Keywords

1 Introduction

The probability of failure in a laminated composite can be quantified by the amount of load it can withstand until its fracture point. Due to the inheritance of randomness and uncertainty in material and geometrical properties, stochastic regime seems to be an efficient way of modelling the failure analysis as compared to the deterministic way. An alternative approach for the analyses necessitating monotonous model evaluation is the utilisation of approximation models, also referred to as metamodels [1]. A result found by Onkar et al. [2], where first-ply failure load is analysed to evaluate the mean and variance of the failure statistics showed that the stochastic finite element has high accuracy. An experimental investigation designed by Reddy and Pandey [3], who are considered as the pioneer of failure analysis of composites conducted computational, as well as numerical investigation on laminated composite, with different ply orientation, ply angle with in-plane and out-of-plane failure load.

Laminated composite plates failure can be seen by debonding or delamination, fibre pull-out, fibre breakage and matrix cracking. These modes of failure are major limitations of laminated composite plate. Delamination is the most common mode of failure among all other kinds of failure. These modes of failure are major limitations of laminated composite. Many researchers investigated on deterministic failure analysis of laminated composite plate. ANN incorporated in analysing the probability of delamination of laminated composite by Chakraborty [4], highlighted that the trained network can predict the delamination for different shape, size and location with less computational time. ANN is applied to predict the kinetic parameters like high-velocity impact on a carbon reinforced fibre composites (CRFC) [5]. In a particular case where split growth in the notched composite is analysed under constant amplitude fatigue by using ANN and power law, ANN proved to be a better predictive tool than power law [6]. End milling process of a Glass fibre reinforced plastic (GFRP) composites ANN provided significant performance increase in analysing the damage factor by developing five learning algorithms and training them which contributed in the reduction of cost as well as time in conducting experiment [7]. Prediction of impact location using ANN during damage based on the kinetic energy of an impact by incorporating the limited strain signatures as inputs provided a warning system in damage initiation [8]. In another instance, prediction for impact resistance of aluminium–epoxy-laminated composites were analysed and it was found that ANN can be used as a substitutive approach to evaluate the effect of bonding strength of laminated composites [9]. ANN also proved to a better option in analysing the failure of a cross-ply composite tube under torsion, as well as axial tension/compression compared to Tsai-Wu theory and tensor polynomial theory [10]. Some of the important works done on the first-ply failure of laminated composites and ANN are hereby mentioned [11,12,13,14,15,16]. The present study deals with uncertainty quantification for first-ply failure of laminated composite plates by incorporating ANN as surrogate model to reduce computational time.

2 Mathematical Formulation

In order to determine the first-ply failure of the laminated composite five failure criteria are taken into consideration as mentioned earlier. The finite element (FE) model is designed based on the failure criteria followed by this the surrogate model is implemented using ANN.

2.1 Failure Criteria for Laminated Composite

In the present work, a three-layered laminated composite as shown in Fig. 1, is considered to study the failure analysis. The orientation of the laminate is [45°, −45°, 45°]. The five failure criteria are employed to analyse the first-ply failure load of the laminate and design the finite element model for the same.

Fig. 1
figure 1

a Isometric view of the laminate. b Layer thickness representation of the laminated plate

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Maximum Stress Theory. This theory involves two forms of stress (normal stress and shear stress) theories. It specifies that, when a material has exceeded its maximum stress enduring capacity in any of its axes, it fails. The mathematical formulation [17], can be expressed as

$$(\sigma_{1}^{c} )_{u} < (\sigma_{1} ) < (\sigma_{1}^{T} )_{u}$$
(1)
$$(\sigma_{2}^{c} )_{u} < (\sigma_{2} ) < (\sigma_{2}^{T} )_{u}$$
(2)
$$(\tau_{12} )_{u} < (\tau_{12} ) < (\tau_{12} )_{u}$$
(3)

where \(\sigma_{1}^{{}}\), \(\sigma_{2}^{{}}\) represent the normal stresses in x-axis and y-axis, respectively. While \(\tau_{12}\) represents the shear stress. \(\sigma_{{}}^{c}\) and \(\sigma_{{}}^{T}\) represent the compressive stress and tensile stress, respectively, through the laminate. Here the suffix ‘u’ is used to signify the ultimate stress point.

Maximum Strain Theory. This theory is based on the maximum normal strain theory of St. Venant and Tresca’s strain equivalent of maximum stress theory for isotropic materials. According to this theory, when the shear and principal strain exceeds the ultimate strain the material tends to rupture or fail. The mathematical deduction [18], for the same can be expressed as

$$(\varepsilon_{1}^{c} )_{u} < (\varepsilon_{1} ) < (\varepsilon_{1}^{T} )_{u}$$
(4)
$$(\varepsilon_{2}^{c} )_{u} < (\varepsilon_{2} ) < (\varepsilon_{2}^{T} )_{u}$$
(5)
$$\gamma_{12} < \varGamma_{12}$$
(6)

where, \(\varepsilon_{1}^{{}}\) and \(\varepsilon_{2}^{{}}\) represent the normal strains in x-axis and y-axis represents while \(\gamma_{12}\) represents the shear strain. \(\varepsilon_{{}}^{c}\) and \(\varepsilon_{{}}^{T}\) represent the compressive strain and tensive strain, respectively through the laminate and \(\varGamma_{12}\) represents the ultimate shear strain.

