Peridynamics for out-of-plane damage analysis of composite laminates

Abstract

A new peridynamic model for predicting the out-of-plane bending and twisting behavior of composite laminates has been proposed, in which fiber bonds and matrix bonds are distinguished for characterizing anisotropy. The peridynamic formulations are obtained based on the principle of virtual displacements using the Total Lagrange formulation, and the equation of motion is reformulated by the interpolation technique. The critical curvature is adopted as the failure criterion, and a micromodulus reduction method is implemented in the PD algorithm. For multi-layer laminated structures, a new single-layer material point model (SLMPM) is proposed, in which the overall micromodulus is integrated according to all plies in laminates. The capability of the developed PD model was demonstrated by the bending examples of composite laminates with different fiber orientations, and damage analysis was further conducted to demonstrate the strong capability of the proposed PD model in replicating the failure process of composite structures. In addition, the computational efficiency of numerical models can be greatly improved due to the SLMPM.

Appendices

Appendix 1: PD material parameters

The geometric relationship between material points x_(k) and x_(j) can be obtained by Taylor expansion as:

$$\phi_{x(j)} = \phi_{x(k)} + \phi_{x,x(k)} \left( {x_{(j)} - x_{(k)} } \right) + \phi_{x,y(k)} \left( {y_{(j)} - y_{(k)} } \right)$$

(77a)

$$\phi_{y(j)} = \phi_{y(k)} + \phi_{y,x(k)} \left( {x_{(j)} - x_{(k)} } \right) + \phi_{y,y(k)} \left( {y_{(j)} - y_{(k)} } \right)$$

(77b)

where the higher-order terms of Taylor expansion are ignored, and the expressions in Eq. (77) can be further written as:

$$\frac{{\phi_{x(j)} - \phi_{x(k)} }}{{\xi_{(j)(k)} }} = \phi_{x,x(k)} \cos \gamma + \phi_{x,y(k)} \sin \gamma$$

(78a)

$$\frac{{\phi_{y(j)} - \phi_{y(k)} }}{{\xi_{(j)(k)} }} = \phi_{y,x(k)} \cos \gamma + \phi_{y,y(k)} \sin \gamma .$$

(78b)

1.1 Lamina subjected to simple twisting

For a lamina subjected to simple twisting, $\kappa_{12} = \kappa$, as shown in Fig. 25. The constitutive relation can be expressed as:

$$\left\{ {\begin{array}{*{20}c} {m_{11} } \\ {m_{22} } \\ {m_{12} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {D_{11} } & {D_{12} } & 0 \\ {D_{12} } & {D_{22} } & 0 \\ 0 & 0 & {D_{66} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} 0 \\ 0 \\ \kappa \\ \end{array} } \right\}\quad {\text{or}}\quad \left\{ {\begin{array}{*{20}c} {m_{11} } \\ {m_{22} } \\ {m_{12} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} 0 \\ 0 \\ {D_{66} \kappa } \\ \end{array} } \right\} .$$

(79)

Fig. 25

Lamina subjected simple twisting

The strain energy density based on CCM at material point x_(k) can be defined as:

$$W_{(k)} = \frac{1}{2}D_{66} \kappa^{2} .$$

(80)

The relative curvature between material points x_(j) and x_(k) in the deformed state can be expressed as:

$$\kappa_{(k)(j)} = \sin \gamma_{(j)(k)} \cos \gamma_{(j)(k)} \kappa .$$

(81)

The strain energy density can be calculated as:

$$W_{(k)} = c_{{{\text{ft}}}} \int_{H} {\delta \left( {\sin \gamma \cos \gamma \kappa } \right)^{2} \xi } {\text{d}}H$$

(82a)

or

$$W_{(k)} = c_{{{\text{ft}}}} h\int_{0}^{\delta } {\int_{0}^{2\pi } {\delta \left( {\sin \gamma \cos \gamma \kappa } \right)^{2} \xi^{2} {\text{d}}\xi {\text{d}}\phi } = \frac{{\pi h\delta^{4} \kappa^{2} c_{{{\text{ft}}}} }}{12}} .$$

