INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 26, 153-165 (1988)

GENERALIZED VISCOELASTIC MODEL FOR CREEP ANALYSIS COUPLED WITH

PLASTIC DEFORMATION

SANG HOON LEE T h e MacNeal-Schwendler Corporation, 815 Colorado Blod., Los Angeles, C A 90041, U.S.A.

SUMMARY

A creep analysis capability, using a generalized viscoelastic model, is introduced to process the creep behaviour coupled with elastoplastic deformation. The formulation is based on the step-by-step time integration of the Kelvin-Maxwell rheological model with non-constant parameters.

The concept of a rheological model is extended to the multiaxial stresses by the Prandtl-Reuss stress-strain relationship, from which the tangential stiffness matrix is formed for Newton's iteration. If the plastic deformation is coupled with creep, the algorithm will seek a solution in two distinct steps. Various choices of empirical creeplawsareavailable and small variations in temperature areallowedasimplemented in thegeneral purpose finite element analysis program.

INTRODUCTION

At the macroscopic level, the creep phenomenon is best observed in a uniaxial test under constant load and a relaxation test under constant strain at constant temperature. A specimen subjected to a constant uniaxial tension at an elevated temperature exhibits three distinct phases in a time frame: primary creep, secondary creep and the tertiary creep to r u p t ~ r e . ~ * ' ~ If the specimen is unloaded after some creep deformation, the elastic strain is immediately recovered and a portion of the creep strain is gradually recovered. The recoverable portion of the anelastic creep deformation is treated by the Kelvin model as primary creep and the non-recoverable portion by the Maxwell model as secondary creep. The tertiary creep, similar to necking in plasticity, is considered as a localized instability phenomenon, which is beyond the scope of this paper.

Elaborating on the Kelvin-Maxwell rheological creep model developed by Badani,' the creep behaviour is coupled with plastic deformation and the variable temperature effect is included. The Kelvin element represents the primary creep behaviour and the Maxwell element describes the secondary creep. For a generalization of the viscoelastic material behaviour, the rheological model parameters are treated as non-linear functions of the effective stress and the temperature. The step- by-step integration is performed assuming that the strain rate varies linearly during each time increment while the rheological model parameters remain constant for a short time interval.

A number of empirical creep laws, recommended by the Oak Ridge National Laboratory (ORNL), are provided along with options for general tabular input of the rheological model parameters as functions of effective stress. When the creep characteristics are specified in terms of empirical creep laws, the program converts the empirical formula to the corresponding rheological model.

0029-5981/88/010153-13$06.50 0 1988 by John Wiley & Sons, Ltd.

Received 27 January 1987 Revised 9 April 1987

154 S. H. LEE

_u

Figure 1. Rheological model

EFFECTS OF STRESS AND TEMPERATURE6

An analytical solution to the Kelvin-Maxwell model subjected to a constant stress (a) is given by

a - - e-(K~/Cp)tl Eirimary - -

K P

where t denotes time and parameters K,, Cp and C, are identified in Figure 1. For a varying stress case, however, an instantaneous strain rate should be employed to have the creep hardening effects properly accounted for. The creep hardening (and softening) effects, assumed to be exhibited by the primary creep, are accounted for by using the strain hardening formulation, wherein creep strain rates are expressed in terms of primary creep strain rather than time, i.e.

where 1

i i r imary =-(a - KpEir imary) CP

Notice that the creep strain rate reverses the sign with proper effects of the cummulative primary creep when the sign of the stress reverses. Due to this virtue, the stress reversal effects are properly taken into account without any special provision in the algorithm.'

The creep strain rate is, in general, a function of absolute temperature as well as stress and strain. Microscopically, the creep deformation is an integrated effect of dislocations of the crystal structure primarily due to thermal activation and stress. The effects of temperature at the microscopic level can be quantified as

EC = A ~ - M I R T

where AH is the energy of activation, R is the gas constant, T is the absolute temperature and A is in strain/unit time. On this basis, the creep model parameters measured at a reference temperature (To) can be used to compute the creep strain rate at the temperature Tin the vicinity of To, allowing small vairations in the ambient temperature. The creep model parameters are corrected as

(3)

where

GENERALIZED VISCOELASTIC MODEL 155

EQUILIBRIUM OF A KELVIN-MAXWELL MODEL FOR A STRESS COMPONENT

Suppose that the Kelvin-Maxwell model is applied to a typical component of a deviatoric stress- strain pair, denoted by (Asl, el). Referring to Figure 1, the state equilibrium equation of the model at any instant is expressed as

