1 Introduction

The phase field model was initially used to solve the interfacial problem, like ferromagnetism, ferroelectrics, and solidification dynamics [1]. Moreover, the phase field model can also be applied in fracture mechanics [2,3,4,5,6]. The method has the advantage that it takes a monolithic approach to simulate crack initiation, branching, bifurcation, and unification. It also overcomes stress singularity, displacement jumps, or interface tracking during the fracture simulation. Differing from other methods, neither remeshing nor finite elements with special shape functions are needed in the phase field model; the simulation is performed on a fixed mesh. The core idea of a phase field fracture model is to introduce an additional field variable to represent cracks. This scalar field variable interpolates smoothly between the values of 0 and 1, representing cracked and undamaged material, respectively. The relevant equations are derived from the total energy of the system by a variational principle (one equation models the equilibrium of the stress field, and a second models the evolution of the crack field). Consequently, contour plots of a scalar field variable used allow for the visualization of the progression of fracture and reproduce the crack situation. A phase field model has been successfully applied for quasi-static [7,8,9] and dynamic cases [6, 10,11,12,13,14]. Further recent model extensions also allow the consideration of ductile fracture [15,16,17,18], anisotropic fracture properties [19,20,21], and the evolution of fracture in various multi-physics scenarios [22,23,24,25].

In manufacturing processes like cutting, the dynamic loads typically do not cause an immediate failure of a tool; instead, tool failure can occur due to fatigue fracture development over numerous loading cycles. Thus, a phase field model which can handle the fatigue scenarios is required. In this paper the application of the model presented is focused on the fatigue failure of manufacturing tools. Since the driving mechanisms of fatigue failure significantly differ from those of classical linear elastic fracture mechanics, it was necessary to make appropriate adjustments to the evolution equation of the fracture field to model fatigue crack growth in manufacturing. Time-resolved simulations are impractical since fatigue failure only happens after a significant number of cycles; hence the evolution equation must be written in the context of cycles. The numerical implementation must be able to consolidate multiple cycles into a pseudo time to achieve the efficiency needed for the use of the model in actual production processes. In this work, we present a phase field model for cyclic fatigue. Since fatigue cracks won't appear until several loading cycles have been completed, fatigue simulations generally consume high computing time. We introduce an adaptive cycle increment algorithm, which provides a moderate computing time without losing accuracy compared to the classical computing strategies.

This paper proceeds as follows: in Sect. 2, a phase field model for cyclic fatigue is presented. In addition, a “cycle”- “time” transfer is proposed to bundle several cycles to a pseudo time domain for efficient computing. An adaptive cycle increment algorithm is then developed to reduce the computational cost without losing accuracy. In Sect. 3, an example of a manufacturing problem is modeled by the phase field fatigue model. In Sect. 4, the conclusions are stated.

2 A Phase Field Model for Cyclic Fatigue

The phase field fracture model introduces an additional field variable to represent cracks [7, 26]. The crack field \(s\) is \(1\) if the material is undamaged and if it is \(0\) where cracks occur. Furthermore, it is postulated that the displacement field \(\mathbf{u}\) and crack field \(s\) locally minimize the total energy of a loaded body \(\Omega \). This yields the equilibrium of the stress field and the evolution of the crack field for fatigue fracturing. The extended total energy \(\mathcal{E}\) with \({\varvec{t}}\) as the external traction and \({\varvec{f}}\) as the volume forces on the body is given by

$$\mathcal{E}={\int }_{\Omega }\psi {\text{d}}V-{\int }_{\partial\Omega }{\varvec{t}}{\text{d}}A-{\int }_{\Omega }{\varvec{f}}{\text{d}}V$$
(1)

where \(\psi \) denotes the total energy density of the body

$$\psi =\left(g\left(s\right)+\eta \right){\psi }^{e}\left({\varvec{\varepsilon}}\right)+{\psi }^{s}\left(s, \nabla s\right)+h\left(s\right){\psi }^{\text{ad}}\left(D\right),$$
(2)

which consists of three parts: elastic part, fracture surface part and additional fatigue part.

