Crack tip fields in a fiber-reinforced hyperelastic sheet: Competing roles of fiber and matrix stiffening
Item Type Article
Authors Liu, Yin;Moran, Brian
Citation Liu, Y., & Moran, B. (2022). Crack tip fields in a fiber-reinforced hyperelastic sheet: Competing roles of fiber and matrix stiffening.
Mechanics Research Communications, 120, 103837. doi:10.1016/
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Crack Tip Fields in a Fiber-reinforced Hyperelastic Sheet: Competing Roles of Fiber and Matrix Stiffening
Yin LIU1*, Brian MORAN1
1Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology, KAUST, Thuwal, 23955-6900, Kingdom of Saudi Arabia
*Corresponding author: [email protected] Tel.:+966 (0)12 808 2030
Abstract
Fiber-reinforced materials are widely observed in biological tissue such as human arteries and tendons.
Characterizing the influence of fibers on crack-tip fields in such nonlinear materials undergoing large deformation is an important precursor to understanding damage evolution and fracture or tearing. In this paper, we present asymptotic crack tip fields in a fiber-reinforced hyperelastic sheet and explore the competing roles of fiber and matrix stiffening.
A generalized neo-Hookean model and a power-law model are employed to characterize the behavior of the matrix and fibers, respectively. We show that the asymptotic crack-tip fields are mainly determined by the phase with largest degree of stiffening. For example, when the stiffening effect of fibers is much larger than that of the matrix, the crack tip fields are determined by the constitutive behavior of fibers, and the wire model and the fabric model reasonably characterize the deformation and stress near the crack tip. On the contrary, if the matrix stiffening is larger than that of the fibers, we show that the crack tip fields approach those of a pure matrix material. The asymptotic results are complemented by finite element simulations which show good agreement. These findings may shed light on material damage at a crack tip in fiber reinforced materials.
© 2015 The Authors. Published by Elsevier Ltd.
Keywords: Fiber-reinforced materials; crack tip fields; asymptotic analysis; generalized neo-Hookean material.
1. Introduction
Fracture or tearing of soft tissue is important in some clinical diseases, such as tears in tendons and ligaments [1], aortic dissection [2], and abdominal aortic aneurysm rupture [3]. Revealing mechanical deformation near a crack or a slit (crack tip fields) in soft tissue can aid in understanding of the failure mechanism and possibly provide important precursor to clinical therapy and guidance for developing medical devices.
Anisotropy due to presence of microscale fibers is an important feature of soft biological tissue. Without considering the effect of viscosity, residual stress etc., the deformation behavior of an anisotropic soft tissue is usually characterized by fiber-reinforced hyperelastic
constitutive models within the framework of continuum mechanics. Widely-used models for the fibers include a composite based model (neo-Hookean fibers) [4], a standard reinforcing model [5–7] and the Holzapfel- Gasser-Ogden (HGO) model [8]. For the crack tip fields in fiber-reinforced hyperelastic materials, the neo- Hookean sheets reinforced by the fibers characterized by the composite-based model and standard reinforced model have been studied [9–11]. The effect of multiple families of nonlinear fibers on the crack tip fields was also investigated [12]. It has been found that the presence of fibers has large influence on the crack tip fields compared to the case for matrix alone. For example, for a neo-Hookean matrix reinforced by the standard reinforcing model, the deformation near the crack tip can
Author name / Mechanics Research Communications 00 (2015) 000–000 2
be divided into three regions based on the fiber orientation angle, and stress peaks are observed at the interfaces between regions due to the stiffening effects of the fibers [11]. The stress peaks decreases significantly for the case with two families of fibers compared to the case for a single family of fibers [12]. An interesting observation in these studies is that in some cases, both the matrix and fiber effects are essential in determining the crack tip fields (e.g., a neo-Hookean matrix reinforced by the standard reinforcing model), while in some cases, the fibers seem fully dominating the asymptotic crack tip fields and we can use a wire or fabric model to approximate the crack tip fields [12]. We speculate that it is the phase (fiber or matrix) with larger stiffening degree dominating the asymptotic crack tip fields. In this paper, we study the competing roles of the fibers and the matrix with different stiffening degrees, and summarize the conditions that determine asymptotic behavior of the crack tip fields.
