1

Determination of Fracture Toughness in mode I and mode II

for copper sheet

Tinghan Huang tinghanhuang@tecnico.ulisboa.pt

Instituto Superior Técnico, Universidade de Lisboa, Portugal

November 2018

Abstract

For present paper, there are two objectives, the first is the determination of the fracture toughness in

copper sheet (1 mm thickness) by the DNTT, the Shear and the Staggered DNTT tests, in order to verify

the possibility of having constant value of the fracture toughness for all mode of fracture mechanics. The

second is to determine the formability limits (FLC-forming limit curve, FFL-fracture forming limit and

SFFL-shear fracture forming limit), using the same tests plus the tensile, the bulge and the nakazima

tests.

Results show that the value of fracture toughness obtained for mode I, mode II and mixed mode

(transition mode between mode I and mode II) were in the same order of magnitude, so it allow to

conclude that the fracture toughness can be considered as property of material. Regarding to the

representation of fracture limits in the principal strain space, the angle between the FFL and SFFL are

88º, this value is very close to 90º and allow to conclude that the perpendicularity between two lines

remains valid.

Keywords: Copper, Sheet Metal Forming Processes, Formability, Necking, Fracture, Fracture

Toughness, Experimentation

1. Introduction

The term of formability is used to characterize the maximum level of deformation that a material can

resist, during a technological process, without the appearance of failure (Rodrigues and Martins [1]).

This can be quantified by the formability limits, which can be characterized by necking and fracture. The

formability limit by necking is represented by the Forming Limit Curve (FLC) while the formability limit

by fracture is represented by the Fracture Forming Limit (FFL) and the Shear Fracture Forming Limit

(SFFL). The FFL is associated with tensile stresses, mode I of fracture mechanics and the SFFL is

related by in-plane shear stresses, mode II of fracture mechanics.

The forming limit curve (FLC), was initially obtained using circle-grid analysis and introduced by Keeler

[2] in the tension domain and by Goodwin [3] in the tension-compression domain. Embury and Duncan

[4] demonstrated that in conventional plastic deformation processes, such as equi-biaxial tension or

deep drawing of a square cup, can occur interaction between fracture with necking. By associating FFL

with the ductile fracture criterion proposed by McClintock [5], Atkins [6] proposed that FFL can be

characterized by a straight line falling from left to right with a slope equal to “−1” associated to the

condition of critical thickness reduction at failure caused by tension (mode I of fracture mechanics). Isik

et al. [7] introduced the SFFL and represented it in the principal strains space as a straight line that is

perpendicular to FFL in fair agreement with the condition of critical distortion at fracture induced by in-

plane shear (mode II of fracture mechanics).

The formability limits can be also represented in the triaxiality plane where the triaxial stress state is

known to influence the amount of plastic strain that a material can support until fracture. This diagram

represents the effective strain (𝜀 ̅) as a function of the triaxiality (η) which is the ratio of the hydrostatic

stress (𝜎𝑚) with the effective stress (�̅�). Considering the Hill 48 [8] plasticity criteria normal anisotropy

and plane stress assumption (𝜎3 = 0), the effective strain, hydrostatic stress and effective stress can be obtained by,

mailto:tinghanhuang@tecnico.ulisboa.pt

2

𝜀 ̅ = √2(2 + �̅�)

3(1 + 2�̅�)2√(𝜀2 − �̅�𝜀3)

2 + (𝜀1 − �̅�𝜀3)2 + �̅�(𝜀1 − 𝑑𝜀2)

2 (1)

𝜎𝑚 =𝜎1 + 𝜎2 + 𝜎3

3 (2)

𝜎 = √𝜎12 + 𝜎2

2 +2�̅�

1 + �̅�𝜎1𝜎2 (3)

where, �̅� is the normal anisotropy coefficient, 𝜀1 and 𝜀2 are the first and second in-plane principal strains

and, 𝜎1 and 𝜎2 are the principal stresses.

The fracture toughness quantifies the amount of energy that the material can absorb until the fracture.

For the present work, the methodology utilized was the essential work of fracture (EWF) and was

originally proposed by Cotterell and Reddel [9].

