A Review of Benchmark Experiments for the Validation of Peridynamics Models
Abstract
Peridynamics (PD), a nonlocal generalization of classical continuum mechanics (CCM) allowing for discontinuous displacement fields, provides an attractive framework for the modeling and simulation of fracture mechanics applications. However, PD introduces new model parameters, such as the socalled horizon parameter. The length scale of the horizon is a priori unknown and need to be identified. Moreover, the treatment of the boundary conditions is also problematic due to the nonlocal nature of PD models. It has thus become crucial to calibrate the new PD parameters and assess the model adequacy based on experimental observations. The objective of the present paper is to review and catalog available experimental setups that have been used to date for the calibration and validation of peridynamics. We have identified and analyzed a total of 39 publications that compare PDbased simulation results with experimental data. In particular, we have systematically reported, whenever possible, either the relative error or the Rsquared coefficient. The best correlations were obtained in the case of experiments involving aluminum and steel materials. Experiments based on imaging techniques were also considered. However, images provide large amounts of information and their comparison with simulations is in that case far from trivial. A total of six publications have been identified and summarized that introduce numerical techniques for extracting additional attributes from peridynamics simulations in order to facilitate the comparison against imagebased data.
Keywords
Peridynamics Benchmark problems Validation Experimental data Fracture mechanics Wave propagation Visualization techniques1 Introduction
Peridynamics [113] provides a novel framework for the modeling and simulation of fracture mechanics applications. Peridynamics (PD) can be viewed as a nonlocal generalization of classical continuum mechanics (CCM), in which the mechanical behavior is modeled in terms of an integral equation of the displacement field rather than by a partial differential equation. The general framework for peridynamics can be further classified into two approaches: (1) bondbased PD [113, 117] and (2) statebased PD [118]. In all cases, the PD formulation allows for discontinuous displacement fields, which may be seen as an attractive feature of the method for the treatment of problems dealing with fracture mechanics. Another advantage of PD is that failure phenomena, like in molecular dynamics or atomistic lattice models, can be explicitly included in the constitutive models. In other words, no other external criterion is needed within the peridynamics framework for modeling the initiation and growth of cracks, in contrast to continuum models. This implies that PD is well suited for the simulation of fracture mechanics applications initially devoid of cracks. We refer to [10, 87] for a general description of PD and to [116] for a review of its advantages.
Although the PD framework appears to be a promising approach for many problems in solid mechanics, a collective effort remains to be done to validate the bondbased and statebased constitutive models in order to gain confidence in the approach’s predictability and reliability. In particular, PD introduces new parameters, such as the socalled horizon, that are a priori unknown and need to be identified. Both calibration and validation processes require experimental data to assess model’s adequacy and predictability [41, 42]. Another challenge is the treatment of boundary conditions in the case of nonlocal models. For example, boundary traction does not naturally appear in the nonlocal formulation [88], which implies that external loads should be applied in a manner different from that in classical continuum mechanics.
The main objective of this survey paper is to review and catalog the available experimental data that have already been used in the literature for the validation of peridynamics models and simulations. Our goals are essentially to overview the experimental setups and observables that have been achieved so far and to summarize the validation results presented by the authors, if available.
The experiments are classified here into three main categories: (1) the first category is concerned with dynamical experiments that involve wave propagation phenomena; (2) the second category lists the experiments that focus on characterizing the initiation and growth of cracks for various materials; (3) the last category gathers the experiments whose outputs are digital images of observed phenomena. This classification was arbitrarily chosen to simplify the presentation of the experiments.
The experimental data has been classified by type: (1) scalar values of observables; (2) scalar functions of input parameters; (3) and digital images. We report the relative error between experiments and predictions for scalar observables. We estimate the socalled coefficient of determination Rsquared [26], also referred to as the Pearson correlation coefficient R, between the data and model predictions for scalar functions, like series of points. Data stored in the form of images provide a large amount of information, which makes comparison with simulations less trivial. It may be necessary to develop advanced visualization methods [60, 61, 140] to take full advantage of the richness of imaging techniques and be able to fully compare the results from nonlocal model simulations such as those obtained from peridynamics or molecular dynamics. We think that these methods could be useful to the community and therefore report on existing methods, e.g., extraction of fragments or crack surfaces.
We hope that this survey will provide a useful resource for those interested in the validation of PD models. We also believe that such a contribution could lead to the definition of benchmark problems and experiments for the analysis of PD simulations. This has been in fact the source of motivation for writing such a paper.
The paper is structured as follows: Section 2 presents some preliminaries, in particular, the approach that we followed to collect the publications relevant to the survey and a brief description of the method to estimate the coefficient of determination Rsquared. Section 3 presents the list of dynamical experiments based on wave propagation phenomena for the validation of crack propagation and Section 4 reviews the experiments used for the validation of crack initiation/propagation. Section 5 reports in a concise fashion the relative error or the coefficient of determination for each of the previous experiments that provide a confidence level in the validity of the peridynamics models. Section 6 describes advanced visualization techniques and methods to extract additional attributes for comparison against experimental data. Section 7 provides conclusions and perspectives on the stateoftheart in the validation of peridynamics models.
2 Preliminaries
2.1 Literature Search Strategy
 1.
A search with keywords “peridynamics + experiment” and “peridynamics + benchmark” resulted in a total of 53 references, of which 39 papers directly dealt with comparison of simulation results against experimental data. Note that we chose to include a search based on the keyword “benchmark” to find works in which authors used experimental data to verify the implementation of their codes. These papers do not necessarily address model validation. Although we will concentrate on the results of the 39 papers most relevant to this study, the remaining 14 articles are also included in the list of references.
 2.
A search with keywords “peridynamics + computer graphics” and “peridynamics + visualization” was also carried out and resulted in five papers, among which three utilize techniques from computer graphics for the comparison with experimental data and two use peridynamics for physicsbased rendering.
2.2 Choice of Validation Metric
Validation processes involve the comparison of metrics and threshold values between experimental data sets and predictions. Many metrics, either statistical or deterministic, have been introduced in the past depending on the nature and availability of data. For consistency throughout the paper, we have chosen to consider two simple metrics.
Some of the references did not provide any Rsquare coefficient. In that case, we chose to extract the values from the experimental and simulation data by using the webbased application WebPlotDigitizer^{1} and computed the R^{2} coefficient using the SciPy package.^{2} Note that the computed R^{2} value depends on the extraction procedure used to obtain the data from the published figures. The procedure that we followed here was first to take a screen shot of the figures and upload them to the WebPlotDigitizer. Then, we selected a range on the curve with tick marks on the xaxis and the yaxis, and provided the corresponding values on the axes. Following this calibration process, we defined uniformly distributed points along the two curves obtained from the experiment and simulation. We acknowledge that the choice of the calibration points and the equidistant points may influence the R^{2} value. For a given data set, we repeated the extraction process three times and observed that the variations would be on the order of 2%. We deemed this difference as acceptable.
We wish to emphasize at this point that accessing raw published experimental data in experimental fracture mechanics is difficult. A study of the longterm availability of raw experimental data was carried out in [35]. The main result of the study was that out of the 187 articles published from 2000 to 2016, only 11 data sets were still available and 42% of the authors who were contacted did actually reply. This situation explains why we had to resort to alternatives means to analyze the published data.
3 Wave Propagation Experiments
Material  Mechanical test  B  S  Exp  Sim 

Plastic polymer (allyl diglycol carbonate or CR39)  Stress wave propagation (halfplane)  ✓  [25]  [93]  
ALON/PMMA  Wave speed (edgeon impact experiment)  ✓  
Aluminum  SplitHopkinson pressure bar  ✓  [64]  
Steel (4340 RC 43)  SplitHopkinson pressure bar  ✓  [46]  [44]  
Sandstone (Bera and Massilion)  Wave dispersion and propagation  ✓  [18] 
3.1 Propagation of Stress Waves in a HalfPlane
3.2 Edgeon Impact Experiment
Zhang [141] compared the simulated and experimentally measured average propagation speed of the primary damage front [90]. The average propagation speed in the simulation was estimated at about 8.7kms^{− 1} while the corresponding value measured in experiments was about 8.4kms^{− 1}. The propagation speed of individual cracks obtained from PD simulations was about 4.8kms^{− 1} and about 4.6 kms^{− 1} for cracks growing in ydirection. In comparison, the value obtained from experiments was about 4.4 kms^{− 1}.
3.3 SplitHopkinson Pressure Bar
Foster et al. [44] also studied the Hopkinson pressure bar experiment [46]. Figure 4 depicts the sketch of the experimental setup. Their objective was to compare the predicted true strain  stress curves with the experimentally measured curves, for various strain rates. The R^{2} coefficient was computed here as 1.00 and 0.97 in the cases of strain rates of 1150 s^{− 1} and 2900 s^{− 1}, respectively.
