On the Use of Maximum Force Criteria to Predict Localised Necking in Metal Sheets under Stretch-Bending

Article On the Use of Maximum Force Criteria to Predict Localised Necking in Metal Sheets under Stretch-Bending Domingo Morales-Palma * ID , Andrés J. Martínez-Donaire ID and Carpóforo Vallellano ID Department of Mechanical Engineering and Manufacturing, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain; [email protected] (A.J.M.-D.); [email protected] (C.V.) *Correspondence: [email protected]; Tel.: +34-954-481355 Received: 10 October 2017; Accepted: 30 October 2017; Published: 2 November 2017 Abstract: The maximum force criteria and their derivatives, the Swift and Hill criteria, have been extensively used in the past to study sheet formability. Many extensions or modifications of these criteria have been proposed to improve necking predictions under only stretching conditions. This work analyses the maximum force principle under stretch-bending conditions and develops two different approaches to predict necking. The first is a generalisation of classical maximum force criteria to stretch-bending processes. The second approach is an extension of a previous work of the authors based on critical distance concepts, suggesting that necking of the sheet is controlled by the damage of a critical material volume located at the inner side of the sheet. An analytical deformation model is proposed to characterise the stretch-bending process under plane-strain conditions. Different parameters are considered, such as the thickness reduction, the gradient of variables through the sheet thickness, the thickness stress and the anisotropy of the material. The proposed necking models have been successfully applied to predict the failure in different materials, such as steel, brass and aluminium. Keywords: sheet-metal forming; stretch-bending; necking; maximum force criterion; bending effect 1. Introduction The maximum force principle has been extensively used in the past to study sheet formability. Considère’s maximum force criterion (MFC) states that diffuse necking is initiated in a tensile test of a bar when the maximum force is reached. The classical models of Swift [1] and Hill [2] are extensions of the MFC for the determination of necking in metal sheets subjected to different stretching conditions in the sheet plane. The former is applicable to the prediction of diffuse necking in the entire domain of the forming limit diagram (FLD). The latter is coupled with the initiation of strain localisation along a narrow band and is limited to predict localised necking in the left side of the FLD. Later, Hora et al. [3] extended the MFC to predict strain localisation for both sides of the FLD by including the contribution of the minor principal strain. The simplicity of necking criteria based on the maximum force principle make them very attractive. However, their predictions do not always agree with experimental results. Many extensions and alternatives to the MFC have been proposed to date, to deal with predicting localised necking. For instance, Bressan and Williams [4] suggested that failure occurs when the shear stress reaches a maximum. Hora et al. [5] proposed a phenomenological criterion, which states that once the maximum force is reached, the loading path gradually evolves towards plane strain and then localisation occurs. Brunet and Morestin [6] included the effects of damage to refine the material model for the prediction of the necking curve in the FLD. Recently, Aretz [7] assumed that localised necking does not necessarily occur when the axial force reaches a maximum, but rather when it reaches a Metals 2017,7, 469; doi:10.3390/met7110469 www.mdpi.com/journal/metals Metals 2017,7, 469 2 of 16 critical value. This method aimed to scale the Hill’s model so that the major strain predicted by the M–K model [ 8 ] or its experimental counterpart at plane-strain conditions (usually referred to as FLD0 ) was matched. The above criteria, which are collectively referred to as maximum force criteria (MFCs), were developed to predict necking in nearly stretching operations, neglecting implicitly the strain and stress gradients through the sheet thickness. However, in sheet metal forming processes such as stretch-bending or stamping with punches of mild or severe radii, it is well known that the occurrence of bending has a “beneficial effect” on the initiation of necking. In practice, the average strain through the sheet thickness (or the strain at the middle of the sheet surface) has been used to characterise failure criteria. However, the predictions of this approach, sometimes called the mid-plane rule (MPR), are in general very conservative. Research carried out to analyse the sheet failure in simultaneous stretching and bending conditions has followed a different approach from the maximum force principle. Tharrett and Stoughton [9] studied the bending