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Nonlinear dynamic response and damage analysis for functionally graded metal shallow spherical shell under low-velocity impact

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Abstract

The nonlinear dynamic response and damage evolution of functionally graded shallow spherical shell under low-velocity impact are investigated in this work. Basing on continuum damage theory, a damage constitutive relation is established for functionally graded material and the Kachanov damage evolution law is adopted to predict the damage propagation in the structure. A modified contact model suitable for non-homogenous material (functionally graded material) is applied to model the contact force in impacting process. The laminated modeling method is adopted to model the functionally graded shell with varying material constants along the thickness by dividing the shell to N plies with the constant material properties for each ply. With the established damage constitutive relations and nonlinear geometric relations of FGM shallow spherical shell with elastic modulus varying as a power-law function, the nonlinear motion equations of FGM shallow spherical under low-velocity impact have been obtained in the term of displacement functions. The problems are solved by using the orthogonal collocation point method, the Newmark method and the iterative method synthetically. Some numerical examples are carried out to validate present impacting model and the calculating methods, and parametrical analysis are presented to discuss the effects of the material properties, the geometrical size and impacting velocity on damage state and dynamic response of the structure when under low-velocity impact.

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Abbreviations

\({\varphi }, \theta , z\) :

Curvilinear coordinates along the meridional, circumferential and radial/thickness directions

\(u,v,w\) :

Displacement of plate

\(N_{\varphi }, N_\theta \) :

Result force

\(M_{\varphi }, M_\theta \) :

Resultant bending moment

\(Q_{\varphi } \) :

Out plane shear force

\(G\) :

Shear modulus

\(E_0 \) :

Elastic modulus of FGM plate on the central plane

\(E_1 \) :

Elastic modulus of impacting sphere

\(\delta \) :

Contact indentation

\(P\) :

Total contact force

\(\alpha \) :

Material index of FGM plate

\(p_\mathrm{c} \) :

Concentrated force in contact area

\(P(r_\mathrm{s} )\) :

Contact pressure in contact area

\(w_1 \) :

Deflection of the FGM substrate in contact model

\(w_2 \) :

Deflection on the isotropic impactor in contact model

\(\mathbf{D}\) :

Damage tensor

\(a\) :

Radius of impactor

\(R\) :

Radius of curvature of shallow spherical shell

\(\rho \) :

Density of the micro-damage on the \(i\)th surface

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Acknowledgments

This study is supported by the Natural Science Foundation of China (No. 11302004); the Equipment research foundation (No. 2012YQ03007502) and Natural Science Foundation of China (No. 11072076).

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Authors and Affiliations

  1. State Key Laboratory of Turbulence and Complex System, College of Engineering, Peking University, Beijing, 100871, People’s Republic of China

    Y. Q. Mao & D. N. Fang

  2. Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    S. G. Ai

  3. Department of Civil and Architecture Engineering, Xiamen University of Technology, Xiamen, 361024, Fujian, People’s Republic of China

    C. P. Chen

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  1. Y. Q. Mao
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Correspondence to Y. Q. Mao.

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Mao, Y.Q., Ai, S.G., Chen, C.P. et al. Nonlinear dynamic response and damage analysis for functionally graded metal shallow spherical shell under low-velocity impact. Arch Appl Mech 85, 1627–1647 (2015). https://doi.org/10.1007/s00419-015-1009-4

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