Finite element implementation and numerical simulation of the
hyperelastic anisotropic behavior of biomaterials
DJERIDI Rachid1,a and OULD OUALI Mohand1,b
1
Laboratoire d’Elaboration et Caractérisation des Matériaux et Modélisation
Université de Tizi-Ouzou, Algeria
a
[email protected] , bould [email protected]
(LEC2M)
Keywords: finite element simulation, jacobian matrix, biomaterial, hyperelastic anisotropic, fiber
reinforced material.
Abstract. The modeling of fiber reinforced material with anisotropic hyperelastic laws –
Theoretical study and finite element analysis is presented. To determine the strain and stress in the
fiber reinforced materials such biomaterial and biological soft tissues, anisotropic hyperelastic
constitutive laws are often used in the context of finite element analysis. The simulation of the used
model for the domain of the study materials has allowed analyze on the influence of the choice of
the material constants. The case of the reinforced materials by one family of fibers is treated, it
represents the major case of this materials. This formwork presents the implementation step of
anisotropic hyperelastic law in a computational code by finite element. Then more, the simulation
of the material deformation constitutes a primary orientation for the choice of material constants for
parametric identification by numerical methods. The numerical simulation of a cube element, we
have allowed as to stud the influence of the anisotropy which is analyzed by fibers direction,
material parameters, material compressibility and material deformation. To take note on the
numerical simulation, the material parameters identification of the used model by the numerical
methods can be oriented by one simple analysis of the material deformation during the uniaxial
tension test. This technique permits an approximation of biomaterial constants hyperelastic
anisotropic model since databank results of simulation in relation with the tension experimental
results.
Introduction
Despite a large number of publications in the last decade, the accurate prediction of the
mechanical behavior of rubberlike materials remains an open issue [1]. These materials are often
used under cyclic conditions, where large deformation viscoelasticity coupled with damage is
relevant. Their mechanical behavior has been described first as hyperelastic, and several forms of
strain energy density have been defined so far, see for instance [2-6].
The status of composite material in engineering application is increasing steadily. Fiber
matrix combinations are employed in the aircraft industry and for the construction of space
structures. Moreover, composites are also used for civil engineering structures like footbridges. The
description of linear elastic behavior of fiber reinforced material is shown in standard text books on
composites. Assuming linear elasticity and small strains Jones [7], Christensen [8] or Gibson [9] for
example present the constitutive relations for deferent types of anisotropy based on engineering
constants.
For the geometrically nonlinear regime the linear elastic approach is often used as to be seen
in the literature, i.e., modeling is accomplished by the so-called de St. Venant Kirchhoff material.
Klee and Stein [10] show that the formulation may be employed just for a small range of
deformation, i.e., up to 5% of strain. General considerations on anisotropic elasticity at finite strains
are given in Truesdell and Noll [11], Suhubi [12] and Lurie [13]. Spencer [27] developed a
constitutive strain energy function in terms of invariants. For this purpose, he extended the basis of
three characteristic quantities of isotropy to five invariants of transverse isotropy. Schroder [14] and
Weiss et al. [15] investigate finite and anisotropic nonlinear elasticity. Moreover, Weiss et al. [16]
show a specific strain energy function applied to finite element simulations of biological material.
Biological soft tissues sustain large deformations, rotations and displacements, have a highly
non-linear behavior, posse’s anisotropic mechanical properties and show a clear time and strain rate
dependency. Their typical anisotropic behavior is caused by several collagen fiber families (usually
one or two coincide at each point) that are arranged in a matrix of soft material named ground
substance [17]. Typical examples of fibered soft biological tissues are blood vessels, tendons,
ligaments, cornea and cartilage [18-22].
Another class of modeled materials by anisotropic behavior law is the thermoforming or the
thermoplastic shuffling charged fibers carried above the glass temperature. Using techniques of
homogenization, it constructed macroscopic behavior laws of type anisotropic viscous fluid [23] or
the materials symmetry properties [24].
The objective of this work is to present an integral methodology to numerically mode the
hyperelastic anisotropy behavior of materials, showing the development of a numerical tool to
implement hyperelastic anisotropy model in the commercial finite element code Abaqus.
The experimental study allows observing the physical phenomena during the
characterization of anisotropic visco-hyperelastic materials parameters. The hyperelastic modeling,
whether it is phenomenological or microphysical, passes by the construction of strain energy
function. This function must present certain mathematical forms witch correspond to the physical
phenomena of materials. In this framework we expose the technique of construction of strain energy
function anisotropic for materials reinforced by fibers.
