# A Unified Abaqus Implementation of the Phase Field Fracture Method Using Only a User Material Subroutine

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## Abstract

**:**

`AT1`,

`AT2`and phase field-cohesive zone models (

`PF-CZM`). Both staggered and monolithic solution schemes are handled. We demonstrate the potential and robustness of this new implementation by addressing several paradigmatic 2D and 3D boundary value problems. The numerical examples show how the current implementation can be used to reproduce numerical and experimental results from the literature, and efficiently capture advanced features such as complex crack trajectories, crack nucleation from arbitrary sites and contact problems. The code developed is made freely available.

## 1. Introduction

`AT2`model [24], based on the Ambrosio and Tortorelli regularization of the Mumford-Shah functional [39], (ii) the

`AT1`model [40], which includes an elastic phase in the damage response, and (iii) the phase field-cohesive zone model

`PF-CZM`[41,42], aimed at providing an explicit connection to the material strength. Our framework also includes two strain energy decompositions to prevent damage in compressive states: the spectral split [29] and the volumetric-deviatoric one [43]—both available in the context of anisotropic and hybrid formulations [44]. Moreover, the implementation can use both monolithic and staggered solution schemes, enhancing its robustness. Two example codes are provided with this work (www.empaneda.com/codes), both capable of handling 2D and 3D analyses without any modification. One is a simple 33-line code, which showcases the simplicity of this approach by adopting the most widely used constitutive choices (

`AT2`, no split). The other one is an extended version, with all the features mentioned above, aimed at providing a unified implementation for phase field fracture. To the authors’ knowledge, the present work provides the simplest Abaqus implementation of the phase field fracture method.

`AT2`,

`AT1`and

`CZ-PFM`models. Then, in Section 3, the details of the finite element implementation are presented, including the analogy with heat transfer and the particularities of the Abaqus usage. The potential of the implementation presented is showcased in Section 4, where several boundary value problems of particular interest are addressed. Specifically, (i) a three-point bending test, to compare with the results obtained with other numerical methods; (ii) a concrete single-edge notched beam, to compare with experimental data; (iii) a notched plate with a hole, to simulate complex crack paths, merging and nucleation; and (iv) a 3D gear, where cracking occurs due to contact between the teeth. Finally, concluding remarks are given in Section 5.

## 2. A Generalised Formulation for Phase Field Fracture

#### 2.1. Kinematics

#### 2.2. Principle of Virtual Work. Balance of Forces

#### 2.3. Constitutive Theory

`AT1`[40],

`AT2`[24] and

`PF-CZM`[41,42] models are then derived as special cases. The total potential energy of the solid reads,

`AT2`,

`AT1`and

`PF-CZM`models.

`AT2`and

`AT1`models were originally formulated using a quadratic degradation function:

`PF-CZM`model typically uses the following degradation function,

`AT2`model: $w\left(\varphi \right)={\varphi}^{2}$ and $c=1/2$. Since ${w}^{\prime}\left(0\right)=0$, this choice implies a vanishing threshold for damage. An initial, damage-free linear elastic branch is introduced in the

`AT1`model, with the choices $w\left(\varphi \right)=\varphi $ and $c=2/3$. Finally, in the

`PF-CZM`case we have $w\left(\varphi \right)=2\varphi -{\varphi}^{2}$ and $c=\pi /4$.

`AT2`model, is often referred to as the isotropic formulation:

`AT1`and

`AT2`models, damage under compression is prevented by decomposing the strain energy density following typically two approaches. One is the so-called volumetric-deviatoric split, proposed by Amor et al. [43], which reads

`PF-CZM`model the driving force for fracture is defined as [41]:

`AT1`and

`PF-CZM`models there is a minimum value of the fracture driving force, which we denote as ${\mathcal{H}}_{min}$. This is needed as otherwise $\varphi \le 0$, as can be observed by setting $\varphi =0$ and solving the balance Equation (17). The magnitude of ${\mathcal{H}}_{min}$ is then given by the solution of (17) for $\mathcal{H}$ under $\varphi =0$. For the

`AT1`case: ${\mathcal{H}}_{min}=3{G}_{c}/(16\ell )$; while for the

`PF-CZM`model: ${\mathcal{H}}_{min}=2{G}_{c}/(\pi a\ell )={f}_{t}^{2}/\left(2E\right)$.

## 3. Finite Element Implementation

#### 3.1. Heat Transfer Analogy

#### 3.2. Abaqus Particularities

## 4. Results

`PF-CZM`model to simulate fracture in a three-point bending experiment and compare the results with those obtained by Wells and Sluys [50] using an enriched cohesive zone model. Secondly, we model mixed-mode fracture in a concrete beam to compare the crack trajectories predicted by the

`AT2`model to those observed experimentally [51]. Thirdly, cracking in a mortar plate with an eccentric hole is simulated to benchmark our predictions with the numerical and experimental results of Ambati et al. [44]. Finally, the

`AT1`model is used in a 3D analysis of crack nucleation and growth resulting from the interaction between two gears.

#### 4.1. Three-Point Bending Test

`PF-CZM`) [41,42] using the exponential degradation function.

`PF-CZM`model the material strength is explicitly incorporated into the constitutive response and, as a consequence, results become largely insensitive to the choice of phase field length scale, which is here assumed to be $\ell =0.1$ mm. The model is discretised using 4-node coupled temperature-displacement plane strain elements (CPE4T in Abaqus notation). As shown in Figure 1b, the mesh is refined in the center of the beam, where the crack is expected to nucleate and grow. The characteristic element is at least five times smaller than the phase field length scale and the total number of elements equals 5820. Results are computed using the monolithic scheme.

