∎

11institutetext: M. Seiler and M. Kästner 22institutetext: Chair of
Computational and Experimental Solid Mechanics, TU Dresden, Germany
22email<EMAIL_ADDRESS>33institutetext: S. Keller and N.
Kashaev and B. Klusemann44institutetext: Institute of Materials Research,
Materials Mechanics, Helmholtz-Zentrum Geesthacht, Germany 55institutetext: B.
Klusemann 66institutetext: Institute of Product and Process Innovation,
Leuphana University of Lüneburg, Germany

# Phase-field modelling for fatigue crack growth under laser-shock-peening-
induced residual stresses

Martha Seiler  Sören Keller  Nikolai Kashaev  Benjamin Klusemann  Markus
Kästner

(Received: date / Accepted: date)

###### Abstract

For the fatigue life of thin-walled components, not only fatigue crack
initiation, but also crack growth is decisive. The phase-field method for
fracture is a powerful tool to simulate arbitrary crack phenomena. Recently,
it has been applied to fatigue fracture. Those models pose an alternative to
classical fracture-mechanical approaches for fatigue life estimation. In the
first part of this paper, the parameters of a phase-field fatigue model are
calibrated and its predictions are compared to results of fatigue crack growth
experiments of aluminium sheet material. In the second part, compressive
residual stresses are introduced into the components with the help of laser
shock peening. It is shown that those residual stresses influence the crack
growth rate by retarding and accelerating the crack. In order to study these
fatigue mechanisms numerically, a simple strategy to incorporate residual
stresses in the phase-field fatigue model is presented and tested with
experiments. The study shows that the approach can reproduce the effects of
the residual stresses on the crack growth rate.

###### Keywords:

Laser shock peening Fatigue crack growth Phase-field modelling Residual
stresses

##  1 Introduction

Fatigue fracture is one of the most common causes of component failure. Still,
the underlying physical phenomena, especially on the microscale
bathias_fatigue_2010 , are not fully understood. Simulating fatigue fracture,
e. g. via the Finite Element Method (FEM), is often very time-consuming, as
several hundred to millions of load cycles have to be simulated. Generally,
the fatigue life of a component until fracture can be divided into the crack
initiation and the crack propagation stage. While crack initiation often takes
the major part, in thin-walled specimen like fuselage shells, crack growth is
crucial for the design process. The growth rate of the long, visible cracks
determines the maintenance interval.

Particularly, residual stresses created by the process of laser shock peening
(LSP) can be used to deliberately influence the fatigue crack growth (FCG)
rate of long cracks, as demonstrated for the aluminium alloy AA2024
hombergsmeier2014fatigue ; tan2004laser . The application of laser shock
peening aims at the introduction of compressive residual stresses in regions
susceptible to fatigue, where the process provides a relatively high
penetration depth as well as surface quality peyre1996laser . These
compressive residual stresses interact with the applied stresses of the
fatigue load cycles and reduce the fatigue crack driving quantity. However,
compressive residual stresses are always accompanied by tensile residual
stresses to meet stress equilibrium. While compressive residual stresses are
expected to retard the fatigue crack growth, tensile residual stresses may
lead to increased fatigue crack growth rates reducing the fatigue life of the
component, as shown by keller_experimentally_2019 . Thus, the efficient
application of residual stress modification techniques needs the precise
prediction of the residual stress field to determine the FCG rate.

In order to estimate the fatigue life of residual stress affected components,
mainly two types of models are used. Empirical concepts based on Wöhler curves
mostly evaluate the fatigue life until crack initiation and often treat
residual stresses as mean stresses landersheim_analyse_2009 . For the
remaining fatigue life during crack growth, models based on fracture mechanics
are often used. In this context, residual stresses can be applied with the
eigenstrain approach. The residual stress field is applied with the help of a
fictitious temperature field benedetti_numerical_2010 ;
keller_experimentally_2019  and considered with effective stress intensity
factors in the fatigue crack growth computations. In contrast to the
previously mentioned approaches, in this contribution, an FEM framework is
applied which covers both crack initiation and growth – the phase-field
method. However, the focus of this paper lies on the fatigue crack growth.

The phase-field method has become a popular tool to simulate fracture
phenomena because of its capability to treat arbitrary crack paths in a
straight-forward way. This is possible due to a second field variable which
describes the crack topology, making mesh alterations due to crack growth
redundant. Originally formulated for static brittle fracture
miehe_thermodynamically_2010  by simply regularising the Griffith criterion
griffith_phenomena_1921  for crack growth, the phase-field method has now been
applied to a large variety of materials and phenomena like e. g. ductile
fracture miehe_phase_2016 ; ambati_phasefield_2015 .

However, the phase-field modelling of fatigue fracture has been addressed only
recently. While some models reduce the crack resistance of the material as a
result of its cyclic degradation carrara_framework_2019 ;
seiler_efficient_2020 ; mesgarnejad_phasefield_2018 , others increase the
crack driving force schreiber_phase_2020 ; haveroth_nonisothermal_2020 . There
are first approaches to cover plastic ulloa_phasefield_2019  and viscous
loew_fatigue_2020  materials. To tackle the crucial problem of computational
time when simulating repetitive loading, Loew et al. loew_accelerating_2020
and Schreiber et al. schreiber_phase_2020  use cycle jump techniques. Seiler
et al. seiler_efficient_2020  approached the problem by incorporating a
classic fatigue concept – the local strain approach (LSA) – into the model,
enabling the simulation of several load cycles within only one increment.
However, this model has been studied only qualitatively.

In this contribution, the phase-field fatigue model of seiler_efficient_2020
is calibrated and validated using FCG experiments of aluminium AA2024.
Moreover, a straightforward strategy to include residual stresses in the model
is presented. Additional FCG experiments are conducted with specimen in which
residual stresses are introduced deliberately with LSP. Those residual
stresses are analysed with the incremental hole drilling method and serve as
an initial state for the fatigue simulation.

The paper is structured as follows: In Section 2, the model formulation is
recapitulated, including the underlying phase-field equations for brittle
fracture and its extension to fatigue, and the incorporation of residual
stresses is explained. Section 3 deals with the LSP experiments for the
creation of residual stresses, the experimental determination of the resulting
residual stress state, as well as the FCG experiments. Section 4 contains the
numerical predictions of the proposed model including the model parameter
calibration and the comparison to experimental results. The conclusion follows
in Section 5.

