The open hole composite laminates with different stacking sequence were tested and the failure modes and strength were recorded. A progressive damage model of the open-hole composite laminate under compression was developed and in-plane damage is modeled by using the three-dimensional continuum shell elements and cohesive element was adopted for delamination failure simulation. Based on the ABAQUS user subroutine UMAT, this model simulates damage initiation, propagation and final failure. With different failure criteria, the strength and failure modes of open-hole laminates under compression were predicted in this model. Compared with the experiment results, the effects of different stacking sequences on the strength and final failure modes of composite laminates are analyzed.

Fiber reinforced composites have been widely used in aerospace and other engineering fields with the advantages of high specific strength and high specific modulus. The usual composite laminates contain holes and notches for joints. It is well known that the compressive strengths of composite laminates are lower than their tensile strength, and composites laminates with holes are much lower. So it is necessary to predict the compressive strengths and failure modes of open-hole laminates, and to guide the structural design of composite materials.

The average stress model proposed by Whitney and Nuismer [1] is the most widely applied empirical model for predicting OHC (Open Hole Compressive) strength. Unnotched strength of the laminates and characteristic distance is required in their models, which can be identified with experiments. The cohesive zone model (CZM) which was firstly proposed by Dugdale [2] and Barenblatt [3] for metallic material is applied to composite laminates later by the study of Soutis et al. [4]. The compressive strength of laminates is predicted by representing the damage zone where the laminates undergo micro-buckling. The model assumes that micro-buckling initiates when the local stress of the longitudinal direction besides the hole reaches the strength of the unopened laminates. The CZM requires two parameters, the unnotched strength and the critical crack closing displacement, which are not material properties and specific for different lay-ups. Based on this, Camanho et al. [5] proposed a finite fracture mechanics model to predict the OHT (Open Hole Tension) strength of laminates, with the unnotched strength and the fracture toughness of the laminates [6]. The model can be applied to predict the OHT strength. Further the corresponding experience and semi empirical formula was put forward. While applied to OHC strength predicting, the model required extra parameters by experiment.

Recently, progressive damage modeling has been an important method to study strength and failure modes of composite laminates. However, the result of this progressive damage models is inaccurate because of the dependence of mesh partition. Considering the degradation of material properties of continuous material, Lee and Kim [7] established a micro-mechanical constitutive model, which suggested that the laminates damage was composed of the interfacial fiber debonding and micro-cracks in the matrix. It is shown that the model can predict accurate results for cross-ply laminates, while it overestimates the stiffness and final failure load for angle-ply and quasi-isotropic laminates. In the previous model, linear constitutive relation is used mostly, which leads to the failure displacement less than the actual displacement observed by experiments. A continuum damage mechanics model of nonlinear constitutive behavior is proposed by McGregor et al. [8] to model the compressive failure of notched composites. However, it is difficult to determine a suitable predefined stress–strain relationship, which leads to the prediction results inaccurate.

Based on the progressive damage criterion and the finite element method, the OHT damage model proposed by Ridha et al. [9] and Wisnom et al. [10] can predict the effects of the specimen size and the orthogonality of the layups by considering the in-plane damage and interface damage. The simulation under any load and the optimal way of laying can also be modeled. From more accurate simulation of the OHT model, it is shown that only taking into account multiple failure modes including in-plane failure and delamination, the prediction results could be accurate enough for the progressive compressive damage model. In recent research, the critical delamination force (CDF) model proposed by Zou et al. [11] shows the better accuracy in simulation. The establishment of CDF model based on fracture mechanics, classical bending plate theory and cohesive element was used for characterizing delamination in simulation with bilinear constitutive law. In the current study, for OHT strength analysis, the following methods are generally used to solve the above problems. The zigzagging method, which deals with the degradation of the material properties in the process of damage, so that the instability of the computation results tends to be gentle and the results, is accelerated to convergence [12,13]. The method using the cohesive elements to characterize the delamination damage [14]. The method that considers local micro-buckling has not yet been accurately modeled, but by defining the damage surface and the damage angle can be a good approximation [15]. Compared with OHC strength study, for which related research is limited, the OHT strength study has achieved further develop. Combined with the progressive damage analysis, the above methods provide a new idea for the present authors to study the compressive damage of composite laminates [16–19].

