Normal stress components during shear tests of metal sheets

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Normal stress components during shear tests of metal sheets A.F.G. Pereira , P.A. Prates , M.C. Oliveira , J.V. Fernandes PII: DOI: Reference:

S0020-7403(18)34138-9 https://doi.org/10.1016/j.ijmecsci.2019.105169 MS 105169

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

13 December 2018 10 September 2019 15 September 2019

Please cite this article as: A.F.G. Pereira , P.A. Prates , M.C. Oliveira , J.V. Fernandes , Normal stress components during shear tests of metal sheets, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105169

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Highlights 

For a strong anisotropy, the normal stresses components should not be neglected.



The specimen geometry and boundary conditions influence the normal stresses.



The rotation of the material axes, affects the stresses evolutions.



The analytically predicted normal stresses agree with those numerically evaluated.

Normal stress components during shear tests of metal sheets A.F.G. Pereira*; P.A. Prates; M.C. Oliveira; J.V. Fernandes CEMMPRE — Department of Mechanical Engineering, University of Coimbra, Rua Luís Reis Santos, Pinhal de Marrocos, 3030-788 Coimbra, Portugal *

Corresponding author

e-mail: [email protected]; telephone: +351 239 790 716/00

Abstract The normal stress components that arise during the shear test of metal sheets are numerically investigated. Simulations of the shear test are used to study the influence of the material anisotropy, specimen geometry, boundary conditions and loading direction on those components. For anisotropic materials, the normal stresses can reach magnitudes similar to those of the shear component. These components arise due to the imposed boundary conditions and the stress disturbance at the free ends of the specimen and its evolution is affected by the rotation of the material axes. An analytical approach to estimate these components is proposed. The predicted normal components agree with those evaluated numerically and their use allows the correct determination of the equivalent stress vs. strain curve.

Keywords Shear Test; Normal Stresses; Specimen Geometry; Boundary Conditions; Anisotropy; Metal Sheets

1. Introduction: The design of sheet metal components and their forming operations are commonly performed with finite element analysis. In this context, the modelling and characterization of the material plastic behaviour are crucial for the accuracy of the numerical results. In order to characterize the plastic behaviour of the materials under

different loading conditions, various experimental tests are generally used for the identification of the constitutive parameters [1–4]. One example is the shear test, which is an important characterization tool for the accurate evaluation of the materials response under shear loading that generally occurs in sheet metal forming processes [5]. Moreover, the shear test allows reaching relatively large strain values without the occurrence of plastic instability and has the possibility of reversing the loading direction for the study and characterization of the Bauschinger effect. Several experimental approaches and specimen geometries have been proposed over the years to performed in-plane shear tests [3,6–9] and the related in-plane torsion tests [10–12]. The stress distribution in the shear test is generally characterized by a heterogeneous zone near the free ends of the specimen and a homogeneous zone in the centre of the gauge area [9,13]. When the lack of stress homogeneity is restricted to a small zone of the specimen, the shear stress is simply evaluated by dividing the shear loading by the shear area [9,13]. Previous analysis of the in-plane shear test results commonly neglected the occurrence of normal stress components, although magnitudes similar to those of shear stresses have been reported in some numerical studies [9,14,15]. According to Yoon [14] and Bouvier [13], these normal components emerge from the imposed boundary conditions, i.e. constraints on the variation of the width and length of the gauge area, which generates lateral forces. Rahmaan [16] pointed out that the normal stress components are dependent on the constitutive model and the objective stress rate adopted in the numerical formulation. Nonetheless, Rahmaan [16] showed that, in case of isotropic materials, the normal stress components were much lower than the shear component (bellow 0.4% for large strains), and therefore could be neglected. Yin [9] compare the numerically evaluated stress and strain distributions, for three distinct shear test devices and an isotropic material (von Mises). The authors reported differences in the normal stress distributions between the specimens used in each device. Additionally, the normal stress components can reach values up to 25% of the shear stress magnitude in the centre of the gauge area, depending on the specimen geometry. Yoon [14] conducted numerical simulations of shear tests at 45º and 90º with the rolling direction of an anisotropic sheet (AA1050-O). Although the numerical model assumed an isotropic hardening law and a constant shape for the yield surface, the numerical results displayed an apparent anisotropic hardening behaviour. Yoon [14] stated that this contradictory hardening behaviour results from the existence of significant values of the normal stresses (up to 102% of the shear component) due to the

imposed boundary conditions. In fact, the normal stresses are influenced by the loading direction, leading to the apparent anisotropic hardening. Also, Kohar [15] conducted numerical shear tests in an anisotropic material (AA6063-T6) using either a crystal plasticity model and a phenomenological model with microstructural evolution. For both models, the normal components reach a magnitude of about 42% of the shear stress, at a shear strain of 40%. In order to experimentally apply shear and normal loading combinations, Mohr and Oswald [17] proposed a shear test device using a butterfly specimen with a dual-actuator system. The experimental apparatus permits the application of a shear loading and simultaneously the control of the normal force perpendicular to the shear direction. Consequently, it is possible to keep the normal force and thus the normal stress component perpendicular to the shear direction, equal to zero, during the shear loading. Nevertheless, during the test it is not guaranteed that the other normal stress component (along the shear direction) presents a negligible magnitude. Moreover, the testing setup was validated using numerical simulations of an isotropic material, modelled with von Mises yield criterion, for which the magnitude of the normal components is expected to remain small [16]. Although, the normal stress components can reach magnitudes similar to those of shear, few studies on the influence of the normal components in the shear test results are available in literature due do the difficulty/impossibility of experimentally measure these components. Moreover, when studying the effect of the normal stress components, the numerical works were mainly focused on isotropic materials, although the metal sheets are generally anisotropic. As it will be shown, depending on the anisotropy of the materials and on the test characteristics (geometry and boundary conditions), the normal components can have a visible influence on the results of the shear test. In the present work, the impact of the material anisotropy, specimen geometry, boundary conditions and loading direction will be numerically evaluated. Based on these results an analytical approach to estimate the normal stress components during the in-plane shear test is proposed.

