computational aspects of incrementally objective algorithms for large deformation plasticity

*Correspondence to: Jacob Fish, Departments of Civil, Mechanical and Aerospace Engineering, Rensselaer PolytechnicInstitute, Troy, NY 12180-3590, U.S.A. E-mail: [email protected]

CCC 0029—5981/99/060839—13$17.50 Received 19 November 1997Copyright ( 1999 John Wiley & Sons, Ltd. Revised 9 February 1998

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 44, 839—851 (1999)

COMPUTATIONAL ASPECTS OFINCREMENTALLY OBJECTIVE ALGORITHMS FOR

LARGE DEFORMATION PLASTICITY

JACOB FISH* AND KAMLUN SHEK

Departments of Civil, Mechanical and Aerospace Engineering, Rensselaer Polytechnic Institute, ¹roy, N½ 12180, º.S.A.

SUMMARY

A methodology for computationally efficient formulation of the tangent stiffness matrix consistent withincrementally objective algorithms for integrating finite deformation kinematics and with closest pointprojection algorithms for integrating material response is developed in the context of finite deformationplasticity. Numerical experiments illustrate an excellent performance of the proposed formulation incomparison with other algorithms. Copyright ( 1999 John Wiley & Sons, Ltd.

KEY WORDS: finite deformation; plasticity; objectivity; consistent tangent

1. INTRODUCTION

The notion of consistency between the tangent stiffness matrix and the integration algorithmemployed in the solution of the incremental problem has been introduced by Nagtegaal1 andSimo and Taylor.2 Within the framework of closest point projection algorithms3—5 and in thecontext of small deformation plasticity, Simo and Taylor2 demonstrated the crucial role of theconsistent tangent stiffness matrix in preserving the quadratic rate of asymptotic convergence ofiterative solution schemes based upon the Newton method. Consistent formulations have beensubsequently developed for finite deformation plasticity6—8 within the framework of multiplica-tive decomposition of the deformation gradient and hyperelasticity.

It is commonly perceived that the tangent stiffness matrix consistent with the incrementallyobjective algorithms for integrating finite deformation kinematics and with implicit algorithmsfor integrating material response is absolutely necessary to optimize the overall computationalcost. This notion stems from the common perception that the overall CPU time in implicitmethods is governed by the cost of solving a linearized system of equations. This is certainly truein the asymptotic range in the case of skyline direct solvers (CPºJNB2; N, B being theproblem size and the bandwidth), multifrontal methods9 (CPºJNb, b"1)2—1)7) and precon-ditioned conjugate gradient methods10 (CPºJNb, b"1)17—1)33) . Multilevel methods, on theother hand, possess an optimal rate of convergence (b"1), i.e., CPU time is proportional to thenumber of discrete unknowns. When multilevel methods are employed as linear solvers within

Table I. CPU time for stiffness matrix evaluation and solution of linear system of equations

CPU(s) SUN ULTRA SPARC

Problem Element type Equations Stiffness eval Linear solver

Turbine blade11 Tetra-10 207,840 380 820Ring-strut11 Tetra-4 102,642 61 126Turbine nozzle11 Tetra-10 131,565 212 462Joint of cyl.12 ANS-814 186,245 492 517Concrete canoe11 ANS-814 132,486 390 497Automobile11 DMT-DKT15 265,128 395 1116Hybrid car13 HANS (p"6)16 67,120 1850 980

the Newton method, the computational complexity of solving a linearized system of equationsbecomes comparable to that of stiffness matrix evaluation. Table I compares the cost of stiffnessmatrix evaluation with the cost of solving a linear system of equation11 for several industry andmodel problems.

