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Effects of translaminar edge crack and fiber angle on fracture toughness and crack propagation behaviors of laminated carbon fiber composites

  • Ahmet Murat Asan ORCID logo EMAIL logo , Mete Onur Kaman , Serkan Dag , Serkan Erdem and Kadir Turan
Published/Copyright: May 27, 2024
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International Journal of Materials Research
From the journal International Journal of Materials Research Volume 115 Issue 6

Abstract

In this study, the translaminar fracture toughness of carbon fiber laminated composites with different layer sequences was investigated experimentally and numerically for different crack directions. In the numerical study, first of all, the critical stress intensity factor was determined by using the M-integral method. Three-dimensional model and M-integral analysis were achieved in the ANSYS finite element package program. The non-local stress fracture criterion was used to in order to find failure curves of the materials. Then, in order to find the crack propagation directions numerically, the solid model was transferred to the LS-DYNA program and progressive failure analysis was performed. Fracture toughness decreased by 9.92 % with the change of crack angle from 15° to 90°. As the fiber angle changed from 0° to 45°, it decreased by 9.17 %. The biggest error between the experimental and numerical study results was found at α = 45°, with a rate of 12.3 %.

Keywords: M-integral; Translaminar fracture toughness; Laminated composites; Progressive failure analysis
Corresponding author: Ahmet Murat Asan, Mechanical Engineering, Firat University, Elazig, Türkiye, E-mail:

  1. Research ethics: Not applicable.

  2. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Competing interests: The authors state no competing interests.

  4. Research funding: This work was supported by the Scientific and Technological Research Council of Türkiye, TUBITAK (Grant No. 120M596).

  5. Data availability: The raw data can be obtained on request from the corresponding author.

Appendix A

[R] transformation matrix of direction cosines defines as

(A.1) R = t 11 2 t 12 2 t 13 2 t 12 t 13 t 13 t 11 t 11 t 12 t 21 2 t 22 2 t 23 2 t 22 t 23 t 23 t 21 t 21 t 22 t 31 2 t 32 2 t 33 2 t 32 t 33 t 33 t 31 t 31 t 32 2 t 21 t 31 2 t 22 t 32 2 t 23 t 33 t 22 t 33 + t 23 t 32 t 23 t 31 + t 21 t 33 t 21 t 32 + t 22 t 31 2 t 31 t 11 2 t 32 t 12 2 t 33 t 13 t 32 t 13 + t 33 t 12 t 33 t 11 + t 31 t 13 t 31 t 12 + t 32 t 11 2 t 11 t 21 2 t 12 t 22 2 t 13 t 23 t 12 t 23 + t 13 t 22 t 13 t 21 + t 11 t 23 t 11 t 22 + t 12 t 21

for rotating the material stiffness and compliance matrices. With these angles, the direction cosines t ij , (i, j = 1–3) are found as

(A.2) t = Cos [ θ z ] Cos [ θ y ] − Sin [ θ z ] Sin [ θ x ] Sin [ θ y ] − Sin [ θ z ] Cos [ θ x ] Cos [ θ z ] Sin [ θ y ] + Sin [ θ z ] Sin [ θ x ] Cos [ θ y ] Sin [ θ z ] Cos [ θ y ] + Cos [ θ z ] Sin [ θ x ] Sin [ θ y ] Cos [ θ z ] Cos [ θ x ] Sin [ θ z ] Sin [ θ y ] − Cos [ θ z ] Sin [ θ x ] Cos [ θ y ] − Cos [ θ x ] Sin [ θ y ] Sin [ θ x ] Cos [ θ x ] Cos [ θ y ]

in the matrix form. Here, θ x , θ y and θ z are Euler angles. 38

Appendix B

Mechanical properties of the plate E a , E b , v ab , v ba ,G ab stacked at 0°, are substituted in Equation (B.1) to obtain the coefficients of the rotated stiffness matrices (Equation (B.2)). 41

Q 11 = E a 1 − v a b v b a

Q 12 = v a b E b 1 − v a b v b a

(B.1) Q 16 = 0

Q 22 = E b 1 − v a b v b a

Q 26 = 0

Q 66 = G 12

Q 11 ̄ = Q 11 c 4 + Q 22 s 4 + 2 ( Q 12 + 2 Q 66 ) s 2 c 2

Q 12 ̄ = ( Q 11 + Q 22 − 4 Q 66 ) s 2 c 2 + Q 12 ( s 4 + c 4 )

