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Physics of large radius air bending
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Procedia Manufacturing 29 (2019) 161–168 Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia
18th International Conference on Sheet Metal, SHEMET 2019 18th International Conference on Sheet Metal, SHEMET 2019
Physics of large radius air bending Physics of large radius air bending
a,bInternational Conference b 2017, 28-30 June Manufacturing Engineering Society MESIC Richard Aerens *, Vitalii Vorkovb, Joost2017, R. Duflou a,b Vigo (Pontevedra),b Spain 2017, Richard Aerens *, Vitalii Vorkov , Joost R. Dufloub a Sirris, Celestijnenlaan 300, 3001, Heverlee, Belgium a Deparment of Mechanical Engineering, KUCelestijnenlaan Leuven, Celestijnenlaan 300, 3001, Heverlee, Sirris, 300, 3001, Heverlee, Belgium Belgium, member of Flanders Make b Deparment of Mechanical Engineering, KU Leuven, Celestijnenlaan 300, 3001, Heverlee, Belgium, member of Flanders Make b
Costing models for capacity optimization in Industry 4.0: Trade-off between used capacity and operational efficiency
Abstract Abstract a a,* b b A.bending Santana , P.modeled Afonso , A. Zanin , R.byWernke Conventional or small radius air has been and analytically described many authors. This process is traditionally called 3-point or bending. Whenair using a large punch for design reasonsdescribed or because is necessary bendingishigh strength Conventional small radius bending been modeled and analytically byitmany authors.when This process traditionally ahasradius University of Minho, 4800-058 Guimarães, Portugal steels, a new phenomenon arises.The between the punch and the plate, which is located atwhen the tip of thehigh punch at the called 3-point bending. When using a contact large radius for design reasons or because it is necessary bending strength b area punch Unochapecó, 89809-000 Chapecó, SC, Brazil outset of the forming operation, splitscontact into two areas duringthe thepunch bending bywhich slidingisaway from the tip tip of of the the punch punch.atThis steels, a new phenomenon arises.The area between andprocess the plate, located at the the phenomenon is the so-called "multi-breakage" follows the process better described a 4-point scheme in outset of the forming operation, splits into twoeffect. areas It during thethat bending processis by sliding away by from the tip bending of the punch. This which the contact of applied forces are effect. continuously changing theis bending operation. different phenomena phenomenon is thepoints so-called "multi-breakage" It follows that theduring process better described by The a 4-point bending scheme of in importance for thispoints type of of applied air bending are addressed in theduring current First, the characteristic which the contact forcesprocess are continuously changing thecontribution. bending operation. Thetheoretical different phenomena of Abstract equation, required determining the plate profileare andaddressed the punchinforce, is established. Further, this the initial model ischaracteristic enhanced by importance for thisfortype of air bending process the current contribution. First, theoretical the effectsrequired of the tool coefficient variations. The introduction theseiseffects allows equation, forindentation, determiningthe theplate platethinning profile and the friction punch force, is established. Further, this initial of model enhanced by Under theof ofmodel "Industry 4.0",thinning production be variations. pushed to increasingly interconnected, to develop anconcept analytical of the plate large radius airand bending basedcoefficient onwill physical considerations. Although the developed model is the effects the tool indentation, theprocesses friction Thebeintroduction of these effects allows information time and, necessarily, much efficient. In this context, optimization notdevelop the ultimate one, itonis amodel anreal important step thebending completebased understanding andconsiderations. the robust prediction ofcapacity large radius air bending. to an based analytical of the basis largetowards radius air onmore physical Although the developed model is goes beyond traditional aim of step capacity maximization, contributing and alsothe forrobust organization’s profitability andbending. value. not the ultimatethe one, it is an important towards the complete understanding prediction of large radius air © 2018The Authors. Published and by Elsevier B.V. improvement approaches suggest capacity optimization instead of Indeed, lean management continuous © 2019 The Authors. Published by Elsevier B.V. This is an open access articleof under the CCoptimization BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) © 2018The Authors. by Elsevier B.V. maximization. The Published study capacity and costing models is an important research topic that deserves This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. and discusses a mathematical This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) contributions from both the practical and theoretical perspectives. This paper presents Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. Selection peer-review under responsibility the organizing of SHEMET 2019. model forand capacity management based on of different costingcommittee models (ABC and TDABC). A generic model has been Keywords:air bending; large radius punch; multi-breakage effect
developed and it was used to analyze idle capacity and to design strategies towards the maximization of organization’s Keywords:air bending; large radius punch; multi-breakage effect value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity optimization might hide operational inefficiency. 1. Introduction
© 2017 The Authors. Published by Elsevier B.V. 1. Introduction Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference When using a large radius punch, one can observe that the contact area between the punch and the plate, which is 2017.
Whenlocated using aatlarge radius one can thatduring the contact area between punch and the plate, is initially the tip of thepunch, tool, splits intoobserve two zones the bending process.the These contact zones arewhich moving initially located at the tip of the tool, splits into two zones during the bending process. These contact zones are moving towards the tool sides during the bending operation. This phenomenon is the so-called "multi-breakage" effect. This Keywords: Cost Models; ABC; TDABC; Capacity Management; Idle Capacity; Operational Efficiency towards the tool sides during the bending operation. This phenomenon is the so-called "multi-breakage" effect. This
1. Introduction
* Corresponding author. Tel.: +32 2 376 31 01. E-mail address: [email protected] * The Corresponding author. Tel.: +32 31 01. cost of idle capacity is 2a376 fundamental information for companies and their management of extreme importance E-mail address: [email protected] in modern production systems. In general, it is defined as unused capacity or production potential and can be measured 2351-9789© 2018 The Authors. Published by Elsevier B.V. in several ways: tons ofunder production, available hours of manufacturing, etc. The management of the idle capacity This is an open access the CC by BY-NC-ND license(https://creativecommons.org/licenses/by-nc-nd/4.0/) 2351-9789© 2018 Thearticle Authors. Published Elsevier B.V. * Paulo Afonso. Tel.: +351 253 510 761; fax: +351 253 604 741committee of SHEMET 2019. Selection under responsibility of the organizing This is an and openpeer-review access article under the CC BY-NC-ND license(https://creativecommons.org/licenses/by-nc-nd/4.0/) E-mail address: [email protected] Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under of the scientificbycommittee the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2019responsibility The Authors. Published Elsevier of B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. 10.1016/j.promfg.2019.02.121
162
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2
name originates from the fact that the profile of the bent plate seems to "break", giving it a somewhat polygonal shape, as described by S. Benson in [1,2]. During large radius bending, the loading scheme is continuously changing. This is the reason why the laws of classical bending must be reviewed. Some authors already addressed this problem. S. Benson [1] contributed an empirical formula for the resulting radius of the bent plate versus the punch radius and the plate thickness for mild cold-rolled steels and a bending angle of 90°. Pouzols [3] provides a short explanation for the multi-breakage effect, but neither converts this into equations nor into a numerical model. Using the finite element method, Burchitz [4] models the multi-breakage effect for the case of a frictionless contact between the plate and the tooling. However, in the case of a contact with friction, the plate wraps around the punch. Recently, Vorkov [5-11] extensively studied the multi-breakage phenomenon, conducting a large number of large radius bending tests with five materials: St 37, AISI 304, AlMg3 and two high-strength steels Strenx 700 MC and Strenx 1300. This work resulted in a reference database which can be used to verify theories and models [21]. He also developed a model based on the assumption that the plate profile can be approximated by a circular arc and two straight segments [11]. Despite this approximation, the circular approximation model delivers satisfactory results. In this contribution, the foundation of a model based on the physics of the multi-breakage phenomenon is developed. First, the theoretical characteristic equation, which allows deriving the plate profile and the punch force, is established. Then this first model is enhanced by the effects of the tool indentation, the plate thinning and the advanced friction interaction. 2. Forces and moments in case of multi-breakage Consider the free body diagram of a plate experiencing multi-breakage (Fig. 1). There are two distinct zones; the first one is between the tool tip (point A) and the contact point with the punch (point C), and the second one is between point C and the contact point with the die (point M).
