Nonlinear low-velocity impact analysis of functionally graded shallow spherical shells in thermal environments

Abstract

This article presents the nonlinear dynamic response of functionally graded (FG) shallow spherical shells in thermal environments subjected to low-velocity impact by an elastic ball. The material properties of a FG shallow spherical shell vary continuously through the thickness according to a power law distribution of the volume fraction of the constituents. The temperature field is considered to vary along the thickness direction due to the steady state heat transfer. Based on the higher order shear deformation theory, the governing equations of motion for the shell, which account for geometric nonlinearity is obtained using Hamilton’s principle. The contact force between the shell and the impactor is relative to local deformation and calculated using a numerical method. Then, the governing equations of motion are solved numerically by the Chebyshev collocation method and Newmark scheme. This is a complete model that can not only fully model the dynamic behavior of the shell but also fully model the impactor’s dynamic behavior. In the numerical example, the effects of material properties, temperature, initial impact velocity and mass of the impactor on the dynamic behavior of the shells, and contact force are discussed in detail.

Introduction

The concept of functionally graded (FG) materials (FGMs) was first introduced in 1984 by the material scientists in the Sendai area of Japan. ¹ FGMs refer to heterogeneous composite materials whose composition varies continuously from one surface of the materials to the other surface, which results in continuously varying material properties. The continuity of material properties can avoid the large interlaminar stresses compared with the usual composite materials ² and survives the environments with high-temperature gradients while maintaining their structural integrity. ³ Therefore, FGMs have gained potential applications in aerospace, biomedical, and civil engineering. The low-velocity impact phenomena exist extensively in these application areas, thereby making the study of FG structures to low-velocity impact greatly significant.

In contrast to the impact problem dealing with the structures made of the usual materials and composite laminates, the study of FGMs with similar problem have received little attention. Apetre et al. ⁴ studied the low-velocity impact response of sandwich beams with FG core, in which the contact problem is solved using the assumed contact stress distribution method. Gong et al. ⁵ developed an Spring-Mass (S-M) model for the calculation of contact force between an FG cylindrical shell and an elastic ball and investigated the elastic response of the FG cylindrical shell using the Navier method. The low-velocity impact problem of FG circular plates was investigated using the contact model of composite laminates by Chelu and Librescu. ⁶ Larson and Palazotto ⁷ studied FG circular plates to low-velocity impact using the classical rule-of-mixtures approach and designed experiments to characterize the impact response of FG plates in Larson’s doctoral dissertation. ⁸ Apalak and coworkers ^{9 –11} studied the impact performance of aluminum (Al)/silicon carbide FG circular plates by experiments and the explicit finite element code LS-DYNA. Based on the contact law proposed by Giannakopoulos and Suresh, ^12,13 Mao et al. ¹⁴ presented a contact force formula for FGMs by assuming that the pressure distribution in contact area follows the Hertzian law and investigated the nonlinear dynamic response of FG shallow spherical shells to low-velocity impact in thermal environments. Shariyat and Jafari ¹⁵ studied the nonlinear low-velocity impact response of FG circular plates employing an amendatory Hertz-type contact law. Khalili et al. ¹⁶ investigated the impact response of FG plates with temperature-dependent properties based on classical plate theory and S-M model. Yalamanchili and Sankar ¹⁷ investigated two kinds of FG beams based on Galerkin method by assuming contact stress distribution method and found that the soft- to hard-graded materials can increase impact resistance. Since FGMs are mainly designed for temperature environments, the thermal problem has received much attention. Apalak and Gunes ¹⁸ investigated the thermal elastic residual stresses occurring in FG plates using three-dimensional layered finite element. Sankar and Tzeng ¹⁹ obtained a closed-form solution for the thermal stress analysis of FG beams. As the shear deformation effects are pronounced for the moderately thick structures subjected to transverse load, the classical shell theory is not adequate to model the behavior of the shells. Hence, many suitable higher order shear deformation theories (HSDTs) have been proposed to obtain the realistic variation of the transverse shear strains and stresses through the thickness of the shells. These theories can be classified into two major classes on the basis of assumed fields ²⁰ : (1) stress-based theory and (2) displacement-based theory. Donnell ²¹ and Reissner ²² developed the HSDT for the plates and shells based on series expansions. Reddy ²³ obtained the parabolic shear stress distribution through the thickness of the plates and shells. However, only limited literatures relating to the application of the higher shear deformation theory to study the dynamic response of FG shells subjected to the transverse impact loads are available.

In this article, the nonlinear low-velocity impact analysis of FG shallow spherical shells in thermal environments is conducted using a third-order shear deformation theory (TSDT). The temperature varies along the thickness of the shell and is obtained by solving the steady state heat transfer equation. The contact force is expressed by a developed contact law and solved by a numerical method. The governing equations of motion are solved numerically by the Chebyshev collocation method and Newmark scheme. The dynamic behavior of the shells during the impact procedure is analyzed in the numerical examples.

Basic equations

An FG shallow spherical shell clamped with in-plane immovable on its edge to low-velocity impact is considered, as shown in Figure 1, in which R, a, and h are the curvature radius of the middle curved surface, radius of the base circle, and the thickness of the shell, respectively. Arbitrary point in the shell can be determined by the orthogonal curvilinear coordinates $\left(\mathit{\varphi },\mathit{\theta }\right)$ on the middle plane and z coordinate along the inner normal direction of the middle plane. The complementary angle of latitude ϕ is measured from the axisymmetric axis, and the longitude θ is measured from any radius of a parallel circle. $z=-h/\phantom{h2}2$ and $z=h/\phantom{h2}2$ denote the outer and inner surfaces of the shell, respectively. In the given orthogonal curvilinear coordinates, the principal radii of curvature are $R_1=R_2=R$ , the Lamé coefficients of the middle curved surface are $A=R$ and $B=R\hspace{0.17em}\mathrm{s}\mathrm{i}\mathrm{n}\hspace{0.17em}\mathit{\phi }$ and the principal curvatures are ${\mathit{\kappa }}_1={\mathit{\kappa }}_2=-1/\phantom{1R}R$ .

Figure 1.

Geometrical configuration of FG shallow spherical shell to low-velocity impact. FG: functionally graded.

Properties of FGMs

The material properties of an FG shallow spherical shell are assumed to vary as a function of position. But the Poisson’s ratio ν depends weakly on position change and is assumed a constant. ^24,25 The elastic modulus E(z), thermal conductivity k(z), mass density ρ(z), and coefficient of thermal expansion α(z) vary from the outer surface to the inner surface as ²⁶

$P\left(z\right)=\left(P_{\mathrm{m}}-P_{\mathrm{c}}\right)V_{\mathrm{m}}+P_{\mathrm{c}},$

1

where P _m and P _c denote the material properties of metal and ceramic, respectively. V _m denotes the volume fraction of metal and can be expressed as a simple power function:

$V_{\mathrm{m}}={\left(\frac{h+2z}{2h}\right)}^n,$

2

where n is the volume fraction index. According to the distribution, the outer surface $\left(z=-h/\phantom{h2}2\right)$ of the shallow spherical shell is pure ceramic, and the inner surface $\left(z=h/\phantom{h2}2\right)$ is pure metal.

