# Decoupling Strength and Grid Resolution in Peridynamic Theory

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## Abstract

The primary mechanism of damage evolution of brittle materials in peridynamic theory is based on individual pair-wise bond strain. A critical value for bond strain is derived based on linear elastic fracture mechanic theory. It is a function of the horizon, the radius of non-locality of the material. The horizon must be larger than the material point spacing but not too large which would significantly slow the simulation. Typical sizes used for the horizon range between three and six times the material point spacing, if a constant multiple is used it could be said that the critical strain is a function of the model’s material point grid resolution. This critical strain function ensures that the fracture toughness is independent of the horizon and grid resolution. This works well when modeled materials have pre-existing cracks. However, since the peridynamic fracture toughness manifests as a critical strain it imposes an artificial strength onto the modeled material that may not represent the real strength of the material. When the material strength is less than the model’s critical strain certain flaw insertion methods can be used to capture the strength behavior. However, if the material strength is larger than the critical strain then the flaw size that represents that strength is too small to model. To address this discrepancy, multi-resolution models have been previously used, having differently sized horizons, such that the region that is expected to nucleate failure is represented correctly while the bulk material is modeled with a coarser grid and larger horizon. Such a multiscale approach could be designed from the beginning of the simulation or exhibit adaptive refinement during crack propagation. Such multiscale approaches add significant complexity to the simulation framework and fundamental model descriptions. This paper introduces a method that is able to decouple the model strength from the horizon and grid resolution by using a refinement overlay technique. This overlay is virtual and does not change the explicit material resolution as adaptive techniques do.

## Keywords

Strength Peridynamics Fracture Toughness Resolution Grid Mesh## 1 Introduction

The strength of brittle materials is strongly process and handling dependent. It is not an inherent material property. Handling changes the surface structure by introducing scratches, dents, etc [1, 2]. Strength can also be sample size dependent. Leonardo da Vinci found that the strength of iron wires varied inversely with their length [3]. This phenomenon is also observed in glass fibers [4]. This implies that the strength derives from microscopic flaws in the material and is a statistical quantity. On the other hand, the fracture toughness of brittle materials does not change with handling. Toughness is a size-independent material property [3]. Thus, we can conclude that fracture toughness and strength are two different parameters.

Modeling methods, both analytical and numerical, use these two parameters differently. Stress analysis is used to determine strength failure, while fracture mechanics is used for crack propagation determination. The latter requires an initial flaw considered as a pre-crack in order to calculate a stress intensity factor (SIF) [5].

There are modeling frameworks that use both of these parameters independently. Numerical modeling methods like the finite element method (FEM) have implemented certain modeling techniques that can handle these two damage parameters. For instance, the software package ABAQUS [5] has three possible initiation criteria for cohesive damage: max principal stress or strain, max nominal stress or strain, and quadratic nominal stress or strain. ABAQUS can also analyze crack tips to determine SIFs; it can even propagate a crack with cohesive separation [6] or using extended FEM (XFEM) [7]. There are drawbacks to these approaches as they have no mechanism for crack instabilities. Crack branching or crack interaction cannot occur in XFEM, without additional criteria, but can when using cohesive elements, although the resulting fracture pattern is highly mesh dependent [8]. Crack path instabilities are thought to play a significant role in crack branching as it manifests as fracture surface roughening that occurs before branching in brittle materials [9]. Most, if not all, FEM fracture propagation models can not handle this phenomenon. The velocity of crack growth also changes the stress distribution around the crack tip and can exhibit a bifurcation of the max tangential stress location which is hypothesized to facilitate crack branching [10]. FEM would need to choose a specific criterion in order to model a crack branching event.

