Non-linear elastic analysis of delamination in two-dimensional functionally graded multilayered beams

Abstract

An analytical study of delamination fracture in two-dimensional functionally graded multilayered non-linear elastic split cantilever beam configurations is carried out. The beam is made of an arbitrary number of adhesively bonded longitudinal vertical layers, which have different thicknesses and material properties. Besides, the material is two-dimensional functionally grade in the cross-section of each layer. A vertical delamination crack is located arbitrary along the beam width. The beam is loaded eccentrically in tension by a force applied at the free end of the right-hand crack arm. The delamination fracture is studied in terms of the strain energy release rate. A comparison with the J-integral approach is carried out for verification. The results, yielded by the two analyses, are identical. The solutions obtained are used to evaluate the influences of crack location, material gradients and non-linear behaviour of material on the delamination fracture in the split cantilever beam under consideration. The distribution of the J-integral value along the crack front is analyzed too. It is found that the strain energy release rate can be reduced by appropriate selection of the material gradients in the design stage of two-dimensional functionally graded multilayered beams.

Keywords

Multilayered beam two-dimensional functionally graded material delamintion crack material non-linearity

1. Introduction

Recently, functionally graded materials have been widely applied in modern industry due to the fact that they can be manufactured by mixing of their constituent materials with a desired continuous variation of properties in certain directions [1,3,5,6,10,12,22,23,25]. These novel inhomogeneous materials provide maximum benefits of their constituents. By tailoring the microstructure and composition of functionally graded materials during manufacturing, one can get optimum performance of structural members and components which are subjected to non-uniform service requirements. Since the microstructure of functionally graded materials is heterogeneous, fracture is the most common type of failure. Therefore, fracture behaviour of functionally graded materials remains a topic of active research [8,9,13–15,17–19,24,26,27].

Studies of fracture behaviour of graded materials have been reviewed in [24]. Cracks oriented parallel or perpendicular to the gradient direction have been analyzed by applying methods of linear-elastic fracture mechanics. A range of specific problems have been discussed, including fracture in functionally graded materials under fatigue crack loading conditions.

Cracked functionally graded three-point bending beams have been investigated by applying the compliance approach [26]. An equivalent homogeneous beam of variable depth for the cracked beam has been proposed assuming linear-elastic behaviour of the material. The rate of change of the beam compliance with the crack length has been analyzed. It has been found that the method developed is particularly suitable for investigating cracked functionally graded structural components under concentrated loads.

Cracked functionally graded beams have been analyzed in [14]. It has been assumed that the modulus of elasticity of the functionally graded material varies in the thickness direction (i.e., the material is one-dimensional functionally graded). Clamped–clamped and clamped–free beams have been studied. Free vibration behaviour has been investigated for different crack location and crack depth ratio. Beams having a transversal crack are modelled as two sub-beams connected by a mass-less rotational spring. The effect of open crack on the modes of vibration of functionally graded beams has been captured and discussed.

A longitudinal crack in a functionally graded beam configuration has been investigated analytically with taking into account the non-linear behaviour of the material in [18]. It has been assumed that the material is one-dimensional functionally graded. Results of parametric studies of non-linear longitudinal fracture behaviour have been reported.

Multilayered structural members and components are manufactured by bonding of layers of different materials. One of the characteristic properties of multilayered systems is their high strength to weight and high stiffness to weight ratios. Multilayered systems are extensively used in structural applications where high performances are required. However, these systems have relatively law interlaminar strength. Thus, delamination fracture, i.e. separation of layers, is usually the earliest failure mode when these structures are subjected to various external loadings.

Delamination fracture behaviour of multilayered functionally graded beams which exhibit material non-linearity has been analyzed in [19]. It has been assumed that the material is one-dimensional functionally graded in each layer. A solution to the strain energy release rate has been derived.

It can be summarized that various crack problems in functionally graded structures and materials have been analyzed mainly assuming linear-elastic mechanical behaviour of the material [8,9,13]. Therefore, the main purpose of the present paper is to analyze the delamination fracture of a beam configuration with taking into account the non-linear mechanical behaviour of material. A multilayered spilt cantilever beam configuration made of vertical layers of material that is functionally graded along the width and height of layers is studied in the present paper in contrast to the previous articles which are focussed on delamination fracture in beams made of vertical layers of materials which are functionally graded along the width of the layers only [20,21]. Beside, the present paper considers a beam loaded by a longitudinal force in contrast to the previous articles which deal with delamination analysis of beam configurations loaded in pure bending with respect to the horizontal centroidal axis of the beam cross-section [20,21]. The fracture is studied in terms of the strain energy release rate in the present paper. The non-linear solution obtained is verified by analyzing the delamination fracture with the help of the J-integral. The effects of crack location along the width of the beam cross-section, the material gradients and the non-linear behaviour of material on the delamination fracture are investigated. The distribution of the J-integral value along the delamination crack front is investigated too.

2. Non-linear elastic analysis of delamination fracture

The present paper analyses the non-linear elastic delamination behaviour of the two-dimensional functionally graded multilayered split cantilever beam configuration shown schematically in Fig. 1. The beam is made of an arbitrary number of vertical layers. In the cross-section of each layer, the material is two-dimensional functionally graded. Also, each layer has individual thickness and material properties. Perfect adhesion is assumed between layers. The beam has a rectangular cross-section of width, b, and height, h. A delamination crack of length, a, is located arbitrary between layers. The widths of the right-hand and left-hand crack arms are denoted by b ₁ and b ₂, respectively. The beam is clamped in section, B. The external loading consists of one longitudinal force, F, applied at the free end of the right-hand crack arm (Fig. 1).

Fig. 1.

Geometry of the two-dimensional functionally graded multilayered split cantilever beam loaded by a longitudinal force.

The delamination fracture is studied in terms of the strain energy release rate. By assuming an elementary increase, da, of the delamination crack length, the strain energy release rate, G, is written as [18] $\begin{eqnarray}\displaystyle G=\displaystyle \frac{dU^{\ast }}{hda}, & & \displaystyle\end{eqnarray}$ (1) where dU ^∗ is the change of the beam complementary strain energy.

