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A theory for analysis of thermoplastic materials
A THEORY
FOR ANALYSIS OF THERMOPLASTIC MATERIALS
DAVID H. ALLEN~ Engineering Science and Mechanics Department. Virginia Polytechnic Institute and State University, Blacksburg. VA 24061, U.S.A. and WALTER E. HAI%ER Associate Professor. Aerospace Engineering Department. Texas A&M University, College Station, TX ‘77843,U.S.A. (Received 8 Mari 1980) AIrstract-The purpose of this paper is to present a theoretial model for predicting the behavior of elasticplastic materials subjected to cyclic mechanical and thermal loading. The considerable literature in this area gives ample evidence of its significance. Solutions are sought in the areas of laser technology. nuclear reactor design, and turbine analysis, to name but a few. There are four secttons contained in this paper: (1) the formulation of a special purpose constitutive law; (2) review of a variational principle for use with the constitutive law; (3) a discussion of the application of the model to a computer code: and (4) a comparison of several experiments to results obtained in the theoretical model.
stress space. which for temperature dependent combined isotropio-kinematic hardening is:
INTRODUCTION In this paper the classical isothermal rate independent
theory of plasticity is extended to obtain a nonisothermal theory for thermoplastic materials. This extension is performed in two parts: first, it is assumed that material properties are temperature dependent; and second. an uncoupled rate dependent strain tetm is included to account for creep. In order to account for the Elauschinger effect during cyclic loading the theory includes a combined isotropiokinematic workhardening rule. The resulting incremental constitutive law includes a term attributable to nonisothermal loading and not contained in the isothermal theory. The addition of this temperature dependent stress increment into the constitutive law necessitates the derivation of a variational principle which differs from the wellknown isothermal principle. Thus, a virtual work equation is presented for use with the finite element method. This section is followed by a short discussion of the computational procedure used to implement the constitutive law to a computer. In the final section results of several example problems are presented. The impetus of these sample problems is twofold in nature. First, examples will be utilized to clarify certain computational aspects of the constitutive theory, and second, experimental data will be compared to theoretical results as a means of verificatton
F(S,j-a,j)~k’(S
d%‘, T),
(1)
where S,=second ~ol&~rc~off stress tensor aij=coo~dinates of the yield surface center in stress space, dg = uniaxial plastic strain increment, T = temperature, and l dip is a state variable representing the plastic strain history dependence in the yield function. Note that the explicit temperature dependence on the r.h.s. of eqn (1) precludes kinematic hardening due to temperature changes Cl]. A schematic representation of eqn (1) is shown in Fig 1. ~ffe~ntiating eqn (1) givesthe follo~g consistency condition during plastic loading: r(
rFdS,,(q
4
&:da,,-2k 1,
$- d:‘-2/+2+
d”j-_-O,(2)
where the term (~F/SSij) represents ~F(Si,-q,j)/liS, evaluated at Sj-aij which can be seen to be equtvalent to ~?F@S,~--tl,~t. Since during neutral loading the plastic strain increment and db,j are zero, it is apparent that a statement governing loading is: T
$
dS,-2k; dT>O ‘I In addition to the statement of consistency during loading given by (2), the following associated flow rule is employed :
of the theory.
PRESENTATIONOF THE CONSTITIJTIVETHEORS Recall that in the incr~ental theory of plasticity the
workhardening
rule is defined by a yield function in where dEE represents the plastic strain increment tensor.
tFormerly at Texas A&M University. 1’9
DAVID
H
and
ALLEN
WALTER E HAISLER
by Yamada [4, 51 m that the term containing dT IS written as a separate portion of the stress increment in the latter’s derivatton. It w11lbe shown. however. that the two are mathemati~aliy equivalent. UnfortuMtel~. in the form presented in eqn (8) it is not possible to determine the stress increment due to the occurrence of several undetermined parameters on the r.h.s. Yamada has accounted for these terms bt introducing the plasuc work rate. We have chosen a siightly different method for determining the effect of these unknowns. As presented in a previous papeiby us [6], we assume that there exists a scalar parameter C.called the hardenmg modulus. which when multiplied by the plastic strain mcrement and subtracted from the stress increment wdl be parallel to a tangent to the yield surface. Mathematically. this may be stated as:
Fig. I. Yield surface as a functwn of stress and temperature.
The stress at any time is assumed to be given by: S,, =
Dij,,(E,,
-
Ei, - Ei, - Ezn)7
(5)
where DZlnnis the elastic constitutive tensor at time E. In~emen~tion of eqn (5) was shown in our previous paper [2] to give: dS,,=D:,‘;P,‘(dE,,-Deb,-dEC,,-Deb,) +dD. r,lnn(E’mn-Ep’!?I”--EC’ Inn-Er’)mn.
If one employs the no~aI~ty condition in eqn (IO) and substitutes the result into the constitutive relation leqn 8). the result is:
16)
where D:,;, = etastic modulus tensor at time t + At, dDij,,=change in the elastic constitutive tensor (due to a temperaturechange) during time step At. E,,= total strain tensor, EL,= plastic strain tensor, Ez, = creep strain tensor. EL, = thermal strain tensor. and all superscripts t indicate me~urement at time t. The superscript t+ At on the elastic constitutive tensor is necessary due to the fact that the time step is finite rather than infinitesimal. Note that incrementation of the constitutive law results in a stress rate which is not rotationally invariant unless an infinitesimal strain measure is used. Therefore, the resulting theory is applicable eniy to small deformations. The final expression required to complete the constitutive law is the hardening rule. According to Ziegler’s modification [3] a tensorially correct statement is: d~,j=d&Si,-=i_$,
It can be seen from an examination of the consistency condition (eqn 2). that in order for the above statement to be valid. the foltowing must hoid:
(7)
where p is a scalar to be determined by the consistency condition (eqn 2). The constitutive law is thus obtained by substituting eqn (4t into eqns (2) and (6) and solving for the stress increment tensor. The resulting equation is: The above constitutive law differs from thai obtained
x(dE,,-dE;,-dE,r,)
x tE~~-E~“-E~~-E~~) i-d&,,&;,
- EgB-E:= -EL:).
