Experimental program and numerical assessment for determination of stress triaxiality and J3 effects on AA6101-T4

Journal Pre-proofs Experimental program and numerical assessment for determination of stress triaxiality and J3 effects on AA6101-T4 L. Malcher, L.L.D. Morales, V.A.M. Rodrigues, V.R.M. Silva, L.M. Araújo, G.V. Ferreira, R.S. Neves PII: DOI: Reference:

S0167-8442(19)30549-X https://doi.org/10.1016/j.tafmec.2020.102476 TAFMEC 102476

To appear in:

Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

27 September 2019 3 January 2020 3 January 2020

Please cite this article as: L. Malcher, L.L.D. Morales, V.A.M. Rodrigues, V.R.M. Silva, L.M. Araújo, G.V. Ferreira, R.S. Neves, Experimental program and numerical assessment for determination of stress triaxiality and J3 effects on AA6101-T4, Theoretical and Applied Fracture Mechanics (2020), doi: https://doi.org/10.1016/ j.tafmec.2020.102476

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Experimental program and numerical assessment for determination of stress triaxiality and J3 effects on AA6101-T4 L. Malcher1, L.L.D. Morales2, V.A.M. Rodrigues, V.R.M. Silva, L.M. Araújo, G.V. Ferreira and R.S. Neves [email protected]

University of Brasília, Faculty of Technology, Department of Mechanical Engineering Campus Darcy Ribeiro, Asa Norte, Brasília – DF – Brazil [email protected]

University Center of the Federal District – UDF, Department of Mechanical Engineering Setor de Edifícios Públicos Sul, Eq 704/904, Conj. A - Asa Sul, Brasília – DF – Brazil

Abstract: This article proposes the study of the mechanical behavior of AA6101-T4 at fracture for different stress states. Firstly, an experimental program has conducted including the design and manufacture of six different types of specimens to perform mechanical tests considering some levels of stress triaxiality and normalized third invariant. The specimens are the smooth and notched cylindrical bars under tensile loading and rectangular cross-section specimens under pure shear and combined tensile/shear loading conditions. Furthermore, the Gao’ model assuming non-linear isotropic hardening has used to describe the mechanical behavior of the material in axisymmetric and tridimensional problems until the facture, regarding an implicit numerical integration scheme in a finite elements framework. Posteriorly, reaction curves obtained numerically have compared with curves observed in the experimental tests. Then, it has proposed a discussion about the influence of the stress triaxiality and normalized third invariant on the level of the accumulated plastic strain at fracture. Finally, the equivalent plastic strain at fracture has been analyzed as a fracture indicator in order to observe the correct fracture onset. 1 – INTRODUCTION The approach based on the second invariant of the deviatoric stress tensor, 𝐽2, has been a formulation widely used to describe the behavior of ductile materials during the elastoplastic regime and near to fracture. The approach is commonly represented by von Mises’s criterion and regards that the hydrostatic stress effect is negligible on the description of the plastic flow rule in ductile materials, which according to literature, is a parameter responsible for control the size of the elastic domain. Furthermore, according to von Mises’s theory, the third invariant effect, 𝐽3, is also ignored. However, for some researchers, it is a parameter used in the definition of the Lode angle or Azimuth angle, which is responsible for the shape of the yield surface (Bardet, 1990; Bai, 2008; Brünig et al., 2013; De Sá et al., 2015). Over the last three decades, the importance of both stress triaxiality and third invariant in the description of mechanical behavior at the moment of fracture have been recognized, and details studies have conducted by several authors (Bai and Wierzbicki, 2007; Bai, 2008; Driemeier et al., 2010; Mirone et al., 2010; Gao et al., 2011, Malcher et al., 2012, Brünig et al., 2013; De Sá et al., 2015).

According to literature, the following contributions can be highlighted: Richmond and Spitzing (1980) proposed a study to determine the effects of hydrostatic pressure on the yielding of aluminum alloys. After ten years, Bardet (1990), showed the dependence of the Lode angle into some constitutive models, and Wilson (2002) conducted studies on notched specimens manufactured of 2024-T351 aluminum alloy under tensile load and confirmed the importance of these effects. Moreover, Brunig et al. (1999) proposed a constitutive model with three invariants that could be applied in the description of ductile fracture. According to Mirone et al. (2010), the failure in ductile materials can be better described by the accumulated plastic strain, stress triaxiality and Lode angle parameters. Furthermore, Driemeier et al. (2010) conducted an experimental program regarding fracture in shear loading conditions. At the begging of the present decade, Gao et al. (2011) proposed a new criterion and redefining the concept of equivalent stress, which is a function of 𝐼1(𝝈) 𝐽2(𝑺) and 𝐽3(𝑺). Therefore, Malcher et al. (2012) presented an assessment of three constitutive models considering the effect of the stress triaxiality and the Lode angle on the mechanical behavior of ductile materials near the fracture. Cavalheiro and Malcher (2017) used the Gao’ model to present the effect of the third invariant on fracture onset regarding several stress states and discussed some limitations of the constitutive formulation. Brünig et al. (2018) carried out experiments related to damage and fracture at negative stress triaxiality. Wang and Qu (2018) have analyzed the ductile fracture by extended unified strength theory, including both stress state parameters. Regarding these contributions, it is possible to conclude that the elastoplastic behavior and fracture onset of ductile materials are often followed by plastic strain field, which is highly influenced by some stress state parameters, such as the stress triaxiality and the third invariant. The stress triaxiality controls the size of the yield surface and the third invariant in its shape. According to stress triaxiality and third invariant levels, the material may need a larger or smaller plastic strain field to fracture. Nevertheless, the 𝐽2 approach is not accurate enough to describe the physical phenomena and more advanced constitutive models must be formulated and applied for a wide range of stress states. In this contribution, it is discussed the accuracy in description of the mechanical behavior of the material at fracture, based on the Gao model and the influence of both stress state parameters on AA6101-T4 alloy observed during the experimental tests program. According to literature, the stress triaxiality and the third invariant in the normalized form can be mathematically defined, by Equations (1)-(2) (Brunig et al., 2008; Bai, 2008; Zadpoor et al., 2009; Tvergaard, 2008; Nahshon et al., 2008, Malcher et al., 2012, De Sá et al., 2015). 𝐼1 𝜂= (1) 3 3𝐽2 𝜉=