The analysis developed in the present study holds for non-linear elastic behaviour of the material. The analysis can also be applied for elastic-plastic behaviour if the beam considered undergoes active deformation, i.e. if the external loading increases only [4,11]. It should also be noted that the present analysis is based on the small strains assumption.

The beam complementary strain energy is derived by integrating the complementary strain energy density in the volume of the right-hand crack arm and the un-cracked beam portion (a ≤ x ₃ ≤ l) $\begin{eqnarray}\displaystyle U^{\ast }=a\displaystyle \mathop{\sum }_{i=1}^{i=n_{R}}\displaystyle \int _{-\frac{h}{2}}^{\frac{h}{2}}\left(\int _{y_{1i}}^{y_{1i+1}}u_{0R_{i}}^{\ast }dy_{1}\right)dz_{1}+(l-a)\mathop{\sum }_{1=i}^{i=n}\int _{-\frac{h}{2}}^{\frac{h}{2}}\left(\int _{y_{2i}}^{y_{2i+1}}u_{0U_{i}}^{\ast }dy_{2}\right)dz_{2}, & & \displaystyle\end{eqnarray}$ (2) where $u_{0R_{i}}^{\ast }$ and $u_{0U_{i}}^{\ast }$ are the complementary strain energy densities in the i-th layer of the right-hand crack arm and un-cracked beam portion (a ≤ x ₃ ≤ l), respectively. Axes, y ₁ and z ₁, and coordinates, y _1i and y _1i+1, are shown in Fig. 2, y ₂ and z ₂ are the centroidal axes of the cross-section un-cracked beam portion (z ₂ is directed downwards), n _R and n are the numbers of layers in the right-hand crack arm and the un-cracked beam portion, respectively.

Fig. 2.

The geometry and loading of the free end of the right-hand crack arm (n ₁ − n ₁ is the neutral axis).

Fig. 3.

Schematic of a non-linear stress-strain curve (u ₀ and $u_{0}^{\ast }$ are the strain energy and the complementary strain energy densities, respectively).

Fig. 4.

Schematic of two three-layered functionally graded split cantilever beam configurations.

The following power-law stress-strain relation is used in order to describe the material behaviour [16]: $\begin{eqnarray}\displaystyle {\sigma}_{i}=L_{i}{\varepsilon}^{m_{i}},\quad i=1,∼2,\ldots ,∼n & & \displaystyle\end{eqnarray}$ (3) where 𝜎_i is the longitudinal normal stress in the i-th layer, 𝜀 is the longitudinal strain, L _i and m _i are material properties in the same layer. The material is two-dimensional functionally graded in the cross-section of each layer, i.e. L _i varies continuously along the width and height of each layer. The following law is applied to describe the distribution of the material property, L _i, in the cross-section of the i-th layer: $\begin{eqnarray}\displaystyle L_{i}(y_{3},∼z_{3})=L_{D_{i}}+\displaystyle \frac{L_{H_{i}}-L_{D_{i}}}{(y_{3i+1}-y_{3i})^{3}}(y_{3}-y_{3i})^{3}+L_{K_{i}}\displaystyle \frac{16z_{3}^{4}}{h^{4}}, & & \displaystyle\end{eqnarray}$ (4) where $\begin{eqnarray}\displaystyle y_{3i}\leq y_{3}\leq y_{3i+1},\quad -\displaystyle \frac{h}{2}\leq z_{3}\leq \displaystyle \frac{h}{2},\quad i=1,∼2,\ldots ,∼n. & & \displaystyle\end{eqnarray}$ (5) In (4), L _{D
_i} and L _{H
_i} are the values of L _i in points D _i and H _i, respectively (Fig. 1). The material gradient along the height of the i-th layer is governed by the material property, L _{K
_i}. Axes, y ₃ and z ₃, and coordinates, y _3i and y _3i+1, are shown in Fig. 1. Formula (4) indicates that the material property, L _i, is distributed symmetrically with respect to axis, y ₃.

The complementary strain energy density is equal to the area OQR that supplements the area OPQ, enclosed by the stress-strain curve, to a rectangle (Fig. 3). Therefore, the complementary strain energy density in the i-th layer of the right-hand crack arm, when the power-law stress-strain relation (3) is used, can be expressed as [19] $\begin{eqnarray}\displaystyle u_{0R_{i}}^{\ast }=L_{i}∼\displaystyle \frac{m_{i}{\varepsilon}^{m_{i}+1}}{m_{i}+1}. & & \displaystyle\end{eqnarray}$ (6) The strain energy density, u _{0R
_i}, is equal to the area OPQ (Fig. 3). Thus, u _{0R
_i} can be written as [19] $\begin{eqnarray}\displaystyle u_{0R_{i}}=\displaystyle \frac{L_{i}{\varepsilon}^{m_{i}+1}}{m_{i}+1}. & & \displaystyle\end{eqnarray}$ (7)

The Bernoulli’s hypothesis for plane sections is applied to analyze the distribution of strains since the span to height ratio of the beam considered is large. Concerning application of the Bernoulli’s hypothesis in the present analysis, it should also be noted that since the beam is loaded in eccentric tension (Fig. 1) the only non-zero strain is the longitudinal strain, 𝜀. Therefore, according to the small strain compatibility equations, 𝜀 is distributed linearly along width of the beam cross-section. Thus, 𝜀 in the right-hand crack arm (Fig. 2) is written as $\begin{eqnarray}\displaystyle {\varepsilon}=(y_{1}-y_{1n_{1}}){\kappa}_{1}, & & \displaystyle\end{eqnarray}$ (8) where y _1n
₁ is the coordinate of the neutral axis, n ₁ − n ₁, in the right-hand crack arm cross-section, 𝜅₁ is the curvatures of right-hand crack arm. The neutral axis shifts from the centroid since the beam is functionally graded and multilayered (Fig. 2).

