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22.04.2020 | Original Article | Ausgabe 1/2021 Open Access

# Fatigue modeling for wrought magnesium structures with various fatigue parameters and the concept of highly strained volume

Zeitschrift:
Continuum Mechanics and Thermodynamics > Ausgabe 1/2021
Autoren:
A. Nischler, J. Denk, O. Huber
Wichtige Hinweise
Communicated by Andreas Öchsner.

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## 1 Introduction

Innovative materials for lightweight constructions, e.g., in the automotive industry, will become increasingly important in the future because they can contribute to reduce CO $$_2$$ emissions [ 18, 22]. Magnesium (Mg) alloys are counted among these lightweight materials due to their excellent strength-to-weight ratio compared to conventional steel and aluminum alloys [ 18]. This work presents an approach for the fatigue modeling of cyclically loaded structural components made of the wrought AZ31B Mg alloy, as this is an important aspect in the design of lightweight components.
During the twin-roll cast process of Mg alloys at higher temperature, slip occurs predominantly on the basal planes, which leads to their alignment parallel to the direction of metal flow [ 2]. Consequently, a strong basal texture develops during the production, which means that most of the basal planes $$\{0001\}$$ are approximately parallel to the sheet plane [ 21, 29]. At ambient temperature, tensile stress in the sheet plane direction mainly activates basal $$\left\langle a \right\rangle$$ slip, whereas compressive stress leads to extensive $$\{10{\bar{1}}2\}\left\langle 10{\bar{1}}1\right\rangle$$ tension twinning [ 2, 5]. As a result of the basal texture and the dominant twin formation under compressive stress in the sheet plane directions, clearly delimited macroscopic bands of twinned grains (BTGs) are formed [ 5, 8]. Within these BTGs, the compressive strain is considerably higher than outside the BTGs, leading to a discontinuous strain field [ 9]. In addition, [ 10] shows that common fatigue parameters such as strain amplitude, Smith–Watson–Topper or energy-based fatigue parameters, taken from inside BTGs, have higher values, which corresponds to the observation that the initial macroscopic crack is always within BTGs. Further challenges in fatigue modeling of basal textured wrought Mg alloys are anisotropic and asymmetric yield strengths, almost ideal plastic material behavior during twinning and sigmoidal shaped stress–strain hystereses [ 20, 27, 31].
In recent years, several works have been published on fatigue modeling, that deal with the complex deformation behavior of Mg alloys [ 5, 19, 23, 26, 27]. All these researches use a total strain energy density fatigue parameter, first introduced by Ellyin et al. [ 12]. Each work uses different approaches to determine the positive elastic strain energy density and achieves a good agreement between the predicted and experimental life for uniaxial unnotched specimens. Only a few investigations were carried out concerning the fatigue modeling of notched specimens [ 8, 30, 32], bending specimens [ 7, 15, 28] and the fatigue behavior under dominant multiaxial loading [ 1, 24]. The mentioned researches investigate the influence of stress concentrations, stress gradients and different types of loading on fatigue life separately from each other and achieve good results in fatigue modeling for a specific specimen type (uniaxial unnotched specimen, notched specimen, bending specimen, axial and torsional loaded tubular specimen). Furthermore, the discontinuous strain field (BTGs), important for fatigue modeling [ 10], has not been taken into account by the mentioned researches.
To consider all these aspects in one fatigue model, Denk et al. [ 10] introduced the concept of highly strained volume (CH $$\varepsilon$$V), inspired by the highly stressed volume from the early work of Kuguel [ 17]. They made in [ 10] fatigue tests on unnotched, notched and bending specimens and determined the size of the highly strained volume and the effective strain amplitude within the highly strained volume to predict the number of cycles to failure.
This work is an extension to [ 10], where further fatigue parameters such as the effective strain amplitude separated into an effective elastic and plastic strain amplitude, the Smith–Watson–Topper fatigue parameter and energy-based fatigue parameters are used. In addition, an extended stress–strain hysteresis model, based on Dallmeier et al. [ 4], is proposed that enables the computation of stress–strain hystereses of arbitrary load ratios required to calculate the mentioned fatigue parameters. Finally, the different fatigue parameters are evaluated and compared with each other.

## 2 Material and experimental procedure

The study for the fatigue tests was conducted with a commercial 3-mm-thick twin-roll cast AZ31B sheet with a strong basal texture. Detailed information on the microstructure of the tested material is given in [ 6]. Different types of unnotched uniaxial specimens, notched uniaxial specimens and unnotched bending specimens were tested with constant amplitude loading. Figure  1 and Table  1 show and list the geometrical dimensions of the different used specimens. Additionally, the stress concentration factors $$K_\mathrm {t}$$ of the notched uniaxial specimens are listed in Table  1. Further details on the specimen manufacturing can be found in [ 6].
Table 1
Dimensions for the unnotched uniaxial and unnotched bending specimens as well as the stress concentration factors $$K_\mathrm {t}$$ for the notched uniaxial specimens
Unnotched uniaxial specimens
$$l_1$$ (mm)
$$l_2$$ (mm)
$$l_3$$ (mm)
$$l_4$$ (mm)
$$R_1$$ (mm)
T
100
6
3
15
25
S
100
15
5
15
25
M
134
12
10
25
25
L
110
30
7
15
25
H
130
60
10
25
25
Notched uniaxial specimens
Stress concentration factors $$K_\mathrm {t}$$
K2
1.86
K3
3.14
K4
4.26
K5
5.48
Unnotched bending specimens
$$l_1$$ (mm)
$$l_2$$ (mm)
$$l_3$$ (mm)
$$l_4$$ (mm)
$$R_1$$ (mm)
B30
110
12
3
15
30
B45
110
12
4.5
15
30
B60
110
12
6
15
30
The fatigue data for the unnotched uniaxial specimens were gathered strain-controlled with extensometer strain ratios $$R_{\varepsilon ,\mathrm {ext}}=2,-\infty ,-1,{-0,5}$$. In contrast, the notched uniaxial specimens and unnotched bending specimens were exclusively stress-controlled tested with the nominal stress ratios $$R_{\sigma ,\mathrm {n}}=-1,{-0,5}$$ and $$R_{\sigma ,\mathrm {n}}=-1$$, respectively. An anti-buckling device is used to prevent the specimen from buckling when subjected to compressive stress. The in situ measured strain fields were obtained using the GOM ARAMIS™ 12M digital image correlation (DIC) photogrammetry system.

