Experiments and numerical simulation of damage and fracture of the X0-specimen under non-proportional loading paths

Engineering Fracture Mechanics xxx (xxxx) xxxx

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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Experiments and numerical simulation of damage and fracture of the X0-specimen under non-proportional loading paths ⁎

Steffen Gerke , Moritz Zistl, Michael Brünig Institut für Mechanik und Statik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany

A R T IC LE I N F O

ABS TRA CT

Keywords: Ductile materials Damage and fracture Numerical simulations Biaxial experiments Non-proportional loading paths

The paper deals with new non-proportional biaxial experiments and corresponding numerical simulations to analyze the load-history dependence on damage and fracture behavior of ductile metals. The numerical simulations are based on a phenomenological, thermodynamically consistent anisotropic continuum damage model considering the effect of stress triaxiality and Lode parameter on damage behavior. The model as well as the corresponding material parameter identification is discussed and a full set of material parameters is given. The new biaxial experiments take into account non-proportional and corresponding proportional loading paths with special focus on stress states changes from shear to tension dominated cases. The corresponding strain fields of critical regions of the X0-specimen have been analyzed by digital image correlation technique and indicate good accordance with the numerically predicted ones. The evolution of the calculated plastic and damage fields elucidate the relevance of both independent mechanisms. The results are supported by scanning electron microscopy images of the fracture surfaces. This experimental–numerical technique applied to the non-proportional biaxial experiments provides new insights of the stress and load-history dependent damage and fracture processes.

1. Introduction Ductile sheet metals are of outstanding relevance as raw material in many engineering applications. Within the subsequent fabrication process frequently large deformations including multiple stages and possibly reverse loading occur. In addition, environmental and economic requirements have to be met and thus higher performance ratios have to be achieved. Consequently, material degradation and material failure (i.e. damage and fracture) have to be controlled during the fabrication process and a detailed knowledge of the stress state and loading history dependent damage and fracture processes is required. To facilitate this information and enable numerical predictions of the fabrication process well controlled experiments with carefully designed specimens have to be performed. These tests mandatorily need to include different stress states and non-proportional loading to characterize the material behavior comprehensive way. Furthermore, a phenomenological material model has to be provided which can reflect different material degradation processes and facilitate corresponding numerical simulations. During loading and deformation often localization of inelastic strains takes place. They usually indicate damage and fracture processes and the evolution of these mechanisms on the micro-scale generally leads to macro-cracks and finally to fracture of structural elements. This degradation process is mainly influenced by two aspects: firstly, by the stress state acting in a material point. For example, nucleation, growth and coalescence of micro-voids occur during tension loading with high positive stress triaxialities

⁎

Corresponding author. E-mail address: steff[email protected] (S. Gerke).

https://doi.org/10.1016/j.engfracmech.2019.106795 Received 3 June 2019; Received in revised form 27 September 2019; Accepted 23 November 2019 0013-7944/ © 2019 Published by Elsevier Ltd.

Please cite this article as: Steffen Gerke, Moritz Zistl and Michael Brünig, Engineering Fracture Mechanics, https://doi.org/10.1016/j.engfracmech.2019.106795

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f da damage condition f pl yield condition F1, F2 applied forces g da damage potential function H hardening modulus I1, J2 , J¯2 , J3 invariants of (deviatoric) stress tensors n hardening exponent T1, T2 , T3 principal stresses u displacement ¯,N N normalized stress tensors ¯, T T stress tensors ¯ ̇ pl , Ḣ da strain rate tensors Ḣ , Ḣ el , H Ael , Ada strain tensors

Nomenclature

α, α¯ , λ ,̇

damage mode parameters damage rule parameters rate of internal variables stress triaxiality η η1…η4 elastic-damage moduli γ internal plastic variable μ internal damage variable ω Lode parameter σ , σ0 equivalent damage stresses σeq von Mises equivalent stress σm mean stress c , c0 yield stresses E , ν , G , K elastic material parameters

β β¯ γ̇, μ̇

whereas formation and growth of micro-shear-cracks is the predominant process during shear and compression loading with small positive or negative stress triaxialities. Combination of these basic damage mechanisms on the micro-level occurs for moderate positive stress triaxialities and no damage formation has been observed in ductile metals for high negative stress triaxialities. Secondly, the damage and fracture behavior strongly depends on the loading path which is often non-proportional in general applications and may even be characterized by load reversal. These two aspects have to be taken into account at the development of corresponding continuum models as well as at the experimental studies to characterize these processes. The stress state within the region of interest can be controlled by the applied load and the geometry of the specimen. Thus, many experiments with different specimens taken from metal sheets have been proposed by various research groups. Dog bone specimens (Fig. 1(a)) are commonly used to identify the elastic–plastic material parameters. They are characterized by a homogeneous stress state throughout the experiment till necking takes place. With differently notched specimens (b) influence of the stress state on the

Fig. 1. Different specimen geometries for in plane testing of thin sheet metal, adopted from literature, see references in text. 2

