SIMULATIONS OF THE PENETRATION OF LIMESTONE TARGETS
USING TWO DIMENSIONAL MULTIMODAL FOURIER ANALYSIS

M. Brun¹, A. Combescure¹, C. Baillis², A. Limam¹, E. Buzaud³
(e-mails: Michael.brun@insa-lyon.fr, Alain.Combescure@insa-lyon.fr,
Christophe.Baillis@ec2ms.fr, Ali.Limam@insa-lyon.fr,
Eric.buzaud@dga.defense.gouv.fr)

¹Institut National des Sciences Appliquées de Lyon, 69621 Villeurbanne, France
²EC2-MS, 66 bd Niels BOHR, BP 2132, 69603 Villeurbanne, France
³Centre d’Etudes de Gramat, 46500 Gramat, France

Abstract

This communication is devoted to the presentation of recent progresses in dynamic fracture simulation. It is shown that X-FEM method is a nice frame for the simulation of dynamic fracture. The main reason is that it can be proved mathematically that the use of X-FEM permits to simulate dynamic crack propagation in brittle or ductile materials with a perfect control of energy. There is no proof of energy conservation for the standard methods like remeshing or element erosion techniques. This could explain that this type of technique is rarely predictive. Moreover in case of softening material behaviour the results are mesh dependant because of the physical and numerical localization which lead to a faster rupture as the mesh size decreases: there is no convergence of the numerical methods. This can be controlled using non local space models or delayed damage models, but in all cases the mesh has to be very fine to capture localisation and fracture. The X-FEM technology which may be combined with cohesive zones give results of the same quality wit a coarser mesh because all the localized fracture line is concentrated in the velocity jump region. The simulations are compared to original dynamic fracture experiments with complex geometries mixed loadings and stop and restart of the crack.

Fig. 1 -A typical dynamic fracture experiment

The experiments are done on PMMA and aluminium specimens impacted by an Hopkinson’s bar system. Cases of interacting cracks shall also be presented. In a second part some very recent results obtained by SPH shall be presented. A very new formulation of SPH shells up to fracture shall be shown with interesting prospective in case of impact on fluid structure systems.

                                    Fig. 2 - typical SPH shell perforation simulation.

Keywords : high strain-rates, impact, computational mechanics, dynamic fracture, X-FEM, SPH shells, experiments

Abstract

This presentation reports on recent advances on meso-scale simulation of concrete and, in particular, on the simulation of reinforced concrete structures subjected to penetration. First, a recently developed meso-scale model for concrete is reviewed. Concrete is modeled by a threedimensional discrete particle model, called Lattice Discrete Particle Model (LDPM), which simulates concrete mesostructure. Each particle represents a coarse aggregate piece with its surrounding mortar. Number and size of particles considered in the simulations are established according to the given grain size distribution. The position of particle centers is obtained by a try-andreject procedure that randomly places the particles throughout the volume of interest, fulfilling the constraints given by previously placed particles and the external boundaries. Three-dimensional conforming Delaunay triangulation and a dual domain tessellation define the topology and the geometry of the connections among the particles. The deformation of each strut, connecting two adjacent aggregate pieces, is defined by adopting rigid body kinematics characterized by the displacement and rotation vectors at the lattice nodes. Each strut can transmit both axial and shear forces. The adopted constitutive law simulates fracture, friction, and cohesion at the meso-level and takes into account the strain rate dependence of concrete fracturing behavior. This dependence, which is crucial for correctly simulating the effect of dynamic loadings, is due to two different physical mechanisms: 1) the dependence of the fracture process on the rate of crack opening, and 2) the viscoelastic deformation of cement paste. The first mechanism is described by the activation energy theory applied to ruptures that occur along a crack, and the second by the microprestress-solidification theory of concrete creep.
 
Second, an algorithm to couple LDPM with steel rebars is presented. Steel rebars are modeled as three dimensional elastic-plastic beam elements and they are constrained to the adjacent discrete particles through an ad hoc bond element. Constitutive equations of this bond element provide the relationships for computing interface forces given the relative displacements between particles and rebars. These equations implement the complex physical mechanisms taking place in the thin concrete layer surrounding steel rebars, including the formation of oblique cracks at rib locations, dilation due to slippage, frictional interaction, etc. Third, an extensive calibration and validation activity recently performed under both quasi-static and dynamic loading condition is discussed in order to evaluate the predictive capability of the developed model.

Abstract

To investigate the tensile strength of a micro-concrete (maximum grain size: 5 mm) under high loading rates (i.e. in the range of 10 to 100 per second), spalling tests were designed and performed in LPMM laboratory before to be compared with predictions given by an anisotropic damage model. Spalling tests have been used for approximately one decade to characterise the dynamic strength of concrete materials [Klepaczko and Brara, 2000; Schuler et al., 2006; Weerheijm and Van Doormaal, 2007]. During such test, a cylindrical specimen of concrete is placed at the end of a Hopkinson bar (Fig. 1). A compressive pulse is generated by projectile-impact at the opposite end. A part of this pulse is transmitted to the specimen and a part is reflected in the Hopkinson bar. When the transmitted pulse reaches the free end of the specimen, it is reflected as a positive pulse. Superposition of both signals induces a tensile loading within the core of the specimen that leads to its failure.

