Shock and Vibration

Volume 2016, Article ID 4593749, 13 pages

http://dx.doi.org/10.1155/2016/4593749

## Improved Element Erosion Function for Concrete-Like Materials with the SPH Method

Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic

Received 21 December 2015; Revised 27 February 2016; Accepted 7 March 2016

Academic Editor: Matteo Aureli

Copyright © 2016 Jiří Kala and Martin Hušek. 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.

#### Abstract

The subject of the paper is a description of a simple test from the field of terminal ballistics and the handling of issues arising during its simulation using the numerical techniques of the finite element method. With regard to the possible excessive reshaping of the finite element mesh there is a danger that problems will arise such as the locking of elements or the appearance of negative volumes. It is often necessary to introduce numerical extensions so that the simulations can be carried out at all. When examining local damage to structures, such as the penetration of the outer shell or its perforation, it is almost essential to introduce the numerical erosion of elements into the simulations. However, when using numerical erosion, the dissipation of matter and energy from the computational model occurs in the mathematical background to the calculation. It is a phenomenon which can reveal itself in the final result when a discrepancy appears between the simulations and the experiments. This issue can be solved by transforming the eroded elements into smoothed particle hydrodynamics particles. These newly created particles can then assume the characteristics of the original elements and preserve the matter and energy of the numerical model.

#### 1. Introduction

When a projectile flying at high speed collides with a concrete surface, its kinetic energy drops to zero but the internal energy of the system increases sharply. The response of the concrete surface to this type of load often takes the form of irreversible (plastic) deformations. If the kinetic energy of the projectile is sufficiently great and the body of the projectile is sufficiently stiff, penetration of the concrete surface occurs. This type of failure is also accompanied by the chipping off of the concrete and the development of dynamically propagating cracks. The successful execution of numerical simulations of this phenomenon is very difficult, however, particularly when the finite element method (FEM) is used. In order for the results of FEM simulations to correspond with the results of experiments, it is necessary to combine suitable material models with numerical model failure techniques, but it is also essential to avoid numerical problems which often negatively affect the results of the simulations.

Today, thanks to constantly ongoing research into concrete structures, extensive concrete material model databases are available and often implemented in commercial programs such as LS-DYNA [1]. The options and conditions for the use of a given selected material model are, however, often open to debate [2]. On top of that, before the execution of the simulation an assumption often has to be made regarding the type of failure that will be decisive so it is possible to select a material model at all [3]. The response of the concrete also depends on the character of the load [4–7], which makes the choice even more complicated. Generally, in the case of high-speed loading it is necessary to use a material model which takes strain rate into consideration [8–10]. In such cases, equations of state (EOS) are often used to enable the successful description of the material model due to the fact that bodies behave in a similar manner to fluids when under high-speed loading. As far as the software is concerned, an algorithm has to be accessible which will interpret such information appropriately for the computational process. This is enabled, for example, by the previously mentioned LS-DYNA program [1] and also AUTODYN [11]. The HJC model [12, 13] (named after its authors Holmquist, Johnson, and Cook) can then be a suitable material model, as it has the above-mentioned properties; it considers the strain rate and utilizes EOS for description.

However, simulations also need to include a numerical technique which will enable the simulation of continuum failure; in the case of the FEM method, this takes the form of the failure of the finite element mesh. If this technique was not used, the simulation of, for example, the chipping off of material or the perforation of the loaded structure would not be possible. Additionally, numerical problems (the locking of elements, negative volumes, etc.) could arise due to the influence of excessive deformations (distortion of elements). The numerical erosion of finite elements can be such a technique. Even though numerical erosion is primarily used to filter out problem elements from the calculation, it can also be suitably used to aid in the simulation of cracks, penetration or perforation, and also the fragmentation of matter. The technique of combining the FEM and element erosion is often used in high-speed simulations, particularly in the simulation of penetration and perforation [14–19]. However, the criterion of element erosion is not unambiguous [20] and can affect the results of the simulation [11]. For example, the damage parameter is selected as an erosion parameter in [14], while in [15] it is maximum tensile stress, in [16, 18] it is geometric strain, in [17] it is fracture strain, and in [19] the erosion parameter is a combination of the damage parameter value and the maximum principal strain. Despite the use of advanced numerical techniques, the results of simulations are still being compared, most frequently with the values from analytical relationships obtained from an extensive amount of experiments [21, 22]. In the case of specific simulations, results are compared directly with experiments such as [23]. Unfortunately, as a result of the heterogeneity of the structure of concrete, agreement between simulations [24] and experiments cannot be guaranteed despite the use of complex modelling techniques.

