Computational modeling of the initiation and propagation of complex fracture is central to the disci- pline of engineering fracture mechanics. This review focuses on two promising approaches: phase- ﬁeld (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are brieﬂy summarized for both models. Validation against experimental data is essential for all com- putational methods to demonstrate predictive accuracy. To that end, The Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture mod- eling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized. APREPRINT - JANUARY 26, 2022 1 Introduction Fracturing phenomena in natural and engineered systems is studied extensively experimentally, theoretically, and computationally. Here we focus on two promising ap- proaches: phase-ﬁeld (PF) and peridynamics (PD) for the computational modeling of fractures in materials. Both approaches are distinct from earlier ones as they seek to predict crack path as functions of specimen loading and geometry. Ideally, these approaches attempt the compu- tational predictions after an initial calibration using engi- neering constants such as Young’s modulus and fracture toughness speciﬁc to the material. These ambitious ap- proaches are called free fracture models. This review is intended as a snapshot capturing in broad strokes the mod- eling details, assumptions, experimental data sets, and numerical simulations necessary for validation. These methods have the potential to address fundamental issues in complex fracturing with minimal introduction of phe- nomenological modeling assumptions and numerical tun- ing parameters. However, systematic comparative analy- sis for these models, together with validation studies on the set of experiments, are rare. In this review, we attempt to initiate such a comparative analysis and, when possible, invoke validation studies from the experimental literature. As an example of an engineering fracture mechanics ap- plication, Hattori et al. (2017) [1] presented a compre- hensive comparison of various numerical approaches for the hydraulic fracturing of shale and showed the advan- tages as well as limitations of many numerical approaches including peridynamics (PD) and phase-ﬁeld (PF). How- ever, this comparative analysis for various models lacked validation studies on the same set of hydraulic fracturing experiments in order to evaluate predictive capabilities of numerical models. Our review is motivated by the recent workshops on phase-ﬁeld, peridynamics, and experimen- tal fracture mechanics held at The Banff International Re- search Station: Hydraulic Fracturing: Modeling, Simula- tion, and Experiment1, and the Workshop on Experimen- tal and Computational Fracture Mechanics2[2]. 1.1 Other review papers One of the ﬁrst overview papers of peridynamic and non- local modeling was written by Du and Lipton3in the SIAM news, volume 47 (2014). However, this article was quite brief. In 2019, Javili et al. [3] published a review emphasizing the applications of PD. In the same year Diehl et al. [4] published a review with the focus on benchmarking PD against experimental data. In 2021, Isiet et al. [5] published a review on the usage of PD for impact damage. In the same year, Hattori et al. [6] pub- lished a review on the usage of PD in reinforced concrete 1https://www.birs.ca/events/2018/ 5-day-workshops/18w5085 2http://wfm2020.usacm.org/ 3https://sinews.siam.org/Details-Page/ peridynamics-fracture-and-nonlocal-continuum-modelsstructures. Zhou and Wang [7] published a review on the usage of PD in geomaterials in 2021. To summarize, ex- cept for the ﬁrst PD review in 2019, all reviews focused on some speciﬁc topic. Hidayat et al. [8] published a review on the connection of meshfree PD between other meshfree methods in 2021. In addition, the following three books about PD are available [9–11]. In what follows, we now list PF review papers and mono- graphs. The ﬁrst summary on variational modeling of fracture was by the founding authors Bourdin, Francfort and Marigo [12]. Although they explicitly state on page 7 that they are not attempting to review and access ex- isting literature, their work describes on 148 pages the current state-of-the-art in the year 2008. The ﬁrst review papers (that have explicitly this purpose) with regard to computational/engineering aspects of phase-ﬁeld (varia- tional) fracture were published by Rabczuk [13] and Am- bati, Gerasimov, and De Lorenzis in 2015 [14]. In the latter study, despite review aspects, a new formulation for stress splitting is proposed therein. Due to the ongoing increasing popularity, shortly after, various other reviews, monographs, and news articles appeared. A short SIAM news article about the latest developments and future per- spectives was published by Bourdin and Francfort in 2019 [15]. An