effect on sheet failure and proposed a necking criterion referred to as the concave-side rule (CSR). This criterion establishes that the failure is initiated when the strain at the inner surface (concave side) of the sheet reaches a critical value at the bending zone, which matches the limit strain of in-plane stretching. This criterion notably improved the predictions of sheet failure in some materials compared to the MPR. In a later study, authors recognised that stresses are less sensitive to strain path changes, and consequently they reformulated the CSR in terms of stresses [10]. In a previous work, the present authors proposed a natural improvement of the CSR [ 11 ]. The model assumes that necking should be controlled by the development of damage in a material volume located at the inner side of the sheet. The study concludes that the critical size of this volume (represented by a certain critical distance from the sheet surface) should be related to the material microstructure. This work analyses the maximum force principle under stretch-bending conditions, proposing an analytical deformation model to characterise the stretch-bending process under plane-strain conditions. Different generalisations of the MFC for stretch-bending conditions are presented and discussed. As a result, the study suggests that the effect of the existence of strain and stress gradients on sheet thickness may be assessed by applying the MFCs locally at a given material’s critical distance from the inner surface. The proposed necking models are successfully applied to describe the failure by necking in sheets under stretch-bending conditions in different materials, such as 1008 AK steel, 70/30 brass, 6010 aluminium alloy and 7075-O aluminium alloy. 2. Maximum Force Criteria Considère’s analysis of the uniaxial tension of a metal bar states that diffuse necking appears when the maximum force is applied, that is, dF1= 0, with F1=σ1l2l3 . Swift extended this model to biaxial loading of a metal sheet of thickness l3=t, represented schematically in Figure 1, assuming a simultaneous maximum of both components of the force dF1= 0, dF2= 0, with F1=σ1l2t and F2=σ2l1t[1]. Figure 1. Schema of a stretched element. Metals 2017,7, 469 3 of 16 Hill also formulated a criterion to predict localised necking in metal sheets [ 2 ]. A simplified formulation of this model states that the localisation is initiated when the load per unit of width reaches its maximum: d f1= 0, with f1=σ1t . The application of the Hill criterion is restricted to the left-hand side of the FLD, that is, uniaxial-stretching. To predict localised necking in both sides of the FLD, Hora extended Considère’s model to biaxial loading of a metal sheet and considered that the major principal stress ( σ1 ) is a function of both major (ε1) and minor (ε2) principal strains in the plane of the sheet [3]. The above models of plastic instability lead to the formulation of the MFCs in a single and unified expression: 1 σY dσY dεeq =1 Z(1) where the left-hand side is a material property, sometimes referred to as the non-dimensional strain-hardening characteristic, whereas Z is the critical value for the subtangent of the stress–strain curve [ 12 ]. The former is evaluated using a stress–strain constitutive equation and the latter, by a yield function under plane-stress conditions, which can be characterised by the following parameters: α=σ2 σ1 ,β=dε2 dε1 ,ϕ=σeq σ1 ,ρ=dεeq dε1 (2) The expressions for the above parameters are given in Appendix Afor both Hosford and Mises yield criteria. It is not intended in this article to review the formulation of these well-known models. There are several works in the literature that analyse these and provide specific expressions of Equation (1) (e.g., [ 13 – 15 ]). Most of these assume a Hollomon law ( σY=Kεn eq ) to evaluate the aforementioned material characteristic and a Mises yield function to evaluate 1 /Z . For instance, the Hill necking criterion [2] under proportional load transforms Equation (1) to 1 σY dσY dεeq =n εeq and 1 Z=1+β ρ→n εeq =1+β ρ(3) which leads to algebraic expressions to evaluate the principal strains: ε1=n 1+β,ε2=βn 1+βand ε1+ε2=n(4) Figure 2presents an illustrative example of the application of the Swift [ 1 ], Hill [ 2 ] and Hora [ 3 ] criteria to predict necking in AA7075-O metals sheets. Both sides of Equation (1) have been evaluated by assuming a Hollomon law and a Mises yield criterion, respectively. Critical values of 1 /Z are represented in Figure 2a as functions of the strain path β under proportional loading conditions. Figure 2b presents necking predictions in the FLD along with the experimental data of localised necking reported by Martínez-Donaire et al. [16] for AA7075-O sheets of 1.6 mm thickness. The coefficient of the Hollomon law was found to be n= 0.21. As can be observed, both Hill and Hora criteria for localised necking reproduce the experimental