27𝐽3 3

2(3𝐽2)

2

(2)

where 𝜂 represents the stress triaxiality, 𝐼1 is the first invariant of Cauchy stress tensor, 𝐽2 is the second invariant of the deviatoric stress tensor, 𝜉 is the third invariant in the

normalized form and 𝐽3 is the third invariant of the deviatoric stress tensor. The invariants 𝐼1, 𝐽2 and 𝐽3 are defined as: 𝐼1 = tr (𝝈)

(3)

1 𝐽2 = 𝑺:𝑺 2

(4)

𝐽3 = det(𝑺)

(5)

with, 𝑺 = 𝝈 ― 𝑝𝑰 and represents the deviatoric stress tensor. The Lode angle is an alternative form to represents the effect of the third invariant (see Bai, 2008; Malcher et al., 2012), and can be defined as: 1 𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑠(𝜉) 3

(6)

where 𝜃 represents the Lode angle. The Lode angle can be normalized (𝜃) within the range of ― 1 ≤ 𝜃 ≤ 1, regarding its definition according to axis of the principal stresses. 𝜃=1―

6𝜃 . 𝜋

(7)

2 – MATHEMATICAL FORMULATION AND NUMERICAL STRATEGY The mathematical aspects for understand the Gao’ formulation with isotropic hardening can be observed as follow. The Gao’ criterion is mathematically represented by Eq. (8). 𝜙 = 𝑐[

𝑎𝐼16

+

27𝐽23

1 6

+ 𝑏𝐽3] ― 𝜎𝑦0 ― 𝐻𝐼𝜀𝑝 ,

(8)

where 𝜙 represents the yield function, 𝜎𝑦0 is the initial yield stress of the material, 𝐻𝐼 is the isotropic hardening module, which is a function of the accumulated plastic strain, 𝐻𝐼( 𝜀𝑝), and 𝜀𝑝 represents the accumulated plastic strain, which plays role of the isotropic hardening internal variable. The terms 𝑎 and 𝑏 are material parameters and needs to be calibrated, according to tensile and shear tests, respectively. The term 𝑐 is calculated based on Eq. (9), as:

[

]

4 𝑏+1 𝑐= 𝑎+ 728

―

1 6

.

(9)

In this setting, the plastic flow vector is calculated by associative plasticity, according to Eq (10). 5

∂𝜙 1 ― ∂𝛼 𝑵≡ = 𝑐 (𝛼) 6 , ∂𝝈 6 ∂𝝈 where 𝑵 represents the flow vector and 𝛼 is a parameter defined as:

(10)

𝛼 = 𝑎𝐼16 + 27𝐽23 + 𝑏𝐽3 .

(11)

Moreover, the derivative of 𝛼 in relation to the Cauchy stress tensor is determined as: ∂𝐼1 ∂𝐽2 ∂𝐽3 ∂𝛼 + 81𝐽22 +𝑏 , = 𝑎6𝐼15 ∂𝝈 ∂𝝈 ∂𝝈 ∂𝝈

(12)

with, ∂𝐼1 ∂𝝈 ∂𝐽2 ∂𝝈 ∂𝐽3 ∂𝝈

=𝐼 (13)

=𝑺

= det 𝑺(𝑺 ―𝑇:𝕀𝑑) 1

where, 𝐼 represents the second order identity tensor, 𝕀𝑑 = 𝕀4 ― 3𝐼 ⊗ 𝐼 is the fourth order deviatoric identity tensor and 𝕀4 is the fourth order identity tensor. In this setting, based on the flow vector, the plastic flow rule is defined according to Eq. (14). 5

[

]

1 ― ∂𝛼 , 𝜺 ≡ 𝛾𝑵 = 𝛾𝑐 (𝛼) 6 ∂𝝈 6 𝑝

(14)

where 𝜺𝑝 represents the evolution equation for the plastic strain tensor and 𝛾 is called as plastic multiplier. At the end, the evolution equation for the equivalent plastic strain is calculated according to equivalency of the total plastic work: 𝑝

𝜀 =𝛾

𝝈:𝑵 . 𝜎𝑦

(15)

Box 1 contains all elements that compose the mathematical model of Gao with non-linear isotropic hardening. Box 1: Mathematical model for Gao with isotropic hardening. i)

Additive split of the total strain: 𝜺 = 𝜺𝑒 + 𝜺𝑝

ii)

Hooke’s law: 𝝈 = 𝔻𝑒:𝜺𝑒

iii)

Yield function: 𝜙 = 𝑐(𝛼)6 ― 𝜎𝑦0 ― 𝐻𝐼𝜀𝑝

1

[

𝑐= 𝑎+

]

4 𝑏+1 728

―

1 6

𝛼 = 𝑎𝐼16 + 27𝐽23 + 𝑏𝐽3 iv)

[

1

Plastic flow rule: 𝜺𝑝 = 𝛾 𝑐6(𝛼)

5

― 6 ∂𝛼 ∂𝝈

]

