# Fatigue crack growth in anisotropic aluminium sheets – phase-field modelling
and experimental validation

Martha Kalina  Vanessa Schöne  Boris Spak  Florian Paysan  Eric Breitbarth
Markus Kästner<EMAIL_ADDRESS>Chair of Computational and
Experimental Solid Mechanics, TU Dresden, Dresden, Germany  Institute of
Materials Research, German Aerospace Center (DLR), Köln, Germany  Dresden
Center for Computational Materials Science (DCMS), TU Dresden, Dresden,
Germany

###### Abstract

Fatigue crack growth is decisive for the design of thin-walled structures such
as fuselage shells of air planes. The cold rolling process, used to produce
the aluminium sheets this structure is made of, leads to anisotropic
mechanical properties. In this contribution, we simulate the fatigue crack
growth with a phase-field model due to its superior ability to model arbitrary
crack paths. A fatigue variable based on the Local Strain Approach describes
the progressive weakening of the crack resistance. Anisotropy regarding the
fracture toughness is included through a structural tensor in the crack
surface density. The model is parameterised for an aluminium AA2024-T351 sheet
material. Validation with a set of experiments shows that the fitted model can
reproduce key characteristics of a growing fatigue crack, including crack path
direction and growth rate, considering the rolling direction.

###### keywords:

Phase-field , Fracture , Fatigue crack growth , Aluminium sheets , Anisotropy

††journal: International Journal of Fatigue

##  1 Introduction

Understanding the fatigue crack behaviour of materials is essential for
assessing the service life and damage tolerance of thin-walled structures such
as aircraft fuselages. The so-called damage tolerance concept for ensuring
structural integrity relies on the ability of a defective structure to
withstand the expected loads until this defect can be detected and repaired by
scheduled inspection or non-safety-relevant failure [1]. To evaluate the
service life, Wöhler curves provide information on crack initiation and damage
accumulation [2]. Then, the range of stable fatigue crack growth (FCG) is
determined by crack propagation tests under consideration of small-scale
yielding and relates the crack tip stress to the crack growth rate [3]. These
experimentally determined crack propagation curves serve as basis for fitting
FCG describing models, for example FASTRAN [4] or NASGRO [5], to predict the
fatigue life and fatigue crack behavior of structures under variable loads
[6]. The unified two-parameter driving force model for fatigue crack growth
analysis [7] establishes a correlation between the maximum stress intensity
factor, the stress intensity range and the elasto-plastic stress-strain field
of the crack tip. Therefore, the UniGrow model can also account for overload
effects or residual stresses from cyclic plasticity near the crack tip [8].
Additional modifications extend the UniGrow model to the range of short
fatigue crack growth [9].

The fatigue crack propagation behavior of ductile materials is controlled by
an interaction between damaging mechanisms in front of the crack tip and
extrinsic shielding mechanisms due to the plastic wake [10]. Damaging
mechanism include, for example, intergranular fracture or micro-void
nucleation in front of the crack tip, which are material-specific and are
influenced by the crack driving force as well as the crack tip stress field
[11]. Shielding mechanisms such as crack deflection or crack closure reduce
the crack tip loadings and thus have a retarding effect [12].

The aluminum alloy AA2024-T351 studied here is widely used for airplane
fuselages, which can be explained by its excellent damage tolerance behaviour
[13]. During the manufacturing process, the base cast material is rolled along
the longitudinal axis. Material properties show a rolling direction-dependent
anisotropy due to grain elongation and an orientation of the texture [14, 15],
as can be seen in the EBSD scan of AA2024-T351 in Fig. 1. As a consequence,
many investigators reported anisotropic strength properties [16, 17, 18].
Regarding the static crack resistance, Amstutz et al. [19] tested AA2024-T351
sheet material with a thickness of 2.3 mm in L-T (crack perpendicular to grain
elongation) and T-L (crack parallel to grain elongation) orientation under
Mode-I loading. They pointed out that the resulting fracture surface depends
on the orientation. Specimen in L-T orientation showed a higher crack
resistance compared to the T-L orientation. This is in agreement with tests by
Johnston and Newman [20] on 1.6 mm thin AA2024-T351 sheets.

Regarding FCG behavior, Bergner [21] performed constant-amplitude Mode-I
fatigue tests on MT160 specimen with a thickness of 1.6 mm, made from AA2024
alloys. His tests revealed a dependency of the crack propagation behavior, not
only on the temper condition, but also on the orientation, showing an
increased crack propagation rate for cracks in T-L orientation. This was
confirmed by Da Cunha and de Lima [22]: A slightly extended lifetime can be
seen for L-T oriented specimens. This also applies for crack growth behavior
under variable test amplitudes representing sequences of aircraft flight load
spectra as the test setup was given by Kocańda and Jasztaal [23]. They tested
3 mm thick Middle Tension (MT) test sheets. L-T-oriented specimens endure a
higher number of load sequences until failure.

1000 µm
---

001
---
direction
---
T = Transverse
---
S =
---
111
---
101
---
Thickness
---
L = Rolling
---
Figure 1: EBSD scan of a rolled 2 mm thick AA2024-T351 sheet. Grain elongation
along the rolling direction L is visible. Grain length to height ratio for
LT/TS/LS: 2.1/2/3.3.

Traditional methods to predict and simulate FCG such as the NASGRO equations,
cohesive zone models [24] and XFEM [25] involve certain drawbacks. Usually,
the crack path has to be known a priori or is intricately described with
additional elements or enriched ansatz functions. The phase-field method,
though, is a powerful framework to model crack initiation, propagation and
arbitrary crack paths in a straight-forward way. This is particularly
important for anisotropic materials under realistic loads where the crack path
cannot be previously known or predicted easily. The crack geometry is
described with an additional field variable – the phase-field variable – in a
regularised manner, forming a coupled problem. See [26] for an overview of the
original phase-field models for brittle static fracture. Later, the method was
extended to ductile [27] and rate-dependent [28] material behaviour. A
comparison of different approaches to model anisotropic static fracture is
given in [29].

More recently, the method has also been applied to fatigue fracture, see [30]
for a comparison of various phase-field fatigue models. They include brittle
and ductile material behaviour, high and low cycle fatigue as well as various
material classes such as metals, elastomers and concrete. However, to the best
of the authors knowledge, no phase-field fatigue model for anisotropic
material behaviour has been published yet. In order to model the direction-
dependent fracture behaviour of rolled aluminium sheet material, we introduce
an extension to the phase-field fatigue model by Seiler et al. [31] to
anisotropic fracture toughness, based on the anisotropic static phase-field
formulation by Teichtmeister et al. [32]. Thereby, we establish a versatile
framework allowing to simulate fatigue crack initiation and growth independent
of the structure’s geometry under arbitrary loading conditions, including
crack deflection and change in the crack growth rate due to direction-
dependent material qualities. The model is here parameterised for an aluminium
material, yet it can be applied to any sheet metal material.

