This paper presents a nonlinearly elastic anisotropic microplane formulation in 3D for computational constitutive modeling of arterial soft tissue in the passive regime. The constitutive modeling of arterial (and other biological) soft tissue is crucial for accurate finite element calculations, which in turn are essential for design of implants, surgical procedures, bioartificial tissue, as well as determination of effect of progressive diseases on tissues and implants. The model presented is defined at a lower scale (mesoscale) than the conventional macroscale and it incorporates the effect of all the (collagen) fibers which are anisotropic structural components distributed in all directions within the tissue material in addition to that of isotropic bulk tissue. It is shown that the proposed model not only reproduces Holzapfel’s recent model but also improves on it by accounting for the actual three-dimensional distribution of fiber orientation in the arterial wall, which endows the model with advanced capabilities in simulation of remodeling of soft tissue. The formulation is flexible so that its parameters could be adjusted to represent the arterial wall either as a single material or a material composed of several layers in finite element analyses of arteries. Explicit algorithms for both the material subroutine and the explicit integration with dynamic relaxation of equations of motion using finite element method are given. To circumvent the slow convergence of the standard dynamic relaxation and small time steps dictated by the stability of the explicit integrator, an adaptive dynamic relaxation technique that ensures stability and fastest possible convergence rates is developed. Incompressibility is enforced using penalty method with an updated penalty parameter. The model is used to simulate experimental data from the literature demonstrating that the model response is in excellent agreement with the data. An experimental procedure to determine the distribution of fiber directions in 3D for biological soft tissue is suggested in accordance with the microplane concept. It is also argued that this microplane formulation could be modified or extended to model many other phenomena of interest in biomechanics.

1.
Castaneda-Zuniga
,
W. R.
,
Formanek
,
A.
,
Tadavarthy
,
M.
,
Vlodaver
,
Z.
,
Edwards
,
J. E.
,
Zollikofer
,
C.
, and
Amplatz
,
K.
, 1980, “
The Mechanism of Balloon Angioplasty
,”
Radiology
0033-8419,
135
, pp.
565
571
.
2.
Humphrey
,
J. D.
, 2003, “
Continuum Biomechanics of Soft Biological Tissues
,”
Proc. R. Soc. London, Ser. A
1364-5021,
459
, pp.
3
46
.
3.
Humphrey
,
J. D.
, and
Canham
,
P. B.
, 2000, “
Structure, Mechanical Properties, and Mechanics of Intracranial Saccular Aneurysms
,”
J. Elast.
0374-3535,
61
, pp.
49
81
.
4.
Greenwald
,
S. E.
, and
Berry
,
C. L.
, 2000, “
Improving Vascular Grafts: The Importance of Mechanical and Hemodynamic Properties
,”
Sol. Energy
0038-092X,
190
(
3
), pp.
292
299
.
5.
Liu
,
S. Q.
, 1999, “
Biomechanical Basis of Vascular Tissue Engineering
,”
Crit. Rev. Biomed. Eng.
0278-940X,
27
(
1–2
), pp.
75
148
.
6.
Nerem
,
R. M.
, and
Ensley
,
A. E.
, 2004, “
The Tissue Engineering of Blood Vessels and the Hear
,”
American J. of Transplantation
,
4
, pp.
36
42
.
7.
Holzapfel
,
G. A.
,
Gasser
,
T. C.
, and
Ogden
,
R. W.
, 2000, “
A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models
,”
J. Elast.
0374-3535,
61
, pp.
1
48
.
8.
Humphrey
,
J. D.
, 1995, “
Mechanics of the Arterial Wall: Review and Directions
,”
Crit. Rev. Biomed. Eng.
0278-940X,
23
, pp.
1
162
.
9.
Humphrey
,
J. D.
, 2002,
Cardiovascular Solid Mechanics. Cells, Tissues and Organs
,
Springer-Verlag
,
New York
.
10.
von Maltzahn
,
W.-W.
,
Besdo
,
D.
, and
Wiemer
,
W.
, 1981, “
Elastic Properties of Arteries: A Nonlinear Two-Layer Cylindrical Model
,”
J. Biomech.
0021-9290,
14
, pp.
389
397
.
11.
Tözeren
,
A.
, 1984, “
Elastic Properties of Arteries and Their Influence on the Cardiovascular System
,”
ASME J. Biomech. Eng.
0148-0731,
106
, pp.
182
185
.
12.
Wuyts
,
F. L.
,
Vanhuyse
,
V. J.
,
Langewouters
,
G. J.
,
Decraemer
,
W. F.
,
Raman
,
E. R.
, and
Buyle
,
S.
, 1995, “
Elastic Properties of Human Aortas in Relation to Age and Atherosclerosis: A Structural Model
,”
Phys. Med. Biol.
0031-9155,
40
, pp.
1577
1597
.
13.
Demiray
,
H.
