A three-dimensional failure criterion for composites based on a new theoretical model of kink-band formation

Abstract

A theoretical model for predicting kink-band formation in unidirectional composites is developed. In contrast to previous models, the proposed model takes into account the shear deformation in the non-misaligned regions when determining the fiber misalignment angle, and introduces an analytical approach for determining the orientation of the kink-band plane based on the applied stress state, instead of the conventional approach based on maximizing a failure index. Based on this model, a three-dimensional failure criterion for composites is further developed, which considers the effect of fiber misalignment under all stress states and therefore overcomes the discontinuity in the failure envelope produced by Pinho’s criterion. Using the proposed model and criterion, the failure behavior of unidirectional composites under various stress states is analyzed. A notable finding is that, under the $σ_{11} - τ_{21}$ stress state, both longitudinal tensile stress and low longitudinal compressive stress can enhance the longitudinal shear stress at failure. The proposed criterion is validated against six sets of experimental data, with prediction errors of 1.77–11% and the lowest error in four of the six cases.

Keywords

composites strength failure criterion fiber kinking

Highlights

• A theoretical model is proposed to predict the formation of kink bands.

• The model incorporates the effects of all stress components on kink-band formation.

• A three-dimensional failure criterion accounting for fiber misalignment is developed.

• The influence of fiber misalignment is considered under any stress states.

Introduction

The failure mechanism of composite materials has always been a focal topic of interest among researchers. Compared to other failure modes, the failure mechanism of fiber kinking is more complex. Experimental results indicate that localized kink bands are observed for unidirectional (UD) fiber-reinforced composites after high longitudinal compressive failure.^1–3 For the mechanism of kink-band formation, some researchers follow Rosen’s hypothesis. Rosen⁴ modeled the fibers as initially straight, linearly elastic plates embedded in a linearly elastic matrix, and analyzed elastic micro-buckling under the assumption of perfectly aligned fibers. The shear mode of microbuckling proposed by Rosen depends on the matrix shear modulus and is independent of any initial imperfection. In contrast, Argon⁵ assumed the existence of an initial fiber misalignment, which generates a shear stress between fibers under axial compression. This shear stress further rotates the fibers and amplifies itself, leading to a local shear instability and the formation of a kink band. The critical compressive stress predicted by Argon’s model depends on the matrix shear yield stress and the initial misalignment angle, rather than on the elastic shear modulus. Some subsequent studies have shown that Argon’s hypothesis seems more reasonable. For instance, based on the experimental observations, Schultheisz and Waas⁶ pointed out that the formation of kink bands seems to be triggered by local microstructural defects, such as fiber misalignments and matrix cracking. Furthermore, Sun et al.⁷ used X-ray computed tomography (CT) to observe the formation of kink bands and concluded that the presence of micro-voids leads to stress concentrations in the matrix and reduces the lateral support of the fibers, eventually leading to kinking failure. These experiments support Argon’s hypothesis.

Numerous studies have indicated that the initial fiber misalignment in composites primarily originates from manufacturing-induced imperfections.^7,8 Localized fiber misalignment results in shear stresses in the matrix under longitudinal compression. The shear stresses induce additional fiber rotation, which in turn leads to a further increase in the shear stresses. This closed-loop effect will ultimately lead to failure. At present, extensive research has been conducted on the magnitude and distribution characteristics of initial fiber misalignment.^9–11 Recently, Zheng et al.⁸ performed a statistical analysis of fiber orientations in unidirectional composites using CT, revealing that the misalignment angles approximately follow a normal distribution and exhibit similar magnitudes across different directions. Varandas et al.¹² developed a micromechanical model that treats fiber misalignment as a stochastic process to investigate its influence on compressive failure behavior. Their results showed that, although individual fibers exhibit different initial misalignment directions, after peak load, the fibers within the kink bands tend to align in a consistent orientation.

Extensive research has been conducted on fiber kinking failure in composites. Gu et al.¹³ performed longitudinal compression tests on carbon fiber-reinforced composites under different strain rates, and analyzed the failure morphology using scanning electron microscopy (SEM). The results showed that under dynamic loading, failure was dominated by kink bands and longitudinal splitting, whereas under quasi-static loading, fiber kinking was the primary failure mode. Takahashi et al.¹⁴ carried out uniaxial compression tests on carbon fiber reinforced plastics (CFRP) and employed CT to investigate the stress-strain response and failure process. They reported that the local fiber bending deformation occurred before the peak load. Clay et al.¹⁵ used CT to observe the evolution of kink bands in notched composite specimens under longitudinal compression, and found that kink bands primarily formed in the central 0° ply region.

The aforementioned studies primarily investigated the effect of fiber misalignment on the longitudinal compressive failure of composites through experimental approaches. In this context, incorporating fiber kinking into failure criteria has emerged as an important focus of recent research. Although numerous failure criteria for composites have been proposed historically,¹⁶ and new ones continue to emerge in recent years,^17–20 most of them do not account for fiber kinking. Consequently, their ability to predict this specific failure mode remains limited. This highlights the necessity of incorporating fiber kinking into failure criteria. In existing failure criteria that account for fiber kinking, the initial fiber misalignment angle is typically treated as a fixed value, rather than a randomly distributed variable as observed in real structures. Dávila and his workers^21,22 proposed a method to solve the fiber misalignment angle for a generic stress state. Pinho et al.²³ argued that the initial misalignment angle is a material property that can represent initial defects, and proposed a three-dimensional (3D) failure model. In addition, many macroscopic failure criteria considering fiber misalignment have been proposed in Refs. 24–26. Meanwhile, a number of studies have focused on developing numerical models that incorporate the fiber kinking failure.^27–30 For instance, Davidson and Waas²⁷ distinguished the mesoscale and microscale misalignment of fibers in the micromechanical model and studied the effect of fiber misalignment on failure under longitudinal compression. Ding et al.²⁸ introduced a three-dimensional elasto-plastic damage model that accounts for fiber kinking failure. Divse et al.²⁹ proposed a progressive damage model incorporating fiber kinking behavior. Xue and Kirane³¹ developed a microplane model to simulate the initiation and propagation of kink bands under longitudinal compression. In addition, the finite element analyses on the influence of fiber misalignment have been carried out in Refs. 32–36.

Although various failure criteria incorporating fiber misalignment have been developed, they share a common theoretical basis. Following the framework originally established by Argon.⁵ Dávila and his co-workers^21,22 developed the first widely adopted analytical failure criterion that incorporates this mechanism. Most subsequent criteria, including those of Pinho et al.²³ and Refs. 24–26, have been developed on the basis of this work. Although these criteria differ in the choice of the matrix failure function, they adopt similar approaches in handling fiber misalignment and determining the orientation of the kink-band plane. As a result, the previous failure criteria considering fiber kinking share several notable limitations. Specifically, they neglect the influence of shear deformation in the non-misaligned regions on the fiber misalignment angle and incorrectly determine the angle of the kink-band plane using failure indices; in addition, they consider the effect of fiber misalignment only when σ₁₁ < 0, leading to discontinuities in the predicted failure envelopes. To overcome these limitations, this study develops a theoretical model for kink-band formation, based on which a three-dimensional failure criterion is proposed. Through theoretical analysis and experimental validation, the effectiveness of the proposed model and criterion is demonstrated.

The theoretical model for kink-band formation

Following previous studies,^21–24 the present study assumes that the initial fiber misalignment angle θ_i is a fixed value rather than a randomly distributed variable. In real composites, in addition to fiber misalignment, various manufacturing-induced imperfections such as fiber waviness, voids, and matrix defects also influence the failure behavior. Therefore, θ_i is treated here as an effective parameter representing the combined effect of these imperfections, and is calibrated from the experimentally measured longitudinal compressive strength $X_{C}^{*}$ . This treatment is consistent with that adopted in Refs. 21–24. Based on experimental findings of Zheng et al.,⁸ the present study considers that the initial fiber misalignments are approximately equal in the transverse direction, through the thickness direction, or in any direction in between the two. Under external loads, localized kink bands form in UD composites, and within these kink bands, the fibers tend to align in a consistent orientation.

Figure 1 shows a 3D model with a kink-band plane for UD composites. The coordinate system 123 is aligned with the composite material axes, where the 1 axis is parallel to the fiber direction and the 2 and 3 axes represent the transverse directions. The coordinate system $1^{ψ} 2^{ψ} 3^{ψ}$ is associated with the kink-band plane, which is obtained by rotating the coordinate system 123 around the 1 axis by an angle $ψ$ , and $ψ$ is defined as the angle of the kink-band plane ranging from 0° to 360°. The coordinate system $1^{m} 2^{m} 3^{m}$ is associated with the localized misaligned regions, which is obtained by rotating the coordinate system $1^{ψ} 2^{ψ} 3^{ψ}$ around the $3^{ψ}$ axis by an angle $θ$ , and $θ$ is defined as the fiber misalignment angle greater than 0. In the present study, the localized regions with fiber misalignment are called localized misaligned regions (also known as kink bands), and other regions without such misalignment are called non-misaligned regions. As shown in Figure 1(b), under external loads the fibers in the non-misaligned regions rotate by θ_ψ, indicating that the rotation occurs throughout the entire material rather than only within the localized misaligned regions.

Figure 1.

3D model with a kink-band plane. (a) The 3D model; (b) the rotational components in the kink-band plane.