Geometric relations

Based on the TSDT, the displacement components $\left(u,v,w\right)$ of arbitrary point in an FG shallow spherical shell along the coordinates $\left(\mathit{\phi },\mathit{\theta },z\right)$ at any time t are expressed as ²⁷

$\begin{array}{rl}u\left(\mathit{\phi },\mathit{\theta },z,t\right)& =u_0+z{\mathit{\varphi }}_u+z^2{\mathit{\psi }}_u+z^3{\mathit{\eta }}_u\\ v\left(\mathit{\phi },\mathit{\theta },z,t\right)& =v_0+z{\mathit{\varphi }}_v+z^2{\mathit{\psi }}_v+z^3{\mathit{\eta }}_v\\ w\left(\mathit{\phi },\mathit{\theta },z,t\right)& =w_0,\end{array}$

3

where $\left(u_0,v_0,w_0\right)$ are the displacement components of the corresponding point of the middle plane along the $\left(\mathit{\phi }\mathrm{,}\mathit{\theta }\mathrm{,}z\right)$ directions, respectively, and $\left({\mathit{\varphi }}_u,{\mathit{\varphi }}_v\right)$ are the rotation angles of the normal to middle plane along the $\left(\mathit{\phi },\mathit{\theta }\right)$ direction, respectively. The functions $\left({\mathit{\psi }}_u,{\mathit{\eta }}_u,{\mathit{\psi }}_v,{\mathit{\eta }}_v\right)$ are determined by the conditions that the transverse shear stresses vanish on the outer and inner surfaces.

In the case of the axisymmetric deformation of a shallow spherical shell, the circumferential displacement $v=0$ , and all the variables are independent of θ. According to the von Karman nonlinear theory, the strain components $\left({\mathit{\epsilon }}_{\mathit{\phi }},{\mathit{\epsilon }}_{\mathit{\theta }},{\mathit{\epsilon }}_{\mathit{\phi }z}\right)$ of arbitrary point in the shallow spherical shell can be expressed as

${\mathit{\epsilon }}_{\mathit{\phi }}=\frac{\mathrm{\partial }u}{R\mathrm{\partial }\mathit{\phi }}-\frac{w}{R}+\frac{1}{2}{\left(\frac{\mathrm{\partial }w}{R\mathrm{\partial }\mathit{\phi }}\right)}^2,\hspace{0.17em}{\mathit{\epsilon }}_{\mathit{\theta }}=\frac{u}{R}cot\hspace{0.17em}\mathit{\phi }-\frac{w}{R},\hspace{0.17em}{\mathit{\epsilon }}_{\mathit{\phi }z}=\frac{\mathrm{\partial }u}{\mathrm{\partial }z}+\frac{\mathrm{\partial }w}{R\mathrm{\partial }\mathit{\phi }}.$

4

The functions $\left({\mathit{\psi }}_u,{\mathit{\eta }}_u\right)$ will be determined using the conditions that the transverse shear stresses vanish on the outer and inner surfaces. And these conditions are equivalent to the requirement that the corresponding strains are zero on these surfaces, that is, ${\mathit{\epsilon }}_{\mathit{\phi }z}{|}_{z=-h/2}={\mathit{\epsilon }}_{\mathit{\phi }z}{|}_{z=h/2}=0$ . Employing equations (3) and (4), equation (5) is obtained as:

${\mathit{\psi }}_u=0,\hspace{0.17em}{\mathit{\eta }}_u=-\frac{4}{3h^2}\left({\mathit{\varphi }}_u+\frac{\mathrm{\partial }w}{R\mathrm{\partial }\mathit{\phi }}\right).$

5

Introducing the coefficients $\left(f_1,f_2\right)$ in the following equation gives

$f_1(z)=-\frac{4}{3h^2}{z}^3,\hspace{0.17em}{f}_2(z)=z-\frac{4}{3h^2}{z}^3.$

6

In the analysis of the shallow spherical shell, introducing a new coordinate r along the direction of the radius of the parallel circle, using the following approximate relations $r\approx R\mathit{\phi },\hspace{0.17em}\mathrm{s}\mathrm{i}\mathrm{n}\hspace{0.17em}\mathit{\phi }\approx \mathit{\phi },\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}\mathrm{c}\mathrm{o}\mathrm{s}\hspace{0.17em}\mathit{\phi }\approx 1$ , and substituting equations (3) and (5) into equation (4) yields equation (7) as:

$\begin{array}{rl}{\mathit{\epsilon }}_r& =u_{0,r}+f_1{w}_{0,rr}+f_2{\mathit{\varphi }}_{u,r}-{\displaystyle \frac{w_0}{R}+\frac{(w_{0,r}{)}^2}{2}}\\ {\mathit{\epsilon }}_{\mathit{\theta }}& ={\displaystyle \frac{u_0+f_1{w}_{0,r}+f_2{\mathit{\varphi }}_u}{r}}-\frac{w_0}{R}\\ {\mathit{\epsilon }}_{rz}& =f_{1,z}{w}_{0,r}+f_{2,z}{\mathit{\varphi }}_u+w_{0,r}\end{array}$

7

where the subscript “,” denotes the partial derivative.

Constitutive relations

For an FG shallow spherical shell, the material properties vary through the thickness direction and the temperature effect, and the elastic constitutive relations can be written as

$\left\{\begin{array}{c}{\mathit{\sigma }}_{\mathrm{r}}\\ {\mathit{\sigma }}_{\mathit{\theta }}\\ {\mathit{\sigma }}_{\mathrm{r}\mathrm{z}}\end{array}\right\}=\left[\begin{array}{ccc}{C}_{11}(z)& C_{12}(z)& 0\\ C_{21}(z)& C_{22}(z)& 0\\ 0& 0& C_{44}(z)\end{array}\right]\left(\left\{\begin{array}{c}{\mathit{\epsilon }}_{\mathrm{r}}\\ {\mathit{\epsilon }}_{\mathit{\theta }}\\ {\mathit{\epsilon }}_{\mathrm{r}\mathrm{z}}\end{array}\right\}-\left\{\begin{array}{c}{\mathit{\alpha }}_1(z)\\ {\mathit{\alpha }}_2(z)\\ 0\end{array}\right\}T\right),$

8

in which ${\mathit{\sigma }}_{\mathrm{r}}$ , ${\mathit{\sigma }}_{\mathit{\theta }}$ , and ${\mathit{\sigma }}_{\mathrm{r}\mathrm{z}}$ are the stress components of any point in the shell, α ₁ and α ₂ are coefficients of thermal expansion along the r and θ directions, T is the temperature increment from the stress-free temperature, and $C_{ij}(i,j=1,2,4)$ are the elastic stiffness coefficients and expressed as

$C_{11}=C_{22}=\frac{E\left(z\right)}{1-{\mathit{\nu }}^2},\hspace{0.17em}{C}_{12}=C_{21}=\mathit{\nu }{C}_{11},\hspace{0.17em}{C}_{44}=\frac{E\left(z\right)}{2\left(1+\mathit{\nu }\right)}.$

