A universal integration algorithm for rate-dependent elastoplasticity

Pergamon 00457949(95)00240-5

Computers & Structures Vol. 59, No. 6, pp. 1173-l 184, 1996 CopyrIght b 1996 Elsevm Science Lfd

Printed in Great Britain. All rights reserved OO45-7949/96 $15.00 + 0.00

A UNIVERSAL INTEGRATION ALGORITHM FOR

RATE-DEPENDENT ELASTOPLASTICITY

P. A. Fotiut and S. Nemat-Nasser

Center of Excellence for Advanced Materials, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA 92093, U.S.A.

(Received 26 August 1994)

Abstract-An algorithm is developed for integrating rate-dependent constitutive equations of elstoplastic- ity including isotropic and kinematic hardening, as well as thermal softening and non-coaxiality of the plastic strain rate and the driving stress. The method is unconditionally stable and accurate for large time steps and UN possible ranges of rate-dependency. Under a constant loading rate the algorithm gives exact results at arbitrary step sizes for rate-independent materials without hardening, and in proportional loading for rate-independency with hardening, and linear viscosity without hardening. The present method is an extension of a recently proposed integration algorithm for stiff equations to domains of high rate-sensitivity like, for example, in power-law creep. The algorithm employs a plastic predictor-elastic corrector scheme, which, in general, requires less numerical effort in the return mapping process than the assumption of an elastic predictor. Numerical examples underline the efficiencv of this integration algoriihm in comparison to-gradient techniques and an kxtended radial return method for rate-dependent plasticity. Copyright 0 1996 Elsevier Science Ltd.

A key factor in the economic computation of elas-

1. INTRODUCl’ION

tic-plastic solids by finite elements is the numerical integration of nonlinear constitutive equations. Al- gorithms, such as the forward gradient method [l-4], which approximate the nonlinear stress-strain relation by a tangent at the beginning of the time step, are known to be limited in their step size in order to yield satisfactory results. These limits are prescribed by either stability requirements [5,6] or, for unconditionally stable algorithms, by the accuracy of the result.

In order to achieve accurate solutions for large time increments, it is apparently not sufficient to linearize at the beginning of the time interval, t = t,, but it is necessary to take into account the entire nonlinear structure of the constitutive equation, and to solve these iteratively at the end of the time step, t = tb = t, + At. This may be done, for example, by the Newton method, where the nonlinear flow rule or, if a yield surface exists, the yield function fb =f(tb) = 0 is linearized at each iteration step, and

an improved estimate is found by equating the linearized flow condition to zero [7,8]. Such an algorithm, which is, thus, stable and accurate, is the well-known backward Euler scheme, which, for

t Present address: Department of Civil Engineering, Technical University of Vienna, A-1040 Vienna, Austria.

$ Although implicit in its architecture (the flow condition is satisfied at the end of the step), the result is obtained in a single step, which makes the performance equivalent to that of an explicit method.

J2 -plasticity, reduces to the radial return method [9-121. In this case the nonlinear equation to be solved at the end of the increment is a scalar one, which makes this method attractive for numerical computations.

The solution of the nonlinear flow condition at t = tb can be understood as a step by step return mapping from an elastic predictor (trial) state back onto the (yield) surfacef, = 0. Since the elastic-plastic tangent modulus is usually very small compared to the elastic modulus, the elastic predictor state is located far outside the surface fh = 0, and following the return path from there may require numerous iterations. This can be avoided by employing a technique initially proposed by Nemat-Nasser and Chung [13] and subsequently extended by Nemat- Nasser [ 141 and Nemat-Nasser and Li [15]. Here, instead of an elastic predictor a plastic predictor is used, based on the fact that in the plastic regime most of the total strain is plastic rather than elastic. Hence, the solution of the plastic consistency condition fb = 0

can be approached by linearizing f at the plastic predictor state, which is defined by the assumption that the plastic strain at the end of the time step is equal to the initial value plus the increment of total strain (contrary to the elastic predictor, where this is assumed for the elastic strain). For stiff equations such a plastic predictor is already very close to the exact solution, so that it suffices to use only one correction step in solvingf, = 0. This correction term can be derived as a function of known quantities at t = to and the total strain increment, and the algorithm becomes therefore explicit:. Such a method

1174 P. A. Fotiu and S. Nemat-Nasser

works very well as long as the total strain is every- where either dominantly elastic or dominantly plastic. At higher rate-dependencies there are also major portions of the response where elastic and plastic strains are of the same order. Then, more iterations will be necessary to achieve plastic consistency at t = t,. Hence, we will adopt an iterative procedure in our formulation, but still start from a plastic predictor, since this minimizes the necessary iterations when the plastic strain magnitude becomes close to that of the total strain,

In the estimate of the stress and plastic strain increment we use a technique, which still allows the plastic strain rate to LVZ~~ within the step, i.e. we obtain the increments from the solution of a first- order differential equation. As a consequence, the response of linear viscous materials, loaded at a constant strain rate, can be modeled exactly by this method. An algorithm employing similar ideas has been proposed by Chulya and Walker [16]. In most other theories a constant estimate of the plastic strain rate is taken.