We use a generalized neo-Hookean (GNH) model as the constitutive model for the matrix and a power law model for the fibers so that the stiffness of each phase can be separately changed. We develop a finite element procedure to simulate the crack tip fields in a long notched strip under uniaxial tension. The finite element results are completed by analytical results for some special cases, such as the crack-tip fields for the GNH model [13–15], for a neo-Hookean matrix reinforced by neo-Hookean fibers [9], nonlinear fibers characterized by the standard reinforcing model [11] and multiple families of fibers [12]. Various case studies confirm our hypothesis that the asymptotic crack-tip fields are mainly determined by the phase showing the largest degree of stiffening.
This paper is organized as follows. In Section 2, we introduce general governing equations for a fiber- reinforced GNH material in 2D plane stress. In Section 3, asymptotic crack tips are analyzed based on the stiffening behavior of the matrix and the fibers. In Section 4, we show and analyze theoretical results and finite element results for the crack tip fields in a long notched strip with different stiffening parameters. Conclusions are given in Section 5.
2. Governing equations in plane stress
We conduct the theoretical analysis for the crack tip fields within the framework of continuum mechanics.
The deformation gradient is defined by F y x, where x and y are coordinates of a material point in the reference configuration and the current configuration, respectively. The material is assumed to be an incompressible hyperelastic material with a strain energy
function composed of a generalized neo-Hookean matrix [16] and two families of nonlinear fibers characterized by a power law model, that is,
1
matrix
2 2
a b
4a 4b
fibers
1 ( 3) 1
2
( 1) ( 1) ( 1)
2 2
n
N N
W b I
b n
I I p J
(1)
where is the shear modulus, b and n0.5 are constants describing the stiffening degree of the matrix, I1 is the first invariant of the right Cauchy Green tensor, i.e., I1trC, M, Ma, b are modulus ratios of the two families of fibers, N1, 2, ... is an integer determining the stiffening behavior of the fibers, p is a Lagrange multiplier enforcing the condition of incompressibility of the material, i.e., the Jacobian J det F1 . The invariants I4M, M a, b are defined by
2
4M M M M
I A CA , M a, b (2) and AM , Ma, b are unit vectors denoting spatial orientation of the fibers.
The first Piola-Kirchhoff stress is then given by
1 1
2 1 T
4 a,b
1 ( 3)
2 ( 1)
n
N
M M M M
M
W b
n I
N I p
P F
F
FA A F
(3)
The material model in Eq. (1) reduces to some widely used models for specific parameters. For example, when
1
n , the matrix becomes a neo-Hookean material, and when N1 , the model for the fibers becomes the standard reinforcing model [5]. The stiffening degree of the matrix (or fibers) increases with the exponent n (or
N).
Fig. 1 Sketch of a crack in a 2D plane and associated coordinate system
We then specialize the 3D constitutive model in 2D plane stress in the x x1- 2 plane (Fig. 1). The crack faces are
initially aligned with the x1 axis and the origin of the coordinate system x x1- 2 is at the initial crack tip. The fibers are located in the x x1- 2 plane and thus
{sin , cos , 0}T
M M M
A , where M is the fiber
orientation angle with respect to the x2 axis. In this setting, I4M, M a, b are given by
2 4
2 2
12 11 22 21
2 2
1 2
( cos sin ) ( cos sin )
M M
M M M M
M M
I
F F F F
(4)
Using the out-of-plane equilibrium equation in the plane stress, i.e., P330, we obtain
1 2
1 ( 1 3) b n
p I
n
(5)
where (detF)1 , , 1, 2 is the out-of-plane stretch. Thus, from Eq. (3), the in-plane stress is given by
1
2 1
1
2 1
4 ,
a,b
1 ( 3) ( )
2 ( 1) , , 1, 2
n
N M M
M M u u
M
P b I F F
n
N I y A A
(6)where A1M sinM and A2M cosM , M a, b . In the next section, we identify dominant terms in P near the crack tip for different constitutive parameters using the asymptotic conditions.