(82b)

The PD strain energy density at material point x_(k) must be equal to the counterpart from CCM, Eqs. (80) and (82b), as follows

$$c_{{{\text{ft}}}} = \frac{{6D_{66} }}{{\pi h\delta^{4} }}.$$

(83)

1.2 Lamina subjected to uniaxial bending

For a lamina subjected to uniaxial bending in the fiber direction, $\kappa_{11} = \kappa ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa_{22} = 0$, as shown in Fig. 26. The constitutive relation can be expressed as:

$$\left\{ {\begin{array}{*{20}c} {m_{11} } \\ {m_{22} } \\ {m_{12} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {D_{11} \kappa } \\ {D_{12} \kappa } \\ 0 \\ \end{array} } \right\}$$

(84)

where D₁₁ and D₁₂ represent the bending stiffness coefficients of orthotropic materials which can be expressed as:

$$\left[ {\mathbf{D}} \right] = \left[ {\begin{array}{*{20}c} {D_{11} } & {D_{12} } & 0 \\ {D_{12} } & {D_{22} } & 0 \\ 0 & 0 & {D_{66} } \\ \end{array} } \right]$$

(85)

where

$$D_{ij} = \frac{1}{12}q_{ij} t^{3} \quad \left( {i,j = 1,2,6} \right).$$

(86)

Fig. 26

Lamina subjected uniaxial bending in the fiber direction

The dilatation and strain energy density based on CCM at material point x_(k) can be defined as:

$$\theta_{(k)} = \kappa \quad W_{(k)} = \frac{1}{2}D_{11} \kappa^{2} .$$

(87a, b)

The relative curvature between material points x_(j) and x_(k) in the deformed state can be expressed as:

$$\kappa_{(k)(j)} = \left( {\cos^{2} \gamma_{(j)(k)} } \right)\kappa .$$

(88)

Due to uniaxial bending deformation, the dilatation can be calculated as

$$\theta_{(k)} = d\int_{H} {\delta \left( {\cos^{2} \gamma } \right)} \kappa {\text{d}}H$$

(89a)

or

$$\theta_{(k)} = \frac{{\pi dh\delta^{3} \kappa }}{2}.$$

(89b)

The dilatation at material point x_(k) must be equal to the counterpart from CCM, Eqs. (87a) and (89b), as follows:

$$d = \frac{2}{{\pi h\delta^{3} }}.$$

(90)

The strain energy density of the lamina subjected to uniaxial bending can be calculated as:

$$W_{(k)} = a\kappa^{2} + c_{{\text{f}}} \sum_{j = 1}^{J} {\delta \left( {\cos^{2} \gamma_{(k)(j)} \kappa } \right)^{2} \xi_{(k)(j)} V_{(j)} } + c_{{{\text{ft}}}} \int_{H} {\delta \left( {\cos^{2} \gamma \kappa } \right)^{2} \xi_{(k)(j)} {\text{d}}H}$$

(91a)

or

$$W_{(k)} = a\kappa^{2} + c_{{\text{f}}} \delta \kappa^{2} \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } + \frac{{\pi h\delta^{4} \kappa^{2} }}{4}c_{{{\text{ft}}}} .$$

(91b)

Substituting Eq. (83) into Eq. (91b) and the integral value can be obtained as:

$$W_{(k)} = a\kappa^{2} + c_{{\text{f}}} \delta \kappa^{2} \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } + \frac{{3D_{66} \kappa^{2} }}{2}.$$

(92)

The strain energy density at material point x_(k) must be equal to the counterpart from CCM, Eqs. (87b) and (92), as follows:

$$a + \delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } c_{{\text{f}}} = \frac{1}{2}\left( {D_{11} - 3D_{66} } \right).$$

(93)

Similarly, for a lamina subjected to uniaxial bending in the transverse direction, $\kappa_{11} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa_{22} = \kappa$, as shown in Fig. 27. The expression can also be obtained as:

$$a + \delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} c_{t} } = \frac{1}{2}\left( {D_{22} - 3D_{66} } \right).$$

