Ductile fracture experiments with locally proportional loading histories

Ductile fracture experiments with locally proportional loading histories

International Journal of Plasticity xxx (2015) 1e27 Contents lists available at ScienceDirect International Journal of Plasticity journal homepage: ...

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International Journal of Plasticity xxx (2015) 1e27

Contents lists available at ScienceDirect

International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Ductile fracture experiments with locally proportional loading histories Christian C. Roth a, c, Dirk Mohr a, b, c, *  Solid Mechanics Laboratory (CNRS-UMR 7649), Department of Mechanics, Ecole Polytechnique, Palaiseau, France Impact and Crashworthiness Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA c ETH Zurich, Department of Mechanical and Process Engineering, Zurich, Switzerland a

b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 July 2015 Available online xxx

Basic ductile fracture experiments for sheet metal (or flat coupons extracted from bulk material) are presented to characterize the onset of fracture at different stress states. Special emphasis is placed on designing the experiments such that the stress triaxiality and the Lode angle parameter remain constant while the specimen is loaded all the way to fracture. A new in-plane specimen with two parallel gage sections is proposed to determine the strain to fracture for approximately zero stress triaxiality. A FEA based methodology is shown to identify the optimal specimen geometry as a function of the material's ductility and strain hardening. A tension specimen with a central hole is investigated in detail with regard to determining the strain to fracture for uniaxial tension. It is found that the required hole-to-ligament width ratio decreases as a function of the material ductility and increases as a function of the strain hardening exponent. The bending of a wide strip is pursued to prevent the necking prior to fracture under plane strain tension conditions, while an Erichsen-type of punch test is used to characterize the material response for equibiaxial tension. It is worth noting that the strain to fracture can be directly determined from surface strain measurements in the cases of shear, plane strain tension and equibiaxial tension loading, thereby removing the need to perform finite element simulations for extracting the loading path to fracture. © 2015 Elsevier Ltd. All rights reserved.

Keywords: A. Ductility B. Finite strain C: Optimization Fracture experiments

1. Introduction There is a constant quest for reliable experimental data characterizing the effect of stress state on ductile fracture. Different stress states may be achieved through different initial specimen geometries or by applying different combinations of loading to the specimen boundaries. Examples for the first approach are the works of Bao and Wierzbicki (2004), Brünig et al. (2008), Gao et al. (2010) or Driemeier et al. (2010). Example for the second approach are the tension-torsion experiments of Barsoum and Faleskog (2007a), Faleskog and Barsoum (2013), Haltom et al. (2013) and Papasidero et al. (2015), the internal pressuretension testing of tubes (Kuwabara et al., 2005; Korkolis and Kyriakides, 2009), the tension-shear loading of butterfly specimens (Wierzbicki et al., 2005; Mohr and Henn, 2007; Mae et al., 2007; Dunand and Mohr, 2011, Abedini et al., 2015) and the biaxial loading of cruciform-like specimens (e.g. Abu-Farha et al., 2009; Brenner et al., 2014).

* Corresponding author. ETH Zurich, Department of Mechanical and Process Engineering, Zurich, Switzerland. E-mail address: [email protected] (D. Mohr). http://dx.doi.org/10.1016/j.ijplas.2015.08.004 0749-6419/© 2015 Elsevier Ltd. All rights reserved.

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The main result of ductile fracture experiments on isotropic materials is the so-called loading path to fracture, i.e. the evolution of the equivalent plastic strain as a function of the stress triaxiality and the Lode parameter. The mechanical fields within the specimen gage section are heterogeneous (at the macroscopic level) in most experiments and the local stress state cannot be calculated based on force measurements using analytical formulas. A hybrid experimental-numerical approach is therefore often required (e.g. Mohr and Henn, 2007; Bai and Wierzbicki, 2008; Dunand and Mohr, 2010; Gruben et al., 2011; Brünig and Gerke, 2011; Lou et al., 2012, 2014; Fourmeau et al., 2013; Malcher et al., 2012). This introduces additional uncertainty in the determined loading paths to fracture related to the employed finite strain plasticity model. A noteworthy predicament of ductile fracture experiments is that their outcome is often no longer a direct experimental observation, but a combined numerical-experimental result. In micromechanical studies, ductile fracture has been thoroughly investigated for monotonic proportional loading, i.e. for loading histories during which the stress state remains constant up to the point of fracture initiation (e.g. Tvergaard, 1981; Koplik and Needleman, 1988; Barsoum and Faleskog, 2007b, 2011; Scheyvaerts et al., 2011; Danas and Ponte Castaneda, 2012; Tekoglu et al., 2012; Dunand and Mohr, 2014; Brünig et al., 2014). Moreover, knowledge of the strain to fracture as a function of the stress state for proportional loading is also a main ingredient of fracture initiation models that fall into the category of damage indicator models (e.g. Wilkins et al., 1980; Bai and Wierzbicki, 2010; Lou et al., 2014; Mohr and Marcadet, 2015). Another important predicament of ductile fracture experiments is that proportional loading conditions at a material point are extremely difficult to achieve. In most ductile fracture experiments, the local stress state actually evolves throughout loading even if the ratios of the forces acting on the specimen boundaries are kept constant. This stress state evolution is due to changes in the specimen geometry that are almost inevitable in experiments involving large deformations. For example, Ebnoether and Mohr (2013) have shown that in a conventional flat uniaxial tension specimen, the stress triaxiality can increase after the onset of necking from 0.33 to values as high as 0.8 at the instant of fracture initiation. The above experimental predicaments partially prohibit the progress in the field since the direct validation of fracture models for proportional loading through experimental results becomes almost impossible. Early works represented the results from fracture experiments in terms of either the average stress triaxiality (e.g. Bao and Wierzbicki, 2004) or the stress state at the instant of fracture initiation (Barsoum and Faleskog, 2007a). However, more recent considerations for non-proportional loading (Benzerga et al., 2012; Marcadet and Mohr, 2015; Papasidero et al., 2015) raise doubt about the meaningfulness of the representation of experimental data in terms of average or final stress triaxialities. In this work, an attempt is made to present fracture experiments that feature (i) a constant stress state up to the onset of fracture, and (ii) that allow for the direct determination of the strain to fracture from experimental measurements without any numerical simulations. In addition, we focus on stress states that are particularly useful for identifying the plane stress fracture envelope for proportional loading: (1) pure shear, (2) uniaxial tension, (3) plane strain tension, and (4) equi-biaxial tension. New specimen geometries are identified for the first two stress states through constrained shape optimization, while V-bending and punch experiments are considered for the latter two stress states. To facilitate the fracture characterization in an industrial environment, all experiments are designed such that they can be performed in a universal testing machine. It is emphasized that we take a 3D continuum mechanical point of view on failure. We are interested in determining the intrinsic failure response at a material point which is assumed to depend on the history of the local mechanical field variables only. For mechanical reasons, all specimens are flat and are thus extracted from steel sheets. However, we are not concerned with sheet metal mechanics where it is common practice to distinguish between in-plane and bending properties. To avoid any confusion, it might be worth thinking of all proposed fracture specimens as thin specimens that have been extracted from a bulk material. 2. Plasticity model Numerical simulations play a central role in developing, analyzing and validating ductile fracture experiments. We therefore begin our presentation with a brief summary of the plasticity and fracture initiation models that are employed in the sequel. We calibrated both models based on experiments on specimens extracted from a 1 mm thick dual phase steel DP780 provided by US Steel. The material serves as model material in the present work; it is composed of a ferrite matrix with martensite inclusions and features an average grain size of about 8 mm. 2.1. Plasticity model formulation A rate-independent non-associated quadratic plasticity model (Mohr et al., 2010) is used to model the material response. This particular model had been proposed based on the results from combined tension-shear experiments on DP590 and TRIP780 steels. It also provided a remarkably accurate description of multi-axial experiments on DP780 steel specimens (Mohr and Marcadet, 2015). Its isotropic yield function is written in terms of the von Mises equivalent stress, s, and a deformation resistance k,

f ½s; k ¼ s  k ¼ 0:

