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Bearing failure of composite laminate is very complicated due to the complexity of different failure mechanisms and their interactions. In this paper, an elasto-plastic damage model is built up to describe the process of failure in composite laminates subjected to bearing load. Non-linear behavior of composite before failure is taken into consideration by using a modified Sun-Chen one parameter plasticity model. LaRC05 failure criteria are employed to predict the initiation of failure and the evolution of failure is described by a CDM based stiffness degradation model. Both theory and some application issues like parameter determination are discussed according to phenomenon of experiments. The model is firstly validated by several experiment results of unidirectional laminate and then applicated into the progressive analysis of bearing failure in pin-loaded multidirectional laminates, both intralaminar and interlaminar damage are taken into consideration. The result of finite element analysis is compared with experiment results; it shows good agreements in both mechanical response and progress of failure, so the model can be evaluated to be effective and practical in bearing failure analysis of composite laminates.

At present, the application of composite materials in aircrafts has been transited gradually from secondary parts to main parts. In the main parts, the connections between different components are mostly in the form of mechanically fastened joints, so the strength of mechanically fastened joints in composite laminates has great influence on the strength and integrity of aircraft structures.

The failure of mechanically fastened joints in composite materials can be divided into three patterns macroscopically: shear-out, net-tension and bearing, in which bearing failure shows the best carrying capacity [

Xiao and Ishikawa [

Gamble et al. [_{s} lay-up laminate tensile test.

Camanho et al. [

Nguyen [

Atas et al. [_{s} lay-up composite laminate was simulated to find a non self-similar delamination propagation with an eclipse crack tip distribution. Delamination initiated at interface of 0 and 90 layers where a maximum average shear stress exists. But no progressive damage model was incorporated into this analysis which means that the effect of intralaminar damage was ignored.

Shen et al. [

Frizzell and McCarthy [

Although there already have been many significant achievements in this area, a lot of unsolved problems still exist [

Wang et al. [

f * = [ ( σ 22 − σ 33 ) 2 + 4 σ 23 2 + 2 a 66 Γ 2 ( σ 31 2 + σ 12 2 ) + ( Γ − 1 ) ( σ 22 + σ 33 ) ] / Y 2 T Γ (1)

where Y 2 T is transverse tensile yield strength, a 66 is a parameter in Sun-Chen model, m is the ratio of tensile yield strength and compressive yield strength. Γ is a parameter which characterizes the non-symmetry of material behavior under tension and compression and Γ = 2 m / ( m + 1 ) .

The effective stress can be expressed as follows:

σ ¯ = σ ˜ 3 D + 3 2 ( Γ − 1 ) ( σ 22 + σ 33 ) (2)

σ ˜ 3 D = 3 2 [ ( σ 22 − σ 33 ) 2 + 4 σ 23 2 + 2 a 66 Γ 2 ( σ 31 2 + σ 12 2 ) ] (3)

Generalized Hill yield function is taken as the plastic potential function and following the incremental derivation method of plastic work per unit volume, the relationship between plastic strain increment and effective plastic strain increment can be deduced:

d ε i j p = ∂ σ ¯ ∂ σ i j d ε ¯ p (4)

Substituting Equation (2) into Equation (4), the expression of plastic strain components in incremental form is derived:

{ d ε 11 p d ε 22 p d ε 33 p d ε 23 p d ε 31 p d ε 12 p } = { 0 3 ( σ 22 − σ 33 ) 2 σ ˜ 3 D + 3 2 ( Γ − 1 ) 3 ( σ 33 − σ 22 ) 2 σ ˜ 3 D + 3 2 ( Γ − 1 ) 6 σ 23 σ ˜ 3 D 3 Γ 2 a 66 σ 31 σ ˜ 3 D 3 Γ 2 a 66 σ 12 σ ˜ 3 D } d ε eff p (5)

From Equation (4), it can be easily found that plasticity flows in all material directions except the longitudinal direction which is supposed to have a linear mechanical behavior and the flow of plasticity is closely related to current stress state.