∂𝛼 = 𝑎6𝐼15𝐼 + 81𝐽22𝑺 + 𝑏det 𝑺(𝑺 ―𝑇:𝕀𝑑) ∂𝝈 𝑝

𝝈:𝑵

and evolution ratio for equivalent plastic strain: 𝜀 = 𝛾 𝜎𝑦 v)

Loading/unloading rule: 𝛾 ≥ 0 ,𝜙 ≤ 0 ,𝛾𝜙 = 0

The constitutive equations of the Gao’ model, presented previously, were treated and implemented within a finite element framework called Hyplas (see De Souza Neto et al., 2008), regarding an implicit numerical integration algorithm (see Simo and Hughes). The following system of residual equations were solved for the return maps algorithm:

{

𝑒 𝑹𝝈𝑛 + 1 = 𝝈𝑛 + 1 ― 𝝈𝑡𝑟𝑖𝑎𝑙 𝑛 + 1 + Δ𝛾𝔻 :𝑵𝑛 + 1 𝝈𝑛 + 1:𝑵𝑛 + 1 𝑅𝜀𝑝𝑛 + 1 = 𝜀𝑝𝑛 + 1 ― 𝜀𝑝𝑛 ― Δ𝛾

(16)

𝜎𝑦

1

𝑅Δ𝛾 = 𝑐(𝑎𝐼1 𝑛 + 16 + 27𝐽2 𝑛 + 13 + 𝑏𝐽3 𝑛 + 1)6 ― 𝜎𝑦0 ― 𝐻𝐼𝜀𝑝𝑛 + 1

The Newton-Raphson (N-R) procedure was also chosen to solve it, motivated by the quadratic rates of convergence achieved, which results in return mapping procedures computationally efficient (see Simo and Hughes, 1998; De Souza Neto et al., 2008, Malcher et al., 2009, Andrade et al., 2009), see Box 2: Box 2: The N-R algorithm for solution of the return mapping system of equations. (0) 𝑡𝑟𝑖𝑎𝑙 1) Initialize iteration counter, 𝑘: = 0, set initial guess for 𝝈(0) = 0, 𝑛 + 1 = 𝝈𝑛 + 1, ∆𝛾 (0)

𝜀𝑝𝑛 + 1

[

= 𝜀𝑝𝑛 and corresponding residual:

( ( (

) ) )

𝑹𝝈𝑛 + 1 𝝈𝑛 + 1, 𝜀𝑝𝑛 + 1, Δ𝛾 𝑅𝜀𝑝𝑛 + 1 𝝈𝑛 + 1, 𝜀𝑝𝑛 + 1, Δ𝛾 𝑅Δ𝛾 𝝈𝑛 + 1, 𝜀𝑝𝑛 + 1, Δ𝛾

][

]

𝑒 𝝈𝑛 + 1 ― 𝝈𝑡𝑟𝑖𝑎𝑙 𝑛 + 1 + Δ𝛾𝔻 :𝑵𝑛 + 1 𝝈𝑛 + 1:𝑵𝑛 + 1 𝜀𝑝𝑛 + 1 ― 𝜀𝑝𝑛 ― Δ𝛾

𝜎𝑦

=

1 6―𝜎

𝑐(𝑎𝐼1 𝑛 + 16 + 27𝐽2 𝑛 + 13 + 𝑏𝐽3 𝑛 + 1)

𝑦0

― 𝐻𝐼𝜀𝑝𝑛 + 1

2) Perform Newton-Raphson iteration

[

]

∂𝑹𝝈𝑛 + 1

∂𝑹𝝈𝑛 + 1 ∂𝑹𝝈𝑛 + 1

∂𝝈𝑛 + 1 ∂𝑅𝜀𝑝𝑛 + 1

∂𝜀𝑝𝑛 + 1 ∂𝑅𝜀𝑝𝑛 + 1

∂Δ𝛾 ∂𝑅𝜀𝑝𝑛 + 1

∂𝝈𝑛 + 1 ∂𝑅Δ𝛾

∂𝜀𝑝𝑛 + 1 ∂𝑅Δ𝛾

∂Δ𝛾 ∂𝑅Δ𝛾

∂𝝈𝑛 + 1

∂𝜀𝑝𝑛 + 1

∂Δ𝛾

𝑘

[ ]

𝛿𝝈𝑛 + 1 . 𝛿𝜀𝑝𝑛 + 1 𝛿Δ𝛾

𝑘+1

[

𝑹𝝈𝑛 + 1(𝝈𝑛 + 1, 𝜀𝑝𝑛 + 1, Δ𝛾)

𝑝 = ― 𝑅𝜀𝑝𝑛 + 1(𝝈𝑛 + 1, 𝜀𝑛 + 1, Δ𝛾) 𝑅Δ𝛾(𝝈𝑛 + 1, 𝜀𝑝𝑛 + 1, Δ𝛾)

]

𝑘

New guess for 𝝈𝑛 + 1, 𝜀𝑝𝑛 + 1 and ∆𝛾: 𝝈𝑛 + 1(𝑘 + 1) = 𝝈𝑛 + 1(𝑘) + 𝛿𝝈𝑛 + 1(𝑘 + 1) ;

(𝑘 + 1)

𝜀𝑝𝑛 + 1

(𝑘)

= 𝜀𝑝𝑛 + 1

(𝑘 + 1)

+ 𝛿𝜀𝑝𝑛 + 1

;

∆𝛾(𝑘 + 1) = ∆𝛾(𝑘) + 𝛿Δ𝛾(𝑘 + 1) 1

3) Check for convergence: Φ = 𝑐(𝑎𝐼1 𝑛 + 16 + 27𝐽2 𝑛 + 13 + 𝑏𝐽3 𝑛 + 1)6 ― 𝜎𝑦(𝜀𝑝𝑛 + 1) , IF |Φ| ≤ 𝜖𝑡𝑜𝑙 THEN return to stress update, ELSE go to step (2). where: ∂𝑅𝝈𝑛 + 1

