Progressive failure analysis of notched composite plate by utilizing macro mechanics approach

Abstract

In this study, gradual and sudden reduction methods were combined to simulate a progressive failure in notched composite plates using a macro mechanics approach. Using the presented method, a progressive failure is simulated based on a linear softening law prior to a catastrophic failure, and thereafter, sudden reduction methods are employed for modeling a progressive failure. This combination method significantly reduces the computational cost and is also capable of simultaneously predicting the first and last ply failures (LPFs) in composite plates. The proposed method is intended to predict the first ply failure (FPF), LPF, and dominant failure modes of carbon/epoxy and glass/epoxy notched composite plates. In addition, the effects of mechanical properties and different stacking sequences on the propagation of damage in notched composite plates were studied. The results of the presented method were compared with experimental data previously reported in the literature. By comparing the numerical and experimental data, it is revealed that the proposed method can accurately simulate the failure propagation in notched composite plates at a low computational cost.

Graphical abstract

Keywords

progressive failure / notched composite plate / Hashin failure criterion / macro mechanics approach / finite element method

1 Introduction

The use of composite structures in different industries has increased significantly over the past few years. One of the most significant parameters in the usage of composite structures is the prediction of the initiation and propagation of failure in these materials. The propagation of damage in composite structures is extremely complex and depends on a large number of parameters, including the mechanical properties, stacking sequence, volume fraction, and layer orientation. Moreover, in composite structures, various failure mechanisms can be identified in the microstructure of a material at the same time. Owing to these difficulties, precise forecasting of progressive failure in composite laminates is complex and time consuming. In the past few years, many researchers have studied progressive failure modeling in composite structures [1–9]. In this regard, a progressive failure method for composite materials with elastic-brittle behavior was developed by Matzenmiller et al. [10]. They utilized five failure variables to reduce the elastic moduli in their model. To determine the proper input parameters in the numerical solution, they conducted sensitivity and uncertainty analyses. These effects have also been studied elsewhere [11,12]. Lapczyk and Hurtado [13] presented a progressive failure method for composite structures. With their proposed method, the plane stress equations are employed, and the behavior of the damaged structure is assumed to be linear. Four damage modes, including tension and compression in both the fiber and matrix, were simulated separately in the aforementioned model. Voyiadjis et al. [14] presented a micromechanical method for modeling progressive damage in a composite plate. This micromechanical method can simulate the damage behavior of a composite plate with good accuracy. Nobeen et al. [15] proposed a micromechanical method for calculating the progressive failure in braided fiber composites. Praud et al. [16] developed a hybrid micromechanical method for predicting the anisotropic progressive failure in unidirectional composites. Similar progressive failure models were developed by Barbero et al. [17], Wang et al. [18], and Papanikos et al. [19]. Damage and fracture propagation utilizing the screened Poisson equation and local mesh refinement have been studied by Areias et al. [20–22]. In addition, anisotropic softening elements for a finite strain fracture of 2D problems were presented by Areias et al. [23]. Efficient remeshing techniques for simulating crack propagation in numerical solutions have also been presented [24,25], and Rabczuk et al. [26–28] described a 3D mesh-free method for arbitrarily evolving cracks.

The study of progressive failure in notched composite plates is extremely limited in the literature in comparison with un-notched composite plates. In this regard, Moure et al. [29] investigated progressive failure in a composite plate with a hole under in-plane loads using a discrete damage method, which predicts the matrix damage evolution and fiber failure. The progressive damage in notched composite plates subjected to compression was studied by Su et al. [30]. In their study, in-plane and out-of-plane deformations were simulated with continuum shell elements. In addition, a two-dimensional study of progressive failure and residual strengths in notched composite structures was presented by Chang and Lessard [31], who employed the Yamada-Sun and Hashin failure criteria in their study.

Regarding the above literature review, a few studies have been conducted on the progressive failure of notched composite plates. In addition, the effects of the mechanical properties and different stacking sequences on the progressive failure of notched composite structures have not been considered in the analysis of such structures. Therefore, in this study, gradual and sudden reduction methods are combined for a simulation of progressive failure in notched composite plates using a macro mechanics approach. In the proposed method, the Hashin failure criterion and bilinear softening law/sudden reduction rules were employed for the initial and progressive failure modeling of notched composite structures, respectively. In addition, the effects of the mechanical properties and different stacking sequences on the progressive failure of notched composite plates were investigated. The results of the employed model were also verified based on experimental results reported in the literature. The advantages of the proposed method over other competitive methods are a low computational cost, high prediction accuracy, simultaneous predictions of the first and last ply failure (LPF), and dominant failure mode prediction in notched composite plates.