3. Asymptotic analysis
The asymptotic solutions for the deformed coordinates , 1, 2
y may be generally given by [12,17,18]
1( , ) 2( , ) ..., 1, 2, as 0 y p f r q f r r (7) where p and q are the amplitude coefficients determined by far-field boundary conditions, f r1( , ) and f r2( , ) are the first and second order eigenmodes and r and are polar coordinates in the x x1- 2 plane (Fig. 1). In a mode I crack, the eigenmodes for y do not change, but the first order eigenmode f r1( , ) does not come into play in y1. Thus, the expressions for y (in a mode I crack) become
1 1 2
2 2 1 2 2
( , ) ...
( , ) ( , ) ...
y q f r
y p f r q f r
(8)
Because the leading (first) order term f r1( , ) is more singular than f r2( , ) as r0, we obtain that
2 1
y y as r0 (9)
This relation in a mode I crack allows us to neglect the minor terms related to y1 in expressions for P21 and P22 in the asymptotic analysis ((see Eqs. (15)) and to obtain a decoupled eigenvalue problem solely related to y2 from the equilibrium equation P21,1P22,20 (see, e.g., [14,15]). From the decoupled eigenvalue problem, the first order eigenmode f r1( , ) can be obtained. Due to this, we limit the analysis in the following on the quantities related to y2 which is mainly determined by the leading order eigenmode f r1( , ) as r0.
Following Knowles and Sternberg [19], we assume that the components of the deformation gradient related to y2 are singular near to the crack tip, i.e.,
2 ~ ( q)
F O r q1 as r0 (10)
Using the asymptotic conditions in Eqs. (9) and (10), we obtain, in a mode I crack,
~O r( q)
as r0 (11)
2 2 2
1~ 21 22~ ( q)
I F F O r as r0 (12)
2 2
4 ~ ( 22cos M 21sin M) ~ ( q)
I F F O r as r0 (13)
Using the relations (10)~(13), the asymptotic strain energy function and stress as r0 in Eqs. (1) and (6) are expressed by, respectively,
2 2
a b
1 4a 4b
~ 2 2 2
n
n N N
W b I I I
b n
(14)
1
1 2 1
2 1 2 4 2,
a,b
matrix fibers
~ 2 , 1, 2
n
n N M M
M M u u
M
P b I F N I y A A
n
(15)
We see from Eq. (15) that the asymptotic behavior of P2 as r0 is determined by the phase which is more singular (or shows larger stiffening degree) than the other. Three scenarios can be identified based on the value of the exponents n and 2N, i.e.,
1 1
1 2
1 1 2 1
2 1 2 4 2,
a,b
2 1
4 2,
a,b
, if 2 , (I)
~ 2 , if ~ 2 , (II)
2 , if 2 , (III)
n n
b n
n n N M M
b
M M u u
n
M
N M M
M M u u
M
I F n N
P I F N I y A A n N
N I y A A n N
(16)
Author name / Mechanics Research Communications 00 (2015) 000–000 4
which represent the crack tip fields dominated by the matrix, both the matrix and fibers, and the fibers, respectively. Note that the feasible values of N are integers, i.e., N 1, 2,....
In case I for n2N , where the matrix stiffening is larger than that of the fibers, we see from Eqs. (14) and (15), W ~O r( 2qn) and P2 ~O r( (2n1)q) as r0, and the asymptotic crack tip fields approach those of a pure generalized neo-Hookean matrix [14,15]. In case III for
2
n N, the stiffening effect of fibers is larger than that of the matrix, and the crack tip fields tend to be determined by the constitutive behavior of fibers, i.e.,
~ ( 4qN)
W O r and P2 ~O r( (4N1)q). In this case, a model with pure fiber effect may reasonably characterize the deformation near the crack tip, such as the wire model for a single family of fibers [11] and the fabric model for two families of fibers [12]. We next conduct finite element analysis to verify these observations on the asymptotic fields.