∂𝑵𝑛 + 1 ; = 𝕀 + Δ𝛾𝔻𝑒: ∂𝝈𝑛 + 1 ∂𝝈𝑛 + 1 ∂𝑅𝜀𝑝𝑛 + 1

(

∂𝑅𝝈𝑛 + 1 ∂𝜀𝑝𝑛 + 1

=𝟎;

∂𝑅𝜀𝑝𝑛 + 1

)

∂𝑵𝑛 + 1 Δ𝛾 ; =― 𝑵𝑛 + 1 + 𝝈𝑛 + 1: ∂𝝈𝑛 + 1 𝜎𝑦 ∂𝝈𝑛 + 1 ∂𝑅𝜀𝑝𝑛 + 1 ∂Δ𝛾

∂𝑅Δ𝛾 ∂𝝈𝑛 + 1

=―

∂𝜀𝑝𝑛 + 1

∂Δ𝛾

= 1 + Δ𝛾

= 𝔻𝑒:𝑵𝑛 + 1

𝝈𝑛 + 1:𝑵𝑛 + 1 𝐻𝐼 ; 𝜎𝑦2

𝝈𝑛 + 1:𝑵𝑛 + 1

(17)

(18)

𝜎𝑦

∂𝑅Δ𝛾

= 𝑵𝑛 + 1 ;

∂𝑅𝝈𝑛 + 1

∂𝜀𝑝𝑛 + 1

∂𝑅Δ𝛾

= ― 𝐻𝐼 ;

∂Δ𝛾

=0

(19)

and, ∂𝑵𝑛 + 1

[

5 𝑐 ― = ― (𝛼) 6 ∂𝝈𝑛 + 1 6

11 6

5

∂𝛼 ∂𝛼 ― ⊗ + (𝛼) 6𝔾 ∂𝝈 ∂𝝈

]

(20)

regarding, ∂(𝐽22𝑺) ∂𝐼15 ∂(det 𝑺 𝑺 ―𝑇) :𝕀𝑑 𝔾 = 𝑎6 ⊗ 𝐼 + 81 +𝑏 ∂𝝈 ∂𝝈 ∂𝝈

(21)

3 – EXPERIMENTAL DATA AND CALIBRATION PROCEDURE 3.1 – Experimental results Experimental tests were performed for smooth and notched cylindrical specimens, subjected to tensile loading, rectangular cross-section specimens, subjected to pure shear and combined tensile/shear loading, both manufactured from AA6101-T4. The tests performed on cylindrical specimens were done under strain control, using a strain gauge of 25 mm of gauge length. However, the tests performed on rectangular cross-section specimens were done under displacement control, due to the limitation of the gauge length of the strain gauge. In these cases, a length of 100 mm between symmetric parts of the specimen was controlled, in order to determine the axial strain. Figure 1 illustrates the experimental control done during the monotonic tests in cylindrical and rectangular specimens until the fracture.

(a) (b) Figure 1: a) Strain control for cylindrical specimens with strain gauge of 25 mm of gauge length, b) Displacement control for the combined load specimen. 3.1.1 – Geometries of specimens The cylindrical specimens were designed to generate an initial stress triaxiality, in the critical node, of 0.33, 0.50 and 0.60, representing a region of high level of stress triaxiality (see Figure 2a, 2b and 2c) and regarding a solid cross-section. Moreover, for the rectangular specimens, it was assumed the geometries proposed by Driemeier et al. (2010) and Yan and Zhao (2018), which representing levels of initial stress triaxiality of zero, for the pure shear (see Figure 2d), 0.10 and 0.23, for two configurations of combined tensile/shear loading (see Figures 2e and 2f). Nevertheless, the initial values of stress triaxialities were measured at the critical point of the specimens, which represents the region of maximum accumulated plastic strain at fracture. The geometries of the manufactured specimens can be observed in Figure 2 and the experimental tests were performed in a universal test machine MTS-312 with a capacity of 100 kN. The same displacement rate of 1 mm/min was assumed for tests in both cylindrical and rectangular specimens, regarding that both ones have similar area on the critical solid cross-section, 55.24 mm2 and 44.25 mm2, respectively.

(a)

(b) Section A-A

(c)

Section B-B

(d)

Section A-A

(e)

Section B-B

(f)

Section B-B Figure 2: Geometries of manufactured specimens. The cylindrical specimens were designed for a total length of 120 mm, as well as, a useful length of approximately 40 mm. However, the relation between the notch radius and the initial stress triaxiality was obtained according to recommendation of Bridgman (Bridgman, 1952). Figure 3 schematically shows the dimensions of the cylindrical specimen in the critical region, as well as, Eq. (22) shows the mathematical relation between notch radius, 𝑅, stress triaxiality, 𝜂 and radius of the specimen, 𝑎.

Figure 3: Relation between geometry and stress triaxiality of cylindrical specimens in the critical section (Bai, 2018).

(

)

1 𝑎 𝜂 = + ln 1 + 3 2𝑅

(22)

Furthermore, for the rectangular specimens, the loading history follows the schematic conditions shown in Figure 4. Then, the stress triaxialities of 0.10 and 0.23 were obtained by a rotation, on the critical region of the specimen, at 𝛼 = 30 ° and 𝛼 = 60 °, which leads to combined tensile and shear loads by the components of 𝐹𝑦.