Since the merits of the present method are not directly related to the plastic spin, we consider here only small deformation plasticity. The constitutive equations include nonlinear isotropic and kinematic hardening as well as thermal softening, and are based on J,-plasticity. Also included is the effect of non- coaxiality, which is an important factor in dilatant and frictional plasticity [ 171, and in the micromechan- ical modeling of granular and geological materials [l&20]. The performance of the algorithm is tested in numerous examples covering a broad range of material parameters to demonstrate its universal applicability.

2. CONSTITUTIVE EQUATIONS

Let the plastic strain rate 8’ be given by

ip = jy + aajl, (1)

where p is a normalized tensor,

and a prime denotes the deviatoric component of the corresponding tensor. The stress difference d is given

as

a=z-j, (3)

where r is the Cauchy stress and fi denotes the back stress. In J,-plasticity, p is a gradient to the yield function (or normal to the yield surface, if a yield surface is defined). Since p : Jo = 0, the second term on the right of eqn (I), involving the non-coaxiality factor X, is tangential to the yield surface. If c( = 0, j, is equal to the effective plastic strain rate,

a = 0: $=Jgq,

and y is the accumulated plastic strain,

y= ‘i’dt s 0

(5)

We define effective stresses by

and we will also use the effective total strain rate i, given by

@=W, (7)

With eqn (7), the deviator of the total strain rate can be written as

According to J,-plasticity, we consider a general form of the flow rule

f = fJ - gtJ> 1’> T) = 0, (9)

where T denotes the temperature. In models with a yield surface, eqn (9) describes the dynamic yield condition, and g(y, j = 0, T) is the corresponding static yield stress. If no yield surface is defined, eqn (9) can be viewed simply as a plastic constraint relation. The loading-unloading conditions may be given by the Kuhn-Tucker relations.

f<O, ?j 20, “/$=o, (10)

which must hold at any time. For the back stress we assume the following evol-

ution equation:

where H is the kinematic hardening modulus, which, in general, depends on plastic strain and temperature, and may also be rate-dependent.

At high strain rates, temperature effects due to plastic deformation become significant and lead to a decrease of the flow stress (thermal softening). We neglect here internal heat flow and consider only adiabatic heating. It is thereby assumed that a frac- tion of the plastic work l&i’= 73 is converted into

heat,

f = c,zlj, (12)

where c0 is a material parameter. Note that the non-coaxiality term does not contribute to the plastic work, since 6’ and B are both orthogonal to ic.

In the elastic regime we consider only isotropic material behavior for simplicity,

r’ = 2G(c’ -c”),

where G is the elastic shear modulus.

(13)

Algorithm for rate-dependent elastoplasticity 1175

We will approach the solution in two steps. First, the magnitudes of stress and plastic strain are determined at the end of the time step, and then we estimate the components of these tensors. Thereby, we will use the following expression for the effective stress rate

6 =t?‘:p =2Gi’:p -3G+H-j, (14)

which is found by introducing eqn (3) for b’, eqn (11) for b and eqn (I 3) for +‘, before multiplying by ~1.

3. DETERMINATION OF THE STRESS AND PLASTIC

STRAIN MAGNITUDES

Assuming L’ to be constant, we integrate eqn (14) over the time interval At = tb - t,. This gives

s 65

A0 =o,--~7,=2G r’:p({) dr - K By, (15) 1.

where

K(Ay, AT) = ;

We emphasize that, although p, = pe (0 = 1) is the return direction of the radial return method, [(0 = 1) according to eqn (21) is not the rotation-factor of the radial return method. The latter is given by (a derivation can be found in the Appendix)

f?(y, v, T) = s

H dy . (16)

Here and in what follows, a subscript a (6) always indicates a variable at the beginning (end) of the time step. Now we write for the integral in eqn(l5),

(a>

s fh

2Gi’ : p(5) d< = 3G5P At, (17) 10

with the (stress-) rotation-factor [ given by

where po=p(t = t,+QAt), 0~0 < 1, is an inter- mediate value of k. We estimate pii as the return mapping direction from an elastic predictor (Fig. la),

CrCZ+ or,tl = J(WJ’ + 2er,cos l/I, + 1’

09)

with

r, = 3Gt! At

(1 + 3G+, cos&=ftl:pa. (20)

Introducing eqn (19) into eqn (18) yields

A comparison between relations (21) and (22) for several values of 8 and $, is given in Fig. I b. Numerical results show that for larger time steps the stress tensor usually rotates faster than assumed by the radial return algorithm. Figure 1 b demonstrates that ((0 = 1) approaches unity faster than the other estimates, and therefore leads to greater accuracy in numerical calculations with large time steps.

In proportional loading p and f are collinear, and we have 9, = 0, [ = 1. For a non-zero angle I(/, , the rotation-factor [ will always depend on At. However,

for large At, and hence, large r,, it will become nearly constant and close to 1 (Fig. lb).

With the above results we are now able to rewrite eqn(l5) as

i= or, + cos J/,

J(W,>2 f 2w,cos II/, + 1 . (21)

o,=o,f3G&At --KAY. (23)

(b)

- Eqn. (21), e = 1 -- Eqn. (21). @ = 0.5 _- ERR, EW (22)

0.0 I.0 2.0 3.0 m

Fig. 1. (a) Direction of pe in the evaluation of rotation- factor [. (b) Rotation-factor c as a function of r,.