3.4 Wave Dispersion in Bera and Massilion (Dry) Sandstone
Butt et al. [18] compared the predicted and experimental dispersion curves. For both sand stones, the experimental dispersion data measured at a confining pressure of 20 MPa [133] was used to fit the peridynamic model parameters. The R^{2} coefficient between the fitted material parameters and the experimental data used for the fitting was estimated at 1.00 and 0.93 for the Massilion sandstone and Berea sandstone, respectively.
4 Crack Initiation and Propagation Experiments
Applications of bondbased and statebased peridynamics for the comparison with experimental data
Material  Mechanical test  B  S  Exp  Sim 

Composite  Flexural test with an intial crack  ✓  [75]  [2]  
Composite  Damage growth prediction (sixbolt specimen)  ✓  [120]  [96]  
Composite  Damage prediction (centercracked laminates)  ✓  [70]  
Composite  Dynamic tension test (prenoteched rectangular plate)  ✓  [58]  
Steel  Crack growth (KalthoffWinkler)  ✓  ✓  
Aluminum/Steel  Fracture (compact tension test)  ✓  
Aluminum  Taylor impact test  ✓  
Aluminum (6061T6)  Ballistic impact test  ✓  [132]  [127]  
Concrete  Lapsplice experiment  ✓  [48]  [48]  
Concrete  3point bending beam  ✓  ✓  
Concrete  Failure in a Barazilian disk under compression  ✓  [51]  [54]  
Concrete  Anchor Bolt Pullout  ✓  [128]  [83]  
Glass  Dynamic crack propagation (prenotched thin rectangular plate)  ✓  
Glass  Impact damage with a thin polycarbonate backing  ✓  [59]  
Glass  Single crack paths (quenched glass plate)  ✓  [71]  
Glass  Multiple crack paths (quenched glass plate)  ✓  [71]  
Glass  Crack tip propagation speed  ✓  [15]  
PMMA  Fast cracks in PMMA  ✓  [39]  [2]  
PMMA  Tensile test  ✓  [124]  [32]  
Sodalime glass  Impact on a twoplate system  ✓  [130] 
4.1 Composites
 1.
Dynamic crack growth around stiff inclusion
The dynamic growth of a crack around a stiff inclusion was numerically studied with peridynamics by Agwai et al. [2]. In [75], an 140 mm×42 mm×8 mm epoxy, embedded with a central stiff glass inclusion (fiber) of diameter d = 4 mm and featuring an initial crack, is impacted at a velocity of 5.3ms^{− 1} using a pneumatic hammer with a hemispherical tip. Figure 6 depicts the setup. Experiments were conducted for specimens with weakly and strongly bonded fibers. Agwai et al. [2] qualitatively compared the predicted crack paths for weak and strong interfaces in the visualized simulation results with those observed in the images during the experiments. The objective of the study was primarily to investigate the influence of the bonding interface on the crack path. The simulations qualitatively captured the different failure modes that were observed in the experiment.
 2.
Damage growth in a doublelap joint test using a sixbolt specimen
Oterkus et al. [96] simulated the initiation and propagation of damage in a sixbolt carbon fiber/epoxy system (HTA7/6376) specimen [120] during a fatigue experiment. The fatigue behavior of a doublelap bolted joint with multiple fasteners was induced by subjecting the specimen to a cyclic tensile load \(^{\sigma _{\min }}/_{\sigma _{\max }}=1\) until failure. Figure 7 depicts the sample’s geometry.
Oterkus et al. [96] quantitatively compared the simulated and predicted failure modes using photo micro graphs. Failure essentially occurred in the matrix near the bolt holes. The failure modes predicted in the simulations were consistent with the experimental observations in the images.
 3.
Progressive damage in centercracked laminates
Kilic et al. [70] simulated the crack initiation and growth in centercracked laminates using peridynamics and compared their results with the experimental observations in [6, 69, 134]. Figure 8 shows the rectangular plate utilized in the simulations of dimension 10.16 mm×5.08 mm having an initial crack of length 1.27 mm. The plate is made of four plies of thickness 0.0413 mm each so that its total thickness is 0.1651 mm. The uniform tension applied in experiment [134] is gradually applied by prescribing the velocity boundary conditions v = ± 1.27 × 10^{− 7} mm per time step at the nodes located within a distance of 0.097 mm from both vertical ends. Note that they used a 15 times smaller inplane dimension than the one in the experiment [106], due to limitation in the computational resources. For lamina the elastic properties of T800/39002 prepreg tape reported by [106] were chosen. Lamina orientations of fibers are 0^{∘} fibers are running parallel to the xaxis and 90^{∘} fibers are running parallel to the yaxis. The stacking sequences [0^{∘}/90^{∘}/0^{∘}] and [0^{∘}/45^{∘}/0^{∘}] were considered for threeply laminates. Kilic et al. [70] obtained an asymmetric delamination pattern due to the presence of a 45^{∘} ply.
Moreover, the multiple splitting around the crack tip was also observed in [6]. Finally, Kilic et al. [70] predicted that the crack propagated along the direction parallel to the fibers within the plies oriented at 45^{∘} and 90^{∘}. This behavior was also detected in [134].
 4.
Dynamic tensile test on a prenotched rectangular plate
A dynamic tensile test performed on a prenotched rectangular 0^{∘} UD composite plate (M55J/M18 carbon/epoxy) [12, 65] was considered in Hu et al. [58]. Figure 9 shows a diagram of the 200 mm×100 mm plate with an initial crack of length 25 mm subjected to a uniform tensile load σ of ±40 Pa. The simulations showed a symmetric path of splitting fracture mode and bond breaking only in the matrix, which is consistent with the experiments [12].
4.2 Steel and Aluminum Materials
 1.
KalthoffWinkler experiment
The KalthoffWinkler experiment consists in striking a plate featuring two initial cracks of length 50 mm in the middle with a steel impactor, as sketched in Fig. 10. The observable in this experiment is the crack angle, along with the fact that the cracks propagate through the free surface [66, 67, 68]. The crack propagates from the initial crack tip at an angle around 68^{∘} with respect to the initial crack direction.
Silling [114] correctly predicted the crack angle with respect to that measured in the experiment and observed that the cracks propagated all the way to the free surface. Amani et al. [3] and Zhou et al. [144] reported that the crack angle predicted by the statebased PD model matched the angle observed in the experiments. Gu et al. [52] also showed that the crack angle and final crack path were qualitatively in good agreement with the experimental observations, even if the crack initiation time obtained in the simulations, 28μs, was slightly lower than that measured in the experiments, 29μs.
 2.
Compact tension experiments
Compact tension (CT) tests are used to study fracture in metals [89, 91, 131] and are standardized by ASTM E64700 [5] and ISO 75396 [62]. Figure 11a shows a plate specimen of dimension 126 mm×121 mm with four circular cutouts. For this configuration, Yolum et al. [135] simulated the crack mouth opening (in mm) with respect to the applied force (in kN) and compared their results with the experimental results of [77, 89]. For aluminum alloy (D16AT) CT specimens with four and eight circular cutoffs, the R^{2} coefficient was computed as 1.00 in both cases. For a plate devoid of cutoff, the R^{2} coefficient, using the experimental results provided in [131], is 1.00. Zhang [141, 142] simulated the crack paths observed in [91] on a modified CT test [142], whose configuration is shown in Fig. 11b. The dimensions of the plate are in this case 40 mm×40 mm×8 mm. Subsequently, Zhang [141] simulated the position (X, Y ) (in mm) of the crack path and the fatigue life for three values of the PD horizon, i.e., δ = {0.6,1.2,2.4} mm. In the case of the modified CT test, the R^{2} coefficient for the crack path positions and fatigue life is given by R^{2} = 0.99 and R^{2} = 0.99 for δ = 0.6, and R^{2} = 0.97 and R^{2} = 0.99 for δ = 1.2, respectively. Zhang reported that the case δ = 2.4 produces a short line in the plots due to a different crack path. Therefore, this case is not considered for the R^{2} correlation.
 3.
Taylor impact experiments
A Taylor impact test [4, 21], in which a cylindrical projectile is shot on a hard target, was considered by Amani et al. [3] and by Foster et al. [43, 45]. The experiment is designed for low velocity impacts in order to induce relatively small deformations. Amani et al. [3] qualitatively compared the post test deformation using dynamic structural light (DSL) images from [4] with the deformation obtained from their simulations. Moreover, Foster et al. [45] compared the normalized length and diameter of the damage as a function of the impact velocity for the same experiment. The coefficient of determination was estimated as R^{2} = 1.00 and R^{2} = 0.99 for the normalized length and normalized diameter, respectively. The true stress, as a function of the Lagrangian strain, was also estimated and compared to the experimental results of [21]. The R^{2} coefficient is in this case equal to 0.96.
 4.