## 3 Proposed fatigue model

This section provides the theoretical approach for the concept of highly strained volume (CH $$\varepsilon$$V) in combination with two effective strain amplitude fatigue parameters, the Smith–Watson–Topper fatigue parameter, and two energy density fatigue parameters. A uniaxial stress–strain hysteresis model is proposed to compute the stress–strain data required to determine four of a total of five fatigue parameters.

### 3.1 The concept of highly strained volume

Figure  2a and b shows the components of the principal strain fields at the lower load level (LLL) $$\varepsilon ^*_{3}(x_{1},x_{2})|_{\mathrm {LLL}}$$ and at the upper load level (ULL) $$\varepsilon ^*_{1}(x_{1},x_{2})|_{\mathrm {ULL}}$$ at half of fatigue life $$N_{\mathrm {f}}/2$$ for an unnotched uniaxial specimen. Note that the principal strains are sorted according to the convention $$\varepsilon ^*_{1}\ge \varepsilon ^*_{2}\ge \varepsilon ^*_{3}$$ and that proportional cyclic loading is assumed. In the middle part of the specimen, the anti-buckling device covers the strain field.
Caused by the basal texture, the twin formation and the almost ideal plastic material behavior during twinning, a discontinuous strain field with macroscopic bands of twinned grains (BTG) is formed [ 9]. Figure  2a shows the resulting BTG in which the compressive strain is significantly higher (blue color) compared to the regions outside the BTG and the BTG is a clearly delimited subdomain. Its volume fraction, related to the gauge volume, is approximately 0.375. Figure  2b illustrates that even after a subsequent tensile load the BTG remains in place and shows still a compressive strain.
The investigations revealed that initial macroscopic cracks are always located within the BTG [ 9], as it can be seen in the image detail of Fig.  2b. It should be noted that the image detail in Fig.  2b is taken from this position, but at a later stage. The red stripe indicates a large measured tensile strain, which results from an opening crack. This finding can be explained with the help of Fig.  3, which shows stress–strain hystereses (SSH) obtained via DIC at certain strain increments from inside and outside the BTG. The normal strain $$\varepsilon _{22}=\frac{\partial u_2}{\partial x_2}$$ refer to the coordinate system of Fig.  2, and thus means the normal strain in the $$x_2$$ direction. Inside the BTG, the total strain amplitude of the SSH is 17% and the hysteresis area is 70% larger compared to the SSH outside the BTG. Hence all of the common fatigue parameters like the strain amplitude, the Smith–Watson–Topper fatigue parameter and an energy density-based fatigue parameter are higher within the BTG. Since the extensometer strain amplitude $$\varepsilon _{\mathrm {a,ext}}={0.35}{\%}$$ is small, the stress–strain hystereses in Fig.  3 do not have distinct sigomidal shapes.
In summary, the strongly discontinuous strain field and the higher fatigue parameter values within BTGs are the main motivation for using the CH $$\varepsilon$$V [ 10]. Figure  4 illustrates the separation of the highly strained region (HSR) from a general strain field $$\varepsilon _{ij}(x_{1},x_{2})|_{\mathrm {LLL}}$$ at the LLL. Starting with the components of a general strain tensor $$\varepsilon _{ij}(x_{1},x_{2})|_{\mathrm {LLL}}$$, the components of the principal strain field $$\varepsilon ^*_{ij}(x_{1},x_{2})$$ and the corresponding transformation matrix $$a^*_{ij}(x_{1},x_{2})|_{\mathrm {LLL}}$$ have to be calculated. Furthermore, the maximum $$\varepsilon ^*_{1,\mathrm {max}}|_{\mathrm {LLL}}$$ and the minimum value $$\varepsilon ^*_{3,\mathrm {min}}|_{\mathrm {LLL}}$$ of the entire principal strain field at the LLL $$\varepsilon ^*_{ij}(x_1,x_2)|_\mathrm {LLL}$$ are determined. Subsequently, the HSR is separated using a threshold strain
