Dynamic response and damage analysis of fiber-reinforced composite laminated plates under low-velocity oblique impact

Abstract

Fiber-reinforced composite laminates (FRCL) is susceptible to the external impacting. Understanding the crack propagation and structural mechanical properties of the damaged FRCL under low-velocity oblique impact is of great value in practical application. A new analytical dynamic model is developed in this work to research the dynamic response and damage property of FRCL under oblique impacting. The displacement field and strain–displacement relations of the FRCL are established by utilizing higher-order shear plate theory. The matrix damage and fiber rupture in FRCL under oblique impacting are captured by an internal variable-based continuum damage constitutive relation. To accurately predict the oblique impacting force, an analytical dynamic impacting model is proposed basing on a developed contact model, where normal and tangential contact is coupled and solved simultaneously. The whole initial boundary value problem is iteratively solved by synthetically using finite differential method and Newmark-$\beta$ method. The solving convergence and accuracy of the model is demonstrated and validated. Simulations show that the matrix damage is more easily to appear in FRCL under shear force due to oblique contact when under oblique impacting, and the damage profile is different from normal impacting. The dynamic responses of the FRCL plate under oblique impacting differ also greatly from normal impacting. The current research provides a theoretical basis for FRCL design and its engineering application when under low-velocity impacting.

Appendices

Appendix 1: Detailed forms of coefficients for the damage constitutive relations (Eq. 7 in the text)

Consider that the damage of the fiber-reinforced composite material with orthotropic property is induced by cracks in the matrix. The local coordinate system $o-123$ is adopted, in which axis1 is parallel to fibrous direction, axis2 is normal to fibrous direction and axis3 is normal to the mid-surface by adopting the fundamental assumption on the shell $\varepsilon _{13} =\varepsilon _{23} =0$ and applying Voigt notation to describe strains and damage variables, the Helmholtz free energy can be given as

$$\begin{aligned} \rho \psi= & {} C_1 ^{0}\varepsilon _1 ^{2}+C_2 ^{0}\varepsilon _1 \varepsilon _2 +C_3 ^{0}\varepsilon _2 ^{2}+C_4 ^{0}\varepsilon _6 ^{2}\nonumber \\&+\,C_5 ^{0}\varepsilon _3 ^{2}+C_6 ^{0}\varepsilon _1 \varepsilon _3 +C_7 ^{0}\varepsilon _2 \varepsilon _3 \nonumber \\&+\,\sum _{m=1}^n {[C_1 ^{m}\varepsilon _1 ^{2}\omega _1 ^{m}} +C_2 ^{m}\varepsilon _1 ^{2}\omega _2 ^{m}+C_3 ^{m}\varepsilon _1 ^{2}\omega _3 ^{m}\nonumber \\&+\,C_4 ^{m}\varepsilon _2 ^{2}\omega _1 ^{m} +C_5 ^{m}\varepsilon _2 ^{2}\omega _2 ^{m}+C_6 ^{m}\varepsilon _2 ^{2}\omega _3 ^{m} \nonumber \\&+\,C_{15} ^{m}\varepsilon _3 ^{2}\omega _1 ^{m}+C_{16} ^{m}\varepsilon _3 ^{2}\omega _2 ^{m}+C_{17} ^{m}\varepsilon _3 ^{2}\omega _3 ^{m}\nonumber \\&+\,C_7 ^{m}\varepsilon _6 ^{2}\omega _1 ^{m}+C_8 ^{m}\varepsilon _6 ^{2}\omega _2 ^{m}+C_9 ^{m}\varepsilon _6 ^{2}\omega _3 ^{m} \nonumber \\&+\,C_{10} ^{m}\varepsilon _1 \varepsilon _2 \omega _1 ^{m}+C_{11} ^{m}\varepsilon _1 \varepsilon _2 \omega _2 ^{m}+C_{12} ^{m}\varepsilon _1 \varepsilon _2 \omega _3 ^{m}\nonumber \\&+\,C_{18} ^{m}\varepsilon _1 \varepsilon _3 \omega _1 ^{m}+C_{19} ^{m}\varepsilon _1 \varepsilon _3 \omega _2 ^{m} \nonumber \\&+\,C_{20} ^{m}\varepsilon _1 \varepsilon _3 \omega _3 ^{m}+C_{21} ^{m}\varepsilon _2 \varepsilon _3 \omega _1 ^{m}+C_{22} ^{m}\varepsilon _2 \varepsilon _3 \omega _2 ^{m}\nonumber \\&+\,C_{23} ^{m}\varepsilon _2 \varepsilon _3 \omega _3 ^{m}+C_{13} ^{m}\varepsilon _1 \varepsilon _6 \omega _6 ^{m} \nonumber \\&+\,C_{14} ^{m}\varepsilon _2 \varepsilon _6 \omega _6 ^{m}+C_{24} ^{m}\varepsilon _3 \varepsilon _6 \omega _6 ^{m}]+P_0 \nonumber \\&+\,P_1 (\varepsilon _p ,\omega _q ^{m})+P_2 (\omega _q ^{m}) \end{aligned}$$

