Fibrous soft tissues are biopolymeric materials that are made of extracellular proteins, such as different types of collagen and proteoglycans, and have a high water content. These tissues have nonlinear, anisotropic, and inelastic mechanical behaviors that are often categorized into viscoelastic behavior, plastic deformation, and damage. While tissue's elastic and viscoelastic mechanical properties have been measured for decades, there is no comprehensive theoretical framework for modeling inelastic behaviors of these tissues that is based on their structure. To model the three major inelastic mechanical behaviors of tissue's fibrous matrix, we formulated a structurally inspired continuum mechanics framework based on the energy of molecular bonds that break and reform in response to external loading (reactive bonds). In this framework, we employed the theory of internal state variables (ISV) and kinetics of molecular bonds. The number fraction of bonds, their reference deformation gradient, and damage parameter were used as state variables that allowed for consistent modeling of all three of the inelastic behaviors of tissue by using the same sets of constitutive relations. Several numerical examples are provided that address practical problems in tissue mechanics, including the difference between plastic deformation and damage. This model can be used to identify relationships between tissue's mechanical response to external loading and its biopolymeric structure.

References

References
1.
Woo
,
S. L.-Y.
,
Simon
,
B. R.
,
Kuei
,
S. C.
, and
Akeson
,
W. H.
,
1980
, “
Quasi-Linear Viscoelastic Properties of Normal Articular Cartilage
,”
ASME J. Biomech. Eng.
,
102
(
2
), pp.
85
90
.
2.
Huang
,
C. Y.
,
Mow
,
V. C.
, and
Ateshian
,
G. A.
,
2001
, “
The Role of Flow-Independent Viscoelasticity in the Biphasic Tensile and Compressive Responses of Articular Cartilage
,”
ASME J. Biomech. Eng.
,
123
(
5
), pp.
410
417
.
3.
Connizzo
,
B. K.
, and
Grodzinsky
,
A. J.
,
2017
, “
Tendon Exhibits Complex Poroelastic Behavior at the Nanoscale as Revealed by High-Frequency AFM-Based Rheology
,”
J. Biomech.
,
54
, pp.
11
18
.
4.
Maher
,
E.
,
Creane
,
A.
,
Lally
,
C.
, and
Kelly
,
D. J.
,
2012
, “
An Anisotropic Inelastic Constitutive Model to Describe Stress Softening and Permanent Deformation in Arterial Tissue
,”
J. Mech. Behav. Biomed. Mater
,
12
, pp.
9
19
.
5.
Caro-Bretelle
,
A.
,
Gountsop
,
P.
,
Ienny
,
P.
,
Leger
,
R.
,
Corn
,
S.
,
Bazin
,
I.
, and
Bretelle
,
F.
,
2015
, “
Effect of Sample Preservation on Stress Softening and Permanent Set of Porcine Skin
,”
J. Biomech.
,
48
(
12
), pp.
3135
3141
.
6.
Abrahams
,
M.
,
1967
, “
Mechanical Behaviour of Tendon In Vitro—A Preliminary Report
,”
Med. Biol. Eng.
,
5
(
5
), pp.
433
443
.
7.
Rigby
,
B. J.
,
1964
, “
Effect of Cyclic Extension on the Physical Properties of Tendon Collagen and Its Possible Relation to Biological Ageing of Collagen
,”
Nature
,
202
(
4937
), pp.
1072
1074
.
8.
Natali
,
A.
,
Pavan
,
P.
,
Carniel
,
E.
,
Lucisano
,
M.
, and
Taglialavoro
,
G.
,
2005
, “
Anisotropic Elasto-Damage Constitutive Model for the Biomechanical Analysis of Tendons
,”
Med. Eng. Phys.
,
27
(
3
), pp.
209
214
.
9.
Von Forell
,
G. A.
, and
Bowden
,
A. E.
,
2014
, “
A Damage Model for the Percutaneous Triple Hemisection Technique for Tendo-Achilles Lengthening
,”
J. Biomech.
,
47
(
13
), pp.
