Sandwich Structures Subjected to UNDEX Loading

Abstract

Sandwich structures have high energy absorption and blast mitigation capabilities due to their configuration. In the present chapter, a computational model using time domain spectral element method or spectral element method (SEM) developed by the authors for a sandwich beam with composite facesheets and a compliant core is explained. This computational model is developed based on extended high order sandwich panel theory (EHSAPT) in which higher order displacement functions are taken for the core in order to incorporate core compression. A brief explanation about the underwater explosion (UNDEX) phenomena and the loading is presented. The sandwich model and analytical model developed by Hayman (1995) are used to study the fluid–structure interaction (FSI) and response of sandwich structures to the UNDEX loading. Significant differences in the FSI responses of homogeneous and sandwich structures is shown. These studies will result in the design of an optimal configuration for blast resistant sandwich structures.

Appendix

The matrices of Eqs. (7) and (8) are given as,

$$ [Z^{t,b}]_{2 \times 4}=\begin{bmatrix} 1&0&z&0\\ 0&1&0&z \end{bmatrix} $$

$$ [Z^{c}]_{3 \times 12}=\left[ \begin{array}{cccccccccccc} 1 &{} 0 &{} 0 &{} z &{} 0 &{} 0 &{} z^2 &{} 0 &{} 0 &{} z^3 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} z &{} 0 &{} 0 &{} z^2 &{} 0 &{} 0 &{} z^3 &{} 0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} z &{} 0 &{} 0 &{} z^2 &{} 0 &{} 0 &{} z^3 \end{array} \right] $$

$$\begin{aligned}&[\partial ^{t}]_{4 \times 3} = \begin{bmatrix} \dfrac{\partial }{\partial x}&0&\left( c+\dfrac{f_t}{2}\right) \dfrac{\partial }{\partial x}\\ 0&\dfrac{\partial }{\partial x}&-1\\ 0&0&-\dfrac{\partial }{\partial x}\\ 0&0&0 \end{bmatrix} {{{,}}}&[\partial ^{b}]_{4 \times 3} = \begin{bmatrix} \dfrac{\partial }{\partial x}&0&-\left( c+\dfrac{f_b}{2}\right) \dfrac{\partial }{\partial x}\\ 0&\dfrac{\partial }{\partial x}&-1\\ 0&0&-\dfrac{\partial }{\partial x}\\ 0&0&0 \end{bmatrix} \end{aligned}$$

$$ [\partial ^{c}] _{12 \times 9} = \begin{bmatrix} 0&0&0&\frac{\partial }{\partial x}&0&0&0&0&0\\ 0&\frac{1}{2c}&0&0&0&0&0&-\frac{1}{2c}&0\\ 0&0&0&0&\frac{\partial }{\partial x}&1&0&0&0\\ 0&0&0&0&0&\frac{\partial }{\partial x}&0&0&0\\ 0&\frac{1}{c^2}&0&0&\frac{-2}{c^2}&0&0&\frac{1}{c^2}&0\\ \frac{1}{c^2}&\frac{1}{2c}\frac{\partial }{\partial x}&\frac{ft}{2c^2}&-\frac{2}{c^2}&0&0&\frac{1}{c^2}&-\frac{1}{2c}\frac{\partial }{\partial x}&-\frac{f_b}{2c^2}\\ \frac{1}{2c^2}\frac{\partial }{\partial x}&0&\frac{f_t}{4c^2}\frac{\partial }{\partial x}&-\frac{1}{c^2}\frac{\partial }{\partial x}&0&0&\frac{1}{2c^2}\frac{\partial }{\partial x}&0&-\frac{f_b}{4c^2}\frac{\partial }{\partial x}\\ 0&0&0&0&0&0&0&0&0 \\ \frac{3}{2c^3}&\frac{1}{2c^2}\frac{\partial }{\partial x}&\frac{3f_t}{4c^3}&0&-\frac{1}{c^2}\frac{\partial }{\partial x}&-\frac{3}{c^2}&-\frac{3}{2c^3}&\frac{1}{2c^2}\frac{\partial }{\partial x}&\frac{3f_b}{4c^3}\\ \frac{1}{2c^3}\frac{\partial }{\partial x}&0&\frac{f_t}{4c^3}\frac{\partial }{\partial x}&0&0&\frac{-1}{c^2}\frac{\partial }{\partial x}&\frac{-1}{2c^3}\frac{\partial }{\partial x}&0&\frac{f_b}{4c^3}\frac{\partial }{\partial x}\\ 0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0 \\ \end{bmatrix} $$