2 Constitutive model

The initiation and propagation of damage is a significant issue in the study of composite plates. A composite plate usually exhibits elastic-brittle behavior, which shows no substantial plastic deformation prior to damage occurrence. Therefore, a plastic deformation can be ignored in the failure modeling of the composite plates. Failure modeling in the composite plate was divided into two steps. In the first step, damage initiation models are based on the appropriate failure criteria. In the second step, damage models are propagated using a macro-mechanics approach.

2.1 Damage initiation modeling of composite plate

The initiation of damage in the composite plate indicates the beginning of the stiffness reduction in the specific layer. Many failure criteria are available for the prediction of damage initiation in composite materials [32–37]. In this study, the Hashin failure criterion is selected. Hashin’s failure criterion is widely used in numerous industries such as aerospace and automobile industries. Hashin’s failure criterion is widely reported in the literature [38–42] and in most cases, it has a good prediction of the onset of failure in composite laminates.

Hashin’s failure criterion is expressed as follows:

The fiber tension ({\sigma }_x\ge \mathrm{0}) is expressed as

(1){\left({\displaystyle \frac{{\sigma }_x}{X_{\mathrm{T}}}}\right)}^{\mathrm{2}}+\alpha {\left({\displaystyle \frac{{\tau }_{\mathit{xy}}}{S}}\right)}^{\mathrm{2}}\ge \mathrm{1.}

The fiber compression is written as

(2){\left({\displaystyle \frac{{\sigma }_x}{X_{\mathrm{C}}}}\right)}^{\mathrm{2}}\ge \mathrm{1.}

The matrix tension ({\sigma }_y\ge \mathrm{0}) is formulated as

(3){\left({\displaystyle \frac{{\sigma }_y}{Y_{\mathrm{T}}}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{{\tau }_{\mathit{xy}}}{S}}\right)}^{\mathrm{2}}\ge \mathrm{1.}

Finally, the matrix compression is expressed as

(4){\left({\displaystyle \frac{{\sigma }_y}{\mathrm{2}S}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{{\tau }_{\mathit{xy}}}{S}}\right)}^{\mathrm{2}}+\left[{\left({\displaystyle \frac{Y_{\mathrm{C}}}{\mathrm{2}S}}\right)}^{\mathrm{2}}-\mathrm{1}\right]{\displaystyle \frac{{\sigma }_y}{Y_{\mathrm{C}}}}\ge \mathrm{1.}

In the above equations, σ_x, σ_y, and τ_xy represent the normal stress in the x-direction, normal stress in the y-direction, and in-plane shear stress components, respectively. In addition, X, Y, and S represent the strengths in the x-direction, strengths in the y-direction, and in-plane shear strengths, respectively. Subscripts “T” and “C” refer to tensile and compressive conditions. In addition, the coefficient α controls the effect of shear stress in the fiber tension failure mode.

2.2 Progressive failure modeling of composite plate

Once damage initiation has been predicted, progressive failure methods are employed to forecast the LPF in the composite plate. Progressive failure methods have been applied to reduce the stiffness of the damaged plies. There are many degradation theories in the literature for the simulation of progressive failure [43–48]. Such theories can be categorized into sudden and gradual methods. With sudden degradation methods, certain mechanical properties of the damaged ply immediately decrease to zero. Although the accuracy degradation methods is usually inadequate, these methods are widely used owing to their low computational cost. By contrast, with gradual degradation methods the mechanical properties of the failed ply gradually decrease when using a macro mechanics approach. These methods can precisely simulate failure in composite materials. However, the computational cost of these methods is high. Consequently, the most accurate results with minimum computational cost are achieved through a combination of gradual and sudden reduction methods. In this regard, gradual and sudden reduction methods are combined for the simulation of progressive failure in notched composite plates.

With a method using a macro mechanics approach, the behavior of the composite plate is considered linearly elastic prior to failure initiation. After failure initiation and before catastrophic failure (fiber breakage), the behavior of the composite plate was calculated as follows:

(5)\sigma =C_{\mathrm{d}}\epsilon ,

where C_{\mathrm{d}} and \epsilon are the damaged elasticity and strain matrix, respectively, and C_{\mathrm{d}} is defined as follows [18]:

(6)C_{\mathrm{d}}={\displaystyle \frac{\mathrm{1}}{D}}\left[\begin{array}{lll}(\mathrm{1}-d_{\mathrm{f}})E_x& (\mathrm{1}-d_f)(\mathrm{1}-d_{\mathrm{m}}){\upsilon }_{\mathit{yx}}{E}_{\mathit{x}}& \mathrm{0}\\ (\mathrm{1}-d_{\mathrm{f}})(\mathrm{1}-d_{\mathrm{m}}){\upsilon }_{\mathit{xy}}{E}_y& (\mathrm{1}-d_{\mathrm{m}})E_{\mathit{y}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& (\mathrm{1}-d_{\mathrm{s}})GD\end{array}\right],

where d_{\mathrm{f}}, d_{\mathrm{m}}, and d_{\mathrm{s}} represent the failure variables related to the fiber, matrix, and shear damage modes, respectively; E_x and E_y represent the Young’s moduli in the fiber and matrix directions, respectively; G refers to the shear modulus; {\upsilon }_{\mathit{xy}} and {\upsilon }_{\mathit{yx}} correspond to the Poisson’s ratios; and D is also defined as follows:

(7)D=1-(1-d_{\mathrm{f}})(1-d_{\mathrm{m}}){\upsilon }_{\mathit{x}\mathit{y}}{\upsilon }_{\mathit{y}\mathit{x}}\mathrm{.}

Before a catastrophic failure, the failure variables d_{\mathrm{f}}, d_{\mathrm{m}}, and d_{\mathrm{s}} are determined by the failure variables d_{\mathrm{f}}^{\mathrm{t}}, d_{\mathrm{f}}^{\mathrm{c}}, d_{\mathrm{m}}^{\mathrm{t}}, and d_{\mathrm{m}}^{\mathrm{c}}, associated with different failure modes, in the following manner [18]:

(8)

The failure variables in the above equations can be varied between zero and 1. After a catastrophic failure, the failure variables immediately increase to 1. The failure variable in the employed method will progress based on the stress-displacement behavior, as shown in Fig. 1. Based on this figure, after damage initiation and before a catastrophic failure, the propagation of damage is defined through a linear equation, and the mechanical properties of the damaged plies are reduced based on linear material softening rules. After a catastrophic failure, the mechanical properties of the damaged plies immediately decrease to zero.

In the employed method, the equivalent displacement and stress before a catastrophic failure in various failure modes are calculated as follows:

The fiber tension ({\sigma }_x\ge \mathrm{0}) [18] is expressed as

(9)\begin{array}{l}{\delta }_{\mathrm{e}\mathrm{q}}^{\mathrm{f}\mathrm{t}}=L_c\sqrt{{\langle {\epsilon }_{\mathit{x}}\rangle }^{\mathrm{2}}+\beta {\epsilon }_{\mathit{xy}}^{\mathrm{2}}},\\ {\sigma }_{\mathrm{e}\mathrm{q}}^{\mathrm{f}\mathrm{t}}={\displaystyle \frac{\langle {\sigma }_{\mathit{x}}\rangle \langle {\epsilon }_{\mathit{x}}\rangle +\beta {\tau }_{\mathit{xy}}{\epsilon }_{\mathit{xy}}}{{\delta }_{\mathrm{e}\mathrm{q}}^{\mathrm{f}\mathrm{t}}/L_{\mathrm{c}}}},\end{array}

The fiber compression [18] is written as

(10)\begin{array}{l}{\delta }_{\mathrm{e}\mathrm{q}}^{\mathrm{f}\mathrm{c}}=L_{\mathrm{c}}\langle -{\epsilon }_x\rangle ,\\ {\sigma }_{\mathrm{e}\mathrm{q}}^{\mathrm{f}\mathrm{c}}={\displaystyle \frac{\langle -{\sigma }_{\mathit{x}}\rangle \langle -{\epsilon }_{\mathit{x}}\rangle }{{\delta }_{\mathrm{e}\mathrm{q}}^{\mathrm{f}\mathrm{c}}/L_{\mathrm{c}}}}\mathrm{.}\end{array}

The matrix tension ({\sigma }_{\mathit{y}}\ge \mathrm{0}) [18] is composed as

(11)\begin{array}{l}{\delta }_{\mathrm{e}\mathrm{q}}^{\mathrm{m}\mathrm{t}}=L_{\mathrm{c}}\sqrt{{\langle {\epsilon }_{\mathit{y}}\rangle }^{\mathrm{2}}+{\epsilon }_{\mathit{xy}}^{\mathrm{2}}},\\ {\sigma }_{\mathrm{e}\mathrm{q}}^{\mathrm{m}\mathrm{t}}={\displaystyle \frac{\langle {\sigma }_{\mathit{y}}\rangle \langle {\epsilon }_{\mathit{y}}\rangle +{\tau }_{\mathit{xy}}{\epsilon }_{\mathit{xy}}}{{\delta }_{\mathrm{e}\mathrm{q}}^{\mathrm{m}\mathrm{t}}/L_{\mathrm{c}}}}\mathrm{.}\end{array}