9

Nonlinear governing equations of motion

The nonlinear governing equations of motion for an FG shallow spherical shell are derived from Hamilton’s principle that requires ²⁸

$\mathit{\delta }{\int }_0^t\left(T_{\mathrm{e}}-V_{\mathrm{e}}-U_{\mathrm{e}}\right)dt=0,$

10

where δ is the variable operator, T _e is the kinetic energy of the shallow spherical shell, V _e is the potential of the external load, and U _e is the total elastic strain energy. The Lamé coefficients of any point in the shallow spherical shell are $H_1=A_1\left(1-z/\phantom{1-z{R}_1}{R}_1\right),\hspace{0.17em}{H}_2=A_2\left(1-z/\phantom{1-z{R}_2}{R}_2\right)$ . And there exists the following appropriate relations $1-z/\phantom{1-z{R}_1}{R}_1\approx 1$ , for the shallow spherical shell. In the absence of the rotary inertias and in-plane inertias, the nonlinear governing equations of motion for the shallow spherical shell can be written as

$\begin{array}{rl} & (r{N}_r{)}_{,r}-N_{\mathit{\theta }}=0\\ & (r{R}_r{)}_{,r}-R_{\mathit{\theta }}-r{K}_{rz}=0\\ & (r{w}_{,r}{N}_r{)}_{,r}+r(N_r+N_{\mathit{\theta }})/R-(r{P}_r{)}_{,rr}+P_{\mathit{\theta },r}+(r{Q}_{rz}{)}_{,r}+rq=I_{0}r{w}_{0_{,tt},}\end{array}$

11

where $N_r,N_{\mathit{\theta }},Q_{rz},K_{rz},R_r,R_{\mathit{\theta }},P_r$ and $P_{\mathit{\theta }}$ are the forces and moments, which are calculated from the following expressions:

$\begin{array}{rl}(N_r,R_r,P_r)& ={\int }_{-h/2}^{h/2}{\mathit{\sigma }}_r(1,f_1,f_2)dz\\ (N_{\mathit{\theta }},R_{\mathit{\theta }},P_{\mathit{\theta }})& ={\int }_{-h/2}^{h/2}{\mathit{\sigma }}_{\mathit{\theta }}(1,f_1,f_2)dz\\ Q_{rz}& ={\int }_{-h/2}^{h/2}(f_{1,z}+1){\mathit{\sigma }}_{rz}dz\\ K_{rz}& ={\int }_{-h/2}^{h/2}{f}_{2,z}{\mathit{\sigma }}_{rz}dz,\end{array}$

12

and I ₀ is defined as

$I_0={\int }_{-h/2}^{h/2}\mathit{\rho }(z)dz.$

The forces and moments $N_r^T,N_{\mathit{\theta }}^T,R_r^T,R_{\mathit{\theta }}^T,P_r^T,P_{\mathit{\theta }}^T$ induced by the temperature field can be calculated as

$\begin{array}{rl} & (N_r^T,R_r^T,P_r^T)={\int }_{-h/2}^{h/2}(C_{11}{\mathit{\alpha }}_1+C_{12}{\mathit{\alpha }}_2)T(1,f_1,f_2)dz\\ & (N_{\mathit{\theta }}^T,R_{\mathit{\theta }}^T,P_{\mathit{\theta }}^T)={\int }_{-h/2}^{h/2}(C_{21}{\mathit{\alpha }}_1+C_{22}{\mathit{\alpha }}_2)T(1,f_1,f_2)dz.\end{array}$

13

Introducing the coefficients $A_{ij},A_{ijk},A_{ijkl},G_{11},G_{12}$ and $G_{22}$ , which are defined as

$\begin{array}{rl}(A_{ij},A_{ijk},A_{ijkl})& ={\int }_{-h/2}^{h/2}(C_{ij},C_{ij}{f}_k,C_{ij}{f}_k{f}_l)dz\hspace{0.17em}(i,j,k,l=1,2)\\ G_{11}& ={\int }_{-h/2}^{h/2}{C}_{44}(\hspace{0.17em}{f}_{1,z}+1{)}^{2}dz\\ G_{12}& ={\int }_{-h/2}^{h/2}{C}_{44}(\hspace{0.17em}{f}_{1,z}+1)f_{2,z}dz\\ G_{22}& ={\int }_{-h/2}^{h/2}{C}_{44}\hspace{0.17em}{f}_{2,z}^{2}dz.\end{array}$

14

Introducing the following dimensionless parameters:

$\begin{array}{rl}\bar{r}& =\frac{r}{a},{\bar{u}}_0=\frac{a{u}_0}{h^2},{\bar{\mathit{\varphi }}}_u=\frac{a}{h}{\mathit{\varphi }}_u,{\bar{w}}_0=\frac{w_0}{h},{\mathit{\lambda }}_1=\frac{a}{h},{\mathit{\lambda }}_2=\frac{a}{R},\hspace{0.17em}{\bar{A}}_{ij}=\frac{A_{ij}}{E_{m}h},{\bar{A}}_{ijk}=\frac{A_{ijk}}{E_m{h}^2}\mathrm{,}\\ {\bar{A}}_{ijkl}& =\frac{A_{ijkl}}{E_m{h}^3},{\bar{G}}_{11}=\frac{G_{11}}{E_{m}h},{\bar{G}}_{12}=\frac{G_{12}}{E_{m}h},{\bar{G}}_{22}=\frac{G_{22}}{E_{m}h}\mathrm{,}\hspace{0.17em}{\bar{N}}_r^T=\frac{N_r^T}{E_{m}h},{\bar{N}}_{\mathit{\theta }}^T=\frac{N_{\mathit{\theta }}^T}{E_{m}h},{\bar{R}}_r^T=\frac{R_r^T}{E_m{h}^2},\\ {\bar{R}}_{\mathit{\theta }}^T& =\frac{R_{\mathit{\theta }}^T}{E_m{h}^2},{\bar{P}}_r^T=\frac{P_r^T}{E_{m}h},{\bar{P}}_{\mathit{\theta }}^T=\frac{P_{\mathit{\theta }}^T}{E_{m}h},\bar{q}=\frac{q{a}^4}{E_m{h}^4},{\bar{I}}_0\hspace{0.17em}\mathrm{=}\hspace{0.17em}\frac{I_0}{{\mathit{\rho }}_{m}h},\mathit{\tau }=t\sqrt{\frac{E_m{h}^2}{{\mathit{\rho }}_m{a}^4}}.\end{array}$

15

Substituting equation (12) into equation (11), and using equations (8), (7), and (15), the dimensionless nonlinear governing equations of motion for the shallow spherical shell can be expressed as

