STABLE AND PHYSICAL RESPONSE OF CONVEX HYPERELASTIC MODELS FOR FIBRE REINFORCED MATERIALS ỨNG xử tự NHIÊN và ổn ĐỊNH của các mô HÌNH SIÊU đàn hồi lồi mô tả các vật LIỆU COMPOSIT cốt sợi

STABLE AND PHYSICAL RESPONSE OF CONVEX HYPERELASTIC MODELS FOR FIBRE-REINFORCED MATERIALS ỨNG XỬ TỰ NHIÊN VÀ ỔN ĐỊNH CỦA CÁC MÔ HÌNH SIÊU ĐÀN HỒI LỒI MÔ T Ả CÁC VẬT LIỆU COMPOSIT CỐT S

STABLE AND PHYSICAL RESPONSE OF CONVEX HYPERELASTIC

MODELS FOR FIBRE-REINFORCED MATERIALS

ỨNG XỬ TỰ NHIÊN VÀ ỔN ĐỊNH CỦA CÁC MÔ HÌNH SIÊU ĐÀN HỒI LỒI

MÔ T Ả CÁC VẬT LIỆU COMPOSIT CỐT SỢI

Minh Tuan Duong 1a , Thi Thanh Hai Tran 1b

1Hanoi University of Science and Technology, Hanoi, Vietnam

a tuan.duongminh@hust.edu.vn; b hai.tranthithanh@hust.edu.vn

ABSTRACT

Fibre-reinforced hyperelastic models characterising rubbers, fibre-reinforced elastomers, and biological soft tissues (e.g arteries), can be unstable and lead to unphysical behaviour This problem is due to the ill-conditioned constitutive matrix of the model Thus, the paper analyzes this numerical instability by investigating a hyperelastic model in which the fibres characterised by a 2D Fung-type potential and an isotropic term representing the matrix, namely the modified Fung-type one (MFH) The unrealistic response of the convex MFH might take place in physiological range of soft tissues or in interested range of hyperelastic materials Several tension tests are conducted with the material constants of the MFH obtained from curve fitting to experimental data of porcine arterial samples by G A Holzapfel, which show poor stress solutions associated with unrealistic thickness thickening

of the tested specimens Consequently, the paper proposes a novel Fung-type potential which solves this numerical instability and can ensure physical response in the physiological deformation range of the artery

Keywords: Unphysical response, Ill-conditioning, FEM, Hyperelastic models,

Fibre-reinforced materials

TÓM T ẮT

Ứng xử phi tự nhiên có thể xảy ra đối với các mô hình siêu đàn hồi dùng cho mô tả cao

su, vật liệu dẻo cốt sợi hay mô sinh học Vấn đề này xảy ra là do ma trận cấu thành của mô hình vật liệu ở dạng điều kiện yếu Vì vậy bài báo phân tích sự mất ổn định số này bằng việc

khảo sát một mô hình siêu đàn hồi trong đó các sợi được mô tả bằng hàm năng lượng biến

dạng Fung 2D và chất nền được mô hình hóa bằng hàm biến dạng đẳng hướng, gọi là mô hình

sửa đổi dạng Fung (MFH) Ứng xử phi tự nhiên của mô hình lồi MFH này có thể xảy ra trong vùng biến dạng sinh học của mô hay vùng biến dạng được quan tâm Kết quả mô phỏng của

một số thí nghiệm kéo đơn dùng mô hình này với các hệ số vật liệu được xác định từ các mẫu

lấy từ động mạch của lợn được thí nghiệm bởi G A Holzapfel đều thiếu chính xác và không phù hợp với kết quả lý thuyết đồng thời kèm theo độ dầy của các mẫu trong các thí nghiệm tăng lên một cách phi vật lý Do đó, bài báo đề xuất một mô hình mới dạng Fung giải quyết được vấn đề mất ổn định số trên và đảm bảo ứng xử tự nhiên của mô hình mới này trong vùng

biến dạng sinh học của động mạch

Từ khóa: ứng xử phi tự nhiên, điều kiện yếu, FEM, mô hình siêu đàn hồi, vật liệu dẻo

c ốt sợi

1 INTRODUCTION

Fibre-reinforced materials play an important role in a wide range from biomechanics [1] (e.g biological soft tissue: arteries) to aircraft industry (fibre-reinforced elastomers for flexible aircraft structures) [2] Constitutive equations are not only capable of capturing the

material behavior of composite materials accurately, but must ensure numerical stability Herein, hyperelastic materials are mainly considered in biomechanics such as soft tissues, however this instability problem can occur in any fibre-reinforced composite models used in flexible aircraft structures or engineering Obviously, the modified Fung-type model (MFH) [3] and the Holzapfel model [1] are convex [4] and are typical examples subjected to unphysical response in numerical simulation with thickness thickening of specimen strips in uniaxial tension tests Gasser et al [5] claimed that the problem was caused by fibre rotation