[C] (At} + [ K ] {Ae} = (As} where

and AS)^ = ( Asl, 0)

Introducing a linearly varying strain rate during the time interval, the strain rate increment can be expressed as

(6) 2 At

{At) = - [Adt) - t(t)At]

by which equation (5) is reduced to

[ C + K]{Ae) = 2 [C] {t) + {As)

Defining the stiffness of the primary and secondary creep elements by

2c 2cs At 2-x k , = K p + L and k -

(7)

the equivalent creep stiffness (k,) for a deviatoric stress-strain pair can be determined by

Solving equation (7) for Ael gives

k,Ae, = As' t Asl (9) where As' is a pseudo-incremental stress, determined by

- 62) + - 6, k, cp k l 1

The pseudo-incremental stress represents the change in deviatoric stress component due to creep relaxation. This value of As' can be converted to the equivalent pseudo-incremental strain (Ae'), which represents the change in deviatoric strain component due to creep. By virtue of creep volume constancy, deviatoric strains ( A d ) and the strain rates (6, and 6,) in equation (10) are identical to the ordinary strain components. Hence the pseudo-incremental strain can be expressed as

in which the first and the second terms are contributions of the Maxwell and the Kelvin elements,

156 S. H. LEE

respectively. Notice that as At approaches zero, equation (1 1) reduces to

A&' N At i l = At C,",,,,

which makes it easier to comprehend that equation.

ADAPTATION OF RHEOLOGICAL MODEL TO MULTIAXIAL STRESS STATE

The concept of the rheological model for a stress-strain pair is extended to the general multiaxial stresses with the aid of effective stress and effective strain rate by adopting the Prandtl-Reuss stress-strain relationship, i.e.

where creep strain rates

{ 2 } T = (i:, i;, i;, jL;y, ?fz, g g ) deviatoric stresses

= a;, a:, 22,,, 2Tyz, 2Tzx) and the effective stress (6) and the effective creep strain rate (2) are defined analogously to those in plasticity with the von Mises yield criterion.

Implied by this relationship there is a unique set of rheological parameters (Rp, c, and c,) based on the effective stress, which can be related to all the deviatoric stress components and the corresponding creep strain rates. In terms of effective creep model parameters, equations (8) and (1 1) may be rewritten as follows:

k - - ( K p + x ) . 2 - 2cp k ' = T z 4 cs ' - 3

and

The creep strain rates in equation (2) may be rewritten likewise, i.e. in terms of deviatoric stresses

3 {it"otal) =- {sI + { i i r i m a r y l

2CS where -

2CP CP

3 K ( i ir imary I = t ~ ) - 2 {EErimary 1

Then the pseudo-incremental strain in equation (1 la) is reduced

In the absence of plastic expressed as

deformation, the total strain increment (elastic and creep) may be

GENERALIZED VISCOELASTIC MODEL 157

where D , and D , are material matrices for elasticity and creep, respectively. However, the total strain increment must be corrected with a pseudo-incremental strain {Ad}, i.e.

{ A E , + A&'} = {A& - A d )

{Ag} = [D,,] {A& - Ad}

(1 5 )

(16)

Combining equations (14) and (15), we obtain

where the elastic-creep tangent matrix may be conveniently obtained for an isotropic material by

K + + k , , K - f k , , K - f k , , 0 0 K + + k , , K - + k , , 0 0

K + j k , , $kcc 0 0 0 0 !] (17)

3 k e c 0 3 k e c

C D e c l =

with K being a bulk modulus and k,, the stiffness of the elastic-creep component defined by

1 1 1 k,, - 2G + k, ---

where G denotes the shear modulus.

COUPLING OF PLASTICITY

When the plastic deformation is involved in the creep analysis, the plastic strain increment should be included in the total strain increment. The plastic strain increment may be obtained by7

{A$} = D,'{Ao} (18) where

with the plasticity modulus H = da/dP and the function ( f ) defining the effective stress (8). Introducing equation (18) into equation (14), the elastoplastic-creep stress-strain relations are established as

[D,-' + D c ~ ' + D ~ ' ] { A ~ } = { A ~ - A d } (19)

from which the stress increment {Ag} can be obtained. However, equation (19) may be rearranged as

(Aa} = [D,,] (A& - AE'} - AACD,,] - {XI with

where the effective plastic strain increment is denoted by AEp. The creep deformation tends to relax the stress gradient in the absence of further increments in

158 S . H. LEE

external loads. In the creep-dominant process, therefore, plastic deformation can be induced only by creep deformation to alleviate stresses in the neighbouring material. For this reason, the material routine employs a solution scheme which seeks a solution in two distinct steps when the plastic deformation is coupled with creep. First, it solves for the incremental stress components with an elastic-creep material as in equation (16). Then, if the new stress state (a = a,,, + Ao) exceeds the current yield stress, i.e.

