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Modeling of delamination in fiber-reinforced composite using high-dimensional model representation-based cohesive zone model

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Abstract

Prediction of delamination failure is challenging when the researchers try to achieve the task without overburdening the available computational resources. One of the most powerful computational models to predict the crack initiation and propagation is cohesive zone model (CZM), which has become prominent in the crack propagation studies. This paper proposes a novel CZM using high-dimensional model representation (HDMR) to capture the steady-state energy release rate (ERR) of a double-cantilever beam (DCB) under mode I loading. The finite element models are created using HDMR-based load and crack length response functions. Initially, the model is developed for 51-mm crack size DCB specimens, and the developed HDMR-based CZM is then used to predict the ERR variations of 76.2-mm crack size DCB model. Comparisons have been made between the available unidirectional composite (IM7/977-3) experimental data and the numerical results obtained from the 51-mm and 76.2-mm initial crack size DCB specimens. In order to demonstrate the efficiency of the proposed model, the results of the second-order nonlinear regression model using RSM are used for the comparison study. The results show that the proposed method is computationally efficient in capturing the delamination strength.

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Authors and Affiliations

  1. Department of Civil Engineering, National Institute of Technology Karnataka, Surathkal, 575 025, India

    B. Kesava Rao & A. S. Balu

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  1. B. Kesava Rao
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  2. A. S. Balu
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Correspondence to A. S. Balu.

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Technical Editor: Paulo de Tarso Rocha de Mendonça, Ph.D..

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Appendix: A

Appendix: A

1.1 Development of HDMR equations using five sample points (n = 5)

Consider the first-order HDMR expression to develop the response surface equations,

$$\tilde{f}\left( {\mathbf{x}} \right) = \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{n} {\upphi_{j} (} } x_{i} )f(c_{1} \ldots ,c_{i - 1} \ldots ,x_{i}^{j} \ldots ,c_{i + 1} \ldots ,c_{N} ) - (N - 1)f_{0}$$

For N = 5 and n = 5: \(\tilde{f}\left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{5} {\sum\nolimits_{j = 1}^{5} {\upphi_{j} (x_{i} )f(c_{1} \ldots ,c_{i - 1} \ldots ,x_{i}^{j} \ldots ,c_{i + 1} \ldots ,c_{N} ) - (N - 1)f_{0} } }\)

The expanded form of the above expression is given as:

$$\begin{aligned} & \tilde{f}\left( {\mathbf{x}} \right) = \sum\limits_{j = 1}^{5} {\upphi_{j} } (x_{1} )f(x_{1}^{j} ,c_{2} ,c_{3} , \ldots ,c_{5} ) + \sum\limits_{j = 1}^{5} {\upphi_{j} } (x_{2} )f(c_{1} ,x_{2}^{j} ,c_{3} , \ldots ,c_{5} ) + \cdots + \sum\limits_{j = 1}^{5} {\upphi_{j} } (x_{5} )f(c_{1} ,c_{2} , \ldots ,x_{5}^{j} ) - (5 - 1)f(c_{1} ,c_{2} , \ldots ,c_{5} ) \\ & {\text{For}}\;i = 1\;{\text{and}}\;j = \;1{-}5 \\ \end{aligned}$$
$$\begin{aligned} & {\text{Expansion of }}1 \\ &\upphi_{1} (x_{1} )f(x_{1}^{1} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{2} (x_{1} )f(x_{1}^{2} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{3} (x_{1} )f(x_{1}^{3} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{4} (x_{1} )f(x_{1}^{4} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) \\ & \quad +\upphi_{5} (x_{1} )f(x_{1}^{5} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) \\ \end{aligned}$$
$$\begin{aligned} & {\text{Similarly}}\;{\text{from}}\;i = 2\;{\text{to}}\; 5\;{\text{and}}\;j = 1{-}5\;{\text{expression}}\;{\text{are}}\;{\text{given}}\;{\text{by}} \\ & {\text{Expansion of }}2 \\ &\upphi_{1} (x_{2} )f(c_{1} ,x_{2}^{1} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{2} (x_{2} )f(c_{1} ,x_{2}^{2} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{3} (x_{2} )f(c_{1} ,x_{2}^{3} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{4} (x_{2} )f(c_{1} ,x_{2}^{4} ,c_{3} ,c_{4} ,c_{5} ) \\ & \quad { + \phi }_{5} (x_{2} )f(c_{1} ,x_{2}^{5} ,c_{3} ,c_{4} ,c_{5} ) \\ \end{aligned}$$
$$\begin{aligned} & {\text{Expansion}}\,{\text{of}}\,3 \\ &\upphi_{1} (x_{3} )f(c_{1} ,c_{2} ,x_{3}^{1} ,c_{4} ,c_{5} ) +\upphi_{2} (x_{3} )f(c_{1} ,c_{2} ,x_{3}^{2} ,c_{4} ,c_{5} ) +\upphi_{3} (x_{3} )f(c_{1} ,c_{2} ,x_{3}^{3} ,c_{4} ,c_{5} ) +\upphi_{4} (x_{3} )f(c_{1} ,c_{2} ,x_{3}^{4} ,c_{4} ,c_{5} ) \\ & \quad + \,\upphi_{5} (x_{3} )f(c_{1} ,c_{2} ,x_{3}^{5} ,c_{4} ,c_{5} ) \\ \end{aligned}$$
$$\begin{aligned} & {\text{Expansion}}\,{\text{of}}\, 4\\ &\upphi_{1} (x_{4} )f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{1} ,c_{5} ) +\upphi_{2} (x_{4} )f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{2} ,c_{5} ) +\upphi_{3} (x_{4} )f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{3} ,c_{5} ) +\upphi_{4} (x_{4} )f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{4} ,c_{5} ) \\ & \quad + \,\upphi_{5} (x_{4} )f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{5} ,c_{5} ) \\ \end{aligned}$$
$$\begin{aligned} & {\text{Expansion}}\,{\text{of}}\, 5\\ &\upphi_{1} (x_{5} )f(c_{1} ,c_{2} ,c_{3} ,c_{4} ,x_{5}^{1} ) +\upphi_{2} (x_{5} )f(c_{1} ,c_{2} ,c_{3} ,c_{4} ,x_{5}^{2} ) +\upphi_{3} (x_{5} )f(c_{1} ,c_{2} ,c_{3} ,c_{4} ,x_{5}^{3} ) +\upphi_{4} (x_{5} )f(c_{1} ,c_{2} ,.c_{3} ,c_{4} ,x_{5}^{4} ) \\ & \quad + \,\upphi_{5} (x_{5} )f(c_{1} ,c_{2} ,c_{3} ,c_{4} ,x_{5}^{5} ) \\ \end{aligned}$$

In order to obtain the HDMR expression for the desired response, the functions \(f(c_{1} \ldots ,c_{i - 1} \ldots ,x_{i}^{j} \ldots ,c_{i + 1} \ldots ,c_{N} )\) are evaluated using ABAQUS. Table 7 shows the component functions and corresponding load and crack length values. The responses (load and crack length) for the function evaluations in the above expansion are as below:

Table 7 Cut-HDMR sampling and corresponding FEA results for 51 mm initial crack
Full size table

The number of function evaluations can be obtained by using Eq. (13).