2. Numerical Model: Shear Test The study of the normal stress components is focused in a shear test device of the type proposed by G’Sell [6] for polymers and later adapted for metallic materials by Rauch and G’Sell [8]. This shear test has two clamping areas that move parallel to each other, in order to generate shear strain in a single gauge area. The specimen geometry

has been widely used [3,18–21], due to the homogeneous deformation achieved and the highly simplified geometry of the specimen. In order to avoid buckling and premature failure, and to generate homogeneous shear stress distributions over a large measurable area, Hu [18] and Bouvier [13] recommended a length to width ratio greater than 10 and a width to thickness ratio between 2 and 10. Following these recommendations, the geometry presented in Figure 1 will be used in this work. The sheet thickness is 1 mm and a length of 120 mm was adopted to ensure that the free end effects do not propagate to the central part of the specimen. The numerical model of the shear test is based on a previous work [3]. The model considers only the gauge region of the shear specimen, as shown in Figure 1. This region was discretized with a total number of 11376 solid elements, with two elements through the thickness of 1 mm. The numerical simulation is performed by imposing a displacement, along the Ox axis, on the moving nodes, which are located on the plane defined by the red line (see Figure 1). The nodes located on the plane defined by the blue line (see Figure 1) have their movement restricted in all directions. To study the influence of the boundary conditions on the normal stress components, two distinct boundary conditions are applied to the moving nodes. In the more general case, where the moving grip can only move along the loading direction (Ox axis), the displacement of the moving nodes (located under the red line) is prevented in all other directions. The test with these boundary conditions will hereafter be designated by constrained test. For the cases where the moving grip can also move perpendicular to the loading direction (similar to what occurs in the test apparatus proposed by Mohr and Oswald [17]), the displacement of the moving nodes (located under the red line) is allowed along the Ox and Oy axes. In this case, the test will hereafter be designated by unconstrained test. The numerical simulations were carried out with the implicit finite element home program, DD3IMP (Deep Drawing 3D Implicit Code), using hexahedral tri-linear solid elements with eight nodes, combined with a selective reduced integration technique [22–24]. The formulation adopted assumes that the Jaumann stress rate is used to deduce the differential form of the elastoplastic behaviour law. It is worth mentioning that for high shear strains values and so high rotations, the numerical results are strongly influenced by the objective stress rate used in the numerical formulation [25,26]. Several objective stress rates have been proposed in literature [27,28]. Nevertheless, the proper choice of the stress rate is still a subject of research and discussion [26,27,29,30]. In this work, the shear test results are computed only up to a

level of 40% of shear strain (around 20% in equivalent plastic strain), and therefore it is assumed that the shear test results are independent of the objective stress rate adopted [26]. The constitutive model used in this work assumes that the materials are orthotropic and follow an isotropic hardening law. However, it should be noted that, for large shear strains, the hardening evolution can be anisotropic and the orthotropic symmetry is generally lost, due to texture evolution [31]. This is in some way discarded by limiting the shear strains analysed to 40% (also to avoid that the results depend on the objective stress rate, as mentioned above). Therefore, the following assumptions are taken for the materials behaviour: (i) the elastic behaviour is isotropic and defined by the generalised Hooke’s law; (ii) the plastic behaviour is described by orthotropic yield criteria, either the Hill’48 [32] or by the extension of the Cazacu [33] yield criterion using two linear transformations proposed by Plunkett [34], hereinafter referred to as CPB2x; (iii) the isotropic hardening is modelled by the Swift hardening law [35]. The Hill’48 yield surface is described by the following equation:

[ (̂

(̂

)

̂

)

̂

(̂

̂

) (1)

̂

̂

where , ,

] ̂

, ,

and

are anisotropic parameters; ̂

, ̂

, ̂

, ̂

, ̂

and

are the components of the Cauchy stress tensor, ̂, defined in the orthotropic ̂

coordinate system of the material (

( )̅ represents the hardening law,

) and

which is a function of the equivalent plastic strain, .̅ In the case of the CPB2x yield criterion, the yield surface is described by: [(| | (|

)

|

(|

)

where ,

and

principal values of and

(|

|

) |

(| )

| (|

) |

are material parameters; and

(2)

) ]

( = 1 to 3) and

( = 1 to 3) are the

, respectively. The fourth-order tensors

operating on the deviatoric stress tensor, , of ̂, are given respectively by:

(3) [

]

(4) [

]

where,

(

= 1 to 6) and

that makes

(

= 1 to 6) are material parameters.

is a constant,

(in Equation (2)) equal to the flow stress in uniaxial tension along the

rolling direction, when it is defined as follows: [(|

|

)

(|

|

)

(|

|

) (5)

(|

|

)

(|

|

with,

( ⁄ )

( ⁄ )

( ⁄ )

( ⁄ )

( = 1 to 3).

)

(|

( ⁄ )

|

) ]

(

= 1 to 3) and

( ⁄ )

The hardening is described by the Swift law: (

where

)̅

,

and

(6)

are the parameters of the hardening law;

is the initial

yield stress.