This paper is structured as follows: Chapter 2 presents the FCG experiments on
MT specimen which show a dependency on the rolling direction. Chapter 3
introduces the phase-field model for anisotropic fatigue fracture to simulate
these phenomena. The experimental determination of model parameters –
including elastic, cyclic and fracture parameters – is presented in Chapter 4.
Finally, the phase-field model is parameterised in Chapter 5 and subsequently
used for numerical studies of the fracture phenomena previously observed in
the MT experiments. The paper ends with a conclusion.

##  2 Fatigue fracture in aluminium sheets

###  2.1 Experimental setup

An aluminum alloy A2024-T351 (AlCu4Mg1) was used. Rolled sheets with a
thickness of 2 mm were procured by a commercial supplier and test specimens
were machined. The fatigue crack behavior was determined using MT160
specimens, see Fig. 2a, in three orientations:

  * 1.

$\theta=0$°: T-L orientation, crack growth parallel to grain elongation (Load
in T-, FCG in L-direction)

  * 2.

$\theta=90$°: L-T orientation, crack growth perpendicular to grain elongation
(Load in L-, FCG in T-direction)

  * 3.

$\theta=45$°: crack growth 45° to grain elongation

The crack propagation tests were carried out according to ASTM E674-15 [3].
The specimen is loaded at constant force amplitude with $F_{\max}=15$ kN and
load ratios $R=$ $F_{\min}/F_{\max}=\\{0.1,0.3,0.5\\}$ on a uniaxial servo-
hydraulic testing machine. Using a sinusoidal load with constant amplitude and
a test frequency of 20 Hz, fatigue crack propagation was initiated from a
centric initial notch with a length of $2a=16$ mm, sawed symmetrically into
the specimen. The potential difference was measured between two pins located
10 mm below and above the fatigue crack ($I_{\mathrm{DC-POT}}$ = 100 A =
const., $U_{\mathrm{DC-POT}}$ = 60 mV) to calculate the crack length by means
of the Johnson equation [33]. To determine the crack growth rate, the secant
method was used. The range of stress intensity factors $\Delta K$ were
calculated according to ASTM E647-15 and the formula provided by Feddersen
[34], leading to fatigue crack propagation curves
($\mathrm{d}a/\mathrm{d}N-\Delta K$) shown in Fig. 3.

Additionally, full-field displacement and strain information at the surface
were captured using a GOM Aramis 12M 3D digital image correlation (DIC)
system. Using a black and white stochastic pattern and a facet size of
25$\times$25 pixel with a facet distance of 16 pixels (1 pixel = 0.045 mm), it
is possible to achieve a spatial resolution of 0.72 mm. Images were taken each
$\Delta a=0.5$ mm crack propagation for maximum, mean and minimum load of the
cycle. For calculating the displacements, we use the GOM Aramis Professional
V2020 software. Based on this displacement field, a trained artificial neural
network is used to determine the crack tip position [35, 36] and the crack
path.

$x_{1}$
---
$x_{2}$
---

direction
---

$\theta$
---

Rolling
---

$a$
---
$F$
---
$F$
---

$400$ mm
---

a)
---
b)
---
$F$
---
$F$
---
$t=2$ mm
---
$t=25$ mm
---
$160$ mm
---
$W=50$ mm
---

$0.6\,W$
---

$\theta$
---
$x_{1}$
---
$x_{2}$
---
$a$
---
Figure 2: (a) MT specimen. (b) CT specimen.

###  2.2 Fatigue crack propagation data

The $\mathrm{d}a/\mathrm{d}N-\Delta K$ FCG curves for $R=0.1$ are shown in
Fig. 3a. The highest crack propagation rate can be found for $\theta=0$° (T-L)
which corresponds to crack growth parallel to grain elongation. The crack
propagation rates for $\theta=45$° and $\theta=90$° (L-T) are almost similar.
When comparing the resulting crack path for the right side of the cracked MT
specimen, it becomes apparent that the resulting path depends on the
orientation, though. $\theta=0$° shows a predominant straight path with
growing crack length, whereas the crack path of $\theta=90$° is characterized
by a zig-zag path $\pm 1$ mm around the initial crack path. The path of
$\theta=45$° reveals the highest crack path slope where it is clearly visible
that the crack path is strongly influenced by the rolling direction.

FCG rates are lower in L-T direction since there are more grain boundaries to
cross perpendicular to the elongated grains, blocking dislocations [37].
Instead, cracks often tend to follow grain boundaries since these are weaker
areas due to the precipitation of the supersaturated mixed crystal and the
collision of several crystals, leading to crystallographic defects.
Furthermore, the rolling process changes the texture, making the crystallites
align in rolling direction. The oriented slip systems favour cracks parallel
to the rolling direction [14, 15].

Regarding the Paris curves for different R-ratios $R=\\{0.1,0.3,0.5\\}$ and
$\theta=0$° in Fig. 3b, the crack propagation rates are also very similar when
being plotted over the range of the stress intensity factor $\Delta K$, with
$\theta=0$° showing the highest crack propagation rate.

In the following, these results are to be reproduced with phase-field
simulations.

$10^{1}$
---
$\Delta K$/MPa$\sqrt{\mathrm{m}}$
---
$10^{-6}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
$10^{0}$
---

$\mathrm{d}a/\mathrm{d}N$/mm
---

20
---
40
---
60
---
$a/$mm
---
-8
---
-6
---
-4
---
-2
---
0
---
2
---
4
---
6
---
8
---

$a_{y}/$mm
---

$\theta=0^{\circ}$
---
$\theta=45^{\circ}$
---
$\theta=90^{\circ}$
---
a)
---
b)
---
$a$
---
$a_{y}$
---
$\theta$
---
$F$
---
Rolling
---
direction
---
AA2024-T351
---
$R=0.1$
---
$\theta=0^{\circ}$
---
$\theta=45^{\circ}$
---
$\theta=90^{\circ}$
---
$10^{1}$
---
$\Delta K$/MPa$\sqrt{\mathrm{m}}$
---
$10^{-6}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
$10^{0}$
---

$\mathrm{d}a/\mathrm{d}N$/mm
---

$R=0.5$
---
$R=0.3$
---
$R=0.1$
---
$t$
---
$F$
---
$\theta=0^{\circ}$
---
AA2024-T351
---
Figure 3: Experimental results of FCG experiments with MT specimens. (a)
Variation of the angle between crack and sheet rolling direction $\theta$. The
fatigue crack propagation rate (upper) varies little. The crack path (lower)
inclines for $\theta=45^{\circ}$. (b) Variation of the load ratio $R$ between
load minimum and maximum of a load cycle for $\theta=0$°. Increased $R$-ratio,
i. e. increasing load minimum while keeping maximum load constant, reduces the
fatigue crack propagation rate. $a$ is the crack length in $x_{1}$-direction,
$a_{y}$ the crack position in $x_{2}$-direction.

