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Inverse identification of the work hardening law from circular and elliptical bulge tests
Journal Pre-proof Inverse identification of the work hardening law from circular and elliptical bulge tests A.F.G. Pereira (Investigation)Formal Analysis)Writing - Original Draft)Writing- Review and Editing), P.A. Prates (Funding acquisition)Formal Analysis)Writing- Review and Editing), M.C. Oliveira (Software)Formal Analysis)Writing- Review and Editing), J.V. Fernandes (Funding acquisition) (Supervision)Writing- Review and Editing)
PII:
S0924-0136(19)30546-1
DOI:
https://doi.org/10.1016/j.jmatprotec.2019.116573
Reference:
PROTEC 116573
To appear in:
Journal of Materials Processing Tech.
Received Date:
25 February 2019
Revised Date:
14 November 2019
Accepted Date:
20 December 2019
Please cite this article as: Pereira AFG, Prates PA, Oliveira MC, Fernandes JV, Inverse identification of the work hardening law from circular and elliptical bulge tests, Journal of Materials Processing Tech. (2019), doi: https://doi.org/10.1016/j.jmatprotec.2019.116573
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Inverse identification of the work hardening law from circular and elliptical bulge tests
A.F.G. Pereira* [email protected]; P.A. Prates; M.C. Oliveira; J.V. Fernandes CEMMPRE — Department of Mechanical Engineering, University of Coimbra, Rua Luís Reis
*
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Corresponding author : telephone: +351 239 790 716/00
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Santos, Pinhal de Marrocos, 3030-788 Coimbra, Portugal
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Abstract
An inverse identification strategy is proposed to characterize the hardening behaviour of metal sheets up to high strains, regardless of the material anisotropy. The
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Levenberg-Marquardt method is used to minimize the gap between the experimental and numerical, pressure vs. pole height curves, of bulge tests with circular and elliptical
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dies, by iteratively updating the work hardening and the Hill’48 parameters of the
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numerical model. The optimization of the Hill’48 parameters is used only to ensure that the yield surface is conveniently described in a region close to the stress paths that occur in the circular and elliptical bulge tests, in order to improve the identification of the hardening parameters. The strategy aims to be accurate and simple from an experimental point of view, using only the results of pressure vs. pole height. The results are compared with those of the membrane theory procedure standardized in ISO
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16808 (2014) and experimentally validated for two materials, DP600 steel and Al5754 aluminium alloy. For anisotropic materials, the proposed methodology represents a clear improvement when compared to the membrane theory procedure since it avoids the equibiaxial stress state assumption.
Keywords Hydraulic Bulge Test; Elliptical Die; Work Hardening; Inverse Analysis;
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Levenberg-Marquardt Method
1. Introduction
Sheet metal forming processes are among the most common and relevant metal
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working operations because of their high manufacture rate and low costs for high
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volume productions. Finite element methods are commonly used for simulating and optimizing sheet metal forming processes, in order to reduce the time and costs
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associated with the process development. Nevertheless, the accuracy of the numerical results depends on the appropriate characterization and modelling of the mechanical
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behaviour, using constitutive models that allow an approximate description of the experimental tests results.
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The hydraulic bulge test has become an important characterization tool, mainly due to the possibility of evaluating the response of materials under biaxial strain path
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and up to high levels of plastic deformation. For the bulge test with circular die, the determination of the biaxial stress vs. strain curve is commonly performed using the membrane theory, assuming an equibiaxial stress state, as in the procedure recently standardized in ISO 16808 (2014). This analysis requires the evaluation of the sheet thickness and the radius of curvature at the pole as well as the hydraulic pressure. Mulder et al. (2015) studied the assumptions and simplifications to be used in the 2
analysis of the circular bulge test results, through the membrane theory, and proposed some recommendations to improve the results accuracy. The authors conclude that, for anisotropic materials, the test conditions force the material, as much as possible, towards an equibiaxial strain path, leading to deviations (up to 25%) from the equibiaxial stress path, which is in agreement with the work by Yoshida (2013). Although the recommendations of Mulder et al. (2015) improved the accuracy of the membrane theory results, they did not study in detail the effect of the equibiaxial stress
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assumption. Reis et al. (2017a) studied the equibiaxial stress assumption and showed that it could lead to significant errors in the determination of the biaxial stress vs. strain curve from the circular bulge test of prominently anisotropic materials. Reis et al.
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(2017a) suggested an empirical equation to estimate the stress path based on the strain
path at the pole, which can be assessed by digital image correlation technique. The use
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of this equation, within the membrane theory, improves the accuracy of the biaxial
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stress vs. strain curves for the majority of materials. Nevertheless, there is no guarantee that all materials will closely follow the proposed equation. The bulge test with elliptical die has also been used to study the materials
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response under stress paths near the biaxial region. Several approaches can be found in literature to analyse the results of elliptical bulge tests using the membrane theory. Rees
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(1995) proposed equations to evaluate the hardening curve of the material taking into
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account the anisotropy. These equations proved to be valid only up to 20% of plastic deformation and thus limiting the analysis to that level of deformation. Lăzărescu et al. (2012) proposed an alternative analytical model that only use the pressure and pole height results. Although this model significantly simplifies the analysis of the bulge test results, some discrepancies are observed between analytical and experimental results. Based on geometrical assumptions, Ragab and Habib (1984) developed equations to
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calculate the hoop and longitudinal stresses at the pole of the bulge. Williams and Boyle (2016) show that these equations accurately predict the hoop stress, but fail regarding the longitudinal stress. Therefore, Williams and Boyle (2016) use a yield criterion, calibrated with tensile and bulge data, and the associated flow rule to calculate the longitudinal stress component based on the strains measured at the pole. Chen et al. (2016) also use the associated flow rule to evaluate the stresses at the pole, but the yield criterion is calibrated based on the results of the bulge, tensile and plane strain tests. The
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approaches of Chen et al. (2016) and Williams and Boyle (2016) to evaluate the stresses, from the strains at the pole, are valid whenever the selected yield criterion accurately describes the material behaviour in the biaxial region. However, this
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description can be compromised due to the choice of the yield criterion and the
(tensile test and plane strain tests).
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calibration of its parameters using tests with stress states outside the biaxial region
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Chamekh et al. (2006) and Reis et al. (2017b) proposed inverse analysis methodologies that bypass the assumptions and simplifications associated with the membrane theory. Chamekh et al. (2006) describe an inverse strategy, based on
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Artificial Neural Networks (ANN), to identify the material parameters of a stainless steel (AISI 304), by comparing the results of the pressure vs. pole height evolutions of
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circular and elliptical bulge tests. The authors concluded that the ANN can
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instantaneously predict accurate results, once it is trained. Despite these results, Prates et al. (2016) infers that the strategy presented substantial errors in the determination of the work-hardening. Reis et al. (2017b) proposed an alternative inverse analysis strategy, consisting of the search for the best coincidence between numerical and experimental pressure vs. pole height results of a circular bulge test. The strategy allows the accurate identification of the Swift hardening law parameters but cannot be extended
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to other types of laws, such as the Voce law. Moreover, this inverse analysis study focused on anisotropic materials with relatively weak anisotropy in the sheet plane. Bambach (2011) proposed a re-identification strategy, for the Voce law, that resorts to the membrane theory to identify an initial estimate for the hardening parameters and then uses an inverse analysis to improve the solution. This strategy was applied to an isotropic material and the results show that the initial estimate, obtained through the membrane theory, is very similar to the final solution. As a consequence, the efficiency
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of the inverse analysis was not properly evaluated, particularly with respect to convergence and accuracy in the case of anisotropic materials, where the assumption of an equibiaxial stress state in the membrane theory can lead to quite imprecise results
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(Reis et al., 2017a).
To accurately identify the parameters of the work hardening law, a new strategy
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is proposed that overcomes the limitations of the existing inverse methodologies and
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avoids the assumptions and simplifications commonly associated with the membrane theory. The strategy follows an inverse analysis methodology making use of numerical and experimental results of pressure vs. pole height, obtained from the bulge tests with
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circular and elliptical dies. The work hardening parameters are firstly optimized, by comparing the pressure vs. pole height curves of the bulge test with a circular die, under
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isotropy assumption. Then, the work hardening parameters are identified simultaneously
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with the optimisation of two Hill’48 parameters, using the pressure vs. pole height curves of bulge tests with circular and elliptical dies. The Hill’48 parameters are optimized only to ensure that the yield surface is accurately described in the biaxial region and thereby improve the identification of the hardening parameters, avoiding the assumption of equibiaxial stress state. The strategy will be initially tested using fictitious materials and then it will be applied to experimental cases.
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2. Numerical Model Numerical models of the elliptical and circular bulge tests were built in order to perform the study of the proposed identification strategy. The geometry of the selected tools was based on the bulge test apparatus described by Santos et al. (2010), which will also be used to obtain the experimental results. Figure 1 schematically shows the geometries of the bulge tools, where 𝑅𝐶 is the die profile radius, 𝑅𝐷 is the radial
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position of the drawbead, 𝑅𝑆 is the blank radius and 𝑅𝑀 is the radius of the circular die, which also corresponds to the length of the major semi-axis of the elliptical die. The
length of the minor semi-axis of the elliptical die was set to 55 mm; for smaller values,
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a pre-failure of the sheet was observed in some materials near the die curvature, as
previously reported by Rees (2000). The remaining geometrical parameters, 𝑅𝐶 , 𝑅𝐷 and
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𝑅𝑆 , are the same for the circular and elliptical dies.
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Figure 1- Representation of the bulge test, with identification of the main dimensions of the tools in agreement with the work by Santos et al. (2010). The dimensions characterize the circular die apparatus as well as the major semiaxis of the elliptical die.
In order to simplify the numerical model, only a quarter of the tools geometry
was taken into account due to the material and geometric symmetries. Additionally, the drawbead geometry was neglected and its effect was replaced by a boundary condition that restricts the radial displacement of the nodes placed in the drawbead position (𝑅𝐷 = 95mm). The discretisation of the blanks was based on a previous study of Reis et al. 6
(2017b), such that the blank plane was divided into 5 zones, each one with an appropriate number of elements, as shown in Figure 2. The circular and elliptical blanks were respectively discretised with a total number of 10584 and 11216, 8-node hexahedral solid elements, with two layers of elements through the sheet thickness of 1mm. The simulations were carried out assuming an incremental increase of the applied pressure on the inner surface of the sheet. The friction was modelled by the Coulomb’s law with a friction coefficient of 0.02, as in the work by Reis et al. (2017b). All the
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numerical simulations were performed with the in-house code DD3IMP on a computer equipped with an Intel® Core™ i7–6700K Quad-Core processor (4.4 GHz). The contact with friction is treated by an augmented Lagrangian approach and all the non-linearities
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are solved, in a single iterative loop, using a static implicit iterative Newton-Raphson scheme. More details on the numerical formulation can be found in Menezes and
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Teodosiu (2000).
(a) 2
2
20
2
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20
40
42
20
20
2
35
5
48
40
40
50
2
30
40
35
20
5
x
20
45
40
2
Oz displacement[mm]
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20
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(b) 5
y
38
5
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Figure 2 – (a) Discretisation of the blank used in the circular and elliptical bulge tests showing the number of elements along Ox and Oy for the different zones; b) Representation of the discretisation zones, showing their dimensions and the displacement along Oz, in mm, for an isotropic material.
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The constitutive model of the materials used in this work assumes: (i) the isotropic elastic behaviour defined by the generalised Hooke’s law, with a Young's modulus, 𝐸 = 210 GPa, and a Poisson's ratio, 𝜈 = 0.30; (ii) the plastic behaviour described by the orthotropic yield criterion, Hill’48 (Hill, 1948) or by the yield criterion proposed by Drucker et al. (1948) extended to orthotropy by means of a linear transformation (Cazacu and Barlat, 2001), hereafter referred to as Drucker+L; (iii) an isotropic hardening modelled by the Swift (Swift, 1952) or Voce (Voce, 1948) laws.
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The Hill’48 yield surface is described by the following equation:
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𝐹(𝜎yy − 𝜎zz ) + 𝐺(𝜎zz − 𝜎xx )2 + 𝐻(𝜎xx − 𝜎yy ) + 2𝐿𝜏yz 2 + 2𝑀𝜏xz 2 + 2𝑁𝜏xy 2 = 𝑌 2 , (1)
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where 𝜎xx , 𝜎yy , 𝜎zz , 𝜏xy , 𝜏xz and 𝜏yz are the components of the Cauchy stress tensor, 𝛔,
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defined in the orthotropic coordinate system of the sheet, such that the Ox, Oy and Oz axes are parallel to the rolling, transverse and normal directions, respectively; 𝐹, 𝐺, 𝐻,
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𝐿, 𝑀 and 𝑁 are anisotropy parameters; and 𝑌 represents the yield stress and its evolution during deformation. In the case of the Drucker+L yield criterion, the yield
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surface is described by:
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3 2 1 1 𝑌 6 [ tr((𝐋: 𝛔)2 )] − 𝑐 [ tr((𝐋: 𝛔)3 )] = 27 ( ) , 2 3 3
(2)
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where 𝑐 is a weighting isotropy parameter, ranging between −27/8 and 9/4, to ensure the convexity of the yield surface, and L is a linear transformation operator (Cazacu and Barlat, 2001), written as:
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C2 C3 / 3 C3 / 3 C2 / 3 L 0 0 0
C3 / 3
C3 C1 / 3
C2 / 3
0
0
C1 / 3
0
0
C1 / 3
C1 C2 / 3
0
0
0
0
C4
0
0
0
0
C5
0
0
0
0
0 0 0 , 0 0 C6
(3)
where, 𝐶1 , 𝐶2 , 𝐶3 , 𝐶4 , 𝐶5 and 𝐶6 are the anisotropy parameters. In both criteria, the yield stress evolution during deformation, 𝑌 = 𝑌(𝜀̅), can be described by either the Swift or
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Voce hardening laws, respectively written as:
𝑌 = 𝐾(𝜀0 + 𝜀̅)𝑛 ,
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(4)
(5)
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𝑌 = 𝑌0 + (𝑌𝑆𝑎𝑡 − 𝑌0 )(1 − 𝑒 −𝐶𝑌𝜀̅ ),
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where 𝜀̅ is the equivalent plastic strain; 𝜀0 , 𝐾, 𝑛, 𝑌0 , 𝑌𝑆𝑎𝑡 and 𝐶𝑌 are the material
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parameters, where 𝑌0 is the initial yield stress (in case of Swift law: 𝑌0 = 𝐾(𝜀0 )𝑛 ).
3. Identification Procedure
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An inverse strategy is proposed to identify the work hardening behaviour described by Swift or Voce laws, for example. The proposed strategy requires the
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pressure vs. pole height results of: (i) a circular bulge test; (ii) an elliptical bulge test with the major axis parallel to the rolling direction of the metal sheet (elliptical 0º); (iii) optionally, a third bulge test with the major axis perpendicular to the rolling direction (elliptical 90º) will also be used. The strategy consists on minimizing the gap between the experimental and numerical curves of pressure vs. pole height in bulge tests, by iteratively updating the material parameters in the finite element simulations. The 9
minimization procedure is performed using a gradient-based optimisation algorithm, the Levenberg-Marquardt method proposed by Marquardt (1963), and three distinct objective functions, 𝐹(𝐀)Circ , 𝐹(𝐀)Elli0° and 𝐹(𝐀)Elli90° , defined as follows:
𝑚
𝐹(𝐀)Circ
1 2 exp = √∑ (𝑃(ℎ𝑖 )Circ − 𝑃(ℎ𝑖 )num Circ ) , 𝑚
(6)
𝑚
𝐹(𝐀)Elli0°
1 2 exp = √∑ (𝑃(ℎ𝑖 )Elli0° − 𝑃(ℎ𝑖 )num Elli0° ) , 𝑚
𝑚
1 2 exp = √∑ (𝑃(ℎ𝑖 )Elli90° − 𝑃(ℎ𝑖 )num Elli90° ) , 𝑚
(8)
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𝐹(𝐀)Elli90°
(7)
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𝑖=1
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𝑖=1
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𝑖=1
where 𝐹(𝐀)Circ, 𝐹(𝐀)Elli0° and 𝐹(𝐀)Elli90° are respectively the objective functions
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associated with the circular, elliptical 0º and elliptical 90º tests; 𝐀 is the set of exp
parameters to be optimized; 𝑚 is the total number of points in each test; 𝑃(ℎ𝑖 )Circ , exp
exp
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𝑃(ℎ𝑖 )Elli0° and 𝑃(ℎ𝑖 )Elli90° are the experimental values of the pressure for the circular,
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num num elliptical 0º and elliptical 90º tests, respectively; 𝑃(ℎ𝑖 )num Circ , 𝑃(ℎ𝑖 )Elli0° and 𝑃(ℎ𝑖 )Elli90°
are the numerically simulated values of the pressure for the circular, elliptical 0º and elliptical 90º tests, respectively; in each test, the difference between experimental and
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numerical pressures is evaluated for the same value of pole height, ℎ𝑖 1. For each curve, 𝑚=1000 points uniformly distributed along the range of pole heights were used. The Levenberg-Marquardt method requires the knowledge of the Jacobian matrix (sensitivity matrix), which is updated after each iteration to improve the convergence of the method. Typically, the sensitivity matrix is computed using finite differences, which requires at least an additional numerical simulation per optimized parameter and test, increasing the computational cost. To overcome this inconvenience,
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the analytical approach proposed by Prates et al. (2019) was used to determine the
sensitivity matrix of the work hardening parameters. During the optimization procedure, a damping factor value is updated after each iteration, which according to Marquardt
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(1963) improves the convergence of the method.
