A numerical approach for nonclassical plasticity

Comjurns & S~ucrures Vol. 47. No. 2. pp. 3lM20, 1993 Rimed in Great Britain.

A NUMERICAL

004s7949193 $6.00 + 0.00 0 1993 Pcrgamon Press Ltd

APPROACH FOR NONCLASSICAL PLASTICITY

X. PENG and J. FAN

Department of Engineering Mechanics, Chongqing University, Chongqing 630044, People’s Republic of China

(Received 16 December 1991)

Abstract-A new incremental form of constitutive equation and the corresponding nonlinear FEM approach are developed based on an endochronic constitutive equation without using a yield surface; a kind of nonclassical theory of plasticity. Compared with the numerical method proposed elsewhere, the new algorithm greatly reduces the error induced in the numerical process, especially in the region of initial extremely small inelastic strain with its slope of stressplastic strain curve being sufficiently large. The residual stress field of an autofrettaged thick-walled cylinder and the cyclic stress responses of a notched plate are then analyzed.

INTRODUCTION

One of the essential promises of classical mathematical theory of plasticity (CP) is the assumption of a priori existence of a yield surface and time-indepcn- dent plasticity. Although CP has, and will still have, a contribution to solving different kinds of engineering problems, to the extent that this assumption is mostly an idealization, one should be in a good position to develop theories, alternative to CP, which do not make use of this idealization u priori [l]. Recently, Fan [2] proposed the classification of ‘nonclassical plasticity’ (NCP) distinguished from classical plasticity (CP). By definition, NCP, does not make use of either yield surface idealization or time- independent assumption a priori. Valanis [3], in 1971, developed a theory of viscoplasticity, which used a unified point of view to describe elastic-plastic behav- ior of materials during loading and unloading. This is the theory of viscoplasticity described for the first time without using the concept of yield surface and loading function. This model was modified in 1980 to be based on a more solid physical foundation [4]. Bodner and Partom [5] proposed in 1975 a relation of elastic-viscoplasticity, in which the inelastic strain rate was expressed as a function of state and internal variables. This relation does not require either a yield criterion or a loading-unloading rule. Furthermore, Hart [6], Krempl[7j, Fan and his co-workers [8-lo], to name a few, also took this direction.

The unavoidable common difficulty for different theories is how to treat the initial extremely small inelastic strain with its stress-plastic strain slope being infinite. In Bodner-Partom’s model [5], it was mathematically described by introducing a singularity at the origin of deviatoric stress space and a highly decaying function to ensure the plastic strain rate developed very slowly in the vicinity of the origin.

While in endochronic plasticity [ 111, weak singularity is introduced into the kernel function of its inte- gration-type constitutive equation. The difficulties caused by the singularity was certainly a serious challenge, since a more realistic constitutive equation is appreciated only if it can be conveniently used in numerical analysis. Otherwise, as pointed out by Lamba and Sidebottom [12], the involved numerical complexity may make the advantages lose their value. Fan [ 11, Valanis and Fan [ 11, 131 circumvented these difficulties by introducing three internal variables with one particularly describing the initial inelastic deformation. The corresponding kernel function to that variable is expressed by a highly decaying expo- nential function which takes the unit-impulse function as its limit. In this way, a differential constitutive equation and an incremental matrix form were derived and a nonlinear finite element approach was developed.

There exists a serious problem that the introduced singularity or highly decaying property may induce considerable error and even deteriorate convergence in the numerical process. This may partly account for why most nonclassical theories of plasticity have not been successfully and extensively applied to nonhomogeneous field analysis of complex boundary value problems as they should. It can be proved that, for endochronic plasticity, a remarkable error might be induced if the incremental constitutive equation is derived directly from the differential one. A new incremental form is, therefore, derived from the integral constitutive equation. Which can reduce to a large extent the numerical error and make the increment of loading become much larger. This new equation is then implemented into a tangent stiffness finite element algorithm and applied to the stress analysis of an autofrettaged thick-walled cylinder and a double-notched plate.

313

314 X. PENG and J. FAN

Although the proposed method is developed for the endochronic theory of plasticity, the authors wish that it could provide some available message or concept to other theories of NCP for them to be applied to nonhomogeneous inelastic stress and strain analysis.

