A theoretical and numerical comparison of three finite strain finite element formulations for elastic-viscoplastic materials

Computers & Structure.~ Vol 16. No. I-,.4, pp. 215-222. 1983 Pnnted In Great Britain.

004~-7~9/83/010215-08803.00/0 ~ 1983 Pergamon Press L~d.

A THEORETICAL AND NUMERICAL COMPARISON OF THREE FINITE STRAIN FINITE ELEMENT FORMULATIONS FOR ELASTIC-VISCOPLASTIC MATERIALS R. A. LEMASTER Sverdrup Technology, Inc., Tullahoma, TN 37388, U.S.A. A~traet--Three fundamentally different finite strain constitutive formulations for describing elastic-viscoplastic materials are considered. These are classified as: (1) hyperelastic-viscoplastic, (2) hypoelastic-viscoplastic, and (3) updated hyperelastic-viscoplastic. The methods differ primarily in the kinetic and kinematic quantities used to characterize the material constitutive equations. A discussion of the theoretical differences between the three descriptions is presented, and numerical examples are given which provide a quantitative comparison of the three material models. I NTRODUCTION

In recent years, a number of elastic-viscoplastic constitutive equations have been developed which appear to be capable of reproducing certain types of inelastic material behavior[l--6]. In particular, the fundamentally different phenomena of rate sensitivity to dynamic loading, and the combined plastic-creep response at elevated temperatures can be modeled by elastic-viscoplastic constitutive equations. In addition, several state variable elastic-viscoplastic constitutive theories are based on microstructure or dislocation movement phenomena[4,5]. This coupling of microstructure physics with continuum-based constitutive equations represents an important trend which should lead to constitutive equations which treat in a consistent manner the interaction between the physical phenomena currently referred to as elastic, anelastic, plastic, and creep. The rate form of the inelastic strain and state variable evolution equations made elastic-viscoplastic constitutive equations very attractive numerically. For the particular case where the strains and rotations can be considered infinitesimal, it is possible to obtain the inelastic material response from an implicit algorithm using a single "stiffness" matrix. Zienkiewicz[7] has discussed small strain finite element formulations for elastic-viscoplastic materials, and several numerical integration schemes applicable to the rate evolution laws have been discussed by Argyris[8]. In this paper, three fundamentally different finite strain constitutive formulations for elastic-viscoplastic materials are considered. These are classified as: (1) hyperelastic-viscoplastic, (2) hypoelastic-viscoplastic, and (3) updated hyperelastic-viscoplastic. The hyperelastic-viscoplastic constitutive theory has been proposed by Kanchi[9] and Nagarajan[10] to treat problems where rotational or "geometric" effects are important. For the case of large deformations involving finite strains and/or rotations, the hypoelastic-viscoplastic material description is considered to be the most theoretically consistent, and is similar to finite strain elastic-plastic theories presented by Hutchinson[Ill, Key [12], Hibbitt, [ 13], and McMeeking [ 14]. The updated hyperelastic-viscoplastic material description is an approximation to the hypoelastic-viscoplastic material description and is based on the work of Bathe[15]. In many practical dynamic problems involving finite

inelastic deformations, the finite inelastic strains are confined to localized regions. In other cases, the strains and/or rotation may be of an intermediate magnitude, where possibly more than one set of constitutive equations provides a satisfactory material description. Objectives of this effort are to: (1) consider the theoretical basis of the various material models, (2) determine under what conditions the different descriptions yield comparable results, and (3) evaluate the relative numerical efficiency of the different descriptions with respect to accuracy versus computing time. In the following section, the general structure of elastic-viscoplastic constitutive equations is reviewed, and then is specialized to the Perzyna[3] generalization of the Malvern[1] and Sokolovsky[6] viscoplastic constitutive equations. In subsequent sections, the three constitutive expressions are developed, followed by numerical examples and a discussion of the results. ELASTIC-V[SCOPLASTICC O N S T I T U T I V E EQUATIONS

The mathematical structure of elastic-viscoplastic constitutive equations may be summarized by the following equations [16]; ,e -n @j = ~i~+ E~ + ~. T