The strain energy density

$${\psi }^{e}\left({\varvec{\varepsilon}}\right)=\frac{1}{2}{\varvec{\varepsilon}}:{\mathbb{C}}\left({\varvec{\varepsilon}}\right)$$
(3)

is the elastic energy stored inside of a body with \(g\left(s\right)\) as a degradation function, which models the loss of stiffness of the broken material. The tensor \({\varvec{\varepsilon}}\) is the infinitesimal strain, defined by

$${\varvec{\varepsilon}}=\left[\begin{array}{ccc}{\varepsilon }_{xx}& {\varepsilon }_{xy}& {\varepsilon }_{xz}\\ {\varepsilon }_{yx}& {\varepsilon }_{yy}& {\varepsilon }_{yz}\\ {\varepsilon }_{zx}& {\varepsilon }_{zy}& {\varepsilon }_{zz}\end{array}\right]=\left[\begin{array}{ccc}\frac{\partial {u}_{x}}{\partial x}& \frac{1}{2}\left(\frac{\partial {u}_{y}}{\partial x}+\frac{\partial {u}_{x}}{\partial y}\right)& \frac{1}{2}\left(\frac{\partial {u}_{x}}{\partial x}+\frac{\partial {u}_{z}}{\partial z}\right)\\ \frac{1}{2}\left(\frac{\partial {u}_{y}}{\partial x}+\frac{\partial {u}_{x}}{\partial y}\right)& \frac{\partial {u}_{y}}{\partial y}& \frac{1}{2}\left(\frac{\partial {u}_{y}}{\partial z}+\frac{\partial {u}_{z}}{\partial y}\right)\\ \frac{1}{2}\left(\frac{\partial {u}_{z}}{\partial x}+\frac{\partial {u}_{x}}{\partial z}\right)& \frac{1}{2}\left(\frac{\partial {u}_{z}}{\partial y}+\frac{\partial {u}_{y}}{\partial z}\right)& \frac{\partial {u}_{z}}{\partial z}\end{array}\right]$$
(4)

The crack surface density

$${\psi }^{s}\left(s,\nabla s\right)={\mathcal{G}}_{c}\left(\frac{{\left(1-s\right)}^{2}}{4\upepsilon }+\upepsilon {\left|\nabla s\right|}^{2}\right)$$
(5)

is the energy required to separate the material to generate a crack, which is assumed to be proportional to the crack surface. The parameter \({\mathcal{G}}_{c}\) denotes fracture resistance and can be related to fracture toughness. The numerical parameter \(\upepsilon \) – not to be confused with strain tensor - models the width of the smooth transition zone between the broken and unbroken material.

The fatigue energy density

$${\psi }^{\text{ad}}\left(D\right)=q<D-{D}_{c}{>}^{b}\hspace{1em}{\text{with}}\hspace{1em}D={D}_{0}+{\text{d}}D$$
(6)

is introduced to account for the accumulated fatigue driving forces, which is associated with a fatigue damage parameter \(D\). This parameter \(D\) models the damage related to fatigue, inspired by Miner rule [28], which is accumulated during the simulation. The parameter \({D}_{0}\) is the previous damage and

$${\text{d}}D={\frac{\mathrm{d}N}{{n}_{D}}\left(\frac{\widehat{\sigma }}{{A}_{D}}\right)}^{k}$$
(7)

is the damage increment, which is associated with the cycle increment \({\text{d}}N\), where the parameters \({n}_{D}\), \({A}_{D}\) and \(k\) are extracted from the Wöhler curve of experiments [29]. This formulation allows the phase field fatigue model to incorporate all the influences from the environment into the fatigue propagation behavior [30]. In the phase field model, the first principal stress \({\widehat{\sigma }}_{1}\) from the undegraded stress field

