkowski [1960, 1967, 19811, and called the “exertion factors”.

Here, in principle, we shall consider the processes prescribed either in stresses, or in strains, but evolution equations depend always on plastic strains, hence both on total strains and on stresses, and final equations are implicit anyway. In stresses the process trajectory will be prescribed by

(51)

VI.2. Radii Ri

Usually it is assumed that eqn (18) is sufficient to describe evolution of Rip responsible for isotropic hardening component, even for general process trajectory. However, according to the remarks of Ilyushin [ 19611, and Lensky [ 19611, a more precise description needs certain parameters of the trajectory (51) to be included into relevant evolution equations. Indeed, analyzing various experimental data we found that such parameters are quite essential, even for RI and R2, and for RJ, Rq, and RS they are necessary.

We propose the following generalizations of (18) and (24) for all five radii:


(52)

with the initial condition R,{ZEp = 0) = ao. Here the coefficients pCi,......@C4 are responsible for primary, secondary . . . cross effects (relative changes of transversal dimensions in subsequent mutually perpendicular directions), whereas dimensionless parameters (or functions along the trajectory) pz,...ps should describe deviations in the plastic range from simple loading, from trajectory lying in a plane, from trajectory lying in a three-dimensional and in a four-dimensional hyperplane, respectively. Without such deviations the subsequent hyperspheres cannot be distinguished, and that is reflected in (52). The parameter 1-1~ is not necessary in (52), but will be used later, in evolution equations for di.

Following Ilyushin [1954], [1961] we could choose as p2, p3, p4, the following quantities, proportional to three curvatures of the trajectory

where

I&, j=2,3,4 pj = 4-1

c= = o’i$idT s (53)

(54)

is the current length of the trajectory in the five-dimensional stress space. Indeed, they describe deviations from a straight-line loading, from trajectory lying in a plane, from trajectory lying in a three-dimensional hyperplane, but the problem is more complicated. Any curvature of the trajectory within the yield surface, and, in particular, within the initial yield hypersphere is without any effect on plastic deformations. For examiple, the trajectory shown in Fig. 3 may be regarded as simple loading, (since it is radial. in the plastic range, though its shape in the elastic range is arbitrary), and just two hyperspheres may be distinguished, as described in Section III.

On the other hand, if a trajectory crosses the initial yield surface at a certain boundary point ab with the angle (a, c+) different from zero, and all three curvatures (53) different from zero, we can distinguish all five directions tii = w& and all five hyperspheres with the radii Ri at the beginning of the plastic part of the process, it means at the point Ub. To this aim we may use the Gram-Schmidt orthogonalization process.

Direction il = wib is always determined by the vector ffb and its coordinates are given in the first row of (17), written now in the form

3 Sllb &,lb + h22b *lb = ‘Nlb(Jj~, 2 7 (55)

0 fJ0

Direction tizb may be defined as perpendicular to i%tb, lying in the plane determined by bb and & (Fig. 4). Writing i+2b in the form of a linear combination

204 T. Kurtyka and M. Zyczkowski

loading trajectory

initial yield surface

Fig. 3. An example of simple loading in the plastic range.

load traje

Fig. 4. Rotated systems of coordinates.

(56)

and requiring %2b . Ub = 0, Ifi = 1, we find

ii2b = -(ub * b.6) + jab12cib

i~blj/l~bt2h+b/2 - (cb . 8)

(57)

Similarly, direction +Jb, may be defined as perpendicular to +b, and 6’~ (or to fib and &b), lying in a three-dimensional hyperplane determined by 66, & and &b. Writing

+3b = (Y306b + a31&, + o&b (58)

and requiring ti3b . bb = 0, ti3b . C+.6 = 0, l+X, = 1 we evaluate a30, (Yap, a32, and finally


where

labI2 t,b . C+b ub . ijb ub . & j&l2 &, ’ tib ub * 6,, bb ’ tib l&l2

In a similar manner we can define +db and +sb, using tib and orthogonality; the prin- ciples are clear, but the calculations are more and more cumbersone.