4. Results and analysis 4.1. The finite element model
We consider a long notched strip under uniaxial tension on its top and bottom surfaces as shown in Fig. 2.
We use eight-node quadrilateral elements with four Gauss integration points to discretize the domain. The mesh scheme near the crack tip are the same as our previous work [9,11,12], which leads to a size of
6
~ 10 H0 for the smallest element, where H0 is the width of the strip (Fig. 2). This mesh scheme has been shown to be fine enough to capture the singular behavior of stress components near the crack tip in various kinds of fiber- reinforced hyperelastic sheets [9,11,12].
Fig. 2 Sketch of a long notched strip under uniaxial tension
The material model given by Eq. (1) or (3) was implemented in an in-house finite element code programed following a total Lagrange formulation. The corresponding tangent moduli are given by
2 1 1 1 1
1 1
2 1 2 1
2 1
2 2
4 a,b
2 1
4 a,b
( )[ (2 )]
2 ( )( )( )
[4 (2 1)( 1) ]
[2 ( 1) ]
ij
ijkl ik jl ji lk jk li
kl
ij ji kl lk
N M M M M
M M i j k l
M
N M M
M M ik j l
M
P g I F F F F
F
g I F F F F
N N I a A a A
N I A A
(17)
where g I1( )1 [1 bn(I13)]n1 , g I2( )1 bn(n1) [1
2
( 1 3)]n
b
n I , aiM , i1, 2 are components of the directional vector of fibers in the current configuration, i.e., aM FAM, and AM is given by Eq. (2). In the next sections, we consider the crack tip fields with varying stiffening degree of the matrix and fibers by choosing different constitutive parameters n and 2N.
4.2. Effect of the stiffening degree in matrix
GNH matrix reinforced by a single family of fibers. In this scenario, we set a 1, b 0, a 0 and N1, and compare the results for different values of n as shown in Fig. 3. For the case n 3 2N2, the matrix dominates the asymptotic crack tip fields, and the deformed coordinate y2 approaches the results of a generalized neo-Hookean matrix, given by [13–15],
1 2
1 1 1
2 2 2
2 2
1 2
2
2 2
2 cos
~ [ ( cos )] sin 1 ,
2 1
1 1, 1 sin
n n k
y r n w k
w
k w k
n
(18)
The finite element result for the stress 22~P F21 21P F22 22 also approaches to the result of a pure GNH matrix for
3
n (Fig. 3b).
In the case that the matrix effect is fully neglected in asymptotic analysis, the asymptotic crack-tip fields are determined by the fibers. For a single family of fibers, we obtain a wire model with the asymptotic stress and equilibrium equations given by, respectively [11],
2 ~ 2 4 2,u u , 1, 2
P I y A A (19)
55 2,11 45 2,12 44 2,22
45 2,1 44 2,2 1 2
2 0
0, at 0, 0
c y c y c y
c y c y x x
(20)
where c44cos2 , c45sin cos and c55sin2. The solution of the eigen problem (20) is given by
2/3
2 ~ ( )
y r f , where the first order eigenmode f( ) can be solved using a 1D finite element method [11]. We can see in Fig. 3a that as n decreases from 3 to 1, the role of fibers tends to become more and more dominant in the asymptotic fields compared to the matrix, because both y2 and 22 get closer to the results obtained by the wire
model (Fig. 3a and b). Specifically, as n decreases from 3 to 1, the deformation in the middle region decreases and the stress peaks at 2 increases. The case for n1 represents a fiber-reinforced neo-Hookean material and the corresponding crack tip fields [11] show good agreement with the finite element results. It is interesting that the location of peak stress in the circumferential direction, 22m, varies with n. As shown in Fig. 3b, for
3
n with matrix dominance, 22m is found at , while for n1 with fiber dominance, 22m is located at
2
. Fig. 3c shows the deformation at the crack tip within the region r103H0 at the stretch s 1.12, which again shows decrease of the deformation in the middle region (or increase in the region near to the crack faces) as the fibers become dominated in the crack tip field fields.
