Normalization of cohesive laws for quasi-brittle materials Johan Tryding ⇑ , Matti Ristinmaa Division of Solid Mechanics, Lund University, SE-221 00 Lund, Sweden article info Article history: Received 29 June 2016 Received in revised form 13 March 2017 Accepted 14 March 2017 Available online xxxx Keywords: Cohesive laws Normalization Concrete Paperboard abstract Analytical relations to describe experimentally measured traction-separation laws are often expressed in dimensionless quantities. The traction and the separation are commonly normalized using the cohesive strength and a length measure, respectively. The ratio between the fracture energy and the cohesive strength is often used as a length measure. An alternative length measure is the ratio between the cohesive strength and the maxi- mum slope of the traction-separation law. A relation between these two length measures are established. To illustrate the implications on cohesive laws, three existing cohesive laws are rewritten using the alternative normalization. As a result it is shown that the number of unknown material parameters can be reduced. One of the derived dimension- less cohesive law is validated against experimental uniaxial tension and compression load-deformation data of different sample sizes and different quasi-brittle materials, i.e. concrete and paperboard. A good fit of the cohesive law is shown to all the investigated data. These findings indicate that the derived normalized cohesive law is independent of material directions, moisture contents and sample size. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction Cohesive crack models utilizing traction-separation laws are successfully applied to a wide range of materials, such as concrete and rocks [1], paper based materials [2], adhesive joints [3] and composite materials [4], to mention a few. The Hillerborg cohesive concept [1] has also been implemented into several commercial general purpose finite element codes. The usage of the cohesive zone concept in industrial applications, is illustrated e.g. in [5–7]. Thus knowing the traction- separation laws for the quasi-brittle material studied, simulation can be used to predict cohesive fracture. For arbitrary loading histories the cohesive law needs to include the characteristics of both the normal and shear cohesive behaviors, cf. [4,8]. Recently in [9,10] a thermodynamically consistent cohesive crack model for generally normal and shear loading was proposed utilizing only a calibration of the traction-separation law for tension loading. The model includes the Helmholtz free energy, loading and plastic potential functions together with a damage potential function calibrated to an analytical cohesive law in uniaxial tension. Analytical traction-separation laws used to model, e.g. concrete data, range from linear, bilinear, exponential to non- linear functions, cf. [1,11–14]. These functions are often expressed as dimensionless analytical cohesive laws, cf. e.g. [5], where the cohesive stress commonly is normalized using the cohesive strength. The normalization of the separation is not as obvious as for the cohesive traction and a frequently used normalization is obtain by relating the separation to the ratio between the fracture energy and the cohesive strength, cf. e.g. [5]. An alternative length measure is found from the ratio http://dx.doi.org/10.1016/j.engfracmech.2017.03.020 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: johan.tryding@tetrapak.com (J. Tryding). Engineering Fracture Mechanics xxx (2017) xxx–xxx Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech Please cite this article in press as: Tryding J, Ristinmaa M. Normalization of cohesive laws for quasi-brittle materials. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.020