Nonlinear analysis model of the progressive damage of aluminum–wood sandwich structures under high-speed impact conditions

Abstract

The impact responses of various protective structures composed of 2A12 aluminum alloy and wood laminates were studied experimentally. The experiments were conducted using different impact energies. By varying the sandwich material thickness and using two different bullet shapes, the effects of the sandwich material’s damage process and the core layer thickness on the protective performance were studied. The multilayer structure’s core layer failure condition was determined using the improved 3D Hashin criterion and a finite element model was established using Abaqus software. Tensile and three-point bending tests were conducted and the progressive damage model was verified statically. The model was then verified dynamically using the Hopkinson bar test. The mechanical properties of the materials under high dynamic strain rates were obtained through action loading testing of the specimens at different loading rates. The loading waveform was analyzed and a stress-strain relationship diagram was drawn at various strain rates. By verifying the experimental data, a numerical model that could capture the deformation and failure details during crushing was established, and the composite target plate impact failure mode and the trajectory change law were described. This study could lead to use of a new impact damage prediction method for laminates.

Introduction

Sandwich structures offer various advantages, including light weight, high specific strength, high specific stiffness, good stability, low manufacturing cost, and ease of maintenance, and these structures have been used widely in various engineering applications.^1,2 A sandwich structure is assembled by combining soft and hard materials. The external material generally has higher hardness and lower ductility than the internal material, and the internal material is composed of one or more flexible layers. These layers and structures have a wide range of applications and have unparalleled benefits for use in high-speed fields, such as sea, air, and aerospace applications. Any increase in distance between the upper and lower panels of the core layer increases the moment of inertia of the entire structure, while the resulting increase in weight is slight, and this makes the structure resilient under bending and buckling loads. ³ The compressibility of the core layer results in the material having an excellent buffering ability, which allows it to be used in protective structures.

In general, impact failure involves two key aspects. The impact reduces the material’s structural stiffness and strength, which are essential properties for maintaining the crashworthiness of the structure and enhance the structure’s safety by improving its energy absorption capacity. Previous studies have shown that the failure modes and energy absorption capacities of sandwich structures are mainly affected by the panel material’s properties and thickness ⁴ and by the core’s density and thickness. ⁵

The most commonly used protective structures are high-strength metals that absorb most of the impact energy, and existing research indicates that^6,7 the failure mode of such metal panels under medium- and low-velocity impact loads is large plastic deformation. Mohan et al. ⁴ performed a comparison study of the mechanical responses and failure modes of structures such as aluminum alloys and steel plates under low-speed impact, and they found that the energy absorbed by these structures was proportional to the strength of the individual metal plate. Sun et al. ⁸ also studied the energy absorption performances and failure modes of aluminum alloy, stainless steel, and carbon fiber sandwich panel structures.

Although high-strength metal armor plates can absorb a great deal of impact energy, volume weight costs and other factors mean that the replacement of these plates with inexpensive multilayer plates such as sandwich structures can improve the rigidity-to-weight and strength-to-weight ratios of a protective structure to a certain extent. As a derivative product of wood, a multilayer board formed via an adhesive layup process can minimize the defects in the wood. Therefore, the design of multilayer core sandwich structures becomes important and these designs require continuous improvement. In recent years, researchers have studied various forms of layered structures, including layered honeycomb structures^9–11 and layered composite honeycomb sandwich cylindrical structures, ¹² by using multiple layers and structures in their structural designs. ¹³ Additionally, Noor et al. ¹⁴ analyzed the nonlinear responses of composite sandwich plates under thermal gradients and mechanical loading conditions. During an impact process, the mechanical behavior of a sandwich structure is mainly dependent on the properties and the structural parameters of the panel and core materials. ¹⁵ Shin et al. ¹⁶ studied the response and damage characteristics of woven/epoxy and aluminum sandwich panels under impact loads. Mohan et al. ⁴ studied the impact responses of various panel materials experimentally, including the effects of the panel materials, the panel thickness, and the foam core size on the energy absorption properties. According to the principle of minimum potential energy, Fatt and Park^17,18 conducted a theoretical study of the impact resistance of composite sandwich plates under the actions of local indentation and global bending.

With regard to dynamic performance, scientists have used equivalent models to predict the natural frequencies and modal vibration types of sandwich structures, ¹⁹ along with the mechanical properties, failure mechanisms and impact resistance properties of the different sandwich structure levels. ²⁰ Zhu and Sun ²¹ divided the low-speed impact response process of a sandwich layer structure into five stages and then established a prediction model for the contact force, impact displacement, and energy absorption properties. Samlal and Santhanakrishnan ²² assessed the impact behavior and the main failure modes of sandwich plates experimentally. Mohammadkhani et al. ²³ studied the effects of steel wire on the energy absorption performances of sandwich structures from both experimental and numerical perspectives. Chauhan et al. ²⁴ studied the effects of silicon pins on the impact behavior of polymer interlayers, and their results showed that pin strengthening has a significant impact on the maximum contact force and the elastic recovery of interlayers after impact. Nejad et al. ²⁵ used the finite element method to discuss the effects of polymer needle size on the maximum peak load and energy absorption characteristics of sandwich structures under low-speed indentation conditions and verified their results through experimental testing. Klaus et al. ²⁶ studied a sandwich structure composed of a carbon fiber plate and an aramid fiber core and determined a relationship between the levels of damage and deformation caused by an impact and the bending strengths of the damaged samples. Basily and Elsayed ²⁷ compared the energy absorption capacity per unit volume of folded honeycomb structures under various impact velocities.

Research on the high-speed impact damage to the core layer, which is generally an anisotropic material layer, involves additional influencing factors,^28,29 including the fiber and matrix types, and the laying mode. These different factors lead directly to different damage forms and different energy absorption mechanisms. Nunes et al. ³⁰ conducted experimental high-speed impact research on eight different glass/epoxy composite laminates and an analysis of the damage shapes showed that there was obvious anisotropy in the high-speed impact damage to the composite laminates. The damage areas were measured and it was found that their shapes could be characterized approximately by an ellipse. In a high-speed penetration experiment on composite laminated thick plates, Cheng et al. ³¹ reported three typical damage forms caused by high-speed impact damage to laminates: (1) crushing, which mainly occurred on the impact contact fronts of composite laminated plates, with matrix crushing and local shear fracture of the fibers as the main failure modes; (2) tensile fracture of fibers, which mainly occurred in the middle layer of composite laminates, where fiber tensile fracture was the main failure form; and (3) delamination, which occurs mainly at the back of the composite laminate, where the area of delamination damage is much larger than the fiber fracture area and the impact hole area. Yew and Kendrick ³² used ultrasonic C-scanning and section staining to detect damage in composite laminates after high-speed impacts. Their results showed that the shape of the in-plane damage to a composite laminate caused by high-speed impact was not symmetrical or circular, which indicated that there was obvious anisotropy in the high-speed impact damage to the composite. Material thickness is one of the most important factors that affect high-speed impact damage.

It is particularly important to analyze the effects of impact loads on structures by using air cannons to impart such impact loads on target plates. It is difficult to test the importance of each factor because of differences in materials, sample sizes, impact energy, and boundary conditions, and accurate effects can be extracted from existing research results. In addition to these issues, the coupled effects of the bullet type and size parameters on the high-speed impact responses of laminated plates under high pressure and high speed conditions remain unexplored and must be investigated in detail.

The purpose of this work is to bridge this research gap by determining the effects of the bullet size parameters on the low-speed impact behavior of foam core sandwich structures. Core sandwich structures with differing position and size parameters are established by varying the thickness of the core layer. Impact experiments are conducted under high-energy and high-pressure conditions. The effects of the bullet size parameters on the maximum impact force are studied in detail, and the degrees of damage to several samples are compared. To overcome the limitations of the experimental conditions, the typical failure modes of both the panel and the core layer are analyzed by establishing an appropriate finite element model. Based on the calculations, the results of this study can be used to aid in the design of core sandwich structures made from various wooden materials.

Establishment of the progressive damage model of the sandwich core layer

Transverse isotropic constitutive equation of the core layer

The elastic and plastic properties of conventional wood differ in the longitudinal, tangential, and radial directions. This naturally manifests as orthotropic behavior. On a macroscopic level, wood can be regarded as a continuous, uniform material; therefore, transverse isotropic material behavior can reproduce the main behavior of wood effectively. When the compression response of wood is considered, four main stages can be distinguished. First, as the elastic part softens, the wood cell wall begins to collapse; then, the platform begins to collapse; third, the wood cell continues to be squeezed; and finally, the system undergoes densification. On a microscopic level, wood has a honeycomb structure. ¹⁰ Microbuckling of the tracheid (cell wall) occurs during longitudinal compression and this mechanism can be compared with the macroscopic buckling that occurs in square metal tube structures.