##  3 Phase-field model for fatigue crack propagation in anisotropic material

The fatigue crack propagation experiments with anisotropic aluminium material
shall be simulated with a phase-field model. In order to do so, an existing
phase-field model for fatigue fracture by Seiler et al. [31] is extended to
cover anisotropic material behaviour. In the following, the original model is
recapitulated first, the extension to anisotropy is presented thereafter.

###  3.1 Isotropic material behaviour

The underlying isotropic phase-field fatigue model [31] is formulated in a
small strain setting with linear elastic material behaviour. The considered
domain in $\mathcal{B}\in\mathbb{R}^{n}$ with a boundary $\partial\mathcal{B}$
and material points defined by their location $\boldsymbol{x}$ at time $t$.
The strain $\boldsymbol{\varepsilon}(\boldsymbol{x},t)$ is defined by
$\boldsymbol{\varepsilon}=1/2\,(\nabla\boldsymbol{u}+\nabla\boldsymbol{u}^{\top})$
with displacement $\boldsymbol{u}(\boldsymbol{x},t)$. Fractured material can
be described in a regularised manner with the phase-field variable
$d(\boldsymbol{x},t)$, thereby $d=0$ indicates intact material and $d=1$ fully
broken material. The regularisation width is controlled by the length scale
parameter $\ell$, chosen as a compromise between a narrow crack profile
approximating the ideal sharp crack ($\ell\rightarrow 0$) and a preferably
coarse mesh. The cyclic loading and damage history of the material point is
described by the accumulating fatigue variable $D\in[0,1]$.

#### Energy functional

The total energy density

$W(\boldsymbol{\varepsilon},d,\nabla
d;D)=g(d)\psi^{\mathrm{e}}(\boldsymbol{\varepsilon})+\alpha(D)\,G_{\mathrm{c}}\gamma_{\mathrm{iso}}(d,\nabla
d)$ (1)

consists of an elastic part with the free energy density

$\psi^{\mathrm{e}}=\frac{1}{2}\lambda\,\mathrm{tr}\,^{2}(\boldsymbol{\varepsilon})+\mu\,\mathrm{tr}\,\left(\boldsymbol{\varepsilon}\right)$
(2)

degraded by the degradation function $g(d)=(1-d)^{2}$ 111While a split of the
elastic energy density into a degraded tensile and an undegraded compressive
part can be included in a straightforward manner, it is not applied in this
case due to the numerical examples not requiring it. and the regularised
fracture surface density

$\gamma_{\mathrm{iso}}(d,\nabla d)=\frac{d^{2}}{2\ell}+\frac{\ell}{2}|\nabla
d|^{2}.$ (3)

The fracture toughness $G_{\mathrm{c}}$ is the parameter characterising the
materials resistance against cracks. The fracture energy is degraded by the
fatigue degradation function $\alpha\in[\alpha_{0},1]$ being

$\alpha(D)=(1-\alpha_{0})(1-D)^{\xi}+\alpha_{0}$ (4)

with parameters $\alpha_{0}$ and $\xi$, describing the ongoing weakening of
the material due to cyclic stressing in form of progressively reducing the
fracture toughness.

The stress is defined as
$\boldsymbol{\sigma}=g(d)\,\partial\psi^{\mathrm{e}}(\boldsymbol{\varepsilon})/\partial\boldsymbol{\varepsilon}$.
Given the volume force $\bar{f}$ and the traction vector $\bar{\tau}$, the
total energy of the body is

$\displaystyle\Pi(\boldsymbol{\varepsilon},d,\nabla d;D)=$
$\displaystyle\mathop{\int}_{{\cal B}}W(\boldsymbol{\varepsilon},d,\nabla
d;D)\;\mathrm{d}v$ $\displaystyle-\mathop{\int}_{{\cal
B}}\bar{f}\cdot\boldsymbol{u}\;\mathrm{d}v-\mathop{\int}_{\partial{\cal
B}}\bar{\tau}\cdot\boldsymbol{u}\;\mathrm{d}a.$ (5)

#### Derivation of model equations through variational principle

The model equations of the coupled problem can be derived by the means of the
variational principle

$\\{\boldsymbol{u},d\\}=\arg\left\\{\min_{\boldsymbol{u}\in{\cal
W}_{\bar{u}}}\min_{\boldsymbol{d}\in{\cal
W}_{d}}\Pi(\boldsymbol{\varepsilon},d,\nabla d;D)\right\\}$ (6)

with conditions

$\displaystyle{\cal W}_{\bar{u}}$
$\displaystyle=\left\\{\boldsymbol{u}\in\mathbb{R}^{3}\,|\,\boldsymbol{u}=\bar{\boldsymbol{u}}\text{
on }\partial{\cal B}^{\mathrm{D}}\right\\}$ (7) $\displaystyle{\cal W}_{d}$
$\displaystyle=\\{d\in\mathbb{R}\,|\,\dot{d}\geq 0\\}$ (8)

for prescribed displacements $\bar{\boldsymbol{u}}$ on Dirichlet boundaries
$\partial\mathcal{B}$ and the irreversibility condition for the phase-field.
The variational derivative $\delta_{u}\Pi$ with respect to $\boldsymbol{u}$
eventually yields the balance of momentum

$\nabla\cdot\boldsymbol{\sigma}+\bar{\boldsymbol{f}}=\boldsymbol{0}\text{ in
}{\cal B}$ (9)

with the boundary condition
$\boldsymbol{\sigma}\cdot\boldsymbol{n}=\boldsymbol{\bar{t}}$ on
$\partial{\cal B}^{\mathrm{N}}$. Likewise, $\delta_{d}\Pi$ yields the weak
form of the phase-field problem

$\displaystyle\delta_{d}\Pi=$ $\displaystyle\int_{{\cal
B}}\left\\{\left[g^{\prime}(d)\psi^{\mathrm{e}}+\alpha(D)\frac{G_{\mathrm{c}}}{\ell}d\right]\delta
d\,\right.$ $\displaystyle\quad+\alpha(D)G_{\mathrm{c}}\ell\,\nabla
d\cdot\delta\nabla d\Big{\\}}\,\mathrm{d}v=0,$ (10)

further demanding $\dot{d}\geq 0$. Its local form is the phase-field evolution
equation

$\displaystyle 2(1-d)\max_{\tau\in[0,t]}\psi^{\mathrm{e}}(\tau)=$
$\displaystyle\,G_{\mathrm{c}}\alpha(D)\left(\ell\Delta
d-\frac{d}{\ell}\right)$ $\displaystyle+G_{\mathrm{c}}\ell\nabla
d\cdot\nabla\alpha(D)$ (11)

with the boundary condition $\nabla d\cdot\boldsymbol{n}=0$. Here, in order to
ensure $\dot{d}\geq 0$, the history variable [38] $\max_{t}\psi^{\mathrm{e}}$
is applied.