Projectile Hopkinson bar Specimen

Fig. 1 – Experimental device used for spalling tests

A specific methodology was applied to process experimental data. First, a laser extensometer was used to measure the axial velocity on the rear face of the specimen (Fig. 1). An example of experimental data is shown on Figure 2. The spalling strength is deduced from equation (1) in which r and C0 are

respectively the density and the speed wave of the concrete and DV corresponds to the difference between the maximum and the rebound velocities. This equation was checked by numerical simulation of spalling tests in which an arbitrary failure criterion was used.

1

s _spall = r C₀ DV

² (1) Moreover, by recording the incident and reflected pulses from gauges located on the Hopkinson bar, the transmitted pulse is reconstructed. It was used in a numerical simulation that involves the specimen alone assuming a purely elastic behaviour for the concrete. This calculation allows deducing the state of stress and strain rate before initiation of damage in the specimen. As failure is characterised by a rebound in the velocity on rear face (Fig. 2), the failure time may be also obtained. Thus, the strain rate at failure is also taking out from the numerical analysis. Finally, several tests were performed with dry and wet specimens and compared with data available in the literature.

3,0E-04 3,5E-04 4,0E-04 4,5E-04 5,0E-04

Fig. 2 – Velocity on the rear face measured by laser extensometer during spalling test

An anisotropic damage model was developed based on a micromechanical description of dynamic fragmentation process [Denoual and Hild, 2000; Forquin and Hild, 2007]. In this work, this model is used to simulate the damage of concrete specimens during spalling tests. The model allows explaining the increase of strength with loading rate. Moreover, the different experimental data (velocity measured on rear face of specimens by the laser, damage pattern) are compared with data obtained from the numerical simulations. Finally, the modelling is used to highlight the possible roles plaid by microstructure on the dynamic response of concrete materials under such high loading rates.

Keywords : micro-concrete, spalling tests, tensile strength, anisotropic damage model.

REFERENCES

1. Klepaczko, J.R. and Brara, A. (2001). "An experimental method for dynamic tensile testing of concrete by spalling." Int. J. Impact Eng., 25, 387-409.

2. Weerheijm, J. and Van Doormaal, J.C.A.M., (2007). "Tensile failure of concrete at high loading rates: New test data on strength and fracture energy from instrumented spalling tests." Int. J. Impact Eng., 34, 609-626.

3. Schuler, H., Mayrhofer, C., Thoma, K. (2006). "Spall experiments for the measurement of the tensile strength and fracture energy of concrete at high strain rates." Int. J. Impact Eng., 32, 1635–1650.

4. Denoual, C., and Hild, F. (2000). "A Damage Model for the Dynamic Fragmentation of Brittle Solids." Comp. Meth. Appl. Mech. Eng., 183, 247-258.

5. Forquin, P., Hild, F. (2007). "Dynamic Fragmentation of an Ultra-High Strength Concrete during Edge-On Impact Tests." ASCE J. of Eng. Mechanics, accepted manuscript.

ANALYTICAL MODELS AND EXPERIMENTAL RESULTS FOR SOME
PENETRATION PROBLEMS

M. J. Forrestal, D. J. Frew, T. L. Warren
(e-mails: mmforrestal@comcast.net, djfrew@sandia.gov, tlwarre@msn.com)

1805 Newton PL., NE, Albuquerque, NM, 87106, USA
National Labs., Albuquerque, NM, 87185, USA
3804 Shenandoah Pl., Albuquerque, NM 87111, USA

Abstract

For this presentation, we focus on experimentally verified, analytical models that predict the penetration depth of rigid projectiles launched into aluminium, limestone, and concrete targets. In addition, we present projectile deceleration-time measurements for concrete targets and compare these results with our models. Forces on the projectiles from the targets predicted by these models have been used by some investigators to solve the more detailed problems of oblique impact and structural and response [1,2].

Recently, we presented closed-form, penetration equations for rigid, ogive-nose rods launched into aluminium targets [3]. These equations approximated the two-dimensional target response with results from a one-dimensional, spherically symmetric, cavity-expansion analysis. The aluminium target materials were described as incompressible, elastic- plastic with power-law strain hardening. Predictions showed excellent agreement with data for striking velocities to 1,800 m/s.

In [4], we present penetration depth data for ogive-nose, rod projectiles launched into limestone targets. The rod projectiles had a length-to-diameter of ten and diameters of 7.1, 12.7, and

25.4 mm. We also presented an empirical penetration equation that was guided by cavity-expansion models [3] and post-test observations. This equation contains a target resistance parameter R that is determined from penetration depth versus striking velocity data. While this methodology provides a convenient procedure for data analysis, the material response mechanisms are not modelled. We found that a constant value of R accurately predicted penetration depth versus striking velocity for a fixed projectile. However, the value of R decreased as the projectile diameter increased, and we suggested this was due to strain-rate effects. To further investigate rate effects in limestone, we presented split Hopkinson pressure bar data [5] that showed the uniaxial strength doubled as the rates varied from quasi-static to 200 per second. Clearly, more fundamental research is required to better understand the material behaviour of limestone targets.

For our latest work on concrete targets [6,7], we measure projectile deceleration and final depth. An 83-mm-diameter powder gun launched 76.2-mm-diameter, 530-mm-long,and 13 kg projectiles to striking velocities between 140 and 460 m/s. The projectiles were designed to contain a single-channel data recorder, and acceleration during launch and deceleration during penetration were measured. The projectiles lost small amounts of mass from abrasion and experienced relatively small deformations. Therefore, rigid-body deceleration data provide a measure of net force. Data were analyzed with the model previously discussed for limestone targets.