The matrix compression [18] is formulated as

(12)\begin{array}{l}{\delta }_{\mathrm{e}\mathrm{q}}^{\mathrm{m}\mathrm{c}}=L_{\mathrm{c}}\sqrt{{\langle -{\epsilon }_{\mathit{y}}\rangle }^{\mathrm{2}}+{\epsilon }_{\mathit{xy}}^{\mathrm{2}}}\\ {\sigma }_{\mathrm{e}\mathrm{q}}^{\mathrm{m}\mathrm{c}}={\displaystyle \frac{\langle -{\sigma }_{\mathit{y}}\rangle \langle -{\epsilon }_{\mathit{y}}\rangle +{\tau }_{\mathit{xy}}{\epsilon }_{\mathit{xy}}}{{\delta }_{\mathrm{e}\mathrm{q}}^{\mathrm{m}\mathrm{c}}/L_{\mathrm{c}}}}\end{array},

where L_{\mathrm{c}} represents the characteristic length, and the symbol \langle \rangle refers to the Macaulay bracket operator. The characteristic length is utilized to reduce the mesh dependency during the softening regime under the progressive damage simulation, and is regarded as the square root of the element reference surface area for shell and membrane elements. Thus, the constitutive model is transformed from the stress–strain state into a stress-displacement space to alleviate the mesh dependency. This effect of the characteristic length has been addressed in several studies [2,13,49].

With the proposed method, the variable prior to a catastrophic failure for a specific failure mode is defined as follows [18]:

(13)d={\displaystyle \frac{{\delta }_{\mathrm{eq}}^{\mathrm{f}}({\delta }_{\mathrm{eq}}-{\delta }_{\mathrm{eq}}^0)}{{\delta }_{\mathrm{eq}}({\delta }_{\mathrm{eq}}^{\mathrm{f}}-{\delta }_{\mathrm{eq}}^0)}},

where {\delta }_{\mathrm{eq}}^0 represents the initial equivalent displacement at which the initiation criterion for the mentioned mode was reached. In addition, {\delta }_{\mathrm{eq}}^{\mathrm{f}} represents the displacement at which the material fully fails in this mode.

3 FEM implementation of the constitutive model

In this section, the FEM implementation of the constitutive model proposed in this paper for the numerical modeling of a progressive failure in a notched composite plate is described. The constitutive models employed in the previous section were conducted in a FORTRAN routine, and an innovative executable ANSYS file was produced, considerably reducing the time of the solution. The algorithm utilized for the numerical modeling of a progressive failure in a notched composite plate is categorized into four steps:

1) a finite element simulation of the considered notched composite plate (4 node structural Shell 181 was utilized);

2) stress analysis of composite plate under applied load;

3) failure study of composite plate based on Hashin failure criterion;

4) decrease of mechanical properties of damaged plies by utilizing described progressive failure theory.

In other words, in the employed model, the applied loads in the composite plate increase step-by-step. In each step, a stress calculation is conducted, and the damage is cheeked utilizing the Hashin failure criterion. If no damage occurs, the applied load increases for the next step; otherwise, the mechanical properties of the damage plies are decreased based on the theory described in Section 2. This process continues until the LPF occurs. A flowchart of the progressive failure method employed for notched composite plates is shown in Fig. 2.

Standard specimens were used for the simulations of progressive failure in the notched composite plate. The geometry of the specimens considered in the finite element modeling is shown in Fig. 3.

The loading and boundary conditions implemented during the finite-element simulations are illustrated in Fig. 4. As can be seen, the 6-DOF of all nodes in the outer areas of the left tabs were constrained.

To investigate the effects of the stacking sequence on the propagation of damage in the notched composite plates, different lay-up configurations were modeled using the ANSYS code. The stacking sequences considered for the progressive failure simulation of the notched composite plates are shown in Table 1.

To investigate the effects of mechanical properties on the propagation of failure in notched composite plates, carbon/epoxy and glass/epoxy composite plates were modeled using the ANSYS code. The mechanical properties of the carbon/epoxy and glass/epoxy composites are listed in Table 2. Fracture energies are defined as the dissipated energy owing to a failure in each failure mode. In the numerical simulation of a progressive failure, the values of {\delta }_{\mathrm{eq}}^{\mathrm{f}} for the various modes were determined from the respective fracture energy values. Four fracture energies were defined for the composite materials. A discussion on the measurement of the fracture energies and an evaluation of these parameters through the use of standard tests is given elsewhere [50–52].

A mesh refinement in the numerical solution of a progressive failure has an important effect on the predicted strength of the materials. An accurate prediction is achieved when the strength of the materials is independent of the mesh size. A convergence analysis should be conducted to determine the optimum size of the elements to ensure the accuracy of the numerical solution. In this regard, the force-displacement behavior of the glass/epoxy notched composite plate with a lay-up configuration {[{\mathrm{45}}_{\mathrm{8}}]}_{\mathrm{T}} and different numbers of elements is illustrated in Fig. 5. As can be seen, the force-displacement behavior of the glass/epoxy notched composite plate is independent of the number of elements after 8456 elements and the numerical solution after 8456 elements converge. Therefore, in this study, the number of elements was considered to be 8456.