Wood is a material that has two axisymmetric orthogonal planes and it is considered to be approximately anisotropic, with three mutually orthogonal elastic symmetric planes (1, 2 and 3) representing the three axes of the material; plane 1 represents the direction of the length of the wood fiber, plane 2 represents the direction tangential to the annual ring, and plane 3 represents the axis along the radial direction. Figure 1 shows the directions of the log fibers.

Figure 1.

Directions of the log fibers.

At this time, analysis indicates that the radial and tangential Young’s moduli of the wood fibers are approximately equal, and the following assumptions are therefore made to simplify the numerical calculations. The stress state Δ at a specific point in the core layer material is shown in Figure 2, where:

$${\sigma }_2=\sigma ,{\sigma }_3=-\sigma ,{\sigma }_4={\sigma }_5={\sigma }_6=0$$

(1)

Figure 2.

Material unit fiber directions.

In this case, the strain potential energy density W is determined as follows:

$$\mathrm{W}=\frac{1}{2}{S}_{22}{\sigma }^2-S_{23}{\sigma }^2+\frac{1}{2}{S}_{22}{\sigma }^2=(S_{22}-S_{23}){\sigma }^2$$

(2)

In Figure 2, the coordinate system 2-3 is rotated by 45° to become system 2′-3′. The stress components under the new coordinate system are given as follows:

$${\sigma }_1\prime ={\sigma }_2\prime ={\sigma }_3\prime =0,{\sigma }_4\prime =-\sigma ={\tau }_{23},{\tau }_{31}\prime ={\tau }_{12}\prime =0$$

(3)

At this time, the force of the unit body is in a pure shear stress state, which is given by:

$$\mathrm{W}=\frac{1}{2}{S}_{44}{\sigma }_4=\frac{1}{2}{S}_{44}{\sigma }^2$$

(4)

The simultaneous acquisition is then calculated as follows:

$$2(S_{22}-S_{23}){\sigma }^2=S_{44}{\sigma }^2$$

(5)

$$C_{44}=\frac{1}{2}(C_{22}-C_{23})$$

(6)

The stiffness matrix can be written as shown:

$$C=\left[\begin{array}{c}\begin{array}{ccc}{C}_{11}\hfill & C_{12}\hfill & C_{12}\hfill \\ \multicolumn{1}{c}{C_{12}}& C_{22}\hfill & C_{23}\hfill \\ \multicolumn{1}{c}{C_{12}}& C_{23}\hfill & C_{22}\hfill \end{array}\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\begin{array}{ccc}{C}_{44}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& C_{66}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & C_{66}\hfill \end{array}}\end{array}\right]$$

(7)

The flexibility matrix after deformation can then be expressed as follows:

$$\left[S\right]=\left[\begin{array}{cc}\begin{array}{ccc}{S}_{11}\hfill & S_{12}\hfill & S_{12}\hfill \\ \multicolumn{1}{c}{S_{12}}& S_{22}\hfill & S_{23}\hfill \\ \multicolumn{1}{c}{S_{12}}& S_{23}\hfill & S_{22}\hfill \end{array}\hfill & \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}& \multicolumn{1}{c}{\begin{array}{ccc}2(S_{22}-S_{23})\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& C_{66}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & C_{66}\hfill \end{array}}\end{array}\right]$$

(8)

The transition from the 3D state to the 2D flat state can be expressed as shown in the following. The equation above is converted into a form in which the stress is expressed using the strain:

$$\left[\begin{array}{c}{\sigma }_1\hfill \\ \multicolumn{1}{c}{{\sigma }_2}\\ \multicolumn{1}{c}{{\tau }_{12}}\end{array}\right]=\left[\begin{array}{ccc}{Q}_{11}\hfill & Q_{12}\hfill & 0\hfill \\ \multicolumn{1}{c}{Q_{12}}& Q_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & Q_{66}\hfill \end{array}\right]\left[\begin{array}{c}{\epsilon }_1\hfill \\ \multicolumn{1}{c}{{\epsilon }_2}\\ \multicolumn{1}{c}{{\gamma }_{12}}\end{array}\right]$$

(9)

$$T=\left[\begin{array}{ccc}\mathrm{co}{s}^2\theta \hfill & \mathrm{si}{n}^2\theta \hfill & 2\cos \theta \sin \theta \hfill \\ \multicolumn{1}{c}{\mathrm{si}{n}^2\theta }& \mathrm{co}{s}^2\theta \hfill & -2\cos \theta \sin \theta \hfill \\ \multicolumn{1}{c}{-\cos \theta \sin \theta }& \cos \theta \sin \theta \hfill & \mathrm{co}{s}^2\theta -\mathrm{si}{n}^2\theta \hfill \end{array}\right]$$

(10)

The transformation relationship between the stress in the material coordinate system and that in the global coordinate system is then given as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{c}{\sigma }_1\hfill \\ \multicolumn{1}{c}{{\sigma }_2}\\ \multicolumn{1}{c}{{\tau }_{12}}\end{array}\right]=T\left[\begin{array}{c}{\sigma }_x\hfill \\ \multicolumn{1}{c}{{\sigma }_y}\\ \multicolumn{1}{c}{{\tau }_{\mathrm{xy}}}\end{array}\right],\left[\begin{array}{c}{\epsilon }_1\hfill \\ \multicolumn{1}{c}{{\epsilon }_2}\\ \multicolumn{1}{c}{\frac{{\gamma }_{12}}{2}}\end{array}\right]=T\left[\begin{array}{c}{\epsilon }_x\hfill \\ \multicolumn{1}{c}{{\epsilon }_y}\\ \multicolumn{1}{c}{\frac{{\gamma }_{\mathrm{xy}}}{2}}\end{array}\right]}\end{array}$$

(11)

At this time, on further exploration, it can be determined that the plywood in this case is a four-layer wood rotating cut sheet that was laid and bonded by the orthogonal laying method. From a macroscopic perspective, the multilayer board has the same Young’s modulus (isotropic) in the plane 1-2 direction, as shown in Figure 3.

Figure 3.

Fiber laying angles for the multilayer board.

By taking the Δ element in Figure 3 and transforming it according to the steps described above, the stiffness matrix shown in Figure 3 can be expressed as follows:

$$C=\left[\begin{array}{c}\begin{array}{ccc}{C}_{11}\hfill & C_{12}\hfill & C_{13}\hfill \\ \multicolumn{1}{c}{C_{12}}& C_{11}\hfill & C_{23}\hfill \\ \multicolumn{1}{c}{C_{13}}& C_{23}\hfill & C_{33}\hfill \end{array}\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\begin{array}{ccc}{C}_{44}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& C_{44}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & C_{66}\hfill \end{array}}\end{array}\right]$$

(12)

$$C_{66}=\frac{1}{2}(C_{11}-C_{12})$$

(13)

Three-dimensional progressive damage model of the core layer

Assuming that the top plate and the bottom plate conform to the completely plastic law of yield strength rigidity, poplar wood is modeled here as an anisotropic nonmetallic material by using the single-layer wood model with consideration of the stratified damage criterion.

The Hashin damage criterion is given by the following:

• (1) Fiber failure

$${\left(\frac{{\sigma }_{11}}{X_T}\right)}^2+{\left(\frac{{\tau }_{12}}{S_{12}}\right)}^2+{\left(\frac{{\tau }_{13}}{S_{13}}\right)}^2\ge 1{\sigma }_{11}\ge 0$$

(14)

$${\left(\frac{{\sigma }_{11}}{X_C}\right)}^2\ge 1{\sigma }_{11}\le 0$$

(15)

• (2) Matrix failure

$$\begin{array}{c}{\left(\frac{{\sigma }_{22}+{\sigma }_{33}}{Y_T}\right)}^2+\frac{1}{S_{23}^2}({\tau }_{23}^2-{\sigma }_{22}{\sigma }_{33})+{\left(\frac{{\tau }_{12}}{S_{12}}\right)}^2+{\left(\frac{{\tau }_{13}}{S_{13}}\right)}^2\ge 1\hfill \\ {\sigma }_{22}+{\sigma }_{33}>0\hfill \end{array}$$

(16)

$$\begin{array}{c}\frac{{\sigma }_{22}+{\sigma }_{33}}{Y_C}[{\left(\frac{Y_C}{2S_{23}}\right)}^2-1]+{\left(\frac{{\sigma }_{22}+{\sigma }_{33}}{2S_{23}}\right)}^2+\frac{{\tau }_{23}^2-{\sigma }_{22}{\sigma }_{33}}{S_{23}^2}\hfill \\ +{\left(\frac{{\tau }_{12}}{S_{12}}\right)}^2+{\left(\frac{{\tau }_{13}}{S_{13}}\right)}^2\ge 1\hfill \\ {\sigma }_{22}+{\sigma }_{33}<0\hfill \end{array}$$