Fig. 1. Forces in case of multi-breakage effect.
The punch reaction Fs1, the total punch force Fs, and the internal axial force FxA, which is a traction force in this configuration, are obtained according to the system equilibrium. It is expected that the friction interaction is different between the punch and plate, and the die and the plate. Therefore, it is necessary to distinguish two friction coefficients: one for the contact with the die µm and another one for the contact with the punch µs. For convenience we will use the axis system shown in Fig. 1. All the forces and moments are expressed as a function of the reaction force Fm. 𝛴𝛴𝛴𝛴 = 0 →
The punch force
𝐹𝐹𝑠𝑠1 = 𝐹𝐹𝑚𝑚 ∙ G
with
G=
cos 𝜃𝜃+𝜇𝜇𝑚𝑚 sin 𝜃𝜃 cos 𝜓𝜓−𝜇𝜇𝑠𝑠 sin 𝜓𝜓
𝐹𝐹𝑠𝑠 = 2 ∙ 𝐹𝐹𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 (1 + 𝜇𝜇𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡 𝜃𝜃)
(1) (2)
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𝛴𝛴𝛴𝛴 = 0 →
𝐹𝐹𝑥𝑥𝑥𝑥 = −𝐹𝐹𝑚𝑚
(1−𝜇𝜇𝑠𝑠 𝜇𝜇𝑚𝑚) sin(𝜃𝜃−𝜓𝜓)−(𝜇𝜇𝑠𝑠 +𝜇𝜇𝑚𝑚) cos(𝜃𝜃−𝜓𝜓) cos 𝜓𝜓−𝜇𝜇𝑠𝑠 sin 𝜓𝜓
163 3
(3)
When the friction forces are null, FxA is negative; the plate is thus in a compression mode at this location (Fig.1). The bending moments in the segments M-C and C-A are calculated according to:
𝑀𝑀𝑃𝑃(𝑀𝑀−𝐶𝐶) = 𝐹𝐹𝑚𝑚 [𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 𝑦𝑦 sin 𝜃𝜃 + 𝜇𝜇𝑚𝑚 (𝑥𝑥 sin 𝜃𝜃 − 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 + 𝑠𝑠0 ⁄2)]
or
𝐹𝐹𝑚𝑚 ∙ Λ1 (𝑥𝑥, 𝑦𝑦)
(4)
As shown in Fig.1, in case of multi-breakage, there must be a point where the tangents to the punch and to the internal profile are equal (i.e. in point C). It should be noted that the radius of curvature in point A is smaller than the radius of curvature in point C, as can be seen in comparison with the punch of constant curvature in Fig. 1. 3. Analytical solution with constant thickness and usual friction coefficient The plate can be divided into two zones: the first one which is deformed elastically and the second one which experiences an elastoplastic deformation. For the first zone, the curvature and the corresponding profile is found according to the classical beam theory. For the second zone, the curvature is driven by the so-called unit moment. The unit moment σ* is the bending moment M per unit width b and per unit squared thickness s0 /2. This parameter is a function of the bending strain εb which is the halve thickness s0 /2 multiplied by the plate curvature 1/Rm. 𝜎𝜎 ∗ =
𝑀𝑀
(6)
𝑏𝑏 𝑠𝑠02
𝜀𝜀𝑏𝑏 =
𝑠𝑠0
2𝑅𝑅𝑚𝑚
(7)
The unit moment can be measured with a dedicated setup [12,13] or can be computed. Consider that the constitutive law of the material is classical Swift hardening: 𝜎𝜎 = 𝐶𝐶(𝜑𝜑 + 𝜑𝜑0 )𝑛𝑛
(8)
in which C is the hardening coefficient, 𝜑𝜑 the true strain, 𝜑𝜑0 the initial strain, and n the hardening exponent.
For this material model, after having accepted some simplifying work hypotheses, the corresponding unit moment can be computed according to [13]. The results of this computation can be written as a Swift-like equation:
with
𝐶𝐶" =
1
2
( )
2(𝑛𝑛+2) √3
𝜎𝜎 ∗ = 𝐶𝐶"(𝜀𝜀𝑏𝑏 + 𝜀𝜀𝑑𝑑 )𝑛𝑛
𝑛𝑛+1
and
𝐶𝐶
(9) 𝜀𝜀𝑑𝑑 = 𝜀𝜀0 ∙ (1 + 𝑛𝑛⁄2)1⁄𝑛𝑛
The advantage of measuring the unit moment is that simplifying hypotheses are not required, and reliable values are directly available. Moreover, the strain range of the measured unit moment is larger than the one of the computed unit moment based on tensile testing. As was shown in [13], the measured unit moment curves are often close to the calculated ones. See, as an example, the case of St52 in Fig. 2. However, for some materials (e.g. aluminum plates) differences of up to 10% could be observed, as was documented in [13]. In Equation 9, substituting the expressions for σ* and εb by Equation 6 and 7 respectively, the plate curvature can be written as: 1 𝑅𝑅𝑚𝑚
2
𝑀𝑀
1⁄𝑛𝑛
= 𝑠𝑠 [(𝐶𝐶"𝑏𝑏𝑠𝑠2 ) 0
0
− 𝜀𝜀𝑑𝑑 ]
(10)
Further, replacing the curvature1/Rm by its Cartesian form, and replacing M by the expression of Equation 4 or 5, we obtain the differential equation for air bending with or without multi-breakage for a Swift model.
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164
𝑦𝑦"
− (1+𝑦𝑦′2 )3⁄2 =
2 𝐹𝐹𝑚𝑚 ∙Λ(𝑥𝑥,𝑦𝑦) 1⁄𝑛𝑛 ) [( 𝑠𝑠0 𝐶𝐶"𝑏𝑏𝑠𝑠02
− 𝜀𝜀𝑑𝑑 ]
4
(11)
The solution of this equation is the profile of the half bent plate for a given value of Fm and a half bending angle θ (Fig. 1). This input parameter is in the initial conditions.
True stress (N/mm²)
a
700 600 500 400 300
measurement
200
regression (Swift)
100 0 0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
True strain
Fig. 2 Tensile curve of St52 (a) and its corresponding unit moment curve (b).