\begin{aligned} \varepsilon _{\mathrm {thr}}|_{\mathrm {LLL}} = {\left\{ \begin{array}{ll} 0.8\,\varepsilon ^*_{1,\mathrm {max}}|_{\mathrm {LLL}}, &{} \quad \mathrm {if}~ |\varepsilon ^*_{1,\mathrm {max}}|_{\mathrm {LLL}}|\geqslant |\varepsilon ^*_{3,\mathrm {min}}|_{\mathrm {LLL}}|\\ 0.8\,\varepsilon ^*_{3,\mathrm {min}}|_{\mathrm {LLL}}, &{}\quad \mathrm {if}~ |\varepsilon ^*_{3,\mathrm {min}}|_{\mathrm {LLL}}|>|\varepsilon ^*_{1,\mathrm {max}}|_{\mathrm {LLL}}|\, . \end{array}\right. } \, \end{aligned}
(1)
Best results for fatigue modeling are achieved by determining the HSR with the principal strain field $$\varepsilon ^*_{ij}(x_{1},x_{2})$$ and a threshold value of $${0.8}\,\varepsilon ^*_{1,\mathrm {max}}|_{\mathrm {LLL}}$$ or $${0.8}\,\varepsilon ^*_{3,\mathrm {min}}|_{\mathrm {LLL}}$$ [ 6]. Further evaluation procedures for the HSR, such as different threshold values, are described in [ 6]. All of the strain fields are evaluated at $$N_{\mathrm {f}}/2$$ since a stabilized material behavior is achieved [ 9]. Figure  3 shows the separated HSR that meets $$\varepsilon ^*_{ij}(x_{1},x_{2})|_{\mathrm {LLL}} \gtreqless \varepsilon _{\mathrm {thr}}|_{\mathrm {LLL}}$$, which predominantly originates from the BTG. For the remaining principal strain field at the upper load level (ULL) $$\varepsilon ^*_{ij}(x_{1},x_{2})|_{\mathrm {ULL}}$$, the same measuring facettes k are used as for the selected HSR.
A high compressive strain does not necessarily cause material failure. In contrast, the strain amplitude has a significant influence on the fatigue lifetime. Therefore, the effective strain amplitude tensors
\begin{aligned} {\tilde{\varepsilon }}^{\,*}_{ij,\mathrm {a}}(x_{1},x_{2}) = \frac{1}{2} \left( {\tilde{\varepsilon }}^{\,*}_{ij}(x_{1},x_{2})|_{\mathrm {ULL}}-{\tilde{\varepsilon }}^{\,*}_{ij}(x_{1},x_{2})|_{\mathrm {LLL}} \right) \end{aligned}
(2)
are calculated from the separated HSR, whereby effective quantities from the HSR are indicated with . To ensure that the coordinate systems of $${\tilde{\varepsilon }}^{\,*}_{ij}(x_{1},x_{2})|_{\mathrm {ULL}}$$ and $${\tilde{\varepsilon }}^{\,*}_{ij}(x_{1},x_{2})|_{\mathrm {LLL}}$$ have the same orientation, a transformation
\begin{aligned} {\tilde{\varepsilon }}^{\,*}_{ij}(x_{1},x_{2})|_{\mathrm {ULL}} = {\tilde{\varepsilon }}_{kl}(x_{1},x_{2})|_{\mathrm {ULL}} \, {\tilde{a}}^{\,*}_{ik}(x_{1},x_{2})|_{\mathrm {LLL}} \, {\tilde{a}}^{\,*}_{jl}(x_{1},x_{2})|_{\mathrm {LLL}} \end{aligned}
(3)
of the components of the effective strain tensor $${\tilde{\varepsilon }}_{kl}(x_{1},x_{2})|_{\mathrm {ULL}}$$ is performed and $${\tilde{\varepsilon }}^{\,*}_{ij}(x_{1},x_{2})|_{\mathrm {ULL}}$$ is used in Eq.  2. The components of $${\tilde{\varepsilon }}^{\,*}_{ij}(x_{1},x_{2})|_{\mathrm {ULL}}$$ are not sorted according to the convention $${\tilde{\varepsilon }}^{*}_{1}|_{\mathrm {ULL}}\ge {\tilde{\varepsilon }}^{*}_{2}|_{\mathrm {ULL}}\ge {\tilde{\varepsilon }}^{*}_{3}|_{\mathrm {ULL}}$$, otherwise the wrong components are subtracted in Eq.  2. The highly strained region with high strain amplitudes (HSRA) is separated by defining a threshold strain amplitude
\begin{aligned} {\tilde{\varepsilon }}^{\,*}_{\mathrm {a,thr}}=0.7\,{\tilde{\varepsilon }}^{\,*}_{\mathrm {a,max}} \,, \end{aligned}
(4)
with the maximum effective principal strain amplitude component
\begin{aligned} {\tilde{\varepsilon }}^{\,*}_{\mathrm {a,max}} = \mathrm {max} \left[ {\tilde{\varepsilon }}^{\,*}_{ij,\mathrm {a}} (x_1,x_2)\right] \end{aligned}
(5)
of the entire strain field. The definition of the threshold strain amplitude $${\tilde{\varepsilon }}^{\,*}_{\mathrm {a,thr}}$$ ensures that only those regions are selected for the HSRA that are not only twinned but also untwinned after load reversal. This was determined by analyzing strain fields and micrographs [ 6]. Additionally, good fatigue modeling results were obtained with a threshold strain amplitude $${\tilde{\varepsilon }}^{\,*}_{\mathrm {a,thr}}$$ of 70% of the maximum effective principal strain amplitude component $${\tilde{\varepsilon }}^{\,*}_{\mathrm {a,max}}$$. Figure  3 shows the HSRA remaining at the end of the calculation algorithm, which is further used to determine fatigue parameters, such as the highly strained volume $$V_{\varepsilon }$$.