Next, we require plastic consistency at the endpoint In Ref. [ 151, Nemat-Nasser and Li approximated g, tb, i.e. as

f* = Oh - &%> ?ib> Tb) = 0,

and make a Taylor expansion of g,,

(24) gh = gtr, + @> Ay/At, T, + AT), (32)

&J = g(y, + AY, li*, T, + AT) + (d, - i*)g’> (25)

where

and y * is a reference strain rate, which should be close to jb in order to render a single term expansion like eqn (25) sufficient. We choose y * to be the mean value of the plastic strain rate within the time step

?*A$ (27)

This ensures consistency in the sense that g,, as given by eqn (25), approaches g, when At +O. For this to be true, v* has to approach 3, for vanishing At, which is satisfied by the choice eqn (27).

We may write eqn (23) at each time instant 5 = t - t, in the interval 0 < 5 < At,

c(t)=o,+3GM -Q(5), (28)

where y(c) = ~(5) - y,. Introducing now eqns (28) and (25) into eqn (24) results in a first-order differential equation for y,

g’$+Ky=3G&<+o,+g’j*-g(1’*), (29)

where it is understood that both g and g’ are taken at the end of the time step, and, therefore, depend on yb and T, or on Ay and AT. If, for the moment, we consider all the coefficients in eqn (29) as constant, the differential equation can be solved analytically. The solution with the initial condition y(< = 0) = 0 reads as follows:

x[l -exp( -$<)I. (30)

Now we set 5 = At, which yields y(At) = Ay, and with eqn (27) we obtain from eqn (30)

K(Ay, AT)Ay = 3G@ At - g(Ay, AT) - ~~

_ AY) n’(AttAT)]

K(AY> AT) At

&(A?> AT)

from which we get instead of eqn (31),

K(Ay, AT) Ay = 3G@ At - g(Ay, AT) + 0”. (33)

In obtaining eqn (33) the effective plastic strain

rate was directly replaced by its average value within the time interval, while eqn (30) also includes vari- ations of i, and thus, is more general than eqn (33). If j stays nearly constant during the load step, both formulas will give equally good results. On the other hand, when abrupt changes in loading occur, eqn (33) introduces errors in the estimate of the effective plastic strain.

When temperature effects are neglected, eqns (31) and (33) arc nonlinear equations in the single un- known Ay, which can be solved numerically. To include thermal softening, it is necessary to introduce an estimate for AT into eqn (31) or (33). To this end, we integrate eqn (I 2) which yields

+ (B - 4,)) : ip d5 I

=co I Ai +pU:At”+ ‘*(R-A,)jp:rdr 1 L J I<,

= c,,[Ag + Ay/?, : prc + ; (AR - na AY )I,

where

J

(34)

A2 = gh - R,, i;l = g dy,

and

&?h = E(y, + A;‘, ArlAt, T*),

R,, = ii(y, + bp, Ay/At, T*).

(35a)

(35b)

(36)

Since eqn (31) or (33) is solved iteratively, T* = T,

is used as a first estimate, and in each iteration, T*

is set to the value of T, + AT from the preceding iteration. Because the influence of temperature on the plastic work is usually small, such an approach converges fast. In obtaining the right-hand side of eqn (34) the assumption was made that the increment


b - /I, is collinear with c throughout the entire time and use the plastic corrector increments interval, and the plastic strain increment is rep-

resented by AEP = AYpn, with pn given by eqn (19). A,,‘“’ = _K*(“) AY’“’ , K*(“) = 3G + (3/2)W’,

Again, it is recommended to use 0 = 1 in eqn (19), because of a better performance at large time steps.

Finally, having found Ay and AT, we obtain the A+“) = & ,$ A$” _Y’“‘,

[ 1 r-l updated variables

to obtain

y,, = yO + Ay, oh = cro + 3G& At - K(Ay, AT) Ay,

T,= T, f AT. (37)

3.1. Iterative return mappingfrom theplasticpredictor state

n > 1: A$“‘= j-W

K*(n) + g’“’ + g’$/Ar ’ ,1 91

AY (1) = f(l) + g$’ +, K*(l) + g(l) + g’!‘/At ’

7)’ 9

(4) Calculate updated stresses and strains

The concept of a plastic predictor is based on the fact that in the plastic domain the plastic strain becomes close to the total strain. In real materials this applies regardless of the rate-sensitivity or the hardening characteristics. These features only influence the instant when the plastic strain approaches the total strain. In rate-independent materials, this hap- pens nearly instantaneously after exceeding the yield limit, whereas increased rate-dependency or substantial hardening in the early stages of plastic deformation delay this effect.

In the following, we outline the difference between the elastic predictor and the plastic predictor method in the return mapping process. In A and B below, we will show only the basic steps, and for the sake of simplicity, we do not include non-coaxiality and temperature dependency. Also, we restrict ourselves to an approximation of the flow rule according to eqn (32). A detailed consideration of the iterative solution of eqn (31) follows thereafter.

,,(“+ 1) = a(“) _ K*‘“‘Ay’“‘,

q’(“+ 1) = u’(n) - (2/3)K*‘“‘AY’“‘$“‘,

Y (n+ 1) = Y’“’ + Ay’“‘, l;(n+ 1) = Y’“‘+ A+“‘,

H’“+” = H(Y@+i), j@+i)).