However, the value of Fm is still unknown. In order to simultaneously obtain Fm and the bent profile, the following iterative procedure is applied. We start with an approximated value of Fm, which can easily be computed using a circular approximation [7,10]. Then with this first value of 𝐹𝐹𝑚𝑚 , the profile is computed numerically point by point starting from point M (x = 0, y = 0, y' = tan θ) by means of Equation 11 with Λ = Λ1 (Equation 4) until the tangency condition is fulfilled, i.e. the slope of the curve is equal to the slope of the punch. When this point is found (i.e. Point C), the values of xC, yC and ψ are known and fixed. Further, the remainder of the profile is computed with Λ = Λ2 (Equation 5) till x = xA. At this point, a resulting slope 𝑦𝑦′𝐴𝐴 is obtained. The desired slope value should be equal to zero. Therefore the procedure described above is iteratively repeated with adjusted values of Fm until 𝑦𝑦′𝐴𝐴 ≈ 0. This iterative procedure has been implemented in a software code described in Section 6. Based on the abovementioned equations, however, the solution procedure does not converge to a multi-breakage solution. In this case the plate is assumed to wrap around the tool from the point where the inner radius of curvature is equal to the punch radius, i.e. a "wrap-around solution". So, in order to obtain more realistic results, it was decided to take into account the thinning of the plate and the effective friction coefficients. The remainder of this article discusses how these two aspects contribute to a possible multi-breakage solution. 4. Plate thinning There are two factors which affect the local plate thickness: the tool indentation into the material and the natural plate thinning. These factors are examined hereafter. 4.1.Indentation of a punch For conventional bending, the tool indentation is important and easy to measure, as reported in [13]. For large radius bending, this effect is less pronounced, however it is always present at the outset of the forming process. In order to describe this effect, a model of indentation of a cylinder in a cylinder is necessary and, moreover, a cylinder in a cylinder with a variable radius. Obviously, a ready to use solution is not available. Therefore, starting from the same working hypotheses as those used for spherical indentation, namely the "plastic similarity regime" [14-16], although this theory is questioned by modern finite element calculations, the authors developed a simple model allowing to compute the depth of penetration h of a cylinder in a cylinder:
Richard Aerens et al. / Procedia Manufacturing 29 (2019) 161–168 Richard Aerens / Procedia Manufacturing 00 (2018) 000–000
ℎ=
2
𝐹𝐹 ′ 𝑘𝑘 1+𝑛𝑛 ( 𝑠𝑠 ) 8 𝜎𝜎𝑚𝑚 𝑅𝑅𝑒𝑒𝑒𝑒
𝑅𝑅𝑒𝑒𝑒𝑒
ℎ=
or
1
8𝑅𝑅𝑒𝑒𝑒𝑒
2
𝐹𝐹𝑠𝑠′ 𝑘𝑘 1+𝑛𝑛 ) 1−𝑛𝑛 ( 𝜎𝜎𝑚𝑚 1+𝑛𝑛
165 5
(12)
where 1/Req = 1/Rp – 1/Ri with Rp is punch radius and Ri the average internal radius of the plate in the contact area, σm is the ultimate strength, 𝐹𝐹𝑠𝑠′ is the punch force per unit width, and k a dimensionless coefficient. Due to the difference between the considered hypothesis of indentation and reality, this formula has been adapted in order to obtain the best fit. This was achieved by introducing correction coefficients C1, C2, and C3.
ℎ=
2
𝐶𝐶 𝐹𝐹𝑠𝑠′ 𝐶𝐶1 2 1+𝑛𝑛 ) 1−𝑛𝑛 ( 𝜎𝜎𝑚𝑚 𝐶𝐶3 1+𝑛𝑛
1
8𝑅𝑅𝑒𝑒𝑒𝑒
(13)
Based on the finite element simulations of 3 mm thick plates of St14 bent with a die of 24 mm width and a punch with 5 mm radius, the following correction coefficients have been obtained: C1 = 1.60, C2 = 0.637, C3 = 0.736 .It should be mentioned that this solution should be improved by a more in-depth specific study. Contrary to a simple indentation, for which the center of force is in the middle of the punch, in the case of large radius bending, the centers of force are moved to the sides of the contact area. From FEM simulation, it appears that we can assume that the force is applied at 80% of the contact length (Fig. 3). 4.2. Natural thinning During a bending operation, a metal plate made of a conventional material thins. This phenomenon has been described by Hill [17] and Proska [18]. They developed a differential equation which allows to compute the theoretical thinning ratio η: actual thickness/original thickness. Tan [19] performed thinning measurements with pure moment bending tests for five materials: X5CrNi1810 (n=0.57), St05 (n=1.38), AA5052 (n=1.159), QStE 340 (n=0.201), QStE 550 (n=0.187). He plotted the results together with the computed values versus the bending strain εb. Since the computed values are not very reliable, only a synthesis of his measurements is reproduced in Fig. 4. Pouzols [3] made thinning measurements with classical bending tests for three materials: AlMg3 (n=0.23 [13]) thickness 6 mm, St37 (n=0.17 [13]) thickness 6.2 mm and X5CrNi1810 (n=0.47 [13]) thickness 2 mm, 4 mm, 6 mm. The punch radius was 6 mm. The results are also plotted in Fig. 4. One can notice that the thinning of X5CrNi 18 10 measured by Pouzols [3] is much more important than the one measured by Tan [19]. Feng [20] developed his own formula for the thinning. This formula is independent of the material characteristics. The author refers to the measurements of two additional authors which are also plotted in Fig. 4. Observing the results of those measurements, we can conclude that they all show a parabolic trend and that the influence of n appears but weakly. So, in order to have a tool for introducing the thinning in the model, a simple regression formula for thinning is established. This equation is based on the parabolic trend and the principle that the thinning increases theoretically with the hardening exponent n and does not exist when n = 0: ∆𝑠𝑠 𝑠𝑠0
= 𝑛𝑛𝜀𝜀𝑏𝑏 2
𝑜𝑜𝑜𝑜
𝜂𝜂 =
𝑠𝑠
𝑠𝑠0
= 1 − 𝑛𝑛𝜀𝜀𝑏𝑏 2
In Fig. 4, the bold green and red lines represent the "boundary" curves for n=0.1and n=0.57 respectively.
(14)
Richard Aerens / Procedia Manufacturing 00 (2018) 000–000
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Richard Aerens et al. / Procedia Manufacturing 29 (2019) 161–168
166
Fig. 3. Center of force during the indentation (finite element simulation of St14 3mm thick, punch radius 5 mm, die of 24 mm).
5. The effective friction coefficients
Fig. 4. Thinning ratio 𝜂𝜂 versus bending strain 𝜀𝜀𝑏𝑏 measured by Tan[19], Pouzols[3]and Feng[20].
In order to see the influence of the friction coefficients, reconsider solving the problem of the plate with constant thickness. This hypothesis implies that the curvature depends only on the bending moment. So, to have a multibreakage effect, the radius of curvature in point A should be smaller than the radius of curvature in point C. This implies that, at least at a certain time, the bending moment in point A must be larger than in point C. Considering the simple relation between the moments in points A and C: 𝑀𝑀𝐴𝐴 = 𝑀𝑀𝐶𝐶 − 𝐹𝐹𝑥𝑥𝑥𝑥 ∙ (𝑦𝑦𝐴𝐴 − 𝑦𝑦𝐶𝐶 )
(15)
Since (yA - yC) has always a positive value, this condition will be automatically fulfilled when the internal axial force FxA has a negative value i.e. compression occurs. Equation 3 shows that FxA depends on the friction coefficients and the angle ψ. The force FxA should be a compression if the friction coefficients were null. Often, a friction coefficient value of 0.1 is selected. Note however that higher friction coefficients need to be used in this case, as can be justified considering the friction marks left on a steel sheet by the die. Observing those marks by means of a microscope shows that the friction is associated with micro-plowing. Friction coefficients have been measured by Aerens [13]. Conclusion of this study was that a friction coefficient at the die µm = 0.20 can be applied for materials having a fine surface quality (cold rolled material). For materials with a coarse surface quality (e.g. hot rolled material), one must apply µm = 0.25. Even a value of µm = 0.30 has been recorded for St37. It is important to note that, at the outset of the forming process, the external surface of the plate does not slide inwards into the die cavity, but outwards. This effect is more pronounced for dies with large radius shoulders. This effect is translated in the formulas by a negative value of µm as long as the external profile length A'M' (Fig. 1) is increasing. Taking this effect into account also contributes to the revealing of the multi-breakage effect. For the friction coefficient at the punch µs, the situation is more complicated. With a large punch radius, no microplowing has been observed, which is a valid reason to consider a friction coefficient µs smaller than the friction coefficient on the die shoulders µm. From finite element simulations performed for a few multi-breakage cases in ANSYS, it was concluded that one can consider that the effective friction coefficient at the die µs can be approximated as µs = (0.5 … 0.7)·µm. 6. The updated analytical model Finally, considering the analytical solution presented in Section 3 including now the tool indentation, the natural thinning of the plate and the effective friction coefficients is implemented as described below.
