# Fracture and mode mixity analysis of shear deformable composite beams

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## Abstract

To analyse delaminated composite beams with high accuracy under mixed-mode I/II fracture conditions first-, second-, third- and Reddy’s third-order shear deformable theories are discussed in this paper. The developed models are based on the concept of two equivalent single layers and the system of exact kinematic conditions. To deduce the equilibrium equations of the linearly elastic system, the principle of virtual work is utilised. As an example, a built-in configuration with different delamination position and external loads are investigated. The mechanical fields at the delamination tip are provided and compared to finite element results. To carry out the fracture mechanical investigation, the *J*-integral with zero-area path is introduced. Moreover, by taking the advantage of the *J*-integral, a partitioning method is proposed to determine the ratio of mode-I and mode-II in-plane fracture modes. Finally, in terms of the mode mixity, the results of the presented evaluation techniques are compared to numerical solutions and previously published models in the literature.

## Keywords

Delamination Energy release rate Mixed-mode I/II fracture*J*-integral Higher-order beam theories

## 1 Introduction

The application of the linear elastic fracture mechanics is one possibility to characterise the interlaminar fracture resistance of conventional high performance composites with inherent brittleness [8, 9]. The critical value of the energy release rate \(G_c\), which can be considered as a material property, is suitable to characterise the strength of an interface layer between two elastic plies [19]. Like any other material parameters, mechanical tests must be carried out to be determined. In the case of fracture mechanical investigation, as linear elastic fracture mechanics works only if the location, size and shape of the crack are known, the experiments must be carried out on different type of pre-cracked specimens including mode-I [23], mode-II [8], mixed-mode I/II [58], mode-III [14, 25], mixed-mode I/III [1], mixed-mode II/III [43, 50] and mixed-mode I/II/III [57] fracture conditions. Finally, these experiments must be evaluated based on very simple mechanical structures where the delamination zone is known a priori, such as delaminated beam and plate structures.

To increase the accuracy of the evaluation techniques and to capture more precisely the complex fracture mechanical behaviour of these mechanical systems, the researchers have been inspired to introduce novel, higher-order beam and plate theories [33, 53, 54]. The first-order shear deformable beam theory (FSDT) assumes independent rotation around the bending axes [46]. As a next step, the second-order shear deformable theory (SSDT) describes the in-plane displacement as a quadratic function in terms of the through-thickness coordinate. The third-order shear deformable theory (TSDT) captures the longitudinal displacement in the form of cubic function [12]. And finally but not least, in order to satisfy dynamic boundary conditions at the top and bottom surfaces of the beam, Reddy’s third-order shear deformable theory is also developed [41, 47].

In the following, to analyse a delaminated composite beam under mixed-mode I/II fracture conditions, the basic idea of higher-order beam theories is utilised with the semi-layerwise approach [51]. The concept of the proposed description can be seen in Fig. 1. where beam elements with an artificial delamination are depicted. To perform fracture mechanical investigation, the *J*-integral with zero-area path is applied [42, 50]. Moreover, a partitioning method is proposed to determine the mode-I and mode-II in-plane fracture modes without any semi-analytical considerations [29]. Finally, the obtained results of the mode partitioning are compared to previously published evaluation methods in the literature [4, 21, 34, 56].

Applications already exist in the literature which uses the *J*-integral to evaluate delaminated composite beams [3, 30, 44] under pure mode-I or pure mode-II fracture conditions. The effect of reinforcement direction and delamination length on the energy release rate have already been investigated in numerous studies, but *J*-integral- based mode mixity evaluation, which undeniably requires complicated mechanical models, has not been extensively studied.

## 2 Mechanical model: the method of two equivalent single layers (ESLs)

*i*denotes the index of actual ESL, \(z^{(i)}\) is the local thickness coordinate of the

*i*th ESL and always coincides with the local midplane, \(u_0\) is the global, \(u_{0i}\) is the local membrane displacement, \(\theta \) denotes the rotation of the cross section about the

*Y*axes, \(\phi \) means the second-order, and \(\lambda \) expresses the third-order polynomial terms. Moreover, \(w_{(i)}\) represents the separate transverse deflection functions. The displacement field of FSDT and SSDT can be obtained by substituting \(\phi =0\) and \(\lambda =0\) into the Eq. (1), respectively. In the case of the undelaminated part, the kinematic continuity between ESLs can be imposed by the system of exact kinematic conditions [50].