(17)

• (3) Delamination failure

$${\left(\frac{{\sigma }_{33}}{Z_T}\right)}^2+{\left(\frac{{\tau }_{13}}{S_{13}}\right)}^2+{\left(\frac{{\tau }_{23}}{S_{23}}\right)}^2\ge 1{\sigma }_{33}>0$$

(18)

$${\left(\frac{{\sigma }_{33}}{Z_C}\right)}^2+{\left(\frac{{\tau }_{13}}{S_{13}}\right)}^2+{\left(\frac{{\tau }_{23}}{S_{23}}\right)}^2\ge 1{\sigma }_{33}<0$$

(19)

where the subscripts 1, 2, and 3 represent the coordinate directions (i.e., 1 is the fiber direction, 2 represents the direction perpendicular to the fiber direction in the laminate surface, and 3 represents the thickness direction of the laminate overlay); $X_p$ , $Y_p$ , and $Z_p$ are the strengths of the anisotropic monolayer material in directions 1, 2, and 3, respectively; the subscript P = T represents the tensile strength; the subscript P = C is the compressive strength; and S _ij represents the shear strength in the corresponding plane. The mechanical parameters of the poplar plants are given in Table 1.

Table 1.

Initial parameters of the aspen wood (wood and composite piles).

Parameter	Value	Parameter	Value
$\rho$	$400\mathrm{kg}/{\mathrm{m}}^3$	$G_{13}$	$160\mathrm{MPa}$
$E_1$	$13\mathrm{GPa}$	$X_T$	$40\mathrm{MPa}$
$E_2$	$9\mathrm{GPa}$	$X_C$	$15\mathrm{MPa}$
$E_3$	$7\mathrm{GPa}$	$Y_T,Z_T$	$10\mathrm{MPa}$
${\nu }_{12}$	0.42	$Y_C,Z_C$	$26\mathrm{MPa}$
${\nu }_{13}$	0.24	$S_{12}$	15 $\mathrm{MPa}$
${\nu }_{23}$	0.24	$S_{13}$	8 $\mathrm{MPa}$
$G_{12}$	$670\mathrm{MPa}$	$S_{23}$	7 $\mathrm{MPa}$
$G_{13}$	$260\mathrm{MPa}$

Three-dimensional damage model based on exponential evolution

The multilayer board in this case is refined into orthogonal layers of anisotropic materials, the laminate is divided into nine layers, and the bonding behavior between adjacent layers is established. Each part of the material is assigned a coordinate system. Therefore, each layer is considered to be orthotropic, and a constitutive model is defined in the elastic phase that includes nine elastic constants. For the orthotropic continuous damage mechanics, the relationship between the effective stress tensor and the actual stress tensor is established by introducing a damage factor tensor after the damage initiation. The relationship is defined as follows. In damage mechanics, there are microscopic defects in the material that expand continuously under the action of an external load, causing the mechanical properties of the material to deteriorate, and the damage state variable is therefore introduced into the material point stiffness matrix. The fibers in the material carry most of the load within the structure, and when the fibers are destroyed, the matrix material around the fibers then fails. Therefore, it is assumed that the fiber damage affects the matrix and that shear damage occurs, and this damage is considered to be irreversible.

The main models studied here were based on the built-in damage models for fiber-reinforced composites available in the Abaqus/Standard and Abaqus/Explicit software (Dassault Systemes).

Schematics of the four types of material damage constitutive model were given in the literature, ³³ and are as shown in Figure 4.

Figure 4.

Four material damage constitutive relationship models.

During the progressive damage process, the effective elastic matrix consists of four damage variables, denoted by $d_f$ , $d_d$ , $d_m$ , and $d_S$ .

In the user subroutine UMAT, the stress is updated according to the following equation:

$$\mathbf{\sigma }=C^d:\epsilon$$

(20)

$$C=\left[\begin{array}{ccc}{C}_{11}\hfill & C_{12}\hfill & C_{13}\hfill \\ \multicolumn{1}{c}{C_{12}}& C_{22}\hfill & C_{23}\hfill \\ \multicolumn{1}{c}{C_{13}}& C_{23}\hfill & C_{33}\hfill \end{array}\right]$$

(21)

According to the hypothesis of energy equivalence, the residual energy function of the damaged material takes the same form as that of the undamaged material if the stress is replaced by the effective stress. The damage stiffness matrix of the material points can then be obtained as follows:

$$\begin{array}{c}{C}^d=\hfill \\ \left[\begin{array}{ccc}(1-d_f)C_{11}\hfill & (1-d_S)C_{12}\hfill & (1-d_f)(1-d_d)C_{13}\hfill \\ \multicolumn{1}{c}{(1-d_S)C_{12}}& (1-d_m)C_{22}\hfill & (1-d_f)(1-d_m)C_{23}\hfill \\ \multicolumn{1}{c}{(1-d_f)(1-d_d)C_{13}}& (1-d_f)(1-d_m)C_{23}\hfill & (1-d_d)C_{33}\hfill \end{array}\right]\hfill \end{array}$$

(22)

The Linde exponential attenuation model is given as follows.

Fiber damage begins when the following criteria apply:

$$f_f=\sqrt{\frac{{\epsilon }_{11}^{f,t}}{{\epsilon }_{11}^{f,c}}{\left({\epsilon }_{11}\right)}^2+({\epsilon }_{11}^{f,t}-\frac{{\left({\epsilon }_{11}^{f,t}\right)}^2}{{\epsilon }_{11}^{f,c}})}{\epsilon }_{11}>{\epsilon }_{11}^{f,t}$$

(23)

where ${\epsilon }_{11}^{f,t}={\sigma }_L^{f,t}/C_{11},{\epsilon }_{11}^{f,c}={\sigma }_L^{f,c}/C_{11}$ , and $C_{\mathrm{ij}}$ are the components of the elastic matrix in the undamaged state. When the criteria above are met, the fiber damage variable $d_f$ then evolves according to the following equation:

$$d_f=1-\frac{{\epsilon }_{11}^f}{f_f}{e}^{(-C_{11}{\epsilon }_{11}^f(f_f-{\epsilon }_{11}^f)L^c/G_f)}$$

(24)

where $L^c$ is the feature length of the material point correlation, and G is the rupture energy.

$$f_m=\sqrt{\frac{{\epsilon }_{22}^{f,t}}{{\epsilon }_{22}^{f,c}}{\left({\epsilon }_{22}\right)}^2+({\epsilon }_{22}^{f,t}-\frac{{\left({\epsilon }_{22}^{f,t}\right)}^2}{{\epsilon }_{22}^{f,c}}){\epsilon }_{22}+{\left(\frac{{\epsilon }_{22}^{f,t}}{{\epsilon }_{22}^{f,c}}\right)}^2{\left({\epsilon }_{12}\right)}^2}>{\epsilon }_{22}^{f,t}$$

(25)

where ${\epsilon }_{22}^{f,t}={\sigma }_T^{f,t}/C_{22},{\epsilon }_{22}^{f,c}={\sigma }_T^{f,c}/C_{22}$ , and ${\epsilon }_{12}^f={\tau }_{\mathrm{LT}}^f/C_{44}$ .

The evolution law of the matrix damage variable $d_m$ is given as follows:

$$d_m=1-\frac{{\epsilon }_{22}^f}{f_m}{e}^{(-C_{22}{\epsilon }_{22}^{f,t}(f_m-{\epsilon }_{22}^{f,t})L^c/G_m)}$$

(26)

The evolution law of the layered damage $d_d$ is given as follows:

$$f_d=\sqrt{\frac{{\epsilon }_{33}^{f,t}}{{\epsilon }_{33}^{f,c}}{\left({\epsilon }_{33}\right)}^2+({\epsilon }_{33}^{f,t}-\frac{{\left({\epsilon }_{33}^{f,t}\right)}^2}{{\epsilon }_{33}^{f,c}}){\epsilon }_{22}+{\left(\frac{{\epsilon }_{33}^{f,t}}{{\epsilon }_{13}^f}\right)}^2{\left({\epsilon }_{13}\right)}^2}>{\epsilon }_{33}^{f,t}$$

(27)

$$d_d=1-\frac{{\epsilon }_{33}^f}{f_m}{e}^{(-C_{33}{\epsilon }_{33}^{f,t}(f_m-{\epsilon }_{33}^{f,t})L^c/G_m)}$$

(28)

$$d_s=1-\sqrt{d_m{d}_f}$$

(29)

Finite element model of progressive damage

Geometric model and boundary conditions

Geometry

A finite element (FE) model was established using the Abaqus/Explicit technique to simulate the dynamic responses of sandwich plates to out-of-plane impact loads. The FE model of the sandwich structure during high-speed impact is shown in Figure 5.