(I)

~ = bii(o,~, q
(2)

q'~' = q~(o',, q',~', T)

(3)

where e~, ., e~j ., and ~.r are the elastic, inelastic, and thermal strain rates, respectively; e,, is the stress tensor, T the temperature, and q~,~' are state variables. The number of state variables differs in the different constitutive models and they can be scalars or tensors. These state variables are assumed to completely characterize the present deformation state of the material, and the history dependence of the rate of inelastic strain is taken into account by their current values. In the present investigation, the Perzyna[3] generalization of the Malvern[1] and Sokolovsky[6] elasticviscoplastic overstress model is considered. In this theory, the inelastic strain rate tensor is given by ~ = .~ < &(F) > ~of 215

(4)

2t6

g ,\ L~M~sr~

where y is a material viscosity parameter, F is the overstress function given by F =

]'(o'"') ~ - t. r'(~;})

~5)

In eqn (5), f(o "~s) is the dynamic yield surface stress, F(e~) is a function which takes into account the previous inelastic deformation of the material and 4) is a function chosen to match experimental data. In this paper. [(c/~) is taken to be the Von-Mises stress given by

component form as

The viscoplastic rate of Green's strain is found by replacing the kinetic and kinematic quantities in eqn ~4) by the 2nd Piola stress and Green's strain. 112) The overstress function in eqn (12) is given by

f(o "°) = "v (3L_)

i6)

and F(e~) is taken to be d~(i"), where d~ is the static stress-plastic strain curve, and g" is the integral of the equivalent plastic strain. Although an overstress model is employed in this study, the constitutive formulations may be applied to state variable models with little additional numerical difficulty. ItYII~g]igI.,ASTIC-VlSCOPLASTIC MATZilLALDgSCRIFI'ION The appropriate kinetic and kinematic variables employed in the hyperelastic-viscoplastic constitutive equations are the 2nd Piola stress tensor, S, and Green's strain tensor, E. Following Green and Nagdhi[17, 18], it is postulated that the total Green's strain may be separated into elastic and inelastic contributions, E = E ~' ÷ E 'p',

(7)

In eqn (7), only the total Green's strain is given a kinematic interpretation, and the elastic and inelastic components are obtained from constitutive postulates and the laws of thermodynamics. The 2nd Piola stress tensor is related to the elastic component of the total Green's strain tensor by the expression a~b

(8)

S = po OEi~,

where po is the density in the undeformed configuration, and d is the strain energy function. Equation (8) represents a hyperelastic or elastic potential constitutive relationship between the stress and elastic strains. With regard to the present constitutive development, it is desirable to express the constitutive variables in terms of rates, and eqn (7) is written as

F = /(su)

1.

(13)

Several observations concerning the material behavior described by eqns (11)-(13) should now be considered. First. the equivalent dynamic stress. [(S), is expressed in terms of the components of the 2nd Piola stress tensor, which are defined per unit area of the reference configuration. This is theoretically inconsistent, since a mathematical change in the reference configuration will alter the viscopiastic strain rate[19]. This statement has no significance when a state variable constitutive expression is used which does not rely on the concept of a yield function[4, 5]. Furthermore. in the limit of vanishingty small viscoplastic strain rates, the viscoptastic material response should approach that obtained from classical inviscid plasticity which is correctly expressed in terms of kinetic and kinematic quantities referred to the current configuration. In particular, mvariance of the loading and unloading elastic moduli requires that the stress and strain measures he expressed in terms of the current configuration. Also, the distinction between the initial tension and compression stress-strain curves usually is not necessary when the stress and strain measures are referred to the current configuration. Hill[20] is very distinct about referring the stress and strain measures used in his book to the current configuration. An excellent discussion of the experimental observations underlying incremental plasticity theories is given by Lee[21]. Based on the above discussion, the hyperetastic-viscoplastic material description is considered to be applicable to problems involving small strains with large rotations where area changes may be considered negligible.

