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

Fig. 1. Peridynamic body B modeled as a sum of material points at x k with horizon δ , point family H k , and bond ξ to the neighboring point x j in reference configuration and in deformed configuration with ordinary state-based force density T (see Madenci and Oterkus (2014))

fracture because here, a non-local formulation of the material is stated. The method was initially introduced by Silling Silling (2000); Silling et al. (2007) and uses an integral equation to describe the relative displacements and forces between material particles. Because this equation is also defined for material discontinuities, peridynamics can con veniently be employed to model the spontaneous formation of cracks and fragments. In this paper, we show that peridynamic simulations are capable of computing pneumatic fracture in a qualitative and quantitative correct manner. To this end, we choose a standard ordinary state-based peridynamic model with a linear solid material and extend it to brittle fracture with traction on crack surfaces. In opposite to related works of driven fracture Nadimi and Miskovic (2015); Nadimi et al. (2016); Ouchi et al. (2015), we focus here on an assessment of the quality of the method - on one hand qualitatively, in terms of crack pattern and shape of the fragments, and on the other hand quantitatively, regarding the correspondence to analytical models of linear elastic fracture mechanics. Our results may later serve as a reference for further developments of peridynamics, in particular, regarding nonuniform discretizations and nonlinear material responses. The remaining of the paper is organized as follows: In Section 2, we shortly introduce the peridynamic theory used for our calculations. Then, in Section 3, we introduce the peridynamic approach to pressure-driven crack growth, compare sample computations with the analytical results of a penny-shaped crack model, and validate in this way exemplarily the correctness of our approach. Section 4 is devoted to the formation of cracks inside a concrete cylinder, whereby we first describe our pneumatic fracture experiments, and then we compare the shape of the fragments with our computational results. Finally, we summarize our results in Section 5. All numerical results were obtained using an in-house stand-alone C ++ code. We start presenting the basic equations employed in our calculations. In peridynamics the body B of a continuum is represented by N material points x k ∈ B , k = 1 , 2 , ..., N , each assigned with a mass density ρ k and a volume V k (see Fig. 1). The framework is non-local, i.e., each material point interacts with other points within a neighborhood of radius δ , the horizon . The horizon δ limits the so called point family H k of each material point to the set H k = x j ∈ B 0 < | x j − x k | ≤ δ ∀ x k ∈ B . (1) The bonds within the horizon are defined as the relative position vector in the reference configuration, ξ : = x j − x k . (2) In the current configuration, the material points are denoted by y k , and then a bond in deformed configuration reads y j − y k = ξ + η , (3) where u k = u ( x k , t ) is the material point displacement vector and η : = u j − u k . According to Silling (2000), the bond strain ε b is introduced in analogy to the strain in a rod as ε b = | y j − y k | − | x j − x k | | x j − x k | . (4) 2. Peridynamic theory