Nonlinear Transient Response of Functionally Graded Material Sandwich Doubly Curved Shallow Shell Using New Displacement Field

Abstract

In this paper, the nonlinear transient dynamic response of functionally graded material (FGM) sandwich doubly curved shell with homogenous isotropic material core and functionally graded face sheet is analyzed using a new displacement field on the basis of Reddy’s third-order shear theory for the first time. The equivalent material properties for the FGM face sheet are assumed to obey the rule of simple power law function in the thickness direction. Based on Reddy’s theory of higher shear deformation, a new displacement field is developed by introducing the secant function into transverse displacement. Four coupled nonlinear differential equations are obtained by applying Hamilton’s principle and Galerkin method. It is assumed that the FGM sandwich doubly curved shell is subjected to step loading, air-blast loading, triangular loading, and sinusoidal loading, respectively. On the basis of double-precision variable-coefficient ordinary differential equation solver, a new program code in FORTRAN software is developed to solve the nonlinear transient dynamics of the system. The influences of core thickness, volume fraction, core-to-face sheet thickness ratio, width-to-thickness ratio and blast type on the transient response of the shell are discussed in detail through numerical simulation.

Appendix A

$$\begin{aligned} L_{11}= & {} \chi _{11} +\chi _{12} d_{11} +\chi _{13} d_{22} , \quad L_{12} =\chi _{14} d_{12} , \quad L_{13} =\chi _{15} d_{111} +\chi _{16} d_{122} +\chi _{17} d_1 \nonumber \\ L_{14}= & {} \chi _{18} +\chi _{19} d_{11} +\chi _{110} d_{22} , \quad L_{15} =\chi _{111} d_{12} , \quad L_{16} =\chi _{112} d_1 +\chi _{113} d_{111} +\chi _{114} d_{122} \nonumber \\ N_1 ({\bullet })= & {} \chi _{115} d_2 d_{12} +\chi _{116} d_1 d_{22} +\chi _{117} d_1 d_{11} \end{aligned}$$

(a)

$$\begin{aligned} L_{21}= & {} \chi _{21} d_{12} , \quad L_{22} =\chi _{22} +\chi _{23} d_{22} +\chi _{24} d_{11} , \quad L_{23} =\chi _{25} d_{112} +\chi _{26} d_{222} +\chi _{27} d_2 \nonumber \\ L_{24}= & {} \chi _{28} d_{12} , \quad L_{25} =\chi _{29} +\chi _{210} d_{11} +\chi _{211} d_{22} , \quad L_{26} =\chi _{212} d_2 +\chi _{213} d_{112} +\chi _{214} d_{222} \nonumber \\ N_2 ({\bullet })= & {} \chi _{215} d_1 d_{12} +\chi _{216} d_2 d_{11} +\chi _{217} d_2 d_{22} \end{aligned}$$

(b)

$$\begin{aligned} L_{31}= & {} \chi _{31} d_1 +\chi _{32} d_{122} +\chi _{33} d_{111} , \quad L_{32} =\chi _{34} d_2 +\chi _{35} d_{222} +\chi _{36} d_{112} \nonumber \\ L_{33}= & {} \chi _{37} +\chi _{38} d_{11} +\chi _{39} d_{1122} +\chi _{310} d_{2222} +\chi _{311} d_{1111} +\chi _{312} d_{22} \nonumber \\ L_{34}= & {} \chi _{313} d_1 +\chi _{314} d_{122} +\chi _{315} d_{111} , \quad L_{35} =\chi _{316} d_2 +\chi _{317} d_{222} +\chi _{318} d_{112}\nonumber \\ L_{36}= & {} \chi _{319} +\chi _{320} d_{1122} +\chi _{321} d_{2222} +\chi _{322} d_{1111} +\chi _{323} d_{22} +\chi _{324} d_{11} \nonumber \\ N_3 ({\bullet })= & {} \chi _{325} (d_{12} w_0 )^{2}+\chi _{326} d_{11} w_0 d_{11} w_1 +\chi _{327} d_{11} w_0 d_{12} w_0 +\chi _{328} d_{11} w_0 d_{22} w_0 +\chi _{329} d_{11} w_0 d_{22} w_1 \nonumber \\&\,+\chi _{330} (d_2 w_0 )^{2}d_{11} w_0 +\chi _{331} (d_2 w_0 )^{2}+\chi _{332} (d_2 w_0 )^{2}d_{22} w_0 +\chi _{333} d_2 v_0 d_{11} w_0\nonumber \\&+\,\chi _{334} d_1 \phi _y d_{12} w_0 +\chi _{335} w_1 d_{11} w_0 +\chi _{336} w_1 d_{22} w_0 +\chi _{337} d_2 \phi _x d_{12} w_0 +\chi _{338} d_2 v_0 d_{22} w_0\nonumber \\&+\,\chi _{339} d_2 \phi _y d_{11} w_0 +\chi _{340} d_2 \phi _y d_{22} w_0 +\chi _{341} d_1 \phi _x d_{11} w_0 +\chi _{342} d_1 \phi _x d_{22} w_0 +\chi _{343} (d_1 w_0 )^{2}d_{11} w_0\nonumber \\&+\,\chi _{344} (d_1 w_0 )^{2}+\chi _{345} (d_1 w_0 )^{2}d_{22} w_0 +\chi _{346} d_1 u_0 d_{11} w_0 +\chi _{347} d_1 u_0 d_{22} w_0 +\chi _{348} d_1 w_0 d_1 w_1\nonumber \\&+\,\chi _{349} d_1 w_0 d_2 w_0 d_{12} w_0 +\chi _{350} d_1 w_0 d_{11} u_0 +\chi _{351} d_1 w_0 d_{22} \phi _x +\chi _{352} d_1 w_0 d_{11} \phi _x\nonumber \\&+\,\chi _{353} d_1 w_0 d_{12} v_0 +\chi _{354} d_1 w_0 d_{12} \phi _y +\chi _{355} d_1 w_0 d_{22} u_0 +\chi _{356} d_1 w_0 d_{111} w_1\nonumber \\&+\,\chi _{357} d_1 w_0 d_{112} w_0 +\chi _{358} d_1 w_0 d_{122} w_0 +\chi _{359} d_1 w_0 d_{122} w_1 +\chi _{360} d_{22} w_0 d_{22} w_1 +\chi _{361} d_2 w_0 d_2 w_1\nonumber \\&+\,\chi _{362} d_2 w_0 d_{12} \phi _x +\chi _{363} d_2 w_0 d_{11} v_0 +\chi _{364} d_2 w_0 d_{22} v_0 +\chi _{365} d_2 w_0 d_{112} w_0 +\chi _{366} d_2 w_0 d_{112} w_1\nonumber \\&+\,\chi _{367} d_2 w_0 d_{222} w_1 +\chi _{368} d_{11} w_1 d_{22} w_0 +\chi _{369} d_2 u_0 d_{12} w_0 +\chi _{370} d_1 v_0 d_{12} w_0 +\chi _{371} w_0 d_{11} w_0\nonumber \\&+\,\chi _{372} d_{12} w_0 d_{12} w_1 +\chi _{373} w_0 d_{22} w_0 +\chi _{374} d_2 w_0 d_{11} \phi _y +\chi _{375} d_2 w_0 d_{12} u_0 \end{aligned}$$