(1)

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The non-associated plastic flow is determined through the stress derivative of a flow potential g½s,

dεp ¼ dl

vg½s : vs

(2)

dεp denotes the increment in the plastic strain vector (in the material coordinate system)

 T εp ¼ εp11 εp22 εp33 2εp12 2εp23 2εp13 ;

(3)

with the 1-, 2-, and 3-directions corresponding to the rolling, transverse and thickness directions, respectively. s denotes the Cauchy stress vector,

s ¼ ½s11 s22 s33 s12 s23 s13 T :

(4)

and dl  0 is a scalar plastic multiplier. The flow potential is defined through the quadratic form

g½s ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðGsÞ$s

(5)

with

2

1 6 G12 6 6 ð1 þ G12 Þ G¼6 60 6 40 0

G12 G22 ðG22 þ G12 Þ 0 0 0

ð1 þ G12 Þ ðG22 þ G12 Þ 1 þ 2G12 þ G22 0 0 0

0 0 0 G33 0 0

0 0 0 0 3 0

3 0 07 7 07 7: 07 7 05 3

(6)

Note that g½s corresponds to a special case of the Hill'48 yield function which accounts for the direction dependence of the Lankford ratios through the anisotropy coefficients G12, G22 and G33. The case of associated plastic flow is recovered for G12 ¼ 0.5, G22 ¼ 1 and G33 ¼ 3. The equivalent plastic strain increment dεp is defined as work-conjugate to the von Mises equivalent stress. The material's strain hardening behavior is modeled through a linear combination of a power and an exponential law,

      kε εp ¼ akS εp þ ð1  aÞkV εp

(7)

with the weighting factor a 2 [0,1], the power law (Swift, 1952)

   n kS εp ¼ A εp þ ε0 ;

(8)

as defined through the Swift parameters {A,ε0,n}, and the exponential law (Voce, 1948),

    kV εp ¼ k0 þ Q 1  ebεp :

(9)

as defined through the Voce parameters {k0,Q,b}. 2.2. Plasticity model parameter identification Uniaxial tension experiments on flat dogbone specimens (Fig. 1a) are performed along three different in-plane directions (0 , 45 and 90 with respect to the rolling directions). All specimens are tested on a hydraulic testing machine at a crosshead displacement of 2 mm/min which results in a pre-necking strain rate of about ε_ p ¼ 103 =s. The strains along the axial and width directions are measured using virtual extensometers with an initial gage length of 7 mm and 5 mm, respectively. Evaluation of the slopes of the logarithmic plastic width strain versus the logarithmic plastic thickness strain (as calculated assuming plastic incompressibility) along the thickness direction (Fig. 2b) yields the Lankford coefficients r0 ¼ 0.77, r45 ¼ 0.84 and r90 ¼ 0.90. Using the analytical relationships

G12 ¼ 

r0 ; 1 þ r0

G22 ¼

r0 1 þ r90 1 þ 2r45 r0 þ r90 and G33 ¼ ; r90 1 þ r0 r90 1 þ r0

(10)

the anisotropy coefficients G12 ¼ 0.44, G22 ¼ 0.92 and G33 ¼ 2.8 are determined. The true stress-strain curves for the three different specimen orientations all lie on top of each other (Fig. 2a) which supports the assumption of an isotropic yield function (Eq. (1)). In a first step, the Swift parameters {A,ε0,n} and the Voce parameters {k0,Q,b} are determined from fitting expressions (8) and (9) to the true stress versus logarithmic plastic axial strain curves (Fig. 2c). Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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Fig. 1. Specimens used in this study: (a) uniaxial tension (UT), (b) miniature punch, (c) bending strip, (d) notched tension with radius r ¼ 20 mm (NT20), (e) tension with a central hole (CH), (f) in-plane shear. Blue solid dots highlight the position of the virtual extensometer for relative displacement measurements.

The difference between the Voce and Swift approximations becomes significant at large strains. The determination of the weighting factor a therefore requires the finite element analysis of the specimen response in the post-necking range where very large strains are reached. To allow for the post-necking analysis of the specimen response through an eighthmodel of the specimen, a notched tension experiment (Fig. 1d) is selected. The measured forceedisplacement curve for a 10 mm wide specimen with two R ¼ 20 mm notches is shown by solid dots in Fig. 2d. The corresponding simulation result using the finite element model of Dunand and Mohr (2010) and a weighting factor of a ¼ 0.7 (obtained through computational minimization) is shown as solid line. Note that equivalent plastic strains as high as εp ¼ 0:71 prevail within the gage section of that specimen. A summary of all plasticity model parameters for the DP780 steel is provided in Table 1. 3. Fracture initiation model 3.1. Fracture initiation model formulations A HosfordeCoulomb fracture initiation model is used to predict the onset of fracture. Its backbone is a localization criterion for proportional loading in stress-space of the form, Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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Fig. 2. Plasticity of DP780 steel: (a) True stress vs. logarithmic strain curves, and (b) plastic logarithmic width strain vs. thickness strain; (c) inter- and extrapolation of the isotropic hardening function, (d) forceedisplacement curve of notched tension experiment used for inverse identification of the mixed Swift-Voce law.

Table 1 Plasticity model parameters for the DP780 steel examined. A [MPa]

ε0 [e]

n [e]

k0 [MPa]

Q [MPa]

b [e]

1315.40

0.28E-4

0.146

349.54

536.36

93.07

a [e]

E [GPa]

n [e]

r [kg/m3]

0.70

194

0.33

7850

sHF þ cðsI þ sIII Þ ¼ b*

(11)

with the ordered principal stresses sI  sII  sIII, and the Hosford equivalent stress defined as

sHF ¼

 1 ðsI  sII Þa þ ðsII  sIII Þa þ ðsI  sIII Þa 2

1 a

:

(12)

This criterion is transformed into the mixed stress-strain space (Mohr and Marcadet, 2015) leading to an expression of the equivalent plastic strain at the onset of fracture for proportional loading as a function of the stress triaxiality h and the Lode angle parameter q, Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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pr 

εf

11n 0

1 a    1 1 þ cð2h þ f1 þ f3 ÞA h; q ¼ bð1 þ cÞn @ ðf  f2 Þa þ ðf2  f3 Þa þ ðf1  f3 Þa 2 1

(13)

with the model parameters {a,b,c} and the transformation constant n ¼ 0.1.1 The functions fi are Lode angle parameter dependent trigonometric functions that are associated with the transformation from principal stresses to Haigh-Westergaard coordinates,

hp  i   2 1q ; f1 q ¼ cos 3 6

(14)

hp  i   2 f2 q ¼ cos 3þq ; 3 6

(15)

hp  i   2 f3 q ¼  cos 1þq : 3 6

(16)

