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An analytical approach with the help of numerical simulations based on the equivalent constraint model (ECM) was proposed to investigate the progressive failure behavior of symmetric fiber-reinforced composite laminates damaged by transverse ply cracking. A fracture criterion was developed to describe the initiation and propagation of the transverse ply cracking. This work was also concerned with a statistical distributions of the critical fracture toughness values with due consideration given to the scale size effect. The Monte Carlo simulation technique coupled with statistical analysis was applied to study the progressive cracking behaviors of composite structures, by considering the effects of lamina properties and lay-up configurations. The results deduced from the numerical procedure were in good agreement with the experimental results obtained for laminated composites formed by unidirectional fiber reinforced laminae with different orientations.

Laminated polymer composites are of considerable interest in a spectrum of structural applications on account of their superior mechanical performance, such as light weight, high specific strength and stiffness. To ensure the structural reliability of composite materials, it is an urgent task to comprehensively understand their failure mechanisms under thermo-mechanical loads and to develop a methodology that enables their failure behaviors to be predicted.

Performance in service of a composite structure is influenced by the progressive occurrence and interaction of some or all of many multi-mechanisms of damage, such as matrix cracking, delamination, fiber fracture and fiber/matrix debonding under loading and environment conditions. The first damage that occurs in laminates is transverse matrix cracking, the formation of intralaminar cracks running parallel to the fibers in the off-axis piles at a much lower stress [

The detailed mechanisms of transverse ply cracking in laminates have been a subject of active research efforts. A large variety of analyses have been conducted to investigate the damage phenomena of composite laminates: different modifications of the shear-lag model [

Failure processes in real composites are of a great variety, which is why many papers investigating the strengths of particular types of composite laminates are of quite special interest. In order to present detailed reviews, a worldwide failure exercise was launched by Hinton

In the previous papers [

A schematic diagram of (

In the present study, we summarize our research work and propose an analytical approach that has the capacity of predicting the progressive failure of material structures damaged by transverse matrix cracking subjected to quasi-static loading, appropriate for the mechanical analysis of fiber reinforced composite laminates [

Micro-cracking has been a mechanism of failure extensively studied in many investigations. This mechanism occurs under transverse tensile or longitudinal shear loading conditions. A known analytical modeling of mechanical properties due to transverse cracking was performed by using the suggested ECM to determine the in-plane stress distributions in the damaged layers [^{(k)} are the degraded stiffness matrices of the _{i}_{i}_{i}

Our simulation reproduces the initial crack and its propagation in the thickness direction using the energy criterion. Indeed, the potential energy method has been presented by Zhang _{c}

The energy release rate, _{1}, _{2}, …, _{n}_{n}

For illustration purposes, a Cartesian coordinate system is introduced, in which the _{j}_{j}

Schematic of the lamina containing matrix cracking.

By applying an opening displacement of the transverse crack (cracking opening displacement, COD) solution for a crack, one can obtain the total energy release rate due to matrix cracking, as follows:
_{n}_{1} and:
_{n}_{−1} ≤ _{n}_{j}_{−1} ≤ _{n}_{j}

The parameters, _{n}_{c}_{c}

The origins of transverse cracking are the inherent material defects, such as the microcracks, voids, debonded fibers, areas of high fiber volume fraction,

In the present paper, the crack multiplication can be simulated by dividing the initial gage length into equal elements along the direction of the length, 2_{c}

Based on the Weibull statistics, the fracture toughness is given as:
_{0} represents the scale parameter or the characteristic quantity of the material. The Weibull modulus, or shape parameter _{c}_{ci}_{0} = 350 J/m^{2}. This finding indicates that _{c}

The Weibull distribution of fracture toughness.

In fiber-reinforced composites, the largest portion of the loads is resisted by the fibers. When matrix cracking occurs, the internal loads must redistribute to other areas of the structures and may cause a structural collapse [_{1}, σ_{2} and τ_{12} are longitudinal stress, transverse stress and shear stress, respectively. _{t}_{c}_{t}_{c}

It should be mentioned that in the present study, the ultimate failure of the materials takes place when the stress state of the primary load-bearing lamina satisfies the condition mentioned [

The simulation is conducted by controlling the stress loading of the model.

Step 1: The ply material and ply orientation are selected to form a laminate. The laminate properties can be determined indirectly from mechanics analysis based on the classical laminate theory. The critical strain energy release rate, _{c}

Flow chart for the analysis of the progression of transverse cracking.