A NEW INCREMENTAL ENDOCHRONIC CONSTITUTIVE EQUATION

The following are the formulae concerning endochronic constitutive equations for plastically in- compressible and isotropic materials in the case of isothermal and small deformation [4]

(1)

ds dz =f~, ds = IldeP, I/ (2)

ekk = 3Kc,, (3)

dS.. de?. = de.. - 2 v ” 2G

de, = dc, - j dc, 6,j, dSij = da, - f da, a,, (5)

where uij and cij denote the components of stress and strain tensors, Sj is the component of the deviatoric stress tensor, eij and es are components of deviatoric strain and plastic strain tensors, s and z are intrinsic time measure and intrinsic time scale, respectively, and_f(z) is the hardening function which is defined as follows [lo]:

F = j?(d -f). (6)

Suppose that the material in hand has been subjected to some plastic deformation history which is characterized by intrinsic time scale z,,, then the stress response at the surrounding time can be expressed as

s z dep. E,e-%(:-Z”-?dz’. dz’ (7)

,=I r”

By noticing that

E e-“,(‘-““~d I

I dz’ ’

=e -a, A2 s 2. de P E e-“,“n -“‘/dz,, (8) I

0 dz’

where AZ = z - z, , and using the mean value theorem for the second term on the right-hand side of eqn (7), one obtains

S,(z) = i q(z) (9) ,=I

S!C)(z) = SW(z ) ema, Az 'I 1, n

deP, E,

+= ,“+@A,% -(l-e-+% (10)

where 0 satisfies 0 < 0 < 1. In most cases deP/dz can be approximately replaced by AeP/Az since the inte- gration region AZ is small. The stress increment can then be approximately expressed as

A,‘$,= i (1 -ema,&) :z -~$l(z,) > . (11) ,=I I

By setting

k = (1 -e-“,&) , u, AZ

A = f: k,E,, B, = i - k,a,S$)(z,) (13) ,=, ,=I

one derives the following simple expression

AS,, = A Ae$ + Bij AZ. (14)

This expression is exact if the error caused by the replacement of de$/dz by AeP,/Az can be neglected. Letting AZ tends to zero, one has

lim A = i E, = p(O), AZ-O ,=O

lim Bij = - i C&;)(Z) = h,(z) (15) b-0 ,=I

then eqn (14) reduces to

AS, = ~(0) Ae$ + h,(z) AZ. (16)

This is just the incremental form of endochronic constitutive equation in [l, 131. We will find that

the relation eqn (16) is correct only for a very small AZ, and the error increases rapidly as AZ increases. We now analyze the error between the expressions in eqns (14) and (16). In order to dis- tinguish these two quantities, we let AS2 and AS’ be the AS in eqns (14) and (16) respectively. With the definition

h,(z) = i - tx,sy(z) (17)

A numerical approach for nonclassical plasticity 315

one derives

+ Ae$ E,

c( Az (1 -e-“rA’). (18) r

From eqns (16)-(18), it is not difficult to find the following relation

AS$‘)(z) = emzrh[Er Ae$ - a,S$‘)(z,,) AZ]. (19)

On the other hand eqn (14) can be rewritten as

By comparing eqn (19) with (20) and assuming that both S’ and S* are exact at time z,,, one obtains

In order to consider the error quantitatively, we choose the case of uniaxial loading where the expression for AS*(‘) in eqn (14) is completely exact if the hardening function is chosen as constant, for instance, unit. The corresponding relative error is

a;= AS I(r) _ A*(r) e-l, AZ

=-- AS*“’ k, ‘. (22)

In incremental computation process, SQ)(z”) is often employed to take the place of S$)(z) in eqn (17) by the assumption that the difference between S$)(z,) and S$)(z) is small. In uniaxial case, the error caused by this approach is defined as

(23)

where ASIn is still determined by eqn (16). It is easy to find the following relation by substituting eqn (16)

Combining eqns (2), (4), (5), (26) and (28) we derive

into (20) the following incremental elastoplastic constitutive equation in a matrix form

Thus eqn (23) becomes

sp+ 1.