$${\widehat{\sigma }}_{1}={\left[{\mathbb{C}}{\varvec{\varepsilon}}\right]}_{1}$$
(8)

is used as the fatigue driving force for high cycle fatigue. It is noted that it is not claimed that this choice of the driving force is suitable for all materials. Other effective stress quantities, e.g. the von-Mises stress, might be more suitable for ductile material and low cycle fatigue [31, 32]. Moreover, a mean stress corrector can be applied to include the mean stress effect on the fatigue crack propagation [27, 33]. The parameter \({D}_{c}\) is a damage threshold, which models the crack nucleation process. With the Macauley brackets (\(<\bullet >\)), the additional fatigue energy \({\psi }^{\text{ad}}\) will not contribute when the damage \(D\) is below this threshold. After the crack nucleation stage, the parameters \(q\) and \(b\) are parameters controlling how intense the additional fatigue energy drives the crack. A discussion of different choices of the parameters \(q\) and \(b\) can be found in [34]. The degradation function \(h\left(s\right)\) - similarly as \(g\left(s\right)\) - models the loss of the stiffness of broken material due to cyclic fatigue. A discussion of different choices of the degradation functions can be found in [34, 36].

With the variational principle of Eq. (1), four coupled equations are derived

$${\text{div}}\frac{\partial \psi }{\partial \nabla \mathbf{u}}+{\varvec{f}}=0$$
(9)
$$\frac{\partial \psi }{\partial s}-{\text{div}}\frac{\partial \psi }{\partial \nabla s}=0$$
(10)
$$\frac{\partial \psi }{\partial \nabla s}\cdot \mathbf{n}=0\hspace{1em}{\text{on}}\,\,\, \partial {\Omega }_{\nabla s}$$
(11)
$$\left(\frac{\partial \psi }{\partial \nabla {\varvec{u}}}\right)\mathbf{n}={\varvec{t}}\hspace{1em}{\text{on}}\,\,\, \partial {\Omega }_{t}$$
(12)

Equation (9) describes the equilibrium condition of the stress field; Eq. (10) described the evolution behavior of the crack field; Eq. (11) and Eq. (12) are the Neumann boundary conditions for the crack field and displacement field. Those equations define the fatigue fracture problem.

The phase field fatigue model can reproduce the most important fatigue properties. In the following evaluation, the material parameters are taken from [26] with a CT specimen [37] as a numerical example. The crack growth rate is depicted in Fig. 1 for various maximum stress amplitude values. It is to observe that even though different stress amplitudes for the simulation are applied, the rate of crack growth can be described with the same Paris’ law. The result matches Paris’ law with \(m=5.54\) very well. Radhakrishnan [38] shows that in some materials the constant \(C\) and the slope \(m\) depend on the stress ratio \(R\). The stress ratio \(R\) is defined as the ratio between the minimum stress and the maximum stress. At high positive mean stress, a decrease in fatigue life is associated with multiple crack initiation sites at the specimen surface. Fatigue limit is highly affected by the tensile mean stress and stress ratio since the maximum stress approaches near yield stress and it causes cyclic ratcheting [39]. Figure 2 displays the effect of mean stress on the crack growth rate, which reflects the fact that higher mean stress increases the rate of crack growth [40]. Figure 3 reports the effect of the loading sequence on the crack growth rate. Results show that a high-low loading sequence results in short fatigue life. This phenomenon is called the loading sequence effect [41, 42]. It has been shown that the material with a low-high load sequxsence results in a longer fatigue life because the low load level is mostly involved in the crack nucleation and the high load level is contributed to the crack propagation [43]. This effect can be explained by the residual stresses and crack closure near the crack tip [44]. Although Miner's rule does not include the loading sequence effect, the damage quantity \(D\) with a low load level increases slowly, such that it reaches the critical damage state \({D}_{c}\) later than a high load level.

