Numerical simulations of the splitting tensile behaviour of needle-punched carbon/carbon composites

Abstract

Carbon/carbon (C/C) composites are materials developed for applications requiring excellent mechanical and thermal properties at elevated temperatures. Needle-punching has been used to improve the delamination resistance of C/C composites; however, its effect on tensile behaviour has yet to be fully understood. This work studies the splitting tensile behaviour of needle-punched C/C composites numerically using experimentally validated finite element (FE) simulations with a damage initiation criterion. The splitting tensile test, comprising the diametral loading of flattened Brazilian discs (FBDs), was employed for the simulations and validation. A mesoscale model of the composite FBD specimen was built, in which each composite layer was explicitly modelled, including cohesive interfacial behaviour. The composite needle fibre bundles were modelled using truss elements. The FE simulations showed the crucial role that needled fibre bundles play in the C/C composite tensile performance by carrying tensile stresses perpendicular to the loading direction and reducing the levels of localised deformation in the central region of the specimen at the early stages of loading. Furthermore, the damage initiation criterion showed that when the needle fibre bundles are included in the FE simulation, the extension of damaged areas is less than in the simulations without needled fibre bundles. In addition, the numerical results showed that increasing the number of needled fibre bundles reduced the extension of the damage in the matrix.

Keywords

Carbon/carbon composite needle-punched composite splitting tensile test flattened Brazilian disc finite element simulation damage initiation criterion cohesive interface

Introduction

Carbon/carbon (C/C) composites are composite materials comprising carbon fibres reinforcing a carbonaceous matrix, usually carbon or graphite. C/C composites were initially developed for aerospace applications due to their low density, high thermal conductivity, low coefficient of thermal expansion, and excellent mechanical properties at elevated temperatures.¹ Also, C/C composites can maintain high mechanical strength at temperatures up to 3000°C in non-oxidising atmospheres.^2–6 C/C composite applications include rocket nozzles, re-entry vehicles’ leading edges and nose tips, and aircraft brake discs.^1,2,7,8 Other applications include, for instance, brake systems for racing cars,⁹ manufacturing applications such as sintering trays¹ and hot pressing moulds,¹⁰ and biomedical implants.¹¹ However, the use of C/C composites in general engineering industries, i.e., non-aerospace applications, is limited due to the high costs of manufacturing and machining this material,^12,13 which has also limited the scientific research on C/C composites.

C/C composites have also been studied for their use in the nuclear energy generation industry. Popp et al.¹⁴ investigated using C/C composites in hot gas ducts for nuclear reactors and showed that C/C composites were an excellent alternative to metals for this application. Venugopalan et al.¹⁵ found that C/C composites coated with silicon carbide have an improved oxidation resistance, demonstrating that C/C composites are a potential candidate for structural materials for reactor applications. More recently, with the development of Generation IV nuclear reactors,¹⁶ the need for materials with excellent mechanical properties at elevated temperatures has increased. Since then, C/C composites have been further developed and have been identified as candidate materials for their use in components of Generation IV nuclear reactors, such as control rods, hot duct assembly, or heat exchangers,¹⁷ particularly for the very-high-temperature reactor (VHTR) and the molten salt reactor (MSR).^17-20 C/C composites are also being considered for structural components in fusion energy reactors.²¹ However, to design safe components for nuclear applications made of C/C composites, it is crucial to understand the mechanical behaviour, damage resistance, and failure mechanisms under different loading conditions.^13,22

Traditional two-dimensional (2D) C/C composites, i.e., composite laminates, often exhibit limited interlaminar bonding strength,²³ making them prone to delamination. Improved delamination resistance can be achieved by producing three-dimensional (3D) composites using stitching or Z-pinning,²⁴ needle-punching,²⁵ and 3D woven or braided preforms.^24,26,27 The 3D needle-punching process is less complex and relatively cheaper than other 3D preform manufacturing techniques.²⁵ In the needle-punching process, the composite preform containing layers of short-cut fibre web is punched with hook-fitted (barbed) needles, which push the web layer fibres into the cloth layers in the through-thickness direction, effectively attaching adjacent layers and creating a 3D structure.^25,28–30 Zhang et al.³¹ investigated the compressive behaviour of needle-punched C/C composites loaded in the longitudinal direction (load perpendicular to the through-thickness direction). They concluded that the primary failure-initiating mechanisms were fibre cloth layer delamination and debonding of interfaces. Tan et al.¹¹ investigated the tensile behaviour of needle-punched C/C composites and found that the specimens failed in a brittle fashion, exhibiting a complete transverse fracture. The tensile behaviour of C/C composites is usually analysed using rectangular specimens, which typically require a large amount of expensive C/C composite material. As an alternative test to use reduced sample size, the splitting tensile strength test, also known as the Brazilian disc test, can be employed, which is an indirect method to determine the tensile strength of brittle materials such as rocks,³² granite,³³ ceramics³⁴ and graphite.^35,36 It has also been used to test transversely isotropic materials,³⁷ fibre-reinforced polymer composites³⁸ and C/C composites.³⁹ For the Brazilian disc test, a thin circular disc is diametrically compressed to failure, inducing tensile stresses perpendicular to the compression loading direction in the central region of the specimen.³² Flores-Johnson et al.⁴⁰ indirectly investigated the tensile behaviour of needle-punched C/C composites using the splitting tensile strength test on flattened Brazilian disc specimens. They found that the needle fibre bundles enabled different energy-absorbing failure mechanisms and globalised deformation.