### 3.2 Fatigue parameters

Highly strained volume $$V_{\varepsilon }$$The highly strained volume $$V_\varepsilon$$ is determined from the HSRA and the sheet thickness t. For this purpose, the highly strained area $$A_{\varepsilon }$$ is calculated, which is the sum of the areas $$A_k$$ for all measuring facettes k that are included in the HSRA. By multiplying the highly strained area $$A_{\varepsilon }$$ and the measured specimen thickness t, the highly strained volume $$V_\varepsilon$$ is calculated with the equation
\begin{aligned} V_\varepsilon = \sum _{k=1}^{n} A_k\,t\,. \end{aligned}
(6)
Its size depends on the current load and on the specimen geometry. A large highly strained volume $$V_\varepsilon$$ leads to smaller numbers of cycles to failure $$N_\mathrm {f}$$ and vice versa. By defining $$V_\varepsilon$$ as an absolute quantity, geometric size effects, stress gradients and stress concentrations are considered with the fatigue model CH $$\varepsilon$$V. For the unnotched specimens, the minimum and maximum measured highly strained volumes are $$V_{\varepsilon ,\mathrm {min}}={0.97}{\mathrm{mm}^{3}}$$ and $$V_{\varepsilon ,\mathrm {max}}={864}{\mathrm{mm}^{3}}$$, respectively. The notched specimens have values between $$V_{\varepsilon ,\mathrm {min}}={0.192}{\mathrm{mm}^{3}}$$ and $$V_{\varepsilon ,\mathrm {max}}={20.6}{\mathrm{mm}^{3}}$$ and the values for the bending specimens are between $$V_{\varepsilon ,\mathrm {min}}={0.579}{\mathrm{mm}^{3}}$$ and $$V_{\varepsilon ,\mathrm {max}}={9.3}{\mathrm{mm}^{3}}$$.
Fatigue parameters representing the load In the following, the used fatigue parameters in combination with the CH $$\varepsilon$$V are explained. The different measuring facettes k of the HSRA have different effective principal strain amplitudes $${\tilde{\varepsilon }}^{\,*}_{\mathrm {a},k}$$. Thus, a mean value is calculated yielding a unique scalar value for the strain amplitude within the HSRA. Accordingly, the maximum arithmetic mean effective principal strain amplitude is calculated by
\begin{aligned} \tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}} = \mathrm {max} \left[ \frac{1}{n}\sum _{k=1}^{n}{\tilde{\varepsilon }}^{\,*}_{ij,\mathrm {a},k} \right] \,. \end{aligned}
(7)
Quantities marked with represent mean effective measured values within the HSR or HSRA. In addition, the corresponding mean effective strain ratio is calculated by
\begin{aligned} \tilde{{\bar{R}}}_{\varepsilon } = \frac{ \sum _{k=1}^{n} {\tilde{\varepsilon }}^{\,*}_{k}(x_1,x_2)|_{\mathrm {LLL}}}{\sum _{k=1}^{n} {\tilde{\varepsilon }}^{\,*}_{k}(x_1,x_2)|_{\mathrm {ULL}} } \end{aligned}
(8)
which describes the actual effective strain ratio within the HSRA. In Eq. ( 8), the same component is used for the effective principal strains $${\tilde{\varepsilon }}^{\,*}_{k}(x_1,x_2)|_{\mathrm {LLL}}$$ and $${\tilde{\varepsilon }}^{\,*}_{k}(x_1,x_2)|_{\mathrm {ULL}}$$, which is derived from the calculation of the maximum arithmetic mean effective principal strain amplitude $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$ in Eq. ( 7). The mean effective principal strain amplitude $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$ in Eq. ( 7) is the only investigated fatigue parameter that does not require knowledge of the stress state.
Another fatigue parameter is the decomposed mean effective principal strain amplitude $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$, represented by an elastic $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}$$ and a plastic $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {\,pl}}_{\mathrm {a}}$$ portion which is calculated by
\begin{aligned} \tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}} = \tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {\,el}}_{\mathrm {a}} + \tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {\,pl}}_{\mathrm {a}} = \frac{{\tilde{\sigma }}^*_{\mathrm {a}}}{E} + \tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {\,pl}}_{\mathrm {a}} \end{aligned}
(9)
with the effective principal stress amplitude $${\tilde{\sigma }}^*_{\mathrm {a}}$$ and the Young’s modulus E. Another analyzed fatigue parameter is the Smith–Watson–Topper fatigue parameter $$P_{\mathrm {SWT}}$$ that takes into account the influence of the effective mean principal stress $${\tilde{\sigma }}^*_\mathrm {m}$$. It is defined by
\begin{aligned} P_{\mathrm {SWT}} = \sqrt{\left( {\tilde{\sigma }}^*_\mathrm {m}+{\tilde{\sigma }}^*_{\mathrm {a}}\right) \, \tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}} \, E}\, , \end{aligned}
(10)
wherein the mean effective principal strain amplitude $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$ is calculated with Eq. ( 7).
Additionally, the total strain energy density fatigue parameter, introduced in [ 13], is calculated with
\begin{aligned} \varDelta W^{\mathrm {t}} = \varDelta W^{\mathrm {+el}} + \varDelta W^{\mathrm {pl}}\, \end{aligned}
(11)
and used within this paper. The total strain energy density is defined as the positive elastic strain energy density $$\varDelta W^{+\mathrm {el}}$$ and the plastic strain energy density $$\varDelta W^{\mathrm {pl}}$$. Among the different calculation methods for the positive elastic strain energy density [ 5, 16, 19, 27], the equations
(12)
according to [ 5], are used here. In Eq.  12, $${\tilde{\sigma }}^*_{\mathrm {max}}$$ and $${\tilde{\sigma }}^*_{\mathrm {min}}$$ are the effective maximum and minimum principal stress at the upper and lower load level within the HSRA. A variant of the total strain energy density $$\varDelta W^\mathrm {t}$$ for AZ31B Mg alloy is the total strain energy density with a mean stress sensitivity factor $$f_{\mathrm {mss}}={0.25}$$ from [ 5], resulting in
\begin{aligned} \varDelta W^{\mathrm {t}}_{\mathrm {mss}} = f_{\mathrm {mss}}\, \varDelta W^{+\mathrm {el}} + \varDelta W^{\mathrm {pl}}\,, \end{aligned}
(13)
in which $$f_{\mathrm {mss}}$$ adjusts the influence of the mean stress.
It should be noted that for the fatigue parameter $$\tilde{{\bar{\varepsilon }}}^{\mathrm {\,*,el}}_{\mathrm {a}} + \tilde{{\bar{\varepsilon }}}^{\mathrm {\,*,pl}}_{\mathrm {a}}$$ in Eq. ( 9), $$P_{\mathrm {SWT}}$$ in Eq. ( 10), $$\varDelta W^{\mathrm {t}}$$ in Eq. ( 11) and $$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$ in Eq. ( 13) stress–strain hystereses must be determined and therefore the stress state must be known. It is demonstrated in [ 9] that the discontinuous strain field illustrated in Fig.  2 cannot be calculated with current phenomenological material models such as the Cazacu, Plunket and Barlat yield criterion CPB06 [ 3]. Hence, for that lack of an accurate 2D and 3D material model, a phenomenological uniaxial stress–strain hysteresis model was developed [ 4], which is explained in detail in the next section.