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material behavior can be studied as the stress triaxiality increases with decreasing notch radius, see for example [1–3]. A similar effect is achieved by introducing a central hole (c) in the uniaxially loaded specimen [4] but the stress state is more inhomogeneous due to the deformation of the free side of the specimen. Plane strain specimens (d) with notches in thickness direction [5] indicate a rather homogeneous stress state within the notched region which can be seen as an advantage of this geometry. The geometries (a-d) 1 indicate tension dominated stress with stress triaxialities ⩾ 3 defined by the ratio of mean stress and von Mises equivalent stress. It is a big challenge to study shear dominated stress states in a one–dimensional testing device. In this context different geometries with one shear zone (e) have been proposed, see e.g. [2,3,6]. Unfortunately during elongation of the bar this type of specimen shows rotations of the central part leading to tension dominated stress states. This effect can be reduced by adding notches in thickness direction [7] or by the symmetric double shear specimen (f) presented by [8] which can be seen in continuation of the double symmetric specimen presented by [9] and mentioned by [10]. Furthermore, [11] designed a circular notched specimen (j) and by opposed rotation shear loading is forced. The up to now presented specimens can be used for one loading condition and load reversal can only be applied with specimens (g) and (j). For more arbitrary non-proportional loading it is crucial that one specimen can be used under different loading conditions. This idea has been presented by [12] (h) where different loading angles cause different stress states in the region of interest. This idea has been followed up by [13] for the design of the butterfly specimen which can be tested under different angles and loading conditions but requires a special testing device. Moreover, classical cruciform geometries (see for instance [10]) have been designed to analyze the yield surface and the subsequent plastic behavior. These specimens are characterized by an extended central part where at the beginning of the experiment relatively homogeneously distributed stress and strain states are obtained and evaluated, but with ongoing deformation strains tend to localize and a well controlled study of damage and fracture under pre-defined loading conditions is difficult to realize with this type of cruciform specimen. Most specimens (a-g, j) have been designed to characterize the material behavior under one loading condition whereas the geometries (g, j) allow load reversal and consequently cyclic loading. The specimens (h, i, k, l) can be used under different loading conditions facilitating the possibility of non-proportional loading scenarios. Furthermore it is worthy to mention that all discussed geometries (besides (a)) show stress gradients within the region of interest. In this context [14,15] presented new biaxial specimens for sheet metals with moderate thickness. Amongst others the X0-specimen indicated applicability in a wide range of stress states including shear superimposed with compression or tension as well as tension dominated states. The geometry of this cruciform specimen is characterized by two crosswise arranged slotted holes which leave connecting material between neighboring machine clamping supports of the specimen. This connecting material is the region of interest and it is additionally notched in thickness direction from both sides to localize the inelastic strain field and consequently damage and fracture. The loading history dependence of damage and fracture behavior of ductile metals has been shown in several studies: For instance, experiments on the micro-level including in situ X-ray tomography and in situ scanning electron microscopy (SEM) have indicated the effect of void distribution as well as of pre-compression of the tensile specimens on reverse loading [16]. Furthermore, the influence of path dependence on forming limit stresses has been studied (see for instance [17,18]) which indicated a remarkable effect. Tubes or hollow cylindrical specimens facilitate good possibilities to study non-proportional loading, since tension, torsion and internal pressure can be applied. In this context the path dependence could be shown for tension–internal pressure [19] and for tension–torsion [20] combinations. Focussing on sheet metal with in plane loading, [21,22] presented a first approach of two-stage tests where firstly a large material sample is deformed by uni-axial tension while secondly from this sample smaller specimens are cut at a certain angle and subsequently tested, see Fig. 1(m). With this method major influences of the loading history could be detected while the elastic unloading and cutting between the two stages does not reflect the continuous change in loading conditions of most fabrication processes and it does not seem favorable to machine the specimen after the first stage. Standard cruciform specimens have been used by [23,24] to study the crack propagation of a previously introduced crack with varying inclination. In [25,26] a special testing device with two actuators for tension and shear loading to study non-proportional loading is discussed while this testing device is similar to the one used for the recently proposed butterfly specimens [27,8]. Consequently, the need of new experimental approaches to study the effect of loading-history on damage and fracture is evident and the here presented new non-proportional load cases tend to fill this gap. Overall the non-proportional testing of sheet metals to analyze the effect of the loading history on damage and fracture behavior is a challenging task. Especially the localization tendencies of these processes and the risk of buckling under compression loads complicate the proposal of new experimental techniques. Gerke et al. presented in [28] experiments with the X0-specimen under proportional and non-proportional loading conditions including shear-compression and shear-tension stress states and Brnig et al. [29] presented experiments and corresponding numerical simulations with the H-specimen. The loading-path dependence of the damage and fracture processes could be clearly shown and the X0-specimen has proven to be suitable to analyze in detail the influence of the stress state and loading path on the damage and fracture behavior of ductile sheet metals. In the present paper, new non-proportional load cases with changes from shear to tension dominated stress states with the X0specimen are discussed in detail. In this context the employed phenomenological continuum damage model is briefly presented and the corresponding material parameter identification is discussed in its present form. Experimental results including load-displacement data, strain fields reported by digital image correlation as well as images of the fracture surfaces obtained by scanning electron microscopy and numerical results of corresponding numerical simulations of the biaxial experiments indicating the stress fields are presented. Based on the given experimental and numerical results stress-state-dependent damage and resulting fracture mechanisms are elucidated for the new non-proportional load cases. Consequently, the present paper is organized as follows: the next section gives a brief overview of the phenomenological description of the damage behavior of ductile metals by a continuum damage model proposed by [30]. Subsequently the material parameters are motivated and a full set of parameters is given for the aluminum alloy EN AW 6082 (AlSiMgMn). The main part of the paper focuses on biaxial experiments under proportional and non-proportional loading 3