There is another modeling method that can likely incorporate all of these parameters and overcome several of these limitations of FEM; it is the peridynamic model of solid mechanics [11]. Peridynamics (PD) is a non-local mesh-free approach to solving the equations of motion. The simplest PD material model is the prototype microelastic brittle (PMB) description, a bond-based approach which connects the material points via pair-wise linear “springs” such that one material point is connected to all neighboring points within a given radius, called the *horizon*. Under certain boundary or loading conditions, these grid points will move to minimize the elastic energy by integrating Newton’s equation of motion. These material point interactions (springs) can break in a brittle manner (useful for modeling the failure of glass or ceramics) when they reach a certain pair-wise engineering strain, *s*_{0}. This critical strain can be derived based on linear elastic fracture mechanics; the 2D version can be simplified to Eq. 1 for plane stress and Eq. 2 for plane strain [12, 13].

*G*

_{0}is the fracture energy release rate (also known as Griffith energy) related to the material fracture toughness and

*δ*is the horizon or the radius of material point interactions,

*K*is the Bulk modulus, and

*E*is Young’s modulus. As can be seen this critical bond strain acts as a material strength, in that it may allow the initiation of bonds to break and flaws to nucleate, as have been studied in the literature [15, 16, 17]. The failure stress,

*f*

_{t}, can be calculated based on the critical bond strain via Eq. 4, assuming that the critical bond strain is equal to the uniaxial tensile strain [12].

The horizon, *δ*, is a model parameter that dictates the degree of non-locality of the modeled material. As this parameter approaches zero, the model recovers the continuum solution. If the material being modeled exhibits length-scale effects, due to its microstructure, rate effects, plasticity, or to match dispersion curves, the horizon can be chosen to correspond to such length parameters [18]. All materials have a length scale, those materials that appear homogeneous and isotropic simply have very small length scales, and this is why classical continuum mechanics has been so successful. Glass for example has a vibrational length scale on the order of nanometers [19]. One may choose to use a horizon larger than the material’s in an attempt to save degree of freedom and hence computational expense. Although care must be taken to ensure that trailing waves dispersed by the non-local region are not affecting the evolution of damage [18].

The horizon is commonly expressed as a multiple of the grid spacing, typically ranging from about three to six; this multiple is given the variable, *m*. It has been shown that if this value is too small it can affect the crack propagation direction due to the bond orientation distribution being non-uniform in angular space, thus care must be taken to allow a large enough value but not too large as to render the model computationally intractable [20, 21]. From this expression, the horizon, *δ* = *m**d**x*, varies directly with the grid spacing, *dx*, which then means that the critical strain also varies with the grid spacing. Since any material point pair that extends beyond this strain will break, it is a strength parameter in the PD model. In order to define both a strength and fracture toughness, the horizon, *δ*, becomes a fixed value. This approach was used by Gerstle et al. to model concrete structures; however, since the tensile strength of the concrete modeled was 0.4 ksi the resulting horizon size was 31 inches, allowing macroscopic and very large model sizes [12, 22].

If the horizon is smaller than that which reproduces the material strength, the model strength becomes larger than the real material. In this case, another method, shown by Zhang and Qiao, can be used to nucleate a flaw [23]. For a given load, a crack size can be found in which the stress intensity is equal to the toughness; this flaw size (if larger than the horizon) can be supplied to the model to simulate the same strength as the real material.

*s*

_{0}, failure stress,

*f*

_{t}, and the horizon,

*δ*, as expressed in Eq. 1. For a strengthened glass, such as ion exchanged or tempered the required horizon would need to be much smaller, about 24 micron assuming 2D plane stress, precisely because of its higher strength [24, 25, 26].

The clear drawback of this failure relationship is that it constrains the horizon, and thus grid size, usable for simulation. Two examples illustrate this shortcoming. If a model does not begin with an explicit initial flaw then its nucleation will be entirely determined by the peridynamic model strength. If the model includes a region of residual tensile stress, care must be made to prevent it from exceeding the critical strain. If residual tension remains too close to the value of the critical strain, the dynamics of the fracture could cause elastic stress waves capable of pulverizing the material as if it contained micro-voids. Needless to say this is quite unphysical; the obvious solution to these problems is to reduce the horizon (and likely the grid spacing with it). In some cases, this is not feasible as the resulting grid spacing may be too small, causing a very large number of material points to simulate, so much so that computing clusters may not be able to handle it.

