## Abstract

Unlike micromechanics failure models that have a well-defined crack path, phase-field fracture models are capable of predicting the crack path in arbitrary geometries and dimensions by utilizing a diffuse representation of cracks. However, such models rely on the calibration of a fracture energy (G_{c}) and a regularization length-scale (l_{c}) parameter, which do not have a strong micromechanical basis. Here, we construct the equivalent crack-tip cohesive zone laws representing a phase-field fracture model, to elucidate the effects of G_{c} and l_{c} on the fracture resistance and crack growth mechanics under mode I K-field loading. Our results show that the cohesive zone law scales with increasing G_{c} while maintaining the same functional form. In contrast, increasing l_{c} broadens the process zone and results in a flattened traction-separation profile with a decreased but sustained peak cohesive traction over longer separation distances. While G_{c} quantitatively captures the fracture initiation toughness, increasing G_{c} coupled with decreasing l_{c} contributes to a rising fracture resistance curve and a higher steady-state toughness—both these effects cumulate in an evolving cohesive zone law with crack progression. We discuss the relationship between these phase-field parameters and process zone characteristics in the material.

## 1 Introduction

The ability to model and predict crack initiation and propagation in both brittle and ductile solids is pivotal to achieving optimal structural design for fracture resistance. This is especially apparent with the advancement of additive manufacturing technologies which are now capable of rapidly producing material structures of varying complexities [1–3]. Micromechanical failure models typically rely on a local approach, where the fracture event is localized within a well-defined fracture process zone, which is embedded within a continuum constitutive model representing the background material [4,5]. The micromechanisms for fracture, such as void growth and coalescence within the fracture process zone, are modeled either discretely or through damage constitutive models, such as the Gurson model or cohesive zone laws [6–8]. Such micromechanical models have enabled fundamental studies to elucidate the coupling between complex mechanisms of fracture within the process zone and plastic deformation of the background material [9–11]. A major limitation, however, is that the process zone (and hence the crack path) has to be established a priori [12–17], which complicates the modeling of convoluted crack patterns, including crack branching and merging.

The phase-field approach to fracture departs from the discontinuous description of failure in local micromechanics-based models. The formulation is based on the variational approach of the classical Griffith energy balance for brittle fracture and regularizes the topology of the sharp crack as a diffuse damage zone instead of a discontinuity [18–23]. As such, the model is able to handle topologically complex fractures and has been widely adopted to solve challenging fracture problems, including hydrogen-assisted cracking [24,25], fracture in viscoelastic materials, biomaterials, or anisotropic solids with different material symmetry [26–30], as well as dynamic brittle fracture with complex branching [31,32].

To-date, there is no clear micromechanical basis for the parameters used in the phase-field fracture models. In particular, the amount of crack regularization is controlled through a prescribed length-scale parameter, *l*_{c}, which some have perceived as purely a mathematical construct to allow the Griffin crack (Fig. 1(a), left) to be smeared over a diffused continuum zone (Fig. 1(a), right) [33,34]. In the phase-field formulation, *l*_{c} arises through a degradation function to describe the material behavior as it transistions between fully intact and damage states. Classical approach for phase field assumes a stifffness-based degradation function, usually in the form of a hyperbole function, as it allows for an accurate reproduction of linear elastic fracture mechanics response [35–37]. In recent years, an energy-based degradation function has been proposed where certain mechanisms of damage can be incorporated directly into the phase-field model [38]. For all these degradation functions, the phase-field solution should converge to a discrete crack solution in the limit as *l*_{c} approaches zero [33]. Others have argued that *l*_{c} represents a specific material property that is closely connected to the critical stress for crack nucleation [39–42]. Because of this diffuse crack representation, the fracture energy density term *G*_{c} in the phase-field model only approximately relates to the classical Griffith critical energy release rate in the limit of *l*_{c} → 0 [43].

To provide mechanistic insights into the above phase-field parameters, one approach is to extract a micromechanics representation of an equivalent fracture process zone of the phase-field fracture model through a cohesive zone law. This cohesive zone law constitutes the relationship between interfacial tractions in equilibrium with the surrounding body and the cohesive separations compatible with the deformation fields of the surrounding body [9,10]. A general view is that the cohesive strength (peak traction) and the cohesive energy (area under the traction-separation relation) are the two primary material parameters governing the macroscopic fracture behavior [13–16]. However, the functional form of the cohesive zone law has been reported to reflect the fracture micromechanisms [44,45]. This has led to the development of inverse techniques to systematically uncover the exact functional form of the cohesive zone laws governed by different failure processes under both monotonic fracture and cyclic fatigue [7,46–49].