For each of the tested specimens the evolution of force 𝑣𝑠 displacement curve was obtained, and the

total energy was calculated by integrating these evolutions. Assuming that the total energy 𝑊𝑇 is the

sum of the energy of plastic deformation 𝑊𝑝 and an energy related to formation of new surface 𝑊𝑒, the

total energy per unit of area can be express as follows:

𝑊𝑇

𝐴=

𝑊𝑒

𝐴 +

𝑊𝑝

𝐴 = 𝑅𝑇+

𝑊𝑝

𝐴 (4)

Where 𝐴 = 𝑙 × 𝑡 is the area of the ligament and 𝑅𝑇 is the fracture toughness, which is defined as the

amount of energy per unit of area that is required to create a new surface. The determination of 𝑅𝑇

involves extrapolating the total energy per unit area 𝑤𝑇, by application of limiting conditions in which the length (𝑙) of the ligament approaches zero. Graphically this corresponds to the 𝑦-interception of a

straight line. The Figure 1 a) represent the evolution of the tensile force with displacement for test

specimens with different lengths of ligament 𝑙, while Figure 1 b) show the methodology used for the

fracture toughness 𝑅𝑇 determination.

(a) (b)

Figure 1 – a) Schematic evolution of the tensile force with displacement for test specimens with different

ligaments l. b) Determining fracture toughness 𝑅𝑇 from extrapolation of the amount of energy per unit of area w

that is needed to create a new surface. [10]

Cotterell et al. [11] conducted tests in a staggered specimen with varying angles, where the specimens

fractured by transition mode between modes I and II (mixed mode). The material used in the tests was

a steel alloy with a thickness of 1.6 mm.

Based on the results obtained by Mai and Cotterell [12], Atkins and Mai [13] concluded that for the

material with high workhardening exponent (n), 𝑅𝑇 is constant for all stagger angle (𝛼).

The aims of this paper are: (i) The determination of the fracture toughness of a material with high value

of n by the DNTT, the Shear and Staggered DNTT tests, in order to confirm the conclusion proposed by

Atkins and Mai [13], (ii) determine the formability limits, using the same tests plus the tensile, the bulge

and the nakazima tests.

𝑅𝑇

3

2. Experimentation

The fracture toughness tests specimens were cut using wire electric discharge machining, for another

tests, the specimens were cut using CNC machining center. In Table 1 is presented a schematic

representation, geometry and dimensions of the specimens of all sheet formability tests performed for

determination of the formability limit and fracture toughness.

Table 1 - Schematic representation of the experimental sheet formability tests used to determine the formability limit (Adapted from J.P. Magrinho, M.B Silva and P.A.F. Martins [14]).

Test Dimensions (mm) State of Stress

State of Strain Preparation

of specimens

Number of specimens

Tensile

lo

wo

lc

𝑙𝑐 = 80 𝑙0 = 50

𝑤0 = 12.5

𝜎1 > 0 𝜎2 = 𝜎3 = 0

𝜀1 > 0 𝜀2 = 𝜀3 < 0

Circle Grid 𝑑 = 2 𝑚𝑚

or Sparkle pattern

17

Hydraulic bulge

die

Blank holder Draw bead

dDie

od

𝑑0 = 175 𝑑𝑑𝑖𝑒 = 100 (Circular)

𝑑1: 𝑑2 =100: 64, 100: 80

(Elliptic)