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with the X0-specimen and corresponding numerical simulations applying the previously discussed continuum damage model. Finally, the main results are discussed and perspectives to future work are given. 2. Continuum damage model In this section the theoretical formulation of the phenomenological continuum damage model is presented, see [30,31]. The description of inelastic deformation, i.e. the evolution of plastic and anisotropic damage behavior can be adequately described by this approach and the numerical simulations within this paper have been realized with its numerical implementation. The thermodynamically consistent framework is based on the introduction of damaged and corresponding fictitious undamaged configurations. el ¯ ̇ pl and damage Ḣ da strain rate tensors are introduced. The damage strain rate tensor In the kinematic description elastic Ḣ , plastic H allows a meaningful representation of the volume fraction of micro-defects and takes into account the influence of their current shape and distortion on the macroscopic material behavior. In particular the onset of plastic yielding is described by the yield criterion

f pl (J¯2 , c ) =

J¯2 − c = 0,

(1) 1 2

¯ ·devT ¯ of the effective Kirchhoff stress tensor and the equivalent yield stress c. with the second deviatoric invariant J¯2 = devT In addition, the isochoric effective plastic strain rate pl ¯ = γ̇ N ¯ ¯ ̇ pl = λ ̇ ∂f = λ ̇ 1 devT H ¯ ∂T 2 J¯2

(2)

¯ = ⎛ 1 ⎞ devT ¯ represents the normalized is taken to describe the plastic deformation. Here λ ̇ is a non-negative scalar-valued factor, N ⎝ 2J¯2 ⎠ 1 deviatoric stress tensor and γ ̇ = 2 λ ̇ is chosen to be the equivalent plastic strain rate characterizing the amount of plastic deformation. The constitutive equations of anisotropically damaged ductile metals have to take into account the deterioration of the elastic behavior due to damage. With the elastic Ael and the damage strain tensor Ada the Kirchhoff stress tensor can be expressed as

( )

T = 2(G + η2 trAda) Ael

(

)

2

+ ⎡ K − 3 G + 2η1 trAda trAel + η3 (Ada ·Ael) ⎤ 1 ⎣ ⎦ + η3 trAelAda + η4 (AelAda + AdaAel),

(3)

where the constitutive parameters η1, …, η4 correspond to the deteriorating effects on the elastic properties. Furthermore, the onset of damage in ductile metals is characterized by the stress-state-dependent damage condition

f da = αI1 + β J2 − σ = 0

(4)

expressed in terms of the first and second deviatoric invariants, I1 and J2 , of the Kirchhoff stress tensor (Eq. (3)). The equivalent damage stress measure σ represents the material toughness to micro-defect propagation. In Eq. (4) the variables α and β are damage mode parameters corresponding to different damage mechanisms acting on the micro-level and are chosen with respect to the current stress state acting on the material point. The stress state can be uniquely described by the stress intensity

σeq =

3J2

(5)

also known as the von Mises equivalent stress, the stress triaxiality

η=

σm I1 = σeq 3 3J2

(6)

and the Lode parameter

ω=

2T2 − T1 − T3 T1 − T3

with T1 ⩾ T2 ⩾ T3.

(7)

In this context σm is the mean stress and T1, T2 and T3 are the principal components of the Kirchhoff stress tensor. Furthermore, the damage strain rate tensor is given by the damage rule

β¯ ⎞ α¯ Ḣ da = μ̇ ⎛⎜ 1+ N⎟ 2 ⎠ ⎝ 3

(8)

where μ̇ is the equivalent damage strain rate measure characterizing the increase in irreversible damage strains due to damage and 1 devT is related to the normalized deviatoric stress tensor. In Eq. (8) α¯ and β¯ are kinematic variables describing the portion

N=

( ) 2J2

of volumetric and isochoric damage-based deformations. These parameters are stress state dependent and micro-mechanically motivated and correspond to the different damage and fracture processes on the micro-scale discussed above. 4