3354
3360
.
10.
Coleman
,
B. D.
, and
Gurtin
,
M. E.
,
1967
, “
Thermodynamics With Internal State Variables
,”
J. Chem. Phys.
,
47
(
2
), pp.
597
613
.
11.
Horstemeyer
,
M. F.
, and
Bammann
,
D. J.
,
2010
, “
Historical Review of Internal State Variable Theory for Inelasticity
,”
Int. J. Plast.
,
26
(
9
), pp.
1310
1334
.
12.
Reese
,
S.
, and
Govindjee
,
S.
,
1998
, “
A Theory of Finite Viscoelasticity and Numerical Aspects
,”
Int. J. Solids Struct.
,
35
(
26–27
), pp.
3455
3482
.
13.
Holzapfel
,
G. A.
, and
Simo
,
J. C.
,
1996
, “
A New Viscoelastic Constitutive Model for Continuous Media at Finite Thermomechanical Changes
,”
Int. J. Solids Struct.
,
33
(
20–22
), pp.
3019
3034
.
14.
Zhang
,
W.
, and
Sacks
,
M. S.
,
2017
, “
Modeling the Response of Exogenously Crosslinked Tissue to Cyclic Loading: The Effects of Permanent Set
,”
J. Mech. Behav. Biomed. Mater.
,
75
, pp.
336
350
.
15.
Peña
,
E.
,
2011
, “
Prediction of the Softening and Damage Effects With Permanent Set in Fibrous Biological Materials
,”
J. Mech. Phys. Solids
,
59
(
9
), pp.
1808
1822
.
16.
Peña
,
E.
,
2014
, “
Computational Aspects of the Numerical Modelling of Softening, Damage and Permanent Set in Soft Biological Tissues
,”
Comput. Struct.
,
130
, pp.
57
72
.
17.
Fereidoonnezhad
,
B.
,
Naghdabadi
,
R.
, and
Holzapfel
,
G.
,
2016
, “
Stress Softening and Permanent Deformation in Human Aortas: Continuum and Computational Modeling With Application to Arterial Clamping
,”
J. Mech. Behav. Biomed. Mater.
,
61
, pp.
600
616
.
18.
Weisbecker
,
H.
,
Pierce
,
D. M.
,
Regitnig
,
P.
, and
Holzapfel
,
G. A.
,
2012
, “
Layer-Specific Damage Experiments and Modeling of Human Thoracic and Abdominal Aortas With Non-Atherosclerotic Intimal Thickening
,”
J. Mech. Behav. Biomed. Mater.
,
12
(
2
), pp.
93
106
.
19.
Dorfmann
,
A.
, and
Ogden
,
R.
,
2004
, “
A Constitutive Model for the Mullins Effect With Permanent Set in Particle-Reinforced Rubber
,”
Int. J. Solids Struct.
,
41
(
7
), pp.
1855
1878
.
20.
Li
,
W.
,
2016
, “
Damage Models for Soft Tissues: A Survey
,”
J. Med. Biol. Eng.
,
36
(
3
), pp.
285
307
.
21.
Mullins
,
L.
,
1969
, “
Softening of Rubber by Deformation
,”
Rubber Chem. Technol.
,
42
(
1
), pp.
339
362
.
22.
Diani
,
J.
,
Fayolle
,
B.
, and
Gilormini
,
P.
,
2009
, “
A Review on the Mullins Effect
,”
Eur. Polym. J.
,
45
(
3
), pp.
601
612
.
23.
Schmidt
,
T.
,
Balzani
,
D.
, and
Holzapfel
,
G.
,
2014
, “
Statistical Approach for a Continuum Description of Damage Evolution in Soft Collagenous Tissues
,”
Comput. Methods Appl. Mech. Eng.
,
278
, pp.
41
61
.
24.