Figure 5.

Finite element (FE) model of the sandwich structure during high-speed impact.

As a result of the dynamic/explicit use of very small time increments, this FE model can be applied to nonlinear large deformation problems. The VUMAT subroutine was developed to model the wood laminates. The calculation flow is shown in Figure 6.

Figure 6.

Progressive damage analysis process for an aluminum–wood sandwich structure.

Boundary conditions

Fully built-in (U1 = U2 = U3 = UR1 = UR2 = UR3 = 0).

A convergence study was conducted to determine the minimum element size.

Contact

The contact algorithm is the most fundamental part of an FE analysis for influence/contact problems. Three types of contact algorithm are used to model the various contacts and they affect the collision conditions during the process. The erosion contact type is calculated through the keyword options and the contact definitions used in the FE model. Selection of different parameters and settings based on the different specimen structures and the material differences between the interlayers are key to defining the internal contact conditions in model calculations. The system program allows the remaining elements inside the sandwich structure to continue contact operations after the failure unit is removed by the external bullet impact. The contact between the upper and lower fixed plates, the outer layer (front panel and back plate), the core layer, and the bullet is defined as surface-to-surface contact with a friction coefficient of 0.26 between the bullet target plates.

Grid sensitivity analysis

The layers and component parts of the structural target plates were meshed using C3D8R (cubic, eight-node, 3D elements, reduced integration unit) elements, which were studied via the FE calculations. Enhanced hourglass control was enabled to prevent distortion of these elements. The double/triple sandwich structure has an overall cutting size of 200×200 mm² and a sandwich layer thickness of 5 mm. A Johnson–Cook (J-C) model hexahedral mesh was used in the FE model of the aluminum plate, which could be deformed into a three-dimensional solid. The mechanical connection was fixed using bolts that were applied between the aluminum plate and the board layer, and a cohesive force unit was added between the core layer of the wood and the adjacent wood layer. The grid properties of the impacted target plate are given in Table 2.

Table 2.

FE meshing of laminates.

Specimen	Material	Type of grid	Direction of material	Size of mesh	Grid layers	Physical thickness
Bullet	15CR	C3D10M	Isotropic	0.25 mm
Aluminum alloy	2A12	C3D8R	Isotropic	0.25 mm	4	1 mm
Plywood	Wood	C3D8R	Anisotropic	0.25 mm	7	1.6 mm
Cohesive	MUF	COH3D8	Isotropic	0.1 mm	1	0.1 mm

MUF: Melamineureaformaldehyde resin.

To optimize the use of computational resources, an impact calculation model was used to perform a grid sensitivity analysis in conjunction with the main content of this paper. ³⁴ Additionally, based on consideration of the localized nature of the impact damage, the mesh size of the impacted area was divided into various sizes to determine an appropriate value. Figure 7 shows the residual velocity curves (V_r) of bullets when penetrating a target plate for the different grid sizes (3 mm, 2 mm, 1 mm, 0.25 mm, and 0.1 mm) used to conduct the impact simulations. The incident projectile body had an initial velocity of 100 m/s, which resulted in a corresponding residual velocity (V_r) of approximately 37.11 m/s in the relevant experiments.

Figure 7.

Mesh sensitivity analysis.

An analysis of the velocity–operation time–residual velocity curves that were realized using FE software indicates that the overall results show a consistent decreasing trend in calculations and that they are relatively close, except in the case where a mesh size of 3 mm was used. After consideration of all the relevant factors, a mesh size of 0.25 mm was selected.

To analyze the influence of the different forms of grid division on the calculation results, the final grid divisions consisted of front and rear panels totaling approximately 39,200 elements, discrete rigid bodies consisting of bullets totaling approximately 617 elements, and core layer materials containing cohesive units totaling approximately 333,200 elements.

Model of the aluminum alloy and cohesive layer behavior

Because the surface damage modes of 2A12 are mainly plastic deformation and failure, 2A12 was adjusted here in accordance with the study of Hou et al., ³⁵ and the J–C model was used to define the failure model.^36,37 The equivalent fracture strain is given as follows:

$${\sigma }_{\mathrm{eq}}=(A+B{\epsilon }_{\mathrm{eq}}^n)[1+\mathrm{Cln}\left({\stackrel{·}{\epsilon }}_{\mathrm{eq}}^{*}\right)](1-T^{*m})$$

(30a)

$${\epsilon }_f=[D_1+D_2\exp \left(D_3{\sigma }^{*}\right)][1+D_4\ln \left({\stackrel{·}{\epsilon }}_{\mathrm{eq}}^{*}\right)](1+D_5{T}^{*})$$

(30b)

$${\sigma }^{*}=-\frac{{\sigma }_m}{{\sigma }_{\mathrm{eff}}}$$

(30c)

$$T^{*}=(T-T_r)/(T_m-T_r)$$

(30d)

where $A$ represents the yield strength of the material at the reference strain rate and the reference temperature; $B$ and n are strain strengthening coefficients; $C$ is the strain rate sensitivity coefficient; m is the temperature softening coefficient; ${\sigma }_m$ is the hydrostatic pressure; ${\sigma }_{\mathrm{eff}}$ is the equivalent von Mises stress; ${\stackrel{·}{\epsilon }}_{\mathrm{eq}}^{*}$ = ${\stackrel{·}{\epsilon }}_{\mathrm{eq}}/{\stackrel{·}{\epsilon }}_0$ ; ${\stackrel{·}{\epsilon }}_0$ is the reference strain rate of 1 s⁻¹; ${\stackrel{·}{\epsilon }}_{\mathrm{eq}}$ is the plastic strain rate; $D_1$ , $D_2$ , $D_3$ , $D_4$ , and $D_5$ are invalid parameters; $T_m$ is the melting point of the material of 863 K; $T_r$ is the reference temperature of 293 K; and $T$ is the current temperature.

The parameters of the aluminum alloy are given in Table 3.

Table 3.

2A12-T4 alloy parameters (aluminum skin).

Young’s modulus	Density	Poisson’s ratio	A	B	n	m	C	${\epsilon }_p$
69 GPa	2770 kg/m3	0.33	380 MPa	476 MPa	0.28	2.71	0.0083	0.163
${\stackrel{·}{\epsilon }}_0$	${\mathrm{D}}_1$	${\mathrm{D}}_2$	${\mathrm{D}}_3$	${\mathrm{D}}_4$	${\mathrm{D}}_5$
1.1×10−3 s-1	0.116	0.14	1.55	−0.002	0.003

Multilayer plywood structures are constructed using orthogonal layers of wood, and models of layered structures include models of the stretching–traction between the wood layers. The parameters of the cohesively bonded surfaces used in this study are given in Table 4.

Table 4.

Material parameters of the cohesive element. ³⁸

Parameter	Value	Parameter	Value
${\mathrm{k}}_{\mathrm{n}}$	1e6 N/mm3	${\mathrm{t}}_{\mathrm{t}}^0$	60 MPa
${\mathrm{k}}_{\mathrm{s}}$	1e6 N/mm3	${\mathrm{G}}_{\mathrm{I}}^{\mathrm{c}}$	765 J/m2
${\mathrm{k}}_{\mathrm{t}}$	1e6 N/mm3	${\mathrm{G}}_{\mathrm{II}}^{\mathrm{c}}$	1250 J/m2
${\mathrm{t}}_{\mathrm{n}}^0$	60 MPa	${\mathrm{G}}_{\mathrm{III}}^{\mathrm{c}}$	1250 J/m2
${\mathrm{t}}_{\mathrm{s}}^0$	60 MPa	$\eta$	1

The Benzeggagh–Kenane fracture criterion ³⁹ is implemented here, as shown in equations (31) and (32).

$$G_C=G_n^c+(G_s^c-G_n^c){\left(\frac{t_s}{t_s^0}\right)}^{\eta }$$

(31)

$$G_s=G_{\mathrm{II}}+G_{\mathrm{III}},G_T=G_I+G_s$$

(32)

where $t_n$ , $t_s$ , and $t_t$ represent the normal traction and two shear traction forces, respectively; $k_n$ , $k_s$ , and $k_t$ are the interlayer stiffnesses of modes $I$ , $\mathrm{II}$ , and $\mathrm{III}$ , respectively; $t_n^0$ , $t_s^0$ , and $t_t^0$ are the strengths in the corresponding directions; $G_I$ , $G_{\mathrm{II}}$ , and $G_{\mathrm{III}}$ are the fracture toughness values of modes $I$ , $\mathrm{II}$ , and $\mathrm{III}$ , respectively; and $\eta$ is the mixed-mode parameter.