(94)

Fig. 27

Lamina subjected uniaxial bending in the transverse direction

1.3 Lamina subjected to biaxial bending

For a lamina subjected to biaxial bending, $\kappa_{11} = \kappa ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa_{22} = \kappa$, as shown in Fig. 28. The constitutive relation can be expressed as:

$$\left\{ {\begin{array}{*{20}c} {m_{11} } \\ {m_{22} } \\ {m_{12} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {D_{11} } & {D_{12} } & 0 \\ {D_{12} } & {D_{22} } & 0 \\ 0 & 0 & {D_{66} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} \kappa \\ \kappa \\ 0 \\ \end{array} } \right\}\quad {\text{or}}\quad \left\{ {\begin{array}{*{20}c} {m_{11} } \\ {m_{22} } \\ {m_{12} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\left( {D_{11} + D_{12} } \right)\kappa } \\ {\left( {D_{12} + D_{22} } \right)\kappa } \\ 0 \\ \end{array} } \right\}.$$

(95a, b)

Fig. 28

Lamina subjected biaxial bending

The strain energy density based on CCM at material point x_(k) can be defined as:

$$W_{(k)} = \frac{1}{2}\left( {D_{11} + 2D_{12} + D_{22} } \right)\kappa^{2} .$$

(96)

The relative curvature between material points x_(j) and x_(k) in the deformed state can be expressed as:

$$\kappa_{(k)(j)} = \left( {\cos^{2} \gamma_{(j)(k)} + \sin^{2} \gamma_{(j)(k)} } \right)\kappa .$$

(97)

The strain energy density of the lamina subjected to biaxial bending can be calculated as

$$W_{(k)} = 4a\kappa^{2} + c_{{\text{f}}} \kappa^{2} \delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } + c_{{{\text{ft}}}} \frac{{2\pi h\delta^{4} \kappa^{2} }}{3} + c_{{\text{t}}} \kappa^{2} \delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } .$$

(98)

Substituting Eq. (83) into Eq. (98) and the integral value can be obtained as:

$$W_{(k)} = 4a\kappa^{2} + c_{{\text{f}}} \kappa^{2} \delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } + 4D_{66} \kappa^{2} + c_{{\text{t}}} \kappa^{2} \delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } .$$

(99)

The strain energy density at material point x_(k) must be equal to the counterpart from CCM, Eqs. (96) and (99), as follows:

$$\frac{1}{2}\left( {D_{11} + 2D_{12} + D_{22} - 8D_{66} } \right) = 4a + c_{{\text{f}}} \delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } + c_{t} \delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } .$$

(100)

The PD material parameters of the lamina can be obtained by solving Eqs. (93–94) and (100), as:

$$a = \frac{1}{2}\left( {D_{12} - D_{66} } \right)\quad c_{f} = \frac{{\left( {D_{11} - D_{12} - 2D_{66} } \right)}}{{2\delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } }}\quad c_{t} = \frac{{\left( {D_{22} - D_{12} - 2D_{66} } \right)}}{{2\delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } }}.$$

(101)

In bond-based peridynamics (BBPD), the material parameter a associated with dilatation needs to vanish. Therefore, these elastic constants of lamina are limited as:

$$D_{12} = D_{66} \quad {\text{and}}\quad D_{22} = 3D_{12} .$$

(102)

The nonvanishing PD parameters, c_f and c_ft, can be presented as:

$$c_{{\text{f}}} = \frac{{\left( {D_{11} - D_{22} } \right)}}{{2\delta \sum_{j = 1}^{J} {\xi_{(k)(j)} V_{(j)} } }}\quad {\text{and}}\quad c_{{{\text{ft}}}} = \frac{{6D_{66} }}{{\pi h\delta^{4} }}$$

(103)

where c_f and c_ft represent the bending micromodulus of PD bonds in the fiber and arbitrary directions, respectively.

