Keywords

1 Introduction

Two or more components are combined structurally to form composite materials to enhance stiffness, thermal properties, strength and produce light weight materials. Fiber-reinforced composites consisting of fiber and matrix phases are widely applied in aerospace industry. Lamina with unidirectional fibers or woven fibers are stacked together in a sequence to acquire a laminated fiber-reinforced composite. The mechanical behavior of such composites depends on orientation of the fiber, properties and volume fractions of the matrix and fiber constituents. The interface between the two components in a lamina and also between the different laminae plays a key role in the structural stability of the composite. Due to the definite inhomogeneity and presence of varied interfaces, understanding fracture in composites is much more complex when compared to homogeneous, isotropic materials such as steel.

Crack initiation and crack propagation for a material can be explained by two criteria: (i) strength criteria and (ii) energy criteria. In the case of strength criteria, failure occurs when the stresses reaches a critical value in a laminate. The failure mechanism in composites is studied using this approach in [1]. The transverse cracking effect cannot be explained accurately using this criteria. On the other hand, energy criteria states that the crack propagates only when the energy release rate exceeds the critical energy release rate which is a material property [2].

Numerical methods can be employed to model the complex crack phenomena in composites. Stress intensity factors can be evaluated using FEM analysis [3]. Cohesive zone modeling is used to understand both the Mode I and Mode II failures in laminated composites in [4]. There are other methods like extended finite element method XFEM [5], Peridynamics [6] which can model composite fracture/failure. Smeared crack models are widely applied in solving complex problems involving crack initiation, crack branching and multiple cracking phenomena [7]. It considers regularization of the crack represented by a decreasing exponential function. The approach can also be applied to model brittle fracture [8], ductile fracture [9], dynamic fracture [10] and cohesive fracture [11]. This approach is further extended to anisotropic materials [13] to model directional fracture [14], intralaminar and translaminar fracture [15]. It can also be applied to nanocomposites [16], composites with varied stiffness [17] and hyper elastic materials [18].

Delamination under mode I loading is shown in Fig. 1. It ruptures the structure and also deteriorate the compressive strength of the laminate leading to excessive buckling. Delamination grows from pre-existent discontinuities such as notches, stress concentrations. For the complete structural assessment, knowledge on delamination initiation, propagation and the material parameters that impact these events is inevitable.

Fig. 1
figure 1

Unidirectional fiber-reinforced composite lamina in a Undeformed state and b Deformed state after delamination under Mode I loading

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In the present work, anisotropic crack density functional having a structural tensor corresponding to the fiber orientations of the composite is considered which is proposed and implemented in [14, 19, 20]. The anisotropic model is adopted to analyze the delamination in unidirectional fiber-reinforced composite. The crack driving force has two components corresponding to the matrix and fiber phases which determine the crack propagation. The effect of the interface between the fiber and matrix is not considered in the present analysis as at a macroscopic level, the properties of the composite are mostly influenced by the individual phases (matrix and fiber) than the interface (matrix-fiber interface). The main aim of this paper is to understand the influence of fiber orientation and anisotropy parameter \(\alpha\) on the crack propagation and the mechanical response of the whole system.

2 Methodology

A solid body \({\it \Omega}\) with a sharp crack \({\it \Gamma}_{c}\) is considered as shown in Fig. 2. The global energy functional can be written as (body forces and surface tractions are neglected).

Fig. 2
figure 2

Solid domain \({\it \Omega}\) having a sharp crack \({\it \Gamma}_{c}\) and b the smeared crack \({\it \Gamma }_{s}\)

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$$E={\int }_{\Omega }\psi \left({\varvec{\varepsilon}}\left({\varvec{u}}\right)\right)d\Omega + {\int }_{\Omega }{G}_{c}dA$$
(1)

\(\psi\) is the energy storage functional, \({G}_{c}\) is the critical energy release rate and \(\gamma (s,\nabla s)\) is the crack density function. The sharp crack \({\Gamma }_{c}\) is now diffused to \({\Gamma }_{s}\) which is defined as

$${\Gamma }_{s}= {\int }_{a}\gamma \left(s,\nabla s\right) d\Omega , \mathrm{where} \, \gamma \left(s,\nabla s\right)=\frac{1}{2{l}_{s}} {s}^{2}+\frac{{l}_{s}}{2} (\nabla s\cdot {\varvec{A}} \nabla s)$$
(2)

where \(s\) represents the phase field variable whose value ranges between 0 and 1 and \({l}_{s}\) represents the smeared width of the crack. A is the second order anisotropic structural tensor which is defined as

$${\varvec{A}}={\varvec{I}}+\alpha {\varvec{f}}\otimes {\varvec{f}}$$
(3)

I is the second order identity tensor, \(\alpha\) is the anisotropy parameter and \({\varvec{f}}\) is the unit vector corresponding to the fiber orientation in the composite. Equation (1) can be written in terms of the crack density function \(\gamma \left(s,\nabla s\right)\) defined in Eq. (2) as

$$E={\int }_{\Omega }\psi \left({\varvec{\varepsilon}}\left({\varvec{u}}\right)\right)d\Omega + {\int }_{\Omega }{G}_{c} \gamma \left(s,\nabla s\right) d\Omega$$
(4)

The crack driving force can be written as

$$H=\frac{{\psi }^{f}}{{G}_{f}}+ \frac{{\psi }^{mI}}{{G}_{mI}}+ \frac{{\psi }^{mII}}{{G}_{mII}}$$
(5)

where \({\psi }^{f}\), \({\psi }^{mI}\) and \({\psi }^{mII}\) are the energy storage functional corresponding to fiber and matrix failure mechanisms.