##  2 Model framework

The model used in this publication is extensively described and qualitatively
studied in seiler_efficient_2020 . Therefore, the model formulation is only
outlined briefly here, starting with the basis of the framework, the phase-
field method for brittle fracture, as well as its extension to fatigue. Then,
going one step further, the incorporation of residual stresses in the model is
described in detail.

###  2.1 Phase-field method for fracture

$2\ell$
---
$d=0$
---
$\Gamma_{\ell}$
---
$\Omega$
---
$d=1$
---
$\Gamma$
---
$\Omega$
---
a)
---
b)
---
Figure 1: Fractured domain $\Omega$ with crack surface $\Gamma$. (a) Sharp
representation of crack topology. (b) Regularised representation: The crack is
described by the phase-field variable $d=1$, while $d=0$ represents undamaged
material. The crack is regularised over the length scale $\ell$.  $\Delta
D=1/N$
---
$D=\sum\Delta D$
---

$N$

$P_{\mathrm{SWT}}$

$\sigma^{*}$
---
$\varepsilon^{*}$
---
Linear elastic
---
simulation
---
$\sigma,\varepsilon$
---
CSSC
---
Neuber’s
---
rule
---
Stress-strain
---
path
---
Damage
---
parameter
---
Damage
---
accumulation
---
Wöhler
---
curves
---
$\sigma^{*}$
---
$\varepsilon^{*}$
---
$\sigma_{\mathrm{m}}^{*}$
---
$\varepsilon_{\mathrm{a}}^{*}$
---
$\sigma_{\mathrm{a}}^{*}$
---

$P_{\mathrm{SWT}}$

$\sigma^{*},\varepsilon^{*}$
---
Figure 2: Scheme of local strain approach (LSA), with which the fatigue life
variable $D$ is determined at every material point.

The phase-field method for brittle fracture is based on the Griffith criterion
griffith_phenomena_1921  for crack growth, which requires the energy release
rate to be equal to the critical energy release rate or fracture toughness
$\mathcal{G}_{\mathrm{c}}$. This criterion was brought to a variational form
in francfort_revisiting_1998  and regularised for convenient numerical
implementation in bourdin_variational_2008 . During the regularisation, an
additional field variable $d\in[0,1]$ is introduced. The diffuse description
of the crack smoothly bridges the entirely intact ($d=0$) and totally broken
($d=1$) state. In this way, the crack topology can be described without any
mesh modifications, allowing for a straightforward modelling of arbitrary
crack paths. See Fig. 1 for a graphical explanation of the regularisation.

Using the regularisation length scale $\ell$, the regularised energy
functional $\Pi_{\ell}$ in the domain $\Omega$ can be written as

$\Pi_{\ell}=\int_{\Omega}g(d)\,\psi^{\mathrm{e}}(\boldsymbol{\varepsilon})\,\mathrm{d}V+\int_{\Omega}\mathcal{G}_{\mathrm{c}}{\frac{1}{2\ell}(d^{2}+\ell^{2}|\nabla
d|^{2})}\,\mathrm{d}V.$ (1)

In the small strain linear elastic setting. The elastic strain energy density
is

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

with the Lamé constants $\lambda$ and $\mu$, the elasticity tensor
$\mathbb{C}$ and the strain

$\boldsymbol{\varepsilon}=\frac{1}{2}\left(\nabla\boldsymbol{u}+(\nabla\boldsymbol{u})^{\top}\right).$
(3)

The degradation function $g(d)=(1-d)^{2}$ models the loss of stiffness due to
the developing crack, coupling displacement field $\boldsymbol{u}$ and phase-
field $d$. Consequently, the stress is given by

$\boldsymbol{\sigma}=g(d)\frac{\partial\psi^{\mathrm{e}}}{\partial\boldsymbol{\varepsilon}}=g(d)\,\mathbb{C}:\boldsymbol{\varepsilon}.$
(4)

The governing equations of the coupled problem obtained from the variation
$\delta\Pi_{\ell}=0$

$\boldsymbol{0}=\mathrm{div}\,\boldsymbol{\sigma}\qquad\qquad\mathcal{G}_{\mathrm{c}}\left(d-\ell^{2}\Delta
d\right)=(1-d)2\ell\underbrace{\psi^{\mathrm{e}}(\boldsymbol{\varepsilon})}_{\mathcal{H}}$
(5)

are subject to the boundary conditions
$\boldsymbol{n}\cdot\boldsymbol{\sigma}=\tilde{\boldsymbol{t}}$,
$\boldsymbol{u}=\tilde{\boldsymbol{u}}$ and $\boldsymbol{n}\cdot\nabla d=0$
with $\bar{\boldsymbol{t}}$ and $\bar{\boldsymbol{u}}$ being the prescribed
tractions and displacements on the corresponding boundaries, respectively.

To ensure crack irreversibility, in Eq. (5), the crack driving force
$\mathcal{H}$ in each point $\boldsymbol{x}$ is set to its temporal maximum
miehe_phase_2016

$\mathcal{H}(\boldsymbol{x},t)=\max_{s\in[t_{0};t]}\psi^{\mathrm{e}}(\boldsymbol{\varepsilon}(\boldsymbol{x},s)).$
(6)

###  2.2 Extension to fatigue

#### Fatigue degradation

In order to incorporate fatigue into the phase-field framework, the fracture
toughness $\mathcal{G}_{\mathrm{c}}$ is reduced when the material degradation
due to repetitive stressing precedes. This process is described by introducing
a local lifetime variable $D$. An additional scalar fatigue degradation
function $\alpha(D):[0,1]\rightarrow[\alpha_{0},1]$ with $\alpha_{0}>0$ is
introduced, which lowers the fracture toughness $\mathcal{G}_{\mathrm{c}}$
locally. The energy functional then reads

$\displaystyle\begin{split}\Pi_{\ell}=&\int_{\Omega}g(d)\,\psi^{\mathrm{e}}(\boldsymbol{\varepsilon})\,\mathrm{d}V\\\
&+\int_{\Omega}\alpha(D)\,\mathcal{G}_{\mathrm{c}}\frac{1}{2\ell}(d^{2}+\ell^{2}|\nabla
d|^{2})\,\mathrm{d}V,\end{split}$ (7)

which leads to the modified phase-field evolution equation

$\mathcal{G}_{\mathrm{c}}\left(\alpha\,d-\nabla\alpha\cdot\ell^{2}\nabla
d-\alpha\,\ell^{2}\Delta d\right)=(1-d)\,\mathcal{H}\,2\ell$ (8)

with the dependency $\alpha(D)$ dropped for brevity.

