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Research Papers

Strain-Level Dependent Nonequilibrium Anisotropic Viscoelasticity: Application to the Abdominal Muscle

[+] Author and Article Information
Marcos Latorre

Escuela Técnica Superior de Ingeniería
Aeronáutica y del Espacio,
Universidad Politécnica de Madrid,
Plaza Cardenal Cisneros, 3,
Madrid 28040, Spain
e-mail: m.latorre.ferrus@upm.es

Francisco J. Montáns

Escuela Técnica Superior de Ingeniería
Aeronáutica y del Espacio,
Universidad Politécnica de Madrid,
Plaza Cardenal Cisneros, 3,
Madrid 28040, Spain
e-mail: fco.montans@upm.es

1Corresponding author.

Manuscript received April 4, 2017; final manuscript received July 3, 2017; published online August 16, 2017. Assoc. Editor: Jonathan Vande Geest.

J Biomech Eng 139(10), 101007 (Aug 16, 2017) (9 pages) Paper No: BIO-17-1141; doi: 10.1115/1.4037405 History: Received April 04, 2017; Revised July 03, 2017

Soft connective tissues sustain large strains of viscoelastic nature. The rate-independent component is frequently modeled by means of anisotropic hyperelastic models. The rate-dependent component is usually modeled through linear rheological models or quasi-linear viscoelastic (QLV) models. These viscoelastic models are unable, in general, to capture the strain-level dependency of the viscoelastic properties present in many viscoelastic tissues. In linear viscoelastic models, strain-level dependency is frequently accounted for by including the dependence of multipliers of Prony series on strains through additional evolution laws, but the determination of the material parameters is a difficult task and the obtained accuracy is usually not sufficient. In this work, we introduce a model for fully nonlinear viscoelasticity in which the instantaneous and quasi-static behaviors are exactly captured and the relaxation curves are predicted to a high accuracy. The model is based on a fully nonlinear standard rheological model and does not necessitate optimization algorithms to obtain material parameters. Furthermore, in contrast to most models used in modeling the viscoelastic behavior of soft tissues, it is valid for the large deviations from thermodynamic equilibrium typically observed in soft tissues.

