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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 328651, 10 pages
Numerical Simulation of Force Enhancement by Cellular Material under Blast Load
School of Automotive Engineering, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
Received 18 June 2013; Accepted 26 August 2013
Academic Editor: Lei Zhang
Copyright © 2013 Chang Qi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The cause of force enhancement phenomenon when using cellular material, such as metal foam, for blast protection is discussed using both finite element method (FEM) and analytical method. Finite element (FE) models of cellular material under blast load are presented, in which the blast load is modeled by CONWEP blast function and the cellular material is modeled by the homogenized modeling method and the multiunit-cell modeling method. A one-dimensional analytical model is also presented for comparison purpose. Utilizing these models, an aluminum foam bar under blast load is simulated and the mechanism of both force attenuation and enhancement are depicted. The relationship between blast load intensity and the length of the foam bar is analyzed based on the simulation. It is found that the time of momentum transfer between the compacted foam bar and the protected structure is very short compared to the total time of the blast event, which causes force enhancement. Corresponding countermeasures are proposed based on this finding. The study will not only provide new modeling methods for the simulation of cellular material subjected to blast load but will also be beneficial to understand the mechanisms of force attenuation and enhancement, so as to seek for countermeasures.
Cellular materials, such as metal foams, are widely used to attenuate the effects of blast load upon structures due to their high energy absorption capability compared to relatively low density, a characteristic that is very preferable for light-weight applications. The porous nature of the material also helps in heat dissipation as well as provides damping to the shockwave. Besides all these benefits, an undesired phenomenon observed when using cellular material for blast protection is that, under certain conditions, the peak force transmitted to the protected structure can be even higher than when the cellular material is not used. This unexpected phenomenon, the so-called “force enhancement”, has been mentioned in several publications. The initial framework for investigation of force enhancement phenomenon was established by Monti  as early as 1970. Gel’fand et al.  first demonstrated pressure amplification by foam material using the experimental method. The stress or force enhancement phenomenon was observed by Reid et al. [3, 4] in wood and packed ring systems, and by Song et al.  in plastic foams. Skew et al.  demonstrated a substantial increase in the back wall pressure when a slab of porous polyester and polyether foams was mounted to the back wall of a shock tube. Mazor et al.  and Ben-Dor et al.  found that the actual blast pressure acting on the structural surface is a function of the response of the surface itself since this influences the states of the gaseous phase. Hanssen et al.  conducted full-scale free-field blast-loaded pendulum tests; an increase of the swing angle of the blast-loaded pendulum was observed when a foam panel was attached. Hanssen attributed this angle (energy) increase to the continuous transformation of the shape of the initially planar panel surface into a concave shape during the blast. Ouellet et al.  conducted both shock tube experiments and free-field blast trials on three polymeric foams of varying thickness and density and concluded that three different regimes of amplification and attenuation of foam transmitted overpressure can be identified.
Numerical simulation by Olim et al.  based on a two-phase flow model (a dust-gas model) supported the experimental results of Skew et al. . This model treats the solid phase as suspension dusts in the gas phase. This is applicable to foam of low density. Li and Meng  attributed the stress or force enhancement of cellular material to the formation of a shock wave when a critical impact velocity is reached for intensive loads. They showed that stress enhancement may occur during its propagation through a cellular material, which was demonstrated using a one-dimensional mass-spring model. Ma and Ye [13, 14] first considered the coupling effects of the foam claddings and the protected main structure using a one-dimensional analytical model.
Despite all the above mentioned efforts, there is little information available in literature that gives a detailed explanation of the force enhancement phenomenon of cellular material under blast load or high-speed impact load; the physical background of this phenomenon still needs to be discovered. Due to the lack of theoretical support, no practical solutions have been proposed to prevent this undesired phenomenon from happening. As a result, application of cellular material for blast protection design is still limited at the present time.
To find the reason of force enhancement when using cellular material, such as aluminum foam, for blast protection, detailed investigation of the physical process is a premise. In this study, finite element method (FEM) is applied for this purpose. Two types of modeling methods are proposed to model the cellular material based on the explicit program LS-Dyna. The effect of using aluminum foam for blast protection is simulated with various blast load intensities. Simulation results of the finite element models are compared with the data from a one-dimensional analytical model originally proposed in the literature, which is then employed to investigate the root cause of force enhancement.