(c)

$$\begin{aligned} L_{41}= & {} \chi _{41} +\chi _{42} d_{22} +\chi _{43} d_{11} , \quad L_{42} =\chi _{44} d_{12} , \quad L_{43} =\chi _{45} d_{111} +\chi _{46} d_{122} +\chi _{47} d_1 \nonumber \\ L_{44}= & {} \chi _{48} +\chi _{49} d_{11} +\chi _{410} d_{22} , \quad L_{45} =\chi _{411} d_{12} , \quad L_{46} =\chi _{412} d_1 +\chi _{413} d_{111} +\chi _{414} d_{122} \nonumber \\ N_4 ({\bullet })= & {} \chi _{415} d_1 d_{11} +\chi _{416} d_1 d_{22} +\chi _{417} d_2 d_{12} \end{aligned}$$

(d)

$$\begin{aligned} L_{51}= & {} \chi _{51} d_{12} , \quad L_{52} =\chi _{52} +\chi _{53} d_{22} +\chi _{54} d_{11} , \quad L_{53} =\chi _{55} d_{112} +\chi _{56} d_{222} +\chi _{57} d_2 \nonumber \\ L_{54}= & {} \chi _{58} d_{12} , \quad L_{55} =\chi _{59} +\chi _{510} d_{11} +\chi _{511} d_{22} , \quad L_{56} =\chi _{512} +\chi _{513} d_{222} +\chi _{514} d_{112}\nonumber \\ N_5 ({\bullet })= & {} \chi _{415} d_2 d_{11} +\chi _{416} d_2 d_{22} +\chi _{417} d_1 d_{12} \end{aligned}$$

(e)

$$\begin{aligned} L_{61}= & {} \chi _{61} d_{122} +\chi _{62} d_{111} +\chi _{63} d_1 , \quad L_{62} =\chi _{64} d_2 +\chi _{65} d_{222} +\chi _{66} d_{112} \nonumber \\ L_{63}= & {} \chi _{67} d_{2222} +\chi _{68} d_{1122} +\chi _{69} d_{11} +\chi _{610} d_{1111} +\chi _{611} +\chi _{612} d_{22} \nonumber \\ L_{64}= & {} \chi _{613} d_1 +\chi _{614} d_{122} +\chi _{615} d_{111} , \quad L_{65} =\chi _{616} d_{222} +\chi _{617} d_2 +\chi _{618} d_{112}\nonumber \\ L_{66}= & {} \chi _{619} +\chi _{620} d_{2222} +\chi _{621} d_{1122} +\chi _{622} d_{11} +\chi _{623} d_{1111} +\chi _{624} d_{22}\nonumber \\ N_6 ({\bullet })= & {} \chi _{625} (d_2 )^{2}+\chi _{626} d_2 d_{112} +\chi _{627} d_2 d_{222} +\chi _{628} d_{11} d_{22} +\chi _{629} (d_{11} )^{2}+\chi _{630} (d_{22} )^{2}\nonumber \\&+\,\chi _{631} d_{11} d_{12} +\chi _{632} (d_1 )^{2}+\chi _{633} d_1 d_{112} +\chi _{634} d_1 d_{111} +\chi _{635} d_1 d_{122} +\chi _{636} (d_{12} )^{2} \end{aligned}$$

(f)

with

$$\begin{aligned} d_i =\frac{\partial }{\partial \alpha _i }, d_{ij} =\frac{\partial ^{2}}{\partial \alpha _i \partial \alpha _j }, d_{ijl} =\frac{\partial ^{3}}{\partial \alpha _i \partial \alpha _j \partial \alpha _l }, d_{ijlm} =\frac{\partial ^{4}}{\partial \alpha _i \partial \alpha _j \partial \alpha _l \partial \alpha _m }\quad (i,j,l,m=1,2) \end{aligned}$$