Failure of Granular Materials under Impact- Multiscale Simulations and High-Speed Experiments -
Martin SteinhauserFraunhofer Ernst-Mach-Institute for High-Speed Dynamics (EMI), D-79104 Freiburg, Germany
A mesoscopic discrete particle model for simulating fracture andfailure of brittle materials is presented. Within the framework ofparticle dynamics simulations (Discrete Element Method) [1] amacroscopic solid state ceramic tile (typically (10 x 10 x 1) cm3)is modeled as a network of overlapping particles in 2 dimen-sions (2D). This two-dimensional model version presented herehas only three adjustable material parameters but is able to re-produce many salient features of the investigated ceramics undercompressive, tensile and shock impact load. Our three model pa-rameters are customized to the typical tensile strength, Young’smodulus and the compressive strength of the ceramics underinvestigation. Using Lennard-Jones type potentials the classicNewtonian equations of motion are integrated and uni-axial quasi-static load simulations are performed. Subsequently, shock loadsimulations are performed in a standard experimental set-up, theedge-on impact configuration [2,3]. The obtained simulation re-sults are compared with the results of high-speed impact exper-iments on aluminum oxide (Al2O3) and silicon carbide (SiC) ce-ramics [2]. Although the model particles have micrometer size weare able to simulate a specimen of macroscopic dimensions. Fordetails, see [4].
Simulation Model
A fundamental requirement for the coarse-grained model is tohave very few parameters describing the essential physical be-havior of the investigated system. Three basic properties aremodeled, namely, first, the resistance to pressure, second, thecohesive forces, and then the microscopic failure.
Figure 2: The simulation model using discrete el-ements. Overlapping spheres (in 3D), respectivelydisks (in 2D)
Resistance Against Pressure
Resistance against pressure is introduced by a Lennard-Jones-type repulsive potential ϕijrep which acts on every pair of discs{ij} only when dij(t), the mutual distance at time t, is smallerthan dij(t) ≡ d0
ij, the initial separation, cf Figure 2. Then for(0 < dij < d0
ij):
ϕijrep
(dij)= γR0
3
(d0ij
dij
)12
− 2
(d0
ij
dij
)6
+ 1
, (1)
whereas for dij ≥ d0ij the repulsive potential vanishes. The pa-
rameter γ in Eq. (1) scales the energy density and the prefactorR0
3 ensures the correct scaling behavior of the calculated totalstress Σijσ
ij = ΣijFij/A which is independent of N , see Fig-
ure 3.
Cohesive Potential
The cohesive potential ϕijcoh(dij)
is modeled by a harmonic func-tion, given that there are no irreversible changes of state when thematerial is submitted to small external forces. Each pair {ij} ofdiscs can thus be visualized as being connected by a spring, theequilibrium length of which equals the initial distance d0
ij. Thus,for dij > 0:
ϕijcoh
(dij)= λR0
(dij − d0
ij)2
. (2)
In Eq. (2) λ (which has dimension [energy/length]) determinesthe strength of the potential and the prefactor R0 ensures properscaling behavior of the material response.The total potential ϕtot of our model consequently reads:
ϕtot = Σij
(ϕijrep + ϕ
ijcoh
). (3)
A MA
U IIaL L00L
Model MA BModel M
Q
L
Figure 3: Sketch of the proposed model system and itsscaling properties. For sake of simplicity, only the two-dimensional case is shown. Left: The original system MAwith edge length L0, constant particle radii R0 and the sub-system QA ⊂ MA (shaded area) with edge length L. Right:The scaled system MB with particle radii S0 = aR0 andedge length L0 = aL. Both models represent the samemacroscopic solid.
Failure Criteria
Failure is included in our model by introducing two breakingthresholds for the springs with respect to compressive and totensile failure, respectively. A tensile failure criterium is reachedwhen the overlap between two particles vanishes, i.e. when thedistance dij of disc centers exceeds the sum of their radii:
dij > (2R0). (4)
Failure under pressure load occurs in our model when the actualmutual particle distance is less by a factor α (with 0 < α < 1) thanthe initial mutual particle distance, i.e. when
dij < α · d0ij. (5)
Note that particle pairs without a spring after pressure or tensilefailure still interact via the repulsive potential and cannot movethrough each other.
Initial Configurations
Using polydisperse particle distributions directly influences char-acteristic properties of the material, e.g. the coordination numberof the starting configuration. In the simple approach chosen here,mono-disperse disks are used and the initial coordination numberof the system is adjusted via a compactness or density parameterΘ as simulation input, see Figure 4.
Figure 4: Sample configuration for a system with N =10000 elements for different values of density Θ. The colorcode displays the coordination numbers; Blue: 0 neighbors,Green: 4 neighbors, Yellow: 6 neighbors, Red: 8 neigh-bors.
Crack Initiation and Failure
Uni-axial load simulations are performed until failure in the ma-terial occurs. Results of these simulations are displayed in thepicture series of Figure 5, which shows the dynamics of crackpropagation.
(a) (b)
(c) (d)
Figure 5: Crack initiation and propagation in the materialupon uni-axial tensile load. Color code is: Green: force-free bonds. Red: tension. (a) After applying a force at bothsides of the specimen, local tensions occur. (b) Initiationof a crack tip with local tensions concentrated around thistip. (c) Propagation of this crack through the material. (d)Failure.
Shock Impact
The discrete particle model was tested for the case of dynamicmaterial behavior where high strain rates occur. Numerically,this corresponds to dynamic non-equilibrium simulations. Dy-namic experiments for materials characterization at very highstrain rates are often done with a standard set-up, the edge-onimpact (EOI), see e.g. [3,4], where the edge of a target ceramicspecimen is hit by a projectile at high speed. Figure 6 displays acomparison between Simulations and Experiments with SiC.
(d) (e) (f)
(g)
(a) (b) (c)
(h) (i)
Figure 6: (a-c) Results of an edge-on-impact (EOI) sim-ulation at v = 150 m/s with SiC using N = 105 parti-cles. The material is hit at the left edge. A pressure wave(color-coded in blue, unloaded material: green) propagatesthrough the system. The time interval between the individ-ual snapshots from left to right is 2 µs in each case. (d-f)The EOI experiment with a SiC specimen. The time inter-val between the photographs is comparable to that in thetop row. The arrows indicate the location of the wave frontpropagating in the material. (g-i) The same computer sim-ulation as in (a-c), now displaying the occurring damage inthe material with respect to broken bonds.
References
[1] P.A. Cundall, O.D.L. Strack,Geotechnique29, 47-65 (1997)[2] M.O. Steinhauser, K. Grass, K. Thoma, and A. Blumen, Eu-
rophys. Lett., 73, 62 (2006)[3] M.O. Steinhauser, K. Grass, E. Strassburger, and A. Blumen,
Int. J. Plasticity, 25, 161 (2008)[4] M.O. Steinhauser,Computational Multiscale Modeling of Flu-
ids and Solids - Theory and Applications, Springer, Heidel-berg, Berlin, New York, 2008
Presented at the International Conference on Multiscale Materials Modeling 2008 (MMM 2008), 27-31 October 2008, Tahallassee, Florida, U. S. A.
