revisiting single point incremental forming

Manuscript

REVISITING SINGLE POINT INCREMENTAL FORMING & FORMABILITY/FAILURE DIAGRAMS BY MEANS OF FINITE ELEMENTS AND EXPERIMENTATION Silva M. B.(1), Skjoedt M.(2), Bay N.(2) and Martins P. A. F.(1, *)) (1)

IDMEC, Instituto Superior Tecnico, TULisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal

(2)

Technical University of Denmark, Department of Mechanical Engineering, DTU - Building 425, DK-2800, Kgs. Lyngby, Denmark

(*) Corresponding author: Fax: +351-21-8419058 E-mail: [email protected]

ABSTRACT In a previous work [1], the authors presented an analytical framework, built upon the combined utilization of membrane analysis and ductile damage mechanics, that is capable of modelling the fundamentals of single point incremental forming (SPIF) of metallic sheets. The analytical framework accounts for the influence of major process parameters and their mutual interaction to be studied both qualitatively and quantitatively. It allows concluding that the likely mode of material failure in SPIF is consistent with stretching, rather than shearing being the governing mode of deformation. The study of the morphology of the cracks combined with the experimentally observed suppression of neck formation enabled the authors to conclude that traditional forming limit curves are inapplicable to describe failure. Instead, fracture forming limit curves should be employed to evaluate the overall formability of the process. The aim of this paper is twofold: (i) to assess the mechanics of deformation of SPIF, namely the distribution of stresses and strains derived from the analytical framework against numerical estimates provided by finite element modelling and (ii) to assess the forming limits determined by the analytical framework against experimental values. It is shown that agreement between analytical, finite element and experimental results is good, implying that the previously proposed analytical framework can be utilized for explaining the mechanics of deformation and the forming limits of SPIF.

Keywords: Single point incremental forming, membrane analysis, forming limits, finite element method, experimentation.

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NOTATION   - circumferential stress   - meridional stress

 t - thickness stress  Y - yield stress

 - half cone angle of the component  - draw angle between the inclined wall and the initial flat configuration of the sheet

t - thickness of the sheet t 0 - initial thickness of the sheet rtool - radius of the SPIF tool

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1. INTRODUCTION The idea of using incremental forming to produce sheet metal parts was first documented in the patent of Leszak [2] and reflected an extension of the conventional spinning process. Single point incremental forming (SPIF) was developed from the incremental forming concept and was proven to be feasible by Kitazawa et al. [3] in manufacturing rotational symmetric parts in aluminium using a special purpose CNC machine-tool. The capability study of using an ordinary CNC milling machine-tool, instead of a special purpose CNC machine-tool, which was later performed by Jeswiet et al. [4] and Micari et al. [5] was the starting point for the successful and rapid development of the process. In the past years, most studies on SPIF have concerned experimental investigations on applications and forming limits, and besides that a limited number of finite element studies have been carried out. The keynote paper by Jeswiet et al. [6] presents a comprehensive state-of-the-art review and includes a thorough list of references on the most significant, published research work in the field. Results from finite element investigations are utilized to address the mechanics of deformation and failure is related to both the sine law and the spinnability relation due to Kegg [7] (that is, giving emphasis to the importance of axis-parallel shear as in case of shear spinning). In the last years the governing mode of deformation in SPIF has been subject of controversy in the metal forming community [8]. Some authors claim that deformation takes place by stretching instead of shearing while others claim the opposite, but assertions are mainly based on ‘similarities’ with well-known processes of stamping and shear spinning rather than experimental evidence from SPIF itself. However, as recently shown by the authors [1] the examination of the likely mode of material failure at the

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transition zone between the inclined wall and the corner radius of the sheet is consistent with stretching, rather than shearing, being the governing mode of deformation in SPIF. As regards the forming limits of SPIF there are two different views. The commonly accepted view considers (i) that formability is limited by necking; (ii) that the forming limit curve (FLC) in SPIF is significantly raised against conventional FLC’s being utilized in the analysis of sheet metal forming processes (e.g. stamping, deep drawing ...etc) [5] and (iii) that the raise in formability is due to a large amount of through thickness shear [9, 10] or, instead, due to serrated strain paths arising from cyclic, local plastic deformation [11]. This approach (which hereafter is referred to as the ‘necking line of attack’ - NLA) is adopted in most of the numerical and experimental contributions to the understanding of formability in SPIF that were published in the past years. The alternative, and non-traditional, view of formability in SPIF recently proposed by the authors [1] and supported by Cao et al. [12] and Emmens and Boogaard [13] considers (i) that formability is limited by fracture without experimental evidence of previous necking, (ii) that the suppression of necking in conjunction with the low growth rate of accumulated damage is the key mechanism for ensuring the high levels of formability in SPIF and (iii) that FLC’s, that give the loci of necking strains, are not relevant and should be replaced by the fracture forming limits (FFL’s). This approach will be hereafter referred as the ‘fracture line of attack’ – FLA. As it will be shown later in the presentation, when suppression of necking is included in the analysis many shortcomings of the commonly accepted view on deformation and failure are removed. In particular, the experimental evidence that forming limits can be approximated, in the principal strain space, by lines on the form 1   2  q placed well above conventional FLC’s is something difficult to explain under the assumption that failure is limited by previous necking. In fact, if through thickness shear or serrated strain