data in their respective ranges of application reasonably well. Figure 2also presents the failure predictions of a modified Hill criterion based on Aretz’s approach [ 7 ]. Aretz assumed that necking is initiated when the axial force reaches a critical value, rather than its maximum value. The fundamental idea of this approach lies in the calibration of the critical load, which is obtained by scaling the prediction of Hill’s model (4) to fit the major strain at necking under plane-strain conditions (FLD0). Accordingly, Equation (1) is turned into n εeq =1+β ρ n FLD0 (5) Metals 2017,7, 469 4 of 16 which leads to ε1=FLD0 1+β,ε2=βFLD0 1+βand ε1+ε2=FLD0(6) −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 β = ε 2/ ε 1 0.0 0.2 0.4 0.6 0.8 1.0 1/ Z Swift Hill Hora Aretz −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 Minor strain, ε 2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Major strain, ε 1 β = −1 β = −0.5 β =0 β =1 FLD 0 S ift Hill Hora Aretz Experiments Figure 2. Necking predictions for AA7075-O sheets: ( a ) critical values of 1 /Z , and ( b ) forming limit diagram (FLD) with necking curves and experimental data [16]. 3. Stretch-Bending Deformation Model Under stretch-bending conditions, strain and stress gradients through the sheet thickness are induced in the metal sheet. In this situation, the MFCs must to be reformulated to take into account these gradients. The following deformation model is an extension of a stretch-bending model developed in a previous work [ 11 ]. This model assumed that the sheet adapts to the punch geometry in an earlier stage mainly by bending followed by a dominant stretching process until the sheet failure is reached. By simplicity, the evolution of the sheet curvature at the earlier stage is neglected, assuming this to be fixed and equal to the punch curvature. The model assumes that the deformation process is controlled by the reduction of sheet thickness. In order to be able to integrate analytically the radial equilibrium equation, which relates principal stresses through the sheet thickness, only a plane-strain condition ( dε2= 0) is assumed hereafter. The effect on sheet failure of the through-thickness stress is also taken into account. Figure 3shows the model variables in a sheet element located at the dome of the punch that is subjected to stretch-bending. The undeformed sheet dimensions are l0 and the initial thickness is t0 . In the deformed configuration, t is the current thickness, and the punch radius R matches the radius of curvature of the inner surface. In this configuration, θ is the bent angle, r is the radius of curvature of a given layer, z=r−(R+t/2) is the position of the layer measured from the middle surface, and ru is the radius of curvature of the unstretched surface. We note that all layers on the thickness are stretched ( ru<R ) when stretching dominates over bending. The current length of a generic layer, which initially had an undeformed length of l0=ruθ, is l=rθ. Metals 2017,7, 469 5 of 16 Figure 3. Variables in a deformed sheet element. The principal true strain distributions through the sheet thickness can be written as ε1=ln r ru,ε2=0, ε3=−ε1(7) where rucan be obtained from the material incompressibility condition of the sheet element: l0·t0= (R+t/2)θ·t(8) leading to ru= (R+t/2)·t/t0(9) Substituting Equation (9) into Equation (7) , the major strain can be expressed as a function of the variables tand r, and the parameters t0and R, as follows: ε1=ln r R+t/2 −ln t t0 (10) We note that the major strain at the outer and inner sides of the sheet ( ε1,out and ε1,in , as shown in Figure 3) are given by Equation (10) by substituting r=R+tand r=R, respectively. The distribution of principal stresses in the sheet thickness are obtained from the radial equilibrium equation [17]: rdσ3 dr =σ1−σ3(11) Given that dε1=dr/r from Equation (7) , it is convenient to change the variable r for ε1 in Equation (11) and write the radial equilibrium equation as dσ3 dε1 =σ1−σ3(12) The anisotropic non-quadratic yield criterion proposed by Hosford is assumed. This provides simplified expressions for the equivalent stress and strain under plane-strain conditions (see Appendix B): σeq =σ1−σ3 C(13) εeq =Cε1(14) where C is a function of the anisotropic parameters of the material. For instance, if the longitudinal or circumferential direction (axis 1) is aligned with the rolling direction of the sheet, Cis given by Metals 2017,7, 469 6 of 16 C=     (1+r0)1+r 1 a−1 90 a−1 r0+1+r 1 a−1 90 a−1     1 a (15) A material behaviour following the Hollomon power law is assumed: σY=Kεn eq (16) Thus, setting σeq =σY and substituting Equations (13) , (14) and (16) into Equation (12) , the radial equilibrium condition becomes dσ3 dε1 =K0εn 1(17) where K0=K·Cn+1. Integrating the differential Equation (17) , being σ3,out = 0 at the outer surface of the sheet, the distribution of σ3through the sheet thickness is obtained as σ3=−K0εn+1 1,out −εn+1 1 n+1(18) The circumferential stress gradient through the sheet thickness can be now determined from Equations (13), (14), (16) and (18) as σ1=K0 εn 1−εn+1 1,out −εn+1 1 n+1!