$$\begin{aligned} & {\text{Function}}\,{\text{Evaluations}}\,{\text{in}}\,{\text{Expansion-1}} \\ & f(x_{1}^{1} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) = f(100, \, 4, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {127.26;} & {0.1522} \\ \end{array} \\ & f(x_{1}^{2} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) = f(250, \, 4, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {171.92;} & {0.1523} \\ \end{array} \\ & f(x_{1}^{3} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) = f(400, \, 4, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {208.67;} & {0.1489} \\ \end{array} \\ & f(x_{1}^{4} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) = f(550, \, 4, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {243.50;} & {0.1440} \\ \end{array} \\ & f(x_{1}^{5} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) = f(700, \, 4, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {275.26;} & {0.1417} \\ \end{array} \\ & {\text{Similarly,}}\;{\text{all}}\;{\text{the}}\;{\text{function}}\;{\text{evaluations}}\;{\text{are}}\;{\text{carried}}\;{\text{out}}\;{\text{up}}\;{\text{to}}\,i = 5\,{\text{and}}\,j = 1{-}5 \\ \end{aligned}$$
$$\begin{aligned} & {\text{Function}}\;{\text{Evaluations}}\;{\text{in}}\;{\text{Expansion-2}} \\ & f(c_{1} ,x_{2}^{1} ,c_{3} ,c_{4} ,c_{5} ) = f(400,{ 1} . 0, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {312.26;} & {0.0940} \\ \end{array} \\ & f(c_{1} ,x_{2}^{2} ,c_{3} ,c_{4} ,c_{5} ) = f(400,{ 2} . 5, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {243.50;} & {0.1252} \\ \end{array} \\ & f(c_{1} ,x_{2}^{3} ,c_{3} ,c_{4} ,c_{5} ) = f(400,{ 4} . 0, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {208.67;} & {0.1489} \\ \end{array} \\ & f(c_{1} ,x_{2}^{4} ,c_{3} ,c_{4} ,c_{5} ) = f(400,{ 5} . 5, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {187.23;} & {0.1680} \\ \end{array} \\ & f(c_{1} ,x_{2}^{5} ,c_{3} ,c_{4} ,c_{5} ) = f(400,{ 7} . 0, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {170.69;} & {0.1860} \\ \end{array} \\ \end{aligned}$$
$$\begin{aligned} & {\text{Function Evaluations in Expansion-3}} \\ & f(c_{1} ,c_{2} ,x_{3}^{1} ,c_{4} ,c_{5} ) = f(400,{ 4}, \, 0.0100, \, 210, \, 8.5) = \begin{array}{*{20}c} {213.36;} & {0.1510} \\ \end{array} \\ & f(c_{1} ,c_{2} ,x_{3}^{2} ,c_{4} ,c_{5} ) = f(400,{ 4}, \, 0.2325, \, 210, \, 8.5) = \begin{array}{*{20}c} {209.41;} & {0.1495} \\ \end{array} \\ & f(c_{1} ,c_{2} ,x_{3}^{3} ,c_{4} ,c_{5} ) = f(400,{ 4}, \, 0.4550, \, 210, \, 8.5) = \begin{array}{*{20}c} {208.67;} & {0.1489} \\ \end{array} \\ & f(c_{1} ,c_{2} ,x_{3}^{4} ,c_{4} ,c_{5} ) = f(400,{ 4}, \, 0.6775, \, 210, \, 8.5) = \begin{array}{*{20}c} {207.25;} & {0.1489} \\ \end{array} \\ & f(c_{1} ,c_{2} ,x_{3}^{5} ,c_{4} ,c_{5} ) = f(400,{ 4}, \, 0.9000, \, 210, \, 8.5) = \begin{array}{*{20}c} {207.37;} & {0.1482} \\ \end{array} \\ \end{aligned}$$
$$\begin{aligned} & {\text{Function Evaluations in Expansion-4}} \\ & f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{1} ,c_{5} ) = f(400,{ 4}, \, 0.455, \, 190, \, 8.5) = \begin{array}{*{20}c} {203.11;} & {0.1515} \\ \end{array} \\ & f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{2} ,c_{5} ) = f(400,{ 4}, \, 0.455, \, 200, \, 8.5) = \begin{array}{*{20}c} {205.43;} & {0.1550} \\ \end{array} \\ & f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{3} ,c_{5} ) = f(400,{ 4}, \, 0.455, \, 210, \, 8.5) = \begin{array}{*{20}c} {208.67;} & {0.1489} \\ \end{array} \\ & f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{4} ,c_{5} ) = f(400,{ 4}, \, 0.455, \, 220, \, 8.5) = \begin{array}{*{20}c} {209.96;} & {0.1487} \\ \end{array} \\ & f(c_{1} ,c_{2} ,c_{3} ,x_{4}^{5} ,c_{5} ) = f(400,{ 4}, \, 0.455, \, 230, \, 8.5) = \begin{array}{*{20}c} {212.37;} & {0.1417} \\ \end{array} \\ \end{aligned}$$
$$\begin{aligned} & {\text{Function Evaluations in Expansion-5}} \\ & f(c_{1} ,c_{2} ,c_{3} ,c_{4} ,x_{5}^{1} ) = f(400,{ 4}, \, 0.455, \, 210,{ 0}5.00) = \begin{array}{*{20}c} {207.34;} & {0.1472} \\ \end{array} \\ & f(c_{1} ,c_{2} ,c_{3} ,c_{4} ,x_{5}^{2} ) = f(400,{ 4}, \, 0.455, \, 210,{ 0}6.75) = \begin{array}{*{20}c} {209.14;} & {0.1469} \\ \end{array} \\ & f(c_{1} ,c_{2} ,c_{3} ,c_{4} ,x_{5}^{3} ) = f(400,{ 4}, \, 0.455, \, 210,{ 0}8.50) = \begin{array}{*{20}c} {208.67;} & {0.1489} \\ \end{array} \\ & f(c_{1} ,c_{2} ,.c_{3} ,c_{4} ,x_{5}^{4} ) = f(400,{ 4}, \, 0.455, \, 210, \, 10.25) = \begin{array}{*{20}c} {209.63;} & {0.1487} \\ \end{array} \\ & f(c_{1} ,c_{2} ,c_{3} ,c_{4} ,x_{5}^{5} ) = f(400,{ 4}, \, 0.455, \, 210, \, 12.00) = \begin{array}{*{20}c} {209.78;} & {0.1495} \\ \end{array} \\ \end{aligned}$$