In this paper, we explore the relationship between parameters of a phase-field fracture model and the fracture resistance and crack growth mechanics under mode I loading by investigating the equivalent crack-tip cohesive zone laws representing the phase-field fracture model based on a hyperbole stiffness degradation function. Section 2 describes the finite element implementation of the phase-field fracture model, along with the formulation of a small-scale yielding, modified boundary layer model with imposed monotonic *K*_{I} remote displacement loading. In Sec. 3, we systematically study the influence of *G*_{c} and *l*_{c} on both the macroscopic fracture resistance and microscopic fracture processes as quantified through an equivalent crack-tip cohesive zone law. Section 4 discusses the relationship between these phase-field parameters and key process zone characteristics in the material and concludes with a summary.

## 2 Problem Formulation

### 2.1 Phase-Field Fracture Modeling.

*S*in a deformable solid body Ω (Fig. 1(a), left), where the energy balance can be formulated in a variational form as

*ψ*(

*u*) is the elastic strain energy density as a function of displacement, and

*G*

_{c}is the critical energy release rate characterizing the fracture resistance of the material. Minimizing (1) is not mathematically feasible because of the unknown nature of

*S*. The phase-field model overcomes this obstacle by smearing the crack's topology as a diffuse damage zone instead of a discontinuity (Fig. 1(a), right). Specifically, the sharp crack is regularized through a diffuse damage variable,

*ϕ*, representing the damage extent caused by the presence of the crack in the surrounding neighborhood, with the limits

*ϕ*= 0 and 1 representing the intact and fully cracked regions, respectively [50–52]. Accordingly, (1) can be approximated as [34,53]

*g*(

*ϕ*) is a continuous degradation function that monotonically degrades the stiffness of the material as the phase-field approaches the crack phase (

*ϕ*= 1) and is commonly taken as

*S*. Several crack density functionals have been proposed [35,54,55] and we adopt the form [34]

*l*

_{c}is a regularization length-scale parameter, ensuring that (2) converges to (1) in the limit

*l*

_{c}→ 0

^{+};

*l*

_{c}can be interpreted as a material property in the case of

*l*

_{c}> 0

^{+}.

*u*) and the phase field (

*ϕ*)

*b*

_{i}is the body force term, with the Cauchy stress,

*σ*

_{ij}, related to the elastic strain,

*ɛ*

_{ij}, and the fourth-order elasticity (stiffness) tensor,

*C*

_{ijkl}, by

By expressing (5) in a weak form, we discretize (*u*, *ϕ*) in a standard finite element scheme and formulate the residuals and the stiffness matrices. This numerical implementation is described in detail in Ref. [24] and is conducted within a user element subroutine in the commercial finite element software, abaqus.

### 2.2 Boundary Value Problem.

*E*and Poisson's ratio

*ν*, and subjected to remote mode I (

*K*

_{I}) loading under plane strain conditions (Fig. 1(b)). Due to geometrical-symmetry about the

*x*

_{2}-axis, we model only one-half of the geometry, as shown by the finite element mesh in Fig. 1(c). Rather than creating a physical crack, we introduce the crack by setting the phase-field parameter

*ϕ*= 1 on the row of elements along

*x*

_{1}< 0,

*x*

_{2}= 0. The initial crack-tip, located at

*x*

_{1}=

*x*

_{2}= 0, is within a highly refined mesh comprising of uniformly sized elements, each of dimensions

*D*×

*D*, as shown by close-up view of the finite element mesh in Fig. 1(d). We impose roller boundary conditions at the start of the dense mesh region (i.e.,

*x*

_{1}≥ − 20

*D*) along

*x*

_{2}= 0. Along the remote semi-circular boundary of the finite element mesh (Fig. 1(c)), we prescribe the elastic asymptotic in-plane displacement fields