The following equilibrium equations of the right-hand crack arm cross-section are used to determine y _1n
₁ and 𝜅₁: $\begin{eqnarray}\displaystyle N_{1}=\mathop{\sum }_{i=1}^{i=n_{R}}\int _{-\frac{h}{2}}^{\frac{h}{2}}\left(\int _{y_{1i}}^{y_{1i+1}}{\sigma}∼dy_{1}\right)dz_{1}, & & \displaystyle\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle M_{z_{1}}=\displaystyle \mathop{\sum }_{i=1}^{i=n_{R}}\displaystyle \int _{-\frac{h}{2}}^{\frac{h}{2}}\left(\int _{y_{1i}}^{y_{1i+1}}{\sigma}y_{1}∼dy_{1}\right)dz_{1}, & & \displaystyle\end{eqnarray}$ (10) where N ₁ and M _{z
₁} are the axial force and bending moments. The coordinates, y _1i and y _1i+1, are shown in Fig. 2. It is obvious that in the right-hand crack arm $\begin{eqnarray}\displaystyle N_{1}=F,\quad M_{z_{1}}=0. & & \displaystyle\end{eqnarray}$ (11) By using (4), the distribution of material property, L _i, in the i-th layer of the right-hand crack arm is written as: $\begin{eqnarray}\displaystyle L_{i}(y_{1},∼z_{1})=L_{D_{i}}+\displaystyle \frac{L_{H_{i}}-L_{D_{i}}}{(y_{1i+1}-y_{1i})^{3}}(y_{1}-y_{1i})^{3}+L_{K_{i}}\displaystyle \frac{16z_{1}^{4}}{h^{4}}. & & \displaystyle\end{eqnarray}$ (12) Then by substituting of (3), (8) and (12) in (9) and (10), one obtains $\begin{eqnarray}\displaystyle N_{1} & = & \displaystyle \mathop{\sum }_{i=1}^{i=n_{R}}\left\{\left[\frac{L_{D_{i}}h{\kappa}_{1}^{m_{i}}}{m_{i}+1}+\frac{L_{K_{i}}h{\kappa}_{1}^{m_{i}}}{5\left(m_{i}+1\right)}+\frac{{\delta}_{i}h{\kappa}_{1}^{m_{i}}\left(-y_{1i}\right)^{3}}{m_{i}+1}\right]\left({\psi}_{i}^{m_{i}+1}-{\zeta}_{i}^{m_{i}+1}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{i}h{\kappa}_{1}^{m_{i}}y_{1i}^{2}\left[\frac{1}{m_{i}+2}\left({\psi}_{i}^{m_{i}+2}-{\zeta}_{i}^{m_{i}+2}\right)+\frac{y_{1n_{1}}}{m_{i}+1}\left({\psi}_{i}^{m_{i}+1}-{\zeta}_{i}^{m_{i}+1}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{i}h{\kappa}_{1}^{m_{i}}y_{1i}q_{i}\left[\displaystyle \frac{1}{f_{i}+3q_{i}}\left({\psi}_{i}^{\frac{f_{i}+3q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+3q_{i}}{q_{i}}}\right)+\displaystyle \frac{2y_{1n_{1}}}{f_{i}+2q_{i}}\left({\psi}_{i}^{\frac{f_{i}+2q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+2q_{i}}{q_{i}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\displaystyle \frac{y_{1n_{1}}^{2}}{f_{i}+q_{i}}\left({\psi}_{i}^{\frac{f_{i}+q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+q_{i}}{q_{i}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{i}h{\kappa}_{1}^{m_{i}}q_{i}\left[\displaystyle \frac{1}{f_{i}+4q_{i}}\left({\psi}_{i}^{\frac{f_{i}+4q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+4q_{i}}{q_{i}}}\right)+\frac{3y_{1n_{1}}}{f_{i}+3q_{i}}\left({\psi}_{i}^{\frac{f_{i}+3q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+3q_{i}}{q_{i}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left.∼\frac{3y_{1n_{1}}^{2}}{f_{i}+2q_{i}}\left({\psi}_{i}^{\frac{f_{i}+2q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+2q_{i}}{q_{i}}}\right)+\frac{y_{1n_{1}}^{3}}{f_{i}+q_{i}}\left({\psi}_{i}^{\frac{f_{i}+q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+q_{i}}{q_{i}}}\right)\right]\right\},\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle M_{z_{1}} & = & \displaystyle \mathop{\sum }_{i=1}^{i=n_{R}}\left\{\left(L_{D_{i}}{\kappa}_{1}^{m_{i}}h+\frac{L_{K_{i}}{\kappa}_{1}^{m_{i}}h}{5}+{\delta}_{i}{\kappa}_{1}^{m_{i}}h\left(-y_{1i}\right)^{3}\right)+\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼\left[\frac{1}{m_{i}+2}\left({\psi}_{i}^{m_{i}+2}-{\zeta}_{i}^{m_{i}+2}\right)+\frac{y_{1n_{1}}}{m_{i}+1}\left({\psi}_{i}^{m_{i}+1}-{\zeta}_{i}^{m_{i}+1}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{i}{\kappa}_{1}^{m_{i}}hy_{1i}^{2}q_{i}\left[\frac{1}{f_{i}+3q_{i}}\left({\psi}_{i}^{\frac{f_{i}+3q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+3q_{i}}{q_{i}}}\right)+\frac{2y_{1n_{1}}}{f_{i}+2q_{i}}\left({\psi}_{i}^{\frac{f_{i}+2q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+2q_{i}}{q_{i}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\frac{y_{1n_{1}}^{2}}{f_{i}+q_{i}}\left({\psi}_{i}^{\frac{f_{i}+q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+q_{i}}{q_{i}}}\right)\right]+\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{i}{\kappa}_{1}^{m_{i}}hy_{1i}q_{i}\left[\frac{1}{f_{i}+4q_{i}}\left({\psi}_{i}^{\frac{f_{i}+4q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+4q_{i}}{q_{i}}}\right)+\displaystyle \frac{3y_{1n_{1}}}{f_{i}+3q_{i}}\left({\psi}_{i}^{\frac{f_{i}+3q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+3q_{i}}{q_{i}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\frac{3y_{1n_{1}}^{2}}{f_{i}+2q_{i}}\left({\psi}_{i}^{\frac{f_{i}+2q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+2q_{i}}{q_{i}}}\right)+\frac{y_{1n_{1}}^{3}}{f_{i}+q_{i}}\left({\psi}_{i}^{\frac{f_{i}+q_{i}}{q_{i}}}-{\zeta}_{i}^{\frac{f_{i}+q_{i}}{q_{i}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{i}{\kappa}_{1}^{m_{i}}hq_{i}\left[\frac{{\psi}_{i}^{f_{i}+5q_{i}}-{\zeta}_{i}^{f_{i}+5q_{i}}}{f_{i}+5q_{i}}+\frac{4y_{1n_{1}}\left({\psi}_{i}^{f_{i}+4q_{i}}-{\zeta}_{i}^{f_{i}+4q_{i}}\right)}{f_{i}+4q_{i}}+\frac{6y_{1n_{1}}^{2}\left({\psi}_{i}^{f_{i}+3q_{i}}-{\zeta}_{i}^{f_{i}+3q_{i}}\right)}{f_{i}+3q_{i}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left.∼\frac{4y_{1n_{1}}^{3}\left({\psi}_{i}^{f_{i}+2q_{i}}-{\zeta}_{i}^{f_{i}+2q_{i}}\right)}{f_{i}+2q_{i}}+\frac{y_{1n_{1}}^{4}\left({\psi}_{i}^{f_{i}+q_{i}}-{\zeta}_{i}^{f_{i}+q_{i}}\right)}{f_{i}+q_{i}}\right]\right\},\end{eqnarray}$ (14) where 𝛿_i = (L _{H
_i} − L _{D
_i})∕(y _1i+1 − y _1i)³, 𝜓_i = y _1i+1 − y _1n
₁, 𝜁_i = y _1i − y _1n
₁ and f _i∕q _i = m _i (f _i and q _i are positive integers).