The mechanical behaviour of needle-punched C/C composites using finite element (FE) simulations has also been investigated; however, the available work in the open literature is scarce. In addition, developing FE simulations for predicting the mechanical behaviour of C/C composites could reduce the costs of investigating this expensive material. Yu et al.⁴¹ used a reconstructed FE model based on micro-CT images to model a representative volume element (RVE) using solid elements containing all the microscopic characteristics of the C/C composite, including the needled fibre bundles. Using this model, they predicted tensile stress-strain curves with good agreement with the experimental data. Xie et al.⁴² used a multiscale model with solid elements to explicitly simulate the needled fibre bundles and the different layers of a C/C composite subjected to impact loading. They found that the FE model could predict the ballistic impact limit of different composite panel configurations. Employing 3D solid elements to explicitly model the needled fibre bundles in a needle-punched C/C composite can lead to more accurate results; however, these models take more time to build and require extensive computational resources due to the large number of elements used to capture all the composite features. Another approach is using truss or beam elements to model bundles of fibres. For example, truss elements have been used to model polymer-matrix⁴³ and cement-based⁴⁴ fibre-reinforced composites and C/C composites.⁴⁵ Han et al.⁴⁵ investigated the tensile properties of needled C/C composites using solid elements for the different composite layers and beam elements to represent the needled fibre bundles. They found that using mixed solid and beam elements could predict the tensile response of C/C composites.

Based on the literature above, there is still a need for more available data on the tensile behaviour of needle-punched C/C composites. Moreover, a study showing the effects of the needle fibre bundles and their distribution on the splitting tensile behaviour of C/C composites has yet to be reported in the literature. Hence, this paper reports numerical results of the splitting tensile test of needle-punched C/C composites using flattened Brazilian discs (FBDs). Finite element (FE) simulations with experimental validation of FBDs were performed to investigate the effect of the needled fibre bundles on influencing the splitting tensile behaviour and damage of needle-punched C/C composites.

Numerical simulations

Problem description

The primary purpose of this study was to numerically investigate the effect of the needled fibre bundles on the splitting tensile behaviour of needle-punched composites. It is mentioned that the experimental results used to validate the numerical model have been presented elsewhere⁴⁰; however, details of the 3D needle-punched C/C composite used in the validation and construction of the FE model composite numerically investigated here are provided as follows: The studied needle-punched C/C composite had the following layering sequence^19,40: 0° non-woven fibre cloth of T700 carbon fibres with a diameter of 7 μm (Toray Industries Inc., Japan),⁴⁶ short-cut fibre web of HTS40 carbon fibres with a diameter of 7 μm (Toho Tenax Co. Ltd, Japan)⁴⁷ and 90° non-woven cloth of T700 carbon fibres (Figure 1(a)).⁴⁰ The overall fibre volume fraction of the C/C composite reported by the manufacturer is 30%.¹⁹ As schematically shown in Figure 1(a), the needle fibre bundles were parallel to the Z-direction.

The splitting tensile test of flattened Brazilian disc (FBD) specimens^48,49 was employed to indirectly investigate the C/C composites’ tensile behaviour. During the splitting tensile test, the FBD specimen is subjected to a uniform diametral compression (Figure 1(b))⁴⁸; the compression load induces tensile stresses in the centre of the disc specimen in the direction perpendicular to the loading. The tensile stresses then lead to specimen failure by disc splitting. The FBD specimen has two parallel flat ends, which help reduce stress concentration around the loading points that could lead to local cracking. Compared to traditional Brazilian disc samples, the test only requires compression testing platens rather than specialised curved loading fixtures.⁴⁸ The geometry of the FBD specimen is shown in Figure 1(b), in which 2 $α$ = 30°, $D$ = 20 mm, $t$ = 10 mm and 2 $b$ = 5.18 mm.⁴⁰ It is noted that quasi-static displacement-controlled loading was applied to the top loading platen (Figure 1(b)).

Finite element model

Finite element (FE) simulations of the splitting tensile test of the C/C composite FBD specimens were performed using Abaqus/Standard (Version 2022).⁵⁰ The FE simulations were carried out to understand the effect of the needled fibre bundles and the composite layered structure on the mechanical behaviour of the C/C composite. First, a mesoscale model of the composite FBD specimen was built, i.e., each layer was explicitly modelled, which included cohesive interfacial behaviour between the layers; in contrast, the needle fibre bundles were modelled using truss elements. Second, a parametric study using various needled fibre bundle configurations was performed to study the role of bundles on the tensile response.