The shape/interpolation function \(\phi_{j} \left( {x_{i} } \right)\) is evaluated using the Lagrange interpolation:

$$\phi_{j} \left( {x_{i} } \right) = \frac{{\left( {x_{i} - x_{i}^{1} } \right) \ldots \left( {x_{i} - x_{i}^{j - 1} } \right)\left( {x_{i} - x_{i}^{j + 1} } \right) \ldots \left( {x_{i} - x_{i}^{n} } \right)}}{{\left( {x_{i}^{j} - x_{i}^{1} } \right) \ldots \left( {x_{i}^{j} - x_{i}^{j - 1} } \right)\left( {x_{i}^{j} - x_{i}^{j + 1} } \right) \ldots \left( {x_{i}^{j} - x_{i}^{n} } \right)}}$$

Considering the first expansion function, for i = 1 and j = 1–5, the expression of Expansion 1 for load (having 51 mm initial crack) is given by

$$\begin{aligned} {\text{Expansion - 1}} = &\upphi_{1} (x_{1} )f(x_{1}^{1} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{2} (x_{1} )f(x_{1}^{2} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{3} (x_{1} )f(x_{1}^{3} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) \\ & + \,\upphi_{4} (x_{1} )f(x_{1}^{4} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) +\upphi_{5} (x_{1} )f(x_{1}^{5} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ) \\ = & \left( {82.30x_{1}^{4} - 156.37x_{1}^{3} + 106.79x_{1}^{2} - 30.88x_{1} + 3.16} \right)f(100, \, 4, \, 0.455, \, 210, \, 8.5) \\ & + \,\left( { - 329.21x_{2}^{4} + 576x_{2}^{3} - 345.67x_{2}^{2} + 79.83x_{2} - 5.06} \right)f(250, \, 4, \, 0.455, \, 210, \, 8.5) \\ & + \,\left( {493.82x_{3}^{4} - 790.12x_{3}^{3} + 418.51x_{3}^{2} - 81.97x_{3} + 4.75} \right)f(400, \, 4, \, 0.455, \, 210, \, 8.5) \\ & + \,\left( { - 329.21x_{4}^{4} + 477.36x_{4}^{3} - 227.16x_{4}^{2} + 41.31x_{4} - 2.30} \right)f(550, \, 4, \, 0.455, \, 210, \, 8.5) \\ & + \,\left( {82.30x_{5}^{4} - 106.99x_{5}^{3} + 47.53x_{5}^{2} - 8.29x_{5}^{2} + 0.45} \right)f(700, \, 4, \, 0.455, \, 210, \, 8.5) \\ \end{aligned}$$