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3. Material In the present paper the behavior of the aluminum alloy EN AW 6082 (AlSiMgMn) with the temper designation T6 is analyzed. The material is characterized by a smooth transition to inelastic behavior with not very pronounced hardening. Experiments with smooth tension tests cut out in different direction confirmed that in-plane isotropic material behavior can be assumed which could be confirmed by photomicrograph of polished surfaces. The specimens are milled out of sheets with a thickness of 4.0 mm . The basic elastic-plastic material parameters have been determined by experiments with dog-bone shaped specimens as indicated in Fig. 1(a). The central part of the specimen has a width of 5.0 mm and a length of 60 mm. The force signal F of the load cell has been transmitted to the Dantec/Limess digital image correlation (DIC) system equipped with two 6 Mpx cameras and stored with the DIC data sets. The virtual gauge length of the DIC system has been chosen to 25.0 mm located centered at the specimen and Δu represents the length change of the virtual gauge length. Fig. 2 displays the corresponding experimentally (Exp) obtained results. Based on the elastic part of the diagram Young’s modulus E is 69·103 MPa and Poisson’s ratio ν is taken to be to 0.29. Based on standard calculations for isotropic elastic material behavior this leads to the bulk K and shear modulus G declared in the summarizing Table 1 at the end of this section. The plastic hardening behavior is adequately described by the power law n

H γ c = c0 ⎛ 0 + 1⎞ . ⎝ n c0 ⎠ ⎜

⎟

(9)

The corresponding material parameters have been fitted to the c-γ -diagram given in Fig. 3(a) curve (Exp) with the initial yield stress c0 = 163.5 MPa , the hardening modulus H0 = 850 MPa and the hardening exponent n = 0.182 . The corresponding numerical fit is plotted in Fig. 3(a) curve c (γ ) showing excellent agreement with the experimental results. The onset of damage is determined by comparing experimentally obtained curves with numerically predicted ones, see [2] for more details. This leads to an initial damage stress of σ0 = 222.0 MPa . Following [32] the damage softening is assumed to be described by the quadratic function

σ = σ0 − H1 μ2

(10)

without initial slope and a damage softening modulus H1 = 400 MPa , see Fig. 3(b). Furthermore, the damage parameters α and β of the damage condition (4) have to be chosen with respect to the stress state expressed by the stress triaxiality (6) and the Lode parameter (7). The parameter α is related to the hydrostatic stress part expressed by the invariant of the stress tensor I1 and is consequently related to the resistance of initiating void growth. This is reflected in

0 α (η) = ⎧ ⎨ ⎩1/3

for η < 0 , for η > 0

(11)

i.e. for negative stress triaxialities this influence is neglected and for positive stress triaxialities the influence increases with increasing I1. The non-negative parameter β is related to the deviatoric stress part expressed by J2 (second deviatoric invariant) and thus, is related to the resistance of developing micro-shear-cracks. It is chosen with respect to detailed studies [31,33] with an aluminum alloy to

β (η , ω) = β0 (η) + βω (ω) ⩾ 0

(12)

β0 (η) = −1.280η + 0.850

(13)

with

Fig. 2. Force F vs. displacement Δu curves of the unnotched tensile specimen. 5

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Table 1 Material parameters. G [MPa]

K [MPa]

26.74·103

54.76·103

σ0 [MPa] 222

η1 [MPa]

η2 [MPa]

η3 [MPa]

η4 [MPa]

− 10·103

− 2.5·103

− 10·103

− 2.5·103

H1 [MPa] 400

c0 [MPa] 163.5

H0 [MPa] 850

n [−] 0.182

η1̃ [MPa]

η2̃ [MPa]

η3̃ [MPa]

η4̃ [MPa]

− 4·103

− 300·103

+ 600·103

− 75·103

Fig. 3. (a) Plastic hardening c–γ and (b) damage softening σ – μ –diagram.

and

βω (ω) = −0.017ω3 − 0.065ω2 − 0.078ω

(14)

clearly reflecting that with decreasing stress triaxiality the influence of J2 increases and the effect of ω is also taken into account in this study for aluminum alloys. In addition, for metals a cut-off value of stress triaxiality is introduced below which no further damage occurs, see [34]. The stress state dependent damage mode parameters (α¯, β¯ ) of the damage rule (8) have been determined by numerical calculations on the micro-level analyzing deformation and fracture behavior of differently loaded void-containing unit cells [31,33]. Numerical simulations with the micro-mechanically determined parameters α¯ and β¯ with different materials indicate that these parameters might be constant for ductile metals and consequently, they are maintained within this paper. Future work will be dedicated to additional micro-mechanical simulations and to link experimental data to these parameters. In particular, the parameter α¯ corresponds to isotropic volume changes of micro-defects whereas the parameters β¯ corresponds to formation of micro-shear-cracks and governs the amount of anisotropic isochoric damage strains. Thus, the basic microscopic damage and fracture mechanisms (isotropic growth of micro-voids and evolution of micro-shear-cracks) are incorporated in the macroscopic damage rule (8). In particular, the stress-state-dependence of the parameter α¯ is expressed in the form