extensive review paper on theoretical and com- putational aspects was done by Wu et al. in the year 2020 [16]. The authors also discuss the success or fail- ures of several benchmark problems for quasi-static, dy- namic brittle/cohesive fracture. Only considering phase- ﬁeld (variational) fracture, this review paper certainly ex- ceeds in topics, modeling details and numerical aspects and examples of our current work. In the same year, a monograph on multiphysics phase-ﬁeld fracture was pub- lished by Wick [17]. Therein, the focus is brieﬂy on modeling of classical variational phase-ﬁeld fracture, er- ror estimation and adaptivity, and including other physics phenomena via volume or interface coupling techniques. Most recently in 2021, one of the original authors, namely, Francfort, published an article entitled with ‘Variational fracture: Twenty years after’ [18]. 1.2 Purpose and value of this review paper In view of existing (recent) reviews on both approaches, we shall explain the purpose and value of the current re- view paper. First and most importantly, both PD and PF are appearing to be the most prominent approaches for free fracture modeling. Other notable approaches are the displacement-discontinuity method [19], cohesive-zone models [20], boundary elements [21], XFEM/GFEM [22– 24], and the eigen-erosion framework [25–28]. A compar- ative review between XFEM (extended ﬁnite elements), mixed FEM, and phase-ﬁeld models appeared in [29]. The present paper provides for the ﬁrst time a comparison between both approaches (PD and PF). For this reason, the mathematical descriptions remain rather short while fo- cusing on the key ingredients that allow for comparison of PD and PF. It is through this lens we reference the existing 2 APREPRINT - JANUARY 26, 2022 models and numerical methods from the literature. The Sandia fracture challenge is chosen for validation against experimental data. The ﬁrst part of the challenge is to cal- ibrate a free fracture model to simple prototypical prob- lems, e. g. a tensile test, and use this calibration to sim- ulate the crack and fracture phenomena. Thus, the model parameters can not be ﬁtted to the complex scenario and have been instead calibrated using the simple scenario. This additional step of calibration makes the Sandia frac- ture challenge an excellent problem to benchmark phase- ﬁeld models and peridynamic models to assess their per- formance on different kinds of crack and fracture phenom- ena. These numerical, computational, and experimental side-by-side comparisons allow us to identify similarities, common challenges, and speciﬁc aspects to each method. 1.3 Outline The paper is structured as follows: Section 2 introduces the two models and provides a basis on which the mod- els can be compared and contrasted. This summary is an adaptation and extension of the review papers and mono- graph [3, 4, 16, 17] of the two models. Section 3 addresses the fracture physics perspective from the macroscale view. Section 4 attempts to compare the predictive accuracy of the two models for validation against the experimental data. To that end, the Sandia Fracture Challenges data sets were analyzed and computed, the R2correlation of relative errors between the simulations and the experiment are presented. Section 5 compares computational aspects of the two models as well as pointing out challenges and opportunities for development. Finally, Section 6 summa- rizes the modeling capabilities of PD and PF. 2 Overview of models and numerical methodology This section brieﬂy introduces the two methods, peridy- namics (PD) and phase ﬁeld (PF), respectively. A brief overview of the governing equations, material models, discretizations, numerical analysis, and advanced visual- ization methods is given. We introduce the ingredients for the comparison of these two models and provide ref- erences to the extended literature for the interested reader. 2.1 The governing equation of peridynamics Peridynamics (PD), is a non-local adaptation of contin- uum mechanics (CCM). In PD, each material point Xin- teracts with its neighbors inside a ﬁnite interaction zone B(X)with the length , see Figure 1. This type of non- local interaction principle is also seen in molecular dy- namics (MD) [30] and smoothed particle hydrodynamics (SPH) [31, 32] simulations. The important feature of PD fracture modeling is that the interaction between the in- tact material and fractured material is modeled implicitly through a non-local ﬁeld equation that remains the same everywhere in the computational domain. Material dam-B(X)D Figure 1: Sketch for the principle of peridynamics where a material point Xinteracts with its neighbors inside a ﬁnite interaction zone B(X)with the length . age and fracture phenomena are captured through a force versus strain