𝐹�� 𝑦

𝑦

𝐹� 𝐹�� 𝑥

𝐹�

𝑥 𝐹�

𝐹��

𝐹�

𝐹��

(a) shear

(b) mixed shear/tensile

Figure 4: Configuration of forces for a) Pure shear specimen and b) Combined loading specimens. Table 1 shows the notch radium or the rotation angle for all specimens used and the values of the initial stress state parameters. Table 1: Initial stress state parameters for the specimens. Description

R [mm] / 𝛼

Initial stress triaxiality, 𝜂0

Shear specimen Combined specimen 30º Combined specimen 60º Smooth cylindrical bar

0º 30º 60º 10 6

0 0.10 0.23 0.33 0.50 0.60

Notched cylindrical bar

Initial normalized third invariant, 𝜉0 0 0.45 0.88 1 1 1

3.1.2 – Experimental data After experimental tests performed for the AA6101-T4, the reaction force versus displacement curves were obtained and can be presented through Figures 5, 6, 7 and 8. Figure 9 shows the group of six specimens tested until the fracture.

Figure 5: Reaction force versus displacement curve for smooth cylindrical bar specimen.

𝑅 = 10 𝑚𝑚

𝑅 = 6 𝑚𝑚

Figure 6: Reaction force versus displacement curve for notched cylindrical bar specimens, 𝑅 = 10 𝑚𝑚 and 𝑅 = 6 𝑚𝑚.

Figure 7: Reaction force versus displacement curve for rectangular specimen under shear load.

𝛼 = 30º

𝛼 = 60º

Figure 8: Reaction force versus displacement curve for combined specimens under tensile/shear loading, 𝛼 = 30º 𝑎𝑛𝑑 𝛼 = 60º.

𝑅 = 10 𝑚𝑚, 𝑅 = 6 𝑚𝑚

𝛼 = 0º , 𝛼 = 30º , 𝛼 = 60º Figure 9: Cylindrical and rectangular specimens at fracture. Table 2 summarizes the experimental displacement at fracture obtained for all specimens, regarding each combination of initial stress triaxiality and initial normalized third invariant.

Table 2: Displacement at fracture observed experimentally.

Specimen Rectangular shear Rectangular combined, 𝛼 = 30º Rectangular combined, 𝛼 = 60º Cylindrical smooth Cylindrical notched, 𝑅 = 10 𝑚𝑚 Cylindrical notched, 𝑅 = 6 𝑚𝑚

Initial stress triaxiality, 𝜂0 0 0.10 0.23 0.33 0.50 0.60

Initial normalized third invariant, 𝜉0 0 0.45 0.88 1 1 1

Displacement at fracture [mm] 7.0 4.5 2.7 8.9 3.5 2.9

3.2 – Determination of material parameters. Initially, in the calibration procedure for the isotropic hardening curve, two stress states were used to obtain the material parameters. In the first step, the smooth bar specimen under tensile loading was assumed as the first calibration point, which has an initial stress triaxiality of 0.33 and initial normalized third invariant of 1.0. However, in the second step process, the rectangular specimen under pure shear was performed, with both initial stress triaxiality and initial normalized third invariant equal to zero. The first calibration point is in the high stress triaxiality region and the second calibration point in the low stress triaxiality region, as shown in Figure 10.

𝜂0

Figure 10: Schematic distribution of the calibration conditions for mechanical behavior description. The calibration strategy was based on an inverse parametric identification method. This method takes into account the reaction curve obtained experimentally, the finite element modeling to describe the behavior of the calibration point, a multi variable search method based on the gradient approach (Machado, 2019) and a four parameters equation,

proposed by Kleinermann and Ponthot (2003), that considers the description of the isotropic hardening curve of the material, as mathematically represented by Eq. (23). 𝑝

𝜎𝑦 = 𝜎𝑦0 + 𝜔𝜀𝑝 + (𝜎∞ ― 𝜎𝑦0)(1 ― 𝑒 ―𝛿𝜀 )

(23)

where, 𝜎𝑦 represents the yield stress, 𝜎𝑦0 is the initial yield stress, 𝜀𝑝 is the accumulated plastic strain, which represents the internal variable associated with isotropic hardening and the set of parameters (𝜔, 𝜎∞, 𝛿) that represent the fitting of Eq. (23). Tables 3 and 4 contain the elastic and plastic parameters for the AA6101-T4, after the calibration procedure.

Table 3: Elastic parameters for AA6101-T4. Description

Symbol

Young’s modulus Poisson’s ratio Initial yield stress Elongation at break

E [MPa] 𝜈 𝜎𝑦0 [MPa] A [%]

Values Tensile

Shear 65554 0.3

96.13 35.6

81.66 -

Table 4: Hardening curve parameters for AA6101-T4. Description 𝜎𝑦0 / 𝜏𝑦0 [MPa] 𝜔 [MPa] 𝜎∞ [MPa] 𝛿 𝜀𝑝𝑚𝑎𝑥

Values tensile 96.13 63.20 220.87 12.147 1.50

shear 81.66 164.70 107.24 45.45 0.56

It is important to observe in Table 4 that the ratio between 𝜏𝑦0 𝜎𝑦0 = 0.85, which is very different of 1/ 3, in which it was established between the criteria of Mises and Tresca (see Cavalheiro and Malcher, 2017). Regarding the calibration of parameters 𝑎 and 𝑏 for the Gao model, where parameter 𝑎 controls the contribution of the hydrostatic pressure effect and parameter 𝑏 controls the effect of the third invariant of the deviatoric stress tensor, the following experimental tests were assumed: a) tensile test on the cylindrical notched specimen, 𝑅 = 6 𝑚𝑚, for calibration of parameter 𝑎, keeping 𝑏 = 0, and b) pure shear test on the rectangular specimen for calibration of parameter 𝑏, keeping 𝑎 = 0. How was assumed in the calibration of the hardening curve above, the present ones calibrations were also based on an inverse parametric identification, however for an univariable search method. In both cases, the calibration procedure was done until the numerical reaction curve agrees with the experimental data (see Figure 11a and 11b). After calibration procedure, it was