(9)

E = E ' " + E 'oe'

HYPOELASTIC-VlSCOPLASTICMATgRIALDg,.qClUi~ION

and eqn (8) as

a-'q,

S = oo 0-~.775

~:,,, .

(lo)

The combination of eqns (9) and (10) may be written in

tStandard tensor notation is used throughout this paper, where superscripted indices refer to contravariaat components, and subscripted indices refer to covariant components. Upper case indices indicate that the quantity is referred to base vectors in the fixed reference configuration, and lower case indices indicate that the quantity is referred to the fixed current configuration base vectors.

The second constitutive model may be described as a hypoelastic-viscoplastic material description which util. izes the Kirchoff stress, ~, and the rate of deformation, d, as the appropriate kinetic and kinematic variables. The basic hypothesis is that of an additive decomposition of the rate of deformation tensor into elastic and viscoptastic components, d = d (') + d '~°'.

(14)

The Kirchoff stress tensor is related to the Cauchy stress tensor, o-, by the equation r = Po o"

I15)

A theoretical and numerical comparisonof three finite strain finite element formulations where po and p are the material densities in the undeformed and current configurations respectively. For a hypoelastic description, the elastic contribution to the rate of deformation is expressed in terms of the Jauman rate of the Kirchoff stress tensor by the expression ~r"'

=

(16)

d~,'?

C '~

where V ..

• "" r" = ~'" - g i m to,,,'r n j - g i m to,,,,r n i .

(17)

In eqn (17), the superscripted dot signifies a material time derivative, g ' " is the contravariant metric tensor for the fixed coordinate system to which the current configuration is referred, and ~,., is the vorticity tensor. The convected rate of the Kirchoff stress tensor is related to the Jauman rate by the expression o

vt°

""

r 'j+g

im

dm,r

nj

jm

+g

d,,,r

ni

.

(18)

The superscripted symbol (°) in eqn (18) represents the convected rate[19, 22]. The viscoplastic component of the rate of deformation is obtained by replacing the kinetic and kinematic quantities in eqn (14) by the Kirchoff stress and the viscoplastic component of the rate of deformation, ,o~, _

d~

af(r')

- 3'< ¢ ( F ) > ~r--~-r-~.

(19)

The overstress function in eqn (19) is given by F =

fl~.o)

~( d,~.~)

_ 1

.

(20)

The combination of eqns (14), (16), (18) and (19) yields the hypoelastic-viscoplastic constitutive equations

~., =

{C,~,_

gi%,~ _ gj%,,}&,

- C~k~y < ~(F) > 0-~"

(21)

When references to the original configuration, eqn (21) may be written as

based on the work of Lee f23]. In this theory, the total deformation gradient is given by F = F " t U ";

q > ark,'

(22) In the development of the above equations, the Kirchoff stress was chosen as the kinetic variable in eqn (16). The primary reason for choosing the Kirchoff stress over the Cauchy stress is the numerical advantage obtained from working with a symmetric stiffness matrix versus an unsymmetric stiffness. If the Cauchy stress is chosen, an additional term involving the dilatational components of the rate of deformation appears in eqn (18) resulting in an unsymmetric stiffness matrix. This point is discussed by Hutchinson[l l]. Within the last decade, a kinematic theory for finite strain elastic-plastic deformation has been developed CAS

I~:I/4

-

C

(23)

where F ' " is the deformation gradient obtained in going from a stress-free intermediate configuration to the current configuration, and ~0~ is the deformation gradient obtained in going from the reference configuration to the intermediate stress-free configuration. Significant to the current development, when the elastic strains remain small the kinematic postulate (23) is equivalent to the additive decomposition of the rate of deformation into elastic and plastic components[21]. For metals, the elastic strains are bounded by plastic yielding to magnitudes of the order 10-3, Therefore, for elastic-plastic materials, eqn (14) represents a good approximation to the kinematic postulate (23). For elastic-viscoplastic materials, the stresses are permitted to exceed the static yield surfaces, and the decomposition of the rate of deformation into elastic and viscoplastic components may not be as good an approximation to the kinematic postulate in eqn (23). Since the kinetic and kinematic variables in the hypoelastic-viscoplastic constitutive equations are referred to the current configuration they are believed to more closely represent the behavior of real materials experiencing finite strains than do the hyperelastic-viscoplastic constitutive equations. UPDATED HYPERELASTIC-VISCOPLASTICMATERIAL DESCRIPTION The third material model is classified as an updated hyperelastic-viscoplastic material description. It is based on the updated Lagrangian analysis procedure in which the reference configuration is taken to be the current configuration, and is updated each time interval during the analysis. This updating procedure yields the well known "Updated Lagrangian" form of the incremental equations of motion. The rate of Green's strain referred to the current configuration is separated into its elastic and viscoplastic contributions, ~u = i~,~~+ G'; "~.