### 3.3 Extended stress–strain hysteresis model

This section describes the phenomenological uniaxial stress–strain hysteresis model used for calculating fatigue parameters. Dallmeier et al. published in [ 4] a basic stress–strain hysteresis model (SSM), which gives accurate results for a load ratio of $$R_{\varepsilon }=-1$$. In the following, the most important equations as well as an extension of the SSM for $$R_{\varepsilon }\ne -1$$ are shown.
Basic stress–strain hysteresis model Conventional functions for nonlinear stress–strain relationship, such as the Ramberg–Osgood equation, are improper due to the distinct asymmetry between the tensile and compressive yield strength and the sigmoidal shaped stress–strain curves, mainly caused by twinning and detwinning [ 20, 25, 31]. Therefore, the approach
\begin{aligned} \varDelta \varepsilon ^{\mathrm {t}}(\varDelta \sigma )=\varDelta \varepsilon ^{\mathrm {el}}(\varDelta \sigma )+\varDelta \varepsilon ^{\mathrm {psel}}(\varDelta \sigma )+\varDelta \varepsilon ^{\mathrm {pl}}(\varDelta \sigma ) \end{aligned}
(14)
describes the total strain increment $$\varDelta \varepsilon ^{\mathrm {t}}(\varDelta \sigma )$$, which consists of an elastic $$\varDelta \varepsilon ^{\mathrm {el}}(\varDelta \sigma )$$, a pseudoelastic $$\varDelta \varepsilon ^{\mathrm {psel}}(\varDelta \sigma )$$ and a plastic strain increment $$\varDelta \varepsilon ^{\mathrm {pl}}(\varDelta \sigma )$$ as illustrated in Fig.  5a and b. The measured mean effective strain values of the HSRA at the upper $$\tilde{{\bar{\varepsilon }}}|_{\mathrm {ULL}}$$ and lower load levels $$\tilde{{\bar{\varepsilon }}}|_{\mathrm {LLL}}$$ are used as input values for the hysteresis model. The elastic strain increment as a function of the stress increment $$\varDelta \sigma$$ is
\begin{aligned} \varDelta \varepsilon ^{\mathrm {el}}(\varDelta \sigma )=\frac{\varDelta \sigma }{E}\, . \end{aligned}
(15)
For the plastic deformation, caused by twinning and de-twinning, the equation
\begin{aligned} \varDelta \varepsilon ^{\mathrm {pl}}(\varDelta \sigma )=\left[ U(\varDelta \sigma )-U(\varDelta \sigma =0)\right] \,T\,m^{\mathrm {pl}} \end{aligned}
(16)
is given, where the coefficient T describes twice the amount of the plastic strain at the inflection point of $$\varDelta \varepsilon ^{\mathrm {pl}}(\varDelta \sigma )$$ (Fig.  5a) and the memory factor $$m^{\mathrm {pl}}$$ which ensure material memory [ 4]. The substituent $$U(\varDelta \sigma )$$ is
\begin{aligned} U(\varDelta \sigma )=\frac{1}{2}\left[ \tanh \left[ \frac{\varDelta \sigma -|\sigma _{\mathrm {rp}}|+a_{\mathrm {up/down}}\,\sigma _{\mathrm {tw}}}{S}\,a_{\mathrm {up/down}} \right] +1 \right] \end{aligned}
(17)
and gives the stress–strain curve its sigmoidal shape through the $$\tanh$$ function. In Eq. ( 17), $$\sigma _{\mathrm {rp}}$$ is the stress at the reversal point at the beginning of the current stress–strain branch, $$\sigma _{\mathrm {tw}}$$ is the stress and S the slope at the inflection point of $$\varDelta \varepsilon ^{\mathrm {pl}}(\varDelta \sigma )$$. For the factor $$a_{\mathrm {up,down}}$$, a distinction must be made between the following two cases
\begin{aligned} a_{\mathrm {up/down}} = {\left\{ \begin{array}{ll} \frac{1}{2}\left[ \tanh \left[ \frac{\varDelta \sigma -|\sigma _{\mathrm {rp}}|+\sigma _{\mathrm {tw}}}{S}\right] +1 \right] , &{} \mathrm {if}~\varDelta \varepsilon ^{\mathrm {t}} > 0 \\ 1 , &{} \mathrm {if}~\varDelta \varepsilon ^{\mathrm {t}} < 0\, , \end{array}\right. } \end{aligned}
(18)
in which $$\varDelta \varepsilon ^{\mathrm {t}} > 0$$ indicates an ascending stress–strain branch and $$\varDelta \varepsilon ^{\mathrm {t}} < 0$$ a descending stress–strain branch. The pseudoelastic portion is defined as
\begin{aligned} \varDelta \varepsilon ^{\mathrm {psel}}(\varDelta \sigma )=\ln \left[ \frac{\exp ^{\varDelta \sigma -\sigma _{\mathrm {P,up/down}}/50\,{\mathrm {MPa}}}+1}{\exp ^{-\sigma _{\mathrm {P,up/down}}/50\,{\mathrm {MPa}}}+1} \right] \,P\,m^{\mathrm {psel}} \end{aligned}
(19)
with the slope P of the pseudoelastic strain increment $$\varDelta \varepsilon ^{\mathrm {psel}}(\varDelta \sigma )$$, the pseudoelastic cutoff stress $$\sigma _{\mathrm {P,up/down}}$$ for ascending and descending stress–strain reversals and the memory factor $$m^{\mathrm {psel}}$$. The material parameters E, T, $$\sigma _{\mathrm {tw}}$$, S, P, $$\sigma _{\mathrm {P,up}}$$ and $$\sigma _{\mathrm {P,down}}$$ are illustrated in Fig.  5a. The used material parameter values within this work are listed in Table  2. The memory factors $$m^{\mathrm {pl}}$$ and $$m^{\mathrm {psel}}$$ ensure closed hysteresis loops and are determinable after the first hysteresis branch has been calculated. Thus, the additional parameter
\begin{aligned} R_{\mathrm {r}} = \frac{1-m^{\mathrm {psel}}}{1-m^{\mathrm {pl}}} \end{aligned}
(20)
applies for the first branch and defines how much strain is achieved by $$\varDelta \varepsilon ^{\mathrm {pl}}(\varDelta \sigma )$$ and $$\varDelta \varepsilon ^{\mathrm {psel}}(\varDelta \sigma )$$. The definition and the computation of the memory factors $$m^{\mathrm {pl}}$$ and $$m^{\mathrm {psel}}$$ as well as a detailed description of the basic SSM can be found in [ 4].
Table 2