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References

Fung, Y. C. , 2010, Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., Springer, New York.
Humphrey, J. D. , 2002, Cardiovascular Solid Mechanics: Cells, Tissues and Organs, Springer, New York. [CrossRef]
Duenwald, S. E. , Vanderby, R., Jr. , and Lakes, R. S. , 2009, “ Constitutive Equations for Ligament and Other Soft Tissue: Evaluation by Experiment,” Acta Mech., 205(1–4), pp. 23–33. [CrossRef]
Van Loocke, M. , Lyons, C. G. , and Simms, C. K. , 2006, “ A Validated Model of Passive Muscle in Compression,” J. Biomech., 39(16), pp. 2999–3009. [CrossRef] [PubMed]
Van Loocke, M. , Lyons, C. G. , and Simms, C. K. , 2008, “ Viscoelastic Properties of Passive Skeletal Muscle in Compression: Stress-Relaxation Behaviour and Constitutive Modelling,” J. Biomech., 41(7), pp. 1555–1566. [CrossRef] [PubMed]
Van Loocke, M. , Simms, C. K. , and Lyons, C. G. , 2008, “ Viscoelastic Properties of Passive Skeletal Muscle in Compression—Cyclic Behaviour,” J. Biomech., 42(8), pp. 1038–1048. [CrossRef]
Simó, J. C. , 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. [CrossRef]
Simó, J. C. , and Hughes, T. J. R. , 1998, Computational Inelasticity, Springer, Berlin.
Peña, E. , Peña, J. A. , and Doblaré, M. , 2008, “ On Modelling Nonlinear Viscoelastic Effects in Ligaments,” J. Biomech., 41(12), pp. 2659–2666. [CrossRef] [PubMed]
Calvo, B. , Sierra, M. , Grasa, J. , Muñoz, M. J. , and Peña, E. , 2014, “ Determination of Passive Viscoelastic Response of the Abdominal Muscle and Related Constitutive Modeling: Stress-Relaxation Behavior,” J. Mech. Behav. Biomed. Mater., 36, pp. 47–58. [CrossRef] [PubMed]
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. [CrossRef]
Holzapfel, G. A. , 2000, Nonlinear Solid Mechanics: A Continuum Approach for Engineering, Wiley, Chichester, UK.
Kaliske, M. , and Rothert, H. , 1997, “ Formulation and Implementation of Three-Dimensional Viscoelasticity at Small and Finite Strains,” Comput. Mech., 19(3), pp. 228–239. [CrossRef]
Gültekin, O. , Sommer, G. , and Holzapfel, G. A. , 2016, “ An Orthotropic Viscoelastic Model for the Passive Myocardium: Continuum Basis and Numerical Treatment,” Comput. Methods Biomech. Biomed. Eng., 19(15), pp. 1647–1664. [CrossRef]
Reese, S. , and Govindjee, S. , 1998, “ A Theory of Finite Viscoelasticity and Numerical Aspects,” Int. J. Solid Struct., 35(26–27), pp. 3455–3482. [CrossRef]
Haslach, H. W. , 2011, Maximum Dissipation Non-Equilibrium Thermodynamics and its Geometric Structure, Springer, New York. [CrossRef]
Haupt, P. , 1993, “ Thermodynamics of Solids,” Non-Equilibrium Thermodynamics With Applications to Solids W. Muschik , ed., Springer, Vienna, Austria. [CrossRef]
Duenwald, S. E. , Vanderby, R., Jr. , and Lakes, R. S. , 2010, “ Stress Relaxation and Recovery in Tendon and Ligament: Experiment and Modelling,” Biorheology, 47(1), pp. 1–14. [PubMed]
Provenzano, P. , Lakes, R. , Keenan, T. , and Vanderby, R., Jr ., 2001, “ Nonlinear Ligament Viscoelasticity,” Ann. Biomed. Eng., 29(10), pp. 908–914. [CrossRef] [PubMed]
Latorre, M. , and Montáns, F. J. , 2015, “ Anisotropic Finite Strain Viscoelasticity Based on the Sidoroff Multiplicative Decomposition and Logarithmic Strains,” Comput. Mech., 56(3), pp. 503–531. [CrossRef]
Latorre, M. , and Montáns, F. J. , 2016, “ Fully Anisotropic Finite Strain Viscoelasticity Based on a Reverse Multiplicative Decomposition and Logarithmic Strains,” Comput. Struct., 163, pp. 56–70. [CrossRef]
Aranda-Iglesias, D. , Vadillo, G. , Rodríguez-Martínez, J. A. , and Volokh, K. Y. , 2017, “ Modeling Deformation and Failure of Elastomers at High Strain Rates,” Mech. Mater., 104, pp. 85–92. [CrossRef]
Volokh, K. Y. , 2016, Mechanics of Soft Materials, Springer, Singapore. [CrossRef]
Hoo Fatt, M. S. , and Ouyang, X. , 2008, “ Three-Dimensional Constitutive Equations for Styrene Butadiene Rubber at High Strain Rates,” Mech. Mater., 40(1–2), pp. 1–16. [CrossRef]
Sussman, T. , and Bathe, K.-J. , 2009, “ A Model of Incompressible Isotropic Hyperelastic Material Behavior Using Spline Interpolations of Tension-Compression Test Data,” Numer. Methods Biomed. Eng., 25(1), pp. 53–63. [CrossRef]
Latorre, M. , and Montáns, F. J. , 2014, “ What-You-Prescribe-Is-What-You-Get Orthotropic Hyperelasticity,” Comput. Mech., 53(6), pp. 1279–1298. [CrossRef]
Crespo, J. , Latorre, M. , and Montáns, F. J. , 2017, “ WYPiWYG Hyperelasticity for Isotropic, Compressible Materials,” Comput. Mech., 59(1), pp. 73–92. [CrossRef]
Latorre, M. , and Montáns, F. J. , 2013, “ Extension of the Sussman–Bathe Spline-Based Hyperelastic Model to Incompressible Transversely Isotropic Materials,” Comput. Struct., 122, pp. 13–26. [CrossRef]
Latorre, M. , Peña, E. , and Montáns, F. J. , 2016, “ Determination and Finite Element Validation of the WYPiWYG Strain Energy of Superficial Fascia From Experimental Data,” Ann. Biomed. Eng., 45(3), pp. 799–810. [CrossRef] [PubMed]
Romero, X. , Latorre, M. , and Montáns, F. J. , 2017, “ Determination of the WYPiWYG Strain Energy Density of Skin Through Finite Element Analysis of the Experiments on Circular Specimens,” Finite Elem. Anal. Des., 134, pp. 1–15. [CrossRef]
Latorre, M. , Romero, X. , and Montáns, F. J. , 2016, “ The Relevance of Transverse Deformation Effects in Modeling Soft Biological Tissues,” Int. J. Solid Struct., 99, pp. 57–70. [CrossRef]
Latorre, M. , and Montáns, F. J. , 2017, “ WYPiWYG Hyperelasticity Without Inversion Formula: Application to Passive Ventricular Myocardium,” Comput. Struct., 185, pp. 47–48. [CrossRef]
Murphy, J. G. , 2014, “ Evolution of Anisotropy in Soft Tissue,” Proc. R. Soc. A, 470(2161), p. 20130548.
Latorre, M. , and Montáns, F. J. , 2015, “ Material-Symmetries Congruency in Transversely Isotropic and Orthotropic Hyperelastic Materials,” Eur. J. Mech. A, 53, pp. 99–106. [CrossRef]
Dokos, S. , Smaill, B. H. , Young, A. A. , and LeGrice, I. J. , 2002, “ Shear Properties of Passive Ventricular Myocardium,” Am. J. Physiol., 283(6), pp. H2650–H2659.
Latorre, M. , and Montáns, F. J. , 2016, “ Stress and Strain Mapping Tensors and General Work-Conjugacy in Large Strain Continuum Mechanics,” Appl. Math. Modell., 40(5–6), pp. 3938–3950. [CrossRef]
Lubliner, J. , 1985, “ A Model of Rubber Viscoelasticity,” Mech. Res. Commun., 12(2), pp. 93–99. [CrossRef]
Purslow, P. P. , Wess, T. J. , and Hukins, D. W. , 1998, “ Collagen Orientation and Molecular Spacing During Creep and Stress-Relaxation in Soft Connective Tissues,” J. Exp. Biol., 201(Part 1), pp. 135–142. http://jeb.biologists.org/content/201/1/135 [PubMed]