Step 2: A virtual crack is introduced in all positions as possible sites for failure, and the work performed to close the crack surfaces [

Step 3: The equivalent constraint model is employed to analyze the damaged lamina. In the ECM, all the laminae below and above the damaged lamina under consideration are replaced with homogeneous layers [

Step 4: With the stresses calculated, they are substituted in failure criteria Equation (7) to check for failure. When the loads are increased monotonically, the matching strains are computed, and the resulting stresses are substituted in the failure criteria until they are satisfied. The ultimate load of the laminate is thus determined.

A detailed analysis is conducted to assess the predictive capabilities of the present methodology. The mechanical properties of each type of unidirectional laminate used in this work are given in

Six different laminate configurations (denoted by A–F) are studied as shown in ^{−2} mm is chosen for the sake of efficiency [

Basic properties of unidirectional composite material systems.

Fiber Matrix | AS4/3501 | T300/5208 | Glass-fiber epoxy (GFRP) |
---|---|---|---|

_{1} (GPa) |
126 | 132.58 | 44.73 |

_{2} (GPa) |
11 | 10.8 | 12.76 |

_{12} (GPa) |
6.6 | 5.7 | 5.8 |

γ_{12} |
0.28 | 0.24 | 0.297 |

_{T} |
1950 | 1515 | 1062 |

_{C} |
1480 | 1697 | 610 |

_{T} |
48 | 43.8 | 118 |

_{C} |
200 | 43.8 | 118 |

79 | 86.9 | 72 | |

0.125 | 0.132 | 0.144 | |

_{0} (J/m^{2}) |
220 | 228 | 750 |

50 | 50 | 10 |

Laminate configurations of composite structures.

Laminate | Configuration | Laminate | Configuration |
---|---|---|---|

A | [0_{2}/90_{4}]_{S} |
B | [0/90]_{S} |

C | [±15/90_{4}]_{S} |
D | [±30/90_{4}]_{S} |

E | [0/±θ/90]_{S} |
F | [0/90/±θ]_{S} |

Transverse matrix cracking is a progressive damage mode that evolves with the increase in the applied strain, and a measure of accumulated damage is the crack density. Thus, the reduced elastic properties of a laminate, such as a longitudinal Young’s modulus and the major Poisson ratio, are related to the number of cracks formed in the transverse layer. The comparison between the present predictions and experimental data [_{2}/90_{4}]_{S} glass/epoxy laminate.

_{s} lay-up laminate as a function of matrix crack density. It should be mentioned that the numerical result of each case is the average value of 300 data. The results reveal that the theoretical predictions provided by the analytical approach agree well with the experimental values [

Normalized axial modulus as a function of crack density in the 90-degree layer.

Normalized Poisson’s ratio as a function of crack density in the 90-degree layer.

Normalized Poisson’s ratio as a function of crack density in the 90-degree layer.

For the purpose of verifying the accuracy of the present model and demonstrating its predictability for the progression of matrix cracking, _{4}]_{s}, where θ = 15° and 30°. _{0} = 750 J/m^{2} and _{4}] laminates [

Crack density in the 90-degree layer as a function of the stress applied to the composite laminates.

Crack density in the 90-degree layer as a function of the stress applied to the composite laminates.

_{s} and [0/90/±θ]_{s} laminates, respectively. The test results from [

Comparison between the theoretical and experimental results angles for degrees.

It is interesting to note that locating the 90° ply adjacent to the laminate mid-plane will lower the laminate strength. The reason for this is almost certain to arise from the fact that in such a case, the length of the ply cracks in the through-thickness direction is double that for the other laminates, leading to a larger stress concentration in the 0° plies and to fiber failures at lower stresses [

Comparison between the theoretical and experimental results angles for θ.

The objective of this exercise is to approach the modeling of progressive damage and failure in composite laminates. A statistical model based on the computation of the strain energy release rate associated with matrix cracking is proposed to study the stiffness reduction and to predict the ultimate strength of in symmetric laminated composite. The statistical distribution of the critical energy release rate, _{c}

This work was supported by the National Science Foundation of China under grant numbers 11102169 and 11302272 and the Fundamental Research Funds for the Central Universities under grant numbers XDJK2013B019 and XDJK2013D011. The authors would also like to express their gratitude to Professor Junqian Zhang, Shanghai University, Shanghai, China, for helpful discussions.

The authors declare no conflict of interest.

_{n}]

_{s}laminates

_{m}

_{n}

_{s}composite laminates. Part Ι: In-plane stiffness properties

_{m}

_{n}

_{s}composite laminates. Part II: Development of transverse ply crack