The structure of the paper is as follows: in Section 1, the current situation of researches for cellular material under blast load is surveyed. In Section 2, the basic problem is first described; a one-dimensional analytical model is then briefly introduced; some basic theories of finite element modeling of cellular material under blast load are described, including blast load simulation models and cellular material models, the latter include a homogenized model and a model that consists of multiunit-cells. In Section 3, numerical simulations of a case study are conducted based on the proposed models to demonstrate the phenomena of both force attenuation and enhancement; contrastive analysis between the simulation results of different models is carried out; the analytical model is employed to seek the root cause of the force enhancement effect of cellular materials subject to blast loads. At last, some conclusions are deduced in Section 4.
2. Numerical Modeling of Cellular Material under Blast Load
2.1. Problem Description
The studied problem is shown in Figure 1; that is, a fixed end cellular material bar is subjected to a blast pressure pulse . The length and the cross-section area of the bar are and , respectively. The transmitted force on the fixed end is monitored to evaluate the effect of the cellular material under blast load.
2.2. One-Dimensional Analytical Model
A one-dimensional analytical model proposed by Li and Meng  is shown in Figure 2, which has discrete lumped masses connected by identical nonlinear springs, where , , and is the density of the cellular material. The elastic property of the spring is determined by . The input blast pressure pulse is applied on the th lumped mass. The first spring is connected to a rigid wall.
Neglecting the change of cross-section area, equilibrium equations of the system can be written as where compressive stress has a positive value. and is the displacement of th mass in the given direction in Figure 2. The initial conditions are
A complete description of the compressive stress-strain relation of the non-linear spring is shown in Figure 3 and is characterized by the compressive modulus, , plateau stress, , lock-up strain, , and compressive stress-strain relation in the densification range. The non-linear differential equation (1) with the initial condition (2) can be solved numerically to achieve the force transmission in the system.
2.3. Finite Element Modeling of Blast Load
For blast impact simulation, the complexity of the problem lies in the following difficulties: the high speed wave front propagation, the flow of various materials, and the large structural deformation. Existent numerical models developed for blast simulations can be roughly divided into two categories: the numerical models based on the Arbitrary Lagrangian Eulerian (ALE) method and the empirical models for blast pressure approximation.
2.3.1. ALE Model
The ALE methods combine the advantages of Eulerian and Lagrangian methods and allow for a type of “automatic remapping” in the simulation. The time and space distribution of the blast pressure profile are calculated through the Eulerian mesh by utilizing the equation of state (EOS) for high explosives. The mix of the air and explosive reaction products is modeled using multimaterial capabilities (*ALE_MULTIMATEIRAL_GROUP_OPTION) in LS-Dyna. The blast pressure wave traveling through the air interacts with the structure by means of a gas-structure interfacing algorithm in LS-DYNA (*CONSTRAINED_LAGRANGE_IN_SOLID). Physical quantities such as stress, displacement, velocities, and accelerations in the structure are computed. At any given time, the pressure in a high explosive element is given by where is the pressure from the EOS; is called burn fraction, which multiplies the EOS for high explosive, and controls the release of chemical energy for simulating detonations. The Jones-Wilkins-Lee (JWL) EOS model for explosive detonation product is given by where is the pressure field, is the volume of the material at pressure divided by the initial volume of the unreacted explosive, is the internal energy per unit initial volume, and , , , , and are adjustable parameters. For TNT, Mbar, Mbar, , , and . The air is usually modeled to represent the medium in which the blast wave propagates. A linear polynomial EOS is usually used to simulate the proper air behavior, and the pressure is given by where with being the ratio of current density to initial density, and being the constants. For gases to which the gamma law EOS applies, including atmospheric air, and , with as the ratio of specific heats. Therefore, for air, (5) reduces to The units of are the units of pressure.
One drawback of the ALE method is its high computational cost, and it is, therefore, appropriate only for simulating blast events with small standoff distances.
2.3.2. Empirical Model
Compared to the ALE method, the empirical models have a much less computational cost, and it is, therefore, appropriate for simulating blast events with large standoff distances. In this study, the blast loads are simulated using one of the empirical models based on the CONWEP air blast function developed by Kingery and Bulmash . This model, which has been implemented as the *LOAD_BLAST loading card in LS-Dyna, can predict the blast overpressure under certain conditions: the free air detonation of a spherical charge and the surface detonation of a hemispherical charge; the surface detonation approximates the conditions of a mine blast. The model takes into consideration the angle of the incidence of the blast, , the incident pressure, , and the reflected pressure, . The predicted blast overpressure is expressed as with and given by where and are the peak incident overpressure and the peak reflected overpressure, respectively. and are decay coefficients and is the positive phase duration time. The model uses the following inputs to calculate the pressure: equivalent mass of TNT, coordinates of the point of explosion, and the delay time between when the LS-Dyna solution starts and the instant of explosion. The model does not account for shadowing by the intervening objects or the effects of confinement.