The lifetime variable $D\in[0,1]$ is a history variable that is accumulated
strictly locally. For $D=0$, the material has experienced no cyclic loads and
therefore offers full fracture toughness. Consequently, $\alpha(0)=1$ must
hold. For $D=1$, the fracture toughness is reduced to a threshold value
$\alpha_{0}$. Therefore, the fatigue degradation function

$\alpha(D)=(1-\alpha_{0})(1-D)^{\xi}+\alpha_{0}$ (9)

with the parameters $\alpha_{0}$ and $\xi$ is used. For a study of the
influence of the model parameters $\alpha_{0}$ and $\xi$, see Sec. 4.1 and
seiler_efficient_2020 .

#### Local strain approach

The computation of the lifetime variable $D$ follows the LSA
seeger_grundlagen_1996 . This method is generally used for fatigue life
calculations of components, but is implemented here in the material routine of
the FEM framework and therefore executed at each integration point.

The computation scheme is illustrated in Fig. 2. At first, the stresses and
strains from a linear elastic simulation are revaluated using the Neuber rule
neuber_theory_1961 . The von Mises equivalent stress $\sigma$ is projected to
the cyclic stress-strain curve (CSSC) yielding a virtual, revaluated stress-
strain pair $(\sigma^{*},\varepsilon^{*})$ by assuming a constant strain
energy
$\frac{1}{2}\,\sigma\varepsilon=\frac{1}{2}\,\sigma^{*}\varepsilon^{*}$. The
CSSC is thereby described by the Ramberg-Osgood equation
ramberg_description_1943

$\varepsilon^{*}=\frac{\sigma^{*}}{E}+\left(\frac{\sigma^{*}}{K^{\prime}}\right)^{1/n^{\prime}}$
(10)

with the cyclic parameters $K^{\prime}$ and $n^{\prime}$ and Young’s modulus
$E$. It can be determined from standardised cyclic experiments such as the
incremental step test. In this way, the complete virtual stress-strain path
can be derived from the loading sequence. This stress-strain path is divided
into hysteresis loops. For each loop $i$, the damage parameter by Smith,
Watson and Topper smith_stressstrain_1970

$P_{\mathrm{SWT},i}=\sqrt{(\sigma_{\mathrm{a},i}^{*}+\sigma_{\mathrm{m},i}^{*})\varepsilon_{\mathrm{a},i}^{*}E}$
(11)

can be determined from the stress and strain amplitudes
$\sigma_{\mathrm{a}}^{*}$ and $\varepsilon_{\mathrm{a}}^{*}$ and the mean
stress $\sigma_{\mathrm{m}}^{*}$. It quantifies the damaging effect of the
loop. Only the tensile range contributes to $P_{\mathrm{SWT}}$, see Fig. 2.
From strain Wöhler curves (SWC) – also generated with standardised experiments
– the matching virtual load cycle number $N_{i}$ for $P_{\mathrm{SWT},i}$ can
be read. Finally, the fatigue life contribution of the single hysteresis loop
$\Delta D_{i}$ and the full loading path is

$\Delta D_{i}=1/N_{i}\quad\text{and}\quad D=\Sigma_{i}\Delta D_{i}.$ (12)

Note, that the revaluated stresses and strains $\sigma^{*}$ and
$\varepsilon^{*}$ are solely used for the damage calculation and are not used
in the coupled problem in any other way.

In conclusion, the integration of fatigue in the phase-field model with the
LSA is beneficial for the computational cost in three ways:

  1. 1.

Local cyclic plasticity is covered by the Neuber rule, so no elastic-plastic
material model is needed.

  2. 2.

Since only amplitude and mean values of stress and strain enter the
calculation of $D$, the loading path does not have to be resolved in the
simulation, instead the reversal points are sufficient.

  3. 3.

In case of constant load amplitudes and small crack growth rates, several load
cycles can be simulated with only one increment, since the lifetime
contributions are accumulated linearly according to Eq. 12. Especially for
high cycle fatigue, this can save immense computational time.

###  2.3 Incorporation of residual stresses

$\mathcal{H}$

$\sigma_{0}$
---

(a)

(b)

Process
---
$\varepsilon_{0}$
---
Cyclic
---
unloaded
---
$\sigma$
---
$\varepsilon$
---
$\sigma_{0}$
---
$\varepsilon_{0,\mathrm{el}}$
---
$\varepsilon_{\mathrm{p}}$
---
$\varepsilon_{0,\mathrm{el}}$
---
$\varepsilon_{\mathrm{c}}$
---
$\sigma_{\mathrm{c}}$
---
loading
---
Figure 3: (a) Formation of residual stresses $\sigma_{0}$ through plastic
deformation. Remaining strain after unloading is
$\varepsilon_{0}=\varepsilon_{\mathrm{p}}+\varepsilon_{0,\mathrm{el}}$. (b)
Material law for cyclic simulation. Initial state
$(\sigma_{0},\varepsilon_{0,\mathrm{el}})$ undergoes cyclic load with
$(\sigma_{\mathrm{c}},\varepsilon_{\mathrm{c}})$. Crack driving force
$\mathcal{H}$ is strain energy density of total stress-strain state
$(\sigma,\varepsilon)$. Schemes of 1D, undamaged ($d=0$) case.