The material is assumed to satisfy isotropic hardening rule and the effective stress is related to the effective plastic strain with a power law:

ε ¯ p = A σ ¯ n (6)

where A and n are both parameters which describe the plasticity of material. According to the initial yield criterion, subsequent yield surface can be obtained:

Φ ( σ i j , ε ¯ p ) = σ ¯ ( σ i j ) − k ( ε ¯ p ) = 0 (7)

where k is hardening parameter and it associates with plastic deformation which can be expressed as a function of effective plastic strain, then the function of k on effective plastic strain can be deduced:

k ( ε ¯ p ) = σ ¯ ( ε ¯ p ) = A − 1 / n ( ε ¯ p ) 1 / n (8)

The expression of plastic compliance is as follows:

S i j k l p = d ε i j p d σ k l = d ε i j p d ε ¯ p d ε ¯ p d σ ¯ d σ ¯ d σ k l (9)

Combining the plastic compliance with the elastic compliance, incremental stress-strain relationship is derived:

d ε i j = S i j k l ep d σ k l (10)

Using Voigt tensor notation, the non-zero terms in S i j k l ep is expressed explicitly in Equation (11):

{ S 11 = 1 E 1 , S 12 = S 13 = − ν 12 E 1 S 22 = 1 E 2 + 1 H p [ 3 ( σ 22 − σ 33 ) 2 V + 3 2 ( Γ − 1 ) ] 2 S 2 3 = - ν 23 E 2 + 1 H p [ 3 2 ( Γ − 1 ) 2 − 9 ( σ 22 − σ 33 ) 2 4 V 2 ] S 2 4 = 6 σ 23 H p V [ 3 ( σ 22 − σ 33 ) 2 V + 3 2 ( Γ − 1 ) ] S 25 = 3 Γ 2 a 66 σ 31 H p V [ 3 ( σ 22 − σ 33 ) 2 V + 3 2 ( Γ − 1 ) ] S 26 = 3 Γ 2 a 66 σ 12 H p V [ 3 ( σ 22 − σ 33 ) 2 V + 3 2 ( Γ − 1 ) ] S 33 = 1 E 2 + 1 H p [ 3 ( σ 33 − σ 22 ) 2 V + 3 2 ( Γ − 1 ) ] 2 S 34 = 6 σ 23 H p V [ 3 ( σ 33 − σ 22 ) 2 V + 3 2 ( Γ − 1 ) ] S 35 = 3 Γ 2 a 66 σ 31 H p V [ 3 ( σ 33 − σ 22 ) 2 V + 3 2 ( Γ − 1 ) ] S 36 = 3 Γ 2 a 66 σ 12 H p V [ 3 ( σ 33 − σ 22 ) 2 V + 3 2 ( Γ − 1 ) ] S 44 = 1 G 23 + 1 H p ( 6 σ 23 V ) 2 , S 45 = 18 Γ 2 a 66 σ 31 σ 23 H p V 2 S 46 = 18 Γ 2 a 66 σ 12 σ 23 H p V 2 , S 55 = 1 G 12 + 1 H p ( 3 Γ 2 a 66 σ 31 V ) 2 S 56 = σ 12 σ 31 H p ( 3 Γ 2 a 66 V ) 2 , S 66 = 1 G 12 + 1 H p ( 3 Γ 2 a 66 σ 31 V ) 2 (11)

where:

H p = 1 A n σ ¯ n − 1 (12)

V = 3 2 [ ( σ 22 − σ 33 ) 2 + 4 σ 23 2 + 2 Γ 2 a 66 ( σ 31 2 + σ 12 2 ) ] (13)

The determination method of the parameters in this model can be obtaining by referring to Wang et al. [

Matrix failure is also called inter-fiber failure which means a crack parallel to fiber direction has developed through the whole lamina, including matrix cracks and fiber-matrix interface cracks. LaRC05 criteria [

{ f m = ( τ T S T − μ T σ N ) 2 + ( τ L S L − μ L σ N ) 2 ( σ N < 0 ) f m = ( σ N Y T ) 2 + ( τ T S T ) 2 + ( τ L S L ) 2 ( σ N ≥ 0 ) (14)

where τ T and τ L are the transverse shear stress and longitudinal shear stress on the potential fracture plane respectively, σ N is the normal stress on the potential fracture plane. The stress components on the potential fracture plane can be deduced according to the transformation matrices in Appendix 1. The direction of fracture plane is defined in

The parameters can be determined using unidirectional laminate off-axis compression experiment. The off-axis compressive strengths of different off-axis angle are notated as X C ( θ n ) , the stress components on fracture plane are notated as σ N ( θ n , α n ) , τ L ( θ n , α n ) and τ N ( θ n , α n ) respectively. The stress in loading direction is σ x , and the stress state of unidirectional laminate is as follows:

{ σ 11 = cos 2 θ σ x σ 22 = sin 2 θ σ x σ 12 = − sin θ cos θ σ x (15)

Firstly, a small angle θ 1 (typically θ 1 < 30 ∘ ) is taken to determine μ L :

μ L = cot θ − 1 sin 2 θ S L X C ( θ 1 ) (16)

Then two other angles θ 2 ( θ 2 = 90 ∘ ) and θ 3 (typically θ 3 > 45 ∘ ) are used to determinate S T and μ T :

S T = τ T ( θ 2 , α 2 ) + μ T σ N ( θ 2 , α 2 ) (17)

S T = τ T ( θ 3 , α 3 ) 1 − ( τ L ( θ 3 , α 3 ) S L − μ L σ N ( θ 3 , α 3 ) ) 2 + μ T σ N ( θ 3 , α 3 ) (18)

Fiber kinking (

The process of kinking band formation is very complicated, phenomenons like fiber inclination, matrix shear deformation, inter-fiber failure in kinking band and fiber breakage at the boundary of kinking band can be observed [

Before failure initiation, the material is assumed to be continuum, so stress keeps balanced everywhere in the material. Then stress components can be rotated to local framework of kinking band and used to evaluate if failure will happen. Hence the local framework of kinking band needs to be determined firstly.

It’s supposed that kinking plane is always parallel to fiber direction, so the coordinate of kinking plane Ω ψ can be obtained by rotating material coordinate by an angle ψ around axe 1 (

tan ( 2 ψ ) = 2 τ b c σ b − σ c (19)

Fiber misalignment angle φ is the sum of initial misalignment angle φ i and shear strain γ , which means φ = φ i + γ . φ i is deduced from the longitudinal compressive strength of the material. When unidirectional laminate only subject to longitudinal compressive load σ 11 = − X C , the stress state in kinking band is:

{ σ 11 φ c = − X C cos 2 ( φ c ) σ 22 φ c = − X C sin 2 ( φ c ) σ 12 φ c = X C sin ( φ c ) cos ( φ c ) (20)

where the superscripts of stress components represent the coordinate of stress. Applying the matrix failure criteria in Equation (14), the fiber misalignment angle when matrix failure happens in kinking band is derived:

φ c = arctan ( 1 − 1 − 4 ( S L X C + μ L ) S L X C 2 ( S L X C + μ L ) ) (21)

It should be noted that, φ i is not an initial misalignment angle in reality, which means it’s not possible to get the value of φ i by experiment. φ i is actually an effective initial misalignment angle derived from the longitudinal compressive strength X C according to the material’s constitutive model. If different constitutive models are chosen to represent the mechanical behavior of material, different values of φ i will be obtained.

After φ i is determined, fiber misalignment angle φ can be calculated from Equation (22), which means the coordinate of kinking band is determined.

{ Δ γ = S 22 ( σ φ i ) Δ σ 22 φ i + S 66 ( σ φ i ) Δ σ 12 φ i γ = γ ( i − 1 ) + Δ γ φ = τ 12 ψ | τ 12 ψ | ( φ i + γ ) (22)

Then stress components in material coordinate are rotated to kinking band coordinate, LaRC05 criteria is applying to judge the initiation of fiber kinking failure. When the value of f f c exceeds one, failure will happen.

f f c = ( τ 23 φ S T − μ T σ 22 φ ) 2 + ( τ 12 φ S L − μ L σ 22 φ ) 2 + ( 〈 σ 22 φ 〉 + Y T ) 2 (23)

where 〈 • 〉 + is Macauley operator, represent 〈 x 〉 + = max { x , 0 } .