Fig. 3 Comparison of the results for different stiffening degree of the matrix with n1, 2 and 3, which is reinforced by a single family of fibers with N1 (the standard reinforcing model, see Eq. (1)). The wire model refers to the model with no matrix effect [11]. (a)–(c) are normalized deformed coordinates, y2 y2m , normalized stress, 22 22m in the circumferential direction at stretch
s 2.0
, and deformed crack tips within the region
3
10 0
r H at stretch s 1.12, respectively. The finite element results for y2 in (a) are extracted from the nodal points at r 1 104H0 and 22 in (b) from the Gauss integration points at r1.32 10 4H0, where y2m and 22m are corresponding maximum values. The data for y2 ym2 and 22 22m in Fig. 4 and Fig. 5 are produced in the same way. The circles in (a) and (b) are finite element results and the solid lines are analytical results.
GNH matrix reinforced by two families of fibers. In this scenario, we set a b 1, a 0 , b 2 and
1
N , and vary the value of n from 0.6 to 3. The corresponding results are shown in Fig. 4. The fabric model refers to the model where the matrix effect is fully neglected in the asymptotic analysis [12], which leads to asymptotic stresses fully determined by the fibers, i.e.,
2 , a, b
~ 2 M M u uM M, , , 1, 2, as 0
M
P y A A u r
(21)A detailed analysis for the crack tip fields of the fabric model is given in [12]. We can see that for the cases
0.6
n and 1 satisfying n2N , both y2 and 22 approach the results of the fabric model [12], because the fibers dominate the asymptotic crack tip fields (Eq. (16)).
For the case n 2 2N, the normalized y2 is near to that of the fabric model, while slight difference in 22is observed at the location around 2.
Fig. 4 Comparison of the results for different stiffening degree of the matrix with n0.6, 1, 2 and 3, which is reinforced by two families of fibers characterized by the standard reinforcing model (N1, see Eq. (1)). (a)–(c) are normalized deformed coordinates, y2 ym2 , normalized stress, 22 22m , in the circumferential direction and deformed crack tips at stretch s 1.12, respectively. The fabric model refers to the model with no matrix effect [12]. The discretized points are finite element results and the solid curves are analytical results.
For the case n 3 2N , we see the finite element results for y2 agree well with the GNH model (Eq. (18)), which indicates that the asymptotic fields are dominated by the matrix. The stress 22 also show similar
Author name / Mechanics Research Communications 00 (2015) 000–000 6
distribution for the case n3 and the GNH model and their maximum values 22m are both located at the crack faces . Though the matrix dominates the asymptotic crack tip fields in this case, the effect of fibers is non-negligible and influences the value of 22 near the crack faces, and thus causes the discrepancies between the normalized analytical and finite element results (Fig.
4b). We also see that increasing the matrix stiffening behavior does not change the crack-tip deformation too much (Fig. 4a and c), while significantly affects the distribution of the stress.
4.3. Effect of the stiffening degree of fibers
In this section, we consider the influence of varying stiffening degree of fibers on the crack tip fields. The material types considered are a neo-Hookean matrix with
1
n reinforced by four types of reinforced phases, including ① one with no fibers (pure neo-Hookean matrix) [19,20], ② neo-Hookean fibers [9,21], ③ the standard reinforcing model with N1 and ④ the power law model N 2 (Eq. (1)) [11]. A single families of fibers is considered with the fiber orientation angle
a 0
(aligned with the x2 axis). We see from Fig. 5a that the finite element results and analytical results for the deformed coordinate y2 well agree with each other.
Fig. 5 Comparison of the results for a neo-Hookean matrix reinforced by fibers with different stiffening degree, including the case with no fibers (pure neo- Hookean matrix), the neo-Hookean fibers [9], the standard reinforcing model (N1) and the power law model (N 2, see Eq. (1)). (a)-(c) are the results for normalized deformed coordinates, y2 y2m , normalized stress, 22 22m , and deformed crack tips at stretch
s 1.12
, respectively. The discretized points in (a) are finite element results and the solid curves are analytical results.