The geometry of the 3D FE model was based on the FBD specimen described before (Figure 1(b)). A thickness of 0.38 mm was used for both the 0° and 90° fibre cloth layers, while for the short-cut fibre web layer, a thickness of 0.22 mm was employed. These layer thicknesses were selected based on scanning electron microscopy (SEM) measurements of the C/C composite experimentally studied elsewhere.^19,40 The mesh of the C/C composite model is depicted in Figure 2. Due to symmetry, only a quarter model was constructed to reduce the computational cost associated with a full 3D model (Figure 2(a)). Average element sizes of 0.19 × 0.25 × 0.27 mm³ and 0.22 × 0.25 × 0.27 mm³ were used for the fibre cloth layer and short-cut fibre web layers, respectively; however, smaller elements were used in some regions near the outer disc edges (Figure 2(b)). The total number of elements in the model was 84,584. A mesh sensitivity analysis showed that this mesh size was sufficient for convergence. The mesh comprised fully integrated 8-node brick elements (C3D8) and wedge elements (C3D6) for the different composite layers (Figure 2(b)). Wedge elements were only used in some regions near the outer disc edges (Figure 2(b)). To model the needled fibre bundles, 3-node truss elements (T3D3) were used (Figure 2(b)), which were embedded in the composite layer elements (host elements) using the embedded region constraint option. In this option, if a node of an embedded element lies within a host element, its translational degrees of freedom are constrained to the interpolated values of the corresponding degrees of freedom of the host element.⁵⁰ It is mentioned that some composite layers are not shown in Figure 2(b) to illustrate the needled fibre bundles inside the composite specimen. Based on microscopic observations, parallel truss elements were separated by a distance of 1 mm in both X and Y directions (Figure 2(b)) and had a diameter of 0.2 mm ⁴⁰; however, a parametric study using different distances between the needled fibre bundles was performed and is presented in the results and discussion section. Finally, the interface between the different composite layers in the central region (Figure 2(c)) was modelled using 3D cohesive elements (COH3D8) with zero thickness (Figure 2(d)). Perfectly bonded interfaces between the different composite layers were considered in the areas near the outer disc edges, where delamination is unlikely, to reduce the computational cost.

The material models and parameters used to capture the mechanical behaviour of the different layers and needled fibre bundles of the C/C composite are presented in the following sections. The material parameters chosen for the FE model were fixed for all the simulations, and the selection of these parameters is also explained in the following sections. Therefore, no variation of the results due to material model parameters is observed in the simulations but rather due to the variation in the number of needled fibre bundles and their distribution, quantified using the volumetric and cross-sectional damage, which is the main scope of this research. As aforementioned, a mesh sensitivity analysis was conducted using four different element sizes with models ranging from ∼17,000 to ∼131,000 elements. The relative percentage difference $R_{p d}$ of the predicted load at a displacement of 0.07 mm for different mesh sizes was used to assess mesh convergence, which was achieved when $R_{p d}$ < 2%. $R_{p d}$ was calculated as $R_{p d} = 100 \times | F_{f} - F_{c} | / [(F_{f} + F_{c}) / 2]$ , where $F_{f}$ is the predicted load using the finer mesh, and $F_{c}$ is the predicted load using coarser meshes. The mesh sensitivity analysis showed that convergence was achieved even for the coarser mesh (Figure 3). This indicates that no variations in the results due to element size are expected for the mesh size used here, for which $R_{p d}$ < 0.1%.

Material models

Each unidirectional non-woven fibre cloth layer of the C/C composite was modelled as a transversely isotropic elastic material with damage initiation predicted by the LaRC05 criterion.^50,51 The LaRC05 criterion is the only built-in damage initiation criterion in Abaqus/Standard (Version 2022) for 3D solid elements.⁵⁰ This criterion includes four damage initiation mechanisms: matrix cracking, fibre kinking, fibre splitting, and fibre tension,⁵⁰ and requires ten different material input parameters that will be described below. The short-cut fibre web layers and needled fibre bundles were modelled as isotropic and transversely isotropic materials, respectively, with no damage initiation criterion; however, a plasticity model with a single yield stress value was employed to show the onset of damage via the equivalent plastic strain.⁵⁰ It is noted that no damage evolution criterion was implemented in the unidirectional non-woven fibre cloth layers since the primary purpose of the FE simulations was to compare the effect of the needled fibre bundles on the stress distribution and damage initiation at the early stages of loading. Finally, cohesive elements were used to model the interfaces between the different composite layers.

Constitutive response and damage initiation criterion of the unidirectional composite layers

Table 1 shows the material properties of the constituents (fibres and matrix),^41,42,52 in which the subscripts

f

and

m

refer to the fibre and matrix, respectively. The subscript

1

refers to the longitudinal direction of the fibre, while subscripts

2

and

3

refer to the transverse directions of the fibre. Finally, the superscripts

T

and

C

refer to the tensile and compressive strengths, respectively.

Table 1.

Material properties of the constituents of the C/C composite.^41,42,52,53

Properties	T700 carbon fibre	HTS40 carbon fibre	Carbon matrix
Elastic modulus $E$ (GPa)	$E_{f 11} = 237$	$E_{f 11} = 239$	$E_{m} = 7.76$
Elastic modulus $E$ (GPa)	$E_{f 22} = E_{f 33} = 15.8$	$E_{f 11} = 239$	$E_{m} = 7.76$
Shear modulus $G$ (GPa)	$G_{f 12} = G_{f 13} = 23$	$-$	$G_{m} = 3.38$
Shear modulus $G$ (GPa)	$G_{f 23} = 15$	$-$	$G_{m} = 3.38$
Poisson’s ratio $ν (-)$	$ν_{f 12} = ν_{f 13} = 0.27$	$-$	$ν_{m} = 0.2$
Poisson’s ratio $ν (-)$	$ν_{f 23} = 0.3$	$-$	$ν_{m} = 0.2$
Tensile strength $σ^{T}$ (MPa)	$σ_{f 11} = 5300$	$σ_{f 11} = 4900$	$σ_{m}^{T} = 12.3$
Compressive strength $σ^{C}$ (MPa)	$-$	$-$	$σ_{m}^{C} = 57$
Shear strength $τ$ (MPa)	$-$	$-$	$τ_{m} = 16$