For fiber tensile failure, maximum stress criterion is used to predict failure initiation. When the value of f f t exceeds one, failure will happen.

f f t = σ 11 X T (24)

If failure initiates, material stiffness is to be degraded to consider the effect of damage evolution. According to literatures [

C ed = [ ( 1 − d L ) C 11 ( 1 − d L ) ( 1 − d T ) C 12 ( 1 − d L ) ( 1 − d T ) C 13 0 0 0 ( 1 − d T ) C 22 ( 1 − d T ) C 2 3 0 0 0 ( 1 − d T ) C 33 0 0 0 ( 1 − d L ) ( 1 − d T ) C 44 0 0 ( 1 − d L ) ( 1 − d T ) C 55 0 s y m ( 1 − d T ) C 66 ] (25)

where d L and d T are damage variable for fiber failure mode and matrix failure mode respectively.

d L = 1 − exp [ ( − f f t , f c − 1 λ f ) α f ] (26)

d T = 1 − exp [ ( − f m − 1 λ m ) α m ] (27)

where α f , α m , λ f , λ m are parameter which define the mechanical response of material during the process of damage evolution. α define the form of stress-strain relationship and λ define the intense of degradation as shown in

The elasto-plastic damage model developed in Section 2 is firstly validated by two cases of strength prediction. These two cases are provided by literatures [

Case 1: The material system is T300/PR319 (carbon fiber/epoxy). In the test, hydrostatic pressure of 600 MPa is applied to the material and a shear load is applied to the material at the same time. The comparison between the model prediction curve and the test results is shown in

Case 2: The material used in the test was S2-glass/epoxy (high-strength glass fiber/epoxy) system; the material received varying levels of lateral hydrostatic pressure while measuring the axial compressive strength of the material.

The results of the comparison between the model prediction results and the test results are shown in

To verify the validity of the model, the results of off-axis tensile and compression tests of a set of continuous fiber reinforced composite unidirectional laminates were selected [

Material | T300/PR319 | S2-glass/Epoxy |
---|---|---|

E 11 ( GPa ) | 129 | 52 |

E 22 ( GPa ) | 5.6 | 19 |

G 12 ( GPa ) | 1.33 | 6.7 |

G 13 ( GPa ) | 1.33 | 6.7 |

ν 12 | 0.318 | 0.3 |

ν 23 | 0.5 | 0.42 |

a 66 | 3.8 | 0.83 |

A ( MPa − n ) | 3.83E-17 | 1.62E-15 |

n | 5.15 | 5.44 |

α | 1.2 | 1.12 |

X C ( MPa ) | 950 | 1150 |

S L ( MPa ) | 97 | 72 |

η L | 0.082 | 0.25 |

Material properties | Tension | Compression |
---|---|---|

E 11 ( GPa ) | 137.38 | 137.38 |

E 22 ( GPa ) | 8.91 | 8.91 |

G 12 ( GPa ) | 4.41 | 4.41 |

G 13 ( GPa ) | 3.01 | 3.01 |

ν 12 | 0.33 | 0.33 |

ν 23 | 0.48 | 0.48 |

a 66 | 1.2 | 1.2 |

A ( MPa − n ) | 2.33E-12 | 2.09E-13 |

n | 4.6 | 4.78 |

α | 1.0 | 1.7 |

The modified Sun-Chen plasticity model considering the asymmetry of tension and compression was used to calculate the off-axis tensile and compression tests for 30° and 90°. The stress-strain response and test results obtained in the loading direction (shown in

The analysis model developed is implemented by using the UMAT subroutine interface provided by Abaqus. The flow chart of the subroutine is shown is

In this paper, the delamination damage is analyzed by the VCCT method based on fracture mechanics in ABAQUS software. This method is based on linear elastic fracture mechanics (LEFM) to evaluate the strain energy release rate (SERR) at the crack tip. The virtual crack closure technique is based on the crack closure integration method. The basic idea is to assume that the energy released by the crack propagation Δa is equal to the energy required to close the crack. The specific method has been built into ABAQUS software.

When using VCCT method in ABAQUS for numerical simulation analysis, appropriate failure criteria should be selected, among which BK criterion is a criterion that is mostly used currently.

The BK (Benzeggagh-Kenane) criterion is a commonly used failure criterion for judging the delamination of mixed modes. The expression is as follows:

G e q u i v C = G I C + ( G I I C − G I C ) ( G I I + G I I I G I + G I I + G I I I ) η (28)

where G I C , G I I C , G I I I C are fracture toughness for mode I, II, III respectively. G e q u i v C is fracture toughness for mix-mode. η is the mix parameter. When f = G e q u i v / G e q u i v C exceeds 1.0, crack will initiate and G e q u i v is equivalent strain energy release rate of nodes.