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paths arising from cyclic, local plastic deformation, could be capable of increasing the forming limits of AA1050-O to a level of approximately 6 times of that experimentally found by means of tensile, elliptical and circular bulge tests [14] this would mean that the individual effect of stresses and strain paths of SPIF on the FLC’s would be much larger than what is seen in conventional sheet metal forming processes. Moreover, recent experimental results performed by the authors comparing SPIF using forced tool rotation [15] with SPIF using free tool rotation, which most of the time leads to no rotation [14], showed no influence on the overall formability. This observation leads to the conclusion that the influence of circumferential friction resulting from the contact between the tool and the sheet is negligible, a result in close agreement with previous assumptions of the authors concerning frictional effects [1]. This result seems to indicate that the level of through thickness shear arising from the contact with friction between a single point forming tool and a small thickness sheet is not very significant. The present paper is directed at revisiting the analytical framework of SPIF that is based on membrane analysis with in-plane contact friction forces and the forming limits derived from ductile damage mechanics in the light of numerical results provided by finite element modelling and experimental measurements obtained by fundamental materials formability tests as well as formability testing in SPIF. It is shown that the agreement between analytical, finite element and experimental results is good, implying that the previously proposed analytical framework can be easily and effectively utilized for explaining the mechanics of deformation and the forming limits of SPIF.

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2. ANALYTICAL BACKGROUND Figure 1 presents the basic components of SPIF; (i) the sheet metal blank, (ii) the blankholder, (iii) the backing plate and (iv) the rotating single point forming tool. Circle grid analysis combined with observation of smear-mark interference between the tool and the sheet surface allow the classification of all possible tool paths as combinations of the basic modes of deformation that are depicted in Figure 2; (A) flat surfaces under plane strain stretching conditions, (B) rotational symmetric surfaces under plane strain stretching conditions and (C) corners under equal bi-axial stretching conditions. It is worth to notice that in-between these modes of deformation there are other possibilities where neither plane strain stretching nor equal bi-axial stretching appear. The analytical framework of SPIF to be revisited in this section of the paper is focused on the aforementioned extreme modes of deformation that are likely to be found in SPIF. The model was recently published by the authors [1] and due to this the presentation is here only shortly summarized.

2.1 State of stress and strain In SPIF the small localized plastic zone is subjected to normal forces, shear forces and bending moments, so that it conforms to the hemispherical shape of the tip of the pin tool, forming a contact area (A, B or C in Figure 2) between the tool and the part of the sheet placed immediately ahead of the moving tool. The state of stress acting in these areas can be derived from the membrane equilibrium conditions if bending moments are neglected and circumferential, meridional and thickness stresses are assumed to be principal stresses. Further simplifying assumptions are the following: the material is assumed to be rigid-perfectly plastic and

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isotropic and the resultant friction stress acting in the tool-sheet contact interface is assumed to consist of two in-plane components - a meridional component     t due to the downwards movement of the tool and a circumferential component     t due to the circumferential feed combined with the rotation of the tool. This last assumption, which is an untraditional way of modelling friction introduced for convenience, implies that the coefficient of friction    2   2 . Table 1 resumes the strains and stresses along the principal directions that are derived from the analytical framework of SPIF. As mentioned before, the model assumes flat and rotationally symmetric surfaces (A and B in Figure 2) to be formed under plane strain conditions, d   0 , and corners (C in Figure 2) to be formed under equal bi-axial stretching, d   d   0 . The inclined wall of the sheet adjacent to the forming tool is loaded under elastic uniaxial tension conditions. Further details with complete derivation of the equations can be obtained in [1], where it is demonstrated that friction in case of lubrication (μ ≤ 0.1) has minor influence on the stresses and thus can be neglected.

2.2 Forming limits The study of the morphology of the cracks together with measurements of thickness along the cross section of SPIF parts revealed that plastic deformation takes place by uniform thinning until fracture without experimental evidence of localized necking taking place before reaching the onset of fracture [1]. The suppression of localized necking in SPIF is due to the inability of necks to grow. If a neck were to form at the small plastic deformation zone in contact with the incremental forming tool, it would have to grow around the circumferential, bend path that circumvents the tool. This is difficult and creates problems of neck development. Even if

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the conditions for localized necking could be met at the small plastic deformation zone in contact with the tool, growth would be inhibited by the surrounding material which experiences considerably lower stresses. This implies that forming limit curves (FLC’s) of conventional sheet metal forming are inapplicable to describe failure in SPIF. Instead, fracture forming limit curves (FFL’s) showing the fracture strains, placed well above the FLC’s, should be employed. The fracture forming limits in SPIF can be characterized by means of ductile damage mechanics based on void growth models. Assuming the Tresca yield criterion, linear loading paths, and that the damage function f  m   takes the simple form of the triaxiality ratio m  (Ayada et al. [16]), the total amount of accumulated damage for plane strain and equal bi-axial stretching SPIF conditions results in the following critical damage values:

Dc 

f

m 1  r  t  plane strain d   tool  1  2  rtool  t  0



(1)

f

 2 r t  bi axial D c   m d   tool  2 1  3 r  2 t  tool  0

If the critical value of damage D c at the onset of cracking is assumed to be pathindependent, solving equations (1) for  1 results in the following identity:



bi axial plane strain 1 1 bi axial 2





0



 rtool  t  3  rtool  2t  r 5  tool    2  4  rtool  t   rtool  t     t r 3  rtool  2t  3  tool   4  rtool  t   t

 2   6 

(2)

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Equation (2) gives the slope of the fracture forming line (FFL) in the principal strainspace (1 ,  2 ) . For typical experimental values of rtool / t

in the range 3-50

(corresponding to rtool / t 0 in the range 2 to 10 for the investigated material) the slope derived from equation (2) will vary between -1.1 and -1.6. This supports the assumption that the fracture forming limit in SPIF can be approximately expressed as 1   2  q , where  t  q is the thickness strain at the onset of fracture under plane strain conditions and ε1 and ε2 are the major and the minor principal strain in the plane of the sheet respectively. As it will be shown later in section 4, this result is in close agreement with the typical loci of failure strains in conventional sheet forming processes, where the slope of the FFL is often about -1 [17, 18]. It is worth noting that the proposal regarding the suppression of localized necking in SPIF proposed by the authors in [1] is consistent with a recent published work by Emmens and Boogaard [13]. The main difference between the approaches is related to the physics behind the suppression of necking. The authors [1] claim that there is an inability of necks to grow around the circumferential, bend path that circumvents the tool in a zone surrounded by material undergoing elastic deformation while Emmens and Boogaard [13] claim that suppression of necking is a consequence of repeated bending and unbending loading cycles arising from the tool path.

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3. FINITE ELEMENT BACKGROUND Numerical modelling of SPIF was performed using the commercial finite element computer program LS-DYNA (version ls971s). LS-DYNA is based on an explicit-dynamic elasto-plastic formulation and has been reported by the authors as well as other researchers to be capable of taking into account the practical non-linearities in the geometry and material properties that are typical of SPIF to produce good predictions of shape, strain and thickness distributions throughout the SPIF parts [14, 19].

3.1 Modelling conditions The FE model of the sheet blanks is built upon an initial course mesh of 26 x 26 shell elements (type 16 in LS-DYNA) each having a side length about 9.7 mm for a typical part size of 253 x 253 mm. This shell type is not a fully 3D stress element since stresses in the direction normal to the shell surface is always zero ( 33  0 in the local coordinate system). The element is therefore only suitable for calculating stresses in the plane of the shell and the output for the third principal stress is not expected to provide a particularly accurate estimate. Adaptive mesh refinement is utilized throughout the computation in order to limit the interference between the sheet and the contours of the forming tool and of the backing plate as well as to obtain high levels of accuracy both in terms of geometry and distribution of field variables. The adaptive mesh refinement procedure consists of three refinement operations ending up by splitting the original elements into 64 new elements that have 1/8th of the initial element size. This is known to generate incompatible meshes that must be treated by multipoint-constraints in the transition region. The mesh refinement is done ahead of the tool in order to insure a fully refined mesh in the zone of plastic deformation. A full integration scheme is used with five integration points over the sheet thickness.

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The description of the forming tool and of the backing plate is performed by means of surface meshes. Both active tool components are considered rigid and a large number of elements are utilized for modelling its geometry in order to reduce the level of roughness that is artificially introduced by the overall discretization procedure. The movement of the tool in the FE model is identical to that in the actual SPIF process including the rotation and the helical path, which is defined by means of a large number of discrete points. The number of points is determined by the tolerance setting in the CAM program. A tolerance of 0.01 mm was used resulting in about 400 points per meter for the hyperbolic cone and 8 points per meter (the sides are straight) for the pyramid. Vertical step-down was set equal to 0.5 mm and the coefficient of friction adopting Amonton-Coulomb’s law was assumed to be zero (μ = 0) in order to make a fair comparison with the analytical model. Acceleration of the overall CPU time was performed by means of a load-factoring (or time scaling) procedure that changed the rate of loading by an artificially increase in the velocity of the single point forming tool by a factor of 1500 times as compared to the real forming velocity. The time step for performing the explicit (central difference) time integration scheme was set to 3.0x10-7s by use of mass scaling. As a precaution LSDYNA uses 0.9 times this value to guarantee stability. In practical terms this results in an average time step per increment of approximately 2.7x10-7s. The material of the sheet was considered as rigid-perfectly plastic,  Y  100 MPa, in order to allow comparison between the results predicted by LS-DYNA and those calculated by means of the analytical framework developed by the authors. No anisotropy effects were taken into consideration. Finite element simulation of SPIF under the above mentioned modelling conditions is computationally very intensive and, therefore, a full-scale model (that is, a model that

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does not take advantage of the existing symmetric conditions of the benchmark SPIF parts under investigation) required 120 to 240 hours of CPU in a 900 MHz computer or 30 to 60 hours of CPU in a 4-core processor computer.