Keywords : penetration equations, aluminium limestone concrete targets.

REFERENCES

1. Forrestal,M.J., Warren, T. L., “Penetration Equations for Ogive-Nose Rods Into Aluminum Targets”, Int. J. Impact Eng., 2008, In press.

2. Frew, D. J., Forrestal, M. J., Hanchak, S. J., “Penetration Experiments With Limestone Targets and Ogive-Nose Steel Projectiles”,J. Appl. Mech., 2001, Vol. 67, pp. 841-845.

3. Frew, D. J., Forrestal, M. J., Chen, W. “A Split Hopkinson Pressure Bar Technique to Determine Compressive Stress-Strain Data For Rock Materials”, Exp. Mech., 2001, Vol. 30, pp. 41-46.

4. Forrestal, M. J., Frew, D. J., Hickerson, J. P.,Rohwer, T. A. “Penetration of Concrete Targets With Deceleration-Tine Measurements”, Int. J. Impact Eng., 2003, Vol. 28, pp. 479-497.

5. Frew, D. J., Forrestal, M. J., Cargile, J. D. “The Effect of Concrete Target Diameter on Projectile Deceleration and Penetration Depth”, Int. J. Impact Eng., 2006, Vol. 32, pp. 723-734.
6. Brun, M.,Combescure, A., Baillis, C., Limam, Buzaud, E., “Simulations of the Penetration of Limestone Targets Using Two-Dimensional Multimode Fourier Analysis”, Int. J. Impact Eng., 2008,Vol. 35, pp. 251-268.  
7. Warren, T.L., Hanchak S.J., Poormon K.L., “Penetration of Limestone Targets by Ogive-Nose VAR Steel Projectiles     at Oblique Angles”, Int. J. Impact Eng., 2004, Vol. 30, pp.1307-1331.

SIMULATING PROJECTILE PERFORATION THROUGH CONCRETE
SLABS:
HOW SIGNIFICANT IS THE TENSILE BEHAVIOR OF CONCRETE?

Andreas O. Frank, Mark D. Adley, and James D. Cargile (e-mails: andreas.o.frank@usace.army.mil, mark.d.adley@usace.army.mil, james.d.cargile@usace.army.mil)

U.S. Army Engineer Research and Development Center
Geotechnical and Structures Laboratory
Impact & Explosive Effects Branch
Attn: CEERD-GM-I
3909 Halls Ferry Road
Vicksburg, MS 39180-6199, USA

Abstract

The Geotechnical and Structures Laboratory of the U.S. Army Engineer Research and Development Center (ERDC) have conducted a significant amount of projectile penetration research. These efforts have included numerous projectile penetration experiments using the ERDC 83-mm ballistic research facility [1], extensive material property experiments that characterize target mechanical behavior and provide data for fitting constitutive models [2], and numerous fully-firstprinciple calculations of various penetration events [3]. Our research is focused around gathering data and gaining insight into the processes of penetration mechanics. One of the primary end goals of our research is to develop more accurate and robust numerical methods. In this paper, we shall discuss various modeling and simulation issues for predicting the perforation of concrete slabs by projectiles. Projectile perforation events can be simu lated by using fully “first-principle” methods such as “hydrocodes”. This is accomplished by discretizing both the projectile and the target into a finite number of pieces and allowing the first-principle physics; i.e., conservation of mass, momentum, and energy, to dictate the perforation event. Over the past few decades, many hydrocode techniques have been developed. Typically, these codes differ in how they discretize the computational domain by using either an Eulerian, Lagrangian, or “mixed” formulation. The mixed formulation might be an Arbitrary Lagrange Euler (ALE) formulation or a coupled formulation that links a Lagrangian code with an Eulerian code. However, all of these codes require some sort of constitutive model to evaluate the responses of the target material; i.e., stress versus strain relationship. We shall focus our efforts on two different concrete constitutive models, the relatively simple Holmquist-Johnson-Cook material model [4] and the more complex Microplane M4 material model [5,6]. In order to simulate projectile perforation events, we have used the Lagrangian hydrocode EPIC [7]. Our simulations have been based on a series of perforation experiments conducted at ERDC [8]. The experiments were conducted with a 2.3-kg ogival nose projectile launched into three unreinforced concrete slabs of varying thicknesses (ie, target thickness to projectile diameter ratios of T/D= 2.5, 4.25, and 5.0). All the experiments were conducted under “near” normal impact conditions and conventional velocities (ie, ~300 m/sec). The target slabs were all constructed from a standardized conventional concrete mix (WES-5000). Some of the mechanical properties for WES-5000 concrete were experimentally determined at ERDC and are used in order generate material fits to the constitutive models, including quasi-static unconfined compression, triaxial compression, hydrostatic compression, uniaxial compression, and “direct pull” tensile data. All the experiments resulted in complete perforation of the concrete slabs and provide data for the perforation velocity of the projectile as well as the final impact and exit crater shapes (ie, a measure of the damage in the slab). Our results indicate that the Microplane M4 concrete model provides excellent results for these projectile perforation events (see Figure 1). Although, the HJC model provides good perforation velocities it has more difficulty in predicting the crater profiles, especially the impact crater. Due to the exceptional agreement with the experimental data, we have used the Microplane M4 model as a guide in order to help us more carefully examine the significance of the tensile behavior of the concrete slabs during these perforation events. This has emphasized the importance of properly modeling the tensile behavior and highlighted some of the issues concerning the HJC model for perforation calculations. Specifically the effects of strain rate and/or tensile damage can play a critical role in accurate predictions. We shall provide a detailed discussion of these issues, as well as some suggestions for improvements to the HJC model. For example, 1) a failure surface that is dependent on 3rd invariant of the stress deviator, 2) a failure surface that provides increased strength at higher strain rates along the tensile hydrostatic axis as well as the deviatoric axis, and 3) a more complex damage evolution. We have implemented some of these modifications into the HJC concrete model and shall provide preliminary results from our calculations.