In this paper, to study the effect of stacking sequence on strength and failure modes of OHC laminates, a compressive experiment was carried out with OHC laminates of different stacking sequence. A progressive damage finite element model under compression was developed in ABAQUS to predict the OHC strength and failure behavior of composite laminates. In this model, the laminates were modeled with three-dimensional continuum shell elements for ply and cohesive elements for interlayer. The stress–strain relationship after damage initiation was discretized with zigzag method to stabilize the computation and achieve a convergent solution. Tsai-Wu failure criterion and Hoffman failure criterion were used for the matrix damage initiation, respectively. UMAT subroutine is adopted to calculate the composite properties degradation. The predictions were compared with experimental results to verify model accuracy. From the predictions, the failure mechanism could be studied.

2Compressive experiments2.1SpecimensThe composites laminates were made by carbon fiber prepreg. The specimen dimensions were 150mm×31.5mm×2.4mm and the size of effective damage area was 50mm×31.5mm×2.4mm, as schematically shown in Fig. 1. The diameter of the circular hole in the middle of the specimen was 6.3mm, and the ratio of the hole diameter to laminates width was 1:5. To research the effect of different stacking sequence, [453/03/−453/903]s, [45/90/−45/0]3s and [453/903/−453/03]s laminates were designed for testing. The specimen was loaded on the Zwick-Z010 universal testing machine with a fixed loading speed of 4mm/min. The [453/03/−453/903]s and [453/903/−453/03]s correspond to orthorhombic laminates and the [45/90/−45/0]3 corresponds to continuous laminates.

2.2Experiment resultsThe failure strength of each specimen is shown in Table 1. The failure mode and delamination was recorded in each specimen for analyzing the influence of different layup sequence on failure mechanisms.

Due to the different stacking sequence, three different laminates showed different failure modes, which can be seen in Fig. 2. Cracks were observed at 45° along the tangential direction of the hole in the [453/03/−453/903]s laminates, which propagated along the 45° fiber direction. In [45/90/−45/0]3s laminates cracks propagated through the hole in the laminate transverse direction, and local micro-buckling and stacking could be seen in the damage area. The failure modes of [453/903/−453/03]s laminates were more complex. They are mixed with a variety of failure modes, not only including cracks along 45° tangential direction of the hole, transverse cracks and local micro-buckling, but also a new form of crack, which extended along the laminate longitudinal direction.

The delamination of [453/03/−453/903]s laminates is shown in Fig. 3(a). The delamination was not obvious, local cracks appeared only in the damage area and the crack length was short. The delamination area is also limited to the interface between 45° plies and 0° plies. In [45/90/−45/0]3s laminates, the delamination is more obvious. The large span of cracks could be seen in the effective damage area, but the delamination area is still limited, which is shown in Fig. 3(b). The delamination of [453/903/−453/03]s laminates is the most obvious, in which cracks breakthrough the both sides of the laminates, and crack transition can be observed around the hole shown in Fig. 3(c).

According to the experimental results, the failure strength of the three laminates are as follows: [45/90/−45/0]3s>[453/903/−453/03]s>[453/03/−453/903]s. It can be known that the strength of orthorhombic laminates is bigger than the one of continuous laminates. It is can be seen that in Fig. 4 that the strength of interior ply in [453/903/−453/03]s laminate is 0°, which is bigger than the strength of interior ply in [453/03/−453/903]s laminates whose interior ply is 90°. For orthorhombic laminates with symmetric laminates, its stability was better than continuous laminates. Each ply served as an integral, which could bear compressive load with only slightly transverse delamination, so it was more difficult to cause instability, which could result in final failure.

Compressive failure modes of composite laminates were mainly divided into brittle fracture and push-in failure mode, which can be both observed in the experiments. In the [453/03/−453/903]s laminates, brittle fracture occurred in 45° direction of laminates. This is due to delamination expanded between 45° and 0° plies, while the compressive carrying capacity of 0° laminates was significantly bigger than the 45° plies. Therefore, the instability caused by the different load of each ply leaded 45° plies to bear shear stress when the laminate was subjected to longitudinal compressive stress, resulting in a crack at 45° along the tangential direction of the hole. A certain degree of bending can be observed in the laminates. The main failure mode of brittle failure with matrix cracking was shown in Fig. 5.

