Fiber Bundle Models for Composite Materials F. Raischel, F. Kun and H.J. Herrmann Abstract In this paper we will outline the advantages of using fiber bundles in modeling disordered materials. We will present the main aspects of classical fiber bundle models (FBMs) and highlight their ability to capture some essential aspects of composite materials. We then present some extensions of the classical FBM, modifying the way of interaction of fibers, failure law, deformation state and constitutive behavior. The subclasses of FBMs which we generate in this way, i.e., the variable range of interaction model, the continuous damage FBM, the beam model, and the plastic FBM, will turn out to overcome some of the limitations of classical FBMs. We will discuss the characteristic dynamics of these model systems and give references to experiments. 1 Introduction From the viewpoint of statistical physics, unidirectional fi- brous composites can be reduced to just a few characteristic properties: firstly, they contain load-carrying fibers embed- ded in a matrix material which carries a negligible amount of load, but provides for the load transfer between broken and unbroken fibers. Secondly, the material is highly anisotropic, so a characteristic picture can be obtained by considering an uniaxial loading condition, with the load being applied in the fibers’ direction. Thirdly, the load bearing capacity of the fibers is highly disordered, necessitating a description in terms of disordered materials. Finally, a complete dynam- ical description of the failure process needs to account for the load distribution processes that occur when a fiber rup- tures and its load has to be carried by the remaining intact fibers. Fiber bundle models (FBMs) —as we will show in the following— naturally provide these characteristics, and recent developments and additions have made them more applicable to a broad variety of experimental situations. The damage and fracture of disordered materials is a very important scientific and technological problem which has at- tracted intensive research over the past decades. One of the first theoretical approaches to the problem was the fiber bun- dle model introduced by Peires in 1927 to understand the strength of cotton yarns [1]. In his pioneering work, Daniels provided the probabilistic formulation of the model and car- ried out a comprehensive study of bundles of threads assum- ing equal load sharing after subsequent failures [2]. In order to capture fatigue and creep effects, Coleman proposed a time dependent formulation of the model [3], assuming that the strength of loaded fibers is a decreasing function of time. Later on these early works initiated an intense research in both the engineering [4] and physics [5; 6; 7] communi- ties making fiber bundle models one of the most important theoretical approaches to the damage and fracture of dis- ordered materials [8]. Whereas from the engineering point of view, FBMs are a starting point to develop more realis- tic micromechanical models of the failure of fiber reinforced composites, physicists are primarily concerned with embed- ding the failure and breakdown of materials into the general framework of statistical physics and clarifying its analogy to phase transitions and critical phenomena. In this article we first present the basic formulation of the classical fiber bundle model. We then discuss limitations of the model to describe the fracture of fiber reinforced com- posites and propose extensions which make the model more realistic. 2 The Classical Fiber Bundle Model The disordered solid is represented as a discrete set of par- allel fibers of number N , organized on a regular lattice, see Fig. 1a. The fibers can solely support longitudinal deforma- tion which allows to study only loading of the bundle parallel to fibers. When the bundle is subjected to an increasing ex- ternal load F , the fibers behave linearly elastic until they break at a failure load σ i th ,i =1,...,N , as it is illustrated in Fig. 1b. The elastic behavior of fibers is characterized by the Young modulus E, which is identical for all fibers. The strengths σ i th are independent identically distributed random variables with the probability density p(σ th ) and distribution function P (σ th ). The randomness of breaking thresholds is assumed to represent the disorder of heterogeneous ma- terials. A widely used distribution in FBMs is the Weibull distribution P (σ th )=1 - exp  -  σ th λ  m  , (1) where m and λ denote the Weibull index and scale param- eter, respectively. After a fiber has failed, its load has to be shared by the remaining intact fibers. Historically, two extremal cases of 1