At m _i = 1 the power-law stress-strain relation (3) transforms into the Hooke’s law. This means that at m _i = 1 Eq. (14) should transform into the formula for curvature of linear-elastic beam. Indeed, by substituting of m _i = 1, n _R = 1, L _{D
_i} = L _{H
_i} = E and L _{K
_i} = 0 (here E is the modulus of elasticity) in (14), one derives $\begin{eqnarray}\displaystyle {\kappa}_{1}=\displaystyle \frac{12M_{z_{1}}}{Ehb_{1}^{3}}, & & \displaystyle\end{eqnarray}$ (15) which is exact match of the formula for curvature of a linear-elastic homogeneous beam.

Equations (13) and (14) should be solved with respect to y _1n
₁ and 𝜅₁ by using the MatLab computer program.

By substituting of (8) and (12) in (6), the complementary strain energy density in the i-th layer of the right-hand crack arm is written as $\begin{eqnarray}\displaystyle u_{0R_{i}}^{\ast }=\left[L_{D_{i}}+\displaystyle \frac{L_{H_{i}}-L_{D_{i}}}{(y_{1i+1}-y_{1i})^{3}}(y_{1}-y_{1i})^{3}+L_{K_{1}}\displaystyle \frac{16z_{1}^{4}}{h^{4}}\right]∼\frac{m_{i}(y_{1}-y_{1n_{1}})^{m_{i}+1}{\kappa}_{1}^{m_{1}+1}}{m_{i}+1}, & & \displaystyle\end{eqnarray}$ (16) where $\begin{eqnarray}\displaystyle i=1,∼2,\ldots ,∼n_{R}. & & \displaystyle\end{eqnarray}$ (17)

The distribution of the complementary strain energy density, $u_{0U_{i}}^{\ast }$ , in the i-th layer of the un-cracked beam portion can be determined by using (16). For this purpose, y ₁, z ₁, y _1i, y _1i+1, y _1n
₁ and 𝜅₁ have to be replaced with y ₂, z ₂, y _2i, y _2i+1, y _2n
₂ and 𝜅₂, respectively (y _2n
₂ is the coordinate of the neutral axis in the cross-section of the un-cracked beam portion, 𝜅₂ is the curvature of the un-cracked beam). Equilibrium equations (13) and (14) can be used to determine y _2n
₂ and 𝜅₂. For this purpose, n _R, M _{z
₁}, 𝛿_i, 𝜓_i, 𝜁_i, y _1i, y _1i+1, y _1n
₁ and 𝜅₁ have to be replaced respectively with n, −F (b∕2 − b ₁∕2), 𝛿_Ui, 𝜓_Ui, 𝜁_Ui, y _2i, y _2i+1, y _2n
₂ and 𝜅₂, where 𝛿_Ui = (L _{H
_i} − L _{D
_i})∕(y _2i+1 − y _2i)³, 𝜓_Ui = y _2i+1 − y _2n
₂ and 𝜁_Ui = y _2i − y _2n
₂.