*θ*= tan

^{−1}(

*x*

_{2}/

*x*

_{1}) for points on the remote boundary. The energy release rate or

*J*-integral is related to the mode I stress intensity factor

*K*

_{I}by

## 3 Results

From dimensional analysis, both the macroscopic fracture resistance, Γ, in (8), and the spatial distribution of microscopic field quantities, *σ*_{ij}/*E* and *ϕ*, depend on the following dimensionless geometric-material parameters (Γ/*ED*, *ν*, *G*_{c}/*ED*, *l*_{c}/*D*). In this study, we set *ν* = 0.3 and direct attention to the phase-field related parameters (*G*_{c}/*ED*, *l*_{c}/*D*). Note that *l*_{c} was originally introduced as a mathematical construct to transform a discrete crack surface into smooth continuum gradient parameters to represent a smeared (diffuse) crack (Fig. 1(a)), implying that (2) → (1) and *G*_{c} approaches the fracture toughness only in the limit of *l*_{c} → 0. However, convergence studies show that larger (finite) values of *l*_{c} are often required for the phase-field fracture models to match experimental results [24–28,56–58], suggesting that (*l*_{c}, *G*_{c}) can be regarded as phenomenological material properties. In the following, we will parametrically vary *G*_{c}/(*ED*) from 0.005 to 0.03 and *l*_{c}/*D* from 4 to 14 to elucidate the effects of these phase-field parameters on both the microscopic crack growth processes and the macroscopic fracture resistance. For all our calculations, we define the current crack-tip location as the furthest distance along *x*_{2} = 0 where *ϕ* = 1.

### 3.1 Fracture Resistance and Crack Growth Process.

Figure 2 shows the effects of *G*_{c} and *l*_{c} on the fracture resistance (*R*-) curves. For all cases, *G*_{c} (value denoted by square symbols) quantitatively captures the fracture initiation toughness, Γ_{ini}, as defined by the energy release rate or *J*-integral (Γ in (8)) at the first instance of crack growth (i.e., at an elemental crack length of Δ*a* = *D*). The initial crack growth is characterized by the rapid rise in Γ. The rate of increase of Γ slows down significantly for crack growth beyond about 10*D*, but the continued rise of Γ with crack length Δ*a* beyond this point suggests the continued toughening of the background material, albeit at rates that depend on both *l*_{c}/*D* and *G*_{c}/(*ED*). As shown in Fig. 2(b), increasing *l*_{c}/*D* from 4 to 14 at a prescribed *G*_{c}/(*ED*) = 0.015 decreases the rate of increase of Γ by ∼50%. Comparatively, reducing *G*_{c}/(*ED*), and hence Γ_{ini}, from 0.03 to 0.005 at a fixed *l*_{c}/*D* = 4 in Fig. 2(a) decreases the rate of increase of Γ by several-folds.

To elucidate the toughening mechanisms associated with the phase-field parameters, *G*_{c} and *l*_{c}, we show in Fig. 3 contours of the von Mises stress, *σ*_{e}, and the phase-field damage variable, *ϕ*, for three combinations of *G*_{c}/(*ED*) and *l*_{c}/*D*, and at three crack growth instances, $\Delta a/D=5,40,and80$. Comparison between *G*_{c}/(*ED*) = 0.03 versus 0.015 at *l*_{c}/*D* = 4 shows that the stresses at the crack-tip are notably larger at higher *G*_{c}/(*ED*) across all Δ*a*/*D*, with the maximum von Mises stress increasing by nearly two-fold. Interestingly, the contours for *ϕ* remain almost unchanged. By contrast, increasing *l*_{c}/*D* from 4 to 14 but at a fixed *G*_{c}/(*ED*) = 0.015 dramatically increases the width of the diffuse damage zone *ϕ*. This smearing of damage over a wider region also corresponds with an almost two-fold decrease in the maximum von Mises stress.

As previously shown in Fig. 2, *G*_{c}/(*ED*) is a quantitative measure of the fracture initiation toughness. In addition, the higher von Mises stress-fields surrounding the propagating crack at higher *G*_{c} in Fig. 3 suggests that increasing *G*_{c} also increases the stress-carrying capacity ahead of the growing crack, which ultimately leads to improved fracture resistance over the entire transient crack growth regime. On the other hand, *l*_{c}/*D* can be interpreted as a measure of the thickness of the fracture process zone defined by *ϕ* > 0. As shown in Fig. 3, a larger *l*_{c}/*D* corresponds to a larger and more diffused process zone, while a smaller *l*_{c}/*D* results in a narrower process zone. Since the material stiffness monotonically degrades with *g*(*ϕ*) in (3) for *ϕ* > 0, the reduced stress-carrying capacity in the process zone ahead of the crack-tip at higher *l*_{c}/*D* results in lower fracture resistance. These effects of *G*_{c} and *l*_{c} are qualitatively in agreement with the stress–strain response of a homogeneously deformed 1D material with $\u2207\varphi =0$, which shows that the failure stress scales with $Gc/lc$ [23,42,59].