111)
The simplicity ofthe above formulation can be seen when one uses the normality condition in conjunction wrth eqn (9) to obtain $ e=L . g
dS, dEP,
2 do =-L* 3 dBP
(13)
Thus. for isothermal loading it can be Seen that the hardening modulus is simply two-thirds the instantan-
A theory for analysis of thermoplastic materials eous slope of the uniaxial stressplastic strain diagram. However, for nonisothetmal loadings this is not the case. To see this, note that since the uniaxial stress is a function of both the plastic strain history and temperature, an increment of uniaxial stress is given by:
131
resulting constitutive law is reduced from a set of nonlinear coupled fvst order di&ent.ial equations by using a first order Rungs-Kutta process. THE VARIATIONAL PRINCIPLE
constitutive law may be applied to the conservation of momentum via an appropriate variationat principle. We present here briefly an incremental principle utilized in several nonlinear programs such as AGGIE I [S]. NONSAP [9-j and ADINA [lo]. First consider the virtuai work expression within a total Lagrangian description [ 1l] : The
(13) Combining eqns (12) and (13) gives: (14) where H’ is the instantaneous slope of the stressplastic strain diagram, Substituting the above equation into the constitutive relation (e.qn 1l), then rearranging and employing the normality condition. gives dS,-
~~ddE,-dEf,-dE~,)+dP,j,
where D!+bl
Cl,,=D
--
ijmn
ZF dF
--
‘luWa,, Es,,
t+At
,+.a,,
Dwmn .? .
ggg+p+$Lg$ w
P9
P4
83
and
dP,=
-
S’+A’6E’fArdV= TSu PoFduk dV (19) k k dA+ iJ ir I V0 I VO I A where $$ *’ = the 2nd Piola-KirchhoR stress tensor at time r+At referred to the initial configuration at (15) time t-0, 6g;b’= the variation in the Green-Lagrange strains at t + At referred to the initial configuration V,,, T,= the surface tractions at time t+Ar referred to the surface of the configuration A, &A,= the variation (16) in the displacements. po= the local density in the initial configuration and Fk= the body force per unit mass at time r + At referred to the initial configuration V,. We consider finite strain measure here for two reasons: this is the formulation already contained within the above mentioned nonlinear computer codes: and the constitutive law presented herein may be extended to encompass finite strain using an appropriate hypoelasticity theory [12,13-j. The virtual work equation is incremental&d using: S:;” = S;, + AS,,
(20)
and E:;“*=E:,+AE,,
and the terms H’ and &r/o”Tmay be determined from uniaxial strestitram data a1 time t. It can be seen that Ciimn is superscripted at time t since the hardening parameters are determined there. Note that the first two terms in dP,, represent the change in the effective modulus tensor during the load increment multipLed by the elastic strain tensor at time t, and the last term represents a correction term which results from using the hardening modulus, H’, at time t. The translation of the yield surface in stress space may now be obtained by substituting eqn (7) into eqn (2) and solving for dp. The resulting relation is:
(21)
The stress in~~ent AS, is then substitute using eqn (15) and the strain increment is implaced by incremantaliting the Green-Lagrange kinematic relation [Z]. The resulting variational principle is then linearized by neglecting terms nonlinear in the displacement increment. and this approximation is accounted for later by using an appropriate iterative technique [ 14,153. The linearized variational principle is then cast in a finite element matrix formulation by using an assumed displa~ment function. The resulting equations of motion are: [N][~~‘+*‘),s([K’L]+[K:
&Au)=fR+“‘)
--{F’;,
(22)
where
_I
dfi”
+$dSv-2kgd7-2k$dZP ” 21:
(L-a,,)
.
(18)
-jF&
Equation (18) thus guarantees that the state of stress will remain consistent with the yield surface during loading even under nonisothermal conditions. To summarize, then, the constitutive law is obtained by solving, in the following order, eqns (17), (16). (15), (18) and (7). It should also be noted here that the above constitutive theory satisfies thermodynamic restrictions [7], where the entropy generation term is introduced as an uncoupled rate independent (plastic) and rate dependent (creep) dashpot in a Maxwell model with nonlinear temperature dependent modulus and viscosity. The
(25)
and -{F’;
=
[BJr({S) -[C](AE=f
-[C]fAEr: + (AP},dv;
(27)
DAVID H ALLEN and WALTER E. HAISLER
12
and [&I and E&d are the linear strain dispiacement transformation matrix and the nonlinear strain displacement transformation matrix [2. 111.respectively. Also. H is the matrix relatmg nodal displacements to the displacement field [I I]. COMPUTER CODE IMPLEMENTATIOFi
The constitutive law and variational pnnciple presented herein have been implaced in the finite element code AGGIE I [8]. This code has the capability to model both geometric and material nonlinear structural response. and contains two and.three dimensional isoparametric elements [ 161. Under certain loading conditions the constitutive law (eqn 15) may be specialized for computational efficiency [2, 171. Thus, the theory presented herein has been implanted in the code as four different material models: quasiisothermal eiastic. nonisothe~al elastic. quasi-isothermal elasticplastic, and nonisothermal elastic-plastic, where quasiisothermal is defined to tiean that although the component undergoes thermal loading the material properties may be assumed to be unaffected in the temperature range considered. Utilizing the appropriate model has been shown to give significant computational savings L-171. The amount of input data required will depend on the model being used. For the nonisothermal elasticplastic model (eqn 15) with negligible creep the required input data are shown in Fig. 2. If significant creep is expected it will be necessary to input additional data which will depend on the creep model being used. Currently, there are three creep models within the code [S]: a microphenomenological equation of state approach. an intepolation scheme using creep vs time curves. and a nonlinear viscoelasticity model. Normally, it will be necessary to input either creep curves at specified loads and temperatures, or a creepcompliance tensor [18]. One will recall that in order to solve the thermal
l
, T
1
T2
/
T3
i
T4
Fig 2 Matenals input data.
-T
stress problem one must characterze the temperature distribution within the medium. It is assumed in this theory that the mechanical deformation and heat transfer problem may be solved a prrori. AGGIE I does not have the capability to solve this problem. Therefore. an additional input requirement is that the spa&l and time dependent temperature distribution be input to the code on a nodal basis. The procedure for utilizing the constitutive law is to assume a strain increment for each integration point based on information from the previous load step. One may then check for yielding by assuming eqn (6) holds and applying the resulting stress tensor to eqn I 1). If yielding 1s not predicted then eqn (6) is correct. If yielding occurs, then the stress state is updated using eqn I 15)and the yield surface location is updated using eqn (71. A complete outline of this procedure IS contamed in Ref. [19]
EXAMPLE PROBLEMS I. Nonisothermul
elasric axial bnr subjecred tuneous mechanical load and heat itzput
to s~mu/-
The first example demonstrates the capability of the code to predict the static response of elastic materials with strongly temperature dependent material properties to simultaneous mechanical and thermal loading, A significant factor in the accuracy of the theory is the correct determination of the thermal strain increment during a nonisothermal load step. Suppose one assumes that the thermal strzhrinerement is given by: dET=Y+“(7;L-T;,).
