HAL Id: hal-00501746
https://hal.archives-ouvertes.fr/hal-00501746
Submitted on 12 Jul 2010
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A constitutive model for brous tissues considering
collagen ber crimp
F. Cacho, P.J. Elbischger, J.F. Rodríguez, M. Doblaré, G.A. Holzapfel
To cite this version:
F. Cacho, P.J. Elbischger, J.F. Rodríguez, M. Doblaré, G.A. Holzapfel. A constitutive model for brous
tissues considering collagen ber crimp. International Journal of Non-Linear Mechanics, Elsevier, 2007,
42 (2), pp.391. �10.1016/j.ijnonlinmec.2007.02.002�. �hal-00501746�
www.elsevier.com/locate/nlm
Author’s Accepted Manuscript
A constitutive model for fibrous tissues considering
collagen fiber crimp
F. Cacho, P.J. Elbischger, J.F. Rodríguez, M. Doblaré,
G.A. Holzapfel
PII: S0020-7462(07)00055-8
DOI: doi:10.1016/j.ijnonlinmec.2007.02.002
Reference: NLM 1340
To appear in: International Journal of Non-
Linear Mechanics
Received date: 30 December 2006
Revised date: 6 February 2007
Accepted date: 6 February 2007
Cite this article as: F. Cacho, P.J. Elbischger, J.F. Rodríguez, M. Doblaré and G.A.
Holzapfel, A constitutive model for fibrous tissues considering collagen fiber crimp, Inter-
national Journal of Non-Linear Mechanics (2007), doi:10.1016/j.ijnonlinmec.2007.02.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As
a service to our customers we are providing this early version of the manuscript. The
manuscript will undergo copyediting, typesetting, and review of the resulting galley proof
before it is published in its final citable form. Please note that during the production process
errors may be discovered which could affect the content, and all legal disclaimers that apply
to the journal pertain.
Accepted manuscript
A constitutive model for fibrous tissues
considering collagen fi ber crimp
F. Cacho
a,b
, P.J. Elbischger
c
,J.F.Rodr´ıguez
b
M. Doblar´e
b
and G.A. Holzapfel
d,e
a
Institute for Structural Analysis, Computational Biomechanics
Graz University of Technology, Austria
b
Group of Structural Mechanics and Materials Modeling
Aragon Institute of Engineering Research (I3A), University of Zaragoza, Spain
c
Medical Information Technology, Carinthia Tech Institute
University of Applied Sciences, Klagenfurt, Austria
d
Department of Solid Mechanics, School of Engineering Sciences
Royal Institute of Technology (KTH), Stockholm, Sweden
e
Institute for Biomechanics, Graz University of Technology, Austria
Abstract
A micromechanically-based constitutive model for fibrous tissues is presented. The
model considers the randomly crimped morphology of individual collagen fibers, a
morphology typically seen in photomicrographs of tissue samples. It describes the
relationship between the fiber endpoints and its arc-length in terms of a measur-
able quantity, which can be estimated from image data. The collective mechanical
behavior of collagen fibers is presented in terms of an explicit expression for the
strain-energy function, where a fiber-specific random variable is approximated by a
Beta distribution. The model-related stress and elasticity tensors are provided. Two
representative numerical examples are analyzed with the aim of demonstrating the
peculiar mechanism of the constitutive model and quantifying the effect of param-
eter changes on the mechanical response. In particular, a fibrous tissue, assumed
to be (nearly) incompressible, is subject to a uniaxial extension along the fiber di-
rection, and, separately, to pure shear. It is shown that the fiber crimp model can
reproduce several of the expected characteristics of fibrous tissues.
Key words: Collagen fiber, constitutive model, fiber crimp, micromechanics,
statistical distribution.
Corresponding author. Institute for Biomechanics, Center for Biomedical
Engineering, Graz University of Technology, Kronesgasse 5, 8010 Graz, Austria.
E-mail address: gh@biomech.tu-graz.ac.at (G.A. Holzapfel).
Preprint submitted to Elsevier Science 6 February 2007
Accepted manuscript
1 Introduction
The central role of collagen as the major structural protein of mammalian
tissue, comprising approximately one-third of the total protein in mammalian
organisms, has motivated a significant effort towards determining its mechan-
ical properties at all levels, ranging from single monomers [1,2] and long-chain
polymers [3,4] to a structural element within a (macroscopic) biological tissue
[5–8].