3. Analysis of the Normal Stress Components The influence of the material anisotropy, specimen geometry, boundary conditions and loading direction on the magnitude of the normal stress components during the shear test are evaluated with resource to numerical simulations of the shear test. The effect of neglecting the normal stress components in the determination of the equivalent stress vs. strain curve, obtained from the shear test, is also evaluated.

3.1. Material Anisotropy To evaluate the influence of the material anisotropy on the magnitude of the normal stress components, several materials with distinct orthotropic behaviours are analysed. Although, the parameters of these materials (DP500, AA2090-T3, HSS-Y350 and FeP06t) have been previously identified in the literature [26,34,36–38], they should be understood as fictitious materials in the sense that the identified parameters may not fully describe the materials behaviour (e.g. due to the small number and type of tests used in the identifications). The Hill’48 and the CPB2x yield criteria and the Swift law describe the anisotropy and the hardening behaviour of the studied materials, whose constitutive parameters are indicated in Tables 1 and 2. Representative values for the elastic parameters were considered: Young's modulus, ratio,

G a, and Poisson's G a,

in case of the DP500, FeP06t and HSS-Y350 steels; and

and

, in case of the AA2090-T3 aluminium alloy. For a clear understanding of

the anisotropic behaviour, the normalized yield stress in uniaxial tension, in simple shear,

⁄

⁄

, and

, are plotted as a function of the angle with the rolling direction,

, in Figure 2 (a) and (b), respectively. Note that

corresponds to the test global

reference system. These figures show that the DP500 material is essentially isotropic, while the HSS-Y350 has a slight anisotropic behaviour. In contrast, the AA2093-T3 and the FeP06t materials have both pronounced anisotropic behaviour, although the FeP06t steel exhibits similar tensile stresses at 0º and 90º with the rolling direction, which is not the case for the AA2093-T3 aluminium. Figure 3 plots the normal components, normalized by the correspondent shear stress (equal to

(Figure 3 (a)) and

(Figure 3 (b)),

as a function of the shear strain,

), for the material in Tables 1 and 2, during a constrained shear test

performed in the rolling direction. These components are numerically obtained at the Gauss point located at the centre of the specimen, and are defined in the global reference system (coincident with the axes in Figure 1). As observed in Figure 3, the magnitude of the normal stress components generally increases with the shear strain and its evolution depends on the orthotropic behaviour of the material. For the quasiisotropic materials, the DP500 and the HSS-Y350 steels, the magnitude of the normal components is negligible when compared with the shear stress magnitude (below 5.5%), which agrees with the numerical results reported in literature for isotropic materials

[16]. For the materials with a pronounced anisotropic behaviour, AA2090-T3 aluminium and FeP06t steel, the magnitude of the normal stress components can reach significant values when compared to the shear component. In particular, for the AA2090-T3, the normal component

reach magnitudes up to 71.4% (in compression)

of the shear component. The appearance of relatively high normal components is connected with the anisotropy of the material. Figure 3 also show that, in case of the FeP06t material, the components

and

are approximately symmetric due to

similar behaviour in tension at 0º and 90º (see Figure 2), while for the AA2090-T3 material, with different behaviour at 0º and 90º, the normal components

and

have distinct magnitudes, respectively, 67% (in compression) and 16% (in tension) of the shear component at a shear strain of 40%. The effect of neglecting the normal components in the evaluation of the equivalent stress vs. strain curve is now analysed for the most anisotropic materials, AA2090-T3 and FeP06t. The equivalent stress vs. strain curve of these materials is obtained based on the shear test results at the central Gauss point. The equivalent stress, ̅, is evaluated neglecting or not the normal stress components of the Cauchy stress tensor, ̂, in the respective yield criterion (Equations (1) or (2)). At each moment of deformation, the increment of equivalent plastic strain,

,̅ is computed from the plastic

work equivalence, in which the equivalent stress takes part:

̅

̅

where,

(

)

,

and

reference system;

(7)

are the stress components of the Cauchy tensor in the global ,

and

are the strain increments in the global reference

system. Figure 4 shows the equivalent stress vs. strain curves evaluated considering or not the normal components, for the two most anisotropic materials, the AA2090-T3 aluminium (Figure 4 (a)) and the FeP06t steel (Figure 4 (b)). For comparison, the hardening curves used as input in the numerical simulations are also shown. Figure 4 shows that, in case of the material AA2090-T3 with relatively high anisotropy and magnitude of the normal stress components (in compression, up to 71.4% of the shear stress), the accurate determination of the equivalent stress vs. strain curve depends on whether or not the normal stress components are taken into account. In the case where the normal components are neglected, the determination of the

equivalent stress vs. strain curve can reach errors up to 10.3%, whereas if they are considered the errors are less than 0.12%. For the FeP06t material, with moderate magnitude of the normal stress components (up to 11% of the shear stress), the equivalent stress vs. strain curves determined in both conditions are similar (difference inferior to 1.3%) and close to the input hardening curve. In summary, the material anisotropy has a great influence on the magnitude of the normal stress components. For approximately isotropic materials, the appearance of normal stress components can be neglected, whereas for considerably anisotropic materials such components should be taken into account, since they can reach magnitudes similar to those of the shear component. In the latter case, neglecting the normal components can have a significant impact on the determination of equivalent stress vs. strain curve and, consequently, on the use of these curves for the identification of the material parameters.