For a given stress state ( $σ_{11}$ , $σ_{22}$ , $σ_{33}$ , $τ_{23}$ , $τ_{21}$ , $τ_{31}$ ) in the frame 123, the stress state in the frame $1^{ψ} 2^{ψ} 3^{ψ}$ is obtained by

σ_{1^{ψ} 1^{ψ}} = σ_{11}

(1a)

σ_{2^{ψ} 2^{ψ}} = \frac{σ_{22} + σ_{33}}{2} + \frac{σ_{22} - σ_{33}}{2} \cos (2 ψ) + τ_{23} \sin (2 ψ)

(1b)

σ_{3^{ψ} 3^{ψ}} = \frac{σ_{22} + σ_{33}}{2} - \frac{σ_{22} - σ_{33}}{2} \cos (2 ψ) - τ_{23} \sin (2 ψ)

(1c)

τ_{2^{ψ} 1^{ψ}} = τ_{21} \cos (ψ) + τ_{31} \sin (ψ)

(1d)

τ_{2^{ψ} 3^{ψ}} = - \frac{σ_{22} - σ_{33}}{2} \sin (2 ψ) + τ_{23} \cos (2 ψ)

(1e)

τ_{3^{ψ} 1^{ψ}} = τ_{31} \cos (ψ) - τ_{21} \sin (ψ)

(1f)

The stress state in the frame $1^{m} 2^{m} 3^{m}$ is obtained by

{σ_{1^{m}}}_{1^{m}} = \frac{{σ_{1^{ψ}}}_{1^{ψ}} + {σ_{2^{ψ}}}_{2^{ψ}}}{2} + \frac{{σ_{1^{ψ}}}_{1^{ψ}} - {σ_{2^{ψ}}}_{2^{ψ}}}{2} \cos (2 θ) + τ_{2^{ψ} 1^{ψ}} \sin (2 θ)

(2a)

σ_{2^{m} 2^{m}} = \frac{{σ_{1^{ψ}}}_{1^{ψ}} + {σ_{2^{ψ}}}_{2^{ψ}}}{2} - \frac{{σ_{1^{ψ}}}_{1^{ψ}} - {σ_{2^{ψ}}}_{2^{ψ}}}{2} \cos (2 θ) - τ_{2^{ψ} 1^{ψ}} \sin (2 θ)

(2b)

σ_{3^{m} 3^{m}} = σ_{3^{ψ} 3^{ψ}}

(2c)

τ_{2^{m} 1^{m}} = - \frac{{σ_{1^{ψ}}}_{1^{ψ}} - {σ_{2^{ψ}}}_{2^{ψ}}}{2} \sin (2 θ) + τ_{2^{ψ} 1^{ψ}} \cos (2 θ)

(2d)

τ_{2^{m} 3^{m}} = τ_{2^{ψ} 3^{ψ}} \cos (θ) - τ_{3^{ψ} 1^{ψ}} \sin (θ)

(2e)

τ_{3^{m} 1^{m}} = τ_{3^{ψ} 1^{ψ}} \cos (θ) + τ_{3^{ψ} 2^{ψ}} \sin (θ)

(2f)

Fiber misalignment angle

Previous studies^21–25 argued that the fiber misalignment angle for generic loading conditions is given by

θ = θ_{i} + θ_{m}

(3)

where

θ_{i}

is the initial misalignment angle and

θ_{m}

is an additional rotational component caused by shear stress in localized misaligned regions.

However, the rotational component in non-misaligned regions is neglected in equation (3). In fact, under generic loading conditions, there are rotational components in both misaligned and non-misaligned regions. Therefore, the present study defines the effective rotational component, $θ_{e}$ , as follows:

θ_{e} = θ_{m} - θ_{ψ}

(4)

and the fiber misalignment angle,

θ

, is given by

θ = θ_{i} + θ_{e} = θ_{i} + θ_{m} - θ_{ψ}

(5)

where

θ_{ψ}

is the rotational component in non-misaligned regions.

For a given composite, $θ_{i}$ is equal to a constant, while $γ_{m}$ and $γ_{ψ}$ depend on the constitutive law of the material. UD composites usually exhibit nonlinear responses under longitudinal shear stresses, which may be caused by the microcracks in the matrix or the plastic deformation of the matrix.³⁷ The shear constitutive law of the material can be expressed as

τ_{21} = f_{C L} (γ_{21})

(6)

Hahn and Tsai³⁸ proposed a polynomial to describe the nonlinear shear response of materials:

γ_{21} = f_{C L}^{- 1} (τ_{21}) = \frac{τ_{21}}{G_{12}} + β τ_{21}^{3}

(7)

where the parameter

β

can be obtained by fitting stress-strain test data.

Although equation (7) accounts for shear nonlinearity, it does not consider the effect of the transverse stress σ₂₂ on the shear deformation. Experimental studies by Puck and Schürmann^39,40 show that transverse stress σ₂₂ significantly influences the longitudinal shear nonlinear behavior of Glass/Epoxy composites. However, due to the lack of corresponding experimental data for other material systems, it remains challenging to establish a generalized shear stress-strain constitutive law that incorporates the effect of σ₂₂. Therefore, in the present study, equation (7) is still employed as the nonlinear constitutive law for the materials.

Following the shear constitutive law given by equation (7) and using the small angle approximation, equation (5) can be expressed as

θ = θ_{i} + θ_{m} - θ_{ψ} \approx θ_{i} + γ_{m} - γ_{ψ} = θ_{i} + \frac{τ_{2^{m} 1^{m}}}{G_{12}} + β τ_{2^{m} 1^{m}}^{3} - \frac{τ_{2^{ψ} 1^{ψ}}}{G_{12}} - β τ_{2^{ψ} 1^{ψ}}^{3}

(8)

In equation (8), the approximation is based on the small-angle assumption. Since the fiber misalignment angle and the corresponding shear strain in composites are both small, the resulting error can be neglected. This approximation is consistent with previous studies.^21–25

Substituting equation (2d) into equation (8) gives

\begin{array}{l} θ = θ_{i} + \frac{- \frac{{σ_{1^{ψ}}}_{1^{ψ}} - {σ_{2^{ψ}}}_{2^{ψ}}}{2} \sin (2 θ) + τ_{2^{ψ} 1^{ψ}} \cos (2 θ) - τ_{2^{ψ} 1^{ψ}}}{G_{12}} \\ + β [{(- \frac{{σ_{1^{ψ}}}_{1^{ψ}} - {σ_{2^{ψ}}}_{2^{ψ}}}{2} \sin (2 θ) + τ_{2^{ψ} 1^{ψ}} \cos (2 θ))}^{3} - τ_{2^{ψ} 1^{ψ}}^{3}] \end{array}

(9)

where

σ_{1^{ψ} 1^{ψ}}

σ_{2^{ψ} 2^{ψ}}

, and

τ_{2^{ψ} 1^{ψ}}

are given by equations (1a), (1b), and (1d), respectively. The fiber misalignment angle,

θ

, can be obtained by solving equation (9).

Angle of the kink-band plane

Pinho et al.²³ suggested that in the kink band, the fibers rotate consistently in the same direction forming kink-band plane, and the angle of the kink-band plane, $ψ$ , can be numerically calculated within the range of 0° to 360° to maximize the kinking failure index. This requires a complex calculation process. To simplify the calculation, Pinho et al. proposed an alternative method. They noted that the transverse tensile stress would promote the expansion of microcracks in the matrix and promote fiber misalignment, while the transverse compressive stress causes these microcracks to close and inhibits fiber misalignment. Therefore, $ψ$ can be solved by Ref. 23.

\tan (2 ψ) = \frac{2 τ_{23}}{σ_{22} - σ_{33}}

(10)

This method for determining $ψ$ has been widely adopted in the development of failure criteria and numerical simulation studies, as reported in Refs. 20,28,29,35. However, both methods proposed by Pinho et al. have limitations. Specifically, equation (10) only considers the influence of transverse stress ( $σ_{22}$ , $σ_{33}$ , $τ_{23}$ ) on fiber misalignment but ignores the influence of longitudinal shear stress $τ_{21}$ . Regarding the numerical method that determines $ψ$ by maximizing the failure index, the following example illustrates its shortcomings.

For the E-glass/MY750 composite from Ref. 41, under the combined longitudinal and transverse compressive stresses with a stress ratio of $σ_{11} / σ_{22} = 3$ , $ψ$ should be 90°, since transverse compressive stress inhibits the formation of kink bands. Nevertheless, when using numerical calculation to solve for $ψ$ that maximizes the kinking failure index, Pinho’s failure criterion predicts $ψ$ = 0° rather than 90°. This result contradicts the physical understanding that transverse compressive stress inhibits fiber misalignment. Thus, the two methods proposed by Pinho et al. are inconsistent with each other.

The reason for the contradiction is that the angle of the kink-band plane should not be calculated by the kinking failure index. Sun et al.⁷ studied the kinking failure of UD composites under longitudinal compression. In order to understand the mechanism of kink-band formation, they interrupted the compressive loading before the failure of the specimen and found kink bands could be observed in the test specimens clearly. This implies that under external loads, the formation of the kink bands and the increase in the fiber misalignment angle in UD composites represent internal deformations of the materials, which are independent of whether the materials fail. In other words, before the material fails, the kink bands may have formed. Therefore, it is unreasonable to use the kinking failure index to calculate the angle of the kink-band plane.

Based on the above analysis, the present study argues under the external loads, the localized misaligned fibers will tend to rotate toward the direction where the fiber misalignment angle is maximum. Finally, the kink bands form in the plane with the largest fiber misalignment angle.