in which the fibres needed to rotate into the loading direction in a soft ground matrix, and hence, this process resulted in thickness thickening in the middle of the simulated specimen Indeed, this statement is wrong because the problem is a numerical instability For example, this unrealistic phenomenon also takes place in an equi-biaxial test using the MFH model and the Holzapfel model in which the fibre directions are unchanged under deformation, see [6] Therefore, it is needed to analyze this numerical problem to ensure the correct simulations This paper presents further solutions to this problem, especially for the MFH model To this end, we propose a novel model comprising a new strain energy component which is capable

of solving the unphysical response of the MFH model in the interested deformation range

2 HYPERELASTIC MATERIAL MODEL

The stress tensors of homogeneous hyperelastic materials are directly derived (by means

of the Clausius-Planck form of the second law of thermodynamics) from a given scalar-valued strain-energy function

S = 2

where Eis the Cauchy-Green tensor, S is the second Piola-Kirchhoff stress tensor and the left Cauchy tensor C = F FT with the deformation gradient tensor F The Cauchy stress

J

= FSF , in which J is the determinant of the deformation gradient (volume ratio) These types of equations are known as constitutive equations The constitutive equations for composite materials are expressed in a finite element model as in the following

2.1 Finite element formulation

The total Lagrangian formulation is used since considered tensors in the previous section are described in the reference configuration A variational theorem for finite elasticity may be written using hyperelastic materials as

0

in which W C( ) is the strain-energy function in the reference configuration Ω , 0 Γ is the 0

traction boundary surface, u is the nodal displacement vector; b and 0 T are the body force and the specified traction in the reference configuration Ω , respectively The variational equation 0 [7] for the weak form of the problem might be rewritten for the finite deformation as

0

where B and N are the displacement deformation matrix and the shape function matrix, respectively In general, solving nonlinear problems needs to adopt load increments through a load factor λ The principle of virtual work with the external load factor λ reads as

[ int ext ]

δΠ u δu =δu ⋅R u λ =δu ⋅ u −λ u

where R u( , )λ denotes the out-of-balance force or residual force which must be equal to zero to ensure equilibrium Thus, R u( , )λ can be evaluated as

int ext ( , )λ =F ( )−λF ( )=0,

where the internal force F and the external force int F of an element, which can be ext

introduced from the first integral of the principle of virtual work (3) reads as

0 int

T dV

Ω

= ∫

F B S ;

0

The system in (5) is nonlinear discrete governing equation because of both geometrically and physically nonlinearity from large deformation and the constitutive relation

of the material, respectively

As a result, the total classical tangent stiffness matrix K must be defined as t

t∆

K =

Taking the derivative of the internal force vector with respect to the displacements yields

int( )

where D is the constitutive matrix derived from the hyperelastic model

2.2 Stability of the MFH model

Gasser et al [5] have been discussed the unrealistic behaviour of the Holzapfel model, however they could not show the real reason In this section, we point out the unphysical response of the MFH model as a typical example when the model is used to describe reinforcing fibres embedded in a soft material Numerically, this problem is due to the ill-conditioned constitutive matrix of the model [6]

Figure 1: Fibre families of an artery (without intima) and a strip for tension test

When a 2D Fung potential models fibres in a plane O zθ then the model with the fibre orientation vectors  0, 0

a g in Figure 1, the MFH model [3] is written as

( 3) exp ( ) 1 ; ( ) 2

W = µ I − +C Q −  Q =c Eθθ +c E +c E Eθθ

where E rr, Eθθ, and E zz are principal components of the Green deformation strain E

with respect to reference configuration in Figure 1; the invariant I of the right Cauchy-Green 1

tensor C :I1= C; tr µ equivalent to the small strain shear modulus; c 1, c2, and c are 3

dimensionless parameters so that the angle between two fibre families along the hoop

1 2

2 tan(2 ) c

c c

− , tan(2αl)(c1−c2)≤2c3 ≤tan(2αu)(c1−c2) whereαland αu are the lower and upper limits for the fibre angle, respectively The component W aniso contributes mechanical properties only in the fibre plane O zθ as an exponential function, whereas the polynomial function W iso( )I1 =0.5 (µ I1− characterizes 3) the material response in all directions This can result in the same ill-conditioning problem as described above even the model is convex The constraints for the convexity of the model are derived as c c c1, 2, 3 > 0; 2

1 2 3 0

c c −c >

Figure 2: Curve fitting and stress resutls of uniaxial tension tests for the MFH model

Figure 3: Unphysical response (left), different stretch solutions (right) of the MFH

