The authors gratefully acknowledge financial support provided by the Faculty of Civil Engineering, Brno University of Technology under project No. FAST-T-J-23-8329

Eliáš, J.

Fleischerova, B.

DSpace at University of West Bohemia

Development of meso-scale discrete modelfor simulations of ballistic experimentsB. Fleischerováa, J. EliášaaInstitute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology,Veveřı́ 331/95, 602 00 Brno, Czech RepublicFiber-reinforced concrete panels are used for ballistic protection. During their tests, the materialis loaded by a high-velocity projectile. This dynamic loading causes significant damage andfracturing in concrete. The cracks results in discontinuity in the displacement field.In case of heterogeneous material with strain softening constitutive response, the discretemeso-scale material representation can conveniently replace the most common finite elementmethod. Discrete models allow straightforward description of the discontinuity in the displace-ment field and inhomogeneities of the material are represented by an assembly of intercon-nected discrete units. A number of discrete models were developed by many scientists all overthe world [1, 3].The meso-scale models directly represent material heterogeneity provide material lengthscale and fluctuation of stresses. They are computationally expensive, but reducing kinematicsby discrete representation helps to substantially decrease their computational burden.The presented meso-scale discrete model similar to the Lattice Discrete Particle Model [1]is built in programming language C++. The concrete is described as a two-phase material(aggregates and mortar). Individual rigid particles represent larger aggregates and surroundingmatrix and are created by placing spheres randomly into the domain with a minimal mutualdistance corresponding to maximum diameter of the real mineral grains. Each two neighboringparticles are connected via bonds, which define a contact face. The stiffness of the bonds aremodified during the deformation according to the nonlinear constitutive functions defined invectors.The Voronoi tesselation is used to build the geometry of the rigid particles and ensure theperpendicularity between bonds and contact faces. The particles have polyhedral geometry andat least three vertices define contacts for 3D case. For 2D case, the particles are polygons andcontacts are line segments prescribed by two vertices. Edges of Delaunay simplices defineconnections between the particles and therefore also the mechanical elements. The elementsrepresents both aggregates and cement matrix behavior.Transient (dynamic) regime is used to simulate the ballistic experiments. A time discretiza-tion based on the finite difference method is usually used. Implicit time integration scheme,specifically the generalized-α method, is currently utilized by the model [2]. Unconditionallystability and reduced numerical damping are the main advantages of this time integration al-gorithm. The numerical energy dissipation is controlled by user-specified parameter, spectralradius ρ∞. The transient regime was verified on the simple example of the cantilever beam forboth 2D as well as 3D case. The model was loaded by vertical force at the free end. The resultsare shown by the authors in [4].49The extension of the model to include also the explicit integration scheme is under develop-ment. Explicit methods are more suitable for short time periods with high-rate loading, but thestability of the calculation depends on the time step length.Concrete, as a natural brittle material, performs well in compression. On the other hand, itsefficiency in tension is significantly lower. Therefore, the reinforcement is usually required toimprove poor performance in tension. In recent decades, short randomly distributed fibers ofsmall diameter are very popular to use. Nowadays, fiber-matrix interaction is broadly reportedby many works. Although each work has its own contribution to this scientific field, most ofthem refer to the work of Naaman et al. [5].The model simulates behavior of a fiber crossing a single crack. Bridging forces from theleft and the right side of the fiber has to be equal. The equilibrium is satisfied during the wholesimulations. Newton iterative method is used to find the equilibrium between right and leftbridging force. Contribution of the fibers in the elastic regime is neglected.The mechanical behavior of the fiber can be divided into two main stages. First, there isthe debonding stage, where the fiber is still bonded with the matrix at some length. The secondis the pulling-out stage, where the fiber is fully debonded from the matrix. The stages andtypical curves of pullout vs. bridging force diagram are shown in Fig. 1. Non-zero parameterβf describe softening or hardening of the material. The border between the debonding and thepulling-out stage is represented by critical pullout of the fiber νcrit. During the whole simulation,the rupture condition as well as the full pullout of the fiber (ν = νcrit) have to be controlled.The bridging force immediately decrease to zero for the rest of the computation in both cases.0 1 2 30.00.10.20.3bridgingforceF(ν)[N]pullout of the fiber ν [mm]F0νcrit νlimβf = 0βf > 0βf < 00 1 2 30.00.10.20.3F0νcrit νlimβf = 0βf > 0βf < 0(a) Fracture energy Gd > 0 (b) Fracture energy Gd = 0Fig. 1. Typical bridging force F(ν) versus pullout ν relationshipsWhen the crack is closing, the linear unloading-reloading functions towards the origin(F (ν) = 0, ν = 0) are used to find the equilibrium between left and right side of the fiber.When the crack starts to open again, the same linear functions are used unless the crack widthreaches the maximum from its previous opening.The transient regime with fiber-reinforced concrete slabs are currently being tested. Themodel is loaded by prescribed displacement at one point. The convergence of the simulationsas well as the obtained forces are analyzed. Boundary conditions of the model correspond tothe real ballistic experiments. Also explicit integration scheme is being validated. Some partialresults will be shown during the conference.50AcknowledgmentThe authors gratefully acknowledge financial support provided by the Faculty of Civil Engi-neering, Brno University of Technology under project No. FAST-J-23-8329.References[1] Bolander, J. E., Eliáš, J., Cusatis, G., Nagai, K., Discrete mechanical models of concrete fracture,Engineering Fracture Mechanics 257 (2021) No. 108030.[2] Chung, J., Hulbert, G. M., A time integration algorithm for structural dynamics with improvednumerical dissipation: The generalized-α method, Journal of Applied Mechanics 6 (2) (1993)371-375.[3] Cusatis, G., Bažant, P., Cedolin, L., Confinement-shear lattice model for concrete dam-age in tension and compression, Journal of Engineering Mechanics 129 (12) (2003), doi:10.1061/(ASCE)0733-9399(2003)129:12(1439).[4] Fleischerová, B., Eliáš, J., Development of discrete model for dynamic simulations of concretebehavior, AIP Conference Proceedings 2887 (1) (2023) No. 020007.[5] Naaman, A. E., Namur, G. G., Alwan, J. M., Najm, H. S., Fiber pullout and bond slip. I: Analyticalstudy, Journal of Structural Engineering 117 (9) (1991) 2769-2790.51

Development of meso-scale discrete model for simulations of ballistic experiments

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Development of meso-scale discrete model for simulations of ballistic experiments

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