Figures

Grahic Jump Location
Fig. 1

(a) Stress–strain behavior under instantaneous loads; equilibrated, relaxed response, and isochronic response and (b) relaxation tests

Grahic Jump Location
Fig. 3

(a) Left: stress relaxation predictions for the experimental data of Fig. 6(b) of Ref. [10] (longitudinal and transverse specimens). (b) Right: stress relaxation predictions using the model in Ref. [10] (redrawn from Fig. 6(b) of Ref. [10]).

Grahic Jump Location
Fig. 5

Predictions for the relaxation tests performed at different stress levels. Experimental data redrawn from Fig. 4(b) of Ref. [10].

Grahic Jump Location
Fig. 2

(a) Left: predictions for the peak and relaxed behaviors in Fig. 6(a) of Ref. [10]. (b) Right: WYPiWYG strain energy terms for the instantaneous (equilibrated + nonequilibrated) and equilibrated responses.

Grahic Jump Location
Fig. 4

(a) Left: WYPiWYG hyperelastic predictions for the instantaneous (equilibrated + nonequilibrated) and relaxed (equilibrated) responses of the experiments in Fig. 4(a) of Ref. [10]. (b) Right: WYPiWYG stored energy terms for the instantaneous (equilibrated + nonequilibrated) and relaxed (equilibrated) responses of the experiments in Fig. 4(a) of Ref. [10].

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