Crack growth behaviour in biaxial stress fields: Calculation of K-factors for cruciform specimens

Journal Pre-proofs Crack growth behaviour in biaxial stress fields: calculation of K-factors for cruciform specimens Carl H. Wolf, Andreas Burgold, Sebastian Henkel, Meinhard Kuna, Horst Biermann PII: DOI: Reference:

S0167-8442(19)30567-1 https://doi.org/10.1016/j.tafmec.2020.102521 TAFMEC 102521

To appear in:

Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

9 October 2019 20 December 2019 8 February 2020

Please cite this article as: C.H. Wolf, A. Burgold, S. Henkel, M. Kuna, H. Biermann, Crack growth behaviour in biaxial stress fields: calculation of K-factors for cruciform specimens, Theoretical and Applied Fracture Mechanics (2020), doi: https://doi.org/10.1016/j.tafmec.2020.102521

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier Ltd.

Crack growth behaviour in biaxial stress fields: calculation of K-factors for cruciform specimens Carl H. Wolfa,1 , Andreas Burgoldb , Sebastian Henkela , Meinhard Kunab , Horst Biermanna a Institute

of Materials Engineering, Technische Universit¨ at Bergakademie Freiberg, Gustav-Zeuner-Str. 5, 09599 Freiberg, Germany b Institute of Mechanics and Fluid Dynamics, Technische Universit¨ at Bergakademie Freiberg, Lampadiusstraße 4, 09599 Freiberg, Germany

Abstract The aim of this study is to propose a simplified method for calculating the Mode I stress intensity factor K for cruciform specimens under planar biaxial loading. It is assumed that two cracks have to grow with a similar crack growth rate. The straight crack paths of the two single cracks with the length a should also be similar. The calculations are carried out on an aluminum and a steel specimen, respectively. For different load cases and materials, the stresses resulting from the forces are first considered. It was found that the elastic constants E and ν have only a small influence of less than 3 %. In addition, the coupling between the forces of the load axes, which should be minimized by slotted arms, is considered. Moreover, K-factors are calculated by finite elements (FE) for different crack lengths. These K-values together with the transmission factor allow to find a K-factor formula for cruciform specimens, which is based on the prescribed forces. Finally, the results of the FE-calculation of the exact straight crack paths were compared to experimentally determined crack paths. Furthermore, a comparison of the fatigue crack growth rates versus K-factor range for different loading is shown. Keywords: Fracture mechanics, fatigue crack growth, cruciform specimen, in-phase loading, K-solution, geometry factor, biaxial testing.

∗ Corresponding

author Email address: [email protected] (Carl H. Wolf)

Preprint submitted to Theoretical and Applied Fracture Mechanics

February 11, 2020

Nomenclature CT . . . . . . . . . . . compact tension FE . . . . . . . . . . . finite elements RP . . . . . . . . . . . reference point RP+ x . . . . . . . . . . reference point for pulling in x-direction at positive x-value RP− x . . . . . . . . . . reference point for pulling in x-direction at negative x-value RP+ y . . . . . . . . . . reference point for pulling in y-direction at positive y-value RP− y . . . . . . . . . . reference point for pulling in y-direction at negative y-value Tx . . . . . . . . . . . . T-stress resulting from a loading with force Fx Ty . . . . . . . . . . . . T-stress resulting from a loading with force Fy T . . . . . . . . . . . . . crack-parallel T-stress ∆T . . . . . . . . . . . T-stress range ∆Tx . . . . . . . . . . T-stress range resulting from a loading with force range ∆Fx ∆Ty . . . . . . . . . . T-stress range resulting from a loading with force range ∆Fy A5 . . . . . . . . . . . . elongation to fracture E . . . . . . . . . . . . . Young’s modulus F . . . . . . . . . . . . . force Fa . . . . . . . . . . . . force amplitude Fm . . . . . . . . . . . mean force Fx . . . . . . . . . . . . force in x-direction Fy . . . . . . . . . . . . force in y-direction ∆Fx . . . . . . . . . . force range in x-direction ∆Fy . . . . . . . . . . force range in y-direction ∆G . . . . . . . . . . . strain energy release rate K . . . . . . . . . . . . stress intensity factor KCS . . . . . . . . . . stress intensity factor for cruciform specimen Keq . . . . . . . . . . . equivalent stress intensity factor KI . . . . . . . . . . . . Mode I stress intensity factor KI,x . . . . . . . . . . Mode I stress intensity factor resulting from a loading with force Fx KI,y . . . . . . . . . . Mode I stress intensity factor resulting from a loading with force Fy KII . . . . . . . . . . . Mode II stress intensity factor KII,x . . . . . . . . . Mode I stress intensity factor resulting from a loading with force Fx 2

KII,y . . . . . . . . . Mode I stress intensity factor resulting from a loading with force Fy ∆K . . . . . . . . . . stress intensity factor range ∆KCS . . . . . . . . stress intensity factor range for a cruciform specimen ◦

0 ∆KCS . . . . . . . . stress intensity factor range for a 0◦ -specimen ◦

45 ∆KCS . . . . . . . . stress intensity factor range for a 45◦ -specimen

∆KI . . . . . . . . . . Mode I stress intensity factor range ∆KII . . . . . . . . . Mode II stress intensity factor range ∆Keq . . . . . . . . . equivalent stress intensity factor range ∆K∞ . . . . . . . . . stress intensity factor range for a thin plate of infinite size N . . . . . . . . . . . . load cycle R . . . . . . . . . . . . . load ratio Rm . . . . . . . . . . . ultimate tensile strength Rp0.2 . . . . . . . . . yield strength W . . . . . . . . . . . . width of the measuring area Y (a/W, α) . . . . crack-length and crack-orientation dependent geometry factor Y fit (a/W, 45◦ ) crack-length and crack-orientation dependent geometry function for the 45◦ -specimen Yx (a/W, α) . . . crack-length and crack-orientation dependent geometry factor for a loading in x-direction ◦

Yy (a/W, 0 ) . . crack-length dependent geometry factor for a loading in ydirection for the 0◦ -specimen Yy (a/W, α) . . . crack-length and crack-orientation dependent geometry factor for a loading in y-direction Yyfit

◦

(a/W, 0 ) . crack-length and crack-orientation dependent geometry function for a loading in y-direction for the 0◦ -specimen

a . . . . . . . . . . . . . crack length bx . . . . . . . . . . . . scaling factor for a loading in x-direction by . . . . . . . . . . . . scaling factor for a loading in y-direction ∆b . . . . . . . . . . . . scaling factor ∆bx . . . . . . . . . . scaling factor range for a loading in x-direction ∆by . . . . . . . . . . scaling factor range for a loading in y-direction cx . . . . . . . . . . . . coupling of the forces in x-direction cy . . . . . . . . . . . . coupling of the forces in y-direction d . . . . . . . . . . . . . nominal thickness of the specimen in the measuring area 3

da/dN . . . . . . . . crack growth rate f . . . . . . . . . . . . . frequency fx . . . . . . . . . . . . unit force in x-direction fy . . . . . . . . . . . . unit force in y-direction h . . . . . . . . . . . . . diameter of the hole in the middle of the specimen k . . . . . . . . . . . . . unit stress intensity factor kI . . . . . . . . . . . . unit Mode I stress intensity factor kI,x . . . . . . . . . . . unit Mode I stress intensity factor resulting from a loading with unit force fx kI,y . . . . . . . . . . . unit Mode I stress intensity factor resulting from a loading with unit force fy kII . . . . . . . . . . . . unit Mode II stress intensity factor kII,x . . . . . . . . . . unit Mode II stress intensity factor resulting from a loading with unit force fx kII,y . . . . . . . . . . unit Mode II stress intensity factor resulting from a loading with unit force fy kx . . . . . . . . . . . . unit Mode I stress intensity factor resulting from a loading with unit force fx ky . . . . . . . . . . . . unit Mode I stress intensity factor resulting from a loading with unit force fy k∞ . . . . . . . . . . . unit stress intensity factor for a thin plate of infinite size ◦