(28)
This assumption can introduce sign&ant error mto the analysis. The proper definition is given by: dsr=,‘+~9?;,-TR)-r*(Z;,-TRf =r’+“‘f?;,-I;,)+(~t+A’-~cW?;,-7k),
(29)
where TR is the reference temperature at which the thermal strain is zero. It can be seen that the second term in eqn (29) represents the error incurred by using eqn (28). Mathematically, eqn (29) may be interpreted as representing a chain rule differentiation. To illustrate the error which may be incurred by utilizing eqn (281, the code is now compared to an experiment. An aluminum (606LT6f axial bar with material properties as shown in Fig. 3 is subjected to the thermomechanical load history shown in Fig. 4. Due to the relatively short time period and low stress level. creep strain is assumed to be negligible. Analytical results are compared in Fig. 4. Experimental results as well as the theoretical result denoted CREEPARHS [20] are due to Stone. Tbree solutions were performed m AGGIE I using a single plane stress ¶metric element and 26 load steps. In the first analysis it is assumed that material properties are not temperature dependent and room temperature data are used. In the second solution, properties are assumed to vary with temperature. and the definition of the thermal strain Increment given by eqn (28) is used. Finally, the third analysis employs temperature dependent material properties as welf as the definition of the thermal strain increment given in (29). It is found that all theoretical results except the last are m error by approx. 18, or more. Thus. the importance of temperature dependent material properties as well as a correct definition of the
A theory for analysis of thermoplastic materials
133
0; I.78 U,T 42
t
0
_
.
.
I
SlNaN
Fig. 5. No~soth~al
,v
.&I
Fig. 3. Material data for Al 6061-T6 experimental test samples.
*.OOS-
f
s 0.006
-
Z. ?i
Ik
EXPElNEWT
0.004-
0.002
-
CREEP&NNS ABOliE I-eantmt ASWE I-Con APBIE I-Eq”
-
/
o.oooy, / 0
2
, 4
I,,
I,
6
Tw2 &l
--
-
0
----
s
-.-.-
11 -
,
10
-
*
-
,
I2
-
I
14
Fig. 4. Comparison of analysis to experiment for uniaxiai elastic thermomechanical loading. thermal strain increment are illustrated by this example problem. II. Nonisothmal elastic-plastic axial bar In this example an axial bar with material properties shown in Fig. 5 is subjected to the load history shown in the same figure. Attempts at experimental verification of this problem have failed due to the fact that the relatively long heat up time required with the equipment available at this institution induces significant creep. Additional equipment is on order and it is hoped that experimental verification will be forthcoming in the near future. This example is incIuded not as a verification of the theory, but rather as a corroboration of the computational ef5ciency of the model. One will recall that in our previous papers [2,19], we
t
ttbt
TIME
elastic-plastic ax2 bar with thermomechanical bad history.
proposed that to obtain the effective modulus tensor CiJrnnfor a load step one should use the elastic constitutive tensor Dtinurat the end of the load step. This was shown to be mathematically correct and is computationally supported by this example. In theexample, an axial bar is loaded isothermally to some plastic state and is then simultaneously subjected to a mechanical load and spatially constant slow heat input. According to the classical incremental theory of plasticity it is required that the state of stress and strain move to a point on the uniaxial stress-strain diagram for the temperature at the end of the step. This requirement must be satislkd in order to remain consistent with the yield surface during plastic Ioading Using the eIastic modulus proposed by us the theory will predict this result exactly. If one utilizes the elastic modulus at the start of the load step the state of stress and strain will converge incorrectIy to a point denoted by the head of the dashed line in Fig, 5. This point corresponds to a horizontal tran~ation from (e’, &) to the stress-strain curve at the temperature at the end of the step, followed by a translation parallel to the stress strain curve at time Thus, utilizing the elastic modulus at the start of a load step does not satisfy the consistency condition. Further, if one empIoys quibble iteration and correctly up dates the elastic modulus during the iteration procedure, the solution will converge to the correct solution, but in exactly twice the computation time encountered in our theory. Therefore it is suggested that using the elastic wnstitutive tensor at the end of the step is not only consistent but also wm~~tioMlly eBicient. III. Nonisothermal elastic-plastic axial bar with signiJi cant creep
problem illustrates the ability of the theory to predict the response of materials near their ultimate strength. An aluminum (6061-T6) axial bar is subjected to the load history shown in Fig. 6, such that ultimate failure of the specimen occurs. Experimental and This
DAVIDH. ALLENand WALTER E.
I34
HASLER
i’Z& ,,,y Jc----12 6” ,’ 1, ** \ I’ o- loo-’ tl t2 13 0.03
-.“---
-----
s-
AGGIE I - no creep AGGk5 I CAEEPARHS
wtrncr*4p 4 ;1
-12 98
T
14
b
TIME
,\
‘\ \
\
0.01
CREEPARHS [ZO] results are due to Stone. Two analyses were performed using AGGIE I. In the first, creep is assumed to be negligible. In the second, linear interpolation of isothermal creep data at a load of 20 ksi is used. It is seen from results plotted in Fig. 5 that &he theory produces amurate results even near the ultimate strength of the material. IV. ~o~~ff~h~rnu~eiastic-plastic axial bar subjected to cyclic load history One of the primary purposes of this research has been to obtain a theory which can effectively model the response of elastioplastic media to cyclic mechanicat and thermal loading. A&ho@ the literature contains abundant verification tools for isothermal cyclic loading histories we have been unable to obtain experimental data to verify the nonisothermal problem. Therefore, WE have undertaken to worm certain no~~th~~ tests using axial bars on the MTS system. Although results of our experiments are incomplete at this time, analytical results of a cyclic test are presented herein. Al~ou~ the theory is capable of modeling creep response, this study is meant to verify the time independent behavior of the material. Therefore, we have chosen the test case shown in Fig.3. Note that the specimen is heated at zero load after prestrain so that no creep occurs. This problem thus tests the expansion and translation of the yield surface in stress space caused by a temperature change, In order to verify the applicability of the theory it is necessary to perform three materials data tests. Isotherma stressstrain cmvzs are generated at room temperature and 200°F. In addition, an isothermal cyclic load test is performed to determine the ratio of isotropic to kinematic hardening (fl) used in the model. Theoretical results are presented in Fig. 7. It is seen that the combined hardening model produces results which differ signifkantly from both kinematic and isotropic nonisothermal hardening theory as well as isothermal isotropic hardening theory, indicated by the dashed line in Fig. 7. A schematic repletion of the fransformation of the yield surface is shown in Fig. 8. A future paper will compare experimental results to the analytical solution presented here.