(45)

Consider that the cracks in the matrix under the bending deformation have the same direction, then $n=1$, and the stress can be given as

$$\begin{aligned} \sigma _p =\frac{\partial (\rho \psi )}{\partial \varepsilon _p }=[c_{pq}^0 +c_{pq} ]\varepsilon _p \end{aligned}$$

(46)

where $[C_{pq}^{\;0} ]$ and $[C_{pq} ]$ are symmetric matrices having the forms as follows

$$\begin{aligned}{}[C_{pq}^{\;0} ]= & {} \left[ {{\begin{array}{llll} {2C_1^0 }&{} {C_2^0 }&{} {C_6^0 }&{} 0 \\ &{} {2C_3^0 }&{} {C_7^0 }&{} 0 \\ &{} &{} {2C_5^0 }&{} 0 \\ &{} &{} &{} {2C_4^0 } \\ \end{array} }} \right] \nonumber \\ {[C_{pq}]}= & {} \left[ {{\begin{array}{llll} {2C_1 \omega _1 +2C_2 \omega _2 +2C_3 \omega _3 }&{} {C_{10} \omega _1 +C_{11} \omega _2 +C_{12} \omega _3 }&{} {C_{18} \omega _1 +C_{19} \omega _2 +C_{20} \omega _3 }&{} {C_{13} \omega _6 } \\ &{} {2C_4 \omega _1 +2C_5 \omega _2 +2C_6 \omega _3 }&{} {C_{21} \omega _1 +C_{22} \omega _2 +C_{23} \omega _3 }&{} {C_{14} \omega _6 } \\ &{} &{} {2C_{15} \omega _1 +2C_{16} \omega _2 +2C_{17} \omega _3 }&{} {C_{24} \omega _6 } \\ &{} &{} &{} {2C_7 \omega _1 +2C_8 \omega _2 +2C_9 \omega _3 } \\ \end{array} }} \right] \qquad \qquad \end{aligned}$$

(47)

in the above expressions, superscript $m=1$ is omitted.

For the fiber-reinforced composite plate with matrix cracks that is vertical to the fibrous direction, in all damage variables only $\omega _{\;1}$ is not zero, then coefficient matrix in (46) is simplified as

$$\begin{aligned}&\left[ {C_{pq}^{\;0} +C_{pq} } \right] \nonumber \\&\quad =\left[ {{\begin{array}{llll} {2C_1^0 +2C_1 \omega _1 }&{} {C_2^0 +C_{10} \omega _1 }&{} {C_6^0 +C_{18} \omega _1 }&{} 0 \\ &{} {2C_3^0 +2C_4 \omega _1 }&{} {C_7^0 +C_{21} \omega _1 }&{} 0 \\ &{} &{} {2C_5^0 +2C_{15} \omega _1 }&{} 0 \\ &{} &{} &{} {2C_4^0 +2C_7 \omega _1 } \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$

(48)