#### Fatigue variable

The fatigue variable $D$ is a local variable computed at each material point
with the Local Strain Approach (LSA) [39]. Starting point is the von Mises
equivalent stress $\sigma$ defined by the linear-elastic material model. It is
then projected to the cyclic stress-strain curve (CSSC) assuming constant
strain energy $\sigma\varepsilon=\sigma^{*}\varepsilon^{*}$, yielding a
virtual stress-strain pair $(\sigma^{*},\varepsilon^{*})$ [40]. In this way, a
virtual stress-strain history considering cyclic plastic material behaviour
can be determined for the complete loading sequence. This stress-strain path
is segmented into hysteresis loops. The damaging effect of each loop $i$ is
evaluated by a damage parameter [41]

$P_{\mathrm{SWT},i}=\sqrt{(\sigma_{\mathrm{a},i}^{*}+\sigma_{\mathrm{m},i}^{*})\,\varepsilon_{\mathrm{a},i}^{*}\,E}$
(12)

according to its stress amplitude $\sigma_{\mathrm{a}}$, mean stress
$\sigma_{\mathrm{m}}$ and strain amplitude $\varepsilon_{\mathrm{m}}$. The
damage parameter is then compared to a strain Wöhler curve, yielding a fatigue
contribution $\Delta D_{i}$ for this single load cycle. Fatigue contributions
from all passed load cycles can then be accumulated linearly to the fatigue
variable $D=\sum_{i}\Delta D_{i}$.

The LSA represents a simplified way to include cyclic plasticity into the
model, due to the revaluation of the stress-strain path to the CSSC with the
Neuber rule. Please see [31] for a detailed discussion on this simplification.
CSSC and strain Wöhler curve are material characteristics determined from
standardised experiments, here described in Sec. 4.2. Please see [31, 42] for
a more detailed description of the integration of the Local Strain Approach in
the phase-field fatigue model as well as its implementation in a material
routine of a finite element code.

###  3.2 Extension to transversal anisotropy

effective $G_{\mathrm{c}}$
---
$\chi/^{\circ}$
---

$G_{\mathrm{c}}/$MPa$\cdot\mathrm{m}$
---

0
---
90
---
180
---
270
---
360
---
0.016
---
0.018
---
0.020
---
0.022
---
0.024
---
effective $G_{\mathrm{c}}$
---
$G_{\mathrm{c}}$ from 90° (L-T) sheet
---
$G_{\mathrm{c}}$ from 0° (T-L) sheet
---
$G_{\mathrm{c}}$ from 45° sheet
---
Rolling
---
$\theta/^{\circ}$
---
0°
---
30°
---
60°
---
90°
---
120°
---
150°
---
180°
---
210°
---
240°
---
270°
---
300°
---
330°
---
direction
---
$\chi$
---
a)
---
b)
---
Figure 4: Effective fracture toughness $G_{\mathrm{c}}$ for anisotropic phase-
field model according to [32], (a) displayed over angle from rolling direction
$\chi$. (b) Polar plot includes arbitrary rolling direction of sheet material.
Effective $G_{\mathrm{c}}$ was fitted to experimental results for angle to
rolling direction $\theta=\\{0^{\circ},45^{\circ},90^{\circ}\\}$.

The anisotropy due to the process-induced elongation of the grains and
oriented texture in the aluminium material could manifest in three ways:

  * 1.

Anisotropic elastic properties

  * 2.

Ansiotropic static fracture resistance, i. e. direction dependent fracture
toughness $G_{\mathrm{c}}$

  * 3.

Anisotropic fatigue resistance, i. e. direction dependent CSSC and strain
Wöhler curves

All three are studied in experiments described in Sec. 4 for the given
aluminium AA2024-T351. It turns out that for elastic and fatigue parameters,
no significant direction dependence can be detected. Therefore, only the
fracture toughness will be modelled as anisotropic. However, the extension to
anisotropic elastic behaviour is straightforward. An approach to determine the
fatigue variable given direction dependent fatigue parameters is described in
A.

#### Anisotropic crack surface density

The 2 mm thin aluminium sheets are simulated in 2D with a plane stress state.
Given the 2D problem and only one existing preferred direction – the rolling
direction – the material behaviour can be classified as transversely
isotropic. The approach adopted here to include transversely isotropic
fracture resistance into the phase-field model was described first by
Teichtmeister et al. [32] for static fracture. They introduce a structural
tensor $\mathbf{A}$ into the gradient term of the crack surface density

$\gamma_{\mathrm{aniso}}(d,\nabla d)=\frac{d^{2}}{2\ell}+\frac{\ell}{2}\nabla
d\cdot\mathbf{A}\cdot\nabla d.$ (13)

The structural tensor is a tensor of second order and is defined by a
direction vector $\mathbf{a}$, which is the rolling direction. The extent of
anisotropy is characterised by the anisotropy parameter $\beta$, which has to
be calibrated to represent the direction-dependent variation in crack
resistance of the material. With both, the structural tensor is defined as

$\mathbf{A}=\mathbf{I}+\beta(\mathbf{a}\otimes\mathbf{a}).$ (14)

Given the rolling direction is described by angle $\theta$, the direction
vector is

$[a_{l}]=\left[\begin{array}[]{c}\cos\theta\\\ \sin\theta\\\
0\end{array}\right].$ (15)

Consequently, the structural tensor is

$[A_{lk}]=\left[\begin{array}[]{ccc}1+\beta\cos^{2}\theta&\beta\sin\theta\cos\theta&0\\\
\beta\sin\theta\cos\theta&1+\beta\sin^{2}\theta&0\\\ 0&0&1\end{array}\right].$
(16)

The resulting effective fracture toughness depending on the direction
according to [32] is displayed in Fig. 4.