Appendix 2: Surface corrections

Similar to other PD models, the PD material parameters need to be corrected for those material points located in the boundary region. These correction factors can be determined by comparing the strain energy densities obtained from PD and CCM, and the corresponding moment–curvature relations can be modified.

When a lamina is subjected to constant curvature in the fiber and transverse direction, ${{\partial \theta_{\alpha } } \mathord{\left/ {\vphantom {{\partial \theta_{\alpha } } {\partial x_{\alpha } }}} \right. \kern-0pt} {\partial x_{\alpha } }} = \kappa$ with $\left( {\alpha = 1,2} \right)$, the deformation field at material point x can be described as:

$${\mathbf{u}}_{1}^{{\text{T}}} \left( {\mathbf{x}} \right) = \left\{ {\frac{{\partial \theta_{1} }}{{\partial x_{1} }}x_{1} \quad 0} \right\}\quad {\text{and}}\quad {\mathbf{u}}_{2}^{{\text{T}}} \left( {\mathbf{x}} \right) = \left\{ {0\quad \frac{{\partial \theta_{2} }}{{\partial x_{2} }}x_{2} } \right\}.$$

(104)

The PD strain energy density associated with x_(k) can be decomposed as:

$$W_{\alpha }^{{{\text{PD}}}} \left( {{\mathbf{x}}_{(k)} } \right) = W_{\alpha f}^{{{\text{PD}}}} \left( {{\mathbf{x}}_{(k)} } \right) + W_{{\alpha {\text{ft}}}}^{{{\text{PD}}}} \left( {{\mathbf{x}}_{(k)} } \right)$$

(105)

where $W_{{\alpha {\text{f}}}}^{{{\text{PD}}}}$ and $W_{{\alpha {\text{ft}}}}^{{{\text{PD}}}}$ represent strain energy densities of fiber bonds and matrix bonds, respectively. Each term in Eq. (105) can be calculated as:

$$W_{{\alpha {\text{f}}}}^{{{\text{PD}}}} \left( {{\mathbf{x}}_{(k)} } \right) = c_{f} \delta \sum_{j = 1}^{J} {\kappa_{(k)(j)}^{2} V_{(j)} }$$

(106a)

$$W_{{\alpha {\text{ft}}}}^{{{\text{PD}}}} \left( {{\mathbf{x}}_{(k)} } \right) = c_{{{\text{ft}}}} \delta \sum_{j = 1}^{\infty } {\kappa_{(k)(j)}^{2} V_{(j)} } .$$

(106b)

The corresponding strain energy density in CCM associated with material point x_(k) can also be defined as:

$$W_{\alpha }^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right) = \frac{1}{2}D_{\alpha \alpha } \kappa^{2} \quad \left( {\alpha = 1,2} \right)$$

(107)

which can be decomposed as:

$$W_{\alpha }^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right) = W_{{\alpha {\text{f}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right) + W_{{\alpha {\text{ft}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right)$$

(108)

where $W_{{\alpha {\text{f}}}}^{{{\text{CM}}}}$ and $W_{{\alpha {\text{ft}}}}^{{{\text{CM}}}}$ represent strain energy densities in the fiber direction and arbitrary directions, respectively.

For a lamina subjected to uniaxial bending in the fiber direction, each component of strain energy density can be expressed as:

$$W_{{1{\text{f}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right) = \frac{1}{2}\left( {D_{11} - D_{12} - 2D_{66} } \right)\kappa^{2}$$

(109a)

$$W_{{1{\text{ft}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right) = \frac{3}{2}D_{66} \kappa^{2} .$$

(109b)

For a lamina subjected to uniaxial bending in the transverse direction, each component of strain energy density can be expressed as:

$$W_{{2{\text{f}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right) = 0$$

(110a)

$$W_{{2{\text{ft}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right) = \frac{3}{2}D_{66} \kappa^{2} .$$

(110b)