In [45/90/−45/0]3s laminates, the load on each ply was relatively uniform, and the delamination between each single plies occurs in the transverse area through the hole. As shown in Fig. 6, during the loading process, the in-plane damage expanded in the delamination area, and the crack propagated along the transverse direction of laminate. The crack propagation was symmetrical in the transverse direction of the laminates, micro-buckling and push-in damage were the main failure modes in the [45/90/−45/0]3s laminates.

Compared with [453/03/−453/903]s laminates, the interior ply of [453/903/−453/03]s laminates was 0° unidirectional ply. Due to the micro-buckling and stacking of 0° unidirectional ply, the 45° ply under the stress pointed to the outside of the laminates, which resulted in buckling damage. Meanwhile, delamination damage occurred between 45°and 90° plies, and the shear stress caused by instability of the laminates contributed to brittle fracture of 45° plies. These two failure modes mixed together are shown in Fig. 7(a). Longitudinal cracks shown in Fig. 7(b) and (c) were due to the stress concentration caused by in-plane shear stress around the hole, which made the laminates produce longitudinal split along load direction. Similar experimental phenomena and results analysis were consistent with previous studies [20]. Brittle failure, delamination damage occurred in the laminates under compression, and the plies with different direction had different failure mode.

3Failure theoryThe compressive failure of composite laminates includes a variety of failure modes, mainly composed of in-plane damage and interface damage. The in-plane damage is divided into fiber fracture and matrix cracking. In this progressive damage, the damage initiation is determined by different criteria, and the damage evolution process is characterized by the energy conservation criterion. The final failure is determined by the critical energy release rate.

3.1Fiber failureMax-stress criterion is used to determine fiber damage initiation,

where Xc is the ply compressive strength in the fiber direction.When the loading stress reaches the compressive strength, the fiber damage begins to develop. In the damage process, the material constants such as E11, E22, G12 and ν12 degrade. Its degradation is determined based on the assumption that the total energy required for failing an element is equal to the energy required to create a crack passing through it. The strain energy released by a failed element is the area under the stress–strain curve, multiplied by a certain length called the characteristic element length lc. When this energy equals to the critical energy release rate of the composite material in the fiber direction Ψfc, this element is totally failed,

The product of ɛ11lc could also be regarded as the effective displacement for fiber damage.

3.2Matrix damageIn the experiment, the fiber failure mode is mainly fiber fracture, while the matrix failure modes include matrix cracking, buckling, stacking and mixing of the above modes. Because of the significant effect of matrix cracking on failure modes and strength of laminates, it is necessary to select different matrix failure criteria for analyzing. From the simulation results, the influence of different criteria on progressive damage of laminates can be obtained, which contributes to choose better criterion in further research.

3.2.1Tsai-Wu failure criterionWhen the initiation of matrix damage is determined by Tsai-Wu failure criterion, the following equation holds:

where Xt and Xc is the tensile and compressive strength in the fiber direction, respectively. Yt and Yc is the tensile and compressive strength in the transverse direction, respectively. S12 is the in-plane shear strength.3.2.2Hoffman failure criterionWhen the initiation of matrix damage is determined by Hoffman failure criterion, the following equation holds:

In order to characterize the matrix damage in the mixed failure mode accurately, it is necessary to consider not only the normal stress σ22 but also impact of the shear stress τ12. Therefore, the effective stress and displacement are defined,

The failure displacement could also be determined by the effective stress and displacement based on the energy criterion,

where the matrix mixed-mode fracture energy Ψmc is determined by the mixed-mode fracture energy criterion developed by Benzenggagh and Kenane,where Ψmc and Ψsc are the model I and model II fracture toughness, respectively. The parameter η is a material property determined by experiment.3.3Combination of fiber and matrix damageBased on both Tsai-Wu criterion and Hoffman criterion, it is assumed that during the damage process, the fiber damage occurs meanwhile matrix damage occurs. The two damage modes can be combined by the following formula,

where Γij represents material constants such as Eij, Gij, the subscript 0 indicates the undamaged material properties. As E11, E12, G12 and other material constants will continue to degrade during damage progresses, so the above formula introduces the degradation factor df, dm to characterize fiber and matrix damage.The stiffness degradation of fiber and matrix sometimes results in numerical instability and non-convergence during finite element analysis. To alleviate this problem, the linear curve after material damage in the stress–strain relationship could be displaced by a zigzagging curve, illustrated in Fig. 8. Based on this method, the tangent modulus could be positive at all times in the stress–strain curve, which make convergence easier to achieve.