#### Governing equations

Implementing the anisotropic crack surface density (13) in the total energy
functional (3.1) and repeating the variational principle, (6) yields the
modified weak form of the phase-field problem

$\displaystyle\delta_{d}\Pi=$ $\displaystyle\mathop{\int}_{{\cal
B}}\left[g^{\prime}(d)\psi^{\mathrm{e}}_{+}(\boldsymbol{\varepsilon}^{\mathrm{e}})+{\alpha(D)}\frac{G_{\mathrm{c}}}{\ell}d\right]\delta
d\;\mathrm{d}v$ $\displaystyle+\mathop{\int}_{{\cal
B}}{\alpha(D)}G_{\mathrm{c}}\ell\,\nabla d\cdot\mathbf{A}\cdot\delta\nabla
d\;\mathrm{d}v=0.$ (17)

Subsequently, the evolution equation for the phase-field variable can be
identified as

$\displaystyle 2(1-d)\max_{\tau\in[0,t]}\psi^{\mathrm{e}}(\tau)$
$\displaystyle=G_{\mathrm{c}}{\alpha(D)}\left(\ell\mathbf{A}\colon\Delta
d-\frac{d}{\ell}\right)$ $\displaystyle+\,G_{\mathrm{c}}\ell\nabla
d\cdot\mathbf{A}\cdot\nabla\alpha(D)$ (18)

with the boundary condition

$\nabla d\cdot\mathbf{A}\cdot\boldsymbol{n}=0.$ (19)

##  4 Experiments for parameter identification

Input parameters for the phase-field fatigue model are determined from various
experiments. The same material is used for the cyclic characterisation as for
the fatigue crack growth experiments. Sheet material and properties are
identical, see Sec. 2. Only CT specimens are cut from 25 mm thick plate
material of AA2024-T351.

###  4.1 Elastic parameters

Monotonic tensile tests with dog-bone shaped specimen cut from the aluminium
sheet 0°, 30°, 45°, 60° and 90° angle w. r. t. the rolling direction were
carried out. The determined Young’s modulus and Poisson’s ratio are displayed
in Fig. 5. No clear dependency of both values on the orientation becomes
apparent. Therefore, for the simulations, the standardised values of $E=70$
MPa and $\nu=0.33$ are used instead. This is also in accordance with norm SEP
1240 [43], which recommends using standardised values for elastic parameters
due to the scatter in most experiments. The norm is intended for steel
material, though.

a)
---
b)
---
0°
---
30°
---
45°
---
60°
---
90°
---
0.250
---
0.275
---
0.300
---
0.325
---
0.350
---
0.375
---
0.400
---
0.425
---

$\nu$
---

Experiment
---
Norm
---
0°
---
30°
---
45°
---
60°
---
90°
---
64
---
66
---
68
---
70
---
72
---
74
---
76
---

$E/$MPa
---

Experiment
---
Norm
---
Orientation
---
Orientation
---
Figure 5:  (a) Young’s modulus and (b) Poisson’s ratio tested on tensile
specimen cut from sheets with varying angle to rolling direction. Since no
clear dependency on the angle is apparent, a constant value is set as
recommended in norm SEP 1240 [43].

###  4.2 Cyclic parameters

In order to determine the fatigue variable $D$ with the Local Strain Approach,
cyclic stress-strain curves (CSSC) and strain Wöhler curves are needed as an
input for the model. Both are determined for dog-bone shaped specimens cut
from the aluminium sheet in 0°, 45° and 90° angle to the rolling direction.
Tests are performed according to SEP 1240 [43]. Even though intended for steel
material, this norm has led to good results in the past also for aluminium
[44]. Since the load ratio is $R_{\varepsilon}=-1$, i e. symmetric loading in
tensile and compressive direction, a Teflon-coated support system against
buckling was installed.

Fit
---
$0^{\circ}$
---
$0^{\circ}$
---
$45^{\circ}$
---
$45^{\circ}$
---
$90^{\circ}$
---
$90^{\circ}$
---
b)
---
a)
---

$\varepsilon_{\mathrm{a}}$
---

Fit
---
$0^{\circ}$
---
$45^{\circ}$
---
$90^{\circ}$
---
$N$
---
$10^{2}$
---
$10^{3}$
---
$10^{4}$
---
$10^{5}$
---
$10^{6}$
---
Single
---
level tests
---
0.000
---
0.002
---
0.004
---
0.006
---
0.008
---
$\varepsilon_{\mathrm{a}}$
---
0
---
100
---
200
---
300
---
400
---

$\sigma_{\mathrm{a}}/$MPa
---

Incremental
---
step test
---
$10^{-3}$
---
$10^{-2}$
---
Compression
---
Tension
---
Figure 6: (a) cyclic stress-strain curve (CSSC) from incremental step test
(IST) and (b) S-N (Wöhler) curve recorded with single level tests. Tensile
specimen cut from sheet material with varying angle to rolling direction were
tested. The stress-strain tuples are evaluated seperately for tensile and
compressive range. Since no dependency on neither angle nor loading direction
is apparent, a fit (blue line) was done over all data sets for both curves.

#### Cyclic stress-strain curve

The CSSC was determined with an Incremental Step Test (IST), i. e. with
periodically rising and falling amplitudes [45]. The maximum strain amplitude
was 0.8 % for 0° and 90° orientation and 0.65 % for 45° orientation. Tests for
all orientations are evaluated separately for tensile and compressive range.
As customary, the stress-strain characteristic after half the lifetime is
selected for the fit. Results are displayed in Fig. 6b. Obviously, the stress-
strain behaviour depends on neither the sheet orientation nor the loading
direction. Therefore, all data is used for a single fit. The cyclic-stress
strain curve according the the Ramberg-Osgood ansatz [46]

$\varepsilon_{\mathrm{a}}=\frac{\sigma_{\mathrm{a}}}{E}+\left(\frac{\sigma_{\mathrm{a}}}{K^{\prime}}\right)^{1/n^{\prime}}$
(20)

is parametrised with $K^{\prime}=750$ MPa and $n^{\prime}=0.0728$, and is
shown in blue in the graph.

Here, we stick to the usual realisation of the Local Strain approach with only
one CSSC. However, should the material show significant deviations in either
tensile/compressive range or a drastic change of the behaviour over the
specimens’ liftime, a modification of the Local Strain approach by Kühne et
al. [47] can be implemented. It allows for a tension-compression asymmetry and
a transient evolution of the stress-strain behaviour.

#### Strain Wöhler curve

The strain Wöhler curve was recorded with single level tests with various
strain amplitudes. Results are shown in Fig. 6a. Since no dependence on the
rolling direction is apparent, only one single fit using all data was done.
The fit yielded the parameters of the strain Wöhler curve

$\varepsilon_{\mathrm{a}}=\frac{\sigma^{\prime}_{\mathrm{f}}}{E}\,(2N)^{b}+\varepsilon^{\prime}_{\mathrm{f}}\,(2N)^{c}$
(21)

as $\sigma^{\prime}_{\mathrm{f}}=1206$ MPa,
$\varepsilon^{\prime}_{\mathrm{f}}=0.9975$, $b=-0.152$ and $c=-0.860$ and is
marked in blue in Fig. 6a.