0 ∆kCS . . . . . . . . . unit stress intensity factor range for a 0◦ -specimen ◦

45 ∆kCS . . . . . . . . unit stress intensity factor range for a 45◦ -specimen

l . . . . . . . . . . . . . . lenght of the start notch s . . . . . . . . . . . . . unit normal stress sx . . . . . . . . . . . . unit normal stress in x-direction resulting from equibiaxial loading with unit forces fx and fy sx (fx ) . . . . . . . . unit normal stress in x-direction resulting from a loading with unit force fx sx (fy ) . . . . . . . . unit normal stress in x-direction resulting from a loading with unit force fy spath (fy ) . . . . . unit normal stress in x-direction resulting from a loading with x unit force fy along a path across the specimen sy . . . . . . . . . . . . unit normal stress in y-direction resulting from equibiaxial loading with unit forces fx and fy 4

sy (fx ) . . . . . . . . unit normal stress in y-direction resulting from a loading with unit force fx sy (fy ) . . . . . . . . unit normal stress in y-direction resulting from a loading with unit force fy spath y

(fy ) . . . . . unit normal stress in y-direction resulting from a loading with unit force fy along a path across the specimen

spath . . . . . . . . . . unit normal stress in y-direction resulting from equibiaxial y loading with unit forces fx and fy t . . . . . . . . . . . . . . unit T-stress tx . . . . . . . . . . . . unit T-stress resulting from a loading with unit force fx ty . . . . . . . . . . . . . unit T-stress resulting from a loading with unit force fy tP . . . . . . . . . . . . period time x . . . . . . . . . . . . . x-coordinate of the Cartesian coordinate system y . . . . . . . . . . . . . y-coordinate of the Cartesian coordinate system α . . . . . . . . . . . . . angle between x-axis or both loading axes and initial notch orientation ν . . . . . . . . . . . . . Poisson’s ratio σ . . . . . . . . . . . . . stress σx . . . . . . . . . . . . normal stress in x-direction resulting from equibiaxial loading with forces Fx and Fy σx (Fx ) . . . . . . . normal stress in x-direction resulting from a loading with force Fx σx (Fy ) . . . . . . . normal stress in x-direction resulting from a loading with force Fy σy . . . . . . . . . . . . normal stress in y-direction resulting from equibiaxial loading with forces Fx and Fy σy (Fx ) . . . . . . . normal stress in y-direction resulting from a loading with force Fx σy (Fy ) . . . . . . . normal stress in y-direction resulting from a loading with force Fy ∆σx . . . . . . . . . . normal stress range in x-direction resulting from equibiaxial loading with forces Fx and Fy ∆σy . . . . . . . . . . normal stress range in y-direction resulting from equibiaxial loading with forces Fx and Fy

5

∆σy (Fx ) . . . . . normal stress range in y-direction resulting from a loading with force Fx ∆σy (Fy ) . . . . . normal stress range in y-direction resulting from a loading with force Fy

1.

Introduction Many components are subjected to multi-axial stress conditions during their

service life due to loadings in different directions or their geometry [1, 2], e.g. the outer skin of the aircraft [3, 4], blade integrated disks in aircraft engines [5] or thin walled structural components as pressure vessels [6]. However, the estimation of fatigue crack growth is often based on uniaxial data [7–10]. In addition, biaxial fatigue crack growth investigations are an important link between uniaxial small-scale tests on the one hand and complex component tests on the other hand. Difficulties in investigating fatigue crack growth under planar biaxial loading arise from the different specimen types, as there are many different testing machines and no standardized specimen design. However, five common specimen designs can be found in literature: (i) thin specimen design with large radii between the loading arms, cf. Fig. 1a and e.g. Refs. [11–13], (ii) thin specimen design with small radii between the loading arms, cf. Fig. 1b and e.g. Refs. [14–25], (iii) thin specimen design with small radii between the loading arms and a thickness-reduced measuring area, cf. Fig. 1c and e.g. Refs. [5, 11, 26–31], (iv) specimen designs with slotted arms, cf. Fig. 1d and e.g. Refs. [3, 4, 11, 23, 32], and (v) specimen design with thickness-reduced measuring area and slotted arms, cf. Fig. 1e and e.g. Refs. [6, 18, 22, 23, 26, 33–49]. As a result of the different specimen designs, the K-solution has to be computed for each specimen type by FE-calculations considering the individual crack paths and the material. Thus, different evaluations of the crack growth rate are made in the literature for cruciform specimens, for example: (i) crack length a versus the number of load cycles N , cf. e.g. Refs. [5, 19], 6

(a)

(b)

(d)

(e)

(c)

Figure 1: Schematic illustration of the most commonly used cruciform specimen designs for fatigue crack growth tests. (a) Thin specimen with large radii between the loading arms. (b) Thin specimen design with small radii between the loading arms. (c) Thin specimen design with small radii between the loading arms and a thickness-reduced measuring area. (d) Specimen with slotted arms. (e) Specimen with thicknessreduced measuring area and slotted arms. The initial crack with length 2a is plotted horizontally in the specimen’s center.

(ii) crack growth rate da/dN versus the number of load cycles N , cf. e.g. Ref. [37], (iii) crack growth rate da/dN versus the crack length a, cf. e.g. Ref. [46], (iv) crack growth rate da/dN versus the strain energy release rate ∆G, cf. e.g. Refs. [14, 21], and (v) crack growth rate da/dN versus the stress intensity factor K or rather its range ∆K, cf. e.g. Refs. [3, 6, 12–18, 30, 31, 33–35, 40–45, 47–49]. The aim of the present investigation is first to analyse the stress distribution of the specimens used in recent works (cf. e.g. Refs. [6, 42–44]). Subsequently, a simple model is developed to calculate the stress intensity factors for straight crack paths. Two cases are considered: (i) a crack growing parallel to one loading axis and (ii) a crack growing with an angle of 45◦ to both loading axes. Finally, a comparison of the calculated stress intensity factors is carried out between ideally straight crack paths and experimentally determined crack tip coordinates.

7

σ [MPa]

600

400

200

0

steel aluminum (L-T) aluminum (T-L) 0

20

ε [%]

40

60

Figure 2: Stress-strain curves of the investigated materials. Table 1: Important mechanical properties.

material

Rp0.2 [MPa]

Rm [MPa]

A5 [%]

steel steel

252

624

60

aluminum (L-T) aluminum (T-L) Standard*

282 275 240

319 321 290

18.1 14.5 9.0

*

Minimum values according to DIN EN 485-2 [50] for plates with a thickness of 6 mm to 12 mm.

2.

Materials and Methods

2.1.

Materials

The experimental investigations were performed on a powder metallurgically produced austenitic stainless steel X5CrMnNi16-7-7, called steel in the following. In recent work, the aluminum alloy EN AW-AlMgSi1Cu in the condition T651, i.e. solution annealed, quenched, controlled strained and artificially aged, called aluminum in the following, has also been investigated, cf. e.g. Refs. [6, 42, 44]. The chemical compositions of both materials are given in Ref. [44]. The stress-strain curves and selected mechanical properties of the investigated materials are given in Fig. 2 and Tab. 1, respectively. Further mechanical properties are given in Refs. [42–44]. The materials were selected to have roughly comparable yield strength and strongly different strain hardening behaviour, c.f. the Rm -values in Tab. 1. 2.2.

Specimens

The design of the examined cruciform specimens, cf. Figs. 1e, 3a and 3b, corresponds to Brown and Miller [34]. This type of specimen is also used by 8

(a)

(b)

W

a

a

2

·a

2·a

Figure 3: Design of the cruciform specimens and uncracked specimens with the paths for the FE-calculations. (a) Specimen with centered notches aligned parallel to one loading axis, called 0◦ -specimen. (b) Specimen with centered notches aligned with an angle of α = 45◦ to both loading axes, called 45◦ -specimen. The width W of the measuring area is W = 130.8 mm. The drilled hole has a diameter of h = 4 mm. The distance between the notch tips is 2a = 10 mm. The thickness of the measuring area is d ≈ 2 mm.

many other researchers [6, 18, 22, 23, 26, 33, 35–49]. The specimens have a squared measuring area and slotted arms. The squared measuring area is reduced in thickness and has a width of W = 130.8 mm. The load application arms are slotted in order to achieve uncoupling of the forces of the two loading axes, cf. M¨ onch and Galster [51]. A hole with a diameter of h = 4 mm was first drilled in the middle of the measuring area. Subsequently, two start notches of l = 3 mm each were inserted by electro discharge machining and served as initial crack starter. Thus, the total initial crack length is 2a = 10 mm, cf. Fig. 3. For small cracks, a central part-through crack in a plate of infinite size (cf. e.g. Griffith [52]) is a good approximation. Exact dimensions of the specimens were given in a recent work [6]. Depending on the alignment of the initial notches or rather initial cracks, a large number of tests can be carried out. If the start notches are aligned parallel to one loading axis, the specimen is called 0◦ -specimen in the following, cf. Fig. 3a. This specimen design is used for investigations on the influence of the T-stress, cf. e.g. Refs. [6, 42]. If the notches are aligned at an angle of 45◦ to both load axes, the specimen is called 45◦ -specimen in the following, cf. Fig. 3b. This specimen design is used to carry out both equibiaxially loaded tests without T-stress, cf. e.g. Refs. [32, 43, 44], and tests with phase-shifted

9

loadings, cf. e.g. Refs. [42–44]. The manufacturing of the specimens took place in several steps. For the steel-specimens, the base plate of the specimens had to be manufactured first. For this purpose, circular blanks made of steel were inserted into base plates made of a commercially available AISI304 by means of a interference fit. Afterwards, the circular, hot pressed blanks were welded to the base plates using electron beam welding. The production of the specimens out of the base plates was then carried out by milling. The initial notches were finally inserted by electro-discharge machining. A detailed description of the specimen production was given in a recent work [44]. 2.3.