Ballistic impact test
A ballistic impact test was simulated by Tupek et al. [127] using statebased peridynamics and compared to the experimental results of [132], in which a hardsteel spherical projectile of diameter 13.97 mm, impacted an extruded 6061T6 aluminum sandwich panel (Fig. 12) with a velocity of 370ms^{− 1}, 530ms^{− 1}, or 900ms^{− 1}. The observable in the experiment was the residual velocity versus the impact velocity. The predicted values of the observable lead to a R^{2} coefficient of 0.99.
4.3 Glassy Materials
 1.
Crack paths in a quenched glass plate
Experiments with single crack path [13, 103, 136] and multiple crack paths [102, 137] in a quenched glass plate were numerically studied by Kilic et al. [71]. In the experimental setup, a thermal stress was applied to the plate by placing it between two heat reservoirs. Depending on the specimen size and temperature difference, three types of crack propagation path were observed in the experiments: straight crack, oscillating crack, and branched crack as shown in Fig. 13. The rectangular thin plate of dimension 24 mm×64 mm×0.4 mm featuring a single initial crack, similar to the one used in the experiment [136], is shown in Fig. 14a. Kilic et al. [71] predicted straight cracks for small temperature differences and observed that the crack would oscillate first and then propagate in a straight direction for intermediate temperatures, and that crack branching would occur at higher temperatures. In other words, the PD simulations were able to capture most of the observed crack behaviors with respect to the temperature differences. Figure 14b shows the plate of dimension 48 mm×64 mm×0.4 mm for the setup with multiple cracks. Kilic et al. [71] numerically studied a plate with 2, 3, 5, 6, 7, 10, and 16 initial cracks and observed that there is an upper limit on the number of cracks that can propagate inside a specific plate. This is consistent with the experimental results in [102].
 2.
Crack propagation speed in a precracked glass plate
Gu et al. [52] used the bondbased PD model to study the propagation speed of a crack in a precracked glass plate subjected to a step tensile loading. Gu et al. [52] considered the plate of dimension 100 mm×40 mm with an initial crack of length 50 mm, as shown in Fig. 15. Gu et al. [52] computed the crack propagation speed (in ms^{− 1}) as a function of time (μs) and compared it against the maximal fracture velocity v_{F} = 1580 ms^{− 1} obtained in the experiment in [15]. The maximal propagation speeds obtained in the simulations were as follows: 2037 ms^{− 1} for a uniform mesh with 16281 discrete PD nodes, 2341 ms^{− 1} for a uniform mesh with 4141 discrete PD nodes, and 2699 ms^{− 1} for an adaptive mesh with 414 discrete PD nodes up to 7227 discrete PD nodes. Zhou et al. [144] and Ha et al. [53] computed a maximum fracture speed of 1157 ms^{− 1} and 1679 ms^{− 1}, which are 1.45% lower and 6% larger than the experimental value, respectively.
 3.
Crack growth in a prenotched glass sheet
Crack growth in a prenotched glass sheet [100] under tensile loading was studied by Agwai et al. [2]. Figure 15 depicts the prenotched plate of dimension 100 mm×40 mm and the initial crack of length 50 mm. The PD simulations and experiments both put in evidence that the crack splits into two branches, which grow until they eventually reach the right boundary.
 4.
Fast crack growth in an edgecracked plate under tension
Fast growth of a crack in an edgecracked PMMA plate under tension [39] was considered by Agwai et al. [2]. The plate had dimension of 380 mm×440 mm and the initial crack was 4 mm long, as shown in Fig. 16. The crack velocity predicted over the time period up to 80μs was compared with the velocity obtained in the experiment, ranging from 0 to 1000ms^{− 1}. The corresponding R^{2} coefficient was evaluated in this case as R^{2} = 0.72.
 5.
Impact damage on a thin glass plate with polycarbonate backing
Impact damage on a thin glass plate with poly carbonate backing [8, 20, 40] was studied by Hu et al. [59]. They qualitatively compared the damage patterns in the glass layer under three projectile speeds. Hu et al. [59] reported that the main fracture patterns observed experimentally were correctly captured.
 6.
Tensile test
A tensile test [124] was simulated by Diehl et al. [32] using a bondbased PD model. The objective was to predict the Poisson ratio of a PMMA sample under a timedependent loading for different values of the horizon parameter m ∈{4,5,…,12,13}. Thus, the neighborhood B_{δ}(x_{i}) of a discrete PD node x_{i} contains 2m + 1 nodes in [x_{i} − δ, x_{i} + δ] in each direction [11]. The R^{2} coefficient of the Poisson ratio as a function of time was obtained as 0.65. Note that the parameters of the exponential model considered in this study depended on the energy release rate and shear modulus, and not on the horizon, as in many other models. The horizon in the model was determined by the failure strain, which can be measured during the tensile test.
4.4 Concrete
 1.
Lapsplice experiment
Gerstle et al. [48] presented simulation and laboratory results of a reinforced concrete lapsplice benchmark problem. Gerstle et al. [48] quantitatively compared the crack pattern obtained in the laboratory experiment and in the simulations. The crack pattern observed in the experiment consisted of two or three longitudinal splitting cracks, each one being approximately coplanar with the axis of the concrete cylinder. However, the crack patterns predicted by the PD simulations were quite different and the authors suggested that such a difference could be explained by variations in the loading rates.
 2.
3point bending beam experiment
Gu et al. [51] performed some simulations of a 3point bending beam experiment [63]. Figure 17 illustrates the plate (320 mm×70 mm) used in the experiment. The velocity applied at the top of the beam was v = 0.075mmmin^{− 1}. The objectives of the study were twofold: (1) to accurately predict the crack pattern observed in the experiment [63], which was actually the case; (2) to predict the crack mouth opening displacement with respect to the load. The experimental and simulation data give in this case a coefficient of determination of R^{2} = 0.61.
 3.
Crack mouth opening displacement and load point displacement
Aziz [7] performed simulations of the crack mouth opening displacement versus the applied load (CMODload) for three different configurations (D3, D6, and D9) as shown in Fig. 18 and Table 4 [19]. The values of the R^{2} coefficient are collected in Table 5. Here, we observe that the values of R^{2} vary depending on the length and position of the notch. The crack mouth opening displacement is usually insensitive to concrete damage at the loaded point and support. Alternatively, the load point displacement, a quantity that is very sensitive to the damage in concrete at the load points, was also computed with respect to the applied load (LPDload). We observe in Table 5 that the coefficient R^{2} suffers a nonnegligible drop for D9. In a different exercise, the damage model was calibrated with respect to the experimental CMODload diagrams and simulations for the CMODload were repeated. The new values of R^{2} are also reported in Table 5.
 4.
Failure in a Brazilian disk
Failure in a Brazilian disk in compression [54] was considered by Gu et al. [51]. Figure 19 shows the diagram of a Brazilian disk of diameter d =100 mm with an initial crack of 30 mm at the center of the disk subjected to a compressive velocity boundary condition v = 0.05 ms^{− 1}. The objective of this problem was to reproduce the crack pattern observed in the experiment when the crack propagates through the disk. Gu et al. [51] reported that the numerical results were similar to the experimental ones. They also concluded that nonordinary statebased PD could be important in the future for the prediction of damage processes.
 5.
Anchor bolt pullout experiment
The Anchor bolt pullout experiment [128], shown in Fig. 20, was simulated by Lu et al. [83]. In this experiment, the concrete structure is fixed with two anchors and the bolt is pulled out. Measurements considered for three configurations [128] are shown in Table 6. For these three configurations, the peak loads were predicted and compared to the values obtained in the experiments (see Table 7). For one of these configurations, the loaddisplacement response (in kN versus μm) was recorded and simulated: the calculation of the corresponding coefficient of determination yielded R^{2} = 0.92. Finally, the failure mode was quantitatively assessed with respect to the experimental observations and the conclusion was that the crack direction and crack branching obtained in the PD simulations were in good agreement with those observed in the experiments.
The dimensions in mm of the three different configurations of the beam D1–D3 shown in Fig. 18
Configuration  Depth (d)  Span (l)  Noth depth (a_{0}) 

D1  76.2  228.6  25.4 
D2  152.4  457.2  59.8 
D3  228.6  685.8  76.2 
Coefficient of determination R^{2} for configurations (D3, D6, and D9) associated with the CMODload and LPDload of a 3point bending beam
Coefficient of determination R^{2}  

Configuration  CMODload  CMODload (Calibrated model)  LPDload 
D3  0.85  0.37  0.87 
D6  0.89  0.59  0.83 
D9  0.77  0.77  0.51 
4.5 Other Materials
 1.
Rupture phenomena in biomembranes
The rupture phenomenon in biomembranes [49] was considered by Taylor et al. [125]. The objective in this experiment was to compare the fractal dimension of ruptures in actual membranes with that extracted from PD simulations. The fractional dimension was postprocessed from the computer simulations via image processing techniques and the reticular cell counting (box counting) method. The predicted fractional dimensions, namely 1.7, 1.63, and 1.56, computed for different shear moduli, were close to the fractional dimension, 1.7, obtained in the experiment.
 2.