Fig. 7. Elastic-plastic axial bar subjected to cycric that* mechanicltf load history.
Fig 8. Nonisothermal comb&d hardening (/3-0.5, of yield surface for eiastie-plastic axial bar.
We have proposed herein a theory for modelling the response of robotic materi& The theory.has been shown in this report to be adequate in predicting response of many solid media. In addition, it has been shown that certain ~pu~tio~y simpliied forms of the theory are correctly in place in the computer code AGGIE I. However, the theory may be inadequate in modelfine; certain physical phenomena. Among theSe are rate dependence, instability near ultimate strength, finite stress and strain, phase changes. and violation of other a~umptions in the theory such as the normality condition. Research is currently underway to incorporate the above additions to the theory.
A theory for analysis of thermoplastic materials REFERENCES 1. A. Phillips and R. Kasper, On the foundations of thermoplasticity-an experimental investigation. J. Appl. Mech. 891-896 (Dec. 1973). 2. D. H. Allen and W. E. Haisler, The application ofthermal and creep effects to the combined isotropic-kinematic hardening model for inelastic structural analysis by the finite element method. Office of Naval Research, No. 3275-79-3 (May 1979). 3. H. Ziegler. A modification of Prager’s hardening rule. Q. Appl. Math. 17.55-56 (1959). 4. Y. Yamada and T. Sakurai, Basic formulation and a computer program for large deformation analysis. Pressure Vessel Technoloar: Part I. DD.341-352. ASME. New York (19771. “. *’ 5. Y. Yamada. Constitutive modelling of inelastic behavior and numerical solution of nonlinear problems by the finite element method. Compur. Srrucrures 8. 535-543 (1978). 6. D. H. Allen, An investigation into the combined isotropic-kinematic workhardening rule. Office of Naval Research. No. 3275-79-2 (May 1979). 7. D. H. Allen, A survey of current temperature dependent elastic-plastic-creep constitutive laws for applicability to finite element computer codes. 21st AIAA/ASCE,/ ASME SDM Conference, Seattle (May 1980). 8. W. E. Haisler. AGGIE I. A finite element program for nonlinear structural analysis. Office of Naval Research. No. 3275-77-l (June 1977). 9 K. 1. Bathe and E. L. Wilson. NONSAP-A nonlinear structural analysis program. Nucl. Engng and Design (June 19741. 10. M. D. Snvder and K. J. Bathe, Formulation and NumerIcal Solut;on q/ Thermo-elastic-Plastic lems. NTIS (1977).
and Creep Prob-
11. B. Hunsaker, Jr.. The application of combined kinematic-isotropic hardening and the mechanical sublayer model to small stram inelastic structural analysis by the finite element method. Dissertation. Texas A&M University (Aug. 19761. 12. C. A. Truesdell and W. Nell, The non-linear field theories of mechanics. In Encwlopedia of Phpsics. Vol. 3. Part 3. Springer-Verlag. Berlin (1964). 13. S. W. Key, 1. H. BifBe and R. D. Krieg. A study of the computational and theoretical differences of two finite strain elastic-plastic constitutive models. Formulations and Computatronal Algorirhms in Finire Element Analysis: U.S.-Germany Symposium. MIT, New York (1976). 14. J. A. Stricklin. W. E. Haisler and W. A. Von Riesemann, Formulation. computation, and solution procedures for material and/or geometric nonlinear structural analysis by the fimte element method. Sandia Laboratories. No. SC-CR-73-3102 (July 1972). 15. G. Strang and H. Matthies. Numerical computations in nonlinear mechanics. Pressure Vessels and Pzptng Con-
135
ference. ASME, New York (1979). 16. 0. C. Zienkiewicz, The Finite Element Method. McGrawHill. New York (1977). 17. D. H. Allen and W. E. Haisler, The prediction of response of solids to thermal loading using the finite element code AGGIE I. Office of Naval Research. No. 3275-80-l (June 1980). 18. D. R. Sanders and W. E. Haisler, An incremental form of the single-integral nonlinear viscoelastic theory for elastioplastic-creep finite element analysis. Pressure Vessels and Piping Conference. ASME, New York (1979). 19. D. H. Allen and W. E. Haisler, Thermoplastic analysis using the finite element code AGGIE 1. Office of Naval Research No. 3275-79-4 (Dec. 1979). 20. S. J. Stone, Effects of material behavior on the response of rapidly heated structures. McDonnell Douglas Astronautics Company, MDC G8432 (Jan. 1980). 21. C. E. Pugh. J. M. Corum, K. C. Lin and W. L. Greenstreet. Currently recommended constitutive equations for inelastic desigd analysis of FFTF components. ORNL TM-3602, Oak Ridge National Laboratories (Sept. 1972). 22. W. E. Haisler, Numerical and experimental comparison of plastic work hardening rules. Trans 4th lnr. Conj. on Structural Mechanics in Reactor 7echnology. San Francisco (15-19 Aug. 1977). 23. W. E. Haisler, B. 1. Hunsaker and J. A. Stricklin, On the use of two hardening rules of plasticity in incremental and pseudo force analysis. Presented at the Winter Annual Meetinn of the ASME. New York; also in Constuutlve E&&ions in Viscoplasticity Computational nnd Engineering Aspects. AMD Vol. 20, pp. 139-170 (5-10 Dec. 1976). 24. M. Tanaka. Large deflection analysis of elastic-plastic circular plates with combined isotropic and kinematic hardening. lngenieur-Archiv 41,342-356 (1972). 25. B. Hunsaker. Jr., An evaluation of four hardening rules of the incremental theory of plasticity. Thesis, Texas A&M University (Dec. 19731. 26. W. E. Haisler and D. R. Sanders, Elastic-plastic-creep large strain analysis at elevated temperature by the finite element method. Office of Naval Research. No. 2375-78-l (Apr 1978). 27. D. R. Sandersand W. E. Haisler. An incremental form of the single-integral nonlinear viscoelastic theory for elastic-plastic-creep finite element analysis. Office of Naval &search, No. 3275-79-l (Apr. 197%. 28. K. H. Bathe. H. Ozdemir and E. L. Wilson. Static and dynamic geometric and material nonlinear analysis. Structural Engineering Laboratory, University of California. Berkeley. California Rep. UCSESM 74-4 (Feb. 1974). 29. P. Sharifi and D. N. Yates. Nonlinear thermo-elasticplastic and creep analysis by the finite-element method. AlAA .I 12, 1210-1215 (1974).