(19) and its derivative is given by dσ1 dε1 =K0nεn−1 1+εn 1(20) On the other hand, the yield stress is found from Equations (12)–(14) and (17) as σY=1 C dσ3 dε1 =K0 Cεn 1(21) Differentiating Equations (13) and (14) and combining with Equations (17) and (20), one obtains dσY dεeq =1 C2dσ1 dε1−dσ3 dε1=K0 C2nεn−1 1(22) Finally, dividing the last two expressions, the non-dimensional strain-hardening function in stretch-bending for a certain layer on the thickness is given by 1 σY dσY dεeq =n Cε1 (23) We note that this expression is equivalent to Equation (3) for in-plane stretching. 4. Maximum Force Criteria in Stretch-Bending In this section, MFCs are generalised to stretch-bending conditions using the proposed deformation model. The analysis focuses on reformulating the Hill and Aretz necking criteria for metal sheets presented above. The axial force (per unit of width) can be calculated as: Metals 2017,7, 469 7 of 16 f1= R+t ZR σ1dr=σ1·t(24) where σ1 is the average major stress on thickness. Using the average strains ε1 and ε3=−ε1 to characterise the sheet deformation under plane-strain conditions, the derivative of the axial force is given by d f1 dt =σ1+tdσ1 dt =σ1+dσ1 dε3 =σ1−dσ1 dε1 (25) Thus, the Hill necking criterion (d f1=0) yields 1 σ1 dσ1 dε1 =1 (26) which in the actual formulation becomes 1 σY dσY dεeq =1 C(27) As can be seen, the right-hand side of Equation (27) is equal to the Hill criterion for in-plane stretching under plane-strain conditions, given by setting β= 0 in Equation (3) . We note that C=ρ in this situation. However, the left-hand side needs to be averaged over the sheet thickness. Similarly, the modification of the Hill criterion proposed by Aretz can be expressed as 1 σY dσY dεeq =n C·FLD0 (28) by introducing the correction factor n/FLD0. Hereafter, the above necking models will be referred to as Hill-based MFC-SB (Equation (27) ) and Aretz-based MFC-SB (Equation (28) ), where SB stands for stretch-bending. Both criteria are numerically evaluated using the deformation model described in Section 3. Computational Implementation In order to provide a self-contained document, some aspects of the computational implementation are summarised in this section. It is useful to use dimensionless parameters and variables in the deformation model. Thus, those previously defined in Figure 3become τ=t t0 ,ζ=z t,κ=R+t/2 t=τ t0/R+1 2(29) where τ is the thickness reduction, ζ is the non-dimensional position of a layer measured from the middle surface, κ is the relative curvature of the middle surface of the sheet, and t0/R is the bending ratio. Using these variables, the major strain given by Equation (10) is expressed as ε1=1+κ ζ τ(30) The Riemann integral is used to evaluate the average value of variables σY and εeq through the sheet thickness, that is, X={σY,εeq},X= 1/2 Z −1/2 Xdζ=1 N N ∑ j=0 X, with ζ=−1 2+j N(31) Metals 2017,7, 469 8 of 16 where the function Xis given by Equations (14) and (21), respectively. As previously stated, the deformation model neglects the bending process that occurs in an earlier stage of the stretch-bending operation and assumes that thickness reduction τ is the only variable. Thus, the iterative calculation procedure consists of finding τ until a necking criterion is reached, for a given value of the bending ratio t0/R . To avoid negative values of variables, the initial step assumes that major strain is positive through the whole thickness. Because of the strain gradient, this condition is always satisfied when ε1,in ≥0. In this limit, that is, ε1,in =0, this leads to a thickness reduction: τ0=R t0sR 2t0−1 (32) At the end of every thickness decrement ( ∆τ< 0), the non-dimensional strain-hardening function is calculated as λ(i+1)=1 σ(i+1) Y ∆σ(i+1) Y ∆ε(i+1) eq =1 σ(i+1) Y σ(i+1) Y−σ(i) Y ε(i+1) eq −ε(i) eq ,i={0, 1, 2...}(33) where ε(i+1) eq and σ(i+1) Y are determined by Equation (31) . Accordingly, the iterative procedure is established as follows: 1. Compute ε(0) eq and σ(0) Yat initial stage, where τ(0)=τ0. 2. Compute the non-dimensional strain-hardening function λ(i+1)for τ(i+1)=τ(i)+∆τ. 3. If λ(i+1)≤1/Z, then necking is attained; else repeat step 2. Depending on the selected Hill- or Aretz-based MFC-SB criterion, Z is C or C·FLD0/n , respectively. Typical values for discrete parameters are N=100 and ∆τ=−0.001. 