Both the contours of *σ*_{e} and *ϕ* in Fig. 3 are evolving with Δ*a*/*D*, inferring that the process zone is still developing with crack propagation. This change in near-tip condition with the transition from crack initiation to crack growth is responsible for the rising *R*-curves in Fig. 2. We show in Fig. 4 the evolution of the near-tip process zone by centering the contours of *ϕ* about the current crack-tip. Similar contours of *ϕ* (process zone sizes) are observed between *G*_{c}/(*ED*) = 0.03 and 0.015 with *l*_{c}/*D* = 4. In both cases, the damage contours are continuously expanding at a near constant rate as the crack propagates from Δ*a* = 5*D* to 80*D*. In contrast, the contours of *ϕ* for *G*_{c}/(*ED*) = 0.015 with *l*_{c}/*D* = 14 are nearly two- to three-fold larger. A comparatively smaller increase in process zone size is also observed as the crack grows from Δ*a*/*D* = 5*D* to 40*D* versus Δ*a*/*D* = 40*D* to 80*D*, indicating that the crack growth is reaching its steady-state.

### 3.2 Crack-Tip Cohesive Zone Laws.

Unlike local micromechanics fracture models where damage is confined to a narrow process zone ahead of the crack [5], diffuse nonlocal approaches such as the phase-field fracture model dissipate damage over a wider region and across many elements as shown in Figs. 3 and 4 [12,60,61]. To provide a homogenized view of the diffuse crack-tip process in the phase-field fracture model, we construct the equivalent local traction-separation relationship constituting the cohesive zone law embedded within a linear elastic background material. Figures 5 and 6 show the effects of *G*_{c}/(*ED*) and *l*_{c}/*D* on the distributions of the crack-tip cohesive tractions, *t*_{2}, cohesive separations, *δ*_{2}, and the phase-field damage parameter, *ϕ*, at Δ*a*/*D* = 25. Both *t*_{2}(*x*_{1}) and *ϕ*(*x*_{1}) are obtained directly from the phase-field finite element calculations along *x*_{2} = 0. To obtain the equivalent cohesive separation distributions in a linear elastic background material, we impose the measured *t*_{2}(*x*_{1}) along *x*_{2} = 0 of the same finite element mesh with the same *K*-field displacements as boundary conditions, albeit with a linear elastic (i.e., non-phase-field) constitutive relationship of the same *E* and *ν*, and compute *δ*_{2}(*x*_{1}) = 2*u*_{2}(*x*_{1}) along *x*_{2} = 0. In this fashion, we effectively project the diffuse damage *ϕ* within the finite process zone onto a zero-thickness cohesive zone embedded within a linear elastic body, as shown schematically in the inset of Fig. 5(b).

Increasing *l*_{c}/*D* with the same *G*_{c}/(*ED*) (Fig. 6(a)) is found to reduce the peak cohesive tractions. All post-peak traction distributions, however, converge to the same path further ahead of the crack-tip. In contrast, increasing *G*_{c}/(*ED*) with the same *l*_{c}/*D* (Fig. 5(a)) proportionally increases both the peak and post-peak traction distributions, while maintaining the same traction distribution profile. Increasing *G*_{c}/(*ED*) causes a pronounced increase in the crack-tip separations versus the effects of *l*_{c}/*D* (compare Fig. 5(b) versus Fig. 6(b)). However, increasing *l*_{c}/*D* also increases the length of the crack-tip cohesive zone, as evidenced by both the larger non-zero separations (Fig. 6(b)) and the slower decay of damage (Fig. 6(c)) further ahead of the crack-tip. In comparison, increasing *G*_{c}/(*ED*) has negligble influence on the cohesive zone size—the separation and damage distributions (Figs. 5(b) and 5(c)) both decay to zero at nearly identical distances ahead of the crack-tip.