FOR ANALYSIS OF THERMOPLASTIC MATERIALS
DAVID H. ALLEN~ Engineering Science and Mechanics Department. Virginia Polytechnic Institute and State University, Blacksburg. VA 24061, U.S.A. and WALTER E. HAI%ER Associate Professor. Aerospace Engineering Department. Texas A&M University, College Station, TX ‘77843,U.S.A. (Received 8 Mari 1980) AIrstract-The purpose of this paper is to present a theoretial model for predicting the behavior of elasticplastic materials subjected to cyclic mechanical and thermal loading. The considerable literature in this area gives ample evidence of its significance. Solutions are sought in the areas of laser technology. nuclear reactor design, and turbine analysis, to name but a few. There are four secttons contained in this paper: (1) the formulation of a special purpose constitutive law; (2) review of a variational principle for use with the constitutive law; (3) a discussion of the application of the model to a computer code: and (4) a comparison of several experiments to results obtained in the theoretical model.
stress space. which for temperature dependent combined isotropio-kinematic hardening is:
INTRODUCTION In this paper the classical isothermal rate independent
theory of plasticity is extended to obtain a nonisothermal theory for thermoplastic materials. This extension is performed in two parts: first, it is assumed that material properties are temperature dependent; and second. an uncoupled rate dependent strain tetm is included to account for creep. In order to account for the Elauschinger effect during cyclic loading the theory includes a combined isotropiokinematic workhardening rule. The resulting incremental constitutive law includes a term attributable to nonisothermal loading and not contained in the isothermal theory. The addition of this temperature dependent stress increment into the constitutive law necessitates the derivation of a variational principle which differs from the wellknown isothermal principle. Thus, a virtual work equation is presented for use with the finite element method. This section is followed by a short discussion of the computational procedure used to implement the constitutive law to a computer. In the final section results of several example problems are presented. The impetus of these sample problems is twofold in nature. First, examples will be utilized to clarify certain computational aspects of the constitutive theory, and second, experimental data will be compared to theoretical results as a means of verificatton
F(S,j-a,j)~k’(S
d%‘, T),
(1)
where S,=second ~ol&~rc~off stress tensor aij=coo~dinates of the yield surface center in stress space, dg = uniaxial plastic strain increment, T = temperature, and l dip is a state variable representing the plastic strain history dependence in the yield function. Note that the explicit temperature dependence on the r.h.s. of eqn (1) precludes kinematic hardening due to temperature changes Cl]. A schematic representation of eqn (1) is shown in Fig 1. ~ffe~ntiating eqn (1) givesthe follo~g consistency condition during plastic loading: r(
rFdS,,(q
4
&:da,,-2k 1,
$- d:‘-2/+2+
d”j-_-O,(2)
where the term (~F/SSij) represents ~F(Si,-q,j)/liS, evaluated at Sj-aij which can be seen to be equtvalent to ~?F@S,~--tl,~t. Since during neutral loading the plastic strain increment and db,j are zero, it is apparent that a statement governing loading is: T
$
dS,-2k; dT>O ‘I In addition to the statement of consistency during loading given by (2), the following associated flow rule is employed :
of the theory.
PRESENTATIONOF THE CONSTITIJTIVETHEORS Recall that in the incr~ental theory of plasticity the
workhardening
rule is defined by a yield function in where dEE represents the plastic strain increment tensor.
tFormerly at Texas A&M University. 1’9
DAVID
H
and
ALLEN
WALTER E HAISLER
by Yamada [4, 51 m that the term containing dT IS written as a separate portion of the stress increment in the latter’s derivatton. It w11lbe shown. however. that the two are mathemati~aliy equivalent. UnfortuMtel~. in the form presented in eqn (8) it is not possible to determine the stress increment due to the occurrence of several undetermined parameters on the r.h.s. Yamada has accounted for these terms bt introducing the plasuc work rate. We have chosen a siightly different method for determining the effect of these unknowns. As presented in a previous papeiby us [6], we assume that there exists a scalar parameter C.called the hardenmg modulus. which when multiplied by the plastic strain mcrement and subtracted from the stress increment wdl be parallel to a tangent to the yield surface. Mathematically. this may be stated as:
Fig. I. Yield surface as a functwn of stress and temperature.
The stress at any time is assumed to be given by: S,, =
Dij,,(E,,
-
Ei, - Ei, - Ezn)7
(5)
where DZlnnis the elastic constitutive tensor at time E. In~emen~tion of eqn (5) was shown in our previous paper [2] to give: dS,,=D:,‘;P,‘(dE,,-Deb,-dEC,,-Deb,) +dD. r,lnn(E’mn-Ep’!?I”--EC’ Inn-Er’)mn.
If one employs the no~aI~ty condition in eqn (IO) and substitutes the result into the constitutive relation leqn 8). the result is:
16)
where D:,;, = etastic modulus tensor at time t + At, dDij,,=change in the elastic constitutive tensor (due to a temperaturechange) during time step At. E,,= total strain tensor, EL,= plastic strain tensor, Ez, = creep strain tensor. EL, = thermal strain tensor. and all superscripts t indicate me~urement at time t. The superscript t+ At on the elastic constitutive tensor is necessary due to the fact that the time step is finite rather than infinitesimal. Note that incrementation of the constitutive law results in a stress rate which is not rotationally invariant unless an infinitesimal strain measure is used. Therefore, the resulting theory is applicable eniy to small deformations. The final expression required to complete the constitutive law is the hardening rule. According to Ziegler’s modification [3] a tensorially correct statement is: d~,j=d&Si,-=i_$,
It can be seen from an examination of the consistency condition (eqn 2). that in order for the above statement to be valid. the foltowing must hoid:
(7)
where p is a scalar to be determined by the consistency condition (eqn 2). The constitutive law is thus obtained by substituting eqn (4t into eqns (2) and (6) and solving for the stress increment tensor. The resulting equation is: The above constitutive law differs from thai obtained
x(dE,,-dE;,-dE,r,)
x tE~~-E~“-E~~-E~~) i-d&,,&;,
- EgB-E:= -EL:).