I

It is easily found that if AZ +O, lim & = lim 6:” = 0, and when the increment AZ is large, the errors S; and ST may be very large, which are shown in Table 1 for some typical values of CI, and AZ.

It can be observed from Table 1 that for a, = 50,000, the errors S; and 8:” have been over 10% even though the increment AZ is only 0.005%. This kind of sensitivity of stress response to the length of incremental loading leads to a great difficulty in numerical process. In most cases, especially in the analysis of those problems involving large strain and large deformation where the local incremental intrinsic time scale AZ usually reaches 0.1% or over, the original incremental constitutive equation may result in a serious error, while the present one can give a satisfactory result.

ELASTOPLASTIC MATRIX AND FINITE ELEMENT APPROACH

Substituting eqn (4) into (14) yields an incremental elastoplastic constitutive equation

AS, = 2G,, Ae, + T, B,, AZ, (26)

where

1 , 2G,=AT, (27)

and AZ can be expressed as follows due to eqns (1) and (2)

(28)

where

[D,] = [D,] + 2(G; Gp) [D,]. (30)

Table 1. Relative errors S; and S: (%)

a, Ar O.o005% 0.001% 0.002% 0.010% 0.020% 0.050%

50,000 6’” 8;

13.0 27.1 58.2 403.4 900.0 2400.0 50,000 - 12.0 -22.9 -41.8 -96.6 - 100.0 - 100.0

5000 6:” 9;

1.26 2.52 5.08 27.1 58.2 403.4

5000 -1.24 -2.48 -4.92 -22.9 -41.8 -96.6

500 6:” 500 8:

0.125 0.25 0.50 2.52 5.08 27.1

-0.125 -0.25 -0.50 -2.48 -4.92 -22.9


For three-dimensional problem

{Ac} = (Au,, , AR,, Au,,, Aa,,, Ao,,, b,lr (31)

{AC> = (Acn, b, AC,, , SAC,,, 2Ac,, 9 24 S (32)

IDzl=C,(B,,,B*2,B,3,B,2rB23,B,,)’,

(Ml, A&, A@, , W,, Ae%, W, 1 (33)

[Del =

where

c, c, c, 0 0 0

c, c, 0 0 0

c, 0 0 0

G, 0 0

G* 0 sym. GP

2Gf2(z) AZ

TP H = * + 2Gf2(z) AZ

AcP, B,).

1 (34)

T,

The proof for eqns (29) and (30) is shown in the Appendix.

For axisymmetric problem, eqns (31)-(34) are reduced to

{u> = (Au,, Au,, A~zr AQ,,)’ (35)

{AC} = (AC,, Ac,, Acr, SAC,)’ (36)

P21= G(B,, Bo, 4, B,z)r(Aef, AePo, Aep, Ae;?). (38)

For plane problems, eqns (31) and (32) become

{A0 } = (Ao,, Ao, , Aa,,, Y (39)

{Ac > = (At,, At,. , 2At.y.v)‘. (40)

In the case of plane strain, we have

c, c2 0

P,l =

[ 1 c, 0 (41)

sym. GP

[D21=C~(B,,B,,Bxy)'(AeP,,Ae:,AeP,). (42)

In the case of plane stress

(43)

where

D,;= D,- D,,D,, g = 11, 12, 14,21,

22,24,41,42,44 (44)

D, = De, + D,ij. (45)

D,, and D2i, are the components of [D,] and [D2] denoted by eqns (33) and (34).

Using the principle of virtual work, it is easy to derive

in which

[ICI;- ‘{Au}; = {AP}n (46)

[K]:; ’ = f j=l s

[B]qD& ‘[B] dY (47) v,

W'h,={J'b,- f [W’iaJ,-,dV. (48) ,=I

{AP}, represents for n th increment of loading, {u}_ , the stress vector up to (n - 1)th increment and {P}” the total load up to nth increment of loading. The calculation of {BP), in eqn (48) has included the nonequilibrium nodal force caused by the error in computational process. V, denotes the volume of element j.