3.2. Geometry of the specimen and boundary conditions The influence of the specimen geometry on the distributions of the normal stresses, is now analysed using the example of the FeP06t material (see Table 1), for shear tests performed in the rolling direction. Based on the numerical model previously described, several specimen geometries with different ratios of width ( ) to thickness ( ) and length ( ) to width were investigated. In all the geometries the width is kept fixed and equal to 3 mm to permit that the ratios

and

can be changed

independently of one another. The finite element discretization of each geometry was adapted from Figure 1, such that the number of elements per square millimetre of sheet surface is approximately the same. The length of the refined zone, near the free ends, was kept fixed (3 mm, see Figure 1). For each geometry, the components

and

were evaluated on the centre line parallel to the Ox axis (see Figure 1). The stress distributions along this line will be analysed below, for a shear strain of 40%. The normal component along the specimen thickness,

, is not shown, since it is

approximately null for all the cases studied. For the constrained shear tests, Figure 5 and Figure 6 show, respectively, the distributions of the normal stress components

and

, as a function of the distance

from one end of the specimen, normalized by its length, , for geometries with values of the width to thickness ratio,

, of 1, 3 and 10 (in all cases the length is

) and length to width ratio, is

, of 5, 11.5, 20 and 40 (in all cases the thickness

). The results of the distributions of the normal stress components are

marginally affected by the ratio influenced by the ratio

(see Figure 5 (a) and Figure 6 (a)) and strongly

(see Figure 5 (b) and Figure 6 (b)). For

ratios lower

than 40, the normal stress disturbance observed at the free ends spreads to the central part of the specimen. The increase of the

ratio promotes the stabilization of the

values of the normal stress components (

MPa and

MPa, for

= 40) along the length of the specimen (see Figure 5 (b) and Figure 6 (b)). These results make it clear that the stress disturbance reaches the central part of the specimen only if a low value of the ratio

is chosen. Consequently, the occurrence of the

normal stress components in the centre of the specimen for large values of

is not

associated with the stress disturbance observed near the free ends of the specimen. To understand the influence of the boundary conditions in the appearance and magnitude of the normal stress components, the unconstrained shear test is also analysed. That is, the moving nodes in Figure 1 are also allowed to move along Oy, as occurs in the test apparatus proposed by Mohr and Oswald [17], in which the grip can move perpendicularly to the shear direction. The Cauchy stress components were again evaluated for the same values of the geometric ratios,

and

, previously analysed

for the constrained shear tests. The results are shown in Figure 7 and Figure 8 for the stress components

and

, respectively.

The normal stress components are scarcely influenced by the

ratio (see

Figure 7 (a) and Figure 8 (a)), although some differences can be observed between the specimen with the smallest ratio and the other two. In contrast, the

ratio strongly

affects the normal stress distributions as shown in Figure 7 (b) and Figure 8 (b). For the smaller values of

ratios (namely,

= 5 and 11.5), the stress disturbance spreads

to the central region of the specimen, influencing the entire distribution of

(see

Figure 7 (b)) and the peripheral values of

ratio

(see Figure 8 (b)). As the

increases, the disturbed region becomes more confined to the free ends of the specimens, leading to the stabilization of the normal components values ( MPa and

MPa) for a large central region of the specimen (see Figure 7 (b) and

Figure 8 (b)). These results show that the stress disturbance reaches the central part of the specimen only if low values of the ratios

and

are chosen. Comparing the

normal components of the constrained test (Figure 5 and Figure 6) with those obtained

in the unconstrained test (Figure 7 and Figure 8), general differences in the distributions of the normal stress components are noticeable. In the unconstrained test, the normal component

is approximately zero for a large part of the central region of the

specimen, due to the free movement in the Oy direction. However, the use of the unconstrained test leads to greater values of the normal stress component

than those

obtained with the constrained test. In summary, the values of the normal stress components in the central region of the specimen are only related to the stress disturbance observed at the free ends of the specimen if small

and

ratios are chosen. For large

and

ratios, the

origin of the normal stress components is related with the imposed boundary conditions, since the unconstrained test led to

due to the free movement in the Oy

direction. Moreover, the use of this test when compared to the constrained one does not guarantee smaller values of the normal stress components of the shear test, since in the analysed case, the components of geometric ratios

are greater in the unconstrained test. In both tests,

and

should be used to ensure that the measured

results in the central region of the specimen are not influenced by the spread of the stress disturbance at the free ends of the specimen. Nonetheless, for materials with orthotropic behaviour there will always exist at least a non-null normal stress component,

, in the central region of the specimen.

3.3. Loading Direction and Sign The influence of the loading direction in the sheet plane and the sign of the relative displacements of the grips, on the results of the shear tests is studied. The loading direction is defined by the angle, , between the test and rolling directions as shown in Figure 9. The sign of the relative displacements of the grips is noted

(Figure

9 (a)), if the imposed shear stress tends to rotates the specimen clockwise, and ̅ (Figure 9 (b)), if the rotation is anticlockwise. The study focus on the evolution of shear and normal stress components during the shear test, exemplifying with the case of the AA2093-T3 material (Table 2). Rauch [39] demonstrated that the shear stress vs. shear strain curves for tests performed at

, 90

, 90

and 180

are identical at yielding, but evolve

distinctly with the shear strain. To clarify this point, Figure 10 shows the normalized yield stresses in simple shear,

⁄

, as a function of the initial angle with the rolling

direction , between 0 and 180º. Two sets of loading directions are marked in this figure: (i)

(

̅)

(

0 , 90 and 180 ; (ii)

̅)

20 , 70 , 110 and 160 . Figure

11 (a) and (b) show the shear stress vs. shear strain curves numerically evaluated at the central Gauss point for the first and second sets, respectively. For the first set, the curves are quite identical (Figure 11 (a)), whereas for the second set (Figure 11 (b)) the shear stress evolves differently with the shear strain, showing a higher hardening for the tests ̅

20 , ̅

110 and

70 ,

110 and ̅

160 than for the tests ̅

20 ,

70 ,

160 .

