Objective stress rates are often used in commercial finite element (FE) programs. However, deriving a consistent tangent modulus tensor (also known as elasticity tensor or material Jacobian) associated with the objective stress rates is challenging when complex material models are utilized. In this paper, an approximation method for the tangent modulus tensor associated with the Green-Naghdi rate of the Kirchhoff stress is employed to simplify the evaluation process. The effectiveness of the approach is demonstrated through the implementation of two user-defined fiber-reinforced hyperelastic material models. Comparisons between the approximation method and the closed-form analytical method demonstrate that the former can simplify the material Jacobian evaluation with satisfactory accuracy while retaining its computational efficiency. Moreover, since the approximation method is independent of material models, it can facilitate the implementation of complex material models in FE analysis using shell/membrane elements in abaqus.

References

References
1.
Gasser
,
T. C.
,
Ogden
,
R. W.
, and
Holzapfel
,
G. A.
,
2006
, “
Hyperelastic Modelling of Arterial Layers With Distributed Collagen Fibre Orientations
,”
J. R. Soc. Interface
,
3
(
6
), pp.
15
35
.
2.
Fan
,
R.
, and
Sacks
,
M. S.
,
2014
, “
Simulation of Planar Soft Tissues Using a Structural Constitutive Model: Finite Element Implementation and Validation
,”
J. Biomech.
,
47
(
9
), pp.
2043
2054
.
3.
Holzapfel
,
G. A.
, and
Gasser
,
T. C.
,
2001
, “
A Viscoelastic Model for Fiber-Reinforced Composites at Finite Strains: Continuum Basis, Computational Aspects and Applications
,”
Comput. Methods Appl. Mech. Eng.
,
190
(
34
), pp.
4379
4403
.
4.
Genet
,
M.
,
Rausch
,
M. K.
,
Lee
,
L. C.
,
Choy
,
S.
,
Zhao
,
X.
,
Kassab
,
G. S.
,
Kozerke
,
S.
,
Guccione
,
J. M.
, and
Kuhl
,
E.
,
2015
, “
Heterogeneous Growth-Induced Prestrain in the Heart
,”
J. Biomech.
,
48
(
10
), pp.
2080
2089
.
5.
Lubarda
,
V. A.
, and
Hoger
,
A.
,
2002
, “
On the Mechanics of Solids With a Growing Mass
,”
Int. J. Solids Struct.
,
39
(
18
), pp.
4627
4664
.
6.
Nordsletten
,
D.
,
McCormick
,
M.
,
Kilner
,
P. J.
,
Hunter
,
P.
,
Kay
,
D.
, and
Smith
,
N. P.
,
2011
, “
Fluid–Solid Coupling for the Investigation of Diastolic and Systolic Human Left Ventricular Function
,”
Int. J. Numer. Methods Biomed. Eng.
,
27
(
7
), pp.
1017
1039
.
7.
Martin
,
C.
, and
Sun
,
W.
,
2013
, “
Modeling of Long-Term Fatigue Damage of Soft Tissue With Stress Softening and Permanent Set Effects
,”
Biomech. Model. Mechanobiol.
,
12
(
4
), pp.
645
655
.
8.
Haj-Ali
,
R.
,
Dasi
,
L. P.
,
Kim
,
H. S.
,
Choi
,
J.
,
Leo
,
H.
, and
Yoganathan
,
A. P.
,
2008
, “
Structural Simulations of Prosthetic Tri-Leaflet Aortic Heart Valves
,”
J. Biomech.
,
41
(
7
), pp.
1510
1519
.
9.
Kim
,
H.
,
Lu
,
J.
,
Sacks
,
M. S.
, and
Chandran
,
K. B.
,
2008
, “
Dynamic Simulation of Bioprosthetic Heart Valves Using a Stress Resultant Shell Model
,”
Ann. Biomed. Eng.
,
36
(
2
), pp.
262
275
.
10.
Shi
,
Y.
,
Yao
,
J.
,
Young
,
J. M.
,
Fee
,
J. A.
,
Perucchio
,
R.
, and
Taber
,
L. A.
,
2014
, “
Bending and Twisting the Embryonic Heart: A Computational Model for C-Looping Based on Realistic Geometry
,”
Front. Physiol.
,
5
, p.
297
.
11.
Sun
,
W.
,
Li
,
K.
, and
Sirois
,
E.
,
2010
, “
Simulated Elliptical Bioprosthetic Valve Deformation: Implications for Asymmetric Transcatheter Valve Deployment
,”
J. Biomech.
,
43
(
16
), pp.
3085
3090
.
12.
Venkatasubramaniam
,
A.
,
Fagan
,
M.
,
Mehta
,
T.
,
Mylankal
,
K.
,
Ray
,
B.
,
Kuhan
,
G.
,
Chetter
,
I. C.
, and
McCollum
,
P. T.
,
2004
, “
A Comparative Study of Aortic Wall Stress Using Finite Element Analysis for Ruptured and Non-Ruptured Abdominal Aortic Aneurysms
,”
Eur. J. Vasc. Endovasc. Surg.
,
28
(2), pp.