Material parameters used for the phenomenological stress–strain hysteresis model for 3-mm-thick twin-roll cast AZ31B sheet metal
Material parameter
Value
Material parameter
Value
E
44200 MPa
P
0.0015
T
0.06
$$\sigma _{P,\mathrm{up}}$$
155 MPa
$$\sigma _{\mathrm {tw}}$$
$$-$$ 160 MPa
$$\sigma _{P,\mathrm{down}}$$
165 MPa
S
40 MPa
$$R_{\mathrm {r}}$$
0.3
Extended stress–strain hysteresis model for arbitrary load ratios As shown in Fig.  3, the effective strain ratio $${\tilde{R}}_{\varepsilon }$$ within the BTG differs significantly from the controlled extensometer strain ratio $$R_{\varepsilon ,\mathrm {ext}}$$. Consequently, an extension for the basic SSM is necessary to calculate hystereses for arbitrary load ratios. For this purpose, the cyclic stress–strain curves (CSSCs) for strain ratios $$R_{\varepsilon ,{\mathrm {ext}}}=2$$, $$\pm \infty$$ and 0 are determined experimentally with 3-mm-thick wrought AZ31B uniaxial specimens and included into the SSM additionally to the CSSC for the strain ratio $$R_{\varepsilon ,{\mathrm {ext}}}=-1$$. The CSSCs result from the maximum and minimum stress–strain values measured at the upper and lower load levels of the cyclically stable stress–strain hystereses at $$N_\mathrm{f}/2$$. Figure  6 illustrates the determined CSSCs for the different strain ratios, wherein the extensometer strain $$\varepsilon _{22,\mathrm {ext}}$$ is shifted by the mean strain $$\varepsilon _{\mathrm {m}}$$ in order to compare them more easily.
To minimize the experimental effort, only four strain ratios are determined. However, any strain ratio can occur in the BTG. Thus, the CSSCs are interpolated using a weighting coefficient $$W_{\mathrm {R}}$$ for arbitrary strain ratios. The value range of the weighting coefficient is $$0 \le W_{\mathrm {R}} \le 1$$, in which the weighting coefficient is 0 for the first adjacent cyclic stress–strain curve $$\mathrm {CSSC}_{1}$$ and 1 for the second adjacent cyclic stress–strain curve $$\mathrm {CSSC}_{2}$$. In between, it is determined for the different strain ratio ranges $$R_{\varepsilon }$$ as follows
As an example, if the strain ratio is $$R_\varepsilon ={3}$$, the value lies between the strain ratios $$R_\varepsilon ={2}$$ and $$R_\varepsilon =\pm \infty$$ for which the CSSCs are measured experimentally. The first adjacent cyclic stress–strain curve CSSC $$_1$$ would thus be the one with the strain ratio $$R_\varepsilon ={2}$$ and the second adjacent cyclic stress–strain curve CSSC $$_2$$ the one with the strain ratio $$R_\varepsilon =\pm \infty$$. In this case, the weighting coefficient $$W_{\mathrm {R}}$$ can be calculated with Eq. ( 21).
It should be mentioned that the mean effective strains at the upper and lower load levels $$\tilde{{\bar{\varepsilon }}}|_{\mathrm {ULL}}$$ and $$\tilde{{\bar{\varepsilon }}}|_{\mathrm {LLL}}$$, used in Eqs. ( 21), ( 22) and ( 23), are known from the experimentally measured strain fields. The associated maximum and minimum stresses $$\sigma _{\mathrm {max}}$$ and $$\sigma _{\mathrm {min}}$$ are then determined with the extended SSM. Accordingly, the maximum stress for the SSH can be calculated with
\begin{aligned} \sigma _{\mathrm {max}} = \left( 1-W_{\mathrm {R}}\right) \,\sigma _{\mathrm {max}}|_{\mathrm {CSSC}_1} + W_{\mathrm {R}}\,\sigma _{\mathrm {max}}|_{\mathrm {CSSC}_{2}} \end{aligned}
(24)
and the minimum stress for the SSH with
\begin{aligned} \sigma _{\mathrm {min}} = \left( 1-W_{\mathrm {R}}\right) \,\sigma _{\mathrm {min}}|_{\mathrm {CSSC}_1} + W_{\mathrm {R}}\,\sigma _{\mathrm {min}}|_{\mathrm {CSSC}_{2}}\, . \end{aligned}
(25)
Model verification To verify the extended SSM, stress–strain hystereses within the BTGs are experimentally determined for four uniaxial specimens and compared with the numerically calculated stress–strain hystereses, which is illustrated in Fig.  7. For the experiment, the strain field is measured with DIC and the mean effective normal strain $$\tilde{{\bar{\varepsilon }}}_{22}$$ within the HSR is determined. The normal stress $$\sigma _{22}$$ is calculated by dividing the measured force by the measured specimen cross-sectional area. According to Fig.  7a and b, the measured and numerically simulated stress–strain hystereses are in good agreement with each other. To evaluate the extended SSM quantitatively, the ratios between the experimentally measured and numerically calculated maximum stresses $$\sigma _{\mathrm {max}}$$, stress amplitudes $$\sigma _{\mathrm {a}}$$, elastic $$\varDelta W^{+\mathrm {el}}$$ and plastic strain energy densities $$\varDelta W^{\mathrm {pl}}$$ are calculated. These results, for different mean effective strain ratios $$\tilde{{\bar{R}}}_{\varepsilon }$$, are listed in Table  3.
The results show that the simulated maximum stresses and stress amplitudes have a maximum deviation of 7% from the experimental values. Furthermore, the ratios of the positive elastic strain energy density $$\varDelta W^{+\mathrm {el}}|_{\mathrm {exp}}/\varDelta W^{+\mathrm {el}}|_{\mathrm {sim}}$$ are acceptable with a maximum deviation of 15%. The largest deviations are obtained for the plastic strain energy density ratios $$\varDelta W^{\mathrm {pl}}|_{\mathrm {exp}}/\varDelta W^{\mathrm {pl}}|_{\mathrm {sim}}$$, with a maximum deviation of 28.9%.