5. Critical-Distance Rule for Necking In a recent research work [ 11 , 18 , 19 ], the present authors developed a mesoscopic approach to predict failure in stretch-bent sheets. The proposed necking model combines the concepts of CSR and critical distance to predict the failure of the sheet. The model assumes that necking is controlled by the development of damage in a certain material volume located at the inner side of the sheet. The size of the critical volume is assumed to be a material constant, which can be related with the microstructure of the material. According to the above idea, the less-stressed material at the inner zone is responsible for containing the plastic instability of the entire sheet thickness. In previous works, the sheet failure was characterised in terms of principal strains [ 19 ] or principal stresses, which was more appropriate to analyse non-proportional strain paths [ 11 ]. In both cases, the experimental data were successfully analysed. Following the MFC concepts, in the present work, the use of the non-dimensional strain-hardening function (left-hand side of Equation (1) ) is proposed to assess necking initiation under stretch-bending conditions. Figure 4shows schematically the evolution of the strain-hardening function through the sheet thickness at the onset of necking. To rationalise the present model, the sheet can be assumed to be formed by the superposition of layers or fibres in the thickness, all having the mechanical behaviour of the base material. For a given layer located at a distance d measured from the inner side of the sheet, alocal stability index can be defined as lsi(d) = 1 σY dσY dεeq d 1/Z(34) where 1 /Z depends on the failure criteria; for example, in the case of using the Aretz approach, 1 /Z is n/(C·FLD0). Metals 2017,7, 469 9 of 16 Thus, layers exhibiting lsi less than unity are assumed to be layers that are not able to resist the plastic instability of the sheet thickness. We note, from Figure 4, that this condition is first reached at the upper side and propagates downwards in thickness during the forming process. Otherwise, fibres having a lsi value greater than unity are considered to be fibres that contain the plastic instability of the entire thickness. As can be seen, this material extends from the bottom layer to a depth dcr , called here the critical distance, at which lsi takes the unit value. This material volume prevents the sheet from necking. According to the above description, the failure by necking of the sheet will occur when, at a certain critical distance dcr from the inner side of the sheet, the local stability index becomes equal to the unit value. As is discussed in the next section, the value of dcr is influenced only slightly by the bending ratio in stretch-bending; thus it can be considered in practice a material property to be determined experimentally. This criterion is called here the critical-distance rule (CDR) by analogy to those previously proposed in the literature, such as the MPR, CSR and convex-side rule (CxSR). Figure 4. Critical-distance rule. 6. Practical Application and Discussion This section evaluates the capability of the failure criteria described above to predict localised necking under stretch-bending conditions. The experimental results analysed were found from the literature and have already been used in a previous work [11]. Briefly, the experiments involved stretch-bending tests under plane-strain conditions using cylindrical punches of different radii. The materials were 1008 AK steel, 70/30 brass, and 6010 aluminium, from experimental work conducted by Tharrett and Stoughton [ 9 , 20 ], and 7075-O aluminium, from the research carried out by Martínez-Donaire et al. [ 16 ]. All specimens failed in the zone in contact with the punch, under simultaneous bending and streching conditions. The material properties are reported in Table 1. Table 1. Mechanical properties and material constants. 1008 AK Steel 70/30 Brass AA6010 AA7075-O t0(mm) 1.04 0.81 0.89 1.6 σ0(MPa) 187.0 112.5 202.0 102.3 K(MPa) 556.8 809.1 543.9 400.3 n0.24 0.50 0.24 0.25 r01.740 0.870 1.590 0.812 r90 1.800 0.730 1.760 1.317 a6 8 8 8 FLD00.358 0.358 0.166 0.251 Figure 5presents the experimental results provided in the mentioned references. Figure 5a–c depicts the major strains measured on the convex side of the sheet ε1,out as a function of the current bending ratio t/R . Instead, Figure 5d shows ε1,out versus the initial bending ratio t0/R . The cases for which a visible neck was observed are represented as solid circles. Otherwise, open circles are used. The values of FLD0(see Table 1) are represented as solid stars. Metals 2017,7, 469 16 of 16 References 1. Swift, H.W. Plastic instability under plane strain. J. Mech. Phys. Solids 1952,1, 1–18. 2. Hill, R. On discontinuous plastic states, with special reference to localized necking in thin sheets. J. Mech. Phys. Solids 1952,1, 19–30. 3. Hora, P.; Tong, L.; Reissner, J. A prediction method for ductile sheet metal failure using FE-simulation. In Proceedings of the NUMISHEET, Dearborn, MI, USA, 29 September–3 October 1996; pp. 252–256. 4. Bressan, J.D.; Williams, J.A. The use of a shear instability criterion to predict local necking in sheet metal deformation. Int. J. 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