168
176
.
13.
ABAQUS, 2011, “
ABAQUS/Standard: User's Manual
,” Dassault Systèmes Simulia, Johnston, RI.
14.
Simo
,
J.
, and
Hughes
,
T. J. R.
,
1998
, Computational Inelasticity, Springer, New York.
15.
Prot
,
V.
,
Skallerud
,
B.
, and
Holzapfel
,
G.
,
2007
, “
Transversely Isotropic Membrane Shells With Application to Mitral Valve Mechanics. Constitutive Modelling and Finite Element Implementation
,”
Int. J. Numer. Methods Eng.
,
71
(
8
), pp.
987
1008
.
16.
ABAQUS,
2011
, “
ABAQUS Theory Manual, ABAQUS 611 Documentation
,” Dassault Systèmes Simulia, Johnston, RI.
17.
Miehe
,
C.
,
1996
, “
Numerical Computation of Algorithmic (Consistent) Tangent Moduli in Large-Strain Computational Inelasticity
,”
Comput. Methods Appl. Mech. Eng.
,
134
(3–4), pp.
223
240
.
18.
Sun
,
W.
,
Chaikof
,
E. L.
, and
Levenston
,
M. E.
,
2008
, “
Numerical Approximation of Tangent Moduli for Finite Element Implementations of Nonlinear Hyperelastic Material Models
,”
ASME J. Biomech. Eng.
,
130
(
6
), p.
061003
.
19.
Liu
,
H.
, and
Sun
,
W.
,
2015
, “
Computational Efficiency of Numerical Approximations of Tangent Moduli for Finite Element Implementation of a Fiber-Reinforced Hyperelastic Material Model
,”
Comput. Methods Biomech. Biomed. Eng.
,
19
(11), pp.
1
10
.
20.
Tanaka
,
M.
, and
Fujikawa
,
M.
,
2011
, “
Numerical Approximation of Consistent Tangent Moduli Using Complex-Step Derivative and Its Application to Finite Deformation Problems
,”
Trans. Jpn. Soc. Mech. Eng., Ser. A
,
77
(
773
), pp.
27
38
.
21.
Holzapfel
,
G. A.
,
2000
,
Nonlinear Solid Mechanics: A Continuum Approach for Engineering
,
Wiley
,
Chichester, UK
.
22.
Davis
,
F. M.
,
Luo
,
Y.
,
Avril
,
S.
,
Duprey
,
A.
, and
Lu
,
J.
,
2015
, “
Pointwise Characterization of the Elastic Properties of Planar Soft Tissues: Application to Ascending Thoracic Aneurysms
,”
Biomech. Model. Mechanobiol.
,
14
(
5
), pp.
967
978
.
23.
Schiavone
,
A.
, and
Zhao
,
L. G.
,
2015
, “
A Study of Balloon Type, System Constraint and Artery Constitutive Model Used in Finite Element Simulation of Stent Deployment
,”
Mech. Adv. Mater. Modern Processes
,
1
(1), pp. 1–15.
24.
Smoljkić
,
M.
,
Vander Sloten
,
J.
,
Segers
,
P.
, and
Famaey
,
N.
,
2015
, “
Non-Invasive, Energy-Based Assessment of Patient-Specific Material Properties of Arterial Tissue
,”
Biomech. Model. Mechanobiol.
,
14
(
5
), pp.
1045
1056
.
25.
Sun
,
W.
,
Chaikof
,
E. L.
, and
Levenston
,
M. E.
,
2008
, “
Development and Finite Element Implementation of a Nearly Incompressible Structural Constitutive Model for Artery Substitute Design
,”
ASME
Paper No. SBC2008-193164.
26.
Sun
,
W.
, and
Sacks
,
M. S.
,
2005
, “
Finite Element Implementation of a Generalized Fung-Elastic Constitutive Model for Planar Soft Tissues
,”
Biomech. Model. Mechanobiol.
,
4
(2), pp.
190
199
.
27.
Sacks
,
M.
,
1999
, “
A Method for Planar Biaxial Mechanical Testing That Includes In-Plane Shear
,”
ASME J. Biomech. Eng.
,
121
(
5
), pp.
551
555
.
28.
Sacks
,
M. S.
,
Smith
,
D. B.
, and
Hiester
,
E. D.
,
1997
, “
A Small Angle Light Scattering Device for Planar Connective Tissue Microstructural Analysis
,”
Ann. Biomed. Eng.
,
25
(
4
), pp.
678
689
.
29.
Tanaka
,
M.
,
Fujikawa
,
M.
,
Balzani
,
D.
, and
Schröder
,
J.
,
2014
, “
Robust Numerical Calculation of Tangent Moduli at Finite Strains Based on Complex-Step Derivative Approximation and Its Application to Localization Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
269
, pp.
454
470
.
30.
Zulliger
,
M. A.
,
Fridez
,
P.
,
Hayashi
,
K.
, and
Stergiopulos
,
N.
,
2004
, “
A Strain-Energy Function for Arteries Accounting for Wall Composition and Structure
,”
J. Biomech.
,
37
(
7
), pp.
989
1000
.
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