Testing system

The tests were performed on a 250 kN servo-hydraulic, planar-biaxial tension-compression testing system with two perpendicular loading axes in forcecontrolled mode (Fm = 29.14 kN and Fa = 23.76 kN for R = 0.1 and Fm = 39.68 kN and Fa = 13.23 kN for R = 0.5) at a test frequency of f = 20 Hz, i.e. the period time tP was tP = 0.05 s. The loading function was sinusoidal, cf. Fig. 4a. (b)

50 40 30 20 10 0 −10

Fy [kN]

50

F [kN]

(a)

Fy Fx uniaxial Fx equibiaxial 0

1 4

1 2

load cycle

3 4

40 30 20 10 0

time tP

uniaxial equibiaxial 0 10 20 30 40 50 Fx [kN]

Figure 4: Course of the forces (a) and (b) load path during one load cycle for the uniaxial and equibiaxial experiments.

Parallel to the front side of the specimens, a high-resolution camera system with a measuring area of 118.8 mm×118.8 mm was positioned. The images were recorded during the entire experiment in intervals of 1500 to 10 000 load cycles and were taken at mean force. For this purpose the experiments were interrupted. When the experiment was finished, the cracks were evaluated using the image processing system ImageJ according to Ref. [36]. The determined crack 10

tip coordinates were used for the FE-calculation. Furthermore, FE-calculations were performed for ideally straight crack paths for the 0◦ -specimens, cf. Fig. 3a, and for the 45◦ -specimens, cf. Fig. 3b, for steel and aluminum. In addition, the crack length was measured using crack gages (Krak Gage B 20, Russenberger Pr¨ ufmaschinen AG, Neuhausen am Rheinfall, Switzerland). These were applied to both notches on the specimen surface.

The actual

crack length was measured by the indirect potential method using the RUMUL FRACTOMAT from Russenberger Pr¨ ufmaschinen AG (Neuhausen am Rheinfall, Switzerland). Detailed information of the crack length measurement by means of the high-resolution camera system and the crack gages can be found in a recent work [43]. 2.4. 2.4.1.

Forces, stresses, stress intensity factors and geometry factor Scaling of the forces, stresses and stress intensity factors

unit force: sx , sy , kI , kII , t fy

fx Fx

sx (fx ), sy (fx ), kI,x , kII,x , tx

sx (fy ), sy (fy ), kI,y , kII,y , ty fy Fy

Fy

y x

fy Fy scaled: σx , σy , KI , KII , T

fx fx = Fx Fx

y

fx x

Fx

+

y x

fy σx (Fx ), σy (Fx ), KI,x , KII,x , Tx

Fy

σx (Fy ), σy (Fy ), KI,y , KII,y , Ty

Figure 5: Schematic illustration to demonstrate the principle of superposition as well as the scaling. The solutions of the boundary value problem with different boundary conditions (fx , fy or rather Fx , Fy ) are added in order to achieve the combined solution. The input and output variables of the FE-calculation and the respective scaled and superimposed variables for scaling with scaling factors bx or rather by are presented in addition. Scaling with ∆bx or rather ∆by would lead to the respective ranges. The drawn stresses present the normal stresses in the middle of the specimen, i.e. at the coordinate origin (x = 0, y = 0).

The analysis of the specimens is based on a linear boundary value problem, cf. Fig. 5. This allows to apply the concepts of scaling and superposition. Therefore, the FE-calculations have been performed with unit forces fx = 1 kN 11

or fy = 1 kN, cf. Ref. [43], which are applied at the respective loading arms. The unit forces can be scaled to the applied forces Fx and Fy by scaling factors bx or by or rather to the applied force ranges ∆Fx and ∆Fy by scaling factors ∆bx or ∆by , cf. Eqs. (1) and (2).

Fx = fx ·

bx

and

∆Fx = fx · ∆bx

and

Fy = fy ·

by

(1)

∆Fy = fy · ∆by

(2)

Furthermore, the results of the FE-calculations can be scaled analogously to Eqs. (1) and (2). It means: (i) stresses, (ii) Mode I and Mode II stress intensity factors, and (iii) the crack-parallel T-stresses can also be scaled up by the factors bx or by or their ranges ∆bx or ∆by , cf. Fig. 5. To calculate the stress intensity factors KI and KII at the crack tip, the resulting stress intensity factors KI,x and KI,y or rather KII,x and KII,y of the FE-calculation of the two loading axes must be added up, cf. Eq. (3). The same holds for the the stresses σx and σy and the T-stress T and the ranges of all quantities, i.e. ∆KI , ∆KII , ∆σx , ∆σy and ∆T, cf. Fig. 5.

KI =

KI,x +

KI,y

and

KII =

KII,x +

KII,y

(3)

If there is a Mode I and a Mode II portion of the crack opening Mode present, i.e. KI 6= 0 and KII 6= 0 or rather ∆KI 6= 0 and ∆KII 6= 0, the equivalent stress intensity factor Keq is calculated as proposed by Richard et al. [53] and its range ∆Keq as used in [44, 54, 55]. Ideally, a force in x-direction should produce a normal stress exclusively in x-direction and a force in y-direction a normal stress exclusively in y-direction. To achieve this, the effect of the individual forces must not influence each other. This should be achieved by the slotted arms of the specimens. However, there is a small remaining coupling of the two axes, which can be quantified by the coupling factors cx and cy , cf. Eq. (4). 12

σx (Fy ) σy (Fx ) cx = and cy = σx (Fx ) σy (Fy ) 2.4.2.

(4)

Geometry factor

Due to the assumption of straight crack paths and the considered loading conditions (see below), the Mode II terms of the two loading axes x and y abolish each other. This results in a first simplification, i.e. only the Mode I geometry factors have to be calculated. For a clearer representation, the indexing

I

 

is

therefore omitted in the following, i.e. if no index is given, Mode I is meant. In general, the stress intensity factors for a specific crack configuration can be expressed by using a geometry-dependent function Y (a/W, α)

K =σ·

√

π · a · Y (a/W, α)

(5)

whereby σ represents a nominal stress and α the angle between the x-axis and the initial notch orientation. Considering unit forces, the crack-length dependent geometry factors are defined as the relation between the stress intensity factors of the cruciform specimen kx and ky and the stress intensity factor for a thin plate of infinite size k∞ , cf. Eq. (6).

Yx (a/W, α) =

kx k∞

and

Yy (a/W, α) =

ky k∞

(6)

This reference stress intensity factor k∞ , cf. Eq. (7) is related to the stress s perpendicular to the crack plane in the center of the uncracked cruciform specimen.