Fig. 1.
figure 1

Different maximum load amplitude [35].

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Fig. 2.
figure 2

Different mean stress [35].

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Fig. 3.
figure 3

Different loading sequences.

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2.1 A Time-Cycle Transformation in the Phase Field Fatigue Model

As the discussion in the previous section, the phase field fatigue model is more suitable for high cycle fatigue. Speaking of high cycle fatigue, the number of cycles to failure is usually around tens of millions or even more. Thus, it is not feasible to simulate the accumulated cycles one after another.

The first step of an efficient integration concept is proposed by Chaboche [45] with a non-linear cumulative damage model, where cycles with similar loading are bundled into blocks. The “time”- “cycle” transfer of the phase field model is similar to this idea. It is to assume a constant block size of cycle number per time \(\frac{\mathrm{d}N}{\mathrm{d}t}\) representing a certain evolution of fatigue damage [26]. Thus, the individual single loading cycle is not used in the proposed phase field fatigue model; rather, the cycle is converted into continuous pseudo “time” as illustrated in Fig. 4. The red line in Fig. 4 represents the envelope loading, which approximates the actual discrete cyclic loading. In addition, several load cycles are combined into one block in order to reduce the overall number of load cycles: in one simulation step, the incremental change in pseudo “time” is connected to a specific number of load cycles.

Fig. 4.
figure 4

A “cycle” - “time” transformation

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In addition, for irregular loading sequences, the rain flow algorithm is used to convert a loading sequence of varying stress into an equivalent set of constant amplitude stress [19, 46].

However, the cycle number increment is usually determined by a trade-off between the computing time of simulation and the accuracy of the result. The choice of the number of the cycle increment is critical in the phase field fatigue model, not only because it determines the simulation time, but also because it has a strong influence on the crack topology [34].

The damage parameter D is introduced in the phase field model to model material damage caused by fatigue. Additionally, the “cycle”- “time” transform captures the loading with similar fatigue damage influence together. To reduce the computational effect, the adaptive cycle number adjustment algorithm (ACNAA) works by associating the cycle number increment with the damage increment. The simulation of fatigue fracture is divided into three stages based on the damage state (see Fig. 5):

  1. 1.

    \(D<{D}_{c}\): The fatigue energy term disappears at this point, so it can be viewed as a pure static mechanical state. The cycle increment should be as large as possible in order to reach the critical fatigue state as quickly as possible.

  2. 2.

    \(D\approx {D}_{c}\): The material is about to break at this point, and the cycle number increment \(\mathrm{d}N\) should be chosen so that the damage increment \(\mathrm{d}D\) is small enough to simulate the transient process.

  3. 3.

    \(D>{D}_{c}\): The fatigue crack begins to propagate. The damage increment \(\mathrm{d}D\) is regulated at this stage to achieve a moderate growth rate of the fatigue energy.

Our method has been shown to reduce computing time to nearly 3% when compared to constant cycle number increments with \(\mathrm{d}N=5\) [34]. The reason is that the huge computing time involved in the crack nucleation is dramatically reduced. Additionally, the adaptive cycle number adjustment method is also suitable for parallel computing [35]. With parallel computing (e.g. MPI), an additional significant decrease in computing time can be obtained, which keeps a 3D simulation within a reasonable time limit [35].

Fig. 5.
figure 5

A flowchart illustrates the idea of ACNAA [35]. \({\mathrm{d}D}_{\alpha }, {\mathrm{d}D}_{\beta }, {\mathrm{d}D}_{\gamma }\) are suitable numerical parameters.