and the strain displacement matrix, $[B^{x}]$ in Eq. (11) is given as

$$ [B^{t,c,b}] = [\partial ^{t,c,b}] \; [N^{t,c,b}] $$

where

$$ [N^t]_{3\times 72} = \begin{bmatrix} N_{1} [I]_{3\times 3} ,\; \left[ 0\right] _{3\times 6}\! ,\; N_{2} [I]_{3\times 3} ,\; \left[ 0\right] _{3\times 6} ,\; \ldots ,\; N_{8} [I]_{3\times 3} ,\; \left[ 0\right] _{3\times 6} \end{bmatrix} $$

$$ [N^c]_{9\times 72} = \begin{bmatrix} N_{1} [I]_{9\times 9} ,\; N_{2} [I]_{9\times 9} ,\; \ldots ,\; N_{8} [I]_{9\times 9} \end{bmatrix} $$

$$ [N^b]_{3\times 72} = \begin{bmatrix} \left[ 0 \right] _{3 \times 6} ,\; N_{1}[I]_{3\times 3} ,\; \left[ 0\right] _{3 \times 6} ,\; N_{2} [I]_{3\times 3} ,\; \ldots ,\; \left[ 0 \right] _{3 \times 6} \!,\; N_{8}[I]_{3\times 3} \end{bmatrix} $$

where, [0] is the matrix with zeros of size subscripted by the given dimension, [I] is the identity matrix of size subscripted by the given dimension, $N_{1}, N_{2}, \ldots , N_{8}$ are the Lagrange interpolation functions of 7th degree in $\zeta$ and can be found in standard text on FEM/SEM [26].

The stiffness matrix $C^x$ is given as

$$\begin{aligned}&[C^{t,b}]_{4 \times 4} = \begin{bmatrix} [A^{t,b}]\;[B^{t,b}]\\ [B^{t,b}]\;[D^{t,b}] \end{bmatrix}&[C^c]_{12 \times 12} = \begin{bmatrix} [A^c]&[B^c]&[D^c]&[E^c] \\&[D^c]&[E^c]&[F^c] \\&[F^c]&[G^c] \\ {{\text {sym.}}}&&[H^c] \end{bmatrix} \end{aligned}$$

and

$$\begin{aligned}&[A_{ij}^x]=\sum \limits _{k=1}^{N_x}[\bar{Q}_{ij}^x]_k\,(h_{k-1}-h_{k}),&[B_{ij}^x]=\frac{1}{2}\sum \limits _{k=1}^{N_x}[\bar{Q}_{ij}^x]_k\,(h_{k-1}^2-h_{k}^2),\\&[D_{ij}^x]=\frac{1}{3}\sum \limits _{k=1}^{N_x}[\bar{Q}_{ij}^x]_k\,(h_{k-1}^3-h_{k}^3),&[E_{ij}^x]=\frac{1}{4}\sum \limits _{k=1}^{N_x}[\bar{Q}_{ij}^x]_k\,(h_{k-1}^4-h_{k}^4),\\&[F_{ij}^x]=\frac{1}{5}\sum \limits _{k=1}^{N_x}[\bar{Q}_{ij}^x]_k\,(h_{k-1}^5-h_{k}^5),&[G_{ij}^x]=\frac{1}{6}\sum \limits _{k=1}^{N_x}[\bar{Q}_{ij}^x]_k\,(h_{k-1}^6-h_{k}^6),\\&[H_{ij}^x]=\frac{1}{7}\sum \limits _{k=1}^{N_x}[\bar{Q}_{ij}^x]_k\,(h_{k-1}^7-h_{k}^7) \end{aligned}$$

where superscript “x” represents “t”, “b”, or “c” indicating top face sheet, bottom face sheet and core respectively. $N_{x}$ indicates the number of layers in the considered face sheet or the core and $[\bar{Q}_{ij}]$, is the transformed stiffness matrix of the layer “k” and is given in [27].