It can be seen that for linear problems solved with multilevel methods11,13 the cost of stiffnessevaluation ranges from 40 per cent to 190 per cent of the cost of solving a linear system ofequations. Note that the cost of element stiffness matrix evaluation has been shown to beproportional to p4 (p being the polynomial order) for 2D problems.17

For problems with geometric and material non-linearities the relative cost of matrix computa-tions is significantly larger. Thus, in the realm where the computational complexity of multilevelsolvers is comparable to that of matrix evaluation, stress integration and consistent linearizationprocedures could potentially become a bottle neck in implicit computations. For large deforma-tion plasticity the following solution strategies are commonly employed:

1. Large increments, consistent tangent6,18,192. Large increments, approximate tangent20,213. Small increments, approximate tangent22

Each of the three strategies has certain advantages and disadvantages. The first strategyadvocating ‘consistent linearization at any cost’ has the advantage of maintaining quadraticasymptotic rate of convergence while advancing solution in large increments. On the negativeside, the cost of stress updates and consistent tangent evaluation may often overshadow the entirecomputational cost in particular in the context of multilevel solvers. The second strategy attemptsto reduce the cost of matrix evaluations while advancing the solution in large increments at theexpense of suboptimal rate of convergence and consequently increased number of iterations. Thethird strategy is based on reducing the number of iterations within an increment by employingsmaller load steps with approximate and inexpensive tangent, but at the cost of increasing thenumber of load steps.

The approach advocated here is based on selecting both the simple and accurate integrator23that can be consistently linearized to provide a closed form computationally inexpensive tangentstiffness matrix. We show that for moderately large rotation increments the second-orderaccurate incrementally objective integrator23 is comparable in terms of accuracy to the higher-order integrator21 and at the same time the overall computational cost is drastically reduced incomparison with approaches employing similar integrators but with inconsistent tangents.22

840 J. FISH AND K. SHEK

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 839—851 (1999)

The manuscript is organized as follows. Section 2 summarizes constitutive equations of finitedeformation plasticity based on objective stress rates, additive split of rate of deformation, andassociative flow rule. Attention is restricted to materials, such as metals, for which the notion ofhypoelasticity is valid. Integration schemes based on the Hughes—Winget incrementally objectivealgorithm and the closest point projection algorithm,3,4 originally proposed by Wilkins5 are thenbriefly outlined in Section 3. In Section 4 we present a systematic approach for derivation of thetangent stiffness matrix consistent with the integration schemes outlined in Section 3. A numberof numerical examples, illustrating the excellent performance of the proposed formulation andcomparing it with ABAQUS,22 complete the manuscript.

2. RATE CONSTITUTIVE EQUATIONS

The following notation is employed: the left superscript denotes the configuration, such thatt`*tK denotes the current configuration at time t#*t, whereas tK is the configuration at time t.For simplicity, we will omit the left superscript for the current configuration, i.e., K,t`*tK.A comma followed by a superscript variable x

idenotes a partial derivative with respect to that

subscript variable (i.e. f,xi,Lf /Lx

i). Summation convention for repeated subscripts is employed.

Subscript pairs with regular and square parentheses denotes the symmetric and antisymmetricgradients, respectively. The material time derivative is denoted by a superposed dot. For example,vi"xR

iis the velocity component; and the components of the rate of deformation, e5

ij, and spin,

u5ij, are defined as

e5ij,v

(i,xj )"

1

2 ALv

iLx

j

#

Lvj

LxiB , u5

ij,v

*i,xj +"

1

2 ALv

iLx

j

#

Lvj

LxiB (1)

We consider a class of finite-deformation constitutive equations in the rate form commonly usedin computational plasticity

p5ij"p°

ij#p0

ij, where p0

ij""0

ikpkj!p

ik"0

kj(2)

where pij

represents the Cauchy stress; p°ij

is an objective rate of Cauchy stress, which representsthe material response due to deformation; "0

ij"RQ

ikR~1

kjrepresents the rate of rotation R

ij. We

refer to Reference 2 for a comprehensive discussion on various choices of Rij

and "0ij. For

subsequent discussion we consider the choice: "0ij"u5

ij.