3.1 Viscous regularization

The material modeling with a softening response and stiffness reduction has frequently led to rigorous convergence problems in implicit studies [54]. Some of these problems will be facilitated by utilizing the viscous regularization method developed by Duvant and Lions [55]. With this method, the traction-separation law allows stresses to be outside the limits set by the law. The use of viscous regularization in addition to the characteristic length can facilitate the convergence of results under a progressive damage simulation quite effectively. In a viscous regularization method, the viscous failure variable is expressed as follows:

(14){\dot{d}}_v={\displaystyle \frac{\mathrm{1}}{\eta }}(d-d_v)

where \eta represents the viscosity coefficient and d corresponds to the failure variable. As the fundamental theory, the solution of the viscous model relaxes to that of the viscid case as {\displaystyle \frac{t}{\eta }}\to \infty, where t denotes time. The viscous regularization parameter should be chosen carefully. Extremely small values for this parameter lead to an overestimation of the actual strength of the material, and large values for this parameter lead to a high computational cost and more difficulties in convergence. The optimized value for this parameter was obtained from a comparative study of the viscous regularization parameter. The influence of the viscous regularization parameter on the force–displacement curves of a glass/epoxy notched composite plate with a lay-up configuration {[{\mathrm{90}}_{\mathrm{8}}]}_{\mathrm{T}} is illustrated in Fig. 6. This figure shows that the viscous regularization parameter should be selected as small as possible to avoid an overestimation, and large enough to provide a proper computational cost. In this study, a value of \eta =\mathrm{0.001} was chosen.

4 Results and discussion

4.1 Prediction of first ply failure

To predict the first ply failure (FPF), a unit load is employed, and the calculated stresses are fed into the Hashin failure criterion. The results of the FPF prediction, including the critical load and ultimate strength for FPF in the glass/epoxy and carbon/epoxy-notched composite plates with different lay-up configurations are shown in Table 3. According to this table, in both the glass/epoxy and carbon/epoxy composite plates, the configurations {[{\mathrm{0}}_{\mathrm{8}}]}_{\mathrm{T}} and {[{\mathrm{90}}_{\mathrm{8}}]}_{\mathrm{T}} have the maximum and minimum resistance against FPF, respectively. In addition, it is shown that except for the configuration {[{\mathrm{0}}_{\mathrm{8}}]}_{\mathrm{T}}, cross-ply and quasi-isotropic configurations have a maximum strength against the FPF in the glass/epoxy and carbon/epoxy composite plates, respectively.

4.2 Prediction of last ply failure

After predicting the FPF, the proposed progressive failure model was employed for the prediction of the LPF. The results of the LPF prediction, including the critical load and ultimate strength for the LPF in the glass/epoxy and carbon/epoxy-notched composite plates with different lay-up configurations are shown in Table 4. As shown in Table 4, in all cases, the configurations {[{\mathrm{0}}_{\mathrm{8}}]}_{\mathrm{T}} and {[{\mathrm{90}}_{\mathrm{8}}]}_{\mathrm{T}} have a maximum and minimum resistance against LPF, respectively. Moreover, it is shown that a 0° ply has significant effects on the LPF in comparison with other angles. For example, the ultimate strength of the configuration {[\mathrm{0}/\mathrm{90}/\mathrm{0}/\mathrm{90}]}_{\mathrm{S}}with four 0° plies for both glass/epoxy and carbon/epoxy notched composite plates is approximately half that of the configuration {[{\mathrm{0}}_{\mathrm{8}}]}_{\mathrm{T}}with eight 0° plies. In addition, it was found that, excluding the configuration {[{\mathrm{0}}_{\mathrm{8}}]}_{\mathrm{T}}, cross-ply and quasi-isotropic configurations have a maximum strength against the LPF in glass/epoxy and carbon/epoxy-notched composite plates, respectively.

A comparison of the first and LPFs obtained from the method proposed in this study with those of other proposed methods [5] for T300/1304-C laminates is presented in Table 5. According to this table, good agreements between the first and LPFs obtained from this study and other methods proposed in the literature were achieved. Therefore, the method proposed in this study can accurately predict the first and LPFs of notched composite plates.

The contour plots of different failure variables in the glass/epoxy-notched composite plate are shown in Figs. 7–12. By analyzing the failure variables in the composite plate, the failure mode can be determined. In this regard, according to Fig. 7, the dominant failure mode in the lay-up configuration {[{\mathrm{0}}_{\mathrm{8}}]}_{\mathrm{T}} is fiber breakage. The failure mode in the lay-up configuration {[{\mathrm{45}}_{\mathrm{8}}]}_{\mathrm{T}} is matrix cracking. In this configuration, no damage to the fiber was observed. The failure modes in the lay-up configuration {[{\mathrm{90}}_{\mathrm{8}}]}_{\mathrm{T}} were matrix cracking and shear modes. As shown in Figs. 10 and 11, matrix cracking is the dominant failure mode in the lay-up configurations {[\mathrm{0}/\mathrm{90}/\mathrm{0}/\mathrm{90}]}_{\mathrm{S}} and {[\mathrm{30}/-\mathrm{30}/\mathrm{30}/-\mathrm{30}]}_{\mathrm{S}}. Finally, in the lay-up configuration {[\mathrm{0}/\mathrm{90}/\pm \mathrm{45}]}_{\mathrm{S}}, a combination of fiber, matrix, and shear failure modes was simultaneously observed.