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3 Phase Field Model in the Context of Manufacturing Process

3.1 Application in the Cold Forging Process

In the past decades, cold forging has gained a lot of attention and has become a economic production method for complex geometries with net-shaped or near-net-shaped surfaces. The cold forge is characterized by the circumstance that the forming of the workpiece begins at room temperature and without external heating. The major advantages of cold forging are close dimensional tolerances, good surface finish quality, and interchangeability as well as reproducibility due to its simple process [47, 48]. During the cold forging process, the material of a metal billet is put into a container (called a die). The material, compressed by a ram, flows through the container and is formed into the desired shape. In general, the cold forging process involves 5 steps (see Fig. 6):

  1. a.

    lubrication: the workpiece is lubricated to avoid sticking to the die and to maintain a low temperature.

  2. b.

    insertion: the workpiece is inserted onto a die with the shape of the final part.

  3. c.

    stroke: a great force is stroked onto the workpiece to create the desired form.

  4. d.

    flash: the excess metal around the dice is trimmed.

  5. e.

    removing: the workpiece is removed from the die.

Fig. 6.
figure 6

Cold forging process: (a: lubrication; b: insertion; c: stroke; d: flash; e: removing).

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3.2 Modeling Cold Forging Process Using Phase Field Method

In this paper, the cold forging process is modeled by the phase field method, and the fatigue life, where the crack propagation behavior are the main focus. The die geometry is adopted from Lang et al. [49] shown in Fig. 7. To reduce the computational cost, a 2D slice from the die cross-section is extracted for the finite element simulation. The opening angle \(\mathrm{\alpha }\) and the die length \(L\) can be seen as design parameters of the die. In this paper, we evaluate two different die geometries, which are listed in Table 1.

Table 1. Die geometry
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This design of the die enables high stresses at the fillet radius to generate the fatigue crack initiation and crack growth after a short number of production cycles [49]. The material of the die is AISI 2D [50, 51]. The simulation loading settings are motivated by the experiments of Dalbosco et al. [52]. One contribution of his work for this application is the different assumptions regarding the interference between the workpiece and the die.

Fig. 7.
figure 7

The cold forging tool geometry presented in [49] and a 2D slice for finite element simulation.

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In our first simulation setting (see Fig. 8a) the entire inner face of the die is assumed compressed with a constant distributed load, and the bottom of the cold forging tool is fixed by Dirichlet boundary conditions. In a different design of the die (see Fig. 8b), there is no interference from the point of transition radius until the bottom of the die in the second example. This is caused by a lack of material apposition, resulting in stress vacancy along this area of the die. For the sake of simplicity, we assume a constant load only applying it to the inner face of the die. Moreover, as an alternative design, it is also considered that the inner wall can shrunk less due to the lower shrink-fit of the die material on this part as shown in Fig. 8c. As shown in this last Figure, only the fillet of the die is under the tension loading.

Fig. 8.
figure 8

a: both fillet and inner wall are loaded; b: only inner wall is loaded; c: only fillet is loaded.

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3.3 Phase Field Fatigue Model in Cylindrical Coordinate System

In the cartesian coordinate system, the positions of points are determined with respect to three mutually perpendicular planes, giving the length-, width- and height coordinates. For a suitable computational cost, a 2D slice from the cross-section of the die is chosen for the finite element calculation Fig. 9a. This simplification in a sense of a cartesian coordinate system is to assume that the width of the body is infinite and all the derivatives regarding \(z\)-direction are zero. However, the cold forging die does not have an endless width, rather say, it is symmetric around its axis. Thus, a proper way to simulate the cold forging process with less computational resources is to bring this 2D slice cross-section of the die into a cylindrical coordinate system to exploit its rotational symmetry. A cylindrical coordinate system is specified by a radial position, an angular position, and a height position as shown in Fig. 9b.

Fig. 9.
figure 9

a: cartesian coordinate system; b: cylindrical coordinate system.

