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Metamodeling-assisted probabilistic first ply failure analysis of laminated composite plates—RS-HDMR- and GPR-based approach

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Abstract

In structural applications, laminated composites are typically the best choice for providing a high strength-to-weight ratio. The composite laminates, on the other hand, are susceptible to the first ply failure (FPF), which can result in delamination, matrix cracking, and fiber breaking. As a result, it is critical to map the FPF of laminated composites against the uncertainty in material properties. In this paper, we presented a framework based on coupled statistical modeling and failure criteria to perform sensitivity analysis corresponding to the FPF of laminated composites. The practically relevant randomness in material properties (elastic modulus, shear modulus, Poisson’s ratio, and mass density) is enforced by utilizing the Monte Carlo random sampling method. The FPF of a laminated composite subjected to random material properties is evaluated using five failure criteria: maximum strain, maximum stress, Tsai-Hill, Tsai-Wu, and Hoffman. Such a random sampling-based dataset is used to train and validate the random sampling high dimensional model representation (RS-HDMR) metamodel and Gaussian process regression (GPR) model. To ensure sound generalization capabilities, the models are rigorously cross-validated. With sufficient confidence in the constructed models, the models are further utilized to perform the variance-based sensitivity analysis. It is worth mentioning that observations from both models in terms of parameters with the highest sensitivity (for the first-order polynomial function) are comparable. The RS-HDMR metamodel is further used to perform the second-order polynomial function-based sensitivity analysis, wherein the sensitivity index for the most sensitive parameter is observed to be very low when compared with the observations of first-order polynomial function-based sensitivity analysis. The numerically quantifiable outcomes of the present study will serve its purpose in the bottom-up design of the laminated composites.

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The data that support the findings of this study are available from the authors upon reasonable request.

Abbreviations

\(\Pi\) :

Total potential energy

\(U\) :

Total strain energy

\(V\) :

Total work potential due to external forces

\(d\phi\) :

Incremental displacement

\(q_{z}\) :

Load intensity (transverse) of laminate

\(\sigma\) :

Stress component

\(\varepsilon\) :

Strain vector

\(\delta_{x}, \delta_{y}, \delta_{z}\) :

Displacement terms

\( \theta _{x} \;\text{and}\;\theta _{y} \) :

Rotational components at matrix level

\( \phi _{x} \;\text{and}\;\phi _{y} \) :

Rotational components at the nodal level

\(k_{x}, k_{y} \,\text{and}\, k_{xy}\) :

The curvature of the reference plane

\(\tilde{D}\) :

Displacement resultant

\(\tilde{M}\) :

Moment resultant

\(\tilde{T}\) :

Transverse shear resultant

[A], [B] and [D]:

Extensional, flexural extensional, and flexural stiffness matrix

\(\psi_{j}^{e}\) :

Shape function

u :

Displacement

\( \varepsilon _{x}^{0} ,\varepsilon _{x}^{0} \;\text{and}\;\gamma _{{xy}}^{0} \) :

Strain in the reference plane

\( \xi \;\text{and}\; { \upeta} \) :

Axis of the principal coordinate system

\(k\) :

Stiffness

\(\varsigma \,\text{and}\, \varphi\) :

Local natural coordinates of the element

\(P_{0} ,P_{1} ...P_{7}\) :

Generalized degree of freedom (DOF)

\(\overline{s}_{i}\) :

Shape function of node

[J]:

Jacobian matrix

\(\theta_{x} \,\text{and}\, \theta_{y}\) :

The rotational component in x and y directions in each node

\(u_{0}, v_{0} \,\text{and}\, w_{0}\) :

Displacement variables for laminated plate

\(\uptheta \) :

Ply-orientation angle

\(E_{1}\) :

Longitudinal elastic modulus of elasticity

\(E_{2}\) :

Transverse elastic modulus of elasticity

\(G_{12}\) :

Shear modulus along a longitudinal direction (first and second plane)

\(G_{13}\) :

Shear modulus along the transverse direction (first and third plane)

\(G_{23}\) :

Shear modulus in the second and third plane

\(\uprho \) :

Mass density

\(\upmu \) :

Poisson’s ratio

\(N_{s}\) :

Sample size

\(y_{i}\) :

Actual model

\(\mathop {y_{i} }\limits^{\vartriangle }\) :

Predicted model

\(\overline{y}_{i}\) :

Mean data

\(J(x)_{{{\text{actual}}}}\) :

Actual response of the model

\(J(x)_{{{\text{predicted}}}}\) :

Predicted response model

\(V_{t}\) :

Total variance

\(V_{P}\) :

Partial variance

\(S_{a}\) :

Global sensitivity index

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Acknowledgements

During this work, the authors gratefully acknowledge the Aeronautics Research and Development Board (Sanction no: ARDB/01/105885/M/I).

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology Silchar, Silchar, India

    Subrata Kushari, Kritesh Kumar Gupta,  Vaishali & Sudip Dey

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  1. Subrata Kushari
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Correspondence to Kritesh Kumar Gupta.

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Kushari, S., Gupta, K.K., Vaishali et al. Metamodeling-assisted probabilistic first ply failure analysis of laminated composite plates—RS-HDMR- and GPR-based approach. J Braz. Soc. Mech. Sci. Eng. 44, 374 (2022). https://doi.org/10.1007/s40430-022-03674-w

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