(5) Compute

g @I+ 1) = g(Y’“+ I), jPl+ 1’)

and check if

If’“+ ‘)I = (0 (n + 1) _ g’” + 1)) < to/,

where to1 is a prescribed tolerance limit.

If no: go to (3) and repeat steps (3)-(5)

if yes:

A. Elastic predictor-plastic corrector scheme. (1) Find a first trial state assuming At’ to be a ob = f~@+ I), Yb = Y(‘+ ‘I, Yb = g = Y(‘+ ‘)a etc.

purely elastic deformation; from Hooke’s law obtain the B. Plastic predictor-elastic corrector scheme.

(1) Find a first trial state assuming AC’ to be a Elastic predictor: purely plastic deformation; from the Jlow function

obtain the ,s’(‘) = a: + 2Gi’ At,

Plastic predictor:

y(I) = y ‘7

+ [ & i(‘) = @, g(l) = g(y”‘,p’),

y (1) - m-y,, j”‘=do, g”‘=g,, H”‘=ff.

(2) Calculate

ff”’ = fQ,‘l’, y”‘),

Qo),o a, yeU)= e y je(‘)=rJ, a,

KC’) = 3G + (3/2) (fl” - E-i,)/(c Ae),

(3) Linearize p”’ along the plastic corrector path where i’ is the elastic effective strain rate, defined by 1” = <p - j.

f:“’ =f(“) +f,$’ A@ +f,‘;’ Ay ‘“’ +f$’ A$“’ zz 0,

f’“’ = 1 j-p) = _‘,p f’“’ = _ (4 ,c ’ ,, r) ’ ,r g,;, )

(2) Calculate

f’“’ = o(“) _ p _# 0,


(3) Linearize f(“) along the elastic corrector path where

fl”’ =,f’“’ +f;$’ Aa’“’ +fJ;J Aye’“’ +f$j &+) = 0,

f’“’ = 1 0 ) f,‘.Jj = g,‘;‘, f,‘$ = g;;‘,

and use the elastic corrector increments

AY e(a) = @ At - A$“),

AO’“’ = 3G@ At - K’“’ AY’“’

K’“’ = 3G + (3/2)(p) - RJ/(y(“) - Y,),

Introduce this into fp) = 0 and solve for Ay(“),

Ay cn) = 3G + g’.? + g$‘/At

K’“’ + $ + g’;‘/At [P At

,7 ,r

(4) Update stresses and

f (n)

+ K'"' + g,‘;’ + g;;’ /At

strains

a’“+ ‘) = a’“’ + 3G5e At _ K’“’ by’“‘,

Y @+ ‘) = y’“’ + Ay’“‘, y’“+ I) = y(n) + Ai’“’

Y + + ‘1 = ye’“’ + Aye(n), ej%n + I) = je(n) + Aj e(n),

r[l -exp(-Sk)]. (39)

The derivatives of f*, however, become more complicated. Due to their form, the functionsf orf+ are not best suited for an iteration procedure. In the following, we discuss a simple modification of the constraint relation, which improves the convergence behavior.

3.2. Modified constraint relations for the plastic pre-

dictorPelastic corrector method

In the finding of the root of f(Ay) = 0 or f*(Ay) = 0, the elastic predictor always starts from By = 0. The first estimate of the plastic predictor, on the other hand, can be either greater (in loading) or smaller (at relaxation) than the root. The Newton scheme is always convergent for the elastic predictor. In case of the plastic predictor, it can happen for small time increments and Ay /de << 1 that the cor- rected result becomes negative, Y”’ < 0. This can be remedied by changing eqns (31) and (33) to

where

eqn (31):

Ay = h (AY, AT), (40)

h(Ay AT) = (3G51 At + 10, +s’ ArlAtlV -ew-K Atlg’)lj AY K Ay + [g + g’(3G/K)&][l - exp( - K At/g’)] ’

-

(5) Compute

g (n+ 1) = g(y’“+ I), Y(n+ 1))

and check if

If(n+ll)l=I~(n+lI)_g(n+l)l<tol

if no: go to (3) and repeat steps (3)-(5)

if yes:

(Th=b(n+l), Yh=Y(n+I), j~_+n+‘), etc.

In principle, the procedure in B can also be used for the approximation eqn (31). Then, however, the constraint relation has to be different,

f*(n) = @) _ g*@!), (38)

(414

eqn (33):

,f’(AY AT) = 13G5C At + 0,) AY KAy+g

(41b)

A typical variation of h(bi) is shown in Fig. 6, where it becomes obvious that a tangential linearization is always convergent if y”’ - Y, is greater than the root. Equation (41) is mainly iterated in Ay, and the dependence on AT is included according to the procedure outlined after eqn (36). Instead of a New- ton-type iteration we may use a secant relation,

C(d _ 1) by’“’ + h’“‘)

c@) = 1 _ (h’“- 1) _ h’“‘)/(Ay’“- 1) _ Ay’“‘)

(42)


with the starting values Ay(‘) = [ de, Ayc2) = h(l). To avoid iteration to the second root Ay = 0 when Ay(‘)

is smaller than the main root, we set the limit c@) > 0 1 if Ay@) < h(“). 1

3.3. Special cases Differentiating eqn (44) with respect to time yields

We consider the performance of the algorithm in the two extreme cases (a) rate-independent plasticity and (b) linear viscosity without hardening/softening.