$\begin{array}{rl} & ({\bar{A}}_{11}-{\bar{A}}_{21})[{\bar{u}}_{0,\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_0+2^{-1}({\bar{w}}_{0,\bar{r}}{)}^2]\hspace{0.17em}\mathrm{+}\hspace{0.17em}({\bar{A}}_{11\mathrm{1}}-{\bar{A}}_{\mathrm{211}}){\bar{w}}_{0,\bar{r}\bar{r}}+({\bar{A}}_{11\mathrm{2}}-{\bar{A}}_{\mathrm{212}}){\bar{\mathit{\varphi }}}_{u,\bar{r}}\\ & \hspace{0.17em}+({\bar{A}}_{12}-{\bar{A}}_{22})({\bar{r}}^{\hspace{0.17em}-1}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_0)\hspace{0.17em}\mathrm{+}\hspace{0.17em}({\bar{A}}_{1\mathrm{21}}-{\bar{A}}_{\mathrm{221}}){\bar{r}}^{\hspace{0.17em}-1}{\bar{w}}_{0,r}+({\bar{A}}_{1\mathrm{22}}-{\bar{A}}_{\mathrm{222}}){\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_u\\ & \hspace{0.17em}-{\mathit{\lambda }}_1^2({\bar{N}}_r^T-{\bar{N}}_{\mathit{\theta }}^T)+\bar{r}\{{\bar{A}}_{11}({\bar{u}}_{0,\bar{r}\hspace{0.17em}\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,r}+{\bar{w}}_{0,\hspace{0.17em}\bar{r}}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}})\hspace{0.17em}\mathrm{+}\hspace{0.17em}{\bar{A}}_{11\mathrm{1}}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}+{\bar{A}}_{11\mathrm{2}}{\bar{\mathit{\varphi }}}_{u,\bar{r}\hspace{0.17em}\bar{r}}\\ & \hspace{0.17em}+{\bar{A}}_{12}({\bar{r}}^{\hspace{0.17em}-1}{\bar{u}}_{0,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,r})\hspace{0.17em}\mathrm{+}\hspace{0.17em}{\bar{A}}_{1\mathrm{21}}({\bar{r}}^{\hspace{0.17em}-1}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{w}}_{0,\bar{r}})\\ & \hspace{0.17em}+{\bar{A}}_{1\mathrm{22}}({\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_{u,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{\mathit{\varphi }}}_u)\}=0,\end{array}$

16

$\begin{array}{rl} & ({\bar{A}}_{11\mathrm{2}}-{\bar{A}}_{\mathrm{212}})[{\bar{u}}_{0,\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_0+2^{-1}({\bar{w}}_{0,\bar{r}}{)}^2]\hspace{0.17em}\mathrm{+}\hspace{0.17em}({\bar{A}}_{11\mathrm{12}}-{\bar{A}}_{\mathrm{2112}}){\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}+({\bar{A}}_{11\mathrm{22}}-{\bar{A}}_{\mathrm{2122}}){\bar{\mathit{\varphi }}}_{u,\bar{r}}\\ & \hspace{0.17em}+({\bar{A}}_{1\mathrm{22}}-{\bar{A}}_{\mathrm{222}})({\bar{r}}^{\hspace{0.17em}-1}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_0)\mathrm{+}({\bar{A}}_{1\mathrm{2}1\mathrm{2}}-{\bar{A}}_{\mathrm{22}1\mathrm{2}}){\bar{r}}^{\hspace{0.17em}-1}{\bar{w}}_{0,\bar{r}}+({\bar{A}}_{1\mathrm{222}}-{\bar{A}}_{\mathrm{2222}}){\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_u\\ & \hspace{0.17em}-{\mathit{\lambda }}_1^2({\bar{R}}_r^T-{\bar{R}}_{\mathit{\theta }}^T)+\bar{r}\{{\bar{A}}_{11\mathrm{2}}({\bar{u}}_{0,\bar{r}\hspace{0.17em}\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,r}+{\bar{w}}_{0,\bar{r}}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}})+{\bar{A}}_{111\mathrm{2}}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}+{\bar{A}}_{112\mathrm{2}}{\bar{\mathit{\varphi }}}_{u,\bar{r}\hspace{0.17em}\bar{r}}\\ & \hspace{0.17em}+{\bar{A}}_{122}({\bar{r}}^{-1}{\bar{u}}_{0,\bar{r}}-{\bar{r}}^{-2}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,r})+{\bar{A}}_{1212}({\bar{r}}^{-1}{\bar{w}}_{0,\bar{r}\bar{r}}-{\bar{r}}^{-2}{\bar{w}}_{0,\bar{r}})\\ & \hspace{0.17em}+{\bar{A}}_{1222}({\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_{u,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{\mathit{\varphi }}}_u)\}-\bar{r}({\mathit{\lambda }}_1^2{\bar{G}}_{12}{\bar{w}}_{0,\bar{r}}+{\mathit{\lambda }}_1^2{\bar{G}}_{22}{\bar{\mathit{\varphi }}}_u)=0,\end{array}$