The fiber misalignment angle can be regarded as a function of the stress state, the initial misalignment angle, and the angle of the kink-band plane:

θ = \underset{0 ° \leq ψ < 180 °}{Max} f (θ_{i}, ψ, σ_{11}, σ_{22}, σ_{33}, τ_{21}, τ_{31}, τ_{23})

(11)

For a given stress state and initial misalignment angle θ_i, the angle of kink-band plane, ψ, is incrementally varied over the range 0° ≤ ψ < 180° with a step size of 1°. For each ψ, a candidate fiber misalignment angle is computed; the maximum value over all sampled ψ is taken as the final fiber misalignment angle θ, as defined by equation (11). The corresponding ψ identifies the angle of the kink-band plane.

At this point, the theoretical model for kink-band formation has been established. For a given composite material under external loading, the fiber misalignment angle, $θ$ , can be determined using equation (9), and the angle of the kink-band plane, $ψ$ , can be obtained using equation (11). Compared to previous studies,^21–25 the established theoretical model accounts for the influence of shear deformation in the non-misaligned regions on the fiber misalignment angle and the influence of each stress component on the angle of the kink-band plane. Therefore, the model is expected to improve the accuracy of predicting the orientation of the kink bands in composites under various loading conditions.

Development of three-dimensional failure criterion

Failure modes in non-misaligned regions

Fiber tensile failure in non-misaligned regions

The present study adopts the maximum stress criterion to predict fiber tensile failure, and the failure index (FI) for fiber tensile failure in non-misaligned regions is given by

F I_{F T} = \frac{σ_{11}}{X_{T}} \leq 1

(12)

Matrix tensile/compressive failure in non-misaligned regions

The matrix failure criteria of UD composites have been studied extensively.^{21–23,40,42,43} Among them, Puck’s matrix failure criterion is widely used in engineering. However, it contains a large number of empirical parameters, which limits its further development. To address this limitation, a matrix failure criterion independent of empirical parameters has recently been proposed for brittle composites characterized by $Y_{C} / Y_{T} > \sqrt{2}$ , providing the failure indices for matrix tensile and compressive failure in non-misaligned regions⁴⁴:

\begin{array}{l} F I_{M T} = \frac{2 Y_{T} Y_{C} + 2 Y_{C}^{2} - 2 Y_{T}^{2} - Y_{T} \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{3} + Y_{T} Y_{C}^{2} + 2 Y_{T}^{2} Y_{C}} σ_{n} + \frac{3 Y_{T}^{2} - Y_{C}^{2} + Y_{T} \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{4} + Y_{T}^{2} Y_{C}^{2} + 2 Y_{T}^{3} Y_{C}} σ_{n}^{2} \\ + \frac{4 Y_{T} + 2 Y_{C} + \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{3} + Y_{T} Y_{C}^{2} + 2 Y_{T}^{2} Y_{C}} τ_{n t}^{2} + \frac{1}{S_{L}^{2}} τ_{n l}^{2} = 1, σ_{n} \geq 0 \end{array}

(13)

and

\begin{array}{l} F I_{M C} = \frac{2 Y_{T} Y_{C} + 2 Y_{C}^{2} - 2 Y_{T}^{2} - Y_{T} \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{3} + Y_{T} Y_{C}^{2} + 2 Y_{T}^{2} Y_{C}} σ_{n} + \frac{4 Y_{T} + 2 Y_{C} + \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{3} + Y_{T} Y_{C}^{2} + 2 Y_{T}^{2} Y_{C}} τ_{n t}^{2} \\ + \frac{1}{S_{L}^{2}} τ_{n l}^{2} = 1, σ_{n} < 0 \end{array}

(14)

where

Y_{T}

and

Y_{C}

are the transverse tensile and compressive strengths;

S_{L}

is the longitudinal shear strength;

σ_{n}

τ_{n t}

and

τ_{n l}

are three stress components on the potential fracture plane, which are given by

{\begin{cases} σ_{n} = σ_{22} \cos^{2} φ + σ_{33} \sin^{2} φ + 2 τ_{23} \sin φ \cos φ \\ τ_{n t} = (σ_{33} - σ_{22}) \sin φ \cos φ + τ_{23} (\cos^{2} φ - \sin^{2} φ) \\ τ_{n l} = τ_{31} \sin φ + τ_{21} \cos φ \end{cases}

(15)

where φ is the fracture angle that maximizes the failure index and can be determined through numerical search within the range of

0 ° \leq φ < 180 °

, as was done in Refs. 39,40.

Failure modes in localized misaligned regions

Experimental observations^7,45 have shown that failure in materials under high longitudinal compressive stress is caused by fiber kinking in localized misaligned regions. Based on this, Pinho et al.²³ proposed that the failure in localized misaligned regions should be considered when σ₁₁ < 0. However, their approach neglects the effect of fiber misalignment under other stress states. This limitation results in a discontinuous failure envelope when positive and negative initial fiber misalignment angles (± $θ_{i}$ ) are simultaneously considered. Figure 2 shows the $σ_{11} - τ_{21}$ failure envelope predicted by Pinho’s criterion for the $σ_{11} - τ_{21}$ stress state. To reflect the random direction of manufacturing-induced misalignment, both +θ_i and −θ_i are explicitly evaluated at every stress state with σ₁₁ < 0, and the more critical failure load is taken; for σ₁₁ ≥ 0, fiber misalignment is not considered, consistent with Ref. 23. As shown in Figure 2, a discontinuity appears at σ₁₁ = 0, which is physically impossible. This discontinuity reveals an intrinsic limitation of Pinho’s criterion when the random direction of θ_i is rigorously incorporated. To address this issue, the present study proposes that the effect of fiber misalignment should be considered under all stress states, and failure in misaligned regions and non-misaligned regions are competitive.

Figure 2.

Failure envelope of $σ_{11} - τ_{21}$ predicted by Pinho’s criterion.

Matrix tensile/compressive failure in localized misaligned regions

The failure indices for matrix tensile and compressive failure in localized misaligned regions are consistent with the forms of equations (13) and (14), respectively:

\begin{array}{l} F I_{M T}^{m} = \frac{2 Y_{T} Y_{C} + 2 Y_{C}^{2} - 2 Y_{T}^{2} - Y_{T} \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{3} + Y_{T} Y_{C}^{2} + 2 Y_{T}^{2} Y_{C}} σ_{n}^{m} + \frac{3 Y_{T}^{2} - Y_{C}^{2} + Y_{T} \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{4} + Y_{T}^{2} Y_{C}^{2} + 2 Y_{T}^{3} Y_{C}} {(σ_{n}^{m})}^{2} \\ + \frac{4 Y_{T} + 2 Y_{C} + \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{3} + Y_{T} Y_{C}^{2} + 2 Y_{T}^{2} Y_{C}} {(τ_{n t}^{m})}^{2} + \frac{1}{S_{L}^{2}} {(τ_{n l}^{m})}^{2} = 1, σ_{n}^{m} \geq 0 \end{array}

(16)

and

\begin{array}{l} F I_{M C}^{m} = \frac{2 Y_{T} Y_{C} + 2 Y_{C}^{2} - 2 Y_{T}^{2} - Y_{T} \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{3} + Y_{T} Y_{C}^{2} + 2 Y_{T}^{2} Y_{C}} σ_{n}^{m} + \frac{4 Y_{T} + 2 Y_{C} + \sqrt{12 Y_{T}^{2} + 8 Y_{T} Y_{C}}}{Y_{T}^{3} + Y_{T} Y_{C}^{2} + 2 Y_{T}^{2} Y_{C}} {(τ_{n t}^{m})}^{2} \\ + \frac{1}{S_{L}^{2}} {(τ_{n l}^{m})}^{2} = 1, σ_{n}^{m} < 0 \end{array}

(17)

where

σ_{n}^{m}

τ_{n t}^{m}

and

τ_{n l}^{m}

are given by

{\begin{cases} σ_{n}^{m} = σ_{2^{m} 2^{m}} \cos^{2} φ^{m} + σ_{3^{m} 3^{m}} \sin^{2} φ^{m} + 2 τ_{2^{m} 3^{m}} \sin φ^{m} \cos φ^{m} \\ τ_{n t}^{m} = (σ_{3^{m} 3^{m}} - σ_{2^{m} 2^{m}}) \sin φ^{m} \cos φ^{m} + τ_{2^{m} 3^{m}} (\cos^{2} φ^{m} - \sin^{2} φ^{m}) \\ τ_{n l}^{m} = τ_{3^{m} 1^{m}} \sin φ^{m} + τ_{2^{m} 1^{m}} \cos φ^{m} \end{cases}

(18)

where

φ^{m}

can be determined through numerical search within the range of

0 ° \leq φ^{m} < 180 °

. The superscript m in equations (16)–(18) represents localized misaligned regions.