We adopt an additive split of the rate of deformation, e5ij, into elastic rate of deformation, e5 %

ij,

and plastic rate of deformation, e5 1ij, which gives

e5ij"e5 %

ij#e5 1

ij, p°

ij"¸

ijkl(eRkl!eR 1

kl) (3)

where ¸ijkl

are components of the elastic constitutive tensor.We consider the yield function ' defined by

' (pij, a

ij, ½ )"1

2(p

ij!a

ij)P

ijkl(p

kl!a

kl)!1

3½2 (4)

where ½ is the yield stress; aij

the back stress corresponding to the center of the yield surface in thedeviatoric stress space; P

ijklthe projection operator, satisfying P

ijklPklmn

"Pijmn

. For von Misesplasticity the projection operator is defined as follows:

Pijkl

"Iijkl

!13dijdkl, where I

ijkl"1

2(d

ikdjl#d

ildjk

) (5)

ALGORITHMS FOR LARGE DEFORMATION PLASTICITY 841


and dij

is the Kronecker delta. For simplicity we assume the associative flow rule

e5 1ij"

L'Lp

ij

j0 "&ijj0 , where &

ij"P

ijkl(p

kl!a

kl) (6)

and j is a plastic parameter to be determined by plastic consistency condition (4). The evolutionof the yield stress and the back stress are given in the rate form

½Q "2bH

3½j0 (7)

a°ij"

2(1!b)H

3Pijkl

(pkl!a

kl)j0 (8)

where b is a material-dependent parameter, (0)b)1). The extreme values b"0 and b"1 referto Ziegler—Prager kinematic and pure isotropic hardening, respectively; H is a hardeningparameter defining the ratio between the rate of effective stress and the rate of effective plasticstrain.

3. INTEGRATION OF RATE CONSTITUTIVE EQUATIONS

In this section, we briefly outline the Hughes—Winget incrementally objective integrationscheme23 in the context of finite deformation analysis in which the stress objectivity is preservedfor finite rotation increments. We then briefly summarize the closest point projection schemeclosely related to radial return algorithms for integrating material response.2—5

3.1. Incrementally objective integration algorithms

There are several incrementally objective integration schemes. One of the most popularapproaches is known as the corotational method, where all the fields of interest are transformedinto the corotational system.3,24 In such a corotational system, the form of constitutive equationsis analogous to that of small deformation theory and is consistent with the generalized notion ofhyperelasticity provided that an appropriate choice of the rotation tensor, R, is made.25 Analternative approach developed by Hughes and Winget19 is based on the additive incrementalsplit of material and rotational response. In the present manuscript we focus on the latter.

The Hughes—Winget algorithm,23 for integrating the rate constitutive equations arising fromthe finite deformation can be summarized as follows:

pij,t`*tp

ij"tp

ij#*p

ij, tp

ij"R

iktp

klR

jl(9)

aij,t`*ta

ij"ta

ij#*a

ij, ta

ij"R

ikta

klR

jl(10)

where *pij

and *aij

denote the stress and back stress increments resulting from the materialresponse (see Section 3.2), and R

ijis obtained by applying the generalized midpoint rule:23

Rij"d

ij#(d

ik!1

2*u

ik)~1*u

kj(11)



To maintain the second-order accuracy23 strain and rotation increments are obtained using themidpoint rule:

*eij"

1

2 AL*u

iLt`*t@2x

j

#

L*uj

Lt`*t@2xiB , *u

ij"

1

2 AL*u

iLt`*t@2x

j

!

L*uj

Lt`*t@2xiB (12)

where *uiis a displacement increment component and

t`*txi"tx

i#*u

i, t`*t@2x

i,1

2(tx

i#t`*tx

i) (13)

3.2. Closest point projection scheme

For integrating the material response given in the rate form [(6), (7) and (8)], the BackwardEuler integration scheme, which can be interpreted as the closest point projection algorithm,4 isemployed

e1ij"te1

ij#&

ij*j (14)

½"t½#

2bH

3½*j N ½"

3t½

3!2bH*j(15)

aij"ta

ij#

2(1!b)H

3P

ijkl(p

kl!a

kl)*j (16)

pij"ptr

ij!¸

ijkl&

kl*j, ptr

ij"tp

ij#¸

ijkl*e

kl(17)

where *j,t`*tj!tjThe process is regarded elastic if

(ptrij!a

ij)P

ijkl(ptr

kl!a

kl)!