The strain energy release rate is derived by substituting of $u_{0R_{i}}^{\ast }$ , $u_{0U_{i}}^{\ast }$ and (2) in (1). The result is $\begin{eqnarray}\displaystyle G & = & \displaystyle \mathop{\sum }_{i=1}^{i=n_{R}}\frac{m_{i}{\kappa}_{1}^{m_{i}+1}}{h\left(m_{i}+1\right)}\left\{\left[\frac{L_{D_{i}}h}{m_{t_{i}}+1}+\frac{L_{K_{i}}h}{5\left(m_{t_{i}}+1\right)}+\frac{{\delta}_{i}h(-y_{1i})^{3}}{m_{t_{i}}+1}\right]({\psi}_{i}^{m_{t_{i}}+1}-{\zeta}_{i}^{m_{t_{i}}+1})\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{i}hy_{1i}^{2}\left[\frac{1}{m_{t_{i}}+2}({\psi}_{i}^{m_{t_{i}}+2}-{\zeta}_{i}^{m_{t_{i}}+2})+\frac{y_{1n_{1}}}{m_{t_{i}}+1}({\psi}_{i}^{m_{t_{i}}+1}-{\zeta}_{i}^{m_{t_{i}}+1})\right]+\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{i}hy_{1i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{2y_{1n_{1}}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\frac{y_{1n_{1}}^{2}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{i}hq_{t_{i}}\left[\frac{1}{f_{t_{i}}+4q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{3y_{1n_{1}}}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left.∼\frac{3y_{1n_{1}}^{2}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{y_{1n_{1}}^{3}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼\mathop{\sum }_{i=1}^{i=n}\frac{m_{i}{\kappa}_{2}^{m_{i}+1}}{h(m_{i}+1)}\left\{\left[\frac{L_{D_{i}}h}{m_{t_{i}}+1}+\frac{L_{K_{i}}h}{5(m_{t_{i}}+1)}+\frac{{\delta}_{Ui}h(-y_{2i})^{3}}{m_{t_{i}}+1}\right]({\psi}_{Ui}^{m_{t_{i}}+1}-{\zeta}_{Ui}^{m_{t_{i}}+1})\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{Ui}hy_{2i}^{2}\left[\frac{1}{m_{t_{i}}+2}({\psi}_{Ui}^{m_{t_{i}}+2}-{\zeta}_{Ui}^{m_{t_{i}}+2})+\frac{y_{2n_{2}}}{m_{t_{i}}+1}\left({\psi}_{Ui}^{m_{t_{i}}+1}-{\zeta}_{Ui}^{m_{t_{i}}+1}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{Ui}hy_{2i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{2y_{2n_{2}}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\theta}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.∼\frac{y_{2n_{2}}^{2}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{Ui}^{\frac{ft_{i}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{Ui}hq_{t_{i}}\left[\frac{1}{f_{t_{i}}+4q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{3y_{2n_{2}}}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\left.\frac{3y_{2n_{2}}^{2}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{y_{2n_{2}}^{3}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\right\},\end{eqnarray}$ (18) where m _{t
_i} = m _i +1, f _{t
_i}∕q _{t
_i} = m _{t
_i} (f _{t
_i} and q _{t
_i} are positive integers). It should be noted that at m _i = 1, n _R = 1, n = 1, L _{D
_i} = L _{H
_i} = E and L _{K
_i} = 0 formula (18) calculates $\begin{eqnarray}\displaystyle G=\displaystyle \frac{F^{2}}{8Ebh^{2}}, & & \displaystyle\end{eqnarray}$ (19) which is exact match of the strain energy release rate in linear-elastic homogeneous split cantilever beam derived by applying the formula published in [7].

Formula (18) is verified by analyzing the delamination fracture behaviour of the two-dimensional functionally graded multilayered split cantilever beam by applying the J-integral approach [2]. The J-integral is solved by using the integration contour, 𝛤, that coincides with the beam contour (Fig. 1). It is obvious that the J-integral has non-zero values only in segments 𝛤₁ and 𝛤₂ of the integration contour (𝛤₁ and 𝛤₂ coincide with the free end of the right-hand crack arm and the clamping, respectively). Therefore, the solution of the J-integral is written as $\begin{eqnarray}\displaystyle J=J_{{\Gamma}_{1}}+J_{{\Gamma}_{2}}, & & \displaystyle\end{eqnarray}$ (20) where J _𝛤₁ and J _𝛤₂ are the J-integral values in segments 𝛤₁ and 𝛤₂, respectively.

In segment, 𝛤₁, the J-integral is written as $\begin{eqnarray}\displaystyle J_{{\Gamma}_{1}}=\mathop{\sum }_{i=1}^{n_{R}}\int _{y_{1i}}^{y_{1i+1}}\left[u_{0R_{i}}\cos {\alpha}-\left(p_{xi}\frac{\partial u}{\partial x}+p_{yi}\frac{\partial v}{\partial x}\right)\right]ds, & & \displaystyle\end{eqnarray}$ (21) where 𝛼 is the angle between the outwards normal vector to the contour of integration and the crack direction, p _xi and p _yi are the components of stress vector in the i-th layer of the right-hand crack arm, u and v are the components of displacement vector with respect to the crack tip coordinate system xy (x is directed along the crack), ds is a differential element along the contour.

In segment, 𝛤₁, of the integration contour, the J-integral components are found as $\begin{eqnarray}\displaystyle p_{xi}=-{\sigma}_{i}=-L_{i}{\varepsilon}^{m_{i}},\quad p_{yi}=0, & & \displaystyle\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle ds=dy_{1},\quad \cos {\alpha}=-1, & & \displaystyle\end{eqnarray}$ (23) where coordinate, y ₁, varies in the interval [−b ₁∕2, b ₁∕2].

The strain energy density, u _{0R
_i}, is obtained by formula (7). The partial derivative, ∂u∕∂x, that is needed to solve (21) is determined by using the following formula from mechanics of materials: $\begin{eqnarray}\displaystyle \displaystyle \frac{\partial u}{\partial x}={\varepsilon}=(y_{1}-y_{1n_{1}}){\kappa}_{1}, & & \displaystyle\end{eqnarray}$ (24) where y _1n
₁ and 𝜅₁ are obtained from Eqs (13) and (14).

The J-integral solution in segment, 𝛤₁, is derived by substituting of (3), (8), (12), (22), (23) and (24) in (21). The result is $\begin{eqnarray}\displaystyle J_{{\Gamma}_{1}} & = & \displaystyle \mathop{\sum }_{i=1}^{i=n_{R}}\frac{m_{i}{\kappa}_{1}^{m_{i}+1}}{m_{i}+1}\left\{\left[\frac{L_{D_{i}}}{m_{t_{i}}+1}+\frac{16L_{K_{i}}z_{1}^{4}}{h^{4}(m_{t_{i}}+1)}+\frac{{\delta}_{i}(-y_{1i})^{3}}{m_{t_{i}}+1}\right]({\psi}_{i}^{m_{t_{i}}+1}-{\zeta}_{i}^{m_{t_{i}}+1})\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{i}y_{1i}^{2}\left[\frac{1}{m_{t_{i}}+2}({\psi}_{i}^{m_{t_{i}}+2}-{\zeta}_{i}^{m_{t_{i}}+2})+\frac{y_{1n_{1}}}{m_{t_{i}}+1}({\psi}_{i}^{m_{t_{i}}+1}-{\zeta}_{i}^{m_{t_{i}}+1})\right]+\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{i}y_{1i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{2y_{1n_{1}}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\frac{y_{1n_{1}}^{2}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+4q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{3y_{1n_{1}}}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left.∼\frac{3y_{1n_{1}}^{2}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{y_{1n_{1}}^{3}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\right\},\end{eqnarray}$ (25) where the coordinate, z ₁, varies in the interval [−h∕2, h∕2].

