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This paper uses the multimaterial ALE finite element method and the penalty function coupling method to analyze the effects of the three container shapes of cylindrical, truncated cone and drum on the impact response of flexible fluidfilled containers when the volume is constant but the liquid storage amounts are different. For containers of all three shapes, the maximum principal impact stress of the shape increases with an increase in the liquid storage amount. The maximum principal impact stress of the drumshaped container is the smallest, that of the truncated coneshaped container is the largest, and the response time of the maximum principal impact stress of the cylindrical container is the shortest.
Flexible FluidFilled Container, Shape, Impact Response, Ale Method, LiquidSolid Coupling
With the wide use of flexible fluidfilled containers in storage and transportation, water conservancy, aerospace and other fields, the impact response of flexible fluidfilled containers has also drawn the attention of a great number of researchers. The impact of flexible fluidfilled containers is not just a matter of structural mechanics, but rather is a nonlinear, complex liquidsolid coupling problem involving large deformations. Impact dynamic analysis must therefore consider not only the deformation of the flexible container, but also the effects of moving liquid loads on the container.
M.Anghileri [1] et al. used a variety of numberical methods to simulate the process where a liquidfilled container is dropped onto the ground and compared the data with the test results, and in the end achieved satisfactory results. Reed P E [2] analyzed the dropping process of waterfilled plastic containers and built an equivalent massspring model to predict the characteristic pulse time and pressure distribution on the container wall. Cao et al. [3] studied the interactions between the container and the fluid using the penalty function, and the results showed that the dynamic stress and deformation of a flexible liquidfilled container are proportional to the drop height. Li Zheng [46] completed a 3D modeling of the drop impact of a largedeformation in a flexible fluidfilled container. Wang Hui [78] used four different numberical methods to simulated the process where a flexible liquidfilled container is dropped onto the ground and compared the advantages and disadvantages of these various methods. Li Qiang [9] completed the numerical simulation and calculation of the drop impact process of liquid storage PET bottles. Zhang Weiwei [10] simulated and studied the process where flexible fluidfilled containers are dropped into water. Other people also have done some related research [1112]. The above research has mainly focused on theoretical and numerical methods for determining the impacts on flexible fluidfilled containers, while little research has been conducted on the effects of the shape of the fluidfilled container on the impact response. This paper intends to study the effects of the container shape on the impact response. We used the multimaterial ALE (Abitrary LagrangianEulerian) modeling method and the penalty function to carry out a numerical simulation of the impact response process and compared the effects of cylindrical, truncated cone and drum shapes on the impact response of the containers.
In large deformation fluidsolid coupling, the Lagrange method can easily cause distortion in describing the grid cells and makes it impossible to complete the calculation, while the Euler method is likely to cause problems in describing the cells such as increased computation space domain and difficulty in tracking the material contact boundary. ALE description is a kind of motion mode independent of the material and space grid. It introduces a reference domain in addition to the material domain and the space domain and resolves on the reference grid, which avoids the serious deformation of the material under the Lagrange grid and eliminates the complex problems caused by the moving boundary under the Euler grid, providing a good method for solving the liquidsolid coupling problem [1314]. In the ALE method, a third arbitrary reference coordinate, other than Lagrange and Euler coordinates, is introduced, and the material change rate associated with the reference coordinate can be described with the following equation:
$\frac{\partial f({{X}_{i}},t)}{\partial t}=\frac{\partial f({{x}_{i}},t)}{\partial t}+{{w}_{i}}\frac{\partial f({{x}_{i}},t)}{\partial t}$ (1)
where X_{i} is the Lagrange coordinate, x_{i} is the Euler coordinate, and w_{i} is the relative velocity between the moving velocity v_{i} under the Lagrange coordinate and the moving velocity u_{i} under the Euler coordinate; i.e., w_{i}=v_{i}u_{i}. Considering the substitutional relationship between the time derivative of material and the time derivative of the reference geometric configuration, the governing equation of the ALE algorithm governing the Newtonian fluid flow in the fixed domain can be obtained from the following conservation equations:
Mass conservation equation:
$\frac{\partial \rho }{\partial t}=\rho \frac{\partial {{v}_{i}}}{\partial {{x}_{i}}}{{w}_{i}}\frac{\partial \rho }{\partial {{x}_{i}}}$ (2)
Momentum conservation equation:
$\rho \frac{\partial {{v}_{i}}}{\partial t}={{\sigma }_{ij,j}}+\rho {{b}_{i}}\rho {{w}_{i}}\frac{\partial {{v}_{i}}}{\partial {{x}_{j}}}$ (3)
Energy conservation equation:
$\rho \frac{\partial E}{\partial t}={{\sigma }_{ij,j}}{{v}_{i,j}}+\rho {{b}_{i}}{{v}_{i}}\rho {{w}_{i}}\frac{\partial E}{\partial {{x}_{j}}}$ (4)
where σ_{ij} is the Cauchy stress tensor, σ_{ij}=pδ_{ij}+μ(v_{i,j}+v_{j,i}); ρ is the density; p is the pressure; μ is the coefficient of kinetic viscosity; δ_{ij} is the Kronecker function; E is the energy per unit mass; b_{i} is the fluid volume force.
