2. Literature Review
2.2. Armour and Protective Systems
2.2.5. Blast Testing and Modelling
Experimental tests are known to be more expensive and time consuming than their numerical and analytical counterparts, and this is no truer than for blast testing. Detailed analysis can be performed using numerical models which would otherwise be limited due to the need of expensive equipment, short time frames, and dangerous and difficult to manage environments. Numerical modelling of blast within finite element software such as LS-DYNA can instead be performed in several ways. Empirical methods are the fastest to run but are incapable of simulating more complex scenarios (e.g. reflection and shadowing) and are limited to their range of interpolated values. Fluid-structure interaction can be modelled using methods such as the Arbitrary Lagrangian-Eulerian (ALE) method, which allows for a more comprehensive analysis of a problem.
Experiments
Physical blast tests are conducted in a variety of ways, including the outputs to be measured (e.g.
impulse [112]). This includes use of an actual explosive charge which provides the realistic loading case, but is the most difficult to repeat due to the large innate variations in explosions. On the other hand, shock tubes use high pressure gases to load structure in a controlled and repeatable way with a planar shock wave [113]. Examples for blast tests are:
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• High explosive charges, e.g. TNT, suspended charges [78], ballistic pendulums [55].
• Shock tubes [114].
Soil blast experiments require the use of an explosive charge, but the method in which the soil is accounted for changes. Examples include:
• Buried charges [18].
• Covered charges [11, 35].
Buried charges are representative of field conditions but require extensive preparation and display potential large variation in soil properties across the pit. Even in highly controlled cases, large impulse variations up to 16.6% have been observed in soil blast tests [115]. Flying plate tests are often conducted using this method.
Empirical
The empirical methods within LS-DYNA includes the ConWep method to model air blast [116]
(keyword Load_Blast_Enhanced) and Initial_Impulse_Mine (IIM) method to simulate soil blast.
ConWep applies an evolving pressure to a structure but does not consider the surrounding environment it is found within. This is based on the empirical models of Kingery and Bulmash [5, 6].It has data for hemispherical surface blast, spherical air blast, air blast with ground reflection, etc., but cannot account for buried charges. Both methods have been used in the investigation of soil blast loading on vehicles, where it was found they substantially underestimated the damage imparted to the vehicle [117].
ConWep has been described as adequate for modelling land mines and vehicle response [118].
However the fundamental issue is that it is applying a blast pressure reminiscent of air blast and not soil blast, and scaling factors (e.g. 2.3 times the explosive mass [19]) have been necessary to recreate experimental results. Furthermore, ConWep is unable to account for the confinement effect of the soil, and the approximate setup of a surface laid charge cannot recreate the focusing mechanisms of the soil [119]. Additionally, while scaling can be used to give better agreement, there is no guarantee the loading distribution will be correct.
Alternatively, the IIM method is used to simulate landmine blast based on the experiments conducted by Westine et al. [120] and further developed by Tremblay [121]. These equations take into account the soil density, burial depth, charge dimensions, and spatial configuration of the
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loaded structure. An instantaneous velocity is applied to the nodes of the loaded structure based on the conservation of momentum (i.e. the mass of the structure). However, the instantaneous velocity method has been shown to have shortcomings when used for air blast loading, including overestimating the energy absorption [53]. The IIM method is only valid for a certain range of parameters [121]. Furthermore, the predictions of the IIM method have been found to be inaccurate and also require scaling factors [122]. The predictions for a 0.5 kg TNT charge significantly overestimated the experimental findings of the author [123].
Fluid Dynamics – ALE
The Arbitrary Lagrangian-Eulerian method has been successfully used to model both free air blast [124, 125] as well as soil blast [19, 126-129]. It is a fluid structure interaction method which models the explosive and other relevant materials (e.g. air, soil) as Eulerian elements. This decouples the material from the mesh and allows it to expand and handle large deformations which would otherwise cause errors. The Lagrangian element tie the material to the mesh and is widely used in structural mechanics for its ability to handle interfaces between materials [130]. Figure 2.8 depicts the mesh and particle motion for the three methods, Lagrangian, Eulerian, and ALE. The ALE equations for mass, momentum, and energy are as follows [130]:
𝑀𝑎𝑠𝑠: 𝜕𝜌 where 𝜌 is the density, 𝒗 the material velocity vector, 𝝈 the Cauchy stress tensor, 𝒃 the specific body force vector, E the specific total energy, and 𝝌 denotes the reference coordinates.
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Figure 2.8: Lagrangian, Eulerian and ALE mesh and particle motion in one dimension [130].
Reflection and shadowing can be accounted for, as well as the evolution of loading with time. In order for the blast to travel towards a structure it must travel through a material domain (e.g. air).
This can result in large domain sizes to capture the complete flow of the blast towards and around a structure. The pros and cons of the Lagrangian, Eulerian, and ALE methods are discussed in Table 2.2 [131].
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Table 2.2: Pros and cons of methods for modelling high explosives.