The J-integral solution in segment, 𝛤₂, of the integration contour (Fig. 1) can be obtained by (25). For this purpose, n _R, y _1i, y _1i+1, 𝛿_i, 𝜓_i, 𝜁_i, y _1n
₁ and 𝜅₁ are replaced with n, y _2i, y _2i+1, 𝛿_Ui, 𝜓_Ui, 𝜁_Ui, y _2n
₂ and 𝜅₂, respectively. Besides, the sign of (25) must be set to “minus” because the integration contour is directed upwards in segment, 𝛤₂.

Finally, J _𝛤₁ and J _𝛤₂ are substituted in (20). The solution of the J-integral is written as $\begin{eqnarray}\displaystyle J & = & \displaystyle \mathop{\sum }_{i=1}^{i=n_{R}}\frac{m_{i}{\kappa}_{1}^{m_{i}+1}}{m_{i}+1}\left\{\left[\frac{L_{D_{i}}}{m_{t_{i}}+1}+\frac{16L_{K_{i}}z_{1}^{4}}{h^{4}\left(m_{t_{i}}+1\right)}+\frac{{\delta}_{i}\left(-y_{1i}\right)^{3}}{m_{t_{i}}+1}\right]\left({\psi}_{i}^{m_{t_{i}}+1}-{\zeta}_{i}^{m_{t_{i}}+1}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{i}y_{1i}^{2}\left[\frac{1}{m_{t_{i}}+2}\left({\psi}_{i}^{m_{t_{i}}+2}-{\zeta}_{i}^{m_{t_{i}}+2}\right)+\frac{y_{1n_{1}}}{m_{t_{i}}+1}\left({\psi}_{i}^{m_{t_{i}}+1}-{\zeta}_{i}^{m_{t_{i}}+1}\right)\right]+\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{i}y_{1i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{2y_{1n_{1}}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.∼\frac{y_{1n_{1}}^{2}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+4q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{3y_{1n_{1}}}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left.∼\frac{3y_{1n_{1}}^{2}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{y_{1n_{1}}^{3}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼\mathop{\sum }_{i=1}^{i=n}\frac{m_{i}{\kappa}_{2}^{m_{i}+1}}{m_{i}+1}\left\{\left[\frac{L_{D_{i}}}{m_{t_{i}}+1}+\frac{16L_{K_{i}}z_{1}^{4}}{h^{4}\left(m_{t_{i}}+1\right)}+\frac{{\delta}_{Ui}\left(-y_{2i}\right)^{3}}{m_{t_{i}}+1}\right]\left({\psi}_{Ui}^{m_{t_{i}}+1}-{\zeta}_{Ui}^{m_{t_{i}}+1}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{Ui}y_{2i}^{2}\left[\frac{1}{m_{t_{i}}+2}\left({\psi}_{Ui}^{m_{t_{i}}+2}-{\zeta}_{Ui}^{m_{t_{i}}+2}\right)+\frac{y_{2n_{2}}}{m_{t_{i}}+1}\left({\psi}_{Ui}^{m_{t_{i}}+1}-{\zeta}_{Ui}^{m_{t_{i}}+1}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{Ui}y_{2i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{2y_{2n_{2}}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\theta}_{U2}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\frac{y_{2n_{2}}^{2}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{Ui}^{\frac{ft_{i}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{Ui}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+4q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{3y_{2n_{2}}}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left.∼\frac{3y_{2n_{2}}^{2}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{y_{2n_{2}}^{3}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\right\}.\end{eqnarray}$ (26) Formula (26) expresses the distribution of the J-integral value along the delamination crack front. The average value of the J-integral along the delamination crack front is written as $\begin{eqnarray}\displaystyle J_{AV}=\displaystyle \frac{1}{h}\int _{-\frac{h}{2}}^{\frac{h}{2}}Jdz_{1}. & & \displaystyle\end{eqnarray}$ (27)

The fact that the J-integral solution obtained by substituting of (26) in (27) matches exactly the formula for the strain energy release rate (18) is a verification of the fracture analysis of the two-dimensional functionally graded multilayered split cantilever beam configuration developed in the present paper with taking into account the non-linear behaviour of the material.