Because all cracks are parallel to the coordinate plane $2-3$, its effect on stiffness in this coordinate plane can be neglected. Then the last matrix can be further simplified as

$$\begin{aligned}&\left[ {C_{pq}^{\;0} +C_{pq} } \right] \nonumber \\&\quad =\left[ {{\begin{array}{llll} {2C_1^0 +2C_1 \omega _1 }&{} {C_2^0 +C_{10} \omega _1 }&{} {C_6^0 +C_{18} \omega _1 }&{} 0 \\ &{} {2C_3^0 }&{} {C_7^0 +C_{21} \omega _1 }&{} 0 \\ &{} &{} {2C_5^0 }&{} 0 \\ &{} &{} &{} {2C_4^0 +2C_7 \omega _1 } \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$

(49)

Letting $\sigma _3 =0$, the constitutive relation of the composite uni-ply shell with damage in the status of plane stress can be obtained as follows

$$\begin{aligned} \left\{ {{\begin{array}{l} {\sigma _1 } \\ {\sigma _2 } \\ {\sigma _6 } \\ \end{array} }} \right\}= & {} \left[ {C_{pq}^{\;0} +C_{pq} } \right] \left\{ {{\begin{array}{l} {\varepsilon _1 } \\ {\varepsilon _2 } \\ {\varepsilon _6 } \\ \end{array} }} \right\} \end{aligned}$$

(50)

where

$$\begin{aligned} \left[ {C_{pq}^{\;0} } \right]&=\left[ {{\begin{array}{lll} {2C_1^0 -\frac{(C_6 ^{0})^{2}}{2C_5 ^{0}}}&{} {C_2^0 -\frac{C_6 ^{0}C_7 ^{0}}{2C_5 ^{0}}}&{} 0 \\ &{} {2C_3^0 -\frac{(C_7 ^{0})^{2}}{2C_5 ^{0}}}&{} 0 \\ &{} &{} {2C_4^0 } \\ \end{array} }} \right] \nonumber \\ \left[ {C_{pq} } \right]&=\left[ {{\begin{array}{lll} {2C_1 -\frac{C_6 ^{0}C_{18} }{C_5 ^{0}}}&{} {C_{10} -\frac{C_6 ^{0}C_{21} +C_7 ^{0}C_{18} }{2C_5 ^{0}}}&{} 0 \\ &{} {-\frac{C_7 ^{0}C_{21} }{C_5 ^{0}}}&{} 0 \\ &{} &{} {2C_7 } \\ \end{array} }} \right] \omega _{\;1}\nonumber \\&\mathop =\limits ^{def} \left[ {{\begin{array}{lll} {d_{11} }&{} {d_{12} }&{} 0 \\ &{} {d_{22} }&{} 0 \\ &{} &{} {d_{66} } \\ \end{array} }} \right] \omega _{\;1} \end{aligned}$$

(51)

Obviously, $[C_{pq}^{\;0} ]$ is the stiffness matrix without damage, and

$$\begin{aligned} \left[ {C_{pq} ^{0}} \right] =\left[ {{\begin{array}{lll} {\frac{E_1 }{1-v_{12} v_{21} }}&{} {\frac{E_1 v_{21} }{1-v_{12} v_{21} }}&{} 0 \\ &{} {\frac{E_2 }{1-v_{12} v_{21} }}&{} 0 \\ &{} &{} {G_{12} } \\ \end{array} }} \right] \mathop =\limits ^{def} \left[ {{\begin{array}{lll} {C_{11} }&{} {C_{12} }&{} 0 \\ &{} {C_{22} }&{} 0 \\ &{} &{} {G_{12} } \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$

(52)

Appendix 2: Detailed solving of normal and tangential contact force at each iterative step

We have to consider four cases to obtain the convergent contact force when iterative solve Eqs. (28) and (35) by applying Eqs. (23)–(27) and (29)–(34) in the main text.