#### 4.2. Mixed-Mode Fracture of a Single-Edge Notched Concrete Beam

`AT2`model with the experimental observations by Schalangen [51]. Schalangen subjected a concrete beam to the loading configuration shown in Figure 3. The beam is supported at four locations, and each support is connected to a girder beam through a rod. The cross-sections of the outer rods are smaller than those of the inner rods, to ensure an equal elongation. The load is applied to the center of the girder beams and then transferred through the rods to the concrete beam. The resulting fracture is stable and mixed-mode.

`AT2`model. To prevent failure of elements under compression, the strain energy density is divided into tensile and compressive parts employing the strain spectral decomposition proposed by Miehe et al. [29], using the anisotropic formulation (24). The material properties of the concrete beam are taken to be: Young’s modulus $E=35$ GPa, Poisson’s ratio $\nu =0.2$, and toughness ${G}_{c}=0.1$ N/mm. The phase field length scale is assumed to be equal to $\ell =2$ mm and, consequently, the characteristic size of the elements along the potential crack propagation region equals 0.5 mm (see Figure 4b). The rods are modelled using truss elements, while the concrete beam is discretised with a total of 28,265 linear quadrilateral coupled temperature-displacement plane strain elements. The results obtained are presented in Figure 5. Both experimental (Figure 5a) and numerical (Figure 5b) results are shown. A very good agreement can be observed, with the crack initiating in both cases at the right corner of the notch and deflecting, following a very similar trajectory, towards the right side of the bottom support.

#### 4.3. Notched Plate with an Eccentric Hole

`AT2`phase field model is considered, with no split applied to the strain energy density. We discretise the plate with 56,252 linear plane stress coupled displacement-thermal elements (CPS4T, in Abaqus notation). The characteristic element length in the regions surrounding the notch and the hole is five times smaller than the phase field length scale.

#### 4.4. 3D Analysis of Cracking Due to the Contact Interaction between Two Gears

`AT1`model and no split is used for the strain energy density. The model is discretised with more than 120,000 three-dimensional coupled temperature-displacement brick elements. The results obtained are shown in Figure 9, in terms of phase field $\varphi $ contours. Cracking initiates from the root of one of the teeth from the smaller gear and propagates towards the opposite root until the rupture of the gear teeth.

## 5. Conclusions

`AT1`,

`AT2`and

`PF-CZM`. In addition, several strain energy splits are considered, in the framework of both hybrid and anisotropic formulations.

`PF-CZM`version leads to an excellent agreement with the enriched cohesive zone model analysis by Wells and Sluys [50] of crack nucleation and growth in a beam subjected to three-point bending. Secondly, we validated the crack trajectories predicted by the

`AT2`model with the experimental observations by Schalangen [51] on a concrete beam exhibiting mixed-mode fracture. Thirdly, we simulated the failure of a mortar plate with an eccentric hole to showcase the capabilities of the framework in capturing the interaction between cracks and other defects, as well as the nucleation of secondary cracks. The simulations agree qualitatively and quantitatively with the results obtained by Ambati et al. [44]. Finally, we used the

`AT1`version to model cracking due to the interaction between gears to showcase the capabilities of the model in dealing with 3D problems incorporating complex computational features, such as contact and geometric non-linearity. The codes developed have been made freely available, with examples and documentation at www.empaneda.com/codes.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Weak Formulation and Finite Element Implementation

**B**-matrices, containing the derivative of the shape functions, such that:

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**Figure 1.**Three-point bending test: (

**a**) geometry, dimensions and boundary conditions, (

**b**) finite element mesh, and (

**c**) phase field contour at the end of the analysis.

**Figure 2.**Three-point bending test: force versus displacement response. The results obtained with the present phase field fracture framework are compared with the results computed by Wells and Sluys [50] using an enriched cohesive zone model.

**Figure 3.**Mixed-mode fracture of a concrete beam: experimental testing configuration, following Ref. [51].

**Figure 4.**Mixed-mode fracture of a concrete beam: (

**a**) geometry, dimensions (in mm) and boundary conditions, and (

**b**) finite element mesh.

**Figure 5.**Mixed-mode fracture of a concrete beam: (

**a**) Experimental crack patterns [51], and (

**b**) predicted crack trajectory, as given by the phase field contour.

**Figure 6.**Notched plate with an eccentric hole: (

**a**) geometry, dimensions (in mm) and boundary conditions, (

**b**) experimental observation [44], and predicted phase field ϕ contours at (

**c**) u = 0.4 mm and (

**d**) u = 2 mm.

**Figure 7.**Notched plate with an eccentric hole: force versus displacement curve, with several snapshots of several cracking events superimposed.

**Figure 9.**Cracking in interacting gears: phase field contours, (

**a**) overall view at an advanced stage of cracking, and detail at (

**b**) 0.028 + 2 × 10

^{−7}rad, (

**c**) 0.028 + 5 × 10

^{−7}rad and (

**d**) 0.028 + 9 × 10

^{−7}rad.

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**MDPI and ACS Style**

Navidtehrani, Y.; Betegón, C.; Martínez-Pañeda, E. A Unified Abaqus Implementation of the Phase Field Fracture Method Using Only a User Material Subroutine. *Materials* **2021**, *14*, 1913.
https://doi.org/10.3390/ma14081913

**AMA Style**

Navidtehrani Y, Betegón C, Martínez-Pañeda E. A Unified Abaqus Implementation of the Phase Field Fracture Method Using Only a User Material Subroutine. *Materials*. 2021; 14(8):1913.
https://doi.org/10.3390/ma14081913

**Chicago/Turabian Style**

Navidtehrani, Yousef, Covadonga Betegón, and Emilio Martínez-Pañeda. 2021. "A Unified Abaqus Implementation of the Phase Field Fracture Method Using Only a User Material Subroutine" *Materials* 14, no. 8: 1913.
https://doi.org/10.3390/ma14081913