Elastic instability of the matrix in localized misaligned regions

According to the studies of Pinho et al.,⁴⁶ another failure mode in localized misaligned regions is the elastic instability of the fibers in the localized misaligned regions, which occurs only in the presence of longitudinal compressive stress. To characterize this failure mode, the following expressions are obtained from equations (2d) and (7):

τ_{2^{m} 1^{m}} = - \frac{{σ_{1^{ψ}}}_{1^{ψ}} - {σ_{2^{ψ}}}_{2^{ψ}}}{2} \sin [2 (θ_{i} + γ_{m} - γ_{ψ})] + τ_{2^{ψ} 1^{ψ}} \cos [2 (θ_{i} + γ_{m} - γ_{ψ})]

(19)

and

γ_{m} = \frac{τ_{2^{m} 1^{m}}}{G_{12}} + β τ_{2^{m} 1^{m}}^{3}

(20)

Equation (19) is the shear stress in the localized misaligned region induced by longitudinal compressive stress, and equation (20) is the shear stress–strain constitutive law. The elastic instability mode is characterized by the relationship between these two curves, and the corresponding failure condition is determined from the following analysis. As shown in Figure 3(a), the curve given by equation (19) represents the shear stress in localized misaligned regions induced by longitudinal compressive stress of $σ_{11} = - X_{C} / 2$ , and the curve given by equation (20) represents the shear stress-strain material law. The intersection of two curves represents the equilibrium position. As the compressive stress increases, the curve of equation (19) shifts up until it is tangent to the curve of equation (20) in Figure 3(b) when $σ_{11} = - X_{C}$ . It is assumed that the matrix in the localized misaligned regions has not failed when the two curves are tangent. Then, a slight increase in compressive load will result in the two curves not touching each other. From a physical point of view, this leads to unstable rotation of the fibers and causes eventual failure. Therefore, in this case, the failure is caused by the elastic instability of the fiber, rather than the matrix failure in localized misaligned regions.

Figure 3.

The curve of equations (19) and (20). (a) $σ_{11} = - X_{C} / 2$ ; (b) $σ_{11} = - X_{C}$ .

When elastic instability occurs, the curve of equation (19) intersects the curve of equation (20) and they have the same slope at the intersection point. Hence, the failure stresses corresponding to the elastic instability can be solved by

{\begin{cases} f_{C L} (γ_{m}) = - \frac{{σ_{1^{ψ}}}_{1^{ψ}} - {σ_{2^{ψ}}}_{2^{ψ}}}{2} \sin [2 (θ_{i} + γ_{m} - γ_{ψ})] + τ_{2^{ψ} 1^{ψ}} \cos [2 (θ_{i} + γ_{m} - γ_{ψ})] \\ \frac{\partial f_{C L} (γ_{m})}{\partial γ_{m}} = - ({σ_{1^{ψ}}}_{1^{ψ}} - {σ_{2^{ψ}}}_{2^{ψ}}) \cos [2 (θ_{i} + γ_{m} - γ_{ψ})] - 2 τ_{2^{ψ} 1^{ψ}} \sin [2 (θ_{i} + γ_{m} - γ_{ψ})] \end{cases}

(21)

where the function of the shear constitutive law,

f_{C L} (γ_{m})

, is given by equation (7). It is difficult to obtain the analytical solution of equation (21) directly, so a numerical calculation process is used to solve it as per Ref. 46.

Correction of basic strengths

The strength is defined as a single stress that can cause failure in the absence of other stresses. The proposed failure criteria contain four strengths, i.e., $Y_{T}$ , $Y_{C}$ , $S_{L}$ , and $X_{T}$ , which need to be measured experimentally. Nevertheless, the strengths measured experimentally may not be “true”. For example, under pure longitudinal shear stress, if the failure occurs first in the non-misaligned regions, the experimentally determined longitudinal shear strength is considered “true”. If the failure occurs first in the localized misaligned regions due to the transverse tensile stress $σ_{2^{m} 2^{m}}$ and the longitudinal shear stress $τ_{2^{m} 1^{m}}$ , and the longitudinal shear strength measured experimentally is not “true” and requires correction. In the paper, $Y_{T}^{*}$ , $Y_{C}^{*}$ , $S_{L}^{*}$ and $X_{T}^{*}$ are defined as the strengths measured experimentally, and $Y_{T}$ , $Y_{C}$ , $S_{L}$ , and $X_{T}$ are defined as the “true” strengths.

For ease of subsequent derivation, the parameter R is defined as.

R = \frac{\sqrt{3 Y_{T}^{2} + 2 Y_{T} Y_{C}} - Y_{T}}{2}

(22)

The parameter R is determined by the transverse tensile and compressive strengths. For brittle composites where $Y_{C} / Y_{T} > \sqrt{2}$ , the range of R is $0 < {R < Y}_{C} / 2$ .

Equations (13), (14), (16) and (17) can be rewritten as

F I_{M T} = \frac{Y_{C} - 2 R}{R^{2}} σ_{n} + \frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} σ_{n}^{2} + \frac{1}{R^{2}} τ_{n t}^{2} + \frac{1}{S_{L}^{2}} τ_{n l}^{2} = 1, σ_{n} \geq 0

(23)

F I_{M C} = \frac{Y_{C} - 2 R}{R^{2}} σ_{n} + \frac{1}{R^{2}} τ_{n t}^{2} + \frac{1}{S_{L}^{2}} τ_{n l}^{2} = 1, σ_{n} < 0

(24)

F I_{M T}^{m} = \frac{Y_{C} - 2 R}{R^{2}} σ_{n}^{m} + \frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} {(σ_{n}^{m})}^{2} + \frac{1}{R^{2}} {(τ_{n t}^{m})}^{2} + \frac{1}{S_{L}^{2}} {(τ_{n l}^{m})}^{2} = 1, σ_{n}^{m} \geq 0

(25)

F I_{M C}^{m} = \frac{Y_{C} - 2 R}{R^{2}} σ_{n}^{m} + \frac{1}{R^{2}} {(τ_{n t}^{m})}^{2} + \frac{1}{S_{L}^{2}} {(τ_{n l}^{m})}^{2} = 1, σ_{n}^{m} < 0

(26)

The transverse tensile strength

Under uniaxial transverse tension, the failure of the matrix may occur in both the localized misaligned regions and the non-misaligned regions. The failure region can be determined by comparing the magnitude of the failure indices.

For non-misaligned regions, under uniaxial transverse tensile stress $σ_{22}$ , the stresses acting on an inclined plane are given by equation (15):

{\begin{cases} σ_{n} = σ_{22} \cos^{2} φ \\ τ_{n t} = - σ_{22} \sin φ \cos φ \\ τ_{n l} = 0 \end{cases}

(27)

In the stress state given by equation (27), the matrix tensile failure index of non-misaligned regions, ${F I}_{M T}$ , reaches the maximum when $φ = 0 °$ .⁴⁴ Substituting equation (27) and $φ = 0 °$ into equation (23) gives

F I_{M T} = \frac{Y_{C} - 2 R}{R^{2}} σ_{22}

(28)

For localized misaligned regions, under uniaxial transverse tensile stress $σ_{22}$ , the angle of the kink-band plane by solving equation (11) is 0°. The stress state of the matrix in localized misaligned regions is given by equations (1a)–(1f) and (2a)–(2f):

{\begin{cases} σ_{2^{m} 2^{m}} = \frac{σ_{22}}{2} + \frac{σ_{22}}{2} \cos (2 θ) \\ τ_{2^{m} 1^{m}} = - \frac{σ_{22}}{2} \sin (2 θ) \end{cases}

(29)

Under the combination of transverse tensile stress $σ_{2^{m} 2^{m}}$ and longitudinal shear stress $τ_{2^{m} 1^{m}}$ , the plane with $φ^{m} = 0 °$ is most likely to fail.⁴⁴ The stresses acting on the potential fracture plane are obtained by substituting equation (29) and $φ^{m} = 0 °$ into equation (18):

{\begin{cases} σ_{n}^{m} = σ_{22} \cos^{2} θ \\ τ_{n t}^{m} = 0 \\ τ_{n l}^{m} = - σ_{22} \sin θ \cos θ \end{cases}

(30)

The matrix tensile failure index of localized misaligned regions, ${F I}_{M T}^{m}$ , is obtained by substituting equation (30) into equation (25):

F I_{M T}^{m} = \frac{Y_{C} - 2 R}{R^{2}} σ_{22} \cos^{2} θ + \frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} {(σ_{22} \cos^{2} θ)}^{2} + \frac{1}{S_{L}^{2}} {(σ_{22} \sin θ \cos θ)}^{2}

(31)

The failure region can be determined by comparing equations (28) and (31):

\begin{array}{l} F I_{M T} - F I_{M T}^{m} = \frac{Y_{C} - 2 R}{R^{2}} σ_{22} - \frac{Y_{C} - 2 R}{R^{2}} σ_{22} \cos^{2} θ \\ - \frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} {(σ_{22} \cos^{2} θ)}^{2} - \frac{1}{S_{L}^{2}} {(σ_{22} \sin θ \cos θ)}^{2} \end{array}

(32)

Equation (32) can be rewritten as

F I_{M T} - F I_{M T}^{m} = (\frac{Y_{C} - 2 R}{σ_{22}} - \frac{Y_{C} - 2 R}{Y_{T}} + 1 - \frac{R^{2}}{S_{L}^{2}}) \frac{1}{R^{2}} σ_{22}^{2} \sin^{2} θ + (\frac{Y_{C} - 2 R}{2 Y_{T}} + \frac{R^{2}}{S_{L}^{2}} - \frac{1}{2}) \frac{σ_{22}^{2}}{R^{2}} \sin^{4} θ

(33)