*i*th ESL in the

*X*-direction, and \(\gamma _{xz(i)}\) is the shear strain of the

*i*th layer in the \(X{-}Z\) plane. The in-plane strains can be decomposed into constant, linear and higher-order terms with respect to the through-thickness coordinate:

*i*th ESL,

*b*defines the width of the beam, \(z^{i}_{m}\) and \(z^{i}_{m+1}\) are the local bottom and top coordinates of the

*m*th layer in the

*i*th ESL [49]. Furthermore, \(\overline{\varvec{C}}_{(i)}^{(m)}\) is the stiffness matrix of the

*m*th layer within the

*i*th ESL expressed as:

### 2.1 Description of the undelaminated portion

*i*refers to the ESL number, the summation index

*j*defines the component in \(\psi _{j}\) primary parameter vector, and

*w*(

*x*) expresses the transverse deflection of the beam [49].

#### 2.1.1 Reddy’s third-order beam theory

#### 2.1.2 Third-order beam theory

#### 2.1.3 Second-order beam theory

#### 2.1.4 First-order beam theory

#### 2.1.5 Equilibrium equations

### 2.2 Description of the delaminated portion

#### 2.2.1 Reddy’s third-order beam theory

#### 2.2.2 Third-order beam theory

#### 2.2.3 Second-order beam theory

#### 2.2.4 First-order beam theory

#### 2.2.5 Equilibrium equations

Elastic properties of the composite layers

\(E_{11}\) | \(E_{22}\) | \(E_{33}\) | \(G_{23}\) | \(G_{13}\) | \(G_{12}\) | \(\nu _{23}\) | \(\nu _{13}\) | \(\nu _{12}\) | |
---|---|---|---|---|---|---|---|---|---|

[GPa] | [GPa] | [GPa] | [GPa] | [GPa] | [GPa] | [–] | [–] | [–] | |

0 | 148 | 9.65 | 9.65 | 4.91 | 4.66 | 3.71 | 0.27 | 0.25 | 0.25 |

\(\pm \,30\) | 82.4 | 82.4 | 82.4 | 6.61 | 6.61 | 6.61 | 0.4 | 0.4 | 0.4 |

## 3 Example: built-in configuration of a delaminated beam

As an example a built-in beam with asymmetric delamination is considered. The geometry of the structure is depicted by Fig. 3, where *l* denotes the total length, *a* represents the length of the delamination, *b* is the beam width and 2*h* is the total thickness of the beam with \([\pm 30/0_2/\pm 30/\overline{0}]_S\) lay-up. The corresponding thicknesses of the sub-laminates are denoted by \(t_1\) and \(t_2\). The material properties of the transversely isotropic and cross-ply layers are given by Table 1 [49, 52]. The structure of the ELSs in each and every case is determined according to Fig. 1.

### 3.1 Continuity conditions of the displacement field

### 3.2 Continuity conditions of the stress resultants

### 3.3 Boundary conditions

## 4 Displacements and stress distributions

Regarding “Case I” delamination scenario, when the top and bottom sub-laminates are subjected to \(F_\mathrm{t}=10\) N and \(F_\mathrm{b}=-10\) N external forces, the *u* in-plane displacement fields and the *w* deflections are depicted by Fig. 5. As we can see, by using any kind of higher-order theories, the results show good agreement with the finite element solution in connection with *u* in-plane displacement field. In terms of the *w* deflections the agreement between the numerical and the analytical results is also quite good. The application of the higher-order beam theories results only a bit smaller function values than the FEA solution. As it is also illustrated by the highlighted part of the figure, the contribution of the higher-order theories to the deflection becomes smaller and smaller by increasing the order of theory. Thus, the deflection improvement of the SSDT, TSDT and Reddy’s TSDT compared with the FSDT is negligible. From this point of view, application of the higher-order theories is not necessary. In order to present the essence of the higher-order beam theories, Fig. 6 depicts further results regarding the \(\sigma _x\) normal and \(\tau _{xz}\) shear stresses. The stresses are provided at the delamination tip, separately at the undelaminated \(X=+0\) [mm] and the delaminated \(X=-0\) [mm] portions. As we can see, certain stress discontinuities take place at the crack tip which result different \(\sigma _x\) normal and \(\tau _{xz}\) shear stress distributions on the sides. Based on Fig. 6a, apart from the singular nature of the numerical solution, \(\sigma _x\) normal stresses are in good agreement with each other. Although, if we take a look at the delaminated part, the Reddy’s TSDT solution shows a little perturbation. It can only be explained by the rigorously imposed traction-free conditions. The application of the higher-order beam theories becomes quite important only if the \(\tau _{xz}\) shear stresses are investigated. Based on Fig. 6b, by increasing the order of the solutions, the \(\tau _{xz}\) shear stress can be described with a more and more accurate way. If the exact description of the \(\tau _{xz}\) shear stress at the crack tip is important, the application of higher-order theories becomes inevitable. It is worth giving attention to \(\tau _{xz}\) shear stress distribution which is obtained by the Reddy’s TSDT. As it was discussed previously, it is already able to satisfy the traction-free boundary conditions all along the top and bottom surfaces of the beam. Although, it is important to emphasise, it cannot satisfy the boundary conditions at the bottom surface of the top sub-laminate and at the top surface of the bottom sub-laminate. According to our computational experience, if these conditions had been strictly imposed it would have resulted in over-constrained shear strain distribution around the crack tip.