The expression (18) is verified also by deriving the strain energy release rate by considering the energy balance. For this purpose, a small increase, 𝛿a, of the crack length is assumed and the energy balance is written as $\begin{eqnarray}\displaystyle F{\delta}u_{F}=\displaystyle \frac{\partial U}{\partial a}{\delta}a+Gh{\delta}a, & & \displaystyle\end{eqnarray}$ (28) where u _F is the longitudinal displacement of the application point of the force, F, U is the strain energy in the beam. By using (28), the strain energy release rate is obtained as $\begin{eqnarray}\displaystyle G=\displaystyle \frac{F}{h}\displaystyle \frac{\partial u_{F}}{\partial a}-\frac{1}{h}\displaystyle \frac{\partial U}{\partial a}. & & \displaystyle\end{eqnarray}$ (29) By applying the integrals of Maxwell-Mohr, u _F is expressed as $\begin{eqnarray}\displaystyle u_{F}={\varepsilon}_{C_{1}}a+{\varepsilon}_{C_{2}}(l-a)+{\kappa}_{2}\frac{b-b_{1}}{2}(l-a), & & \displaystyle\end{eqnarray}$ (30) where 𝜀_{C
₁} = −𝜅₁ y _1n
₁, 𝜀_{C
₂} = −𝜅₂ y _2n
₂. The strain energy is obtained by replacing of $u_{0R_{i}}^{\ast }$ and $u_{0U_{i}}^{\ast }$ with u _{0R
_i} and u _{0U
_i} in formula (2). By substituting of u _F and U in (29), one derives $\begin{eqnarray}\displaystyle G & = & \displaystyle \frac{F}{h}\left({\varepsilon}_{C_{1}}-{\varepsilon}_{C_{2}}-{\kappa}_{2}\frac{b-b_{2}}{2}\right)-\mathop{\sum }_{i=1}^{i=n_{R}}\frac{{\kappa}_{1}^{m_{i}+1}}{h(m_{i}+1)}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼\left\{\left[\frac{L_{D_{i}}h}{m_{t_{i}}+1}+\frac{L_{K_{i}}h}{5\left(m_{t_{i}}+1\right)}+\frac{{\delta}_{i}h\left(-y_{1i}\right)^{3}}{m_{t_{i}}+1}\right]\left({\psi}_{i}^{m_{t_{i}}+1}-{\zeta}_{i}^{m_{t_{i}}+1}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{i}hy_{1i}^{2}\left[\frac{1}{m_{t_{i}}+2}\left({\psi}_{i}^{m_{t_{i}}+2}-{\zeta}_{i}^{m_{t_{i}}+2}\right)+\frac{y_{1n_{1}}}{m_{t_{i}}+1}\left({\psi}_{i}^{m_{t_{i}}+1}-{\zeta}_{i}^{m_{t_{i}}+1}\right)\right]+\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{i}hy_{1i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{2y_{1n_{1}}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\frac{y_{1n_{1}}^{2}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{i}hq_{t_{i}}\left[\frac{1}{f_{t_{i}}+4q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{3y_{1n_{1}}}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left.∼\frac{3y_{1n_{1}}^{2}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{y_{1n_{1}}^{3}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{i}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\mathop{\sum }_{i=1}^{i=n}\frac{{\kappa}_{2}^{m_{i}+1}}{h\left(m_{i}+1\right)}\left\{\left[\frac{L_{D_{i}}h}{m_{t_{i}}+1}+\frac{L_{K_{i}}h}{5\left(m_{t_{i}}+1\right)}+\frac{{\delta}_{Ui}h\left(-y_{2i}\right)^{3}}{m_{t_{i}}+1}\right]\left({\psi}_{Ui}^{m_{t_{i}}+1}-{\zeta}_{Ui}^{m_{t_{i}}+1}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼3{\delta}_{Ui}hy_{2i}^{2}\left[\frac{1}{m_{t_{i}}+2}\left({\psi}_{Ui}^{m_{t_{i}}+2}-{\zeta}_{Ui}^{m_{t_{i}}+2}\right)+\frac{y_{2n_{2}}}{m_{t_{i}}+1}\left({\psi}_{Ui}^{m_{t_{i}}+1}-{\zeta}_{Ui}^{m_{t_{i}}+1}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼3{\delta}_{Ui}hy_{2i}q_{t_{i}}\left[\frac{1}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{2y_{2n_{2}}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\theta}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.\frac{y_{2n_{2}}^{2}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{Ui}^{\frac{ft_{i}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼{\delta}_{Ui}hq_{t_{i}}\left[\frac{1}{f_{t_{i}}+4q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+4q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{3y_{2n_{2}}}{f_{t_{i}}+3q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+3q_{t_{i}}}{q_{t_{i}}}}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\left.∼\frac{3y_{2n_{2}}^{2}}{f_{t_{i}}+2q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+2q_{t_{i}}}{q_{t_{i}}}}\right)+\frac{y_{2n_{2}}^{3}}{f_{t_{i}}+q_{t_{i}}}\left({\psi}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}-{\zeta}_{Ui}^{\frac{f_{t_{i}}+q_{t_{i}}}{q_{t_{i}}}}\right)\right]\right\}.\end{eqnarray}$ (31) The strain energy release rate obtained by (31) is exact match of that calculated by (18). This fact is a conformation of the correctness of the analysis of the strain energy release rate developed in the present paper.

3. Numerical results

The effects of material gradients, crack location and non-linear behaviour of the material on the delamination fracture are evaluated. For this purpose, calculations of the strain energy release rate are performed by using formula (18). In order to investigate the influence of the crack location along the width of the beam cross-section on the fracture behaviour, two three-layered split cantilever beam configurations are considered (Fig. 4). A delamination crack is located between layers 2 and 3 in the beam shown in Fig. 4a. A beam with delamination crack between layers 1 and 2 is also analyzed (Fig. 4b). The width of each layer is t _l = 0.003 m. It is assumed that b = 0.009 m, h = 0.015 m, m ₁ = m ₂ = m ₃ = 0.6, f ₁ = f ₂ = f ₃ = 6, q ₁ = q ₂ = q ₃ = 10, m _{t
₁} = m _{t
₂} = m _{t
₃} = 1.6, f _{t
₁} = f _{t
₂} = f _{t
₂} =16, q _{t
₁} = q _{t
₂} = q _{t
₃} = 10 and F = 50 N. The strain energy release rate is presented in non-dimensional form by using the formula G _N = G∕(L _{D
₃} h). The material gradient along the width of layer 3 is characterized by L _{H
₃}∕L _{D
₃} ratio. It should be specified that L _{D
₃} is kept constant in the calculations. Thus, L _{H
₃} is varied in order to generate various L _{H
₃}∕L _{D
₃} ratios. The strain energy release rate in non-dimensional form is plotted against L _{H
₃}∕L _{D
₃} ratio in Fig. 5 at L _{K
₃}∕L _{D
₃} = 0.6 for the two three-layered beam configurations. It is also assumed that L _{D
₁}∕L _{D
₃} = 1.2, L _{H
₁}∕L _{D
₁} = 1.5, L _{K
₁}∕L _{D
₁} = 2, L _{D
₂}∕L _{D
₃} = 1.4, L _{H
₂}∕L _{D
₂} = 1.6, L _{K
₂}∕L _{D
₂} = 1.8. Figure 5 shows that the strain energy release rate decreases with increasing of L _{H
₃}∕L _{D
₃} ratio (this is due to the increase of the bending stiffness). The curves in Fig. 5 indicate also that the strain energy release rate decreases when the crack location changes from this shown in Fig. 4a to that in Fig. 4b. This finding is attributed to the increase of the right-hand crack arm stiffness.

