Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Kink band predictions in fiber composites using periodic boundary conditions A.B. Jensen, J. Thesbjerg, J.L. Wind, H.M. Jensen ⁎ Department of Engineering, Aarhus University, Denmark ABSTRACT A finite element based scheme for modelling kink band formation in fiber composites was developed. The model is computationally efficient by requiring only a few number of fiber/matrix layers representing the microstructure of the composite material. Yet, in comparison to finite element models including many fiber/matrix layers, the model is accurate by specifying periodic boundary conditions along the sides of the representative volume elements. These periodic boundary conditions allow for a rotation of the kink band after its initiation as observed in full-scale finite element calculations and experiments. Three different computational schemes for determining the kink band rotation have been suggested and compared. 1. Introduction A critical failure mode leading to loss of load carrying capacity in composite materials under compression parallel to the fibers or layers is the formation of kink bands. The development of efficient numerical methods for studying kink band initiation and propagation in fiber composites have been the objective in a number of previous works. Finite element studies based on individually discretized, planar fiber and matrix models were reported in [1] and [2]. The models were ex- tended to 3D simulations in [3]. It was in these studies established that a large number of layers had to be included in such computational models to eliminate effects from the sides of the representative volume element, which are taken as free edges. A drawback of such methods is obviously the computational effort needed to discretize large volumes of fiber/matrix layers in realistic simulations of composite structures and to avoid effects of boundary conditions imposed in the simulations of infinite kink band development. The study in [4] of kink band for- mation in realistic composite structures emphasizes the complexity of the problem and the need for developing efficient numerical schemes for predicting onset of failure. An approach for overcoming the computational challenges involved in individually discretized fiber-matrix models for realistic geometries is described in [5]. Here, a coupling between a sub-model and a super element was introduced. Another approach is to introduce an effective constitutive model representing the composite material in the finite element formulation. Such a constitutive model for composite materials including non-local effects due to fiber bending stiffness was introduced in [6] and applied in finite element simulations of kink band formation in materials with variations in the local fiber orientation to simulate realistic composite structures. Input parameters in the model for- mulated in [6] are effective overall constitutive models for the com- posite material. The model incorporates a characteristic length scale and, thus, is capable of predicting the kink band width. In [7] and [8], a constitutive model for a composite material in- cluding the constitutive responses of the fibers and the matrix was in- troduced and applied to study kink band initiation in perfect as well as imperfect structures. Input parameters to this model are individual constitutive models for the fibers and the matrix, along with volume fractions and continuity and equilibrium conditions at interfaces. The constitutive model was in [9] used to study the post buckling response and kink band broadening observed in ductile composite materials [10] and [11]. The constitutive model in [7] was in [12] implemented as a user defined material model in a commercial finite element software package, and was used to replicate the simple model predictions in [7] and [8]. The results based on this formulation were compared to the layer-wise individually discretized finite element calculations of [1] and [2] and were also used to study more realistic composite structures [13] where such approaches are computationally more efficient than individually discretized fiber-matrix layer models. Recently, a mixture of the two approaches in [6] and [7] was for- mulated in [14]. A micro-mechanically non-local constitutive model for the composite incorporating a length scale to account for fiber bending stiffness effects was formulated. An implementation of the model in a finite element scheme was also done in [14] and demonstrated the capabilities of such models to predict the onset and evolution of a kink band. The results were in good agreement with individually discretized fiber-matrix layer models. Another recent contribution [15] introduces a length scale to model of kink band formation for damage growth by https://doi.org/10.1016/j.compstruct.2018.09.055 Received 23 January 2018; Received in revised form 6 August 2018; Accepted 18 September 2018 ⁎ Corresponding author. E-mail address: hmj@eng.au.dk (H.M. Jensen). Composite Structures 207 (2019) 331–339 Available online 20 September 2018 0263-8223/ © 2018 Elsevier Ltd. All rights reserved. T