13

k∞ = s ·

√

π·a

(7)

By means of FE, see section 2.5, the stress s and the kx , y and k∞ -values for certain crack length are calculated. Subsequently, the Yx (a/W, α)- or Yy (a/W, α)values for certain crack length a are found using Eq. (6). Afterwards, these values will be interpolated for the range of used crack lengths. Finally, the stress intensity factors for the cruciform specimens KCS can be calculated by Eq. (8) using the determined geometry functions. The same is valid for the stress intensity factor range ∆KCS . KCS = [σx · Yx (a/W, α) + σy · Yy (a/W, α)] ·

√

π·a

= [sx · bx · Yx (a/W, α) + sy · by · Yy (a/W, α)] ·

√

π·a

(8)

Some simplifications could be done if (i) both crack tips with similar crack growth rates, (ii) the crack paths of the two crack tips were symmetrical and (iii) when a certain loading was performed at a certain orientation of the initial notches or rather crack path. On the one hand a simplification can be done when the central notches are aligned parallel to one loading axis (0◦ -specimen), cf. Fig. 3a, and a cyclic loading perpendicular to the direction of the initial notches plus a static loading in direction of the initial notches are performed. On the other hand, a simplification can be done when the central notches are aligned with an angle of α = 45◦ to both loading axes (45◦ -specimen), cf. Fig. 3b, and an equibiaxial loading is performed. This simplifications will be described below. 2.4.3.

0◦ -specimen with static loading in x-direction and cyclic loading in y-direction

Calculation of the cyclic stress intensity factor. The stress intensity factor range ◦

0 ∆KCS for 0◦ -specimens (α = 0, cf. Fig. 3a) can be calculated by Eq. (8) using

14

the stress range ∆σy . Due to the static loading in x-direction and cyclic loading in y-direction, i.e. ∆Fx = 0 and ∆Fy 6= 0, Eq. (8) can be simplified, cf. Eq. (9). ◦

0 ∆KCS = ∆σy ·

√

π · a · Yy (a/W, 0◦ ) = sy · ∆by ·

√

π · a · Yy (a/W, 0◦ )

(9)

In consequence, the geometry factor Yy (a/W, 0◦ ) can be calculated using Eq. (6), whereby k∞ is obtained by plugging in the normal stress in y-direction sy in the center of the uncracked specimen into Eq. (7):

Yy (a/W, 0◦ ) =

sy ·

k √y . π·a

(10)

Remark: If the start notches run parallel to the y-axis and a cyclic loading in x-direction and a static loading in y-direction is applied, i.e. ∆Fx 6= 0 and ∆Fy = 0, then y has to be exchanged by x in Eqs. (9) and (10). Experimental loading. The experiment was performed with a specimen made of steel. Two different R-ratios have been tested. The details of test conditions were given in a recent work, cf. Ref. [44]. To demonstrate the forces acting during one load cycle, Fig. 4a shows the course of the forces Fx (dark gray line) and Fy (black line) during one load cycle for the case of R = 0.1. The load path, i.e. the ratio of the forces to each other, is shown in Fig. 4b (dark gray line). 2.4.4.

Equibiaxially loaded 45◦ -specimen

Calculation of the cyclic stress intensity factor. In the case of equibiaxial loading of the 45◦ -specimen, cf. Fig. 3b, with equal and synchronous forces or rather force ranges, i.e. ∆Fx = ∆Fy and ∆bx = ∆by = ∆b, exists a hydrostatic stress state in the center of the specimen.

σ = σx (Fx )

+ σx (Fy ) = σy (Fx )

+ σy (Fy )

s = sx (fx )

+

+

sx (fy ) = sy (fx ) 15

sy (fy )

(11)

Since this is the opening normal stress for the 45◦ -crack, from Eq. (5) one ◦

45 finds the stress intensity factor for cyclic loading ∆KCS , cf. Eq. (12).

◦

45 ∆KCS = [∆σy (Fx ) + ∆σy (Fy )] ·

= [sy (fx ) + sy (fy )] · ∆b ·

√ √

π · a · Y (a/W, 45◦ )

π · a · Y (a/W, 45◦ )

(12)

Hereby, Y (a/W, 45◦ ) describes the geometry function and can be calculated by Eq. (13). Due to symmetry, the shear crack opening Modes II induced by the two load axes cancel each other, i.e. pure Mode I crack opening occurs.

◦

Y (a/W, 45◦ ) =

45 ∆KCS (kx + ky ) · ∆b k √ = = ∆K∞ k∞ · ∆b [sy (fx ) + sy (fy )] · π · a

(13)

Experimental loading. The experiment was performed with a specimen made of steel. The complete test conditions were given in a recent work, cf. Ref. [44]. To understand the forces acting during one load cycle, Fig. 4a shows the course of the forces Fx (dotted light gray line) and Fy (black line) during one load cycle. The load path, i.e. the ratio of the forces to each other, is shown in Fig. 4b (dotted light gray line). 2.5.

FE-calculations

In order to obtain the geometry factors Yx (a/W, α) and Yy (a/W, α), cf. Eq. (6), there are several quantities, which have to be calculated with finite elements. On the one hand, the stress state in the center (x = 0, y = 0) of the cruciform specimen without crack is needed (sx (fx ), sx (fy ), sx , sy (fx ), sy (fy ) and sy ) in the K-formulas, cf. Eqs. (8), (9) and (12), and in the reference solution k∞ , cf. Eq. (7). On the other hand, kx -values and ky -values, cf. Eqs. (10) and (13), are evaluated for different crack lengths for the cases of the 0◦ -crack and the 45◦ -crack. The finite element analyses are carried out with the commercial software ABAQUS. A two dimensional model of the cruciform specimen is created as16

suming plane stress conditions. In the measurement area and in the region of thickness transition the thickness of 2 mm is prescribed, whereas the outer frame is provided with a thickness of 12 mm. Consequently, there is a jump in thickness at the interface between these regions. The model is used to calculate stresses and stress intensity factors in a sufficiently large distance from this jump, which is admissible. The material behavior is modeled as linear elastic. The material parameters are Young’s modulus E = 192 GPa and Poisson’s ratio ν = 0.24 for steel and E = 72 GPa and ν = 0.34 for aluminum, respectively. The finite element model of a cruciform specimen without hole and crack is depicted in Fig. 6a. The boundary conditions are visualized in Fig. 6b and are applied for all simulations. The nodes at the front end of each loading arm are coupled to one reference point (RP), respectively. During loading unit forces {fx , fy = 0} or {fx = 0, fy } are prescribed to the corresponding reference points, cf. Fig. 6b. The coupling is realized by the ABAQUS option equation“ ” and leads to the following kinematic constraints. Firstly, the front ends of the loading arms are enforced to remain straight. The nodes have the same displacement in the loading direction of the corresponding loading arm and it is exactly the displacement of the RP. Secondly, no displacement perpendicular to the loading direction of the corresponding loading arm is allowed for the nodes at the front ends. Therewith, lateral contraction of the front ends is impeded. These constraints are reasonable and reflect the clamping conditions of the cruciform specimens, because the arms are fixed by clamping jaws. Consequently, the lateral contraction of the hole specimen is constrained to some extend during loading, which is the reason for a small influence of Poisson’s ratio on the results. It is important to note the limited character of these constraints, which − belong only to the front ends. Pulling in y-direction (RP+ y and RPy ) will led

to contraction of the specimen in x-direction because the reference points RP+ x and RP− x together with the front ends of their loading arms can move along x-direction (compare Fig. 6b). The components of stress are evaluated in the measurement area of the specimen without crack. Furthermore, stress intensity factors are computed by the

17

(a)

(b)

fy

RP+y

y

RP−x

x

RP+x

RP−y

1

(c)

fx

fx

fy

(d)

1

y

y x

x

7

7

7 7 Figure 6: (a) Visualization of the finite element model for the case without crack in comparison to the full geometry. The clamping areas are not considered in the FE-model. (b) Sketch of the coupling conditions and boundary conditions prescribed in the FE-model. (c) and (d) Considered crack paths (c) along the x-axis and y-axis and (d) at an angle of 45◦ to both loading axes.

ABAQUS built-in interaction integral technique for a number of cracked configurations. Fig. 7 shows an example of the mesh inside the measurement area of a 45◦ -crack (2a = 10 mm). For all calculations fully integrated quadrilateral elements with quadratic shape functions (8 nodes, 9 integration points) are used. No special or collapsed crack tip elements are utilized, but a highly refined mesh is applied in the crack tip region. The K-factor results of some crack configurations are checked against results obtained with quarter-point-elements.

18

(a)

(b)

Figure 7: (a) Example of an 45◦ -crack of length 2a = 10 mm in the center of the specimen. (b) Associated finite element mesh used for the numerical computation of stress intensity factors.

3.

Results and Discussion

3.1.