Crushing of brittle ice against vertical structures
The PD simulation of brittle ice [119] crushed by a vertical structure was performed by Liu et al. [81]. In the experiment, a thin layer of ice of dimension 1000 mm×1000 mm×60 mm is cut in the middle by a rotating cylindrical structure. The mean force and the peak force in the ice obtained in the simulations were compared with those measured in the experiments for several values of the penetration velocity. Table 8 lists these values. We observe that the relative error strongly depends on the velocity of the cylinder.
 3.
Crosssectional nanoindentation
Oterkus et al. [95] studied the crack patterns for crosssectional nanoindentation (CSN) [105]. Figure 21 sketches the rectangular silicon substrate specimen with three layers of copper metallization embedded in inter layer dielectric (ILD) and etch stops (ES) [94]. In the experiment, the silicon substrate was indented by a Berkovich diamond, which induced a Vshaped crack. Note that in the simulation, an initial 40^{∘} Vshaped crack was introduced similarly as in the experiment. Along the normal to the crack faces, a velocity of ± 0.5 ms^{− 1} was applied. The crack paths observed in the simulation and experiment showed that the main crack propagated through the silicon and the first layer of copper metallization embedded in interlayer dielectric.
 4.
Dynamic crack propagation in functionally graded materials
Experiments of dynamic crack propagation in functionally graded materials (epoxy/sodalime glass) (FGM) [1, 73, 74] were used for comparison with PD simulations by Cheng et al [24]. The experimental setup consisted of a plate of dimension 152mm×43mm with an initial crack of length 8.6mm subjected to 3point loading, as shown in Fig. 22. First, the crack pattern observed in the experiment [74] was quantitatively compared with that obtained in the simulation. The conclusion of this study was that, despite the fact that the dynamic loading used in the simulations was different from that in the experiments, a close resemblance between the predicted and experimental crack paths could still be observed. Second, the evolution of the crack length over time [73] was considered for two cases: (1) precrack on the stiffer side (E_{1} > E_{2}) and (2) precrack on the more compliant side (E_{1} < E_{2}). When using the linear curve fit of elastic modulus and density to compute the peridynamics micro modulus, the R^{2} coefficient is 0.99 for E_{1} < E_{2} and 0.99 for E_{1} < E_{2}. When using the piecewise linear variation measured from the experiments, the R^{2} coefficient is now given by 0.98 for E_{1} < E_{2} and 0.99 for E_{1} < E_{2}.
Configuration  W (mm)  L (mm)  a (mm)  d (mm)  b (mm)  t (mm) 

1  300  300  100  50  15  5 
2  600  600  200  100  30  10 
3  900  900  300  150  45  15 
Configuration  Simulation (kN)  Experiment (kN)  Relative error (%) 

1  17.04  13.4  27.2 
2  28.86  14.5  17.8 
3  37.48  33.6  11.6 
Mean forces and peak forces in ice obtained from experiments and PD simulations. Table is adapted from [81] and completed with values of the relative error
Mean force (kN)  Peak force (kN)  

v(mms^{− 1})  Sim  Exp  𝜖_{rel}(%)  Sim  Exp  𝜖_{rel}(%) 
50  13848  5.3859  74.29  44379  130527  66.00 
130  13739  17309  20.63  46342  52479  11.69 
210  14324  19136  25.15  49742  38522  29.13 
5 Summary of Validation Results for Peridynamics Modeling
Relative error between the observable measured in the experiment and obtained in the simulation
Material  Mechanical test  Observable  Rel. error (%)  Exp  Sim 

ALON  Edgeon impact experiment  Avg. propagation speed of primary wave front  3.6  [90]  [141] 
ALON  Edgeon impact experiment  Wave propagation speed  10  [34]  
Steel  KalthoffWinkler experiment  Crack initiation time  3.4  [87]  [52] 
Steel  KalthoffWinkler experiment  Crack propagation speed  14.2  [17]  
Ice  Crushingbrittle ice by a rotating cylinder  Mean force at 50 mms^{− 1}  74.3  [119]  [81] 
Ice  Crushingbrittle ice by a rotating cylinder  Mean force at 130 mms^{− 1}  20.6  [119]  [81] 
Ice  Crushingbrittle ice by a rotating cylinder  Mean force at 210 mms^{− 1}  25.1  [119]  [81] 
Ice  Crushingbrittle ice by a rotating cylinder  Peak force at 50 mms^{− 1}  66  [119]  [81] 
Ice  Crushingbrittle ice by a rotating cylinder  Peak force at 130 mms^{− 1}  11.7  [119]  [81] 
Ice  Crushingbrittle ice by a rotating cylinder  Peak force at 210 mms^{− 1}  29.1  [119]  [81] 
Sodalime glass  Precracked glass (step tensile loading)  Max. crack propagation speed  6.3  [15]  [53] 
Sodalime glass  Precracked plate (step tensile loading)  Max. crack propagation speed  26.8  [15]  [144] 
Sodalime glass  Precracked plate (step tensile loading)  Max. crack propagation speed (16 281 nodes)  28.9  [15]  [52] 
Sodalime glass  Precracked plate (step tensile loading)  Max. crack propagation speed (4141 nodes)  48.2  [15]  [52] 
Sodalime glass  Precracked plate (step tensile loading)  Max. crack propagation speed (refined)  70.8  [15]  [52] 
Concrete  Anchor Bolt Pullout (Configuration 1)  Peak loads  27.2  [128]  [83] 
Concrete  Anchor Bolt Pullout (Configuration 2)  Peak loads  17.8  [128]  [83] 
Concrete  Anchor Bolt Pullout (Configuration 3)  Peak loads  11.6  [128]  [83] 
R^{2} correlation between the series of observables between experiment and simulation
Material  Mechanical test  Observable  R ^{2}  Exp  Sim 

Aluminum  SplitHopkinson pressure bar  Strain vs time  0.99  [23]  [65] 
Sandstone (Berea)  Wave dispersion  Dispersion curves at confining pressure of 10 MPa  0.84  [133]  [18] 
Sandstone (Massilion)  Wave dispersion  Dispersion curves at confining pressure of 10 MPa  0.97  [133]  [18] 
Sandstone (Berea)  Wave dispersion  Dispersion curves at confining pressure of 20 MPa  0.93  [133]  [18] 
Sandstone (Massilion)  Wave dispersion  Dispersion curves at confining pressure of 20 MPa  1  [133]  [18] 
Sandstone (Berea)  Wave dispersion  Dispersion curves at confining pressure of 40 MPa  0.95  [133]  [18] 
Sandstone (Massilion)  Wave dispersion  Dispersion curves at confining pressure of 40 MPa  0.96  [133]  [18] 
Columbia resin CR39  Propagation of waves in a halfplane  Displacement vs position at 60 μs  0.74  [25]  [93] 
Columbia resin CR39  Propagation of waves in a halfplane  Displacement vs position at 92 μs  0.35  [25]  [93] 
Columbia resin CR39  Propagation of waves in a halfplane  Displacement vs position at 139 μs  0.15  [25]  [93] 
Aluminum (6061T6)  Taylor impact test  Norm diameter/length; strain vs stress  0.96  [45]  
Aluminum (6061T6)  Ballistic impact test  Residual vel vs impact vel  0.99  [132]  [127] 
Steel (4340 RC 43)  SplitHopkinson pressure bar  Strain vs stress  0.97  [46]  [44] 
Aluminum (D16AT)  Compact tension test  Force vs CMOD  1  [135]  
SAE 1020 steel  Compact tension test  Crack path position  0.97  [91]  [141] 
Concrete  3point bending  Load vs CMD  0.61  [63]  [51] 
Concrete  3point bending (D3)  Load vs CMOD  0.85  [19]  [7] 
Concrete  3point bending (D6)  Load vs CMOD  0.89  [19]  [7] 
Concrete  3point bending (D9)  Load vs CMOD  0.77  [19]  [7] 
Concrete  3point bending (D3 LPDload)  Load vs LPD  0.87  [19]  [7] 
Concrete  3point bending (D6 LPDload)  Load vs LPD  0.83  [19]  [7] 
Concrete  3point bending (D9 LPDload)  Load vs LPD  0.51  [19]  [7] 
Concrete  Anchor bolt pullout  Pullout load vs displacement  0.92  [128]  [83] 
PMMA  Fast crack growth  Crack velocity vs time  0.72  [39]  [2] 
PMMA  Tensile test  Poisson ratio vs time  0.65  [124]  [32] 
FEM (Epoxy/Sodalime glass)  3point loading  Crack length vs time  0.99  [73]  [24] 
Figure 23 suggests that the relative error is consistently smaller than 20% for aluminum and steel, which shows that simulations can predict well the experimental results. For concrete materials, the relative error ranges between 10 and 30%. For glassy materials, it is between 3 and 71%. In the case of crushingbrittle ice by a rotating cylinder, the relative error lies in the range between 14 and 74% for three different rotating velocities.