Tsai-Hill (Energy-Based Criterion) Theory. Tsai-Hill theory for the failure of laminate is a combination of two energy principle, the first one is the distortion energy (which is responsible for change of the shape) and the second one is dilation energy (which causes volumetric changes in the material). The failure of the material takes place when the following equation [19], holds true.

$$f\left( {\sigma_{ij} } \right) = F(\sigma_{2} - \sigma_{3} )^{2} + G(\sigma_{3} - \sigma_{1} )^{2} + H(\sigma_{1} - \sigma_{2} )^{2} + 2L\sigma_{4}^{2} + 2M\sigma_{5}^{2} + 2N\sigma_{6}^{2} = 1$$
(7)

where, F, G, H, L, M and N signify strength parameters of the material and \(\sigma_{4} ,\sigma_{5} ,\sigma_{6}\) are the shear stress components.

Tsai-Wu (Interaction Tensor Polynomial) Theory. The Tsai-Wu failure criterion is a special case of the general quadratic failure criteria developed by Gol’denblat and Kopnov. It can be written in a scalar form as [20]

$$F_{i} \sigma_{i} + F_{ij} \sigma_{i} \sigma_{j} \ge 1$$
(8)

where, \(F_{i}\) and \(F_{ij}\) are the first order and fourth order strength tensors of the material. Here \(\sigma_{i}\) denotes the difference between compressive and tensile induced stress. The term \(\sigma_{i} \sigma_{j}\) defines an ellipsoid along with the stress space.

Tsai-Hill’s Hoffman Failure Criteria. The Tsai-Hill’s Hoffman criterion is a special condition of Tsai-Hill failure criteria. In Hoffman’s failure criteria the difference between the strength of tension and compression is considered which is ignored in the case of Tsai-Hill failure criteria which is significant if brittle materials are considered. The modified criteria are established by adding the odd functions of the principal stress components (\(\sigma_{1}^{{}}\), \(\sigma_{2}^{{}}\) and \(\sigma_{3}^{{}}\)) in the actual expression of Tsai-Hill criteria [21]. Thus

$$\begin{aligned} & C_{1} (\sigma_{2} - \sigma_{3} )^{2} + C_{2} (\sigma_{3} - \sigma_{1} )^{2} + C_{3} (\sigma_{1} - \sigma_{2} )^{2} + C_{4} \sigma_{1} \\ & \quad \quad + \;C_{5} \sigma_{2} + C_{6} \sigma_{3} + C_{7} \sigma_{4}^{2} + C_{8} \sigma_{5}^{2} + C_{9} \sigma_{6}^{2} = 1 \\ \end{aligned}$$
(9)

Here \(C_{1}\)–\(C_{9}\) denote the material parameters.

2.2 ANN-Based Surrogate Model

Artificial neural network (ANN) is analogous to the working phenomenon of a human brain based on which the present computational model is being prepared. The algorithm in ANN acquires its working procedure based on the input and internal hidden neuron configuration also known as weights after which the output of the data is compared with known correct values. The modelling of intricate association between input and output data involves non-linear statistical data modelling tool, which is further incorporated and executed through ANN. The input and output data are channelised through a training process which continues until a significant reduction in error is achieved. The input data moves forward on a layer basis, training the data in hidden networks, and simultaneously it is supervised to reduce the error through the back-propagation algorithm. The hidden layers can be more than one according to the design required. The principal objective of using ANN is its ability to compensate the computational time by developing an efficient model similar to the finite element model. The structure of input and output in an ANN is shown in Fig. 2.

Fig. 2
figure 2

Structure of an ANN

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ANN procures their result by following the patterns and relationships in data and learn through experience, instead of following a fixed set of programmable arrangement. The working process of an ANN is followed by a series of steps [22], as specified here. At first, the input data (\(x_{n}\)) is fed to the neurons in the input unit followed by calculating the output (\(y_{n}\)) from the hidden layer of neurons utilising the transfer function

$$y_{n} = \sum {W_{nm} x_{n} + \psi }$$
(10)
$$H_{m} = \frac{1}{{1 = \exp ( - \alpha y_{n} )}}$$
(11)

where \(W_{nm}\) denotes the connection weight for the neurons n and m, Ψ is the bias or threshold value for neuron m that can be observed as the non-zero offset in the data, \(H_{m}\)  is the output of neuron m, and α signifies the non-linear parameter for the neuron’s operation. Following this stage, compute the parameter to be studied (Pk) for the output neuron in a similar manner as mentioned in Eqs. (10) and (11). Since the algorithm will face an error at this stage, the error for each weight in the output value Pk and target output tk is analysed by Eq. (13). This process is known as backpropagation.