The iterative process for each increment of loading or unloading can be stated as follows: with the result obtained by the calculation of the (i - 1)th iteration of the n th increment of loading, one can calculate [IV]:-’ by eqn (47), and then {Au}: by eqn (46) and then

{AC 1: = Pl(Au jb, (49)

{Au}: by eqn (30). Then it is easy to calculate {Ae}:, {Ae’};, AZ; and {B}; at each Gaussian integral point. The iteration process will continue until the following inequality is satisfied

N ERR = max

]Az,: - AZ;- ‘1

AZ:, < ERO. (50) ,=I

where N is the total number of Gaussian integral points and ERO the tolerant error. Such a measure is adopted as a criterion of convergence instead of other measures because the intrinsic time scale z takes a crucial role in the endochronic plasticity. After the iterative process is finished, the derived incremental result is added to the result up to the (n - I)th increment of loading and one, therefore, obtains {a},, {c},, {eP},, IS},,, z, and {u}” and then starts the next increment of loading.


ANALYSIS TO THE RESIDUAL STRESS OF AIJTOFRE~AGED THICK-WALLED CYLINDER

Autofrettaged thick-walled cylinders are widely used in various industrial fields. The analysis of residual stress at the inner skin is very im~rtant, but how to exactly predict the residual stress remains unsolved at present, although various methods have been proposed. Since the difference between the theoretical and experimetnal residual stress at the inner skin is remarkable, some reduced coefficients have often been introduced to correct the theoretical result, some of which even take a value of up to about S-60%. Obviously such a method can hardly be acceptable and reliable.

In this section, the residual stress of autofrettaged thick-walled cylinders is analyzed with the proposed numerical approach and compared with the experimental result. These cylinders are made of high- strength steel 35CrNiMoV, with the inner and outer diameters of 22 and 55 mm, respectively. The material constants were determined as follows:

E,.2,3 = (3.80 x 106, I,24 x 10s, 4.95 x 103) kg/mm2

~1,,~,~ = (5.00 x 104, 8.00 x lo’, 4.50 x 102)

f(z) SX 1.

The internal autofrettaged pressures are 6789, 7055, and 7245 atm, respectively. The eight-node isoparametric element with 2 x 2 Gaussian inte- gration points is adopted in finite element analysis. Thirty increments were taken for the calculation during loading and 20 increments during unIoading. The residual hoop stress uf at inner skin of these cylinders is shown in Table 2 together with the experimental result [14] measured by the Sachs boring hole method. The comparison between calculated and experimental results are in good agreement. In which ERR is defined as

in which (a,“X,, and (o&p denote the calculated and the experimental values of hoop stress, respectively. The largest iterative number in this computational process is five and average number is less than four. If the computational process is composed of 20 increments for loading and 10 for unloading, the induced additional error is less than 3.7% for the above three cases. If the process is further reduced to

Table 2. Residual stress 08” at the inner skin

Internal pressure (a;)& C@eRLD ERR (%)

678.9 MPa -231.7 MPa -253.8 MPa 8.71 705.5 MPa -281.6 MPa -283.4 MPa 0.64 724.5 MPa -318.2 MPa -305.5 MPa 4.80

80mm -i

50mm

Fig. 1. Finite element of the double-notched plate.

10 increments for loading and five for unloading, the further additional error for the residual a, at inner skin is less than 3.2% for the three cases. In this process, the largest iterative number is nine and the average is about 5.5. These results indicate that the numerical process is stable and rapidly convergent.

ANALYSIS TO THE STRESS AND STRAIN FIELDS OF A ~UBL~E~~NOTCHED PANEL

Using the present numerical algorithm, the stress and strain fields of a double-edge-notched panel are analyzed. The material of this panel is assumed to be 304 stainless steel that appeared in [lS, 161, and the material constants and parameters are shown in [lo].

The upper and right quarter of this panel is taken for the analysis due to the symmetry of the problem, which is shown in Fig. 1. This part is separated into 36 elements with 13.5 nodes in total. The eight-node isoparametric element with 2 x 2 Gaussian points is adopted in finite element approach. This panel is assumed to be subjected to cyclic zero to tension axial straining at its upper bound and the loading direction is perpendicular to the notch line. In each straining cycle there are 17 increments for loading and 13 increments for unloading. The analysis covers both plane stress and plane strain problems.

m

2 to

= 0 0

= -to

ba -20

-30

-40

-50 t

Fig. 2. uyff relation at point A (plane stress).