Residual stresses result from plastic deformations which occur during the
production process, e. g. due to forming, tempering or surface treatment. The
stress remaining after unloading is the residual stress
$\boldsymbol{\sigma}_{0}$. The associated strain

$\boldsymbol{\varepsilon}_{0}=\boldsymbol{\varepsilon}_{0,\mathrm{el}}+\boldsymbol{\varepsilon}_{\mathrm{p}}$
(13)

consists of an elastic part $\boldsymbol{\varepsilon}_{0,\mathrm{el}}$ and a
plastic part $\boldsymbol{\varepsilon}_{\mathrm{p}}$, see Fig. 3(a). While the
total residual strain $\boldsymbol{\varepsilon}_{0}$ is geometrically
compatible, this does not apply to its components
$\boldsymbol{\varepsilon}_{\mathrm{p}}$ and
$\boldsymbol{\varepsilon}_{0,\mathrm{el}}$.

Only the elastic part $\boldsymbol{\varepsilon}_{0,\mathrm{el}}$ of the
residual strain $\boldsymbol{\varepsilon}_{0}$ is relevant for the fatigue
life simulation. The plastic forming process is treated as completed and is
not modelled. The plastic part $\boldsymbol{\varepsilon}_{\mathrm{p}}$ is not
of further interest in the fatigue crack simulation, because it is assumed
that the yielding process does not change the crack resistance properties of
the material zerbst2016fatigue . All material points are assigned the same
material parameters initially, regardless of their (plastic) history.
Therefore the total stress-strain state
$(\boldsymbol{\sigma},\boldsymbol{\varepsilon})$ in the model is the sum of
the initial state
$(\boldsymbol{\sigma}_{0},\boldsymbol{\varepsilon}_{0,\mathrm{el}})$ and the
stress-strain state caused by the cyclic loading
$(\boldsymbol{\sigma}_{\mathrm{c}},\boldsymbol{\varepsilon}_{\mathrm{c}})$.
Hence, the total strain is

$\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{0,\mathrm{e}}+\boldsymbol{\varepsilon}_{\mathrm{c}}$
(14)

as displayed in Fig. 3(b). Thereby, the strain
$\boldsymbol{\varepsilon}_{\mathrm{c}}$ is

$\boldsymbol{\varepsilon}_{\mathrm{c}}=\frac{1}{2}\left(\nabla\boldsymbol{u}+(\nabla\boldsymbol{u})^{\top}\right).$
(15)

The regularised energy functional is, analogously to Eq. (7),

$\displaystyle\begin{split}\Pi_{\ell}=&\int_{\Omega}g(d)\,\psi^{\mathrm{e}}(\boldsymbol{\varepsilon}_{0,\mathrm{e}}+\boldsymbol{\varepsilon}_{\mathrm{c}})\,\mathrm{d}V+\\\
&+\int_{\Omega}\alpha(D)\,\mathcal{G}_{\mathrm{c}}\frac{1}{2\ell}(d^{2}+\ell^{2}|\nabla
d|^{2})\,\mathrm{d}V.\end{split}$ (16)

Consequentially, the stress is

$\boldsymbol{\sigma}=g(d)\left(\boldsymbol{\sigma}_{0}+\boldsymbol{\sigma}_{\mathrm{c}}\right)=g(d)\left(\boldsymbol{\sigma}_{0}+\mathbb{C}:\boldsymbol{\varepsilon}_{\mathrm{c}}\right).$
(17)

and the evolution equation remains

$\mathcal{G}_{\mathrm{c}}\left(\alpha\,d-\nabla\alpha\cdot\ell^{2}\nabla
d-\alpha\,\ell^{2}\Delta d\right)=(1-d)\,\mathcal{H}\,2\ell.$ (18)

The crack driving force is the temporal maximum of the strain energy density
of the total stress strain state

$\displaystyle\mathcal{H}(t)$
$\displaystyle=\max_{s\in[t_{0},t]}\left(\psi^{\mathrm{e}}(\boldsymbol{\varepsilon}_{0,\mathrm{el}}+\boldsymbol{\varepsilon}_{\mathrm{c}}(s))\right)$
(19)
$\displaystyle=\max_{s\in[t_{0},t]}\left(\frac{1}{2}(\boldsymbol{\varepsilon}_{0,\mathrm{el}}+\boldsymbol{\varepsilon}_{\mathrm{c}}(s)):\mathbb{C}:(\boldsymbol{\varepsilon}_{0,\mathrm{el}}+\boldsymbol{\varepsilon}_{\mathrm{c}}(s))\right).$
(20)

The initial state at the time $t_{0}$ is hereby
$\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{0,\mathrm{el}}$,
$\boldsymbol{\sigma}=g(d)\,\boldsymbol{\sigma}_{0}$.

The superposition of residual stress state and the stress state caused by
external loading is actually also common in fracture-mechanical fatigue
computations based on stress intensity factors larue_predicting_2007 . Note
that since $\boldsymbol{\varepsilon}_{0,\mathrm{el}}$ is not necessarily
geometrically compatible, the total strain $\boldsymbol{\varepsilon}$ is not,
either.

$\varepsilon_{0,\mathrm{el}}$

$\sigma_{0}$
---
$\sigma_{\mathrm{m}}^{*}$
---
$\sigma_{\mathrm{m}}^{*}$
---
$\sigma_{0}$
---
$\sigma_{\mathrm{m}}^{*}$
---

$\varepsilon_{0,\mathrm{el}}$

$(\sigma,\varepsilon)$
---
$(\sigma^{*},\varepsilon^{*})$
---
$(\sigma,\varepsilon)$
---
$(\sigma^{*},\varepsilon^{*})$
---
$(\sigma,\varepsilon)$
---
$(\sigma^{*},\varepsilon^{*})$
---

$P_{\mathrm{SWT}}$

$P_{\mathrm{SWT}}$

$P_{\mathrm{SWT}}$

residual stress
---
residual stress
---
residual stress
---
No
---
Tensile
---
Compressive
---
Figure 4: Comparison of virtual stress-strain path with and without residual
stress $\sigma_{0}$. Constitutive stress-strain behaviour ( ) for 1D,
undamaged ($d=0$) case. For the computation of the lifetime variable $D$, the
virtual stress-strain path ( ) is used, which is determined from the CSSC ( ).
Mainly, the residual stress shifts the virtual mean stress
$\sigma_{\mathrm{m}}^{*}$, which controls the damage parameter
$P_{\mathrm{SWT}}$ (schematically).