The effective elastic properties and strengths of non-woven fibre cloth unidirectional composite layers can be estimated using the micromechanics equations from Chamis.⁵⁴ This set of equations was selected because of its simplicity in calculating transverse properties requiring only the fibre volume fraction in addition to the constituent’s material properties and being successfully used to calculate the properties of C/C composites non-woven cloth unidirectional layers.^42,55 The micromechanics equations are shown in equations (1) and (2), respectively:

E_{l 11} = V_{f} E_{l 11} + ({1 - V}_{f}) E_{m}

(1)

E_{l 22} = E_{l 33} = E_{m} / (1 - \sqrt{V_{f}} (1 - E_{m} / E_{f 22}))

G_{l 12} = G_{l 13} = G_{m} / (1 - \sqrt{V_{f}} (1 - G_{m} / G_{f 12}))

G_{l 23} = G_{m} / (1 - \sqrt{V_{f}} (1 - G_{m} / G_{f 23}))

ν_{l 12} = ν_{l 13} = V_{f} ν_{f 12} + ({1 - V}_{f}) ν_{m}

X_{T} = V_{f} σ_{f 11}

(2)

Y_{T} = (1 - (\sqrt{V_{f}} {- V}_{f}) (1 - E_{m} / E_{f 22})) σ_{m}^{T}

Y_{C} = (1 - (\sqrt{V_{f}} {- V}_{f}) (1 - E_{m} / E_{f 22})) σ_{m}^{C}

S_{L} = (1 - (\sqrt{V_{f}} {- V}_{f}) (1 - G_{m} / G_{f 12})) τ_{m}

where

V_{f}

is the fibre volume fraction, the subscript

l

refers to the composite layer,

X_{T}

and

Y_{T}

are the longitudinal and transverse tensile strengths,

Y_{C}

is the transverse compressive strength, and

S_{L}

is the longitudinal shear strength of the non-woven composite unidirectional layers. A double-subscript notation was used in equations (1) and (2) and the material properties in Tables 1 and 2 to be consistent with the notation used in the equations developed by Chamis.⁵⁴ It is, however, clarified that, for instance,

E_{f 11}

represents the elastic modulus in the 1-direction.

Table 2.

Effective material properties for the different layers and needled fibre bundles of the C/C composite, which were estimated using micromechanics formulas.

	Properties	Non-woven cloth layer	Short-cut fibre layer	Needled fibre bundle
Elastic properties	Elastic modulus (GPa)	$E_{l 11} = 55.2$	$E = 22.7$	$E_{1} = 39.5$
	Elastic modulus (GPa)	$E_{l 22} = E_{l 33} = 10.76$	$E = 22.7$	$E_{1} = 39.5$
	Shear modulus (GPa)	$G_{l 12} = G_{l 13} = 6.34$	$G = 8.1$	$G_{12} = G_{13} = 4.6$
	Shear modulus (GPa)	$G_{l 23} = 5.87$	$G = 8.1$	$G_{23} = 4.2$
	Poisson’s ratio (−)	$ν_{l 12} = ν_{l 13} = 0.21$	$ν = 0.4$	$ν_{12} = ν_{13} = 0.21$
	Poisson’s ratio (−)	$ν_{l 23} = 0.23$	$ν = 0.4$	$ν_{23} = 0.23$
Strength	Tensile strength (MPa)	$X_{T} = 795, Y_{T} = 10.75$	$-$	$-$
	Compressive strength (MPa)	$X_{C} = 397.5, Y_{C} = 49.8$	$-$	$-$
	Shear strength (MPa)	$S_{L} = 12.6, S_{T} = 18.8$	$-$	$-$
	Initial yield stress (MPa)	$-$	$σ_{y} = 71.6$	$σ_{y} = 71.6$

The effective elastic properties and strengths of the non-woven composite unidirectional layers shown in Table 2 were estimated using equations (1) and (2), the mechanical properties in Table 1 and $V_{f}$ = 0.3. Due to carbon fibre mechanical property degradation during heat treatment in C/C composite manufacturing, a decrease in $E_{f 11}$ and $σ_{f 11}$ of up to 30% and 50%, respectively, is expected.⁵⁶ Therefore, $E_{f 11}$ and $σ_{f 11}$ were reduced to 166 GPa and 2650 MPa, respectively; these values were used in equations (1) and (2) to calculate the properties for the simulations shown in Table 2. In contrast, the transverse properties should not degrade since the matrix properties dominate them.⁴¹ Additionally, in Table 2, $ν_{l 23}$ was obtained from Xie et al.⁴² It has been reported that direct in-situ testing of the longitudinal compressive strength $X_{C}$ is difficult; however, some studies measured $X_{C}$ as ∼40–60% of the tensile strength $X_{T} .$ ⁴¹ Here, $X_{C}$ was taken as 50% of the $X_{T}$ value, that is, $X_{C}$ = ${0.5 X}_{T}$ .⁴¹ Similarly, the transverse shear strength $S_{T}$ is a material property that is difficult to measure experimentally; however, Davila et al.⁵⁷ developed a method to estimate it as $S_{T}$ = ${0.378 Y}_{C}$ ,⁵⁷ based on the assumption that the angle of the fracture plane $α_{0}$ of a unidirectional composite loaded in transverse compression is typically ∼53°.⁵⁸ It has been reported that $α_{0}$ is typically in the range of 51°–55° for carbon composites.⁵¹

The LarC05 damage initiation criterion was used to model the damage initiation of the composite unidirectional layers. This criterion requires the longitudinal and transverse strengths

X_{T}

X_{C}

Y_{T}

Y_{C}

S_{L}

S_{T}

as input parameters. In addition, it requires the following input parameters, which were obtained from Gouskos and Iannucci⁵⁹ and are provided in Table 3: the fracture plane angle for pure compression

α_{0}

, the initial misalignment angle

φ_{0}

, and the longitudinal and transverse shear friction coefficients

η_{L}

and

η_{T}

. Regarding the values in Table 3, the selection of

α_{0}

was based on the typical angle of the fracture plane of carbon composites, as explained in detail before.