The multimaterial ALE description can contain multiple materials in a cell, and by tracking the boundaries of each material, it exchanges and transports materials between the corresponding cell, but it needs to effectively track the interfaces of each material. In this paper, we use the VOF (Volume of Fluid) method to track the locations of the material interfaces. This method determines the material interfaces using the function of material volume ratio in an ALE cell, rather than by tracking the movement of particles on the material surface. To solve the multimaterial ALE equation, we use the separator splitting technique; that is, dividing the calculation at each step into two stages. First, we perform the Lagrange process, where the grid moves with the material. In this process, the equilibrium equations for the calculation speed and the internal energy variation caused by internal and external forces are as follows:
$\rho \frac{\partial {{v}_{\text{i}}}}{\partial t}={{\sigma }_{ij,j}}+\rho {{b}_{i}}$ (5)
$\rho \frac{\partial E}{\partial t}={{\sigma }_{ij}}{{v}_{i,j}}+\rho {{b}_{i}}{{v}_{i}}$ (6)
In the Lagrange process of calculation, as there is no material flowing across the cell boundary, the mass automatically remains conserved. The second stage of calculation, referred to as the convective term, calculates the mass transport, internal energy, and momentum across the cell boundary, which can be regarded as remapping the displacement grid of the Lagrange process back to its original position or any other position.
Hourglass viscosity is used to control the zero energy mode of the grid. Impact viscosity with linear and quadratic terms is used to obtain the shock wave. To this end, the pressure term is added into the energy equation (the second one listed among the equilibrium equations). The central difference method is used to solve the equation based on time increment using the timeexplicit method. The secondorder Van Leer method is used to calculate the convective term. In this paper, both the simple average and equipotential average methods are used to calculate the smoothness of the grid.
3.1 Numerical simulation
Based on the multimaterial ALE finite element method, we use the explicit finite element software LSDYNA to carry out the numerical simulation. We built a finite element model according to the actual size of a cylindrical flexible fluidfilled container and the experimental process. The model consists of a cylindrical flexible container, the water and air in the container and a concrete floor. The flexible fluidfilled container is modeled using the LagrangianBelytschkoTsay (BT) membrane element, which is a superelastic material model. The water and air in the container are described with the hexahedral element and the null material model based on the multimaterial ALE description. The Gruneisen equation is used as the state equation of water. The pressure is defined as below:
$P=\frac{{{\rho }_{0}}{{C}^{2}}\mu \left[ 1+\left( \text{1}\frac{{{\gamma }_{\text{0}}}}{\text{2}} \right)\mu \text{}\frac{\alpha }{\text{2}}{{\mu }^{\text{2}}} \right]}{{{\left[ \text{1}\left( {{S}_{1}}1 \right)\mu {{S}_{2}}\frac{{{\mu }^{2}}}{\mu +1}{{S}_{3}}\frac{{{\mu }^{3}}}{{{\left( \mu +1 \right)}^{2}}} \right]}^{2}}}+\left( {{\gamma }_{0}}+\alpha \mu \right)E$ (7)
where C is the intercept of the u_{s}u_{p} (shock wave velocity particle velocity) curve; S_{1}, S_{2} and S_{3} are the slope coefficients of the u_{s}u_{p} curve; γ_{0} is the Gruneisen parameter; α is the firstorder volume correction value; E is the intrinsic energy of material; volume rate of change $\mu \text{=}\frac{\rho }{{{\rho }_{\text{0}}}}\text{1}$, where ρ is the current density and ρ_{0} is the initial density. The parameters for the material equation of state (EOS) are shown in Table 1. For the air, the linear polynomial state equation is used. The pressure is defined as a function of internal energy of per unit volume, E. The equation is defined as below:
$P\text{=}{{C}_{0}}+{{C}_{1}}\mu +{{C}_{2}}{{\mu }^{2}}+{{C}_{3}}{{\mu }^{3}}+\left( {{C}_{4}}+{{C}_{5}}\mu +{{C}_{6}}{{\mu }^{2}} \right)E$ (8)
where C_{0}, C_{1}, C_{2}, C_{3}, C_{4}, C_{5} and C_{6} are constants. The parameters for the material EOS are shown in Table 2. The physical parameters of the concrete floor are shown in Table 3. The structural mechanical properties of the composite material of the outer container wall are the key to the numerical simulation. According to the woven form, the composite material can be regarded as the transversely isotropic material. In this paper, the mechanical properties of the material are calculated by the method proposed in Reference [15], as shown in Table 4.