111)
The simplicity ofthe above formulation can be seen when one uses the normality condition in conjunction wrth eqn (9) to obtain $ e=L . g
dS, dEP,
2 do =-L* 3 dBP
(13)
Thus. for isothermal loading it can be Seen that the hardening modulus is simply two-thirds the instantan-
A theory for analysis of thermoplastic materials eous slope of the uniaxial stressplastic strain diagram. However, for nonisothetmal loadings this is not the case. To see this, note that since the uniaxial stress is a function of both the plastic strain history and temperature, an increment of uniaxial stress is given by:
131
resulting constitutive law is reduced from a set of nonlinear coupled fvst order di&ent.ial equations by using a first order Rungs-Kutta process. THE VARIATIONAL PRINCIPLE
constitutive law may be applied to the conservation of momentum via an appropriate variationat principle. We present here briefly an incremental principle utilized in several nonlinear programs such as AGGIE I [S]. NONSAP [9-j and ADINA [lo]. First consider the virtuai work expression within a total Lagrangian description [ 1l] : The
(13) Combining eqns (12) and (13) gives: (14) where H’ is the instantaneous slope of the stressplastic strain diagram, Substituting the above equation into the constitutive relation (e.qn 1l), then rearranging and employing the normality condition. gives dS,-
~~ddE,-dEf,-dE~,)+dP,j,
where D!+bl
Cl,,=D
--
ijmn
ZF dF
--
‘luWa,, Es,,
t+At
,+.a,,
Dwmn .? .
ggg+p+$Lg$ w
P9
P4
83
and
dP,=
-
S’+A’6E’fArdV= TSu PoFduk dV (19) k k dA+ iJ ir I V0 I VO I A where $$ *’ = the 2nd Piola-KirchhoR stress tensor at time r+At referred to the initial configuration at (15) time t-0, 6g;b’= the variation in the Green-Lagrange strains at t + At referred to the initial configuration V,,, T,= the surface tractions at time t+Ar referred to the surface of the configuration A, &A,= the variation (16) in the displacements. po= the local density in the initial configuration and Fk= the body force per unit mass at time r + At referred to the initial configuration V,. We consider finite strain measure here for two reasons: this is the formulation already contained within the above mentioned nonlinear computer codes: and the constitutive law presented herein may be extended to encompass finite strain using an appropriate hypoelasticity theory [12,13-j. The virtual work equation is incremental&d using: S:;” = S;, + AS,,
(20)
and E:;“*=E:,+AE,,
and the terms H’ and &r/o”Tmay be determined from uniaxial strestitram data a1 time t. It can be seen that Ciimn is superscripted at time t since the hardening parameters are determined there. Note that the first two terms in dP,, represent the change in the effective modulus tensor during the load increment multipLed by the elastic strain tensor at time t, and the last term represents a correction term which results from using the hardening modulus, H’, at time t. The translation of the yield surface in stress space may now be obtained by substituting eqn (7) into eqn (2) and solving for dp. The resulting relation is:
(21)
The stress in~~ent AS, is then substitute using eqn (15) and the strain increment is implaced by incremantaliting the Green-Lagrange kinematic relation [Z]. The resulting variational principle is then linearized by neglecting terms nonlinear in the displacement increment. and this approximation is accounted for later by using an appropriate iterative technique [ 14,153. The linearized variational principle is then cast in a finite element matrix formulation by using an assumed displa~ment function. The resulting equations of motion are: [N][~~‘+*‘),s([K’L]+[K:
&Au)=fR+“‘)
--{F’;,
(22)
where
_I
dfi”
+$dSv-2kgd7-2k$dZP ” 21:
(L-a,,)
.
(18)
-jF&
Equation (18) thus guarantees that the state of stress will remain consistent with the yield surface during loading even under nonisothermal conditions. To summarize, then, the constitutive law is obtained by solving, in the following order, eqns (17), (16). (15), (18) and (7). It should also be noted here that the above constitutive theory satisfies thermodynamic restrictions [7], where the entropy generation term is introduced as an uncoupled rate independent (plastic) and rate dependent (creep) dashpot in a Maxwell model with nonlinear temperature dependent modulus and viscosity. The
(25)
and -{F’;
=
[BJr({S) -[C](AE=f
-[C]fAEr: + (AP},dv;
(27)
DAVID H ALLEN and WALTER E. HAISLER
12
and [&I and E&d are the linear strain dispiacement transformation matrix and the nonlinear strain displacement transformation matrix [2. 111.respectively. Also. H is the matrix relatmg nodal displacements to the displacement field [I I]. COMPUTER CODE IMPLEMENTATIOFi
The constitutive law and variational pnnciple presented herein have been implaced in the finite element code AGGIE I [8]. This code has the capability to model both geometric and material nonlinear structural response. and contains two and.three dimensional isoparametric elements [ 161. Under certain loading conditions the constitutive law (eqn 15) may be specialized for computational efficiency [2, 171. Thus, the theory presented herein has been implanted in the code as four different material models: quasiisothermal eiastic. nonisothe~al elastic. quasi-isothermal elasticplastic, and nonisothermal elastic-plastic, where quasiisothermal is defined to tiean that although the component undergoes thermal loading the material properties may be assumed to be unaffected in the temperature range considered. Utilizing the appropriate model has been shown to give significant computational savings L-171. The amount of input data required will depend on the model being used. For the nonisothermal elasticplastic model (eqn 15) with negligible creep the required input data are shown in Fig. 2. If significant creep is expected it will be necessary to input additional data which will depend on the creep model being used. Currently, there are three creep models within the code [S]: a microphenomenological equation of state approach. an intepolation scheme using creep vs time curves. and a nonlinear viscoelasticity model. Normally, it will be necessary to input either creep curves at specified loads and temperatures, or a creepcompliance tensor [18]. One will recall that in order to solve the thermal
l
, T
1
T2
/
T3
i
T4
Fig 2 Matenals input data.
-T
stress problem one must characterze the temperature distribution within the medium. It is assumed in this theory that the mechanical deformation and heat transfer problem may be solved a prrori. AGGIE I does not have the capability to solve this problem. Therefore. an additional input requirement is that the spa&l and time dependent temperature distribution be input to the code on a nodal basis. The procedure for utilizing the constitutive law is to assume a strain increment for each integration point based on information from the previous load step. One may then check for yielding by assuming eqn (6) holds and applying the resulting stress tensor to eqn I 1). If yielding 1s not predicted then eqn (6) is correct. If yielding occurs, then the stress state is updated using eqn I 15)and the yield surface location is updated using eqn (71. A complete outline of this procedure IS contamed in Ref. [19]
EXAMPLE PROBLEMS I. Nonisothermul
elasric axial bnr subjecred tuneous mechanical load and heat itzput
to s~mu/-
The first example demonstrates the capability of the code to predict the static response of elastic materials with strongly temperature dependent material properties to simultaneous mechanical and thermal loading, A significant factor in the accuracy of the theory is the correct determination of the thermal strain increment during a nonisothermal load step. Suppose one assumes that the thermal strzhrinerement is given by: dET=Y+“(7;L-T;,).