###  4.3 Fracture toughness

The fracture toughness tests were carried out in accordance to ASTM E1820-01
[48]. In total twelve CT specimens, see Fig. 2b, were tested in three
directions, with rolling directions as indicated in the top-left sketch in
Fig. 7. The specimen geometry was $W=50$ mm, $a_{0}=12.5$ mm according to the
norm, with a thickness of $t=25$ mm ensuring mostly plane strain conditions.
In the phase-field modelling community, $G_{\mathrm{c}}$ is treated as a
material parameter, not a thickness-dependent structural parameter. Plane
strain conditions are therefore chosen in order to obtain a parameter
characterising the behaviour of material, ruling out influences of thickness
and boundary effects as far as possible.

Tests were performed on a standard uniaxial servo hydraulic test rig. To
determine the crack length during fatigue precracking, the direct current
potential drop (DCPD) was used. Starting from an initial V-formed Chevron
notch, a fatigue precrack was initialized with a constant $\Delta
K=10\,\mathrm{MPa\sqrt{m}}$ and cyclic loading until an $a/W$ ratio of
0.45…0.5 was met. The test procedure for determing $K_{Ic}$ was displacement-
controlled with a displacement rate of 1 mm/minute. During this test sequence,
the crack mouth opening displacement (CMOD) as well as the applied load $F$
were recorded. $K_{Ic}$ was determined according to the secant method based on
the load vs. CMOD curve. Fig. 7 shows the measured plane strain fracture
toughness $K_{Ic}$ related to the specimen orientation, as well as the
corresponding critical energy release rate commonly used in phase-field models

$G_{\mathrm{c}}=\frac{1-\nu^{2}}{E}K_{Ic}.$ (22)

With a mean fracture toughness of $34.76\,\mathrm{MPa\sqrt{m}}$ the $\chi=0$°
orientation has the lowest fracture toughness and the highest value of
$44.11\,\mathrm{MPa\sqrt{m}}$ can be measured for specimen in $\chi=90$°. The
experimentally determined values for $\chi=90$° and $\chi=0$° are comparable
to references given in literature [49, 50], whereas no references exist for
fracture toughness in 45° orientation.

The mean value of each orientation was used to fit the anisotropy parameter of
the phase-field model according to eq. (14), yielding $\beta=1.625$. For an
illustration of the calibrated effective fracture toughness, see Fig. 4.

0°
---
45°
---
90°
---
$\chi$
---
34
---
36
---
38
---
40
---
42
---
44
---
46
---

$K_{Ic}/\mathrm{MPa\sqrt{m}}$
---

Orientation | $K_{Ic}$ | $G_{\mathrm{c}}$
---|---|---
0° (T-L) |  34.7 MPa $\sqrt{\mathrm{m}}$ | 0.0153 MPa m
45° |  38.6 MPa $\sqrt{\mathrm{m}}$ | 0.0189 MPa m
90° (L-T) |  44.1 MPa $\sqrt{\mathrm{m}}$ | 0.0248 MPa m

Figure 7: Measured fracture toughness $K_{Ic}$ values for varying angle
between rolling direction an crack $\chi$. Mean values displayed in table
below, as well as corresponding $G_{\mathrm{c}}$.

##  5 Numerical results

In the following, the developed model is used to simulate the FCG according to
the experiments with MT specimen described in Sec. 2. For the elastic, cyclic
and fracture parameters, the values determined experimentally in Sec. 4 are
used, if not stated otherwise. All model parameters are summarised in Tab. 1.
The MT geometry is meshed with quadratic elements. The mesh is refined in the
relevant areas with a minimum mesh size of $h_{\min}=0.3$ mm. The coupled
problem is solved with a staggered approach with convergence control. Viscous
regularisation with a viscous damping parameter $\eta$ is applied in order to
compensate for convergence issues due to fast-developing cracks. See [30] for
a general phase-field fatigue model structure with viscous regularisation.
Here, the viscous damping parameter is chosen according to preliminary studies
with static and cyclic loading, precluding its influence on the results.

Elastic |  $E=70$ MPa, $\mu=0.33$
---|---
Fracture |  $G_{\mathrm{c}}=0.0153$ MPa m, $\ell=0.5$ mm,
|  $\eta=1\cdot 10^{-9}$, $\beta=1.625$
Cyclic stress- |  $K^{\prime}=750$ MPa, $n^{\prime}=0.0728$
strain curve |
Strain Wöhler |  $\sigma^{\prime}_{\mathrm{f}}=1206$ MPa, $\varepsilon^{\prime}_{\mathrm{f}}=0.9975$,
curve |  $b=-0.152$, $c=-0.860$
Fatigue function |  $\alpha_{0}=0.001$, $\xi=1450$
Table 1: Parameters of phase-field fatigue model.

###  5.1 Numerical studies of static loading

Firstly, the MT specimen is loaded with a static load in order to study the
effect of the anisotropic crack surface density without the influence of
fatigue damage accumulation. This is a purely numerical experiment, it is not
compared to experimental data. The characteristic length is chosen to be
$\ell=1$ mm, yielding a ratio $h_{\min}/\ell\approx 1/3$, surpassing the
recommendation [51] of $h_{\min}/\ell<1/2$. A total displacement of 2 mm is
applied. The viscosity parameter is set to $\eta=1\cdot 10^{-6}$. Apart from
the experimentally determined anisotropy parameter $\beta=1.625$, $\beta=0.4$
is tested for comparison.

The phase-field plots are shown in Fig. 8. Only the center part of the MT
specimen is displayed. The rolling direction, which was varied between
$\theta=0$° to 45° from the horizontal direction, is marked with a dashed
black line. As expected, for horizontal rolling direction ($\theta=0$°), the
crack path is horizontal as well, independent of the anisotropy parameter
$\beta$. Crack inclination increases with steeper rolling direction angles.
The effect is stronger for a higher anisotropy parameters $\beta$. Still, the
original difference in effective $G_{\mathrm{c}}$ between 0° and 90° direction
of about 30 % is still quite low compared to other typical anisotropic
materials, like e. g. fibre reinforced composites. This explains that the
crack does not follow the rolling direction (which is the direction of lowest
effective fracture resistance) strictly. See [32] for a study on the degree of
anisotropy in anisotropic static phase-field models.