Now, there is an essential difference between stress trajectories starting from the centre of coordinate system, as discussed by Ilyushin, and starting from the boundary point ‘mb. In the first case a certain curvature is necessary to describe deviation from simple loading, whereas in the second case - an angle between ub and &. In the first case a torsion of the trajectory is necessary to describe its deviation from a plane, whereas in the second - a curvature out of the plane determined by ob and &. So, we propose as p2 the absolute value of sin of the angle (a, &), not only at the point u = bb, but at any point of the trajectory:

p2 = Isin(u,&)I = j/w = !

1 - M u u

and after rearrangements in the five-dimensional space

(61)

The above formula is useful for a trajectory prescribed in stresses, it means with stress components as control variables (exertion factors). A similar formula may be expressed in total (deviatoric) strain components, or even partly in stresses and partly in strains for mixed control of the process. Indeed, for deviatoric materials we have always p2 = 0 for a simple process, for any independent combination of stresses and strains as control variables.

Some authors use similar parameters like p2, defined in other quantities: Benallal et al. 119881 use here the translation (backstress) vector a, whereas Schmitt et al. [1994] - the plastic strain vector +‘. However, these quantities cannot serve as control variables, they cannot be governed from outside, and so they seem to be less convenient in applic,ations.

Consequently, as the parameter ,u3 responsible for an out-of-plane trajectory we propo;se the absolute value of sin of the angle between ii and the plane determined by u and &-. It is equal to cosin of the angle between 6 and 63, calculated as (59) for a current point of the trajectory


Substituting (59) we obtain, after rearrangements,

(63)

(64)

where A3 (a, c+, ii) is given by (60) for a current point U. Formula (62), regarded as a hypothesis, may be used both for differentiable trajec-

tories or trajectories with corners. In the latter case we obtain a jump-like change of p2, but continuous change of R3. On the other hand, (64) cannot be used for trajectories with corners, since then ti behaves like a Dirac’s function.

In some cases it is not difficult to replace (64) by a formula allowing for corners. Consider, for example, a trajectory consisting of segments of straight lines, then for each segment (n) and suitably chosen time-like parameter we have &,, = COIZSL The trajectory starts to be an out-of-plane if for three consecutive segments

\I A3(k(n)r +z+l), b.(n+2))

P(n) I2

+ o 7

k(n) . +l,l)

b(n) . +2+1) P(n+l) I2

where A3(bn, &cn+i), r+~~+~)) is defined as (60). As it was mentioned above, the trajectory within the initial yield surface has no effect on plastic deformations, so instead of &i we should substitute bb like the trajectory were straight, starting from the centre of coordinates.

The parameter ~4 may be defined as the absolute value of the angle between ii: and the hyperplane determined by 6, &, and ii; similarly pus. Final formulae will not be quoted here.

VI.3. Translations ai

The coordinates of the vector a describe translation of the surface, (backstresses or residual microstresses) characteristic for kinematic hardening. So, we propose here an evolution equation generalizing well-known equations for kinematic hardening and present the vector a as a combination of several other vectors employed to describe plastic hardening

(66)

The coefficients Ci are, in general, some functions of the form

Ci = Ci(P, Q, k, &Li, Lp, LJ?) (67)

and should be chosen in such a way as to obtain i independent of the scale of time. Hence Ci and C, must be homogenous functions of the time derivatives of the degree 0,


whereas Cs, and Cd and Cs - of the degree 1. The classical evolution laws for kinematic Ihardening may be obtained from (66) as follows: assuming Cz = C3 = C4 = Cs = 0 we arrive at the rule proposed by Melan [1938], Ishlinsky [1954], and Prager [1956]; assuming C1 = Cs = C, = Cs = 0 we obtain the second proposal of Melan [1938], whereas assuming C1 = Cz = C3 = 0, Cd = -Cs = C&, arrive at the rule proposed by Ziegler [1959].

Many other evolution equations of kinematic hardening are also described by (66). Assuming C, = C3 = Cd = 0, Cs = C&,, (Cs < 0) we obtain the well-known equation proposed by Armstrong and Frederick [1966]; assuming C, = Cd = Cs = 0, Cs = (7&, (Cs < 0) we obtain a similar law proposed by Eisenberg and Phillips [1968], Mroz (?t al. [1976]; assuming C3 = Cd = Cs = 0 we obtain the proposal of Phillips and Lee [1979], Voyiadjis and Foroozesh [ 19901, generalized by Voyiadjis and Kattan [ 1990, 19911 to finite strains; assuming just C z = C3 = 0 we may derive a more complicated proposal by Trampczynski and Mroz [1992]. A general approach to evolution laws for kinem,atic hardening was also discussed by Zbib and Aifantis [1988]. A triple analogy for tree variants of kinematic hardening in stress and in strain spaces was proposed by Zyczkowski [ 19771.