$ sin + = d-cos$ -q:ir’

(47) 0

3.3.1. Rate-independent plasticity. In this case g is

independent of $, and therefore g’ = 0. Then eqn (31) is identical to eqn (33) and the only approximation involved is the estimate of the rotation-factor [, and, if temperature effects are included, in the approximation of the back stress [see the text after eqn (36)]. Hence, in proportional loading (5 = l), the root of eqn (40) is the exact result, which can be found independent of the step length. With { estimated by eqn (22), the results for ab and yb would be identical to those of the radial return method for rate-independent plasticity. The tensor components, however, are evaluated in a different way, as shown in Section 4.

3.3.2. Linear viscosity without hardening or soften-

ing. Here g’ = const., g’j = g(f) and K = 3G. Hence, the solution of eqn (29) is

y(5) = [Pt - + [ 1 - exp( -F <)I. (43) g

Again, the only approximation in this solution is due to [, and in proportional loading eqn (43) is the exact solution.

4. COMPUTATION OF STRESS AND STRAIN COMPONENT’S

Having determined ob and yb, our next task is the evaluation of the components of the tensors u, T, /Y, and LQ at t = t,.

Krieg and Krieg [lo] have shown that in ideal plasticity the rotation of the stress tensor can be evaluated exactly. Since their analysis depends on the constancy of the yield surface radius a,,, their results are also valid for purely kinematic hardening. An extension to isotropic hardening has been given by Schreyer et al. [ll].

Following Krieg and Krieg [lo], we derive a differential equation for the angle tj between the directions of tr’ and Z, i.e. $ is defined by

cos * = $j :/l. (44)

We rewrite eqn (14) in the form

I_+ = 3GP cos $ - K*y, K* = 3G + ;H, (45)

and find for the product rl :a’,

#j:b’= 1 + 3Ga cos2 ij

1+3Gc( 3GP - K*j, cos i,b. (46)

and with eqns (45) and (46) this can be rewritten as

3GP * + (1 + 3Gcc)a sin ti = 0. (48)

If the coefficient of sin tj stays constant during the time step, eqn (48) can be integrated analytically. Thus, we set a = B,,, and obtain the following solution:

ijb = 2 arctan[Ctan($,/2)],

3GC At C=exp(-r,), rb= (1 +3Ga)a,’ (49)

where $,, tib are the angles between q and ai, a;, respectively, and the rotation angle of 6’ is given by $ = $, - $b. With this result, pb can be expressed by a linear combination of the tensor c, and 11,

with

2c z, =

1 + c2 + (1 - C2)cos II/, ’

1-c~+(l-c)~cos1(1~ z2 =

1 - c2 + (1 - C2)cos $, . (51)

With the orientation pb given by eqn (50) we are

now able to update the tensor components at the end of the time step,

2 u;=jBb pb, Pb=~,+@b--ohb.

7; = 0; + Bb, (524

rb = z; + B(tr(Lb) - 3K(Tb - T,))I@‘, (52b)

Acp = (Ay - CI Aa)p, + (312)~~ Aa’

= AYPb + c(0, tab - Co), (52c)

where B = 2(1 - v)/3(1 - 2v) is the bulk modulus, v is Poisson’s ratio, K denotes the coefficient of linear thermal expansion, TO is a reference temperature and 1”) is the second-order unit tensor.


5. EXAMPLES

In the following examples, we use a flow rule given

by

O<N<l, O<nfl, (53)

where c,, is the initial yield stress, i, is a reference strain rate, and yO, N, n and 1 are material constants. Rate-independent behavior is included in eqn (53) with n = 0. Kinematic hardening is assumed to follow a power law,

j=mH,y”-‘tp, O<m < 1, (54)

where Ho and m are material parameters. With such a model we obtain the following expressions, which are needed in the iteration process:

~(AY, AU = no (, +cg?j’($L)”

xexp[-i(T,+A.T- To)],

K(Ag)=3G+iH,,(“+~i)“-:‘_. (55b)

In all subsequent examples we use the following material constants: 2G/a, = 400, v = 0.3, c,~crO/TO = 1, KT~ = 0. The rotation-factor [ is evaluated from eqn (21) with @ = 1.

In Figs 2 and 3 we compare the results of the present universal algorithm (CIA) with those of an extended radial return technique (RR), described in the Appendix. External loading consists of uniaxial straining i,, = -2~~~ = -2t,,, followed by simple shear i,,, which represents a 90” rotation of the loading direction, $, = 7c/2. Only isotropic hardening is considered, and the material parameters and loading rates are:

For n = 0 (rate-independent plasticity) and n =O.l: N = 0.2, y0 = 0.002, t,, = 2oooi,, 0 Q i,t < 2’ 10-S; i ,~=800i”,2~10~5~i”r <4.10--5.

For n =0.3: N =O.l, ‘r” = 0.005, i,, = SOi,( 0 <P,t < 10m-3; i,Z = 6Oi,, 1O-3 < i,t < 2. lo-‘.