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167 7
From finite element simulations it can be observed that when the indentation stops, the multi-breakage is initiated. The switching occurs when the mean radius of the plate equals the punch radius plus half of the plate thickness, which is a typical wrap-around situation. This progress has been translated in a computer program which works as follows: In the first phase, a classical 3-point bending situation is computed taking into account the indentation of the tool and the natural thinning of the plate. By doing so, an array of 250 thicknesses along the profile is calculated. This phase continues until the occurrence of the wrap-around effect. Further, with the predefined thickness array, the program switches over to the second phase where 4-point bending model is computed in case of its presence. In this phase, the natural thinning is still taken into account. Knowing the profile, the bending moment and the forces, the program subsequently computes the bend allowance, the springback, and the punch stroke with the formulas presented earlier.
Fig. 5.Example of finite element simulation. Die opening: 24 mm. Punch radius: 5 mm. Thickness: 3 mm. Material: St14. Bending angle: 88°.
Fig. 6.Computed profile based on the data from Fig. 5.
For several but not all cases, the program can correctly model the multi-breakage effect during the bending process. As an example, the case of St14 treated in Section 4.1 is presented here. The indentation (Fig. 3) is followed by the multi-breakage (Fig. 5). The corresponding profile computed with the program is shown in Fig. 6. The gap of 0.057 mm indicates that the multi-breakage effect is well represented by the model. 7. Tests and results In order to evaluate the performance of the model, a number of simulations has been done for computing the bending forces, the springback, and the bend allowance. The results have been compared to those provided by the circular approximation model. The tests have been conducted by Vorkov and are discussed in detail in [11]. In this contribution, we focus on the tests with St37 for which the constitutive law can be approximated by a Swift law. In total, there are 174 tests with plate thicknesses of 2, 4, 6, and 8 mm, punch radii of 10, 20, 30 and 40 mm, die width of 40, 50, 60, and 80 mm, and product angles of about 90 till 155°. Depending on the case, the simulation led to a multi-breakage solution or a wrap-around solution. The accuracy of results are evaluated by three indicators: the coefficient of determination R², the average absolute error εav and the relative error εrel as defined in [11]. The results are presented in Table 1. As it can be seen, the developed model in general outperforms the circular approximation model. Table 1. Overview of the comparison of the experimental data with the circular approximation and developed models
R² εav
Circular approximation model
Developed model
Bending force 0.98 104.1 N
Bending force 0.98 60.2 N
Springback 0.79 1.83°
Bend allowance 0.91 1.35 mm
Springback 0.81 0.82°
Bend allowance 0.92 1.29 mm
Richard Aerens et al. / Procedia Manufacturing 29 (2019) 161–168 Richard Aerens / Procedia Manufacturing 00 (2018) 000–000
168
εrel (%)
8.7
9.6
-6.7
5.0
4.3
8
-6.4
8. Conclusions Solving the characteristic differential equation for a material represented with a Swift hardening law, for constant thickness and classical friction coefficients, does not allow modeling the multi-breakage effect correctly. Therefore the bending model should consider the effect of tool indentation at the outset of the forming process, the natural thinning of the plate, and the effective friction coefficients. Result of this enhanced model is that for a number of cases the computed profile effectively predicts the multi-breakage phenomenon. At this stage of the study when the calculation does not provide a multi-breakage solution, the plate is assumed to wrap partially around the punch. This combined solution yields better results than the circular approximation. However, some further refinements are still necessary in order to improve the prediction accuracy. In particular, more advanced modeling of the indentation phase can be addressed. References [1] S. D. Benson SD, Press brake technology, A guide to precision sheet metal bending, Society of Manufacturing Engineers, Dearborn, Michigan, (1997). [2] S.D. Benson, A digital handbook for precision sheet metal fabrication, (2015), Retrieved from http://www.theartofpressbrake.com/a-digitalhandbook-for-precision-sheet-metal-fabrication/ on 18.09.2018. [3] V. Pouzols, Optimisation d'opérations industrielles de pliage par la méthode des éléments finis. Thèse de doctorat de l''Université de Grenoble, France, (2011). [4] I.A. Burchitz, Improvement of sprinback prediction in sheet metal forming, University of Twente, Enschede, The Netherlands, (2008). [5] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, The multi-breakage phenomenon in air bending process, Key Eng Mater, 611 (2014) 10471053. [6] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, Springback prediction of high-strength steels in large radius air bending using finite element modeling approach, Procedia Eng, 81 (2014) 1005-1010. [7] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, On the identification of a loading scheme in large radius air bending, Key Eng Mater, 639 (2015) 155-162. [8] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, Experimental investigation of large radius air bending, Int J Adv Manuf Technol, 92 (9) (2017) 3553-3569. [9] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, Two regression approaches for prediction of large radius air bending, Int J Mater Form., (2018) 1-12. [10] V. Vorkov, A. Konyukhov, D. Vandepitte, J.R. Duflou, Contact modeling of large radius air bending with geometrically exact contact algorithm, Journal of Physics: Conference Series, 734 (3) (2016) 032076. [11] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, Analytical prediction of large radius bending by circular approximation, J Manuf Sci Eng, 140 (12) (2018) 121010. [12] R. Aerens, Characterisation of material by bending, Proceedings of the fifth International Conference on Sheet Metal: SheMet'97, (1997) 251262. [13] R. Aerens, S. Masselis, Air bending. Scientific and Technical Research Center of the Metal Fabrication Industry (CRIF/WTCM/SIRRIS) MC 110, Leuven, Belgium, (2000). [14] S.D. Mesarovic, N.A. Fleck, Spherical indentation of elastic–plastic solids, Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 455 (1987):2707-2728. [15] R. Hill, B. Storåkers, A. Zdunek, A theoretical study of the Brinell hardness test, Proceedings of the Royal Society of London A Mathematical and Physical Sciences, 423 (1865) 301-330. [16] I.M. Hutchings, The contributions of David Tabor to the science of indentation hardness, J Mater Res, 24 (3) (2009) 581-589. [17] R. Hill, The mathematical theory of plasticity. The Clarendon Press, Oxford, (1950). [18] F. Proksa, Plastisches biegen von blechen. Der Stahlbau, 28 (2) (1959) 29-36. [19] Z. Tan, Analysis and modelling of plastic bending processes, Luleå tekniska universitet, Sweden, (1994). [20] F. Yang, L. Yi, R. Feng, Thinning coefficient of sheet metal bending Forging & Stamping Technology, 3 (2009) 17. [21] V. Vorkov, Complete experimental data for large radius bending, (2017).