It should be noted that the second Melan’s proposal C, # 0 is often criticized since it does not conform to the postulate of continuous description of a neutral process. However, it may be employed without violation of any postulate if C, contains a multiplier of the type cos(n, b), since then it vanishes for a neutral process. For initial neutral surface (yield surface) in the form of a hypersphere we have cos(r+, b)

= J- 1 - & (61). For subsequent yield surfaces this relation in general does not hold exactly, but may be used as an approximation. Hence we included the parameters of the trajectory /.+ to the arguments of the multipliers Ci in (66).

VI.4. Rotations Qij

In Section VI.2 we constructed the initial rotated system of coordinates &j, determined by $bb, and also subsequent systems based on u, &‘, ii, 6. Denote directional cosines of these systems with respect to (1) by P,. Experimental data show that the projecting directions Qii, identical with P, for u = Ub, in general do not remain identical during the process, but follow Pij with a certain delay. So, one may be tempted to propo;se the following type of evolution equations

However, such equations determine 25 quantities Qii with only 10 of them being independent, and, in general, they are contradictory. In some particular cases we may formulate the counterparts of (68) just for independent quantities. For example, in a two- dimensional case the rotation is determined by one angle cp, and instead of (68) we may propo,se

where ‘pp is determined by U, and ‘pQ gives the angles of projection. Similar equations may be proposed for Euler’s angles in a three-dimensional case.

T. Kurtyka and M. Zyczkowski 208

Another possible approach makes use of the substitution (28), since then the antisymmetric tensor fi has just 10 independent components. According to the general form (42) we may propose the evolution equations

ti = A&,, flij = Aijafq (70)

The antisymmetric tensor A must vanish for simple loading and should describe deviation from simple loading. So, we may construct it as follows

A =fX~,o, Pkr &J[q) ~3 m(2) - m(2) 8 m(l)], (71)

where the symbol @I denotes dyadic product of two vectors mCl) and mC2). Such vectors should be colinear for simple loading, since then A = 0. Hence, any two of the vectors used in (66) may be employed in (71), since all of them are colinear for simple loading.

More effective formulation of (71) needs further research, both theoretical and experimental. Existing data on rotation are very scarce. Though some evolution equations for rotation are “hidden” in general equations for affine hardening listed in Sec- tion I, and some experiments were devoted to the analysis of rotation (e.g. Skrocki [1984]), the correlation between theory and experiments is still rather loose. We mention here just a proposal of Zilauts and Lagzdin [1992].

VI.5 Distortions di

Simple loading is characterized by just one distortional parameter, di. The remaining four di depend essentially on the trajectory, and namely on the parameters p) Forma- tion of the distortional parameters di takes place one step behind Ri. Indeed, the first deviation from simple loading, characterized by p2, results in formation of R3 and of d2. Hence, we propose the following evolution equations generalizing (27):

i=~~~~(P,a,~k,z~~,)(R~-d~)~p, i= 1,2...5, (72)

where the bracket ensures (13), and we introduced b1 = 1 for uniformity of notation. There is no summation over i in (72); the functions fdi may be different for each i.

The derivatives 5; in the rotating frame of coordinates should be understood as the corotational derivatives

5; = !& = Qij i$’ + Qij 3; . (73)

However, if the rotation is rather slow, we could approximately use ordinary directional derivatives,

3; = $& = Qjj 3; (74)

Moreover, assuming associated flow rule, we may use direct differentiation of (12) with respect to &, arriving at a simple formula

(75)

where A is a factor of proportionality.


VI.6. Renumeration of axes

Numeration of axes in the five-dimensional auxiliary stress space, introduced by Ilyushin [1954] by formulae (1) is quite arbitrary: any other numeration would be equally good. On the other hand, numeration of axes of a rotating system &_, introduced for the point u = Ub by the formulae (55), (57), (59) and similar for w& and wsb, is unique provided the derivatives &, ii, ii: are different from zero and the above-mentioned formulae are not degenerated.