Pros Cons
Lagrangian
Efficient, fewer computations per time step Element distortion and small timestep Simple to use code Susceptible to grid tangling and inaccuracies Good strength modelling
Eulerian
Static grid with no distortions Longer computation times
Capable of handling large deformations Fine zoning required for accuracy
Material mixing within cells Diffusion of material boundaries (leakage) Void cells required for areas where material may flow, thus requiring very large meshes ALE
Wide range of applications Longer computation times
Clearly defined boundaries Fine zoning required for accuracy
Material mixing within cells Diffusion of material boundaries (leakage) Void cells required for areas where material may flow, thus requiring very large meshes
The ALE method combines the best of the Lagrangian and Eulerian solvers, providing an easily deformable mesh with strength modelling for structures subjected to blast loading. The Eulerian elements allow multiple fluids to exist in one element, e.g. explosive, air, and soil, however it is recommended to not have any more than three materials in one element at any time [89]. LS-DYNA works by calculating the history variables based on a Lagrangian mesh before mapping the new solution onto the old configuration, hence retaining the Eulerian mesh [132]. This is known as the advection step and has different formulations available (1st order accurate Donor Cell
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scheme [133] or 2nd order accurate van Leer scheme [134]). The stresses of the element are determined by the volume fraction of each material such that the composite stress is [132]:
𝜎∗ = ∑ 𝜂𝑘𝜎𝑘
where 𝜂 is the volume fraction and 𝜎 is the individual material stresses.
The use of the ALE method requires specific material models and equations of states to model the explosives, such as the High_Explosive_Burn material in LS-DYNA [89]. The required inputs are the density of the explosive, the detonation velocity and the Chapman-Jouguet pressure. The detonation velocity will determine the detonation time for each element based on the initial time of explosive [89] and determines the type of explosion that will occur. Detonation occurs for materials which ignites at rates above the material’s speed of sound, resulting in an instantaneous detonation accompanied by high pressures and temperatures at the shock wave front, while deflagration is the other type of explosion and commonly seen in vapour cloud explosions [10].
The Chapman-Jouguet pressure defines the pressure required for steady state detonation [135].
The Jones-Wilkins-Lee (JWL) equation of state for adiabatic expansion of high explosive [136] is used to define the pressure, 𝑝, of the detonation products based on:
𝑝 = 𝐴 (1 − 𝜔
𝑅1𝑉) 𝑒−𝑅1𝑉+ 𝐵 (1 − 𝜔
𝑅2𝑉) 𝑒−𝑅2𝑉 +𝜔𝐸
𝑉 , (𝐸𝑞 2.24) where 𝑉 is the relative volume and 𝐸 the internal energy per initial volume, and 𝐴, 𝐵, 𝑅1, 𝑅2, and 𝜔 are material constants [89]. Finally, it is important to consider the mesh size in ALE formations.
For example, the pressures from a blast have been reported to increase by 50% through an element height reduction of the air mesh from 6.275 mm to 2.23 mm [137].
SPH
Finally, the smoothed particle hydrodynamics (SPH) method has been used to model soil blast loading [128, 129, 138] and similar scenarios [139] modelling both the soil and explosive using SPH, and the loaded structure with Lagrangian elements. Research comparing the ALE and SPH methods found that SPH method simulated soil blast loading slightly better than the ALE method (5% vs 9.7% error respectively), but that ALE was better for air blast loading [128]. Meanwhile,
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other research has shown the SPH method overpredicting the impulse from soil blast and ALE underpredicting it, with the SPH results ultimately closer [129].
The origin of the SPH modelling technique dates back to 1977, when Lucy [140] and Gingold and Monaghan [141] developed the method to be able to work with the large deformation problems encountered in astrophysics. SPH has since been expanded to deal with a wide variety of applications, including but not limited to hypervelocity impact [142].
SPH is a mesh free method and has several advantages over traditional finite element modelling techniques, including the ALE method. The modelling of blast requires expansive void domains to account for material propagation [131]. This is not the case for SPH as the blast energy travels with the particles themselves through a free domain, saving computational resources and time [128, 129]. Other advantages over the ConWep method have also been demonstrated. ConWep blasts will interact with several parts simultaneously (e.g. multiple layers of a laminate), while the SPH particles only interact with the affected surface until erosion takes place, being both more accurate and realistic [143]. SPH does suffer from the same drawbacks as ConWep regarding blast flow around structures which cannot be properly recreated.
SPH does not use a conventional grid for the mesh as the nodes also define the integration points.
Instead the particles make up the computational framework, and is governed by the following calculation method [89]:
𝜫𝒉𝒇(𝒙) = ∫ 𝒇(𝒚)𝑾(𝒙 − 𝒚, 𝒉)𝒅𝒚, (𝐸𝑞 2.25)
𝑊(𝐱, h) = 1
ℎ(𝐱)𝑑𝜃(𝐱), (𝐸𝑞 2.26) where 𝜫𝒉𝒇(𝒙) is the particle approximate according to the kernel function 𝑊. Additionally, 𝑑 is the number of space dimensions, ℎ the smoothing length, and 𝜃(𝒙) is a function describing the kernel’s smoothing. The kernel, also called a shape function, is used to approximate a solution between nodal points. While a low-order function (linear) may be sufficient for finite element problems, SPH requires a higher-order function (Gauss, cubic B-spline) in order to properly integrate a solution [144]. Further information about the SPH method (e.g. the need for a uniform particle spacing) can be found in [89].
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