The Levenberg-Marquardt method is used to identify the work hardening
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parameters 𝑌0 , 𝐾 and 𝑛, in the case of the Swift law, or 𝑌0 , 𝑌𝑆𝑎𝑡 and 𝐶𝑌 , in the case of
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the Voce law. Since the bulge test results are also influenced by the material anisotropy (as shown by Reis et al. (2017a)), the parameters F and G of the Hill’48 criterion are
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also optimized. In this way, the identification of the hardening parameters takes into account that the stress state can be other than equibiaxial, which occurs in anisotropic materials. The Hill’48 parameters should accurately describe the shape of the yield
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surface in a region close to the stress paths that occur in the circular and elliptical bulge
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tests, even if they do not allow describing the entire yield surface. In this sense, the optimized Hill’48 parameters should not be used outside the scope of this work. The parameters of the Hill’48 criterion follow the condition 𝐺 + 𝐻 = 1, and so the
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The assessment of the pressure values for the same pole height was accomplished by fitting a 6th degree polynomial for each pressure vs. pole height curve, as in the work by Reis et al. (2017b). The interpolation excludes the initial part of the curve, more susceptible to experimental errors, and the final part, susceptible to failure initiation and propagation, which is not numerically modelled.
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identified hardening curve is comparable to the uniaxial tensile curve along the Ox axis. The remaining parameters of the Hill’48 criterion, associated to the shear stresses, were set to 𝐿=𝑀=N =1.5 (von Mises), since the pressure vs. pole height results of the circular and the elliptical bulge tests are not sensitive to their values. According to the results of Reis et al. (2017b), the pressure vs. pole height curves of the bulge test are more sensitive to the work hardening parameters than to those of the yield criterion. For this reason and also to improve the efficiency, it is
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proposed a sequential strategy consisting of the following three stages:
1st stage – An initial estimate for the hardening parameters of the Swift or
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Voce laws and for the parameters of Hill’48 yield criterion is required, in order to start the optimization procedure. The hardening parameters can
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be obtained, for example, from the results of the tensile test, or the typical values of the material under study can be adopted. Regarding the
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anisotropy parameters, it can be assumed 𝐹 = 𝐺 = 𝐻 = 0.5 (von Mises). It was observed that the initial estimate does not influence the overall
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quality of the identification, although it affects the required number of
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iterations.
2nd stage – The Levenberg-Marquardt method is used to identify the set of work hardening parameters, 𝐀 = [𝑌0 , 𝐾, 𝑛] or 𝐀 = [𝑌0 , 𝑌𝑆𝑎𝑡 ,𝐶𝑌 ], that
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minimizes the objective function 𝐹(𝐀)Circ (equation 6). The parameters of the Hill’48 criterion [𝐹 and 𝐺] are kept constant and equal to the initial estimate. This stage ends when the relative variation of all parameters from one iteration to the next one is inferior to 5% or when the objective 1% function 𝐹(𝐀)Circ is less than 𝐹Circ , which is defined by:
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𝑚
1% 𝐹Circ
0.01 2 √∑(𝑃(ℎ𝑖 )exp = ) , Circ 𝑚
(9)
𝑖=1
In this way, the identification stops when the objective function reaches values within the range of experimental pressure errors, which according
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to the ISO 16808:2014 standard (ISO 16808, 2014) should be less than 1% of the experimental pressure value. The stopping criterion based on
the relative variation of all parameters from one iteration to the next one, avoids an infinite loop, whenever the objective function is not able to
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1% reach a value of 𝐹Circ . A value of 5% was empirically chosen to
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1% guarantee that the stopping criterion based on the value of 𝐹Circ occurs
first, when the selected hardening law conveniently describes the
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material behaviour. During this stage, the sensitivity matrix is calculated analytically (Prates et al., 2019), so that only one simulation per iteration
3rd stage – The Levenberg-Marquardt method is used to improve the
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is required, making this stage computationally efficient.
identification of the work hardening parameters obtained in the 2nd stage.
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To this end, the set of parameters of the work hardening and the Hill’48 yield criterion 𝐀 = [𝐾, 𝑌0 , 𝑛, 𝐹, 𝐺], in case of Swift law, or 𝐀 = [𝑌0, 𝑌𝑆𝑎𝑡 , 𝐶𝑌 , 𝐹, 𝐺], in case of Voce law, is optimized (with 𝐻 following the condition 𝐺 + 𝐻 = 1). This is achieved by minimizing the gap between experimental and numerical curves of pressure vs. pole height in circular and elliptical, at 0º, tests or in circular and two elliptical, at 0º and 90º, 13
tests. In the first case, the sum of the objective functions 𝐹(𝐀)Circ and 𝐹(𝐀)Elli0° is minimized, while in the second case the minimization focuses on the sum of the functions, 𝐹(𝐀)Circ, 𝐹(𝐀)Elli0° and 𝐹(𝐀)Elli90° . The identification stops when the relative variation of all parameters from one iteration to the next one is inferior to 5% or when the objective functions 𝐹(𝐀)Circ , 𝐹(𝐀)Elli0° and, in the case of using a third test, 1% 1% 1% 𝐹(𝐀)Elli90° are respectively inferior to 𝐹Circ , 𝐹Elli0° and 𝐹Elli90° . The
context of the elliptical test, at 0º and 90º:
𝑚
0.01 2 √∑(𝑃(ℎ𝑖 )exp = ) , Elli0° 𝑚
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1% 𝐹Elli0°
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1% 1% 1% amounts 𝐹Elli0° and 𝐹Elli90° have the same meaning as 𝐹Circ , but in the
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𝑖=1
(10)
𝑚
1% 𝐹Elli90°
0.01 2 √∑(𝑃(ℎ𝑖 )exp = ) , Elli90° 𝑚
(11)
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𝑖=1
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During this optimization stage, the sensitivity matrix of the work hardening parameters is calculated analytically (Prates et al., 2019),
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while the sensitivity of the Hill’48 yield criterion parameters, F and G, is computed using forward finite differences, so that each iteration requires 6 or 9 simulations to compute the sensitivity matrix, in case of using two or three tests, respectively, making each iteration computationally more expensive than in the 2nd stage. Nevertheless, the computational cost is
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significantly mitigated in this stage, since the estimate at the end of the 2nd stage is already close to the final. During the identification procedure, the Hill’48 parameters fulfil the condition 𝐺 + 𝐻 = 1, i.e. the hardening curve is comparable to the uniaxial tensile curve along the Ox axis. In order to obtain the hardening curve comparable to the biaxial stress vs. strain curve, obtained for the stress path observed at the pole of the circular bulge test, the material parameters must be converted using the following equations proposed by
𝑛+1
∗ ; 𝑛∗ = 𝑛; 𝑌0 ∗ = 𝑌0 √𝑄; 𝑌𝑆𝑎𝑡 = 𝑌𝑆𝑎𝑡 √𝑄; 𝐶𝑌∗ = 𝐶𝑌 √𝑄,
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𝐾 ∗ = 𝐾(√𝑄)
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Prates et al. (2015):
(13)
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𝐹 ∗ = 𝑄 𝐹 ; 𝐺 ∗ = 𝑄𝐺; 𝐻 ∗ = 𝑄𝐻; 𝐿∗ = 𝑄𝐿; 𝑀∗ = 𝑄𝑀; 𝑁 ∗ = 𝑄𝑁,
(12)
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where 𝐾, 𝑛, 𝑌0 , 𝑌𝑆𝑎𝑡 , 𝐶𝑌 , 𝐹, 𝐺, 𝐻, 𝐿, 𝑀 and 𝑁 are the set of material parameters in the ∗ condition 𝐺 + 𝐻 = 1; 𝐾 ∗ , 𝑛∗ , 𝑌0 ∗ , 𝑌𝑆𝑎𝑡 𝐶𝑌∗ , 𝐹 ∗ , 𝐺 ∗ , 𝐻 ∗ , 𝐿∗ , 𝑀∗ and 𝑁 ∗ are the equivalent
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set of material parameters determined for the stress path at the pole of the circular bulge test, for which the condition 𝐺 + 𝐻 = 1 is not fulfilled. 𝑄 is the conversion factor given
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by Reis et al. (2017a):
2
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𝑄=
(𝜎yy ⁄𝜎xx ) − (𝜎yy ⁄𝜎xx ) + 1 2
(𝐹 + 𝐻)(𝜎yy ⁄𝜎xx ) − 2𝐻(𝜎yy ⁄𝜎xx ) + (𝐺 + 𝐻)
,
(14)
where 𝜎yy ⁄𝜎xx is the average stress path evaluated at the pole of the circular bulge test from the numerical simulation performed considering the identified parameters. All parameters presented in the following sections refer to the stress path at the pole of the 15
circular bulge test, but for convenience of the presentation the asterisks are omitted. In this way, the parameters no longer respect the condition 𝐺 + 𝐻 = 1, but another condition that depends on the stress path, at the pole of the circular bulge test, which varies with the anisotropic behaviour.
4. Parameters Identification To illustrate the previously described inverse identification strategy, several
ro of
cases are considered in this section. In each case, computer generated results of fictitious materials are used as “experimental” results. This allows the precise
knowledge of their mechanical behaviour, and so the suitable comparison between the
-p
“experimental” and identified parameters. Fictitious materials whose anisotropic
re
behaviour is described by von Mises, Hill’48 (as in the identification strategy) and Drucker+L yield criteria are considered as illustrative cases; their work hardening
lP
behaviour is described by the Swift and Voce laws. The aim is to examine the influence of the anisotropy and hardening behaviours on the results of the identification
na
procedure.
Table 1 shows the constitutive parameters of two Swift materials, one isotropic
ur
(von Mises: S_H Iso) and the other anisotropic (Hill'48: S_H Ani). Table 2 is identical to Table 1 but for two Voce materials: von Mises - V_H Iso and Hill’48 - V_H Ani. The
Jo
initial set of parameters used in the identifications is also shown (S_Ini and V_Ini). Figure 3(a) shows the work hardening curves of the materials in Tables 1 and 2. The choice of the materials was focused on isotropic materials with quite close hardening laws, Swift and Voce; the same is true for the anisotropic materials. Figure 3(b) shows the normalized yield surfaces in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ). The stress paths of the
16
circular and elliptical, at 0º and 90º, tests are also shown in this figure, revealing that they are very similar for all tests, in each material.
Table 1 - Parameters of two fictitious materials described by the Swift hardening law and the Hill’48 criterion: S_H Iso (isotropic material) and S_H Ani (anisotropic material). The initial set of parameters used in both identifications (S_Ini) is also shown. 𝜎yy ⁄𝜎xx is the respective average stress path observed at the pole of the circular bulge test.
Designation
𝑌0 [MPa] 𝐾 [MPa]
S_H Iso
200.00
S_H Ani S_Ini
𝐹
𝐺
𝐻
𝑁
1277.56
0.350
0.500
0.500
0.500
1.500
1.500
1.000
282.27
2034.23
0.350
0.165
1.329
0.663
1.509
4.110
1.633
100.00
200.00
0.100
0.500
0.500
0.500
ro of
𝐿=𝑀
𝜎𝑦𝑦 ⁄𝜎xx
𝑛
1.500
1.500
1.000
Designation
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa]
V_H Iso
305.30
1077.58
4.12
0.500
V_H Ani
431.43
1522.79
5.82
0.166
V_Ini
100.00
200.00
𝐻
0.500
0.500
𝐿=𝑀
𝑁
𝜎yy ⁄𝜎xx
1.500
1.500
1.000
1.335
0.667
1.502
4.120
1.646
0.50
0.500
0.500
1.500
1.500
1.000
Jo
ur
na
2.00
𝐺
re
𝐹
lP
𝐶𝑌
-p
Table 2 - Parameters of two fictitious materials described by the Voce hardening law and the Hill’48 criterion: V_H Iso (isotropic material) and V_H Ani (anisotropic material). The initial set of parameters used in both identifications (V_Ini) is also shown. 𝜎yy ⁄𝜎xx is the respective average stress path observed at the pole of the circular bulge test.
17
S_H Iso
S_H Ani
(a)
S_H Iso
S_H Ani
V_H Iso
V_H Ani
(b) V_H Iso
V_H Ani
1.5
1600 1400 1200
1.0 𝑌0
600
𝜎yy
Pa
800
0.5 𝑃
𝑌
Pa
1000
400
0.0
0 0
0.1
0.2
0.3
𝜀̅
0.4
0.5
0.6
0.0
ro of
200 0.5
𝜎xx
1.0
1.5
𝑌0
-p
Figure 3 - Fictitious materials described by the Hill’48 yield criterion (Tables 1 and 2): (a) work hardening; (b) yield surface in the normalized plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) and the stress paths of the circular and elliptical tests; the central stress paths refers to the circular tests and those immediately adjacent refers to the elliptical tests.
Figure 4 represents the range of pressure vs. pole height curves taken into
re
account in the identification, i.e. neglecting the initial and final regions of the curves, for
lP
the circular and elliptical bulge tests of the fictitious Swift (Figure 4(a)) and Voce (Figure 4(b)) materials (see Tables 1 and 2, respectively). The level of pressure vs. pole
na
height curves is higher for the elliptical than for the circular bulge tests; in case of isotropic materials the curves of the elliptical tests are equal at 0º and 90º. The circular
ur
pressure vs. pole height curves of the materials used as initial estimates (see Tables 1 and 2) are also shown in these figures; initial estimates far enough from the solution
Jo
were chosen to test the proposed strategy.
18
S_H Iso_Circ S_H Iso_Elli0º S_H Iso_Elli90º S_Ini_Circ
V_H Iso_Circ V_H Iso_Elli0º V_H Iso_Elli90º V_ Ini_Circ
28
24
24
20
20
16
Pa
16
12
𝑃
12
Pa
28
Pa
(b)
𝑃
(a)
S_H Ani_Circ S_H Ani_Elli0º S_H Ani_Elli90º
𝑃
8
V_H Ani_Circ V_H Ani_Elli0º V_H Ani_Elli90º
8
0
0 0
10
20
30
ℎ mm
40
50
60
0
10
ro of
4
4
20
ℎ mm
30
40
50
-p
Figure 4 - Circular and elliptical (0° and 90°) pressure vs. pole height curves of the fictitious materials described by (a) Swift (Table 1); (b) Voce (Table 2). The pressure vs. pole height curves of the circular test, corresponding to the initial estimates, S_Ini (Table 1) and V_Ini (Table 2), are also represented (blue lines).
The work hardening behaviour was identified by the inverse procedure described
re
in the previous section and compared with the hardening of the reference fictitious
lP
materials in Tables 1 and 2. The identification procedure was performed using: (i) the circular bulge and the elliptical bulge at 0º, the results of which are hereinafter designated by the abbreviation “2Tests”; (ii) the circular bulge and the two elliptical
na
bulge tests at 0º and 90º, whose results will be designated by the abbreviation “3Tests”. The membrane theory was also used to determine the biaxial stress vs. strain results
ur
from the circular bulge test, following the recommendations of the ISO 16808:2014
Jo
standard, i.e. in the same way that is usually done experimentally. The numerical evaluation of the pressure, radius of curvature and principal strains at the pole of the bulge test are given in the work by Reis et al. (2017a). In detail, the numerical evaluation of the radius of curvature is performed by fitting the equation of a circle to the nodes located at a distance lower than 15 mm from the bulge pole (at half thickness). The radius of curvature was analysed for both Oxz and Oyz planes (see axis
19
in Figure 1), and the principal strains were evaluated at the Gauss point located closer to the bulge pole. The biaxial stress vs. strain results evaluated by the membrane theory procedure (ISO 16808:2014) were used to fit the hardening laws, which parameters are referred to as “ em” in Tables 3 and 4.
Table 3 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “ em”, for the fictitious materials, “S_H Iso” and “S_H Ani”, described by the Hill’48 criterion and the Swift law; the parameters of these fictitious materials are also included for comparison. The stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the corresponding numerical simulation is also indicated.
𝑌0 [MPa] 𝐾 [MPa]
S_H Iso
200.00
S_H Iso|2Tests
𝐻
1277.56
0.350
0.500
0.500
0.500
1.500
1.500
1.000
203.95
1276.40
0.349
0.500
0.500
0.500
1.500
1.500
1.000
S_H Iso|Mem
186.66
1301.01
0.360
-
-
-
-
-
-
S_H Ani
282.27
2034.23
0.350
0.165
1.329
0.663
1.509
4.110
1.633
S_H Ani|2Tests 285.42
2032.09
0.351
0.112
1.419
0.782
3.301
3.301
1.630
S_H Ani|3Tests 290.67
2056.12
0.370
0.277
1.113
0.391
2.257
2.257
1.605
S_H Ani|Mem
1836.16
-
-
-
-
-
-
-p
Jo
ur
na
0.354
𝑁
ro of
𝐺
lP
𝐹
217.38
𝐿=𝑀
𝜎𝑦𝑦 ⁄𝜎xx
𝑛
re
Designation
20
Table 4 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “ em”, for the fictitious materials, “V_H Iso” and “V_H Ani”, described by the Hill’48 criterion and the Voce law; the parameters of these fictitious materials are also included for comparison. The stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the corresponding numerical simulation is also indicated.