Let us take an example model of tempered glass with both consistent strength and toughness. From [25], the strength is about 157 MPa, using Eq. 3, this would require a horizon of *δ* = 14.6 micron, using Young’s modulus of *E* = 72 GPa and *G*_{0} = 6 J/m^{2} as fracture energy. If the grid spacing is one fourth of the horizon (*m* = 4) then the model sample size would only reach a cube with an edge length of 3.65 mm if the model could run with one billion material points. Only a large high performance computing cluster and an efficiently parallel code could handle this.

In many cases, the entire domain of the model does not require the same horizon size. This may be the case if the failure initiation occurs in a well defined region; the surrounding material may not be expected to fail and may then be permitted a larger horizon and thus a coarse grid resolution; this is common practice for Finite Element techniques [27]. There has been progress made to utilize this technique in peridynamic applications. Static problems have been solved using an adaptive multi-horizon approach, the triggering for grid refinement performed by the non-local strain energy density [28]. Additional contributions to the energy based triggering metric to include damage criteria has also been made [13]. Further advancements have come to include dynamic fracture capabilities in adaptive grid-refinement methods [29]. Similar methods have been used in the molecular dynamics (MD) community looking to reach larger length scales [30, 31, 32]. A well established method to couple non-local material models with local models, such as coupling MD to FEM [33, 34], have also reached the peridynamic community [35, 36].

Such grid refinement techniques that scale the horizon across material domains is probably the most direct way to address the strength-fracture relationship of peridynamics. However, they increase the complexity of the models and they introduce scale boundaries that must be properly treated so as ghost forces do not arise [37]. They are also susceptible to wave reflection of high frequencies [38, 39]. Although they reduce the required number of material points modeled, it may not be enough if the entire model geometry may exhibit fractures. In the following section, a new method will be described that decouples the peridynamic strength from the material toughness without the complexity of the aforementioned methods.

## 2 Method

The question now becomes: Is there a way to leverage the benefits of using a smaller horizon without the drawback of increased degree of freedom and computational expense? Is there a way perhaps to double check a given bond strain to see if it’s in a uniform stress field (strength) or near a stress intensity (crack) and to treat them differently?

In a typical peridynamic code, each material point pair, say between points I and J, the pair-wise bond strain, *s*_{IJ}, is checked against the critical bond strain, *s*_{0}, to determine whether the bond should break. If it should, it is first checked to determine whether it is within a user-defined geometric domain that is specified to perform this additional check. This domain is most logically to be the surfaces of the model, as they are the regions likely to experience strength failure, the bulk of the material model should not require this double check, unless the material is assumed to have implicit porosity or some heterogeneity which manifests at smaller length scales than the Horizon. In this work, the domain that implements this method is taken as the entire model merely for testing of the method. If the pair is within this domain then, a smaller implicit grid will be overlaid between these two material points, existing only during this pair-wise calculation for pair I and J. This Peridynamic Refinement Overlay (PDRO) produces virtual material points on a sub-grid of the undeformed material. The virtual points acquire displacement vectors by interpolating the surrounding material points’ displacement vectors. The specific points sampled are those that are within the Horizon of both point I and J, including those that have damage values. In this work, the interpolation is performed using the non-local differential operator as derived by Madenci [40] and utilized by Shojaei [41] although other forms of vector field interpolation could be used as well. The sub-grid strains between the primary material point pair IJ are calculated and the maximum of the sub-grid strains are compared to the sub-grid critical strain, \(s_{0}^{\prime }\) to complete the second check. If the max subgrid strain exceeds the critical sub-grid strain then the primary bond IJ will break, otherwise it remains intact. Further details about this method are described below.