3.2 Post-processing of results After finishing the finite element simulations of SPIF, calculated values of thickness were taken from different sets of shell elements located in the small plastic contact zone between the single point forming tool and the sheet surface. The calculated thickness values were utilized for determining the analytical distribution of the stress field from the equations listed in Table 1. Furthermore, values of the components of thickness, circumferential and meridional strains and stresses calculated by finite elements were taken from selected areas of the formed geometry for comparison with the results provided by the analytical model (Figure 3). All values were obtained by averaging finite element results from the five integration points in thickness and at different, arbitrary locations corresponding to the contact areas A, B and C in Figure 2 and the inclined wall surface of the sheet adjacent to the forming tool. Due to large variation in the five integration points directions of the principal stresses could not be verified, i.e. only size is compared. The elements of each set are positioned along the meridional direction and are located at a safe distance from the backing plate (more than 10 times the initial sheet thickness apart from it).

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4. EXPERIMENTS The introduction to this section describes the experimental techniques utilized for obtaining the material forming limits and the material fracture forming limits. Subsequently the procedure applied in the SPIF tests is outlined. All specimens were made from AA1050-H111 sheet blanks with 1, 1.5 and 2 mm thickness.

4.1 Forming and fracture forming limit diagrams Formability of the sheet material was evaluated by means of tensile tests (using specimens cut at 0, 45 and 90 degrees with respect to the rolling direction) and bi-axial, circular (100 mm) and elliptical (100/63 mm) hydraulic bulge tests (Figure 4). The experimental technique utilized for obtaining the FLC involved electrochemical etching a grid of circles with 2 mm initial diameter on the surface of the sheets before forming and measuring the major and minor axis of the ellipses that result from the plastic deformation of the circles during the formability tests. The FLC was constructed by taking the strains  1 , 2

 at failure from grid-elements placed just outside the neck (that

is, adjacent to the region of intense localization) since they represent the condition of the uniformly-thinned sheet just before necking occurs. The experimental technique is described elsewhere [20] and the resulting FLC is plotted in Figure 4. Although the effect of sheet metal thickness normally is to raise the FLC’s (that is, the thicker the sheet, the higher the FLC) it was decided to build the curve included in Figure 4 from the entire set of formability tests on the three different sheet thicknesses investigated. The arguments for doing this is that the necking strains turned out to be similar for specimens with 1, 1.5 and 2 mm thickness and that the FLC depicted in Figure 4 will only be utilized for qualitative analysis in the forecoming sections of the paper.

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The intersection of the FLC with the major strain axis is found to occur at 1  0.07 in fair agreement with the value of the strain hardening exponent of the stress-strain curve obtained by means of tensile tests,

  140  0.041 MPa

(3)

The experimental FFL is more difficult to construct than the FLC. Application of grids even with very small circles in order to obtain strains in the necking region after it forms and, therefore, close to the fracture, provides strain values that cannot be considered the fracture strains. Moreover, such grids create measurement problems and suffer from sensitivity to the initial size of the circles used in the grids due to the inhomogeneous deformation in the neighbourhood of the crack. As a result of this, the experimental procedure for constructing the FFL requires measuring of thickness at fracture in order to obtain the ‘gauge length’ strains. The adopted procedure involved measuring the length increase parallel to the crack using the grid technique in order to obtain the strain in this direction. The thickness strain was determined by measuring the sheet thickness in a microscope at several places along the crack and the third fracture strain component, in the plane of the sheet and direction perpendicular to the crack, was determined by volume constancy knowing the two other strains. This procedure is, besides being time consuming, increasingly difficult to execute as the thickness of the specimens becomes smaller. In the present investigation ‘gauge length’ strains and the corresponding FFL were determined only from the materials tests and SPIF formability tests performed on specimens with 2 mm initial thickness. This is acceptable because, in the strict fracture mechanics sense, the position of the FFL in the principal strain space will shift upwards or downwards on the positive strain axis only as a function of the ductility of the material.