Conclusions in the case of series of observables, based on Fig. 24, are as follows. For aluminum and steel, the coefficient R^{2} is in general greater than 0.9, which indicates a very good agreement between simulations and experiments. For concrete materials, R^{2} ranges between 0.5 and 0.9 while for glassy materials, it is between 0.6 and 0.72. In the case of functionally graded materials (FGM) and for sandstone, the coefficient reaches values as high as 0.99 and 0.93, respectively. We thus observe that PD simulations usually agree well with experiments involving metallic materials, such as steel and aluminum, and sandstone.
6 Extraction of Additional Attributes from Nonlocal Simulation Data
In peridynamics, information, such as reconstructed stress or strain, is usually provided at the node level. However, quantities measured in experiments are often given at a larger scale. Here, advanced visualization techniques can be utilized to extract missing information or to postprocess the solutions in order to be able to compare predictions and experiments.
Overview of applications of bondbased (B) and statebased (S) peridynamics simulations using visualization algorithms to assess fracture in solids
6.1 Visualization of Fragmentation
6.2 Visualization of Fracture Progression
6.3 Physically Based Modeling and Rendering
Another interesting application of peridynamics could be for physically based modeling and rendering of visual effects [78, 99]. Peridynamics, thanks to the comparative simplicity of the models, could indeed provide a good balance between artistic rendering and accuracy in describing physical behaviors. For example, Levine et al. [78] used bondbased PD for the animation of brittle fracture. Levine et al. [78] animated the impact of a projectile through a glass plate and the fracturing of the Welsh dragon/vase on the floor. Levine et al. [78] reported that as in most physicsbased animation methods, tuning parameters can be a tedious task.
Chen et al. [22] used statebased PD for the animation of fractures in elastoplastic solids. Chen et al. [22] visualized the isotropic brittle, anisotropic brittle, isotropic ductile, and anisotropic ductile fracture behavior of a sphere travelling through a wall. Next, Chen et al. [22] visualized the complex crack pattern when shooting a bullet through jello, the fragments obtained in the case of the Stanford bunny^{3} falling to the ground, and the crack pattern in a thin sheet when pulled apart. Chen et al. [22] concluded that the horizon needed sometimes to be finetuned in order to get wellsynchronized results.
7 Concluding Remarks and Perspectives
Applications for validation of bondbased PD and statebased PD were reviewed and summarized in Tables 1 and 3. Overall, we identified 22 applications for bondbased PD and 9 applications for statebased PD. In the case of applications dealing with initiation and propagation of cracks, we chose to classify the contributions with respect to the type of materials, namely composites, aluminum/steel, glassy materials, concrete, and other materials. The KalthoffWinkler experiment was used to assess the crack initiation time, the crack propagation speed, and the crack angle. We thus believe that the KalthoffWinkler experiment could be a valuable benchmark for the validation of PD models, since it provides three different types of measurement at once. In other words, it is challenging to simultaneously predict the three observables.
We also indicated, to some extent, whether the comparative studies between simulations and experiments were quantitative or qualitative. In most cases, we were able to compute either the relative error, for scalar observables, or the coefficient of determination R^{2}, for series of observables, to assess the predictability of the PD models.
The major results of this investigation are collected in Tables 9 and 10 and plotted in Figs. 23 and 24. We note that the best correlations were obtained for edgeon impact (EOI) experiment experiments involving aluminum and steel materials.
It is commonly accepted that the size of the PD horizon should be set to three or four times the mesh size. Nevertheless, some authors have studied the influence of the horizon δ and mesh size on the predictions. For example, Zhang [141] investigated the influence of the horizon on the position of the crack path: the values of R^{2} were estimated at 0.97 and 0.99 for δ = 1.2 mm and δ = 0.6 mm, respectively. Gu et al. [52] looked at the effect of mesh size on the maximal speed of crack propagation: the values obtained in the simulations were 2341 ms^{− 1} and 2037 ms^{− 1} for uniformly distributed 4141 nodes and 16281 nodes, respectively. Unsurprisingly, simulations are clearly influenced by the choice of the discretization and the value of the horizon.
Applications of computer graphics have usually two objectives: (1) to gain more realistic visual effects using physicsbased modeling and rendering; (2) to reconstruct from the solutions largescale quantities or features, such as number of fragments and shape of the crack surfaces. In all reconstruction approaches, the approximation of damage in PD is based on the notion that damage arises at broken bonds. Unfortunately, validation of these models against experimental data is still lacking at this time. There is maybe one exception in which extraction of crack surfaces in the KalthoffWinkler experiment was used with partial success to measure the fracture growth velocity. We believe that advanced visualization techniques could be useful to gain more insight on the effect of initial positions of the discrete PD nodes on quantities of interest, such as the fragment size or alignment of crack surfaces. They could also help investigate optimal ratios between the horizon and mesh size.

Boundary conditions: One of the major challenges pertains to the treatment of boundary conditions when dealing with nonlocal models [37, 50, 85, 86]. This is an issue that one also finds in molecular dynamics simulations or in smooth particles hydrodynamics. To overcome this issue, several approaches for coupling PD with the finite element method (FEM) or classical continuum mechanics (CCM), e.g., forcebased blending [108, 109], Arlequin approach [55], or discrete coupling [47, 72, 82, 84, 111, 138, 139] have been developed. One could use a coupling approach in order to enforce boundary conditions, that is, by using CCM or FEM along the boundaries and PD in the region of interest where cracks and fractures arise.

Discretization parameters: For the discrete case, the choice of the nodal spacing and the horizon influences the various convergence scenarios [38, 126]. One issue is defining the proper ratio between the horizon and mesh size as numerical results are certainly sensitive [30] to these two parameters. One approach is to adjust the horizon, so that the peridynamic results produce the same dispersion curves as those measured for a specific material [115]. Another approach is to determine the horizon by Griffith’s brittle failure criterion, based on critical failure stress which can be found experimentally [32].

Computational costs: The computational costs for nonlocal models, like PD, are significant. Some cited references were not able to simulate the full size of the experimental geometry, due to computational limitations. Several massive parallel peridynamics implementations are available: Peridigm [80, 97] and PDLammps [98] based on the Message Passing Interface (MPI), peridynamicHPX [31] based on the C+ + standard library for parallelism and concurrency (HPX) [56], acceleration card codes based on OpenCL [92] or on CUDA [27, 33].

Calibration versus validation: Simulations are usually compared to experimental data for calibration/fitting or validation. In the first case, the material parameters are calibrated, such that they fit observable quantities in a given experiment, e.g., reproduce the same dispersion curve or the crack growth velocity. In the second case, the calibrated parameters are used for validation against other experimental scenarios, e.g., different loading values or different positions of the initial crack. In the majority of the cited references, the material parameters were calibrated against one experiment and only a few addressed validation.