The initial misalignment angle of the material generally does not exceed a few degrees. Under transverse tensile stress of $σ_{22} = Y_{T}^{*}$ , the fiber misalignment angle hardly changes, i.e., ${θ \approx θ}_{i}$ . Applying ${θ \approx θ}_{i}$ and $σ_{22} = Y_{T}^{*}$ to equation (33) yields

\begin{array}{l} F I_{M T} - F I_{M T}^{m} \approx (\frac{Y_{C} - 2 R}{Y_{T}^{*}} - \frac{Y_{C} - 2 R}{Y_{T}} + 1 - \frac{R^{2}}{S_{L}^{2}}) \frac{1}{R^{2}} σ_{22}^{2} \sin^{2} θ_{i} \\ + (\frac{Y_{C} - 2 R}{2 Y_{T}} + \frac{R^{2}}{S_{L}^{2}} - \frac{1}{2}) \frac{σ_{22}^{2}}{R^{2}} \sin^{4} θ_{i} \end{array}

(34)

Since $\sin θ_{i}$ is a value close to 0, the term of $\sin^{4} θ_{i}$ can be ignored in equation (34):

F I_{M T} - F I_{M T}^{m} \approx (\frac{Y_{C} - 2 R}{Y_{T}^{*}} - \frac{Y_{C} - 2 R}{Y_{T}} + 1 - \frac{R^{2}}{S_{L}^{2}}) \frac{1}{R^{2}} σ_{22}^{2} \sin^{2} θ_{i}

(35)

For a large number of composites in Refs. 41,47,48, the values of parameter R, solved by using equation (22), are all lower than the longitudinal shear strength, i.e., $R < S_{L}$ . Since $R < S_{L}$ , ${R < Y}_{C} / 2$ and $Y_{T}^{*} \leq Y_{T}$ , we have

F I_{M T} - F I_{M T}^{m} > 0

(36)

Equation (36) means that for composites where $R < S_{L}$ , the matrix in the non-misaligned regions fails first under uniaxial transverse tension, and thus $Y_{T} = Y_{T}^{*}$ . The conclusion is applicable to the vast majority of brittle composites. However, for materials with $R \geq S_{L}$ , this conclusion may not be applicable and $Y_{T}$ can be determined using the following method.

It is assumed that the matrix in the non-misaligned regions fails first under uniaxial transverse tension and $Y_{T} = Y_{T}^{*}$ . Substituting $Y_{T} = Y_{T}^{*}$ and $R \geq S_{L}$ into equation (35) gives

F I_{M T} - F I_{M T}^{m} \leq 0

(37)

Equation (37) means that the matrix in the localized misaligned regions fails first under uniaxial transverse tension, which contradicts the initial assumption. Hence, the matrix in the localized misaligned regions must be the first to fail under uniaxial transverse tension, i.e., ${F I}_{M T}^{m} = 1$ when $σ_{22} = Y_{T}^{*}$ . Substituting $σ_{22} = Y_{T}^{*}$ and ${θ \approx θ}_{i}$ into equation (31) obtains

F I_{M T}^{m} \approx \frac{Y_{C} - 2 R}{R^{2}} Y_{T}^{*} \cos^{2} θ_{i} + \frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} {(Y_{T}^{*} \cos^{2} θ_{i})}^{2} + \frac{1}{S_{L}^{2}} {(Y_{T}^{*} \sin θ_{i} \cos θ_{i})}^{2} = 1

(38)

The true transverse tensile strength, $Y_{T}$ , can be solved by equation (38).

The transverse compressive strength

Under uniaxial transverse compressive stress, the angle of the kink-band plane by solving equation (11) is 90°. Hence, the stress state of the local misaligned regions and the non-misaligned regions is identical, and $Y_{C} = Y_{C}^{*}$ .

The longitudinal shear strength

Under pure longitudinal shear stress $τ_{21}$ , the matrix tensile failure index of non-misaligned regions, ${F I}_{M T}$ , reaches the maximum when $φ = 0 °$ , and equation (23) becomes

F I_{M T} = \frac{1}{S_{L}^{2}} τ_{21}^{2}

(39)

For localized misaligned regions, under pure longitudinal shear stress, the angle of the kink-band plane is 180° by solving equation (11). The stress state in localized misaligned regions is given by

{\begin{cases} σ_{2^{m} 2^{m}} = τ_{21} \sin (2 θ) \\ τ_{2^{m} 1^{m}} = - τ_{21} \cos (2 θ) \end{cases}

(40)

Under the combination of transverse tensile stress $σ_{2^{m} 2^{m}}$ and longitudinal shear stress $τ_{2^{m} 1^{m}}$ , the plane with $φ^{m} = 0 °$ is most likely to fail.⁴⁴ The stresses acting on the potential fracture plane are solved by substituting equation (38) and $φ^{m} = 0 °$ into equation (18):

{\begin{cases} σ_{n}^{m} = τ_{21} \sin (2 θ) \\ τ_{n t}^{m} = 0 \\ τ_{n l}^{m} = - τ_{21} \cos (2 θ) \end{cases}

(41)

Substituting equation (41) into equation (25) to obtain the matrix tensile failure index of localized misaligned regions:

F I_{M T}^{m} = \frac{Y_{C} - 2 R}{R^{2}} τ_{21} \sin (2 θ) + \frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} {[τ_{21} \sin (2 θ)]}^{2} + \frac{1}{S_{L}^{2}} {[τ_{21} \cos (2 θ)]}^{2}

(42)

The failure region can be determined by comparing equations (39) and (42):

F I_{M T} - F I_{M T}^{m} = \frac{1}{S_{L}^{2}} τ_{21}^{2} - \frac{Y_{C} - 2 R}{R^{2}} τ_{21} \sin (2 θ) - \frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} {[τ_{21} \sin (2 θ)]}^{2} - \frac{1}{S_{L}^{2}} {[τ_{21} \cos (2 θ)]}^{2}

(43)

Equation (43) can be rewritten as

F I_{M T} - F I_{M T}^{m} = - \frac{Y_{C} - 2 R}{R^{2}} τ_{21} \sin (2 θ) - (\frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} - \frac{1}{S_{L}^{2}}) τ_{21}^{2} \sin^{2} (2 θ)

(44)

Under pure longitudinal shear stress, the fiber misalignment angle hardly changes, i.e., ${θ \approx θ}_{i}$ . Applying ${θ \approx θ}_{i}$ to equation (44) yields

F I_{M T} - F I_{M T}^{m} \approx - \frac{Y_{C} - 2 R}{R^{2}} τ_{21} \sin (2 θ_{i}) - (\frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} - \frac{1}{S_{L}^{2}}) τ_{21}^{2} \sin^{2} (2 θ_{i})

(45)

Since $\sin 2 θ_{i}$ is a value close to 0, the term of $\sin^{2} (2 θ_{i})$ can be ignored in equation (45):

F I_{M T} - F I_{M T}^{m} \approx - \frac{Y_{C} - 2 R}{R^{2}} τ_{21} \sin (2 θ_{i})

(46)

Since ${R < Y}_{C} / 2$ , we have

F I_{M T} - F I_{M T}^{m} < 0

(47)

Equation (47) means that the matrix in localized misaligned regions fails first under pure longitudinal shear stress, and thus $S_{L} > S_{L}^{*}$ .

When $τ_{21} = S_{L}^{*}$ , ${θ \approx θ}_{i}$ and the matrix tensile failure index of localized misaligned regions is given by equation (42):

F I_{M T}^{m} \approx \frac{Y_{C} - 2 R}{R^{2}} S_{L}^{*} \sin (2 θ_{i}) + \frac{Y_{T} - Y_{C} + 2 R}{2 Y_{T} R^{2}} {[S_{L}^{*} \sin (2 θ_{i})]}^{2} + \frac{1}{S_{L}^{2}} {[S_{L}^{*} \cos (2 θ_{i})]}^{2} = 1

(48)

The true longitudinal shear strength, $S_{L}$ , can be solved by equation (48).

The longitudinal tensile strength

Under uniaxial longitudinal tension, it is difficult to determine the failure region by solving

{F I}_{M T} ‐ {F I}_{M T}^{m}

, so the following method is adopted. For the material parameters in Table 1, the proposed criterion indicates that the longitudinal tensile stress required to initiate matrix failure in localized misaligned regions is considerably higher than the longitudinal tensile strength

X_{T}^{*}

. This implies that fiber failure in non-misaligned regions would occur earlier, and therefore

X_{T} = X_{T}^{*}

Table 1.

Material parameters of composites.

Materials	E-glass MY750⁴¹	A-S epoxy⁴¹	T300 BSL914C⁴⁸	AS4 3501-6⁴⁹	IM7 8552⁵⁰
$X_{T}$ (MPa)	1280	1990	1500	2300	2650
$X_{C}^{*}$ (MPa)	800	1500	900	1725	1725
$Y_{T}$ (MPa)	40	38	27	60.2	76.4
$Y_{C}$ (MPa)	145	150	200	275.8	288
$S_{L}$ (MPa)	83.9	73.1	110.4	76.7	91.7
$S_{L}^{*}$ (MPa)	73	70	80	73.4	89
$θ_{i}$ (°)	3.52	1.05	3.33	1.55	1.24
G₂₁ (MPa)	5830	6700	5500	6600	5600
β (MPa⁻³)	5 × 10⁻⁸	6.1 × 10⁻⁸	3.6 × 10⁻⁸	1.4 × 10⁻⁸	2.1 × 10⁻⁸

In Table 1, it should be mentioned that since the failure under longitudinal compression is caused by the matrix failure or elastic instability in localized misaligned regions, the longitudinal compressive strength measured by the experiment, $X_{C}^{*}$ , is the longitudinal compressive stress that causes the failure of localized misaligned regions, rather than the true longitudinal compressive strength. In the present study, the true longitudinal compressive strength, $X_{C}$ , is defined as the longitudinal compressive stress leading to fiber compressive failure in the absence of other stresses, which is difficult to obtain through experiments.