Zigzag approximation of a linear softening curve [9].

In stress–strain curve, the slope of the curve represents the modulus of material. In the elastic stage, the slope of the curve is constant which satisfies Hooke's law. When damage occurs, the modulus of material Eij is discretized by the Zigzag method. At the same time, the area of the curve after discretization is basically consistent with the curve area without discretization, which ensures the accuracy of the energy conservation law to determine the final failure of the material.

In each iterative process, the change of material modulus is small as the strain changes slightly, so the value of material modulus of previous iteration cycle can be used to calculate stress at the current incremental step. Through this method, the degradation factor at current incremental step could be calculated and it forces computation results to a convergent solution.

3.4Delamination damageInterfacial delamination is assumed to initiate if the following quadratic failure criterion is satisfied,

where tn, ts and tt are the normal and two shear tractions, respectively. N and S are the normal and shear strengths of the interface.An energy criterion based on traction separation law is used to determine final failure,

where Ψif is the mixed fractured energy.The effective traction and displacement defined by the following equations,

where δn, δs and δt are the normal and two shear displacements.4Finite element models4.1OHC dataIn order to study the effects of different stacking sequence on the strength of composite laminates, the following three kinds of layup sequence are used, in which C1 and C3 are orthorhombic laminates and C2 is continuous laminate (Fig. 9; Table 2).

The geometric parameters of the effective damage area are shown in Fig. 10.

The FE model is established by ABAQUS, the ply is modeled using continuous shell element. To characterize delamination damage, a 0.01mm thick interlayer is added between each plies modeled by cohesive element. The material properties are as follow in Tables 3 and 4.

Composite ply properties.

Property | |
---|---|

Longitudinal tensile strength, Xt (MPa) | 2806 |

Longitudinal compressive strength, Xc (MPa) | 550 |

Transverse tensile strength, Yt (MPa) | 60 |

Transverse compressive strength, Yc (MPa) | 90 |

Longitudinal shear strength, S12 (MPa) | 90 |

Longitudinal stiffness, E11 (GPa) | 126 |

Transverse stiffness, E22 (GPa) | 8.7 |

Shear stiffness, G12 (GPa) | 5.17 |

Poisson's ratio, ν12 | 0.32 |

Poisson's ratio, ν23 | 0.436 |

Longitudinal toughness, Ψfc (kJ/m2) | 450.8 |

Transverse normal toughness, Ψnc (kJ/m2) | 0.2 |

Transverse shear toughness, Ψsc (kJ/m2) | 1 |

α | 1.39 |

Due to the symmetrical layup of the three laminates, only the half of the specimen was modeled for improving the computational efficiency. In this model, the composite laminates were modeled with 12 unidirectional plies with 0.1mm thick and the interlayer was modeled with 0.01mm thick. Each ply was modeled using a single layer of three-dimensional continuum shell element while cohesive element was adopted for the interface between two plies. In ABAQUS, SC8R and COH3D8 elements were used for the plies and interlayer, respectively.

Ridha et al. [9] pointed out that in numerical calculation, in order to satisfy the accuracy of Eqs. (2) and (11), the element size or the characteristic element length lc should meet the requirement of a certain size to guarantee that the elastic energy stored in the element is less than or equal to the fracture toughness of the material before damage occurs. The element size is usually determined by the fracture toughness of the matrix,

In order to satisfy the size requirement of the characteristic element length lc and improve computational efficiency at the same time, the laminates can be divided into several zones according to failure mode of the experimental specimen. Bias was applied to the seed arrangement processing, and the mesh size was as narrow as possible in the dangerous area, especially around the hole, which could improve the calculation accuracy. The ends of the laminate and other areas, which were not susceptible to damage, the mesh size can be slightly larger. In this model, two different meshing methods were designed for different laminates. The compressive experiment results showed that the cracks in the [453/03/−453/903]s laminates were mainly in the 45° direction, and the cracks in the [45/90/−45/0]3s laminates were extended transversely around the hole, while the failure modes were relatively single. Therefore, the elements were densely concentrated around the hole and transversely through the hole when meshing, which ensured the mesh number enough in the direction of the crack propagation shown in Fig. 11(a). The failure modes of [453/903/−453/03]s laminates were complicated, cracks propagated not only along 45° tangential direction of the hole, but also through the hole transversely and longitudinally. So the mesh size in all damage area was narrow, which was shown in Fig. 11(b). In the above two meshing methods, the mesh size around the hole and the danger area was 0.4mm, and the mesh size of end areas of the laminates and other damage areas was 0.75mm.