𝜎1 ≥ 𝜎2 > 0 𝜎3 = 0

𝜀1 ≥ 𝜀2 > 0 𝜀3 < 0

Circle Grid 𝑑 = 2.5 𝑚𝑚

5

Nakajima

Die

Blank holder Draw bead

Punch

od

or

𝑑0 = 210 𝑟0 = 50, 57.5, 72.5 𝑎𝑛𝑑 80

𝜎1 > 𝜎2 ≥ 0 𝜎3 = 0

𝜀1 ≥ 0 −𝜀1/2 < 𝜀2 < 𝜀1

𝜀3 < 0

Circle Grid 𝑑 = 2.5 𝑚𝑚

8

Double Notched Tension

lo

l

w

𝑤 = 50 𝑙 = 125

𝑙0 = 5, 7.5, 10, 12.5, 15 𝑎𝑛𝑑 20

𝜎1 > 0 𝜎2 < 0 𝜎3 = 0

𝜀1 > 0 𝜀2 = 0 𝜀3 < 0

Sparkle pattern

12

Shear

lo

l

w

lo

𝑤 = 38.1 𝑙 = 125

𝑙0 = 2, 2.72, 3.72, 4.72, 5.72 𝑎𝑛𝑑 6.72

𝜎1 = −𝜎2 𝜎3 = 0

𝜀1 = −𝜀2 𝜀3 = 0

Sparkle pattern

12

Staggered

lo

l

a

low

𝑤 = 50 𝑙 = 125

𝛼 = 30° 𝑡𝑜 60° 𝑙0 = 3, 5, 6, 8 𝑎𝑛𝑑 12

𝛼 = 60° 𝑡𝑜 80° 𝑙0 = 3,4,5,7 𝑎𝑛𝑑 9

𝜎1 > −𝜎2 𝜎3 = 0

𝜀1 > −𝜀2 𝜀3 < 0

Sparkle pattern

60

2.1. Experimental tests

The uniaxial tensile test is used to make the mechanical characterization of the material, where the test

specimens are subjected to a control tensile loading until fracture. The tests were performed in an

universal testing machine INSTRON, model 5900R. In order to determine the anisotropy coefficients,

the specimens were cut out from the supplied sheets at 0º, 45 º and 90º with respect to the rolling

direction. The specimen geometry and parameters used in the tensile tests followed the E8/E8M-09

standard [15]. From the tensile test, it is possible to obtain the material properties as the modulus of

4

elasticity 𝐸, the yield strength 𝜎𝑦, the ultimate tensile strength 𝜎𝑈𝑇𝑆, the elongation at break 𝐴 and the

anisotropy coefficients �̅� and ∆𝑟.

The bulge test is one of the most used methodology to study the material behaviour under biaxial

stretching conditions. In this test, a circular specimen is clamped and then drawn within a die (circular

or elliptical) by hydrostatic pressure caused by the compression of oil on the inner surface of the

specimens. With the same objective, Nakazima test were performed in specimens with different value

of radius 𝑟0 (as shown in Table 2). In this case a hemispherical punch was used to deform the material,

and in order to reduce the friction between the specimen and the punch, was also used a sheet of

polytetrafluoroethylene with 0.23 mm thickness and grease. The Bulge and the Nakazima test were

performed in a hydraulic universal testing machine, Erichsen 145/60. The schematic representation of

these two tests can be found in the Table 2. All specimens were cut from the supplied sheets at 0º with

respect to the rolling direction.

For the determination of the fracture of toughness in present work, were performed three different type

of tests, the double notched test (DNTT), the shear test and the Staggered DNTT test. All tests were

carried on INSTRON, model 5900R. All specimens were cut from the supplied sheets at 0º with respect

to the rolling direction.

The double notched tensile test (DNTT) loaded in tension (as shown in Table 2) was initially proposed

by Cotterell and Reddel [9] to calculate the fracture toughness where fracture occurs by mode I of

fracture mechanics. The shear test loaded in tension presented in Table 2 follows the ASTM standard

B831-05 [16] and this test was used to determine the fracture toughness where fracture is caused by

in-plane shear stress (mode II of fracture mechanics). The staggered DNTT test loaded in tension was

originally developed by Cotterell et al. [11] to estimate the fracture toughness in transition mode between

mode I and mode II where fracture occurs due to out-of-plane shearing stresses.

2.1. Formability charaterization

The determination of FLC, tensile, bulge and nakazima tests were used. For the determination of the in-

plane strain (𝜀1, 𝜀2 ) at the onset of necking, were utilized two methodology. One is the positions

dependent method, the circle grid analysis (CGA) and another is time dependent method, measurement

by a digital 3D image correlation (DIC) system model Q-400 from Dantec Dynamics (Figure 3).

In function of the measurement methodology are used, the specimens need to be prepared previously.

For the circle grid analysis, the procedure of preparation involves etching or imprinting a grid of circles

in the surface of the blank and for the measurement by DIC system, the procedure is making the sparkle

pattern in the surface of the specimen, by using the white and black paints.