Determination of initial fiber misalignment angle

The initial misalignment angle, $θ_{i}$ , can be calibrated by longitudinal compressive strength $X_{C}^{*}$ measured by the experiments. Under longitudinal compression, two potential failure modes may occur: the matrix compressive failure in localized misaligned regions (as defined by equation (17)), and the elastic instability of the fiber in localized misaligned regions (as defined by equation (21)). Both failure modes are driven by the longitudinal shear stress $τ_{n l}^{m}$ . The failure mode under longitudinal compression and the magnitude of the initial misalignment angle can be determined by the following iterative calculation.

Figure 4 shows the curves of equations (19) and (20) under longitudinal compressive stress of $σ_{11} = - X_{C}^{*}$ . $θ_{i}$ is set to increase from 0°. As $θ_{i}$ increases, the curve of equation (19) shifts up until it becomes tangent to the curve of equation (20), and the first intersection of the two curves is the equilibrium point. If the matrix failure in the localized misaligned regions occurs before the curves become tangent (see Figure 4(a)), the failure mode is the matrix compressive failure in localized misaligned regions, caused by $τ_{2^{m} 1^{m}}$ , and $θ_{i}$ is determined accordingly. In contrast, if the matrix in the localized misaligned regions does not fail when the two curves are tangent (see Figure 4(b)), the failure mode is the elastic instability of the fiber, and $θ_{i}$ is determined by equation (21).

Figure 4.

The curves of equations (19) and (20) when $σ_{11} = ‐ X_{C}^{*}$ . (a) Matrix failure in the localized misaligned regions; (b) elastic instability of the fiber in the localized misaligned regions.

At this point, a three-dimensional failure criterion for UD composites is established. For a given composite, the basic strengths $Y_{C}$ and $X_{T}$ can be measured directly by the experiments; $Y_{T}$ is equal to $Y_{T}^{*}$ when $R < S_{L}$ , and is determined by equation (38) when $R \geq S_{L}$ ; $S_{L}$ is solved by equation (48). Under a given stress state, the angle of the kink-band plane, $ψ$ , is given by equation (11), and the fiber misalignment angle, $θ$ , is obtained by solving equation (9). After determining $ψ$ and $θ$ , the stress states in localized misaligned regions can be given by equations (2a)–(2f). Finally, the failure modes can be determined using the failure indices specified in equations (12)–(14), (16), and (17), along with the function for elastic instability provided in equation (21).

Failure mechanism analysis

This section applies the proposed criterion to analyze the failure of the material under various stress states. The proposed criterion includes six failure modes: fiber tensile failure in the non-misaligned regions (FT), matrix tensile failure in the non-misaligned regions (MT), matrix compressive failure in the non-misaligned regions (MC), matrix tensile failure in the localized misaligned regions (MT^m), matrix compressive failure in the localized misaligned regions (MC^m), and elastic instability of the fiber in the localized misaligned regions (EI).

$σ_{11} - τ_{21}$ failure evaluation

Under the $σ_{11} - τ_{21}$ stress states, the stress state on the kink-band plane exhibits symmetry, i.e., the stress state on the plane with the angle ψ is identical to that on the plane with the angle 360° − ψ. Therefore, it is only necessary to consider ψ within the range of 0° to 180°. Figure 5(a) shows the $σ_{11} - τ_{21}$ failure envelope and the corresponding failure modes for T300/BSL914C, and Figure 5(b) and (c) show the trends of $ψ$ and $θ$ with $σ_{11}$ .

(1) In Figure 5(a), point P₁ corresponds to longitudinal tensile failure. As the stress state transitions along the envelope from point P₁ to P₂, $τ_{21}$ increases while $σ_{11}$ remains constant, and the failure mode is FT. During this transition, it can be observed from Figure 5(b) and (c) that ψ sharply increases, while θ changes very little. Since the longitudinal tensile stress causes the fiber misalignment angle θ to decrease, θ becomes less than the initial fiber misalignment angle $θ_{i}$ . In addition, an interesting observation is that $τ_{21}$ can exceed the experimentally measured longitudinal shear strength $S_{L}^{*}$ . This is because the failure mode under pure longitudinal shear stress ( $τ_{21} = S_{L}^{*}$ ) is MT^m. High longitudinal tensile stress decreases the fiber misalignment angle, thus reducing the likelihood of the occurrence of MT^m. Consequently, under the combination of high longitudinal tensile stress and longitudinal shear stress, $τ_{21}$ can exceed $S_{L}^{*}$ .

(2) As the stress state transitions along the envelope from point P₂ to P₃, $τ_{21}$ initially varies slightly, then decreases rapidly when $σ_{11}$ approaches 0, and increases gradually when $σ_{11} < 0$ . The corresponding failure mode is MT^m. As illustrated in Figure 5(b) and (c), ψ rapidly increases to 180° and θ increases slowly. It should be mentioned that under the combination of low longitudinal compressive stress and longitudinal shear stress, matrix compressive stress in localized misaligned regions inhibits the occurrence of MT^m. Hence, $τ_{21}$ can exceed $S_{L}^{*}$ and reach $S_{L}$ at point P₃. In the stress state corresponding to point P₃, θ on the plane with ψ = 0° is equal to that on the plane with ψ = 180°, and thus there are two points related to P₃ in Figure 5(b).

(3) As the stress state transitions along the envelope from point P₃ to P₄, the magnitudes of the longitudinal compressive stress, $| σ_{11} |$ , gradually increase, and shear stresses $τ_{21}$ remains equal to $S_{L}$ . The related failure mode is MT. It can be seen from Figure 5(b) and (c) that ψ is equal to 0°, and θ rapidly increases. Since the matrix compressive stress in localized misaligned regions inhibits the occurrence of MT^m, the failure mode is MT caused by $τ_{21}$ , and $τ_{21} = S_{L}$ .

(4) As the stress state transitions along the envelope from point P₄ to P₅, the magnitude of the longitudinal compressive stress, $| σ_{11} |$ , increases, and $τ_{21}$ decreases. The related failure mode is EI. It can be observed from Figure 5(b) and (c) that ψ equals 0° and θ remains constant.

Figure 5.

Evaluation for T300/BSL914 C under $σ_{11} - τ_{21}$ stress states. (a) $σ_{11} - τ_{21}$ failure envelope considering fiber misalignment. (b) Trend of $ψ$ with $σ_{11}$ . (c) Trend of $θ$ with $σ_{11}$ . (d) $σ_{11} - τ_{21}$ failure envelope without considering fiber misalignment.

To sum up, under the $σ_{11} ‐ τ_{21}$ stress states, four possible failure modes (FT, MT^m, MT, and EI) can occur. In addition, it has been observed from Figure 5 that under superimposed longitudinal tensile stress or moderate longitudinal compressive stress, the longitudinal shear stress $τ_{21}$ at failure can exceed the experimentally measured longitudinal shear strength $S_{L}^{*}$ , resulting in a concave shape in the failure envelope. Although this behavior may seem counterintuitive, it actually reflects the complex influence mechanism of localized fiber misalignment on failure. As the stress $σ_{11}$ approaches zero from negative values, the angle ψ, reflecting the orientation of the kink-band formation, transitions from 0° to 180° (see Figure 5(b)). Meanwhile, $σ_{1^{m} 1^{m}}$ and $σ_{2^{m} 2^{m}}$ change from compressive stresses to tensile stresses (as shown in Figure 6), which promotes failure at lower shear stress levels, thereby leading to a local concave region of the envelope. If the effect of fiber misalignment is neglected, the failure envelope would follow the ideal convex shape shown in Figure 5(d). A similar concave trend is also observed in Pinho’s criterion (as shown in Figure 2); however, Pinho’s criterion exhibits discontinuities at $σ_{11} = 0$ . In contrast, the present criterion considers the effect of fiber misalignment under all stress states, thereby avoiding discontinuities in the failure envelope.

Figure 6.

The stress states in localized misaligned regions. (a) ψ = 0°; (b) ψ = 180°.

Vogler and Kyriakides² carried out tests on AS4/PEEK under the combination of longitudinal shear and compressive stresses. As depicted in Figure 7, the failure mode under high longitudinal compressive stress is different from that under low longitudinal compressive stress. Under high longitudinal compressive stress, fiber breakage occurs at the upper and lower edges of the localized kink bands; while under low longitudinal compressive stress, very few of the fibers inside the localized kink bands are broken. In addition, the fiber misalignment angle under high longitudinal compressive stress is greater than that under low longitudinal compressive stress. In order to distinguish between these two failure modes, Pinho et al.²⁶ proposed that for all types of composite materials, when $σ_{11} \leq - X_{C} / 2$ , fiber kinking failure occurs; when $σ_{11} > - X_{C} / 2$ , the matrix failure in the localized misaligned regions results in fiber splitting. However, this approach is only applicable to the $σ_{11} - τ_{21}$ stress state.

Figure 7.