17

$\begin{array}{rl} & \bar{r}{\bar{w}}_{0,\bar{r}}\{{\bar{A}}_{11}({\bar{u}}_{0,\overline{\hspace{0.17em}}r\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}}+{\bar{w}}_{0,\bar{r}}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}})+{\bar{A}}_{111}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\overline{\hspace{0.17em}}r\bar{r}}+{\bar{A}}_{112}{\bar{\mathit{\varphi }}}_{u,\bar{r}\hspace{0.17em}\bar{r}}\\ & \hspace{0.17em}+{\bar{A}}_{12}({\bar{r}}^{\hspace{0.17em}-1}{\bar{u}}_{0,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}})+{\bar{A}}_{121}({\bar{r}}^{\hspace{0.17em}-1}{\bar{w}}_{0,\bar{r}\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{w}}_{0,\bar{r}})\\ \hspace{0.17em}+{\bar{A}}_{122}({\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_{u,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{\mathit{\varphi }}}_u)\}\\ & \hspace{0.17em}+({\bar{w}}_{0,\bar{r}}+\bar{r}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}})\{{\bar{A}}_{11}[{\bar{u}}_{0,\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_0+2^{-1}({\bar{w}}_{0,\bar{r}}{)}^2]+{\bar{A}}_{111}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}+{\bar{A}}_{112}{\bar{\mathit{\varphi }}}_{u,\bar{r}}\\ & \hspace{0.17em}+{\bar{A}}_{12}({\bar{r}}^{\hspace{0.17em}-1}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_0)+{\bar{A}}_{121}{\bar{r}}^{\hspace{0.17em}-1}{\bar{w}}_{0,\bar{r}}+{\bar{A}}_{122}{\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_u-{\mathit{\lambda }}_1^2{\bar{N}}_r^T\}\\ & \hspace{0.17em}+\bar{r}\{({\bar{A}}_{11}+{\bar{A}}_{21})[{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{u}}_{0,\bar{r}}-{\mathit{\lambda }}_1^2{\mathit{\lambda }}_2^2{\bar{w}}_0+2^{-1}{\mathit{\lambda }}_1{\mathit{\lambda }}_2({\bar{w}}_{0,\bar{r}}{)}^2]+({\bar{A}}_{111}+{\bar{A}}_{211}){\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}\\ & \hspace{0.17em}+({\bar{A}}_{112}+{\bar{A}}_{212}){\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{\mathit{\varphi }}}_{u,\bar{r}}+({\bar{A}}_{12}+{\bar{A}}_{22})({\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{r}}^{\hspace{0.17em}-1}{\bar{u}}_0-{\mathit{\lambda }}_1^2{\mathit{\lambda }}_2^2{\bar{w}}_0)\\ \hspace{0.17em}+({\bar{A}}_{121}+{\bar{B}}_{221}){\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{r}}^{\hspace{0.17em}-1}{\bar{w}}_{0,\bar{r}}\\ & \hspace{0.17em}+({\bar{A}}_{122}+{\bar{A}}_{222}){\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_u-{\mathit{\lambda }}_1^3{\mathit{\lambda }}_2({\bar{N}}_r^T+{\bar{N}}_{\mathit{\theta }}^T)\}\\ & \hspace{0.17em}-2\{{\bar{A}}_{111}({\bar{u}}_{0,\bar{r}\hspace{0.17em}\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}}+{\bar{w}}_{0,\bar{r}}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}})+{\bar{A}}_{1111}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}+{\bar{A}}_{1112}{\bar{\mathit{\varphi }}}_{u,\bar{r}\hspace{0.17em}\bar{r}}\\ & \hspace{0.17em}+{\bar{A}}_{121}({\bar{r}}^{\hspace{0.17em}-1}{\bar{u}}_{0,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}})+{\bar{A}}_{1211}({\bar{r}}^{\hspace{0.17em}-1}{\bar{w}}_{0,\bar{r}\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{w}}_{0,\bar{r}})\\ \hspace{0.17em}+{\bar{A}}_{1212}({\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_{u,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{\mathit{\varphi }}}_u)\}\\ & \hspace{0.17em}-\bar{r}\{{\bar{A}}_{111}[{\bar{u}}_{0,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}+{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}^{\hspace{0.17em}2}+{\bar{w}}_{0,\bar{r}}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}]+{\bar{A}}_{1111}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}+{\bar{A}}_{1112}{\bar{\mathit{\varphi }}}_{u,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}\\ & \hspace{0.17em}+{\bar{A}}_{121}({\bar{r}}^{-1}{\bar{u}}_{0,\bar{r}\hspace{0.17em}\bar{r}}-2{\bar{r}}^{\hspace{0.17em}-2}{\bar{u}}_{0,\bar{r}}+2{\bar{r}}^{\hspace{0.17em}-3}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}})\\ \hspace{0.17em}+{\bar{A}}_{1211}({\bar{r}}^{-1}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}-2{\bar{r}}^{-2}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}+2{\bar{r}}^{-3}{\bar{w}}_{0,\bar{r}})\\ & \hspace{0.17em}+{\bar{A}}_{1212}({\bar{r}}^{-1}{\bar{\mathit{\varphi }}}_{u,\bar{r}\hspace{0.17em}\bar{r}}-2{\bar{r}}^{\hspace{0.17em}-2}{\bar{\mathit{\varphi }}}_{u,\bar{r}}+2{\bar{r}}^{\hspace{0.17em}-3}{\bar{\mathit{\varphi }}}_u)\}+\{{\bar{A}}_{121}({\bar{u}}_{0,\bar{r}\hspace{0.17em}\bar{r}}-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}}+{\bar{w}}_{0,\bar{r}}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}})\\ & \hspace{0.17em}+{\bar{A}}_{2211}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}\hspace{0.17em}\bar{r}}+{\bar{A}}_{1212}{\bar{\mathit{\varphi }}}_{u,\bar{r}\hspace{0.17em}\bar{r}}+{\bar{A}}_{222}({\bar{r}}^{\hspace{0.17em}-1}{\bar{u}}_{0,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{u}}_0-{\mathit{\lambda }}_1{\mathit{\lambda }}_2{\bar{w}}_{0,\bar{r}})\\ \hspace{0.17em}+{\bar{A}}_{2211}({\bar{r}}^{\hspace{0.17em}-1}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}-{\bar{r}}^{-2}{\bar{w}}_{0,\bar{r}})\\ & \hspace{0.17em}+{\bar{A}}_{2212}({\bar{r}}^{\hspace{0.17em}-1}{\bar{\mathit{\varphi }}}_{u,\bar{r}}-{\bar{r}}^{\hspace{0.17em}-2}{\bar{\mathit{\varphi }}}_u)\}+{\mathit{\lambda }}_1^2({\bar{G}}_{11}{\bar{w}}_{0,\bar{r}}+{\bar{G}}_{12}{\bar{\mathit{\varphi }}}_u)+{\mathit{\lambda }}_1^2\bar{r}({\bar{G}}_{11}{\bar{w}}_{0,\bar{r}\hspace{0.17em}\bar{r}}+{\bar{G}}_{12}{\bar{\mathit{\varphi }}}_{u,\bar{r}})\\ \hspace{0.17em}+\bar{r}\bar{q}={\bar{I}}_0\bar{r}{\bar{w}}_{0,\mathit{\tau }\mathit{\tau }}.\end{array}$

18

Consider the shallow spherical shell clamped with in-plane immovable at $\bar{r}=1$ , the boundary conditions are

$\begin{array}{rl} & \bar{r}=0:{\bar{u}}_0=0,\hspace{0.17em}{\bar{w}}_{0,\bar{r}}=0\mathrm{,}\hspace{0.17em}{\bar{\mathit{\varphi }}}_u=0\mathrm{,}\hspace{0.17em}r{w}_{0,r}{N}_r-(r{P}_r{)}_{,r}+P_{\mathit{\theta }}+r{Q}_{rz}=0\\ & \bar{r}=1:{\bar{u}}_0=0\mathrm{,}\hspace{0.17em}{\bar{w}}_0=0\mathrm{,}\hspace{0.17em}{\bar{\mathit{\varphi }}}_u=0\mathrm{,}\hspace{0.17em}{\bar{w}}_{0,\bar{r}}=0.\end{array}$

19

The dimensionless initial conditions are

$\mathit{\tau }=0:\hspace{0.17em}{\bar{w}}_0=0\mathrm{,}\hspace{0.17em}{\bar{w}}_{0,\mathit{\tau }}=0.$

20

Temperature field

This article considers the influence of one-dimensional steady state temperature field on the behavior of FG shallow spherical shells. The temperature of a shallow spherical shell is assumed to be different on the outer surface $\left(z=-h/\phantom{h2}2\right)$ and inner surface $\left(z=h/\phantom{h2}2\right)$ but constant in-plane. The temperature distribution along the thickness direction can be determined by the following one-dimensional steady state heat transfer equation as ³

$\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(k(z)\frac{\mathrm{\partial }T(z)}{\mathrm{\partial }z}\right)=0,$

21

in which $k(z)$ is the thermal conductivity. The temperature of the shell on the outer and inner surfaces are $T_0$ and $T_{\mathrm{i}}$ , respectively. And the stress-free temperature is assumed to be zero. Then, the temperature field can be obtained by solving equation (21) as follows

$T(z)=T_o+(T_i-T_o){\left({\int }_{-h/2}^{h/2}k(z{)}^{-1}dz\right)}^{-1}{\int }_{-h/2}^{z}k(z{)}^{-1}dz.$

22

Contact force

Consider an FG shallow spherical shell impacted by an elastic ball on the top $\left(\bar{r}=0\right)$ along the normal direction, the local deformation at the contact region is shown in Figure 2. $s(t)$ and $w_0(0,t)$ are the displacements of the ball and transverse deflection in the center point of the shell, respectively, and the depth of indentation is $\mathit{\delta }(t)=s(t)-w_0(0,t)$ . The contact force can be expressed by the Hertzian contact law as ²⁹

Figure 2.