Similarly, all the five HDMR functions are evaluated, and summation of all gives the HDMR approximation equation for load response \(\left( {Y_{1} } \right)\) as below:

$$\begin{aligned} Y_{1} = &\, 47.5396 - 05877x_{1}^{4} + 1.0598x_{1}^{3} - 0.7370x_{1}^{2} + 0.4673x_{1} + 9.9177 \times 10^{ - 5} x_{2}^{4} - 0.0023x_{2}^{3} + 0.0209x_{2}^{2} - 0.0990x_{2} \\ & + \,0.1035x_{3}^{4} - 0.2012x_{3}^{3} + 0.1341x_{3}^{2} - 0.0403x_{3} + 2.4750 \times 10^{4} x_{4}^{4} - 2.0773 \times 10^{4} x_{4}^{3} + 6.5261 \times 10^{3} x_{4}^{2} - 909.2747x_{4} \\ & - \,2.63389 \times 10^{ - 5} x_{5}^{4} + 9.1993 \times 10^{ - 4} x_{5}^{3} - 0.0117x_{5}^{2} + 0.0645x_{5} \\ \end{aligned}$$

Similarly, the HDMR approximation equations are developed for the crack length of 51 mm, load and crack length of 76.2 mm initial crack, i.e., \(Y_{2} ,Y_{3} {\text{ and }}Y_{4}\).

$$\begin{aligned} Y_{2} = & - 228.5620 + 0.1728x_{1}^{4} - 0.1259x_{1}^{3} - 0.0520x_{1}^{2} + 0.0268x_{1} + 4.9382 \times 10^{ - 6} x_{2}^{4} - 7.9012 \times 10^{ - 5} x_{2}^{3} - 0.0025x_{2}^{2} + 0.0285x_{2} \\ & - \,0.017x_{3}^{4} + 0.0188x_{3}^{3} + 0.0023x_{3}^{2} - 0.0081x_{3} - 1.1750 \times 10^{5} x_{4}^{4} + 9.8933 \times 10^{4} x_{4}^{3} - 3.1196 \times 10^{4} x_{4}^{2} + 4.3659 \times 10^{3} x_{4} \\ & + \,3.4208 \times 10^{ - 5} x_{5}^{4} - 0.0012x_{5}^{3} + 0.0149x_{5}^{2} - 0.0799x_{5} \\ \end{aligned}$$
$$\begin{aligned} Y_{3} = & - 22.0774 - 0.5832x_{1}^{4} + 0.9932x_{1}^{3} - 0.6504x_{1}^{2} + 0.3909x_{1} + 3.2169 \times 10^{ - 4} x_{2}^{4} - 0.0049x_{2}^{3} + 0.0287x_{2}^{2} - 0.0933x_{2} \\ & - \,0.4705x_{3}^{4} + 1.0580x_{3}^{3} - 0.8292x_{3}^{2} + 0.2623x_{3} - 1.1104 \times 10^{4} x_{4}^{4} + 9.4199 \times 10^{3} x_{4}^{3} - 2.9908 \times 10^{3} x_{4}^{2} + 421.3760x_{4} \\ & - \,9.5827 \times 10^{ - 6} x_{5}^{4} + 3.1867 \times 10^{ - 4} x_{5}^{3} - 0.0038x_{5}^{2} + 0.0196x_{5} \\ \end{aligned}$$
$$\begin{aligned} Y_{4} = & 26.4019 + 0.4031x_{1}^{4} - 0.5955x_{1}^{3} + 0.2859x_{1}^{2} - 0.0722x_{1} - 2.8026 \times 10^{ - 4} x_{2}^{4} + 0.0038x_{2}^{3} - 0.0193x_{2}^{2} + 0.0583x_{2} \\ & + \,0.5120x_{3}^{4} - 1.1566x_{3}^{3} + 0.9152x_{3}^{2} - 0.2991x_{3} + 1.3142 \times 10^{4} x_{4}^{4} - 1.1143 \times 10^{4} x_{4}^{3} + 3.5367 \times 10^{3} x_{4}^{2} - 498.0339x_{4} \\ & + \,9.5560 \times 10^{ - 6} x_{5}^{4} - 3.1326 \times 10^{ - 4} x_{5}^{3} + 0.0036x_{5}^{2} - 0.0169x_{5} \\ \end{aligned}$$

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Kesava Rao, B., Balu, A.S. Modeling of delamination in fiber-reinforced composite using high-dimensional model representation-based cohesive zone model. J Braz. Soc. Mech. Sci. Eng. 41, 254 (2019). https://doi.org/10.1007/s40430-019-1761-4

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