## Abstract

A split Hopkinson pressure bar (SHPB) device in two-by-two form, including the bar bundle form and the single cylindrical bar form, was designed in response to the demand for the dynamic mechanical experiments for brittle materials such as concrete, rock, etc. The stress waveforms generated through a projectile impacting two different types of incident bars have been studied based on the one-dimensional stress wave theory and numerical simulation method. At last, based on the established two types of mesoscale concrete models with random convex polyhedral aggregates, we performed comparison analysis of SHPB numerical simulations for concrete materials with bar bundle and single cylindrical bar separately, so as to provide technical support for the manufacture and development of this experimental device. The results showed that the introduced two-by-two form SHPB device expanded the scope of practical application, and the wave dispersion effect existing in a large-diameter bar can be evidently reduced when we employed the bar bundle form, indicating its applicability to the dynamic mechanical experiments of concrete-like heterogeneous materials.

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## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51971188 and 51071134), the Science and Technology Major Project of Hunan Province (Grant No. 2019GK1012), and Huxiang High-Level Talent Gathering Program of Hunan Province-Innovative team (Grant No. 2019RS1059). All these supports are acknowledged.

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## Appendix: Establishment of Mesoscale Aggregate Concrete Model and Generation of Random Convex Polyhedral Aggregate

### Appendix: Establishment of Mesoscale Aggregate Concrete Model and Generation of Random Convex Polyhedral Aggregate

The mesoscale aggregate concrete model with spherical aggregates is first introduced. Three mutually independent random numbers, i.e., \(rdm_{1} \), \(rdm_{2} \), and \(rdm_{3}\) uniformly distributed between 0 and 1 are defined. In the three-dimensional Cartesian coordinate system, the center coordinates of the random spherical aggregate can be expressed as:

where \(R_{i} \) is the radius of the generated aggregate, and \(X_{L} \), \(X_{R} \), \(Y_{B} \), \(Y_{T} \), \(Z_{B} \), \(Z_{T} \) are coordinates of the boundaries of the cuboidal specimen.

For the cylindrical specimen, Eq. (A.1) is corrected as:

where \(R_{i} \) is the radius of the cylindrical specimen, \(Z_{B} \) and \(Z_{T} \) are coordinates of the bottom and top boundaries of the cylindrical specimen, respectively. After the coordinates of the aggregate center are generated, the *i*-th aggregate to be placed must have no overlap with the previous *(i-1)*-th one, that is, the distance between two ceners is greater than the sum of the aggregates’ radii, which can be expressed as:

In this paper, we adopt a relatively simplified method to guarantee that the polyhedra have no overlaps with others, that is, to generate the polyhedral aggregates randomly within a spherical aggregate. First of all, a basic octahedral aggregate is generated randomly within a sphere, as shown in Fig. 23. Then, a method referred to as the “outer radial extension method” is utilized to extend the eight faces of the basic aggregate for one time, that is, to define a spatial straight line through the spherical center (*a*, *b*, *c*) and the triangular surface centroid \(({x}',{y}',{z}')\):

Substituting Eq. (A.4) into the spherical equation \((x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}\), we have

Set \(t>0\), so the intersection point will be on the top of the extended surface. Connect the intersection point with the corresponding three vertices of the triangular surface, constituting the concave-convex irregular polyhedra, as shown in Fig. 24. After that, the convex polyhedral aggregate is generated by the connection of the eight salient points, as shown in Fig. 25. Finally, we can scale the aggregate down to 90% and perform Boolean operation in ANSYS software to generate the interfacial layer, as shown in Fig. 26. However, to reduce the amount of mesh cells and shorten calculation time, we adopt a rather simplified method dealing with the interfacial layer in this paper.

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Guo, R., Ren, H., Zhang, L. *et al.* Research of an SHPB Device in Two-by-Two Form for Impact Experiments of Concrete-Like Heterogeneous Materials.
*Acta Mech. Solida Sin.* (2021). https://doi.org/10.1007/s10338-021-00218-y

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### Keywords

- SHPB
- Two-by-two form
- Numerical simulation
- Projectile
- Incident stress waveform