The use of the sub-grid allows a smaller horizon to be used when calculating the critical strain for the sub-grid. Trying to use a smaller horizon with the existing coarse grid would reduce the number of material points within the horizon and would lower the accuracy of the material response and may result in grid orientation dependence of the fracture response [20]. Certain impressions about the local stress field can be found by comparing the primary pair strain to the sub-grid strains. If the sub-grid strain is the same or smaller than the primary grid strain (*s*_{ij} <= *s*_{IJ}), then it can be assumed that the local displacement field is fairly homogeneous, if the sub-grid strain is the same as the primary grid stain *and* exceeds the critical sub-grid strain (with the smaller horizon) (\(s_{ij}>s_{IJ} \& s_{ij}>s_{0}^{\prime }\)), then it would break, representative of a uniform stress and would indicate strength failure. If the sub-grid strain is larger than the primary grid strain (*s*_{ij} > *s*_{IJ}), then it indicates that the local displacement field is being intensified and could be around a crack tip and should then be permitted to fail, if the sub-grid strain exceeds the critical sub-grid strain.

*n*.

*n*is the divisor of the points I-J separation

*dx*. This is illustrated in Fig. 3. Four sub-grid positions are constructed, i, i’, j’, and j; i and j exist at a distance of

*d*

*x*/

*n*from I and J respectively and i’ and j’ at the same distance from one another with their average position in the center. Thus, there are six locations and three sub-pairs; I-i, i’-j’, j-J. Four sub-grid points are used rather than being equally spaced between I and J, as the latter would restrict the possible sub-grid spacing. With these additional sub-grid positions, any value, greater than zero, of

*n*can be used. Equation 5 show the relationships between the material point separation

*dx*and the sub-grid separation

*d*

*x*

^{′}in terms of

*n*and the critical strain,

*s*

_{0}and critical sub-grid strain, \(s_{0}^{\prime }\), due to the reduction in horizon size for the sub-grid horizon,

*δ*

^{′}.

*n*, can be defined by the user such that the sub-grid critical strain, \(s_{0}^{\prime }\), is the target material failure strain (strength). Thus, it can be seen that the modeled material strength can be tailored independently of the material fracture toughness, since smaller horizons result in larger critical strain values for a constant energy release rate. Consider a single edge notched crack tip in a plate, as in Fig. 4, the stresses around the crack tip follow the relations in Eq. 6 [3]. One can see that as the distance from the crack tip becomes smaller the local stress increases nonlinearly. The same relation is seen for the critical strain in Eq. 7 with respect to the horizon,

*δ*. This is why a max sub-grid strain that is larger than the primary grid strain indicates a local stress intensity field, and that fracture using PDRO should be independent for crack growth.

If the sub-grid displacements are accurately interpolated from the material point displacement field, then when the I-J strain is critical around a crack tip, so will the sub-grid strain be critical on the sub-grid scale. By changing or tailoring the sub-grid spacing via the point separation divisor, *n*, it is seen that the critical sub-grid strain and the sub-grid horizon changes but keeps the material fracture toughness, *K*_{IC} (or critical energy release rate, *G*_{0},) constant.

Any interpolation method could be used to find the sub-grid displacement vectors. However, in an effort to provide an accurate and robust interpolation method, the peridynamic differential operator, as derived by Madenci [40] and utilized by Shojaei [41], is chosen in this work. Although calculating derivatives of the displacement field is unnecessary for the PDRO method, the zeroth order derivative, or reproducibility condition, provides a convenient function useful for interpolation. Each displacement field component is assumed to be describable by a Taylor series expansion. An orthogonal function, \(g^{p_{x}p_{y}}(\xi )\), is multiplied by the series and integrated over the points in the horizon, about the position being interpolated, **x**. *ξ* represents the relative position vector of a material point with a defined displacement field value to the point being interpolated, **x**. Due to the orthogonality condition of the function, one can obtain the derivatives. Equation 8 shows this relation up to the second derivative, where *p*_{x} and *p*_{y} denote the order of differentiation with respect to *x* and *y* directions.