Case A normal contact loading, tangential contact loading, i.e., $\Delta q_N>0,\Delta q_T >0$, then Eqs. (28) and (35) in the main text are directly used to calculate the contact force as

$$\begin{aligned} \begin{array}{ll} q_N^{J+1}&{} =q_N^J +\Delta q_N \\ q_T^{J+1} &{}=q_T^J +\mu \Delta q_N+(K_T )_{12}\\ &{}\quad \times \left( \Delta \delta _T -\frac{\mu \Delta q_N }{(K_{T,0} )_{q_N =q_N^J } }\right) \ \,\Delta q_T \ge \mu q_N\\ q_T^{J+1} &{}=q_T^J +(K_{T,0} )_{q_N =q_N^J } \Delta \delta _T\quad \Delta q_T <\mu q_N\\ \end{array} \end{aligned}$$

(53)

here $(K_T )_0 =(K_{T,0} )_{q_N =q_N^J }$, $(K_T )_{12} =(K_{T,0} )_{q_N =q_N^{J+1} } \left( {1-\frac{q_T^J }{\mu q_N^{J+1} }} \right) ^{1/3}$.

Case B normal contact unloading, tangential contact loading, i.e., $\Delta q_N <0,\Delta q_T >0$, then from the backward Euler method, we have the following contact force by Eqs. (28), (31) and (35) in the main text,

$$\begin{aligned} q_N^{J+1}= & {} q_N^J +\Delta q_N (\Delta q_N <0) \nonumber \\ q_T^{J+1}= & {} q_T^J +(K_{T,0} )_{q_N =q_N^{J+1}} \left( 1-\frac{q_T^J }{\mu q_N^{J+1} }\right) \left( \Delta \delta _T\right. \nonumber \\&\left. -\frac{\mu \Delta q_N }{(K_{T,0} )_{q_N =q_N^J } }-\frac{-\mu \Delta q_N }{(K_T )_{12} }\right) \end{aligned}$$

(54)

here $(K_T )_{12} =(K_{T,0} )_{q_N =q_N^J } (1-\frac{q_T^J +\mu \Delta q_N }{\mu q_N^{J+1} })^{1/3}$.

Case C normal contact loading, tangential contact unloading, i.e., $\Delta q_N >0,\Delta q_T <0$, we have the following contact force by Eqs. (28), (33) and (35) in the main text,

$$\begin{aligned} \begin{array}{lll} q_N^{J+1} &{}=q_N^J +\Delta q_N \\ q_T^{J+1} &{}=q_T^J -\mu \Delta q_N&{}\quad \Delta q_T \le -\mu \Delta q_N \\ q_T^{J+1} &{}=q_T^J -\mu \Delta q_N +(K_T )_{12}\\ &{}\quad \times \left( \Delta \delta _T -\frac{-\mu \Delta q_N }{(K_{T,0} )_{q_N =q_N^J }}\right) &{}\quad \Delta q_T >-\mu \Delta q_N \\ \end{array}\nonumber \\ \end{aligned}$$

(55)

here, $(K_T )_{12} =(K_{T,0} )_{q_N =q_N^{J+1} } \left( {1-\frac{q_T^{*} -q_T^J }{2\mu q_N^{J+1} \;}} \right) ^{1/3}$ and $q_T^{*}$ is the maximum tangential contact force.

Case D normal contact unloading, tangential contact unloading, i.e., $\Delta q_N<0,\Delta q_T <0$, we have the following contact force by Eqs. (28), (33) and (35) in the main text,

$$\begin{aligned} q_N^{J+1}= & {} q_N^J +\Delta q_N\;(\Delta q_N \;<0) \nonumber \\ q_T^{J+1}= & {} q_T^J +(K_T )_{23} \left( \Delta \delta _T -\frac{-\mu \Delta q_N }{(K_{T,0} )_{q_N =q_N^J } }\right. \nonumber \\&\left. -\frac{\mu \Delta q_N }{(K_T )_{12} }\right) \end{aligned}$$

(56)

here, $(K_T )_{12} =(K_{T,0} )_{q_N =q_N^J } \left( {1-\frac{q_T^{*} -q_T^J }{2\mu q_N^{J+1} }} \right) ^{1/3}$, $(K_T )_{23} =(K_{T,0} )_{q_N =q_N^{J+1} } \left( {1-\frac{q_T^{*} -q_T^J }{2\mu q_N^{J+1} }} \right) ^{1/3}$.