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However, comparing this assumption and related experimental procedure with equation (2) one may question if the FFL should be treated as a ‘material dependent line’ or a ‘material plus process dependent line’. In the second case the FFL would shift upwards or downwards depending on process variables such as the thickness and the radius of tooling (equation (2)), which directly affect the level of accumulated damage. Yet, because the distance between the FLC and the FFL is so big it is reasonable to neglect any influence from process variables and to treat the FFL as a ‘material dependent line’. The experimental FFL is plotted in Figure 4 and can be approximated by a straight line

1  0.79  2  1.37 falling from left to right, which is close to the condition of constant through-thickness strain at fracture (given by a slope of ‘-1’). The large distance between neck formation (FLC) and fracture (FFL) indicates that AA1050-H111 is a very ductile material that allows a considerable through-thickness strain between neck initiation and fracture. At the onset of local instability implying transition from the FLD representing diffuse necking towards the FFL a sharp bend occurs in the strain path when testing is done with conventional bulge tests, Figure 4, The strain paths of bi-axial circular and elliptical bulge formability tests show a kink after neck initiation towards vertical direction, corresponding to plane strain conditions, as schematically plotted by the grey dashed line for the circular bulge formability test. The strain-paths of tensile formability tests also undergo a significant change of strain ratio from slope -2 to a steeper one although not to vertical direction. The absence of a sharp kink of the strain path into vertical direction in formability testing in tension but a less abrupt bend instead is due to the fact that major and minor strains after the onset of necking do not coincide with the original pulling direction. A comprehensive analysis on the direction of the strain-paths in the tension-compression strain quadrant can be found in the work of Atkins [21].

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4.2 Single point incremental forming The experiments were performed in a Cincinnati Milacron machining centre equipped with a rig, a backing plate, a blankholder for clamping the sheet metal blanks and a rotating, single point forming tool (Figure 1). The forming tool had a diameter of ø12 mm and a hemispherical tip was made of cold working tool steel (120WV4-DIN) hardened and tempered to 60 HRc in the working region. The speed of rotation was 35 rpm and the feed rate 1000 mm/min. The tool path was helical with a step size per revolution equal to 0.5 mm and was generated with the program HeToPaC [22]. The lubricant applied between the forming tool and the sheet was diluted cutting fluid. The experiments were designed with the objective of measuring the strain values at fracture and the tests were performed in truncated conical and pyramidal shapes characterized by stepwise varying drawing angles  with the depth (Figure 5). A grid with 2 mm circles was electrochemically etched on the surface of the sheets allowing the principal strains to be measured. The procedure considers the thickness strain  t to be a principle strain and assumes the sheet blank to be deformed by membrane forces so that it conforms to the shape of the tool path. The experiments were done in random order and at least two replicates were produced for each combination of thickness and geometry in order to provide statistical meaning.

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5. RESULTS AND DISCUSSION The first part of this section examines the mechanics of deformation of SPIF derived from the analytical framework and compares the theoretical distribution of strains and stresses with the numerical estimates provided by finite element analysis. The second part is focused on the assessment of the forming limits proposed by the analytical framework against experimental measurements of SPIF parts at failure.

5.1 Stress and strain fields Figure 6 shows the analytical and finite element estimates of the stress and strain fields in the small localized plastic zone during SPIF of a truncated conical shape of a rigid perfectly plastic metal sheet (  Y  100 MPa) with 1 mm initial thickness. The distribution of the normalized effective stress computed by finite element analysis (Figure 6a) confirms the incremental and localized characteristics of the deformation. Plastic deformation occurs only in the small radial slice of the component being formed under the tool. The surrounding material experiences elastic deformation and, therefore, is subjected to considerably lower stresses. These observations together with the progressive decrease of the applied stresses along the inclined wall of the SPIF part (from the transition point C to point D near the backing plate, Figure 3) is in good agreement with the predictions derived from the analytical framework, (Table 1, section 2). The comparison between the finite element estimates and the analytical values of the meridional ( 1    ), circumferential (  2    ) and thickness (  3   t ) stresses in the plastic deformation zone in three different locations placed along the small plastic

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deformation region further corroborates the applicability of the analytical framework developed by the authors (Figure 6c). Major differences between numerical and analytical estimates are found to occur in the distribution of thickness stress  t and may have two likely explanations. Firstly, shell elements are known to experience difficulties when the amount of through-thickness stress resulting from contact pressure in small radii is significant (as described in section 3.1). Secondly, the analytical model considers a smooth flat transition between the corner and the undeformed bottom region of the SPIF part while the experimental observations and the finite element calculations show a small concave depression (that is, a ‘circumferential valey’) at the transition region. The finite element evolution of thickness along the small plastic deformation region placed under the forming tool shows that reduction in thickness (down to a value of

t  0.45 mm) tends to balance the rise in the meridional stress   . This result is in good agreement with the analytical framework that considers the product   t to be constant in the plastic deformation region [1]. The distribution of major and minor true strains in the principal strain space obtained by means of finite element analysis confirm that SPIF of a truncated conical shape is performed under plane strain conditions because all strains lie close to the major strain axis (Figure 6b). Results shown in Figures 7 and 8 also indicate a good agreement between finite elements and analytical results for the SPIF of pyramidal shapes. It is worth to notice that the sides of the pyramids are formed under plane strain conditions (Figure 7b) while the corners are shaped with strain-paths deviating towards equal bi-axial stretching conditions (Figure 8b). This last observation is important to the overall validation of the