The inertia matrix, $[R^{x}]$ in Eq. (11) is given as

$$ [R^{t,b,c}] = [\partial Z^{t,c,b}]^T [CZ^{t,c,b}] [\partial Z^{t,c,b}] $$

where

$$\begin{aligned}&[\partial Z^t]_{4 \times 3} = \begin{bmatrix} 1&0&\left( c+\dfrac{f_t}{2}\right) \\ 0&1&0 \\ 0&0&-1 \\ 0&0&0 \end{bmatrix} {{{,}}}&[\partial Z^b]_{4 \times 3} = \begin{bmatrix} 1&0&-\left( c+\dfrac{f_b}{2}\right) \\ 0&1&0 \\ 0&0&-1 \\ 0&0&0 \end{bmatrix} {{{,}}} \end{aligned}$$

$$ [\partial Z^c]_{8 \times 9}=\begin{bmatrix} 0&0&0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ 0&\frac{1}{2c}&0&0&0&0&0&-\frac{1}{2c}&0\\ \frac{1}{2c^2}&0&\frac{f_t}{4c^2}&-\frac{1}{c^2}&0&0&\frac{1}{2c^2}&0&-\frac{f_b}{4c^2}\\ 0&\frac{1}{2c^2}&0&0&-\frac{1}{c^2}&0&0&\frac{1}{2c^2}&0 \\ \frac{1}{2c^3}&0&\frac{f_t}{4c^3}&0&0&-\frac{1}{c^2}&-\frac{1}{2c^3}&0&\frac{f_b}{4c^3}\\ 0&0&0&0&0&0&0&0&0 \end{bmatrix} $$

and

$$\begin{aligned}&[CZ^{t,b}]_{4 \times 4} = \begin{bmatrix} [AZ^{t,b}] \; [BZ^{t,b}] \\ [BZ^{t,b}] \; [DZ^{t,b}] \end{bmatrix} {{{,}}}&[CZ^c]_{8 \times 8} = \begin{bmatrix} [AZ^c]&[BZ^c]&[DZ^c]&[EZ^c] \\&[DZ^c]&[EZ^c]&[FZ^c] \\&[FZ^c]&[GZ^c] \\ {{\text {sym.}}}&&[HZ^c] \end{bmatrix} \end{aligned}$$

where,

$$\begin{aligned}&[AZ^x]=\sum \limits _{k=1}^{N_x} \rho ^x_k\,(h_{k-1}-h_{k}) [I_{2\times 2}],&[BZ^x]=\frac{1}{2}\sum \limits _{k=1}^{N_x} \rho ^x_k\,(h_{k-1}^2-h_{k}^2) [I_{2\times 2}],\\&[DZ^x]=\frac{1}{3}\sum \limits _{k=1}^{N_x} \rho ^x_k\,(h_{k-1}^3-h_{k}^3) [I_{2\times 2}]&[EZ^x]=\frac{1}{4}\sum \limits _{k=1}^{N_x} \rho ^x_k\,(h_{k-1}^4-h_{k}^4) [I_{2\times 2}],\\&[FZ^x]=\frac{1}{5}\sum \limits _{k=1}^{N_x} \rho ^x_k\,(h_{k-1}^5-h_{k}^5) [I_{2\times 2}],&[GZ^x]=\frac{1}{6}\sum \limits _{k=1}^{N_x} \rho ^x_k\,(h_{k-1}^6-h_{k}^6) [I_{2\times 2}],\\&[HZ^x]=\frac{1}{7}\sum \limits _{k=1}^{N_x} \rho ^x_k\,(h_{k-1}^7-h_{k}^7) [I_{2\times 2}] \end{aligned}$$

The superscript “x” represents “t”, “b”, or “c” indicating top face sheet, bottom face sheet and $N_{x}$ indicates the number of layers in the considered face sheet or core and $\rho _{k}$ the density of the layer “k”.