F(o) = f(a) - 6 ( P ) > 0

where F is a yield function, a correction is made on a previously obtained incremental stress based on equation (20) and the plastic strains are computed. The algorithms for this process is depicted by the flow diagram in Figure 2.

IMPLEMENTATION

The quasi-static non-linear analysis capability for the generalized viscoelastic model coupled with the plastic deformation is implemented into the general-purpose non-linear static analysis method in MSC/NASTRAN. The solution method employs an adaptive Newton's iteration with accelerated convergence schemes such as the BFGS updates and the line search process. A non- linear static analysis can readily be converted to the quasi-static analysis and vice versa by adding or subtracting a few related data. All the potentially non-linear elements (ROD, BEAM, QUAD4, TRIA3, HEXA and PENTA) are applicable to creep analysis, but not all the non-linear elements in the model need to be composed of creep material.

The material routines are implemented to adopt a subincremental scheme by which the computations in material routines are carried out in small increments within a global incremental step. The creep routine subdivides an incremental time step such that the effective creep strain increments and effective stress increments are bounded by a fraction of the total creep strain and effective stress, respectively. If the plastic deformation occurs during the creep process, the creep routine calls the plasticity routine. The plasticity routine again subdivides the given increment into smaller increments if the increment in the yield stress exceeds a certain fraction of the current yield stress. The fraction which defines the bounds is a user specified parameter with a default value of 0.2. By virtue of this subincremental scheme within the material routines, the non-proportional loading or unloading does not present any difficulty. The non-proportional loading cases including neutral loading have been examined and validated by controlling either the stress or strain path in two-dimensional problems with the elasto-plastic material.

Users are allowed to either specify an empirical creep law or provide direct input of rheological parameter values as functions of the effective stress associated with the temperature at which the data are acquired. If the creep behaviour is prescribed by a creep law, creep law parameters will be converted to rheologicalparameters whenever anew stress stateiscomputedforthecreepanalysis. Rheological parameters are corrected for changes in temperature based on equation (4) if the operating temperature is different from the temperature at which creep behaviour is characterized. The empirical creep laws provided in MSCPASTRAN are in the following equation forms with user specified coefficients (a - 9):

and

for all combinations of types 1 and 2 for A , R and K , which are expressed as in Table 1.

E C ( 0 , t ) = aabtd (21)

E' (a, t ) = A (a) [ 1 - e - R(a)t] + K (a) t (22)

Conversion of creep laws in the form of equation (22) to the rheological model is performed by

Q Start

Lo

J B Yes

E

d

.- 1 C

CJF. and C,IF,

c

Compute k, and k , Compute {Ad} Form [DJ matrix

4

No

1 Solve elastic-creep problem for {Au} 1 t

Compute AE-'(AE') Compute A 8

Compute {A60,.Il Yes Compute n Compute A C C Compute Ar Update c - ' = E - < + AE" Store [Dec]

Update {G) = l e i } + {A&;} Compute {A$iMrJJ

Trial stress u' = u + A u

t -------------------

I Plasticity Routine

Compute { d } = D,,{a} a' A 0 , Compute A). = __

H + aTd '11 hardening Error Correction if required

I L

I I I I 1 I I I I I I I I I I I I I I I I I I I I

Table I.

Parameter Type 1 Type 2

A (4 a d a exp (ba)

K ( 4 e[sinh(fo)le eexp(fo) N o ) c exp (do) Cd

160 S. H. LEE

simply matching equivalent terms in equation (2). However, conversion of the popular and classical creep law in equation (21) to the rheological model is not so simple, owing to lack of similarity. In this case, equivalent rheological parameters are computed by matching the total creep strain and their first and second derivatives of equations (2) and (21).

The material routine computes the elastic-creep tangential matrix for the formation of a global stiffness matrix, which is used for the iteration combined with the BFGS update and line searches. An equilibrium condition is achieved within the convergence tolerance at every time increment. The numerical solution scheme employed in the present creep formulation is unconditionally stable owing to the implicit method adopted for integration. However, a limit should be placed on the time increment to achieve an accurate solution. The time increment should be selected so that the strain and/or stress do not change excessively in a single step.