Hence, for the uniaxial bending in the fiber direction, the correction factor components at material point x_(k) can be defined as:

$$S_{{1{\text{f}}(k)}} = \frac{{W_{{1{\text{f}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right)}}{{W_{{1{\text{f}}}}^{{{\text{PD}}}} \left( {{\mathbf{x}}_{(k)} } \right)}} = \frac{{\frac{1}{2}\left( {D_{11} - D_{12} - 2D_{66} } \right)\kappa^{2} }}{{c_{{\text{f}}} \delta \sum_{j = 1}^{J} {\kappa_{(k)(j)}^{2} V_{(j)} } }}$$

(111a)

$$S_{{1{\text{ft}}(k)}} = \frac{{W_{{1{\text{ft}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right)}}{{W_{{1{\text{ft}}}}^{{{\text{PD}}}} \left( {{\mathbf{x}}_{(k)} } \right)}} = \frac{{\frac{3}{2}D_{66} \kappa^{2} }}{{c_{{{\text{ft}}}} \delta \sum_{j = 1}^{\infty } {\kappa_{(k)(j)}^{2} V_{(j)} } }}.$$

(111b)

When a lamina is subjected to uniaxial bending in the transverse direction, the correction factor components at material point x_(k) can be defined as:

$$S_{{2{\text{f}}(k)}} = 1$$

(112a)

$$S_{{2{\text{ft}}(k)}} = \frac{{W_{{2{\text{ft}}}}^{{{\text{CM}}}} \left( {{\mathbf{x}}_{(k)} } \right)}}{{W_{{2{\text{ft}}}}^{{{\text{PD}}}} \left( {{\mathbf{x}}_{(k)} } \right)}} = \frac{{\frac{3}{2}D_{66} \kappa^{2} }}{{c_{{{\text{ft}}}} \delta \sum_{j = 1}^{P} {\kappa_{(k)(j)}^{2} V_{(j)} } }}.$$

(112b)

These correction factors in Eqs. (111–112) can be written in a vector form as:

$$g_{l(k)} \left( {{\mathbf{x}}_{(k)} } \right) = \left\{ {g_{1l} \left( {{\mathbf{x}}_{(k)} } \right),g_{2l} \left( {{\mathbf{x}}_{(k)} } \right)} \right\}^{{\text{T}}} = \left\{ {S_{1l(k)} ,S_{2l(k)} } \right\}^{{\text{T}}} \quad \left( {l = {\text{f}},{\text{ft}}} \right).$$

(113)

Since these correction factors are only based on deformation in the fiber and transverse directions. To solve the correction factors in any direction, a unit relative position vector between material points x_(k) and x_(j) needs to be introduced, which can be expressed as:

$${\mathbf{n}} = \frac{{\left( {{\mathbf{x}}_{(j)} - {\mathbf{x}}_{(k)} } \right)}}{{\left| {{\mathbf{x}}_{(j)} - {\mathbf{x}}_{(k)} } \right|}} = \left\{ {n_{1} ,n_{2} } \right\}^{{\text{T}}} .$$

(114)

Therefore, the correction factor between material points x_(k) and x_(j) can be obtained as:

$$\overline{g}_{l(k)(j)} = \left\{ {\overline{g}_{l(k)(j)1} ,\overline{g}_{l(k)(j)2} } \right\}^{{\text{T}}} = \frac{{{\mathbf{g}}_{l(k)} + {\mathbf{g}}_{l(j)} }}{2}.$$

(115)

The total correction factor between material points x_(k) and x_(j) can be determined by projecting the correction factor components on the relative position vector as:

$$G_{l(k)(j)} = \left( {\left[ {{{n_{1} } \mathord{\left/ {\vphantom {{n_{1} } {\overline{g}_{l(k)(j)1} }}} \right. \kern-0pt} {\overline{g}_{l(k)(j)1} }}} \right]^{2} + \left[ {{{n_{2} } \mathord{\left/ {\vphantom {{n_{2} } {\overline{g}_{l(k)(j)2} }}} \right. \kern-0pt} {\overline{g}_{l(k)(j)2} }}} \right]^{2} } \right)^{ - 1/2} .$$

(116)