The Hosford exponent controls the deviatoric stress measure (12) with the limiting cases of Tresca (a ¼ 1 or a ¼ ∞) and von Mises (a ¼ 2 or a ¼ 4); the friction coefficient c  0 controls the influence of the normal stress, while the parameter b  0 controls the overall level of the fracture strains; it has been defined based on b* such that it corresponds to the strain to fracture for uniaxial tension (and equi-biaxial tension). Fig. 3a shows a 3D plot of the HosfordeCoulomb fracture initiation model for proportional loading in the space pr fh; q; εf g. It describes a monotonic dependency of the strain to fracture on the stress triaxiality and a convex shaped dependency on the Lode angle parameter with a minimum for generalized shear (q ¼ 0, i.e. stress states for which the intermediate principal stress is equal to the average of the minimum and maximum principal stresses). The above criterion for proportional loading is micromechanically-motivated, i.e. Dunand and Mohr (2014) have shown through computational localization analysis that a HosfordeCoulomb type of criterion is suitable for predicting the onset of coalescence in a porous solid after proportional loading. To predict fracture initiation after non-proportional loading, Eq. (13) is used in a damage indicator model framework. Let D denote a scalar damage indicator of initial value D ¼ 0 and a maximum value of D ¼ 1 at the instant of fracture initiation, the evolution of the damage indicator is described through the differential equation

dD ¼

dεp  : ε pr h; q f

(17)

For proportional loading, the current value of the damage indicator can be interpreted as the fraction of the fracture strain that has been applied to the material. 3.2. Fracture model parameter identification In view of calibrating the model from experiments on sheet metal, the plane stress representation is most relevant as general 3D stress states cannot be easily generated in experiments. For plane stress conditions, the Lode angle parameter is a function of the stress triaxiality (Fig. 3b),

q¼1



 2 27 1 arccos  h h2  for 2=3  h  2=3 p 2 3

(18)

and the fracture initiation model can be conveniently represented in the strain to fracture versus stress triaxiality plane (Fig. 3c). Note that the apparent non-monotonic relationship between the strain to fracture and the stress triaxiality is deceiving as it is actually due to the underlying Lode angle dependency. Also note that irrespective of the choice of the model parameters, the strain to fracture for uniaxial and equi-biaxial tension are always identical according to the HosfordeCoulomb model. This is due to the fact that the Hosford equivalent stress is equal to the

1 In the so-called “consistent version” of the HosfordeCoulomb model, the transformation from stress to strain space would need to be performed using the material's isotropic hardening law (Mohr and Marcadet, 2015). However, in engineering practice, we recommend using a simple power law with the exponent n ¼ 0.1 to perform this transformation. Small adjustments of the parameters a and b can usually attenuate the effect of this approximation on the strain to fracture.

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Fig. 3. (a) Equivalent plastic strain to fracture as a function of the stress state according to the HosfordeCoulomb model for proportional loading; (b) Lode angle parameter as a function of the stress triaxiality for plane stress conditions (blue, black and red curves), (c) HosfordeCoulomb criterion in the fracture strain versus stress triaxiality plane for states of plane stress. Selected stress states are highlighted through solid dots: shear (SH), uniaxial tension (UT), plane strain tension (PST) and equi-biaxial tension (EBT). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

maximum principal stress for these two stress states. Furthermore, with the minimum principal stress being zero in both cases, the HosfordeCoulomb model (Eq. (11)) reduces to maximum principal stress criterion for uniaxial and equi-biaxial tension. In the following, we will present experimental techniques to characterize the strain to fracture for  pure shear (SH):

εSH f ¼b

pffiffiffi 3

!1

1þc

n

1 a

ð1 þ 2a1 Þ

(19)

 uniaxial tension (UT):

εUT f ¼b

(20)

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 Plane strain tension (PST):

εPST f

pffiffiffi 3 ¼b

!1

1þc

n

1 a

ð1 þ 2a1 Þ þ 2c

(21)

 Equi-biaxial tension (EBT):

εEBT ¼b f

(22)

Note that the strain to fracture for uniaxial tension as well as that for equi-biaxial tension are independent of the model parameters a and c. Moreover, the strains to fracture for uniaxial tension and equi-biaxial tension are the same since the Hosford equivalent stress and the sum of the minimum and maximum principal stress (as normalized by the von Mises stress) are the same for these two stress states. Based on the proportional fracture strain triple fεSH ; εPST ; εEBT ; εPST ; εUT g or fεSH g, it is f f f f f f then straightforward to identify the HosfordeCoulomb model parameters: 1. Determine b from UT or EBT experiment. 2. Determine c from PST and SH experiments:

!n 1 c¼

εPST f εSH f

!n p2ffiffiffi 3

εPST f b

!n þ

εPST f εSH f

(23) 1

3. Determine the real exponent a from solving the simple implicit equation:

 1 pffiffiffi a 1 þ 2a1 ¼ 3ð1 þ cÞ

b εSH f

!n (24)

Note that we recommend limiting the Hosford exponent to the interval 1  a  2 to guarantee uniqueness of the solution of Eq. (24). 4. Equi-biaxial tension Fracture testing of sheet materials for equi-biaxial tension is probably the “most standard” among the four experiments discussed. The punch test configuration is preferred over hydraulic bulge testing to avoid the evacuation of excess fluid after fracture. This advantage of the punch test comes at the expense of the possible effect of tool friction on the experimental results. Here, we present a punch testing procedure which is similar to the so-called Erichsen cupping test. 4.1. Mini-punch testing device Fig. 4 shows a photograph and a schematic of the proposed axisymmetric mini-punch testing device. It is designed as an independent testing fixture which can be inserted into any universal testing machine or press. The main distinctive feature over conventional punch test set-ups is that the punch remains stationary throughout the experiment while the die and clamping plate move downward. This reduces the required focal depth of the DIC camera system, thereby allowing for shorter object distances and ultimately increasing the spatial resolution of the acquired surface strain fields. The main components (Fig. 4a and b) of the mini-punch device are: (a) Stationary part composed of a 18 mm thick aluminum base plate (part ①) and a 12.7 mm diameter cylindrical stainless steel punch (part ②) with a hemispherical head and an Ra-0.4 surface finish; Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

Fig. 4. (a) Photographs of the mini-punch testing device, and (b) schematic of the axisymmetric cross-section: ① base plate, ② punch, ③ bottom plate, ④ linear roller bearing, ⑤ specimen, ⑥ clamping plate, ⑦ steel rods, (8) top plate; (c) Evolution of the equivalent plastic strain and the stress triaxiality for three different punch dimensions.

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(b) Movable part composed of a 35 mm thick bottom plate (part ③) with an embedded linear roller bearing (part ④), a 60 mm diameter disc specimen (part ⑤ and Fig. 1b), a floating clamping plate (part ⑥) applying a clamping force of about 140 kN through eight M6 screws, three vertical 10 mm diameter steel rods (items ⑦), and a 20 mm thick top plate (part ⑧) with a recess to receive a 10 mm diameter steel ball for axial load application.