I Number of cvcles 40

;i

g 30

0 s

b” 20

IO

I I I I I IO 20 30 40 50

Distance from the root of notch (mm)

Fig. 3. Variation of distribution of (r,, vs cyetic number N (plane stress).

For plane stress problem, the cyclic axial displacement at the upper bound of the calculated part was assumed to range between 0 and 0.0714 mm (see Fig. I). Figures 2-4 show the numerical results for the plane stress analysis. Figure 2 shows the stress-strain relation at point A (see Fig. 1) which is close to the root of the notch. The following features are observed: (a) hysteresis loops saturate gradually during the cyclic processes, (b) mean stress decreases gradually and tends to zero while the stress amplitude converges to a steady value, (c) ratcheting occurs as cyclic straining proceeds, but the ratcheting rate decreases because of the decrease of the mean stress. Figure 3 is the variation of the distribution of crY in the region close to the notch line and corresponding to the largest displa~ment of the upper bound, in which it is observed that the stress takes the maximum value near the root of the notch and decreases as the distance from the notch increases. The stress also decreases as the number of cycles increases because of the relaxation of the mean stress. Figure 4 shows the variation of the ~st~bution cp, in which it is seen that plastic deformation only exists in a small region near the root of the notch and the plastic strain increases sharply when the distance to notch root tends to zero, and the ratcheting of the material near the root also develops much faster.

Figures 5-7 involve the results of plane stain problem. In this case the displacement at the upper bound cyclically varies between zero to 0.08 mm. Figure 5 shows the relation of u), versus cv at point A during cyclic process, in which it is also observed

I 6 r Number of cycles

Distance from the root of notch [mm)

Fig. 4. Variation of distribution of ty vs cyclic number N @lane stress).

20

P

= 0 0 4 Z

b” -20

Fig. 5. o;-fv relation at point A (plane strain).

that the stress amplitude becomes saturated and the mean stress relaxes gradually during cyclic process and ratcheting occurs as cyclic straining proceeds. Figure 6 is the variation of the distribution of axial stress a,, in the region close to notch line and corresponding to the maximum displacement at upper bound. Figure 7 is the variation of the distribution of axial strain sY. It is also found that the axial stress and strain take the maximum in the region near the root of the notch.

The average iterative number for each increment is around six and the average error is much smaller than the tolerant one. The computation process also shows that the developed approach is stable and of good convergence.

CONCLUSION

A new incremental endochronic elastoplastic constitutive equation is proposed which greatly reduces the error caused by the original one obtained directly from the differential form of endochronic constitutive equation. A simple elastoplastic matrix is then derived, which includes two parts: elastic-like matrix and plastic matrix. The corresponding finite element approach is developed. The prediction to the residual stress field of autofrettaged thick-walled cylinders are in very good agreement with the experimental result, and the stress and strain fields of double-edge- notched panels are also quite reasonable. The numerical process is stable and quickly convergent,

60

Number of cycles

I I I I I IO 20 30 40 50

Distance from the root of notch (mm)

Fig. 6. Variation of distribution of CT~ vs cyclic number N (plane strain).


Number of cycles

\ I I I I

IO 20 30 40 50

Distance from the root of notch (mt’d

Fig. 7. Variation of distribution of 6,” vs cyclic number N (plane strain).

which critically demonstrates the validity of the developed approach.

This work is of theoretical and practical signifi- cance, because it not only provides some possibility and concept to overcome the mathematical difftculty of NCP appeared in numerical process, but also developed an applicable algorithm for NCP to be applied to nonhomogeneous field analysis.