φ_{0}

is a material property that represents initial microstructural defects that can trigger kink-band formation.⁶⁰ Based on the literature,^59,61 a small value of

φ_{0}

= 2.544° was used here. The selection of

η_{T}

was based on the equation

η_{T} = - 1 / \tan 2 α_{0}

,⁶¹ while the value of

η_{L}

was taken from the literature.⁵⁹

Table 3.

LarC05 damage criterion parameters.

Properties	Non-woven cloth layer
Fracture plane angle (°)	$α_{0} = 53$
Misalignment angle (°)	$φ_{0} = 2.544$
Longitudinal shear friction coefficient	$η_{L} = 0.082$
Transverse shear friction coefficient	$η_{T} = 0.287$

The output variables related to the damage initiation mechanisms of the LarC05 criterion are LARCMCCRT (matrix cracking), LARCFKCRT (fibre kinking), LARCFSCRT (fibre splitting) and LARCFTCRT (fibre tension). A value equal to or greater than 1.0 in any output variable indicates that the particular damage initiation criterion has been satisfied.⁵⁰

Constitutive response of the short-cut fibre composite layers and needled fibre bundles

The constitutive behaviour of the short-cut fibre web layers and needled fibre bundles was modelled using the micromechanics formulas in equation (3), the properties in Table 1, and $V_{f}$ = 0.2. This value of $V_{f}$ was selected by assuming that most of the fibres in the needled fibre bundles came from the short-cut fibre web layers, considering that the short-cut fibres are introduced into the preform during the needle-punching process⁶²; thus, both the short-cut fibre web layers and needled fibre bundles had the same $V_{f}$ . Furthermore, compared to the more densely-packed fibres in the non-woven fibre cloth unidirectional layers, it was assumed that the short-cut fibre web layers had a lower $V_{f}$ of 20%, which has been employed in other investigations.^41,63 Short-cut fibre web layers and needled fibre bundles were considered short-fibre composites; however, while the needled fibre bundles were considered unidirectional short-fibre composites, for the short-cut fibre web layers the random orientation of the fibres was also considered. The Halpin-Tsai model^64–67 was used to estimate the elastic properties of a unidirectional short-fibre composite with circular fibres:

\frac{E_{1}}{E_{m}} = \frac{1 + (2 L / d) η_{1} V_{f}}{1 - η_{1} V_{f}}, \frac{E_{2}}{E_{m}} = \frac{1 + 2 η_{2} V_{f}}{1 - η_{2} V_{f}}, \frac{G_{12}}{G_{m}} = \frac{1 + η_{12} V_{f}}{1 - η_{12} V_{f}}, \frac{G_{23}}{G_{m}} = \frac{1 + {ζ η}_{23} V_{f}}{1 - η_{23} V_{f}}, ν_{12} = ν_{f 12} V_{f} + ν_{m} V_{m}

(3)

where

η_{1} = \frac{E_{f 11} / E_{m} - 1}{E_{f 11} / E_{m} + (2 L / d)}, η_{2} = \frac{E_{f 11} / E_{m} - 1}{E_{f 11} / E_{m} + 2}, η_{12} = \frac{G_{f 12} / G_{m} - 1}{G_{f 12} / G_{m} + 1}, η_{23} = \frac{G_{f 12} / G_{m} - 1}{G_{f 12} / G_{m} + 1}, ζ = \frac{1 + ν_{m}}{3 - ν_{m} - ν_{m}^{2}}

In equation (3), $L$ is the length of the fibre estimated as 6 mm while $d$ = 7 μm is the diameter of the fibre. As aforementioned, reduced fibre tensile properties ( $E_{f 11}$ = 168 GPa and $σ_{f 11}$ = 2450 MPa) were used in equation (3). For the short-cut fibre web layers, $E_{1}$ and $E_{2}$ from equation (3) were used in the following formulas (equation (4))⁶⁶ to estimate the elastic properties of the randomly-oriented composite:

E = (3 / 8) E_{1} + (5 / 8) E_{2}

(4)

G = (1 / 8) E_{1} + (1 / 4) E_{2}

ν = E / 2 G - 1

In addition, the initial yield stress of $σ_{y}$ = 71.6 MPa was used⁶⁸ for both short-cut fibre web layers and needled fibre bundles. The estimated mechanical properties of short-cut fibre web layers and needled fibre bundles, using equations (4) and (3), respectively, are shown in Table 2.