Stress distribution in the uncracked specimen

The stresses resulting from the load with the unit force or forces, respectively, are shown in Figs. 8 and 9. The values for the plots are attached as supplementary material. For a better understanding of Figs. 8b to 8e and 9b to 9e, Figs. 8a and 9a show the position of the FE grid and the loading situation. The colors of the different lines in Figs. 8 and 9 correspond to the colors of the lines in Figs. 6c and 6d. Figs. 8b to 8e and 9b to 9e show only the stress distribution for steel. The stress distribution for aluminum was shown in a recent work, cf. Ref. [6]. For both materials, the stress distribution in the uncracked specimen is very similar. Figs. 8b and 8c show the resulting stresses in x-direction spath (fy ) along x the respective crack sections at a static unit force in y-direction of fy = 1 kN. In the middle of the specimen and along the diagonals, compressive stresses arise, whereas tensile stresses occur at the edges of the x- and y-axes. The normal or rather principal stress in x-direction in the middle of the specimen is sx (fy ) = −0.062 MPa for steel and sx (fy ) = −0.0362 MPa for aluminum.

The resulting stresses spath (fy ) in y-direction at a static unit force in yy

direction of fy = 1 kN are shown in Figs. 8d and 8e. It becomes clear that tensile stresses occur over the entire cross-section of the specimen. The normal or rather principal stress in y-direction in the middle of the specimen is sy (fy ) = 2.9331 MPa for steel and sy (fy ) = 2.9474 MPa for aluminum. 19

(a)

fy y x

fy (b)

(c)

spath (fy ) [MPa] x 0.2

−0.2 −0.062 MPa

spath (fy ) [MPa] x 0.2

−0.2 −0.062 MPa

(d)

(e)

spath (fy ) [MPa] y 2.933 MPa 3

spath (fy ) [MPa] y 2.933 MPa 3

2

2

1

1

Figure 8: Illustration of the stresses for steel under uniaxial tensile loading in y-direction with fy = 1 kN. The spacing of the grid lines is 5 mm each. (a) Considered area of the specimen and visualization of the loading direction. Resulting normal stresses in the specimen along the (b, d) x-axis (cf. Fig. 6c, dark gray line) and y-axis (cf. Fig. 6c, light gray line) and (c, e) the diagonals (cf. Fig. 6d, dark gray lines). (b, c) Resulting normal stresses in x-direction spath (fy ). (d and e) Resulting normal x stresses in y-direction spath (fy ). y

20

(a)

fy y fx

x fx

fy (b)

(c)

spath [MPa] y 2.871 MPa 3

spath [MPa] y 2.871 MPa 3

2

2

1

1

(d)

10

(e)

cy [%]

10

2.11 %

cy [%]

2.11 % 5

5

Figure 9: Illustration of the resulting stresses and the force coupling for steel under equibiaxial tensile loading with fx = fy = 1 kN. The spacing of the grid lines is 5 mm each. (a) Considered area of the specimen and visualization of the loading directions. Resulting normal stresses or rather force coupling in the specimen along the (b, d) x-axis (cf. Fig. 6c, dark gray line), y-axis (cf. Fig. 6c, light gray line), and (c, e) the diagonals (cf. Fig. 6d, dark gray lines). (b, c) Resulting stresses in y-direction spath . (d, e) Remaining coupling in y-direction cy of the forces at an equibiaxial y tensile loading with fx = fy = 1 kN.

21

Due to the symmetry of the unnotched specimen, the amount of the resulting stress sy (fy ) at a force of fy = 1 kN is equal to the resulting stress sx (fx ) at a force of fx = 1 kN. The stresses sx (fy ) at fy = 1 kN and sy (fx ) at fx = 1 kN are also identical. Thus, the resulting stresses for equibiaxial loading are calculated via superposition of the results above for the given cross-sections and are depicted in Figs. 9b and 9c. It becomes clear that the stress distribution is more homogeneous compared to the uniaxial loading, compare Figs. 8b to 8e with Figs. 9b and 9c. The resulting stress in the middle of the equibiaxially loaded, unnotched specimen is sx = sy = 2.8711 MPa for steel and sx = sy = 2.9112 MPa for aluminum. The coupling parameter cy , cf. Eq. (4) is less than 3 % for both materials in the middle of the uncracked specimen, cf. Figs. 9d and 9e as well as Refs. [6, 51]. Further FE-calculations with variation of the Young’s modulus and constant Poisson’s ratio show that the Young’s modulus has no influence on the resulting normal stresses. In contrast, a variation of the Poisson’s ratio at constant Young’s modulus (E = 192 GPa) shows that the normal stress depends on the Poisson’s ratio. However, a comparison of the Poisson’s ratio in the range from ν = 0 to ν = 0.5 for a uniaxial loading in y-direction as well as for equibiaxial loading shows that the differences in the normal stresses sy (fy ) or rather sy are less then 3 %. The values are attached as supplementary material. These small differences in the resulting normal stresses enable an approximate material-independent calculation of the geometry functions Y fit (a/W, 45◦ ) or Yyfit (a/W, 0◦ ). To calculate the K-solution, the stresses sx (fy ) = −0.05 MPa and sy (fy ) = 2.94 MPa should be used. 3.2. 3.2.1.

Stress intensity factors and geometry functions Results of the FE-calculation

The unit Mode I and Mode II stress intensity factors for a uniaxial loading in x- and y-direction kI,x , kI,y , kII,x and kII,y as well as the unit T-stress in xand y-direction tx and ty for the 0◦ -specimen as well as for the 45◦ -specimen for steel and aluminum have been calculated. The results are attached as supplementary material.

22

3.2.2.

Solution for the uniaxially loaded 0◦ -specimens

First, the stress intensity factors k∞ for a thin plate of infinite size for unit force for different crack lengths a were calculated using Eq. (7). Due to the uniaxial loading in y-direction, the crack opening stress s is the normal stress sy (fy ), cf. section 3.1. The results are attached as supplementary material. Afterwards, the corresponding ky -values from section 3.2.1 were selected and the corresponding geometry factors Yy (a/W, 0◦ ) were determined by Eq. (10). Then, the geometry function Yyfit (a/W, 0◦ ) was determined in sections using a polynomial fit. The fit function for a static loading in x-direction with Fx = 0 and a cyclic loading in y-direction with Fy 6= 0 is given in Eq. (14).

Yyfit (a/W, 0◦ ) =

       3 X  n   pn · (a/W )     n=0                      4  X  n   pn · (a/W )   n=0        

   p0     p 1 if a ≤ 10 mm with   p2     p 3

if a > 10 mm with

   p0       p   1 p2      p3     p 4

=

1.1610

=

−6.1922

=

85.2718

=−395.5575

=

1.0216

=

−0.3837

=

3.4468

=

−7.7251

=

10.2447 (14)

The course of the function is shown in Fig. 10a. Furthermore, the course of the fit functions Yyfit (a/W, 0◦ ) is compared with the values of Yy (a/W, 0◦ ) determined by FE. It is shown that the determined fit functions approximate geometry factors with a deviation of less than 3 %. The geometry function Yyfit (a/W, 0◦ ) deviates for small cracks (a → 5 mm) from the value 1.0 for an infinite plate, cf. Eq. (7) and cf. Fig. 10a. The limit value of Eq. (14) for a = 5 mm amounts to Yyfit (a/W, 0◦ ) = 1.027 and agrees very well with the handbook-solution [56, p. 239] for a crack emanating from a circular hole of radius R under uniaxial loading Yy (a/W, 0◦ ) = 1.0252. The rea-

23

(a)

(b)

1.2 Y (a/W, 45°)

Yy (a/W, 0°)

1.2

steel aluminum

1.0

0

1.0

fit ±3 % of fit 0.2 0.4 a/W

0

(c)

(d)

1.2

steel aluminum

0 Figure 10:

Y (a/W, 45°)

Yy (a/W, 0°)

1.2

1.0

steel aluminum

fit ±3 % of fit 0.2 0.4 a/W

fit ±3 % of fit 0.2 0.4 0.6 a/W

steel aluminum

1.0

0

fit ±3 % of fit 0.2 0.4 0.6 a/W

Y (a/W, α)-a plot of (a) 0◦ -specimen and (b) 45◦ -specimen as well as (c) 0◦ -specimen and (d) 45◦ -specimen without centered hole in the specimen. Comparison of the solutions for the exact straight crack path computed with the determined stresses and thus the corresponding kx , ky and k∞ for steel (stars) and aluminum (circles) with the material-independent polynomial fit (thick dark gray line) and the ±3 % deviation of this fit (thin dark gray lines).