2

3½2 K*j(m)/0

(0 (18)

Otherwise the process is plastic. In the case of the plastic process we proceed by subtracting (16)from (17) to arrive at the following result:

pij!a

ij"(I

ijkl#*j)

ijkl)~1 (ptr

kl!taL

kl) (19)

where

)ijkl

"¸ijst

Pstkl

#23

(1!b)HPijkl

(20)

The value of *j is obtained by satisfying the consistency condition (4). Substituting (15) and(19) into (4) produces a non-linear equation for *j. The Newton method is typically used to solvefor *j

*jk`1

"*jk!G

L'L*jH

~1' K*jk

(21)

where k is the iteration count. It can be shown that the derivative L'/L*j required in (21) is givenas

L'L*j

"!&ij(I

ijkl#*j)

ijkl)~1)

klmn(p

mn!a

mn)!

4bh½2

9!6bh*j(22)



The converged value of *j is then used in combination with (19), (15), (16) and (17) to update theyield stress, the back stress, and the Cauchy stress.

4. CONSISTENT LINEARIZATION

While integration of the constitutive equations affects the accuracy of the solution, the formula-tion of the tangent stiffness matrix consistent with the integration procedure employed is essentialto maintain the asymptotic quadratic rate of convergence of the Newton method provided thatthe solution is smooth. In Section 4.1 we derive the Jacobian matrix for the finite deformationelasto-plastic constitutive model, which in Section 4.2 leads to the formulation of the tangentstiffness matrix consistent with the integration procedures outlined in Section 3.

4.1. The Jacobian matrix for the finite deformation elasto-plastic constitutive model

The Jacobian matrix for the finite deformation elasto-plastic constitutive model outlined in theprevious sections is obtained by taking the material time derivative of the stress and the backstress [(17), (16)] at the current configuration (time t#*t):

p5ij"tp0

ij#¸

ijklM*eR

kl!P

klmn(pR

mn!aR

mn)*j!&

klj0 N (23)

a5ij"ta0

ij#

2(1!b)h

3MP

ijpq(pR

pq!aR

pq)*j#&

ijj0 N (24)

Remark 1. It is common in practice (see for example Section 3.2.2 in ABAQUS theorymanual22) to introduce the following two approximations, which assume infinitesimality of thetime step:

tp0ij+p0

ij""

ikpkj!p

ik"

kj, *e5

kl+e5

kl(25)

In the remainder of this section we derive the consistent Jacobian matrix exactly, and in Section 5,we show that the two approximations given in (25) considerably increase the number of iterationsin the Newton method.

We start by subtracting (24) from (23) which yields

p5ij!a5

ij"(I

ijkl#*j)

ijkl)~1M( tp0

kl!ta0

kl)#¸

klmn*eR

mn!)

klmn&

mnj0 N (26)

where tp0ij

and ta0ij

in (26) are computed by taking the material time derivative of (9) and (10), whichyields

tp0ij"Ap

ijmnRQ

mn, ta0

ij"Aa

ijmnRQ

mn(27)

where

Apijmn

"(dim

dkn

Rjl#d

jmdnlR

ik) tp

kl(28)

Aaijmn

"(dim

dkn

Rjl#d

jmdnlR

ik) ta

kl



Taking the material derivative of (11) gives

RQmn"B

mnpq*uR

pq(29)

where

Bmnpq

"(2dmp!*u

mp)~1 (d

qn#R

qn) (30)

Substituting (29) into (27) results in

tp0ij!ta0

ij"¹

(ij ) *pq+*uR

pq(31)

where

¹ijpq

"(Apijmn

!Aaijmn

)Bmnpq

(32)

It is important to note that the derivation of *eRij

and *uRij

appearing in equations (26) and (31),should be consistent with the midpoint integration scheme employed. In the following we focuson such consistent linearization.