We note that for the case of the neo-Hookean fibers, the asymptotic stress and deformed coordinates are given by, respectively, [9]
, ,
( u uo o), , , 1, 2, as 0
P y y a a u r (22)
1/ 2
~ sin2 cos , 1, 2, as 0
y p q r (23) where p and q are amplitude coefficients determined by the far fields, and are the polar coordinates in a scaled coordinate system (i A xij j, see details in [9]).
From Eqs. (22), the terms in P arising from matrix and fibers show the same stiffening degree, and thus both of them are essential to determine the asymptotic crack tip fields. The four cases ①-④ show increasing stiffening degree of the fibers compared to the neo-Hookean matrix, and we see that both y2 and 22 approach the results of the wire model where only the fiber effect is considered in asymptotic analysis (see [11] or Eqs. (19) and (20)).
Specifically, as the stiffening degree of fibers increases, the deformation in the middle region [ 2, ]2 becomes smaller (Fig. 5a) and the stress peaks at 2 become larger (Fig. 5b). The deformed configurations of the crack tip in Fig. 5c also show decreasing deformation in the middle region as the stiffening degree of fibers increases.
5. Conclusions
In this paper, we investigate the stiffening effect of fibers and matrix on the asymptotic crack tip fields in a fiber-reinforced hyperelastic sheet. A generalized neo- Hookean (GNH) model and a power law model are used to characterize the constitutive behavior of the matrix and the fibers, respectively. The constitutive models allow us to vary the stiffness of each phase, separately. We consider several representative cases with varying stiffness either in the matrix or the fibers. The finite element results agree well with the analytical results (if
available). We find that the asymptotic crack-tip fields are mainly determined by the phase with largest degree of stiffening. When the stiffening effect of fibers is much larger than that of the matrix, the crack tip fields are determined by the constitutive behavior of fibers, and the wire model (for a single family of fibers) and the fabric model (for two families of fibers) reasonably characterize the deformation and stress near the crack tip. On the contrary, if the matrix stiffening is larger than that of the fibers, we show that the crack tip fields approach those of a GNH matrix. These observation may provide insights into material failure near the crack tip due to loss of ellipticity (e.g. fiber kinking and splitting) in the process of crack formation [7,22,23]. Though further exploration is required, it may be inferred that, near the crack tip, potential fiber-matrix splitting would occur in the direction parallel to the fiber orientation due to the stress peaks observed for the case dominated by those of one family of fibers (Fig. 5b), which seems consistent with the observation in [23] that surfaces of discontinuity are favored along the fiber direction.
References
[1] R.J. Hodgson, P.J. O’Connor, A.J. Grainger, Tendon and ligament imaging, Br. J. Radiol. 85 (2012) 1157–1172.
doi:10.1259/bjr/34786470.
[2] C.A. Nienaber, R.E. Clough, N. Sakalihasan, T. Suzuki, R.
Gibbs, F. Mussa, M.P. Jenkins, M.M. Thompson, A.
Evangelista, J.S.M. Yeh, N. Cheshire, U. Rosendahl, J.
Pepper, Aortic dissection, Nat. Rev. Dis. Primer. 2 (2016) 1–18. doi:10.1038/nrdp.2016.53.
[3] J. Golledge, P.E. Norman, Atherosclerosis and Abdominal Aortic Aneurysm, Arterioscler. Thromb. Vasc. Biol. 30 (2010) 1075–1077. doi:10.1161/ATVBAHA.110.206573.
[4] Z. Guo, X. Peng, B. Moran, A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus, J. Mech. Phys.
Solids. 54 (2006) 1952–1971.
doi:10.1016/j.jmps.2006.02.006.
[5] N. Triantafyllidis, R. Abeyaratne, Instabilities of a Finitely Deformed Fiber-Reinforced Elastic Material, J. Appl. Mech.
50 (1983) 149–156. doi:10.1115/1.3166983.
[6] G.Y. Qiu, T.J. Pence, Remarks on the Behavior of Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solids, J. Elast. 49 (1997) 1–30.
doi:10.1023/A:1007410321319.