It is assumed that the plastic strains do not enter the crack driving force.
Instead the crack is driven by the elastic strain energy density, which again
only depends on the total elastic strain. This assumption is appropriate for
typical HCF and higher LCF loads which do not exceed the static yield limit.

Fig. 4 depicts how the residual stresses affect the LSA procedure: Due to the
initial state, the Neuber rule yields a shifted stress-strain path. Although
the stress and strain amplitude stay the same, the damage parameter
$P_{\mathrm{SWT}}$ is affected by the altered mean stress
$\sigma_{\mathrm{m}}^{*}$ according to Eq. 11. In this way, tensile residual
stresses increase the damage parameter $P_{\mathrm{SWT}}$ while compressive
residual lead to a decrease.

In summary, residual stresses influence the crack development in two ways:

  1. 1.

They change the peak stress-strain state of a load cycle which is decisive for
the crack development in a cyclic load. In this way, tensile residual stresses
increase, compressive residual stresses decrease the crack driving force
$\mathcal{H}$ which depends on the strain energy density
$\psi^{\mathrm{e}}(\boldsymbol{\varepsilon})$.

  2. 2.

They shift the virtual stress-strain path and therefore influence the damaging
effect and with that the lifetime variable $D$. Compressive residual stresses
reduce $D$, while tensile residual stresses increase it.

The initial residual stress tensor $\boldsymbol{\sigma}_{0}$ remains unchanged
throughout the simulation. Residual stress redistribution due to wide-ranging
plasticising is therefore not covered in the model, since this is only
relevant for very low cycle fatigue with macroscopic plastic deformations
benedetti_numerical_2010 . However, as cracks propagate, the stress state is
rearranged due to degradation of the total stress (Eq. 17) which also affects
the residual stresses. In this way, the residual stress state redistributes
compared to the initial state $\boldsymbol{\sigma}_{0}$ due to FCG.

##  3 Experiments

The fatigue crack growth influenced by residual stresses is investigated with
compact tension (C(T)) specimens, where significant residual stresses are
introduced by LSP. The material under investigation is the aluminium alloy
AA2024 in T3 heat treatment condition with 2 mm and 4.8 mm thickness, which is
a representative aluminium alloy used in the aircraft industry for fuselage
structures dursun2014recent . The previous investigation
keller_experimentally_2019  indicates that LSP allows the introduction of
relatively high and deep residual stresses, where the microstructural changes
in AA2024 does not influence the FCG behaviour significantly. Thus,
differences in the FCG rates between untreated and laser peened material are
mainly linked to the effect of residual stresses. In the following the LSP
treatment, the experimental residual stress determination and the
determination of the fatigue crack growth rate are described. The experimental
data for specimens with thickness of 4.8 mm are taken from kallien2019effect
and keller_experimentally_2019 .

###  3.1 Laser shock peening

LSP uses short-time high-energy laser pulses to generate plasma consisting of
near-surface material, see Fig. 5. The extension of the plasma generates
mechanical shock waves, which cause local plastic deformation of the
subsurface material, see Fig. 5(b). After relaxation of the high dynamic
process, these local plastic deformation lead to a residual stress
distribution, where compressive residual stresses remain in the subsurface
region surrounded by balancing tensile residual stresses, Fig. 5(c). The
penetration depth of these compressive residual stresses is in millimeter
range. The efficiency of the process can be increased by the use of a
confinement layer. A laminar water layer is used during the LSP process in
this study as confinement medium. The LSP treatment is conducted with an
Nd:YAG laser. 5 J laser pulses with the duration of 20 ns (full width at half
maximum) and a $3\,\mathrm{mm}\times 3\,\mathrm{mm}$ square focus are used.
The LSP treatment is performed without pulse overlap in five columns on the
sheet material, as shown in Fig. 6(a). The laser pulse sequence of a
rectangular peening patch with $15\,\mathrm{mm}\times 80\,\mathrm{mm}$ is
applied twice at both sides of the sheet material. The sequence is shot at the
first side twice before the second side was treated twice as well.

Figure 5: LSP laser system containing the pulsed Nd:YAG laser and clamping
system fixing the specimen. Laser pulses are used to vaporize near-surface
material to initiate mechanical shock waves (b). These shock waves cause local
plastic deformations, which lead to a residual stress distribution after the
process (c). A water layer increases the efficiency of the process by
confining the plasma.  Figure 6: Schematic of the specimens used to evaluate
the LSP-induced residual stress state (a). 5 J laser pulses with a
$3\,\mathrm{mm}\times 3\,\mathrm{mm}$ square laser focus are used to treat a
region with size $15\,\mathrm{mm}\times 80\,\mathrm{mm}$. The laser pulse
sequence consists of five columns in which the laser pulses are shot without
overlap. The advancing direction of each column is kept constant. The sequence
is applied twice at both sides of the sheet material. Residual stresses were
experimentally determined by incremental hole drilling (b), where the same
depth profile of residual stresses is assumed below the peening patch.

###  3.2 Experimental residual stress determination

The incremental hole drilling system PRISM from Stresstech is used to
determine the depth profile of the residual stresses. The hole drilling system
uses electronic speckle pattern interferometry to determine material surface
deformation after each increment of an incrementally drilled hole. These
surface deformations are correlated with the residual stress at the respective
increment depth via the integral method schajer2005full . The interested
reader is referred to ponslet2003residual ; ponslet2003residual_PartIII ;
steinzig2003residual  for a detailed explanation of the incremental hole
drilling method using electronic speckle pattern interferometry. A driller
with 2 mm diameter is used to determine the residual stresses up to the depth
of 1 mm. This hole depth allows for the experimental determination of the
through thickness residual stress profile within the specimens with 2 mm
thickness, when the residual stresses are determined from both material sides.
As it is recommended that the material thickness is four times larger than the
hole diameter ponslet2003residual_PartIII , the residual stress determinations
of AA2024 with 2 mm thickness where repeated with a hole diameter of 0.5 mm as
well. The determined residual stresses with the hole diameter of 0.5 mm and 2
mm match. Therefore, we focus on the determined residual stresses with the
hole diameter of 2 mm in the following. A relative small increment size is
used near the material surface, where relatively large residual stress
gradients are expected. Residual stresses were determined within the area of
the peening patch, see Fig. 6(b). The depth profile of residual stresses is
assumed to be the same below the whole peening patch. Thus, the average value
and the standard deviation of at least eight experimentally determined
residual stress profiles for both material sides are depicted in the
following.