Geometrical configuration of local deformation.

$F(t)=k_{\mathrm{c}}{\left[s(t)-w_0(0,t)\right]}^{\mathrm{p}},$

23

where F(t) is the contact force, and $k_{\mathrm{c}}$ and p are the contact coefficients. In the available literatures related to the study of FG structures to low-velocity impact, ^5,16 p is given to be 1.5, and k_c is defined by

$k_{\mathrm{c}}=\frac{4\sqrt{R_{\mathrm{s}}}}{3\mathit{\pi }(k_0+k_{\mathrm{s}})},$

24

where the parameters $R_{\mathrm{s}},k_0,\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}{k}_s$ are

$R_{\mathrm{s}}=\sqrt{\frac{R_{0}R}{\left(R_0+R\right)}},\hspace{0.17em}{k}_0=\frac{1-{\mathit{\nu }}_0^2}{\mathit{\pi }{E}_0},\hspace{0.17em}{k}_{\mathrm{s}}=\frac{1-{\mathit{\nu }}^2}{\mathit{\pi }{E}_{\mathrm{s}}},$

25

where $R_0,{\mathit{\nu }}_0$ and $E_0$ are the radius, Poisson’s ratio, and elastic modulus of the impactor, respectively, and $E_{\mathrm{s}}$ is the equivalent elastic modulus and is expressed approximately as ¹⁶

$E_s=\frac{1}{h}{\int }_{-h/2}^{h/2}E(z)dz.$

26

Suppose the initial displacement between the shallow spherical shell and impact ball is zero, then, the displacement of the impact ball with initial velocity $V_0$ can be written as

$s(t)=V_{0}t-\frac{1}{m_0}{\int }_0^t{\int }_0^{t}F(t)dtdt,$

27

where $m_0$ is the mass of the impact ball. Substituting equation into equation , eliminating the displacement $s(t)$ gives equation (28) as

$F(t)=k_{\mathrm{c}}{\left[V_{0}t-\frac{1}{m_0}{\int }_0^t{\int }_0^{t}F(t)dtdt-w_0(0,t)\right]}^{\mathrm{p}}.$

28

Because the analytic solution of equation (28) is hard to obtain, a small time increment method is adopted to seek a numerical solution here, and it is similar with the method reported by Sankar and Sun ³⁰ where the contact force variation is linear in time during each small time increment. The contact force is regarded as constant during the time increment $\mathrm{\Delta }t$ , and for the Jth time incremental interval $t=J\mathrm{\Delta }t$ , equation can be written as follows

$F(J\mathrm{\Delta }t)=k_c{\left[V_{0}J\mathrm{\Delta }t-\frac{1}{m_0}(\mathrm{\Delta }t{)}^2\sum _{i=1}^J{D}_{J-i+1}{F}_i-w_0(0,J\mathrm{\Delta }t)\right]}^{\mathrm{p}},$

29

where the quantity $m_0^{-1}(\mathrm{\Delta }t{)}^2\sum _{i=1}^J{D}_{J-i+1}{F}_i$ comes from the second term on the right hand side of equation , and the expression $\sum _{i=1}^J{D}_{J-i+1}{F}_i$ may be expressed approximately as

$\sum _{i=1}^J{D}_{J-i+1}{F}_i=2\sum _{i=1}^J(J-1)\sum _{j=1}^i(-1{)}^{i-j}{F}_j+\frac{1}{3}\sum _{i=1}^J(-1{)}^{J-i}{F}_i.$

30

At the beginning of the contact, the response of the shallow spherical shell is mainly due to the appearance of local deformation on the contact area, so that the whole displacement of the shallow spherical shell is neglected. According to equation , we can obtain the initial contact force of the iterative process as

$F_1=k_{\mathrm{c}}(V_0\mathrm{\Delta }t{)}^{\mathrm{p}}.$

31

Solution method

As the nonlinear governing equations of motion for FG shallow spherical shell to low-velocity impact are complicated, approximate theories are employed to numerically solve the problem. The unknown dimensionless displacement functions ${\bar{u}}_0$ , ${\bar{w}}_0$ , and ${\bar{\mathit{\varphi }}}_{\mathrm{u}}$ are separated for space and time, respectively. A set of finite degree polynomials in $\bar{r}$ are used to discretize the displacement functions in space domain, that is,

${\bar{u}}_0(\bar{r})=\sum _{m=1}^{M+2}{\bar{r}}^{m-1}{a}_m\mathrm{,}\hspace{0.17em}{\bar{w}}_0(\bar{r})=\sum _{n=1}^{M+4}{\bar{r}}^{n-1}{b}_n\mathrm{,}\hspace{0.17em}{\bar{\mathit{\varphi }}}_u(\bar{r})=\sum _{l=1}^{M+2}{\bar{r}}^{l-1}{c}_l,$

32

where M is the number of the discrete points ${\bar{r}}_i$ , and the discrete points ${\bar{r}}_i$ in the internal $[0,1]$ are taken at the zeros of a Chebyshev polynomial, which given by

${\bar{r}}_i=\frac{1}{2}\left\{1+\mathrm{c}\mathrm{o}\mathrm{s}\left[\frac{\left(2i-1\right)\mathit{\pi }}{2M}\right]\hspace{0.17em}\right\}\hspace{0.17em}\left(i=1,2,...,M\right).$

33

The inertia terms are discretized using the Newmark scheme, which can be expressed as

$\begin{array}{rl}({\bar{w}}_{0,\mathit{\tau }\mathit{\tau }}{)}_J& =\frac{4}{(\mathrm{\Delta }\mathit{\tau }{)}^2}[({\bar{w}}_0{)}_J-({\bar{w}}_0{)}_{J-1}]-\frac{4}{\mathrm{\Delta }\mathit{\tau }}({\bar{w}}_{0,\mathit{\tau }}{)}_{J-1}-({\bar{w}}_{0,\mathit{\tau }\mathit{\tau }}{)}_{J-1}\\ ({\bar{w}}_{0,\mathit{\tau }}{)}_J& =({\bar{w}}_{0,\mathit{\tau }}{)}_{J-1}+\frac{1}{2}\left[({\bar{w}}_{0,\mathit{\tau }\mathit{\tau }}{)}_{J-1}+({\bar{w}}_{0,\mathit{\tau }\mathit{\tau }}{)}_J\right](\mathrm{\Delta }\mathit{\tau }).\end{array}$

34

The whole equations are solved iteratively. In each iterative step J, the nonlinear items in the nonlinear governing equations of motion (16), (17), (18) and boundary conditions (19) are linearized:

$(x\cdot y{)}_J=(x{)}_J\cdot (y{)}_{J_P},$

35

where $(y{)}_{J_P}$ is the average value of those obtained in the previous two iterations. For the initial iterative step, it can be determined using the quadratic extrapolation, that is,

$(y{)}_{J_P}=A(y{)}_{J-1}+B(y{)}_{J-2}+C(y{)}_{J-3}.$

36

For the different iterative steps, the coefficients A, B, and C are

$\begin{array}{rl} & J=1:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}A=1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}B=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}C=0\\ & J=2:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}A=2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}B=-1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}C=0\\ & J=3:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}A=3,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}B=-3,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}C=1.\end{array}$

37

Substituting equations (32), (34), and (35) into the nonlinear governing equations of motion (16), (17), (18), and boundary conditions (19), and using the initial conditions (20), one can get $3M+8$ linear algebraic equations. At the Jth iterative step, the iteration lasts until the difference of the present value and the former is lesser than 0.01%. The convergent values of $a_{\mathrm{m}}\mathrm{,}\hspace{0.17em}{b}_{\mathrm{n}}\mathrm{,}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}{c}_{\mathrm{l}}$ are then obtained, so the displacement field can be determined, and the current iteration step stops and the J + 1th iterative step starts.

Numerical example

In order to verify the present analysis, the results obtained in this article are compared with the results given by Abrate ³¹ and Shariyat and Jafari. ¹⁵ When an isotropic circular plate with clamped boundary conditions impacted at its center by an elastic ball is considered, the geometrical configuration and material properties are the same as reported by Shariyat and Jafari and Abrate. ^15,31 The central deflection of circular plate and contact force history obtained by Abrate, ³¹ Shariyat and Jafari, ¹⁵ and the present analysis, respectively, are shown in Figures 3 and 4. The three results are in agreement with each other on the whole and just have small difference, Shariyat and Jafari ¹⁵ indicated the difference can be acceptable, which is evident that the present analysis yields acceptable results. The reasons for the difference are that the solution methods of contact force and HSDT are applied in this article.

Figure 3.

Comparison of the central deflection history for a circular plate to low-velocity impact.

Figure 4.

Comparison of contact force history for a circular plate to low-velocity impact.

In the subsequent section, the numerical examples are performed for FG shallow spherical shells whose thickness, radius of the base circle, and curvature radius of middle curved surface are 6, 62, and 260 mm, respectively, impacted by an elastic ball. The material properties of metal and ceramic in FG shallow spherical shells are listed in Table 1. ²⁶ The outer surface of FG shallow spherical shell is assumed to be pure ceramic (zirconia) and the inner surface is assumed to be metal rich (Al). The variation of volume fraction $V_{\mathrm{m}}$ through the thickness of FG shallow spherical shell with different volume fraction indices n is shown in Figure 5. The elastic modulus, Poisson ratio, and radius of impactor are

$E_0=200\hspace{0.17em}\hspace{0.17em}\mathrm{G}\mathrm{P}\mathrm{a}\mathrm{,}\hspace{0.17em}{\mathit{\nu }}_0=0.33,R_0=20\hspace{0.17em}\hspace{0.17em}\mathrm{m}\mathrm{m},\hspace{0.17em}{V}_0=0.5\hspace{0.17em}\hspace{0.17em}\mathrm{m}/\mathrm{s},\hspace{0.17em}{m}_0=0.267\hspace{0.17em}\hspace{0.17em}\mathrm{k}\mathrm{g}.$

Table 1.

Material properties of aluminum and zirconia in FGMs. ²⁶

Materials	E (GPa)	ν	ρ (kg m−3)	α (K−1)	k (W m−1-K−1)
Aluminum	70	0.3	2.7 × 103	23 × 10−6	204
Zirconia	151	0.3	3 × 103	10 × 10−6	2.09

FGM: functionally graded material.

Figure 5.

Variations of volume fraction through the thickness of FG shallow spherical shell with different volume fraction indices. FG: functionally graded.

Unless otherwise specified, the above material and geometric parameters for FG shallow spherical shell and impactor are used in the following numerical examples.

In order to study the effect of volume fraction index n on the dynamic response of FG shallow spherical shells, the shallow spherical shells with different volume fraction indices $n=0,\mathrm{0}\mathrm{.5}\mathrm{,}\mathrm{2}$ subjected to low-velocity impact are considered. Figure 6 shows the central deflection history of FG shallow spherical shells with different volume fraction indices. First, the central deflection increases, reaches the peak value, and then decreases with time. Also, one can see the maximum central deflection decreases as the volume fraction index increases. The reason is that the increasing volume fraction index leads to more rigidity of FG shallow spherical shell and then hardens the shell to deform. Contact force history between FG shallow spherical shell and the impact ball is shown in Figure 7. The maximum contact force increases while contact duration decreases with an increase in volume fraction index. Figure 8 shows the axial stress ${\mathit{\sigma }}_{\mathrm{r}}$ history at the center of outer surface of FG shallow spherical shell with different volume fraction indices. During the contact duration, it can be seen that the axial stress is compressive stress, and the magnitude of axial stress increases from zero to a maximum value and then decreases with time. Figure 9 shows the distribution of axial stress at $\bar{r}=0$ along the shell thickness, 0.1 ms after impact. The axial stresses are compressive near the outer surface and tensile near the inner surface, and the compressive region is larger than the tensile region. Also, the magnitude of axial stresses near the inner and outer surfaces increases as the volume fraction index increases. The variation of axial stress through the shell thickness is linear for pure Al shell $(n=0)$ , while the axial stress upon the middle plane changes much larger for the shell with $n=0.5$ and changes much larger under the middle plane for the shell with $n=2$ . This is because the volume fraction varies steeply with thickness upon the middle plane for the shell with $n=0.5$ and changes steeply with thickness under the middle plane for the shell with $n=2$ , as shown in Figure 5. The transverse shear stress ${\mathit{\sigma }}_{\mathrm{r}\mathrm{z}}$ history at the center of middle plane for FG shallow spherical shell with different volume fraction indices $n=0,\mathrm{0}\mathrm{.5}\mathrm{,}\mathrm{2}$ is shown in Figure 10. It can be observed that the magnitude of transverse shear stress increases as the volume fraction index increases during loading, while decreases as the volume fraction index increases during unloading. Figure 11 shows the distribution of transverse shear stress at $\bar{r}=0$ through the shell thickness, 0.1 ms after impact. From the figure, one can see that the transient transverse shear stresses are zero at $z=\pm \hspace{0.17em}h\hspace{0.17em}/\phantom{h2}2$ , which satisfied the requirement that the transverse shear stresses vanish on the outer and inner surfaces. The maximum transverse shear stress occurs on the middle plane for the pure Al shallow spherical shell $(n=0)$ , while it occurs upon the middle plane for FG shallow spherical shell $(n=0.5\hspace{0.17em}\mathrm{o}\mathrm{r}\hspace{0.17em}2)$ . Similar phenomenon can be found in the study by Gong et al. ⁵ The reason is that the material properties distribute inhomogeneity along the thickness for FG shallow spherical shells $(n=0.5\hspace{0.17em}\mathrm{o}\mathrm{r}\hspace{0.17em}2)$ .