(24)

Assuming the existence of a hyperelastic-type constitutive expression, applicable in the neighborhood of the current configuration, we obtain a '"'j = B

- g'*;'}F.o,, - x " l,x'* I , C " % <

217

ijkt

(e; ~kt.

(25)

Similar to the previous two descriptions, the governing constitutive equations may be written as 6 o = B iSkt(~kl -- ~ ~o~)

~P~ = y < ~(F) > ~ '-~"S ij = '~" + a '

, - . , ~ . _- ' ~,p' o x,l..xjk,+~,S,..

(26) (27) (28) (29)

where 6" is the Truesdell stress rate[24]. The updated hyperelastic-viscoplastic constitutive description is similar to the hypoelastic-viscoplastic

218

R. A LzMAszE~

constitutive equations previously presented. One difference between the two models appears in the elastic coefficients, which differ by a factor of p,/#, C,~kl = Po B.k,. P

(30)

Since the Cauchy stress is used in the Von-Mises yield function instead of the Kirchoff stress, the visco#~tic strain rates given by eqns (19) and (27) will also differ by approximately a factor of Pola. For incompressible inelastic deformation of metals, the ratio polo is approximately equal to one, and the effects of the density change should IX small. The main difference between the hypoclastic-viscoplastic and updated hyperelastic-viscoplastic constitutive equations is in the choice of the stress rate appearing in the constitutive equations. For the hyix~elastic description, the elastic coefficients are modified to account for T A Y L O I N ' $ CYLJNIDER

rigid body rotations. The distinction between the two descriptions arises purely from the choice of the kinetic and kinematic variables selected for use in ,he corastitutive equations.

Numericalimplementation The three material descriptions under conslcleration were implemented using the finite element code FINITE, which is based on a Lagrangmn statement of the principle of virtual work given by

fv S3EdVo= fAoto~vdAo+fv fo6vdV,~.

(3t)

In eqn (31), S is the 2rid Piola stress tensor, 8E is the virtual rate of Green's strain tensor, Ao and Vo are the reference configuration area and volume, respectively; to is the surface traction vector per unit area of the reference configuration, fo is the body force vector per unit volume of the reference configuration, and 3v is the virtual velocity vector. It is often desirable to use an incremental form of eqn (31) given by

V,a

f v d S 8gdVo + fv 'S d(Sl~) d Vo- Lodto 8v dA~ - f vc d|o 3v d Vo = fAo t ~~' to Sv dAo + /vo ~*~%, ~v d Vn

~32) - fvo'SSEdVo. Equations (31) and (32) may be used as a basis for total Lagrangian or updated Lagrangian analysis procedures. For wave propagation analyses, eqn (31) is used in conjunction with an explicit central difference time operator. For dynamic problems where only the "inertial" effects are important, eqn (32) is used in conjunction with the implicit Newmark fl method. For more

Fig. 1.

