1 Introduction

The finite element method is more used to predict complex phenomena may be produced by plastic deformation of metals. SB is generally referred to as the change of part shape that occurs upon removal of constraints after forming (Xia and Cao 2014). After SB, the part reaches an internal equilibrium in the absence of external forces. Residual stresses still exist within the part; however, they are self-balanced (Xia 2007).

In recent years, much research has been devoted to find techniques and methods to reduce SB.

The study of the influence of material properties and processing parameters on SB is very important to design the forming tools. In literature, several experimental and numerical studies were conducted to characterize the influence of these parameters on SB during the bending operation. Wang et al. (1993) developed models for plane strain sheet bending to predict SB, minimum bending ratio, strain and residual stress distributions, etc. The results obtained by Jiang et al. (2015) show that the hardening exponent, bending radius and thickness ratio, friction coefficient, and blank holder force have great influences on the springback angle and stress level.

Knowing that the amount of SB that occurs after the bending operation is an intrinsic property of metals, an accurate experimental characterization of sheet metal is unavoidable towards more accurate prediction of SB. So, modelling mechanical behaviour of metal is crucial to improve simulation reliability. The models obtained, for example, by integrating selected physical phenomena seem a more intricate alternative, but they allow a general description without multiplying parameters and costly identifications (Tabourot et al. 2012). Most works in the field make use of classic phenomenological models while this type of models should include some simplifications by considering the material is homogeneous at the microscopic level, also every model corresponds a well-defined situation (Tabourot et al. 2014, Maati et al. 2015). Indeed, metallic alloys are discrete and heterogeneous (polycristals). There are several possible sources of heterogeneity such as grain size, crystallographic grain orientation, precipitates, dislocations, etc. Thus, taking into account the heterogeneities of the metallic structure makes the model more consistent to achieve specific numerical results. To this end, Tabourot et al. (2015) proposed a CM to investigate the localization phenomenon in the case of a tensile test on C68 grade steel. This paper is mainly devoted to the influence of the local heterogeneities on SB phenomenon through case study. By way of comparison, two types of simulations (with and without considering the material heterogeneities) were performed for predicting SB in L-bending operation. Tests have been carried out on commercially pure titanium sheets.

2 Experimental Traction Data

The true Stress-Strain curve for titanium T40 was obtained via the digital image correlation (Vacher et al. 1999). Figure 1 shows the deformation field on the surface of the specimen oriented in the RD. The acquisition frequency was 2 images per second and the total number of images acquired is limited to 120 images over a test time equal to 60 s. The tensile tests are performed at a constant crosshead speed V = 10 mm/min on titanium specimens with a rectangular cross-section \((1.6 \times 10\;{\text{mm}}^{2} )\).

Fig. 1
figure 1

Deformation εyy field obtained via digital image correlation (DIC)

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The true Stress-Strain curve is illustrated in Fig. 2. The Young’s modulus was determined using the linear regression method on extensometer data in the elastic range. In this study, the Poisson’s ratio has not been identified, it is assumed to be equal to its usual experimental value \(\left( {\upupsilon = 0.34} \right)\).

Fig. 2
figure 2

True Stress-Strain curve obtained via DIC in the RD

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3 Numerical Simulation of Tensile Test Experienced by Sheet Specimens of Titanium T40 Alloy

3.1 Without Considering the Material Heterogeneity

The mechanical behaviour of Titanium T40 was initially modeled by a Young’s modulus and a Poisson’s ratio in the elastic range of the traction curve; and by a reference curve σ(εp) in the plastic range of the traction curve as shown in Fig. 3, σy is the yield stress, typically equal to the elastic limit of the material. The numerical simulation of tensile test without considering the material heterogeneity shows the symmetry of stress and strain relative to the center of the sample as illustrated in Fig. 4. Furthermore, the Fig. 5 shows the evolution of the equivalent plastic strain along path oriented towards the center of the sample (12 selected nodes).

Fig. 3
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Schematic reference curve obtained via DIC in the RD

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Fig. 4
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Symmetry of strain distribution with respect to the centre of the test piece

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Fig. 5
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Evolution of the plastic strain along path oriented towards the centre of the sample

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3.2 By Considering the Material Heterogeneity

At a microscopic level, the metal alloys are discrete and heterogeneous materials. There are many top causes that are responsible for these material heterogeneities such as grain and sub grain structuration, crystallographic orientation of the grains, presence of precipitates and gaps, etc. As mentioned above, it has appeared interesting to introduce such spatial distributions in order to model the elastoplastic behaviour of metallic material. An image showing the microstructure of titanium T40 was taken using an optical microscope (Fig. 6).

Fig. 6
figure 6

Highlighting the microstructure of titanium T40 using an optical microscope (G20), a without chemical attack, b with chemical attack (t = 30 s)

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To render the physical effect due to such properties, it is proposed to quantify and distribute pertinent mechanical properties spatially throughout the material. For example, highlighting the Bauschinger effect is numerically possible using a CM without making use of experimental tests which are often costlier.