We start by taking the material time derivative of the gradient of the displacement incrementwith respect to the position vector at the midstep [see equation (12)]:

d

dt AL*u

iLt`*t@2x

jB"

Lvi

Ltxk

Ltxk

Lt`*t@2xj

#

L*ui

Ltxk

d

dt ALtx

kLt`*t@2x

jB (33)

Linearization of the second term in (33) yields

d

dt ALtx

kLt`*t@2x

jB"!

Ltxk

Lt`*t@2xm

d

dt ALt`*t@2x

mLtx

nB

Ltxn

Lt`*t@2xj

(34)

Combining (33) and (34) gives

d

dt AL*u

iLt`*t@2x

jB"

Lvi

Lt`*t@2xj

!

L*ui

Lt`*t@2xm

d

dt ALt`*t@2x

mLtx

nB

Ltxn

Lt`*t@2xj

(35)

Equation (35) can be further simplified by exploiting the following relation:

d

dt ALt`*t@2x

mLtx

nB"

LLtx

nA

d

dtt`*t@2x

mB"L

LtxnG

d

dt Axm#tx

m2 BH"

1

2

Lvm

Ltxn

(36)

which after substitution into (35) yields

d

dt AL*u

iLt`*t@2x

jB"Adim

!

1

2

L*ui

Lt`*t@2xmB

Lvm

Lt`*t@2xj

(37)

By utilizing the following equality:

dim!

1

2

L*ui

Lt`*t@2xm

"

LLt`*t@2x

mAt`*t@2x

i!

1

2*u

iB"Ltx

iLt`*t@2x

m

(38)



equation (37) can be recast into the following form:

d

dt AL*u

iLt`*t@2x

jB"

Ltxi

Lt`*t@2xm

Lvm

Lt`*t@2xj

(39)

Defining Mijkl

as

Mijkl

,

Ltxi

Lt`*t@2xk

Lxl

Lt`*t@2xj

(40)

yields

d

dt AL*u

iLt`*t@2x

jB""M

ijklvk,xl

(41)

Taking symmetric and antisymmetric part of (41) with respect to indexes ij we get the finalexpressions for *e5

ijand *u5

ij:

*e5ij"M

(ij )klvk,xl

, *u5ij"M

*ij +klvk,xl

(42)

It can be easily seen that for infinitesimally small step size Mijkl

"dikdjl.

We proceed by substituting (42) and (31) into (26), which yields

p5ij!a5

ij"(I

ijkl#*j)

ijkl)~1 M¸%

klmnvm,xn

!)klmn

&mn

j0 N (43)

where ¸%klmn

is the Jacobian matrix for the finite deformation elastic constitutive model given as

¸%klpq

"¹klmn

M*mn+pq

#¸klmn

M(mn)pq

(44)

The plastic parameter, j0 , in (43) is computed by linearizing consistency condition (4), i.e. '0 "0,which yields

&ij(p5

ij!a5

ij)!

4bH½2j09!6bH*j

"0 (45)

Substituting (43) into (45) provides

j0 "Smn

vm,n

(46)

where

Smn"

&ij(I

ijkl#*j)

ijkl)~1¸%

klmn&

ij(I

ijkl#*j)

ijkl)~1)

klpq&

pq#4bH½2/(9!6bH*j)

(47)

The Jacobian matrix for the finite deformation plastic constitutive model, ¸1ijkl

, is obtained bysubstituting (46), (27) and (29) into (23), which yields

p5ij"¸1

ijklvk,xl

(48)

where

¸1ijkl

"Apijmn

Bmnpq

M*pq+kl

#¸ijmnGM(mn)kl

!Pmnpq A

Ipqst*j

#)pqstB

~1(¸%

stkl!)

stuv&

uvSkl

)!&mn

SklH (49)



Finally, the Jacobian matrix for the finite deformation elasto-plastic constitutive model is given as