For n = 1: N = 0.1, yo = 0.005, i,, = 8i”, 0 < i t < IO-‘; i,* = 8i,, lo-* < i,t < 2 lo-‘. -.” L

In the rate-independent case, both methods yield the exact result (within the prescribed tolerance limit) for proportional loading. After changing the loading direction, however, the RR shows considerable devi- ations, while the UA still gives a nearly exact answer.

n = 0, N = 0.2, yO = 0.002

Egt ( 1 o-5)

(b) 3.0

r n=0.1,N=0.2,yo=0.002

I I I 0.0 1.0 2.0 3.0 4.0

eat (10-S)

Fig. 2. Stress components T,, and T,~. Uniaxial straining followed by simple shear. Comparison between universal algorithm (C/A) and radial return (RR). (a) n = 0, rate-inde-

pendent, (b) n = 0.1.

At n = 0.1 and n = 0.3 the situation is similar. At higher rate senstivities (n = 0.3, n = l), it is observed

that the CJA tends to overestimate the stresses on the ascending branches, while the RR always predicts lower stress values. Figure 4 shows the effective stress T as a function of the effective plastic strain y for the case n = 1. In predicting the stresses, the UA is clearly superior to the RR. After changing the loading direction, the UA tends to overestimate y for large steps, while the RR underestimates ^J.

The large step sizes used in Figs 2 and 3 make it

necessary to use a backward routine (i.e. collocation at the endpoint of the increment). In Fig. 5, we compare the effective stress z predicted by the UA with that from the tangent modulus method (TM) proposed by Pierce et al. [3]. The material constants and loading rates in this example are the same as in Fig. 2a. Due to the tangential linearization, the TM overshoots after exceeding the yield limit. If the overshoot is not too big, the TM is able to recover to the correct curve. After changing the loading direction, however, the TM (with 20 steps) shows an increase in the effective stress, contrary to the exact result, which decreases. This is because the TM assumes the plastic strain increment AcP to be


n = 0.3. N = 0.2. y,, = 0.005

(W &Or

n=l.N=0.1.yo=0.005

0.0 0.5 I .o 1.5 2.0

Egt (10-Z)

Fig. 3. Stress components T,, and T,~ Uniaxial straining followed by simple shear. Comparison between universal algorithm (UA) and radial return (RR). (a) n =0.3,

(b) n = 1.

15.0 l- 1 n=l.N=O.l.y,,=0.~5 yoo

0 . -.3.X,

0 RR, 8 steps . RR, 2 steps o UA. 8 steps _... ^ ~.

I . “A. L steps

I I I 0.0 5.0 10.0 IS.0

v(lO_2)

Fig. 4. Effective stress r vs effective plastic strain y. Problem of Fig. 3b. Comparison between universal algorithm (UA)

and radial return (RR).

collinear with the stress a: at the beginning of the time step?,

with

1’, At 3G + @i>, 485 =T

0s (57)

t In the calculations the implicit version (0 = 1) of the TM is used.

6.0

t? n = 0.3. N = 0.1, y,, i 0.005 3

-exact 2.0 D q TM, 20 steps

0 VA, 20 srcp.p . UA, 2 ~feps

I I I 1 0.0 5.0 10.0 15.0

TW2)

Fig. 5. Effective stress r vs effective plastic strain y. Problem of Fig. 3a. Comparison between universal algorithm (UA)

and tangent modulus method (TM).

For $, = z/2, we find for the stress increments

Aa’=2G (

(58)

In the first increment after the change to simple shear, the shear components of Au’ increase efusti- cufly, while the normal components decrease according to the second term on the right in eqn (58). With increasing time steps, ?j dominates over 1, and the second term becomes independent of At. Eventually, the shear components predominate over the normal

(a) 0.025

r n = 0.3, N = 0. I, -y. = 0.005. 8 steps

0.020 - 4 step

Ay(‘)=Ae

0.000 0.005

- Eqn.(4la) -- Eqm(4lb)

J 0.010 0.015 0.020 0.025

(b)

0.020

r n = 0.3, N = 0.1, 7, = 0.005.8 steps

/ 1 step

0.015 -

-- Eqn.(4lb)

I 0.005 0.010 0.015 0.020

Fig. 6. Graphic illustration of the numerical solution of eqn (41a, b) starting from the plastic predictor. Problem of Fig. 3a with 8 steps. (a) Iterative solution for the 4 step (r = l), A = y is close to Ae. (b) Iterative solution for the 1

step (c = 1); Ay is significantly smaller than de.


components, leading to an increase of r. From this we see that, although always stable, the tangent modulus method is limited in its step size by accuracy requirements.

In Fig. 6 we illustrate the variation of the nonlinear functions h(Ay ) according to eqn (41a) and (41b). If the state at t = t, is located well within the plastic range, the plastic predictor c Ae is a close estimate of Ay, and only a single correction is necessary (Fig. 6a). Only when the elastic and the plastic strain increment are of about the same order, slightly more iterations are needed (Fig. 6b). The regions where this occurs, however, are confined to the early stages of yielding or relaxation.

Next, we study the effects of kinematic hardening and thermal softening. We use the following material parameters: N = 0 (no isotropic hardening), I?I = 0.2, Ho/a,, = 1.5, and the rate sensitivities n = 0, n = 0.3, n = 1. The loading consists of three different strain rates:

For n =0 and n=0.3: i,,= -2&=-22i,,= look,, 0 < i,t < 10-3; & = 6Oi,, lo-’ < i,t < 2~10~3;i,3=i3,=i,,=80i,,2~10-3~C,t~3~10~3.