$\beta=1.625$
---
$\theta=0^{\circ}$
---
$\theta=30^{\circ}$
---
$\theta=45^{\circ}$
---
$\beta=0.4$
---
$x_{1}$
---
$x_{2}$
---
direction
---
$\theta$
---
Rolling
---
0
---
1
---
Figure 8: Simulation of static fracture with anisotropic phase-field model.
Shown is a contour plot of the phase-field variable $d$. $\beta$ is the
anisotropy parameter. The bigger $\beta$, the stronger the crack inclines
towards the rolling direction, as anisotropy in the fracture toughness is more
pronounced.

###  5.2 Parametrisation of cyclic model

The cyclic MT experiment used for fitting the model is the $\theta=0^{\circ}$
orientation with $R=0.1$ and a constant cyclic load amplitude of $F_{\max}=15$
kN. The damping parameter is set to $\eta=1\cdot 10^{-9}$ and the
characteristic length to $\ell=0.5$ mm. With all other parameters coming from
the experiments in Sec. 4, only the parameters of the fatigue degradation
$\alpha(D)$ function remain to be fitted.

The results are summarised in Fig. 9. Please note that the crack propagation
rate is now plotted over the crack length $a$ instead of $\Delta K$ as in Fig.
3 in order to allow for a better comparison with phase-field contour plots of
the crack. The figure shows the distribution of the fatigue degradation
function $\alpha$ and the phase-field $d$ at the end of the simulation. They
underline how the degradation of the fracture resistance $G_{\mathrm{c}}$ has
allowed the phase-field crack grow at a sub-critical, but repetitive load. The
calibration to the experimental FCG rate yielded the parameters
$\alpha_{0}=0.001$ and $\xi=1450$, as displayed in the lower graph. The
calibrated model reproduces the experimental data fairly well in the first two
thirds of crack length $a$. At a later stage, the simulation is not able to
capture the transition to unstable, static residual fracture at about $a=50$
mm almost at the outer boundary of the specimen.

0
---
1
---
10
---
20
---
30
---
40
---
50
---
60
---
70
---
$a$/mm
---
$10^{-6}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
$10^{0}$
---

$\mathrm{d}a/\mathrm{d}N$/mm
---

Simulation
---
Experiment
---

Fatigue

$\alpha$

degradation

$d$

Phase-field

Figure 9: Fit of phase-field fatigue model to MT experiment with
$\theta=0^{\circ},R=0.1$. Fit yielded parameters of fatigue degradation
function $\alpha_{0}=0.001,\xi=1450$. Displayed are the fatigue degradation
$\alpha$ and the phase-field at the end of the simulation, as well as the
fatigue crack growth rate plotted over crack length $a$.

###  5.3 Validation

For validation, the fit is used to simulate the FCG experiments that were not
used for parameter identification. All parameters remain identical to those
stated in Sec. 5.2. First, the orientation of the sheet material is varied.
Fig. 10 shows a) the resulting crack path and b) the fatigue crack propagation
rate. Consistent with the experimental results, both the 0° and 90°
orientation lead to a straight crack. The crack in the sheet with 45°
orientation, on the other hand, inclines to about 8.5°. Apart from the jagged
experimental crack geometry, the simulation yields a similar crack path
compared to the experiment. The crack propagation rates differ – as in the
experiments – only slightly. The correct order of crack propagation rates,
with the 0° orientation being the fastest and 90° the slowest – is reproduced
correctly.

20
---
40
---
60
---
$a/$mm
---
-8
---
-6
---
-4
---
-2
---
0
---
2
---
4
---
6
---
8
---

$a_{y}/$mm
---

$\theta=0^{\circ}$
---
$\theta=45^{\circ}$
---
$\theta=90^{\circ}$
---
Experiment
---
Simulation
---
$\theta=0^{\circ}$
---
$\theta=45^{\circ}$
---
$\theta=90^{\circ}$
---
10
---
20
---
30
---
40
---
50
---
60
---
70
---
$a$/mm
---
$10^{-6}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
$10^{0}$
---

$\mathrm{d}a/\mathrm{d}N$/mm
---

a)
---
b)
---
Figure 10: Simulation of fatigue crack growth in MT specimen with different
angles to rolling direction $\theta$ and $R=0.1$, see Fig. 8 for geometry. Fit
of $\theta=0^{\circ}$ was used for all simulations. (a) Simulation reproduces
horizontal crack path for $\theta=0^{\circ},90^{\circ}$ and inclination for
$\theta=45^{\circ}$. (b) In experiments and simulations, crack propagation
rate is the highest for $\theta=0^{\circ}$ and the lowest for
$\theta=90^{\circ}$.  10
---
20
---
30
---
40
---
50
---
60
---
70
---
$a$/mm
---
$10^{-6}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
$10^{0}$
---

$\mathrm{d}a/\mathrm{d}N$/mm
---

$R=0.5$
---
$R=0.3$
---
$R=0.1$
---
Simulation
---
Experiment
---
Figure 11: Fatigue crack growth for different load ratios $R$ and $\theta=0$°.
Fit of $R=0.1$ was used in all simulations. Experiments’ tendency of lower $R$
to have the highest crack propagation rate is reproduced.

The final study addresses the $R$-ratio dependence. As shown in Fig. 11,
additionally, $R=0.3$ and $R=0.5$ are tested with the model fitted to the
$R=0.1$ experiments. The maximum load $F_{\max}=15$ kN is kept constant, while
$F_{\min}$ is increased to 4.5 kN and 7.5 kN, thereby reducing the force
amplitude. The simulation correctly reproduces the tendency of higher
$R$-ratios to lead to lower fatigue crack propagation rates.

The results do not match the experiments exactly, though. For one thing, this
is owed to doing the simulation in 2D. Although the sheet is quite thin with a
thickness of only 2 mm, the plane stress state is only an approximation of the
present stress state. Variation of the grain structure over the thickness,
boundary effects, curved crack fronts and tilting of the crack front angle can
be observed for fatigue fracture in this material. Moreover, as a ductile
material, aluminium can form significant plastic zones, influencing the stress
state at the crack tip. The LSA only uses a simplified plasticity model based
on the cyclic stress-strain curve, see [31] for details. Both the extension to
3D and to elastic-plastic material modelling are out of this paper, though,
and subject of future work.

##  6 Conclusions

FCG behaviour in rolled aluminium sheets shows a dependency of the crack path
w. r. t. the sheet rolling direction, as well as a growth rate that decreases
with increasing load ratio. The cyclic crack growth is modelled with the
phase-field method due to its superior ability to model arbitrary crack paths
in a straightforward way. The fatigue variable of the model is based on the
Local Strain Approach. An extension to anisotropic material behaviour is
proposed by introducing a structural tensor in the regularised crack surface
density. Thereby, the effective fracture toughness is interpolated.