$$\delta_{k} = (t_{k} - P_{k} )P_{k} (1 - P_{k} )$$
(12)

where k is the output neuron. According to the error correction factor, the change in weights and bias are updated in this stage

$$W_{mk}^{\text{new}} = W_{mk}^{\text{old}} + \Delta W_{mk} (p)$$
(13)
$$\Delta W_{mk} (p) = \eta \delta_{k} H_{m} + \mu \Delta W_{mk} (p - 1)$$
(14)

where \(W_{mk}\) signifies the adjusted value of the weight between output neuron k and hidden layer neuron m, p and p−1 refer to the present and previous cycles of correction, respectively. Also, η denotes the learning rate and μ signifies momentum.

Now the error for the hidden layer (\(\delta_{m}\)) is calculated as Eq. (15) and an updated weight (Wnm) is formulated due to the hidden unit based on Eqs. (16) and (17).

$$\delta_{m} = H_{m} \left( {1 - H_{m} \sum {\delta_{k} W_{mk} } } \right)$$
(15)
$$W_{mk}^{\text{new}} = W_{mk}^{\text{old}} + \Delta W_{nm} (p)$$
(16)
$$\Delta W_{mk} (p) = \eta \delta_{m} H_{n} + \mu \Delta W_{nm} (p - 1)$$
(17)

At the end terminating condition is checked after every new sample is calculated and finally a significant reduction in error is achieved.

The working module for the ANN-based random first-ply failure analysis can be depicted by the flowchart in Fig. 3.

Fig. 3
figure 3

Flowchart of the first-ply failure analysis incorporating ANN as the surrogate model

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3 Result and Discussion

In this section, the result obtained for the different failure criteria as mentioned above are discussed in brief. Firstly, a deterministic study is carried out for the analysis of first-ply failure loads with respect to the mentioned failure criteria and then the stochastic analysis is designed based on ANN as the surrogate model.

The laminated composite considered is a three-layered T300/5208 graphite-epoxy laminate with ply orientation of [45°, −45°, 45°]. The mean value of the material properties for the specified material is specified in Table 1.

Table 1 Material properties of T300/5208 graphite-epoxy laminate [23] with ply orientation of [45°, −45°, 45°]
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The plate has a dimension of 0.22 m length, 0.127 m breadth and \(3\; \times \;10^{ - 4}\) m as thickness. The plate is subjected to uniformly distributed load on the top surface in the z-direction. Three mesh size is considered for the verification of the deterministic model of sizes (2 × 2), (4 × 4) and (8 × 8) and they are simultaneously validated with the finite element model of Reddy and Pandey [3]. The mesh plane area is considered of (8 × 8) configuration comprising of 64 elements and 225 total number of nodes. The deterministic validation of the five different failure criteria from Reddy and Pandey [3], is shown in Table 2. The present deterministic validation of the failure modes is also validated by Karsh et al. [16], for the spatial vulnerability study for the first-ply failure for laminated composite.

Table 2 Validation of the present finite element model with experimental results [3] for in-plane loading of different failure criteria for the laminate with [45°, −45°, 45°] ply orientation
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Subsequently, the deterministic model is validated in Table 2, the ANN model is designed based on the parent MCS model. For the ANN model, the main MCS model which is of 10,000 samples sized is compared with the ANN-based MCS for three different sample size which is 64,128 and 256 as depicted in Fig. 3.

As it can be perceived from Fig. 4, that out of the three sample size considered, 256 samples sized data converges with the parent MCS to a great extent for the five different failure criteria considered. For the lesser sample size data, the PDF tends to deviate from the parent MCS model. Thus, it can be concluded that as the sample size increases the ANN model tends to improve its accuracy comparatively.

Fig. 4
figure 4

Probability density functions (PDF) of parent MCS and ANN model considering first-ply failure for a Maximum stress. b Maximum strain. c Tsail-Hill. d Tsai-Wu and e Hoffman failure criteria

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Since the 256 samples sized data of the ANN-based model almost converges with the parent MCS, the scatter plot shown in Fig. 5, shows that the present ANN-based model can substitute the time-consuming MCS model with the 256 samples sized ANN model.

Fig. 5
figure 5

Scatter plot of the 256 samples sized ANN model with respect to the original finite element (FE) model for a max. stress b max. strain. c Tsai-Hill. d Tsai-Wu and e Hoffman failure criteria

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4 Conclusion

The novelty of the present work is that ANN is incorporated along with stochastic finite element modelling for first-ply failure analysis of a three-layered laminated composite. It is concluded that as the sample size increases for the ANN-based model, the accuracy level of the model with respect to the parent MCS also increases. The scatter plot for the efficiently matched ANN model is depicted to conclude that the present parent MCS can be replaced with the efficient ANN-based model.