⎧0 α¯ (η) = 0.571η ⎨ ⎩1

for η ⩽ 0 for 0 < η ⩽ 1.75 , for η > 1.75

(15)

and β¯ is given by

β¯ (η , ω) = β¯0 (η) + fβ (η) β¯ω (ω),

(16)

with

⎧ 0.870 β¯0 (η) = 0.979 − 0.326η ⎨ ⎩0

for η ⩽ 1/3 for 1/3 < η ⩽ 3 , for η > 3

(17)

fβ = −0.025 + 0.038η

(18) 6

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and

1 − ω2 β¯ω (ω) = ⎧ ⎨ ⎩0

for η ⩽ 2/3 . for η > 2/3

(19)

The damage moduli η1, η2 , η3 and η4 introduced in Eq. (3) have been firstly determined in [35] based on 4 different experiments presented by [36]. The here applied parameters differ from the originally determined ones which indicates a material dependence. Numerical simulations with the indicated set of parameters lead to good agreement between simulation and experiment, see for instance Fig. 2 for the uniaxial load case. The corresponding values of η1…η4 are given in Table 1 for non-negative stress triaxialities. For negative negative stress triaxialities a second set of parameters η1̃…η4̃ has been introduced to be able to simulate the experimentally observed behavior for compressive loading in a accurate manner. This approach needs further attention in the future and might lead to stress state dependent functions for η1…η4 . All material parameters are summarized in Table 1. 4. Biaxial experiments and corresponding numerical simulations Biaxial experiments with carefully designed specimens facilitate the possibility to generate different stress states with one specimen geometry and in addition to change within one experiment the stress state, i.e. to perform experiments with non-proportional loading paths. In this context [14] proposed several new geometries and [28] presented experimental results with the X0-specimen under proportional and non-proportional loading conditions. Within this section new experimental results and corresponding numerical ones based on the proposed continuum damage model (Section 2) and the material parameters for the investigated aluminum alloy (Section 3) are presented. 4.1. X0-specimen The cruciform X0-specimen (Fig. 4(a)) is characterized by a central opening and crosswise arranged, inclined by 45° notches. The specimen has outer dimensions of 240 mm by 240 mm and details on the geometry are given in the sketches Fig. 4(b-d). The depth of

Fig. 4. X0-specimen: (a) photograph of complete specimen; (b-d) sketches of details: (b) central part, (c) notch, (d) cross section A-A; (e) notation, (f) mesh, detail of notched region. 7

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the notches is 1 mm , reducing the thickness here from 4 mm to 2 mm at its thinnest point, see Fig. 4(d). First polished micrograph images indicate a homogeneous grain size distribution and consequently the remaining material can be seen as representative. The connectors between the specimen legs have a width of 6 mm and are centrally arranged, quite robust and thus, reduce the liability to fabrication inaccuracies. 4.2. Experimental setup All experiments have been performed with the biaxial test machine LFM-BIAX 20 kN produced by Walter + Bai, Switzerland. It contains four electromechanically, individually driven cylinders with load extrema of ±20 kN. During the experiments the specimens are clamped in the four heads of the cylinders and the machine reports its displacements as well as the applied forces of each cylinder. The influence of the loading history on damage and fracture has been analyzed by experiments with proportional (P) and nonproportional (NP) loading. Corresponding experiments (P and NP) are characterized by the same load ratio before fracture and thus the results can be related to each other. To minimize non-symmetric behavior during the experiments a mainly displacement driven (F + F ) (F + F ) procedure has been used, see [14,28]. With the notation given in Fig. 4e) and the definitions F1 = 1.1 2 1.2 and F2 = 2.1 2 2.2 a brief outline can be given as follows: M of cylinder 1.1 is continuously increased by 0.04 mm/min. 1. The leading machine displacement u1.1 M . 2. The same displacement is applied on the cylinder 1.2 on the opposite side of the same axis as u1.2 F 3. The generated force F1.1 is taken, multiplied by the pre-defined factor ζ = F2 and applied on the cylinder 2.1 as F2.1. This causes the 1

M machine displacement u2.1 , i.e. the cylinder 2.1 is force driven. M on the cylinder 2.2 on the opposite side of the same axis. 4. The same displacement is applied as u2.2

This experimental technique has proven to be very stable but it has to be mentioned that the relation between the machine displacements uiM . j and the nominal displacements ui . j (see Fig. 4(e)) is non-linear and depends on the load case. Consequently, the nominal displacements ui . j are used as corresponding displacement measure. This allows the introduction of the relative displacements Δuref. i = ui .1 − ui .2 as an adequate measure to compare results. The experiments with non-proportional loading have been realized in a similar way by performing an axis switch (as) at the end of the first load step (see Fig. 6 and Fig. 5) and when reaching the final proportional load step, see [28] for details. During the experiments the displacement fields of the specimen surfaces have been monitored with a Q-400 digital image correlation (DIC) system provided by Dantec/Limess. For the setup 6.0 Mpx cameras equipped with 75 mm lenses have been applied and thus 7.85 px/mm of resolution could be reached. The forces Fi . j and the machine displacements uiM . j have been transmitted to the DICsystem and stored with the corresponding DIC data sets. Further details on the DIC setup can be found in [28]. 4.3. Numerical setup All numerical simulations have been realized with Ansys Classic 18.0 augmented by an user defined subroutine based on the presented continuum damage model, see Section 2. Details on the numerical implementation can be found in [35,32]. The implementation considers the yield (1) and the damage condition (4) as well as the plastic (2) and damage rate Eq. (8) while the scalar rate equations are solved based on an inelastic predictor elastic corrector technique. The geometry has been meshed with brick elements of Ansys type Solid185 which have been fully integrated and the B-bar method has been applied. The minimum mesh size has been around 0.2 mm within the notched region, see Fig. 4(f). In the numerical analysis the length of the specimen legs has been reduced to the central region with dimensions of 50.0 mm by 50.0 mm with a total of 65,736 elements.