After the development of anisotropic hyperelastic law, in the second paragraph, we study the
hyperelastic problem by the analytical method. This technique seems direct with the mathematical
equations more or less complex with simplified considerations of the studied problem. The finite
element method (FEM) implemented in the commercial computational code remain in development.
The pertinent step in this technique is to calculate the « jacobian matrix » of the stress towards to
strain of the studied model. Preliminary calculations which are related to the jacobian matrix for the
hyperelastic behavior are presented.
Anisotropic Hyperelastic Behavior Law
In this paragraph, we develop the technique of formulation of behavior anisotropic
hyperelastic law. This fact will lead as to the choice of our model. A huge number of materials
show anisotropic behavior such as composites, bio-materials and even single crystals [25].
Hyperelastic materials exhibit anisotropic effects caused by fiber-reinforcement and filler-particles
(Mullin’s effects). In rubber-like materials, it is instructive to subdivided anisotropy into three
classes: material anisotropy, geometrically induced anisotropy and load induced anisotropy.
Let Ω0xR→R3, x → ℱ(X , t) denote the motion mapping and let F be the associated
deformation gradient. Here X and x defines the respective positions of a particle in the reference Ω
and current Ω configurations such as F = . Further, let J = detF be the Jacobian of the motion.
And C = F F the right Cauchy- Green tensor, B = FF left Cauchy- Green tensor.
There are two approaches to introducing the directional dependence on the deformation into
the strain energy: restrict the way in which the strain energy can depend on the deformation [26], or
introduce a vector representing the material preferred direction explicitly into the strain energy [27].
In the former approach, the strain energy can be expressed as a function of the Lagrangian strain
components in a coordinate system aligned with the fiber direction. Thus, all computations must be
performed in this local coordinate system. In the presentation that follows, we choose the second
approach.
The energy density, that the most used in anisotropic hyperelastic imply power law [28] or
exponential law [29, 30]. It is generally admitted that the anisotropy is due to the collagen fiber [31]
whole the matrix is behaved with an isotropic manner. This is why the energy density is separated
in two parts: isotropic part and anisotropic one [32]. It can write:
W = Wiso + ∑
w
(1)
The hyperelastic materials are characterized by the existence of a strain energy function,W,
from wich the stress can be derived. Writing this energy as function of C (W = W(C)), we show
that W satisfies automatically the Principe of objectivity. The isotropic hypothesis induces the
question of representativity of the energy function W (W(C) = W(QCQ ) ∀Q: orthogonal tensor
), and we can be write the energy density was function of invariants (I , I , I ) ofC, Bor F :
W = W = W with W = W(I (C), I (C), I (C) ), W = W(I (B), I (B), I (B) ). The cauchy
stress,σ, and Piola-Kirchhoff stress, S, is expressed as follows:
σ
= 2J (W + W I )B − W B + W I
S
= 2J (W + W I ) − W C + W I
With W =
i = 1,2,3 and : the identity tensor
(2)
In the case of anisotropic materials (materials reinforced of fibers), the principle is not
enough. So, more than the three invariants, we must add certain number of variables to construct a
free energy model [15]. In the field of biomechanics, this type of representation has been used to
model the material behavior of cardiac muscle [33, 34]. We suppose the fibers orientation which are
locally defined by unit vector, a (i = 1,2,3 ), and a tensor of anisotropy, M = a ⊗ a . Begining by
the tensors, M , we definit the group of symetrie, g, which caracterizes the materials symmetry :
g = {Q/QM Q = M }.
(3)
We demonstrate that [37]:
W(C) = W(C, M ) = W(QCQ , QM Q ) ∀Q ∈ g .