2.4. Finite Element Modeling of Cellular Material
Finite element model of cellular material can be roughly divided into two groups: homogenized model and models that consist of multi-unit-cell.
2.4.1. Homogenized Model
Homogenized model neglects the microstructure of a specific cellular material, only the homogenized macroscopic properties are taken into consideration. Homogenized model uses solid elements for space discretization while the properties of cellular material are represented through material models by assigning appropriate parameters and load-deformation curves or stress-strain curves. In LS-Dyna, homogenized models for cellular material include MAT26, MAT57, MAT63, MAT75, and MAT126.
2.4.2. Multiunit-Cell Model
Multi-unit-cell model of cellular material accounts for material morphology in which the material is approximated by an assembly of unit-cells. The multi-unit-cell model is capable of reflecting the complex loading conditions of the cellular material. The properties of the cellular material are represented by parameters of the unit-cell. Several representative unit-cell models of cellular material found in the literature [16–18] are shown in Figure 4.
In this study, we employed a unit-cell model of cellular material proposed by Santosa and Wierzbicki  and improved by Wang  based on the morphology analysis of cellular material microstructure as illustrated in Figure 5. The unit-cell of cellular material is represented by two hollow hemispheres connected through a cruciform section and a web section as shown in Figure 6. Generally, both cell size and wall thickness of the unit-cell contribute to idealized foam density. We assumed a uniform thickness at every point. Contacts are defined among the unit-cells to represent the internal connections and friction of the cellular material. Denoting the cube width by , diameter of the sphere by , and thickness by , the relative density or solidity ratio of the reference multi-unit-cell foam model with respect to a solid volume of which the foam is made can be expressed as and the lock-up strain, , can be obtained as where
3. Numerical Simulation Case Study
3.1. Problem Description
An aluminum foam bar is attached to a fixed rigid reaction wall; the other end of the foam bar is attached to an aluminum cover plate, which is subject to blast loads with equivalent TNT of 4 kg (Case 1) and 8 kg (Case 2), respectively, as illustrated in Figure 7. It is assumed that the distance from the location of the blast to the loading surface of the cover plate is large enough so that the length of the foam bar and the cover plate can be neglected.
3.2. Modeling Method
Both analytical method and FEM are used for the simulation. The analytical model comprises 3 nonlinear springs to represent the aluminum foam bar and one nonlinear spring to represent the cover plate.
In finite element modeling, due to the distance between the explosive spot and the foam bar, the empirical model is employed to simulate the blast load on the cover plate. Keyword *LOAD_BLAST in LS-Dyna is set to generate the air burst load of the TNT exerted on the cover plate, which is modeled by constant stress 8-nodes brick element. Two methods are used to model the aluminum foam bar: one is the homogenized modeling method using MAT26 in LS-Dyna (model 1), the other one is the multi-unit-cell modeling method with unit-cell shown in Figure 7 (model 2) with mm, mm, and mm, respectively. Quarter symmetry is used in both models to reduce the number of elements; all nodes on the plane of symmetry are constrained to stay on the planes of symmetry. The number of elements is 55200 with the nodes of 61302 in model 1, while the number of elements is 348495 with the nodes of 277397 in model 2.
*CONTAC_AUTOMATIC_SINGLE_SURFACE card is defined in both models to account for the effect of contact during simulation. Material properties used in the finite element models are listed in Table 1. Strain rate effect of aluminum was accounted for by using the Cowper and Symonds model which scales the yield stress with the factor , where is the strain rate, and strain rate parameters are set as and , respectively.
3.3. Simulation Results and Discussions
3.3.1. Case 1 (4 kg TNT)
The simulated deformation and pressure distribution in the system at different times are shown in Figure 8 (model 1) and Figure 9 (model 2), respectively. Both finite element models predicted that the aluminum bar deforms from near the blast point to the distal end, layer by layer, in a manner similar to what happens in real tests .
The displacement of the front panel is shown in Figure 10. FE model 1 underpredicted the displacement compared to the analytical model, while FE model 2 overpredicted the displacement. The simulation error is less than 5%.