4.2. On the choice of the punch diameter (to specimen thickness ratio) Finite element simulations are performed to demonstrate the effect of the punch size on the strain distribution in the specimen. The FEA model describes a quarter of the punch system with first-order solid elements of a length of le ¼ 100 mm (corresponding to 10 elements along the thickness direction). Frictionless contact is assumed between the punch and specimen for the simulations presented in this subsection. Fig. 4c shows the stress triaxiality and equivalent plastic strain distribution at the outer specimen surface for three different punch diameters (D ¼ 12.7 mm, 25.4 mm and 50.8 mm). The simulations have been stopped at a maximum equivalent plastic strain of 1. The plots of the equivalent plastic strain as a function of the normalized radius lie almost on top of each other. Similarly, the stress triaxiality versus normalized radius curves are all identical within the apex regions of a radius of about r/RPunch < 0.8. It is thus concluded from these simulation results that the choice of the small diameter punch has no noticeable disadvantage as compared to commonly used large punch sizes as far as the stress and strain distribution on the specimen surface is concerned (for zero friction). For the 12.7 mm diameter punch, the radial gradients in the surface strain field are of the order of 0.1 mm1 which equates to macroscopic strain variations of less than 1% within a single grain. The simulation results also show that the stress triaxiality decreases from its theoretical value of 2/3 at the apex to a slightly smaller value of 0.65 at a distance of about r=RPunch y0:5. For plane stress conditions, this apparently negligible change in stress triaxiality implies a significant change of the Lode angle parameter from 1 (for perfectly equi-biaxial tension) to 0.572. According to the calibrated2 HosfordeCoulomb fracture initiation model for the DP780 steel (Fig. 3c), the material ductility decreases from 0.72 to 0.59 due to this change in stress state. Despite the equivalent plastic strain maximum at the specimen center, it is thus possible from a theoretical point of view that fracture initiates away from the apex at a strain that is smaller than the maximum strain measured within the specimen. 4.3. Experimental procedure The miniature punch device is positioned in a hydraulic universal testing machine equipped with a 250 kN axial load cell. The disc specimen (Fig. 1b) with eight thru-holes for the clamping screws is extracted from the sheet metal using water-jet cutting. For optimal results, in particular if the flat specimens are extracted from bulk material stock, we recommend machining the specimen surface with a sharp end mill to minimize the effect of machining and rolling on the measured fracture properties (see also discussion in subsection 6.2 on machining artifacts). The experiment is carried out at a constant cross-head velocity of 2 mm/min until the onset of fracture. The surface displacement and logarithmic strain field are determined using stereo digital image correlation (see Table 2 for details). 4.4. Experimental results and validation The measured forceedisplacement curve is shown in Fig. 5a. It increases monotonically with a change in curvature from convex to concave at a punch displacement of about 5 mm. Fracture initiates in a stable manner at a radial distance of about 0.3 mm from the specimen center. With the appearance of visible cracks on the specimen surface (Fig. 5e), the force begins to drop sharply. The repetition of the experiments resulted in nearly identical results, i.e. the measured forceedisplacement curves lie on top of each other and fracture initiated at almost the same displacement of 7.97 mm. Based on the DIC determined principal logarithmic surface strains εI and εII (Hencky strains) and the assumption of incompressibility, we compute the effective strain

2 ε ¼ pffiffiffi 3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε2I þ εI εII þ ε2II :

(25)

The effective strain field is shown in Fig. 5c at the instant of fracture initiation. At the top apex of the punched specimen, we have approximately proportional loading conditions for which the effective strain provides a good approximation of the equivalent plastic strain, εp yε: Not knowing the Lode angle and stress triaxiality histories at the locations of fracture initiation, the maximum strain at the apex ðε ¼ 0:72Þ is retained as a lower bound for the strain to fracture for equi-biaxial tension (h ¼ 2/3, q ¼ 1).

2 The calibration has been performed for the DP780 material using the formulas provided in Section 3 after completing all experiments described in Sections 5e7.

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Table 2 Details on the digital image correlation. All images are acquired using AVT-Pike 505B/C cameras with a type 2/3 CCD sensor (2452  2054 pixels). A 150W cold light source is used in all experiments to illuminate the area of interest. An eight-tap filter is used to achieve sub-pixel accuracy through spatial grayscale interpolation. The strain fields are computed using a Gaussian filter (as built into VIC software). Exper-iment

DIC type

camera(s)

Lenses

Speckle size [mm]

Resol-ution [mm/pixel]

Freq-uency [Hz]

Soft-ware

Subset [pixel]

Step [pixel]

Punch

stereo

10

1

VIC-3D

35

10

stereo

100

16

1

VIC-3D

25

6

CH tension

planar

100

21

0.5

VIC-2D

15

4

In-plane shear (global)

planar

AVT Pike

100

21

0.5

VIC-2D

15

4

In-plane shear (local)

planar

AVT Pike

Nikon 90 mm 1:1 macro Nikon 90 mm 1:1 macro Nikon 90 mm 1:1 macro Nikon 90 mm 1:1 macro Mitutoyo 1x telecentric

100

V-bending

2x AVT Pike (17 azimut, 50 polar) 2x AVT Pike (17 azimut, 50 polar) AVT Pike

<50

3.4

0.5

VIC-2D

27

7

The numerical simulation of the experiment has been performed using the same FE model as in subsection 4.2 with a friction coefficient of m ¼ 0.05 for the contact between the punch and the specimen. The computed forceedisplacement curve (solid line in Fig. 5a) matches the experimental curve (solid dots) after correcting its stiffness through an assumed machine f stiffness of 19 kN/mm. The strain at the apex obtained from the simulation (Fig. 5d) is εp ¼ 0:708, which is remarkably close to that measured experimentally (Fig. 5c). 5. V-bending Bending of a small rectangular plate specimen is a standardized experiment for evaluating the edge bending (and hemming) capacity of sheet materials (VDA 238-100, DIN 50111, ASTM E290, ISO 7438, and JIS Z2248). Here, V-bending experiments are performed to determine the strain to fracture for plane strain conditions. In a typical V-bending experiment a rectangular sheet material coupon (Fig. 1c) is placed on top of two parallel rollers featuring a diameter D that is much larger than the sheet thickness t. The sheet specimen is then loaded through a thin knife-like tool. Provided that the gap between the two rollers is not much larger than twice the sheet thickness, a sharp V-bend is formed. The main modification proposed is keeping the knife stationary, while moving the rollers downwards. As for the punch test, this configuration has the advantage that the surface strain field can be measured by means of stereo DIC without any spatial resolution limitations imposed by the limited depth of focus of the optical system. 5.1. Experimental set-up Fig. 6a shows a 3D sketch of the loading device. It features a base plate (part ①) with four vertical guidance rods (items ②) and the central knife-like support point (item ③). The latter is made from a high strength tool steel and features a 0.8 mm wide initially flat tip (Fig. 4b). The large roller support points (items ④) are rigidly connected with the movable cross-head (item ⑤) of the loading device. The roller diameters and center-to-center distance are 30 mm and 33.8 mm, respectively. A 60 mm long and 20 mm wide specimen is tested (Fig. 1c). Before inserting the specimen (item ⑥) into the testing device, a random speckle pattern is applied to the specimen top surface within a 5 mm wide band. The specimen is initially held in place by the surface friction and dead weight pressure of the cross-head. Two cameras are positioned at an object distance of about 450 mm and an angle of 17 with respect to each other. Note that the rollers feature a chamfer to facilitate the observation of the bent specimen surface. We inserted the loading device into our universal testing machine equipped with a 10 kN load cell, which provides the force and cross-head displacement histories in addition to the local surface strain measurements. However, note that the determination of the strain to fracture per se only requires a simple press, i.e. the load measurement is not needed. 5.2. Experimental results Experiments are performed at a constant cross-head velocity of 2 mm/min until cracks are observed by eye on the specimen outer surface. As detailed in Table 2, stereo digital image correlation is used to obtain the surface strain fields. Fig. 6d shows the recorded forceedisplacement curves for V-bending. The measured surface strain field (Fig. 6c) shows the uniformity of the axial strain εx along the bending axis (y-axis). The maximum width strain measured along that axis does not exceed εy  < 0:003 which confirms the validity of the plane strain assumption on the specimen surface. It is assumed that fracture initiates when a first crack becomes visible on the specimen surface. For the DP780 steel, this instant coincides with the maximum in the forceedisplacement curve. Based on the measurement of the axial logarithmic strain to fracture, εfx , i.e. Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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Fig. 5. Punch testing of disk specimens: (a) forceedisplacement curve of the experiments (solid lines) and the FE simulation (solid dots); (b) loading path to fracture for the element closest to the center, (c) DIC-based effective strain field for the last picture before fracture initiation. (d) computed equivalent plastic strain field at force maximum, (e) fractured specimen right after the force maximum, (f) damage indicator field reaching unity (onset of fracture) along the black line at a radial distance of 1.89 mm from the center.