The localized kink bands under (a) high longitudinal compressive stress, (b) low longitudinal compressive stress.²

Vogler and Kyriakides² suggest that the reason why the fibers inside the kink bands do not break under low longitudinal compression stress is due to the smaller fiber misalignment angle. In addition, according to the experimental observations by Sun et al.,⁷ fibers will break if the fiber misalignment angle exceeds a certain threshold. Therefore, the present study suggests that if the fiber misalignment angle, $θ$ , exceeds a critical value $θ_{c v}$ , fiber breakage will occur in the kink bands, resulting in fiber kinking. Conversely, if $θ$ remains below $θ_{c v}$ , fibers will not exhibit breakage. The critical value, $θ_{c v}$ , is taken as a material property. The studies of Sun et al. show that $θ$ in kink bands is about 20.9∼22.5° for CFRP. Hence, $θ_{c v}$ is assumed to be equal to 22° in the absence of relevant test data.

In the $σ_{11} - τ_{21}$ stress states, Figure 5(a) illustrates that the failure mode is EI when $σ_{11} \leq - 0.45 X_{C}$ , which implies unstable rotation of the fibers, and thus $θ$ can reach $θ_{c v}$ , ultimately resulting in fiber kinking. While the failure mode is MT or MT^m when $0 > σ_{11} > - 0.45 X_{C}$ , and $θ$ is below $θ_{c v}$ , ultimately leading to fiber splitting. Therefore, in the $σ_{11} - τ_{21}$ stress states, localized kink bands with fiber breakage can be observed only under moderate or high longitudinal compressive stress in the experiments. In addition, as shown in Figure 5(c), when the failure mode is EI, unstable rotation of the fibers occurs, leading to eventual failure when $θ = θ_{c v}$ . Therefore, θ remains equal to $θ_{c v}$ in the process from point P₄ to P₅.

$σ_{11} - σ_{22}$ failure evaluation

Figure 8 shows the failure envelope for T300/BSL914 C under the $σ_{11} - σ_{22}$ stress states, along with the trends of $ψ$ and $θ$ with $σ_{11}$ . Under the $σ_{11} - σ_{22}$ stress states, all six failure modes (FT, MT, MT^m, MC, MC^m and EI) can occur. As shown in Figure 8(b), $ψ = 0 °$ when $σ_{22} \geq 0$ , and $ψ = 90 °$ when $σ_{22} < 0$ . This prediction is consistent with the views of Pinho et al., which suggest that the transverse tensile stress promotes fiber misalignment, while the transverse compressive stress inhibits it.²³

Figure 8.

Evaluation for T300/BSL914 C under $σ_{11} - σ_{22}$ stress states. (a) $σ_{11} - σ_{22}$ failure envelope. (b) Trend of $ψ$ with $σ_{11}$ . (c) Trend of $θ$ with $σ_{11}$ .

As shown in Figure 8(c), since transverse tensile stress increases $θ$ , for the same $σ_{11}$ , $θ$ when $σ_{22} \geq 0$ is slightly greater than $θ$ when $σ_{22} < 0$ . Compared to longitudinal tensile/compressive stresses, transverse tensile/compressive stresses have a relatively minor effect on $θ$ . Therefore, the value of $θ$ is primarily determined by $σ_{11}$ . It is worth noting that, in Figure 8(c), when the failure mode is EI, $θ$ represents the fiber misalignment angle at the onset of elastic instability in the localized misaligned region, which is obtained by solving equation (21).

An interesting finding is that in Figure 8, EI occurs only when $σ_{11} \leq - 0.98 X_{C}$ , leading to fiber kinking, which is different from the predicted results under the $σ_{11} - τ_{21}$ stress states. Experimental results for T800/924 indicate that fiber kink bands are observed only when $σ_{11} \leq - X_{C}$ under $σ_{11} - σ_{22}$ stress states. Meanwhile, experimental results for T300/LY556-HY917-DY070 suggest that fiber kinking failure occurs only when $σ_{11} < - X_{C} / 2$ under $σ_{11} - τ_{21}$ stress states.²⁶ Therefore, compared to Pinho’s criterion, the proposed failure model provides accurate predictions for fiber kinking failure caused by EI under both the $σ_{11} - σ_{22}$ and $σ_{11} - τ_{21}$ stress states.

$σ_{22} - τ_{21}$ failure evaluation

Under the $σ_{22} - τ_{21}$ stress states, four possible failure modes (MT, MT^m, MC, and MC^m) can occur. As shown in Figure 9(a), for a majority of the $σ_{22} - τ_{21}$ stress states on the failure envelope, failure initiates in localized misaligned regions; only when the stress state approaches transverse tensile or compressive stress state, failure initiates in non-misaligned regions.

Figure 9.

Evaluation for T300/BSL914C under $σ_{22} - τ_{21}$ stress states. (a) $σ_{22} - τ_{21}$ failure envelope. (b) Trend of $ψ$ with $σ_{22}$ . (c) Trend of $θ$ with $σ_{22}$ .

Under the $σ_{22} - τ_{21}$ stress states, equation (10) proposed by Pinho et al.²³ predicts that $ψ = 0 °$ or $ψ = 180 °$ when $σ_{22} \geq 0$ , and $ψ = 90 °$ when $σ_{22} < 0$ . However, equation (10) ignores the effects of $τ_{21}$ . As shown in Figure 9(b), equation (11) proposed by the present study considers the effects of $τ_{21}$ . It predicts that $ψ$ changes continuously with changes in $σ_{22}$ , and as $σ_{22}$ decreases, $ψ$ reduces from 180° to 90°.

As shown in Figure 9(c), the change in $θ$ under the $σ_{22} - τ_{21}$ stress states is slight. This is because the transverse tensile/compressive strength of the composites is significantly lower than their longitudinal tensile/compressive strength. In addition, the shear strains $γ_{m}$ and $γ_{ψ}$ caused by $τ_{21}$ are close to each other, and thus, the effect of $τ_{21}$ on $θ$ is slight. In particular, under transverse compressive stress corresponding to point P₅, $θ$ is equal to the initial fiber misalignment angle $θ_{i}$ .

Assessment of the proposed failure criterion

In order to verify the applicability of the proposed failure criterion, its predictions are compared with those of Hashin’s criterion, Puck’s criterion, and Pinho’s criterion, as well as with experimental data in Figure 10. The experimental data used in this section are mostly sourced from the WWFE (world-wide failure exercise),^51,52 and the material properties are listed in Table 1.

Figure 10.

Comparison between predicted results and experiments under (a) $σ_{11} - τ_{21}$ stress states for T300/BSL914C,⁵¹ (b) $σ_{11} - σ_{22}$ stress states for E-glass/MY750,⁵¹ (c) $σ_{22} - τ_{21}$ stress states for IM7/8552,⁵⁰ (d) off-axis tension for AS4/3501-6,⁴⁹ (e) $σ_{11} - σ_{22} {= σ}_{33}$ stress states for A-S/epoxy,⁵² (f) $σ_{22} - σ_{11} {= σ}_{33}$ stress states for E-glass/MY750.⁵²

To facilitate a more intuitive evaluation of the predictive accuracy of different failure criteria, Table 2 presents a comparison of the average prediction errors for four criteria against the experimental results shown in Figure 10 (Detailed data can be found in Table A.1–A.6 in Appendix A). Among the six loading conditions, the present criterion outperforms the other three in four cases (the

σ_{11} ‐ τ_{21}

test, the

σ_{22} ‐ τ_{21}

test, the off-axis tension test and

σ_{11} - σ_{22} {= σ}_{33}

test). In the

σ_{11} - σ_{22}

test, Puck’s criterion yields the lowest prediction error, while the present criterion shows slightly higher errors, primarily due to its conservative predictions under combined longitudinal and transverse tensile stresses. In the

σ_{22} - σ_{11} {= σ}_{33}

stress states, Hashin’s and Puck’s criteria yield lower errors, while the prediction errors of the present criterion and the Pinho criterion are slightly larger. Overall, the present criterion provides generally reliable and accurate predictive performance.

Table 2.

Comparison of average errors of predicted results.

Materials	Present	Puck	Hashin	Pinho
T300/BSL914C ( $σ_{11} - τ_{21}$ test)	7.18%	7.51%	7.94%	7.71%
E-glass/MY750 ( $σ_{11} - σ_{22}$ test)	3.1%	2.95%	3.36%	3.36%
IM7/8552 ( $σ_{22} - τ_{21}$ test)	3.63%	5.96%	9.18%	6.94%
AS4/3501-6 (off-axis tension)	1.77%	2.93%	4.4%	4.4%
A-S/epoxy ( $σ_{11} - σ_{22} {= σ}_{33}$ test)	11%	11.3%	14.88%	11.17%
A-S/epoxy ( $σ_{22} - σ_{11} {= σ}_{33}$ test)	8.4%	6.86%	5.99%	10.8%

Failure evaluation under $σ_{11} - τ_{21}$ stress states

The tubes made from prepreg T300/BSL914 C carbon/epoxy were tested under combined axial tension or compression and torsion.⁵¹ Figure 10(a) shows the test data and the $σ_{11} - τ_{21}$ failure envelopes of four failure criteria. Under high longitudinal tensile stress, the experimental results indicate that $τ_{21}$ can exceed the experimentally measured longitudinal shear strength $S_{L}^{*}$ . According to the present failure criterion, this is because the longitudinal tensile stress reduces θ, thus inhibiting the occurrence of MT^m, whereas Hashin’s, Puck’s and Pinho’s criteria cannot provide a reasonable explanation for this experimental phenomenon. Under low longitudinal tensile stress, the experiments show that the longitudinal shear failure stress is lower than $S_{L}^{*}$ , but the present criterion cannot predict this result. When $σ_{11} < 0$ , experimental results show that moderate or low longitudinal compressive stress can inhibit longitudinal shear failure. According to the present failure criterion, this is because the matrix compressive stress in localized misaligned regions inhibits the occurrence of MT^m, and thus $τ_{21}$ can exceed $S_{L}^{*}$ , while the other criteria argue that $τ_{21}$ cannot exceed $S_{L}^{*}$ . To sum up, the present criterion provides more reasonable predicted results than the other criteria.