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18th International Conference on Sheet Metal, SHEMET 2019 18th International Conference on Sheet Metal, SHEMET 2019
Physics of large radius air bending Physics of large radius air bending
a,bInternational Conference b 2017, 28-30 June Manufacturing Engineering Society MESIC Richard Aerens *, Vitalii Vorkovb, Joost2017, R. Duflou a,b Vigo (Pontevedra),b Spain 2017, Richard Aerens *, Vitalii Vorkov , Joost R. Dufloub a Sirris, Celestijnenlaan 300, 3001, Heverlee, Belgium a Deparment of Mechanical Engineering, KUCelestijnenlaan Leuven, Celestijnenlaan 300, 3001, Heverlee, Sirris, 300, 3001, Heverlee, Belgium Belgium, member of Flanders Make b Deparment of Mechanical Engineering, KU Leuven, Celestijnenlaan 300, 3001, Heverlee, Belgium, member of Flanders Make b
Costing models for capacity optimization in Industry 4.0: Trade-off between used capacity and operational efficiency
Abstract Abstract a a,* b b A.bending Santana , P.modeled Afonso , A. Zanin , R.byWernke Conventional or small radius air has been and analytically described many authors. This process is traditionally called 3-point or bending. Whenair using a large punch for design reasonsdescribed or because is necessary bendingishigh strength Conventional small radius bending been modeled and analytically byitmany authors.when This process traditionally ahasradius University of Minho, 4800-058 Guimarães, Portugal steels, a new phenomenon arises.The between the punch and the plate, which is located atwhen the tip of thehigh punch at the called 3-point bending. When using a contact large radius for design reasons or because it is necessary bending strength b area punch Unochapecó, 89809-000 Chapecó, SC, Brazil outset of the forming operation, splitscontact into two areas duringthe thepunch bending bywhich slidingisaway from the tip tip of of the the punch punch.atThis steels, a new phenomenon arises.The area between andprocess the plate, located at the the phenomenon is the so-called "multi-breakage" follows the process better described a 4-point scheme in outset of the forming operation, splits into twoeffect. areas It during thethat bending processis by sliding away by from the tip bending of the punch. This which the contact of applied forces are effect. continuously changing theis bending operation. different phenomena phenomenon is thepoints so-called "multi-breakage" It follows that theduring process better described by The a 4-point bending scheme of in importance for thispoints type of of applied air bending are addressed in theduring current First, the characteristic which the contact forcesprocess are continuously changing thecontribution. bending operation. Thetheoretical different phenomena of Abstract equation, required determining the plate profileare andaddressed the punchinforce, is established. Further, this the initial model ischaracteristic enhanced by importance for thisfortype of air bending process the current contribution. First, theoretical the effectsrequired of the tool coefficient variations. The introduction theseiseffects allows equation, forindentation, determiningthe theplate platethinning profile and the friction punch force, is established. Further, this initial of model enhanced by Under theof ofmodel "Industry 4.0",thinning production be variations. pushed to increasingly interconnected, to develop anconcept analytical of the plate large radius airand bending basedcoefficient onwill physical considerations. Although the developed model is the effects the tool indentation, theprocesses friction Thebeintroduction of these effects allows information time and, necessarily, much efficient. In this context, optimization notdevelop the ultimate one, itonis amodel anreal important step thebending completebased understanding andconsiderations. the robust prediction ofcapacity large radius air bending. to an based analytical of the basis largetowards radius air onmore physical Although the developed model is goes beyond traditional aim of step capacity maximization, contributing and alsothe forrobust organization’s profitability andbending. value. not the ultimatethe one, it is an important towards the complete understanding prediction of large radius air © 2018The Authors. Published and by Elsevier B.V. improvement approaches suggest capacity optimization instead of Indeed, lean management continuous © 2019 The Authors. Published by Elsevier B.V. This is an open access articleof under the CCoptimization BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) © 2018The Authors. by Elsevier B.V. maximization. The Published study capacity and costing models is an important research topic that deserves This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. and discusses a mathematical This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) contributions from both the practical and theoretical perspectives. This paper presents Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. Selection peer-review under responsibility the organizing of SHEMET 2019. model forand capacity management based on of different costingcommittee models (ABC and TDABC). A generic model has been Keywords:air bending; large radius punch; multi-breakage effect
developed and it was used to analyze idle capacity and to design strategies towards the maximization of organization’s Keywords:air bending; large radius punch; multi-breakage effect value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity optimization might hide operational inefficiency. 1. Introduction
© 2017 The Authors. Published by Elsevier B.V. 1. Introduction Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference When using a large radius punch, one can observe that the contact area between the punch and the plate, which is 2017.
Whenlocated using aatlarge radius one can thatduring the contact area between punch and the plate, is initially the tip of thepunch, tool, splits intoobserve two zones the bending process.the These contact zones arewhich moving initially located at the tip of the tool, splits into two zones during the bending process. These contact zones are moving towards the tool sides during the bending operation. This phenomenon is the so-called "multi-breakage" effect. This Keywords: Cost Models; ABC; TDABC; Capacity Management; Idle Capacity; Operational Efficiency towards the tool sides during the bending operation. This phenomenon is the so-called "multi-breakage" effect. This
1. Introduction
* Corresponding author. Tel.: +32 2 376 31 01. E-mail address: [email protected] * The Corresponding author. Tel.: +32 31 01. cost of idle capacity is 2a376 fundamental information for companies and their management of extreme importance E-mail address: [email protected] in modern production systems. In general, it is defined as unused capacity or production potential and can be measured 2351-9789© 2018 The Authors. Published by Elsevier B.V. in several ways: tons ofunder production, available hours of manufacturing, etc. The management of the idle capacity This is an open access the CC by BY-NC-ND license(https://creativecommons.org/licenses/by-nc-nd/4.0/) 2351-9789© 2018 Thearticle Authors. Published Elsevier B.V. * Paulo Afonso. Tel.: +351 253 510 761; fax: +351 253 604 741committee of SHEMET 2019. Selection under responsibility of the organizing This is an and openpeer-review access article under the CC BY-NC-ND license(https://creativecommons.org/licenses/by-nc-nd/4.0/) E-mail address: [email protected] Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under of the scientificbycommittee the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2019responsibility The Authors. Published Elsevier of B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the organizing committee of SHEMET 2019. 10.1016/j.promfg.2019.02.121
162
Richard Aerens et al. / Procedia Manufacturing 29 (2019) 161–168 Richard Aerens / Procedia Manufacturing 00 (2018) 000–000
2
name originates from the fact that the profile of the bent plate seems to "break", giving it a somewhat polygonal shape, as described by S. Benson in [1,2]. During large radius bending, the loading scheme is continuously changing. This is the reason why the laws of classical bending must be reviewed. Some authors already addressed this problem. S. Benson [1] contributed an empirical formula for the resulting radius of the bent plate versus the punch radius and the plate thickness for mild cold-rolled steels and a bending angle of 90°. Pouzols [3] provides a short explanation for the multi-breakage effect, but neither converts this into equations nor into a numerical model. Using the finite element method, Burchitz [4] models the multi-breakage effect for the case of a frictionless contact between the plate and the tooling. However, in the case of a contact with friction, the plate wraps around the punch. Recently, Vorkov [5-11] extensively studied the multi-breakage phenomenon, conducting a large number of large radius bending tests with five materials: St 37, AISI 304, AlMg3 and two high-strength steels Strenx 700 MC and Strenx 1300. This work resulted in a reference database which can be used to verify theories and models [21]. He also developed a model based on the assumption that the plate profile can be approximated by a circular arc and two straight segments [11]. Despite this approximation, the circular approximation model delivers satisfactory results. In this contribution, the foundation of a model based on the physics of the multi-breakage phenomenon is developed. First, the theoretical characteristic equation, which allows deriving the plate profile and the punch force, is established. Then this first model is enhanced by the effects of the tool indentation, the plate thinning and the advanced friction interaction. 2. Forces and moments in case of multi-breakage Consider the free body diagram of a plate experiencing multi-breakage (Fig. 1). There are two distinct zones; the first one is between the tool tip (point A) and the contact point with the punch (point C), and the second one is between point C and the contact point with the die (point M).