Model verification

In this section, the experimental materials, the sample sizes, and the forming process are introduced first, and the definitions of the position parameters and the size parameters are then explained in detail. Tensile and three-point bending experiments are conducted on both the core material and the entire structure, and the important mechanical properties of the structure, e.g., the equivalent failure stress, the equivalent strength, the loading displacement, and the maximum load, are obtained. Based on the experimental data, the three-dimensional Hashin damage FE model is then statically verified.

In addition, a dynamic Hopkinson bar experiment is performed on the core material of the aluminum–wood sandwich structure. A separate Hopkinson pressure bar device is used to conduct dynamic loading tests on specimens at different loading rates, determine the mechanical properties of the materials at high dynamic strain rates, analyze the loading waveform, plot the stress-strain relationships at various strain rates, and analyze the failure patterns of the specimens. Based on the experimental data obtained, the 3D Hashin damage FE model is then dynamically verified.

Static verification of the 3D progressive damage model

The quasi-static mechanical properties of the sandwich structures were studied via three-point bending experiments and the three-dimensional progressive damage model was verified via experiments performed under quasi-static loading conditions. According to their size divisions, the samples produced were sandwich structural beams. For the purposes of distinguishing their parameters, the lower corners were marked as follows: f – surface; c– core. The preparation of the three-point bending specimen and its relevant dimensions and loading characteristics are shown in Figure 8.

Figure 8.

Three-point bending forces of the honeycomb structure for (a) double-layer structure and (b) three-layer structure.

From the experimental images and the data analysis results, the entire loading process for three-point bending of the aluminum–wood sandwich structure shows that the loading curves corresponding to the two samples both follow a trend of initial linear elastic deformation and subsequent plastic nonlinear bending and buckling. The aluminum alloy plate gradually experienced a large plastic deformation that began in the early stages and the deformation was relatively obvious, whereas the plywood began to degenerate directly and fracture occurred, as shown in Figure 9.

Figure 9.

Four-stage analysis results for (a) sample 3 and (b) sample 6.

The equivalent stress ${\sigma }_{\mathrm{eq}}$ of the specimen is given as follows:

$${\sigma }_{\mathrm{eq}}=3P_{\max }L/2b{H}^2$$

(34)

$$I_{\mathrm{eq}}=\frac{b{H}^3}{12}$$

(35)

The overall equivalent bending modulus $E_{\mathrm{feq}}$ is determined using:

$$f={\epsilon }_{\mathrm{fi}}{L}^2/6H$$

(36)

$$E_{\mathrm{feq}}=\mathrm{\Delta }\sigma /\mathrm{\Delta }\epsilon$$

(37)

$$D_{\mathrm{eq}}=E_{\mathrm{feq}}\times I_{\mathrm{eq}}$$

(38)

where $\sigma$ is the panel stress (MPa); $P$ is the total load at the loading point (N); $L$ is the three-point bending span of the sandwich structure (mm); $h_c$ is the core layer height of the sandwich structure sample (mm); $b$ is the structural material sample width (mm); $f$ is the deflection generated by loading (mm); $h_f$ is the panel thickness (mm); $H$ is the total specimen thickness (mm); and $\mathrm{\Delta }$ is the is the deformation caused by loading in accordance with Hooke’s law, which can be captured from Figure 9. The calculation results are presented in Table 5.

Table 5.

Three-point bending data and calculation results of the experimental samples.

No.	$f$ (mm)	$P_{\max }$ (N)	$H$ (mm)	$l$ (mm)	$f_{\mathrm{pmax}}$	$\mathrm{\Delta }p$ (N)	$\mathrm{\Delta }f$ (mm)	${\epsilon }_{\mathrm{fi}}$	${\sigma }_{\mathrm{eq}}$ (MPa)	$D_{\mathrm{eq}}$ (MPa)
1	13.0	673.075	6	88	11.1	455.12	3.6	0.0195	35.25	1424.49
2	16.40	1690.0	11	88	4.5	1218	1.42	0.0132	26.33	1568.45
3	11.99	4090.45	16	88	3.6	2231	1.11	0.0146	30.13	1194.2
4	31.44	1042.88	7	88	17.3	531	3.04	0.0165	40.13	1239.4
5	20.48	2067.31	12	88	9.6	724	1.10	0.011	27.07	930.409
6	16.76	3461.11	17	88	4.1	1677	1.09	0.0143	22.58	764.97

Typical specimens (3 and 6) were selected for the analysis. According to the displacement-load curves of the specimens, specimens 3 and 6 both undergo four stages during bending under pressure—comprising elasticity, plasticity, layered cracking coupling damage, and overall fracture—as illustrated in Figure 9.

In Figure 9, I is the elastic stage, II is the plastic stage, III is the crack breeding and merging stage, and IV is the damage stage. Figure 9(a) shows that the rising section of stage II remains smooth until the maximum value is reached, and the experimental records show that no obvious damage occurred in specimen 3 until the moment P_max was reached. At the beginning of stage III, the invisible cracks suddenly begin to merge until they trigger internal defects in the specimen, e.g., natural wood defects, bonding defects, and uneven gluing. In stage IV, the cracks grow slowly but steadily, and the specimen obviously collapses and undergoes shear deformation. In Figure 9(b), phase III is divided into a crack growth region and a crack combination region. The test records show that a small amount of longitudinal cracking initially occurs in specimen 6 at stage 3, after which core material panel debonding occurs. The central part of the load is obviously compressed. The multilayer adhesive tissue in the core layer is damaged and slips under pressure until the cracks in each layer grow and merge in an irregular fashion.

After the material properties and the boundary conditions are defined, the elastic modulus and the yield stress can be estimated, appropriate mesh types can be generated, a mesh convergence analysis can be applied, and the effects of mesh element refinement can be studied.^40,41 According to the load-displacement curve obtained from experiment 4, the results produced by the numerical model are similar to the experimental characteristics. Comparisons show that the coupling zone of delamination and matrix cracking occurs in both the model and the experiment, and the matrix fails rapidly in both cases. The elastic increase stage and the damage expansion pattern are basically consistent with the experimental results. Because of the interactions between the nonlinear behavior of the material and the algorithm superposition, this phenomenon is difficult to predict at a later stage, and the subsequent loading of the core material has basically failed based on experimental observation. The difference between the experimental and simulation errors is less than 6%. A comparison of the numerical model calculations with the experimental results is shown in Figure 10.

Figure 10.

Comparison of the load–displacement curve calculation results.

According to the FE analysis above, the three-dimensional progressive damage model can simulate the mechanical properties of the laminate core accurately, predict the stress-strain relationship for the material to a certain extent, and effectively predict the entire process: elasticity, initial damage, fiber damage, matrix damage, and subsequent failure when the material is bent. In addition, the accuracies of both the three-position damage model and the three-dimensional Hashin criterion when considering layered damage by exponential degradation have been proven, and these results can be used to perform FE numerical simulations of stacked materials under quasi-static conditions.

Dynamic verification of the 3D progressive damage model

During high-speed impact, rapid impact load propagation inside the core layer deforms and damages the core layer. To characterize the dynamic mechanical behavior of the wooden core layers accurately, it is necessary to conduct in-depth research into the damage and the mechanical properties of the wooden core layers under high dynamic strain rate loading. A separate Hopkinson test device was used in the experiments. This test method is mainly used to measure the dynamic compression properties of materials. According to stress wave theory, the stress wave propagates between the end face of the specimen and the transmission bar until it becomes balanced. The loading strain rate of the specimen can then be derived from the relationships between the specimen, the incident bar, and the transmission bar. The principle of the Hopkinson pressure bar is illustrated in Figure 11.

$$\stackrel{`}{\epsilon }=\frac{C_0}{H}({\epsilon }_i-{\epsilon }_r-{\epsilon }_t)$$

where the subscripts $i$ , r, and $t$ represent the incident, reflected, and transmitted signals, respectively; $C_0$ is the internal wave velocity of the impact rod (7075 aluminum alloy); and $H$ is the initial sample length, where H/2R=0.54.

Figure 11.

Schematic diagram of the Hopkinson pressure rod device.

By integrating the strain rate with respect to time, the strain of the material during a wave oscillation period can be obtained as:

$$\epsilon =\frac{C_0}{H}{\int }_0^t({\epsilon }_i-{\epsilon }_r-{\epsilon }_t)\mathrm{dt}$$

(39)

The sensor electrical signal and waveform conversion process is given as follows:

$$\epsilon \left(t\right)=\frac{\mathrm{\Delta }U}{k·k\prime V}$$

(40a)

$${\sigma }_S(t)=\frac{\mathrm{AE}}{A_0}{\epsilon }_t$$

(40b)

where $A$ is the pressure rod diameter (40 mm); $A_0$ is the specimen diameter (30 mm); $E$ is the elastic modulus of the pressure rod (71 GPa); $C_0$ is the elastic wave velocity in the pressure rod; $\rho$ is the test rod density; $H$ is the specimen length (15 mm); $\mathrm{\Delta }U$ is the oscilloscope value; $k$ is the strain gauge coefficient; $k\prime$ is the magnification (400 times for plywood/1000 times for poplar); and $V$ is the strain gauge voltage (2 V/4 V).