The model is parameterised for AA2024-T351 sheet material using experimentally
determined fracture toughness values in three different orientations. Elastic
and cyclic properties do not exhibit a direction dependency. The parameterised
model reproduces the influence of the load ratio on the fatigue crack
propagation rate as well as the crack angle depending on the sheet
orientation.

So far, the simulation of the 2 mm thick sheets are performed with a 2D model
assuming a plane stress state. This raises the question of the correct choice
of the fracture toughness $G_{\mathrm{c}}$. In the phase-field community,
$G_{\mathrm{c}}$ is mostly seen as a material parameter. In accordance with
that, this paper determines $G_{\mathrm{c}}$ with thick specimen,
approximating a plane strain state, as this yields a universal material value.
If thin sheets are used for the determination, the parameter turns out to be
thickness-dependent. Therefore, it could be appropriate to view
$G_{\mathrm{c}}$ as a structural parameter instead, which has to be calibrated
to the thickness of the specimen – if 2D simulations are used.

Although the model is able to reproduce the main characteristics of crack
propagation behaviour, certain phenomena cannot be covered with a 2D model.
This includes three dimensional effects as variation of the crack front, for
example crack tunnelling or slant crack growth, that affect crack closure
behaviour and therefore lead to variation in crack propagation rates. Hence,
the development of 3D models is inevitable in the future. The model presented
here uses a simplified cyclic plasticity model for damage calculation.
However, the effect of the plastic zone should be studied in more detail,
using a proper elastic-plastic material model, especially if higher loadings
$\Delta K$ are investigated.

## Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via the
project Experimental analysis and phase-field modelling of the interaction
between plastic zone and fatigue crack growth in ductile materials under
complex loading (grant numbers KA 3309/12-1, BR 6259/2-1). Furthermore,
funding came from the Federal Ministry for Economic Affairs and Climate Action
on the basis of a decision by the German Bundestag, within the aerospace
program LuFo-VI of the project Untersuchung der Prozess-Struktur-
Eigenschaftsbeziehungen des Rollprofilierens auf die Performanceverbesserung
für Blechbauteile aus Aluminium (Funding ID 20W2102D). The authors are
grateful to the Centre for Information Services and High Performance Computing
(ZIH) TU Dresden for providing its facilities for high throughput
calculations.

## Highlights

  1. 1.

Development of the first phase-field model for fatigue fracture in anisotropic
media

  2. 2.

Acquisition of set of direction-dependent material parameters for a
AA2024-T351 aluminium material, including fracture toughness and cyclic
parameters

  3. 3.

Parametrization of the model for anisotropic aluminum sheet material

  4. 4.

Reproduction of experimentally observed effects in the crack path and crack
growth rate

##  Appendix A Anisotropic fatigue damage calculation

The cyclic material behaviour of AA2024-T351 is not direction dependent, as
shown in Sec. 4.2. However, should the desired material show a cyclic
anisotropy, the following anisotropic fatigue concept can be implemented.

The parameters of the CSSC (20) $K^{\prime}$ and $n^{\prime}$ and the strain
Wöhler curve (21)
$\sigma^{\prime}_{\mathrm{f}},\varepsilon^{\prime}_{\mathrm{f}},b$ and $c$ are
determined for 0° and 90° orientation individually. All parameters – here
represented by $p$ – are then interpreted as direction dependent parameters
$K^{\prime}(\chi),n^{\prime}(\chi),\dots$ and interpolated over $\chi$ with
the cosine ansatz

$p(\chi)=\frac{p_{0^{\circ}}-p_{90^{\circ}}}{2}\cos(2\chi)+p_{0^{\circ}}+\frac{p_{90^{\circ}}-p_{0^{\circ}}}{2},$
(23)

as shown in Fig. 12 on the left. $p_{0^{\circ}}$ and $p_{90^{\circ}}$ are the
experimental values of the parameter in 0° and 90° orientation, respectively.
The angle $\chi$ is measured from the rolling direction. This yields a family
of curves for each the CSSC and the strain Wöhler curve, see Fig. 12. The
interpolation should be checked with experimentally determined curves in
between 0° and 90°. In this way, the material resistance against fatigue can
be described for arbitrary directions in the 2D plane.

$0^{\circ}$
---
$15^{\circ}$
---
$30^{\circ}$
---
$45^{\circ}$
---
$60^{\circ}$
---
$75^{\circ}$
---
$90$
---
$10^{2}$
---
$10^{4}$
---
$N$
---
$\varepsilon_{\mathrm{a}}$
---
0
---
180
---
360
---
$\chi/^{\circ}$
---
$p_{90^{\circ}}$
---
$p_{0^{\circ}}$
---

$p$
---

$\sigma_{\mathrm{a}}$/kN/mm2
---

$\varepsilon_{\mathrm{a}}$
---

$10^{-2}$
---
Cyclic stress-strain curve
---
Cyclic parameters
---
Strain Wöhler curve
---
0.00
---
0.25
---
0.50
---
0.75
---
1.00
---
0.0
---
0.1
---
0.2
---
0.3
---
0.4
---
Figure 12:  Extension of model to anisotropic cyclic parameters: All
parameters of cyclic stress-strain curve and strain Wöhler curve, represented
by $p$, are interpolated with sine function between their value in and
perpendicular to rolling direction. This leads to an array of curves.

The fatigue variable at each material point is then determined with the
critical plane approach [52]. Thereby, the 2D spaces is sampled by a discrete
number of planes at each point, see Fig. 13. For each plane, the stress state
and the materials’ resistance is determined in the particular direction. They
are the input to the Local Strain Approach, instead of e. g. the equivalent
stress in the standard procedure. Finally, the fatigue variable is then
determined as the maximum of all fatigue values in the set of directions at
this particular point. The quality of the result depends, among others, on the
amount of planes used in the sampling.

$F$
---
$x_{1}$
---
$x_{2}$
---
plane $i$
---
rolling direction
---
planes
---
$\sigma_{11,\mathrm{el}}$
---
$\sigma_{12,\mathrm{el}}$
---
$\sigma_{22,\mathrm{el}}$
---
$\sigma_{i,\mathrm{el}}$
---
$\tau_{i,\mathrm{el}}$
---
$\Delta\varphi$
---
$\theta$
---
$\chi_{i}$
---
$\phi_{i}$
---
Figure 13:  Extension of model to anisotropic cyclic parameters: Damage
concept at each material point involves critical plane approach. 2D space is
sampled by finite number of planes $i$, for which damage is calculated
individually, considering stress state and cyclic parameters (see Fig. 12) for
that direction. Maximum of damages of all directions finally enters fatigue
degradation function.

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