As stated, the modeling of blast-loaded structures generally involves an interaction between both a solid and a fluid sub-domain. As this is considered a complex and also computationally costly issue to solve, different simplified modeling techniques may be applied depending on the problem. Generally, there are three different approaches; A pure Lagrangian approach, the uncoupled Euler-Lagrange (UEL), and lastly the fully coupled Euler-Lagrange (CEL) approach. For this thesis, only the pure Lagrangian and the fully coupled Euler-Lagrangian methods have been applied.
2.6.1 The Lagrangian approach
The idea behind the pure Lagrangian approach is to run the simulation as a structural FEM analysis, with a predefined loading. The estimation of the load and the response
2.6. Modeling of blast-loaded structures of the structure is being considered completely separate. The load is applied to the structure as a pressure-time curve and a common approach is to represent the blast load through the Friedlander equation. This method is used in Chapter 4 and 5, and also for most of the models described in Chapter 7.
Friedlander curve fit
Based on estimating some key-parameters for the blast wave, the Friedlander method aims to describe the pressure experienced by the structure as a function of time [24].
The most straightforward way of doing this is through a curve fit on actual pressure measurements taken from experiments, which is the applied method in this thesis.
Alternatively, it is common to find the parameters for the Friedlander curve by apply-ing the semi-empirical approach derived by Kapply-ingery and Bulmash[25]. This method is based on extensive experimental work, including different scaled blast scenarios.
The Kingery and Bulmash method accounts for the weight of the explosive and the distance from the source.
P(t) =Pa+Pr 1− t t+
!
exp −bt t+
!
(2.17) Wheret+is the positive time duration,tis the total time,Pr is the reflected pressure, Pa the atmospheric pressure and b is the exponential decay coefficient.
t+
Pr
Pso
P
t
Measured Pressure Friedlander Fit
Figure 2.9: Plot showing measured pressure and the corresponding Friedlander fit, when only considering the positive phase of the pressure-time history.
In Figure 2.9 the pressure measured at a sensor close to the structure has been fitted to Eq. (2.17) for the positive phase of the event (t < t+). The result is an analytic expression for the pressure-time relation experienced by the structure.
Chapter 2.
2.6.2 Euler-Lagrange methods
The main idea of a Euler-Lagrange approach is to separately describe the struc-ture through a Lagrangian description, and the fluid sub-domain through a Eulerian (CFD) description. Now the Eulerian description of the shock propagation serves as a numerical prediction of the blast load experienced by the structure. At this point, there are two separate ways to go, i.e., the uncoupled and the coupled approach. For the uncoupled method the fluid and structure sub-domains are treated separately, and they are not allowed to interact. For the coupled methods, both the structural Lagrangian description and the Eulerian fluid description are put together in the same analysis. Here the two domains are allowed to interact, and the structural part will serve as a deformable boundary condition for the fluid sub-domain.
2.6.3 Loading regimes
In structural dynamics, the duration of a loading is often related to the response time, or natural frequency of the structure. By doing this, the loading may be classified into one out of three possible loading-regimes; an impulsive loading, a dynamic loading or a quasi-static loading. If the applied load has a duration significantly longer than the response time of the structure, meaning that the structure has reached its maximum deflection before the load has dissipated markedly, the load is said to be quasi-static.
In this regime, the response is governed by the stiffness of the structure K and the maximum applied load, Pmax through the static equilibrium equation.
The loading is often categorized as quasi-static as long as the following inequality holds
ωntd>40 (2.18)
Where the ωn is the natural frequency of the structure and td is the duration of the applied loading.
On the other hand, if the duration of the applied load is much shorter than the response time, meaning that the loading has been both applied and removed before the structure has experienced any significant displacements, the load is classified as impulsive. The resulting maximum displacement is now dependent on the impulse from the load, as conservation of momentum is the governing equation. The response is defined as being in the impulsive domain as long as the following inequality holds
ωntd<0.4 (2.19)
The dynamic regime is defined as the region in-between the impulsive and quasi-static regime, i.e. (0.4 < ωntd <40). The displacements within this regime are dependent on the loading history and the behavior is far more complex to describe. Further derivations of the above criterion can be found in [1].
2.6. Modeling of blast-loaded structures
2.6.4 Fluid-structure interaction algorithms in Europlexus
There are several algorithms available when it comes to how to solve the mutual interaction between the solid and the fluid. The method applied in upcoming sim-ulations will be a method often referred to as the embedded approach. This is the only approach that will be described in any detail. Aune[3] thoroughly describes and evaluates several methods in his Ph.D. thesis.
The main idea behind the embedded approach is to implement the interaction in a way that makes the fluid and the solid mesh independent of each other. This has significant advantages when it comes to modeling large deformation and failure in the plates, as many of the other methods encounter problems in these highly nonlinear cases. ALE methods where the fluid mesh is fitted to the structure imply that the fluid mesh need to deform with the structure, making the solution prone to distorted fluid elements and possibly introducing the need for re-meshing algorithms.
(a) (b)
Figure 2.10: Illustrating mesh dependency in FSI techniques. Both images are taken from [3](a) Illustration of FSI techniques were the fluid mesh is fitted to the solid mesh is using ALE. (b) Illustration of the embedded approach. For this technique the two meshes are completely independent.
When using the embedded approach, the Lagrangian solid mesh is immersed in the Eulerian fluid mesh, but the two meshes are independent of each other. Because the meshes are independent, the embedded approach needs a tracking algorithm that searches for solid and fluid elements that are supposed to interact. This is done by the definition of an influence domain and is applied as a spherical influence radius around all surface nodes on the discretized solid structure. These spheres are further joined to cover the entire solid structure. It should be noted that choosing the influence domain can be challenging. A too small influence domain may introduce spurious flux across the solid structure, whereas a too big influence domain will link too much of the fluid to the motion of the structure.
Chapter 2.
(a) (b) (c)
Figure 2.11: Illustrating a FSI algorithm in Europlexus[26](a) Illustration of the influence domain around the solid structure. (b) Calculating the pressure drop force that is imposed on the structure. (c) Detailed image of the coupling
Instead of imposing certain conditions on the particle velocities, the pressure force is calculated in the fluid mesh and transmitted to the structure. This is illustrated in Figure 2.11(b). With reference to this figure, the two volumes V1 and V2, with pressures p1 and p2 are separated by the coupled face f. The pressure drop force is then calculated as
f∆p = (p1 −p2)Lnf (2.20)
Where L is the length/area of the face and nf is the face normal. The force is distributed from what is marked as point S∗ in Figure 2.11(c) to the neighboring nodes at the structure. To prevent leakage, the flux of mass and energy is set to zero across the fluid elements that are separated by the structural element.
It should be noted that although this FSI formulation is robust when handling large deformations and potential failure, it can run into trouble if the fluid mesh is coarse relative to the shape of the deformed plate.