the surface strain at the specimen center at the instant of fracture initiation, the corresponding equivalent plastic strain (“fracture strain”) for plane strain conditions is estimated as

2 εpf ypffiffiffiεxf : 3

(26)

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Fig. 6. (a) Drawing of the V-bending device with ① base plate, ② guidance rods, ③ knife-like punch, ④ rollers, ⑤ upper cross-head, ⑥ specimen (red); the chamfers in the rollers and the upper plate are introduced to free the view for two cameras. (b) basic design principle, (c) DIC contour of the logarithmic strain in x-direction. Besides the visual examination of the image, fracture can also be determined from the loss of correlation (see gray default pixel near center of zoomed area). (d) Recorded forceedisplacement curves for two experiments and the evolution of the log. strain in x-direction. The first visible crack is denoted by the arrow. The four black dots in both force and strain plot denote the time points of the four images in Fig. 8. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

According to the non-associated anisotropic flow rule, the stress triaxiality for plane strain tension loading along the rolling direction is

G22  G12 : hPST ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 G22 þ G212 þ G12 G22

(27)

The corresponding Lode angle parameter follows from application of Eq. (18). For the present material, we have hPST ¼ 0.57 and qPST ¼ 0:06. In the case of an isotropic flow potential function (i.e. G22 ¼ 1 and G12 ¼ 0.5), the stress triaxiality is hPST ¼ pffiffiffi 1= 3y0:58 and qPST ¼ 0. The repeatability of the experimental procedure is good with a variation of less than 2% in the strain to fracture measured in two experiments εfp ¼ f0:518; 0:509g. At the same time, the corresponding measured forceedisplacement curves lie on top of each other (Fig. 6d). 5.3. Finite element analysis A finite element simulation is performed to gain more insight into the strain and stress state distribution in a V-bending experiment. The results shown in Fig. 7 are obtained using a quarter model of the experimental set-up with eight first-order solid elements along the sheet thickness direction, 200 elements along the bending axis (y-direction) and 60 elements along the x-direction, resulting in an element size of approximately le ¼ 100 mm in the bent zone; the rollers and the knife are modeled as rigid bodies, and a roller-to-sheet friction coefficient of 0.1 is assumed. Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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Fig. 7. Results from the numerical simulation of a V-bending experiment: (a) plot of the equivalent plastic strain field, (b) distribution of the equivalent plastic strain (black), the stress triaxiality (red) and the Lode angle parameter (blue) on the specimen surface along the y-direction (bending axis), (c) evolution of the equivalent plastic strain against the stress triaxiality for an element in the middle of the top surface.

The simulation results show that the mechanical fields are uniform at a distance greater than 7 mm from the specimen edges. The stress triaxiality is close to 1/3 (uniaxial tension) at the specimen boundary and about 0.55 (close to plane strain tension) at the specimen center. The latter deviates slightly from its theoretical value which is attributed to elastic strains. The plane strain constraint needs to be satisfied for the total strains; due to the positive elastic strains, the flow rule must thus be satisfied for a slightly negative (instead of a zero) plastic strain increment. As a result, the flow rule drives the plane stress state from plane strain tension towards uniaxial tension, i.e. the stress triaxiality for zero total strain is lower than that of zero plastic strain along the y-direction. The equivalent plastic strain is lower at the free edge than at the specimen center which can be explained by the equivalent plastic strain definition. If the latter is evaluated for UT and PST for the same axial strain, pffiffiffi UT we obtain a factor of εPST p =εp ¼ 2= 3 ¼ 1:15. Fig. 7b also shows the variation of the Lode angle parameter which is very significant, i.e. it increases from 0 at the specimen center to 1 at the free edges. However, different from the stress state Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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gradients encountered in a punch experiment, the location of fracture initiation is not expected to shift according to the HosfordeCoulomb model due to two benign effects: firstly, the ductility is expected to increase, and secondly, the equivalent plastic strain actually decreases towards the free edge. 5.4. Remarks  Effect of edge condition: It is worth noting that the long edges of the bending specimen need to be machined with a sharp tool to avoid premature fracture from the edges. It is known that pre-damage induced during water-jet and shear cutting results in a significant reduction of ductility (e.g. Wang, 2015), thereby increasing the fracture initiation from the edges and possibly overwriting the above benign effects. For example, we had performed some preliminary experiments on specimens with shear-cut edges where fracture initiated from one of the free edges (instead of the specimen center).  Maximum achievable strain: Assuming that the neutral bending axis coincides with the sheet mid-plane, the equivalent plastic strain in a bent sheet of thickness t and inner radius of curvature R is



2 Rþt : ε ¼ pffiffiffi ln R þ t=2 3

(28)

In the limit of R / 0, the maximum achievable equivalent strain is about 0.8. In practice, the neutral axis shifts towards the compression side, thereby generating strains greater than predicted by (28). For our 2R ¼ 0.8 mm wide tool and a sheet thickness of t ¼ 1.06 mm, the maximum achievable strain according to (28) is 0.52. Even though slightly larger strains can be produced in practice (as the neutral axis will be slightly below the center of the sheet), we have not been able to achieve fracture in a V-bending experiment on a highly ductile 1 mm thick 22MnB5 steel (before hot-stamping).  Detection of fracture initiation: The detection of the instant of fracture initiation by eye based on the acquired photographs requires careful inspection. The photographs shown in Fig. 8 demonstrate this difficulty. We found that the correlation coefficient maps computed by the DIC software are useful in detecting fracture initiation provided that a sufficiently small step and subset size are chosen. It is also worth noting that the difficulty in detecting the instant of fracture initiation in a V-bending experiment is material dependent. For example, a recent V-bending test series on complex phase steels showed the successive emergence of microcracks and the progressive loss of specimen load carrying capacity. For such materials, both the experimental technique and the fracture modeling framework may not be applicable. 6. Tension with a central hole As discussed in the introduction, the stress state in dogbone specimens changes throughout loading and the experimental results are not suitable for evaluating the fracture response after uniaxial tension. Dunand and Mohr (2010) advocated using a tensile specimen with a central hole to characterize the material ductility under uniaxial tension. Experiments on specimens with a central hole (CH) therefore potentially provide insight into the fracture response for uniaxial tension provided that (a) fracture initiates near the hole boundary, and (b) that the effect of hole machining defects on the material's ductility is negligible. 6.1. Evolution of the stress state in a CH specimen A finite element simulation of a tensile experiment on a 20 mm wide CH specimen with a 2.25 mm radius hole has been performed to gain some insight into the evolution of the mechanical fields. One eighth of the specimen is modeled using a fine

Fig. 8. Detection of the instant of fracture initiation in a V-bending experiment; emerging crack on the specimen surface. The numbers below the images denote the acquisition time in seconds.

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solid element mesh with eight first-order solid elements along the thickness direction. The evolution of the specimen geometry and equivalent plastic strain fields are shown in Fig. 9 next to the forceedisplacement curve. In addition, the equivalent plastic strain is plotted as a function of the stress triaxiality for the integration point at which the highest equivalent plastic strain is observed in the simulation (i.e. the point where the onset of fracture will be assumed). In the purely elastic regime, the stress triaxiality is about 0.35. With the onset of plasticity, it decreases to the theoretical value of 0.33 for uniaxial tension (point ①). Between points ① and ②, the equivalent plastic strain distribution is still more or less uniform along the thickness direction, and the stress-triaxiality therefore remains constant up to a strain of 0.4. Note that

Fig. 9. (a) Forceedisplacement curves of the experiment (solid dots) and the simulation (solid line) for a central hole specimen with R ¼ 2.25 mm. (b) Computed evolution of the equivalent plastic strain as a function of the stress triaxiality; (c) effect of central hole size on the evolution of the equivalent plastic strain field: CH-specimen with R ¼ 2.25 mm (left column), and CH-specimen with R ¼ 4 mm (right column). A small arrow highlights the location of the highest strained element.