𝐺
𝐻
4.12
0.500
0.500
0.500
1.500
1.500
1.000
1064.36
4.30
0.500
0.500
0.500
1.500
1.500
1.000
301.48
1114.44
3.86
-
-
-
-
431.43
1522.79
5.82
0.166
1.335
0.667
1.502
4.120
1.646
V_H Ani|2Tests 425.35
1556.38
5.02
0.288
1.035
0.391
2.138
2.138
1.542
V_H Ani|3Tests 429.82
1583.67
5.04
0.246
1.185
0.465
2.475
2.475
1.626
V_H Ani|Mem
1442.59
5.01
-
-
-
-
V_H Iso
305.30
1077.58
V_H Iso|2Tests 306.92 V_H Iso|Mem V_H Ani
𝑁
-
-
-p
388.51
𝐶𝑌
ro of
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa]
𝐿=𝑀
𝜎yy ⁄𝜎xx
𝐹
Designation
The parameters identified with these methodologies are given in Tables 3 and 4
re
for the Swift and Voce materials, respectively. The identification of the isotropic
lP
materials used the circular test and only one elliptical test, since the results of the experimental pressure vs. pole height curves are the same for both cases, at 0 and 90°. In addition, it was not necessary to perform all the optimization planned in the 3rd stage
na
since, at the beginning of this stage, the set of work hardening and yield criterion parameters already fulfils the stopping criteria.
ur
Figure 5 shows the differences between the hardening behaviours corresponding
Jo
to the parameters identified by inverse analysis and by the membrane theory (Tables 3 and 4) and the reference fictitious parameters (Table 1 and 2). Based on these results, Figure 6 plots the absolute value of the relative difference in stress, |𝑅𝑑 | = |(𝑌𝑟𝑒𝑓 − 𝑌𝑖𝑑𝑒 )⁄𝑌𝑖𝑑𝑒 |, between the reference material, 𝑌𝑟𝑒𝑓 , and the identified curves, 𝑌𝑖𝑑𝑒 , by inverse analysis and by using the membrane theory. The proposed strategy, using two or three tests, allows the accurate identification of the work hardening
21
parameters, regardless of the anisotropy behaviour and type of work hardening law. The worst inverse identification (see case “V_H Ani|2Tests”) has a maximum error of 4.6 % with an average value of 3.2%. In contrast, the membrane theory only provides an accurate identification for the isotropic materials, with average errors lower than 0.9%. In the case of the anisotropic materials, the membrane theory supplies poor identifications with average errors close to 10.7% for Voce and Swift laws, which is in agreement with the results by Reis et al. (2017a). In this context, the proposed strategy
ro of
seems to be more appropriate than the membrane theory to identify the work hardening behaviour of anisotropic materials. Moreover, the yield criterion (Hill’48) parameters
evaluated with the proposed strategy only accurately describe the yield surface close to
(a)
S_H Iso S_H Iso|Mem S_H Iso|2Tests
1600
lP
800 600 400
Jo
ur
600
Pa
𝑌
800
𝑌
na
Pa
1000
1000
0
(b)
1200
1200
200
V_H Iso V_H Iso|Mem V_H Iso|2Tests
1400
1400
400
V_H Ani V_H Ani|Mem V_H Ani|2Tests V_H Ani|3Tests
re
S_H Ani S_H Ani|Mem S_H Ani|2Tests S_H Ani|3Tests
-p
the stress paths of the bulge tests, as shown in Figure 7.
0
0.1
0.2
0.3
𝜀̅
200 0 0.4
0.5
0.6
0
0.1
0.2
𝜀̅
0.3
0.4
0.5
Figure 5 - Hardening behaviour corresponding to the parameters identified (Tables 3 and 4) and to the reference fictitious parameters (Table 1 and 2), for the materials described by: (a) Swift law; (b) Voce law.
22
S_H Ani|Mem
(a)
S_H Iso|Mem
S_H Ani|2Tests
(b)
S_H Iso|2Tests
S_H Ani|3Tests
14
10
10
8
8
V_H Iso|2Tests
𝑑
6
6
4
4
2
2 0
0 0
0.1
0.2
0.3
𝜀̅
0.4
0.5
0
0.6
0.1
ro of
𝑌
Pa
𝑑
12
V_H Iso|Mem
V_H Ani|2Tests V_H Ani|3Tests
14
12
V_H Ani|Mem
0.2
𝜀̅
0.3
0.4
0.5
(a)
S_H Ani
S_H Iso
S_H Ani|2Tests
S_H Iso|2Tests
V_H Iso
V_H Ani|2Tests
V_H Iso|2Tests
V_H Ani|3Tests
1.5
lP
1.5
1.0
ur
0.5
0.5 𝜎 xx
𝑌0
0.5
0.0 1.0
1.5
0.0
0.5 𝜎 xx
1.0
1.5
𝑌0
Jo
0.0
𝜎yy
na
𝜎yy
𝑌0
𝑌0
1.0
0.0
V_H Ani
re
S_H Ani|3Tests
(b)
-p
Figure 6 - Relative difference in hardening behaviour when comparing the materials corresponding to the parameters identified (Tables 3 and 4) and to the reference fictitious parameters (Tables 1 and 2). Materials described by: (a) Swift law; (b) Voce law.
Figure 7 – Normalized yield surface in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) corresponding to the parameters identified (Tables 3 and 4) for the fictitious materials (Table 1 and 2). (a) Swift law; (b) Voce law. The stress paths of the circular bulge tests are also represented.
The Drucker+L yield criterion is now used to describe the behaviour of the fictitious materials, in order to illustrate cases where the material behaviour is not described by the Hill'48 criterion, which is used in the identification. This seeks to 23
represent the common case where the yield criterion that better describes the experimental material behaviour is unknown. Table 5 shows the constitutive parameters of two Swift materials, one with a flattened yield surface (Drucker+L: S_D c=2) in the region near the stress path observed in the bulge tests and the other with a sharpened yield surface (Drucker+L: S_D c=-2) in this region. Table 6 is identical to Table 5 but for two Voce materials: V_D c=2 (flattened yield surface) and V_D c=-2 (sharpened yield surface). Figure 8 shows the work hardening curves (Figure 8(a)) and the
ro of
normalized yield surfaces in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) (Figure 8(b)) of the materials in Tables 5 and 6. The stress paths of the circular and the elliptical tests at 0º and 90º are
also shown in Figure 8(b). Figure 9 represents the range of the pressure vs. pole height
-p
curves for the circular and elliptical tests of the reference fictitious materials (Tables 5
and 6) taken into account in the identification procedure; the curves of the circular bulge
re
test of the materials used as initial estimate, S_Ini (see Table 1) and V_Ini (see Table 2)
lP
are also shown.
na
Table 5 - Parameters of two fictitious materials described by the Swift hardening law and the Drucker+L criterion: S_D c=2 (flattened yield surface) and S_D c=-2 (sharpened yield surface). 𝜎yy ⁄𝜎xx is the respective average stress path observed at the pole of the circular bulge test.
𝑌0 [MPa]
𝐾 [MPa]
S_D c = 2
257.53
781.60
ur
Designation
277.08
853.34
𝐶1
𝐶2
𝐶3
𝐶4 = 𝐶5
𝐶6
𝑐
𝜎yy ⁄𝜎xx
0.200 0.754 1.554 1.155 1.359
1.458 2.000 1.374
0.200 0.738 1.521 1.129 1.330
1.426 -2.000 1.671
Jo
S_D c = -2
𝑛
24
Table 6 - Parameters of two fictitious materials described by the Voce hardening law and the Drucker+L criterion: V_D c=2 (flattened yield surface) and V_D c=-2 (sharpened yield surface). 𝜎yy ⁄𝜎xx is the respective average stress path observed at the pole of the circular bulge test.
𝜎yy ⁄𝜎xx
Designation
𝑌0 [MPa]
𝑌𝑆𝑎𝑡 [MPa]
V_D c = 2
310.86
661.68
7.05
0.755 1.557 1.156 1.289
1.459 2.000 1.378
V_D c = -2
333.37
709.61
7.56
0.737 1.518 1.127 1.383
1.423 -2.000 1.664
S_D c = 2
𝐶𝑌
𝐶1
𝐶2
𝐶3
S_D c = -2
(a)
𝐶4 = 𝐶5
𝐶6
𝑐
S_D c = 2
S_D c = -2
V_D c = 2
V_D c = -2
(b) V_D c = 2
V_D c = -2
ro of
1.5
800 700 600
1.0
300 200 100
re
𝑃
0.5
-p
𝑌
Pa
400
𝜎yy
𝑌0
Pa
500
0.0
0 0.1
0.2
𝜀̅
0.3
0.4
0.5
lP
0
0.0
0.5
𝜎xx
1.0
1.5
𝑌0
Jo
ur
na
Figure 8 - Fictitious materials described by the Drucker+L yield criterion (Tables 5 and 6): (a) work hardening; (b) yield surface in the normalized plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) and the stress paths of the circular and elliptical tests.
25
S_D c = 2_Circ S_D c = 2_Elli0º S_D c = 2_Elli90º S_Ini_Circ
V_D c = 2_Circ V_D c = 2_Elli0º V_D c = 2_Elli90º V_ Ini_Circ
(b) 12
10
10
8
8
Pa
6
V_D c = -2_Circ V_D c = -2_Elli0º V_D c = -2_Elli90º
6
𝑃
Pa
12
𝑃
Pa
(a)
S_D c = -2_Circ S_D c = -2_Elli0º S_D c = -2_Elli90º
4 𝑃
4
0
0 0
10
20
30
ℎ mm
40
50
0
ro of
2
2
10
20
ℎ mm
30
40
50
-p
Figure 9 - Circular and elliptical (0° and 90°) pressure vs. pole height curves of the fictitious materials described by (a) Swift (Table 5); (b) Voce (Table 6). The pressure vs. pole height curves of the circular test, corresponding to the initial estimates, S_Ini (Table 1) and V_Ini (Table 2), are also represented (blue lines).
The results of the inverse identification and membrane theory of the reference
re
fictitious materials in Table 5 and 6 are shown in Tables 7 and 8 and Figures 10, 11, 12
lP
and 13. Also in these cases, the proposed strategy, using two or three tests, provided accurate identifications of the work hardening parameters, although in general the
na
identifications with three tests have slightly higher quality. For instance, the worst identification using two tests (V_D c=-2|2Tests) has an average error of 3.1% (see Figure 13), while the worst identification with three tests (V_D c= -2|3Tests) has an
ur
average error of 1.1% (see Figure 13). The best identification with three tests arises
Jo
from a good description of the yield surface near the bulge stress paths, as can be seen in Figures 10 and 11. In fact, the stress paths of the circular test corresponding to the identified yield criterion parameters are more accurate when using three tests. Also for these materials, the procedure based on the membrane theory (ISO 16808:2014) does not provide accurate results for the work hardening behaviour. This is due to the strong anisotropy of these materials, for which the stress paths of the circular bulge test are
26
distant from the equibiaxial, particularly in the case of the fictitious materials “S_D c= 2” and “V_D c= -2” with stress paths of 1.671 and 1.664, respectively (see Tables 5 and 6). In these cases, the membrane theory provides poor identifications with average errors of about 10%. For the cases of the reference fictitious materials, “S_D c= 2” and “V_D c= 2”, with stress paths of 1.374 and 1.378 (see Tables 5 and 6), the identification with the membrane theory is better, with average errors of about 4%.
𝑌0 [MPa] 𝐾 [MPa]
S_D c = 2
257.53
781.60
𝑛
𝐶1
𝐶2
𝐶3
𝐶4 = 𝐶5
0.200 0.754 1.554 1.155 1.359 𝐹
𝐺
𝐻
𝜎yy ⁄𝜎xx
𝑐
𝐶6
1.458 2.000 1.374
-p
Designation
ro of
Table 7 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “ em”, for the fictitious materials, “S_D c = 2” and “S_D c = -2”, described by the Drucker+L criterion and the Swift law; the parameters of these fictitious materials are also included for comparison. The stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the corresponding numerical simulation is also indicated.
𝐿
𝑀
𝜎yy ⁄𝜎xx
𝑁
793.98
0.219 0.369 0.731 0.349 1.620
1.620 1.620 1.290
S_D c = 2|3Tests 257.40
783.33
0.202 0.113 1.146 1.148 3.440
3.440 3.440 1.383
S_D c = 2|Mem
231.80
759.07
Designation
𝑌0 [MPa] 𝐾 [MPa]
S_D c = -2
277.08
lP
re
S_D c = 2|2Tests 257.01
0.205 -
na
𝑛
ur
853.34
-
𝐶1
-
𝐶2
𝐶3
𝐶4 = 𝐶5
0.200 0.738 1.521 1.129 1.330 𝐹
𝐺
𝐻
-
𝜎yy ⁄𝜎xx
𝑐
𝐶6
1.426 -2.000 1.671
𝐿
𝑀
𝜎yy ⁄𝜎xx
𝑁
849.53
0.206 0.282 1.097 0.420 2.275
2.275 2.275 1.614
S_D c = -2|3Tests 276.95
854.79
0.206 0.283 1.136 0.405 2.311
2.311 2.311 1.656
773.23
0.203 -
-
Jo
S_D c = -2|2Tests 280.69
S_D c = -2|Mem 229.63
-
-
-
-
-
27
Table 8 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “ em”, for the fictitious materials, “V_D c = 2” and “V_D c = -2”, described by the Drucker+L criterion and the Voce law; the parameters of these fictitious materials are also included for comparison. The stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the corresponding numerical simulation is also indicated.
Designation
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa] 𝐶𝑌
V_D c = 2
310.86
661.68
𝐶1
𝐶2
𝐶3
𝐶4 = 𝐶5
7.05 0.755 1.557 1.156 1.289 𝐹
𝐺
𝐻
𝜎yy ⁄𝜎xx
𝑐
𝐶6
1.459 2.000 1.378
𝐿
𝑀
𝜎yy ⁄𝜎xx
𝑁
V_D c = 2|2Tests 308.40
715.63
5.06 0.369 0.736 0.330 1.599
1.599 1.599 1.298
V_D c = 2|3Tests 315.83
690.63
6.16 0.164 1.074 1.017 3.138
3.138 3.138 1.398
V_D c = 2|Mem 297.13
652.51
6.39 -
-
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa] 𝐶𝑌
V_D c = -2
333.37
𝐶2
𝐶3
𝐶4 = 𝐶5
7.56 0.737 1.518 1.127 1.383 𝐺
𝐻
-
𝜎yy ⁄𝜎xx
𝑐
𝐶6
1.423 -2.000 1.664
𝐿
𝑀
𝜎yy ⁄𝜎xx
𝑁
V_D c = -2|2Tests326.44
719.94
6.21 0.352 0.909 0.260 1.754
1.754 1.754 1.524
V_D c = -2|3Tests331.99
715.60
6.91 0.269 1.143 0.438 2.372
2.372 2.372 1.636
V_D c = -2|Mem 304.67
660.39
6.49 -
re
𝐹
-
-p
709.61
𝐶1
-
ro of
Designation
-
-
-
-
-
-
Jo
ur
na
lP
-
28
S_D c = 2
S_D c = 2|2Tests
V_D c = 2
(a)
V_D c = 2|2Tests
(b) S_D c = 2|3Tests
V_D c = 2|3Tests
1.5
1.0
1.0
𝜎yy
𝜎yy
𝑌0
𝑌0
1.5
0.5
0.5
0.0
0.5
𝜎xx
1.0
ro of
0.0
0.0
0.0
1.5
0.5 𝜎 xx
𝑌0
1.0
1.5
𝑌0
S_D c = -2
S_D c = -2|2Tests
(a)
V_D c = -2
V_D c = -2|2Tests
re
(b)
S_D c = -2|3Tests
V_D c = -2|3Tests
1.5
lP
1.5
1.0
ur
0.5
Jo
0.0
0.5
𝜎xx
𝑌0
𝜎yy
𝜎yy
na
𝑌0
𝑌0
1.0
0.0
-p
Figure 10 – Normalized yield surface in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) corresponding to the parameters identified (Tables 7 and 8) for the fictitious materials of Tables 5 and 6: (a) “S_D c = 2”; (b) “V_D c = 2”. The stress paths of the circular bulge tests are also represented.
0.5
0.0 1.0
1.5
0.0
0.5 𝜎 xx
1.0
1.5
𝑌0
Figure 11 – Normalized yield surface in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) corresponding to the parameters identified (Tables 7 and 8) for the fictitious materials of Tables 5 and 6: (a) “S_D c = -2”; (b) “V_D c = -2”. The stress paths of the circular bulge tests are also represented.
29
S_D c = 2 S_D c = 2|Mem S_D c = 2|2Tests S_D c = 2|3Tests
(a) 800
S_D c = -2 S_D c = -2|Mem S_D c = -2|2Tests S_D c = -2|3Tests
V_D c V_D c V_D c V_D c
(b) 800 700
600
600
500
500
V_D c V_D c V_D c V_D c
= -2 = -2|Mem = -2|2Tests = -2|3Tests
Pa
Pa
700
=2 = 2|Mem = 2|2Tests = 2|3Tests
400
𝑌
𝑌
400
300
200
200
100
100
𝑌
Pa
300
0
0.1
0.2
𝜀̅
0.3
0.4
ro of
0
0
0
0.5
0.1
0.2
𝜀̅
0.3
0.4
(a)
S_D c = -2|Mem
S_D c = 2|2Tests
S_D c = -2|2Tests
S_D c = 2|3Tests
S_D c = -2|3Tests
V_D c = 2|2Tests
V_D c = -2|2Tests
V_D c = 2|3Tests
V_D c = -2|3Tests
10
lP
10
8
𝑑
8 6
na
𝑑
14
V_D c = -2|Mem
12
12
4 2 0
ur
𝑌
Pa
(b)
V_D c = 2|Mem
re
14
S_D c = 2|Mem
-p
Figure 12 – Hardening behaviour corresponding to the parameters identified (Tables 7 and 8) and to the reference fictitious parameters (Tables 5 and 6), for materials described by: (a) Swift law; (b) Voce law.
0
0.1
0.2
𝜀̅
0.3
0.4
6 4 2 0
0.5
0
0.1
0.2
𝜀̅
0.3
0.4
Jo
Figure 13– Relative difference in hardening behaviour when comparing the materials corresponding to the parameters identified (Tables 7 and 8) for and to the reference fictitious parameters (Tables 5 and 6). Materials described by: (a) Swift laws; (b) Voce laws.
In summary, the proposed inverse analysis strategy allowed to accurately identify the parameters of the Swift and Voce hardening laws, regardless of the material anisotropy. The Hill’48 yield criterion proved to be flexible enough to describe the yield surface of materials in the region close to the stress paths that occur in the circular and 30
elliptical bulge tests, even if the behaviour of the materials is described by a different yield criterion (Drucker+L for the illustrative cases). The description of the entire yield surface is nonessential, as the pressure vs. pole height curves depend mainly on the most deformed zones of the bulge (i.e., between the pole and the die), for which the stress path is similar to that at the pole, as already shown by Reis et al. (2017a). The inverse analysis procedure requires the use of at least two bulge tests, a circular and an elliptical. Nevertheless, the evaluation of the work hardening parameters is usually more
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accurate when a circular and two elliptical bulge tests, perpendicular to each other, are used. In general, this also improves the assessment of the stress path in the circular tests. The proposed strategy shows to be more accurate than the membrane theory, mainly
-p
when the materials behaviour is strongly anisotropic, since the equibiaxial stress state is not imposed.
re
When the optimization in the 3rd stage is continued further than the conditions
lP
assumed by the adopted stopping criteria, the accuracy of the proposed strategy can be improved, regardless of the yield criterion that describes the material behaviour. This reinforces the idea that the strategy efficiency is not dependent on the anisotropy of the
na
material and depends only on the accurateness of the numerical model and on the experimental determination of the pressure vs. pole height curves. From the
ur
experimental point of view, the proposed strategy is not so much exposed to
Jo
experimental errors as the membrane theory procedure (ISO 16808:2014), since the experimental evaluation of the sheet thickness and radius of curvature at the pole during the tests is not required. Additionally, the proposed methodology does not require the use of digital image correlation technique, which according to Koç et al. (2011) can raise accuracy issues at high temperatures.