[7] M. El Hamdaoui, J. Merodio, R.W. Ogden, Loss of ellipticity in the combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube, Int. J. Solids Struct. 63 (2015) 99–108.
doi:10.1016/j.ijsolstr.2015.02.043.
[8] G.A. Holzapfel, T.C. Gasser, R.W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elast. Phys. Sci.
Solids. 61 (2000) 1–48.
[9] Y. Liu, B. Moran, Large deformation near a crack tip in a fiber-reinforced neo-Hookean sheet, J. Mech. Phys. Solids.
143 (2020) 104049. doi:10.1016/j.jmps.2020.104049.
[10] L. Di Stasio, Y. Liu, B. Moran, Large deformation near a crack tip in a fiber-reinforced neo-Hookean sheet with discrete and continuous distributions of fiber orientations, Theor. Appl. Fract. Mech. 114 (2021) 103020.
doi:10.1016/j.tafmec.2021.103020.
[11] Y. Liu, B. Moran, Crack tip fields in a neo-Hookean sheet reinforced by nonlinear fibers, J. Mech. Phys. Solids. 152 (2021) 104406. doi:10.1016/j.jmps.2021.104406.
[12] Y. Liu, B. Moran, Effects of multiple families of nonlinear fibers on finite deformation near a crack tip in a neo- Hookean sheet, Eur. J. Mech. - ASolids. 90 (2021) 104324.
doi:10.1016/j.euromechsol.2021.104324.
[13] Y. Liu, B. Moran, Hodograph transformation for crack-tip fields in hyperelastic sheets: higher order eigenmodes and asymptotic path-independent integrals, Int. J. Fract. 229 (2021) 77–94. doi:10.1007/s10704-021-00542-x.
[14] R. Long, V.R. Krishnan, C.-Y. Hui, Finite strain analysis of crack tip fields in incompressible hyperelastic solids loaded in plane stress, J. Mech. Phys. Solids. 59 (2011) 672–695.
doi:10.1016/j.jmps.2010.12.005.
[15] P.H. Geubelle, W.G. Knauss, Finite strains at the tip of a crack in a sheet of hyperelastic material: I. Homogeneous case, J. Elast. 35 (1994) 61–98. doi:10.1007/BF00115539.
[16] J.K. Knowles, The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids, Int. J.
Fract. 13 (1977) 611–639. doi:10.1007/BF00017296.
[17] J.K. Knowles, E. Sternberg, Finite-deformation analysis of the elastostatic field near the tip of a crack: Reconsideration and higher-order results, J. Elast. 4 (1974) 201–233.
doi:10.1007/BF00049265.
[18] J.K. Knowles, E. Sternberg, An asymptotic finite-
deformation analysis of the elastostatic field near the tip of a crack, J. Elast. 3 (1973) 67–107. doi:10.1007/BF00045816.
[19] J.K. Knowles, E. Sternberg, Large deformations near a tip of an interface-crack between two Neo-Hookean sheets, J.
Elast. 13 (1983) 257–293. doi:10.1007/BF00042997.
[20] Y. Liu, B. Moran, Asymptotic path-independent integrals for the evaluation of crack-tip parameters in a neo-Hookean material, Int. J. Fract. 224 (2020) 133–150.
doi:10.1007/s10704-020-00452-4.
[21] Z. Guo, X. Peng, B. Moran, Large deformation response of a hyperelastic fibre reinforced composite: Theoretical model and numerical validation, Compos. Part Appl. Sci. Manuf.
38 (2007) 1842–1851.
doi:10.1016/j.compositesa.2007.04.004.
[22] M.E. Hamdaoui, J. Merodio, R.W. Ogden, Deformation induced loss of ellipticity in an anisotropic circular cylindrical tube, J. Eng. Math. 109 (2018) 31–45.
doi:10.1007/s10665-017-9904-z.
[23] M.E. Hamdaoui, J. Merodio, R.W. Ogden, Two-phase piecewise homogeneous plane deformations of a fibre- reinforced neo-Hookean material with application to fibre kinking and splitting, J. Mech. Phys. Solids. 143 (2020) 104091. doi:10.1016/j.jmps.2020.104091.