The Hopkinson bar experimental data acquired for the poplar samples are shown in Figure 12.

Figure 12.

Hopkinson data acquired for poplar wood.

The relationships among the incident wave ${\epsilon }_i$ , the reflected wave ${\epsilon }_r$ , and the transmitted wave ${\epsilon }_t$ in a single propagation period are shown in Figure 13. In Figure 13(a), the reflected wave ε_r shows a reduction in its strength, which is caused by a sudden drop in strength produced by the unevenness of the prestressed laminates and the effects of the natural defects in the internal layer of the plywood; $\mathrm{\Delta }{U}_1$ is the residual stiffness value, and it is further compacted because of the fluffy honeycomb property of the wood. The integrity of the laminates is maintained because of the cross-laying process and the presence of the adhesive layer. Figure 13(b) shows the Hopkinson experimental data acquired for the poplar wood. At this instant, the strength increase is $\mathrm{\Delta }{U}_2$ . Based on observation of the reflected wave ε_r for the entire image, it can be concluded that there is a specific law of plywood compaction; i.e., the strength of the plywood after compaction does not exceed its initial strength.

Figure 13.

Incident wave ${\epsilon }_i$ , reflected wave ${\epsilon }_r$ , and transmitted wave ${\epsilon }_t$ : (a) plywood and (b) poplar wood.

In Figure 14, $\mathrm{\Delta }t$ is the propagation time for the incident wave in the specimen relative to the thickness H. The wave velocity $C_1$ for the plywood and the wave velocity $C_2$ for the poplar can then be calculated accordingly. The amplitude from the waveform analysis indicates that the poplar board has a good absorption performance for vibration stress waves; however, because of the low strength of the poplar, a high strain gauge voltage and long magnification times are added during the experiment, and the specific values obtained are reflected as follows. Therefore, the overall strength and hardness properties of the poplar sample are not as good as the corresponding properties of plywood.

Figure 14.

Elemental model of the Hopkinson pressure rod device.

In summary, in this section, the dynamic mechanical property parameters of the core layer are determined via the Hopkinson dynamic compression bar experiment, and the strain rate effect of the material is explored to provide a reference for the numerical simulation performed in experiment 5.

The FE model was established with Abaqus/Explicit for the Hopkinson simulation calculations, and the effectiveness of this numerical model under high dynamic strain rate loading was verified through experimental comparison and debugging of the material parameters. The FE model of the Hopkinson pressure bar is shown in Figure 14.

In Figure 14, 1 and 2 represent the incident rod and the transmission rod, respectively, and 3 represents the sample. The minimum size for the grid encryption in the dotted box is 0.1 mm. The FE model uses 1/4 modeling to reduce the amount of computation required. The incident rod, the transmission rod, and the specimen are all classified using the C3D8R unit type, and the material parameters are listed in Tables 1 and 2. The numerical model calculation results are shown in Figure 15.

Figure 15.

Numerical model calculation results.

Through a combination of the FE analysis above and experimental comparisons, the material data damage model can simulate the mechanical properties of the specimens accurately at high strain rates, and can also predict the stress-strain relationships of materials and the mechanical behavior of materials effectively under impact conditions through comparisons with the experimental results. In addition, the accuracy of both the three-position damage model and the three-dimensional Hashin criterion when considering layered damage by exponential degradation under high dynamic strain rate loading conditions have been verified, and these results can be applied to FE numerical simulations of stacked materials. The numerical simulation results and experimental matching of the data are presented in Figure 16.

Figure 16.

Numerical simulation and experimental comparison of the data: (a) Incident bar and (b) Reflection bar.

High-speed impact experimental research and discussion

First, the experimental materials, the sample size, and the forming process are introduced, and the definitions of the position parameters and the size parameters are explained in detail. Then, the high-speed impact test conditions and the test system are introduced. Finally, the experimental results are statistically analyzed, and the sandwich structure’s damage state is observed and analyzed.

High-speed impact sample preparation

In this study, 2A12T4 aluminum alloy (Northeast Light Alloy Co., Ltd., China) and poplar multilayer board with adhesive urea formaldehyde resin (Dong Yang Plywood Company, China) were used to manufacture the test specimens. The impact area of the high-speed impact sample is 200×200 mm². The thicknesses of both the front and rear panels were 1 mm for the aluminum alloy, and the thicknesses of the middle sandwiches were 5 mm, 10 mm, and 15 mm for the wood multilayer core layers. In addition, the laminate uses a symmetrical laminate [0°/90°/0°] orthogonal laying process. In this paper, three types of aluminum alloy multilayer panel core sandwich structure were prepared, as listed in Table 6. The specimens were named AWA-3, AWA-6, and AWA-9, where A represents the aluminum alloy surface layer, W represents the wood multilayer panel core layer, and the number represents the number of wood layers in each case. The distributions of the layers in the sample core are shown in Figure 1, and the sample sizes and the performance parameters are shown in Table 1 and Table 6.

Table 6.

Basic layouts of the target composite materials (wood and composite plies).

Nomenclature	Layer configuration	Hybrid configuration
AWA-3, three-layer type	[0/90/0]°	Symmetric layup, sandwich hybrid
AWA-6, six-layer type	[0/90/0/90/0/90]°	Symmetric layup, sandwich hybrid
AWA-9, nine-layer type	[0/90/0/90/0/90/0/90/0]°	Symmetric layup, sandwich hybrid

The sandwich structures were prepared by the forming method. First, the urea formaldehyde resin is sprayed between the aluminum alloy layer and the multilayer plate layer; the aluminum alloy and the wood core layer are then bonded, and the bonding force is 5 MPa. The sandwich structure impact sample has dimensions of 200×200 mm², and the surface is drilled and polished for ease of installation. The sandwich combination shows excellent anti-impact and anti-damage properties, and also offers the advantages of light weight and high stiffness. The 2A12T4 layer is fixed to the core material to act as a panel cover. The adhesion strength between the layers is 5 MPa (non-adhesion or structural damage).

A bullet with diameter D, mass M, and initial velocity V strikes the center of the target plate. It is assumed here that the bullet is rigid when compared with the sandwich structure, and that the circular arc S/D=3 (where S is the bullet head arc radius) at the tip of the bullet has a 3CRH-shaped oval shape. The installation of the impact sample is shown schematically in Figure 17.

Figure 17.

Schematic diagram of the specimen size and installation.

High-speed impact test facility

Prior to testing, the sample core layer was stored at room temperature with a controlled water content of 8%. A light gas gun was used to conduct a high-speed impact test (test A) on a mixed sandwich board. As shown in Figure 18, the test machine mainly comprises a high-pressure launcher, an impact protection device, and a recording and detection device.

Figure 18.

High-speed impact device schematic.

The bullet velocity measurement method uses a high-speed camera and a calibration distance to analyze the frame-by-frame rate and the individual pixels to calculate the bullet’s velocity. The bullet was fabricated by computer numerical control (CNC) machine programming, carbon steel turning, and heat treatment with a Rockwell hardness of 58 HRC. When the bullet was launched through the barrel, the sandwich plate was fixed at the middle of the experimental chamber by the impact; the two sides were blocked by bulletproof glass to ensure both safety and transparency, and thus the camera collected the data. The barrel diameter was 12.7 mm and its length was 1.8 m. The control valve was controlled manually using a spherical valve, the camera modes use 0.5 s front and rear delays, and the high-speed camera resolution was 1024×768 pixels at 30,000 fps.

Different impact velocities lead to different failure mechanisms. At low velocities, local invasion phenomena and the overall structural response of the sandwich plate are the main mechanisms. At medium velocities, both local invasion and the overall structural response of the sandwich plate may occur simultaneously to cause failure. At high velocities, the structural response of the sandwich plate can largely be ignored. Because the hardness and the stiffness of the bullet material are much greater than those of the sandwich structure materials, the impact models of the 1-mm-thick aluminum alloy plate and the 5-mm-thick sandwich used in the experiments are simplified to include the process of the rigid bullet striking the thin-plate target. In most cases, only a single failure mechanism, e.g., dish diffusion, ductile reaming, adiabatic shear, punching, bending, or stretching, is generally considered when most thin plates are penetrated.