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the in-plane gradients in the plastic strain field also induce a non-uniform thickness distribution. As the thickness changes become more pronounced (at point ②, the thickness at the hole boundary is 0.81 mm as compared to 1.0 mm at the outer edge), gradients along the thickness-direction become apparent in the mechanical fields. In particular, an out-of-plane stress develops due to the curvature of the specimen surface. At point ③, the strain at the specimen top surface is 0.52 while a strain of 0.6 prevails at the specimen center. This effect becomes more pronounced as the specimen is loaded further. At the instant of specimen failure (point ④), a maximum equivalent plastic strain of 0.7 is reached at a stress triaxiality of 0.35. The spatial gradients in the mechanical fields around the highest strained material point are dh=dxy0:61=mm and dεp =dxy  0:15=mm at the instant of fracture initiation. A second simulation for a CH specimen with a larger hole (4 mm radius) showed different results (right column of Fig. 9). In particular, the highest strained material point moves away from the edge towards the hole ligament center. In very simple terms, for large holes, the two hole ligaments may be seen as two parallel uniaxial tension specimens. The stress state evolution in large diameter CH specimens therefore resembles that of dogbone specimens, i.e. the stress triaxiality is only constant and close to 1/3 up to the point of the onset of through-thickness necking; thereafter, it constantly increases at a rate of about dh=dεp y0:13 (Fig. 9b). The results from a simulation with a very small 1 mm radius hole are very similar to those obtained above. The main difference is that the gradients in the mechanical fields increase. Practical arguments regarding the machining of small diameter holes certainly impose a minimum hole size. Furthermore, the size lu of the uniformly loaded (e.g. stress triaxiality and plastic strain variations of less than 5%, which yields lu ¼ 100 mm for the present example) vicinity of the point of stress initiation should be large as compared to the size of the machining affected zone, lMAZ, and the grain size, lg,

  lu > > max lMAZ ; lg :

(29)

Given that the gradients in the mechanical fields decrease with the size of the hole, the determination of the “optimal” hole diameter corresponds to finding the largest hole diameter for which the highest strained point is still sufficiently close to the hole boundary, such that the stress state remains approximately uniaxial throughout the entire loading history. 6.2. Size of the machining affected zone (MAZ) The pre-damaged zone around the hole boundary is referred to as “Machining Affected Zone (MAZ)”. The following four procedures for introducing a 5 mm diameter hole into a 1 mm thick DP780 steel sheet have been considered:  Water-jet cutting: The edge imperfections introduced by water-jet cutting are visible to the naked eye. In particular, the water-jet produces a slanted surface with a difference of approximately 0.12 mm between the radius on the top and the bottom surface of the sheet.  Drilling: Microscopic analysis of the drilled holes reveals that the material is plastically deformed with a zone of 10 mm around the edge (Fig. 10a).  Drilling followed by reaming: In the case of reaming after drilling, the width of the MAZ is reduced to about 5 mm (Fig. 10b).  CNC milling: Using a sharp 4 mm diameter four-flute end mill, the MAZ size could be reduced to less than lMAZ ~ 2 mm (Fig. 10c). As shown by Dunand and Mohr (2010), the fracture strains obtained from experiments on water jet cut specimens can be more than 50% lower than that obtained from CNC milled specimens. EDM machining has been disregarded due to the known deep (>10 mm) penetration of wire atoms into the substrate material. Similarly, laser cutting has not been considered due to the unavoidable significant material property changes within the solidified molten material near the hole boundary. Based on the above considerations, it is thus recommended to introduce the central hole through CNC milling to minimize the effect of MAZ on the fracture strains determined from experiments on CH specimens. 6.3. Parametric study on optimal hole diameter As stated above, the largest hole diameter is considered as “optimal” for a CH specimen of given width as long as the stress state at the location of fracture initiation remains close to uniaxial tension all the way to fracture. The optimal hole diameter is expected to be a function of the material hardening response and the material ductility due to the important effect of through-thickness necking on the mechanical fields in a CH specimen. A parametric study is thus performed to shed some light on the relationship between the “optional” radius and the material properties. Using the same boundary conditions and mesh densities as in the above examples, we prepared FE models for:  33 different hole radii, varying in 0.25 mm increments from r ¼ 1 mm to r ¼ 9 mm  42 different hardening behaviors: using an elasto-plastic J2-plasticity material model, the following grid of Swift hardening parameters was considered: A ¼ 500, 700, …, 1500 MPa and n ¼ 0.01, 0.05, 0.10, …, 0.30; the same reference strain of ε0 ¼ 0.002 has been used in all simulations.  10 different fracture strains: εf ¼ 0:1; 0:2; ::: ; 1:0 (the effect of stress state is neglected here); Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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Fig. 10. SEM images of the Machining Affected Zone (MAZ) for different machining techniques: (a) drilling, (b) reaming, and (c) CNC milling.

In sum, 33  6  7  10 ¼ 13860 simulations are carried out. For a given set fA; n; εf gi of material parameters, finite element simulations are performed for 10 different radii rj to determine the average stress triaxiality at the ðiÞ highest strained point. Subsequently, we report the optimal radius Ropt as the maximum among all radii for which the average stress triaxiality is closest to 1/3. In all simulations, we observed that the stress triaxiality is always greater than 1/3. In other words, we did not observe any loading path with an average close to 1/3 that featured stress triaxialities below and above 1/3. The deviation of the average stress triaxiality from the target 1/3 is thus also a meaningful measure of the proportionality of the loading paths, i.e. an average triaxiality of 1/3 can only be achieved if the stress state is indeed constant throughout loading. The influence of A on the optimal radius turned out to be small, changing it maximally by 0.25 mm. Fig. 11 therefore shows the obtained function Ropt ¼ Ropt ½n; εf  for A ¼ 1100 MPa only. The main observations are that:

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Fig. 11. Results of the parameter study for the size of the central hole. Surface plot of the optimal radius versus assumed equivalent plastic strain to fracture and hardening exponent.