Fig. 1 -Snapshots in time showing a comparison between the Microplane M4 and HJC concrete models. Shown are the projectile exit velocities, as well as impact and exit craters for EPIC perforation calculations with varying slab thickness, as follows; A) T/D=2.5, B) T/D=4.25, and C) T/D=5.0. Also shown are the experimental perforation data in terms of exit velocities and estimated crater shapes (dotted lines).

Keywords : computational mechanics, concrete material modeling, impact and penetration.

REFERENCES

1.  Frew, D. J., Cargile, J. D., and Ehrgott, J. Q., “WES Geodynamics and Projectile Penetration Research Facilities,” Proceedings of the Symposium on Advances in Numerical Simulation Techniques for Penetration and Perforation of Solids, 1993 ASME Winter Annual Meeting, New Orleans, LA , 28 November-4 December 1993.

2. Akers, S. A., Adley, M. D., and Cargile, J. D., “Comparison of Constitutive Models for Geologic Materials used in Penetration and Ground Shock Calculations,” Proceedings of the 7th International Symposium on the Interaction of Conventional Munitions with Protective Structures, Mannheim FRG, April 1995.

3. Adley, M. D., Cargile, J. D., Akers, S. A., and Rohani, B., “Numerical Simulation of Projectile Penetration into Concrete,” Proceedings of the 14th U.S. Army Symposium on Solid Mechanics, Myrtle Beach, SC, 16-18 October 1996.
4. Holmquist, T.J., Johnson, G.R., and Cook, W.H., "A Computa¬tional Constitutive Model for Concrete Subjected to Large Strains, High Strain Rates, and High Pressures," Presented at the Fourteenth Interna¬tional Symposium on Ballistics, Quebec City, Canada, September 1993.
5. Bazant Z.P., Caner F.C., Carol I., Adley M.D., and Akers S.A., “Microplane Model M4 for Concrete. I: Formulation with Work-Conjugate Deviatoric Stress,” Journal of Engineering Mechanics, American Society of Civil Engineers, September 2000, pp 944, 953.
6. Caner F.C. and Bazant Z.P., “Microplane Model M4 for Concrete. II: Algorithm and Calibration,” Journal of Engineering Mechanics, American Society of Civil Engineers, September 2000, pp 954-961.
7. Johnson G.R., Stryk R.A., and Beissel S.R., “User Instructions for the 2001 Version of the EPIC Code,” Alliant Techsystems Inc., Hopkins, MN, April 2001.
8. Cargile, J. D. “Development of a Constitutive Model for Numerical Simulations of Projectile Penetration into Brittle Geomaterials,” Technical Report SL-99-11, U.S. Army Engineer Research and Development Center, Vicksburg, MS, September 1999.

Acknowledgment

The research reported herein was conducted as part of the U.S. Army Corps of Engineers Survivability and Protective Structures Technical Area, Hardened Combined Effects Penetrator Warheads Work Package, Work Unit “HPC Prediction of Weapon Penetration, Blast and Secondary Effects.” Permission to publish was granted by Director, Geotechnical and Structures Laboratory.

This abstract is approved for public release; distribution is unlimited.

DYNAMIC INDENTATION RESPONSE OF ADVANCED STRUCTURAL
MATERIALS

G. Subhash
(e-mails: subhash@ufl.edu)

Mechanical and Aerospace Engineering
University of Florida, Gainesville, FL 32611, USA

Abstract

Determination of high strain rate response of materials is of significant value in analysis and design of components subjected to dynamic loads as in impact, crash, high speed machining, dynamic wear etc. Traditional techniques such as split Hopkinson pressure bar require elaborate data processing to obtain the material rate sensitivity under dynamic loads. The specimen size requirement and specimen preparation technique are also considerably more demanding. A more simplified ‘dynamic indentation technique’, whose principle of operation is similar to that of traditional static indentation technique, has been developed for determination of rate sensitive behavior of materials. Unlike the traditional static indentation technique where the loading duration is of 15 seconds (strain rate on the order of 0.0001/s), the dynamic indentation technique utilizes elastic stress wave propagation phenomena in a slender rod and delivers indentation load in 100 microsecond duration resulting in nominal strain rates on the order of 1000/s. This technique has been used to capture the rate sensitive hardness of a range of traditional metals and ceramics. The method is then extended to determination of the rate sensitive behavior of bulk metallic glasses (BMGs or amorphous metals), fine grained boron carbide (B4C) ceramics and ultra high temperature ceramic (UHTC) composites (ZrB2-SiC). In BMGs, the evolution of shear band patterns and their spacing as a function of applied loading rate are investigated [1-3]. The negative rates sensitivity of hardness and fracture strength are then rationalized. In B4C, the loss of hardness and fracture toughness under high loading rates are detected and then the reason for the loss of strength is identified with the help of Raman spectroscopy as the dynamic stress-induced structural phase transformation [4,5]. In UHTC, stress induced microplasticity (slip lines) and microcracking are investigated. Suitable micromechanical models that capture the observed behavior have been developed to rationalize the microstructural deformation mechanisms. It will be shown that the dynamic indentation hardness method can be effectively used for rapid screening of materials for their rate sensitive behavior.