In the methodology circle grid analysis, a computer-aided measuring Grid Pattern Analyzer GPA-100

model from ASAME was used. The measurement procedure consists in measuring the grid points (e.g

a circle or ellipses) along predefined directions that crossed the crack region perpendicularly. at time by

computerized camera system and then using the following equations,

𝜀1 = ln (𝑎

𝑑) 𝜀2 = ln (

𝑏

𝑑) (5)

Where a and b are the lengths of the major and minor axes of the ellipses that resulted from plastic

deformation of the original grid of tangent circles, and d the diameter of the original grid point. The

maximum strain pairs at the onset of necking were obtained by means of a mathematical interpolation

of the adjacent and along a perpendicular direction to the crack. The schematic of procedure can be

found in Figure 2. More details are given in Cristino et al. [17].

5

Figure 2 – Schematic procedure of determining the in-plane strains at the onset of necking [17].

For the tests with the DIC system, the onset necking point strains was determined based on the method

proposed by Martínez-Donaire [18], which consist in detection of the onset of necking by the time

evolution of the major strain 𝜀1 and its first time derivative 𝜀1̇.

Figure 3 – Dic system model Dantec Dynamics, Modelo Q-400.

In order to obtain the fracture forming limit line (FFL), the tensile test, the bulge test, the nakazima test

and DNTT test were used. The shear fracture forming limit line (SFFL) was obtained by means of shear

tests. The staggered DNTT tests were used to determine the fracture strains in the transition region

between the fracture forming limit line and shear fracture forming limit line. The procedure to determine

the fracture strains consisted in measuring the thickness ang width of the specimens before and after

the fracture at several locations along the crack. Equipment used for this purpose were a micrometer,

two different microcopes, one is Mitutoyo microscope, Model TM-505B, and another one is Metallurgical

microscope Motic model BA310 MET-H, and Mitutoyo profile projector model PJ300.

3. Results and discussion

3.1. Mechanical characterization

The mechanical characterization of copper was performed by means of tensile tests, at room

temperature, the stress-strain curve was approximated by the Ludwik- Hollomon equation:

𝜎 = 427,54𝜀0,262 [𝑀𝑃𝑎] (6)

Table 2 presents the main properties obtained from the tensile tests performed for copper align at 0º,

45º and 90 º with the rolling direction.

Table 2 - Mechanical Properties for Copper

alignment with the rolling direction

E (GPa) 𝝈𝒆 (MPa) 𝝈𝒓 (MPa) A (%) 𝒓

0º 119.28 131.27 245.46 34.43 0.755

45º 115.14 133.2 236.18 34.96 1.088

90º 139.97 141.33 238.47 36.28 0.895

Mean Value 122.38 134.73 239.07 35.16 �̅� = 0.956

𝛥𝑟 = −0.263

�̅� =𝑟0 + 2𝑟45 + 𝑟90

4 𝛥𝑟 =

𝑟0 + 𝑟90 − 2𝑟452

(7)

3.2. Fracture Toughness in mode I

As shown in Table 1, the specimens used for this study are specimens with length of ligament 𝑙0 = 5,

7.5, 10, 12, 15 and 20 mm, for each length of ligament were performed two tests and the obtained

6

results were similar. The evolution of force with displacement are shown in Figure 4 (a). In order to

facilitate the visualization of the results, one curve by each length of ligament is presented. Figure 4 (b)

represents the total specific work as a function of the ligament length and it can be concluded that the

fracture toughness 𝑅𝑇 for this mode is equal to 189.98 kJ/m2.

(a) (b)

Figure 4 - (a) evolution of the tensile force with displacement curves for test specimens with different ligaments, (b) Energy per area as function of the length of ligament for the DNTT test.

By analysing the Figure 4 (a), it is possible to concluded when the dimension of the ligament increases,

the necessary force to achieve fracture also increases, as expected. For the determination of the fracture

toughness, didn’t consider the value of total specific work of specimens with length of ligament 𝑙0 =

20 mm. As can be seen in the Figure 4 (a), the result for theses specimens are different from the others.

