The objective stress rates used in most commercial finite element programs are the Jaumann rate of Kirchhoff stress, Jaumann rates of Cauchy stress, or Green–Naghdi rate. The last two were long ago shown not to be associated by work with any finite strain tensor, and the first has often been combined with tangential moduli not associated by work. The error in energy conservation was thought to be negligible, but recently, several papers presented examples of structures with high volume compressibility or a high degree of orthotropy in which the use of commercial software with the Jaumann rate of Cauchy or Kirchhoff stress leads to major errors in energy conservation, on the order of 25–100%. The present paper focuses on the Green–Naghdi rate, which is used in the explicit nonlinear algorithms of commercial software, e.g., in subroutine VUMAT of ABAQUS. This rate can also lead to major violations of energy conservation (or work conjugacy)—not only because of high compressibility or pronounced orthotropy but also because of large material rotations. This fact is first demonstrated analytically. Then an example of a notched steel cylinder made of steel and undergoing compression with the formation of a plastic shear band is simulated numerically by subroutine VUMAT in ABAQUS. It is found that the energy conservation error of the Green–Naghdi rate exceeds 5% or 30% when the specimen shortens by 26% or 38%, respectively. Revisions in commercial software are needed but, even in their absence, correct results can be obtained with the existing software. To this end, the appropriate transformation of tangential moduli, to be implemented in the user's material subroutine, is derived.

References

References
1.
Bažant
,
Z.
,
1971
, “
A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies
,”
ASME J. Appl. Mech.
,
38
(
4
), pp.
919
928
.10.1115/1.3408976
2.
Green
,
A.
, and
Naghdi
,
P.
,
1965
, “
A General Theory of an Elastic-Plastic Continuum
,”
Arch. Ration. Mech. Anal.
,
18
(
4
), pp.
251
281
. (Eq. 8.23).10.1007/BF00251666
3.
Červenka
,
V.
, and
Jendele
,
L.
,
2008
,
ATENA Program Documentation—Part 1: Theory
, Cervenka Consulting, www.cervenka.cz
4.
Patzák
,
B.
, and
Bittnar
,
Z.
,
2001
, “
Design of Object Oriented Finite Element Code
,”
Adv. Eng. Softw.
,
32
(
10–11
), pp.
759
767
.10.1016/S0965-9978(01)00027-8
5.
Bažant
,
Z.
,
Gattu
,
M.
, and
Vorel
,
J.
,
2012
, “
Work Conjugacy Error in Commercial Finite Element Codes: Its Magnitude and How to Compensate for It
,”
Proc. Royal Soc. A Math Phys Eng. Sci.
,
468
(
2146
), pp.
3047
3058
.10.1098/rspa.2012.0167
6.
Vorel
,
J.
,
Zant
,
Z. B.
, and
Gattu
,
M.
,
2013
, “
Elastic Soft-Core Sandwich Plates: Critical Loads and Energy Errors in Commercial Codes Due to Choice of Objective Stress Rate
,”
ASME J. Appl. Mech.
,
80
(
4
), p.
041034
.10.1115/1.4023024
7.
Bažant
,
Z.
, and
Beghini
,
A.
,
2005
, “
Which Formulation Allows Using a Constant Shear Modulus for Small-Strain Buckling of Soft-Core Sandwich Structures?
,”
ASME J. Appl. Mech.
,
72
(
5
), pp.
785
787
.10.1115/1.1979516
8.
Bažant
,
Z.
, and
Beghini
,
A.
,
2006
, “
Stability and Finite Strain of Homogenized Structures Soft in Shear: Sandwich or Fiber Composites, and Layered Bodies
,”
Int. J. Solid. Struct.
,
43
(
6
), pp.
1571
1593
.10.1016/j.ijsolstr.2005.03.060
9.
Ji
,
W.
,
Waas
,
A.
, and
Bažant
,
Z.
,
2010
, “
Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs
,”
ASME J. Appl. Mech.
,
77
(
4
), p.
044504
.10.1115/1.4000916
10.
Bažant
,
Z.
, and
Cedolin
,
L.
,
1991
,
Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories
,
1st ed.
,
Oxford University Press
,
New York
.
11.
Hibbitt
,
H.
,
Marcal
,
P.
, and
Rice
,
J.
,
1970
, “
A Finite Strain Formulation for Problems of Large Strain and Displacement
,”
Int. J. Solid. Struct.
,
6
, pp.
1069
1086
.10.1016/0020-7683(70)90048-X
12.
Vural
,
M.
,
Rittel
,
D.
, and
Ravichandran
,
G.
,
2003
, “
Large Strain Mechanical Behavior of 1018 Cold-Rolled Steel Over a Wide Range of Strain Rates
,”
Metal. Mater. Trans. A
,
34
(
12
), pp.
2873
2885
.10.1007/s11661-003-0188-8
13.
Dassault Systèmes
,
2010
, ABAQUS FEA, www.simulia.com
14.
Hughes
,
T.
, and
Winget
,
J.
,
1980
, “
Finite Rotation Effects in Numerical Integration of Rate Constitutive Equations Arising in Large-Deformation Analysis
,”
Int. J. Numer. Meth. Eng.
,
15
(
12
), pp.
1862
1867
.10.1002/nme.1620151210
15.
Fraejis de Veubeke
,
B.
,
1965
,
Displacement Equilibrium Models in the Finite Element Method
,
John Wiley & Sons Ltd.
,
Chichester, UK
.
16.
Jaumann
,
G.
,
1911
, “
Geschlossenes system physikalischer und chemischer differentialgesetze
,”
Sitzungsberichte Akad. Wiss. Wien
,
IIa
, pp.
385
530
.
You do not currently have access to this content.