In order to facilitate the comparison with the existing experimental results, the finite element modeling in this paper is based on the testing program of double lap joint proposed in literature [

The material used in the experiment is IM600/Q133, the mechanical properties of this material are already listed in _{T} = 2700 MPa, X_{C} = 1037 MPa, Y_{T} = 63.7 MPa, Y_{C} = 235 MPa, S_{C} = 140 MPa, G_{IC} = 0.33 N/mm, G_{IIC} = 0.8 N/mm, G_{IIIC} = 0.8 N/mm. The thickness of single lamina is 0.125 mm and the lay-up of specimen is [45/-45/90/0]_{2S} and total thickness of specimen is 2 mm.

Comparing the results of the numerical analysis with the experimental results, the following

In addition to the overall response, the finite element model developed in this paper can also simulate the intralaminar and interlaminar damage evolution progress as in

1) In this paper, an elasto-plastic damage model considering the non-symmetry of composite material behavior under tension and compression, failure judgement and damage evolution is developed to describe the mechanical behavior of composite laminates under both tensile and compressive load.

2) The model is implemented in commercial FEA software ABAQUS through UMAT subroutine interface.

3) The model is validated for strength prediction and mechanical response prediction of unidirectional laminate by experiment results.

4) A progressive failure of composite laminate under bearing load is proceeded using the elasto-plastic damage model. Delamination is taken into account by a fracture mechanics method implemented using the Virtual Crack Closure Technique (VCCT) available in ABAQUS. The numerical simulation results for joint’s progressive damage and mechanical response were compared with the existing experimental data, and the reliability of this model is proved.

5) For further study of this topic, development of a model which can reveal the complex mechanisms of interactions among different damage patterns is highly recommended by the author.

No acknowledgement is declared by the author.

Xue, K. (2018) Progressive Analysis of Bearing Failure in Pin-Loaded Composite Laminates Using an Elasto-Plastic Damage Model. Materials Sciences and Applications, 9, 576-595. https://doi.org/10.4236/msa.2018.97042

The stress and strain in original coordinate ( 1 − 2 − 3 ) are notated as

σ = [ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ] T , ε = [ ε 11 ε 22 ε 33 γ 23 γ 13 γ 12 ] T

The stress and strain in new coordinate ( 1 ′ − 2 ′ − 3 ′ ) are notated as

σ ′ = [ σ ′ 11 σ ′ 22 σ ′ 33 σ ′ 23 σ ′ 13 σ ′ 12 ] T , ε ′ = [ ε ′ 11 ε ′ 22 ε ′ 33 γ ′ 23 γ ′ 13 γ ′ 12 ] T

The relationship between σ and σ ′ as well as ε and ε ′ is as follows.

σ ′ = T σ σ , σ = T σ − 1 σ ′ , ε ′ = T ε ε , ε = T ε − 1 ε ′

And the transformation matrix rotating the stress and strain from original coordinate to new coordinate is expressed as follows.

T σ = [ K 1 2 K 2 K 3 K 4 ] , T σ − 1 = [ K 1 T 2 K 3 T K 2 T K 4 T ] , T ε = [ K 1 K 2 2 K 3 K 4 ] , T ε − 1 = [ K 1 T K 3 T 2 K 2 T K 4 T ]

where

K 1 = [ l 1 2 m 1 2 n 1 2 l 2 2 m 2 2 n 2 2 l 3 2 m 3 2 n 3 2 ] , K 2 = [ m 1 n 1 l 1 n 1 l 1 m 1 m 2 n 2 l 2 n 2 l 2 m 2 m 3 n 3 l 3 n 3 l 3 m 3 ] , K 3 = [ l 2 l 3 m 2 m 3 n 2 n 3 l 1 l 3 m 1 m 3 n 1 n 3 l 1 l 2 m 1 m 2 n 1 n 2 ]

K 4 = [ m 2 n 3 + m 3 n 2 l 2 n 3 + l 3 n 2 l 2 m 3 + l 3 m 2 m 1 n 3 + m 3 n 1 l 1 n 3 + l 3 n 1 l 1 m 3 + l 3 m 1 m 1 n 2 + m 2 n 1 l 1 n 2 + l 2 n 1 l 1 m 2 + l 2 m 1 ]

where l i , m i , n i ( i = 1 , 2 , 3 ) represent the included angle cosines between the axis of original coordinate and new coordinate.