Fig. 5. (a) Force F1 vs. force F2 , (b) force Fi vs. displacement Δuref. i . 8

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The numerical simulations have been performed displacement controlled whereas the displacements at the ends of axis 1 have been predefined and the displacements of the ends of axis 2 have been varied till the load ratio ζ has been reached with a tolerance of 0.5%. To compare the experimentally obtained results with the numerical predicted ones the displacements Δuref. i have been chosen to be an adequate measure to identify corresponding time steps. For this purpose, numerically predicted displacements of nodes at the same location then the red dots in Fig. 4(e) have been extracted. 4.4. Results and discussion Within this paper the final loading ratio 1/ +1 is investigated. For this purpose the proportional load case F1/ F2 = 1/ +1 (P 1/+1) and the non-proportional ones 1/ −0.5 with switch to 1/ +1 (NP 1/−0.5 to 1/+1) and 1/ −1 with switch to 1/ +1 (NP 1/−1 to 1/+1) are analyzed in detail where experimental results are denoted with Exp and numerical ones with Sim. With these load cases different stress states are obtained within the notched regions leading to different damage and fracture mechanisms on the micro-level. Focus will be on the effect of the loading path on deformation, damage and fracture behavior and consequently the results of the proportional load case will be related to the ones with non-proportional loading history. 4.4.1. Forces and displacements The forces and displacements extracted experimentally and numerically of the different load cases illustrate the different loading histories. Fig. 5(a) indicates the experimentally applied forces: for proportional loading (blue line, 1/+1) the forces increase linearly to the maximum load of 7.2 kN where fracture occurs. For the non-proportional loading 1/−0.5 to 1/+1 (green graph) F1 is first increased to 4.6 kN while F2 is simultaneously decreased to −2.3 kN where the axis switch takes place. In continuation F1 is maintained constant at 4.6 kN and F2 is unloaded and in continuation loaded to 4.6 kN where the final proportional loading path is reached. After subsequent proportional loading the specimen fractured at F1 = F2 = 7.7 kN . For the non-proportional path 1/−1 to 1/ +1 (red graph) the axis switch has been realized at F1 = −F2 = 3.2 kN and final fracture occurred at 7.5 kN. The graphs presented in Fig. 5(a) illustrate the principal ideas of these experiments: to be able to compare the different results of each loading path all experiments fracture at the same pre-defined load ratio, here F1/ F2 = 1/ +1. To investigate the influence of non-proportional loading the load ratio ζ has to be kept constant during the first and the last one of the different stages of subsequent load steps. In addition it can be noted, that the maximum loads at fracture change with respect to the loading history. Fig. 6 indicates the corresponding displacements Δuref. i which have been experimentally measured and numerically predicted. All curves have a slight non-linear tendency and it is important to notice that during unloading and reverse-loading of the non-proportional load cases Δuref.1 reduces for about 0.05 mm. Moreover the slope of the final proportional load step differs between all load cases. Furthermore, for (1/−0.5 to 1/+1) a remarkable larger displacement Δuref.1 = 0.5 mm sets in whereas under proportional loading (1/+1) only 0.25 mm is reached. Numerically the load cases (1/+1) and (1/−1 to 1/+1) could be predicted and for (1/ −0.5 to 1/+1) slight differences appear whereas the final displacements are met in an accurate way. Load-displacement curves (F1 vs. Δuref.1 (points) and F2 vs. Δuref.2 (crossed)) for all load cases are shown in Fig. 5(b). For proportional loading (1/+1) both axis as well as experimental and numerical curves coincide very well. For the non-proportional load cases it can be clearly seen that the axis switch has been realized where elevated inelastic deformations have already been evolved. It is important to point out, that the axis switch has to be executed at a point where major inelastic deformations have developed but early enough to avoid fracture in the first two load steps and to reach the final proportional loading step. The difference in displacements mentioned before for (1/−0.5 to 1/+1) can also be identified whereas all curves could be predicted numerically in an excellent way. 4.4.2. Deformation behavior The deformation behavior of the specimens surfaces has been monitored by digital image correlation (DIC) and the reported first