24

son for the deviation from the value 1.0 for an infinite plate lies in the influence of the initial hole. The influence of the drilled hole is studied in Fig. 10c. With increasing crack length a the influence of the hole decreases and the values of the geometry factor coincide. The results of the FE-calculation for a specimen without drilled hole are attached as supplementary material. ◦

0 The cyclic stress intensity factors ∆kCS for the 0◦ -specimen were calculated

in the last step using Eq. (9) for crack lengths in the interval of 5 mm ≤ a ≤ ◦

0 65 mm. For the calculation of ∆kCS , a unit force range of ∆f = 1 kN was used

and a comparison with the determined values of the FE-calculation can be done. ◦

0 The graph of the ∆kCS determined with the fit function Yyfit (a/W, 0◦ ) compared

to ky from the FE-calculation for both materials show deviations of less than 3 %, cf. Fig. 11a. Thus the geometry function Yyfit (a/W, 0◦ ) can be used to ◦

0 calculate the unit cyclic stress intensity factor ∆kCS and by taking advantage

of scaling, cf. section 2.4.1, also for the calculation of the cyclic stress intensity ◦

0 factor ∆KCS for 0◦ -specimens.

3.2.3.

Solution for the equibiaxially loaded 45◦ -specimens

The determination of the geometry function was performed analogously to the determination of the function for the 0◦ -specimens. Due to equibiaxial loading, the normal stresses sy (fx ) and sy (fy ) from section 3.1“ were used to calculate the stress intensity factors k∞ for different ” crack lengths a. The results of k∞ are attached as supplementary material. The corresponding kI -values and the kx - and ky -values from section 3.2.1 and the subsequently determined geometry factor Y (a/W, 45◦ ) using Eq. (13) are also attached as supplementary material. The geometry function Y fit (a/W, 45◦ ) determined by a polynomial fit for equibiaxial loading with Fx = Fy 6= 0 is given in Eq. (15).    p0     3 p X 1 n Y fit (a/W, 45◦ ) = pn · (a/W ) with   p2 n=0     p 3 25

= 0.9730 = 0.4293 =−1.9430 = 3.1962

(15)

(b)

2

√ 45° ∆kCS [MPa m]

√ 0° ∆kCS [MPa m]

(a)

1

0

0

steel aluminum 0° ∆kCS 0° ±3 % of ∆kCS 20 40 a [mm]

1

0

60

(c)

0

steel aluminum 45° ∆kCS 45° ±3 % of ∆kCS

20

40 60 a [mm]

80

(d)

80

40

√ 45° ∆KCS [MPa m]

√ 0° [MPa m] ∆KCS

2

30 20 10 0

0

0° ∆KCS 0° ±3 % of ∆KCS R = 0.1, crack 1 R = 0.1, crack 2 R = 0.5, crack 1 R = 0.5, crack 2 20 40 60 a [mm]

60 40 20 0

0

crack 1 crack 2 45° ∆KCS 45° ±3 % of ∆KCS

20

40 60 a [mm]

80

Figure 11: Calculated cyclic stress intensity factors for (a) 0◦ -specimen and (b) 45◦ -specimen loaded with unit force. Calculated cyclic stress intensity factors for experimentally determined crack paths resulting from (c) uniaxially loaded 0◦ -specimen (cf. Fig. 12a) and (d) equibiaxially loaded 45◦ -specimen (cf. Fig. 12b).

26

The course of the function is shown in Fig. 10b. Furthermore, the course of the fit functions Y fit (a/W, 45◦ ) is compared with the values of Y (a/W, 45◦ ) determined by FE. It is shown that also the determined fit function for 45◦ -specimens approximates the crack-length dependent values with a deviation of less than 3 %. As with 0◦ -specimens, the value of the geometry function Y fit (a/W, 45◦ ) deviates for short cracks from the value 1.0 for an infinite plate, cf. Fig. 10b. The limit value of Eq. (15) for a = 5 mm amounts to Y fit (a/W, 45◦ ) = 0.9866 and is in good agreement with the handbook-solution [56, p. 239] for a crack emanating from a circular hole of radius R under equibiaxial loading Y (a/W, 45◦ ) = 0.9855. The reason is the influence of the initial hole, cf. section 3.2.2 and Fig. 10d. The data of the FE-calculation of the 45◦ -specimen without drilled hole is attached as supplementary material. ◦

45 Again, the cyclic stress intensity factors ∆kCS for the 45◦ -specimen were

calculated using Eq. (12) for crack lengths in the interval of 5 mm ≤ a ≤ 85 mm and compared with the values of the geometry function Y fit (a/W, 45◦ ). The ◦

45 graph of the ∆kCS determined with the fit function Y fit (a/W, 45◦ ) compared

to the k from the FE-calculation for both materials show, that the deviations of the results are less than 3 %, cf. Fig. 11b. The data is attached as supplementary material. Thus the geometry function Y fit (a/W, 45◦ ) can be used to calculate ◦

45 the unit cyclic stress intensity factor ∆kCS and due to the possibility of scaling,

cf. section 2.4.1, also for the calculation of the cyclic stress intensity factor ◦

45 ∆KCS for 45◦ -specimens.

3.2.4.

Comparison of ideally straight and real crack paths

In addition, a comparison of the results of the geometry functions Yyfit (a/W, 0◦ ) and Y fit (a/W, 45◦ ) for ideally straight and real crack paths was performed. The crack paths as well as the measuring points for the FE-calculation are shown in Fig. 12a for the 0◦ -specimen and in Fig. 12b for the 45◦ -specimen. All measured crack tip coordinates are given in a recent work [44]. The results of the FE-calculation are attached as supplementary material. ◦

0 Figs. 11c and 11d show the results of the stress intensity factors ∆KCS and ◦

45 ∆KCS for the performed experiments as well as the results of the calculated

stress intensity factors using the determined fit functions. The comparison of 27

(a)

60

40

20

crack 1 R = 0.1 R = 0.5

crack 2 R = 0.1 R = 0.5

y x

−60

−40

−20

20

40

60

20

40

60

−20

−40 length in [mm] −60

(b)

crack 1

60 1 1

40

20 y −60

−40

x

−20 −20

−40 length in [mm] −6028

crack 2

Figure 12: Crack paths and measuring points for the FE-calculation of the (a) uniaxially loaded specimen and (b) equibiaxially 1 loaded specimen.

1

(b)

10−2

45

10−3

30 15 0

10−4

h

mm cycle

i

60

da dN

a [mm]

(a)

10−5 10−6 10−7

0 2 4 6 N · 105 [number of cycles]

uniaxial uniaxial uniaxial equibiaxial

(CT, ( 0°, ( 0°, (45°,

20

30 40√ 50 60 ∆K [MPa m]

uniaxial uniaxial uniaxial equibiaxial

R = 0.1) R = 0.1) R = 0.5) R = 0.1)

(CT, ( 0°, ( 0°, (45°,

R = 0.1) R = 0.1) R = 0.5) R = 0.1)

Figure 13: Crack growth plots of the unaxially loaded steel-0◦ -specimen and equibiaxially loaded steel-45◦ -specimen compared with a CT specimen. (a) a-N plot and (b) crack growth rate da/dN versus cylcic stress intensity factor ∆K.

stress intensity factors shows deviations of less than 3 %. This confirms that it is possible to calculate the stress intensity factors correctly using the geometry functions Y fit (a/W, 45◦ ) for 45◦ -specimen or Yyfit (a/W, 0◦ ) for 0◦ -specimen within minor errors. 3.2.5.

Comparison of the results of the fit function with the solution of the FE-calculation

As a result of the simple calculation of the K-solution, crack growth rate plots could easily be calculated. Fig. 13a shows measured a-N plots for a compact tension specimen (CT) made of steel compared to the a-N plots for the two tested cruciform specimens. Due to the symmetry of the cracks and comparable crack growth rates, only one crack is shown. It is clearly visible, that the crack of the 0◦ -specimen, loaded with a force ratio of R = 0.1, grows faster compared to a subsequent loading with R = 0.5. Furthermore, the equibiaxially loaded crack of the 45◦ -specimen with a force ratio of R = 0.1 seems to grow slower than the uniaxially loaded crack of 0◦ -specimen. After the computation of the K-solutions for the different crack configurations, all cracks show a comparable crack growth rate, cf. Fig. 13b. This is a reasonable result, which confirms the computed K-factors as well as the utilized geometry functions.