Footnotes
References
 1.AbantoBueno J, Lambros J (2006) An experimental study of mixed mode crack initiation and growth in functionally graded materials. Exp Mech 46(2):179–196CrossRefGoogle Scholar
 2.Agwai A, Guven I, Madenci E (2011) Predicting crack propagation with peridynamics: a comparative study. Int J Fract 171(1):65zbMATHCrossRefGoogle Scholar
 3.Amani J, Oterkus E, Areias P, Zi G, NguyenThoi T, Rabczuk T (2016) A nonordinary statebased peridynamics formulation for thermoplastic fracture. Int J Impact Eng 87(SI):83–94CrossRefGoogle Scholar
 4.Anderson CE, Nicholls AE, Chocron IS, Ryckman RA (2006) Taylor anvil impact. AIP Conf Proc 845(1):1367–1370CrossRefGoogle Scholar
 5.ASTM International (2000) ASTM E64700 standard test method for measurement of fatigue crack growth ratesGoogle Scholar
 6.Awerbuch J, Madhukar MS (1985) Notched strength of composite laminates: Predictions and experiments—a review. J Reinf Plast Compos 4(1):3–159CrossRefGoogle Scholar
 7.Aziz A (2014) Simulation of fracture of concrete using micropolar peridynamics. Ph.D. thesis, University of New MexicoGoogle Scholar
 8.Ball A, McKenzie H (1994) On the low velocity impact behaviour of glass plates. J Phys IV 4(C8):C8–783Google Scholar
 9.Boardman B (1990) Fatigue resistance of steels, vol 1Google Scholar
 10.Bobaru F, Foster JT, Geubelle PH, Silling SA (2016) Handbook of peridynamic modeling. CRC Press, Boca RatonzbMATHGoogle Scholar
 11.Bobaru F, Yang M, Alves LF, Silling SA, Askari E, Xu J (2009) Convergence, adaptive refinement, and scaling in 1d peridynamics. Int J Numer Methods Eng 77(6):852–877zbMATHCrossRefGoogle Scholar
 12.Bogert P, Satyanarayana A, Chunchu P (2006) Comparison of damage path predictions for composite laminates by explicit and standard finite element analysis tools. In: Collection of technical papers  AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, vol 3Google Scholar
 13.Bouchbinder E, Hentschel HGE, Procaccia I (2003) Dynamical instabilities of quasistatic crack propagation under thermal stress. Phys Rev E 68:036,601CrossRefGoogle Scholar
 14.Boudet J, Ciliberto S, Steinberg V (1996) Dynamics of crack propagation in brittle materials. J Phys II 6(10):1493–1516Google Scholar
 15.Bowden F, Brunton J, Field J, Heyes A (1967) Controlled fracture of brittle solids and interruption of electrical current. Nature 216(5110):38–42CrossRefGoogle Scholar
 16.Bowden FP, Field JE (1964) The brittle fracture of solids by liquid impact, by solid impact, and by shock. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 282(1390):331–352CrossRefGoogle Scholar
 17.Bußler M, Diehl P, Pflüger D, Frey S, Sadlo F, Ertl T, Schweitzer MA (2017) Visualization of fracture progression in peridynamics. Comput Graph 67:45–57zbMATHCrossRefGoogle Scholar
 18.Butt SN, Timothy JJ, Meschke G (2017) Wave dispersion and propagation in statebased peridynamics. Comput Mech 60(5):725–738Google Scholar
 19.Chapman S (2011) Clariffication of the notched beam level in testing procedures of ACI 446 committee report 5. Master’s thesis, University of New MexicoGoogle Scholar
 20.Chaudhri MM, Walley SM (1978) Damage to glass surfaces by the impact of small glass and steel spheres. Philos Mag A 37(2):153–165CrossRefGoogle Scholar
 21.Chen W, Song B, Frew DJ (2002) 108 split Hopkinson bar testing of an aluminum with pulse shaping. The Proceedings of the JSME Materials and Processing Conference (M&P) 10.1:58–61CrossRefGoogle Scholar
 22.Chen W, Zhu F, Zhao J, Li S, Wang G (2017) Peridynamicsbased fracture animation for elastoplastic solids. Comput Graph Forum 37(1):112–124CrossRefGoogle Scholar
 23.Chen WW, Song B (2011) Mechanical engineering series. In: Split Hopkinson (Kolsky) bar – design, testing and applications, vol 1. Springer, US, p 388Google Scholar
 24.Cheng Z, Zhang G, Wang Y, Bobaru F (2015) A peridynamic model for dynamic fracture in functionally graded materials. Compos Struct 133:529–546CrossRefGoogle Scholar
 25.Dally J, Thau S (1967) Observations of stress wave propagation in a halfplane with boundary loading. Int J Solids Struct 3(3):293–308CrossRefGoogle Scholar
 26.Devore JL (2012) Probability and statistics for engineering and the sciences, 2nd edn. Springer, New YorkGoogle Scholar
 27.Diehl P (2012) Implementierung eines PeridynamikVerfahrens auf GPU. Diplomarbeit, Institute of Parallel and Distributed Systems, University of StuttgartGoogle Scholar
 28.Diehl P (2017) Modeling and simulation of cracks and fractures with peridynamics in brittle materials. Phd. thesis, Institut für Numerische Simulation, Universit ät BonnGoogle Scholar
 29.Diehl P, Bußler M, Pflüger D, Frey S, Ertl T, Sadlo F, Schweitzer MA (2017) Extraction of fragments and waves after impact damage in particlebased simulations. In: Meshfree methods for partial differential equations VIII. Springer International Publishing, pp 17–34Google Scholar
 30.Diehl P, Franzelin F, Pflüger D, Ganzenmüller GC (2016) Bondbased peridynamics: a quantitative study of mode i crack opening. Int J Fract 201(2):157–170CrossRefGoogle Scholar
 31.Diehl P, Jha PK, Kaiser H, Lipton R, Lévesque M (2018) Implementation of peridynamics utilizing hpx–the c+ + standard library for parallelism and concurrency. arXiv:1806.06917
 32.Diehl P, Lipton R, Schweitzer MA (2016) Numerical verification of a bondbased softening peridynamic model for small displacements: Deducing material parameters from classical linear theory. Tech. rep., Institut für Numerische SimulationGoogle Scholar
 33.Diehl P, Schweitzer MA (2015) Efficient neighbor search for particle methods on gpus. In: Meshfree methods for partial differential equations VII. Springer, pp 81–95Google Scholar
 34.Diehl P, Schweitzer MA (2015) Simulation of wave propagation and impact damage in brittle materials using peridynamics. In: Recent trends in computational engineeringCE2014. Springer, pp 251–265Google Scholar
 35.Diehl P, Tabiai I, Baumann FW, Therriault D, Lévesque M (2018) Long term availability of raw experimental data in experimental fracture mechanics. Eng Fract Mech 197:21–26CrossRefGoogle Scholar
 36.Döll W (1975) Investigations of the crack branching energy. Int J Fract 11(1):184–186CrossRefGoogle Scholar
 37.Du Q (2016) Nonlocal calculus of variations and wellposedness of peridynamics. Handbook of Peridynamic Modeling 63–85Google Scholar
 38.Du Q, Tian X (2015) Robust discretization of nonlocal models related to peridynamics. In: Meshfree methods for partial differential equations VII. Springer, pp 97–113Google Scholar
 39.Eran S, Fineberg J (1999) Confirming the continuum theory of dynamic brittle fracture for fast cracks. Nature 397(6717):333CrossRefGoogle Scholar
 40.Field J (1988) Investigation of the impact performance of various glass and ceramic systems. Tech. rep., Cambridge University (United Kingdom) Cavebdish LaboratoryGoogle Scholar
 41.Fineberg J, Gross SP, Marder M, Swinney HL (1992) Instability in the propagation of fast cracks. Phys Rev B 45:5146–5154CrossRefGoogle Scholar
 42.Fineberg J, Marder M (1999) Instability in dynamic fracture. Phys Rep 313(1):1–108MathSciNetCrossRefGoogle Scholar
 43.Foster JT (2009) Dynamic crack initiation toughness: Experiments and peridynamic modeling. Sandia Report SAN D20097217, Sandia National LaboratoriesGoogle Scholar
 44.Foster JT, Silling SA, Chen WW (2009) State based peridynamic modeling of dynamic fracture. In: Annual conference and exposition on experimental and applied mechanics 2009 society for experimental mechanics  SEM, vol 4, pp 2312–2317Google Scholar
 45.Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Methods Eng 81(10):1242–1258zbMATHGoogle Scholar
 46.Frew DJ, Forrestal MJ, Chen W (2005) Pulse shaping techniques for testing elasticplastic materials with a split Hopkinson pressure bar. Exp Mech 45(2):186CrossRefGoogle Scholar
 47.Galvanetto U, Mudric T, Shojaei A, Zaccariotto M (2016) An effective way to couple fem meshes and peridynamics grids for the solution of static equilibrium problems. Mech Res Commun 76:41–47CrossRefGoogle Scholar
 48.Gerstle W, Sakhavand N, Chapman S (2010) Peridynamic and continuum models of reinforced concrete lap splice compared. fracture mechanics of concrete and concrete structuresrecent advances in fracture mechanics of concreteGoogle Scholar
 49.Gözen I, Dommersnes P, Czolkos I, Jesorka A, Lobovkina T, Orwar O (2010) Fractal avalanche ruptures in biological membranes. Nat Mater 9(11):908CrossRefGoogle Scholar