## 5 *J*-integral

*J*-integral can be applied [42]. Based on the basic definition, considering any arbitrary counterclockwise

*C*contour around the crack tip, the

*J*-integral can be formulated as:

*U*is the strain energy density, \(\sigma _{ij}\) are the components of the stress tensors, \(u_i\) means the displacement vector components, \(\mathrm{d}s\) is the length increment along the contour and \(n_i\) represents the components of the outward unit vector in the given (\(X_1-X_2\)) Cartesian coordinate system. The basic concept is depicted by Fig. 9a. Moreover, considering only linear elastic fracture mechanics, the fundamental property of this integral is the following [2]:

*J*represents the value of the contour integral. Regarding delaminated beams, and based on the discussed semi-layerwise approach, the

*J*-integral can be calculated as a zero-area path integral. The idea is depicted by Fig. 9b, where the \(\varvec{n}\) unit vector always remains parallel to the

*X*-axis. Finally, taking into consideration the actual coordinate system (\(X_1= - X\) and \(X_2=z^{(i)}\)) and the actual number of ESLs, the total value of the energy release rate becomes

### 5.1 Mode partitioning

*J*-integral makes these type of considerations absolutely avoidable and provides a methodical separation technique. For this purpose, we only have to separate the strain and stress fields into symmetrical and asymmetrical function components with respect to the delamination plane:

For “Case I” delamination scenario, the obtained results are depicted by Fig. 12. As can be seen, under different external loadings the FSDT, SSDT and TSDT solutions give results quite close to each other and the Euler–Bernoulli-based evaluation techniques. Furthermore, the agreement with the numerical FEA-VCCT solution is also quite good. Although, the application of Reddy’s TSDT significantly decreases the ratio of the mode-I energy release rate. The difference can only be explained by the rigorously imposed traction-free boundary condition. The caused \(\sigma _x\) perturbation, referring to Fig. 6, is no longer insignificant and it certainly disturbs the ratio of the symmetrical and asymmetrical function components, as well.

Figure 13 gives result of the mode mixity evaluations for “Case II”. In general, we can state that the FSDT, SSDT and TSDT are located between the Williams’ curvature based and the Reddy’s TSDT solutions. The FSDT and the Luo-Tong theory predict almost the same mode mixity values as the FEM-VCCT. Unfortunately, as in the delamination scenario discussed above, the application of Reddy’s TSDT significantly decreases the ratio of the mode-I energy release rate, which can only be attributable to the \(\sigma _x\) perturbation in Fig. 8.

## 6 Conclusion

*J*-integral with zero-area path was introduced. By using symmetric and asymmetric decomposition of the mechanical fields, a well-ordered evaluation technique was proposed.

The results of different higher-order theories, apart form the Reddy’s TSDT, were close to each other and the Luo-Tong solution. Furthermore, these evaluation methods gave results between the Williams’ curvature based and the Bruno-Greco solutions, which defined the most extreme mode ratios. Unfortunately, regarding Reddy’s TSDT it was not true. The rigorously imposed traction-free condition significantly decreased the ratio of mode-I energy release rate and it significantly disturbed the \(\sigma _x\) normal stress distribution in the thinner sub-laminate.

Nevertheless, this paper is the first attempt to perform mode separation by combining the higher-order theories with the *J*-integral. The next step could be the semi-layerwise approach- based finite element development and the consideration of the transverse stretch modelling [52].

## Notes

### Acknowledgements

Open access funding provided by Budapest University of Technology and Economics (BME). This work was supported by the Hungarian Scientific Research Fund (NKFI) under Grant No. 128090. The research reported in this paper was supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of TOPIC research area of Budapest University of Technology and Economics (BME FIKP-NANO).

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