PSI - Issue 35

Kai Friebertshauser et al. / Procedia Structural Integrity 35 (2022) 159–167

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K. Friebertsha¨user and M. Werner and K. Weinberg / Structural Integrity Procedia 00 (2021) 000–000

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For each material point the equation of motion reads ρ k ¨ u ( x k , t ) = L u ( x k , t ) + b ( x k , t ) ∀ x k ∈ B , t ≥ 0 , (5) with acceleration vector ¨ u , an inner force density L u and an external force density b . Here, we use the classical ordinary state-based approach of Silling et al. (2007) because it is well suited to represent linear elastic material behavior. Another reason is that, in contrast to the bond-based formulation, there is no limitation to a Poisson ratio of ν = 1 / 4 to model the material properties of concrete, which are used in the experiments (see Section 4). In the state-based theory the inner force density L u is defined with the help of a force vector state T acting on the bond · as L u ( x k , t ) = H k T [ x k , t ] x j − x k − T [ x j , t ] x k − x j d V j . (6) Presuming a linear solid material, the force vector state T acts in the direction of the bond deformation, T k ξ = t k ξ ξ + η | ξ + η | (7) with the material specific force scalar state t k , t k ξ = κ · θ k m k · w ξ · | ξ | + 15 · µ m k · w ξ · e dev ξ . (8) In this expression κ and µ are bulk and shear modulus, respectively, θ k is the local dilatation, and e dev = ( ε b − 1 3 θ k ) · | ξ | is the deviatoric part of the bond elongation. The parameter m k denotes a weighted volume and w is a bond influence function, see Silling et al. (2007). In the context of this work, material failure is described with a damage model of critical bond elongation, cf. Silling (2000), using the quantities of bond failure d b and damage D k . According to this model, bond failure occurs when the strain (4) exceeds a critical value ε c . An intact bond has the numerical value d b = 1, whereas a broken bond is described by d b = 0, d b = 0 if ε b > ε c 1 otherwise . (9) The critical bond elongation ε c is calculated from the critical energy release rate G c according to Madenci and Oterkus (2014) as ε c = G c 3 µ + 3 4 4 κ − 5 µ 3 δ . (10) Damage of the k -th material point is given by the volume fraction of a damaged point family in comparison with the undamaged state, and defined as D k = 1 − H k d b d V j H k d V j (11) with D k ∈ [0 , 1]. The damage model is used in the influence function w ξ for which w ξ = d b · δ | ξ | (12) applies. Our simulations of pneumatic fracture consider a short period of time, so an explicit time integration scheme is used. We employ the velocity-verlet algorithm of Littlewood (2015), as displayed in Appendix A. In each time step, the acceleration ¨ u ( x k , t ), the velocity ˙ u ( x k , t ), and the displacement u ( x k , t ) are determined for each material point k . The acceleration ¨ u is calculated from the equation of motion (5).