The dissipation of matter from calculations as a result of the deletion of elements can be a significant problem in cases where the numerical erosion technique is used. In the majority of cases, this problem is omitted and left unsolved [14–19]. So, why use the FEM method for high-speed load simulations when so many problems and complications arise during calculations? With regard to the existence of mesh-free methods such as smoothed particle hydrodynamics (SPH), it is possible to realize high-speed load simulations without needing to include numerical erosion or deal with problems concerning the dissipation of matter [25, 26]. There are several reasons for preferring the FEM method, the main one being the low computational requirements compared with the aforementioned SPH method. Even though the SPH method can be a good choice for high-speed simulations of dynamic events, even in this case many numerical difficulties arise (tensile instability, zero-energy modes, etc.) which need to be resolved; see also [27–29]. However, if the strong points of the FEM and SPH methods were combined, a very useful apparatus for dealing with high-speed simulations could be created.

With regard to the facts mentioned above, the aim of the paper is to propose a procedure for the execution of simulations of the high-speed loading of concrete by steel projectiles. This approach will combine the FEM method with the numerical erosion of elements, and its primary focus will be on solving the problem with matter dissipation. This dissipation can be prevented via the transformation of eroded elements into SPH particles. In addition, current procedures for dealing with such simulations which do not introduce either the erosion of elements or the transformation of eroded matter will be compared. The concept of combining the FEM and SPH techniques with the inclusion of element erosion aims mainly at the improvement of simulation techniques in cases of high-speed loading in such a way that possible numerical issues are minimized and the results of simulations correspond better to the results of experiments.

#### 2. The Erosion Problem

The element erosion function, while not a material property or physics-based phenomenon, provides a useful means of simulating the spalling of concrete and provides a more realistic graphical representation of actual impact events. Erosion is characterized by the physical separation of the eroded solid element from the rest of the mesh [30]. Though element removal (erosion) associated with total element failure has the appearance of physical material erosion, it is, in fact, a numerical technique used to permit the extension of the computation. Without numerical erosion, severely crushed elements in Lagrangian calculations would lead to a very small time step, resulting in the use of many computational cycles with a negligible advance in the simulation time. Moreover, Lagrangian elements which have become very distorted have a tendency to “lock up,” thereby inducing unrealistic distortions in the computational mesh [31]. The erosion function allows the removal of such Lagrangian cells from the calculation if a predefined criterion is reached. When a cell is removed from the calculation process, the mass within the cell can either be discarded or be distributed to the corner nodes of the cell. If the mass is retained, the conservation and spatial continuity of inertia are maintained.

However, the compressive strength and internal energy of the material within the cell are lost whether the mass is retained or not. Even though the filtering out of unsuitable (unneeded) elements is more a matter for numerical simulations, it can be connected (to a certain degree) with the physical matter of the material model.

##### 2.1. Residual Compressive Strength with SPH

The moment at which an element of the Lagrangian mesh erodes is in conflict with what happens in real life, however. In reality, the material does not cease to exist but is only crushed and flakes off; see Figure 1. Even though one cannot speak about the strength of the material as such, the particles and wedge-shaped fragments that fly off can create secondary or residual strength. In order to better approximate reality, it is advantageous to maintain the presence of even such particles in the simulation. This can be done via the transformation of eroded elements into SPH particles. Subsequently, these freely moving particles with the characteristics of the original materials will interact with the rest of the computational model and thus better reflect reality.