Fig. 5.

The strain energy release rate in non-dimensional form plotted against L _{H
₃}∕L _{D
₃} ratio (curve 1 – for the beam configuration shown in Fig. 4a, curve 2 - for the beam configuration shown in Fig. 4b).

The influence of the material gradient along the height of layer 3 is elucidated. For this purpose, the strain energy release rate in non-dimensional form is presented as a function of L _{K
₃}∕L _{D
₃} ratio in Fig. 6 at L _{H
₃}∕L _{D
₃} = 1.2. The three-layered split cantilever beam configuration shown in Fig. 4a is analyzed. One can observe in Fig. 6 that the strain energy release rate decreases with increasing of L _{K
₃}∕L _{D
₃} ratio. The influence of non-linear behaviour of the material on the delamination fracture is elucidated too. For this purpose, the strain energy release rate obtained assuming linear-elastic behaviour of the material is also plotted in Fig. 6 for comparison with the non-linear solution. It should be noted that the linear-elastic solution for the strain energy release rate in the two-dimensional functionally graded multilayered beam is derived by substituting of m _i = 1 in formula (18). It can be observed in Fig. 6 that the non-linear behaviour of the material leads to increase of the strain energy release rate.

Fig. 6.

The strain energy release rate in non-dimensional form presented as a function of L _{K
₃}∕L _{D
₃} ratio (curve 1 – at non-linear behaviour of the material, curve 2 – at linear-elastic behaviour of the material).

The distribution of the J-integral value along the delamination crack front is analyzed too. For this purpose, calculations of the J-integral value are carried out by applying formula (26). The coordinate, z ₁, is varied in the interval [0, h∕2]. In order to elucidate the effect of material gradient along the height of the beam cross-section on the distribution of the J-integral value along the delamination crack front, two patterns of material gradient are considered. Pattern 1 is characterized by L _{D
₁}∕L _{D
₃} = 1.2, L _{H
₁}∕L _{D
₁} = 1.5, L _{K
₁}∕L _{D
₁} = 1.2, L _{D
₂}∕L _{D
₃} = 1.4, L _{H
₂}∕L _{D
₂} = 1.6, L _{K
₂}∕L _{D
₂} = 1.2, L _{H
₃}∕L _{D
₃} = 0.6 and L _{K
₃}∕L _{D
₃} = 1.2. Pattern 2 of material gradient is characterized by L _{D
₁}∕L _{D
₃} = 1.2, L _{H
₁}∕L _{D
₁} = 1.5, L _{K
₁}∕L _{D
₁} = −0.4, L _{D
₂}∕L _{D
₃} = 1.4, L _{H
₂}∕L _{D
₂} = 1.6, L _{K
₂}∕L _{D
₂} = −0.4, L _{H
₃}∕L _{D
₃} = 0.6 and L _{K
₃}∕L _{D
₃} = −0.4. The beam configuration shown in Fig. 4a is analyzed. The J-integral value is presented in non-dimensional form by using the formula J _N = J∕(L _{D
₃} h). The distribution of the J-integral value in non-dimensional form along the delamination crack front is shown in Fig. 7 at the two patterns of material gradient. It should be noted that only the lower half of the delamination crack front is shown in Fig. 7 due to the symmetric character of the distribution with respect to the delamination crack front centre. The horizontal axis is defined such that z ₁∕h = 0.0 is at the delamination crack front centre. Thus, z ₁∕h = 0.5 is at the lower surface of the beam. The curves in Fig. 7 indicate that at pattern 1 of material gradient the distribution of the J-integral value is characterized by maximum in the crack front centre and a gradual decrease towards the lower surface of the beam. This finding is attributed to the fact that at pattern 1 of material gradient the material property, L _i, gradually increases towards the lower and upper beam surfaces. Figure 7 shows that at pattern 2 of material gradient the J-integral value has minimum in the crack front centre and gradually increases towards the lower beam surface.

Fig. 7.

Distribution of the J-integral value in non-dimensional form along the delamination crack front (curve 1 – at pattern 1 of material gradient, curve 2 – at pattern 2 of material gradient).

4. Conclusions

The delamination fracture in two-dimensional functionally graded multilayered split cantilever beam configuration is studied analytically in terms of the strain energy release rate with taking into account the non-linear behaviour of the material. The beam is loaded by one longitudinal force applied at the free end of the right-hand crack arm. The solution derived is applicable for a beam which is made of an arbitrary number of adhesively bonded vertical longitudinal layers. A delamination crack is located arbitrary between layers. Each layer has different thickness and material properties. Besides, the material is two-dimensional functionally graded in the cross-section of each layer. The solution derived is verified by analyzing the delamination fracture with the help of the J-integral. The effects of material gradients on the delamination are evaluated. The analysis reveals that the strain energy release rate decreases with increasing of L _{H
₃}∕L _{D
₃} and L _{K
₃}∕L _{D
₃} ratios. The influence of crack location along the width of the beam cross-section is studied too. It is found that the strain energy release rate decreases with increasing the right-hand crack arm thickness. The effect of non-linear behaviour of the material on the delamination is also elucidated. The results obtained indicate that the non-linear behaviour of the material leads to increase of the strain energy release rate. The dependency of the distribution of the J-integral value along the delamination crack front on the material gradients is investigated. It is found that the material gradient along the beam height affects the J-integral value distribution to a great extent. The analysis developed indicates that by appropriate selection of material gradients in the two-dimensional functionally graded multilayered beams it is possible to reduce significantly the strain energy release rate. The results obtained in the present paper can be applied in practical engineering for evaluating the delamination fracture performance in the preliminary design of multilayered functionally graded beam structures which exhibit non-linear mechanical behaviour of the material.

Conflict of interest

None to report.

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