The occurrence of these two behaviours is related to the distinct rotation of the material axes during the shear test, which causes the angle between the test and rolling directions, , to change during deformation from its initial value shown schematically the distinct rotation,

, of the material axes,

induced by the relative displacement of the grips (

. Figure 12 , that are

̅). This rotation can lead to a

higher or lower resistance of the material to shear, which are combined to the isotropic hardening itself, i.e. due to the increased deformation. This leads to the two distinct ̅ equal to 0 , 90 and

behaviours observed in Figure 11 (b). In case of tests at 180 , the increase or decrease of the angle

, similarly increase the resistance of the

material (represented by the green arrows in Figure 10) due to the initial material symmetries in shear. For both sets of tests ( and ( ̅

20 ,

70 , ̅

110

and

20 , ̅

70 ,

110 and ̅

160 )

160 ), the lower and higher hardening

evolutions are represented in Figure 10 by the black and blue arrows, respectively, being identical within each set. Thus, in general, the set of tests , ̅̅̅̅̅̅̅̅̅̅, and ̅̅̅̅̅̅̅̅̅̅̅̅ , schematically represented in Figure 13, have identical shear stress vs. shear strain curves; the same is true for the opposite relative displacements of the grips: ̅,

, ̅̅̅̅̅̅̅̅̅̅ and

.

The evolutions of the normal stress components are now analysed for two sets of tests, obeying to the loading directions indicated in Figure 13: (i) 90 and ̅

180 ; (ii)

(b) shows the evolution of

20 , ̅ and

70 ,

110 and ̅

0, ̅

90 ,

160 . Figure 14 (a) and

with the shear strain, for the first and second set

of shear tests, respectively. Within each set, the normal stress components are distinct although the shear stress evolution is the same (see Figure 11). It can thus be concluded that the normal stresses evolution does not depend exclusively on the material behaviour in shear. Similar evolutions of the normal stresses are only observed for the

and ̅̅̅̅̅̅̅̅̅̅̅̅̅ (e.g.

tests

̅̅̅̅̅̅̅̅̅̅ and

(e.g. ̅

0 and ̅

180 in Figure 14 (a)) as well as for the tests

90 and

90 in Figure 14 (a)), due to the material ⁄

orthotropy. Figure 15 shows the normalized yield stresses in uniaxial tension,

, as

a function of the initial angle with the rolling direction. The normal stress components (see Figure 14) are identical when the rotation of the material axes during the shear tests induces equal evolutions of the normalized yield stresses in uniaxial tension (represented by arrows of the same colour in Figure 15). This is due to the orthotropy of the metal sheets. The results of Figure 14 also show that the normal components have the ( )

following relations:

(̅̅̅̅̅̅̅̅̅̅ ) and

( ̅) and

, the shear tests at

( ( ̅)

) and

( )

(

) and also

(̅̅̅̅̅̅̅̅̅̅ ). In the particular cases of

and ̅ are identical (e.g.

and ̅

, in

Figure 14 (a)) due to the orthotropic symmetry at 0 , 90 and 180 . For the remaining angles the shear tests at

̅ have in general distinct shear and normal stress

evolutions; thus, the use of shear test devices with a double gauge area (e.g. [7] and [40]) that simultaneously promotes shear deformation in opposite directions can compromise the interpretation of the shear results for anisotropic materials. The influence of neglecting the normal components in the determination of the equivalent stress vs. strain curve in shear tests at different angles is now analysed. Figure 16 (a) and (b) show the equivalent stress vs. strain curves, respectively obtained by considering or not the normal components,

and

, in the evaluation of the

equivalent stress. Note that the equivalent stress is evaluated using Equation (2) and the equivalent strain, by Equation (7). For comparison, the input hardening curve described by the Swift isotropic hardening law (Equation (6)) is also shown.Figure 16 (a) confirms that, as long as the normal components are taken into account, the obtained equivalent stress vs. strain curve is identical to the input hardening curve. In contrast, neglecting the normal components can lead to different hardening curves, depending on the angle (Figure 16 (b)). In other words, the material appears to have an anisotropic hardening behaviour, despite of being modelled by an isotropic hardening law (Swift), assuming a constant shape for the yield surface. This apparent anisotropic hardening agrees with the observations by Yoon [14]. The evaluated equivalent stress vs. strain curves for

and

are identical in Figure 16 (b), since these tests show

identical shear stress vs. shear strain evolutions (see Figure 11 (b)). The same behaviour is observed for

and

.

In summary, identical shear stress evolution can be obtained for the tests ̅̅̅̅̅̅̅̅̅̅ ,

,

and ̅̅̅̅̅̅̅̅̅̅̅̅, but the respective normal components evolve

differently with the shear strain. In fact, identical stress tensors only arise for the shear tests

and ̅̅̅̅̅̅̅̅̅̅̅̅, on one hand, and for the tests ̅̅̅̅̅̅̅̅̅̅ and

, on the other

hand. Although the evolution of the shear stress with strain is related to the yield stress distributions in shear, the normal stresses evolution does not depend exclusively on the material behaviour in shear. Both evolutions are influenced by the rotation of the material axes during shear. These conclusions, illustrated for the case of the AA2093-T3 material, are identical for the others less anisotropic materials (DP500, FeP06t and HSS-Y350). In case of the AA2093-T3 material, not taking into account the normal components of the stress tensor can compromise the determination of the equivalent stress vs. strain curve. If so, the equivalent curve evaluated depends on the loading direction and sign, resulting in an apparent anisotropic hardening effect. Furthermore, given the orthotropy of the metal sheets, the use of shear specimens with orientations and ̅ in the range 0º to 45º covers all distinct results of the shear tests.