The attitude of the flying bullet in the experiment is captured using a high-speed camera. Based on the calibrated size and the fixed high-speed camera position with a locking resolution of 30,000 fps, photographs are taken according to the bullet position ΔL and the number of photographs n. The bullet flies for a certain period and images are acquired continuously according to the time–frame–distance conversion. The value of this conversion can be recorded as follows:

$$V=\frac{\mathrm{\Delta }L\times 30000}{n}$$

(41)

The bullets were designed with both flat and oval heads, and have a caliber of 12.7 mm and a mass of 38 g; the size of the bullet is illustrated in Figure 19. According to the installation requirements, the cutting layer is laid in the same manner used for sample production. The core layer plate and the aluminum alloy panel are bonded and fixed to allow them to stand for the adhesion test. In Figure 19 below, the label 1 represents the thickness of the aluminum alloy, and the label 5/10/15 represents the thickness of the new laminate; for example, 1-5 represents a double-layer structure, and 1-5-1 represents a three-layer structure composed of front and rear aluminum core laminates of 5 mm plywood.

Figure 19.

Bullet and sample production images.

Ballistic performance characterization parameters

The ballistic penetration properties of the aluminum–wood laminates under test can be measured by characterizing their ballistic properties, which provide an objective basis for evaluation of the protective properties of these materials.

• (1) Total energy absorption ( $E_a$ ):

$$E_a=\frac{1}{2}{m}_p(V_i^2-V_r^2)$$

(42)

• (2) Energy absorption index of the target plate per unit density (i.e., the ballistic performance indicator (BPI)):

$$\mathrm{BPI}=\frac{E_a}{\rho }$$

(43)

• (3) Energy absorption per unit thickness of the target plate ( $E_h$ ):

$$E_h=\frac{E_a}{h}$$

(44)

• (4) Penetration velocity loss ( $V_0$ ):

For any incident velocity, without consideration of the energy change that occurs during the process of the bullet penetrating the target plate, the penetration velocity loss can be expressed as follows:

$$V_0=\sqrt{V_i^2-V_r^2}$$

(45)

• (5) Critical penetration velocity ( $V_{\mathrm{bl}}$ ):

$$V_{\mathrm{bl}}=\sqrt{\frac{2E_a}{m_p}}$$

(46)

The critical penetration velocity $V_{\mathrm{bl}}$ is the velocity at which the probability that the bullet penetrates the target is 50%. When the bullet penetrates the target plate and $V_r=0$ , $V$ is then called the critical penetration velocity.

Here, $m_p$ is the bullet mass (kg); $V_i$ is the incident bullet velocity; $V_r$ is the residual bullet velocity; ρ is the target plate density (kg/m²); and $h$ is the target plate thickness.

Results and analysis of the high-speed impact experiment

A series of high-speed impact tests were conducted to study the dynamic response and the penetration resistance of the composite sandwich plates. The results of all these tests are summarized in Tables 4 and 5. In this study, the transmission speed was calculated and verified using a high-speed camera to be between 50 m/s and 150 m/s. The effects of two important parameters of the sandwich structure (the core type and the core thickness h_c) on the impact resistance of the panel are considered here. ν_i and ν_r represent the initial impact velocity and the residual velocity of the projectile, respectively. E_i and E_a are used to describe the initial kinetic energy of the projectile and the impact energy absorbed by the sandwich plate during plate penetration, respectively. The sample numbers are as follows: A3: three layers of wood, flat head bullet impact; A6: six-layer planks, flat head bullet impact; A9, nine-layer wood, flat head bullet impact; B3: three layers of wood, R3 bullet impact; B6: six layers of wood, R3 bullet impact; and B9: nine boards, R3 bullet impact. The experimental data are presented in Tables 7 and 8.

Table 7.

Flat-head bullet impact results on the three-layer panel.

A3. Three layers			A6. Six layers			A9. Nine layers
Initial velocity $V_i$ ( $\mathrm{m}/\mathrm{s}$ )	Residual velocity $V_r$ ( $\mathrm{m}/\mathrm{s}$ )	Damage state	Initial velocity $V_i$ ( $\mathrm{m}/\mathrm{s}$ )	Residual velocity $V_r$ ( $\mathrm{m}/\mathrm{s}$ )	Damage state	Initial velocity $V_i$ ( $\mathrm{m}/\mathrm{s}$ )	Residual velocity $V_r$ ( $\mathrm{m}/\mathrm{s}$ )	Damage state
57.26	0	Ricochet	86	0	Ricochet	106.57	4.93	Penetrated
92.13	30.08	Ricochet	133.85	83.4	Penetrated	107.83	19.92	Penetrated
101.98	56.09	Ricochet	125.1	54.6	Penetrated	109.40	28.36	Penetrated
108.46	65.61	Embedded	85	0	Ricochet	115.58	32.26	Ricochet
86.69	0	Penetrated	82	0	Ricochet	119.45	37.47	Embedded
90.77	26.42	Penetrated	108.3	31.59	Penetrated	71.60	0	Ricochet
93.6	41.81	Embedded	92.1	0	Ricochet	107.35	0	Penetrated
91.08	35.52	Penetrated	97.28	0	Embedded	128.14	57.62	Penetrated
122.11	74.58	Ricochet	104	22.33	Penetrated	128.03	45.71	Penetrated

Table 8.

Impact on three-layer panel of R3 bullet.

B3. Three layers			B6. Six layers			B9. Nine layers
Initial velocity $V_i$ ( $\mathrm{m}/\mathrm{s}$ )	Residual velocity $V_r$ ( $\mathrm{m}/\mathrm{s}$ )	Damage state	Initial velocity $V_i$ ( $\mathrm{m}/\mathrm{s}$ )	Residual velocity $V_r$ ( $\mathrm{m}/\mathrm{s}$ )	Damage state	Initial velocity $V_i$ ( $\mathrm{m}/\mathrm{s}$ )	Residual velocity $V_r$ ( $\mathrm{m}/\mathrm{s}$ )	Damage state
67.27	0	Penetrated	75.9	0	Embedded	78.62	0	Penetrated
70.19	23.23	Ricochet	79.13	15.36	Penetrated	91.72	16.81	Penetrated
71.66	21.73	Penetrated	82.6	31.58	Penetrated	93.01	29.26	Penetrated
73.47	33.05	Embedded	91.48	43.98	Penetrated	88.46	0	Ricochet
76.58	42.88	Penetrated	95.8	57.14	Penetrated	101.39	40.98	Embedded
79.68	49.3	Penetrated	109.87	59.7	Penetrated	104.49	49.37	Ricochet
64.89	0	Penetrated	111.49	73.32	Penetrated	101.94	34.53	Penetrated
90.99	57.23	Penetrated	114.77	84.82	Penetrated	108.69	53.07	Penetrated
100.84	78.01	Penetrated	63.1	0	Ricochet	97.01	22.87	Penetrated

For aluminum alloy ductile metal penetration, the kinetic energy that is consumed is mainly used to balance the internal work dissipation. When cylindrical cavities are used for the approximate ductile reamings, the sandwich plate is idealized as a structure composed of multiple thin layers that are independent of each other and lie perpendicular to the direction of penetration. The surface is described by the equation proposed by Lambert and Jonas for ${\mathrm{V}}_{\mathrm{bl}}$ : ⁴²

$$V_r=\beta (V_i^{\psi }-V_{\mathrm{bl}}^{\psi }{)}^{1/\psi }$$

(47)

where $\beta$ and $\mathrm{\Psi }$ are the coefficients of the sandwich plate. The values of $V_i$ and $V_r$ were obtained experimentally for the different sandwich plate types, and β and ψ were also obtained through experiments, with results as shown in Table 9.

Table 9.

Sandwich plate coefficient $\beta \mathrm{and}\mathrm{\Psi }$ values with critical penetration velocities.

Parameters	A3	A6	A9	B3	B6	B9
$\beta$	1.02	0.81	0.92	1.06	0.92	0.81
$\mathrm{\Psi }$	2.42	2.01	2.01	2.02	2.02	2.02
$V_{\mathrm{bl}}$	86 m/s	100 m/s	106 m/s	67 m/s	76 m/s	87 m/s

Using the experimental data in combination with equation (47), the ballistic limit diagrams were then fitted, as shown in Figure 20.

Figure 20.

Ballistic limit diagrams: (a) flat-head bullet and (b) R3 bullet impacts.

Six tests were performed on three samples using flat-head and R3 bullets, and the impact velocity and exit residual velocity values were obtained from the frame count and the images. The impact images are shown in Figure 21.

Figure 21.

Diagrams of the transient penetration processes.

Transient analysis and FE model verification of high-speed impact

The dynamic and static mechanical properties of the materials obtained from the experimental measurement were input into the Abaqus software to perform the calculations. In these numerical calculations, the bullet incident velocity ( ${\mathit{V}}_{\mathit{i}}$ ) was calculated using a velocity similar to that in the experimental curve. The calculated ${\mathit{V}}_{\mathit{i}}-{\mathit{V}}_{\mathit{r}}$ values were then compared with the experimental values, as shown in Figure 22.