Fig. 1. Forces in case of multi-breakage effect.
The punch reaction Fs1, the total punch force Fs, and the internal axial force FxA, which is a traction force in this configuration, are obtained according to the system equilibrium. It is expected that the friction interaction is different between the punch and plate, and the die and the plate. Therefore, it is necessary to distinguish two friction coefficients: one for the contact with the die µm and another one for the contact with the punch µs. For convenience we will use the axis system shown in Fig. 1. All the forces and moments are expressed as a function of the reaction force Fm. 𝛴𝛴𝛴𝛴 = 0 →
The punch force
𝐹𝐹𝑠𝑠1 = 𝐹𝐹𝑚𝑚 ∙ G
with
G=
cos 𝜃𝜃+𝜇𝜇𝑚𝑚 sin 𝜃𝜃 cos 𝜓𝜓−𝜇𝜇𝑠𝑠 sin 𝜓𝜓
𝐹𝐹𝑠𝑠 = 2 ∙ 𝐹𝐹𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 (1 + 𝜇𝜇𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡 𝜃𝜃)
(1) (2)
Richard Aerens et al. / Procedia Manufacturing 29 (2019) 161–168 Richard Aerens / Procedia Manufacturing 00 (2018) 000–000
𝛴𝛴𝛴𝛴 = 0 →
𝐹𝐹𝑥𝑥𝑥𝑥 = −𝐹𝐹𝑚𝑚
(1−𝜇𝜇𝑠𝑠 𝜇𝜇𝑚𝑚) sin(𝜃𝜃−𝜓𝜓)−(𝜇𝜇𝑠𝑠 +𝜇𝜇𝑚𝑚) cos(𝜃𝜃−𝜓𝜓) cos 𝜓𝜓−𝜇𝜇𝑠𝑠 sin 𝜓𝜓
163 3
(3)
When the friction forces are null, FxA is negative; the plate is thus in a compression mode at this location (Fig.1). The bending moments in the segments M-C and C-A are calculated according to:
𝑀𝑀𝑃𝑃(𝑀𝑀−𝐶𝐶) = 𝐹𝐹𝑚𝑚 [𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 𝑦𝑦 sin 𝜃𝜃 + 𝜇𝜇𝑚𝑚 (𝑥𝑥 sin 𝜃𝜃 − 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 + 𝑠𝑠0 ⁄2)]
or
𝐹𝐹𝑚𝑚 ∙ Λ1 (𝑥𝑥, 𝑦𝑦)
(4)
As shown in Fig.1, in case of multi-breakage, there must be a point where the tangents to the punch and to the internal profile are equal (i.e. in point C). It should be noted that the radius of curvature in point A is smaller than the radius of curvature in point C, as can be seen in comparison with the punch of constant curvature in Fig. 1. 3. Analytical solution with constant thickness and usual friction coefficient The plate can be divided into two zones: the first one which is deformed elastically and the second one which experiences an elastoplastic deformation. For the first zone, the curvature and the corresponding profile is found according to the classical beam theory. For the second zone, the curvature is driven by the so-called unit moment. The unit moment σ* is the bending moment M per unit width b and per unit squared thickness s0 /2. This parameter is a function of the bending strain εb which is the halve thickness s0 /2 multiplied by the plate curvature 1/Rm. 𝜎𝜎 ∗ =
𝑀𝑀
(6)
𝑏𝑏 𝑠𝑠02
𝜀𝜀𝑏𝑏 =
𝑠𝑠0
2𝑅𝑅𝑚𝑚
(7)
The unit moment can be measured with a dedicated setup [12,13] or can be computed. Consider that the constitutive law of the material is classical Swift hardening: 𝜎𝜎 = 𝐶𝐶(𝜑𝜑 + 𝜑𝜑0 )𝑛𝑛
(8)
in which C is the hardening coefficient, 𝜑𝜑 the true strain, 𝜑𝜑0 the initial strain, and n the hardening exponent.
For this material model, after having accepted some simplifying work hypotheses, the corresponding unit moment can be computed according to [13]. The results of this computation can be written as a Swift-like equation:
with
𝐶𝐶" =
1
2
( )
2(𝑛𝑛+2) √3
𝜎𝜎 ∗ = 𝐶𝐶"(𝜀𝜀𝑏𝑏 + 𝜀𝜀𝑑𝑑 )𝑛𝑛
𝑛𝑛+1
and
𝐶𝐶
(9) 𝜀𝜀𝑑𝑑 = 𝜀𝜀0 ∙ (1 + 𝑛𝑛⁄2)1⁄𝑛𝑛
The advantage of measuring the unit moment is that simplifying hypotheses are not required, and reliable values are directly available. Moreover, the strain range of the measured unit moment is larger than the one of the computed unit moment based on tensile testing. As was shown in [13], the measured unit moment curves are often close to the calculated ones. See, as an example, the case of St52 in Fig. 2. However, for some materials (e.g. aluminum plates) differences of up to 10% could be observed, as was documented in [13]. In Equation 9, substituting the expressions for σ* and εb by Equation 6 and 7 respectively, the plate curvature can be written as: 1 𝑅𝑅𝑚𝑚
2
𝑀𝑀
1⁄𝑛𝑛
= 𝑠𝑠 [(𝐶𝐶"𝑏𝑏𝑠𝑠2 ) 0
0
− 𝜀𝜀𝑑𝑑 ]
(10)
Further, replacing the curvature1/Rm by its Cartesian form, and replacing M by the expression of Equation 4 or 5, we obtain the differential equation for air bending with or without multi-breakage for a Swift model.
Richard Aerens et al. / Procedia Manufacturing 29 (2019) 161–168 Richard Aerens / Procedia Manufacturing 00 (2018) 000–000
164
𝑦𝑦"
− (1+𝑦𝑦′2 )3⁄2 =
2 𝐹𝐹𝑚𝑚 ∙Λ(𝑥𝑥,𝑦𝑦) 1⁄𝑛𝑛 ) [( 𝑠𝑠0 𝐶𝐶"𝑏𝑏𝑠𝑠02
− 𝜀𝜀𝑑𝑑 ]
4
(11)
The solution of this equation is the profile of the half bent plate for a given value of Fm and a half bending angle θ (Fig. 1). This input parameter is in the initial conditions.
True stress (N/mm²)
a
700 600 500 400 300
measurement
200
regression (Swift)
100 0 0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
True strain
Fig. 2 Tensile curve of St52 (a) and its corresponding unit moment curve (b).