Fig. 6. Relative displacements Δuref.1 vs. Δuref.2 . 9

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principal strains are displayed in Fig. 7 Exp for the non-proportional experiments at axis switch (as) and for all experiments at the end of the test shortly before fracture occurs (end). Due to restrictions of the digital image correlation technique a strip at the edges of the specimen with a width of around 0.4 mm cannot be displayed. At the same displacements Δuref. i the numerically predicted (Sim) first principal strains have been plotted at the same scale. Overall a good accordance between experimental and numerical results can be observed. The experimentally measured strains under proportional loading (P 1/+1 end) show a wide band with elevated strains; maximum values reach 0.13 whereas at the center only 0.10 are indicated. The numerically predicted strains reach 0.16 and are more concentrated in two zones above and below the center. These differences might be caused by anisotropy in thickness direction which is difficult to quantify and might be more relevant under tension conditions within the plastic range. The deformed shape of the edges is predicted in an accurate way. The non-proportional load case at axis switch (NP 1/−0.5 to 1/+1 as) displays a narrow band of elevated strains which is slightly inclined from the top-right to bottom-left. Maximum values at axis switch reach already 0.15. The numerically predicted strains form also in an inclined band which is slightly wider whereas values up to 0.11 are reached. Before fracture (NP 1/−0.5 to 1/+1 end) experimentally the band of elevated strains is further developed with values up to 0.18 and besides the band a wider zone with elevated strain is shown. Numerically two zones with elevated strains (0.16) are predicted and the same tendencies as experimentally observed are calculated. For the non-proportional load case (NP 1/−0.5 to 1/+1 as) the axis switch had to be realized earlier, see Fig. 5, and consequently a narrow band of strains with values up to 0.10 is indicated experimentally. Numerically a similar tendency is predicted whereas the band is slightly wider. Before fracture (NP 1/−0.5 to 1/+1 end) experimentally the band is dissolved whereas its effects can still be seen and numerically the band is still notable whereas two elevated zones with values up to 0.16 are predicted.

4.4.3. Stress state Fig. 8 displays the numerically predicted stress state expressed in stress triaxiality (Eq. 6) and Lode parameter (Eq. 7) for all load cases before fracture (end) and for the non-proportional cases also at axis switch (as). This is done in a comprehensive way by three perspectives: the surface of the notch (S), a longitudinal cross-section (L) and a perpendicular cross-section (C). All plots are shown for the red marked box indicated in Fig. 4e) and it has to be noted, that the upper part of the plots is directed to the outside of the specimen and the lower side towards the specimen center and consequently neither horizontal nor vertical symmetry of deformation develops. The proportional experiment (Fig. 8(a)) shows a rather homogeneous stress distribution with influence of the geometry (0.4 ⩽ η ⩽ 0.65 and − 0.5 ⩾ ω ⩾ −1, see L and C) whereas the distribution appears more likely to be symmetric with respect to a horizontal and vertical symmetry plane. The non-proportional experiments indicate a comparable stress state before fracture (Fig. 8(c, e)) with influence of the loading history. Due to the asymmetric loading the distribution appears asymmetric as well, see (c) L and (e) L. At axis switch (Fig. 8(b, d)) the first loading step is clearly reflected: for (1/−0.5) η reaches values of approximately 0.15 and for (1/−1) values of about 0 are predicted. The Lode parameter indicates for both load cases slightly negative values. Consequently, both non-proportional experiments indicate a remarkable loading history dependence by significant changes in the stress

Fig. 7. First principal strain on specimen surface before axis switch (as) and fracture (end) measured by DIC system and the corresponding numerical simulation. 10

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Fig. 8. Triaxiality η and Lode parameter ω : (a) proportional loading (1/+1) before fracture, (b, c) non-proportional loading (NP 1/−0.5 to 1/+1), (d, e) non-proportional loading (NP 1/−1 to 1/+1); axis switch (as) and before fracture (end); S: surface; L: longitudinal cross-section; C: perpendicular cross-section.

states. The axis switch was performed after the occurrence of elevated inelastic strains, see Fig. 5(b), i.e. the influence of the loading history on the damage and fracture process has to be analyzed in detail. This can be realized by the experimental results as well as by the predicted damage strains expressed by the internal damage variable μ , see Eq. (8).

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4.4.4. Plastic and damage behavior Both inelastic deformation processes can be characterized by their internal variables: plasticity by the equivalent plastic strain γ (Eq. (2)) and damage by the equivalent damage strain μ (Eq. (8)). Fig. 9 displays γ and μ for all load cases just before fracture (end) and for the non-proportional cases additionally at axis switch (as). Under proportional tension loading (a) γ reaches values up to 0.15 whereas at the center of the notch two regions indicate smaller values up to 0.10 . In this central region μ reaches its maximum values

Fig. 9. Plastic deformation (internal plastic variable γ ) and damage deformation (internal damage variable μ ): (a) proportional loading (1/+1) before fracture, (b,c) non-proportional loading (NP 1/−0.5 to 1/+1), (d,e) non-proportional loading (NP 1/−1 to 1/+1); axis switch (as) and before fracture (end); S: surface; L: longitudinal cross-section; C: perpendicular cross-section; gray indicates no plastic deformation or no damage deformations respectively. 12