For n=l:i,,=-2i,,=-2i,,=lOi,, O<i,t<

1.5. lo-*; i,, = 6i,, 1.5. lo-* < i,t < 3. 10-2; i,, = i3, = ii, = 8i,, 3. lo-* < i,t < 5. lo-*.

Figure 7as suggests that for kinematic hardening the UA gives a nearly exact answer for any exponent n, even when thermal softening effects are included (IT, > 0).

Finally, we take into account the non-coaxiality effects (a > 0). Numerical results are presented for low (n = 0.05) and high (n = 0.3) rate-sensitivity and isotropic hardening N = 0.1, y0 = 0.01. The strain rates are: i,, = -2c** = -2i,, = lOOi,, O<i,t <2.10-3; c 12 = 5oi,, 2 10-r < i,t < 4 10-3; i,, = i3, = i,* = 8Oi,, 4. 10m3 < i, t < 6. 10m3.

Figure 8 demonstrates the substantial effects of non-coaxiality on the stress distribution after a change in the loading direction. But still, there seems to be no limitation in the step size for the UA, since even with a single step in each loading interval we can achieve a highly accurate result. Again, a r-y plot (Fig. 9) shows a slight overestimation of the plastic strain measure y for very large steps.

6. DISCUSSION

The universal integration algorithm proposed in this paper provides accurate results, even for large time steps and for the entire range of rate-dependent elastic-plastic material behavior, including nonlinear isotropic and kinematic hardening, thermal softening and non-coaxiality effects. The performance exceeds those of gradient schemes and comparable radial return methods, because the universal algorithm allows a variation of Jo within the time step. This proves to be important in the computation of problems with high rate-sensitivity exponents n > 0.1. In particular, for n = 1 without hardening or softening, the exact

(a) 2.0

1.5 - exact, ATo = 0 --- exact, XT, = 0.8

0.0 1.0 Eat (10-3) *.O

3.0

n = 0.3. m = 0.2. Hdq = I .5

- exact. AT,, = 0 .-- exact, AT,, = 0.2

I I 0.0

I I .o

cot (10-3) 2.0 3.0

n = 1, m = 0.2, H,jq = I.5 - exact, ATo = 0

6.0 --- exact, AT,, = 0.1

bo4.0 3

2.0

0.0 1 .o 2.0 3.0 4.0 5.0

EOt ( 1 O-3)

Fig. 7. Stress components r,, and rn. Uniaxial straining followed by simple shear and combined shear. Comparison between solutions with and without thermal softening. (a) n =O, iT,=O, 0.8. (h) n =0.3, IT,=O, 0.2. (c) n = 1,

m,=o, 0.1.

solution can be recovered. As shown in the examples, inclusion of hardening or softening does not lead to significant errors, even for very large time steps.

For rate-independent or weakly rate-dependent material behavior (n < 0.05) excellent results are obtained. Because of the evaluation of the stress-rotation via the exact analysis of Krieg and Krieg, near-exact solutions for small n are available for arbitrary changes of the loading direction.

In high strain-rate loading, most metals show rate exponents between n = 0.005 and n = 0.05. The performance of the algorithm for such small values of n

does not differ significantly from that for n = 0, as


(a) 1.5

3.0

bo2.0 >

(I = 0.02

1 .o

t 0.0

E0t (I 0-3)

Cb) 4.0 n = 0.3. N = 0.1. y0 = 0.01

Fig. 8. Stress components r,, and r,?. Uniaxial straining followed by simple shear and combined shear. Comparison between solutions with and without coaxiality between stress and plastic strain rate. (a) n = 0.05, a =O, 0.08.

(b) n = 0.3, a = 0, 0.02.

shown in the numerical examples. On purpose we considered problems with--often unrealistically- high values of n (= 0.3, 1) in order to demonstrate that the algorithm can be successfully applied to types of material behavior, where other integration pro- cedures often show severe limitations.

The improvement in accuracy comes with no increase in numerical effort. In fact, we are able to show that the nonlinear equation for y can be solved within a few iterations, and the number of iterations decreases (sometimes even to zero) with increasing step

8.0

n=0.3,N=0.1.yo=0.01

6.0

b” 2

4.0 -cxPct. (1 = 0 ---exact. LI = 0.02

oa=o. 12steps l a=o. 3 steps

2.0 q a = 0.02, ilsteps

=a = 0.02, 3 steps

0.0 IO 20 30 40 50 60 70

rW2)

Fig. 9. Effective stress r vs plastic strain y. Problem of Fig. 8b. Comparison between coaxial (a = 0) and non-coaxial

(a = 0.02) solution.

sizes. This is an important feature of the plastic

predictor-elastic corrector scheme which we em- ployed in this algorithm. However, this type of operator splitting may also be applied to any other type of return mapping algorithms.