¸ijkl

"G¸%ijkl

¸1ijkl

for elastic process,

for plastic process(50)

4.2. Consistent tangent stiffness matrix

We start from the system of non-linear equations arising from the finite element discretization

rA"f */5

A(q

A)!f %95

A"0, f */5

A"P)

NiA,xj

pij

d) (51)

where NkA

is set of C0 continuous shape functions, such that vk"N

kAqRA; the upper case

subscripts denote the degree-of-freedom and the summation convention over repeated indexes isemployed for the degrees-of-freedom and for the spatial dimensions; q

Aand qR

Aare components of

nodal displacement and velocity vectors; and f */5A

and f %95A

are components of the internal andexternal force vectors, respectively.

The consistent tangent stiffness matrix, KAB

, is obtained via consistent linearization of thediscrete equilibrium equations (51). It is convenient to formulate such a linearization procedure as

KAB

,

LLqR

BA

d

dtrAB (52)

For simplicity, assuming that the external force vector is not a function of the solution, theconsistent linearization procedure yields:

d

dtrA"Pt)

d

dt(N

iA, txmF~1mj

pijJ ) d t) (53)

where J is the Jacobian between the configurations t and t#*t ; Fjm

is the deformation gradientdefined as

Fjm"x

j, txm,t`*tx

j, txmand F~1

mj"tx

m,xj,tx

m, t`*txj(54)

Linearization of (53) yields

d

dtrA"Pt)

NiA, txm

MFQ ~1mj

pijJ#F~1

mjpRijJ#F~1

mjpijJQ Ndt) (55)

Substituting (50) into (55) and exploiting the well-known kinematical relations JQ "Jvk,xk

,FQ ~1mj

"

!F~1ml

vl,xj

yields

KAB

"P)N

iA,xjMijkl

NkB,xl

d) (56)

where

¸ijkl

"¸ijkl

#dklpij!d

kjpil

(57)

and ¸ijkl

is defined in (50).



Figure 1. Rotating—stretching bar problem

5. NUMERICAL EXPERIMENTS

Numerical integration and consistent linearization schemes described in Sections 4 and 5 havebeen implemented into ABAQUS22 as a User defined Element (UEL). Note that in ABAQUSfinite deformation plasticity algorithms are similar to those described in Section 4. The keydifference is in the formulation of the tangent stiffness matrix. Hence while the solutions are(almost) identical, the iterative process would have a different character. Three numericalexamples are considered. In the first (Section 5.1) we investigate the accuracy of the Hughes-Winget integration algorithm, whereas the following two Sections (5.2 and 5.3) focus on thecomputational efficiency aspects. For the numerical experiments considered in Sections 5.2 and5.3, the local increments were chosen so that the total error resulting from the numericalintegration will not exceed 3 per cent in the maximal deflection.

5.1. Rotating—stretching bar problem

To investigate the accuracy of the integration algorithm, we consider a rotating—stretching barproblem.21 The total rotation is 90°, whereas the total stretching is only 0)1 per cent. Theprescribed solution is applied in three increments, i.e., rotation increment is 30°. Figure 1 com-pares the accuracy of the Hughes-Winget integration algorithm with the higher order integrationscheme21 and the exact solution. It can be seen that even though rotation increments are verylarge, the second-order accurate Hughes—Winget algorithm provides excellent accuracy in thestress field, resulting in the maximum error not exceeding 6 per cent.

5.2. Rigid body rotation of tetrahedral

In our second numerical experiment, we consider a single tetrahedral element subjected toa rigid body finite rotation. The initial and final configurations are shown in Figure 2.

The material is considered elastic with Young’s modulus, E"21 000, the Poisson’s ratio,v"0)3. The boundary conditions are set in such a way that nodes A and B are held fixed, while



Figure 2. Rotation of tetrahedral element

Figure 3. Deformation of cantilever beam

the horizontal component of the displacement at node C is prescribed, resulting in a rotationangle of approximately 40°.