The orthogonality property of \(g^{p_{x}p_{y}}(\xi )\) is expressed in Eq. 9

The orthogonal functions can be constructed as polynomials of the form in Eq. 10, where \(w_{q_{x}q_{y}}(|{\xi }|)\) are weight functions for each term, however in this implementation are taken to be identical to one another. The unknown terms \(a_{q_{x}}^{p_{x}}\) and \(a_{q_{y}}^{p_{y}}\) can be found by solving the system of linear equations that result upon substitution of Eq. 10 in to Eq. 9. For further details on the calculations involved, please see the current uses in the literature [40, 41].

Upon calculation of the unknown terms \(a_{q_{x}}^{p_{x}}\) and \(a_{q_{y}}^{p_{y}}\) and using them to calculate the orthogonal functions, \(g^{p_{x}p_{y}}(\xi )\), the calculation of any derivative, only up to second order is shown here, can be made by Eq. 8. In this work, only the zeroth order derivative is used for interpolation of each displacement field component. These equations were solved for every sub-grid position constructed.

## 3 Results and Analysis

Three different conditions are explored in this section: the first is to investigate the tailorable material strength claim by using an elastic bar in tension. The second study uses a single edge notched tension specimen to show that material fracture toughness is not changed and captures the analytical prediction. The third case involves the propagation and bifurcation of a crack and the accuracy of replication when using PDRO for various divisor values, *n*.

### 3.1 Strength

*E*= 72 GPa with a density of 2400 kg/m

^{3}. Although the specific kind of material is irrelevant for this demonstration, these material properties align closely with commercial soda-lime glass. The grid sizes ranged from 0.1 to 0.9 mm. The horizon was allowed to vary with the grid size as it was assigned to be equal to 4 times the grid spacing. The smallest grid thus has a horizon of 400 micron which provides a strength comparable to the surface of soda-lime glass.

*n*ranging from 1.111 to 10. These

*n*values correspond to sub-grid spacings already investigated explicitly with high-resolution models that classically exhibit higher PD strengths. This was to compare the model strengths between the explicit fine grid models to the use of the PDRO on the coarse grid model. The use of PDRO on the coarse grid is to enhance its strength beyond that classically allowed by Eq. 2. The remote load strain at failure is plotted in Fig. 6 for the explicit fine grid models (symbol + ) and the coarse grid model using PDRO (symbol ×).

It is seen that the coarse grid is able to achieve greater strengths with the use of PDRO. The strengths targeted with PDRO are extremely close to the native grid spacing; both of which follow closely to the analytical strength value from Eq. 2. The discrepancy with the analytical comes from the definition of strain being a function of the entire family while strength is only dependent upon an individual bond. In uniaxial tension, one bond may have a critical strain while the material strain at that material point would be slightly lower. This example illustrates how a given grid and horizon can achieve greater model strengths through the use of the PDRO method.

### 3.2 Stress Intensity

^{1}/

_{4}

^{th}of the way across the width of the plate. This plate was subjected to a remote stress increasing at a rate of 7.6 GPa/sec, crack propagation was expected at 200 microseconds. The schematic setup can be seen in Fig. 7 and the stress field in Fig. 4. The analytical equations that predict the stress intensity for this geometry are expressed

in Eqs. 11 and 12 [3]. Young’s modulus selected for this material was *E* = 72 GPa with a density of 2400 kg/m^{3}. The grid sizes ranged from 0.1 to 0.9 mm; just as in the last example, this corresponds to a typical soda-lime glass. The horizon was allowed to vary with the grid size as it was defined to be equal to 4 times the grid spacing.