The contour plots of different failure variables in the carbon/epoxy-notched composite plate are shown in Figs. 13–18. The dominant failure modes in the carbon/epoxy-notched composite plate are similar to the occurrence failure modes in the glass/epoxy, as described previously. Figures 7–18 show the weakest ply in each lay-up configuration. Trends of the failure variables in other plies are similar to the presented figures.

The computational cost of the combination method employed in the comparison of the gradual degradation methods for modeling progressive failure in notched composite plates is shown in Table 6. As expected, the method employed reduces the computational cost by up to 46%. The elimination of nonlinear calculations in the softening regime in the progressive damage model after a catastrophic failure and the utilization of the optimum viscous regularization parameter are the main reasons for the reduction in computational cost in the proposed combination method.

The progressions of the fiber, matrix, and shear failure variables in the glass/epoxy and carbon/epoxy-notched composite plates with different lay-up configurations are plotted in Figs. 19 and 20, respectively. By attending these figures, some points can be discovered. The first point is related to the angle ply laminates. In these laminates, regardless of the mechanical properties, fiber failure starts when the plies are completely damaged in the matrix direction. This trend was also observed in the lay-up configurations {[{\mathrm{45}}_{\mathrm{8}}]}_{\mathrm{T}} and {[{\mathrm{90}}_{\mathrm{8}}]}_{\mathrm{T}}. The second point is related to cross-ply laminates. In these laminates, a fiber breakage starts when the matrix has not fully failed. The third point is related to the quasi-isotropic laminates. In these laminates, the three failure variables progressed simultaneously. Finally, in the lay-up configuration {[{\mathrm{90}}_{\mathrm{8}}]}_{\mathrm{T}} matrix, the shear failure variables will progress simultaneously, and the failure variables in the fiber direction will not progress.

A comparison of the predicted damage patterns in the glass/epoxy and carbon/epoxy-notched composite plates with different lay-up configurations is illustrated in Fig. 21. As can be seen, it is clear that damage initiates and propagates near the hole because of stress concentration near the notch. In addition, by comparing the predicted damage patterns in glass/epoxy and carbon/epoxy, it was revealed that the mechanical properties have insignificant effects on the damage pattern in the composite plate. By contrast, the lay-up configuration has significant effects on the damage pattern in the composite plate.

5 Verification of proposed model

In this section, the verification of the method employed in this study for a progressive failure simulation in a notched composite plate is presented. In this regard, experimental results obtained by Wisnom et al. [56] were used. A comparison of the experimental results obtained by Wisnom et al. [56] and the numerical tension strength obtained from the method proposed in this study for the different laminate specimens are presented in Table 7. Based on this table, good agreement between the experimental and numerical results was obtained. Thus, the method proposed in this study can accurately predict the tension strength of notched composite plates.

In addition, the experimental and numerically predicted damage patterns for the carbon/epoxy notched composite laminate with configuration {[{\mathrm{0}}_{\mathrm{8}}]}_{\mathrm{T}} are compared. In this regards, the failure mode observed in the experiment in [56] is a brittle failure straight across the hole. The damage pattern predicted from the model employed in this study is also a brittle failure straight across the hole. As shown in Fig. 7, the damage pattern predicted with the employed model is quite similar to the experimental observation [56]. Therefore, the method employed in this study can also successfully predict the progressive damage pattern in a notched composite plate with extremely high accuracy.

6 Conclusions

In this study, a progressive failure of the notched composite plates was investigated numerically. In this regard, progressive failure in the notched composite was simulated with a linear softening law and sudden reduction rules before and after a catastrophic failure, respectively. This combination method significantly reduces the computational cost compared to other methods for modeling progressive failure in notched composite plates. The performance of the employed method was investigated by modeling a progressive failure in glass/epoxy and carbon/epoxy-notched composite plates. FPF, LPF, and dominant failure modes of the laminates were predicted using the employed method. Four failure modes, including tension and compression in both the fiber and matrix, were considered and simulated separately. In addition, the effects of the mechanical properties and different stacking sequences on the progressive failure of notched composite plates were studied. The results showed that the mechanical properties and lay-up configuration had insignificant and significant effects on the damage pattern in the composite plate. In addition, it was shown that in angle ply laminates, regardless of the mechanical properties, a fiber failure starts when plies are completely damaged in matrix mode. This trend was also observed in the lay-up configurations {[{\mathrm{45}}_{\mathrm{8}}]}_{\mathrm{T}} and {[{\mathrm{90}}_{\mathrm{8}}]}_{\mathrm{T}}. In cross-ply laminates, fiber breakage begins when the matrix does not fully fail. In quasi-isotropic laminates, the three failure variables will progress simultaneously. The results of the employed model were also verified against the experimental data reported in the literature. The results revealed that the employed technique can precisely predict the progressive failure in a notched composite plate at a low computational cost.