The orientation of the rotating system must be equal to the orientation of the fixed system. Hence, for a particular system of loadings it is convenient to renumerate axes of the fixed system. For example, if we consider tension with torsion, we should leave the notation o1 as in (l), but denote r& = si& by ~2; if we consider double shear, then <icy& = si2fi should be denoted b y (TV, and 7Xzfi = sis& by ~2. Then o1 is transformed into c?i, 02 into 82, and so on.

VII. NUMERICAL EXAMPLE: BUI’S EXPERIMENT

Sublsequent yield surfaces depend in an essential manner on the definition of the yield- point stress. This definition is by no means unique: Haythornthwaite [1968] gives six various definitions being in use by experimentators. Some of them are very “sharp”, defining yield-point stress by very small deviations from the linearly elastic behaviour, and some are of “engineering type”, with the yield-point stress connected with remarkable plastic effects. Many papers are devoted to experimental evaluation and comparison of subsequent yield surfaces corresponding to individual definitions: for example, Khan and Wang [ 19931 constructed families of subsequent yield surfaces corresponding to various offset plastic strains and equimodulus surfaces in torsion-tension tests.

As a numerical example of evolution equations we use torsion-compression tests on aluminium specimens, performed by Bui [1966], since the parameters for individual surfaces were identified by Kurtyka [1988] for the distortional hardening model considered in the present paper. The definition of the yield-point stress, based on the increment of plastic strain

A&-’ = 20 x 1O-6 (76)

may be regarded as intermediate: not as sharp as in Phillips experiments, but far from ‘engineering’ definitions.

Bui”s results overcalculated to Ilyushin’s space are shown in Fig. 5. It is seen that the surface Fi does not conform to the sequence, probably due to larger experimental errors.; hence, the evolution equations proposed will be based mostly on Fe, F2, and F3.

A two-dimensional case, as investigated by Bui, may be described by 8 evolution equations, and namely for RI, R2, R3, al, a2, di, d2 and cp. However, Bui gave no data as regards R3 (an additional internal pressure had to be added), and so we propose just 7 evolution equations. They should be completed by 2 constitutive equations, one equation of subsequent yield surfaces, and two equations describing the trajectory. These 12 equations interrelate 14 quantities: 7 internal state variables listed above, and 37, %$I, cl, u2, c?i, 32, ZEp. Total strains 31 and 32 may be calculated later, they are not involved. So, two quantities may be regarded as independent (in our case pi and g2), and the remaining 12 should be expressed in terms of those quantities.

The equations proposed, based on Kurtyka’s identification of individual parameters and subsequent approximations, look as follows:


Fig. 5. Approximation of experimental yield surfaces for the simplified distortional model. Experimental points according to Bui [1966].

ril = 2000+~,

ir2 = 2000 go*,

R2-&. & = -25oo/q/~~ y

(c7=40000(1-p:)2 arctg~+mr-cp 01

(77)


6, = -q = const. = dlcoscp - &sincp + al,

a2 = 61 sincp + ~~cosy.~ + a2,

8 6 Rf 4 + =

2d,&, +R;-&+2dB 2 2 1. -

Of course, the equation 01 = -00 is valid just for the loading trajectory considered by Bui; otherwise it should be replaced by a more general equation f(~i, ~7~) = 0. The multiplier n in the equation for ~3 depends on the initial value pb; in our case n = 1 since (Pb = ‘tr for uniaxial compression.

The authors are fully aware that the proposal (77), based on three subsequent yield surfaces only, may be not general enough, but at least it gives an idea as regards specific forms of such equations.

The: time-like parameter t in (77) may be chosen arbitrarily, since it is subject to cancellation. Some proposals of choice were given by Sobotka [1985]. In the problem under consideration we may choose simply t = a2, since o2 is an independent, mono- tonica.lly increasing parameter.

VIII. CONCLUSIONS

The: model of distortional plastic hardening, proposed by the authors in 1985, has been completed here by relevant evolution equations. They describe all five elements of transformation of the initial yield surface, and are not too complicated as it is seen from the example (77).

Acknowledgement - Grant KBN - 1960/91 is gratefully acknowledged.

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