### 3.3 Dynamic Crack Propagation and Bifurcation

A 2D plane strain peridynamic model was generated to model a linear elastic disk with a single central crack. This disk had a radius of 0.204 meters and was clamped around the circumference and cooled in order to produce a biaxial tensile strain of − *α**d**T* = 5 × 10^{− 5} caused by the thermal contraction strains. Elastic thermal expansion was implemented by subtracting the linear thermal strain from the elastic strain following Eq. 13, where *α* is the linear thermal expansion coefficient and *d**T* is the relative difference in temperature [42]. Here, *ξ*_{I} is the undeformed position of material point *I* and *η*_{I} is the displacement vector of material point *I*.

*E*= 72 GPa with a density of 2400 kg/m

^{3}, corresponding to common soda-lime glass. The explicit time integration was performed by a forward Euler algorithm, no drift in total energy was observed over the few hundred time steps simulated. The time step used in this explicit time integration was about 120 nanoseconds and was defined based on a condition of stability [14]. The grid size chosen was 0.9 mm, yielding 161424 material points. The horizon was set equal to 4 times the grid spacing. The initial center crack was 4 mm long. PDRO was used with different values of the divisor,

*n*, to determine how well it maintained the bifurcation pattern. Figure 9 shows the resulting damage evolution vs. time. Each material point has a damage fraction that is defined as the fraction of broken bonds that were attached to it. A material point at a typical smooth fracture surface has a damage fraction around 0.35 to 0.4; as the number of points within a horizon goes to infinity,

*m*→

*∞*, it would approach 0.5. The total system damage shown in Fig. 9 is simply a sum of all material point damage fractions in the model. The bifurcation pattern is shown via the material points’ damage fraction on the right of the figure with the value of

*n*that was used. These snapshots were taken at 24 microseconds which corresponds to the 200

^{th}time step. The bifurcation event occurs around 20 microseconds, at which time the stress waves from the initial crack opening have not yet reached the outer boundaries of the model, which would occur at about 30 microseconds, excluding them from affecting the fracture events.

*n*≤ 2 will yield the highest accuracy. Unfortunately this only increases the material strength by about 41%. It is not entirely clear why values of

*n*> 2 yield such poor accuracies but could be related to portions of the displacement field between material points I-J that are inadequately sampled. Unfortunately in order to increase the samples would require further interpolation, and then the speed of the calculation may begin to be hampered by the extensive calculations being performed on every critical strain. However, the PDRO method is designed to help the user overcome the obstacle of a restrictive model strength. In most cases the surfaces of materials are the weakest locations; in this case a geometric domain could be applied that only uses the PDRO method within a user-defined region, like the surface the model, for instance. This would not only be computationally efficient but also physically sound. The only conditions under which a crack tip would need to survive the PDRO process is when it is at a surface. This normally occurs during nucleation or at the ends of a crack front in 3D.

## 4 Conclusion

The inherent coupling of peridynamic material strength to the model’s horizon has been shown to be mitigated through the use of a refinement overlay. This acts to provide a refined mesh between critically strained material point pairs, providing an alternative pair strain for bond rupture determination. This method is shown to provide a tailored material strength while maintaining the material fracture toughness and model grid spacing and horizon size. This technique adds three sub-pairs between any real material point pair; this provides dynamic accuracy down to a sub-grid spacing of half the real grid spacing, beyond which causes significant accuracy deterioration for propagating cracks. This technique depends on an accurate and fast interpolation method for the sub-grid displacements, which dictates how efficient this method will be in practice. Since interpolations need only be performed on critical bonds, typically near the surfaces, the fraction of bonds in the model that require this extra treatment minimizes as the model size increases and would increasingly perform better than a full model at a finer resolution and smaller horizon size.

## Notes

### Acknowledgements

The authors thank Jason Harris for being supportive of this work and for guiding discussions, as well as Wei Xu and Sam Zoubi for enthusiastic support.

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