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analytical framework because it confirms the extreme modes of deformation that were necessary to assume during the theoretical developments [1] (Figure 2). However, a closer observation of the finite element results included in Figure 8b allows to conclude that although bi-axial stretching occurs in the corners, it does not appear as balanced, equal bi-axial deformation due to heavier constraints in tangential direction  than in meridional direction  . This is due to the variation in rigidity of the elastic region surrounding the local plastic deformation zone. In fact, the region above the deformation zone is fully deformed, thus having minimum thickness, whereas the elastic zone at the same depth as the deformation zone, but larger or smaller  than that, has larger average thickness. In other words, the deviation from equal bi-axial into non-equal biaxial stretching at the corners of the pyramid is probably due to the fact that stiffness in the circumferential direction is much higher than in the meridional direction.

5.2 Forming limits The principal strain space in Figure 4 includes the strain signatures characteristic of necking as well as those of fracture. The concept of the FFL is, that all possible fracture strains are located on a specific line, which characterizes the material in contradiction to the concept of the FLC, which is sensitive to the strain path. Figure 4 contains three different FFL’s. The black solid line, denoted as ‘FFLexperimental’, is obtained from the fracture strains measured in the experimental tensile and bi-axial bulge tests (refer to section 4.1). The black thin solid line, denoted as ‘FFLconstant thickness strain’, is the theoretical fracture line falling from left to right in the strain space with a slope equal to -1. The black dashed line, denoted as ‘FFL-SPIF’ is the fracture line derived from the critical value of damage D c at the onset of cracking for

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the analytical distribution of stresses and strains that was proposed by the authors (refer to equation (2) in section 2.2). The agreement between the three above-mentioned FFL’s is good and so is the agreement between these lines and the experimental fracture strains measured for the conical and pyramidal shapes made from AA1050-H111 sheets of 1 and 2 mm initial thickness too. The largest deviations are found to occur adopting the ‘FFL-SPIF’ line, which can be explained by the fact that this line was constructed under the assumption of a rigid perfectly plastic material. It is worth to notice that the experimental values of fracture strains for the pyramidal shapes are not placed on the equal bi-axial strain ratio line with slope +1 in the principle strain space. In fact, although the onset of failure is located at corners of the pyramids the values of fracture strains are somewhat deviating towards the plane strain direction. This verifies the earlier observation from the finite element analysis concerning the occurrence of non-equal bi-axial deformation instead of equal bi-axial deformation due to heavier constraints in tangential direction  than in meridional direction  . Finally it is important to make reference to the solid grey square placed exactly on top of the major strain axis. The fracture strain of this point was calculated from the experimental measurement of the maximum depth at fracture by means of a procedure that was comprehensively described by the authors in their previous work [1]. As seen, the agreement between theory and experiments is good as regards the fracture strains of the conical shapes that are formed under plane strain assumptions. Concerning the fracture strains at the corners of the pyramids, the agreement with the FFL’s is good but deviations appear between the experimental measurements and the estimates calculated from the experimental value of maximum depth under equal bi-axial stretching assumptions. The reason for this deviation may, once again, be attributed to the assumption of equal bi-axial stretching.

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Figure 9 presents finite element estimates of the meridional   and circumferential   stresses in three different locations placed along the inclined wall surface of a truncated conical shape. As seen, the meridional stress   is lower than the yield stress and its value decreases along the inclined wall being higher at the transition point C and smaller at point D (C and D are defined in Figure 3). The circumferential stress   is close to zero and, therefore, can be neglected. This result confirms that the inclined wall surface of the sheet adjacent to the forming tool is elastic and further corroborates the applicability of the analytical framework developed by the authors.

6. CONCLUSIONS This paper presents an evaluation of the applicability and accuracy of the analytical framework for single point incremental forming (SPIF) of metal sheets that was recently proposed by the authors [1]. The investigation made use of finite element analysis and of theoretical and experimental techniques targeted at the construction and utilization of fracture forming limit diagrams (FFLD’s) and fracture forming lines (FFL’s). Results confirm that the governing mode of deformation is stretching, that formability is limited by fracture without previous necking and that the analytical framework is capable of successfully and effectively addressing the mechanics of deformation and the forming limits of the process.

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ACKNOWLEDGEMENTS The first author and the corresponding author would like to acknowledge PTDC/EMETME/64706/2006 FCT/Portugal for the financial support. The support provided by Prof. Anthony G. Akins during the investigation is also greatly acknowledged.