29

4.

Summary Cruciform specimens based on the design by Brown and Miller [34] were

examined. It was demonstrated that the coupling of the forces due to clamping effects in the measuring area is less than 3 %. The determined geometry functions Y fit (a/W, 45◦ ) and Yyfit (a/W, 0◦ ) approximate the crack-length dependent geometry values for K-factors with a deviation of less than 3 %. Furthermore, depending on the alignment of the starting notches, an approximate materialindependent K-solution for a straight crack path was found and compared to experimental crack paths for different R-ratios. The deviations of the cyclic stress intensity factors were also less than 3 %. This means that an easy calculation of K-factors and their ranges can be done in the proposed manner, which is now available for other researchers. The transferability of geometry functions Yyfit (a/W, 0◦ ) and Y fit (a/W, 45◦ ) is exactly valid, if the considered cruciform specimens have the same geometry as presented here, cf. Fig. 3 and Henkel et al. [6], or are proportionally scaled in their dimensions. Only the relationship between the applied forces Fx , Fy and the stresses σx and σy in the specimen center have to be adopted. Moreover, even for similar other cruciform specimens with approximate constant stress state in the measuring area, the geometry function can be used, since they are referred to the stress state in the specimen center. The user has to calculate merely the transfer functions with the applied forces. Furthermore, it was shown that the fatigue crack growth measured by CT specimens and cruciform specimens with a different alignment of the initial notches or rather different loading conditions yield comparable crack growth rates. Acknowledgement The authors thank and acknowledge gratefully funding of subprojects B4 and C5 within Collaborative Research Center SFB 799 (Project number 54473466) by German Research Foundation (DFG). In particular, the authors want to thank Stephanie Ackermann for her support in the production of the specimens and Matthias Brensing and Yangxi Qiu for their support during the uniaxial and equibiaxial experiments. The authors would also like to thank Matthias Droste and Tim H¨ uhnerf¨ urst for providing the crack growth data of the CT specimen. 30

References [1] C. D. Hopper, K. J. Miller, Fatigue crack propagation in biaxial stress fields, The Journal of Strain Analysis for Engineering Design 12 (1977) 23–28. doi:10.1243/03093247v121023. [2] P. R. G. Anderson, G. G. Garrett, Fatigue crack growth rate variations in biaxial stress fields, International Journal of Fracture 16 (1980) R111– R116. doi:10.1007/bf00013388. [3] E. Breitbarth, M. Besel, S. Reh, Biaxial testing of cruciform specimens representing characteristics of a metallic airplane fuselage section, International Journal of Fatigue 108 (2018) 116–126. doi:10.1016/j.ijfatigue. 2017.12.005. [4] E. Breitbarth, M. Besel, Fatigue crack deflection in cruciform specimens subjected to biaxial loading conditions, International Journal of Fatigue 113 (2018) 345–350. doi:10.1016/j.ijfatigue.2018.04.021. [5] V. Giannella, G. Dhondt, C. Kontermann, R. Citarella, Combined staticcyclic multi-axial crack propagation in cruciform specimens, International Journal of Fatigue 123 (2019) 296–307. doi:10.1016/j.ijfatigue.2019. 02.029. [6] S. Henkel, C. H. Wolf, A. Burgold, M. Kuna, H. Biermann, Cruciform specimens used for determination of the influence of T-stress on fatigue crack growth with overloads on aluminum alloy Al 6061 T651, Fracture and Structural Integrity 13 (2019) 135–143. doi:10.3221/IGF-ESIS.48.16. [7] A. Burgold, S. Henkel, S. Roth, M. Kuna, H. Biermann, Fracture mechanics testing and crack growth simulation of highly ductile austenitic steel, Materials Testing 60 (2018) 341–348. doi:10.3139/120.111156. [8] D. Martelo, A. Mateo, M. Chapetti, Fatigue crack growth of a metastable austenitic stainless steel, International Journal of Fatigue 80 (2015) 406– 416. doi:10.1016/j.ijfatigue.2015.06.029.

31

[9] H. J. Lim, Y.-J. Lee, H. Sohn, Continuous fatigue crack length estimation for aluminum 6061-t6 plates with a notch, Mechanical Systems and Signal Processing 120 (2019) 356–364. doi:10.1016/j.ymssp.2018.10.018. [10] O. Scott-Emuakpor, T. George, C. Cross, M.-H. H. Shen, Multi-axial fatigue-life prediction via a strain-energy method, AIAA Journal 48 (2010) 63–72. doi:10.2514/1.39296. [11] Z. Zhu, Z. Lu, P. Zhang, W. Fu, C. Zhou, X. He, Optimal design of a miniaturized cruciform specimen for biaxial testing of TA2 alloys, Metals 9 (2019) 823. doi:10.3390/met9080823. [12] R. Yuuki, K. Akita, N. Kishi, The effect of biaxial stress state and changes of the state on fatigue-crack growth behavior,

Fatigue & Fracture of

Engineering Materials and Structures 12 (1989) 93–103. doi:10.1111/j. 1460-2695.1989.tb00516.x. [13] H. Kitagawa, R. Yuuki, K. Tohgo, A fracture mechanics approach to high cycle fatigue crack growth under in-plane biaxial loads, Fatigue & Fracture of Engineering Materials and Structures 2 (1979) 195–206. doi:10.1111/j. 1460-2695.1979.tb01355.x. [14] H. Misak, V. Perel, V. Sabelkin, S. Mall, Crack growth behavior of 7075-t6 under biaxial tension–tension fatigue, International Journal of Fatigue 55 (2013) 158–165. doi:10.1016/j.ijfatigue.2013.06.003. [15] V. Shlyannikov, A. Zakharov, Multiaxial crack growth rate under variable t-stress, Engineering Fracture Mechanics 123 (2014) 86–99. doi:10.1016/ j.engfracmech.2014.02.013. [16] R. Sunder, B. Ilchenko, Fatigue crack growth under flight spectrum loading with superposed biaxial loading due to fuselage cabin pressure, International Journal of Fatigue 33 (2011) 1101–1110. doi:10.1016/j.ijfatigue. 2010.11.018. [17] A. Shanyavskiy, Fatigue cracking simulation based on crack closure effects in al-based sheet materials subjected to biaxial cyclic loads, Engineering

32

Fracture Mechanics 78 (2011) 1516–1528. doi:10.1016/j.engfracmech. 2011.01.019. [18] V. Shlyannikov, A. Tumanov, A. Zakharov, The mixed mode crack growth rate in cruciform specimens subject to biaxial loading, Theoretical and Applied Fracture Mechanics 73 (2014) 68–81. doi:10.1016/j.tafmec.2014. 06.016. [19] E. Lee, R. Taylor, Fatigue behavior of aluminum alloys under biaxial loading, Engineering Fracture Mechanics 78 (2011) 1555–1564. doi:10.1016/ j.engfracmech.2010.11.005. [20] M. Lepore, F. Berto, D. Kujawski, Non-linear models for assessing the fatigue crack behaviour under cyclic biaxial loading in a cruciform specimen,

Theoretical and Applied Fracture Mechanics 100 (2019) 14–26.

doi:10.1016/j.tafmec.2018.12.008. [21] S. Mall, V. Perel, Crack growth behavior under biaxial fatigue with phase difference, International Journal of Fatigue 74 (2015) 166–172. doi:10. 1016/j.ijfatigue.2015.01.005. [22] V. Shlyannikov, A. Zakharov, Generalization of mixed mode crack behaviour by the plastic stress intensity factor, Theoretical and Applied Fracture Mechanics 91 (2017) 52–65. doi:10.1016/j.tafmec.2017.03.014. [23] C. Dalle Donne, K.-H. Trautmann, H. Amstutz, Cruciform specimens for in-plane biaxial fracture, deformation, and fatigue testing, in: Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM International, 2000, pp. 405–422. doi:10.1520/stp13517s. [24] F. Khelil, B. Aour, A. Talha, N. Benseddiq, Numerical investigation of mechanical behavior of cracked cruciform specimens in aluminum alloy 6082T6 subjected to different biaxial loading conditions, Journal of Theoretical and Applied Mechanics 57 (2019) 1021–1037. doi:10.15632/jtam-pl/ 112458. [25] D. Infante-Garc´ıa, G. Qian, H. Migu´elez, E. Giner, Analysis of the effect of out-of-phase biaxial fatigue loads on crack paths in cruciform specimens 33

using XFEM, International Journal of Fatigue 123 (2019) 87–95. doi:10. 1016/j.ijfatigue.2019.01.019. [26] A. Makinde, L. Thibodeau, K. W. Neale, Development of an apparatus for biaxial testing using cruciform specimens, Experimental Mechanics 32 (1992) 138–144. doi:10.1007/bf02324725. [27] I. H. Wilson, D. J. White, Cruciform specimens for biaxial fatigue tests: An investigation using finite-element analysis and photoelastic-coating techniques,