Abstract

Artificially jointed Indiana limestone was manufactured by imposing an orthogonal grid of tensile splitting fractures onto flat thin plates. A stack of these jointed limestone plates was loaded with a spherical explosive charge embedded in solid limestone. The mechanical properties of the solid limestone and the fracture interfaces in the jointed limestone were measured in confined compression tests. Special quartz stress gauges and electromagnetic particle velocity gauges were embedded in the jointed limestone. These measurements correlated very well with the assumption of spherical symmetry. A comparison with velocity measurement from solid limestone demonstrates the large effect the joints have on the stress wave. Posttest determination of the residual strain state in the jointed material indicates a significant amount of dilatation.

Abstract

Dynamic fragmentation processes are observed in brittle materials such as ceramics, concrete glass or rocks when subjected to impact or blast. Under such loadings, a spherical (or cylindrical) divergent wave is generated in the target, leading to a radial motion of the material that induces high levels of radial and hoop stresses. This phenomenon produces a dynamic fragmentation characterized by a high density of oriented cracks. This anisotropic damage may decrease significantly the impact-strength of the target and the quasi-static strength of the loaded structure. Therefore, the knowledge of damage properties (i.e., characteristic time, crack density, orientation of cracks, and role of microstructure) as well as the dynamic strength of the material are important points to be understood. In this work, an anisotropic damage model is developed based on the fragmentation phenomenology assuming unstable micro-cracks initiated from point defects. This model shows how the brittle and probabilistic behavior under quasi-static loading condition transforms into a deterministic and stress-rate-dependent behavior with the increase of loading rate (Figure 1). It also underlines the influence of, on the one hand, some material parameters, and on the other hand, the loading rate and volume size on the fragmentation regime (i.e., single/multiple cracking) and properties (time to damage, crack density, mean failure stress and scatter). Last, it allows one to analyze the relative role of surface defects and flaws in the bulk. Figure 1 shows that very high strain rates are needed for flaws in the bulk to initiate cracks. The present approach is based on non-local Continuum Damage Mechanics to account for both microcracking and mesocracking, the transition between these two scales being described by the model. Comparisons with experimental observations are performed to validate the model and show that the cracking patterns may be reproduced numerically when the dynamic behavior of glass alone under impact loading is studied.

Fig. 1 - Average ultimate tensile strength vs. strain rate for a 10 mm3 element made of soda-lime silicate glass (the surface is endowed with Weibull parameters m = 7, S0 = 100 MPa, Seff = 100 cm2, whereas the bulk is characterized by intrinsic flaws with Weibull parameters m = 30, ? 0 = 3 GPa, Veff = 10 6 mm3).

Keywords : impactBrittle materials, edge-on impacts, Poisson point process, Weibull model.

REFERENCES

1. Denoual C., Hild F., 2000, A Damage Model for the Dynamic Fragmentation of Brittle Solids, Comp. Meth. Appl. Mech. Eng., 183, 247-258.

2. Denoual C., Hild F., 2002, Dynamic Fragmentation of Brittle Solids: A Multi-Scale Model, Eur. J. Mech. A/Solids, 21, 1, 105-120.

1. Brajer X., Forquin P., Gy R., Hild F., 2003, The role of surface and volume defects in the fracture of glass under quasi-static and dynamic loadings, J. Non Cryst. Solids, 316, 42

2. 53.
3. Hild F., Brajer X., Denoual C., Forquin P., 2003, On the Probabilistic-Deterministic Transition Involved in a Fragmentation Process of Brittle Materials, Comput. Struct., 81, 12, 1241-1253.

NUMERICAL MODELING OF PROJECTILE PENETRATION INTO
COMPRESSIBLE RIGID
VISCO-PLASTIC MEDIA

I. R. Ionescu¹, O. Cazacu², T. Perrot³
(e-mails: ioan.ionescu@lpmtm.univ-paris13.fr, cazacu@reef.eng.ufl.edu,
thomas.perrot@wanadoo.fr)

¹LPMTM, University Paris 13, 99, Avenue Jean-Baptiste Clement F-93430 Villetaneuse, France

                            2 REEF, University of Florida, 1350 N. Poquito Road Shalimar FL 32579, USA
                                    ³LAMA, University of Savoie, Campus Scientifique F-73376 Le Bourget-du-Lac, France

Abstract

We develop here computational methods for modeling the penetration of a rigid projectile into porous media. Compressible rigid visco-plastic models (see [1]) are used to capture the solid-fluid transition in behavior at high strain rates, account for damage/plasticity couplings and viscous effects observed in geological and cementitious materials. A hybrid time-discretization is used to model the non-stationary flow of the target material and the projectile -target interaction, i.e. an explicit Euler method for the projectile equation and a forward (implicit) method for the target boundary value problem. At each time step, a mixed finite-element and finite-volume strategy is used to solve the "target" boundary value problem. Specifically, the nonlinear variational inequality for the velocity field is discretized using the finite element method while a finite volume method is adopted for the hyperbolic mass conservation and damage evolution equations. To solve the velocity problem, a decomposition-coordination formulation coupled with the augmented lagrangian method is adopted. Numerical simulations of penetration into concrete were performed. By conducting a time step sensitivity study it was shown that the numerical model is robust and computationally inexpensive. We discussed the evolution of the density changes around the penetration tunnel, of the shape and location of the rigid/plastic boundary, of the compaction zones and of the extent of damage due to air void collapse in such materials. For the constants involved in the model (shear and volumetric viscosities, cut-off yield limit and exponential weakening parameter for friction) that cannot be determined from data, a parametric study was performed.