, 1991, “
A Layered Cylindrical Shell Model for an Aorta
,”
Int. J. Eng. Sci.
0020-7225,
29
, pp.
47
54
.
14.
Rachev
,
A.
, 1997, “
Theoretical Study of the Effect of Stress-Dependent Remodeling on Arterial Geometry Under Hypertensive Conditions
,”
J. Biomech.
0021-9290,
30
, pp.
819
827
.
15.
Leukart
,
M.
, and
Ekkehard
,
R.
, 2003, “
A Comparison of Damage Models Formulated on Different Material Scales
,”
Comput. Mater. Sci.
0927-0256,
28
(
3–4
), pp.
749
762
.
16.
Brocca
,
M.
, and
Bažant
,
Z. P.
, 2000, “
Microplane Constitutive Model and Metal Plasticity
,”
Appl. Mech. Rev.
0003-6900,
53
(
10
), pp.
265
281
.
17.
Carol
,
I.
, and
Bažant
,
Z. P.
, 1997, “
Damage and Plasticity in Microplane Theory
,”
Int. J. Solids Struct.
0020-7683,
34
(
29
), pp.
3807
3835
.
18.
Caner
,
F. C.
,
Bažant
,
Z. P.
, and
Červenka
,
J.
, 2002, “
Vertex Effect in Strain-Softening Concrete at Rotating Principal Axes
,”
J. Eng. Mech. Div., Am. Soc. Civ. Eng.
0044-7951,
128
(
1
), pp.
24
33
.
19.
de Borst
,
R.
, 2002, “
Fracture in Quasi-Brittle Materials: A Review of Continuum Damage-Based Approaches
,”
Eng. Fract. Mech.
0013-7944,
69
(
2
), pp.
95
112
.
20.
Bažant
,
Z. P.
,
Caner
,
F. C.
,
Carol
,
I.
,
Adley
,
M. D.
, and
Akers
,
S. A.
, 2000, “
Microplane Model m4 for Concrete. I: Formulation With Work-Conjugate Deviatoric Stress
,”
J. Eng. Mech. Div., Am. Soc. Civ. Eng.
0044-7951,
126
(
9
), pp.
944
953
.
21.
Caner
,
F. C.
, and
Bažant
,
Z. P.
, 2000, “
Microplane Model m4 for Concrete. II: Algorithm and Calibration
,”
J. Eng. Mech. Div., Am. Soc. Civ. Eng.
0044-7951,
126
(
9
), pp.
954
961
.
22.
Bažant
,
Z. P.
, and
Zi
,
G.-S.
, 2003, “
Microplane Constitutive Model for Porous Isotropic Rocks
,”
J. Geophys. Res., [Solid Earth]
0148-0227,
108
, B2(
2119
).
23.
Brocca
,
M.
,
Bažant
,
Z. P.
, and
Daniel
,
I. M.
, 2001, “
Microplane Model for Stiff Foams and Finite Element Analysis of Sandwich Failure by Core Indentation
,”
Int. J. Solids Struct.
0020-7683,
38
(
44–45
), pp.
8111
8132
.
24.
Brocca
,
M.
,
Brinson
,
L. C.
, and
Bažant
,
Z. P.
, 2002, “
Three-Dimensional Constitutive Model for Shape Memory Alloys Based on Microplane Model
,”
J. Mech. Phys. Solids
0022-5096,
50
(
5
), pp.
1051
1077
.
25.
Carol
,
I.
,
Jirásek
,
M.
, and
Bažant
,
Z. P.
, 2001, “
A Thermodynamically Consistent Approach to Microplane Theory. Part I. Free Energy and Consistent Microplane Stress
,”
Int. J. Solids Struct.
0020-7683,
38
(
17
), pp.
2921
2931
.
26.
Carol
,
I.
,
Jirásek
,
M.
, and
Bažant
,
Z. P.
, 2004, “
A Framework for Microplane Models at Large Strain, With Application to Hyperelasticity
,”
Int. J. Solids Struct.
0020-7683,
41
(
2
), pp.
511
557
.
27.
Belytschko
,
T.
,
Liu
,
W.
, and
Moran
,
B.
, 2004,
Nonlinear Finite Elements for Continua and Structures
,
Wiley
,
New York
.
28.
Ogden
,
R. W.
, 1976, “
Volume Changes Associated With the Deformation of Rubber-Like Solids
,”
J. Mech. Phys. Solids
0022-5096,
24
, pp.
323
338
.
29.
Cescotto
,
S.
, and
Fonder
,
G.
, 1979, “
A Finite Element Approach for Large Strains of Nearly Incompressible Rubber-Like Materials
,”
Int. J. Solids Struct.
0020-7683,
15
, pp.
589
605
.
30.
Eussman
,
T.
, and
Bathe
,
K.
, 1987, “
A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis
,”
Comput. Struct.
0045-7949,
26
, pp.
357
409
.
31.