Delamination Test

Fig. 3
figure 3

a Geometry and b boundary conditions for the single edge notched specimen

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Consider a single edge notched (SEN) specimen of size \(1\times 1\) as depicted in Fig. 3a. The notch extends to the center of the specimen. An element size h = 0.01 mm and internal length scale \({l}_{s}\)= 2 h is adopted. The load is applied in terms of displacement at top and bottom ends as shown in Fig. 3b and \(\theta\) corresponds to the orientation of the fibers. Two models, M1, M2 corresponding to \(\theta ={0}^{\circ }\) and \(\theta =3{0}^{\circ }\) and four cases, A, B, C, D corresponding to \(\alpha\) = 0, \(\alpha\) = 10, \(\alpha\) = 15 and \(\alpha\) = 20 are considered for the analysis. The combinations are given in Table 1.

Table 1 Combinations considered for the analysis
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Remark The unidirectional composite is considered to be orthotropic. The constitutive matrix can be written as.

$${\varvec{C}}={{\varvec{R}}}^{T}{\varvec{Q}}{\varvec{R}}, {\varvec{R}}=\left[\begin{array}{ccc}\mathrm{cos}\theta & \mathrm{sin}\theta & 0\\ -\mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right], {\varvec{Q}}= \left[\begin{array}{ccc}{Q}_{11}& {Q}_{12}& 0\\ {Q}_{12}& {Q}_{22}& 0\\ 0& 0& {Q}_{66}\end{array}\right]$$
(6)
$${Q}_{11}=\frac{{E}_{11}}{1-{\nu }_{12}{\nu }_{21}}, {Q}_{22}=\frac{{E}_{22}}{1-{\nu }_{12}{\nu }_{21}}, {Q}_{12}=\frac{{{\nu }_{12}E}_{22}}{1-{\nu }_{12}{\nu }_{21}}, {Q}_{66}={G}_{12}$$

The longitudinal stiffness, transverse stiffness and shear stiffness of the lamina are taken as \({E}_{11}\) = 114.8 GPa, \({E}_{22}\) = 11.7 GPa and \({G}_{12}\) = 9.66 GPa, respectively. Poisson’s ratio, \({\nu }_{12}\)= 0.21. The critical energy release rates are taken as \({G}_{f}=106.3\times {10}^{-3}\) kN/mm, \({G}_{mI}=0.2774\times {10}^{-3}\) kN/mm, \({G}_{mII}=0.2774\times {10}^{-3}\) kN/mm (refer [19]).

3 Results

From Fig. 4, it is evident that the crack propagation is straight for M1-A and M2-A as the anisotropy parameter \(\alpha =0\) indicates isotropic case. For the anisotropy parameter, \(\alpha =20\), the crack propagation is along the fiber orientation \(\theta\) which is \({0}^{\circ }\) for M1-D and \({30}^{\circ }\) for M2-D. The crack path for different values of anisotropy parameter for \(\theta = {30}^{\circ }\) is plotted in Fig. 4c–f. It can be observed that as the anisotropy parameter \(\alpha\) increases, the crack path is tending toward \({30}^{\circ }\) and the difference is less between \(\alpha =15\) and \(\alpha =20\) when compared to other values of \(\alpha\). The load–displacement curves are plotted for (a) M1-D and M2-D in Fig. 5a, b M1-A, M1-D, M2-A, M2-B, M2-C and M2-D in Fig. 5b.

Fig. 4
figure 4

Evolution of the crack phase field for a M1-A at \(\overline{{\varvec{u}} }=0.00989\) mm b M1-D at \(\overline{{\varvec{u}} }=0.0113\) mm c M2-A at \(\overline{{\varvec{u}} }=0.00663\) mm d M2-B at \(\overline{{\varvec{u}} }=0.00725\) mm e M2-C at \(\overline{{\varvec{u}} }=0.00736\) mm and f M2-D at \(\overline{{\varvec{u}} }=0.00744\) mm where \(\overline{{\varvec{u}} }\) is the failure displacement. The contour plot of the phase field variable s varying from 0 to 1 is shown on the right side of each row.

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Fig. 5
figure 5

Load–displacement plots for a M1-D and M2-D b M1-A, M1-D, M1-C3, M2-A, M2-B, M2-C and M2-D

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From Fig. 5a, it is evident that as the fiber orientation (\(\theta\)) increases, the failure load increases. It can be observed from Fig. 5b that as the anisotropy parameter (\(\alpha\)) increases the failure load increases and the rise is very high for anisotropic cases (\(\alpha = 10, 15, 20\)) when compared to the isotropic case (\(\alpha =0\)).

4 Conclusions

From the results in Sect. 3, it can be inferred that the crack propagates along the fiber direction for the anisotropic case, \(\alpha >0\), because the fracture toughness of the fiber is high compared to the matrix, and the fiber failure is not possible. For the isotropic case, the crack propagates straight irrespective of fiber orientation as the crack propagation is not direction-dependent. From load–displacement plots, it can be observed that the failure load is very high for the anisotropic case and also failure load increases with an increase in fiber orientation and anisotropy parameter.