###  3.3 Fatigue crack growth

C(T) specimens with a width of 100 mm are used to determine the FCG rate
according to ASTM E647 standard. The FCG tests were performed with the servo
hydraulically testing machine from Schenk/Instron and a 25 kN load cell. The
specimen geometry is displayed in Fig. 7. A pre-crack of 5 mm is introduced
extending the initial crack length to 25 mm (20 mm notch and 5 mm pre-crack).
Afterwards, the peening patch, as described in Sec. 3.1, is applied 10 mm in
front of the initial crack front to the LSP treated specimens. The applied
load ratio $R=0.1$ is kept constant during the FCG test. The maximum applied
force of the fatigue load load cycles is 1.65 kN and 4.0 kN for specimens with
2 mm and 4.8 mm thickness, respectively. All FCG experiments are repeated at
least twice.

Figure 7: C(T) specimen with 100 mm width according to ASTM-E647 standard. The
specimen is pre-cracked by 5 mm and subsequently peened with 10 mm distance to
the crack front of the pre-crack. C(T) specimens are treated twice from both
surfaces of the sheet material. The introduction of compressive residual
stresses below the peening patch lead to balancing tensile stresses in the
surrounding material.

###  3.4 Experimental results

#### Residual stresses

The LSP treatment leads to the introduction of significant compressive
residual stresses over the thickness for the investigated AA2024, see Fig. 8
for 2 mm thickness and Fig. 9 for 4.8 mm thickness, respectively.

Since the residual stresses are only determined up to a depth of 1 mm from
each side with the incremental hole drilling technique, numerical simulations
via a LSP process simulation, as described elsewhere keller_crack_2019 ;
keller_experimentally_2019  are used to estimate the residual stress profile
along the entire material cross-section in $z$ direction.

The residual stress components $\sigma_{xx}$ and $\sigma_{yy}$ in surface
plane direction differ, whereby the magnitude of the component perpendicular
to the crack growth direction, $\sigma_{yy}$, is more pronounced. This
difference of the residual stress components might be attributed to
geometrical effects, such that the rectangular peening patch geometry, as
experiments with a square peening patch do not indicate this significant
difference of $\sigma_{xx}$ and $\sigma_{yy}$ in aluminium alloy AA2024
keller_experimentally_2019 .

The residual stress magnitude and gradient differ significantly depending on
the material thickness. The LSP-treated aluminium alloy with 2 mm thickness
contains a lower maximum compressive residual stress of approximately 160 MPa
compared to the compressive maximum of approximately 280 MPa in the 4.8 mm
thick material. While tensile residual stresses occur at mid-thickness for the
thicker material, the residual stress component $\sigma_{yy}$ is completely
compressive along the $z$ direction for the 2 mm thick material.

The resulting residual stress field depends on the order of the applied pulse
sequences on the two sides. These differences are more pronounced for the
thinner material. The non-symmetric residual stress profile is assumed to
result from the interaction of mechanical shock waves initiated at the
secondly peened side at $z=2\,\mathrm{mm}$ and already existing residual
stresses introduced by the LSP treatment of the first side at
$z=0\,\mathrm{mm}$. These interactions result in increased residual stresses
between 0.4 – 0.9 mm depth.

It has to be noted, that the peening patch surrounding material in $x$-$y$
direction contains balancing tensile residual stresses. A detailed analysis
about the overall residual stress field of the C(T) specimen after the LSP
treatment in AA2024 for 4.8 mm thickness can be found in
keller_experimentally_2019 .

$\sigma_{\mathrm{BM},xx}$
---
$\sigma_{\mathrm{BM},yy}$
---
0.0
---
0.5
---
1.0
---
1.5
---
2.0
---
Depth / mm
---
-200
---
-100
---
0
---
100
---

Residual stress / MPa
---

$\sigma_{\mathrm{LSP},xx}$
---
$\sigma_{\mathrm{LSP},yy}$
---
Figure 8: Residual stress profile of the base material (BM) and after LSP in
AA2024 with 2 mm thickness. The average value and standard deviation are
depicted. At least seven depth profiles were experimentally investigated. At
first, LSP was applied at 0 mm and secondly from the other side of the
specimen at 2 mm. The LSP treatment leads to compressive residual stresses
along the entire material thickness. 0
---
1
---
2
---
3
---
4
---
5
---
Depth / mm
---
-300
---
-200
---
-100
---
0
---
100
---

Residual stress / MPa
---

$\sigma_{\mathrm{LSP},xx}$ (Sim.)
---
$\sigma_{\mathrm{LSP},yy}$ (Sim.)
---
$\sigma_{\mathrm{BM},xx}$
---
$\sigma_{\mathrm{BM},yy}$
---
$\sigma_{\mathrm{LSP},xx}$
---
$\sigma_{\mathrm{LSP},yy}$
---
Figure 9: Experimentally determined and numerically calculated (Sim.) residual
stresses after LSP treatment in AA2024 with 4.8 mm thickness. The firstly
treated side is at 0 mm. The numerically determined residual stress profile
shows tensile residual stresses at mid-thickness (1.8-3.0 mm). The residual
stress profiles are taken from kallien2019effect  (BM) and
keller_experimentally_2019  (LSP).

#### Fatigue crack growth

Experimentally determined FCG rates of the untreated material show the typical
exponential correlation between FCG rate $\mathrm{d}a/\mathrm{d}N$ and stress
intensity factor range $\Delta K$ known for the Paris regime, see Fig. 10.
This characteristic FCG behaviour is significantly affected by the introduced
residual stresses for both investigated material thicknesses. For the LSP-
treated samples, the FCG rate increases between the initial crack front and
the peening patch. This increased FCG rate is attributed to balancing tensile
residual stresses, as indicated in Fig. 7. Thereafter, the FCG rate decreases
up to a minimum at $a\approx 49\,\mathrm{mm}$, when the crack front is located
within the area of the peening patch. After the crack front has passed the
peening patch, the FCG rate accelerates, but stays below the FCG rate of
specimens without LSP treatment. This characteristic FCG behaviour is observed
for both material thicknesses.