 the optimal radius increases as a function of the strain hardening exponent: this may be explained by the fact that ligament necking is more likely to occur for low n values; consequently smaller radii are required to promote the localization at the hole boundary;  the optimal radius decreases as a function of the strain to fracture; the same argument as above may be applied: the higher the strain to fracture, the more strain hardening capacity is consumed prior to fracture, and consequently, smaller radii are required to promote the localization at the hole boundary. The strain hardening coefficient of most metallic engineering materials is seldom higher than 0.2, and we therefore recommend a hole radius of about 2 mm to ensure that fracture initiates under uniaxial conditions near the hole boundary. 6.4. Experimental validation Tensile experiments are performed on CH-specimens with two different hole radii: one with a radius of R ¼ 2 mm and one with R ¼ 4 mm (of a different batch of DP780 steel). The specimens were extracted from the steel sheet initially using water jet cutting for the main geometry, followed by CNC milling to enlarge the holes to their final dimensions. Using custom high pressure clamps, the specimen is clamped into a universal testing machine and loaded at a cross-head speed of 0.4 mm/min until fracture. To perform planar DIC (see Table 2 for details), a random black and white speckle pattern is applied onto the specimen surfaces. The solid dots in Fig. 12a show the measured forceedisplacement curves next to those obtained from numerical simulations. The solid lines in Fig. 12b show the evolution of the equivalent plastic strain as a function of the stress triaxiality for the highest strained elements. While the loading path of the specimen with R ¼ 2 mm stays close to 1/3 and fractures at h ¼ 0.41, the one for the specimen with R ¼ 4 mm deviates early and fails at h ¼ 0.53; the specimen with the smaller radius failed at a strain of εf ¼ 0:98, while the specimen with the larger radius could only achieve a maximum strain of εf ¼ 0:79. This result is consistent with the HosfordeCoulomb model which predicts a loss of ductility if the stress triaxiality increases under plane stress conditions between uniaxial and plane strain tension. However, the difference between these two specimens becomes important when the strain at the hole boundary of the R ¼ 4 mm specimen (see dashed loading path in Fig. 12b) is used to estimate the strain to fracture for uniaxial tension. In that case, the fracture strain estimate of the R ¼ 4 mm specimen would be almost 25% lower than that for the R ¼ 2 mm specimen (εUT ¼ 0:98 versus εUT ¼ 0:74). It is thus reemphasized here that an inappropriate choice of hole size and/or machining technique may lead to significant errors (underestimation) in the determined strain to fracture for uniaxial tension. 7. In-plane shear The main challenge of in-plane shear experiments on flat specimens is premature fracture initiation near free gage section boundaries (Mohr and Henn, 2007; Ghahremaninezhad and Ravi-Chandar, 2013). By definition, the shear stress components along the tangent direction t must be zero at a free boundary. For plane stress specimens it is straightforward to show that the stress state at a free boundary is uniaxial tension, i.e. s ¼ sjj t5t. Unless the strain to fracture for pure shear is much lower than that for uniaxial tension, fracture is more likely to initiate under uniaxial tension at the free specimen boundary (or under plane strain tension near the grips) than under pure shear at the specimen center. To address this issue, Mohr and Henn (2004) proposed reducing the gage section thickness such that the strains are significantly higher at the specimen center, thereby increasing the probability of fracture initiation under pure shear near the specimen center. Here, an in-plane shear Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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Fig. 12. Results of the hybrid experimental-numerical approach of two central hole specimens with radius R ¼ 2 mm and R ¼ 4 mm from a different batch of DP780. (a) Force-Displacement curve of the experiment (dotted) and the simulation (line). (b) Loading paths as extracted from the element with the highest equivalent plastic strain at the onset of failure (solid lines) and from the element at the edge of the R ¼ 4 mm specimen (dashed line).

fracture specimen for sheet metal is presented that does not require any thickness reductions. This is achieved through a FEAbased optimization of the specimen shape. 7.1. Specimen design Inspired by the work of Till and Hackl (2013) and Miyauchi (1984), a “smiley” shear specimen comprising two parallel gage sections is designed for fracture testing (Fig. 1f). The main geometric feature of the specimen is a set of notches that define the contours of the shear gage sections. To prevent shear buckling, the gage section width and height shall be of the same order as the sheet thickness. A finite element simulation of a shear experiment3 using the basic specimen geometry (Fig. 14a) confirms the above concerns regarding pre-mature fracture initiation from the free boundaries (Fig. 14c). We therefore create a shape optimization problem for the outer contour of the shear gage section (Fig. 13a). In particular, we consider notch contours of overall width h that. (i) are positioned point-symmetrically at an off-set Dx with respect to the gage section center {xc,yc} (ii) feature a notch radius Rn around the notch centers fxc ±ðDx þ Dxn Þ ; yc ±Dyg with Dy ¼ w/2 þ Rn and Dxn, an additional offset between the centers of the radii; (iii) feature a smooth transition of filet radius Rf from the notch tangent to the parallel notch boundaries.

3

The HosfordeCoulomb parameters {a,b,c} ¼ {1.61, 0.71, 0.062} were chosen for the optimization.

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Fig. 13. (a) Illustration of the parameters of the model. (b) Finite element model of the optimized shear specimen geometry. The boundary zone (red) and central zone (blue) are highlighted. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

As illustrated in Fig. 13, the vector p ¼ fxc ; Dx; h; w; Rn ; Dxn ; Rf g summarizes all parameters characterizing the specimen geometry. Using symmetry boundary conditions, only the right quarter of the shear specimen is modeled (Fig. 13b). The model is created in a manner such that a uniform mesh with an element length of approximately le ¼ 100 mm is achieved in the gage section area. The finite element software Abaqus/Explicit (Abaqus, 2012) is used to perform the numerical simulations. The lower boundary of the specimen is clamped, while a constant vertical velocity is applied to the upper boundary. All simulations are stopped when the damage indicator reaches unity. The scalar cost function F ¼ F½pk  is defined as

Z

max fDi g F½pk  ¼

i2SN

max fDi g i2SG

þ

Z jhjdt þ

T

  qdt;

(30)

T

with Di denoting the damage indicator in the final time step for an element i, and SN and SG denoting two distinct element sets comprising all elements on the free gage section boundary and gage section center respectively (Fig. 13b). The first term therefore ensures that fracture is more likely to initiate near the gage section center than at the free gage section boundary. The second and third terms are added to minimize variations in the stress triaxiality and Lode angle parameter histories at the point of maximum D. Recall that the target values for pure shear are h ¼ 0 and q ¼ 0. The above cost function is minimized using a derivative-free simplex algorithm (Nelder and Mead, 1965). A solution predicting fracture initiation within the gage section center, with small variations in the stress state, is found after about 200 iterations. The minimization has been repeated for ten different starting points p0 which all led to the specimen shown in Fig. 13b. The overall notch width and gage section width are h ¼ 2.04 mm and w ¼ 2.36 mm, with a notch radius of RN ¼ 0.62 mm and an off-set Dx ¼ 0.19 mm. The distribution of the equivalent plastic strain within the gage section of the optimized specimen at the predicted instant of fracture initiation is shown in Fig. 14d. A comparison with the seed geometry simulation result shows that much higher strains could be achieved in the sheared gage section (0.86 vs. 0.74). Another way of illustrating the advantage of the optimized over the seed geometry is to compare the critical strains at the free gage section edge for the same shear strains at the specimen center. For example, for an equivalent plastic strain at the center of 0.74, the seed specimen experiences a strain of 0.61 at the edge (i.e. it fractures) while the optimized specimen features an edge strain of 0.52 only. Please cite this article in press as: Roth, C.C., Mohr, D., Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity (2015), http://dx.doi.org/10.1016/j.ijplas.2015.08.004

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Fig. 14. (a) Finite element model of the geometry proposed by (a) Till&Hackl and (b) the optimized geometry. (c) and (d) show the equivalent plastic strain at the onset of fracture (D ¼ 1) of the specimen proposed by Till&Hackl. Note how much more strain the optimized geometry can endure. (e) The evolution of the equivalent plastic strain as a function of the stress state in the element where fracture commences.

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7.2. Experimental procedure To validate the proposed shear experiment, the optimized specimen is extracted from the sheet metal using wire electricdischarge machine (EDM). It is inserted into our hydraulic tensile testing machine and loaded at constant cross-head speed of 0.2 mm/min. Throughout loading, the specimen is observed with two cameras: a first camera equipped with a 90 mm 1:1 macro lens monitors the relative displacement of the specimen shoulders, while a second camera equipped with a Mitutoyo 1x telecentric lens takes high resolution pictures (3.4 mm/pixel) of the two gage sections only (see Table 2 for details on DIC). 7.3. Results The recorded forceedisplacement curves are shown in Fig. 15a. Note that the force drops after the instant of fracture initiation. The curves for the two experiments lie on top of each other, which confirms the repeatability of the experimental procedure. In addition, Fig. 15b shows the measured effective strain field, which provides a good approximation of the equivalent plastic strain field for nearly proportional loading histories. At the instant of fracture initiation, the strain has reached a maximum value of 0.86. Despite the above efforts in optimizing the specimen geometry, this measurement is still interpreted as a lower bound estimate for the strain to fracture under pure shear loading due to the absence of a clear experimental proof of fracture initiation at the specimen center. The crack propagation after fracture initiation is unstable, i.e. we were unable to stop the experiment with a partial crack only. Assuming a crack velocity of the order of 103 m/s, a high resolution high speed camera with an acquisition frequency above 1 MHz would be required to monitor the crack propagation. However, it can be concluded from Fig. 15c that the observed final crack path is consistent with the assumption of crack initiation under pure shear away from the gage section boundaries.

