A Erman Tekkaya Werner Homberg Alexander Brosius Eds 60 Excellent Inventions in Metal Forming 60 Excellent Inventions in Metal Forming A Erman Tekkaya � Werner Homberg � Alexander Brosius Editors 60 Excellent Inventions in Metal Forming Editors A Erman Tekkaya Institut für Umformtechnik und Leichtbau Technische Universität Dortmund Dortmund, Germany Werner Homberg Lehrstuhl für Umformende und Spanende Fertigungstechnik Universität Paderborn Paderborn, Germany Alexander Brosius Institut für Ferti.
Modelling
Novel Method for Combined Tension and Shear Loading of Thin-Walled Tubes
Christopher P Dick and Yannis P Korkolis
Bulk deformation processes used to create thin sheets and tubes impart specific crystallographic orientations, resulting in plastic anisotropy that influences their plastic flow and failure limits Accurately establishing this plastic anisotropy and calibrating material models is essential for reliable numerical simulations of forming processes, directly impacting the correlation between simulations and experimental results Common methods for characterizing sheet material anisotropy include tension tests at various angles to the rolling direction, biaxial testing of cruciform specimens, hydraulic bulge tests, and various shear tests For tubes, the most effective experiments involve combined tension and torsion or combined tension and internal pressure loading, though these methods require advanced equipment and are complex to execute.
This paper presents two simple experiments for testing tubes using a universal testing machine: the Ring Hoop Tension Test (RHTT) and the Ring Plane-Strain Tension Test (RPST) In both tests, a ring is cut from the parent tube, and a test-section is machined onto it The specimen is then positioned over two lubricated, closely fitting D-shaped mandrels, which are separated in the universal testing machine Care is taken to ensure that the test-section remains on one mandrel during the experiment, allowing it to maintain its curvature and experience only stretching.
University of New Hampshire, Durham, USA e-mail: Yannis.Korkolis@unh.edu
A E Tekkaya et al (eds.), 60 Excellent Inventions in Metal Forming,
The RHTT specimen, depicted in Fig 1, utilizes a combination of oil and Teflon tape to minimize friction with the mandrels, while a random speckle pattern is applied for Digital Image Correlation (DIC) analysis A key concern is the impact of mandrel contact and friction on stress distribution, potentially deviating from ideal uniaxial conditions To address this, finite element analysis (FEA) was employed, modeling the material as a finitely-deforming, rate-independent elastoplastic solid based on J 2 flow theory The simulations incorporated the hardening curve of Al-6061-T4, revealing that hoop strain results, illustrated in Fig 2, indicate uniform hoop stress within the test section and elastic behavior throughout the specimen Notably, the regions between the mandrels experience both stretching and bending, leading to a through-thickness stress gradient Detailed analysis confirmed that contact pressure in the test-section area is nearly uniform, with induced stresses remaining negligible when specimen-mandrel friction is maintained below 0.15–0.2 Furthermore, the study found that contact pressure in a well-lubricated RHTT experiment closely resembles the effects of internal pressure in an inflated tube.
In RHTT experiments conducted on extruded Al-6061-T4 tubes with an outer diameter of 60 mm and a thickness of 3 mm, the hoop stress-strain response was observed to differ from the axial response After considering factors such as tube eccentricity, specimen-mandrel friction, and specimen preparation, the remaining discrepancy was attributed to material anisotropy The Digital Image Correlation (DIC) technique was utilized to analyze local strain fields during the RHTT process, allowing for the accurate determination of the R-values, also known as Lankford coefficients.
Fig 1 The Ring Hoop Tension
Novel Method for Combined Tension and Shear Loading of Thin-Walled Tubes 5
Fig 2 Finite element predic- tion of the hoop stress in the
The RHTT experiment demonstrates that the R-values of the material evolve with plastic deformation, with the hoop direction R-value exceeding 2 Additionally, this experiment can effectively assess the response of welds in both electrical-resistance-welded (ERW) and extruded porthole-die tubes Unlike tube inflation tests, the RHTT facilitates direct uniaxial tension loading of the weld transversely, providing valuable insights into weld performance.
