Contents lists available at ScienceDirect Materials Today Communications journal homepage: www.elsevier.com/locate/mtcomm Bridging the gap between structural levels in concrete technology – Principles of geometric averaging Kai Li a, *, Piet Stroeven b a Key Laboratory for Green & Advanced Civil Engineering Materials and Application Technology of Hunan Province, College of Civil Engineering, Hunan University, Changsha, 410082, China b Faculty of Civil Engineering and Geosciences, Delft University of Technology, 2628CN, Delft, the Netherlands ARTICLE INFO Keywords: Cementitious materials Geometric averaging Cauchy concepts Fibers Cracks Pores ABSTRACT Research on cementitious materials can provide geometric data on structural topics of interest. Such structural data should be subjected to a process called “geometric averaging” to be coupled on the engineering level with properties like strength, stiffness or permeability. The prime purpose of this work is to demonstrate that the mathematics underlying the process of going from the materials structure to engineering properties is governed by simple stereology-based constants. Particularly, Cauchy concepts are referred to for expressions relating the length of a curved line in a plane or in space, alternatively of a curved surface in space, to extensions of total projections on lines, respectively planes, in random directions. The two parameters due to Cauchy, i.e., 2/π and 1/2 will be demonstrated dominantly governing the geometric averaging process. The stereological metho- dology to assess the geometric material parameters and the engineering application of the outcomes are also briefly touched upon in the paper. 1. Introduction A significant part of concrete research is conducted on lower mi- crostructural level than the engineering one. Yet, obtained results can only be used for practical purposes on the highest level. So, a process of structural averaging is necessary, which is commonly denoted as “geometric averaging” for engineering materials. The major subject of interest is concerned with curved surfaces in space, i.e. surfaces of particles, cracks or pores. This covers the range of centimetres, via millimetres to micrometres (and recently even to nanometres). However, curved lineal features in space could also be at issue, i.e. in the case of fiber reinforcement. Existing literature in most cases is presenting methodological details that, unfortunately, sometimes give rise to controversial outcomes. Of course, nowadays, researchers have to be aware that the proper methodology should be based on stereo- logical notions. Moreover, it may be appreciated to know that the mathematical principles for geometric averaging go back to pre-ste- reological times [1,2]. A large number of publications about the proper geometric averaging approaches to such structural entities (surfaces and lineal features of different scales) are already available [3–6]. Therefore, this paper mainly aims at concentrating on the (pre-) ste- reological framework of the methodology, which makes it possible unifying the various approaches. The engineering topics at issue (particle packing, fracture, reinforcement efficiency, permeability) will not be discussed in full depth. The topic covered by this paper will be approached in two steps. Firstly, the curved lineal features will be discussed, followed by the curved surfaces. Hence, complexity is seemingly increasing. However, it will be demonstrated that the involved mathematics are similar and simple in both cases. 2. Analytical approach 2.1. Stereological principles for curved lines The concept of how to assess the length of a curved line in a plane is due to Cauchy [7]. Additionally, it should be mentioned that Steinhaus [8] was reported having used Cauchy’s method already for the same purpose. The curved line is projected in random or sytematic directions, as shown in Fig. 1. The total projections L’ can easily be measured, whereupon the successive measurements are simply averaged. When L is the length of the curved line, its total projection in a direction β can be written as L’(β). Hence, according to Cauchy https://doi.org/10.1016/j.mtcomm.2020.101061 Received 12 October 2019; Received in revised form 12 February 2020; Accepted 9 March 2020 ⁎ Corresponding author. E-mail address: kaili@hnu.edu.cn (K. Li). Materials Today Communications 24 (2020) 101061 Available online 10 March 2020 2352-4928/ © 2020 Elsevier Ltd. All rights reserved. T