 50.Gu X, Madenci E, Zhang Q (2018) Revisit of nonordinary statebased peridynamics. Eng Fract Mech 190:31–52CrossRefGoogle Scholar
 51.Gu X, Wu Q (2016) The application of nonordinary, statebased peridynamic theory on the damage process of the rocklike materials. Math Probl Eng 2016, 9 ppGoogle Scholar
 52.Gu X, Zhang Q, Xia X (2017) Voronoibased peridynamics and cracking analysis with adaptive refinement. Int J Numer Methods Eng 112(13):2087–2109MathSciNetCrossRefGoogle Scholar
 53.Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162(1):229–244zbMATHCrossRefGoogle Scholar
 54.Haeri H, Shahriar K, Marji MF, Moarefvand P (2014) Experimental and numerical study of crack propagation and coalescence in precracked rocklike disks. Int J Rock Mech Min Sci 67:20–28CrossRefGoogle Scholar
 55.Han F, Lubineau G (2012) Coupling of nonlocal and local continuum models by the arlequin approach. Int J Numer Methods Eng 89(6):671–685MathSciNetzbMATHCrossRefGoogle Scholar
 56.Heller T, Diehl P, Byerly Z, Biddiscombe J, Kaiser H (2017) Hpx–an open source c+ + standard library for parallelism and concurrencyGoogle Scholar
 57.Hopkinson B (1914) A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets. Philosophical Transactions of the Royal Society of London Series A, Containing Papers of a Mathematical or Physical Character 213(497–508):437–456Google Scholar
 58.Hu W, Ha YD, Bobaru F (2011) Modeling dynamic fracture and damage in a fiberreinforced composite lamina with peridynamics. Int J Multiscale Comput Eng 9(6)Google Scholar
 59.Hu W, Wang Y, Yu J, Yen CF, Bobaru F (2013) Impact damage on a thin glass plate with a thin polycarbonate backing. Int J Impact Eng 62:152–165CrossRefGoogle Scholar
 60.Humphrey W, Dalke A, Schulten K (1996) VMD – visual molecular dynamics. J Mol Graph 14:33–38CrossRefGoogle Scholar
 61.Ihmsen M, Orthmann J, Solenthaler B, Kolb A, Teschner M (2014) SPH fluids in computer graphics. In: Lefebvre S, Spagnuolo M (eds) Eurographics 2014  state of the art reports. The Eurographics associationGoogle Scholar
 62.International Organization for Standardization (2003) Corrosion of metals and alloys  stress corrosion testing  part 6: Preparation and use of precracked specimens for tests under constant load or constant displacementGoogle Scholar
 63.Jenq YS, Shah SP (1988) Mixedmode fracture of concrete. Int J Fract 38(2):123–142Google Scholar
 64.Jia T (2012) Development and applications of new peridynamic models. Ph.D. thesis, Michigan State UniversityGoogle Scholar
 65.Jose S, Kumar RR, Jana M, Rao GV (2001) Intralaminar fracture toughness of a crossply laminate and its constituent sublaminates. Compos Sci Technol 61(8):1115–1122CrossRefGoogle Scholar
 66.Kalthoff JF (1988) Shadow optical analysis of dynamic shear fracture. Opt Eng 27(10):271,035CrossRefGoogle Scholar
 67.Kalthoff JF (2000) Modes of dynamic shear failure in solids. Int J Fract 101(1):1–31CrossRefGoogle Scholar
 68.Kalthoff JF, Winkler S (1988) Failure mode transition at high rates of shear loading. Impact Loading and Dynamic Behavior of Materials 1:185–195Google Scholar
 69.Kawai M, Morishita M, Satoh H, Tomura S, Kemmochi K (1997) Effects of endtab shape on strain field of unidirectional carbon/epoxy composite specimens subjected to offaxis tension. Compos A: Appl Sci Manuf 28(3):267–275CrossRefGoogle Scholar
 70.Kilic B, Agwai A, Madenci E (2009) Peridynamic theory for progressive damage prediction in centercracked composite laminates. Compos Struct 90(2):141–151CrossRefGoogle Scholar
 71.Kilic B, Madenci E (2009) Prediction of crack paths in a quenched glass plate by using peridynamic theory. Int J Fract 156(2):165–177zbMATHCrossRefGoogle Scholar
 72.Kilic B, Madenci E (2010) Coupling of peridynamic theory and the finite element method. J Mech Mater Struct 5(5):707–733CrossRefGoogle Scholar
 73.Kirugulige M, Tippur HV (2008) Mixedmode dynamic crack growth in a functionally graded particulate composite: experimental measurements and finite element simulations. J Appl Mech 75(5):051,102CrossRefGoogle Scholar
 74.Kirugulige MS, Tippur HV (2006) Mixedmode dynamic crack growth in functionally graded glassfilled epoxy. Exp Mech 46(2):269–281CrossRefGoogle Scholar
 75.Kitey R, Tippur HV (2008) Dynamic crack growth past a stiff inclusion: optical investigation of inclusion eccentricity and inclusionmatrix adhesion strength. Exp Mech 48(1):37–53CrossRefGoogle Scholar
 76.Kolsky H (1949) An investigation of the mechanical properties of materials at very high rates of loading. Proc Phys Soc Sect B 62(11):676CrossRefGoogle Scholar
 77.Kumar S, Singh I, Mishra B (2014) A coupled finite element and elementfree Galerkin approach for the simulation of stable crack growth in ductile materials. Theor Appl Fract Mech 70(Supplement C):49–58CrossRefGoogle Scholar
 78.Levine JA, Bargteil AW, Corsi C, Tessendorf J, Geist R (2014) A peridynamic perspective on springmass fracture. In: Proceedings of the ACM SIGGRAPH/Eurographics symposium on computer animation, SCA ’14. Eurographics association, pp 47–55Google Scholar
 79.Littlewood D, Silling S, Demmie P (2016) Identification of fragments in a meshfree peridynamic simulation. In: ASME 2016 international mechanical engineering congress and exposition. American society of mechanical engineers, pp V009T12A071–V009T12A071Google Scholar
 80.Littlewood DJ (2015) Roadmap for Peridynamic Software Implementation. Tech Rep 20159013, Sandia National LaboratoriesGoogle Scholar
 81.Liu M, Wang Q, Lu W (2017) Peridynamic simulation of brittleice crushed by a vertical structure. Int J Naval Architecture Ocean Eng 9(2):209–218CrossRefGoogle Scholar
 82.Liu W, Hong JW (2012) A coupling approach of discretized peridynamics with finite element method. Comput Methods Appl Mech Eng 245:163–175MathSciNetzbMATHCrossRefGoogle Scholar
 83.Lu J, Zhang Y, Muhammad H, Chen Z (2018) Peridynamic model for the numerical simulation of anchor bolt pullout in concrete. Math Probl Eng 2018:1–10Google Scholar
 84.Madenci E, Barut A, Dorduncu M, Phan ND (2018) Coupling of peridynamics with finite elements without an overlap zone. In: 2018 AIAA/ASCE/AHS/ASC structures, structural dynamics, and materials conference, p 1462Google Scholar
 85.Madenci E, Dorduncu M, Barut A, Phan N (2018) A statebased peridynamic analysis in a finite element framework. Eng Fract Mech 195:104–128CrossRefGoogle Scholar
 86.Madenci E, Dorduncu M, Barut A, Phan N (2018) Weak form of peridynamics for nonlocal essential and natural boundary conditions. Comput Methods Appl Mech Eng 337:598–631MathSciNetCrossRefGoogle Scholar
 87.Madenci E, Oterkus E (2014) Peridynamic theory and its applications, vol 17. Springer, New YorkzbMATHCrossRefGoogle Scholar
 88.Madenci E, Oterkus E (2014) Peridynamic theory and its applications, chap 2 peridynamic theory, vol 17. Springer, Berlin, pp 19–43zbMATHCrossRefGoogle Scholar
 89.Mahanty D, Maiti S (1990) Experimental and finite element studies on mode I and mixed mode (I and II) stable crack growth—I. Exp Eng Fracture Mech 37(6):1237–1250CrossRefGoogle Scholar
 90.McCauley J, Strassburger E, Patel P, Paliwal B, Ramesh K (2013) Experimental observations on dynamic response of selected transparent armor materials. Exp Mech 53(1):3–29CrossRefGoogle Scholar
 91.Miranda A, Meggiolaro M, Castro J, Martha L, Bittencourt T (2003) Fatigue life and crack path predictions in generic 2d structural components. Eng Fract Mech 70(10):1259–1279CrossRefGoogle Scholar
 92.Mossaiby F, Shojaei A, Zaccariotto M, Galvanetto U (2017) Opencl implementation of a high performance 3d peridynamic model on graphics accelerators. Comput Math Appl 74(8): 1856–1870MathSciNetzbMATHCrossRefGoogle Scholar
 93.Nishawala VV, OstojaStarzewski M, Leamy MJ, Demmie PN (2016) Simulation of elastic wave propagation using cellular automata and peridynamics, and comparison with experiments. Wave Motion 60:73–83MathSciNetCrossRefGoogle Scholar
 94.Ocaña I, MolinaAldareguia J, Gonzalez D, Elizalde M, Sánchez J, MartĩnezEsnaola J, Sevillano JG, Scherban T, Pantuso D, Sun B (2006) Fracture characterization in patterned thin films by crosssectional nanoindentation. Acta Mater 54(13):3453–3462CrossRefGoogle Scholar
 95.Oterkus E, Agwai A, Guven I, Madenci E (2009) Peridynamic theory for simulation of failure mechanisms in electronic packages. In: ASME 2009 InterPACK Conference collocated with the ASME 2009 summer heat transfer conference and the ASME 2009 3rd international conference on energy sustainability. American Society of Mechanical Engineers, pp 63–68Google Scholar