The total creep strain increment can be recovered by subtracting {Ace) and {A&*}from the total {A&} in equation (19), i.e.

1 {A&Fota,} = [ D c ] - ' {Ao} + { A d } =-{AS} + { A d )

k C

The primary creep strain increment is saved in the database to take into account creep hardening/softening effects, i.e.

where 2 4 K

prrmaryj - {s} -pP (6" primary ' 1 {&' . 1 -- k , 3 k l

VERIFICATION AND VALIDATION

All aspects of creep capabilities were verified with respect to the algorithm, accuracy and programming errors with a wide variety of problems in MSC/NASTRAN. All types of elements (lD, 2D and 3D) have been tested and known solutions have been reproduced. Various creep laws in the form of equation (22) and the direct input of rheological parameters reproduced analytical solutions accurately with errors less than 0.1 per cent in the creep strain at the end of 70 steps. Creep laws in equation (21), however, produced a cumulative error of 15-1 9 per cent in the creep strain at the end of 70 steps, because of the mismatch between the empirical formula and the rheological model.

The creep analysis capability under variable temperatures is verified using empirical creep laws recommended by ORNL. There are two different creep laws (types 111 and 121) for the same material (type 304 stainless steel) established at different temperatures (1 100°F and 1200"F), as shown in Figure 3. The creep law type 11 1, defining the creep behaviour at 1100"F, is applied at the operating temperature of 1200°F with corrections for variable temperature. The results are compared with the creep law type 121, which predicts proper creep behaviour at 1200°F. The accuracy of the varying temperature case was found satisfactory only for a short period of time or for small temperature variations. It is difficult to define limits on the applicability of the variable temperature since they are material dependent. Nevertheless this feature is considered useful and essential.

Coupling of the plastic deformation with creep is verified by reproducing the isochronous stress-strain curve for stainless steel, type 304, as shown in Figure 4. The data points are obtained in 15 steps to the creep time of 100 hours.

GENERALIZED VISCOELASTIC MODEL 161

Creep Law 121 at 1200°F

with EXP - lo - ' ' >

/ Creep Law 11 1 Defined at 1 100°FI

0 1w m 300 400 M O CREEP TIME (HOURS)

0

Figure 3. Creep capability verification for variable temperature

Original Matorial X MSCINASTRAN

0 ASME Cod0 Prediction - Fiefersnco 1

1 0 0.4 0.8 1.2 1.8 2.0 2.4

TOTAL STRAIN (%)

Figure 4. Isochronous stress-strain curve

The creep behaviour is manifested in the relaxation process under constan strain. Figure 5 compares relaxation predictions by various methods. It is demonstrated that the current method is superior to the initial-strain method. Results from MSCPASTRAN analysis fall between experimental data and the closed-form solution.

The creep response of various elements to the stress reversal was thoroughly investigated. The effects of the creep hardening/softening were exhibited properly. However, the convergence occasionally required a stiffness matrix update and a smaller time increment at the onset of stress reversal.

The solution to the creep behaviour of an infinitely long thick-walled cylinder subjected to interanl pressure was presented by Greenbaum and Rubinsteh2 They employed an incompres- sible material with a simple empirical formula which accounts only for secondary creep, to which an analytical solution exists. MSCPASTRAN reproduced the solution very accurately using nine HEXA elements with axisymmetric and plane strain boundary conditions.

162 S. H. LEE

--- 0 closed Form PI

Initial Strain Method [l]

-m 0 0 20 40 TIME (HOURS) Bo 80 100

Figure 5. Comparison of creep relaxation predictions

L M I D ............ 400. L M I E ............ 500. Level F ............ 800. Level 0 ............ 7M). Level H ............ 8M). W I ............ 800. L e d J .......... 1ooO. We: Slms Unltl in kai

Figure 6. Effective stress distribution after 3 hours of creep

The creep behaviour of a thick-walled pressure vessel with a flat-end closure was also analysed by Greenbaum and Rubinstein2 under an internal pressure of 445 psi with a material obeying an empirical creep law in the form of

E' = (19.8 x 10- 16)(c73.61)(t1.06)

A finite element model of the same pressure vessel was analysed by MSC/NASTRAN using 72 solid elements with 355 active DOFs. Figure 6 shows stress contours at t = 3 hours. For a model five times coarser, this solution represents an excellent agreement with that in Reference 2.