Constitutive response of the cohesive elements

The 3D cohesive elements (COH3D8) used to model the interface between the different composite layers followed a bilinear traction-separation constitutive law,⁵⁰ which assumes an initial linear elastic behaviour, that is, under external loading, the initial traction force of the cohesive interface material increases linearly with the displacement caused by the external force.⁶⁹ Since zero-thickness cohesive elements were used, the interface stiffness in the normal direction $K_{n}$ and transverse directions $K_{s}$ and $K_{t}$ must be defined to describe the elastic response of the material. Following the initial elastic response, material damage starts to occur once a damage initiation criterion is met. Here, the damage initiation of the cohesive elements is assumed to occur when a quadratic interaction function involving the normal traction $t_{n}$ and the shear tractions $t_{s}$ and $t_{t}$ reaches a value of 1, as follows:

{(\frac{t_{n}}{t_{n}^{0}})}^{2} + {(\frac{t_{s}}{t_{s}^{0}})}^{2} + {(\frac{t_{t}}{t_{t}^{0}})}^{2} = 1

(5)

where

t_{n}^{0}

is the normal strength, and

t_{s}^{0}

and

t_{t}^{0}

are the shear strengths.

t_{n}^{0}

represents the peak value of the nominal stress (traction) when the deformation is purely normal to the interface, while

t_{s}^{0}

and

t_{t}^{0}

represent the peak values when the deformation is in the first or second shear direction, respectively (Figure 4(a)).⁵⁰ The corresponding separation (the relative displacements between the top and bottom faces of the cohesive element) of the normal and shear tractions are denoted by

δ_{n}

δ_{s}

and

δ_{t}

, respectively. The normal and shear separations at the peak strengths are denoted by

δ_{n}^{0}

δ_{s}^{0}

and

δ_{t}^{0}

, respectively (Figure 4(a)).

Following damage initiation, damage evolution in the cohesive elements will result in linear material softening; the damage then progressively extends, resulting in a linear reduction of the traction force with further displacement due to stiffness degradation. Here, linear softening was used for the damage evolution law of the cohesive elements using the effective displacement at complete failure $δ_{m}^{f}$ , relative to the effective displacement at damage initiation $δ_{m}^{0}$ (Figure 4(b)).⁵⁰ The effective displacement $δ_{m}$ , employed to describe the damage evolution under a combination of normal and shear deformation across the interface (cohesive element) is defined as⁵⁰:

δ_{m} = \sqrt{δ_{n}^{2} + δ_{s}^{2} + δ_{t}^{2}}

(6)

For linear softening, the evolution of the scalar damage variable

D

is defined as⁷⁰:

D = \frac{δ_{m}^{f} (δ_{m}^{\max} - δ_{m}^{0})}{δ_{m}^{\max} (δ_{m}^{f} - δ_{m}^{0})}

(7)

where

δ_{m}^{\max}

is the maximum value of

δ_{m}

attained during the loading history.⁵⁰

D

represents the overall damage in the material and initially has a value of 0 at the initiation of damage (

δ_{m}

δ_{m}^{0}

) and then increases monotonically with further loading up to a value of 1 at complete failure (

δ_{m}

δ_{m}^{f}

).⁵⁰ A large value of

δ_{m}^{f}

(1 mm) was used in the simulations since the primary purpose of the cohesive element layers was to account for stiffness degradation at the interface compared to perfectly bonded interfaces while avoiding any premature delamination that could occur, considering that the needled fibre bundles modelled using truss elements cannot capture the deflection of delamination cracks, which were observed experimentally at large displacements.⁴⁰ Table 4 shows the material parameters of the cohesive interface used between the different composite layers. The interface stiffness values (

K_{n}

K_{s}

K_{t}

) in Table 4 were chosen based on similar values found in the literature for carbon matrix composites, which ranged from 5 × 10⁴ to 10⁶.^68,71 The normal and shear strength values (

t_{n}^{0}

t_{s}^{0}

t_{t}^{0}

) in Table 4 were selected based on values found in the literature for C/C composites.^72,73

Table 4.

Material properties and cohesive element parameters for the interface between the different composite layers.

Properties
$K_{n}$ , $K_{s}$ , $K_{t}$ (N/mm³)	10⁵
$t_{n}^{0}$ (MPa)	12
$t_{s}^{0}$ , $t_{t}^{0}$ (MPa)	6

Results and discussion

Experimental validation of the FE model

Figure 5 compares the predicted load-displacement curve of the needle-punched C/C composite FBD specimen with the average experimental load-displacement curve with standard deviation (shaded area) obtained from specimens subjected to the splitting tensile test.⁴⁰ It is mentioned that in Flores-Johnson et al.,⁴⁰ three FBD specimens were tested, and good repeatability was observed based on the obtained load-displacement curves and the measured values of the peak load (11.84 ± 0.46 N) and the displacement at peak load (0.17 ± 0.02 mm).⁴⁰ It can be seen in Figure 5 that the FE model predicts the load-displacement curve well; however, it slightly overestimates the stiffness of the sample (indicated by the slope of the load-displacement curve). For instance, the predicted load at a displacement of 0.07 mm is 7.6% higher than the average experimental load. This observation is explained by the numerical model not considering voids and pre-existing cracks, which contribute to the stiffness reduction. In addition, the FE model does not consider localised damage evolution occurring at the early stages of loading. Furthermore, the model is not designed to predict the load-displacement curves for displacements > 0.07 mm, in which large inelastic deformation occurs, as a damage evolution model was not implemented for the 0° and 90° fibre layers. As mentioned, the FE models aim to show the effect of the needled fibre bundles on the mechanical response at low displacements at the early stages of loading. Figure 5 also shows the predicted load-displacement curve of a C/C composite FBD specimen without needled fibre bundles, in which the truss elements have been removed from the simulation. Although it can be seen that the curve predicted by the FE model with needled fibre bundles is slightly higher than the curve of the model without needled fibre bundles, it is evident that the needled fibre bundles do not significantly affect the predicted load-displacement behaviour at low displacements at the early stages of loading. Notwithstanding, the needled fibre bundles greatly influence the strain distribution in the central region of the specimen, as will be discussed below.