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REFERENCES [1] Silva M. B., Skjoedt M., Atkins A. G., Bay N. and Martins P. A. F., Single point incremental forming & formability/failure diagrams, Journal of Strain Analysis for Engineering Design, 2008, 43, 15-36. [2] Leszak E., Apparatus and process for incremental dieless forming, Patent US3342051A, 1967. [3] Kitazawa K., Wakabayashi A., Murata K. and Yaejima K., Metal-flow phenomena in computerized numerically controlled incremental stretch-expanding of aluminum sheets, Keikinzoku / Journal of Japan Institute of Light Metals, 1996, 46, 65-70. [4] Jeswiet J., Incremental single point forming, Trans. of North American Manufacturing Research Institute, 2001, 29, 75-79. [5] Filice L., Fratini L. and Micari F., Analysis of material formability in incremental forming, Annals of CIRP, 2002, 51, 199-202. [6] Jeswiet J., Micari F., Hirt G., Bramley A., Duflou J. and Allwood J., Asymmetric single point incremental forming of sheet metal, Annals of CIRP, 2005, 54, 623-650. [7] Kegg R. L., A new test method for determination of spinnability of metals, Trans. ASME Journal of Engineering for Industry, 1961, 83, 119-124. [8] Emmens W. C. and van den Boogaard A. H., Strain in shear and material behaviour in incremental forming, Key Engineering Materials, 2007, 344, 519-526. [9] Allwood J. M., Shouler D. R. and Tekkaya A. E., The increased forming limits of incremental sheet forming processes, Key Engineering Materials, 2007, 344, 621-628. [10] Jackson K. and Allwood, J. M., The mechanics of incremental sheet forming, Journal of Materials Processing Technology, 2009, 209, 1158-1174. [11] Eyckens P., He S., van Bael A., van Houtte P. and Duflou J., Forming Limit Predictions for the Serrated Strain Paths in Single Point Incremental Sheet Forming, In

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Cesar de Sa J. M. A. and Santos A. D. (Editors), NUMIFORM 2007, American Institute of Physics, Porto, Portugal, 2007, 141-146. [12] Cao J., Huang Y., Reddy N.V., Malhotra R. and Wang Y., Incremental sheet metal forming: advances and challenges. In Yang D. Y., Kim Y. H., Park C. H. (Editors), ICTP 2008 International Conference on Technology of Plasticity, Gyeongju, Korea, 2008, 751752. [13] Emmens W. C. and van den Boogaard A. H., Incremental forming studied by tensile tests with bending. In Yang D. Y., Kim Y. H., Park C. H. (Editors), ICTP 2008 International Conference on Technology of Plasticity, Gyeongju, Korea, 2008, 245-246. [14] Skjoedt M., Silva M. B., Martins P. A. F. and Bay N., Strain paths and fracture in multi stage single point incremental forming. In Yang D. Y., Kim Y. H., Park C. H. (Editors), ICTP 2008 International Conference on Technology of Plasticity, Gyeongju, Korea, 2008, 239-244. [15] Skjoedt M., Bay N., Endelt B., and Ingarao G., Multi stage strategies for single point incremental forming of a cup. In Boisse P. et al. (Editors), ESAFORM 2008, International Conference on Material Forming, Lyon, France, 2008. [16] Ayada M., Higashino T. and Mori K., Central bursting in extrusion of inhomogeneous materials. In Kudo H. (Editor), ICTP 1984 International Conference on Technology of Plasticity, Tokyo, Japan, 1984, 553-558. [17] Atkins A. G., Fracture mechanics and metal forming: damage mechanics and the local approach of yesterday and today, In Rossmanith H. P. (Editor), Fracture Research in Retrospect, A. A. Balkema, Rotterdam, 1997, 327-350. [18] Embury J. D. and Duncan J. L., Formability Maps, Annual Review of Materials Science, 1981, 11, 505-521. [19] Ambrogio G., Costantino I., De Napoli L., Filice L., Fratini L. and Muzzupappa M., Influence of some relevant process parameters on the dimensional accuracy in

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incremental forming: a numerical and experimental investigation, Journal of Materials Processing Technology, 2004, 153-154, 501-507. [20] Rossard C., Mise en forme des métaux et alliages, CNRS, Paris, 1976. [21] Atkins A. G., Fracture in forming, Journal of Materials Processing Technology, 1996, 56, 609-618. [22] Skjoedt M., Hancock M.H. and Bay N., Creating helical tool paths for single point incremental forming, Key Engineering Materials, 2007, 344, 583-590.

26

Figure 1

Figure 1 – Schematic representation of a cross section view of the rotational symmetric single point incremental forming (SPIF) process. The tool rotates while performing a round (or helical) path. Note: Inset shows a picture of the experimental set-up.

Figure 2

Φ θ

Figure 2 – Schematic representation of the volume element resulting from the local contact between the tool and sheet placed immediately ahead of tool and identification of the basic modes of deformation of single point incremental forming: (A) Flat surface under plane strain stretching conditions, (B) Rotational symmetric surface under plane strain stretching conditions, (C) Corner under bi-axial stretching conditions. Note: Insets show images of typical cracks occurring in deformation modes (B) and (C).

Figure 3

Figure 3 – Schematic representation of the stress field in the radial slice of the component being formed that contains the local shell element utilized for the membrane analysis of SPIF. Note: The distribution of stress shown above is valid only for plane-strain conditions because in case of equal bi-axial stretching in the small plastic zone BC located at the corner of the components.