Together, the above cohesive traction (Figs. 5(a) and 6(a)) and separation (Figs. 5(b) and 6(b)) distributions are used to construct the traction-separation relationships in Fig. 7 constituting the crack-tip cohesive zone laws in a linear elastic background material. Generally, the cohesive strength (peak cohesive traction) and the cohesive energy (area under the (*t*_{2}, *δ*_{2}) envelope) are two important variables which control the fracture resistance. While *G*_{c}/(*ED*) corresponds to the fracture initiation toughness, Γ_{ini}, the cohesive energy here quantitatively corresponds to the fracture resistance Γ at the specific crack instant where the cohesive zone law was constructed (Δ*a*/*D* = 25). As shown in Fig. 7(a), increasing *G*_{c}/(*ED*) increases both the cohesive strength and energy, but the functional form of the cohesive traction-separation relationship remains the same. Increasing *l*_{c}/*D* in Fig. 7(b) decreases the cohesive strength, and to a lesser extent, the cohesive energy. More importantly, a significant change in the functional form of the cohesive zone law is observed—from a rapid increase and subsequent decrease in cohesive tractions at short separation distances for *l*_{c}/*D* = 4 to the flattening of the cohesive zone law resulting in a sustained, but lower, peak traction over longer separation distances (i.e., longer cohesive zone) for *l*_{c}/*D* = 14. These changes in shape of the cohesive zone law with *l*_{c}/*D* can be used to represent changes in the fracture micromechanisms.

In the phase-field model, the damage parameter *ϕ* provides a measure of the transition from an undamaged (*ϕ* = 0) to a fully cracked (*ϕ* = 1) material. An alternative “cohesive zone” approach to quantify crack growth is to consider the cohesive energy dissipated as the separation progresses, as quantified by the evolving area encompassed by the traction-separation response, i.e., $\Gamma s(\delta 2)=\u222b0\delta 2t2(\delta 2\u2032)d\delta 2\u2032$, relative to the separation energy Γ. Thus, Γ_{s}/Γ = 0 at *δ*_{2} = 0 represents a fully intact material, while Γ_{s}/Γ = 1 when the cohesive zone is fully developed at *δ*_{2} = *δ*_{0} (inset in Fig. 5(b)) denotes a fully separated material. A comparison of the evolution of *ϕ* and Γ_{s}/Γ with *δ*_{2} across various (*l*_{c}, *G*_{c}) combinations in Fig. 8 demonstrates that the damage or crack growth assessment from both approaches are in good quantitative agreement.

The rising *R*-curves in Fig. 2 suggest that the cohesive zone law is evolving as the crack propagates. Accordingly, we show in Fig. 9 the evolving cohesive zone laws reconstructed from the traction and separation distributions at increasing Δ*a*/*D* across various *l*_{c} and *G*_{c} combinations. Note that the cohesive energy of each of these (*l*_{c}, *G*_{c}, Δ*a*) combinations are quantitatively in perfect agreement with the calculated energy release rate, Γ(Δ*a*/*D*), in Fig. 2, since all of the dissipation energy in the process zone is projected onto an equivalent zero-thickness cohesive zone. While the shape of the cohesive zone law generally remains unchanged, increasing Δ*a*/*D* increases both the peak cohesive traction as well as the total separation, *δ*_{0}, particularly at high *G*_{c}/(*ED*) of 0.025 where a 16% increase in the cohesive strength is observed as the crack propagates from $\Delta a/D=$ 10 to 115. At higher *l*_{c}/*D*, the cohesive zone laws rapidly converge with Δ*a*/*D*, indicating a transition to steady-state crack growth.

## 4 Discussions and Conclusion

In the micromechanics modeling of fracture, a brittle or ductile fracture response often correlates with the extent of plastic dissipation in the background material. In the case of brittle fracture, damage is often confined to the thin process zone ahead of the crack, and the limited plastic dissipation results in a flat *R*-curve representing rapid and unstable crack propagation. In the case of ductile fracture, the development of significant background plastic dissipation results in a rising *R*-curve. The phase-field fracture model we have adopted is widely considered to be “brittle” as we have assumed an elastic background material. We remark that “ductile” phase-field fracture models based on elasto-plastic background materials have also been proposed in the open literature [36,56,57,62,63]. However, our crack growth simulations for this “brittle” phase-field fracture model demonstrate a rising *R*-curve in cases with large *G*_{c} and/or small *l*_{c}, suggesting that this seemingly “brittle” model can effectively simulate the fracture response of ductile materials.

Akin to crack growth in an elasto-plastic material, the macroscopic fracture resistance Γ in our phase-field model can be delineated into two primary contributions: the fracture initiation toughness, Γ_{ini}, and phase-field dissipation energy, Γ_{p}. Our simulation results in Fig. 2 show that *G*_{c} is a direct quantitative measure of Γ_{ini}, in agreement with prior studies [42,53], while both *G*_{c} and *l*_{c} have profound effects on Γ_{p}. Similar to the growth of the plastic zone size during transient crack growth, the transition from crack initiation to steady-state crack growth in our phase-field fracture model is marked by the development of a diffuse damage zone which evolves with Δ*a* (Figs. 3 and 4). This diffuse zone can be treated as an *evolving* fracture process zone in the transient crack growth regime and is distinct from micromechanics fracture models which almost always assume a fixed process zone size.