It is worth noting that the failure envelope is not smooth in the second quadrant. This is caused by the change in the angle of the kink-band plane ψ together with the transition of the failure mode as σ₁₁ approaches zero from compression. It reflects an actual change in the dominant failure mode and is therefore physically reasonable. The failure envelope remains continuous, although it is non-smooth in this region.

Failure evaluation under $σ_{11} - σ_{22}$ stress states

Figure 10(b) shows the $σ_{11} - σ_{22}$ failure envelopes of four failure criteria and the test data for E-glass/MY750, and the majority of the experimental results were obtained from testing nearly circumferentially wound tubes under combined internal pressure and axial load.⁵¹ Under the combination of transverse compressive stress and longitudinal tensile stress, both Hashin’s criterion and Pinho’s criterion degenerate into the maximum stress criterion and overestimate the failure stresses. The predictions of the present criterion and Puck’s criterion are closer to the experimental results than those of Hashin’s criterion.

Under the combination of transverse tensile/compressive stress and longitudinal tensile stress, Puck’s criterion predicts the failure mode is matrix tensile/compressive failure. Puck’s criterion suggests that the longitudinal tensile stress reduces the fracture resistance of the matrix fracture plane.³⁹ Therefore, its predicted results are closer to the experimental data than Hashin’s and Pinho’s criteria. However, in Puck’s approach, the proportion of resistance reduction is determined empirically, which can be difficult to quantify without extensive experience with the specific material system. In contrast, the present criterion does not rely on any empirical parameters, and attributes the promoting effect of longitudinal tensile stress on matrix failure to failure initiation in localized misaligned regions.

Failure evaluation under $σ_{22} - τ_{21}$ stress states

Schaefer et al.⁵⁰ carried out quasi-static tests for IM7/8552 under $σ_{22} - τ_{21}$ stress states. As shown in Figure 10(c), the results predicted by the present criterion are closer to the experimental data than those predicted by Hashin’s, Puck’s and Pinho’s criterion, indicating that the present criterion can better predict the inhibitory effect of moderate transverse compressive stress on longitudinal shear failure. In addition, the present criterion predicts that the failure under moderate transverse compressive stress initiates in localized misaligned regions, while the other criteria predict the failure initiates in non-misaligned regions.

Failure evaluation under off-axis tension

Daniel et al.⁴⁹ performed off-axis tensile tests on AS4/3501-6. As illustrated in Figure 10(d), the predicted strengths of the four failure criteria are close, and their predicted results are in good agreement with the test data. Unlike the other criteria, the present criterion accounts for the influence of fiber misalignment when $σ_{11}$ ≥ 0. It predicts that, at most off-axis angles, failure initiates in localized misaligned regions. This is because under off-axis tensile loading, the matrix is subjected to a combination of transverse tensile stress $σ_{22}$ and longitudinal shear stress $τ_{21}$ , under which the dominant failure mode is MT^m.

Failure evaluation under $σ_{11} - σ_{22} {= σ}_{33}$ stress states

Figure 10(e) shows the $σ_{11} - σ_{22} {= σ}_{33}$ failure envelopes of four failure criteria and the test data for A-S/epoxy.⁵² When $σ_{11} > 0$ , both Hashin’s criterion and Pinho’s criterion degenerate into the maximum stress criterion; Puck’s criterion considers the influence of $σ_{11}$ on matrix failure, and therefore predicts lower strengths compared to Hashin’s and Pinho’s criteria; the present criterion considers the effect of fiber misalignment, leading to predictions that are also lower than those of Hashin’s and Pinho’s criteria. Despite the slight differences among the four criteria, all of them overestimate the experimental data. When $σ_{11} < 0$ , the failure envelope predicted by the present criterion closely aligns with that of Pinho’s criterion. Compared to the predictions of Hashin’s and Puck’s criterion, the present criterion yields results that are more consistent with the experimental data.

Failure evaluation under $σ_{22} - σ_{11} {= σ}_{33}$ stress states

Figure 10(f) shows the $σ_{22} - σ_{11} {= σ}_{33}$ failure envelopes and the test data for A-S/epoxy.⁵² Under high hydrostatic compression, Hashin’s criterion predicts that the failure mode is fiber compressive failure, resulting in a closed failure envelope; Puck’s criterion assumes that longitudinal compressive stress reduces the resistance of the matrix fracture plane, causing matrix failure to initiate under high hydrostatic compression, and thus, the failure envelope is also closed. In contrast to Hashin’s and Puck’s criteria, the present criterion and Pinho’s criterion predict an open failure envelope in the hydrostatic compressive direction. This is because under hydrostatic compressive stress, only normal compressive stresses act on any oblique plane, thereby suppressing material failure. Compared to Pinho’s criterion, the present criterion provides predictions that are more consistent with the experimental results.

Conclusion

This study proposes a theoretical model for predicting kink-band formation, and compared to previous models, it considers the effect of the shear strain in non-misaligned regions on the fiber misalignment angle and the influence of each stress component on the angle of the kink-band plane. Based on this model, a three-dimensional failure criterion for composites is further proposed. In contrast to Pinho’s criterion, it accounts for the effect of fiber misalignment under all stress states, thereby overcoming the discontinuity problem in the failure envelope. Because the present failure criterion suggests that failure in localized misaligned regions and non-misaligned regions is competitive under any stress state, the experimentally measured basic strength may not be true. The present study corrects the experimentally measured basic strength and finds that, for most composites, the true longitudinal shear strength differs from the experimentally measured strength.

Using the proposed failure model, the failure behavior of the material under different stress states is analyzed. An interesting conclusion is that, under the $σ_{11} - τ_{21}$ stress state, both longitudinal tensile stress and low longitudinal compressive stress can increase the longitudinal shear stress at failure. Additionally, the present study suggests that the kink bands with fiber breakage form when the fiber misalignment angle exceeds a critical value. The fiber misalignment angle only increases significantly and exceeds this critical value when the failure mode is elastic instability. For the $σ_{11} - τ_{21}$ stress states, EI occurs under moderate or high longitudinal compressive stress, while for $σ_{11} - σ_{22}$ stress states, elastic instability occurs when $σ_{11}$ approaches $X_{C}^{*}$ . Comparisons with experimental results show that the predictions of the proposed failure model agree well with the test data, and it can reasonably predict the effect of fiber misalignment on failure under different stress states. Despite these results, the proposed model has several limitations. The initial fiber misalignment angle is treated as a fixed effective parameter rather than a randomly distributed variable, and the critical misalignment angle θ_cv is determined from limited experimental data. Further refinement of these aspects could be considered in future studies based on more extensive experimental data.

Footnotes

ORCID iD

Naiyu Liu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Joint Funds of the National Natural Science Foundation of China (Grant Nos. U25B6004, 12202066) and the Fundamental Research Funds for the Central Universities (Nos. NS2024008, NJ2024003).

Declaration of conflicting interests

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

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Appendix

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A three-dimensional failure criterion for composites based on a new theoretical model of kink-band formation

Abstract

Keywords

Highlights

Introduction

The theoretical model for kink-band formation

Fiber misalignment angle

Angle of the kink-band plane

Development of three-dimensional failure criterion

Failure modes in non-misaligned regions

Fiber tensile failure in non-misaligned regions

Matrix tensile/compressive failure in non-misaligned regions

Failure modes in localized misaligned regions

Matrix tensile/compressive failure in localized misaligned regions

Elastic instability of the matrix in localized misaligned regions

Correction of basic strengths

The transverse tensile strength

The transverse compressive strength

The longitudinal shear strength

The longitudinal tensile strength

Determination of initial fiber misalignment angle

Failure mechanism analysis

σ 11 − τ 21 failure evaluation

σ 11 − σ 22 failure evaluation

σ 22 − τ 21 failure evaluation

Assessment of the proposed failure criterion

Failure evaluation under σ 11 − τ 21 stress states

Failure evaluation under σ 11 − σ 22 stress states

Failure evaluation under σ 22 − τ 21 stress states

Failure evaluation under off-axis tension

Failure evaluation under σ 11 − σ 22 = σ 33 stress states

Failure evaluation under σ 22 − σ 11 = σ 33 stress states

Conclusion

Footnotes

ORCID iD

Funding

Declaration of conflicting interests

Data Availability Statement

Appendix

References

$σ_{11} - τ_{21}$ failure evaluation

$σ_{11} - σ_{22}$ failure evaluation

$σ_{22} - τ_{21}$ failure evaluation

Failure evaluation under $σ_{11} - τ_{21}$ stress states

Failure evaluation under $σ_{11} - σ_{22}$ stress states

Failure evaluation under $σ_{22} - τ_{21}$ stress states

Failure evaluation under $σ_{11} - σ_{22} {= σ}_{33}$ stress states

Failure evaluation under $σ_{22} - σ_{11} {= σ}_{33}$ stress states