3.2. Fracture Toughness in mode II

As shown in Table 1, the specimens used for this study are specimens with length of ligament 𝑙0 = 2,

2.72, 3.72, 4.72, 5.72 and 6.72 mm, for each length of ligament were performed two tests the obtained

results were similar. The results for the evolution of force with displacement are shown in Figure 5 a).

In order to facilitate the visualization of the results, one curve by each length of the ligament is presented.

Figure 5 b) represents the total specific work as a function of ligament length and it can be concluded

that the fracture toughness 𝑅𝑙 for this mode is equal to 164.76 kJ/m2.

(a) (b)

Figure 5 - (a) The tensile force with displacement curves for test specimens with different ligaments, (b) Energy per area as function of the length of ligament for the shear test.

By analysing the Figure 5 a), it is possible to concluded that as the dimension of the ligament increases,

the necessary force to achieve fracture also increases, as expected. For the determination of fracture

toughness, weren’t considered the grey points. The reason for this exclusion can be explained by Figure

6.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Fo

rce (

kN

)

Displacement (mm)

L5

L7.5

L10

L12

L15

L20

0

100

200

300

400

500

600

700

800

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5

En

erg

y p

er

are

a (

kJ/m

2)

Ligament (mm)

𝑅𝑇= 177.92 kJ/m2

0,00

0,25

0,50

0,75

1,00

1,25

1,50

0 1 2 3 4 5 6 7

Fo

rce (

kN

)

Displacement (mm)

L2

L2.72

L3.72

L4.72

L5.72

L6.72

0

200

400

600

800

1000

1200

0 1 2 3 4 5 6 7 8

En

erg

y p

er

are

a (

kJ/m

2)

Ligament (mm)

𝑅𝑇= 164,76 kJ/m2

7

Figure 6 - The tensile force with displacement curves for test specimens with length of ligament 𝑙0 =2.72, 3.72 e

4.72 and 6.72 mm.

By analysing Figure 6, it is possible to observe that although all curves have a similar behaviour, but the

curves of specimens with length of the ligament of 4.72 mm and 3.72 mm are closer than the curves of

specimens with length of ligament of 2 mm and 6.72 mm.

3.3. Fracture Toughness in mixed mode (transition mode between mode I and mode II)

As shown in Table 1, for the stagger angle 𝛼 = 30°, 45° 𝑎𝑛𝑑 60°, length of ligament used were 3, 5, 6, 8

and 12 mm, and for the stagger angle 𝛼 = 70°, 80° 𝑎𝑛𝑑 85°, length of ligament used were 3, 4, 5, 7 and

9 mm. For each length of ligament were performed two tests and the obtained results were similiar. The

results for the evolution of force with displacement are shown in Figure 7. In order to facilitate the

visualization of the results, only one curve by each length of ligament for different stagger angle 𝛼 is

presented.

(a) (b) (c)

(d) (e) (f)

Figure 7 - The tensile force with displacement curves for staggered DNTT specimens with inclination of (a) 30º, (b) 45º, (c) 60º, (d) 70º, (e) 80º e (f) 85º.

By analysing Figure 7, it is possible to conclude when the inclination angle is higher, the necessary force

to achieve fracture decreases and the displacement of fracture increase.

0

250

500

750

1000

1250

1500

0 1 2 3 4 5 6 7

Fo

rce (N

)

Displacement (mm)

L2_P1

L2_P2

L3.72_P1

L3.72_P2

L4.72_P1

L4.72_P2

L6.72_P1

L6.72_P2

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fo

rce (

kN

)

Displacement (mm)

30º_L3

30º_L5

30º_L6

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fo

rce (

kN

)

Displacement (mm)

45º_L3

45º_L5

45º_L6

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fo

rce (

kN

)

Displacement (mm)

60º_L3

60º_L5

60º_L6

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fo

rce (

kN

)

Displacement (mm)

70º_L3

70º_L4

70º_L5

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fo

rce (

kN

)

Displacement (mm)

80º_L3

80º_L4

80º_L5

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fo

rce (

kN

)

Displacement (mm)

85º_L3

85º_L4

85º_L5

8

Figure 8 – Energy per area as function of the length of ligament for the staggered DNTT test.