31
In average, the Levenberg-Marquardt algorithm required 7 iterations in the 2nd stage and 4 iterations in the 3rd stage, regardless of using two or three bulge tests. Each numerical simulation of the circular and elliptical tests was respectively completed in 14.15 and 15.40 min, using the blank sheet discretisations shown in Figure 2. Hence, the average computational time of each identification was 7.56 hours, when using two bulge tests, and 10.64 hours for three bulge tests. Due to the high amount of computational time required in the identification, it is suggested in Appendix A an
ro of
optimized discretisation of the blank sheet that significantly reduces the computational time (about ten times), but preserves the accurate prediction of the pressure vs. pole
re
5. Experimental Cases
-p
height curves and so the results of the identification procedure.
The proposed strategy is now applied to the cases of two materials, a steel
lP
(DP600) and an aluminium alloy (Al5754) with initial sheet thicknesses of 0.8 and 1 mm, respectively. Typical values for the elastic parameters were assumed: Young's
na
modulus, 𝐸 = 210 GPa, and Poisson's ratio, 𝜈 = 0.30 in case of DP600 steel; and 𝐸 = 70 GPa, and 𝜈 = 0.30, in case of Al5754 aluminium alloy. The hydraulic bulge test
ur
apparatus developed by Santos et al. (2010), whose geometry and dimensions are shown in Figure 1, was used to obtain the experimental pressure vs. pole height results,
Jo
necessary to the identification procedure. In order to ensure that the boundary conditions of the numerical and experimental tests are as close as possible, it has been ensured that, in the latter case, the clamping force and the drawbead do not allow the occurrence of any draw in, similarly to the numerical model. Figure 14 shows the parts of the experimental pressure vs. pole height curves taken into account in the identifications. In this figure are also represented the circular pressure vs. pole height 32
curves of the materials used as initial estimates, S_Ini (the same as in Table 1) and V_Ini (the same as in Table 2). Figure 14 shows that the pressure vs. pole height curves of the elliptical tests at 0º and 90º are identical. Therefore, equibiaxial stress paths should occur in the circular bulge test and, consequently, it is expected that the membrane theory provides accurate results (see Figure 6), which can be used to test the proposed inverse strategy.
DP600_Elli0º
Al5754_Circ
Al5754_Elli0º
Al5754_Elli90º
V_ Ini_Circ
ro of
DP600_Circ
(a)
(b) DP600_Elli90º
S_Ini_Circ
12
5
Pa
Pa
3 2
𝑃
re
𝑃
𝑃
Pa
6
-p
4
9
3
0 0
10
20
ℎ mm
lP
1
30
40
50
0 0
10
20
ℎ mm
30
40
50
ur
na
Figure 14 - Circular and elliptical (0° and 90°) pressure vs. pole height curves of the materials: (a) DP600; (b) Al5754. The circular pressure vs. pole height curves corresponding to the initial estimates used in the identification, S_Ini (the same as in Table 1) and V_Ini (the same as in Table 2), are also represented (blue lines).
The work hardening parameters of the aluminium alloy (Al5754) and steel
Jo
(DP600) were identified using the proposed strategy with two or three tests. At the end of the identification procedure, the experimental and numerical pressure vs. pole height curves are identical, with an average difference in pressure inferior to 0.0617 MPa, for the DP600 steel, and 0.0289 MPa, for the Al5754 aluminium. The membrane theory was also used for determining the biaxial stress vs. strain curves. Tables 9 and 10 show the results obtained when the Swift and Voce law are used for describing the work 33
hardening, respectively. These tables also show the Swift and Voce laws fitted to the membrane theory results, designated by “Mem”. Note that in Tables 9 and 10 the parameter 𝐻 is negative in three cases. Although this does not make sense from the experimental point of view, Hill (1948) stated that it is mathematically admissible that one and only one of the parameters F, G, H is negative. In fact, the Hill’48 criterion is used in this work only as a mathematical function to describe the shape of the yield surface in a region close to the stress paths that occur in the circular and elliptical bulge
ro of
tests, regardless of the physical meaning of its parameters.
Designation
𝑌0 [MPa] 𝐾 [MPa]
S_DP600|2Tests
417.30
S_DP600|3Tests S_DP600|Mem
𝐹
973.30
0.165
0.533
421.31
951.19
0.160
415.53
966.46
0.156
343.11
0.219
𝐻
0.484
-0.108
0.564
0.876
0.500
-0.047
0.680
1.002
-
-
-
-
-
0.501
0.499
0.243
1.112
0.998
0.500
lP
na
S_Al5754|2Tests 53.57
𝐿 = 𝑀 = 𝑁 𝜎yy ⁄𝜎xx
𝐺
re
𝑛
-p
Table 9 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “Mem”, for the experimental cases, on the assumption that they are described by the Swift hardening law. The observed stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the numerical simulation corresponding to the identified parameters is also indicated.
S_Al5754|3Tests 54.70
343.19
0.216
0.506
0.494
0.292
1.179
0.990
S_Al5754Mem
341.12
0.216
-
-
-
-
-
Jo
ur
13.07
34
Table 10 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “Mem”, for the experimental cases, on the assumption that they are described by the Voce hardening law. The observed stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the numerical simulation corresponding to the identified parameters is also indicated.
Designation
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa]
𝐶𝑌
𝐹
𝐺
𝐻
𝐿 = 𝑀 = 𝑁 𝜎yy ⁄𝜎xx
812.08
8.54
0.476
0.532
0.052
0.875
1.083
V_DP600|3Tests 474.25
807.91
8.59
0.498
0.502
-0.045
0.685
1.009
V_DP600|Mem
484.86
832.62
7.87
-
-
-
-
-
V_Al5754|2Tests 114.13
266.37
9.71
0.501
0.499
0.239
1.107
0.997
V_Al5754|3Tests 114.56
264.12
9.68
0.496
0.504
0.152
0.985
1.009
V_Al5754|Mem
257.05
10.89
-
-
-
-
-
116.46
ro of
V_DP600|2Tests 474.68
-p
Figure 15 compares all stress vs. strain curves of each material with the points obtained from the membrane theory (designated by “ emPoints”). The relative
re
differences in stress determined when comparing the points obtained by the membrane
lP
theory with the stress vs. strain curves obtained by fitting the hardening law to these points and the proposed strategy, are shown in Figure 16. The proposed strategy with two or three tests provides similar results to those of the membrane theory. The relative
na
difference in stress has an average value inferior to 2%, whatever the case, i.e. the identification of the Swift and Voce parameters, provided similar results for DP600 and
ur
Al5754. These errors are similar to those obtained by fitting the hardening laws
Jo
(designated by “ em” in Figure 16) to the set of points obtained from the membrane theory. The relative difference in stress only achieves a significant scale near the yield stress, which can be related with a poor fitting of the hardening law or due to experimental errors at small deformations.
35
S_DP600|Mem S_DP600|2Tests S_DP600|3Tests DP600|MemPoints
(a)
S_Al5754|Mem S_Al5754|2Tests S_Al5754|3Tests Al5754|MemPoints
(b) 300 250
600
200 Pa
750
V_Al5754|Mem V_Al5754|2Tests V_Al5754|3Tests
150
𝑌
450
𝑌
300
100
150
50 0
0 0
0.1
0.2
𝜀̅
0.3
ro of
Pa
900
V_DP600|Mem V_DP600|2Tests V_DP600|3Tests
0
0.4
0.1
𝜀̅
0.2
0.3
Figure 15 - Hardening curves corresponding to the identified parameters (Tables 9 and 10) for the materials: (a) DP600; (b) Al5754.
V_DP600|Mem
S_DP600|2Tests
V_DP600|2Tests
S_DP600|3Tests
V_DP600|3Tests
(b) 10
S_Al5754|Mem
V_Al5754|Mem
S_Al5754|2Tests
V_Al5754|2Tests
S_Al5754|3Tests
V_Al5754|3Tests
re
10
S_DP600|Mem
-p
(a)
8
lP
8 6
𝑑
𝑑
6
na
4
0
ur
2
0
0.1
0.2
𝜀̅
0.3
4 2 0
0.4
0
0.1
𝜀̅
0.2
0.3
Jo
Figure 16 - Absolute value of the relative difference in stress when comparing the stress vs. strain points obtained by the membrane theory with their fitting and the inverse identification results (see Tables 9 and 10) for the materials: (a) DP600; (b) Al5754.
The results of the bulge test are now compared with those obtained from the uniaxial tensile test. For this comparison, the hardening curves are transferred to uniaxial tension based on the procedure recommended in Annex D of ISO 16808:2014. Figure 17 compares the stress vs. strain curves directly determined from the uniaxial tensile test 36
(Figure 17(a): DP600|Tensile; and Figure 17(b): Al5754|Tensile) with those obtained from the biaxial tests, identified through the membrane theory and with the proposed strategy. It can be concluded from this figure that the results obtained from tensile and bulge tests are in agreement; the initial parts of the directly determined tensile curves are in better agreement with the identifications, when using the Swift law in case of the DP600 material, and when using the Voce law in case of the Al5754 material.
(b) 300 250
600
200 Pa
750
150
𝑌
450
V_Al5754|Mem V_Al5754|2Tests V_Al5754|3Tests
re
𝑌
Pa
900
S_Al5754|Mem S_Al5754|2Tests S_Al5754|3Tests Al5754|Tensile
-p
(a)
V_DP600|Mem V_DP600|2Tests V_DP600|3Tests
ro of
S_DP600|Mem S_DP600|2Tests S_DP600|3Tests DP600|Tensile
100
300
50
0 0.1
0.2
𝜀̅
0.3
0.4
0 0
0.1
𝜀̅
0.2
0.3
na
0
lP
150
ur
Figure 17 – Comparison of the hardening laws, directly obtained from the uniaxial tensile test with those obtained from the biaxial tests, identified through the membrane theory and the proposed strategy, for the materials: (a) DP600; (b) Al5754.
6. Conclusions
Jo
An inverse analysis strategy is proposed that allows determining the parameters
of work hardening laws, Swift and Voce, regardless of the material anisotropy. The procedure uses the pressure vs. pole height curves from the circular and elliptical bulge tests. The use of two tests (a circular and an elliptical) is enough to accurately determine the hardening law parameters, although the use of three tests (a circular and two elliptical) allows greater accuracy in the evaluation of the stress path of the circular 37
bulge test. In fact, the identified hardening law has an associated stress path that is of interest to know accurately, for example in the framework of the parameters identification of the yield criterion. The identified hardening behaviour corresponds to the biaxial stress vs. strain curve under the bulge test and can be related with the tensile stress vs. strain curve through the plastic work equivalence. From the experimental point of view, the proposed inverse strategy is not so exposed to experimental errors as the membrane theory procedure (ISO 16808:2014).
ro of
The later requires the evaluation of the curvature radius and the strains measurement at the pole of the bulge test, for which the experimental errors are more critical. In fact, it is worth emphasizing that the efficiency of the proposed identification strategy only
-p
depends on the accurate experimental determination of the pressure vs. pole height
curves and the appropriateness of the numerical model. Moreover, although it needs to
re
resort to one or two elliptical tests besides the circular, the current strategy proves to be
lP
an efficient alternative to the membrane theory procedure (ISO 16808:2014), that leads to significant errors in the hardening identification of strongly anisotropic materials, due to the equibiaxial stress state assumption. The proposed strategy avoids this assumption
na
with the accessory optimization of two of the Hill’ 48 parameters that enables the accurate description of the yield surface in a region close to the stress paths that occur in
ur
the circular and elliptical bulge tests. Nevertheless, the optimized Hill’48 parameters do
Jo
not model the entire yield surface.
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: SAFEFORMING, co-
38
funded by the Portuguese National Innovation Agency, by FEDER, through the program Portugal-2020 (PT2020), and by POCI, with reference POCI-01-0247FEDER-017762; 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 Portugal2020 (PT2020), and by POCI, with reference POCI-01-0145-FEDER-031216. Two of
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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
-p
gratefully acknowledged.
re
Author Contributions:
na
lP
Pereira A.F.G.: Investigation, Formal Analysis, Writing - Original Draft, WritingReviewing and Editing.; Prates P.A.: Funding acquisition, Formal Analysis, WritingReviewing and Editing.; Oliveira M.C.: Software, Formal Analysis, Writing- Reviewing and Editing.; Fernandes V.A.: Funding acquisition, Supervision, Writing- Reviewing and Editing.
ur
Declaration of interests
Jo
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Appendix A
39
The numerical discretisation of the blank sheets shown in Figure 2 was prepared in order to allow accurate results when used to determine the biaxial stress vs. strain curve by the membrane theory procedure (ISO 16808, 2014), which requires the determination of the curvature radius and the principal strains at the pole during the bulge test. In fact, the accuracy of these results requires a fine discretisation (see Figure 2), and so a high computational cost. Since the proposed strategy only requires the results of pressure vs. pole height, it is suggested in Figure A.1 a coarser discretisation
ro of
of the blank sheets that reduces the computational time but preserves the accurate assessment of the pressure vs. pole height curves, as shown in Figure A.2, and so of the inverse strategy identification results. In Figure A.2, the average difference in pressure
-p
between the coarse and refine meshes is approximately 0.025 MPa. The discretisation of the circular and elliptical blank sheets has respectively a total number of 915 and 1065,
re
8-node hexahedral solid elements, with one layer of elements through the sheet
lP
thickness. In average, each numerical simulation of the circular and elliptical tests was completed in approximately 1.40 and 1.71 min, respectively. Hence, each identification required in average a total computational time of 0.79 and 1.13 hours in case of using
na
two or three bulge tests, respectively. The use of these discretisations was tested for some materials and has proved to be about 10 times more efficient than those shown in
Jo
ur
Figure 2, without compromising the inverse strategy identification results.
40
(a)
1
1
2 20
8 20
15
(b)
20
15 5
2
1
7
y 15
20
53.03
34.97
5
8
Oz displacement[mm]
27
34.97 29.11
53.03
38.89 53.03
34.97
7
x
1
5
7
5
ro of
15
Circ_Coarse
Circ_Refine
Elli0º_Coarse
Elli0º_Refine
Elli90º_Coarse
Elli90º_Refine
8 4
Jo
0
na
12
ur
Pa 𝑃
20
0
10
20
30
ℎ mm
40
50
𝑃
24
Pa
lP
28
16
re
-p
Figure A.1 – (a) Alternative discretisation of the blanks used in the circular and elliptical bulge tests showing the number of elements along Ox and Oy for the different zones; b) Representation of the discretisation zones, showing their dimensions and the displacement along Oz, in mm, for an isotropic material.
60
Figure A.2 – Circular and elliptical (0° and 90°) pressure vs. pole height curves of the fictitious material S_H Ani (Table 1), generated with the mesh discretisations presented in Figure 2 and Figure A.1.
41
References Bambach, M., 2011. Comparison of the identifiability of flow curves from the hydraulic bulge test by membrane theory and inverse analysis. Key Eng. Mater. 473, 360– 367. https://doi.org/10.4028/www.scientific.net/KEM.473.360 Cazacu, O., Barlat, F., 2001. Generalization of Drucker’s Yield Criterion to Orthotropy. Math. Mech. Solids 6, 613–630. https://doi.org/10.1177/108128650100600603
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Chamekh, A., BelHadjSalah, H., Hambli, R., Gahbiche, A., 2006. Inverse identification
using the bulge test and artificial neural networks. J. Mater. Process. Technol. 177, 307–310. https://doi.org/10.1016/J.JMATPROTEC.2006.03.214
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Chen, K., Scales, M., Kyriakides, S., Corona, E., 2016. Effects of anisotropy on
material hardening and burst in the bulge test. Int. J. Solids Struct. 82, 70–84.
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https://doi.org/10.1016/J.IJSOLSTR.2015.12.012
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Drucker, D.C., of Applied Mathematics, B.U.D., of Naval Research, U.S.O., of Ships, U.S.N.D.B., 1948. The Relation of Experiments to Mathematical Theories of
na
Plasticity, Technical report (Brown University. Division of Applied Mathematics). Division of Applied Mathematics, Brown University.
ur
Hill, R., 1948. A Theory of the Yielding and Plastic Flow of Anisotropic Metals. Proc. R. Soc. A Math. Phys. Eng. Sci. 193, 281–297.
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https://doi.org/10.1098/rspa.1948.0045 ISO 16808, 2014. Metallic materials - sheet and strip - determination of biaxial stressstrain curve by means of bulge test with optical measuring systems 2014, 36.
Koç, M., Billur, E., Cora, Ö.N., 2011. An experimental study on the comparative assessment of hydraulic bulge test analysis methods. Mater. Des. 32, 272–281. https://doi.org/10.1016/j.matdes.2010.05.057
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Lăzărescu, L., Nicodim, I.P., Comşa, D.S., Banabic, D., 2012. A Procedure for the Evaluation of Flow Stress of Sheet Metal by Hydraulic Bulge Test Using Elliptical Dies. Key Eng. Mater. 504–506, 107–112. https://doi.org/10.4028/www.scientific.net/KEM.504-506.107 Marquardt, D.W., 1963. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math. 11, 431–441. https://doi.org/10.1137/0111030 Menezes, L.F., Teodosiu, C., 2000. Three-dimensional numerical simulation of the
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deep-drawing process using solid finite elements. J. Mater. Process. Technol. 97, 100–106. https://doi.org/10.1016/S0924-0136(99)00345-3
Mulder, J., Vegter, H., Aretz, H., Keller, S., Van Den Boogaard, A.H., 2015. Accurate
J. Mater. Process. Technol. 226, 169–187.
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determination of flow curves using the bulge test with optical measuring systems.