Table 1. Parameters for Water EOS
ρ (g/cm^{3}) 
C(m/s) 
S_{1} 
S_{2} 
S_{3} 
γ_{0} 
α 
0.998 
1650 
1.92 
0.096 
0 
0.35 
0 
Table 2. Parameters for Air EOS (gcmμs)
ρ (g/cm^{3}) 
C_{0} 
C_{1} 
C_{2} 
C_{3} 
C_{4} 
C_{5} 
C_{6} 
E_{0} 
1.29e3 
0 
0 
0 
0 
0.4 
0.4 
0 
2.5e06 
Table 3. Performance parameters of concrete material
Density g/cm^{3} 
Elasticity modulus GPa 
Poisson's ratio 
ρ 
E 
ν 
2.665 
40 
0.3 
Table 4. Performance Parameters of Flexible Container Shell Material
Elasticity modulus GPa 
E_{1} 
E_{2} 
E_{3} 
1.535 
1.535 
1.070 

Shear modulus GPa 
G_{12} 
G_{23} 
G_{13} 
0.389 
0.267 
0.267 

Poisson's ratio 
ν_{12} 
ν_{23} 
ν_{13} 
0.163 
0.379 
0.379 

Yield stress GPa 
σ_{y} 
Density g/cm^{3} 
ρ 
0.2 
1.1 
A finite element model was built to simulate the process where a flexible fluidfilled container, filled to 50% of liquid storage capacity, falls from a height of 100m and hits the cement floor. This model has a total of 160178 cells and 165344 nodes, and is shown in Figure 1.
Figure 1. Finite element model of the cylindrical fluidfilled container
In case there is air resistance and in order to shorten the calculation time and save resources, the height of 100m is converted to a speed of 40m/s. The ground is treated as a rigid body. Ground base grid cells are fully constrained. The gravitational acceleration is 9.81m/s. The coupling between the container and the liquid and the collision between the container and the ground are realized by the penalty function. The numerical simulation takes into account the effect of the air inside the container. As we mainly analyze the maximum principal stress of the container, we regard the cover as a part of the shell in modeling. To ensure stability of calculation, the time step is subject to the minimum cell size of the model, the number of liquidsolid coupling points is set to 4, and the minimum volume parameter of coupling is set to 0.1.
The numerical simulation results are shown in Figure 2 and 3. Figure 2 is the nephogram for the principal stress of the cylindrical container at various moments. Figure 3 shows the form of the liquid in the container at various moments.
Figure 2. Nephogram for the principal stress of the cylindrical container at different moment
Figure 3. Form of the liquid in the container at different moments
At the moment when the container hits the ground, the container stops moving, but the liquid continues to move downward due to the effect of inertia. The nature of the fluid causes the liquid to move to the surrounding areas at the container bottom and creates impact stress on the container wall. Under the stress, the container is deformed and a portion of the potential energy is converted into deformation energy. The deformed container also has a reaction to the fluid, which forces a change in the fluid velocity and acceleration. This is a liquidsolid coupling process.
It can be seen from Figure 2 and Figure 3 that at the moment of 1.3ms, the stress reaches the maximum value in the whole impact process, and the bottom of the container wall experiences the maximum deformation. Meanwhile, the deformation energy of the container also reaches the maximum value. Then, under its own elastic effect, the container is gradually restored to its original state. At this moment, the fluid moves inward driven by the deformation force of the container, while in the vertical direction, under the rebound effect, the fluid begins to move upward, resulting in a cavity at the bottom and further reducing the pressure. Later, the cavity is gradually closed, and the pressure rises again. As time passes, the fluid continues to move upward and squeezes towards both sides, and with the liquid flowing upward, the upper air is compressed. The water inside the container splashes up and begins to surround the air. With the upper air compressed, its volume is reduced, and the cavity at the bottom continues to expand due to the upward movement of the fluid, and the pressure is reduced.