In Figure 8 is represented the energy per area as function of the length of the ligament for the staggered

DNTT test. The value of fracture toughness for different inclination angle can be found in Table 3.

3.4. Comparison of the results of Fracture Toughness for mode I, II and Mixed (transition mode

between mode I and mode II)

In Figure 9 is represented the Energy per area as function of the length of ligament for the DNTT, Shear

and staggered DNTT test.

Figure 9 - Energy per area as function of the length of ligament for the DNTT, Shear and staggered DNTT test.

In the Table 3 is represented the value of the fracture toughness for the DNTT, the Shear and the

staggered DNTT test, and slope of the regression line for the different tests.

Table 3 – Fracture toughness of the all specimens used.

Specimen type Fracture toughness, 𝑹𝒍 (kJ/m2) Slope of regression line

DNTT 177.92 28.52

Staggered DNTT

30º 142.27 42.63

45º 137.44 52.38

60º 124.01 70.60

70º 144.63 79

80º 131.49 101.13

85º 105.15 115.19

Shear 164.76 137.42

By analysing the Table 3, although there is variation of the value of the fracture toughness value for

different tests, however they have the same order of magnitude. It is also possible to conclude that when

we change from the mode I to mode II, the slope of regression line increases progressively.

3.5. Formability Limits

Figure 10 and Figure 11 present the formality limits in the principal strain space and triaxiality space,

respectively.

0

200

400

600

800

1000

1200

0 1 2 3 4 5 6

En

erg

y p

er

are

a (

kJ/m

2)

Ligament (mm)

30º

45º

60º

70º

80º

85º

0

200

400

600

800

1000

1200

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

En

erg

y p

er

are

a (

kJ/m

2)

Ligament (mm)

DNTT

Staggered DNTT_30º

Staggered DNTT_45º

Staggered DNTT_60º

Staggered DNTT_70º

Staggered DNTT_80º

Staggered DNTT_85º

Shear

9

. Figure 10 – Representation of the formability limits in the principal strain space.

The equation obtained for the FFL curve is:

𝜀1 + 0.7𝜀2 = 1.99

(8)

And the equation obtained for the SFFL curve is:

𝜀1 − 1.41𝜀2 = 3.49 (9)

The slope obtained for the FFL is “-0.7”, this value is higher than the slope proposed by Atkins [6]. The

slope obtained for the SFFL is “+1.41”, this value is higher than the slope proposed by Isik et al. [7]. The

angle between the experimental FFL and SFFL is approximately equal to 88°. This angle is in good

agreement with the condition of perpendicularity between the two fracture lines, as defined theoretically.

Figure 11 - Representation of the formability limits in the triaxiality space.

By analysing Figure 11, it is possible to concluded that the results obtained from the Staggered DNTT

tests are localized near to the FFL.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

-1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Uniaxial Tensile Test

Bulge Test

Nakazima Test

DNTT

Shear Test

Staggered DNTT

Principal Strain 2

Pri

ncip

al S

train

1

SFFL

FFL

FLC

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Eff

ecti

ve s

train

Triaxiality

Tensile Test

Bulge Test

Nakazima Test

DNTT

Shear Test

Staggered DNTT

SFFL

FFL

FLC

Pure

She

ar U

nia

xia

lT

ensio

n

Pla

ne

stra

in

Eq

ui-B

iaxia

l

10

4. Conclusion

This paper proposed the determination of the fracture toughness in mode I, II and mixed by means of

DNTT, Shear and Staggered DNTT test. The representation of formability limits by necking and fracture

mode I and II in the principal strain and triaxiality was also determined. From the tensile test, the hardening coefficient obtained was 0.262, this value is comparative large than

a steel alloy, in which was used by Cotterell and Mai [11].

Regarding to the representation of fracture limits in the principal strain space, the angle between the

FFL and SFFL are 88º, this value is very close to 90º and allow to conclude that exist perpendicularity

between two lines.

Results obtained from the DNTT test, the shear test and the staggered DNTT test, show that the value

of fracture toughness obtained from different tests were in the same order of magnitude, so it allow to

conclude that the fracture toughness can be considered as property of material.

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