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https://doi.org/10.1016/j.jmatprotec.2015.06.034
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Prates, P.A., Oliveira, M.C., Fernandes, J. V., 2015. On the equivalence between sets of parameters of the yield criterion and the isotropic and kinematic hardening laws. Int. J. Mater. Form. 8, 505–515. https://doi.org/10.1007/s12289-014-1173-z
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Prates, P.A., Pereira, A.F.G., Oliveira, M.C., Fernandes, J.V., 2019. Analytical sensitivity matrix for the inverse identification of hardening parameters of metal
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sheets. Eur. J. Mech. - A/Solids 75, 205–215.
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https://doi.org/10.1016/j.euromechsol.2019.01.010 Prates, P.A., Pereira, A.F.G., Sakharova, N.A., Oliveira, M.C., Fernandes, J. V., 2016. Inverse Strategies for Identifying the Parameters of Constitutive Laws of Metal Sheets. Adv. Mater. Sci. Eng. 2016. https://doi.org/10.1155/2016/4152963 Ragab, A.R., Habib, O.E., 1984. Determination of the equivalent stress versus equivalent strain rate behaviour of superplastic alloys in biaxial stress systems.
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Mater. Sci. Eng. 64, 5–14. https://doi.org/10.1016/0025-5416(84)90068-5 Rees, D.W.., 1995. Plastic flow in the elliptical bulge test. Int. J. Mech. Sci. 37, 373– 389. https://doi.org/10.1016/0020-7403(94)00071-Q Rees, D.W.A., 2000. Rim and pole failures from elliptical bulging of oriented orthotropic sheet metal. J. Strain Anal. Eng. Des. 35, 109–120. https://doi.org/10.1243/0309324001514062 Reis, L.C., Prates, P.A., Oliveira, M.C., Santos, A.D., Fernandes, J. V., 2017a.
70–93. https://doi.org/10.1016/j.ijmecsci.2017.04.007
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Anisotropy and plastic flow in the circular bulge test. Int. J. Mech. Sci. 128–129,
Reis, L.C., Prates, P.A., Oliveira, M.C., Santos, A.D., Fernandes, J. V., 2017b. Inverse
https://doi.org/10.1007/s12289-016-1296-5
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identification of the Swift law parameters using the bulge test. Int. J. Mater. Form.
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Santos, A.D., Teixeira, P., da Rocha, A., Barlat, F., 2010. On the Determination of Flow
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Stress Using Bulge Test and Mechanical Measurement. AIP Conf. Proc. 1252, 845–852. https://doi.org/10.1063/1.3457644 Swift, H.W., 1952. Plastic instability under plane stress. J. Mech. Phys. Solids 1, 1–18.
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https://doi.org/10.1016/0022-5096(52)90002-1 Voce, E., 1948. The relationship between stress and strain for homogeneous
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deformation. J. Inst. Met. 74, 537–562.
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Williams, B.W., Boyle, K.P., 2016. Characterization of anisotropic yield surfaces for titanium sheet using hydrostatic bulging with elliptical dies. Int. J. Mech. Sci. 114, 315–329. https://doi.org/10.1016/J.IJMECSCI.2016.05.022
Yoshida, K., 2013. Evaluation of Stress and Strain Measurement Accuracy in Hydraulic Bulge Test with the Aid of Finite-element Analysis. ISIJ Int. 53, 86–95. https://doi.org/10.2355/isijinternational.53.86
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PII:
S0924-0136(19)30546-1
DOI:
https://doi.org/10.1016/j.jmatprotec.2019.116573
Reference:
PROTEC 116573
To appear in:
Journal of Materials Processing Tech.
Received Date:
25 February 2019
Revised Date:
14 November 2019
Accepted Date:
20 December 2019
Please cite this article as: Pereira AFG, Prates PA, Oliveira MC, Fernandes JV, Inverse identification of the work hardening law from circular and elliptical bulge tests, Journal of Materials Processing Tech. (2019), doi: https://doi.org/10.1016/j.jmatprotec.2019.116573
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Inverse identification of the work hardening law from circular and elliptical bulge tests
A.F.G. Pereira* [email protected]; P.A. Prates; M.C. Oliveira; J.V. Fernandes CEMMPRE — Department of Mechanical Engineering, University of Coimbra, Rua Luís Reis
*
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Corresponding author : telephone: +351 239 790 716/00
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Santos, Pinhal de Marrocos, 3030-788 Coimbra, Portugal
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Abstract
An inverse identification strategy is proposed to characterize the hardening behaviour of metal sheets up to high strains, regardless of the material anisotropy. The
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Levenberg-Marquardt method is used to minimize the gap between the experimental and numerical, pressure vs. pole height curves, of bulge tests with circular and elliptical
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dies, by iteratively updating the work hardening and the Hill’48 parameters of the
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numerical model. The optimization of the Hill’48 parameters is used only to ensure that the yield surface is conveniently described in a region close to the stress paths that occur in the circular and elliptical bulge tests, in order to improve the identification of the hardening parameters. The strategy aims to be accurate and simple from an experimental point of view, using only the results of pressure vs. pole height. The results are compared with those of the membrane theory procedure standardized in ISO
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16808 (2014) and experimentally validated for two materials, DP600 steel and Al5754 aluminium alloy. For anisotropic materials, the proposed methodology represents a clear improvement when compared to the membrane theory procedure since it avoids the equibiaxial stress state assumption.
Keywords Hydraulic Bulge Test; Elliptical Die; Work Hardening; Inverse Analysis;
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Levenberg-Marquardt Method
1. Introduction
Sheet metal forming processes are among the most common and relevant metal
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working operations because of their high manufacture rate and low costs for high
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volume productions. Finite element methods are commonly used for simulating and optimizing sheet metal forming processes, in order to reduce the time and costs
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associated with the process development. Nevertheless, the accuracy of the numerical results depends on the appropriate characterization and modelling of the mechanical
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behaviour, using constitutive models that allow an approximate description of the experimental tests results.
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The hydraulic bulge test has become an important characterization tool, mainly due to the possibility of evaluating the response of materials under biaxial strain path
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and up to high levels of plastic deformation. For the bulge test with circular die, the determination of the biaxial stress vs. strain curve is commonly performed using the membrane theory, assuming an equibiaxial stress state, as in the procedure recently standardized in ISO 16808 (2014). This analysis requires the evaluation of the sheet thickness and the radius of curvature at the pole as well as the hydraulic pressure. Mulder et al. (2015) studied the assumptions and simplifications to be used in the 2
analysis of the circular bulge test results, through the membrane theory, and proposed some recommendations to improve the results accuracy. The authors conclude that, for anisotropic materials, the test conditions force the material, as much as possible, towards an equibiaxial strain path, leading to deviations (up to 25%) from the equibiaxial stress path, which is in agreement with the work by Yoshida (2013). Although the recommendations of Mulder et al. (2015) improved the accuracy of the membrane theory results, they did not study in detail the effect of the equibiaxial stress
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assumption. Reis et al. (2017a) studied the equibiaxial stress assumption and showed that it could lead to significant errors in the determination of the biaxial stress vs. strain curve from the circular bulge test of prominently anisotropic materials. Reis et al.
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(2017a) suggested an empirical equation to estimate the stress path based on the strain
path at the pole, which can be assessed by digital image correlation technique. The use
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of this equation, within the membrane theory, improves the accuracy of the biaxial
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stress vs. strain curves for the majority of materials. Nevertheless, there is no guarantee that all materials will closely follow the proposed equation. The bulge test with elliptical die has also been used to study the materials
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response under stress paths near the biaxial region. Several approaches can be found in literature to analyse the results of elliptical bulge tests using the membrane theory. Rees
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(1995) proposed equations to evaluate the hardening curve of the material taking into
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account the anisotropy. These equations proved to be valid only up to 20% of plastic deformation and thus limiting the analysis to that level of deformation. Lăzărescu et al. (2012) proposed an alternative analytical model that only use the pressure and pole height results. Although this model significantly simplifies the analysis of the bulge test results, some discrepancies are observed between analytical and experimental results. Based on geometrical assumptions, Ragab and Habib (1984) developed equations to
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calculate the hoop and longitudinal stresses at the pole of the bulge. Williams and Boyle (2016) show that these equations accurately predict the hoop stress, but fail regarding the longitudinal stress. Therefore, Williams and Boyle (2016) use a yield criterion, calibrated with tensile and bulge data, and the associated flow rule to calculate the longitudinal stress component based on the strains measured at the pole. Chen et al. (2016) also use the associated flow rule to evaluate the stresses at the pole, but the yield criterion is calibrated based on the results of the bulge, tensile and plane strain tests. The
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approaches of Chen et al. (2016) and Williams and Boyle (2016) to evaluate the stresses, from the strains at the pole, are valid whenever the selected yield criterion accurately describes the material behaviour in the biaxial region. However, this
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description can be compromised due to the choice of the yield criterion and the
(tensile test and plane strain tests).
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calibration of its parameters using tests with stress states outside the biaxial region
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Chamekh et al. (2006) and Reis et al. (2017b) proposed inverse analysis methodologies that bypass the assumptions and simplifications associated with the membrane theory. Chamekh et al. (2006) describe an inverse strategy, based on
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Artificial Neural Networks (ANN), to identify the material parameters of a stainless steel (AISI 304), by comparing the results of the pressure vs. pole height evolutions of
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circular and elliptical bulge tests. The authors concluded that the ANN can
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instantaneously predict accurate results, once it is trained. Despite these results, Prates et al. (2016) infers that the strategy presented substantial errors in the determination of the work-hardening. Reis et al. (2017b) proposed an alternative inverse analysis strategy, consisting of the search for the best coincidence between numerical and experimental pressure vs. pole height results of a circular bulge test. The strategy allows the accurate identification of the Swift hardening law parameters but cannot be extended
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to other types of laws, such as the Voce law. Moreover, this inverse analysis study focused on anisotropic materials with relatively weak anisotropy in the sheet plane. Bambach (2011) proposed a re-identification strategy, for the Voce law, that resorts to the membrane theory to identify an initial estimate for the hardening parameters and then uses an inverse analysis to improve the solution. This strategy was applied to an isotropic material and the results show that the initial estimate, obtained through the membrane theory, is very similar to the final solution. As a consequence, the efficiency
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of the inverse analysis was not properly evaluated, particularly with respect to convergence and accuracy in the case of anisotropic materials, where the assumption of an equibiaxial stress state in the membrane theory can lead to quite imprecise results
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(Reis et al., 2017a).
To accurately identify the parameters of the work hardening law, a new strategy
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is proposed that overcomes the limitations of the existing inverse methodologies and
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avoids the assumptions and simplifications commonly associated with the membrane theory. The strategy follows an inverse analysis methodology making use of numerical and experimental results of pressure vs. pole height, obtained from the bulge tests with
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circular and elliptical dies. The work hardening parameters are firstly optimized, by comparing the pressure vs. pole height curves of the bulge test with a circular die, under
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isotropy assumption. Then, the work hardening parameters are identified simultaneously
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with the optimisation of two Hill’48 parameters, using the pressure vs. pole height curves of bulge tests with circular and elliptical dies. The Hill’48 parameters are optimized only to ensure that the yield surface is accurately described in the biaxial region and thereby improve the identification of the hardening parameters, avoiding the assumption of equibiaxial stress state. The strategy will be initially tested using fictitious materials and then it will be applied to experimental cases.
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2. Numerical Model Numerical models of the elliptical and circular bulge tests were built in order to perform the study of the proposed identification strategy. The geometry of the selected tools was based on the bulge test apparatus described by Santos et al. (2010), which will also be used to obtain the experimental results. Figure 1 schematically shows the geometries of the bulge tools, where 𝑅𝐶 is the die profile radius, 𝑅𝐷 is the radial
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position of the drawbead, 𝑅𝑆 is the blank radius and 𝑅𝑀 is the radius of the circular die, which also corresponds to the length of the major semi-axis of the elliptical die. The
length of the minor semi-axis of the elliptical die was set to 55 mm; for smaller values,
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a pre-failure of the sheet was observed in some materials near the die curvature, as
previously reported by Rees (2000). The remaining geometrical parameters, 𝑅𝐶 , 𝑅𝐷 and
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𝑅𝑆 , are the same for the circular and elliptical dies.
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Figure 1- Representation of the bulge test, with identification of the main dimensions of the tools in agreement with the work by Santos et al. (2010). The dimensions characterize the circular die apparatus as well as the major semiaxis of the elliptical die.
In order to simplify the numerical model, only a quarter of the tools geometry
was taken into account due to the material and geometric symmetries. Additionally, the drawbead geometry was neglected and its effect was replaced by a boundary condition that restricts the radial displacement of the nodes placed in the drawbead position (𝑅𝐷 = 95mm). The discretisation of the blanks was based on a previous study of Reis et al. 6
(2017b), such that the blank plane was divided into 5 zones, each one with an appropriate number of elements, as shown in Figure 2. The circular and elliptical blanks were respectively discretised with a total number of 10584 and 11216, 8-node hexahedral solid elements, with two layers of elements through the sheet thickness of 1mm. The simulations were carried out assuming an incremental increase of the applied pressure on the inner surface of the sheet. The friction was modelled by the Coulomb’s law with a friction coefficient of 0.02, as in the work by Reis et al. (2017b). All the
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numerical simulations were performed with the in-house code DD3IMP on a computer equipped with an Intel® Core™ i7–6700K Quad-Core processor (4.4 GHz). The contact with friction is treated by an augmented Lagrangian approach and all the non-linearities
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are solved, in a single iterative loop, using a static implicit iterative Newton-Raphson scheme. More details on the numerical formulation can be found in Menezes and
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Teodosiu (2000).
(a) 2
2
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2
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20
40
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20
20
2
35
5
48
40
40
50
2
30
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35
20
5
x
20
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40
2
Oz displacement[mm]
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20
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(b) 5
y
38
5
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Figure 2 – (a) Discretisation of the blank used in the circular and elliptical bulge tests showing the number of elements along Ox and Oy for the different zones; b) Representation of the discretisation zones, showing their dimensions and the displacement along Oz, in mm, for an isotropic material.
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The constitutive model of the materials used in this work assumes: (i) the isotropic elastic behaviour defined by the generalised Hooke’s law, with a Young's modulus, 𝐸 = 210 GPa, and a Poisson's ratio, 𝜈 = 0.30; (ii) the plastic behaviour described by the orthotropic yield criterion, Hill’48 (Hill, 1948) or by the yield criterion proposed by Drucker et al. (1948) extended to orthotropy by means of a linear transformation (Cazacu and Barlat, 2001), hereafter referred to as Drucker+L; (iii) an isotropic hardening modelled by the Swift (Swift, 1952) or Voce (Voce, 1948) laws.
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The Hill’48 yield surface is described by the following equation:
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𝐹(𝜎yy − 𝜎zz ) + 𝐺(𝜎zz − 𝜎xx )2 + 𝐻(𝜎xx − 𝜎yy ) + 2𝐿𝜏yz 2 + 2𝑀𝜏xz 2 + 2𝑁𝜏xy 2 = 𝑌 2 , (1)
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where 𝜎xx , 𝜎yy , 𝜎zz , 𝜏xy , 𝜏xz and 𝜏yz are the components of the Cauchy stress tensor, 𝛔,
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defined in the orthotropic coordinate system of the sheet, such that the Ox, Oy and Oz axes are parallel to the rolling, transverse and normal directions, respectively; 𝐹, 𝐺, 𝐻,
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𝐿, 𝑀 and 𝑁 are anisotropy parameters; and 𝑌 represents the yield stress and its evolution during deformation. In the case of the Drucker+L yield criterion, the yield
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surface is described by:
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3 2 1 1 𝑌 6 [ tr((𝐋: 𝛔)2 )] − 𝑐 [ tr((𝐋: 𝛔)3 )] = 27 ( ) , 2 3 3
(2)
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where 𝑐 is a weighting isotropy parameter, ranging between −27/8 and 9/4, to ensure the convexity of the yield surface, and L is a linear transformation operator (Cazacu and Barlat, 2001), written as:
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C2 C3 / 3 C3 / 3 C2 / 3 L 0 0 0
C3 / 3
C3 C1 / 3
C2 / 3
0
0
C1 / 3
0
0
C1 / 3
C1 C2 / 3
0
0
0
0
C4
0
0
0
0
C5
0
0
0
0
0 0 0 , 0 0 C6
(3)
where, 𝐶1 , 𝐶2 , 𝐶3 , 𝐶4 , 𝐶5 and 𝐶6 are the anisotropy parameters. In both criteria, the yield stress evolution during deformation, 𝑌 = 𝑌(𝜀̅), can be described by either the Swift or
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Voce hardening laws, respectively written as:
𝑌 = 𝐾(𝜀0 + 𝜀̅)𝑛 ,
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(4)
(5)
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𝑌 = 𝑌0 + (𝑌𝑆𝑎𝑡 − 𝑌0 )(1 − 𝑒 −𝐶𝑌𝜀̅ ),
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where 𝜀̅ is the equivalent plastic strain; 𝜀0 , 𝐾, 𝑛, 𝑌0 , 𝑌𝑆𝑎𝑡 and 𝐶𝑌 are the material
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parameters, where 𝑌0 is the initial yield stress (in case of Swift law: 𝑌0 = 𝐾(𝜀0 )𝑛 ).
