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Failure of Granular Materials under Impact - Multiscale Simulations and High-Speed Experiments - Martin Steinhauser Fraunhofer Ernst-Mach-Institute for High-Speed Dynamics (EMI), D-79104 Freiburg, Germany [email protected] Introduction A mesoscopic discrete particle model for simulating fracture and failure of brittle materials is presented. Within the framework of particle dynamics simulations (Discrete Element Method) [1] a macroscopic solid state ceramic tile (typically (10 x 10 x 1) cm 3 ) is modeled as a network of overlapping particles in 2 dimen- sions (2D). This two-dimensional model version presented here has only three adjustable material parameters but is able to re- produce many salient features of the investigated ceramics under compressive, tensile and shock impact load. Our three model pa- rameters are customized to the typical tensile strength, Young’s modulus and the compressive strength of the ceramics under investigation. Using Lennard-Jones type potentials the classic Newtonian equations of motion are integrated and uni-axial quasi- static load simulations are performed. Subsequently, shock load simulations are performed in a standard experimental set-up, the edge-on impact configuration [2,3]. The obtained simulation re- sults are compared with the results of high-speed impact exper- iments on aluminum oxide (Al 2 O 3 ) and silicon carbide (SiC ) ce- ramics [2]. Although the model particles have micrometer size we are able to simulate a specimen of macroscopic dimensions. For details, see [4]. Simulation Model A fundamental requirement for the coarse-grained model is to have very few parameters describing the essential physical be- havior of the investigated system. Three basic properties are modeled, namely, first, the resistance to pressure, second, the cohesive forces, and then the microscopic failure. Figure 2: The simulation model using discrete el- ements. Overlapping spheres (in 3D), respectively disks (in 2D) Resistance Against Pressure Resistance against pressure is introduced by a Lennard-Jones- type repulsive potential ϕ ij rep which acts on every pair of discs {ij } only when d ij (t), the mutual distance at time t, is smaller than d ij (t) ≡ d 0 ij , the initial separation, cf Figure 2. Then for (0 <d ij <d 0 ij ): ϕ ij rep ( d ij ) = γR 0 3 ( d 0 ij d ij ) 12 - 2 ( d 0 ij d ij ) 6 +1 , (1) whereas for d ij ≥ d 0 ij the repulsive potential vanishes. The pa- rameter γ in Eq. (1) scales the energy density and the prefactor R 0 3 ensures the correct scaling behavior of the calculated total stress Σ ij σ ij =Σ ij F ij /A which is independent of N , see Fig- ure 3. Cohesive Potential The cohesive potential ϕ ij coh ( d ij ) is modeled by a harmonic func- tion, given that there are no irreversible changes of state when the material is submitted to small external forces. Each pair {ij } of discs can thus be visualized as being connected by a spring, the equilibrium length of which equals the initial distance d 0 ij . Thus, for d ij > 0: ϕ ij coh ( d ij ) = λR 0 ( d ij - d 0 ij ) 2 . (2) In Eq. (2) λ (which has dimension [energy/length]) determines the strength of the potential and the prefactor R 0 ensures proper scaling behavior of the material response. The total potential ϕ tot of our model consequently reads: ϕ tot =Σ ij ( ϕ ij rep + ϕ ij coh ) . (3) A M A U I I aL L 0 0 L Model M A B Model M Q L Figure 3: Sketch of the proposed model system and its scaling properties. For sake of simplicity, only the two- dimensional case is shown. Left: The original system M A with edge length L 0 , constant particle radii R 0 and the sub- system Q A ⊂ M A (shaded area) with edge length L. Right: The scaled system M B with particle radii S 0 = aR 0 and edge length L 0 = aL. Both models represent the same macroscopic solid. Failure Criteria Failure is included in our model by introducing two breaking thresholds for the springs with respect to compressive and to tensile failure, respectively. A tensile failure criterium is reached when the overlap between two particles vanishes, i.e. when the distance d ij of disc centers exceeds the sum of their radii: d ij > (2R 0 ). (4) Failure under pressure load occurs in our model when the actual mutual particle distance is less by a factor α (with 0 <α< 1) than the initial mutual particle distance, i.e. when d ij <α · d 0 ij . (5) Note that particle pairs without a spring after pressure or tensile failure still interact via the repulsive potential and cannot move through each other. Initial Configurations Using polydisperse particle distributions directly influences char- acteristic properties of the material, e.g. the coordination number of the starting configuration. In the simple approach chosen here, mono-disperse disks are used and the initial coordination number of the system is adjusted via a compactness or density parameter Θ as simulation input, see Figure 4. Figure 4: Sample configuration for a system with N = 10000 elements for different values of density Θ. The color code 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 the picture series of Figure 5, which shows the dynamics of crack propagation. (a) (b) (c) (d) Figure 5: Crack initiation and propagation in the material upon uni-axial tensile load. Color code is: Green: force- free bonds. Red: tension. (a) After applying a force at both sides of the specimen, local tensions occur. (b) Initiation of a crack tip with local tensions concentrated around this tip. (c) Propagation of this crack through the material. (d) Failure. Shock Impact The discrete particle model was tested for the case of dynamic material behavior where high strain rates occur. Numerically, this corresponds to dynamic non-equilibrium simulations. Dy- namic experiments for materials characterization at very high strain rates are often done with a standard set-up, the edge-on impact (EOI), see e.g. [3,4], where the edge of a target ceramic specimen is hit by a projectile at high speed. Figure 6 displays a comparison 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 = 10 5 parti- cles. The material is hit at the left edge. A pressure wave (color-coded in blue, unloaded material: green) propagates through 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 the top row. The arrows indicate the location of the wave front propagating in the material. (g-i) The same computer sim- ulation as in (a-c), now displaying the occurring damage in the 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.