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The total energy of the body reads

$$\mathcal{E}=\int \left[\left(g\left(s\right)+\eta \right){\psi }^{e}\left({{\varvec{\varepsilon}}}^{\text{cyl}}\right)+{\psi }^{s}\left(s,{\nabla }^{\text{cyl}}s\right)+h\left(s, {\nabla }^{\text{cyl}}s\right){\psi }^{\text{ad}}\text{(}{\text{D}}\text{)]d}\right.{V}^{\text{cyl}},$$
(13)

where \({{\varvec{\varepsilon}}}^{\mathrm{cyl}}\) is the strain tensor in the cylindrical coordinates and \({\mathrm{d}V}^{\text{cyl}}\) is the infinite cylinder volume element.

Let \(r\) be the radius, \(\theta \) be the circumferential angle and \(z\) be the height, the transformation between the cartesian coordinates \((x,y,z)\) and cylindrical coordinates \(\left(r,\theta ,z\right)\) can be given as

$$x=r\, \mathrm{cos}\, \theta \quad y=r\, \mathrm{sin}\, \theta \quad z=z, $$
(14)

and the Jacobian matrix transforming the infinitesimal vectors from cartesian coordinates to cylindrical coordinates is given as

$$J=\left[\begin{array}{ccc}\frac{\partial x}{\partial r}& \frac{\partial y}{\partial r}& \frac{\partial z}{\partial r}\\ \frac{\partial x}{\partial \theta }& \frac{\partial y}{\partial \theta }& \frac{\partial z}{\partial \theta }\\ \frac{\partial x}{\partial z}& \frac{\partial y}{\partial z}& \frac{\partial z}{\partial z}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}\theta & \mathrm{sin}\theta & 0\\ -\mathrm{rsin}\theta & r\mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right].$$
(15)

The displacement vector in the cylindrical coordinate system with rotational symmetry properties is given as

$${\mathbf{u}}^{\text{cyl}}={\left[{u}_{r},{ u}_{\theta },{ u}_{z}\right]}^{T}\underset{rot.\,\, sym.}{\Longrightarrow } {[{u}_{r},0,{ u}_{z}]}^{T},$$
(16)

where \({u}_{r}\) and \({u}_{z}\) are the width and height components of the displacement vector.

For rotational sysmmetry, the derivative in angular direction vanishes, thus, the strain tensor is given by.

$${{\varvec{\varepsilon}}}^{\text{cyl}}=\left[\begin{array}{ccc}\frac{\partial {u}_{r}}{\partial r}& 0& \frac{1}{2}\left(\frac{\partial {u}_{r}}{\partial z}+\frac{\partial {u}_{z}}{\partial r}\right)\\ 0& \frac{{u}_{r}}{r}& 0\\ \frac{1}{2}\left(\frac{\partial {u}_{z}}{\partial r}+\frac{\partial {u}_{r}}{\partial z}\right)& 0& \frac{\partial {u}_{z}}{\partial z}\end{array}\right].$$
(17)

It is noted that the entry in the middle \(\frac{{u}_{r}}{r}\) provides an additional contribution into the energy density, which is omitted in the cartesian coordinates system for 2D. The fatigue driving force \({\widehat{\sigma }}^{\mathrm{cyl}}\) can be given with the constitutive law and taking as the first principal stress

$${\widehat{\sigma }}^{\mathrm{cyl}}={\left[{\mathbb{C}}{{\varvec{\varepsilon}}}^{\mathrm{cyl}}\right]}_{1}$$
(18)

where the stiffness tensor \({\mathbb{C}}\) remains the same as it is in the cartesian coordinate system because of its isotropic character. The crack field \(s\) itself does not need to be modified into a cylindrical coordinate system since it is a scalar variable to indicate the broken state of the material. The gradient of the crack field \({\nabla }^{\text{cyl}}s\) in the cylindrical coordinate system is given as

$${\nabla }^{\text{cyl}}s={[\frac{\partial s}{\partial r} 0 \frac{\partial s}{\partial z}]}^{T}.$$