The prescribed displacement is applied in one increment, and it takes 5 iterations usingconsistent tangent and 54 iterations using approximate tangent (the original ABAQUS algo-rithm). With smaller load increments the advantage is less drastic. For example, for the sameloading applied in three increments the number of iterations using the consistent tangent is 4,5 and 5, whereas with the approximate tangent the number of iterations is 15, 16 and 16.

5.3. ¹he 3D beam problem

We next consider a cantilever beam problem as shown in Figure 3. All the degrees-of-freedomat the clamped end are fixed. Uniform loading is applied at the tip of the beam in the tranversedirection. The length, width, and the depth of the beam are 12, 1 and 2, respectively. The elasticconstants are the same as in the previous example. Plasticity parameters are as follows: thehardening modulus, H"1000, the mixed hardening parameter, b"1, and the initial yield stress,



0½"21. The finite element mesh contains 4351 4-node tetrahedral elements totaling 1091 nodes.We consider two cases: an elastic beam (geometric non-linearity only), and an elasto-plastic beam(geometric and material non-linearity). In both cases the magnitude of loading is selected so thatthe maximal deflection at the tip is approximately one third of the beam length. For the problemwith material non-linearity 79 per cent of elements experience plastic deformation.

The loading is applied in one increment. With the consistent tangent stiffness matrix thenumber of iterations is 11 and 16 for elastic and elasto-plastic problems, respectively. With theconventional tangent the number of iterations increases to 26 and 38 for elastic and elasto-plasticproblems, respectively.

To this end we note that the computational cost of the consistent tangent stiffness matrixevaluation was approximately twice as high as that of the approximate tangent, but the overallcomputational cost associated with the formulation employing consistent tangent was still lower.

6. SUMMARY AND FUTURE RESEARCH

A methodology for efficient implementation of the incrementally objective algorithm23 has beendeveloped. The usefulness of the proposed formulation has been demonstrated as the numericalexperiments show significant savings in computational cost.

The scope of the paper was limited to the cases where the notion of hypoelasticity is valid. Thisis appropriate for metals, where elastic strains remain small compared to plastic deformation. Forpolymers, which exhibit large elastic and plastic deformations of comparable magnitude, a differ-ent treatment might be required.

Several questions, however, remained unanswered. First, we have not investigated whethera consistent tangent operator for the incrementally objective corotational formulation with therotation part extracted from the deformation gradient can be derived with the same ease as forthe present formulation. Such a corotational formulation would have a number of advantages,the key one being compatible with the notion of hyperelasticity.25 Secondly, it is important toinvestigate how increasingly complex incrementally objective algorithms would fare against themultiplicative decomposition and hyperelasticity-based algorithms7 in terms of accuracy andcomputational efficiency.

REFERENCES

1. J. C. Nagtegaal, ‘On the implementation of inelastic constitutive equations with special reference to large deformationproblems’, Comput. Meth. Appl. Mech. Engng., 33, 1982.

2. J. C. Simo and and R. L. Taylor, ‘Consistent tangent operators for rate-independent elasto-plasticity’, Comput. Meth.Appl. Mech. Engng., 48, 1985.

3. T. J. R. Hughes, ‘Numerical implementation of constitutive models: rate-independent deviatoric plasticity’, inS. Nemat-Nasser, R. J. Asaro and G. A. Hegemier (eds.), ¹heoretical Foundation for ¸arge Scale Computations forNonlinear Material Behavior, Martinus Nijhoff Publishers, Dordrecht, 1983.

4. M. Ortiz and E. P. Popov, ‘Accuracy and stability of integration algorithms for elasto-plastic constitutive equations’,Int. J. Numer. Meth. Engng., 21, 1985.

5. M. L. Wilkins, ‘Calculation of elastic-plastic flow’, in B. Adler et al. (ed.), Methods in Computational Physics, AcademicPress, NY, Vol. 3, 1964.

6. J. C. Simo, ‘A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicativedecomposition. Part II: computational aspects’, Comput. Meth. Appl. Mech. Engng., 68, 1988.