Acknowledgemenf-Peter A. Fotiu gratefully acknowledges support from an “Erwin Schrodinger” scholarship of the Austrian National Science Foundation FWF. This work was supported in part by AR0 contract DAAL03-86-K- 0169 and AR0 contract DAAL03-92-G-0108 to the University of California, San Diego.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

REFERENCES

J. H. Argyris, L. E. Vaz and K. J. Willam, Improved solution methods for rate problems. Comput. Meth. appl. Mech. Engng 16, 231-277 (1978). K. J. Willam, Numerical solution of inelastic rate processes. Comput. Struct. 8, 51 I-531 (1978). D. Peirce, C. F. Shih and A. Needleman, A tangent modulus method for rate-dependent solids. Comput. Srruct. 18, 875-887 (1984). H. Chen and E. Krempl, An adaptive time-stepping scheme for the viscoplasticity theory based on over- stress. Comput. Struct. 22, 573-578 (1986). I. C. Cormeau, Numerical stability of quasi-static elasto/visco-plasticity. Int. J. numer. Meth. Engng 9, 109-127 (1975). T. J. R. Hughes and R. L. Taylor, Unconditionally stable algorithms for quasi-static elasto/visco-plastic finite element analysis. Comput. Srruct. 8, 169-173 (1978). J. C. Simo and R. L. Taylor, A return mapping algorithm for plane stress elastoplasicity. Int. J. numer. Meth. Engng 22, 6499670 (1986). M. Ortiz and J. C. Simo, An analysis of a new class of integration algorithms for elastoplastic constitutive relations. Int. J. numer. Melh. Engng 23, 353-366 (1986). M. L. Wilkins, Calculation of elastic plastic flow. In: Methods of Computational Physics, Vol. 3 (Edited by B. Alder et al.), pp. 211-262. Academic Press, New York (1964). R. D. Krieg and D. B. Krieg, Accuracies of numerical solution methods for the elastic perfectly plastic model. J. Press. Vessel Tech. 89, 510-515 (1977). H. L. Schreyer, R. F. Kulak and J. M. Kramer, Accurate numerical solutions for elastic-plastic models. J. Press. Vessel Tech. 101, 226234 (1979). P. J. Yoder and R. G. Whirley, On the numerical implementation of elastoplastic models. J. appl. Mech. 51, 283-288 (1984). S. Nemat-Nasser and D. T. Chung. An explicit constitutive algorithm for large strain, large strain rate elastic- viscoplasticity. Comput. Meth. appl. Mech. Engng 95, 205-219 (1992). S. Nemat-Nasser, Rate-independent finite-deformation elastroplasticity: a new explicit constitutive algorithm. Mech. Mafer. 11, 235-249 (1991). S. Nemat-Nasser and Y. F. Li, A new explicit algorithm for finite-deformation elastoplasticity and elastovis- coplasticity: performance evaluation. Comput. Snwt. 44, 937-963 (1992). A. Chulya and K. P. Walker, A new uniformly valid asymptotic integration algorithm for elasto-plastic creep and unified viscoplastic theories including contin- uum damage. Int. J. numer. Melh. Engng 32, 385418 (1991). J. W. Rudnicki and J. R. Rice, Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23,37 l- 394 (1975).


18.

19.

20.

A. J. M. Spencer, Deformation of ideal granular materials. In: Mechanics of Solids (Edited by H. G. Hopkins and M. J. Sewell). Pergamon Press, Oxford (1982). S. Nemat-Nasser, On finite plastic flow of crystalline solids and geomaterials. J. appl. Mech. 50, 11141126 (1983). B. Balendran and S. Nemat-Nasser, Double sliding model for cyclic deformation of granular materials, including dilatancy effects. J. Mech. Phys. Solids 41, 573-612 (1993).,

APPENDIX A RADIAL RETURN METHOD FOR RATE-DEPENDENT

PLASTICITY INCLUDING NON-COAXIALITY OF STRESS AND PLASTIC STRAIN RATE

In some of the numerical examples in this paper we used an extended radial return algorithm for rate-dependent plasticity and non-coaxiality of Cp and u’ (a # 0). We present here the basic equations of this method.

According to the backward Euler method the rate equations are integrated with collocation at the endpoint t = I,, i.e.

Acr = (Ay - a Aa)p,, + (3/2)a ha’. (A.1)

We introduce this into the expression for the stress difference,

Aa’ = 2G(Ac’ - Acr)- (4.2)

and find

= & (2G Aeg - ($K Ay - 2Ga A~)c,,). (A.3)

From this relation we obtain the return direction as

P, + r,, 9 kh =

Jr ?, + 2@,cos *, + I ’ (-4.4)

with r<, and cos ((/, given by eqn (20). By multiplying (A.3)

with ph, we arrive at the following expression for the

effective stress at the end of the time interval:

oh = [(1 + 3Ga)(T; + ZT,cos II/, + 1)“’

- 3Ga]u, - K by. (AS)

Comparison of eqn (A.5) with eqn (15) leads to the esti-

mate of the rotation-factor [ according to eqn (22),

&=&Jr;:- 1,. (‘4.6)

The elastic predictor is now given by

2G1’ AI e’(l) = u; + ___,

I+ 3Gu (4.7)

Finally, having found Ay and A.T from the return map-

ping, the components of oh, c[, etc. are obtained from the

relations (52), but the ~~ estimated by eqn (A.4) instead of

eqn (50).

Documents

A universal integration algorithm for rate-dependent elastoplasticity