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ALlY

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(_~..~), ~o

t

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3o

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ALUMtNUMALLOY~ f

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1

STRAIN RATE ~ . SEG.-1

Fig. 2. Dynamic yield stress vs strain rate.

f lO

A theoretical and numerical comparison of three finite strain finite element formulations detailed treatments of nonlinear finite methods, the reader is referred to [10-15, 24]. NUMERICALEXAMPLES To compare the three material descriptions, two problems are considered which involve both dynamic plasticity and finite strains. The first problem is frequently referred to as Taylor's cylinder and involves a right circular cylinder impacting a rigid wall (Fig. 1). The strain rates and amount of inelastic deformation can be altered by changing the impact velocity. The second problem is a cantilevered beam subjected to a triangular pulse at its free end. TAYLOR'SCYLINDER Taylor's cylinder problem was analyzed using an impact velocity of 8000 in.tsec. The response of the cylinder was computed for a duration of 1.2 x 10-5 sec after impact using a time step of 2.0 × 10-s sec. The material considered is 6061-T6 aluminum; dynamic yield stress data for this material is shown in Fig. 2, and the static stress-strain curve is given in Fig. 3. Figure 4 shows the dimensions of the cylinder, and the axisymetric finiteelement mesh used. Figure 5 gives a time-history plot of the centerline displacement of the free end of the cylinder. As seen in the figure, the hypoelastic-viscoplastic and updated hyperelastic-viscoplastic material description yielded very similar results, while the hyperelastic-viscoplastic approach differed appreciably. Figure 6 shows the results of experimental data obtained by Wiftin [25] for a number of different materials. The predictions of the three material descriptions for the 8000 in./sec impact velocity shown in Figure 6 are seen to fall slightly above the experimental curve, when a value of 44,000 psi is used for yo. This suggests that the strain-rate sensitivity of the material may not satisfy the simple power taw relationship shown in Fig. 2 throughout the entire deformation history. In addition, the parameter yo in Fig. 6 is not well defined, and a slightly higher value will lower the analytical predictions to yield better agreement between experimental and analytical results. Figure 7 shows examples of residual plastic strain contour plots for the hypoelastic-viscoplastic and hyperelastic-viscoplastic material descriptions. The updated hyperelastic-viscoplastic contour plots differ only slightly from the hypoelastic-viscoplastic contour plots.

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The planar mature of the plastic wave can be easily identified in tlmse plots.

CANTILgSqLqlgDBEAMSUIg/gCTEDTOTRIANGULARIMPULSE Taylor's cylinder problem involves finite strains, with the rotations being of less importance. The second problem is a cantilevered beam subj¢cted to a triangular impulse, and dominated by large rotations; the stretching components of the strain are considmred infinitesimal. Figure 8 gives properties for a ficticious mat¢rial used in tim analysis, and shows the time history r¢spons¢ of the cantilever tip. The results of this numerical experiment indicate that the three material descriptions yield almost identical results. The hypoelastic and updated

hypereiastic-viscoptastic descriptions predicted a slightly larger permanent set than did the hyp¢relastic-viscoplastic material description. CONCLUSIONS

The results of this investigation demonstrate that the three material descriptions under consideration yield nearly identical results under the conditions of small strains with large rotations, For the case of finite stretching strains, the computational results obtained from the hypoelastic-viscoplastic and updated hyperelastic-viscoplastic material descriptions were always within 1%. However, the hyperelastic-viscoplastic description yielded a significantly different response. Based on published experimental data, it was not pos-

A theoretical and numerical comparison of three finite strain finite element formulations

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sible to determine which theory yielded the best results. The hypoelastic-viscoplastic material description is considered to be the most theoretically consistent. All three constitutive descriptions required approximately the same amount of computational effort.

TRENDS

The development of constitutive equations to represent the behavior of real materials over a wide range of loading conditions and service environments is a formidable task. Elastic-viscoplasticstate variable constitutive equations offer a convenient mathematical structure for formulating or extending constitutive theories. Much of the experimental data used to develop constitutive descriptions is limited to small strains and usually is uniaxial in nature. Therefore, the extension of these theories to finite strain environments relies heavily on mathematics and invariance principles. Increasingly, numerical calculations are being applied to evaluate the applicability of constitutive descriptions to the behavior of real materials. The macroscopic quantities such as displacement and force are often the only quantities which can be obtained from experiments involving severe environments. Therefore, numerical calculations which relate these macroscopic quantities to constitutive variables are important,

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