(B.2) Q 22 ̄ = Q 11 s 4 + Q 22 c 4 + 2 ( Q 12 + 2 Q 66 ) s 2 c 2

Q 16 ̄ = ( Q 11 − Q 12 − 2 Q 66 ) c 3 s − ( Q 22 − Q 12 − 2 Q 66 ) s 3 c

Q 26 ̄ = ( Q 11 − Q 12 − 2 Q 66 ) s 3 c − ( Q 22 − Q 12 − 2 Q 66 ) c 3 s

Q 66 ̄ = ( Q 11 + Q 22 − 2 Q 12 − 2 Q 66 ) s 2 c 2 + Q 66 ( s 4 + c 4 )

where; c = Cos[θ°] and s = Sin[θ°], Equation (B.3) is obtained for each fiber angle where θ is the fiber angle.

(B.3) Q θ ̄ = Q 11 ̄ Q 12 ̄ Q 16 ̄ Q 12 ̄ Q 22 ̄ Q 26 ̄ Q 16 ̄ Q 26 ̄ Q 66 ̄

The obtained stiffness matrices are substituted in Equation (B.4) and the elongation stiffness matrix is found (Equation (B.4)).

(B.4) [ A i j ] = ∑ k = 1 t Q k ̄ h k − h k − 1

In Equation (B.4), t represents the number of layers and h k represents the thickness of each layer. As a result, the mechanical properties of the plates stacked in different directions are obtained as in Equation (B.5).

E x = 1 h A 11 *

E y = 1 h A 22 *

(B.5) G x y = 1 h A 66 *

υ x y = − A 12 * A 11 *

υ y x = − A 12 * A 22 *

In Equation (B.5), the expressions A i j * are obtained by inverting Equation (B.4) and h is the total thickness of the plate.

Appendix C

The coefficients (A, B, C, D) in K I and K II for DCM are calculated with 42

(C.1) A = Re i μ 1 − μ 2 ( μ 1 p 2 − μ 2 p 1 )

(C.2) B = Re i μ 1 − μ 2 ( p 2 − p 1 )

(C.3) C = Re i μ 1 − μ 2 ( μ 1 q 2 − μ 2 q 1 )

(C.4) D = Re i μ 1 − μ 2 q 2 − q 1

Appendix D

Trigonometric functions λ 1, λ 12 and λ 2 in Equation (19) can be found with using

(D.1) λ 1 = ( Z 11 − Z 21 ) Co s 2 γ + Z 31 Sin 2 γ − Z 11 2 + c 12 c 1 ( Z 11 − Z 21 ) 0.5 Sin 2 γ − 2 Z 31 Co s 2 γ + Z 31 2

(D.2) λ 12 = 2 Z 12 − Z 22 Co s 2 γ + 2 Z 32 Sin 2 γ − 2 Z 12 × 2 Z 11 − Z 21 Co s 2 γ + 2 Z 31 Sin 2 γ − 2 Z 11 + c 12 c 1 Z 12 − Z 22 Sin 2 γ − 4 Z 32 Co s 2 γ + 2 Z 32 × Z 11 − Z 21 Sin 2 γ − 4 Z 31 Co s 2 γ + 2 Z 31

(D.3) λ 2 = Z 12 − Z 22 Co s 2 γ + Z 32 Sin 2 γ − Z 12 2 + c 12 c 1 Z 12 − Z 22 0.5 Sin 2 γ − 2 Z 32 Co s 2 γ + Z 32 2

In these equations, Z 11 … Z 32 are the coefficients in the asymptotic solution of the stress fields near the crack tip in an orthotropic material and are calculated by

(D.4) Z 11 = Re μ 1 μ 2 μ 1 − μ 2 μ 2 Cos ϕ + μ 2 Sin ϕ   − μ 1 Cos ϕ + μ 1 Sin ϕ

(D.5) Z 12 = Re 1 μ 1 − μ 2 ( μ 2 ) 2 Cos ϕ + μ 2 Sin ϕ   − ( μ 1 ) 2 Cos ϕ + μ 1 Sin ϕ