The CM proposed in this study is based on the one hand on the modelling studies carried out by Tabourot et al. (2014), and on the other hand, on the heterogeneous finite element approach like the one proposed by Furushima et al. (2013). To predict the effect of a phenomenon sensitive to material heterogeneities (such as SB), the CM assigns to each grain size a local behaviour law by using a probability distribution function. Literature has a set of local behaviour laws and probability distribution functions such as those described in Fig. 7.

Fig. 7
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Examples of local behaviour laws and probability distribution functions (Bizet 2016)

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Assume that \(\sigma_{y}\) is randomly distributed according to a Rayleigh distribution as given in the following equation (Bizet 2016):

$$f\left( {\frac{x}{{\sigma_{ym} }}} \right) = \frac{2x}{{\sigma_{ym}^{2} }}\exp \left( { - \frac{{x^{2} }}{{\sigma_{ym}^{2} }}} \right)$$
(1)

x varies from 0 to ∞, \(\sigma_{ym}\) is the average yield stress (σymean) defining Rayleigh’s distribution shape (see Fig. 8).

Fig. 8
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Rayleigh’s distribution for the CM model for 10,000 elements

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A saturation stress \(\sigma_{sat}\) is used to delimit the random distribution of \(\sigma_{y}\). By way of comparison; the numerical simulation used two elastoplastic constitutive laws as local laws. A bilinear elastoplastic constitutive law characterized by slop n, it is given by the following relation:

$$\sigma = n\upvarepsilon_{p} +\sigma_{y}$$
(2)

The Hollomon’s law is used to model the plastic portion of true Stress—Strain curve. It is given by the following relation:

$${\bar{\sigma}} = K\left( {\mathop {\upvarepsilon_{p} }\limits^{ - } } \right)^{n}$$
(3)

A Coupled Abaqus/Python algorithm was used to optimize the parameters of both local laws proposed above by fitting numerical curve to experimental data. The true Stress-Strain curve obtained with optimized values (see Table 1) is given in Fig. 9.

Table 1 Numerical parameters introduced in the simulation
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Fig. 9
figure 9

Experimental and numerical true Stress—Strain curves of T40 alloy

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The numerical simulation of tensile test by considering the material heterogeneity shows the asymmetry of stress and strain with respect to the centre of the test piece. I.e. the most loaded element can be located at any point in the test piece. The Fig. 10 shows the evolution of the equivalent plastic strain along path previously selected.

Fig. 10
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Evolution of the equivalent plastic strain along path by using the CM

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To highlight the kinematic hardening effect, a traction-compression test is simulated. Some cycles were presented before buckling deformation (Fig. 11).

Fig. 11
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Numerical tension—compression test on titanium T40 alloy using CM

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4 Numerical Simulation of L-Bending Operation with SB Stage

The software ABAQUS/Standard has been used to setup the numerical model of L-bending operation with SB stage. The tools were assumed to be perfectly rigid. Two mesh element types were used, on the one hand, the part mesh which consisted of 12,000 volume elements, is obtained using C3D8R, and on the other hand, the part mesh which consisted of 1200 shell surface elements, is obtained using CPE4R. In order to take the different interactions between the rigid tools and the blank into account, a friction coefficient \(\left( {\upmu = 0.15} \right)\) is proposed. Once the punch has done the prescribed displacement \(\left( {U_{2} = 18\;{\text{mm}}} \right)\), the simulation removes the punch and the spring back will occur. The numerical description of the L-bending operation with SB stage is illustrated in Fig. 12. Other representative figures resulting from the simulation have been illustrated (Fig. 13a, b).

Fig. 12
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a 3D L-bending numerical model with spring back stage, b 2D L-bending numerical model with springback stage

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Fig. 13
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a Force-displacement curve during the loading step, b a random variation of von Mises stress through the sheet thickness

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A comparison was done between results obtained; firstly, from numerical simulation with and without considering the material heterogeneities and secondly by measuring experimentally the spring back for bending a 90°. As shown in Table 2, the results for the three methods are presented. \(\Delta\uptheta =\uptheta_{ 1} -\uptheta_{ 2}\) denotes the variation in the bending angle of the test piece before and after spring back.

Table 2 SB values of different methods
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According to the numerical simulation results, a good agreement in terms of SB amount is observed between simulation results using Hollomon’s approach as a local law and experimental data. This comparison showed an average relative error of less than 3%, Furthermore, this error increases in the case using bilinear approach as a local law with an average value of 13%, and it increases more if the material heterogeneities is not included.

5 Conclusion

A CM that introduces the material heterogeneities to describe the mechanical behaviour of metals has been presented. An experimental characterization of the material response using plate specimens has been firstly performed in order to obtain the reference curve that will be used as input data in the FEA code.

A python script is used to identify the CM parameters. Numerical simulation of L-bending process with SB stage was performed using a CM which is a combination of a local law and a probability distribution function. An acceptable convergence is observed between the numerical results using Hollomon’s approach as a local law and the measurements of SB, the error increases if the material heterogeneities is not included.

A numerical simulation of traction—compression test was carried out using a CM in order to reproduce the kinematic hardening effect due to cyclic loading; this effect is weakly observable for simple bends and creases.

Finally, this type of physically based models will certainly be more effective to predict complex phenomena generated by the plastic deformation and which are more sensitive to the material heterogeneities (spring back in deep-drawing process for example).