One approach to quantify the phase-field fracture process is to project the diffuse damage into an equivalent crack-tip cohesive zone law which evolves with crack growth. In doing so, the cohesive energy quantitatively equates to the total fracture resistance, Γ(Δ*a*), which encompasses both the contributions of Γ_{ini} and Γ_{p}(Δ*a*). Our results in Figs. 5–7 show that *G*_{c} has a strong influence on both the cohesive strength and cohesive energy but has negligible effects on the shape of the cohesive zone law—this suggests that the underlying crack growth mechanics remain unchanged with *G*_{c}. In contrast, *l*_{c} significantly changes the functional form of the cohesive zone law, and a transition from a sharp traction-separation profile resembling a bilinear cohesive zone model to a flattened traction-separation profile resembling a trapezoidal cohesive zone model is observed with increasing *l*_{c}. This change in the shape of the traction-separation relationship is associated with the increasing size of the diffuse damage zone with *l*_{c}—a larger *l*_{c} creates both a thicker and longer fracture process zone with reduced cohesive strength. By comparison, increasing *G*_{c} has negligible influence on the process zone size.

Our studies, therefore, demonstrate that *l*_{c} quantitatively relates to the size of the fracture process zone. In the limit of *l*_{c} → 0, the process zone physically collapses to a zero-thickness cohesive zone, and the phase-field fracture model would indeed be equivalent to a sharp crack propagating within an elastic medium. The fracture process would always be brittle with Γ = *G*_{c}, since there would be no energy dissipation in the background material. However, convergence studies often lead to the adoption of relatively large values of *l*_{c}/*D* > 10 [24,25,29,62], which introduces a finite thickness fracture process zone. Differing from these numerical studies, our results suggest that *l*_{c} should be calibrated and selected based on the physical thickness of the fracture process zone. For example, a small value of *l*_{c}/*D* should be used to represent the narrow process zone for cracking in a brittle rock-like material such as shale or concrete [64,65], while a larger *l*_{c}/*D* will better represent the more diffused process zone associated with micro-crazing in polymers [66,67]. This would provide *l*_{c}/*D* with a stronger physical basis. The “brittle” phase-field fracture model we adopt does not account for background plasticity. Without loss of generality, our simulations suggest that the background plastic dissipation in elasto-plastic materials constituting Γ_{p} can heuristically be treated as a diffuse damage zone represented by appropriately calibrated values of *l*_{c}/*D*.

The cohesive strength within the process zone depends on both *G*_{c} and *l*_{c}. Uniform deformation studies show that the critical strength in the stress–strain response of a phase-field element scales with $Gc/lc$ [23,42,59], and the cohesive strength of our equivalent crack-tip cohesive zone laws appear to follow similar trends. Conceivably, once *l*_{c} is calibrated to represent the size of the fracture process zone, *G*_{c} can in turn be calibrated to fit the cohesive strength representing the appropriate crack growth mechanisms, such as void growth and coalescence, fiber pull-out, and phase transformation.

In conclusion, we have obtained new physical insights into the role of the energy and length-scale parameters, *G*_{c} and *l*_{c}, in phase-field fracture models, by constructing the equivalent crack-tip cohesive zone laws representative of the diffuse damage process. We demonstrate that *G*_{c} can be perceived as the fracture initiation toughness, and quantitatively controls both the cohesive strength and energy. While *l*_{c} was initially introduced as a mathematical construct, we show that *l*_{c} should be physically interpreted as a measure of the thickness of the diffuse damage (fracture process) zone. By adopting finite values of *l*_{c}, together with appropriately calibrated values of *G*_{c}, the “brittle” phase-field fracture model can be used to simulate a wide variety of material systems, from brittle rock materials with narrow process zones to ductile metals with diffuse background plastic dissipation. We remark that a commonly used stiffness-based hyperbole degradation function is considered in this study. Extracting the equivalent cohesive zone laws for other degradation functions is a subject of future work.

## Acknowledgment

The authors acknowledge the support provided by National Science Foundation under Grant Nos. NSF-CMMI-2009684 and NSF-DMR-18-09696.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.