The observation of the increased FCG rate highlights the importance of the
overall residual stress field for an efficient application of residual stress
modification techniques, such as LSP. Furthermore, tools for FCG rate
calculation need to contain the prediction of this possible increase of the
FCG rate as well. While the material thickness of 2 mm allows the experimental
determination of residual stresses over entire material thickness, the
relatively thin material may lead to buckling during the fatigue testing at
larger crack length. These buckling phenomena are indicated from
$a>50\,\mathrm{mm}$ and may cause the increased scatter of the experimentally
determined FCG rate for the material thickness 2 mm.

20
---
30
---
40
---
50
---
60
---
$a/mm$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
BM
---
LSP
---
20
---
30
---
40
---
50
---
60
---
$a/mm$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---

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

BM
---
LSP
---

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

2 mm
---
4.8 mm
---
peened area
---
peened area
---
Figure 10: FCG rate in AA2024 with 2 mm and 4.8 mm thickness in BM and LSP-
treated material. The introduced residual stress field leads to an accelerated
FCG in front of the peened area and a FCG retardation when the crack front is
located within the peening patch. The different curves, represent the repeated
experiments. The data for the 4.8 mm specimen are taken from
keller_experimentally_2019

##  4 Simulation of fatigue crack growth

In the following, the phase-field model described in Sec. 2 is used to
simulate the fatigue crack growth experiments described in Sec. 3.3. Starting
with an unpeened specimen, model parameters are studied and the model is
calibrated to one fatigue crack growth curve. With the calibrated model
parameters, the other unpeened and peened specimens are simulated.

For all simulations, a staggered solution scheme hofacker_continuum_2012  is
applied to solve the coupled problem with mechanical field and phase-field. A
structured, locally refined mesh with a minimum mesh size of $h_{\min}=0.33$
mm is used. Due to the thin specimens, a plane stress state is assumed. The
plane stress assumption is supported by the experimental determination of the
residual stress component perpendicular to the material surface
($\sigma_{zz}\approx 0\,\mathrm{mm}$) after LSP application via synchrotron
radiation in keller2018experimental . As comparative computations showed, a
tension-compression split, see miehe_phase_2010 , is not necessary for the
simulations due to the simple stress state of the specimen. The characteristic
length is set to $\ell=1$ mm as a compromise between mesh refinement and
accuracy. The elastic, cyclic and fracture-mechanical parameters for the
AA2024-T3 material taken from literature are specified in Tab. 1.

The specimen are loaded by cyclic loading with force ratio
$\tilde{F}_{\mathrm{min}}/\tilde{F}_{\mathrm{max}}=0.1$. The force boundary
condition was kept to $\tilde{F}_{\mathrm{max}}$ throughout the simulation.
This is possible due to the model formulation with the LSA. The damage
parameter $P_{\mathrm{SWT}}$ only needs amplitude and mean stress values as an
input, see Sec. 2.2. Therefore, a simulation of the full loading path is not
necessary for the damage calculation. Due to the constant amplitude loading,
several load cycles $\Delta N$ can be simulated within one increment. $\Delta
N$ is reduced adaptively depending on the number of Newton iterations required
to in the staggered loop, starting with $\Delta N=3000$.

Table 1: Material parameters of aluminium AA2024-T3 used in phase-field simulations. Elastic constants boller_materials_1987  |  $E=74.6$ GPa | $\nu=0.33$
---|---|---
CSSC boller_materials_1987  |  $K^{\prime}=0.453$ GPa | $n^{\prime}=0.201$
SWC boller_materials_1987  |  $\sigma^{\prime}_{\mathrm{f}}=0.314$ GPa | $\varepsilon^{\prime}_{\mathrm{f}}=0.162$
| $b=-0.091$ | $c=-0.452$
Fracture toughness kaufman_fracture_2001  | $\mathcal{G}_{\mathrm{c}}=0.165\,\mathrm{MPa\,m}$ |

###  4.1 Unpeened specimens

#### Model parameters

$10^{1}$
---
$\Delta K$/GPa$\sqrt{\mathrm{mm}}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
$10^{1}$
---
$\Delta K$/GPa$\sqrt{\mathrm{mm}}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
$\xi=500$
---
$\xi=1000$
---
$\xi=1500$
---
$\alpha_{0}=0.0002$
---
$\alpha_{0}=0.001$
---
$\alpha_{0}=0.002$
---

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

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

Figure 11: FCG rate for range of stress intensity factor $\Delta K$ (Paris
curves). Study of the exponent $\xi$ and the threshold $\alpha_{0}$ of the
fatigue degradation function.

The parameters of the fatigue degradation function (9), $\alpha_{0}$ and
$\xi$, are the only model parameters that have to be calibrated apart from the
characteristic length $\ell$. All other parameters – listed in Tab. 1 – are
drawn from standardised experiments. The influence of the fatigue degradation
function is studied on the 4.8 mm thick specimen loaded with
$\tilde{F}_{\mathrm{max}}=4\,\mathrm{kN}$. Fig. 11 shows the results as a
Paris plot, i. e. the crack growth rate over the range of the stress intensity
factor. The variation of the threshold value $\alpha_{0}$ while keeping
$\xi=1000$ shows its influence on the inclination and curvature of the Paris
curve. For $\alpha_{0}=0.002$, the graph is a straight line in the double
logarithmic plot, which is typical for most crack growth experiments. Varying
the exponent $\xi$ while keeping $\alpha_{0}=0.002$ shifts the Paris curve in
vertical direction. For a study of the influence of the model parameters on
crack initiation, the reader is referred to seiler_efficient_2020 .

#### Model calibration

1
---
0
---
$\alpha$
---
$d$
---
$10^{1}$
---
$\Delta K/$GPa$\sqrt{\mathrm{mm}}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
Simulation
---
Experiments
---
a)
---
b)
---
c)
---

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

Figure 12: Fatigue crack growth of a 2 mm thick specimen loaded with
$\tilde{F}_{\mathrm{max}}=1.65\,\mathrm{kN}$. Model parameters are fitted to
experimental results yielding $\alpha_{0}=0.0015$ and $\xi=500$. a) Fatigue
degradation $\alpha$ and b) phase-field variable $d$ after $\approx 251\,500$
load cycles. c) Paris curve.