found 𝑎 = ―0.25 and 𝑏 = ―140. Moreover, due to the non-convexity of the Gao model, already discussed by Cavalheiro and Malcher (2017), a value of 𝑏 = ― 110 was also assumed for the further numerical simulation. This value represents that the Gao’ yield surface calibrated in tensile touches the mises yield surface calibration in pure shear, or the convexity limit of the Gao model (see Figure 11d). Figure 11 presents the results of the calibration procedure based on the reaction curves and the shape of Gao’ yield surface for calibrated values of 𝑎 and 𝑏. It is important to remark that 𝑎 = 0 and 𝑏 = 0 recovers the Mises model. It is also important to highlight that to better understand the effect of the stress triaxiality and the third invariant on AA6101-T4, it was decided to perform simulations: a) for the high stress triaxiality region: activating the volumetric term, 𝑎 ≠ 0, and uncoupling the deviatoric term, 𝑏 = 0, and b) for the low stress triaxiality region: uncoupling the volumetric term, 𝑎 = 0, and activating the deviatoric term, 𝑏 ≠ 0. Regarding this convention, it can be observing the transferability of the model in each stress triaxiality regime.

tensile - notched bar 𝑅 = 6 𝑚𝑚

pure shear rectangular specimen

(b)

𝜎2

𝜎2

(a)

𝜎1

𝜎1

(c)

(d)

Figure 11: Behavior of Gao’ yield surfaces for different values for 𝑎 and 𝑏.

4 – NUMERICAL SIMULATION AND DISCUSSIONS In this part of the work, two type of analyzes were done, in order to better understand the presence of the stress triaxiality and the normalized third invariant effects on AA6101T4, regarding the experimental results performed and the Gao’s model. The first one for the cylindrical specimens in high level of stress triaxiality, uncoupling the effect of the third invariant, and the second one for the rectangular specimens in zero or low level of stress triaxiality, uncoupling the volumetric effect and coupling the third invariant effect. 4.1 – Finite elements mesh The numerical simulations were carried out through an academic development of finite elements, called Hyplas (see De Souza Neto et al., 2008), considering large strain and the implicit integration of the constitutive model, which includes the Gao’ yield criterion, nonlinear isotropic hardening and associative plasticity. A preliminary mesh convergence study was done and only the discretization level with the stabilized results was presented in the work. Regarding the cylindrical specimens, the finite element models were structured considering an axisymmetric problem, eight-node quadrilateral bidimensional finite element and reduced integration, with 675 finite elements and 2146 nodes. Figure 12 illustrates the finite element meshes used for the different cylindrical specimens.

(a)

(b)

(c)

Figure 12: Finite element meshes for cylindrical specimens. (a) Smooth bar, (b) notched bar - 𝑅 = 10 𝑚𝑚 e (c) notched bar - 𝑅 = 6 𝑚𝑚. However, for the rectangular specimens, the finite element simulations were performed considering the three-dimensional problem, due to the complexity of the geometries. It was used 8080 nodes and 6000 eight-node hexahedral finite elements with complete integration. Figure 13 shows the finite element mesh used for the rectangular specimens.

(a) (b) (c) Figure 13: Finite element mesh for rectangular specimens (a) shear, (b) combined 𝛼 = 30º and combined - 𝛼 = 60º . 4.2 – Numerical and experimental reaction curves 4.2.1 – High level of stress triaxiality In order to analyze the numerical response for the reaction curves, prescribed displacements were given, and the simulations were performed until the displacement at fracture, observed experimentally (see Table 2). Regarding the cylindrical specimens two simulations were done for each specimen: case 1 - isotropic hardening curve calibrated for tensile loading (see Table 4), 𝑎 = 0 and 𝑏 = 0, recovering Mises’ model; case 2 isotropic hardening curve calibrated for tensile loading (see Table 4), 𝑎 = ―0.25 and 𝑏 = 0. Figure 14 presents comparison between the numerical results and experimental data for cylindrical bar specimens.

(a) Smooth bar Specimen

(b) Notched bar specimen - R10

(c) Notched bar specimen - R6

Figure 14: Reaction curves for cylindrical specimens. It can be observed, by Figure 14(a), the reaction curve for the smooth bar specimen, obtained numerically, agrees with the reaction curve obtained experimentally. This behavior is justified because this point was assumed as the first calibration point of the material properties. However, the reaction curves obtained numerically for the notched bar specimens (see Figure 14(b) and 14(c)) disagree from experimental curves, with the progressive increase of the notch, which is numerically represented by the increase of the stress triaxiality. However, it is important to highlight that according to Figures 14(b) and 14(c), the term 𝑎 is also different of zero and the stress triaxiality effect is activated. The notch of the specimen increase the value of the stress triaxiality from 0.33 to 0.60 and the term 𝑎 needs to be used (case 2), in order to ability the model to represent correctly mechanical behavior of the aluminum alloy until the fracture. The case 1 represents the behavior of the von Mises’ model or Gao’ model with 𝑎 = 0 and 𝑏 = 0. The difference between case 1 and case 2 is the contribution of the stress triaxiality on the behavior of the material. How the parameter 𝑎 was calibrated assuming a notch radius of 𝑅 = 6 𝑚𝑚, the better agreement of Gao’s model can be observed for several values of stress

triaxiality. Regarding Figure 14(b), where notch radius is not several (𝑅 = 10 𝑚𝑚), the numerical response is in transition between Mises e Gao.