However, the value of Fm is still unknown. In order to simultaneously obtain Fm and the bent profile, the following iterative procedure is applied. We start with an approximated value of Fm, which can easily be computed using a circular approximation [7,10]. Then with this first value of 𝐹𝐹𝑚𝑚 , the profile is computed numerically point by point starting from point M (x = 0, y = 0, y' = tan θ) by means of Equation 11 with Λ = Λ1 (Equation 4) until the tangency condition is fulfilled, i.e. the slope of the curve is equal to the slope of the punch. When this point is found (i.e. Point C), the values of xC, yC and ψ are known and fixed. Further, the remainder of the profile is computed with Λ = Λ2 (Equation 5) till x = xA. At this point, a resulting slope 𝑦𝑦′𝐴𝐴 is obtained. The desired slope value should be equal to zero. Therefore the procedure described above is iteratively repeated with adjusted values of Fm until 𝑦𝑦′𝐴𝐴 ≈ 0. This iterative procedure has been implemented in a software code described in Section 6. Based on the abovementioned equations, however, the solution procedure does not converge to a multi-breakage solution. In this case the plate is assumed to wrap around the tool from the point where the inner radius of curvature is equal to the punch radius, i.e. a "wrap-around solution". So, in order to obtain more realistic results, it was decided to take into account the thinning of the plate and the effective friction coefficients. The remainder of this article discusses how these two aspects contribute to a possible multi-breakage solution. 4. Plate thinning There are two factors which affect the local plate thickness: the tool indentation into the material and the natural plate thinning. These factors are examined hereafter. 4.1.Indentation of a punch For conventional bending, the tool indentation is important and easy to measure, as reported in [13]. For large radius bending, this effect is less pronounced, however it is always present at the outset of the forming process. In order to describe this effect, a model of indentation of a cylinder in a cylinder is necessary and, moreover, a cylinder in a cylinder with a variable radius. Obviously, a ready to use solution is not available. Therefore, starting from the same working hypotheses as those used for spherical indentation, namely the "plastic similarity regime" [14-16], although this theory is questioned by modern finite element calculations, the authors developed a simple model allowing to compute the depth of penetration h of a cylinder in a cylinder:
Richard Aerens et al. / Procedia Manufacturing 29 (2019) 161–168 Richard Aerens / Procedia Manufacturing 00 (2018) 000–000
ℎ=
2
𝐹𝐹 ′ 𝑘𝑘 1+𝑛𝑛 ( 𝑠𝑠 ) 8 𝜎𝜎𝑚𝑚 𝑅𝑅𝑒𝑒𝑒𝑒
𝑅𝑅𝑒𝑒𝑒𝑒
ℎ=
or
1
8𝑅𝑅𝑒𝑒𝑒𝑒
2
𝐹𝐹𝑠𝑠′ 𝑘𝑘 1+𝑛𝑛 ) 1−𝑛𝑛 ( 𝜎𝜎𝑚𝑚 1+𝑛𝑛
165 5
(12)
where 1/Req = 1/Rp – 1/Ri with Rp is punch radius and Ri the average internal radius of the plate in the contact area, σm is the ultimate strength, 𝐹𝐹𝑠𝑠′ is the punch force per unit width, and k a dimensionless coefficient. Due to the difference between the considered hypothesis of indentation and reality, this formula has been adapted in order to obtain the best fit. This was achieved by introducing correction coefficients C1, C2, and C3.
ℎ=
2
𝐶𝐶 𝐹𝐹𝑠𝑠′ 𝐶𝐶1 2 1+𝑛𝑛 ) 1−𝑛𝑛 ( 𝜎𝜎𝑚𝑚 𝐶𝐶3 1+𝑛𝑛
1
8𝑅𝑅𝑒𝑒𝑒𝑒
(13)
Based on the finite element simulations of 3 mm thick plates of St14 bent with a die of 24 mm width and a punch with 5 mm radius, the following correction coefficients have been obtained: C1 = 1.60, C2 = 0.637, C3 = 0.736 .It should be mentioned that this solution should be improved by a more in-depth specific study. Contrary to a simple indentation, for which the center of force is in the middle of the punch, in the case of large radius bending, the centers of force are moved to the sides of the contact area. From FEM simulation, it appears that we can assume that the force is applied at 80% of the contact length (Fig. 3). 4.2. Natural thinning During a bending operation, a metal plate made of a conventional material thins. This phenomenon has been described by Hill [17] and Proska [18]. They developed a differential equation which allows to compute the theoretical thinning ratio η: actual thickness/original thickness. Tan [19] performed thinning measurements with pure moment bending tests for five materials: X5CrNi1810 (n=0.57), St05 (n=1.38), AA5052 (n=1.159), QStE 340 (n=0.201), QStE 550 (n=0.187). He plotted the results together with the computed values versus the bending strain εb. Since the computed values are not very reliable, only a synthesis of his measurements is reproduced in Fig. 4. Pouzols [3] made thinning measurements with classical bending tests for three materials: AlMg3 (n=0.23 [13]) thickness 6 mm, St37 (n=0.17 [13]) thickness 6.2 mm and X5CrNi1810 (n=0.47 [13]) thickness 2 mm, 4 mm, 6 mm. The punch radius was 6 mm. The results are also plotted in Fig. 4. One can notice that the thinning of X5CrNi 18 10 measured by Pouzols [3] is much more important than the one measured by Tan [19]. Feng [20] developed his own formula for the thinning. This formula is independent of the material characteristics. The author refers to the measurements of two additional authors which are also plotted in Fig. 4. Observing the results of those measurements, we can conclude that they all show a parabolic trend and that the influence of n appears but weakly. So, in order to have a tool for introducing the thinning in the model, a simple regression formula for thinning is established. This equation is based on the parabolic trend and the principle that the thinning increases theoretically with the hardening exponent n and does not exist when n = 0: ∆𝑠𝑠 𝑠𝑠0
= 𝑛𝑛𝜀𝜀𝑏𝑏 2
𝑜𝑜𝑜𝑜
𝜂𝜂 =
𝑠𝑠
𝑠𝑠0
= 1 − 𝑛𝑛𝜀𝜀𝑏𝑏 2
In Fig. 4, the bold green and red lines represent the "boundary" curves for n=0.1and n=0.57 respectively.
(14)
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Fig. 3. Center of force during the indentation (finite element simulation of St14 3mm thick, punch radius 5 mm, die of 24 mm).
5. The effective friction coefficients
Fig. 4. Thinning ratio 𝜂𝜂 versus bending strain 𝜀𝜀𝑏𝑏 measured by Tan[19], Pouzols[3]and Feng[20].
In order to see the influence of the friction coefficients, reconsider solving the problem of the plate with constant thickness. This hypothesis implies that the curvature depends only on the bending moment. So, to have a multibreakage effect, the radius of curvature in point A should be smaller than the radius of curvature in point C. This implies that, at least at a certain time, the bending moment in point A must be larger than in point C. Considering the simple relation between the moments in points A and C: 𝑀𝑀𝐴𝐴 = 𝑀𝑀𝐶𝐶 − 𝐹𝐹𝑥𝑥𝑥𝑥 ∙ (𝑦𝑦𝐴𝐴 − 𝑦𝑦𝐶𝐶 )
(15)
Since (yA - yC) has always a positive value, this condition will be automatically fulfilled when the internal axial force FxA has a negative value i.e. compression occurs. Equation 3 shows that FxA depends on the friction coefficients and the angle ψ. The force FxA should be a compression if the friction coefficients were null. Often, a friction coefficient value of 0.1 is selected. Note however that higher friction coefficients need to be used in this case, as can be justified considering the friction marks left on a steel sheet by the die. Observing those marks by means of a microscope shows that the friction is associated with micro-plowing. Friction coefficients have been measured by Aerens [13]. Conclusion of this study was that a friction coefficient at the die µm = 0.20 can be applied for materials having a fine surface quality (cold rolled material). For materials with a coarse surface quality (e.g. hot rolled material), one must apply µm = 0.25. Even a value of µm = 0.30 has been recorded for St37. It is important to note that, at the outset of the forming process, the external surface of the plate does not slide inwards into the die cavity, but outwards. This effect is more pronounced for dies with large radius shoulders. This effect is translated in the formulas by a negative value of µm as long as the external profile length A'M' (Fig. 1) is increasing. Taking this effect into account also contributes to the revealing of the multi-breakage effect. For the friction coefficient at the punch µs, the situation is more complicated. With a large punch radius, no microplowing has been observed, which is a valid reason to consider a friction coefficient µs smaller than the friction coefficient on the die shoulders µm. From finite element simulations performed for a few multi-breakage cases in ANSYS, it was concluded that one can consider that the effective friction coefficient at the die µs can be approximated as µs = (0.5 … 0.7)·µm. 6. The updated analytical model Finally, considering the analytical solution presented in Section 3 including now the tool indentation, the natural thinning of the plate and the effective friction coefficients is implemented as described below.