## 4 Results and discussion

In this section, the CH $$\varepsilon$$V is applied to the uniaxial unnotched, uniaxial notched and the bending specimens. Additionally, the CH $$\varepsilon$$V is applied to each specimen type separately. For all shown results, the HSR and HSRA are evaluated with the criteria according to Eqs. ( 1) and Eq. 4). The fatigue parameters $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$, $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$, $$P_{\mathrm {SWT}}$$, $$\varDelta W^{\mathrm {t}}$$ and $$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$ are calculated with Eqs. ( 7), ( 9), ( 10), ( 11) and ( 13). Fatigue parameters, where the stress state is required, have been determined using the extended SSM.
Table 3
Verification of experimentally determined and simulated hystereses with the ratios of the experimentally and simulated maximum stresses $$\sigma _{\mathrm {max}}$$, stress amplitudes $$\sigma _{\mathrm {a}}$$, positive elastic $$\varDelta W^{+\mathrm {el}}$$ and plastic strain energy densities $$\varDelta W^{\mathrm {pl}}$$ for different mean effective strain ratios $$\tilde{{\bar{R}}}_{\varepsilon }$$
$$\tilde{{\bar{R}}}_{\varepsilon }$$ (-)
$$\tilde{{\bar{\varepsilon }}}$$ (%)
$$\frac{\sigma _{\mathrm {max}}|_{\mathrm {exp}}}{\sigma _{\mathrm {max}}|_{\mathrm {sim}}}$$ (-)
$$\frac{\sigma _{\mathrm {a}}|_{\mathrm {exp}}}{\sigma _{\mathrm {a}}|_{\mathrm {sim}}}$$ (-)
$$\frac{\varDelta W^{+\mathrm {el}}|_{\mathrm {exp}}}{\varDelta W^{+\mathrm {el}}|_{\mathrm {sim}}}$$ (-)
$$\frac{\varDelta W^{\mathrm {pl}}|_{\mathrm {exp}}}{\varDelta W^{\mathrm {pl}}|_{\mathrm {sim}}}$$ (-)
13
0.42
1.04
1.02
1.07
0.711
$$-$$ 1.3
0.49
1.07
1.01
1.15
0.872
$$-$$ 1.5
0.55
1.03
1.00
1.05
0.729
$$-$$ 0.82
0.66
0.97
0.95
0.95
1.03
Application of the concept of highly strained volume From the experiments, the numbers of cycles to failure $$N_\mathrm{f}$$, the highly strained volumes $$V_{\varepsilon }$$ and the various fatigue parameters $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$, $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$, $$P_{\mathrm {SWT}}$$, $$\varDelta W^{\mathrm {t}}$$ and $$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$ are plotted in three-dimensional graphs with logarithmic scale in Figs.  8910 and 11 . In addition to the 3D $$V_{\varepsilon }$$- $$N_{\mathrm {f}}$$ fatigue diagrams, Figs.  8910 and 11 show 2D projection views of the fatigue parameters vs. number of cycles to failure $$N_{\mathrm {f}}$$ for each fatigue parameter. The different specimens (unnotched, notched, bending specimen) are assigned by different markers. The experimental data is approximated with the double power functions
\begin{aligned}{}[\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}},~P_{\mathrm {SWT}},~\varDelta W^{\mathrm {t}},~\varDelta W^{\mathrm {t}}_{\mathrm {mss}}] (N_{\mathrm {f}},~V_{\varepsilon }) = C_1\,N^{d_1}_{\mathrm {f}}\,\left( \frac{V_{\varepsilon }}{\mathrm{mm}^{3}}\right) ^{d_2} \end{aligned}
(26)
and
\begin{aligned}{}[\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}] (N_{\mathrm {f}},~V_{\varepsilon }) = C_1\,N^{d_1}_{\mathrm {f}}\,\left( \frac{V_{\varepsilon }}{\mathrm{mm}^{3}}\right) ^{d_2} + C_2\,N^{d_3}_{\mathrm {f}}\,\left( \frac{V_{\varepsilon }}{\mathrm{mm}^{3}}\right) ^{d_4}\, , \end{aligned}
(27)
wherein the coefficients $$C_1,~C_2$$ and the exponents $$d_1,...,d_4$$ denote material parameters which are used to fit the equations. It should be noted that for the different fatigue parameters the regression coefficients and exponents have different values and units. Finally, the regressions of Eqs. ( 26) and ( 27) lead to regression planes for the fatigue parameters $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$, $$P_{\mathrm {SWT}}$$, $$\varDelta W^{\mathrm {t}}$$ and $$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$ and to a regression surface for the fatigue parameter $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$, plotted with green color in Figs.  8910 and 11. All material parameters for the different fatigue parameters are listed in Table  4. For the nonlinear regression, MATLAB ® and its iterative reweighted least-squares algorithm [ 11, 14] is used to determine the material coefficients and exponents.
Verification of the concept of highly strained volume for different fatigue parameters To verify how accurate the nonlinear regression represents the experimental data for the individual fatigue parameters, the coefficient of determination $$r^2$$ is calculated. Table  5 lists the coefficient of determination $$r^2$$ for the regression surfaces of the different fatigue parameters shown in Figs.  89 and 10.
The best regression is obtained for the fatigue parameter $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$ (Fig.  9) with a coefficient of determination of $$r^2={0.86}$$ corresponding to a high value, proofing good accuracy. Moreover, the fatigue parameters $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$ and $$P_{\mathrm {SWT}}$$, plotted in Fig.  8a and b, achieves good accuracy for regression. Table  3 shows that the maximum stress $$\sigma _{\mathrm {max}}$$ computed with the extended SSM provides accurate results, which is the basis for the good correlation between the regression and the experimental data of both fatigue parameters. The fatigue parameters $$\varDelta W^{\mathrm {t}}$$ and $$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$ reveal with $$r^2={0.82}$$ (Table  5) the largest discrepancy. As discussed in Sect.  3.3, the computation of the plastic strain energy density $$\varDelta W^{\mathrm {pl}}$$ results in larger deviations, leading to smaller $$r^2$$ values. The use of strain energy density fatigue parameters for fatigue modeling thus requires a more accurate computation of stress–strain hystereses. A further reason for the deviation could be the calculation approach of the positive elastic strain energy density $$\varDelta W^{+\mathrm {el}}$$, which considers the mean stress. Roostaei et al. states in [ 27] that wrought Mg alloys show positive mean stress even during fully reversed strain-controlled tests with strain ratio $$R_{\varepsilon }=-1$$. By using their calculation approach of $$\varDelta W^{+\mathrm {el}}$$, they improve their fatigue model especially for the load ratio $$R={0.5}$$. Since the fatigue tests, presented in this paper, were performed at load ratios $$R=2$$, $$-\infty$$, -1 and -0.5, only small positive mean values result. Thus, the influence of the different calculation approaches for $$\varDelta W^{+\mathrm {el}}$$ is considered to be small.
The red surfaces in Figs.  89, and 10 represent the 2-times error bounds with respect to the $$N_{\mathrm {f}}$$-axis. It can be seen that most experimental data lies between the two two-time error bounds. This demonstrates that the CH $$\varepsilon$$V is well suited for fatigue modeling of wrought Mg alloys, even with different specimen types including stress concentrations, stress gradients, different sizes and different load types.
Table 4
Values for the material parameters for the fatigue parameters, determined with a nonlinear regression analysis which contains all specimen types
Fatigue parameter
$$C_1$$
$$C_2$$
$$d_1$$
$$d_2$$
$$d_3$$
$$d_4$$
$$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$
2.57%
$$-$$ 0.188
$$-$$ 0.0120
$$P_{\mathrm {SWT}}$$
665 MPa
$$-$$ 0.149
$$-$$ 0,0150
$$\varDelta W^{\mathrm {t}}$$
$${17.4}\,{\mathrm{MJm}^{-3}}$$
$$-$$ 0.420
$$-$$ 0.0238
$$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$
$${16.2}\,{\mathrm{MJm}^{-3}}$$
$$-$$ 0.369
$$-$$ 0.0286
$$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$
0.914%
2.24%
$$-$$ 0.114
$$-$$ 0.0175
$$-$$ 0.308
0.00578
Table 5
Coefficients of determination $$r^2$$ for different fatigue parameters evaluated from the nonlinear regression analysis which contains all specimen types