3.4 Phase Field Simulation of Cold Forging Process

In the our first analysis, it is assumed that the fillet and the inner wall of the die are completely loaded, two different geometries of the die are investigated. The angle of crack propagation is nearly \(30^\circ \) in Fig. 10a and nearly \(40^\circ \) in Fig. 10b. Those angles of the crack propagation directions can be explained by the mixed energy fracture criterion [53], since the tools are under a mixed mode I/II load situation. The bigger angle of fracture initiation in Fig. 10b can be explained by the dominant influence of shear stress from mode II in comparison to the tension stress from mode I. Furthermore, the first initialized crack can be found after around 3,000 production cycles at the forging tool with an opening angle of \(45^\circ \) and the fatigue life of the second tool \(\left(\mathrm{\alpha }={60}^{\circ }\right)\) is only around 500 cycles of production. This analysis reveals the fact that the dominated shear stress on the inner wall of the die shortens the fatigue life of the tool.

Fig. 10.
figure 10

The simulation of the cold forging process at first crack \({N}_{1}\) and final stage \({N}_{2}\) (a: opening angle \(\mathrm{\alpha }=45^\circ \); b: opening angle \(\mathrm{\alpha }=60^\circ \)).

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For further investigations, the cold forging tools are simulated with different loading assumptions (Fig. 8) as shown in Fig. 11. Results show that loading acts merely on the inner wall of the die and can dramatically increase the fatigue life of the die. In the analysis that was performed itt yielded the highest fatigue life at around 55,000 production cycles for the opening angle of \(45^\circ \) (Fig. 11b). Different loading assumptions lead to different patterns of crack propagation. In Fig. 11a and Fig. 11b, the crack propagates first sloping downward, which is influenced by a mixed mode loading situation. After these stages, the crack curves moves in a nearly horizontal direction because of the mainly vertical tensile stress. In contrast, loads acting only on the inner wall yield almost the same crack propagation patterns, where the angle of crack propagation is around \(70^\circ \). This can be explained by a pure shear mode II loading situation. These crack propagation behaviors from the phase field simulations have been found similarly in reported experiments [52].

Fig. 11.
figure 11

The simulation of the cold forging process by different loading conditions.

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4 Conclusion

In this paper, we presented a phase field model for cyclic fatigue, which is used to analyze manufacturing process namely the cold forging process. The phase field model introduces an additional phase field variable to model the broken material. The entire crack evolution behavior can be derived by considering the total energy of the body. The total energy consists of three parts: an elastic energy part, which represents the energy stored inside of the body; a fracture surface energy part, which represents the energy to generate cracks; and an additional fatigue energy part, which represents the additional driving forces associated with fatigue evolution. Inspired by Miner rule, a damage parameter is introduced to model the accumulative fatigue damage. The phase field fatigue model can reproduce the most important fatigue properties, e.g., the Paris’ law, the mean stress effect, and the loading sequence effect. Moreover, a “cycle”- “time” transfer is presented which would transform the cycle domain into the pseudo time domain for an efficient fatigue simulation. For irregular loading sequences, the rain flow counting algorithm is used to convert the load cycles into several blocks of regular uniform loading. The existing fatigue simulation methods usually suffers from its huge computational demand. In order to further reduce the computational time without losing accuracy, different numerical strategies are proposed. The core idea of the ACNAA is to associate the damage increment with the cycle increment. Additional computing time reduction can be obtained by applying parallel computing.

The main contribution of this work is that we apply the phase field model to the manufacturing problem to predict fatigue life and crack patterns. We used the cold forging process as the demonstrated example since it is an important manufacturing methods for producing parts with complex geometries. To exploit the rotational symmetry property of the problem, a phase field fatigue model for cylindrical coordinates is introduced. Different cold forging die geometries and load conditions in the processing are presented to analyze the fatigue life and crack patterns. Results show that the phase field model can be effectively applied to cold forging process. This enables a physics-based prediction of the lifetime of manufacturing tools and the identification of process parameters relevant to detect the onset of damage.