Fig. 15. Results for the in-plane shear specimen: (a) Comparison of the experimental to the simulated forceedisplacement curve. (b) Surface plot of the effective strain field just before the onset of fracture; (c) fractured gage section.

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The finite element simulation results agree well with the experimental observations. The predicted forceedisplacement curve lies on top of those measured experimentally, while the contour plot of the equivalent plastic strain distribution (Fig. 14d) is also very similar to that obtained from DIC (Fig. 15b). In particular, it predicts nearly the same value of the maximum equivalent plastic strain (0.85 vs 0.86). The simulation estimates of the evolution of the Lode angle parameter and the stress triaxiality are shown in Fig. 14e. The average values are ¼ 0.027 and ¼ 0.069, while fracture initiates at hf ¼ 0.12 and qf ¼ 0:32. With regards to the fracture initiation model for the DP780 steel, these variations are “small”, i.e. the

Fig. 16. Results for the shape optimization for three engineering materials: A 1.4 mm thick DP590 dual phase steel (black), a 2 mm thick AA2198-T8R (red) and a 1.5 mm thick Ti6Al4V (blue). (a) Comparison of the hardening behavior of the three materials. (b) Comparison of the fracture loci for the materials. The solid part of the curves represents the calibrated fracture locus, while the dashed part corresponds to the assumed fracture locus. Optimized geometries for the (c) DP590, the (d) AA2198-T8R and the (e) Ti6Al4V. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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strain to fracture for proportional loading changes by less than 5% within the range of stress states experienced at the location of maximum equivalent plastic strain. 7.4. Influence of the material properties on the shear specimen geometry It is emphasized that the above specimen shape “optimization” has been performed for one specific material (DP780 steel). As for the central hole tension specimen, it is expected that the optimal specimen shape depends on the elasto-plastic and fracture properties of a material. If the identified specimen geometry is used for other materials, it may be expected that the obtained fracture strain is only a lower bound for the material ductility under pure shear. Given the high computational costs associated with the optimization, a comprehensive parametric study is omitted here. However, using the above methodology, we have identified the “optimal” shear specimen geometry for three other engineering materials: (i) a 1.4 mm thick DP590 dual phase steel, (ii) a 2 mm thick AA2198-T8R aluminum alloy, and a 1.5 mm thick Ti6Al4V titanium alloy (Tancogne et al., submitted for publication). The corresponding stress-strain curves and assumed fracture envelopes for plane stress conditions are shown in Fig. 16a and b respectively. The obtained “optimal” specimen geometries are defined through the parameters given in Table 3 and are shown in Fig. 16cee. The main geometrical features changing are the notch radii, the offset between the two notches and the notch width. Even though the DP590 and the Ti6Al4V exhibit a similar strain hardening exponent, their fracture loci differ significantly which leads to a different shear specimen geometries. Different geometries are also obtained for AA2198 and the DP590 which feature similar fracture loci, but differing hardening behavior. 8. Concluding remarks Four basic fracture experiments with approximately proportional loading condition all the way until fracture initiation are analyzed in detail. They provide the strains to fracture for pure shear, uniaxial tension, plane strain tension and equi-biaxial tension. Two original testing devices are presented to perform reliable punch and V-bending experiments. The distinctive feature of both devices is that the location of fracture initiation remains stationary throughout the experiment, thereby increasing the spatial resolution of the stereo DIC surface strain measurements. The in-depth analysis of experiments on tensile specimens with a central hole demonstrated the importance of the size of the chosen hole. If the hole size is not chosen appropriately, significant errors in the estimated strains to fracture for uniaxial tension are expected. A comprehensive parametric study revealed that a 4 mm diameter hole in a 20 mm wide tensile gage section is expected to yield satisfactory results for most engineering materials.

Fig. 17. Evolution of the principal strain paths for the punch (black), bending (red) and shear (blue) experiments as extracted from digital image correlation at the point where fracture occurred. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 3 Geometry parameters for the smiley shear specimens for the DP780 examined, a DP590, an AA2198 and a Ti6Al4V. All dimensions are given in mm. [mm]

xc

Dx

h

w

Rn,

Dx n

Rf

DP780 DP590 AA2198 Ti6Al4V

2.85 3.18 2.99 2.87

0.05 0.11 0.14 0.10

2.04 1.86 1.20 1.48

2.36 2.70 1.50 1.66

0.62 0.55 0.23 0.35

0.14 0.23 0.00 0.00

1.03 1.81 1.61 1.35

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In addition, a new flat shear fracture specimen is proposed. It features two parallel gage sections without any thickness reduction. The FEA based shape optimization elucidated the known issues of in-plane shear specimens: if the geometry is not chosen appropriately, fracture will initiate prematurely from the free gage section boundaries under a tensile stress state. As a result, a poor lower bound estimate of the strain to fracture for pure shear loading is obtained. All proposed experiments have been performed on specimens extracted from 1 mm thick DP780 steel sheets. A simple analytical procedure is shown to identify the three material parameters of the HosfordeCoulomb fracture initiation model based on measured strains to fracture. Compared to other multi-axial experimental techniques such as notched tension, tension-torsion, or Nakazima experiments, the proposed four experiments have the important qualitative advantage of approximately constant stress states throughout the entire loading history up to the instant of fracture initiation (Fig. 17). As a result, important points on the socalled fracture surface (fracture strain as a function of the stress triaxiality and Lode angle parameter) for proportional loading can finally be determined from experiments thereby removing uncertainty related to non-proportional loading effects. The main limitation of the present work is that the fracture strain can be identified for only four distinct stress states. However, based on our current understanding of the ductile fracture process of modern engineering materials, three of the four points are local extrema for plane stress conditions: the V-bending experiment features the most critical stress state (ductility minimum for plane strain tension), while local ductility maxima are expected to prevail for uniaxial tension (CH experiment) and equi-biaxial tension (mini-punch experiment). The result from the pure shear experiment (when compared to that for uniaxial tension) provides insight into the competition of the stress triaxiality and Lode parameter effects which is crucial for extrapolating the fracture surface into the range of negative stress triaxialities. The entire fracture surface cannot be constructed from four experimental points only. However, given that the proposed experimental program includes important extreme points, it is expected that more reliable predictions may be achieved when existing fracture theories are calibrated based on the obtained material data. If ductile fracture is indeed due to the loss of ellipticity at a material point (e.g. Rice, 1976), it can probably only be detected in experiments when the emerging shear/normal localization band reaches a critical length. This process of finite band formation is expected to depend on spatial gradients in the mechanical fields at the instant of fracture initiation which is a remaining experimental uncertainty in our search for the “intrinsic” fracture envelope for proportional loading. Acknowledgments The partial financial support through the MIT Industrial Fracture Consortium is gratefully acknowledged. Thanks are due to Dr. B. Hackl from VoestAlpine for the suggestion of the initial shear specimen geometry and discussion. Thanks are also due to Dr. B. 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