On the basis of the mechanical properties, a number of constitutive mod-
els have been developed in the past in attempts to describe the experimental
data. While at the microscopic level, chain models such as the (Kratky-Porod)
worm-like model are popular [9–11], at the macroscopic level the continuum
theory of finite elastic deformations of solids reinforced with fibers is frequently
the constitutive theory of choice. The basic ideas of the theory are contained
in [12], with further developments on strongly anisotropic solids in [13], and
applications to model, e.g., arterial walls in [14,15]; see also the recent vol-
ume [16]. In such macroscopic models the collagen fibers are assumed to be
continuously arranged in the matrix material, as utilized in [17], and the char-
acteristic nonlinear stiffening is best represented by means of an exponential
function. Effective alternatives are based on limiting chain models, see, for
example, [18], and references therein.
The pioneering work by Lanir [19,20] on the mechanics of fibrous (connec-
tive) tissues as a consequence of its microstructure has influenced much of the
works on microstructural constitutive models. Essentially, the works [19,20]
postulate that the fibers are crimped and that they have different lengths so
that for a given macroscopic deformation in the material each individual fiber
is stretched differently. There is, thus, a distribution in either the stretch of
the fibers or their lengths. This idea has also been adopted subsequently by
means of constitutive models to describe the mechanical response of, e.g., ar-
terial walls ([21] with ideas from [22,14]) or tendons and ligaments [23], just
to name a few. All these constitutive models, however, assume unbounded
statistical distributions for the fiber length (or stretch), which is a bounded
quantity. In addition, in these models no attempt has been made to correlate
the fiber morphology (crimp) with the associated mechanical response in the
form of stress-stretch relationships. It was Lanir who considered the possibility
that the stretch could be nonuniform due to crimping, with a generic distri-
bution along the fiber axis, which he assumed to be Gaussian. Recently, Freed
and Doehring [24] have proposed a model where crimped fibrils in a fascicle
are approximated as a helical spring. Thereby, the collagen fiber waveforms
have a pre-defined arrangement; no statistical distribution is used. In differ-
ent works, such as [25–27], the distribution of the fiber orientations has been
addressed; however, therein, the mechanical properties of the collagen fibers
2
Accepted manuscript
within the tissue were considered to be independent of the degree of crimping.
In this paper a new constitutive model for the macroscopic behavior of fi-
brous tissues is presented. It takes the randomly crimped morphology of the
individual collagen fibers into account. In Section 2 a statistical description
of the fiber crimp is developed, which is used in Section 3 to model the col-
lective behavior of fibers. In Section 4 the mechanical behavior of a fibrous
tissue, assumed to be (nearly) incompressible, is analyzed in detail. The tissue
is subject to a stretch-controlled uniaxial extension along the fiber direction,
and, separately, to pure shear. In particular, the effect of the different model
parameters on the mechanical response is studied. The final section contains a
brief discussion together with a description of some limitations of the proposed
constitutive model.
2 Statistical and constitutive description of a single collagen fiber
In unloaded tissue samples collagen fibers show a wavy structure. In this
section we develop amodelthatincorporates the random crimp of collagen
fibers to be characterized.
2.1 Random crimp of a single fiber
We start by considering a set of randomly generated data in an interval of
length L
0
+ w on the X-axis such that at any point x within that interval
the associated coordinates y and z are independent and normally distributed
random variables with zero mean. Under this condition the data generated
can be regarded as white Gaussian noise, and characterized by the variance
σ
2
. In the following it is assumed that the variances in the Y and Z-directions
are equal, in other words the fiber undulates with equal characteristics in all
directions orthogonal to the X-axis.
The randomness of the data generated may be larger than that of an actual
fiber. By applying a smoothing function or filter h, which averages the coordi-
nates of the points in a neighborhood [−w/2,w/2] of each point, a derived set
of data in the interval [0,L
0
] is obtained. It is implicitly assumed that h and
its first derivative has compact support in [−w/2,w/2]. The resulting data
are also random and normally distributed with zero mean since the filtering
operation does not affect the Gaussian nature of the distribution. As a con-
sequence, the variance of the new random variable is unequivocally related to
that of the white Gaussian noise through the filter.
3