Oakley
,
D. R.
, and
Knight
,
N. F.
, 1995, “
Adaptive Dynamic Relaxation Algorithm for Nonlinear Hyperelastic Structures Part I. Formulation
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
126
, pp.
67
89
.
32.
Gasser
,
T. C.
, and
Holzapfel
,
G. A.
, 2002, “
A Rate-Independent Elastoplastic Constitutive Model for Biological Fiber-Reinforced Composites at Finite Strains: Continuum Basis, Algorithmic Formulation and Finite Element Implementation
,”
Comput. Mech.
0178-7675,
29
, pp.
340
360
.
33.
Ogden
,
R. W.
, 1984,
Non-linear Elastic Deformations
,
Wiley
,
New York
.
34.
Sacks
,
M. S.
, 2003, “
Incorporation of Experimentally Derived Fiber Orientation Into a Structural Constitutive Model for Planar Collagenous Tissues
,”
ASME J. Biomech. Eng.
0148-0731,
125
, pp.
280
287
.
35.
Bažant
,
Z. P.
, 2005 (Private communication).
36.
Holzapfel
,
G. A.
, 2003, “
Structural and Numerical Models for the (Visco)Elastic Response of Arterial Walls and Residual Stresses
,” in
Biomechanics of Soft Tissue in Cardiovascular Systems
,
G. A.
Holzapfel
and
R. W.
Ogden
, eds.
Springer
,
Wien
, pp.
109
184
.
37.
Stroud
,
A. H.
, 1971,
Approximate Calculation of Multiple Integrals
.
Prentice-Hall
,
Englewood Cliffs, NJ
.
38.
Bažant
,
Z. P.
, and
Oh
,
B.-H.
, 1986, “
Efficient Numerical Integration on the Surface of a Sphere
,”
Z. Angew. Math. Mech.
0044-2267,
66
(
1
), pp.
37
49
.
39.
Kulkarni
,
M.
,
Belytschko
,
T.
, and
Bayliss
,
A.
, 1995, “
Stability and Error Analysis for Time Integrators Applied to Strain-Softening Materials
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
124
, pp.
335
363
.
40.
Wood
,
W. L.
, 1967, “
Comparison of Dynamic Relaxation With Three Other Iterative Methods
,”
Engineer (London)
0013-7758,
224
, pp.
683
687
.
41.
Metzger
,
D. R.
, 2003, “
Adaptive Damping for Dynamic Relaxation Problems With Non-Monotonic Spectral Response
,”
Int. J. Numer. Methods Eng.
0029-5981,
56
, pp.
57
80
.
42.
Gradshteyn
,
I. S.
, and
Ryzhik
,
I. M.
, 2000,
Table of Integrals, Series and Products
,
6th ed.
,
Academic Press
,
San Diego, CA
.
43.
Simo
,
J. C.
, and
Taylor
,
R. L.
, 1991, “
Quasi-Incompressible Finite Elasticity in Principal Stretches. Continuum Basis and Numerical Algorithms
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
85
, pp.
273
310
.
44.
Stergiopulos
,
N.
,
Vulliémoz
,
S.
,
Rachev
,
A.
,
Meister
,
J.-J.
, and
Greenwald
,
S. E.
, 2001, “
Assessing the Homogeneity of the Elastic Properties and Composition of the Pig Aortic Media
,”
J. Vasc. Res.
1018-1172,
38
, pp.
237
246
.
45.
Standley
,
P. R.
,
Camaratta
,
A.
,
Nolan
,
B. P.
,
Purgason
,
C. T.
, and
Stanley
,
M. A.
, 2002, “
Cyclic Stretch Induces Vascular Smooth Muscle Cell Alignment Via NO Signaling
,”
Am. J. Physiol. Heart Circ. Physiol.
0363-6135,
283
, pp.
H1907
-
H1914
.
46.
Boerboom
,
R.
,
Driessen
,
N.
,
Bouten
,
C. V. C.
,
Huyghe
,
J. M.
, and
Baaijens
,
F. P. T.
, 2003, “
Finite Element Model of Mechanically Induced Collagen Fiber Synthesis and Degradation in the Aortic Valve
,”
Ann. Biomed. Eng.
0090-6964,
31
(
9
), pp.
1040
1053
.
47.
Wagenseil
,
J. E.
,
Elson
,
E. L.
, and
Okamoto
,
R. J.
, 2004, “
Cell Orientation Influences the Biaxial Mechanical Properties of Fibroblast Populated Collagen Vessels
,”
Ann. Biomed. Eng.
0090-6964,
32
(
5
), pp.
720
731
.
48.
Wang
,
J. H.
,
Jia
,
F.
,
Gilbert
,
T. W.
, and
Woo
,
S. L.
, 2003, “
Cell Orientation Determines the Alignment of Cell-Produced Collagenous Matrix
,”
J. Biomech.
0021-9290,
36
(
1
), pp.
97
102
.
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