References

[1] Cárdenas D, Elizalde H, Marzocca P, Abdi F, Minnetyan L, Probst O. Progressive failure analysis of thin-walled composite structures. Composite Structures, 2013, 95: 53–62

[2] Rafiee R, Torabi M A. Stochastic prediction of burst pressure in composite pressure vessels. Composite Structures, 2018, 185: 573–583

[3] Irisarri F X, Laurin F, Carrere N, Maire J F. Progressive damage and failure of mechanically fastened joints in CFRP laminates—Part II: Failure prediction of an industrial junction. Composite Structures, 2012, 94(8): 2278–2284

[4] Jordan J B, Naito C J, Haque B Z. Progressive damage modeling of plain weave E-glass/phenolic composites. Composites. Part B, Engineering, 2014, 61: 315–323

[5] Lee C S, Kim J H, Kim S, Ryu D M, Lee J M. Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method. Composite Structures, 2015, 121: 406–419

[6] Fakoor M, Mohammad Navid Ghoreishi S. Experimental and numerical investigation of progressive damage in composite laminates based on continuum damage mechanics. Polymer Testing, 2018, 70: 533–543

[7] Vo-Duy T, Nguyen-Minh N, Dang-Trung H, Tran-Viet A, Nguyen-Thoi T. Damage assessment of laminated composite beam structures using damage locating vector (DLV) method. Frontiers of Structural and Civil Engineering, 2015, 9(4): 457–465

[8] Fakoor M, Ghoreishi S M N, Khansari N M. Investigation of composite coating effectiveness on stress intensity factors of cracked composite pressure vessels. Journal of Mechanical Science and Technology, 2016, 30(7): 3119–3126

[9] Fakoor M, Ghoreishi S M N. Verification of a micro-mechanical approach for the investigation of progressive damage in composite laminates. Acta Mechanica, 2019, 230(1): 225–241

[10] Matzenmiller A, Lubliner J, Taylor R. A constitutive model for anisotropic damage in fiber-composites. Mechanics of Materials, 1995, 20(2): 125–152

[11] Hamdia K M, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

[12] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[13] Lapczyk I, Hurtado J A. Progressive damage modeling in fiber-reinforced materials. Composites. Part A, Applied Science and Manufacturing, 2007, 38(11): 2333–2341

[14] Voyiadjis G Z, Taqieddin Z N, Kattan P I. Micromechanical approach to damage mechanics of composite materials with fabric tensors. Composites. Part B, Engineering, 2007, 38(7–8): 862–877

[15] Nobeen N S, Zhong Y, Francis B A P, Ji X, Chia E S M, Joshi S C, Chen Z. Constituent materials micro-damage modeling in predicting progressive failure of braided fiber composites. Composite Structures, 2016, 145: 194–202

[16] Praud F, Chatzigeorgiou G, Chemisky Y, Meraghni F. Hybrid micromechanical-phenomenological modelling of anisotropic damage and anelasticity induced by micro-cracks in unidirectional composites. Composite Structures, 2017, 182: 223–236

[17] Barbero E, Cosso F, Roman R, Weadon T. Determination of material parameters for Abaqus progressive damage analysis of E-glass epoxy laminates. Composites. Part B, Engineering, 2013, 46: 211–220

[18] Wang L, Zheng C, Luo H, Wei S, Wei Z. Continuum damage modeling and progressive failure analysis of carbon fiber/epoxy composite pressure vessel. Composite Structures, 2015, 134: 475–482

[19] Papanikos P, Tserpes K, Labeas G, Pantelakis S. Progressive damage modelling of bonded composite repairs. Theoretical and Applied Fracture Mechanics, 2005, 43(2): 189–198

[20] Areias P, Msekh M, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143

[21] Areias P, Reinoso J, Camanho P, César de Sá J C, Rabczuk T. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Engineering Fracture Mechanics, 2018, 189: 339–360

[22] Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350

[23] Areias P, Rabczuk T, Camanho P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63

[24] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

[25] Areias P, Rabczuk T. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41

[26] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[27] Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799

[28] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455

[29] Moure M, Otero F, García-Castillo S, Sánchez-Sáez S, Barbero E, Barbero E. Damage evolution in open-hole laminated composite plates subjected to in-plane loads. Composite Structures, 2015, 133: 1048–1057

[30] Su Z, Tay T, Ridha M, Chen B. Progressive damage modeling of open-hole composite laminates under compression. Composite Structures, 2015, 122: 507–517