(4)
Therefore, W is a isotrpic function of two tensors, we can then apply it the theoreme of
representation and write W was function of invariants of C , M and their products. The transverse
isotropic models or orthotropic models use then the invariants (I , I , I ) of and the mixed
invariants of C and M (I , I , I ):
I = M . CM , I = M . C M and I = M . M M . CM and W(C) = W I , I , I , I , I , I
(5)
I , , I characterize the anisotropy constitutive response of the fibers: I has a clear physical meaning
since it is the square of the stretch along the fibers. In order to reduce the number of material
parameters and to work with physically motivated invariants, we shall omit the dependency of the
free energy W on I . This hypothesis is usually used in biomechanical modeling [43]. We obtain the
following constitutive equations form:
σ=σ
S = S
+∑
+ ∑
2J − 1 F
M +
M +
(CM + M C ) F
(CM + M C )
(6)
Implementation of Anisotropic Hyperelastic Behavior Law in Computational Code by Finite
Element Method (FEM):
The numerical approach is so, of a great richness which pushes the development calculating
code. We describe the processes to be followed to integrate a behavior law in a calculating code by
finites elements. We will calculate the mechanical fields beginning the model defined by the
derivatives in certain directions of elastic energy density. The user must furnish the set of stress the
« jacobian » matrix of stress towards to a given actual deformation:
σ = FSF
.
(7)
The hyperelastic strain energy can be written also [35]:
W = E: : E .
(8)
The « jacobian» is obtained then:
= σ. + . σ + F[ : (F . . F)]F
With: =
(9)
the fourth order, I: the second order unit tensor.
= a ⊗ Ca + Ca ⊗ a , n = 1, N; I
=
=
C C
+C C e ⊗e ⊗e ⊗e
And for unit vector, , that represents a fiber direction :
∂
1
I =
= (I ) = (a a δ + a a δ + a a δ + a a δ )e ⊗ e ⊗ e ⊗ e
∂C
2
A lot of materials show different behavior for volumetric and for isochoric states of
deformation. Biomaterials as well as elastomers are incompressible or nearly incompressible with
respect to hydrostatic pressure loading. In order to take this observation into account, an additive
division of the constitutive strain energy function is introduced:
W = W(C, M ) + U(J) Where C = F F and F = J F .
(10)
(
)
U J : The penalty term is used for the definition of the compression behavior and it is function
penalty parameter к corresponds to the bulk modulus that controls the volume change
Subsequently, the pseudo-invariants I of the incompressible Right Cauchy Green tensor C
are computed, and to write:
W(C, M ) = W I , I , I , I , I
i = 1,3 .
(11)
An uncoupled structure of the second Piola Kirchhoff stress tensor is derived:
S=S
+S
(12)
And in equivalence, the relations which express the tensors can be got by replacing the GreenCauchy tensor C by the pseudo tensor of Green-Cauchy C also the pseudo invariant and by taking
into account the fact that the deviatoric part is independent from I . Even though the expressions
standards of volumic and isochor tangents are (for applications see [38]):
(C
=J
= J
⁄
S: C
⊗C )+J
I
− (C
(C
⊗C
−2I
⊗ C ) − (S
)
⊗C
(13)
+C
⊗S
)+J
⁄
DEV
.
Numerical Example
In order to study the influence of the parameters sets in the stress–strain response, we
considered a transversely isotropic and hyperelastic material with its constitutive behavior defined
by the initial elastic stored-energy function [39]:
W = U(J) + W(C, M ) = (log(J)) + c (I − 3) + c ( I̅ − 3) +
[e
(̅
)
− 1]
(14)
Where c ≥ 0 is a stress-like material parameter and c ≥ 0 is a dimensionless parameter.
Three sets of elastic material constants were chosen (Table 1).
Set I
Set II
Set III
c
10
10
10
c
c
10
100
10
0.1
0
100
Table 1: material parameters
c
1
1
1
D
0.0036844
0.0036844
0.0036844
In this example we deal with the numerical simulation of a composite rubber block bonded
with two perfectly rigid plates as shown in Fig. 1. Due to the symmetry, only(1 ⁄ 8) of the global
geometry was modeled. The fiber distribution was aligned with the axial direction. The purpose of
this simulation is to demonstrate also the effectiveness the finite element implementation discussed
in section 3. The model used in this study represents the biomaterial modeling.
In the first case of this example, we have submitted the cube to an elongation λ = 2, by
considering teh material SetII compressible and incompressible and for different fibers directions
(Fig. 2). We can notice that the maximal stress of Van Mises varies according to the fibers
orientations: it passes of value which equals 6,021.102 MPa for an orientation α=0° to a value which
equals 5,825.102MPa for an orientation α=45°. This variation is more notable in the incompressible
case where the values are 1,644.104 MPa for α=0° and 4,473.105 MPa. It is also clear that the stress
is increased by the incompressibility of material, which is evident (it is the same for deformation).