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up to 0.15 with a slight central reduction. I.e. with ongoing deformation damage becomes the predominant deformation process at the part with more elevated stress triaxialities, see Fig. 8. For the non-proportional load case (NP 1/−0.5 to 1/+1) the largest plastic deformations (γ = 0.16) are predicted at the surface of the notch in a slightly inclined band as indicated by the first principal strain measurement, Fig. 7 Exp(b). At axis switch first damage evolution started at the double-curved edges with maximum values of μ = 0.004 , i.e. at the central part of the specimen with stress triaxialities of about 0.15 no damage is predicted at this point, see Fig. 8. Before fracture (c) the internal plastic variable has increased to γ = 0.16 at the center of the specimen while at the notched surface only small increases can be noted. Furthermore, damage evolution is predicted in the complete central part with maxima of μ = 0.15 at the notch surface with 0.5 mm from the double-notched edge. From this point the predicted zones of elevated damage are from the top maximum towards the left and from the bottom maximum toward the right which is different to the direction of the band with elevated plastic deformations. Consequently plasticity and damage have the tendency to develop in different areas whereas the damage maximum is located in a zone with elevated plastic deformations which might be the point of crack initiation. The load case (NP 1/−1 to 1/+1) (d, e) indicates a similar behavior as (NP 1/−0.5 to 1/+1) whereas it has to be kept in mind, that the axis switch was performed earlier. Consequently, the plastic strains at axis switch (as) reach γ = 0.08 and first damage takes place with values up to μ = 0.002 . Before fracture (e) plasticity indicates values of γ = 0.13 and damage with values of μ = 0.15 in a small band at the center of the notch. Overall it can be noted, that the inelastic behavior is strongly loading path dependent. The evolution of plastic and damage strains might evolute in different zones of the specimen and specially the damage softening does influence the further evolution of plastic deformations significantly. 4.4.5. Fracture behavior The photos given in Fig. 10 display fractured specimens of all load cases. The specimens fractures instantaneously, i.e. no fracture evolution could be observed with the used equipment. Mostly two notches with the same direction fractured, but also it occurred that two neighboring notches or three notches fractured at the same time without recognizable pattern. The localized strain bands (Fig. 7) correspond to the fracture lines shown in Fig. 10. The small kinks close to the double curved edge of load case (NP 1/−0.5 to 1/+1) might be seen as a result of elevated damage strains as indicated in Fig. 9(c). Furthermore, the global deformation of the specimen center reflects clearly the load case: for (P 1/+1) a rather uniform enlargement, for (NP 1/−0.5 to 1/+1) clear asymmetric tendencies and for (NP 1/−1 to 1/+1) an elongation in one direction can be observed. The fracture surfaces have been analyzed by scanning electron microscopy (SEM) and representative pictures are given in Fig. 11. All load cases indicate fracture due to void nucleation, growth and coalescence whereas differences can be noted. Specially the load case (NP 1/−1 to 1/+1) is characterized by the influence of the shear stress state before axis switch with η approximately 0 and ω approximately 0, see Fig. 8(d); the influence of micro-shear-cracks is visible by smooth regions between pores. This influence is smaller for (NP 1/−0.5 to 1/+1) although the axis switch has been realized at remarkably higher strains (Fig. 7), but at a more elevated stress triaxiality level η = 0.15. Consequently, the SEM images clearly demonstrate that the loading path and the corresponding stress history influence the damage and fracture behavior. 5. Conclusions Initially, a phenomenological continuum damage model (CDM) has been presented and a set of material parameters for the aluminum alloy EN AW 6082 (AlSiMgMn) is given. In continuation a series of new experiments with biaxially loaded X0-specimens under proportional and non-proportional loading has been presented and experimental as well as corresponding numerical results obtained with the CDM have been discussed in detail. In this context focus has been given on the loading path dependence of the inelastic behavior. The experimental and numerical results of the biaxial experiments have shown good agreement and the influence of the loading path on the damage and fracture could be observed. Furthermore, the numerical simulation provided valuable information regarding

Fig. 10. Fractured specimens: (a) proportional loading (1/+1), (b) non-proportional loading (1/−0.5 to 1/+1), (c) non-proportional loading (1/ −1 to 1/+1). 13

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Fig. 11. Pictures taken of fracture surfaces by scanning electron microscopy (SEM): (a) proportional loading (1/+1), (b) non-proportional loading (1/−0.5 to 1/+1), (c) non-proportional loading (1/−1 to 1/+1).

the stress state and its influence on the ongoing material deterioration. Within the notched part of the specimen a big range of stress triaxialities and Lode parameters could be reached and the influence of initially shear dominated loading followed by tension dominated loading could be clearly indicated. Consequently the presented CDM is suitable to analyze the corresponding material behavior and the X0-specimen can be applied to study the material behavior in a well controlled way under different stress states and loading histories. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The research has been supported by the DFG – Deutsche Forschungsgemeinschaft (German Research Foundation) through project number 322157331 which is gratefully acknowledged. Furthermore, the SEM images of the fracture surfaces presented in this paper have been performed at the Institut für Werkstoffe des Bauwesens at the Universität der Bundeswehr München and the special support of Wolfgang Saur is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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