[31] Chang F K, Lessard L B. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part I—Analysis. Journal of Composite Materials, 1991, 25(1): 2–43

[32] Hashin Z, Rotem A. A fatigue failure criterion for fiber reinforced materials. Journal of Composite Materials, 1973, 7(4): 448–464

[33] Tsai S W, Wu E M. A general theory of strength for anisotropic materials. Journal of Composite Materials, 1971, 5(1): 58–80

[34] Sun C, Tao J. Prediction of failure envelopes and stress/strain behaviour of composite laminates1. Composites Science and Technology, 1998, 58(7): 1125–1136

[35] Puck A, Kopp J, Knops M. Guidelines for the determination of the parameters in Puck’s action plane strength criterion. Composites Science and Technology, 2002, 62(3): 371–378

[36] Cuntze R, Freund A. The predictive capability of failure mode concept-based strength criteria for multidirectional laminates. Composites Science and Technology, 2004, 64(3–4): 343–377

[37] He P, Shen Y, Gu Y, Shen P. 3D fracture modelling and limit state analysis of prestressed composite concrete pipes. Frontiers of Structural and Civil Engineering, 2019, 13(1): 165–175

[38] Nali P, Carrera E. A numerical assessment on two-dimensional failure criteria for composite layered structures. Composites. Part B, Engineering, 2012, 43(2): 280–289

[39] Yang B, Yue Z, Geng X, Wang P, Gan J, Liao B. Effects of space environment temperature on the mechanical properties of carbon fiber/bismaleimide composites laminates. Proceedings of the Institution of Mechanical Engineers. Part G, Journal of Aerospace Engineering, 2018, 232(1): 3–16

[40] Liu P, Zheng J. Recent developments on damage modeling and finite element analysis for composite laminates: A review. Materials & Design, 2010, 31(8): 3825–3834

[41] Catalanotti G, Camanho P, Marques A. Three-dimensional failure criteria for fiber-reinforced laminates. Composite Structures, 2013, 95: 63–79

[42] Weng J, Wen W, Zhang H. Study on low-velocity impact and residual strength at high temperatures of composite laminates. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. 2017, 233(3): 1106–1123

[43] Maimí P, Camanho P P, Mayugo J, Dávila C. A continuum damage model for composite laminates: Part I—Constitutive model. Mechanics of Materials, 2007, 39(10): 897–908

[44] Baaran J, Kärger L, Wetzel A. Efficient prediction of damage resistance and tolerance of composite aerospace structures. Proceedings of the Institution of Mechanical Engineers, Part G, Journal of Aerospace Engineering, 2008, 222(2): 179–188

[45] Zhang B, Zhao L. Progressive damage and failure modeling in fiber-reinforced laminated composites containing a hole. International Journal of Damage Mechanics, 2012, 21(6): 893–911

[46] Zhang X. Impact damage in composite aircraft structures-experimental testing and numerical simulation. Proceedings of the Institution of Mechanical Engineers. Part G, Journal of Aerospace Engineering, 1998, 212(4): 245–259

[47] Liu P F, Zheng J. Progressive failure analysis of carbon fiber/epoxy composite laminates using continuum damage mechanics. Materials Science and Engineering A, 2008, 485(1–2): 711–717

[48] Mills A, Jones J. Investigation, manufacture, and testing of damage-resistant airframe structures using low-cost carbon fibre composite materials and manufacturing technology. Proceedings of the Institution of Mechanical Engineers. Part G, Journal of Aerospace Engineering, 2010, 224(4): 489–497

[49] Liu P F, Chu J, Hou S, Xu P, Zheng J. Numerical simulation and optimal design for composite high-pressure hydrogen storage vessel: A review. Renewable & Sustainable Energy Reviews, 2012, 16(4): 1817–1827

[50] Bažant Z P, Pfeiffer P A. Determination of fracture energy from size effect and brittleness number. ACI Materials Journal, 1987, 84(6): 463–480

[51] Lange F, Radford K. Fracture energy of an epoxy composite system. Journal of Materials Science, 1971, 6(9): 1197–1203

[52] Bažant Z P, Kazemi M T. Determination of fracture energy, process zone longth and brittleness number from size effect, with application to rock and conerete. International Journal of Fracture, 1990, 44(2): 111–131

[53] Guide AMUs. Engineering Data User’s Guide. Material Data, Mechanical Material Curve Fitting. Los Angeles: ANSYS, Inc., 2017

[54] Hibbitt H, Karlsson B, Sorensen P. Abaqus Analysis User’s Manual Version 6.10. Providence, RI: Dassault Systèmes Simulia Corp, 2011

[55] Duvant G, Lions J L. Inequalities in Mechanics and Physics. Berlin: Springer Science & Business Media, 2012

[56] Wisnom M, Hallett S, Soutis C. Scaling effects in notched composites. Journal of Composite Materials, 2010, 44(2): 195–210

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