The different effects of the two parameters allow for a convenient calibration
of the model. Here, a fatigue crack growth experiment with the 2 mm thick
specimen loaded with $\tilde{F}_{\mathrm{max}}=1.65\,\mathrm{kN}$, which was
repeated three times, is used for calibration. The fit of the Paris curve
yielded the model parameters $\alpha_{0}=0.0015$ and $\xi=500$ as displayed in
Fig. 12. The figure also shows the distribution of the fatigue degradation
$\alpha$ and the crack indicating phase-field variable $d$ after $251\,500$
load cycles.

#### Test loading case

The calibrated parameters are tested with a different a fatigue crack growth
experiment with a 4.8 mm thick specimen loaded with
$\tilde{F}_{\mathrm{max}}=4\,\mathrm{kN}$, also repeated three times. As
displayed in Fig. 13, the simulation meets the experiments quite well, yet
underestimates the crack growth rate slightly. This could be due to the fact
that the assumption of a plane stress state as it was used in the simulation
is less accurate for a thicker (albeit still thin) specimen.

$10^{1}$
---
$\Delta K/$GPa$\sqrt{\mathrm{mm}}$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
Simulation
---
Experiments
---

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

Figure 13: Fatigue crack growth in a 4.8 mm thick specimen loaded with
$\tilde{F}_{\mathrm{max}}=4\,\mathrm{kN}$. Test loading case with calibrated
model parameters $\alpha_{0}=0.0015$ and $\xi=500$.

###  4.2 Peened specimens

BM (Sim.)
---
LSP (Sim.)
---
BM (Exp.)
---
LSP (Exp.)
---
20
---
30
---
40
---
50
---
60
---
$a/mm$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
2 mm
---
20
---
30
---
40
---
50
---
60
---
$a/mm$
---
$10^{-5}$
---
$10^{-4}$
---
$10^{-3}$
---
$10^{-2}$
---
$10^{-1}$
---
4.8 mm
---
-110
---
200
---
2 mm
---
4.8 mm
---
$\sigma_{0,yy}/\text{MPa}$
---
a)
---
b)
---
c)
---
peened area
---
peened area
---

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

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

$x$
---
$y$
---
Figure 14: Fatigue crack growth in 2 mm and 4.8 mm thick peened and unpeened
specimen loaded with $\tilde{F}_{\mathrm{max}}=1.65\,\mathrm{kN}$ and
$\tilde{F}_{\mathrm{max}}=4\,\mathrm{kN}$. Model parameters
$\alpha_{0}=0.0015$ and $\xi=500$. a) Imposed residual stress component
$\sigma_{0,yy}$ from residual stress measurements after LSP. The component
$\sigma_{0,xx}$ is not shown here. b) and c) Crack growth rate.

Before the crack growth simulation, the initial residual stress state has to
be established. For this purpose, the experimentally determined residual
stresses are mapped to the used mesh. In this context, the integral mean of
the depth profile of the experimentally determined residual stresses
$\sigma_{0,xx}$ and $\sigma_{0,yy}$ is taken to fit the 2D plane stress
simulation. The integrated shear stress as well as the stress in thickness
direction is close to zero. With a preliminary, load-free simulation, an
equilibrium stress state is found which serves as the initial residual stress
$\boldsymbol{\sigma}_{0}$ in the actual simulation. For both, the 2 mm and the
4.8 mm thick specimen, the employed residual stress component $\sigma_{0,yy}$
is depicted in Fig. 14a) exemplarily.

Both peened specimens are now simulated with the parameters fitted in the
previous section. Please note that no additional parameters are modified for
the simulations including the residual stresses. The initial load cycle
increments are $\Delta N=300$ and $\Delta N=1000$, respectively. Fig. 14 shows
the FCG rates. The peened area is shaded. Within this area, there are
dominantly compressive residual stresses, while before and after it the
residual stresses are primarily in the tensile range. Both simulations
reproduce the effect of the peening very well qualitatively: The crack is
accelerated in front and after the peened area while within the peened area it
is inhibited.

The model overestimates the influence of the residual stresses. This is true
for both crack accelerating and inhibiting effects. One reason for the
quantitative gap between experiment and simulation is that crack closure is
not considered in the model. Another reason could be the simplification to a
2D stress state. The residual stresses introduced through LSP have a distinct
profile over the thickness which influence the FCG rate.

The oscillations at the end of the peened area result from the fact that the
very low FCG rates in this area lead to almost zero crack growth in some
increments. The jump at the end of the peened area presumeably stems from the
high residual stress gradient applied in the simulation due to the fact that
measurements of residual stresses are only pointwise.

##  5 Conclusion

This paper revisits a phase-field model for the computationally effective
simulation of fatigue cracks seiler_efficient_2020 . The model is calibrated
and validated with FCG experiments in aluminium metal sheets. It is able to
reproduce different FCG experiments fairly well. In the second part, residual
stresses are introduced into the metal sheets through LSP. A method for the
incorporation of residual stresses into the model is presented. The model is
able to reproduce the crack inhibiting effect of the compressive residual
stresses qualitatively.

Future works will now focus on low and very low cycle fatigue where the
elastic approximation is not valid anymore. Moreover the degradation of
residual stresses due to cyclic plasticity deserves closer attention. A 3D
simulation which considers the distinct crack closure effects over the
thickness of the specimen could also yield more realistic results.

###### Acknowledgements.

The group of M. Kästner thanks the German Research Foundation DFG which
supported this work within the Priority Programme 2013 ”Targeted Use of
Forming Induced Residual Stresses in Metal Components” with grant number KA
3309/7-1. The authors would like to thank M. Horstmann and H. Tek for the
specimen preparation and performing the fatigue tests.

Conflict of interest On behalf of all authors, the corresponding author states
that there is no conflict of interest.

This is a preprint of an article published in Archive of Applied Mechanics.
The final authenticated version is available online at:
https://doi.org/10.1007/s00419-021-01897-2.

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