3. Identification Procedure
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An inverse strategy is proposed to identify the work hardening behaviour described by Swift or Voce laws, for example. The proposed strategy requires the
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pressure vs. pole height results of: (i) a circular bulge test; (ii) an elliptical bulge test with the major axis parallel to the rolling direction of the metal sheet (elliptical 0º); (iii) optionally, a third bulge test with the major axis perpendicular to the rolling direction (elliptical 90º) will also be used. The strategy consists on minimizing the gap between the experimental and numerical curves of pressure vs. pole height in bulge tests, by iteratively updating the material parameters in the finite element simulations. The 9
minimization procedure is performed using a gradient-based optimisation algorithm, the Levenberg-Marquardt method proposed by Marquardt (1963), and three distinct objective functions, 𝐹(𝐀)Circ , 𝐹(𝐀)Elli0° and 𝐹(𝐀)Elli90° , defined as follows:
𝑚
𝐹(𝐀)Circ
1 2 exp = √∑ (𝑃(ℎ𝑖 )Circ − 𝑃(ℎ𝑖 )num Circ ) , 𝑚
(6)
𝑚
𝐹(𝐀)Elli0°
1 2 exp = √∑ (𝑃(ℎ𝑖 )Elli0° − 𝑃(ℎ𝑖 )num Elli0° ) , 𝑚
𝑚
1 2 exp = √∑ (𝑃(ℎ𝑖 )Elli90° − 𝑃(ℎ𝑖 )num Elli90° ) , 𝑚
(8)
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𝐹(𝐀)Elli90°
(7)
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𝑖=1
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𝑖=1
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𝑖=1
where 𝐹(𝐀)Circ, 𝐹(𝐀)Elli0° and 𝐹(𝐀)Elli90° are respectively the objective functions
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associated with the circular, elliptical 0º and elliptical 90º tests; 𝐀 is the set of exp
parameters to be optimized; 𝑚 is the total number of points in each test; 𝑃(ℎ𝑖 )Circ , exp
exp
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𝑃(ℎ𝑖 )Elli0° and 𝑃(ℎ𝑖 )Elli90° are the experimental values of the pressure for the circular,
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num num elliptical 0º and elliptical 90º tests, respectively; 𝑃(ℎ𝑖 )num Circ , 𝑃(ℎ𝑖 )Elli0° and 𝑃(ℎ𝑖 )Elli90°
are the numerically simulated values of the pressure for the circular, elliptical 0º and elliptical 90º tests, respectively; in each test, the difference between experimental and
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numerical pressures is evaluated for the same value of pole height, ℎ𝑖 1. For each curve, 𝑚=1000 points uniformly distributed along the range of pole heights were used. The Levenberg-Marquardt method requires the knowledge of the Jacobian matrix (sensitivity matrix), which is updated after each iteration to improve the convergence of the method. Typically, the sensitivity matrix is computed using finite differences, which requires at least an additional numerical simulation per optimized parameter and test, increasing the computational cost. To overcome this inconvenience,
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the analytical approach proposed by Prates et al. (2019) was used to determine the
sensitivity matrix of the work hardening parameters. During the optimization procedure, a damping factor value is updated after each iteration, which according to Marquardt
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(1963) improves the convergence of the method.
The Levenberg-Marquardt method is used to identify the work hardening
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parameters 𝑌0 , 𝐾 and 𝑛, in the case of the Swift law, or 𝑌0 , 𝑌𝑆𝑎𝑡 and 𝐶𝑌 , in the case of
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the Voce law. Since the bulge test results are also influenced by the material anisotropy (as shown by Reis et al. (2017a)), the parameters F and G of the Hill’48 criterion are
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also optimized. In this way, the identification of the hardening parameters takes into account that the stress state can be other than equibiaxial, which occurs in anisotropic materials. The Hill’48 parameters should accurately describe the shape of the yield
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surface in a region close to the stress paths that occur in the circular and elliptical bulge
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tests, even if they do not allow describing the entire yield surface. In this sense, the optimized Hill’48 parameters should not be used outside the scope of this work. The parameters of the Hill’48 criterion follow the condition 𝐺 + 𝐻 = 1, and so the
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The assessment of the pressure values for the same pole height was accomplished by fitting a 6th degree polynomial for each pressure vs. pole height curve, as in the work by Reis et al. (2017b). The interpolation excludes the initial part of the curve, more susceptible to experimental errors, and the final part, susceptible to failure initiation and propagation, which is not numerically modelled.
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identified hardening curve is comparable to the uniaxial tensile curve along the Ox axis. The remaining parameters of the Hill’48 criterion, associated to the shear stresses, were set to 𝐿=𝑀=N =1.5 (von Mises), since the pressure vs. pole height results of the circular and the elliptical bulge tests are not sensitive to their values. According to the results of Reis et al. (2017b), the pressure vs. pole height curves of the bulge test are more sensitive to the work hardening parameters than to those of the yield criterion. For this reason and also to improve the efficiency, it is
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proposed a sequential strategy consisting of the following three stages:
1st stage – An initial estimate for the hardening parameters of the Swift or
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Voce laws and for the parameters of Hill’48 yield criterion is required, in order to start the optimization procedure. The hardening parameters can
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be obtained, for example, from the results of the tensile test, or the typical values of the material under study can be adopted. Regarding the
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anisotropy parameters, it can be assumed 𝐹 = 𝐺 = 𝐻 = 0.5 (von Mises). It was observed that the initial estimate does not influence the overall
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quality of the identification, although it affects the required number of
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iterations.
2nd stage – The Levenberg-Marquardt method is used to identify the set of work hardening parameters, 𝐀 = [𝑌0 , 𝐾, 𝑛] or 𝐀 = [𝑌0 , 𝑌𝑆𝑎𝑡 ,𝐶𝑌 ], that
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minimizes the objective function 𝐹(𝐀)Circ (equation 6). The parameters of the Hill’48 criterion [𝐹 and 𝐺] are kept constant and equal to the initial estimate. This stage ends when the relative variation of all parameters from one iteration to the next one is inferior to 5% or when the objective 1% function 𝐹(𝐀)Circ is less than 𝐹Circ , which is defined by:
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𝑚
1% 𝐹Circ
0.01 2 √∑(𝑃(ℎ𝑖 )exp = ) , Circ 𝑚
(9)
𝑖=1
In this way, the identification stops when the objective function reaches values within the range of experimental pressure errors, which according
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to the ISO 16808:2014 standard (ISO 16808, 2014) should be less than 1% of the experimental pressure value. The stopping criterion based on
the relative variation of all parameters from one iteration to the next one, avoids an infinite loop, whenever the objective function is not able to
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1% reach a value of 𝐹Circ . A value of 5% was empirically chosen to
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1% guarantee that the stopping criterion based on the value of 𝐹Circ occurs
first, when the selected hardening law conveniently describes the
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material behaviour. During this stage, the sensitivity matrix is calculated analytically (Prates et al., 2019), so that only one simulation per iteration
3rd stage – The Levenberg-Marquardt method is used to improve the
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is required, making this stage computationally efficient.
identification of the work hardening parameters obtained in the 2nd stage.
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To this end, the set of parameters of the work hardening and the Hill’48 yield criterion 𝐀 = [𝐾, 𝑌0 , 𝑛, 𝐹, 𝐺], in case of Swift law, or 𝐀 = [𝑌0, 𝑌𝑆𝑎𝑡 , 𝐶𝑌 , 𝐹, 𝐺], in case of Voce law, is optimized (with 𝐻 following the condition 𝐺 + 𝐻 = 1). This is achieved by minimizing the gap between experimental and numerical curves of pressure vs. pole height in circular and elliptical, at 0º, tests or in circular and two elliptical, at 0º and 90º, 13
tests. In the first case, the sum of the objective functions 𝐹(𝐀)Circ and 𝐹(𝐀)Elli0° is minimized, while in the second case the minimization focuses on the sum of the functions, 𝐹(𝐀)Circ, 𝐹(𝐀)Elli0° and 𝐹(𝐀)Elli90° . The identification stops when the relative variation of all parameters from one iteration to the next one is inferior to 5% or when the objective functions 𝐹(𝐀)Circ , 𝐹(𝐀)Elli0° and, in the case of using a third test, 1% 1% 1% 𝐹(𝐀)Elli90° are respectively inferior to 𝐹Circ , 𝐹Elli0° and 𝐹Elli90° . The
context of the elliptical test, at 0º and 90º:
𝑚
0.01 2 √∑(𝑃(ℎ𝑖 )exp = ) , Elli0° 𝑚
-p
1% 𝐹Elli0°
ro of
1% 1% 1% amounts 𝐹Elli0° and 𝐹Elli90° have the same meaning as 𝐹Circ , but in the
lP
re
𝑖=1
(10)
𝑚
1% 𝐹Elli90°
0.01 2 √∑(𝑃(ℎ𝑖 )exp = ) , Elli90° 𝑚
(11)
na
𝑖=1
ur
During this optimization stage, the sensitivity matrix of the work hardening parameters is calculated analytically (Prates et al., 2019),
Jo
while the sensitivity of the Hill’48 yield criterion parameters, F and G, is computed using forward finite differences, so that each iteration requires 6 or 9 simulations to compute the sensitivity matrix, in case of using two or three tests, respectively, making each iteration computationally more expensive than in the 2nd stage. Nevertheless, the computational cost is
14
significantly mitigated in this stage, since the estimate at the end of the 2nd stage is already close to the final. During the identification procedure, the Hill’48 parameters fulfil the condition 𝐺 + 𝐻 = 1, i.e. the hardening curve is comparable to the uniaxial tensile curve along the Ox axis. In order to obtain the hardening curve comparable to the biaxial stress vs. strain curve, obtained for the stress path observed at the pole of the circular bulge test, the material parameters must be converted using the following equations proposed by
𝑛+1
∗ ; 𝑛∗ = 𝑛; 𝑌0 ∗ = 𝑌0 √𝑄; 𝑌𝑆𝑎𝑡 = 𝑌𝑆𝑎𝑡 √𝑄; 𝐶𝑌∗ = 𝐶𝑌 √𝑄,
-p
𝐾 ∗ = 𝐾(√𝑄)
ro of
Prates et al. (2015):
(13)
re
𝐹 ∗ = 𝑄 𝐹 ; 𝐺 ∗ = 𝑄𝐺; 𝐻 ∗ = 𝑄𝐻; 𝐿∗ = 𝑄𝐿; 𝑀∗ = 𝑄𝑀; 𝑁 ∗ = 𝑄𝑁,
(12)
lP
where 𝐾, 𝑛, 𝑌0 , 𝑌𝑆𝑎𝑡 , 𝐶𝑌 , 𝐹, 𝐺, 𝐻, 𝐿, 𝑀 and 𝑁 are the set of material parameters in the ∗ condition 𝐺 + 𝐻 = 1; 𝐾 ∗ , 𝑛∗ , 𝑌0 ∗ , 𝑌𝑆𝑎𝑡 𝐶𝑌∗ , 𝐹 ∗ , 𝐺 ∗ , 𝐻 ∗ , 𝐿∗ , 𝑀∗ and 𝑁 ∗ are the equivalent
na
set of material parameters determined for the stress path at the pole of the circular bulge test, for which the condition 𝐺 + 𝐻 = 1 is not fulfilled. 𝑄 is the conversion factor given
ur
by Reis et al. (2017a):
2
Jo
𝑄=
(𝜎yy ⁄𝜎xx ) − (𝜎yy ⁄𝜎xx ) + 1 2
(𝐹 + 𝐻)(𝜎yy ⁄𝜎xx ) − 2𝐻(𝜎yy ⁄𝜎xx ) + (𝐺 + 𝐻)
,
(14)
where 𝜎yy ⁄𝜎xx is the average stress path evaluated at the pole of the circular bulge test from the numerical simulation performed considering the identified parameters. All parameters presented in the following sections refer to the stress path at the pole of the 15
circular bulge test, but for convenience of the presentation the asterisks are omitted. In this way, the parameters no longer respect the condition 𝐺 + 𝐻 = 1, but another condition that depends on the stress path, at the pole of the circular bulge test, which varies with the anisotropic behaviour.
4. Parameters Identification To illustrate the previously described inverse identification strategy, several
ro of
cases are considered in this section. In each case, computer generated results of fictitious materials are used as “experimental” results. This allows the precise
knowledge of their mechanical behaviour, and so the suitable comparison between the
-p
“experimental” and identified parameters. Fictitious materials whose anisotropic
re
behaviour is described by von Mises, Hill’48 (as in the identification strategy) and Drucker+L yield criteria are considered as illustrative cases; their work hardening
lP
behaviour is described by the Swift and Voce laws. The aim is to examine the influence of the anisotropy and hardening behaviours on the results of the identification
na
procedure.
Table 1 shows the constitutive parameters of two Swift materials, one isotropic
ur
(von Mises: S_H Iso) and the other anisotropic (Hill'48: S_H Ani). Table 2 is identical to Table 1 but for two Voce materials: von Mises - V_H Iso and Hill’48 - V_H Ani. The
Jo
initial set of parameters used in the identifications is also shown (S_Ini and V_Ini). Figure 3(a) shows the work hardening curves of the materials in Tables 1 and 2. The choice of the materials was focused on isotropic materials with quite close hardening laws, Swift and Voce; the same is true for the anisotropic materials. Figure 3(b) shows the normalized yield surfaces in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ). The stress paths of the
16
circular and elliptical, at 0º and 90º, tests are also shown in this figure, revealing that they are very similar for all tests, in each material.
Table 1 - Parameters of two fictitious materials described by the Swift hardening law and the Hill’48 criterion: S_H Iso (isotropic material) and S_H Ani (anisotropic material). The initial set of parameters used in both identifications (S_Ini) is also shown. 𝜎yy ⁄𝜎xx is the respective average stress path observed at the pole of the circular bulge test.
Designation
𝑌0 [MPa] 𝐾 [MPa]
S_H Iso
200.00
S_H Ani S_Ini
𝐹
𝐺
𝐻
𝑁
1277.56
0.350
0.500
0.500
0.500
1.500
1.500
1.000
282.27
2034.23
0.350
0.165
1.329
0.663
1.509
4.110
1.633
100.00
200.00
0.100
0.500
0.500
0.500
ro of
𝐿=𝑀
𝜎𝑦𝑦 ⁄𝜎xx
𝑛
1.500
1.500
1.000
Designation
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa]
V_H Iso
305.30
1077.58
4.12
0.500
V_H Ani
431.43
1522.79
5.82
0.166
V_Ini
100.00
200.00
𝐻
0.500
0.500
𝐿=𝑀
𝑁
𝜎yy ⁄𝜎xx
1.500
1.500
1.000
1.335
0.667
1.502
4.120
1.646
0.50
0.500
0.500
1.500
1.500
1.000
Jo
ur
na
2.00
𝐺
re
𝐹
lP
𝐶𝑌
-p
Table 2 - Parameters of two fictitious materials described by the Voce hardening law and the Hill’48 criterion: V_H Iso (isotropic material) and V_H Ani (anisotropic material). The initial set of parameters used in both identifications (V_Ini) is also shown. 𝜎yy ⁄𝜎xx is the respective average stress path observed at the pole of the circular bulge test.
17
S_H Iso
S_H Ani
(a)
S_H Iso
S_H Ani
V_H Iso
V_H Ani
(b) V_H Iso
V_H Ani
1.5
1600 1400 1200
1.0 𝑌0
600
𝜎yy
Pa
800
0.5 𝑃
𝑌
Pa
1000
400
0.0
0 0
0.1
0.2
0.3
𝜀̅
0.4
0.5
0.6
0.0
ro of
200 0.5
𝜎xx
1.0
1.5
𝑌0
-p
Figure 3 - Fictitious materials described by the Hill’48 yield criterion (Tables 1 and 2): (a) work hardening; (b) yield surface in the normalized plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) and the stress paths of the circular and elliptical tests; the central stress paths refers to the circular tests and those immediately adjacent refers to the elliptical tests.
Figure 4 represents the range of pressure vs. pole height curves taken into
re
account in the identification, i.e. neglecting the initial and final regions of the curves, for
lP
the circular and elliptical bulge tests of the fictitious Swift (Figure 4(a)) and Voce (Figure 4(b)) materials (see Tables 1 and 2, respectively). The level of pressure vs. pole
na
height curves is higher for the elliptical than for the circular bulge tests; in case of isotropic materials the curves of the elliptical tests are equal at 0º and 90º. The circular
ur
pressure vs. pole height curves of the materials used as initial estimates (see Tables 1 and 2) are also shown in these figures; initial estimates far enough from the solution
Jo
were chosen to test the proposed strategy.
18
S_H Iso_Circ S_H Iso_Elli0º S_H Iso_Elli90º S_Ini_Circ
V_H Iso_Circ V_H Iso_Elli0º V_H Iso_Elli90º V_ Ini_Circ
28
24
24
20
20
16
Pa
16
12
𝑃
12
Pa
28
Pa
(b)
𝑃
(a)
S_H Ani_Circ S_H Ani_Elli0º S_H Ani_Elli90º
𝑃
8
V_H Ani_Circ V_H Ani_Elli0º V_H Ani_Elli90º
8
0
0 0
10
20
30
ℎ mm
40
50
60
0
10
ro of
4
4
20
ℎ mm
30
40
50
-p
Figure 4 - Circular and elliptical (0° and 90°) pressure vs. pole height curves of the fictitious materials described by (a) Swift (Table 1); (b) Voce (Table 2). The pressure vs. pole height curves of the circular test, corresponding to the initial estimates, S_Ini (Table 1) and V_Ini (Table 2), are also represented (blue lines).
The work hardening behaviour was identified by the inverse procedure described
re
in the previous section and compared with the hardening of the reference fictitious
lP
materials in Tables 1 and 2. The identification procedure was performed using: (i) the circular bulge and the elliptical bulge at 0º, the results of which are hereinafter designated by the abbreviation “2Tests”; (ii) the circular bulge and the two elliptical
na
bulge tests at 0º and 90º, whose results will be designated by the abbreviation “3Tests”. The membrane theory was also used to determine the biaxial stress vs. strain results
ur
from the circular bulge test, following the recommendations of the ISO 16808:2014
Jo
standard, i.e. in the same way that is usually done experimentally. The numerical evaluation of the pressure, radius of curvature and principal strains at the pole of the bulge test are given in the work by Reis et al. (2017a). In detail, the numerical evaluation of the radius of curvature is performed by fitting the equation of a circle to the nodes located at a distance lower than 15 mm from the bulge pole (at half thickness). The radius of curvature was analysed for both Oxz and Oyz planes (see axis
19
in Figure 1), and the principal strains were evaluated at the Gauss point located closer to the bulge pole. The biaxial stress vs. strain results evaluated by the membrane theory procedure (ISO 16808:2014) were used to fit the hardening laws, which parameters are referred to as “ em” in Tables 3 and 4.
Table 3 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “ em”, for the fictitious materials, “S_H Iso” and “S_H Ani”, described by the Hill’48 criterion and the Swift law; the parameters of these fictitious materials are also included for comparison. The stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the corresponding numerical simulation is also indicated.