Figure 4 shows the principal stress timehistory curves of the three cells selected from the upper, middle, and lower parts in the outer wall of the container. It can be seen that the principal stress of each of the three parts reaches the maximum value in sequence of the lower, upper and middle parts. The lower part has the highest stress value, followed by the upper part and the middle part. This is also consistent with the results of container rebound deformation in the experiment. It can be seen that the maximum principal stress of the lower part of the container occurs before that of the upper part, with an amplitude greater than that of the latter. The maximum principal stress of each of the two cells reaches the peak and then rapidly falls and oscillates at a smaller amplitude, and gradually reaches a steady state.
Figure 4. Principal stress timehistory curves of the three cells in the upper, middle and lower parts
3.2 Effectiveness verification of numerical simulation
In this paper, we verify the effectiveness of the numerical simulation mainly through experiments. A cylindrical flexible fluidfilled container mainly consists of a container body, a cover and ancillary devices, as shown in Figure 5. It has a height of 430cm and a diameter of 360cm. The body is reinforced with fabric, and the base is made of natural rubber with a thickness of 1.5cm. When the liquid storage amount is 50%, 80% and 100% of capacity respectively, we drop the container from a height of 100m and with a highspeed camera film the dynamic response process of the container after being dropped. Then we process and analyze the data and obtain the relationship between the drop impact velocity and the deformation of the container.
Figure 5. Cylindrical flexible fluidfilled container
We verify the effectiveness of the numerical simulation results mainly by analyzing whether the relationships between the deformation of the container and the time are consistent in the impact processes respectively in the experiment and numerical simulation. If the numerical simulation is the same as or similar to the experiment result, the numerical simulation results are considered acceptable. Specifically speaking, we select several points on the outer wall of the container, and then analyze whether the displacement variations of the points with time are consistent with those in the simulation to determine the correctness of the simulation. As shown in Figure 6, Δd is the variation in the distance between a point on the outer wall of the container and the center of the container; d is the distance between the point and the center line of the container (cylindrical tank) at different moments during the impact process; r is the instantaneous radius of the cylindrical fluidfilled container contacting the ground; t_{0 }is the moment when the container drops and touches the ground. If the change rule of Δd over time is consistent with the numerical simulation, then we consider the simulation results to be accurate. The expression of Δd is as follows:
$\Delta d=\left\{ \begin{matrix} \text{0}(t={{t}_{0}}) \\ dr(t>{{t}_{0}}) \\\end{matrix} \right.$
Figure 6. Verification method for the effectiveness of the numerical simulation
Figure 7 shows the comparison between the variations in the distances from the three points in the upper, middle and lower parts to the centerline and the numerical simulation results. It is found that the numerical simulation results deviate a little from the physical experiment results, with the maximum deviation being 6%. This is probably because the numerical simulation model has simplified the process. But if this simulation is used to analyze the impact responses of flexible containers of different shapes, such deviations are still acceptable. It can accurately reflect the actual drop impact process of the container.
Figure 7. Comparison between test results and numerical simulation results
We built the 3D finite element models for truncated coneshaped and drumshaped flexible fluidfilled containers respectively so as to compare them with the cylindrical container. The three kinds of containers have the same liquid capacity and are made of the same material, and we assume they have no cover or accessory. Calculation results show that the mass of the three differentlyshaped containers with the same volume differs by no more than 9% and that the mass of an empty container accounts for no more than 6% of the total mass of a container containing a liquid storage amount at 50% of capacity. For convenience of comparison, we ignore the response result errors caused by the differences in container mass. The finite element models built, shown in Figure 8, are used to carry out the numerical simulation of the drop impact process of the above three kinds of containers with a liquid storage amount at 50%, 80% and 100% of capacity respectively.