4.2.2 – Low level of stress triaxiality Regarding the rectangular specimens, four simulations were done for each specimen: case 1 - isotropic hardening curve calibrated for tensile loading (see Table 4), 𝑎 = 0 and 𝑏 = 0, recovering Mises’ model; case 2 - isotropic hardening curve calibrated for tensile loading (see Table 4), 𝑎 = 0 and 𝑏 = ―110 (limit of convexity); case 3 - isotropic hardening curve calibrated for tensile loading (see Table 4), 𝑎 = 0 and 𝑏 = ―140 (calibrated); case 4 isotropic hardening curve calibrated for shear loading (see Table 4), 𝑎 = 0 and 𝑏 = 0, recovering Mises’ model. Otherwise, Figure 15 presents a comparison between the numerical results and experimental data for rectangular specimens.

(a)

(b)

Rectangular specimen under shear

Rectangular Specimen combined loading - 𝛼 = 30º

(c)

Rectangular Specimen combined loading - 𝛼 = 60º

Figure 15: Reaction curve for rectangular specimens.

According to Figure 15, it is possible to observe the behavior of the Gao’ model for simulations using the hardening curve calibrated under tensile loading, cases 1, 2 and 3, and for simulations assuming the hardening curve calibrated under shear loading, case 4. Firstly, for cases 1, 2 and 3, it is possible observe the effect of the third invariant of the deviatoric stress tensor. The simulations of case 1 recover the behavior of the von Mises’ model and disagree with the experimental data. However, the simulations of cases 2 and 3, the term 𝑏 is different of zero and the third invariant effect is activated. The optimized value for 𝑏 is between -110 and -140, which can be observed in Figure 11, where the yield surface of cases 2 and 3 reach the yield surface of case 4. The non-convexity of the Gao’ model for values of 𝑏 < ―110, limits the application of the Gao’ model for AA6101-T4, or for materials with high ductility.

4.3 – Evolution for accumulated plastic strain. The point of maximum value of the accumulated plastic strain is assumed as the critical point, and the evolution curve were analyzed, regarding cylindrical and rectangular specimens. In this sense, the critical point is assumed as the point of fracture onset, which in these cases, coincides with the point observed experimentally where starting a macroscopic crack. Figures 16, 17 and 18 have shown the evolution of the accumulated plastic strain as a function of the level of displacement imposed until the fracture. Regarding the cylindrical specimens (Figure 16) the evolution of the accumulated plastic strain was not so different for case 1 and case 2, in both smooth and notched bars. However, it is important to observe the convergence problem in case 2, near of the fracture. For both 𝑅 = 10 𝑚𝑚 and 𝑅 = 6 𝑚𝑚, the Gao’ model was not able to reach the displacement at fracture, observed experimentally. However, for the rectangular specimens (Figures 17-18), all simulations reached the displacement at fracture, imposed. In this setting, the values for the accumulated plastic strain were relatively different, near at fracture, mainly when the term 𝑏 is activated (see cases 2 and 3). In the case 4, simulations were done using the hardening curve calibrated in shear loading condition. Analyzing the accumulated plastic strain as fracture indicator, it can be observed for cylindrical specimens that the fracture starts from the center of the specimen in direction to the surface. However, regarding the rectangular specimen under shear loading, the fracture starts from the surface and propagate in direction to the center of the specimen. In the case of the combined specimens, there are two main behaviors: (a) 𝛼 = 30º, the fracture starts from the surface and propagate to the center and (b) 𝛼 = 60º, the fracture starts from the center and propagate to the surface. Regarding 𝛼 = 30º, there is more contribution of the shear than the tensile loading. On the other side, regarding 𝛼 = 60º, there is more contribution of the tensile than the shear loading.

(a) Smooth bar Specimen

(b) Notched bar Specimen - R10

(c) Notched bar Specimen - R6

Figure 16: Evolution curve for the accumulated plastic strain at the critical node for cylindrical specimens.

Figure 17: Evolution curve for the accumulated plastic strain at the critical node for pure shear specimen.

(a) rectangular specimen,𝛼 = 30º

(b) rectangular specimen,𝛼 = 60º

Figure 18: Evolution curve for the accumulated plastic strain at the critical node for combined loading specimens. Table 5 shows values for the accumulated plastic strain at fracture calculated through the Gao’ model at the critical point, regarding all cases analyzed. Table 5: Values for the accumulated plastic strain at fracture.

Specimen Shear combined - 𝛼 = 30º combined - 𝛼 = 60º Smooth Notched - 𝑅 = 10 𝑚𝑚 Notched - 𝑅 = 6 𝑚𝑚