To obtain the material constants for the MFH model, a curve fitting phase was performed on the adventitial data [8], see Figure 2 All coefficients are not only fitted but checked for the convexity conditions

Table 1: Fitted parameters for the MFH

Simulation results of the uniaxial tension test using the material constants in Table 1 with the shear modulus µ* =22.72 [kPa] are shown in Figure 3, the unphysical behavior occurs at very low stretch, leading to an inaccurate Cauchy stress σz curve of the MFH in Figure 2

Three values of the shear modulus (remaining constants were unchanged) were employed in the simulation, illustrated in Figure 3; the red one is for µ =0.1µ*, the blue one

is simulated with the shear modulus µ µ= * and the last one (violet) uses µ =10µ* It is obvious that the largest shear modulus accounts for the best radial stretch which gradually decreases due to the incompressibility constraint J =detF=1 Subsequently, this value makes the model more stable and the result is, of course, closer to the analytical The numerical stability of the model strongly depends on the isotropic strain-energy function (neo-Hookean one) The unphysical behavior (the radial stretch increasing) is clearly exhibited as the shear modulus decreases

This problem is observed as the constitutive matrix is ill-conditioned and is caused by the large difference between the isotropic energy and the anisotropic energy [6] Thus, we can control properly the balance between the isotropic and the anisotropic strain energy functions Practically, tissues generally have fibre dispersion, in which the fibres are placed not only in the fibre plane but in any direction out-of the fibre plane Therefore, the overall model needs

to be modified in a way such that the isotropic component must be characterized by exponential functions Different from the MFH model, the novel model based on the Fung-type potential was proposed to characterize the fibres in all directions by adopting the principal Green-Lagrange strain component E (out-of the fibre plane) in the exponential 1

term Thereby, it is capable of describing the fibre dispersion in order to remove/mitigate the numerical instability known as the non-physical response

3 NOVEL FUNG-TYPE MODEL

By incorporating the capability of being immune from material instability of a isotropic Fung-type model [6], a novel Fung-type model (MF2) is proposed for treating the ill-conditioning problem of the MFH This model is formulated by taking the fibre dispersion into consideration The isotropic Fung-type model in [6] can be used to model the fibre dispersion when combined with the MFH model On the other hand, the mechanical properties contributed by the fibres are modeled by a 2D Fung-type model [3] To this end, the proposed strain-energy function with the fibre plane in O23 or O zθ is postulated as

1

1 1 11 11 11 2 11 22 22 33 11 33 1 3 22 4 33 5 22 33

(I 3) ( ), ( ) [exp{Q ( )} 1],

2

coup

coup

E E E c E E E E E E c E c E c E E

µ

(10)

3.1 Stability of the model

If the exponent Qcoupis convex then the proposed model is logarithmically convex [9] The exponent Qcoup is a quadratic function of E therefore A coup is the Hessian matrix of

Qcoup Thus, if A coup is positive definite then the exponent Qcoup is convex Specifically,

coup

A is positive definite when constraints on 8 material constants are expressed as

0≤ ≤ν 1, ( , , ,µ C c c c c c, , , )> , 0 2 2

1 1 2 1 1 3 1 (1−ν ){(c −c )(1−ν )+c cν }>0 and

(11)

The constraints in (11) were checked after the material constants of the model were obtained from the curve fitting process Only the satisfying parameters were utilized in the numerical example section The material coefficients of the model become more meaningful compared to those of the general Fung-type strain-energy function [10] Moreover, the weighting coefficient ν1, which might be considered as a fibre dispersion factor, plays an important role to balance the difference between the isotropic strain energy and the anisotropic one If ν1= then the whole model becomes fully isotropic In contrast, 0 ν1= , 1 the term W coup totally turns into an anisotropic one, and the overall model becomes the MFH This material law can be comparable to others taking the fibre dispersion into account [5]

3.2 Implementation into open source code-FEAP

To implement a material model into a finite element package such as FEAP [11] or any numerical schemes, the expressions for the hyperelastic stress and the constitutive matrix are required Soft tissues are, in general, considered to be incompressible or nearly incompressible For the case of nearly incompressibility, the elastic energy can be written as a function of the following modified deformation tensor which are defined in [10] as

;

The Green-Lagrange strain tensors are subsequently defined as

.