Figure 8. Finite element models for three container shapes
4.1 Comparison of three different container shapes under different liquid storage conditions
Figure 9. Timehistory curves of maximum principal stress elements under different liquid storage conditions
Figure 9 shows the timehistory curves of the maximum principal stress elements in the three containers under different liquid storage conditions. It can be seen that regardless of whether the liquid storage amount is at 50%, 80% or 100% of capacity, the maximum impact stress of the drumshaped container is the smallest, indicating that the drumshaped container has better resistance to impact than the other two container shapes. This is mainly because in the impact process, the container is deformed under the fluid load and in turn it reacts on the fluid. Figure 10(a) shows the reaction force of the drumshaped container against the fluid. In this figure, the reaction force F of the container on the fluid is inwardly vertical to the container wall, which can be decomposed into the vertical force F_{1} and the horizontal force F_{2}. Due to the symmetrical structure of the container, the reaction forces in the horizontal direction cancel each other out, and the component force in the vertical direction can partially offset the vertical impact load from the upper liquid, thereby reducing the impact force on the container. Meanwhile, as the structure of the drumshaped container upwardly opens up, there is more space for the liquid that is rebounded upwards, reducing the pressure and dispersing the rebound impact pressure. In contrast, the structure of the truncated coneshaped container is completely different, as shown in Fig. 10 (b). The fluid impact load and the reaction force of the container on the fluid are superposed, increasing the impact force on the container.
Figure 10. Diagram for the reaction force of the container on the fluid
In terms of the response time of the maximum principal stress, whether the liquid storage amount is at 50%, 80% or 100% of capacity, the cylindrical container responds the fastest, the drumshaped container responds the slowest at the storage amount of 50% and 80% of capacity and the truncated coneshaped container responds the slowest at the storage amount of 100% of capacity. Such differences in the response time may have something to do with the container shape, a question which merits further studies.
4.2 Comparison of containers of the same shape but under different liquid storage amounts
Figure 11. Timehistory curves of the maximum principal stress elements in containers of different shapes
Figure 11 shows the stress timehistory curves of the maximum principal stress elements in containers of different shapes. It can be seen from the figure that the maximum principal stress of each container increases with an increase in the liquid storage amount, and the maximum principal stress element reaches the stable stress value faster when the storage amount is increased. When the liquid storage amount is at 100% of capacity, the stable stress values of the cylindrical and truncated coneshaped containers are smaller than those with a liquid storage amount at 50% and 80% of capacity, while that of the drumshaped container with a liquid storage amount at 100% of capacity is roughly similar to those with a liquid storage amount at 50% and 80% of capacity.
Figure 12. Maximum principal stress and response time by container shape and liquid storage amount
Figure12 shows the maximum principal stress and response time by container shape and liquid storage amount. From Figure 12, we can see that when the liquid storage amount changes from 50% to 80% and 100% of capacity, the maximum principal stress of the containers of different shapes also respond at different speeds. The response rateof the cylindrical container becomes gradually faster, that of the truncated coneshaped container does not change significantly, and that of the drumshaped container first gets higher and then rapidly decreases. The maximum stresses of the containers of different shapes also change at different rates. The maximum principal impact stress of the cylindrical container and the truncated coneshaped container gradually increase with an increase in the liquid storage amount, while that of the drumshaped container does not change much when the liquid storage amount is at 50% and 80% of capacity. Changes in the maximum principal impact stress amplitudes of the cylindrical and the truncated coneshaped containers are not significantly different. When the liquid storage amount changes from 50% to 100% of capacity, changes in the maximum principal impact stress amplitudes of the cylindrical, truncated coneshaped and drumshaped containers are 50.8MPa, 49.5MPa and 43.4MPa respectively. The question of why the changes for the cylindrical and truncated coneshaped containers are quite similar merits further studies.
4.3 Position of the maximum principal stress
Figure 13. Stress nephogram at the maximum principal stress time by liquid storage amount and container shape
Figure 13 reveals that the position of the maximum principal stress in the cylindrical container basically remains the same, and the distribution range of the maximum principal stress in the cylindrical container takes the shape of "Λ" , and is distributed most widely when the liquid storage amount is 80% of capacity.The position of the maximum principal stress in the truncated coneshaped container gradually rises vertically with an increase in the liquid storage amount, and the distribution range does not change much.For the drumshaped container, when the liquid storage amount is at 50% or 80% of capacity, the maximum principal stress is located at the bottom edge of the container, while when the storage amount is at 100% capacity, the maximum stress is in the lower outer part of the container, with the widest distribution.
(1) Under the same liquid storage condition, containers of different shapes have significantly different maximum principal impact stresses and maximum principal stress response times. Under all the three liquid storage amount conditions, the drumshaped container has the smallest maximum principal stress, and the cylindrical container requires the shortest maximum principal stress response time.
(2) The maximum principal stresses of all the three container shapes increase with an increase in the liquid storage amount, but at different speeds.
(3) In the same container shape, under different liquid storage conditions, the position of the maximum principal stress is different.
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