constitutive model. Because of this, PD is able to capture fracture as an emergent phenomena aris- ing from the non-local equation of motion. This contrasts with classic fracture theory where, off the crack, the elas- tic interaction is modeled by the equation of elastodynam- ics and the fracture set is a free boundary with motion cou- pled to elastodynamics through a physically motivated ki- netic relation. Other recently developed non-local models exhibiting emergent behavior include the Cucker Smail equation, where swarming behavior emerges from leader- less ﬂocks of birds [33–36]. The equation of motion for bond-based peridynamics pro- posed by Silling [37, 38] reads as %(X)u(t;X) =Z B(X)f(u(t;X0)u(t;X);X0X)dX0+b(t;X), (1) where%2Ris the material density,  u2Rnis the ac- celeration at time t2Rof the material point X2Rn, f:RnRn[0;T]!Rnis the pair-wise force func- tion,b2Rnis an external force density, and u2Rnis the state of deformation at a point in space time, (t;X). The constitutive model for the pair-wise force function f is material dependent and deﬁnes how the internal forces react to the displacement u. For the elastic regime the force interaction between a pair of points XandX0is a so-called bond and contributes to the name of bond-based PD. The pair-wise interaction leads to a constraint on the Poisson ratio [39, 40]. To overcome the Poisson ratio constraint more general non-local interactions are included in the “state-based” peridynamic model. The generic state-based peridynamic equation of motion introduced by Silling et al. [41] reads 3 APREPRINT - JANUARY 26, 2022 as %(X)u(t;X) =Z B(X)(T[X;t]hX0XiT[X0;t]hXX0i)dX0(2) +b(t;X), where the pair-wise force function fis exchanged with the so-called peridynamic force state T:RnRn[0;T]! Rn. The usage of PD states lead to the name state-based PD. A peridynamic state relates to second order tensor and they assign each point in a neighborhood a tensorial quan- tity. However, in general, it is not a linear or continuous function with respect to XX0. For more details, we refer to [41, Section 2]. In peridynamics material damage is part of the consti- tutive modeling and the material is viewed as damaged when the force state at a point Xno longer inﬂuences a material point X0and vice versa. Damage occurs when the difference between deformation states at each point X andX0surpass a threshold. The speciﬁcs of how this oc- curs depends on the material model used. For example, for pairwise force functions fthe force acting between two points is often referred to as a bond. When the pair- wise force is zero, it is said that the bond is broken. Bonds can break irreversibly, or alternatively they can heal under the right conditions, this depends upon the material model used. To ﬁx ideas, we illustrate the constitutive law given by the prototype brittle microelastic (PMB) model introduced in [38]. The deformation of a point Xisu(t;X)and its position in the deformed conﬁguration is X+u(t;X). In this model, the pair-wise force function depends on the strain between two points in the deformed conﬁguration given by S(t;X;X0) := k(X0+u(t;X0))(X+u(t;X))kk X0Xk kX0Xk: (3) The direction vector between two points in the deformed conﬁguration is given by e=(X0+u(t;X0))(X+u(t;X)) k(X0+u(t;X0))(X+u(t;X))k: (4) The force directed between the two points in the deformed conﬁguration is given by the constitutive law f(u(t;X0)u(t;X);X0X) := (5) (t;X;X0)cS(t;X;X0)e; (6) wherec >0is the material dependent stiffness constant, and the scalar damage function is given by (t;X;X0) :=1; S (t;X;X0)sc 0;otherwise(7)wheresc>0is the material dependent critical bond stretch. Once the force between the two points goes to zero it stays zero. It is noted that there are several other deﬁnitions for the damage function using strain-based criteria [38, 42], as well as stress-based [43], or energy- based damage criteria [44, 45], see Figure 4. In numerical implementations crack path evolution is re- covered as a post-processing step after simulation. To identify the crack we introduce the damage ﬁeld d: [0;T]Rn!Rgiven by the density d(t;X)deﬁned by d(t;X) = 1R B(X)(t;X;X0)dX0 R B(X)dX0(8) where the scalar function : [0;T]RnRn!Rin- dicates if the bond between XandX0at timetis broken (= 0) or active (= 1). Locations where d= 0 indi- cate the crack set. The set where d= 0is highly localized and indicates the evolving crack path. This damage local- ization has been demonstrated mathematically by Lipton [46, 47] for peridynamic fracture evolution correspond- ing to the small deformation bond based models. The method uses Gronwall’s inequality together with methods from the non-local image approximation of the Mumford Shah model [48] introduced by Gobbino [49]. For more theory of peridynamic states, we refer to [41, Section 2]. For additional