Alastrué
,
V.
,
Rodríguez
,
J.
,
Calvo
,
B.
, and
Doblaré
,
M.
,
2007
, “
Structural Damage Models for Fibrous Biological Soft Tissues
,”
Int. J. Solids Struct.
,
44
(
18–19
), pp.
5894
5911
.
25.
Kachanov
,
L. M.
,
1968
,
Introduction to Continuum Damage Mechanics
,
Martinus Nijhoff Publishers
,
Dordrecht, The Netherlands
.
26.
Lemaitre
,
J.
,
1984
, “
How to Use Damage Mechanics
,”
Nucl. Eng. Des.
,
80
(
2
), pp.
233
245
.
27.
Green
,
M. S.
, and
Tobolsky
,
A. V.
,
1946
, “
A New Approach to the Theory of Relaxing Polymeric Media
,”
J. Chem. Phys.
,
14
(
2
), pp.
80
92
.
28.
Tobolsky
,
A. V.
, and
Andrews
,
R. D.
,
1945
, “
Systems Manifesting Superposed Elastic and Viscous Behavior
,”
J. Chem. Phys.
,
13
(
1
), pp.
3
27
.
29.
Andrews
,
R. D.
,
Tobolsky
,
A. V.
, and
Hanson
,
E. E.
,
1946
, “
The Theory of Permanent Set at Elevated Temperatures in Natural and Synthetic Rubber Vulcanizates
,”
J. Appl. Phys.
,
17
(
5
), pp.
352
361
.
30.
Rajagopal
,
K.
, and
Wineman
,
A.
,
1992
, “
A Constitutive Equation for Nonlinear Solids Which Undergo Deformation Induced Microstructural Changes
,”
Int. J. Plast.
,
8
(
4
), pp.
385
395
.
31.
Rajagopal
,
K. R.
, and
Srinivasa
,
A. R.
,
2004
, “
On the Thermomechanics of Materials That Have Multiple Natural Configurations—Part I: Viscoelasticity and Classical Plasticity
,”
Z. Angew. Math. Phys.
,
55
(
5
), pp.
861
863
.
32.
Muliana
,
A.
,
Rajagopal
,
K. R.
,
Tscharnuter
,
D.
, and
Pinter
,
G.
,
2016
, “
A Nonlinear Viscoelastic Constitutive Model for Polymeric Solids Based on Multiple Natural Configuration Theory
,”
Int. J. Solids Struct.
,
100–101
, pp.
95
110
.
33.
Scott
,
K. W.
, and
Stein
,
R. S.
,
1953
, “
A Molecular Theory of Stress Relaxation in Polymeric Media
,”
J. Chem. Phys.
,
21
(
7
), pp.
1281
1286
.
34.
Demirkoparan
,
H.
,
Pence
,
T. J.
, and
Wineman
,
A.
,
2009
, “
On Dissolution and Reassembly of Filamentary Reinforcing Networks in Hyperelastic Materials
,”
Proc. R. Soc. A Math. Phys. Eng. Sci.
,
465
(
2103
), pp.
867
894
.
35.
Wineman
,
A.
,
2009
, “
On the Mechanics of Elastomers Undergoing Scission and Cross-Linking
,”
Int. J. Adv. Eng. Sci. Appl. Math.
,
1
(
2–3
), pp.
123
131
.
36.
Meng
,
F.
,
Pritchard
,
R. H.
, and
Terentjev
,
E. M.
,
2016
, “
Stress Relaxation, Dynamics, and Plasticity of Transient Polymer Networks
,”
Macromolecules
,
49
(
7
), pp.
2843
2852
.
37.
Mao
,
Y.
,
Lin
,
S.
,
Zhao
,
X.
, and
Anand
,
L.
,
2017
, “
A Large Deformation Viscoelastic Model for Double-Network Hydrogels
,”
J. Mech. Phys. Solids
,
100
, pp.
103
130
.
38.