Figure 4

2.00 Tensile Test

FFL (SPIF) 1.80

Elliptical Bulge Test Circular Bulge Test

1.60 1.40

FFL (experimental)

SPIF Cone

1 mm

SPIF Pyramid

Major True Strain

2 mm

1.20

SPIF Cone (max. depth)

1.00 1 mm 2 mm

0.80 0.60

FFL (constant thickness strain)

Kink in the strain path

0.40 0.20 0.00 -0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

Minor True Strain Figure 4 – Fracture forming limit diagram containing the forming limit curve (FLC), the fracture limit line (FFL) and the fracture points obtained from conical and pyramidal SPIF parts.

Figure 5

60° 65° 70° 75° 80° 85°

65°

60° 60° 65° 65° 70° 70° 75° 75° 80° 80°

Figure 5 – Geometry of the truncated conical and pyramidal shapes that were utilized in the experiments.

Figure 6

(a) 100

1.2

80

Stress / Yield Stress (%)

1.0

Major Strain

0.8

0.6

0.4

0.2

60

40

20

0 0.53

0.66

0.72

0.86

-20 0.0 -0.6

-0.4

-0.2

0.0

0.2

Minor Strain

(b)

0.4

0.6

-40

Stress 1 (FEM) Stress 2 (Theory)

Stress 1 (Theory) Stress 3 (FEM)

Stress 2 (FEM) Stress 3 (Theory)

Thickness (mm) (c)

Figure 6 – Single point incremental forming of a truncated conical shape (material: rigid perfectly plastic, initial thickness: 1mm). a) Finite element distribution of the normalized effective stress (%). b) Finite element distribution of major and minor true strains in the principal strain space. c) Comparison between finite element and theoretical estimates of the principal stresses for three different values of the final thickness of the cone corresponding to different locations in the plastic deformation zone.

Figure 7

(a) 100 1.2

80

Stress / Yield Stress (%)

1.0

Major Strain

0.8

0.6

0.4

0.2

60

40

20

0 0.65

0.76

0.85

-20 0.0 -0.6

-0.4

-0.2

0.0

0.2

Minor Strain

(b)

0.4

0.6

-40

Stress 1 (FEM) Stress 2 (Theory)

Stress 1 (Theory) Stress 3 (FEM)

Stress 2 (FEM) Stress 3 (Theory)

Thickness (mm) (c)

Figure 7 – Single point incremental forming of a pyramid shape (material: rigid perfectly plastic, initial thickness: 1mm, zone: side of the pyramid). a) Finite element distribution of the normalized effective stress (%). b) Finite element distribution of major and minor true strains in the principal strain space. c) Comparison between finite element and theoretical estimates of the principal stresses for three different values of the final thickness of the pyramid corresponding to different locations in the plastic deformation zone.

Figure 8

(a) 100 1.2

80

Stress / Yield Stress (%)

1.0

Major Strain

0.8

0.6

0.4

0.2

60

40

20

0 0.65

0.77

0.81

-20 0.0 -0.6

-0.4

-0.2

0.0

0.2

Minor Strain

(b)

0.4

0.6

Stress 1 (FEM) Stress 2 (Theory)

Stress 1 (Theory) Stress 3 (FEM)

Stress 2 (FEM) Stress 3 (Theory)

-40

Thickness (mm) (c)

Figure 8 – Single point incremental forming of a pyramid shape (material: rigid perfectly plastic, initial thickness: 1mm, zone: corner of the pyramid). a) Finite element distribution of the normalized effective stress (%). b) Finite element distribution of major and minor true strains in the principal strain space. c) Comparison between finite element and theoretical estimates of the principal stresses for three different values of the final thickness of the pyramid corner corresponding to different locations in the plastic deformation zone.

Figure 9

100

Stress / Yield Stress (%)

80

60

40

20

0 6.5

15

23

-20

Stress 1 (FEM)

Stress 2 (FEM)

-40

Distance to the plastic zone (mm)

Figure 9 – Finite element estimates of the meridional

and circumferential

stresses in three

different locations placed along the inclined wall surface of a truncated conical shape.

Table 1

Assumption

SPIF (flat and rotational symmetric surfaces)

State of strain

d   d t  0 plane strain conditions

SPIF (corners)

equal bi-axial stretching

SPIF (inclined walls)

uniaxial tension

d   0 d t  0

d   d   0 d t  0

d   0 d   d t  0

State of stress

Y

   1 

Hydrostatic stress

 1  t rtool      2  21  1  3   t   3   Y       1 

t

 rtool  t 

 D   c

m 

 rtool

rtool

  t  0



0

t 0  2t 

rC  Y rD

 Y  rtool  t    2  rtool  t 

0

Y

1 2 t

 t   3  2  Y

0

m 

2Y 3

 rtool  t     rtool  2 t 

m 

 3

Table 1 - State of stress and strain in the small localized plastic zones and inclined walls of SPIF.