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Research Papers

# Muscle and Tendon Tissues: Constitutive Modeling and Computational Issues

[+] Author and Article Information
L. A. Spyrou

Department of Mechanical Engineering, University of Thessaly, 38334 Volos, Greece; The Mechatronics Institute, Center for Research and Technology-Thessaly (CE.RE.TE.TH.), 38500 Volos, Greecelspyrou@mie.uth.gr

N. Aravas1

Department of Mechanical Engineering, University of Thessaly, 38334 Volos, Greece; The Mechatronics Institute, Center for Research and Technology-Thessaly (CE.RE.TE.TH.), 38500 Volos, Greecearavas@mie.uth.gr

Equation 9 leads to the more involved formula for the evaluation of $εm$, which is used sometimes incrementally in computational mechanics: $dεm=m⋅D⋅mdt or εm(t)=∫0tm(τ)⋅D(τ)⋅m(τ)dτ$, where $t$ denotes time.

1

Corresponding author.

J. Appl. Mech 78(4), 041015 (Apr 14, 2011) (10 pages) doi:10.1115/1.4003741 History: Received October 28, 2010; Revised February 28, 2011; Posted March 02, 2011; Published April 14, 2011; Online April 14, 2011

## Abstract

A three-dimensional constitutive model for muscle and tendon tissues is developed. Muscle and tendon are considered as composite materials that consist of fibers and the connective tissues and biofluids surrounding the fibers. The model is nonlinear, rate dependent, and anisotropic due to the presence of the fibers. Both the active and passive behaviors of the muscle are considered. The muscle fiber stress depends on the strain (length), strain-rate (velocity), and the activation level of the muscle, whereas the tendon fiber exhibits only passive behavior and the stress depends only on the strain. Multiple fiber directions are modeled via superposition. A methodology for the numerical implementation of the constitutive model in a general-purpose finite element program is developed. The current scheme is used for either static or dynamic analyses. The model is validated by studying the extension of a squid tentacle during a strike to catch prey. The behavior of parallel-fibered and pennate muscles, as well as the human semitendinosus muscle, is studied.

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## Figures

Figure 1

A typical force-length relation for skeletal muscle fiber

Figure 2

A typical active force-velocity relation for skeletal muscle fiber

Figure 3

A typical force-deformation relation for tendon tissue

Figure 4

Tissue fiber in the deformed configuration with its direction defined locally by unit vector m. Also shown the fiber stress σm acting on an infinitesimal fiber segment of length ds.

Figure 5

Active fiber force versus fiber-length at different activation levels (26). The dash line indicates the location of the maximum force on each curve.

Figure 6

Up: Finite element mesh used in the calculations. Due to symmetry, only a quarter is analyzed. The tentacle base is constrained in the axial direction and symmetry boundary conditions are imposed to the interior regions. Down: Cross section of the stalk (9). Longitudinal muscles cover 15% of the tentacle’s cross section.

Figure 7

Evolution of the tentacle length

Figure 8

History of the velocity at the tentacle tip

Figure 9

Parallel-fibered muscle. (a) Undeformed configuration. (b) Concentric contraction. (c) Eccentric contraction.

Figure 10

Dimensionless function fe versus ε0m adopted from Delp (29). The variation of the optimal fiber-length ℓ0am with fa is considered in the model.

Figure 11

Dimensionless function fr versus ε̇m∗=ε̇0m/ε̇max, where ε̇max=5 s−1(29)

Figure 12

Passive muscle fiber force-strain relationship adopted from Delp (29). The variation of the passive tension fp with fa is considered in the model.

Figure 13

Contours of fiber strain εm in the parallel-fibered muscle. (a) Concentric contraction and (b) eccentric contraction. The maximum values of εm appear at the center of the muscle belly during concentric contraction and at the muscle ends during eccentric contraction.

Figure 14

Pennate muscle. (a) Undeformed configuration. (b) Concentric contraction. (c) Isometric contraction.

Figure 15

Human semitendinosus muscle. (a) Undeformed configuration and finite element mesh. (b) Deformed configuration. (c) Contours of axial fiber strain εm (fibers that shorten are shown blue).

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