Figure 6.

Central deflection history of FG shallow spherical shell with different volume fraction indices $n=0,\mathrm{0}\mathrm{.5}\mathrm{,}\mathrm{2}$ subjected to low-velocity impact. FG: functionally graded.

Figure 7.

Contact force history of FG shallow spherical shell with different volume fraction indices $n=0,\mathrm{0}\mathrm{.5}\mathrm{,}\mathrm{2}$ subjected to low-velocity impact. FG: functionally graded.

Figure 8.

Axial stress history at the center of outer surface for FG shallow spherical shell with different volume fraction indices $n=0,\mathrm{0}\mathrm{.5}\mathrm{,}\mathrm{2}$ subjected to low-velocity impact. FG: functionally graded.

Figure 9.

Distribution of axial stress at $\bar{r}=0$ through the shell thickness of FG shallow spherical shell with different volume fraction indices $n=0,\mathrm{0}\mathrm{.5}\mathrm{,}\mathrm{2}$ subjected to low-velocity impact, 0.1 ms after impact. FG: functionally graded.

Figure 10.

Transverse shear stress history at the center of middle plane for FG shallow spherical shell with different volume fraction indices $n=0,\mathrm{0}\mathrm{.5}\mathrm{,}\mathrm{2}$ subjected to low-velocity impact. FG: functionally graded.

Figure 11.

Distribution of transverse shear stress at $\bar{r}=0$ through the shell thickness of FG shallow spherical shell with different volume fraction indices $n=0,\mathrm{0}\mathrm{.5}\mathrm{,}\mathrm{2}$ subjected to low-velocity impact, 0.1 ms after impact. FG: functionally graded.

The effect of temperature field on the behavior of FG shallow spherical shell with volume fraction index $n=\mathrm{0}\mathrm{.5}$ subjected to low-velocity impact is presented. The temperature on the inner surface of the shell is $T_{\mathrm{i}}=0$ , and the various temperatures on the outer surface are taken as T ₀=100, 200, and 300 K. Figure 12 shows the central deflection history of FG shallow spherical shell subjected to low-velocity impact under different temperatures. It can be observed that the central deflection is negative and the magnitude of central deflection increases with the temperature increases. The negative deflection is due to that the temperature on the outer surface of FG shallow spherical shell is larger than the inner surface, and the magnitude of deflection caused by the temperature effect is larger than the positive deflection due to the low-velocity impact load. With the increasing temperature, the higher thermal expansion due to the higher thermal resultants seen in equation will make the temperature effect more prominent influences on the central deflection of FG shallow spherical shell. Also, the contact force increases as the temperature increases, as shown in Figure 13. The reason is that the magnitude of negative central deflection increases with the temperature increases, and then the depth of indentation increases accordingly, so the contact force increases.

Figure 12.

Central deflection history of FG shallow spherical shell to low-velocity impact with different temperatures on the outer surface of the shell T ₀ = 100, 200, and 300 K. FG: functionally graded.

Figure 13.

Contact force history of FG shallow spherical shell to low-velocity impact with different temperatures on the outer surface of the shell T ₀ = 100, 200, and 300 K. FG: functionally graded.

In order to investigate the effect of initial impact velocity on the dynamic behavior of FG shallow spherical shell with volume fraction index $n=\mathrm{0}\mathrm{.5}$ , the various initial impact velocities are taken as V ₀ = 0.5, 1, and 2 m s⁻¹. Figure 14 shows the central deflection history of FG shallow spherical shell impacted by the impactors with the increasing initial velocity. As expected, the higher the initial impact velocity, the larger the central deflection. This can be explained that the higher initial impact velocity carries a higher striking energy, which is transmitted to the shell and affected the dynamic response of the shell remarkably. Figure 15 shows that the initial impact velocity affects contact force markedly. As the initial velocity increases, the contact force increases accordingly, which is similar to the change of central deflection.

Figure 14.

Effect of initial impact velocity on the central deflection history of FG shallow spherical shell to low-velocity impact. FG: functionally graded.

Figure 15.

Effect of initial impact velocity on contact force history of FG shallow spherical shell to low-velocity impact. FG: functionally graded.

The effect of impactor’s mass on the dynamic behavior of FG shallow spherical shell with the volume fraction index $n=\mathrm{0}\mathrm{.5}$ is discussed. The various masses of impactor are taken as $m=m_0\mathrm{,}\mathrm{2}{m}_0,\mathrm{3}{m}_0$ . Figures 16 and 17 show the central deflection and contact force history of FG shallow spherical shell impacted by the impactor with different masses. It can be observed that the central deflection, contact force, and contact duration increase with the impactor’s mass increases. And it is important to notice that the increases in impactor mass have almost no influence on the central deflection, and the contact force in the time domain is approximately 0–0.08 ms. From Figures 14 to 17, it can be seen that both the initial impact velocity and the mass of impactor have great influence on the central deflection and contact force of the shell because the striking energy is relative to the initial impact velocity and mass of impactor, but only the mass of impactor affects contact duration markedly.

Figure 16.

Effect of impactor’s mass on the central deflection history of FG shallow spherical shell to low-velocity impact. FG: functionally graded.

Figure 17.

Effect of impactor’s mass on contact history of FG shallow spherical shell to low-velocity impact. FG: functionally graded.

Conclusion

This article presents a nonlinear higher order dynamic modeling for an FG shallow spherical shell in a thermal environment subjected to low-velocity impact. The temperature field varies along the thickness of the shell and is obtained from the steady state heat transfer equation. A small time increment method is adopted to seek a numerical solution of contact force. Then, the governing equations of motion are solved numerically by the Chebyshev collocation method and Newmark scheme. The main conclusions can be drawn as follows.

The whole impact duration is influenced significantly by the volume fraction index n. The larger volume fraction index contributes to more rigidity of the shell and hardens the shell to deform when the shell is subjected to transverse impact. The magnitude of axial stresses near the inner and outer surfaces increases as the volume fraction index increases, 0.1 ms after impact. The magnitude of transverse shear stress at the center of middle plane increases as the volume fraction index increases during loading while decreases as the volume fraction index increases during unloading. The maximum transverse shear stress through the thickness occurs upon the middle plane for FG shallow spherical shell $\left(n=0.5,\mathrm{2}\right)$ , while it is on the middle plane for the pure metal shallow spherical shell $(n=0)$ . The increasing temperature on the outer surface of FG shallow spherical shell to low-velocity impact can increase the magnitude of central deflection and contact force. As the initial impact velocity and mass of impactor increase, the central deflection and contact force increase because of the increase in the striking energy. But contact duration is affected by the mass of impactor markedly while not much by the initial impact velocity. Therefore, this article will provide theoretical basis to study the low-velocity impact, thermal, and shell problems. Also, the analysis and results could make recommendations for engineers and scientists to improve performance and suitability of those structures such as aerospace platforms, fuselage, and nuclear reactor system, which are used as thermal protection systems and often subjected to impact by foreign objects.