E = (F F - I) = (C - I) E = (F F - I) = (C - I) (13)

Employing the Voigt’s notation, the modified Green-Lagrange strain is expressed in the matrix form as E=[E11,E22,E33, 2E12, 2E23, 2E31]T The novel Fung-type strain-energy function in a form of nearly incompressibility is

coup

The second Piola-Kirchhoff stress vector, which is calculated from W coup is

coup

Ce

where

6 6

3

×

with I6 6× is identity matrix of dimension 6×6; C−1 is a vector that contains components

of tensor − 1

C in order of the Voigt’s notation We only calculate the constitutive matrix, which is derived fromW coup as

2

coup

Derivatives in (17) are calculated as

; 2 2 ;

C

[ ]

( )

2/3 1

6 6

J

J

×

4 NUNERICAL RESULTS

The MF2 can solve the unphysical response of the MFH By utilizing simple tension tests we can investigate the performance of the novel model in detail In these tests, Table 2 show the material constants for the MF2 and the MFH that were obtained from the curve fitting to the adventitial data [8], see Figure 1 and Figure 4 To prevent the ill-conditioning problem of the stiffness matrix at high values of Poisson's ratio (0.4996< <ν 0.5), the Uzawa algorithm for an augmented Lagrangian method is used for all numerical tests [11] The MF2

is also capable of representing compressive state as shown in Figure 5, in which the neo-Hookean, the MFH and the Holzapfel models cannot describe the stress state

Figure 4 Curve fitting for MF2 in uniaxial tests (left) and compression tests (right)

Figure 5 Cauchy stress and stretch results for the uniaxial tension test

Table 2 Fitted parameters for MF2 model of the artery in [8]

In the uniaxial tension test, the numerical results employing the MF2 are more accurate than those of the MFH, see Figure 5 The stretch and Cauchy stress computed for the MF2 are not influenced by the ill-conditioning problem and are in good agreement with the analytical ones, whereas the response of the MFH are severely affected by the ill-conditioned constitutive matrix Moreover, the numerical results for equi-biaxial tests of the MF2 are also stable and are in good agreement with the analytical, see also [6]

5 CONCLUSIONS

Numerical instability of the MFH has been studied for the first time This problem is due to a large difference between the isotropic energy function and the anisotropic one, leading to the ill-conditioned constitutive matrix of the model, so-called the ill-conditioning problem Therefore, this leads to the solution is to reduce the difference by introducing a fibre dispersion factor ν1 By this way, we proposed the novel model MF2, which is very useful to characterize arteries, intestines, and fibre-reinforced materials The coefficients of the material law bear physical meaning compared to those of the Fung-type potential [12] In conclusion, the model MF2 is applicable to any kind of tissues or fibre-reinforced materials showing anisotropy However, the interaction between the fibres and the matrix should be considered and will be future work of this research

REFERENCES

[1] Holzapfel, G A., Gasser T C., & Ogden, R W., A new constitutive framework for

arterial wall mechanics and a comparative study of material models J Elast Phys Sci solids, 2000, vol 61, no 1–3, p 1–48

[2] Campion, R P., Elastomer composites for engineering applications – spontaneous and other

forms of bonding between components, Mater Sci Technol., 1989, vol 5, no 3, p 209–221

[3] Holzapfel, G A., Determination of material models for arterial walls from uniaxial

extension tests and histological structure, J Theor Biol., 2006, vol 238, no 2, p 290–302

[4] Balzani, D., Polyconvex anisotropic energies and modeling of damage applied to arterial walls, PhD Thesis, TU Darmstadt, 2006

[5] Gasser, T C., Ogden, R W., & Holzapfel, G A., Hyperelastic modelling of arterial layers

with distributed collagen fibre orientations, J R Soc Interface, 2006, vol 3, no 6, p 15–35

[6] Duong, M T., Hyperelastic Modeling and Soft-Tissue Growth Integrated with the Smoothed Finite Element Method-SFEM, PhD Thesis, RWTH Aachen University, Germany, 2014

[7] Bathe, K J., Finite Element Procedures MIT Press: Prentice-Hall, 1996

[8] Holzapfel, G A., Sommer, G., Gasser, T C., & Regitnig, P., Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal

thickening and related constitutive modeling, Am J Physiol Hear Circ Physiol, 2005, vol

289, no 5, p H2048–58

[9] Conway, J B., Functions of One Complex Variable 2nd ed New York: Springer Verlag, 2001 [10] Holzapfel, G A., Nonlinear solid mechanics, A Continuum Approach for Engineering:

Chichester John Wiley & Sons, 2000

www.ce.berkeley.edu/projects/feap/

[12] Fung, Y C., Fronek, K., & Patitucci, P., Pseudoelasticity of arteries and the choice of its

mathematical expression, Am J Physiol., 1979 vol 237, no 5, p H620–31

AUTHOR’S INFORMATION

1 TS Minh Tuan DUONG, Hanoi University of Science and Technology, Hanoi, Vietnam

tuan.duongminh@hust.edu.vn Tel: +84947036686, +84438680073

2 TS Thi Thanh Hai TRAN, Hanoi University of Science and Technology, Hanoi, Vietnam

hai.tranthithanh@hust.edu.vn Tel: +84978263926, +84438680073