4. Prediction of the Normal Stress Components As discussed above, the magnitude of the normal stress components that arise during the shear test depends on the material anisotropy, loading direction, specimen geometry and boundary conditions. An analytical approach to predict the normal stress components that appears in the numerical simulation is now presented, based on the knowledge of the shear stress

and the strain increment

in the global reference

system. The proposed approach assumes that the normal stress components at the central part of the specimen are not associated with the stress disturbance observed near the free ends of the specimen, which is guaranteed for geometric ratios of

and

, i.e. the appearance of the normal stress components is strictly due to the boundary conditions of the test and the material orthotropic behaviour. Figures 5, 6, 7 and 8 show that the Cauchy stress tensor at the centre of the shear specimen is given by the following equations, for the constrained and unconstrained tests, respectively:

[

]

(8)

[

]

(9)

where,

is the stress tensor defined in the global reference system (

which can be rotated to the material reference system, ̂, (

, in Figure 12),

, in Figure 12) by:

̂

(10)

where,

is the rotation matrix defined as follows:

[

].

(11)

is the rotation angle between the global and the material reference systems, which is equal to the initial angle with the rolling direction, shear rotation,

, plus the angle induced by the

(see angles in Figure 12). When using the Jaumann stress rate,

is

given by ⁄ , and it is similar to those of other objective stress rates for shear strains, , below 40% [26]. For a shear strain of 40%, the rotation angle

(see Figure 12) is about

11.5º. The incremental strain tensor in the material reference system,

̂, is evaluated

using the associated flow rule:

̂

where,

(̂) ̂

(12)

̂ and ̂ are elements of the incremental plastic strain tensor, ̂, and stress

tensor, ̂, respectively;

is the plastic multiplier and

(̂) is the left-hand side of

equation (1) or (2) respectively for the Hill’48 or C B x yield criteria. The

incremental plastic strain tensor on the global reference system,

, can be obtained by

rotating the tensor ̂: ̂

(13)

Since in the global reference system the component

is known (equal to

enables the evaluation of the plastic multiplier,

. Depending on the boundary

conditions applied to the test, the incremental plastic strain tensor, components

and

⁄ ), this

has: (i) null

for the constrained test (see also stress tensor defined by

equation (8)); or (ii) only null component

for the unconstrained test (see also stress

tensor defined by equation (9)). This enables defining the equations to evaluate both normal stress components,

and

, for the constrained test, or only

unconstrained test, as a function of the shear stress shear strain

and the angle

, for the

(and so of the

see Figure 12).

For the particular case of the Hill’48 criterion this procedure leads to explicit solutions for the normal stress components as follows, for the constrained test:

(

)(

(

(

)

(

(

)

(

)

)(

(

) ( ( ) (

( )

)

) ( ( ) (

) ) )

(

)

(

(14)

) ) )

(15)

while for the unconstrained test:

(

)(

( (

) )

( )

( ) (

( )

) ( )

(

( ) )

(

)

)

(16)

For more complex yield criteria, such as the CPB2x, the equations that arise for each value of shear strain,

, can be solved applying an iterative procedure. This

procedure consists on the iterative update of the normal stress components,

and

(starting from a null value), until the strain components satisfy the boundary conditions of the test, i.e.

and

for the constrained test; and

for the

unconstrained test. In case of the unconstrained test, only the stress component evaluated since the component

is

is null (see Figure 8).

The analytical predictive capability of the normal stress components is now tested in case of constrained and unconstrained shear tests of the AA2090-T3 and the FeP06t materials, which are respectively modelled by the C B x and Hill’48 The normal stress components assessed through the numerical simulation (lines) and by analytical prediction (points) are compared in Figures 17, 18 and 19, for shear test at several orientations, , with the rolling direction. For the FeP06t material, described by Hill’48 yield criterion, the analytical prediction of the normal stress components used equations (14) and (15), for the constrained test, and equation (16), in case of the unconstrained test. For the unconstrained test, the component

is always null, and

therefore it is not represented in Figure 19. Figure 17 and Figure 19 (a) show that the normal stress components analytically predicted are in good agreement with those evaluated from the numerical simulation, for the FeP06t material. In case of the AA2090-T3 material (see Figure 18 and Figure 19 (b)), some differences in normal stress components can be observed at the elastic region and up to shear strains about 15%, for the constrained tests at and