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From finite element simulations it can be observed that when the indentation stops, the multi-breakage is initiated. The switching occurs when the mean radius of the plate equals the punch radius plus half of the plate thickness, which is a typical wrap-around situation. This progress has been translated in a computer program which works as follows: In the first phase, a classical 3-point bending situation is computed taking into account the indentation of the tool and the natural thinning of the plate. By doing so, an array of 250 thicknesses along the profile is calculated. This phase continues until the occurrence of the wrap-around effect. Further, with the predefined thickness array, the program switches over to the second phase where 4-point bending model is computed in case of its presence. In this phase, the natural thinning is still taken into account. Knowing the profile, the bending moment and the forces, the program subsequently computes the bend allowance, the springback, and the punch stroke with the formulas presented earlier.
Fig. 5.Example of finite element simulation. Die opening: 24 mm. Punch radius: 5 mm. Thickness: 3 mm. Material: St14. Bending angle: 88°.
Fig. 6.Computed profile based on the data from Fig. 5.
For several but not all cases, the program can correctly model the multi-breakage effect during the bending process. As an example, the case of St14 treated in Section 4.1 is presented here. The indentation (Fig. 3) is followed by the multi-breakage (Fig. 5). The corresponding profile computed with the program is shown in Fig. 6. The gap of 0.057 mm indicates that the multi-breakage effect is well represented by the model. 7. Tests and results In order to evaluate the performance of the model, a number of simulations has been done for computing the bending forces, the springback, and the bend allowance. The results have been compared to those provided by the circular approximation model. The tests have been conducted by Vorkov and are discussed in detail in [11]. In this contribution, we focus on the tests with St37 for which the constitutive law can be approximated by a Swift law. In total, there are 174 tests with plate thicknesses of 2, 4, 6, and 8 mm, punch radii of 10, 20, 30 and 40 mm, die width of 40, 50, 60, and 80 mm, and product angles of about 90 till 155°. Depending on the case, the simulation led to a multi-breakage solution or a wrap-around solution. The accuracy of results are evaluated by three indicators: the coefficient of determination R², the average absolute error εav and the relative error εrel as defined in [11]. The results are presented in Table 1. As it can be seen, the developed model in general outperforms the circular approximation model. Table 1. Overview of the comparison of the experimental data with the circular approximation and developed models
R² εav
Circular approximation model
Developed model
Bending force 0.98 104.1 N
Bending force 0.98 60.2 N
Springback 0.79 1.83°
Bend allowance 0.91 1.35 mm
Springback 0.81 0.82°
Bend allowance 0.92 1.29 mm
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εrel (%)
8.7
9.6
-6.7
5.0
4.3
8
-6.4
8. Conclusions Solving the characteristic differential equation for a material represented with a Swift hardening law, for constant thickness and classical friction coefficients, does not allow modeling the multi-breakage effect correctly. Therefore the bending model should consider the effect of tool indentation at the outset of the forming process, the natural thinning of the plate, and the effective friction coefficients. Result of this enhanced model is that for a number of cases the computed profile effectively predicts the multi-breakage phenomenon. At this stage of the study when the calculation does not provide a multi-breakage solution, the plate is assumed to wrap partially around the punch. This combined solution yields better results than the circular approximation. However, some further refinements are still necessary in order to improve the prediction accuracy. In particular, more advanced modeling of the indentation phase can be addressed. References [1] S. D. Benson SD, Press brake technology, A guide to precision sheet metal bending, Society of Manufacturing Engineers, Dearborn, Michigan, (1997). [2] S.D. Benson, A digital handbook for precision sheet metal fabrication, (2015), Retrieved from http://www.theartofpressbrake.com/a-digitalhandbook-for-precision-sheet-metal-fabrication/ on 18.09.2018. [3] V. Pouzols, Optimisation d'opérations industrielles de pliage par la méthode des éléments finis. Thèse de doctorat de l''Université de Grenoble, France, (2011). [4] I.A. Burchitz, Improvement of sprinback prediction in sheet metal forming, University of Twente, Enschede, The Netherlands, (2008). [5] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, The multi-breakage phenomenon in air bending process, Key Eng Mater, 611 (2014) 10471053. [6] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, Springback prediction of high-strength steels in large radius air bending using finite element modeling approach, Procedia Eng, 81 (2014) 1005-1010. [7] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, On the identification of a loading scheme in large radius air bending, Key Eng Mater, 639 (2015) 155-162. [8] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, Experimental investigation of large radius air bending, Int J Adv Manuf Technol, 92 (9) (2017) 3553-3569. [9] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, Two regression approaches for prediction of large radius air bending, Int J Mater Form., (2018) 1-12. [10] V. Vorkov, A. Konyukhov, D. Vandepitte, J.R. Duflou, Contact modeling of large radius air bending with geometrically exact contact algorithm, Journal of Physics: Conference Series, 734 (3) (2016) 032076. [11] V. Vorkov, R. Aerens, D. Vandepitte, J.R. Duflou, Analytical prediction of large radius bending by circular approximation, J Manuf Sci Eng, 140 (12) (2018) 121010. [12] R. Aerens, Characterisation of material by bending, Proceedings of the fifth International Conference on Sheet Metal: SheMet'97, (1997) 251262. [13] R. Aerens, S. Masselis, Air bending. Scientific and Technical Research Center of the Metal Fabrication Industry (CRIF/WTCM/SIRRIS) MC 110, Leuven, Belgium, (2000). [14] S.D. Mesarovic, N.A. Fleck, Spherical indentation of elastic–plastic solids, Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 455 (1987):2707-2728. [15] R. Hill, B. Storåkers, A. Zdunek, A theoretical study of the Brinell hardness test, Proceedings of the Royal Society of London A Mathematical and Physical Sciences, 423 (1865) 301-330. [16] I.M. Hutchings, The contributions of David Tabor to the science of indentation hardness, J Mater Res, 24 (3) (2009) 581-589. [17] R. Hill, The mathematical theory of plasticity. The Clarendon Press, Oxford, (1950). [18] F. Proksa, Plastisches biegen von blechen. Der Stahlbau, 28 (2) (1959) 29-36. [19] Z. Tan, Analysis and modelling of plastic bending processes, Luleå tekniska universitet, Sweden, (1994). [20] F. Yang, L. Yi, R. Feng, Thinning coefficient of sheet metal bending Forging & Stamping Technology, 3 (2009) 17. [21] V. Vorkov, Complete experimental data for large radius bending, (2017).