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REFERENCES

J. Fan, Ph.D. dissertation, University of Cincinnati. Manuscript publications, University Microfilms Inter- national, June (1983). J. Fan, Nonclassical plasticity: context and current developments. Proc. 2&d Midwestern Mechanics Conf., University of Missouri-Rolla (1991). K. C. Valanis, A theory of viscoplasticity without a yield surface. Arch. Mech. 23, 517-551 (1971). K. C. Valanis, Fundamental consequences on new intrinsic time measure as a limit of the endochronic theory. Arch. Mech. 32, 171-190 (1980). S. R. Bodner and Y. Partom, Constitutive equations for elastoplastic strain-hardening materials. J. Appl. Mech. 42, 385-389 (1975). E. R. Hart, Constitutive relation for nonelastic deformation of metals. J. Engng. Mat. Tech. 98, 193-202 (1976). E. Krempl, Models of viscoplasticity, some comments on equilibrium stress and drag stress. Acta Mechanica 69, 25-42 (1987). J. Fan and J. Zhang, An endochronic constitutive equation for damaged materials. Science in China 32, 246-256 (1988). J. Fan, On a thermomechanical constitutive theory and its application to CDM, fatigue, fracture and com- posites. Proc. IUTAM Symp. on Thermomechanical Coupling in Solids, pp. 223-237. Elsevier (1987). J. Fan and X. Peng, A physically based constitutive equation for nonproportional cyclic plasticity. J. Engng. Mat. Tech. 113, 254-262 (1991). K. C. Valanis and J. Fan, Endochronic analysis of cyclic elastoplastic strain fields in a notched plate. J. Appl. Mech. 50, 789-793 (1983). H. S. Lamba, Nonproportional cyclic plasticity, T. and A. M. Report, n413, UILU-Engng, 766008, Depart- ment of Theoretical and Applied Mechanics, University of Illinois, Urbana, Ill. K. C. Valanis and J. Fan, A numerical algorithm of endochronic plasticity with experimental verification. Compur. Srruct. 19, 717-724 (1984). W. Cao, Residual stress analysis of an autofrettaged thick-walled cylinder. MS. thesis, Chongqing Univer- sity (1986).

15. E. Tanaka, S. Murakami and M. Ooka, Effect of strain path shape on nonproportional cyclic plasticity. J. Mech. Phys. So&& 33, 559-575 (1985).

16. E. Tanaka, S. Murakami and M. Ooka, Effect of strain amplitudes on nonproportional cyclic plasticity. Acta Mechanica 57, 167-182 (1985).

APPENDIX

By substituting eqns (lH5) into eqn (26), one obtains

(VI + W’,lW~ = ([~,I + 2G[D,W 1, (AlI

where the expressions of {Au}, {AC}, [B,], [Oz] are the same as those in part 2. [I] denotes the unit matrix and

[o,l=C,(B,,,B,,B,,,B,,,B*,,B,,) T

x (At%, de%, At%, 2AeL, 2M’,, 2Ae$, 1. 642)

It will be proved that

I[4+P,lI>o (A3)

and therefore there exists the inverse of ([I] + [D,]) such that

{Aa) = (VI + PII)-'([~,I + ‘=P,lWI.

Keeping in mind that

][I] + [a,bj]l = 1 + i a,b, ,=I

one, therefore, derives

I[I]+[D,]I=l+C,B,~eP,, i,j=l,2,3.

By using the inequality

2ab ga2+b2

and noticing that for strain hardening material

f(z’) <f(z) if z’ Q z.

By setting

E = -C,B,Aeef;

1 de? deP Aep. Aep x_ ~__L!+lj_?

( 2dz’dz’ AzAz > &’

s$k, I

I b,~,e-‘,“_“‘[f2(Z’)+f*(Z)]dz’

r-l 0 Y2(z )

~2 i k,E,(l -e--‘) r-1

<$ i k,E, --r-l

(A4)

(A5)

(A6)

(A7)

(A8)


A* 2G

and then

GP =-ie 649) = PA+ w [DJ. (Al 1)

ThusI[I]+[D,][=l-E>Oandtheinverse

([I]+ [D,])-' = [I] -E 2G

Equation (30) is then derived by noticing H = 1 - E. If material suffers softening during plastic deformation pro-

(AlO) cess, eqn (30) may still be available if the softening is not too serious so that E < 1 is not satisfied.

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A numerical approach for nonclassical plasticity