(see Figure 18 (b)), and for the unconstrained test at

(see Figure 18 (a)) (see Figure 19

(b)), although their trend is well predicted. These differences, only observed for AA2090-T3 aluminium, are certainly related to the low value of Young's modulus of this material, which increases the relative importance of the elastic deformation (not considered in the analytical prediction) in relation to the plastic deformation, at the beginning of deformation. Specifically, the boundary conditions in the numerical simulation impose that certain components of the total (elastic + plastic) strain tensor are zero, but in the analytical prediction only the corresponding components of the plastic strain tensor are considered null. In order to illustrate the implication of correctly predicting the components of normal stress, Figure 20 and Figure 21 show the equivalent stress vs. strain curves evaluated with and without considering the normal stress components, analytically predicted, in the calculation of the equivalent stress (Equations (1) or (2)) and the equivalent strain (Equation (7)). These curves were evaluated, respectively, for the constrained (Figure 20) and unconstrained (Figure 21) shear tests at different angles

with the rolling direction, using the materials FeP06t and AA2090-T3. For comparison the input hardening curves of these materials (Tables 1 and 2) are also shown. Figure 20 (a) and Figure 21 (a) show that the use of the analytically predicted normal stress components allows the correct determination of the stress vs. strain curves. In the particular case of the AA2090-T3 material, the determination of the equivalent stress vs. strain curve is clearly improved when the analytically predicted normal stress components are used, even when these components are slightly different from those numerically evaluated (see Figure 18 and Figure 19 (b)). For this material, the previously mentioned ―apparent anisotropic hardening‖ which is observed in Figure 20 (b) and Figure 21 (b), is no longer observed when the predicted normal stress components are used to determine the equivalent stress vs. equivalent strain curve (see Figure 20 (a) and Figure 21 (a)). In summary the normal stress components that arise during the numerical simulation of shear tests can be analytically predicted in constrained and unconstrained tests. The use of these components in the determination of the equivalent stress vs. strain curve allows the correct determination of this curve.

5. Conclusion

The influence of the material anisotropy, specimen geometry, loading direction and boundary conditions, in the appearance and magnitude of the normal stress components during shears test is evaluated with resource to numerical simulation. The main conclusions of this numerical study are summarized as follows: 

The material anisotropy has a large impact in the magnitude of the normal stress components. For approximately isotropic materials, the appearance of normal stress components can be ignored, whereas for considerably anisotropic materials, these components can reach magnitudes similar to those of the shear component.



The normal stress components in the central region of the specimen are only influenced by the stress disturbance observed near the free ends of the specimen, if small geometric specimen ratios of length to width, thickness,

, are chosen. For

and

, and width to

ratios, the source of the

normal stress components is exclusively related with the imposed boundary conditions and the orthotropic behaviour of the material.



The shear stress vs. shear strain curves are identical for the test orientations , ̅̅̅̅̅̅̅̅̅̅ ,

and ̅̅̅̅̅̅̅̅̅̅̅̅ , but the normal stress components that arise

during these tests evolve differently with the amount of shear strain. In fact, the normal components are identical for the orientations hand, and for the orientations ̅̅̅̅̅̅̅̅̅̅ and

and ̅̅̅̅̅̅̅̅̅̅̅̅, on one

, on the other hand. This is

discussed in terms of the rotation of the material axes. An analytical approach to predict the normal stress components is proposed, which must take into account the boundary conditions imposed during the shear test. The predicted normal stress components agree with those evaluated numerically and their use allows the correct determination of the equivalent stress vs. strain curve. These conclusions were obtained for a given numerical model, and thus the experimental corroboration of the normal components analytically predicted and numerically evaluated would be interesting. However, further work is required to make such a comparison, for instance, concerning the development of experimental equipment to adequately measure the normal components and to ensure that the boundary conditions are well represented. Moreover, the normal stresses were studied for shear strains below 40%. For larger strains it is necessary to consider the effect of the corotational rate, the loss of orthotropy and the strong possibility of textural evolution. In this case, the use of models that consider texture evolution are recommended.

Acknowledgments This work was supported by funds from the Portuguese Foundation for Science and Technology and by FEDER funds via project reference UID/EMS/00285/2013. It was also supported by the projects: RDFORMING co-funded by Portuguese Foundation for Science and Technology, by FEDER, through the program Portugal-2020 (PT2020), and by POCI, with reference POCI-01-0145-FEDER-031243; EZ-SHEET co-funded by Portuguese Foundation for Science and Technology, by FEDER, through the program Portugal-2020 (PT2020), and by POCI, with reference POCI-01-0145-FEDER-031216. Two of the authors, A. F. G. Pereira and P.A. Prates, were supported by grants for scientific research from the Portuguese Foundation for Science and Technology (reference: SFRH/BD/102519/2014 and SFRH/BPD/101465/2014, respectively). All supports are gratefully acknowledged.

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Moving nodes y

3 (18) x

3 (18)

Fixed nodes

114 (280)

Fixed nodes

3 (18)

Figure 1 – Geometry of the shear specimen showing the dimensions, in mm, and the number of elements along Ox and Oy axes, within parentheses.

Figure 2 - Normalized yield stresses as a function of the angle with the rolling direction, for the materials of Tables 1 and 2, in cases of: (a) uniaxial tension and (b) simple shear.

Figure 3 – Normal components normalized by the shear stress, obtained from constrained shear tests in the rolling direction, for the materials of Tables 1 and 2: (a) normal component parallel to the shear direction, ; (b) normal component perpendicular to the shear direction, .

Figure 4 – Equivalent stress vs. strain curves obtained by considering or not the normal stress components for the anisotropic materials: (a) AA2090-T3 and (b) FeP06t. For comparison, the input hardening curves are also shown.

Figure 5 – Distributions of the normal stress component parallel to the shear direction, as a function of the distance from one end of the specimen normalized by its length, , evaluated along the centre line of the specimen (see Figure 1) for different geometric ratios of the constrained shear test of the FeP06t material: (a) equal to of 1, 3 and 10 (in all cases the length is ); (b) equal to 5, 11.5, 20 and 40 (in all cases the thickness is ).