𝑌0 [MPa] 𝐾 [MPa]
S_H Iso
200.00
S_H Iso|2Tests
𝐻
1277.56
0.350
0.500
0.500
0.500
1.500
1.500
1.000
203.95
1276.40
0.349
0.500
0.500
0.500
1.500
1.500
1.000
S_H Iso|Mem
186.66
1301.01
0.360
-
-
-
-
-
-
S_H Ani
282.27
2034.23
0.350
0.165
1.329
0.663
1.509
4.110
1.633
S_H Ani|2Tests 285.42
2032.09
0.351
0.112
1.419
0.782
3.301
3.301
1.630
S_H Ani|3Tests 290.67
2056.12
0.370
0.277
1.113
0.391
2.257
2.257
1.605
S_H Ani|Mem
1836.16
-
-
-
-
-
-
-p
Jo
ur
na
0.354
𝑁
ro of
𝐺
lP
𝐹
217.38
𝐿=𝑀
𝜎𝑦𝑦 ⁄𝜎xx
𝑛
re
Designation
20
Table 4 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “ em”, for the fictitious materials, “V_H Iso” and “V_H Ani”, described by the Hill’48 criterion and the Voce law; the parameters of these fictitious materials are also included for comparison. The stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the corresponding numerical simulation is also indicated.
𝐺
𝐻
4.12
0.500
0.500
0.500
1.500
1.500
1.000
1064.36
4.30
0.500
0.500
0.500
1.500
1.500
1.000
301.48
1114.44
3.86
-
-
-
-
431.43
1522.79
5.82
0.166
1.335
0.667
1.502
4.120
1.646
V_H Ani|2Tests 425.35
1556.38
5.02
0.288
1.035
0.391
2.138
2.138
1.542
V_H Ani|3Tests 429.82
1583.67
5.04
0.246
1.185
0.465
2.475
2.475
1.626
V_H Ani|Mem
1442.59
5.01
-
-
-
-
V_H Iso
305.30
1077.58
V_H Iso|2Tests 306.92 V_H Iso|Mem V_H Ani
𝑁
-
-
-p
388.51
𝐶𝑌
ro of
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa]
𝐿=𝑀
𝜎yy ⁄𝜎xx
𝐹
Designation
The parameters identified with these methodologies are given in Tables 3 and 4
re
for the Swift and Voce materials, respectively. The identification of the isotropic
lP
materials used the circular test and only one elliptical test, since the results of the experimental pressure vs. pole height curves are the same for both cases, at 0 and 90°. In addition, it was not necessary to perform all the optimization planned in the 3rd stage
na
since, at the beginning of this stage, the set of work hardening and yield criterion parameters already fulfils the stopping criteria.
ur
Figure 5 shows the differences between the hardening behaviours corresponding
Jo
to the parameters identified by inverse analysis and by the membrane theory (Tables 3 and 4) and the reference fictitious parameters (Table 1 and 2). Based on these results, Figure 6 plots the absolute value of the relative difference in stress, |𝑅𝑑 | = |(𝑌𝑟𝑒𝑓 − 𝑌𝑖𝑑𝑒 )⁄𝑌𝑖𝑑𝑒 |, between the reference material, 𝑌𝑟𝑒𝑓 , and the identified curves, 𝑌𝑖𝑑𝑒 , by inverse analysis and by using the membrane theory. The proposed strategy, using two or three tests, allows the accurate identification of the work hardening
21
parameters, regardless of the anisotropy behaviour and type of work hardening law. The worst inverse identification (see case “V_H Ani|2Tests”) has a maximum error of 4.6 % with an average value of 3.2%. In contrast, the membrane theory only provides an accurate identification for the isotropic materials, with average errors lower than 0.9%. In the case of the anisotropic materials, the membrane theory supplies poor identifications with average errors close to 10.7% for Voce and Swift laws, which is in agreement with the results by Reis et al. (2017a). In this context, the proposed strategy
ro of
seems to be more appropriate than the membrane theory to identify the work hardening behaviour of anisotropic materials. Moreover, the yield criterion (Hill’48) parameters
evaluated with the proposed strategy only accurately describe the yield surface close to
(a)
S_H Iso S_H Iso|Mem S_H Iso|2Tests
1600
lP
800 600 400
Jo
ur
600
Pa
𝑌
800
𝑌
na
Pa
1000
1000
0
(b)
1200
1200
200
V_H Iso V_H Iso|Mem V_H Iso|2Tests
1400
1400
400
V_H Ani V_H Ani|Mem V_H Ani|2Tests V_H Ani|3Tests
re
S_H Ani S_H Ani|Mem S_H Ani|2Tests S_H Ani|3Tests
-p
the stress paths of the bulge tests, as shown in Figure 7.
0
0.1
0.2
0.3
𝜀̅
200 0 0.4
0.5
0.6
0
0.1
0.2
𝜀̅
0.3
0.4
0.5
Figure 5 - Hardening behaviour corresponding to the parameters identified (Tables 3 and 4) and to the reference fictitious parameters (Table 1 and 2), for the materials described by: (a) Swift law; (b) Voce law.
22
S_H Ani|Mem
(a)
S_H Iso|Mem
S_H Ani|2Tests
(b)
S_H Iso|2Tests
S_H Ani|3Tests
14
10
10
8
8
V_H Iso|2Tests
𝑑
6
6
4
4
2
2 0
0 0
0.1
0.2
0.3
𝜀̅
0.4
0.5
0
0.6
0.1
ro of
𝑌
Pa
𝑑
12
V_H Iso|Mem
V_H Ani|2Tests V_H Ani|3Tests
14
12
V_H Ani|Mem
0.2
𝜀̅
0.3
0.4
0.5
(a)
S_H Ani
S_H Iso
S_H Ani|2Tests
S_H Iso|2Tests
V_H Iso
V_H Ani|2Tests
V_H Iso|2Tests
V_H Ani|3Tests
1.5
lP
1.5
1.0
ur
0.5
0.5 𝜎 xx
𝑌0
0.5
0.0 1.0
1.5
0.0
0.5 𝜎 xx
1.0
1.5
𝑌0
Jo
0.0
𝜎yy
na
𝜎yy
𝑌0
𝑌0
1.0
0.0
V_H Ani
re
S_H Ani|3Tests
(b)
-p
Figure 6 - Relative difference in hardening behaviour when comparing the materials corresponding to the parameters identified (Tables 3 and 4) and to the reference fictitious parameters (Tables 1 and 2). Materials described by: (a) Swift law; (b) Voce law.
Figure 7 – Normalized yield surface in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) corresponding to the parameters identified (Tables 3 and 4) for the fictitious materials (Table 1 and 2). (a) Swift law; (b) Voce law. The stress paths of the circular bulge tests are also represented.
The Drucker+L yield criterion is now used to describe the behaviour of the fictitious materials, in order to illustrate cases where the material behaviour is not described by the Hill'48 criterion, which is used in the identification. This seeks to 23
represent the common case where the yield criterion that better describes the experimental material behaviour is unknown. Table 5 shows the constitutive parameters of two Swift materials, one with a flattened yield surface (Drucker+L: S_D c=2) in the region near the stress path observed in the bulge tests and the other with a sharpened yield surface (Drucker+L: S_D c=-2) in this region. Table 6 is identical to Table 5 but for two Voce materials: V_D c=2 (flattened yield surface) and V_D c=-2 (sharpened yield surface). Figure 8 shows the work hardening curves (Figure 8(a)) and the
ro of
normalized yield surfaces in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) (Figure 8(b)) of the materials in Tables 5 and 6. The stress paths of the circular and the elliptical tests at 0º and 90º are
also shown in Figure 8(b). Figure 9 represents the range of the pressure vs. pole height
-p
curves for the circular and elliptical tests of the reference fictitious materials (Tables 5
and 6) taken into account in the identification procedure; the curves of the circular bulge
re
test of the materials used as initial estimate, S_Ini (see Table 1) and V_Ini (see Table 2)
lP
are also shown.
na
Table 5 - Parameters of two fictitious materials described by the Swift hardening law and the Drucker+L criterion: S_D c=2 (flattened yield surface) and S_D c=-2 (sharpened yield surface). 𝜎yy ⁄𝜎xx is the respective average stress path observed at the pole of the circular bulge test.
𝑌0 [MPa]
𝐾 [MPa]
S_D c = 2
257.53
781.60
ur
Designation
277.08
853.34
𝐶1
𝐶2
𝐶3
𝐶4 = 𝐶5
𝐶6
𝑐
𝜎yy ⁄𝜎xx
0.200 0.754 1.554 1.155 1.359
1.458 2.000 1.374
0.200 0.738 1.521 1.129 1.330
1.426 -2.000 1.671
Jo
S_D c = -2
𝑛
24
Table 6 - Parameters of two fictitious materials described by the Voce hardening law and the Drucker+L criterion: V_D c=2 (flattened yield surface) and V_D c=-2 (sharpened yield surface). 𝜎yy ⁄𝜎xx is the respective average stress path observed at the pole of the circular bulge test.
𝜎yy ⁄𝜎xx
Designation
𝑌0 [MPa]
𝑌𝑆𝑎𝑡 [MPa]
V_D c = 2
310.86
661.68
7.05
0.755 1.557 1.156 1.289
1.459 2.000 1.378
V_D c = -2
333.37
709.61
7.56
0.737 1.518 1.127 1.383
1.423 -2.000 1.664
S_D c = 2
𝐶𝑌
𝐶1
𝐶2
𝐶3
S_D c = -2
(a)
𝐶4 = 𝐶5
𝐶6
𝑐
S_D c = 2
S_D c = -2
V_D c = 2
V_D c = -2
(b) V_D c = 2
V_D c = -2
ro of
1.5
800 700 600
1.0
300 200 100
re
𝑃
0.5
-p
𝑌
Pa
400
𝜎yy
𝑌0
Pa
500
0.0
0 0.1
0.2
𝜀̅
0.3
0.4
0.5
lP
0
0.0
0.5
𝜎xx
1.0
1.5
𝑌0
Jo
ur
na
Figure 8 - Fictitious materials described by the Drucker+L yield criterion (Tables 5 and 6): (a) work hardening; (b) yield surface in the normalized plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) and the stress paths of the circular and elliptical tests.
25
S_D c = 2_Circ S_D c = 2_Elli0º S_D c = 2_Elli90º S_Ini_Circ
V_D c = 2_Circ V_D c = 2_Elli0º V_D c = 2_Elli90º V_ Ini_Circ
(b) 12
10
10
8
8
Pa
6
V_D c = -2_Circ V_D c = -2_Elli0º V_D c = -2_Elli90º
6
𝑃
Pa
12
𝑃
Pa
(a)
S_D c = -2_Circ S_D c = -2_Elli0º S_D c = -2_Elli90º
4 𝑃
4
0
0 0
10
20
30
ℎ mm
40
50
0
ro of
2
2
10
20
ℎ mm
30
40
50
-p
Figure 9 - Circular and elliptical (0° and 90°) pressure vs. pole height curves of the fictitious materials described by (a) Swift (Table 5); (b) Voce (Table 6). The pressure vs. pole height curves of the circular test, corresponding to the initial estimates, S_Ini (Table 1) and V_Ini (Table 2), are also represented (blue lines).
The results of the inverse identification and membrane theory of the reference
re
fictitious materials in Table 5 and 6 are shown in Tables 7 and 8 and Figures 10, 11, 12
lP
and 13. Also in these cases, the proposed strategy, using two or three tests, provided accurate identifications of the work hardening parameters, although in general the
na
identifications with three tests have slightly higher quality. For instance, the worst identification using two tests (V_D c=-2|2Tests) has an average error of 3.1% (see Figure 13), while the worst identification with three tests (V_D c= -2|3Tests) has an
ur
average error of 1.1% (see Figure 13). The best identification with three tests arises
Jo
from a good description of the yield surface near the bulge stress paths, as can be seen in Figures 10 and 11. In fact, the stress paths of the circular test corresponding to the identified yield criterion parameters are more accurate when using three tests. Also for these materials, the procedure based on the membrane theory (ISO 16808:2014) does not provide accurate results for the work hardening behaviour. This is due to the strong anisotropy of these materials, for which the stress paths of the circular bulge test are
26
distant from the equibiaxial, particularly in the case of the fictitious materials “S_D c= 2” and “V_D c= -2” with stress paths of 1.671 and 1.664, respectively (see Tables 5 and 6). In these cases, the membrane theory provides poor identifications with average errors of about 10%. For the cases of the reference fictitious materials, “S_D c= 2” and “V_D c= 2”, with stress paths of 1.374 and 1.378 (see Tables 5 and 6), the identification with the membrane theory is better, with average errors of about 4%.
𝑌0 [MPa] 𝐾 [MPa]
S_D c = 2
257.53
781.60
𝑛
𝐶1
𝐶2
𝐶3
𝐶4 = 𝐶5
0.200 0.754 1.554 1.155 1.359 𝐹
𝐺
𝐻
𝜎yy ⁄𝜎xx
𝑐
𝐶6
1.458 2.000 1.374
-p
Designation
ro of
Table 7 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “ em”, for the fictitious materials, “S_D c = 2” and “S_D c = -2”, described by the Drucker+L criterion and the Swift law; the parameters of these fictitious materials are also included for comparison. The stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the corresponding numerical simulation is also indicated.
𝐿
𝑀
𝜎yy ⁄𝜎xx
𝑁
793.98
0.219 0.369 0.731 0.349 1.620
1.620 1.620 1.290
S_D c = 2|3Tests 257.40
783.33
0.202 0.113 1.146 1.148 3.440
3.440 3.440 1.383
S_D c = 2|Mem
231.80
759.07
Designation
𝑌0 [MPa] 𝐾 [MPa]
S_D c = -2
277.08
lP
re
S_D c = 2|2Tests 257.01
0.205 -
na
𝑛
ur
853.34
-
𝐶1
-
𝐶2
𝐶3
𝐶4 = 𝐶5
0.200 0.738 1.521 1.129 1.330 𝐹
𝐺
𝐻
-
𝜎yy ⁄𝜎xx
𝑐
𝐶6
1.426 -2.000 1.671
𝐿
𝑀
𝜎yy ⁄𝜎xx
𝑁
849.53
0.206 0.282 1.097 0.420 2.275
2.275 2.275 1.614
S_D c = -2|3Tests 276.95
854.79
0.206 0.283 1.136 0.405 2.311
2.311 2.311 1.656
773.23
0.203 -
-
Jo
S_D c = -2|2Tests 280.69
S_D c = -2|Mem 229.63
-
-
-
-
-
27
Table 8 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “ em”, for the fictitious materials, “V_D c = 2” and “V_D c = -2”, described by the Drucker+L criterion and the Voce law; the parameters of these fictitious materials are also included for comparison. The stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the corresponding numerical simulation is also indicated.
Designation
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa] 𝐶𝑌
V_D c = 2
310.86
661.68
𝐶1
𝐶2
𝐶3
𝐶4 = 𝐶5
7.05 0.755 1.557 1.156 1.289 𝐹
𝐺
𝐻
𝜎yy ⁄𝜎xx
𝑐
𝐶6
1.459 2.000 1.378
𝐿
𝑀
𝜎yy ⁄𝜎xx
𝑁
V_D c = 2|2Tests 308.40
715.63
5.06 0.369 0.736 0.330 1.599
1.599 1.599 1.298
V_D c = 2|3Tests 315.83
690.63
6.16 0.164 1.074 1.017 3.138
3.138 3.138 1.398
V_D c = 2|Mem 297.13
652.51
6.39 -
-
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa] 𝐶𝑌
V_D c = -2
333.37
𝐶2
𝐶3
𝐶4 = 𝐶5
7.56 0.737 1.518 1.127 1.383 𝐺
𝐻
-
𝜎yy ⁄𝜎xx
𝑐
𝐶6
1.423 -2.000 1.664
𝐿
𝑀
𝜎yy ⁄𝜎xx
𝑁
V_D c = -2|2Tests326.44
719.94
6.21 0.352 0.909 0.260 1.754
1.754 1.754 1.524
V_D c = -2|3Tests331.99
715.60
6.91 0.269 1.143 0.438 2.372
2.372 2.372 1.636
V_D c = -2|Mem 304.67
660.39
6.49 -
re
𝐹
-
-p
709.61
𝐶1
-
ro of
Designation
-
-
-
-
-
-
Jo
ur
na
lP
-
28
S_D c = 2
S_D c = 2|2Tests
V_D c = 2
(a)
V_D c = 2|2Tests
(b) S_D c = 2|3Tests
V_D c = 2|3Tests
1.5
1.0
1.0
𝜎yy
𝜎yy
𝑌0
𝑌0
1.5
0.5
0.5
0.0
0.5
𝜎xx
1.0
ro of
0.0
0.0
0.0
1.5
0.5 𝜎 xx
𝑌0
1.0
1.5
𝑌0
S_D c = -2
S_D c = -2|2Tests
(a)
V_D c = -2
V_D c = -2|2Tests
re
(b)
S_D c = -2|3Tests
V_D c = -2|3Tests
1.5
lP
1.5
1.0
ur
0.5
Jo
0.0
0.5
𝜎xx
𝑌0
𝜎yy
𝜎yy
na
𝑌0
𝑌0
1.0
0.0
-p
Figure 10 – Normalized yield surface in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) corresponding to the parameters identified (Tables 7 and 8) for the fictitious materials of Tables 5 and 6: (a) “S_D c = 2”; (b) “V_D c = 2”. The stress paths of the circular bulge tests are also represented.
0.5
0.0 1.0
1.5
0.0
0.5 𝜎 xx
1.0
1.5
𝑌0
Figure 11 – Normalized yield surface in the plane (𝜎xx ⁄𝑌0 ; 𝜎yy ⁄𝑌0 ) corresponding to the parameters identified (Tables 7 and 8) for the fictitious materials of Tables 5 and 6: (a) “S_D c = -2”; (b) “V_D c = -2”. The stress paths of the circular bulge tests are also represented.