$$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$
$$P_{\mathrm {SWT}}$$
$$\varDelta W^{\mathrm {t}}$$
$$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$
$$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$
$$r^2$$
0.85
0.85
0.82
0.82
0.86
Verification of the concept of highly strained volume for specific specimen types This paragraph examines the individual influences of the different specimen types. For this purpose the CH $$\varepsilon$$V is applied separately to the fatigue tests of the unnotched, notched, and bending specimens. The verification of the CH $$\varepsilon$$V including all specimen types showed that the fatigue parameter $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$ achieved the best regression. Therefore, only the fatigue parameter $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$ is used for subsequent fatigue modeling.
Figure  11a–c shows the $$V_\varepsilon$$- $$N_\mathrm{f}$$ fatigue diagrams containing the experimental data of the fatigue tests of the unnotched, notched and bending specimens, respectively. The regression analysis is carried out on the basis of Eq. ( 27), with Table  6 listing the determined material coefficients and exponents for the respective specimen types.
Table 6
Values for the material parameters from the nonlinear regression Eq. ( 27) for the unnotched, notched, and bending specimen, determined with fatigue parameter $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$
Specimen type
$$C_1$$ (%)
$$C_2$$ (%)
$$d_1$$
$$d_2$$
$$d_3$$
$$d_4$$
Unnotched specimens
2.15
2.38
$$-$$ 0.209
$$-$$ 0.0304
$$-$$ 0.308
$$-$$ 0.00787
Notched specimens
0.755
2.19
$$-$$ 0.0906
$$-$$ 0.0336
$$-$$ 0.312
0.0488
Bending specimens
1.09
5.27
$$-$$ 0.118
$$-$$ 0.0742
$$-$$ 0.381
$$-$$ 0.122
Table  7 shows the coefficients of determination $$r^2$$ of the regression analyses of the unnotched, notched and bending specimens, where the smallest value is obtained for notched specimens with $$r^2 = {0.88}$$. It was found in [ 8] that the BTGs that form at notches are relatively small and narrow, which brings the strain field measurement with DIC to its resolution limits. Thus, higher measuring uncertainties cause the slightly lower $$r^2$$ value for notched specimens. The best result is obtained for the bending specimens with a coefficient of determination of $$r^2={0.98}$$. Figure  11c shows that the values for the highly strained volumes $$V_{\varepsilon }$$ of the bending specimens cover a smaller range than, e.g., those of the unnotched specimens. The limited variance for this case of the measurement points can be a reason for the good regression. In summary, only small improvements in regressions are achieved for the unnotched and notched specimens by applying the CH $$\varepsilon$$V to only one specific specimen type rather than combining all specimen types in one model. This shows that the CH $$\varepsilon$$V can estimate the lifetime almost as accurately using different specimen types in one model as applying it to a specific specimen type.
Table 7
Coefficients of determination $$r^2$$ for the different specimen types, evaluated with the fatigue parameter $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$

Unnotched specimens
Notched specimens
Bending specimens
$$r^2$$
0.89
0.88
0.98

## 5 Conclusions

A method for fatigue modeling of basal textured AZ31B wrought alloy structures is presented, taking into account stress concentrations, stress gradients, different sizes, different types of loading and macroscopic bands of twinned grains (BTGs). The compressive strain in the BTGs is significantly larger compared to the adjacent regions outside the BTGs. Within the BTGs, all investigated fatigue parameters are higher compared to the adjacent regions, which is consistent with the observation that first macroscopic cracks are observed within the BTGs for all specimens. Due to the fact that failure always occurs within the BTGs, only the highly strained regions with high strain amplitudes (HSRA) are considered using the concept of highly strained volume (CH $$\varepsilon$$V).
The main results of this work are summarized in the following items:
• Five fatigue parameters $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$, $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$, $$P_{\mathrm {SWT}}$$, $$\varDelta W^{\mathrm {t}}$$ and $$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$ are determined exclusively from the HSRA and evaluated with the CH $$\varepsilon$$V.
• For the fatigue parameters $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$, $$P_{\mathrm {SWT}}$$, $$\varDelta W^{\mathrm {t}}$$ and $$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$ stress–strain hystereses must be modeled. Therefore, a stress–strain hysteresis model (SSM) is presented that computes stress–strain hystereses for arbitrary load ratios. Comparisons between experimentally measured and modeled stress–strain hystereses show good agreement.
• In total, the CH $$\varepsilon$$V is applied and evaluated including 30 unnotched, 34 notched and 12 bending specimens using the five fatigue parameters $$\tilde{{\bar{\varepsilon }}}^{\,*}_{\mathrm {a}}$$, $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$, $$P_{\mathrm {SWT}}$$, $$\varDelta W^{\mathrm {t}}$$ and $$\varDelta W^{\mathrm {t}}_{\mathrm {mss}}$$. The best regression is achieved for the fatigue parameter $$\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {el}}_{\mathrm {a}}+\tilde{{\bar{\varepsilon }}}^{\,*,\mathrm {pl}}_{\mathrm {a}}$$ with a coefficient of determination $$r^2=0.86$$, demonstrating the aptitude of the CH $$\varepsilon$$V for fatigue modeling of wrought Mg alloys.
• Considering only one specific specimen type in the fatigue model yields only slightly better regressions for the unnotched and notched specimens. The best regression is obtained for the bending specimens with a coefficient of determination of $$r^2={0.98}$$. One reason for the good regression can be the limited variance of the highly strained volumes $$V_\varepsilon$$ of the 12 tested bending specimens.

## Acknowledgements

Open Access funding provided by Projekt DEAL. The authors acknowledge the financial support of the European Regional Development Fund (ERDF) within the funding program INTERREG V-A Programm Österreich-Bayern 2014-2020 in the project ’AB29 - n2m’ under the contract number CCI-Nr. 2014TC16RFCB004.

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