(28)
This assumption can introduce sign&ant error mto the analysis. The proper definition is given by: dsr=,‘+~9?;,-TR)-r*(Z;,-TRf =r’+“‘f?;,-I;,)+(~t+A’-~cW?;,-7k),
(29)
where TR is the reference temperature at which the thermal strain is zero. It can be seen that the second term in eqn (29) represents the error incurred by using eqn (28). Mathematically, eqn (29) may be interpreted as representing a chain rule differentiation. To illustrate the error which may be incurred by utilizing eqn (281, the code is now compared to an experiment. An aluminum (606LT6f axial bar with material properties as shown in Fig. 3 is subjected to the thermomechanical load history shown in Fig. 4. Due to the relatively short time period and low stress level. creep strain is assumed to be negligible. Analytical results are compared in Fig. 4. Experimental results as well as the theoretical result denoted CREEPARHS [20] are due to Stone. Tbree solutions were performed m AGGIE I using a single plane stress ¶metric element and 26 load steps. In the first analysis it is assumed that material properties are not temperature dependent and room temperature data are used. In the second solution, properties are assumed to vary with temperature. and the definition of the thermal strain Increment given by eqn (28) is used. Finally, the third analysis employs temperature dependent material properties as welf as the definition of the thermal strain increment given in (29). It is found that all theoretical results except the last are m error by approx. 18, or more. Thus. the importance of temperature dependent material properties as well as a correct definition of the
A theory for analysis of thermoplastic materials
133
0; I.78 U,T 42
t
0
_
.
.
I
SlNaN
Fig. 5. No~soth~al
,v
.&I
Fig. 3. Material data for Al 6061-T6 experimental test samples.
*.OOS-
f
s 0.006
-
Z. ?i
Ik
EXPElNEWT
0.004-
0.002
-
CREEP&NNS ABOliE I-eantmt ASWE I-Con APBIE I-Eq”
-
/
o.oooy, / 0
2
, 4
I,,
I,
6
Tw2 &l
--
-
0
----
s
-.-.-
11 -
,
10
-
*
-
,
I2
-
I
14
Fig. 4. Comparison of analysis to experiment for uniaxiai elastic thermomechanical loading. thermal strain increment are illustrated by this example problem. II. Nonisothmal elastic-plastic axial bar In this example an axial bar with material properties shown in Fig. 5 is subjected to the load history shown in the same figure. Attempts at experimental verification of this problem have failed due to the fact that the relatively long heat up time required with the equipment available at this institution induces significant creep. Additional equipment is on order and it is hoped that experimental verification will be forthcoming in the near future. This example is incIuded not as a verification of the theory, but rather as a corroboration of the computational ef5ciency of the model. One will recall that in our previous papers [2,19], we
t
ttbt
TIME
elastic-plastic ax2 bar with thermomechanical bad history.
proposed that to obtain the effective modulus tensor CiJrnnfor a load step one should use the elastic constitutive tensor Dtinurat the end of the load step. This was shown to be mathematically correct and is computationally supported by this example. In theexample, an axial bar is loaded isothermally to some plastic state and is then simultaneously subjected to a mechanical load and spatially constant slow heat input. According to the classical incremental theory of plasticity it is required that the state of stress and strain move to a point on the uniaxial stress-strain diagram for the temperature at the end of the step. This requirement must be satislkd in order to remain consistent with the yield surface during plastic Ioading Using the eIastic modulus proposed by us the theory will predict this result exactly. If one utilizes the elastic modulus at the start of the load step the state of stress and strain will converge incorrectIy to a point denoted by the head of the dashed line in Fig, 5. This point corresponds to a horizontal tran~ation from (e’, &) to the stress-strain curve at the temperature at the end of the step, followed by a translation parallel to the stress strain curve at time Thus, utilizing the elastic modulus at the start of a load step does not satisfy the consistency condition. Further, if one empIoys quibble iteration and correctly up dates the elastic modulus during the iteration procedure, the solution will converge to the correct solution, but in exactly twice the computation time encountered in our theory. Therefore it is suggested that using the elastic wnstitutive tensor at the end of the step is not only consistent but also wm~~tioMlly eBicient. III. Nonisothermal elastic-plastic axial bar with signiJi cant creep
problem illustrates the ability of the theory to predict the response of materials near their ultimate strength. An aluminum (6061-T6) axial bar is subjected to the load history shown in Fig. 6, such that ultimate failure of the specimen occurs. Experimental and This
DAVIDH. ALLENand WALTER E.
I34
HASLER
i’Z& ,,,y Jc----12 6” ,’ 1, ** \ I’ o- loo-’ tl t2 13 0.03
-.“---
-----
s-
AGGIE I - no creep AGGk5 I CAEEPARHS
wtrncr*4p 4 ;1
-12 98
T
14
b
TIME
,\
‘\ \
\
0.01
CREEPARHS [ZO] results are due to Stone. Two analyses were performed using AGGIE I. In the first, creep is assumed to be negligible. In the second, linear interpolation of isothermal creep data at a load of 20 ksi is used. It is seen from results plotted in Fig. 5 that &he theory produces amurate results even near the ultimate strength of the material. IV. ~o~~ff~h~rnu~eiastic-plastic axial bar subjected to cyclic load history One of the primary purposes of this research has been to obtain a theory which can effectively model the response of elastioplastic media to cyclic mechanicat and thermal loading. A&ho@ the literature contains abundant verification tools for isothermal cyclic loading histories we have been unable to obtain experimental data to verify the nonisothermal problem. Therefore, WE have undertaken to worm certain no~~th~~ tests using axial bars on the MTS system. Although results of our experiments are incomplete at this time, analytical results of a cyclic test are presented herein. Al~ou~ the theory is capable of modeling creep response, this study is meant to verify the time independent behavior of the material. Therefore, we have chosen the test case shown in Fig.3. Note that the specimen is heated at zero load after prestrain so that no creep occurs. This problem thus tests the expansion and translation of the yield surface in stress space caused by a temperature change, In order to verify the applicability of the theory it is necessary to perform three materials data tests. Isotherma stressstrain cmvzs are generated at room temperature and 200°F. In addition, an isothermal cyclic load test is performed to determine the ratio of isotropic to kinematic hardening (fl) used in the model. Theoretical results are presented in Fig. 7. It is seen that the combined hardening model produces results which differ signifkantly from both kinematic and isotropic nonisothermal hardening theory as well as isothermal isotropic hardening theory, indicated by the dashed line in Fig. 7. A schematic repletion of the fransformation of the yield surface is shown in Fig. 8. A future paper will compare experimental results to the analytical solution presented here.