Figure 11 depicts the force attenuation effect of aluminum foam under blast load. Force exerted on the rigid reaction wall is equivalent to the product of the plateau stress and the contact area, that is, the cross section area of the foam bar, which is less than the blast load peak. Energy from the blast is fully absorbed by the aluminum foam through plastic deformation. It is seen from Figure 11 that the predicted force transmitted on the reaction wall by both FE models is in very good agreement with the data given by the one-dimensional analytical model. Model 2 based on the multi-unit-cell modeling method predicted an early occurrence of force transmission. This means the model is stiffer due to the strain rate effect.
Figure 12 compares the time history of total energy, absorbed energy, and kinetic energy of the system. The total energy is equal to the work done on the cover plate by the blast load. It is observed that both FE models give results which are in good agreement with the results predicted by the analytical model. The kinetic energy of the system increases initially due to the blast load, then drops eventually to zero.
3.3.2. Case 2 (8 kg TNT)
In case that the blast load is above a critical level, as in Case 2, the blast energy cannot be completely absorbed when the foam bar is fully compacted, peak force exerted on the reaction wall is greater than the blast peak load itself, and the so-called force enhancement occurs as shown in Figure 13. Both FE models predicted force enhancement occurrence successfully. It shows that model 1 over-predicted the enhanced force peak as compared with the analytical model while model 2 predicted the enhanced peak force in very good agreement with the analytical model. It also shows that the occurrence time of force enhancement predicted by both FE models is earlier than the one-dimensional analytical model.
Based on the developed models, for a fixed-end cellular bar of specific configuration (material and cross-section area), a critical length can be identified according to the intensity of the blast load; if the length of the cellular bar is greater than the critical length, blast force can be attenuated; otherwise, force enhancement will occur. On the other hand, for a fixed-length cellular bar, a critical value of the blast load intensity can be identified as well, above which force enhancement will occur. Figure 14 shows the relationship between the critical length of the foam bar and the critical blast load level. For blast loads under the critical curve, the cellular bar will not be fully compacted, and the blast force is attenuated at the fixed end. But, for blast loads above the critical curve, once the cellular bar is fully compacted, force enhancement is expected. This critical curve can be used to determine when “force enhancement” will happen, so as to assist in the appropriate design process.
To further investigate the physical process causing force enhancement, momentum transfer in the system was investigated using the one-dimensional analytical model for both load cases. Figure 15 compares the momentum histories in Case 1. Because node 1 and node 2 are connected by the spring element representing the aluminum cover plate, which can be considered as rigid compared to aluminum foam under blast load, node 1 and node 2 have an identical momentum history curve as shown in Figure 16. It also shows that the momentum transfer between nodes 1 and node 3 occurred at around 0.3 ms, and no momentum transfer occurred between node 3 and node 4, which means that momentum was fully dissipated before the foam bar fully compacted in Case 1. In contrast, momentum transfer occurs between all nodes and is finally transmitted to the rigid reaction wall in Case 2. It is also seen that these momentum transfers occur within very short periods of time compared to the total time of the event. Once the propagated momentum transferred through the system reaches the fixed end, a high magnitude force is expected, and this will cause the force enhancement at the fixed end. Countermeasures which can increase the time duration for momentum transfer between the compacted foam bar and the protected structure are needed to eliminate the undesired force enhancement. Adding well-designed interim isolation (I-I) structure between the cellular material and the protected structure was proven to be effective in preventing force enhancement .
Two finite element models are presented to simulate the effect of using cellular materials for blast protection. Both force attenuation and force enhancement phenomena are simulated successfully. Good agreement between the two FE models and the one-dimensional analytical model has confirmed the correctness and credibility of the models. Critical curve of a fixed-end aluminum foam bar under blast load was identified based on the simulation results of the proposed models, which can be employed to assist in the appropriate design process of cellular material for blast protection applications. Based on the one-dimensional analytical model, force enhancement of cellular material under blast load is related to the momentum transfer between the cellular material and the protected structure within short periods of time compared to the total time of the event. Countermeasures which can increase this time duration are needed to eliminate the undesired force enhancement.
This work is funded by the National Natural Science Foundation of China (nos. 50905024, 51105053), Liaoning Provincial Natural Science Foundation of China (no. 20102026), the Research Fund for the Doctoral Program of Higher Education of China (nos. 20090041120032, 20110041120022), and the Fundamental Research Funds for the Central Universities (DUT13Lk47).
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