Figure 6(a) shows the experimental DIC contour of the strain in the Z direction ( $ε_{Z}$ ),⁴⁰ which is normal to the compressive loading direction (X direction), at a displacement of 0.07 mm corresponding to ∼60% of the average peak load), along with the corresponding predicted $ε_{Z}$ contour using the FE model of the C/C composite FBD specimen (with needled fibre bundles) subjected to the splitting tensile test. It is noted that the FE model has been mirrored with respect to the symmetry plane to show the entire specimen. It can be seen that the FE model can accurately predict the location of the highest tensile strains occurring in the 0° fibre layers (in red colour) and the highest compressive strains at the flat ends of the specimen (in blue colour). Similarly, Figure 6(b) shows the DIC contour of the strain in the X direction ( $ε_{X}$ ),⁴⁰ which is parallel to the compressive loading direction, along with the predicted contour, where it can be seen that the predicted highest compressive strains in the central region agree with the experimental results.

Numerical analysis of the splitting tensile behaviour and damage of C/C composites with and without needled fibre bundles

Figure 7(a) shows the predicted strain $ε_{Z}$ at a displacement of 0.07 mm for the FE model with needled fibre bundles. It can be seen that the needled fibre bundles exhibit high levels of tensile strain, which enables more global distribution of the strain $ε_{Z}$ , which in turn reduces the area with high strain levels (in grey colour) in the middle of the specimen, as seen in the cross-section. It is noted that the cross-section has been mirrored with respect to the symmetry plane to show the entire specimen. In contrast, the FE model without needled fibre bundles (Figure 7(b)) exhibits a larger area of high tensile strain levels, which are visible on the outer surface at the 0° fibre layer located in the centre of the specimen (in grey colour). Figure 7(c) shows the predicted stress in the Z direction ( $σ_{Z}$ ), normal to the applied load, for the FE model with needled fibre bundles. It can be seen that the tensile stress in the needled fibre bundles is high, which reduces the stress levels in other parts of the composite. In contrast, a larger area of high tensile stress is observed in the FE model without needled fibre bundles (Figure 7(d)). Figure 7(e) and (f) show the contours of the damage variable SDEG for the cohesive elements of the FE models with and without needled fibre bundles, respectively. For the composite with needled fibre bundles, more global distribution of the damage in the cohesive elements (larger damaged area) is observed (Figure 7(e)) compared to the composite without needled fibre bundles (Figure 7(f)). In addition, damage in the cohesive elements is not observed in the central region of the specimen with needled fibre bundles (Figure 7(e)). In contrast, areas with higher damage levels (in grey colour) are observed for the composite without needled fibre bundles (Figure 7(f)). Moreover, damage in the cohesive elements in the middle of the specimen is observed for the composite without needled fibre bundles (Figure 7(f)).

Figure 8(a) and (b) show the contours of the LarC05 criterion output variables for the FE models with and without needled fibre bundles, respectively, at a displacement of 0.07 mm. The output variables include LARCMCCRT (matrix cracking), LARCFKCRT (fibre kinking), and LARCFSCRT (fibre splitting). In the contours, the regions where the damage initiation mechanisms have been met (output variable > 1.0) are shown in red and brown colour, while the areas where the criterion has just been met or is close to being met (0.5 $\leq$ output variable $\leq$ 1.0) are shown in orange colour. It is noted that the short-cut fibre web layers appear in white colour (not showing any value) as the LarC05 criterion can only be used for unidirectional fibre-reinforced composites. It is also noted that the fibre tension initiation mechanism (LARCFTCRT) is not presented since this criterion was not met in any of the simulations.

For the predicted matrix cracking initiation criterion (LARCMCCRT), it can be seen in Figure 8 that there is matrix cracking in the regions near the flat ends in both cases (with and without needled fibre bundles). However, the matrix cracking area in the specimen’s centre is much more significant for the model without needle fibre bundles. For the fibre kinking criterion (LARCFKCRT), the regions in which this type of damage is observed are very similar in both cases (with and without needled fibre bundles), as depicted in Figure 8. These areas are observed in the 0° fibre layers near the flat edges, which is explained by the high levels of compressive strain $ε_{X}$ observed in those locations (Figure 6(b)). Finally, for the fibre splitting criterion (LARCFSCRT), it can be seen in Figure 8 that the regions where this criterion has been met (i.e., the 0° fibre layers) are similar in both cases; however, it can be seen that in the central region of the model without the needled fibre bundles, the criterion for the splitting damage has been fully met in three 0° fibre layers (red colour). In contrast, the criterion has only been partially met in the model with needled fibre bundles.