(D.6) Z 21 = Re 1 μ 1 − μ 2 μ 1 Cos ϕ + μ 2 Sin ϕ   − μ 2 Cos ϕ + μ 1 Sin ϕ

(D.7) Z 22 = Re 1 μ 1 − μ 2 1 Cos ϕ + μ 2 Sin ϕ   − 1 Cos ϕ + μ 1 Sin ϕ

(D.8) Z 31 = Re μ 1 μ 2 μ 1 − μ 2 1 Cos ϕ + μ 1 Sin ϕ   − 1 Cos ϕ + μ 2 Sin ϕ

(D.9) Z 32 = Re 1 μ 1 − μ 2 μ 1 Cos ϕ + μ 1 Sin ϕ   − μ 2 Cos ϕ + μ 2 Sin ϕ

Here, ϕ represents the angle in the local polar coordinate system (r, ϕ) in an arbitrary fractured structure. μ 1 and μ 2 are the positive imaginary parts obtained from the solution of Equation (7). The c 12 and c 1 in Equations (D.1–3) are the extensional and sliding compliance that characterize the material weakened by microcracks oriented in the plane of orthotropy of the normal and are expressed by 70

(D.10) c 12 c 1 = K Ic K IIc 2 = E I ′ E II ′

Here, K Ic and K IIc are Mode I and Mode II fracture toughnesses, respectively. E I ′ and E II ′ are generalized elasticity modules and are calculated by 71

(D.11) E I ′ = C 11 ′ C 22 ′ 2 C 22 ′ C 11 ′ + 2 C 12 ′ + C 66 ′ 2 C 11 ′ − 1 / 2

(D.12) E II ′ = C 11 ′ 2 2 C 22 ′ C 11 ′ + 2 C 12 ′ + C 66 ′ 2 C 11 ′ − 1 / 2

In the above equations, C kl and C k l ′ are coefficients dependent on elastic properties and are calculated by

(D.13) C k l ′ = C k l − ( C k 3 C l 3 ) / C 33 ( k , l = 1,2 )

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Received: 2023-10-23
Accepted: 2024-01-17
Published Online: 2024-05-27
Published in Print: 2024-06-25

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Papers
  3. Experimental investigation and thermodynamic analysis of TiC–Fe cermets with Mo additions
  4. Investigations on porous silicon nitride ceramics prepared by the gel-casting method
  5. Catalysis effect of rare earth element Ce on paste boriding treatment of AISI 410 steel
  6. Effect of nitrogen content on the static recrystallization and precipitation behaviors of vanadium–titanium microalloyed steels
  7. Effects of addition of Er and Zr on microstructure and mechanical properties of Al–Cu–Mn–Si–Mg alloy
  8. The quasi-binary phase diagrams of R 2Fe14B–Ce2Fe14B (R = Nd, Pr) systems
  9. Trivalent Gd incorporated Zn2SiO4 phosphor material for EPR and luminescence investigations
  10. Effects of translaminar edge crack and fiber angle on fracture toughness and crack propagation behaviors of laminated carbon fiber composites
  11. Blast protection of underwater tunnels with 3D auxetic materials
  12. News
  13. DGM – Deutsche Gesellschaft für Materialkunde
https://doi.org/10.1515/ijmr-2023-0317
Keywords for this article
M-integral; Translaminar fracture toughness; Laminated composites; Progressive failure analysis

Articles in the same Issue

  1. Frontmatter
  2. Original Papers
  3. Experimental investigation and thermodynamic analysis of TiC–Fe cermets with Mo additions
  4. Investigations on porous silicon nitride ceramics prepared by the gel-casting method
  5. Catalysis effect of rare earth element Ce on paste boriding treatment of AISI 410 steel
  6. Effect of nitrogen content on the static recrystallization and precipitation behaviors of vanadium–titanium microalloyed steels
  7. Effects of addition of Er and Zr on microstructure and mechanical properties of Al–Cu–Mn–Si–Mg alloy
  8. The quasi-binary phase diagrams of R 2Fe14B–Ce2Fe14B (R = Nd, Pr) systems
  9. Trivalent Gd incorporated Zn2SiO4 phosphor material for EPR and luminescence investigations
  10. Effects of translaminar edge crack and fiber angle on fracture toughness and crack propagation behaviors of laminated carbon fiber composites
  11. Blast protection of underwater tunnels with 3D auxetic materials
  12. News
  13. DGM – Deutsche Gesellschaft für Materialkunde
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