 96.Oterkus E, Barut A, Madenci E (2010) Damage growth prediction from loaded composite fastener holes by using peridynamic theory. In: Collection of technical papers  AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, pp 1–14Google Scholar
 97.Parks M, Littlewood D, Mitchell J, Silling S (2012) Peridigm users’ guide. Tech Rep SAND20127800, Sandia National LaboratoriesGoogle Scholar
 98.Parks ML, Lehoucq RB, Plimpton SJ, Silling SA (2008) Implementing peridynamics within a molecular dynamics code. Comput Phys Commun 179(11):777–783zbMATHCrossRefGoogle Scholar
 99.Pharr M, Jakob W, Humphreys G (2016) Physically based rendering: from theory to implementation, 3rd edn. Morgan Kaufmann, San MateoGoogle Scholar
 100.Ramulu M, Kobayashi AS (1985) Mechanics of crack curving and branching — a dynamic fracture analysis. Int J Fract 27(3):187–201CrossRefGoogle Scholar
 101.Raymond S, Lemiale V, Ibrahim R, Lau R (2014) A meshfree study of the Kalthoff–Winkler experiment in 3D at room and low temperatures under dynamic loading using viscoplastic modelling. Engineering Analysis with Boundary Elements 42:20–25MathSciNetzbMATHCrossRefGoogle Scholar
 102.Ronsin O, Perrin B (1997) Multifracture propagation in a directional crack growth experiment. EPL (Europhysics Letters) 38(6):435CrossRefGoogle Scholar
 103.Ronsin O, Perrin B (1998) Dynamics of quasistatic directional crack growth. Phys Rev E 58:7878–7886CrossRefGoogle Scholar
 104.Rosenfeld A, Pfaltz JL (1966) Sequential operations in digital picture processing. J ACM 13(4):471–494zbMATHCrossRefGoogle Scholar
 105.Sanchez J, ElMansy S, Sun B, Scherban T, Fang N, Pantuso D, Ford W, Elizalde M, MartınezEsnaola J, MartınMeizoso A et al (1999) Crosssectional nanoindentation: a new technique for thin film interfacial adhesion characterization. Acta Materialia 47(17):4405–4413CrossRefGoogle Scholar
 106.Satyanarayana A, Bogert P, Chunchu P (2017) The effect of delamination on damage path and failure load prediction for notched composite laminates. In: 48th AIAA/ASME/ ASCE/AHS/ASC structures, structural dynamics, and materials conference, p 1993Google Scholar
 107.Schram S, Meyer H (2005) Simulating the formation and evolution of behind armor debris fields. ARLRP 109, US Army Research LaboratoryGoogle Scholar
 108.Seleson P, Beneddine S, Prudhomme S (2013) A forcebased coupling scheme for peridynamics and classical elasticity. Comput Mater Sci 66:34–49CrossRefGoogle Scholar
 109.Seleson P, Ha YD, Beneddine S (2015) Concurrent coupling of bondbased peridynamics and the navier equation of classical elasticity by blending. Int J Multiscale Comput Eng 13(2)Google Scholar
 110.Shapiro LG (1996) Connected component labeling and adjacency graph construction. In: Kong TY, Rosenfeld A (eds) Topological algorithms for digital image processing, machine intelligence and pattern recognition, vol 19. NorthHolland, pp 1–30Google Scholar
 111.Shojaei A, Mudric T, Zaccariotto M, Galvanetto U (2016) A coupled meshless finite point/peridynamic method for 2d dynamic fracture analysis. Int J Mech Sci 119:419–431CrossRefGoogle Scholar
 112.SIERRA Solid Mechanics Team (2016) Sierra/SolidMechanics 4.40 User’s Guide. Tech. Rep. SAND20162707 O, Sandia National Laboratories, Albuquerque, NM and Livermore, CAGoogle Scholar
 113.Silling SA (2000) Reformulation of elasticity theory for discontinuities and longrange forces. J Mech Phys Solids 48(1):175–209MathSciNetzbMATHCrossRefGoogle Scholar
 114.Silling SA (2002) Peridynamic modeling of the KalthoffWinkler experiment submission for the 2001 Sandia prize in computational scienceGoogle Scholar
 115.Silling SA (2011) A coarsening method for linear peridynamics. Int J Multiscale Comput Eng 9(6):609–622CrossRefGoogle Scholar
 116.Silling SA (2016) Why peridynamics? Handbook of Peridynamic Modeling: 1Google Scholar
 117.Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17):1526–1535CrossRefGoogle Scholar
 118.Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2):151–184MathSciNetzbMATHCrossRefGoogle Scholar
 119.Sodhi DS, Morris CE (1986) Characteristic frequency of force variations in continuous crushing of sheet ice against rigid cylindrical structures. Cold Reg Sci Technol 12(1):1–12CrossRefGoogle Scholar
 120.Starikov R, Schön J (2002) Local fatigue behaviour of cfrp bolted joints. Compos Sci Technol 62(2):243–253CrossRefGoogle Scholar
 121.Strassburger E (2004) Visualization of impact damage in ceramics using the edgeon impact technique. Int J Appl Ceram Technol 1:235–242CrossRefGoogle Scholar
 122.Strassburger E, Patel P, McCauley JW, Tempelton DW (2005) Visualization of wave propagation and impact damage in a polycrystalline transparent ceramic  AlON. In: 22nd international symposium on ballistics, vol 2, pp 769–776Google Scholar
 123.Strassburger E, Patel P, McCauley JW, Templeton DW (2006) Highspeed photographic study of wave propagation and iimpact damage in fused silcia and alon using the edgeon impact (EOI) method. AIP Conf Proc 845:892–895CrossRefGoogle Scholar
 124.Tabiai I (2017) PMMA tensile test until failure, loaded in force.Google Scholar
 125.Taylor M, Gözen I, Patel S, Jesorka A, Bertoldi K (2016) Peridynamic modeling of ruptures in biomembranes. PloS one 11(11):e0165, 947CrossRefGoogle Scholar
 126.Tian X, Du Q (2014) Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J Numer Anal 52(4):1641–1665MathSciNetzbMATHCrossRefGoogle Scholar
 127.Tupek M, Rimoli J, Radovitzky R (2013) An approach for incorporating classical continuum damage models in statebased peridynamics. Comput Methods Appl Mech Eng 263:20–26MathSciNetzbMATHCrossRefGoogle Scholar
 128.Vervuurt A, Van Mier JG, Schlangen E (1994) Analyses of anchor pullout in concrete. Mater Struct 27(5):251–259CrossRefGoogle Scholar
 129.Vogler TJ, Thornhill TF, Reinhart WD, Chhabildas LC, Grady DE, Wilson LT, Hurricane OA, Sunwoo A (2003) Fragmentation of materials in expanding tube experiments. Int J Impact Eng 29(1–10):735–746CrossRefGoogle Scholar
 130.Wang Y (2015) Peridynamic studies of interactions between stress waves and propagating cracks in brittle solids. Ph.D. thesis, University of NebraskaGoogle Scholar
 131.Weng T, Sun C (2000) A study of fracture criteria for ductile materials. Eng Fail Anal 7(2):101–125CrossRefGoogle Scholar
 132.Wetzel JJ (2009) The impulse response of extruded corrugated core aluminum sandwich structures. Master’s thesis, University of VirginiaGoogle Scholar
 133.Winkler KW (1983) Frequency dependent ultrasonic properties of highporosity sandstones. J Geophys Res Solid Earth 88(B11):9493–9499CrossRefGoogle Scholar
 134.Wu E (1967) Application of fracture mechanics to anisotropic plates. Trans ASME J Appl Mech 34(4):967–974CrossRefGoogle Scholar
 135.Yolum U, Taştan A, Güler MA (2016) A peridynamic model for ductile fracture of moderately thick plates. Procedia Structural Integrity 2:3713–3720CrossRefGoogle Scholar
 136.Yuse A, Sano M (1993) Transition between crack patterns in quenched glass plates. Nature 362(6418):329–331CrossRefGoogle Scholar
 137.Yuse A, Sano M (1997) Instabilities of quasistatic crack patterns in quenched glass plates. Physica D: Nonlinear Phenomena 108(4):365–378CrossRefGoogle Scholar
 138.Zaccariotto M, Mudric T, Tomasi D, Shojaei A, Galvanetto U (2018) Coupling of fem meshes with peridynamic grids. Comput Methods Appl Mech Eng 330:471–497MathSciNetCrossRefGoogle Scholar
 139.Zaccariotto M, Tomasi D, Galvanetto U (2017) An enhanced coupling of pd grids to fe meshes. Mech Res Commun 84:125–135CrossRefGoogle Scholar
 140.Zhang F, Wu J, Shen X (2011) SPHbased fluid simulation: a survey. In: Proceedings  2011 international conference on virtual reality and visualization, ICVRV 2011, pp 164–171Google Scholar
 141.Zhang G (2017) Peridynamic models for fatigue and fracture in isotropic and in polycrystalline materials. Ph.D. thesis, University of NebraskaGoogle Scholar
 142.Zhang G, Bobaru F (2016) Modeling The Evolution of Fatigue Failure with Peridynamics. In: Ro J Techn Sci – Appl Mechanics, vol 1, pp 22–40Google Scholar
 143.Zhang JJ, Bentley LR (1999) Change of bulk and shear moduli of dry sandstone with effective pressure and temperature. CREWES Research Report 11:01–16Google Scholar
 144.Zhou X, Wang Y, Qian Q (2016) Numerical simulation of crack curving and branching in brittle materials under dynamic loads using the extended nonordinary statebased peridynamics. Eur J Mech A Solids 60:277–299MathSciNetzbMATHCrossRefGoogle Scholar