Fig. 1 - The finite element mesh obtained after remeshing (left), and the distribution of the deformation rate in 1/sec (right). Note that the boundary between the rigid zone and the viscoplastic one is sharply captured.

Fig. 2 - Left: the distribution of the density (kg/m3) in the target at t=0.925 ms. Right: the distribution of damage in the target at t=0.925 ms.

Fig. 3 -The distribution of the normal stress acting on the planes Ozr in the target. The regions where tensile failure occurs are colored in red. Since the tensile failure could occur only along planes which could not be symmetric with respect to the penetrator centerline, anisotropy is generated in the target. Trajectory deviations may be related to this induced anisotropy.

Keywords : decomposition-coordination formulation, augmented lagrangian method, high velocity penetration in porous materials, compressible rigid visco-plastic fluid, rate-effects, compaction, damage.

Abstract

Predictive simulation of fracture in heterogenous materials has traditionally been an ill-posed problem due to the complex microstructural features that do not lend themselves to simple geometric representations. Planer cracks, penny-shaped cracks, and spherical inclusions can provide useful insight into damage and fracture behavior, however, they can fall short as honest descriptors of real heterogeneous microstructure. The work described here is aimed at developing new ways to characterize the physical microstructure of damage and fracture, so that more realistic ways of simu lating and predicting behavior may be developed. The assumption guiding this work is that observed phenomena such as scale effects will follow naturally from a quantitative relationship between microstructure and properties.

Towards this end, we are examining the fracture of small mortar specimens using a 3D imaging technique called x-ray microtomography. This technique allows us to make quantitative characterizations of internal structure, and the changes in structure that occur due to damage and fracture. Specimens can be examined at multiple levels of damage, and the damage can be related to bulk compliance and energy dissipation. In this work we are examining cylindrical specimens subjected to both end compression and split cylinder configurations. Resulting internal cracking and damage is characterized by several different geometric descriptors based on surface normals and principal curvatures, as well as simple surface area and fractal dimension measurements. We suggest from this work that although the characterization of damage in this way becomes more complex, the basic mechanics of failure may become more transparent, and therefore holds promise for future predictive simulation tools.

AN EMPIRICAL APPROACH OF THE RANDOM RUPTURE
IN LAGRANGIAN NUMERICAL SIMULATIONS.

PETIT Jacques (jacques-m.petit@dga.defense.gouv.fr)

Centre d’études de Gramat
BP 80200, F-46500 GRAMAT, France.

Abstract

The need of rupture predictions, especially the fragments size, is obvious for many applications. The work performed on the constitutive relations during the last decades allows to undertake the treatment of the final localization-rupture stage rather confidently in the field of large strain at high strain rate.

The localizations cannot appear in a lagrangian simulation with fine meshes and regular geometry particularly with ductile materials. Different methods can be used to promote them. Starting from a "macroscopic" constitutive relation, we chose to introduce in it small arbitrary random perturbations specific to each lagrangian mesh.

Using this approach we simulated different configurations (cylindrical implosion, ring expansion) with different materials (copper, nickel, titanium alloy). When the final rupture is processed, it is with simple estimated threshold on plastic strain and erosion. Even though this work has to be strengthened and improved, the results are promising. They reproduce approximately the localizations frequency observed in the corresponding experiments (see figure 1). They also show the effect of constitutive relation and their sensitivity to strain rate.

a – frame camera record of the inner surface b – numerical simulation

Fig; 1 - Cylindrical implosion of titanium alloy sample. Experiment – numerical simulation comparison.

Keywords : high strain-rate, random localization, lagrangian numerical simulation

THE SHOCK RESPONSE OF GEOLOGICAL AND CEM ENTITIOUS
MATERIALS

D.J. Chapman. D.M.Williamson, C.H. Braithwaite, W.G. Proud (e-mails: wgp1000@cam.ac.uk)

Fracture and Shock Physics Group, Cavendish Laboratory,
JJ Thomson Ave., Cambridge, CB3 0HE, UK

Abstract:

Over a period of 11 years the Cavendish Laboratory in association with a number of government and commercial organisations has been studying the high rate response of geological and construction materials over a range of strain rates. Initially the study focussed upon wet and dry cement paste and upon the response of the aggregate material. This was followed by studies on grout and microconcrete. Finally, full-up concrete, concrete with aggregate size as found in standard mixes was characterised. In these studies the level of longitudinal and lateral stress pulses produced by 1-D plate impact was obtained and the principal Hugoniot and variation in shear strength with shock pressure was obtained (figure 1). This required the study and development of appropriate shielding for the stress gauges. VISAR was also used in this system.

)

 

0	0.1	0.2	0.3	0.4	0 .5	0.6	0.7	0	1	2	3	4	5	6	7
		Pa rt icl e Ve locit y (mm µs -1)									Longi tudinal St res s ( GPa )
		(a)									(b)

Fig. 1 - (a)Longitudinal and (b) Lateral Response of Cement Paste under shock wave loading.