This fact leads us to study the relation between the stiffness of hyperelastic material and the
orientation of fibers. Stiffness is defined as the elastic energy which is in play during the
deformation of the cube element which equals to 40%. Precisely, stiffness is the relationship strain
energy deformation for orientation angle α=0°and strain energy deformation for orientation angle
α=0°. The result is represented by the curve of Fig. 3.
In this paragraph, we present the study of choice relation related to parameters of material on
the hyperelastic behavior of this latter. We have stimulated the deformation of a cube element
reinforces of the following fiber its axis (α=0°), with deferential Set, to a deformation of 40%
(Fig. 4 -5). And for a wider comparative study, we have considered the model of Holzapfel-Gasser
and Ogden (HGO)[41,42].The form of the strain energy potential is used for modeling arterial
layers with distributed collagen fiber orientations:
W=
− lnJ + C (I̅ − 3) +
∑
{exp[c 〈E 〉 ] − 1}
(15)
Subsequently, the values of c are 524.6, 168.9212 and 107.473 for HGO(к=0),
HGO(к=0,226) and Set IV respectively. Particularly, the material HGO (к=0,226), the exponential
part of the energy function is composed on term of the fourth invariant (where I = a. a = a . C. a
is the square of the stretch of the fiber; a the fiber direction in the reference configuration; c
scales the exponential stress and c is related to the rate of collagen uncrimping) and a term on the
first invariant which translated an isotropic deformation of the matrix. The analysis of the obtained
results is done the two parts of strain energy: the isotropic part (represents the deviatoric mechanical
contribution of the tissue matrix) and the anisotropic one (that of the fibers) (Eq.1). This fact leads
to compare the ratio or c in front of the constant c . We notice that the constant c is important
then the constant c , the behavior of material to the deformation is anisotropic and the stress of Van
Mises which indicates the contribution of fibers to the mechanical behavior of material. In the case
of SetII and SetIII, the values c and c are identical (with the constant c null for SetIII): we notice
that the two Sets have a near behavior to stress and deformation. This fact which makes the
comparative study of the three materials represented by Set IV, HGO (к=0) and HGO (к=0,226)
that the exponential part of the function of strain energy dominates and represents the contribution
of reinforcement.
Figure 2: Influence of direction anisotropy on the mechanical behavior of compressible material (c)
and incompressible material (I).
Strain energy
30°
20
0°
1,E+05
10°
8,E+04
15
1,E+05
6,E+04
1,E+05
10
8,E+04
4,E+04
6,E+04
5
2,E+04
4,E+04
0
2,E+04
0,E+00
0,E+00
0
2
0
4
5
4
300
2
200
100
0
4
4
90°
6
3
2
2
60°
4
400
0
0
4
Displacement
45°
500
2
6
1
3
2
0
1
0
0
2
4
0
2
4
6
Figure 3: Influence of direction anisotropy (fiber direction) on the stiffness of compressible material
(c/ ° ) and incompressible material (I/ ° ).
Figure 4: Influence of choice material parameters on the anisotropic hyperelastic behavior for the
compressible material case.
Figure 5: Influence of choice material parameters on the anisotropic hyperelastic behavior for the
incompressible material case
Conclusion
This paper has presented constitutive formulations for material that is idealized as
transversely isotropic. The approaches cover material with fibrous microstructure like biological
tissues and all kinds of unidirectional fiber matrix combinations. This investigation focuses on the
homogeneous macroscopic material level in order to provide a formulation suitable for structural
finite element simulations. We expose the implementation processes of hyperelastic behavior law in
commercial computational code Abaqus. The implementation step is summarized in the calculation
of the « jacobian » matrix. The numerical simulation of a cube element, we have allowed as to stud
of the influence of the anisotropy which is analyzed by fibers direction, material parameters,
material compressibility and material deformation. To take note on the numerical simulation, the
material parameters identification of the used model by the numerical methods can be oriented by
one simple analysis on the material deformation during the uniaxial tension test.
A reduction of the stiffness depending on loading, summarized as `damage phenomena, is
not considered in the present paper. Future work will add the micromechanical change of the
material structure due to fiber or matrix rupture and molecular deboning like Mullins-effect for
rubber material. An anisotropic damage model is capable of predicting this behavior. It is remained
to present an anisotropic visco-hyperelastic (Kelvin–Voigt, generalized Maxwell) constitutive
model capable to model fiber-reinforced composite materials undergoing finite strains.
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