7. J. C. Simo and M. Ortiz, ‘A unified approach to finite deformation elasto-plastic analysis based on the use ofhyperelastic constitutive tensor’, Comput. Meth. Appl. Mech. Engng., 49, 1985.



8. B. Moran, M. Ortiz and C. F. Shih, ‘Formulation of implicit finite element methods for multiplicative finitedeformation plasticity’, Int. J. Numer. Meth. Engng., 29, 1990.

9. O. O. Storaasli, VSS code, Computational Structures Branch, NASA Langley, 1996.10. O. Axelsson and V. A. Barker, Finite Element Solution of Boundary »alue Problems, Academic Press, New York, 1984.11. J. Fish and A. Suvorov, ‘Automated Adaptive Multilevel Solver’, Comput. Meth. Appl. Mech. Engng., 149, 267—287

(1997).12. V. Belsky and J. Fish, ‘Towards an ultimate finite element oriented solver on unstructured meshes’, Proc. Computers in

Engineering Conference, ASME, 17—20 September, 1995, Boston.13. J. Fish and R. Guttal, ‘Adaptive solver for the p-version of finite element method’, Int. J. Numer. Meth. Engng., 40,

1767—1784 (1997).14. K. C. Parks and G. Stanley, ‘A curved C0 shell element based on assumed natural coordinate strain’, J. Appl. Mech.,

108, 276—290 (1986).15. J. Fish and T. Belytschko, ‘Stabilized rapidly convergent shell element with drilling degrees-of-freedom’, Int. J. Numer.

Meth. Engng., 33, 149—162 (1992).16. J. Fish and R. Guttal, ‘The p-version of finite element method for shell analysis’, Comput. Mech., 16, 328—340 (1995).17. I. Babuska and T. Scapolla, ‘Computational aspects of the h, p, h—p versions of the finite element method’, in R.

Vichnevetsky and R. S. Stepleman (eds.), Advances in Computer Methods for Partial Differential Equations, IMACSpublications, vol. VI, New Branswick, N.J., 1987.

18. H. P. Hackenberg, ‘Large deformation finite element analysis with inelastic constitutive models including damage’,Comput. Mech., 16, 315—327 (1995).

19. R. I. Borja and E. Alarcon, ‘A mathematical framework for finite strain elastoplastic consolidation. Part 1: Balancelaws, variational formulation, and linearization’, Comput. Meth. Appl. Mech. Engng., 122, 145—171 (1995).

20. B. Nour-Omid and C. C. Rankin, ‘Finite rotation and consistent linearization using projectors’, Comput. Meth. Appl.Mech. Engng., 93, 353—384 (1991).

21. M. M. Rashid, ‘Incremental Kinematics for Finite Element Applications’, Int. J. Numer. Meth. Engng., 36, 3937—3956(1993).

22. ABAQUS Theory Manual, Version 5.4, Habbit, Karlson & Sorensen, Inc., 1994.23. T. J. R. Hughes and J. Winget, ‘Finite rotation effects in numerical integration of rate constitutive equations arising in

large deformation analysis’, Int. J. Numer. Meth. Engng., 15, 1980.24. T. Belytschko and B. J. Hsieh, ‘Nonlinear transient finite element analysis with convected coordinates’, Int. J. Numer.

Meth. Engng., 7, 1973.25. J. C. Simo and K. S. Pister, ‘Remarks on rate constitutive equations for finite deformation problems: computational

implications’, Comput. Meth. Appl. Mech. Engng., 46, 1985.26. S. Choudhry, S. Krishnaswami and T. B. Wertheimer, ‘Hyperelasticity based finite strain plasticity algorithms in

MARC’, Proc. 4th ºS National Congress on Computational Mechanics, 6—8 August 1997.27. J. Fish and R. Guttal, ‘Adaptive solver for the p-version of finite element method’, Int. J. Numer. Meth. Engng., 40,

1767—1784 (1997).



computational aspects of incrementally objective algorithms for large deformation plasticity

Documents