NOZZLE-TO-SPHERICAL SHELL ATTACHMENT AS BENCHMARK PROBLEM

The High-Temperature Structural Design (HTSD) program of ORNL was instigated to develop design methodologies for breeder reactor structures. Under the ORNL program, a nozzle-to- spherical shell attachment model was presented as a benchmark problem with extensive experimental3 and analytical435 results. The model represents a liquid-metal fast breeder reacter component, made of type 304 stainless steel, subjected to successive cycles of loading with internal pressure and end moment at 1100°F.

GENERALIZED VISCOELASTIC MODEL 163

Figure 7. Finite element model by MSC/NASTRAN

5.0 o Experiment .- .-. Pressure Loading - MSCINASTRAN

Figure 8. Total circumferential strain at gauge location during pressure loading

The model depicted in Figure 7 represents this component using 1552 DOFs, which is analysed by MSCPASTRAN with elastoplastic-creep material and geometric non-linear effects. The creep characteristics are defined by a creep law in the form of equation (22) with ORNL recommended values for parameters as follows:

A(0) = 6.985 x 10-6a2'444 R(a) = 7.032 x 10-4exp(0.1072a) K(a) = 6.73 x 10-g[sinh(0.1479a)]3'0

where the stress (a) and the time (t) are measured in ksi and hours, respectively. Typical results at selected points, where the plastic deformation is significant, are compared with

data in References 3,4 and 11 as shown in Figures 8.9(a) and 9(b). It is noted that MSC/NASTRAN used an isotropic, instead of a kinematic hardening rule as employed by Levy.4 Figures 10(a) and lqb ) show effective-stress contours upon unloading at t = 2546 and t = 3400 hours, respectively. These results demonstrate proficient capabilities of the general purpose program for a complex problem with combined non-linearties.

164 S. H. LEE

m TIME (HOURS)

Figure 9(a). Total circumferential strain at gauge location (1) during moment loading

,,: o Eapriment

4.0 -

-1.0 l e o o r n 2 m 2 4 0 0 2 8 w 2 M x ) 3 M ) o 3 2 0 0 3 u ) o :

TIME (HOURS)

Figure 9(b). Total circumferential strain at gauge location (2) during momcnt loading

Numbers Represent Stress in ksi u Figure lO(a). Effective stress contour plot at t = 2546 hours upon loading

GENERALIZED VISCOELASTIC MODEL 165

u Figure 10(b). Effective stress contour plot at t = 3400hours upon loading

CONCLUDING REMARKS

One of the virtues of the present method is the ability to accommodate stress reversal with ease. Effects of creep hardening and softening were properly reflected in the solution to the stress- reversal case with no special efforts. The creep analysis capability in MSC/NASTRAN is a generalization of the viscoelastic-plastic material capability. The validity of the theory and the algorithm is proven and the analysis results show close agreement with experimental data.

REFERENCES

1. M. P. Badani, ‘A new method to solve nonlinear creep problems in structural mechanics using incompatible

2. G . A. Greenbaum and M. F. Rubinstein, ‘Creep analysis of axisymmetric bodies using finite elements’, Nucl. Eng. Des.,

3 . J. A. Clinard ef 01.. ‘Elevated-temperature deformation and failure testing and analysis of nozzle-to-spherical shells:

4. A. Levy, ‘High-temperature inelastic analysis’, Comp. Strucl., 13, 249-256 (1981). 5 . A. Levy and A. B. Piko, ‘On computational strategies for problems involving plasticity and creep’, Int. j. numer. methods

6. S . H. LCC. ‘Application of the rheological model to the creep analysis coupled with plastic deformation’, 8th S M I R T

7. 0. C. Zienkiewicz, The Finite Element Method, McGraw-Hill, London, 1977. 8. H. Kraus. Creep Analysis, Wiley, New York, 1980. 9. N. H. Polakowski and E. J. Ripling, Strength and Struclure oJ’ Engineering Materials, Prentice-Hall, Englewood Cliffs,

isoparametric finite elements’, Ph.D. Dissertation, U.C. Berkeley, 1980.

7, 379-397 (1968).

spccimens NS-2 and NS-1’, Oak Ridge National Laboratory. ORNL-5939, Jan. 1983.

eng., 17, 747-773 (1981).

Conference, Brussels, Aug. 1985.

N.J., 1966. 10. R. C. Juvinall, Stress. Strain and Strength, McGraw-Hill, New York, 1967. 1 1. A. Levy, ‘High-temperature inelastic analysis of a pressure vessel nozzle attachment’, Grumman Aerospace

Corporation, ORNL/Suh-4485/3, June 1980.