Nims
,
R. J.
,
Durney
,
K. M.
,
Cigan
,
A. D.
,
Dusséaux
,
A.
,
Hung
,
C. T.
, and
Ateshian
,
G. A.
,
2016
, “
Continuum Theory of Fibrous Tissue Damage Mechanics Using Bond Kinetics: Application to Cartilage Tissue Engineering
,”
Interface Focus
,
6
(
1
), p.
20150063
.
39.
Ateshian
,
G. A.
,
2015
, “
Viscoelasticity Using Reactive Constrained Solid Mixtures
,”
J. Biomech.
,
48
(
6
), pp.
941
947
.
40.
Nims
,
R. J.
, and
Ateshian
,
G. A.
,
2017
, “
Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues
,”
J. Elast.
,
129
(
1–2
), pp.
69
105
.
41.
Lee
,
E. H.
,
1969
, “
Elastic-Plastic Deformation at Finite Strains
,”
ASME J. Appl. Mech.
,
36
(
1
), pp.
1
6
.
42.
Simo
,
J.
,
1988
, “
A Framework for Finite Strain Elastoplasticity Based on Maximum Plastic Dissipation and the Multiplicative Decomposition—Part I: Continuum Formulation
,”
Comput. Methods Appl. Mech. Eng.
,
66
(
1
), pp.
199
219
.
43.
Lubarda
,
V. A.
,
2004
, “
Constitutive Theories Based on the Multiplicative Decomposition of Deformation Gradient: Thermoelasticity, Elastoplasticity, and Biomechanics
,”
ASME Appl. Mech. Rev.
,
57
(
2
), pp.
95
108
.
44.
Spencer
,
A. J.
,
1984
,
Continuum Theory of the Mechanics of Fibre-Reinforced Composites
,
Springer Vienna
,
Austria
.
45.
Cortes
,
D. H.
, and
Elliott
,
D. M.
,
2014
, “
Accurate Prediction of Stress in Fibers With Distributed Orientations Using Generalized High-Order Structure Tensors
,”
Mech. Mater.
,
75
, pp.
73
83
.
46.
Tanaka
,
F.
, and
Edwards
,
S.
,
1992
, “
Viscoelastic Properties of Physically Crosslinked Networks
,”
J. Non-Newtonian Fluid Mech.
,
43
(
2–3
), pp.
273
288
.
47.
Simo
,
J. C.
, and
Hughes
,
T. J. R.
,
1998
,
Computational Inelasticity: Interdisciplinary Applied Mathematics
,
Springer-Verlag
,
New York
.
48.
Khan
,
A. S.
, and
Huang
,
S.
,
1995
,
Continuum Theory of Plasticity
,
Wiley
,
New York
.
49.
Simo
,
J.
,
1987
, “
On a Fully Three-Dimensional Finite-Strain Viscoelastic Damage Model: Formulation and Computational Aspects
,”
Comput. Methods Appl. Mech. Eng.
,
60
(
2
), pp.
153
173
.
50.
Naghdi
,
P.
, and
Trapp
,
J.
,
1975
, “
The Significance of Formulating Plasticity Theory With Reference to Loading Surfaces in Strain Space
,”
Int. J. Eng. Sci.
,
13
(
9–10
), pp.
785
797
.
51.
Aravas
,
N.
,
1994
, “
Finite-Strain Anisotropic Plasticity and the Plastic Spin
,”
Model. Simul. Mater. Sci. Eng.
,
2
(
3A
), pp.
483
504
.
52.
Dafalias
,
Y. F.
,
1990
, “
The Plastic Spin in Viscoplasticity
,”
Int. J. Solids Struct.
,
26
(
2
), pp.
149
163
.
53.
Krajcinovic
,
D.
,
2000
, “
Damage Mechanics: Accomplishments, Trends and Needs
,”
Int. J. Solids Struct.
,
37
(
1–2
), pp.
267
277
.
54.