Accumulated plastic strain at fracture

Stress triaxiality

Normalized thrid invariant

Case 1

Case 2

Case 3

Case 4

0 0.10 0.23 0.33 0.50 0.60

0 0.45 0.88 1 1 1

0.60 0.56 0.50 1.51 1.10 0.95

0.51 0.47 0.41 1.51 0.89 0.63

0.46 0.42 0.37 -

0.56 0.58 0.47 -

4.4 – Evolution for stress triaxiality and normalized third invariant Through the numerical simulations performed, it is also possible to analyze the evolution of the two stress state parameters highlighted here. Figures 19 and 20 show the evolution of the stress triaxiality and normalized third invariant, from its initial value, until the value at fracture. It can be observed that in both numerical simulations, the final value of the stress triaxiality is high different of their initially projected values. This is justified by the fact that the plastic strain process causes a significant modification in the geometry of the specimen, which leads to significant changes in the stress states. This phenomenon can cause, at the end, an acceleration or deceleration in the process of progressive degradation of the material, causing premature failure predictions for stress states under predominant shear contribution (rectangular specimens) and even optimistic failure predictions for fracture for other stress states under predominant tensile contribution (cylindrical specimens). However, regarding the normalized third invariant, for cylindrical specimens, the parameter keeps equal to one, from the beginning to the end of the process. In this setting, the rate of the normalized third invariant is equal to the rate of the equivalent stress. On the other side, for rectangular specimens, the modification of the geometry can modify the normalized third invariant at the critical point, due to the rate of the 𝐽3 is higher than the rate of 𝐽2. This modification in the geometry of the specimens can be observed through Figure 21a and Figure 22a, which show the contours of the stress triaxiality for the displacement equal to zero and at fracture, regarding the rectangular specimen under pure shear (𝛼 = 0º ) and combined loading (𝛼 = 60º). Figure 21b and Figure 22b also show the contour of the normalized third invariant around the geometry of the specimen to an instant before beginning of the deformation (displacement equal to zero) and to an instant close to the fracture (fracture displacement). In both contours (Figure 21 and 22) it is possible to observe a significant modification in the distribution of the stress state parameters around the geometry of the specimens, as well as, the evolution in the numerical value within the critical point.

Figure 19: Evolution of the stress triaxiality and normalized third invariant for cylindrical specimens.

Figure 20: Evolution of the stress triaxiality and normalized third invariant for rectangular specimens.

“initial step”

“near at fracture” (a) stress triaxiliaty

(b) Normalized third invariant

Figure 21: Contour of the stress triaxiality and normalized third invariant for rectangular specimens, 𝛼 = 0º.

“initial step”

“near at fracture” (a) stress triaxiliaty

(b) Normalized third invariant

Figure 22: Contour of the stress triaxiality and normalized third invariant for rectangular specimens, 𝛼 = 60º.

5 – CONCLUSIONS In this contribution, experimental tests were performed for different specimens, in order to show the influence of the stress triaxiality and the normalized third invariant on the behavior of a ductile material until the fracture. The plastic properties of the AA6101 – T4 alloy were determined based on two stress states or calibration points (tensile and shear). The Gao’ model was assumed and an implicit numerical integration algorithm was proposed in order to describe the effect of both stress triaxiality and the third invariant parameters on the mechanical behavior of the material at fracture by the reaction force curves, evolution curves of the stress state parameters at the critical point and the contour of them around the geometries of the specimens. Regarding the cylindrical specimens, activating the term 𝑎 of the Gao’ yield criterion, the model was able to reproduce numerically, the reaction curve experimentally observed. Otherwise, assuming the rectangular specimens, activating the term 𝑏, the model was also able to reproduce the experimental behavior, regarding some limitation. The terms 𝑎 and 𝑏 introduced the stress triaxiality and normalized third invariant effects in the constitutive formulation. It was possible observe by the difference between reaction curves numerically calculated and experimentally observed that the third invariant effect is more several than the stress triaxiality effect and needs to be accounted in the plastic flow rule for a correct description of the mechanical behavior of ductile material at fracture. Due to

non-convexity of the Gao’ model in some stress state, for example in low stress triaxiality regime, the reaction curve obtained numerically for a specific value of 𝑏 disagree to the experimental data, but within a band of dispersion, regarding 𝑏 = ―110 and 𝑏 = ―140. Furthermore, the accumulated plastic strain was able to identify the critical point or the node of beginning of a crack in both cases. For all specimens performed, the maximum value of 𝜀𝑝 agrees with the point of crack initiation observed experimentally. In general, it was observed that for stress states under predominant tensile contribution, the crack starts into the specimen and propagate in direction to the surface. However, for stress states under predominant shear contribution, the crack starts on the surface and propagate in direction to the center of the specimen. Regarding the evolution of the stress triaxiality and normalized third invariant during the deformation process, the first parameter changes severally from the beginning to the end of the process, due to the high contribution of the hydrostatic pressure in all stress states simulated. For the second parameter, it can be observing several modifications for stress state predominantly shear. Regarding the cylindrical specimens and rectangular specimen with 𝛼 = 60º, where the tensile contribution is higher than shear, the normalized third invariant keeps constant or follow for an asymptotic value.

ACKNOWLEDGMENTS The financial support provided by Federal District Research Support Foundation (FAPDF) is gratefully acknowledged. Lucival Malcher would also like to acknowledge the support from the Brazilian Council for the Scientific and Technological Development—CNPq (contract 311933/2018-1). Leonel L.D. Morales would also like to acknowledge the support from University Center of the Federal District – UDF. REFERENCES Andrade, F. X. C., Pires, F.M.A., De Sá, J.M.A., Malcher, L. (2009). Improvement of the numerical prediction of ductile failure with an integral nonlocal damage model. International Journal of Material Forming, v. 2, p. 439-442. Bardet, J. P. (1990). Lode Dependence for Isotropic Pressure-Sensitive Elastoplastic materials. Jornal of Applied Mechanics, 57:498-506. Bai, Y. (2008). Effect of Loading History on Necking and Fracture. Ph.D Thesis, Massachusetts Institute of Technology. Bai, Y., Wierzbicki, T. (2007). A new model of metal plasticity and fracture with pressure and Lode dependence, International Journal of Plasticity, 24:1071-1096. Bridgman, P. (1952). Studies in Large Plastic and Fracture. London: McGraw-Hill Book Company. Brünig, M., Chyra, O., Albrecht, D., Driemeier, L., Alves, M. (2008). A ductile damage criterion at various stress triaxialities, International Journal of Plasticity, 24: 1731–1755.

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1) 2) 3) 4) 5)

Experimental program for AA6101-T4; Stress triaxiality and normalized third invariant effects; Ductile fracture and finite elements analysis; Accumulated plastic strain at fracture indicator; Gao’ model with isotropic hardening.