Journal of Strain Analysis 6 (1971) 27–37. doi:10.1243/

03093247v061027. [28] T. Ogata, Y. Takahashi, Development of a high-temperature biaxial fatigue testing machine using a cruciform specimen, in: Multiaxial Fatigue and Fracture, Elsevier, 1999, pp. 101–114. doi:10.1016/s1566-1369(99) 80010-7. [29] S. Taira, K. Tanaka, M. Kan, A. Yamada, Effect of non-singular stress on fatigue crack propagation in biaxial stress fields, Proceedings of the 22nd Japan Congress on Materials Research (1979) 130–137. [30] T. Hoshide, K. Tanaka, A. Yamada, Stress-ratio effect of fatigue crack propagation in a biaxial stress field, Fatigue & Fracture of Engineering Materials and Structures 4 (1981) 355–366. doi:10.1111/j.1460-2695.1981. tb01132.x. [31] K. Tanaka, T. Hoshide, A. Yamada, S. Taira, Fatigue crack propagation in biaxial stress fields, Fatigue & Fracture of Engineering Materials and Structures 2 (1979) 181–194. doi:10.1111/j.1460-2695.1979.tb01354.x. ¨ [32] C. D. Donne, Ubertragbarkeit von Risswiderstandskurven von Standardproben auf biaxial belastete, bauteil¨ahnliche Kreuzproben, VDI Verlag, 1997. [33] P. Bold, M. Brown, R. Allen, Shear mode crack growth and rolling contact fatigue, Wear 144 (1991) 307–317. doi:10.1016/0043-1648(91)90022-m.

34

[34] M. Brown, K. Miller, Mode I fatigue crack growth under biaxial stress at room and elevated temperature, in: Multiaxial Fatigue, ASTM International, 1985, pp. 135–135. doi:10.1520/stp36221s. [35] S. Datta, A. Chattopadhyay, N. Iyyer, N. Phan, Fatigue crack propagation under biaxial fatigue loading with single overloads, International Journal of Fatigue 109 (2018) 103–113. doi:10.1016/j.ijfatigue.2017.12.018. [36] S. Henkel, D. Holl¨ ander, M. W¨ unsche, H. Theilig, P. H¨ ubner, H. Biermann, S. Mehringer, Crack observation methods, their application and simulation of curved fatigue crack growth, Engineering Fracture Mechanics 77 (2010) 2077–2090. doi:10.1016/j.engfracmech.2010.04.013. [37] R. Neerukatti, S. Datta, A. Chattopadhyay, N. Iyyer, N. Phan, Fatigue crack propagation under in-phase and out-of-phase biaxial loading, Fatigue & Fracture of Engineering Materials & Structures 41 (2017) 387–399. doi:10.1111/ffe.12690. [38] V. Shlyannikov, T-stress for crack paths in test specimens subject to mixed mode loading, Engineering Fracture Mechanics 108 (2013) 3–18. doi:10. 1016/j.engfracmech.2013.03.011. [39] H. Theilig, D. Hartmann, M. W¨ unsche, S. Henkel, P. H¨ ubner, Numerical and experimental investigations of curved fatigue crack growth under biaxial proportional cyclic loading, Key Engineering Materials 348-349 (2007) 857–860. doi:10.4028/www.scientific.net/kem.348-349.857. [40] A. C. Pickard, Fatigue crack propagation in biaxial stress fields, The Journal of Strain Analysis for Engineering Design 50 (2014) 25–39. doi:10. 1177/0309324714551082. [41] M. O. Wang, R. H. Hu, C. F. Qian, J. C. M. Li, Fatigue crack growth under Mode II loading, Fatigue & Fracture of Engineering Materials and Structures 18 (1995) 1443–1454. doi:10.1111/j.1460-2695.1995.tb00867.x. [42] S. Henkel, E. Liebelt, H. Biermann, S. Ackermann, Crack growth behavior of aluminum alloy 6061 t651 under uniaxial and biaxial planar

35

testing condition,

Fracture and Structural Integrity 9 (2015) 466–475.

doi:10.3221/IGF-ESIS.34.52. [43] C. H. Wolf, S. Henkel, A. Burgold, Y. Qiu, M. Kuna, H. Biermann, Fatigue crack growth in austenitic stainless steel: Effects of phase shifted loading and crack paths, Advanced Engineering Materials 21 (2018) 1800861. doi:10.1002/adem.201800861. [44] C. H. Wolf, S. Henkel, A. Burgold, Y. Qiu, M. Kuna, H. Biermann, Investigation of fatigue crack growth under in-phase loading as well as phaseshifted loading using cruciform specimens, International Journal of Fatigue 124 (2019) 595–617. doi:10.1016/j.ijfatigue.2019.03.011. [45] S. Xiao, M. Brown, Fatigue crack growth in notched AISI 316 stainless steel plates under biaxial loading, in: Mechanical Behaviour of Materials V, Elsevier, 1988, pp. 659–664. doi:10.1016/b978-0-08-034912-1.50089-8. [46] M.-. Wang, R.-H. Hu, C.-J. Zhang, S.-H. Dai, An experimental investigation on the fatigue crack growth in mode II loadig condition, in: Proceedings of the 7th International Conference on Pressure Vessel Technology, ¨ 1992, pp. 1413–1427. volume 2, VdTUV, [47] C. Qian, M.-O. Wang, B.-J. Wu, S.-H. Dai, J. C. M. Li, Mixed-mode fatigue crack growth in stainless steels under biaxial loading, Journal of Engineering Materials and Technology 118 (1996) 349–355. doi:10.1115/ 1.2806817. [48] C.-F. Qian, M.-O. Wang, B.-J. Wu, S.-H. Dai, J. C. M. Li, Symmetric branching of mode II and mixed-mode fatigue crack growth in a stainless steel, Journal of Engineering Materials and Technology 118 (1996) 356–361. doi:10.1115/1.2806818. [49] J. J. Kibler, R. Roberts, The effect of biaxial stresses on fatigue and fracture, Journal of Engineering for Industry 92 (1970) 727–734. doi:10. 1115/1.3427838. [50] DIN EN 485-2:2018-12, Aluminium und Aluminiumlegierungen - B¨ander, Bleche und Platten - Teil 2: Mechanische Eigenschaften; Deutsche Fassung EN 485-2:2016+a1:2018, 2018. doi:10.31030/2893565. 36

[51] E. M¨ onch, D. Galster, A method for producing a defined uniform biaxial tensile stress field, British Journal of Applied Physics 14 (1963) 810–812. doi:10.1088/0508-3443/14/11/319. [52] A. A. Griffith, The theory of rupture, Proceedings of the First International Congress for Applied Mechanics (1924) 55–63. Delft. [53] H. A. Richard, M. Fulland, M. Sander, Theoretical crack path prediction, Fatigue & Fracture of Engineering Materials and Structures 28 (2005) 3–12. doi:10.1111/j.1460-2695.2004.00855.x. [54] H. Richard, B. Schramm, N.-H. Schirmeisen, Cracks on mixed mode loading – theories, experiments, simulations, International Journal of Fatigue 62 (2014) 93–103. doi:10.1016/j.ijfatigue.2013.06.019. [55] H. Richard, A. Eberlein,

Material characteristics at 3d-mixed-mode-

loadings, Procedia Structural Integrity 2 (2016) 1821–1828. doi:10.1016/ j.prostr.2016.06.229. [56] Y. Murakami, Stress intensity factors handbook, Pergamon Press, 1987.

37

     

Biaxial fatigue crack growth investigation  Investigation of the stress distribution in cruciform specimens along the crack paths  Development of simple model for the K‐solution calculation for cruciform specimens  Comparison of FE results with experimentally determined results or rather crack paths