In parallel with this a series of studies were conducted on a wide range of geological materials: sands stones to granites. Some of these materials e.g. Kimberlite, showed properties very similar to Concrete. In addition to plane wave loading, ballistic studies were performed to determine the properties of the materials under ballistic loading. Both the front and rear surfaces of a material under impact loading were examined using high-speed photography, optical correlation techniques and post-impact analysis. The fundamental processes, their order of appearance and their timescale were the important factors.

Finally, studies using x-rays have probed the internal motion of samples using digital image correlation techniques, which has been successfully applied to a variety of material types. Amongst these have been brief studies of the surface and bulk motion of cement and concrete samples under shaped-charge jet impact. These experimental data were obtained as part of a wider programme to predict and model material response, study high-pressure low-rate effects on structural components. This research has been carried out at a numb er of locations, QinetiQ, Imperial College London and the University of Sheffield.

7 6 5 5 2 1 0 0

Particle Velocity (mm µs^-1) Particle Velocity (mm µs^-1)

Fig. 2 -Comparison of Cement Paste respsonse with ^{Fig. 3 - Hugoniot curve and corresponding release}

Hugoniot Stress (GPa)

Stress (GPa)

4 3

published Concrete data (from a variety of authors). Several important phenomena have been found:

states of concrete.

(1)

The vast majority of building and geological materials do not display an obvious Hugoniot Elastic Limit.

(2)

Study of the lateral stress reveals a point above which the material is degraded significantly. The longitudinal behaviour of the material remains predominantly linear above and below this limit, but the shear strength deviates significantly from undamaged material.

(3)

The binder, i.e. cement paste, dominates the shock response of the cement in the stress-particle velocity space (figure 2). The release properties have also been examined (figure 3)

(4)

The spall strength of the materials is small compared to their compressive properties. Cratering processes proceed by a circumferential cracking process in advance of radial cracking in some materials.

(5)

The properties of the aggregate, added to current concrete systems are generally much stronger and tougher than the cement paste or grout and so affect the material response mainly in terms of density.

(6)

It is possible to measure bulk motion within the sample under shaped charge jet loading (figure 4)

(7)

The system is amenable to predictive modelling using approaches such as the Porter-Gould approach.

SMOOTH PARTICLE HYDRODYNAMICS: SOME OBSERVATIONS ON ITS USE
FOR PENETRATION AND PERFORMATION SIMULATION

Abstract

The availability of meshfree methods, and in particular Smooth Particle Hydrodynamics (SPH), in general
purpose commercial analysis software, e.g. LS -DYNA and AUTODYN, provide analysts with another tool for
simulating penetration and perforation analyses. Although traditional Eulerian techniques, and more recently the
so called Multi-Material Arbitrary Lagrange Eulerian (MM-ALE), form a class of well established methods for
treating penetration and perforation, these techniques have not been as popular with analysts as the more
traditional Lagrange solvers with the addition of material erosion.

Lagrange solvers with material erosion have grown in popularity because of the wide availability of Lagrange
solvers in a large number of commercial software packages, and in most of the explicit time integration solvers
in this class, the analysts often has the ability to remove elements from the simulation based on a wide variety of
criteria. The selection of an appropriate ‘erosion criteria’ is always ad hoc, and often ill advised. The most
common such criterion is the so called effective plastic strain, a positive definite quantity that is monotonic in
time. While this criterion has a solid basis in uniaxial stress testing of metals, it is a rare analysis that aims to
simulate a uniaxial tensile test using erosion. Rather, analysts (too) frequently use effective plastic strain as an
erosion criterion for penetration simulations which are dominated by compressive stresses, where this criterion
has no physical basis. It seems analysts have lost sight of erosion criteria as numerical artefacts, i.e. there is no
material characterization test to determine erosion anything.

The use of erosion criteria to calibrate models to given experimental results is folly. First there is no unique
erosion criteria, and thus many numerical ‘solutions’ can reproduce the experimental results, i.e. non-unique
solutions. Second, for a give erosion criteria one must perform mesh refinements to assess the change in the
predicted penetration results. If the ‘answer’ continues to change with mesh refinement then there is no ‘answer.’
The contention is made this will be the case for most mesh refinements with ad hoc erosion criteria.
Element erosion is essentially old “technology” that facilitated getting an (any?) answer. For the most part,
erosion predates the advent of damage mechanics, not to mention the current wider availability of Eulerian and
meshfree solvers.

The Smooth Particle Hydrodynamics (SPH) technique is well suited to penetration and perforation analyses.
First, there is no need for ad hoc erosion criteria, as there is no element distortion requiring the removal of
elements; there are no elements. Second, as the particles are forced to separate along the boundary of the
projectile during the penetration event, they ‘naturally’ form discontinuities in the continuum, much as shear
bands and cracks are formed in penetrated materials. These advantages over Lagrange techniques with erosion,
do come at a cost of increased CPU time for simulations, and perhaps more importantly, resources need to be
expended to train analysts in the use of these techniques.

Case studies of SPH techniques applied to metal plate perforation, both thin and thick plates by both rigid and
highly deformable projectiles, metal tubes, complex geometries consisting of metal part assemblies, brick and
mortar walls, and concrete slabs are presented. These applications are used to illustrate a range of applicability of
SPH techniques, and when available, their predictive comparisons with experimental results. The most
challenging of these illustrations is the case of a highly deformable (aluminum) projectile perforating a metal
(steel) plate. In this case, the prediction of the ballistic limit for the plate depends on accurately modeling both

the deformation of the perforated plate and the highly deformed projectile whose shape determines its interaction with the plate.