Chu
,
B. M.
, and
Blatz
,
P. J.
,
1972
, “
Cumulative Microdamage Model to Describe the Hysteresis of Living Tissue
,”
Ann. Biomed. Eng.
,
1
(
2
), pp.
204
211
.
55.
Jacobs
,
N. T.
,
Cortes
,
D. H.
,
Peloquin
,
J. M.
,
Vresilovic
,
E. J.
, and
Elliott
,
D. M.
,
2014
, “
Validation and Application of an Intervertebral Disc Finite Element Model Utilizing Independently Constructed Tissue-Level Constitutive Formulations That Are Nonlinear, Anisotropic, and Time-Dependent
,”
J. Biomech.
,
47
(
11
), pp.
2540
2546
.
56.
Holmes
,
M. H.
, and
Mow
,
V. C.
,
1990
, “
The Nonlinear Characteristics of Soft Gels and Hydrated Connective Tissues in Ultrafiltration
,”
J. Biomech.
,
23
(
11
), pp.
1145
1156
.
57.
Szczesny
,
S. E.
, and
Elliott
,
D. M.
,
2014
, “
Interfibrillar Shear Stress Is the Loading Mechanism of Collagen Fibrils in Tendon
,”
Acta Biomater.
,
10
(
6
), pp.
2582
2590
.
58.
Lynch
,
H. A.
,
Johannessen
,
W.
,
Wu
,
J. P.
,
Jawa
,
A.
, and
Elliott
,
D. M.
,
2003
, “
Effect of Fiber Orientation and Strain Rate on the Nonlinear Uniaxial Tensile Material Properties of Tendon
,”
ASME J. Biomech. Eng.
,
125
(
5
), pp.
726
731
.
59.
Safa
,
B. N.
,
Lee
,
A. H.
,
Santare
,
M. H.
, and
Elliott
,
D. M.
,
2018
, “
Evaluating Plastic Deformation and Damage as Potential Mechanisms for Tendon Inelasticity Using a Reactive Modeling Framework
,” bioRxiv,
349530
.
60.
Reiser
,
K.
,
McCormick
,
R. J.
, and
Rucker
,
R. B.
,
1992
, “
Enzymatic and Nonenzymatic Cross-Linking of Collagen and Elastin
,”
FASEB J.
,
6
(
7
), pp.
2439
2449
.
61.
Eyre
,
D. R.
,
Weis
,
M. A.
, and
Wu
,
J.-J.
,
2008
, “
Advances in Collagen Cross-Link Analysis
,”
Methods
,
45
(
1
), pp.
65
74
.
62.
Depalle
,
B.
,
Qin
,
Z.
,
Shefelbine
,
S. J.
, and
Buehler
,
M. J.
,
2015
, “
Influence of Cross-Link Structure, Density and Mechanical Properties in the Mesoscale Deformation Mechanisms of Collagen Fibrils
,”
J. Mech. Behav. Biomed. Mater.
,
52
, pp.
1
13
.
63.
Marlena
,
G.
,
Komorowska
,
M.
,
Hanuza
,
J.
,
Mirosław
,
M.
,
Romuald
,
B.
, and
Kobielarz
,
M.
,
2011
, “
Mechanobiology of Soft Tissues: FT-Raman Spectroscopic Studies Biomedical Engineering
,”
Challenges Mod. Technol.
,
2
, pp.
8
11
.https://yadda.icm.edu.pl/baztech/element/bwmeta1.element.baztech-0c304bfb-a63c-4f08-bf72-982c937afd0e
64.
Hanuza
,
J.
,
Maczka
,
M.
,
Gasior-Glogowska
,
M.
,
Komorowska
,
M.
,
Kobielarz
,
M.
,
Bedzinski
,
R.
,
Szotek
,
S.
,
Maksymowicz
,
K.
, and
Hermanowicz
,
K.