Fig. 7. Elastic-plastic axial bar subjected to cycric that* mechanicltf load history.
Fig 8. Nonisothermal comb&d hardening (/3-0.5, of yield surface for eiastie-plastic axial bar.
We have proposed herein a theory for modelling the response of robotic materi& The theory.has been shown in this report to be adequate in predicting response of many solid media. In addition, it has been shown that certain ~pu~tio~y simpliied forms of the theory are correctly in place in the computer code AGGIE I. However, the theory may be inadequate in modelfine; certain physical phenomena. Among theSe are rate dependence, instability near ultimate strength, finite stress and strain, phase changes. and violation of other a~umptions in the theory such as the normality condition. Research is currently underway to incorporate the above additions to the theory.
A theory for analysis of thermoplastic materials REFERENCES 1. A. Phillips and R. Kasper, On the foundations of thermoplasticity-an experimental investigation. J. Appl. Mech. 891-896 (Dec. 1973). 2. D. H. Allen and W. E. Haisler, The application ofthermal and creep effects to the combined isotropic-kinematic hardening model for inelastic structural analysis by the finite element method. Office of Naval Research, No. 3275-79-3 (May 1979). 3. H. Ziegler. A modification of Prager’s hardening rule. Q. Appl. Math. 17.55-56 (1959). 4. Y. Yamada and T. Sakurai, Basic formulation and a computer program for large deformation analysis. Pressure Vessel Technoloar: Part I. DD.341-352. ASME. New York (19771. “. *’ 5. Y. Yamada. Constitutive modelling of inelastic behavior and numerical solution of nonlinear problems by the finite element method. Compur. Srrucrures 8. 535-543 (1978). 6. D. H. Allen, An investigation into the combined isotropic-kinematic workhardening rule. Office of Naval Research. No. 3275-79-2 (May 1979). 7. D. H. Allen, A survey of current temperature dependent elastic-plastic-creep constitutive laws for applicability to finite element computer codes. 21st AIAA/ASCE,/ ASME SDM Conference, Seattle (May 1980). 8. W. E. Haisler. AGGIE I. A finite element program for nonlinear structural analysis. Office of Naval Research. No. 3275-77-l (June 1977). 9 K. 1. Bathe and E. L. Wilson. NONSAP-A nonlinear structural analysis program. Nucl. Engng and Design (June 19741. 10. M. D. Snvder and K. J. Bathe, Formulation and NumerIcal Solut;on q/ Thermo-elastic-Plastic lems. NTIS (1977).
and Creep Prob-
11. B. Hunsaker, Jr.. The application of combined kinematic-isotropic hardening and the mechanical sublayer model to small stram inelastic structural analysis by the finite element method. Dissertation. Texas A&M University (Aug. 19761. 12. C. A. Truesdell and W. Nell, The non-linear field theories of mechanics. In Encwlopedia of Phpsics. Vol. 3. Part 3. Springer-Verlag. Berlin (1964). 13. S. W. Key, 1. H. BifBe and R. D. Krieg. A study of the computational and theoretical differences of two finite strain elastic-plastic constitutive models. Formulations and Computatronal Algorirhms in Finire Element Analysis: U.S.-Germany Symposium. MIT, New York (1976). 14. J. A. Stricklin. W. E. Haisler and W. A. Von Riesemann, Formulation. computation, and solution procedures for material and/or geometric nonlinear structural analysis by the fimte element method. Sandia Laboratories. No. SC-CR-73-3102 (July 1972). 15. G. Strang and H. Matthies. Numerical computations in nonlinear mechanics. Pressure Vessels and Pzptng Con-
135
ference. ASME, New York (1979). 16. 0. C. Zienkiewicz, The Finite Element Method. McGrawHill. New York (1977). 17. D. H. Allen and W. E. Haisler, The prediction of response of solids to thermal loading using the finite element code AGGIE I. Office of Naval Research. No. 3275-80-l (June 1980). 18. D. R. Sanders and W. E. Haisler, An incremental form of the single-integral nonlinear viscoelastic theory for elastioplastic-creep finite element analysis. Pressure Vessels and Piping Conference. ASME, New York (1979). 19. D. H. Allen and W. E. Haisler, Thermoplastic analysis using the finite element code AGGIE 1. Office of Naval Research No. 3275-79-4 (Dec. 1979). 20. S. J. Stone, Effects of material behavior on the response of rapidly heated structures. McDonnell Douglas Astronautics Company, MDC G8432 (Jan. 1980). 21. C. E. Pugh. J. M. Corum, K. C. Lin and W. L. Greenstreet. Currently recommended constitutive equations for inelastic desigd analysis of FFTF components. ORNL TM-3602, Oak Ridge National Laboratories (Sept. 1972). 22. W. E. Haisler, Numerical and experimental comparison of plastic work hardening rules. Trans 4th lnr. Conj. on Structural Mechanics in Reactor 7echnology. San Francisco (15-19 Aug. 1977). 23. W. E. Haisler, B. 1. Hunsaker and J. A. Stricklin, On the use of two hardening rules of plasticity in incremental and pseudo force analysis. Presented at the Winter Annual Meetinn of the ASME. New York; also in Constuutlve E&&ions in Viscoplasticity Computational nnd Engineering Aspects. AMD Vol. 20, pp. 139-170 (5-10 Dec. 1976). 24. M. Tanaka. Large deflection analysis of elastic-plastic circular plates with combined isotropic and kinematic hardening. lngenieur-Archiv 41,342-356 (1972). 25. B. Hunsaker. Jr., An evaluation of four hardening rules of the incremental theory of plasticity. Thesis, Texas A&M University (Dec. 19731. 26. W. E. Haisler and D. R. Sanders, Elastic-plastic-creep large strain analysis at elevated temperature by the finite element method. Office of Naval Research. No. 2375-78-l (Apr 1978). 27. D. R. Sandersand W. E. Haisler. An incremental form of the single-integral nonlinear viscoelastic theory for elastic-plastic-creep finite element analysis. Office of Naval &search, No. 3275-79-l (Apr. 197%. 28. K. H. Bathe. H. Ozdemir and E. L. Wilson. Static and dynamic geometric and material nonlinear analysis. Structural Engineering Laboratory, University of California. Berkeley. California Rep. UCSESM 74-4 (Feb. 1974). 29. P. Sharifi and D. N. Yates. Nonlinear thermo-elasticplastic and creep analysis by the finite-element method. AlAA .I 12, 1210-1215 (1974).