Keywords : Smooth Particle Hydrodynamics, SPH, penetration, perforation, erosion criteria, concrete

Abstract

Current practice in materials-by-design has evolved from a trial and error approach to physically-based approach where computational materials tools are used extensively in the design of new materials. An example is in composite design, where techniques that enable tailoring of microstructure topology has allowed design of structures with interesting extremal properties such as negative thermal expansion and negative Poisson’s ratio. In contrast to composites, techniques that allow tailoring of properties of engineering alloys involve tailoring of preferred orientation of crystals and grain sizes. With recent multi-length scale modeling advances in predicting behavior of materials, it is increasingly becoming possible to devise methodologies to control microstructure formation and generate materials with tailored property distributions. In this talk, I will describe a recently developed optimization strategy for computational design of materials with tailored microstructure-sensitive properties. The topics covered in this talk are listed below:

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A non-linear large strain finite element homogenization technique for prediction of microstructure evolution [1].

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A multi-scale sensitivity analysis technique for designing deformation processes that generate optimal material properties in the final product [2].

During deformation processes, mechanisms such as crystallographic slip and lattice rotation drive the formation of texture and variability in property distributions in such materials. A useful method for designing materials is hence, through control of manufacturing process variables such as die and performs shapes (Fig 1) that affect final microstructure and properties of the material.

In the multi-scale sensitivity analysis technique for controlling properties, sensitivity of microstructure field variables such as slip resistances and crystal orientation changes due to perturbations in process parameters such as forging rates, die and perform shapes are exactly defined by direct differentiation of governing equations. A macro-micro energy equivalence principle is developed to compute sensitivity of stress and various material properties at the macroscopic level from microstructural sensitivity fields. These sensitivities are used within a gradient optimization framework for computational design of forming processes.

This inherently is a complex multi-scale problem involving microstructure analysis and requires extensive use of parallel computing. Since the design has to be such that properties as well as the finished product shapes have to be controlled, these are also complex multi-objective problems. Our novel ‘multi-scale sensitivity analysis’ technique (Ref. 1) allows us to address these issues efficiently. Examples that illustrate efficiency of this approach in allowing control of properties such as stiffness and strength to generate graded metallic armor materials will be shown. Increased use of computer and information technologies has allowed integration of optimization-based design techniques for thermo-mechanical processing with microstructure analysis. Techniques that we have developed can be broadly applied to design of functionally graded armor materials as well as sensors with highly directional electromagnetic, thermal or optical properties.

Fig. 1 - Properties such as strength and stiffness are tailored by controlling process variables such as die and perform shapes. This affects the evolution of the microstructure and associated properties.

Fig. 2. -An example of ontrol of material strength distribution for creating graded polycrystalline materials by controlling the preform shapes during closed die forging processes.

Abstract

Explosives are too often considered from the sole viewpoint of their destination, i.e. detonation. For other purposes, however, they must be considered as “normal materials” and studied from the mechanical and material science standpoints. This is particularly the case for the problem of low velocity impacts, crucial with regard to safety issues, and yet poorly understood.

It is well known that the initiation of impact induced explosions involves “hot spots”, i.e. local temperature peaks resulting from heterogeneous dissipative mechanisms. Understanding such phenomena compels one to recognize that the macroscopic scale is not relevant, and that mechanical processes take place prior to reaction. It is thus required to study micromechanical and hot spot processes as well, using appropriate experimental tools.

Fig. 1 -Concrete like microstructure: left : pristine – center : after dynamic uniaxial compression.

This is illustrated on a pressed HMXbased high explosive, whose microstructural state and evolutions are investigated mainly using optical microscopy on postmortem samples. Given the concrete like initial microstructure (see Fig. 1 left), pressure and strain rate are identified as the main influential parameters on macroscopic response. Hence, several samples are submitted to a variety of laboratory loading histories, recovered and analyzed. This panel also includes an explosive target recovered after a subcritical impact.

It is shown that the main deformation mechanisms are microcracking (see Fig. 1 right) and crystal twinning (see Fig. 2 left), both strongly oriented processes. Small scale reacted spots are also identified in most samples, even in cases of low energy loading cases. Strong relationships between deformation and reaction micromechanisms are deduced from observations. For example, it appears clearly that twinning impedes reaction, which thus appears as strongly linked to microcracking. Finally a thorough study of the recovered target shows that three main reaction mechanisms are at work, including crystallographic transition, solid state reaction, and melting (see Fig. 2 right).

Figure 2 : Effects of deformation and reaction – left : twinned crystals after quasi-static high pressure triaxial compression -right : small scale melting after uniaxial dynamic compression (arrows).

These manifestations are separate mechanisms, as they can be observed independently. Moreover, some of them, small scale melting for example, are found after quasistatic experiments. This indicates clearly that they take place at a very rapid time scale, for which thermal conduction has but little time to occur, even though the time scale of the external load is several orders of magnitude longer. Although the deformation mechanisms are somewhat clarified, much remains to be done and understood. Partially reacted microstructures appear quite complex, and seem to involve several uncoupled mechanisms, suggesting the crucial role of solid state, melting and reaction kinetics.