5Simulation results5.1Results with Tsai-Wu failure criterionThe simulation results with Tsai-Wu failure criterion are shown in Fig. 12 and Table 5.

It can be seen that the simulation strength was basically consistent with the experimental strength, and the error was within 10%. The strength of orthorhombic laminates was larger than the one of continuous laminates, and the strength of the C3 is stronger than the one of C1.

From the simulation results, the failure modes of the model were basically consistent with the experimental results, and the crack propagation direction coincided with the experiment. In C1, cracks appeared in the 45° tangential direction along the hole in 45° layers, and in inner plies fiber transverse fracture occurred in 0°and 90° layers. In C2, the crack of surface layers and inner layers propagated along the hole transversely. It was consistent with the experimental results that the cracks occurred transversely and the local micro buckling was observed around the cracks. It was proved that the cracks in orthorhombic laminates were not only propagated along the fiber direction, but also propagated along the transverse direction. In C3, the cracks of the surface ply not only appeared along the fiber direction, but also orthorhombic with the fiber direction. At the same time, a large area of fiber damage occurred in 0° ply (Figs. 13–15).

When the damage of laminates occurred, elements failed in a short time including the transverse fracture zone initiated by the fiber and the 45° crack of the matrix in the lower half of the laminates. In the damage evolution stage, symmetric damage occurred in the upper half of the laminate consistent with the lower half, and finally the laminate failed (Fig. 16).

Cohesive elements began to fail in elastic deformation stage of the laminates, but only limited to the local damage around the hole. At the beginning of the damage, the affected area around the hole start to expand and the damage became evident. In the short time after damage initiation, the cohesive elements began to fail rapidly through the transverse region of the ply, and the width of the damage zone was basically the same as the diameter of the hole. In the subsequent damage process, the area of the damage area had no obvious change, which indicated that the interlayer had completely failed in the short time after the damage initiated. Meanwhile, in the interlayer between 45°/0°, the expanding angle of the delamination area was the same as the 45° ply, which was developed along the 45° of the hole. In the interlayer between 0°/−45°and −45°/90° ply, delamination damage occurred in the middle of the ply, forming a delamination band, which was as wide as the hole.

5.2Results with Hoffman failure criterionIn order to study the influence of different damage criteria on simulation results, the criterion of matrix damage initiation is replaced by Hoffman criterion. Table 6 and Fig. 17 show the strength predicted by the Hoffman criterion and experimental results.

The influence of changing Tsai-Wu failure criterion with Hoffman failure criterion on laminate strength prediction is quite obvious. However, the energy criterion is still used in the determination of the final failure, so the influence of this change on the failure mode and failure mechanism of the material is little. The crack propagation and failure modes simulated by Tsai-Wu and Hoffman failure criteria were similar, and delamination area was also same with each other.

6ConclusionThe progressive damage model of composite laminate was developed to study the failure mechanism of open hole laminate under compression. The in-plane damage and delamination failure in interfacial layer are captured by this model, and the simulation results show good agreements with experimental results.

The experiment results show that the failure strength of the three laminates are as follows: [45/90/−45/0]3s>[453/903/−453/03]s>[453/03/−453/903]s. The strength of orthorhombic laminates is bigger than the one of continuous laminates.

The prediction with both Tsai-Wu and Hoffman failure criteria could obtain good strength prediction results within the allowable error. When laminate is subjected to the unidirectional compressive loading, the prediction result with Tsai-Wu failure criterion is more accurate. The initiation and propagation of cracks and the final failure modes simulated by two different matrix failure criteria are basically consistent with the experimental results.

Conflicts of interestThe authors declare no conflicts of interest.

This study was supported by the National Natural Science Foundation of China (No. 11402064), the State Key Program of National Natural Science Foundation of China (No. 11734017), the Fundamental Research Fund for the Central Universities (Grant No. HIT. NSRIF. 201623) and the Heilongjiang Postdoctoral Fund (No. LBH-Z13097).