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.
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This study is supported by the Natural Science Foundation of China (no. 11302004) and the Natural Science Foundation of China (no. 11272270).
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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
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
where \([C_{pq}^{\;0} ]\) and \([C_{pq} ]\) are symmetric matrices having the forms as follows
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
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
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
where
Obviously, \([C_{pq}^{\;0} ]\) is the stiffness matrix without damage, and
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
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,
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,
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,
here, \((K_T )_{12} =(K_{T,0} )_{q_N =q_N^J } \left( {1-\frac{q_T^{*} -q_T^J }{2\mu q_N^{J+1} }} \right) ^{1/3}\), \((K_T )_{23} =(K_{T,0} )_{q_N =q_N^{J+1} } \left( {1-\frac{q_T^{*} -q_T^J }{2\mu q_N^{J+1} }} \right) ^{1/3}\).
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Mao, Y., Hong, L., Ai, S. et al. Dynamic response and damage analysis of fiber-reinforced composite laminated plates under low-velocity oblique impact. Nonlinear Dyn 87, 1511–1530 (2017). https://doi.org/10.1007/s11071-016-3130-5
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DOI: https://doi.org/10.1007/s11071-016-3130-5