Figure 6 - Distributions of the normal stress component perpendicular to the shear direction, as a function of the distance from one end of the specimen normalized by its length, , evaluated along the centre line of the specimen (see Figure 1) for different geometric ratios of the constrained shear test of the FeP06t material: (a) equal to of 1, 3 and 10 (in all cases the length is ); (b) equal to 5, 11.5, 20 and 40 (in all cases the thickness is ).

Figure 7 - Distributions of the normal stress component parallel to the shear direction, as a function of the distance from one end of the specimen normalized by its length, , evaluated along the centre line of the specimen (see Figure 1) for different geometric ratios of the unconstrained shear test of the FeP06t material: (a) equal to of 1, 3 and 10 (in all cases the length is ); (b) equal to 5, 11.5, 20 and 40 (in all cases the thickness is ).

Figure 8 - Distributions of the normal stress component perpendicular to the shear direction, as a function of the distance from one end of the specimen normalized by its length, , evaluated along the centre line of the specimen (see Figure 1) for different geometric ratios of the unconstrained shear test of the FeP06t material: (a) equal to of 1, 3 and 10 (in all cases the length is ); (b) equal to 5, 11.5, 20 and 40 (in all cases the thickness is ).

(a) (a) y

’

(b) (b) y

TD

’

TD x

x

’

RD

’

RD

Figure 9 – Schematic representation of the shear tests at: (a) and (b) ̅. is the material reference system and is the global reference system; RD – rolling direction and TD – transverse direction.

Figure 10 – Yield stresses in simple shear, normalised by its value for = 0 , as a function of the initial angle with the rolling direction, for the AA2093-T3 aluminium. The red points and the red crosses indicate the normalized stress value at yielding for the first second set of shear tests, respectively. The arrows specify the stress evolution due to the rotation of the material axes.

Figure 11 – Shear stress vs. shear strain curves numerically evaluated for different angles with the rolling direction: (a) , , ,̅ ,̅ and ̅ ; (b) , , , ,̅ ,̅ ,̅ and ̅ .

(a)

y

’

’ini

(b)

y

y

(c)

’

’

’

’

x

x x

’

’

’

’

’

’

Figure 12 – Rotation of the material axes during the shear test: (a) initial angle, , between the global reference system, , and the material reference system, ; (b) and (c) distinct rotations, , of the material reference system, from its initial position ( and ), for tests designated by and ̅, respectively.

’

TD

’

RD

Figure 13 – Shear tests with the same shear stress vs. shear strain curves.

Figure 14 – Comparison of the normal components evolution along the shear strain for the shear tests: (a) 0 , ̅ 90 , 90 and ̅ 180 ; (b) 20 , ̅ 70 , 110 and ̅ 160 .

Figure 15 – Yield stresses in uniaxial tension, normalised by its value for = 0 , as a function of the initial angle with the rolling direction for the AA2093-T3 aluminium. The red points and the red crosses indicate the normalized stress value at yielding for the first second set of shear tests, respectively. The arrows specify the stress evolution due to the rotation of the material axes.

Figure 16 - Equivalent stress vs. strain curves determined for shear tests at different angles with the rolling direction, assuming that the normal components are: (a) considered and (b) neglected. For comparison the input hardening curve is also shown.

Figure 17 - Normal stress components evolutions during constrained shear tests of the FeP06t material, at , , and : (a) and (b) . The normal components were assessed numerically (lines) and analytically predicted (points).

Figure 18 - Normal stress components evolutions during constrained shear tests of the AA2090-T3 material, at , , and : (a) and (b) . The normal components were assessed numerically (lines) and analytically predicted (points).

Figure 19 - Evolution of the normal component, , during unconstrained shear tests evaluated numerically (lines) and analytically predicted (points) at , , and , for the materials: (a) FeP06t; (b) AA2090-T3.

Figure 20 - Equivalent stress vs. strain curves calculated from constrained shear tests at different angles with the rolling direction by: (a) considering the analytically predicted normal stress components; (b) neglecting the normal stress components. For comparison is also shown the input hardening curves (Tables 1 and 2).

Figure 21 - Equivalent stress vs. strain curves calculated from unconstrained shear tests at different angles with the rolling direction by: (a) considering the analytically predicted normal stress components; (b) neglecting the normal stress components. For comparison is also shown the original hardening law curves (Tables 1 and 2).

Table 1 – Constitutive parameters identified by Zang [37] and Duchêne [26] for the DP500 and FeP06t steels, respectively. Swift Hill’48 [MPa]

[MPa]

DP500 259.300 832.900 0.175 [MPa]

0.488

0.461

0.539

1.589

1.589

1.589

0.265

0.285

0.715

1.280

1.280

1.280

[MPa]

FeP06t 82.815

550.300 0.278

Table 2- Constitutive parameters identified by M.Safaei [38] and Plunket [34] for the AA2090-T3 aluminium and by Hu [36] and by Plunket [34] for the HSS-Y350 steel. Swift [MPa]

CPB2x

[MPa]

279.624 646.015 0.227 0.476 -0.987 -3.070 0.563 2.008 1.117 -0.724 -1.209 -2.098

0

0.476 0.095 0.715 1.940 1.032 2.028 -0.763 1.143 -0.608

12

358.400 709.000 0.146 1.761 1.712 1.664 1.499 1.499 1.499 0.630 0.475 -0.229

0

1.761 4.616 4.352 0.761 0.761 0.761 2.913 2.661 3.358

5

AA2090-T3

[MPa]

[MPa]

HSS-Y350