Influence of the needled fibre bundle configuration on the damage of C/C composites with and without needled fibre bundles

The FE simulation results in the previous section showed the crucial role that needled fibre bundles play in the structural performance of the C/C composites as they redistribute the deformation in the specimen, producing less stress concentration. To further understand the influence of the needled fibre bundle configuration on the mechanical properties, two additional cases were numerically evaluated (N2 and N0.5), as shown in Figure 9(a), i.e., the distance between the needled fibre bundles was varied (and thus the number of bundles); $L_{N}$ indicates the shortest distance between the bundles. In Figure 9(a), N1 corresponds to the configuration of the C/C composite specimens tested experimentally.⁴⁰ Figure 9(b) shows the contours of the LarC05 criterion output variable LARCMCCRT (matrix cracking) for FE models without needled fibre bundles (NN) and with various needled fibre bundle configurations (Figure 9(a)) at a displacement of 0.07 mm. In all cases in Figure 9(b), the left image shows the external specimen surface, while the right image shows the cross-section in the middle of the specimen. It can be seen in the central region of the specimen (Figure 9(b)) that the area where the matrix cracking has occurred or is nearly occurring (output variable ≥ 0.5) reduces with the reduction of the distance between the needled fibre bundles (increase in the number of bundles).

The volumetric damage was calculated as the volume of damaged elements (output variable > 1.0) divided by the total volume of the specimen. In contrast, the cross-sectional damage was calculated as the damaged cross-sectional area in the middle of the specimen divided by the total area of the cross-section. Both the volumetric damage and the cross-sectional damage were plotted against the number of needled fibre bundles of each configuration for a displacement of 0.07 mm (Figure 10) to quantify the matrix damage shown in Figure 9(b). It can be seen in Figure 10 that there is a reduction in the volumetric and cross-sectional damage with the increase in the number of bundles, which follows a somewhat negative linear relationship from 0 to 114 bundles. When the number of bundles is increased from 114 to 429 (configuration N0.5), the volumetric damage is further reduced; however, the reduction is not as significant considering the large increase in the number of bundles. In contrast, Figure 10 shows that increasing the needled fibre bundles from 114 to 429 significantly reduced the cross-sectional damage area from 21.1% to 5.4%.

Discussion

As mentioned, the FE simulations revealed the critical importance of the needled fibre bundles in the structural performance of the C/C composites, redistributing the deformation in the specimen more globally, resulting in less stress concentration. The results also showed that increasing the number of needled fibre bundles reduces the extension of the damage in the matrix, which could improve mechanical performance and avoid catastrophic failure by sudden delamination.

It is acknowledged, however, that the FE model presented here is limited by the assumption that no damage evolution occurs at low displacements at the early stages of loading. The lack of microscopic features, such as voids and defects, also limits the FE model. Therefore, it is recommended that future work uses an FE model that includes a damage evolution criterion coupled with the LarC05 damage initiation criterion to predict and analyse crack propagation at large displacements.⁷⁴ Furthermore, it is acknowledged that a model that considers porosity and pre-existing damage could provide more accurate results, particularly in the inelastic response of the C/C composites; thus, an FE model that includes these characteristics is recommended when microstructural features are important for specific numerical studies.⁷⁵ However, it should be considered that a model that explicitly considers small features, such as needled fibre bundles, voids and cracks, takes more time to build and requires extensive computational resources due to the large number of elements needed to capture all the composite features.³ It is also recommended to include the anisotropy of the matrix in the FE model material properties, which could result from specific manufacturing processes.⁷⁶ Related to the previous recommendation, a parametric study using different material properties for the matrix and interface, resulting from different manufacturing process parameters⁷⁷ or various manufacturing processes, could be of interest to improve the fabrication of C/C composites. For this purpose, a computational methodology evaluating a large number of numerical simulations to understand each material parameter’s range and its effect on results variations is recommended if computationally efficient FE models are available.⁷⁸ Finally, further studies are recommended to investigate the effect of the configuration and number of bundles on the mechanical performance of C/C composites.

It is concluded that the FE model presented here can be used to investigate the splitting tensile behaviour of needle-punched C/C composites for low displacements, and the approach employed in these simulations could be used to model the macroscopic mechanical behaviour of other needle-punched composites.

Conclusions

This work numerically investigated the splitting tensile strength of needle-punched C/C composites through the diametral loading of flattened Brazilian discs (FBD). The finite element (FE) simulations demonstrated the capability of the model to predict the splitting tensile behaviour of the needle-punched C/C composites at the early stages of loading, including the prediction of the regions with the highest compressive and tensile strains observed experimentally. The FE simulations of FBD subjected to diametral loading showed that the needled fibre bundles play a crucial role in the tensile performance of the C/C composites by carrying tensile stresses perpendicular to the loading direction and reducing the levels of localised deformation in the central region of the specimen at the early stages of loading. In addition, the damage initiation criterion showed that when the needle fibre bundles are included in the FE simulation, the extension of damaged areas (matrix cracking) is less than in the simulations without needled fibre bundles. Furthermore, the numerical results also showed that increasing the number of the needled fibre bundles further reduced the extension of the damage in the matrix.

Footnotes

Acknowledgements

The authors thank C. Yang for providing the material, and O. Muránsky for his assistance with the modelling work and experimental results used for the model validation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the Australian Institute of Nuclear Science and Engineering (AINSE) for providing financial support to J. Townsend through the Honours Scholarship.

Correction (February 2025):

The article has been added with the graphical representation of the abstract since its original publication.

ORCID iDs

Emmanuel A Flores-Johnson

Johannes Reiner

Data availability statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.*

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