,
2010
, “
FT-Raman Spectroscopic Study of Thoracic Aortic Wall Subjected to Uniaxial Stress
,”
J. Raman Spectrosc.
,
41
(
10
), pp.
1163
1169
.
65.
Herod
,
T. W.
,
Chambers
,
N. C.
, and
Veres
,
S. P.
,
2016
, “
Collagen Fibrils in Functionally Distinct Tendons Have Differing Structural Responses to Tendon Rupture and Fatigue Loading
,”
Acta Biomater.
,
42
, pp.
296
307
.
66.
Veres
,
S. P.
,
Harrison
,
J. M.
, and
Lee
,
J. M.
,
2014
, “
Mechanically Overloading Collagen Fibrils Uncoils Collagen Molecules, Placing Them in a Stable, Denatured State
,”
Matrix Biol.
,
33
, pp.
54
59
.
67.
Li
,
Y.
, and
Yu
,
S. M.
,
2013
, “
Targeting and Mimicking Collagens Via Triple Helical Peptide Assembly
,”
Curr. Opin. Chem. Biol.
,
17
(
6
), pp.
968
975
.
68.
Zitnay
,
J. L.
,
Li
,
Y.
,
Qin
,
Z.
,
San
,
B. H.
,
Depalle
,
B.
,
Reese
,
S. P.
,
Buehler
,
M. J.
,
Yu
,
S. M.
, and
Weiss
,
J. A.
,
2017
, “
Molecular Level Detection and Localization of Mechanical Damage in Collagen Enabled by Collagen Hybridizing Peptides
,”
Nat. Commun.
,
8
, p.
14913
.
69.
Szczesny
,
S. E.
,
Aeppli
,
C.
,
David
,
A.
, and
Mauck
,
R. L.
,
2018
, “
Fatigue Loading of Tendon Results in Collagen Kinking and Denaturation but Does Not Change Local Tissue Mechanics
,”
J. Biomech.
,
71
, pp.
251
256
.
70.
Lee
,
A. H.
,
Szczesny
,
S. E.
,
Santare
,
M. H.
, and
Elliott
,
D. M.
,
2017
, “
Investigating Mechanisms of Tendon Damage by Measuring Multi-Scale Recovery Following Tensile Loading
,”
Acta Biomater.
,
57
, pp.
363
372
.
71.
Shen
,
Z. L.
,
Dodge
,
M. R.
,
Kahn
,
H.
,
Ballarini
,
R.
, and
Eppell
,
S. J.
,
2008
, “
Stress-Strain Experiments on Individual Collagen Fibrils
,”
Biophys. J.
,
95
(
8
), pp.
3956
3963
.
72.
Fondrk
,
M. T.
,
Bahniuk
,
E. H.
, and
Davy
,
D. T.
,
1999
, “
A Damage Model for Nonlinear Tensile Behavior of Cortical Bone
,”
ASME J. Biomech. Eng.
,
121
(
5
), pp.
533
541
.
73.
Meng
,
F.
, and
Terentjev
,
E.
,
2016
, “
Transient Network at Large Deformations: Elastic–Plastic Transition and Necking Instability
,”
Polymers
,
8
(
4
), p.
108
.
74.
Ju
,
J. W.
,
1990
, “
Isotropic and Anisotropic Damage Variables in Continuum Damage Mechanics
,”
J. Eng. Mech.
,
116
(
12
), pp.
2764
2770
.
75.
Ateshian
,
G. A.
,
2011
, “
The Role of Mass Balance Equations in Growth Mechanics Illustrated in Surface and Volume Dissolutions
,”
ASME J. Biomech. Eng.
,
133
(
1
), p.
011010
.
76.
Maas
,
S. A.
,
Ellis
,
B. J.
,
Ateshian
,
G. A.
, and
Weiss
,
J. A.
,
2012
, “
FEBio: Finite Elements for Biomechanics
,”
ASME J. Biomech. Eng.
,
134
(
1
), p.
011005
.
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