29
S_D c = 2 S_D c = 2|Mem S_D c = 2|2Tests S_D c = 2|3Tests
(a) 800
S_D c = -2 S_D c = -2|Mem S_D c = -2|2Tests S_D c = -2|3Tests
V_D c V_D c V_D c V_D c
(b) 800 700
600
600
500
500
V_D c V_D c V_D c V_D c
= -2 = -2|Mem = -2|2Tests = -2|3Tests
Pa
Pa
700
=2 = 2|Mem = 2|2Tests = 2|3Tests
400
𝑌
𝑌
400
300
200
200
100
100
𝑌
Pa
300
0
0.1
0.2
𝜀̅
0.3
0.4
ro of
0
0
0
0.5
0.1
0.2
𝜀̅
0.3
0.4
(a)
S_D c = -2|Mem
S_D c = 2|2Tests
S_D c = -2|2Tests
S_D c = 2|3Tests
S_D c = -2|3Tests
V_D c = 2|2Tests
V_D c = -2|2Tests
V_D c = 2|3Tests
V_D c = -2|3Tests
10
lP
10
8
𝑑
8 6
na
𝑑
14
V_D c = -2|Mem
12
12
4 2 0
ur
𝑌
Pa
(b)
V_D c = 2|Mem
re
14
S_D c = 2|Mem
-p
Figure 12 – Hardening behaviour corresponding to the parameters identified (Tables 7 and 8) and to the reference fictitious parameters (Tables 5 and 6), for materials described by: (a) Swift law; (b) Voce law.
0
0.1
0.2
𝜀̅
0.3
0.4
6 4 2 0
0.5
0
0.1
0.2
𝜀̅
0.3
0.4
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Figure 13– Relative difference in hardening behaviour when comparing the materials corresponding to the parameters identified (Tables 7 and 8) for and to the reference fictitious parameters (Tables 5 and 6). Materials described by: (a) Swift laws; (b) Voce laws.
In summary, the proposed inverse analysis strategy allowed to accurately identify the parameters of the Swift and Voce hardening laws, regardless of the material anisotropy. The Hill’48 yield criterion proved to be flexible enough to describe the yield surface of materials in the region close to the stress paths that occur in the circular and 30
elliptical bulge tests, even if the behaviour of the materials is described by a different yield criterion (Drucker+L for the illustrative cases). The description of the entire yield surface is nonessential, as the pressure vs. pole height curves depend mainly on the most deformed zones of the bulge (i.e., between the pole and the die), for which the stress path is similar to that at the pole, as already shown by Reis et al. (2017a). The inverse analysis procedure requires the use of at least two bulge tests, a circular and an elliptical. Nevertheless, the evaluation of the work hardening parameters is usually more
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accurate when a circular and two elliptical bulge tests, perpendicular to each other, are used. In general, this also improves the assessment of the stress path in the circular tests. The proposed strategy shows to be more accurate than the membrane theory, mainly
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when the materials behaviour is strongly anisotropic, since the equibiaxial stress state is not imposed.
re
When the optimization in the 3rd stage is continued further than the conditions
lP
assumed by the adopted stopping criteria, the accuracy of the proposed strategy can be improved, regardless of the yield criterion that describes the material behaviour. This reinforces the idea that the strategy efficiency is not dependent on the anisotropy of the
na
material and depends only on the accurateness of the numerical model and on the experimental determination of the pressure vs. pole height curves. From the
ur
experimental point of view, the proposed strategy is not so much exposed to
Jo
experimental errors as the membrane theory procedure (ISO 16808:2014), since the experimental evaluation of the sheet thickness and radius of curvature at the pole during the tests is not required. Additionally, the proposed methodology does not require the use of digital image correlation technique, which according to Koç et al. (2011) can raise accuracy issues at high temperatures.
31
In average, the Levenberg-Marquardt algorithm required 7 iterations in the 2nd stage and 4 iterations in the 3rd stage, regardless of using two or three bulge tests. Each numerical simulation of the circular and elliptical tests was respectively completed in 14.15 and 15.40 min, using the blank sheet discretisations shown in Figure 2. Hence, the average computational time of each identification was 7.56 hours, when using two bulge tests, and 10.64 hours for three bulge tests. Due to the high amount of computational time required in the identification, it is suggested in Appendix A an
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optimized discretisation of the blank sheet that significantly reduces the computational time (about ten times), but preserves the accurate prediction of the pressure vs. pole
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5. Experimental Cases
-p
height curves and so the results of the identification procedure.
The proposed strategy is now applied to the cases of two materials, a steel
lP
(DP600) and an aluminium alloy (Al5754) with initial sheet thicknesses of 0.8 and 1 mm, respectively. Typical values for the elastic parameters were assumed: Young's
na
modulus, 𝐸 = 210 GPa, and Poisson's ratio, 𝜈 = 0.30 in case of DP600 steel; and 𝐸 = 70 GPa, and 𝜈 = 0.30, in case of Al5754 aluminium alloy. The hydraulic bulge test
ur
apparatus developed by Santos et al. (2010), whose geometry and dimensions are shown in Figure 1, was used to obtain the experimental pressure vs. pole height results,
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necessary to the identification procedure. In order to ensure that the boundary conditions of the numerical and experimental tests are as close as possible, it has been ensured that, in the latter case, the clamping force and the drawbead do not allow the occurrence of any draw in, similarly to the numerical model. Figure 14 shows the parts of the experimental pressure vs. pole height curves taken into account in the identifications. In this figure are also represented the circular pressure vs. pole height 32
curves of the materials used as initial estimates, S_Ini (the same as in Table 1) and V_Ini (the same as in Table 2). Figure 14 shows that the pressure vs. pole height curves of the elliptical tests at 0º and 90º are identical. Therefore, equibiaxial stress paths should occur in the circular bulge test and, consequently, it is expected that the membrane theory provides accurate results (see Figure 6), which can be used to test the proposed inverse strategy.
DP600_Elli0º
Al5754_Circ
Al5754_Elli0º
Al5754_Elli90º
V_ Ini_Circ
ro of
DP600_Circ
(a)
(b) DP600_Elli90º
S_Ini_Circ
12
5
Pa
Pa
3 2
𝑃
re
𝑃
𝑃
Pa
6
-p
4
9
3
0 0
10
20
ℎ mm
lP
1
30
40
50
0 0
10
20
ℎ mm
30
40
50
ur
na
Figure 14 - Circular and elliptical (0° and 90°) pressure vs. pole height curves of the materials: (a) DP600; (b) Al5754. The circular pressure vs. pole height curves corresponding to the initial estimates used in the identification, S_Ini (the same as in Table 1) and V_Ini (the same as in Table 2), are also represented (blue lines).
The work hardening parameters of the aluminium alloy (Al5754) and steel
Jo
(DP600) were identified using the proposed strategy with two or three tests. At the end of the identification procedure, the experimental and numerical pressure vs. pole height curves are identical, with an average difference in pressure inferior to 0.0617 MPa, for the DP600 steel, and 0.0289 MPa, for the Al5754 aluminium. The membrane theory was also used for determining the biaxial stress vs. strain curves. Tables 9 and 10 show the results obtained when the Swift and Voce law are used for describing the work 33
hardening, respectively. These tables also show the Swift and Voce laws fitted to the membrane theory results, designated by “Mem”. Note that in Tables 9 and 10 the parameter 𝐻 is negative in three cases. Although this does not make sense from the experimental point of view, Hill (1948) stated that it is mathematically admissible that one and only one of the parameters F, G, H is negative. In fact, the Hill’48 criterion is used in this work only as a mathematical function to describe the shape of the yield surface in a region close to the stress paths that occur in the circular and elliptical bulge
ro of
tests, regardless of the physical meaning of its parameters.
Designation
𝑌0 [MPa] 𝐾 [MPa]
S_DP600|2Tests
417.30
S_DP600|3Tests S_DP600|Mem
𝐹
973.30
0.165
0.533
421.31
951.19
0.160
415.53
966.46
0.156
343.11
0.219
𝐻
0.484
-0.108
0.564
0.876
0.500
-0.047
0.680
1.002
-
-
-
-
-
0.501
0.499
0.243
1.112
0.998
0.500
lP
na
S_Al5754|2Tests 53.57
𝐿 = 𝑀 = 𝑁 𝜎yy ⁄𝜎xx
𝐺
re
𝑛
-p
Table 9 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “Mem”, for the experimental cases, on the assumption that they are described by the Swift hardening law. The observed stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the numerical simulation corresponding to the identified parameters is also indicated.
S_Al5754|3Tests 54.70
343.19
0.216
0.506
0.494
0.292
1.179
0.990
S_Al5754Mem
341.12
0.216
-
-
-
-
-
Jo
ur
13.07
34
Table 10 - Parameters identified by the proposed strategy, using two “2Tests” and three tests “3Tests”, and by the membrane theory “Mem”, for the experimental cases, on the assumption that they are described by the Voce hardening law. The observed stress path, 𝜎yy ⁄𝜎xx , of the circular bulge test observed for the numerical simulation corresponding to the identified parameters is also indicated.
Designation
𝑌0 [MPa] 𝑌𝑆𝑎𝑡 [MPa]
𝐶𝑌
𝐹
𝐺
𝐻
𝐿 = 𝑀 = 𝑁 𝜎yy ⁄𝜎xx
812.08
8.54
0.476
0.532
0.052
0.875
1.083
V_DP600|3Tests 474.25
807.91
8.59
0.498
0.502
-0.045
0.685
1.009
V_DP600|Mem
484.86
832.62
7.87
-
-
-
-
-
V_Al5754|2Tests 114.13
266.37
9.71
0.501
0.499
0.239
1.107
0.997
V_Al5754|3Tests 114.56
264.12
9.68
0.496
0.504
0.152
0.985
1.009
V_Al5754|Mem
257.05
10.89
-
-
-
-
-
116.46
ro of
V_DP600|2Tests 474.68
-p
Figure 15 compares all stress vs. strain curves of each material with the points obtained from the membrane theory (designated by “ emPoints”). The relative
re
differences in stress determined when comparing the points obtained by the membrane
lP
theory with the stress vs. strain curves obtained by fitting the hardening law to these points and the proposed strategy, are shown in Figure 16. The proposed strategy with two or three tests provides similar results to those of the membrane theory. The relative
na
difference in stress has an average value inferior to 2%, whatever the case, i.e. the identification of the Swift and Voce parameters, provided similar results for DP600 and
ur
Al5754. These errors are similar to those obtained by fitting the hardening laws
Jo
(designated by “ em” in Figure 16) to the set of points obtained from the membrane theory. The relative difference in stress only achieves a significant scale near the yield stress, which can be related with a poor fitting of the hardening law or due to experimental errors at small deformations.
35
S_DP600|Mem S_DP600|2Tests S_DP600|3Tests DP600|MemPoints
(a)
S_Al5754|Mem S_Al5754|2Tests S_Al5754|3Tests Al5754|MemPoints
(b) 300 250
600
200 Pa
750
V_Al5754|Mem V_Al5754|2Tests V_Al5754|3Tests
150
𝑌
450
𝑌
300
100
150
50 0
0 0
0.1
0.2
𝜀̅
0.3
ro of
Pa
900
V_DP600|Mem V_DP600|2Tests V_DP600|3Tests
0
0.4
0.1
𝜀̅
0.2
0.3
Figure 15 - Hardening curves corresponding to the identified parameters (Tables 9 and 10) for the materials: (a) DP600; (b) Al5754.
V_DP600|Mem
S_DP600|2Tests
V_DP600|2Tests
S_DP600|3Tests
V_DP600|3Tests
(b) 10
S_Al5754|Mem
V_Al5754|Mem
S_Al5754|2Tests
V_Al5754|2Tests
S_Al5754|3Tests
V_Al5754|3Tests
re
10
S_DP600|Mem
-p
(a)
8
lP
8 6
𝑑
𝑑
6
na
4
0
ur
2
0
0.1
0.2
𝜀̅
0.3
4 2 0
0.4
0
0.1
𝜀̅
0.2
0.3
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Figure 16 - Absolute value of the relative difference in stress when comparing the stress vs. strain points obtained by the membrane theory with their fitting and the inverse identification results (see Tables 9 and 10) for the materials: (a) DP600; (b) Al5754.
The results of the bulge test are now compared with those obtained from the uniaxial tensile test. For this comparison, the hardening curves are transferred to uniaxial tension based on the procedure recommended in Annex D of ISO 16808:2014. Figure 17 compares the stress vs. strain curves directly determined from the uniaxial tensile test 36
(Figure 17(a): DP600|Tensile; and Figure 17(b): Al5754|Tensile) with those obtained from the biaxial tests, identified through the membrane theory and with the proposed strategy. It can be concluded from this figure that the results obtained from tensile and bulge tests are in agreement; the initial parts of the directly determined tensile curves are in better agreement with the identifications, when using the Swift law in case of the DP600 material, and when using the Voce law in case of the Al5754 material.
(b) 300 250
600
200 Pa
750
150
𝑌
450
V_Al5754|Mem V_Al5754|2Tests V_Al5754|3Tests
re
𝑌
Pa
900
S_Al5754|Mem S_Al5754|2Tests S_Al5754|3Tests Al5754|Tensile
-p
(a)
V_DP600|Mem V_DP600|2Tests V_DP600|3Tests
ro of
S_DP600|Mem S_DP600|2Tests S_DP600|3Tests DP600|Tensile
100
300
50
0 0.1
0.2
𝜀̅
0.3
0.4
0 0
0.1
𝜀̅
0.2
0.3
na
0
lP
150
ur
Figure 17 – Comparison of the hardening laws, directly obtained from the uniaxial tensile test with those obtained from the biaxial tests, identified through the membrane theory and the proposed strategy, for the materials: (a) DP600; (b) Al5754.
6. Conclusions
Jo
An inverse analysis strategy is proposed that allows determining the parameters
of work hardening laws, Swift and Voce, regardless of the material anisotropy. The procedure uses the pressure vs. pole height curves from the circular and elliptical bulge tests. The use of two tests (a circular and an elliptical) is enough to accurately determine the hardening law parameters, although the use of three tests (a circular and two elliptical) allows greater accuracy in the evaluation of the stress path of the circular 37
bulge test. In fact, the identified hardening law has an associated stress path that is of interest to know accurately, for example in the framework of the parameters identification of the yield criterion. The identified hardening behaviour corresponds to the biaxial stress vs. strain curve under the bulge test and can be related with the tensile stress vs. strain curve through the plastic work equivalence. From the experimental point of view, the proposed inverse strategy is not so exposed to experimental errors as the membrane theory procedure (ISO 16808:2014).
ro of
The later requires the evaluation of the curvature radius and the strains measurement at the pole of the bulge test, for which the experimental errors are more critical. In fact, it is worth emphasizing that the efficiency of the proposed identification strategy only
-p
depends on the accurate experimental determination of the pressure vs. pole height
curves and the appropriateness of the numerical model. Moreover, although it needs to
re
resort to one or two elliptical tests besides the circular, the current strategy proves to be
lP
an efficient alternative to the membrane theory procedure (ISO 16808:2014), that leads to significant errors in the hardening identification of strongly anisotropic materials, due to the equibiaxial stress state assumption. The proposed strategy avoids this assumption
na
with the accessory optimization of two of the Hill’ 48 parameters that enables the accurate description of the yield surface in a region close to the stress paths that occur in
ur
the circular and elliptical bulge tests. Nevertheless, the optimized Hill’48 parameters do
Jo
not model the entire yield surface.
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: SAFEFORMING, co-
38
funded by the Portuguese National Innovation Agency, by FEDER, through the program Portugal-2020 (PT2020), and by POCI, with reference POCI-01-0247FEDER-017762; 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 Portugal2020 (PT2020), and by POCI, with reference POCI-01-0145-FEDER-031216. Two of
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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
-p
gratefully acknowledged.
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Author Contributions:
na
lP
Pereira A.F.G.: Investigation, Formal Analysis, Writing - Original Draft, WritingReviewing and Editing.; Prates P.A.: Funding acquisition, Formal Analysis, WritingReviewing and Editing.; Oliveira M.C.: Software, Formal Analysis, Writing- Reviewing and Editing.; Fernandes V.A.: Funding acquisition, Supervision, Writing- Reviewing and Editing.
ur
Declaration of interests
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Appendix A
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The numerical discretisation of the blank sheets shown in Figure 2 was prepared in order to allow accurate results when used to determine the biaxial stress vs. strain curve by the membrane theory procedure (ISO 16808, 2014), which requires the determination of the curvature radius and the principal strains at the pole during the bulge test. In fact, the accuracy of these results requires a fine discretisation (see Figure 2), and so a high computational cost. Since the proposed strategy only requires the results of pressure vs. pole height, it is suggested in Figure A.1 a coarser discretisation
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of the blank sheets that reduces the computational time but preserves the accurate assessment of the pressure vs. pole height curves, as shown in Figure A.2, and so of the inverse strategy identification results. In Figure A.2, the average difference in pressure
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between the coarse and refine meshes is approximately 0.025 MPa. The discretisation of the circular and elliptical blank sheets has respectively a total number of 915 and 1065,
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8-node hexahedral solid elements, with one layer of elements through the sheet
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thickness. In average, each numerical simulation of the circular and elliptical tests was completed in approximately 1.40 and 1.71 min, respectively. Hence, each identification required in average a total computational time of 0.79 and 1.13 hours in case of using
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two or three bulge tests, respectively. The use of these discretisations was tested for some materials and has proved to be about 10 times more efficient than those shown in
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Figure 2, without compromising the inverse strategy identification results.
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(a)
1
1
2 20
8 20
15
(b)
20
15 5
2
1
7
y 15
20
53.03
34.97
5
8
Oz displacement[mm]
27
34.97 29.11
53.03
38.89 53.03
34.97
7
x
1
5
7
5
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15
Circ_Coarse
Circ_Refine
Elli0º_Coarse
Elli0º_Refine
Elli90º_Coarse
Elli90º_Refine
8 4
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0
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12
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Pa 𝑃
20
0
10
20
30
ℎ mm
40
50
𝑃
24
Pa
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28
16
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-p
Figure A.1 – (a) Alternative discretisation of the blanks used in the circular and elliptical bulge tests showing the number of elements along Ox and Oy for the different zones; b) Representation of the discretisation zones, showing their dimensions and the displacement along Oz, in mm, for an isotropic material.
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Figure A.2 – Circular and elliptical (0° and 90°) pressure vs. pole height curves of the fictitious material S_H Ani (Table 1), generated with the mesh discretisations presented in Figure 2 and Figure A.1.
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