Figure 22.

Comparison between the results of numerical calculations and experiments: (a) flat-head bullet penetration and (b) R3 bullet penetration.

The residual velocities under different impact velocity conditions were obtained via experiments and FE simulations. Under the different impact conditions, the error between the numerical residual velocity and the experimental residual velocity was maintained at less than 10%. The predictions of the numerical model were thus accurate, and these numerical predictions were strongly correlated with the experimental results.

The results obtained were then used to study and reproduce the mechanical behavior and the damage mode of the core layer under impact conditions. The numerical model can analyze and calculate the impact on the sandwich structure layer by layer and can also obtain the residual velocity of the projectile, the energy absorption rate, and the residual strength of the sandwich structure. The impact stress clouds were obtained for each single-layer plate of the core layer (the images were processed to remove part of the grid for ease of observation), as shown in Figures 23 and 24.

Figure 23.

Cloud charts of impact stress acting on each single-layer plate of the test A9 core layer.

Figure 24.

Cloud charts of impact stress acting on each single-layer plate of the test B9 core layer.

The high-speed impact experiment was recorded using a camera throughout the entire process, and the calibration distance was measured using the differences in the number of bullets between frames based on the steps in the shooting process described above. The velocity was measured using a Photron FASTCAM Viewer to enable observation of the bullet penetration process. Equation (41) was then used to convert the pixels, frames, and times using the mapping principle shown in Figure 25.

Figure 25.

Principle of speed measurement procedure.

Next, as an example, A9-3 shows bullet penetration: the recording times and the frame numbers are shown in Table 10.

Table 10.

A9-3 flat-head bullet mapping data.

Sampling frequency	Frame rate	Initial point	End point	Instantaneous velocity	Time (s)
3	30,000	336	363	109.40	1E–4
3	30,000	363	379	81.87	1.49E–4
3	30,000	379	391	67.13	2.65E–4
3	30,000	391	405	53.04	3.41E–4
3	30,000	405	411	47.41	4.64E–4
3	30,000	411	421	39.23	6.10E–4
3	30,000	421	430	36.19	7.09E–4

Figure 26 shows the frame numbers used for the process where the flat-head projectile penetrated the target.

Figure 26.

Penetration process images.

In the velocity history curve, the bullet contacts the target plate at 76.55 ms and starts to slow because of the resistance of the target plate. At this time, the bullet is under the maximum resistance condition, and the entire target plate is in elastic deformation. As the bullet continues to move forward, the target plate then begins to be damaged until it finally breaks. Experimental group A9-3 was selected for analysis here. The air resistance was ignored when the bullet flew out, and the resistance of the bullet with respect to the target plate was zero. The data are shown in Figure 27.

Figure 27.

A9-3 R3 mapping data: (a) instantaneous velocity and (b) instantaneous acceleration experimental values and the simulation comparison chart.

The transient data from experimental group B9-5, where a pointed projectile penetrated the target plate, are shown in Table 11.

Table 11.

B9-5 R3 pointed bullet mapping data.

Sampling frequency	Frame rate	Initial point	End point	Instantaneous velocity	Time (s)
2	30,000	367	383	101.17	0
2	30,000	383	395	90.38	6.51E–5
2	30,000	395	407	79.53	1.37E–4
2	30,000	407	417	72.15	2.19E–4
2	30,000	417	427	66.04	3.06E–4
2	30,000	427	435	59.75	3.75E–4
2	30,000	435	444	54.35	4.73E–4
2	30,000	444	457	50.87	5.40E–4
2	30,000	457	463	46.54	5.96E–4

Figure 28 shows the frame numbers for the time process of the R3 head projectile penetrating the target.

Figure 28.

Penetration process images.

The acceleration curve can be compared with the simulation model results by drawing the speed–time instantaneous curve and then differentiating the velocity curve, as shown in Figure 29.

Figure 29.

B9-5 R3 mapping data: (a) instantaneous velocity and (b) instantaneous acceleration experimental values and the simulation comparison chart.

Effect of projectile shape on impact behavior

In a high-speed impact, the projectile body eventually breaks through the target. Based on the relationship between the incident velocity and the initial velocity, when the incident velocity $v_i$ increases, the residual velocity $v_r$ also increases gradually. According to equation (47), the fitting data for $v_i-v_r$ tend to remain infinitely equal with this gradual increase in velocity. As the velocity increases, the damage caused by projectile body penetration of the target plate changes from the overall response to local plastic penetration. Using the $v_i-v_r$ curve, the interactions between the target plate and the projectile body can be calculated, with results as shown in Figure 30.

Figure 30.

Kinetic energy attenuation of the impact sandwich structures of groups A and B: (a) experimental group A, and (b) experimental group B.

In Figure 30(a), when the bullet shape is the same in each case, the protection performance of the experimental curve is distributed in order according to the target plate thickness. The protection performance of the 10-mm core layer is similar to that offered by the 15-mm core layer. With a 5-mm increase in thickness, the protection performance improves greatly. As the impact speed increases, the protection characteristics of the 10-mm and 15-mm samples remain approximately the same. In addition, the gap between the results for A3 and A9 widened rapidly. The flat head bullet impact is thus sensitive to the thickness and the size of the core material.

Experimental curves B3, B6, and B9 in Figure 30(b) are all R3 bullet head curves, which are generally arranged in sequence and all transition smoothly with increasing speed. The insensitivity to the thickness parameter increases proportionally for all three curves.

By aligning the data, we can obtain the energy decay rates for different structures and shapes. The antipenetration abilities of samples A6 and A9 with the three-layer structure were approximately the same as those in the control group. Among the double-layer structures, the B3, B6, and B9 structures all showed poor resistance to penetration by pointed bullets. The target plate distancing trend was not obvious in this case.

Figure 31 shows A3B3, A6B6, and A9B9 characteristics for the same bullet difference group thicknesses. A3B3 is the energy absorption curve for the flat head bullets, and R3 bullets penetrate the 5-mm-thick core layer. It is understood from these results that a sandwich structure composed of thin core layers is not sensitive to the bullet shape. The A6, A9, and B9 experimental groups were approximately identical, and the thick versions of the target were insensitive to the bullet size. By comparing the energy absorption curves and differentiating them, the resistance to contact of each target plate can be determined.

Figure 31.

Summary of characteristics for the experimental groups.

When slicing samples of different thicknesses, the core materials with thicknesses of 5 mm and 10 mm cannot perfectly represent the mechanical behavior on impact because of size limitations; therefore, two groups of A3A6 were used for the slicing research, with results as shown in Figure 32 below.

Figure 32.

Sections of the core layers damaged by impact.

Conclusions

In this study, the J-C model of the aluminum alloy and the 3D Hashin criterion for the core material were used to model and fabricate impact specimens of aluminum–wood sandwich structures, and their relevant mechanical properties were evaluated. The overall stability of these sandwich structures increased. Wooden laminates are a type of structural material. In engineering calculations, equivalent model theory is used to simplify multiple laminates into a type of transverse isotropic material, and the simplified model is subsequently used to perform stress analysis. The computational efficiency is greatly improved and the results obtained are more accurate.

Three-point bending experiments were performed on both the sandwich structure and the core layer, and tensile experiments were conducted on the core layer to determine the important mechanical parameters of the structure. The forms of damage to these core laminates included interlayer delamination, normal cracking of the laminates, compressive deformation of the core layer, and steady peeling of the adhesive layer. Natural defects affected the mechanical properties of the core structure strongly during the bending experiments. The results of the Hopkinson pressure bar experiments showed that the wood was brittle at high dynamic strain rates, the strength of the core laminate was approximately 20 times greater than that of the wood at high strain rates, and the secondary compression strengthening phenomenon occurred under loading; this was caused by the strong constraint effect of the interlacing fiber lamination binder on the transverse cracking of the fiber.

The numerical damage model was validated dynamically and statically via three-point bending and Hopkinson experiments. The research showed that the energy absorption performance of the assembled target plate was much greater than that of the single material under high-speed impact conditions. The damage pattern sequence caused by impact included the following five damage types: delamination, backside fiber failure, front and rear cracking, interlayer-derived cracking with penetration, and front cracking. For impacts at low velocity, the core layer damage was mostly large-area matrix cracking plus delamination damage, and was accompanied by large-area deformation.

From an impact comparison group experiment performed using multilayer boards and solid wood core layers, it was found that under the impact of R3 bullets of the same thickness, the impact resistance of the multilayer boards was 16% greater than that of the solid wood alone, with a 10% boost being observed under flathead bullet impact.