Fig 3 Nominal stress-strain responses in the axial and hoop direction of the tube [6]
Fig 4 Evolution of the R-values during deformation, showing axial-hoop anisotropy [6]
3 Ring Plane-Strain Tension Test
The RHTT experiment is utilized to assess the hoop stress-strain response of a tube, while the RPST experiment enables the exploration of the plastic response along various loading paths In the RPST setup, a wide and short test-section is created on a ring and tested using the same equipment as the RHTT experiment The elastic wide shoulders above and below the test-section are stiffer than the plastically deforming section, thereby maintaining tension-under-plane-strain conditions By rotating the orientation of the test-section relative to the tube generatrix, different loading paths can be achieved, allowing for a combination of tension and shear forces on the test-section.
Fig 5 Ring Plane-Strain Tension specimens with inclined test-sections at a 15 o , b 30 o and c 45 o to the tube axis [8]
Novel Method for Combined Tension and Shear Loading of Thin-Walled Tubes 7
The evolution of logarithmic strain fields along the test section indicates that plane-strain conditions dominate the central portion during the majority of the experiment This is illustrated by the extensional and shear strains observed throughout the testing process.
The Digital Image Correlation (DIC) technique confirmed the presence of plane-strain conditions during testing, particularly in the central portion of a specimen with a 45-degree inclined test-section The findings indicate that strain parallel to the test-section increases towards the edges, suggesting a shift from plane-strain to uniaxial tension Notably, the shear strain remains consistent throughout the test-section.
The non-uniform strain distribution in the RPST experiment complicates result analysis by affecting the net force carried by the test section, which cannot be determined by halving the load-cell reading as in the RHTT experiment Additionally, the stresses in the test section cannot be calculated simply as "force/area." These challenges were effectively addressed through Finite Element Analysis (FEA) of the RPST specimens.
The results enable the plotting of constant plastic work contours, illustrated in Figures 7 and 8 Figure 7 presents the plastic work contour in the axial-hoop plane-stress space at fixed shear stress levels, incorporating the Yld2000-2D yield function by Barlat et al [9], which accurately reflects the experimental data Figure 8 depicts the evolution of plastic work contours in the -plane, featuring both the von Mises and Yld2000-2D yield functions While the von Mises function fails to account for anisotropy and its changes during plastic deformation, the Yld2000-2D function is effectively calibrated to align closely with the experimental results.
Fig 7 Plane-stress yield locus of Al-6061-T4 tube, includ- ing the RPST experiments and the Yld2000-2D material model [8]
Fig 8 Yield locus of the Al-
6061-T4 tube on the -plane, including the RPST experi- ments and the Yld2000-2D material model [8]
The RHTT and RPST experiments are effective methods for investigating the hoop response and yield locus of tube materials, utilizing basic testing equipment The RHTT experiment specifically evaluates the performance of weld seams in ERW or porthole-die-extruded tubes, while the RPST experiment generates valuable data under both tension and shear When combined with tube inflation experiments, these methods offer a comprehensive understanding of the anisotropic properties of tube materials.
Novel Method for Combined Tension and Shear Loading of Thin-Walled Tubes 9
This work was supported by the U.S National Science Foundation under the GOALI Grant CMMI 1031169 The extruded Al-6061-T4 tubes were provided by Dr Cedric Xia of Ford Motor Company.
1 Kuwabara, T., Hashimoto, K., Iizuka, E., and Yoon, J.W., 2011, Effect of anisotropic yield func- tions on the accuracy of hole expansion simulations J Mat’s Processing Tech., 211, 475–481.
2 Kuwabara, T.: Biaxial Stress Testing Methods for Sheet Metals In Comprehensive Materials Processing; Van Tyne, C J., Ed.; Elsevier Ltd., 2014; Vol 1, pp 95–111.
3 Korkolis, Y.P and Kyriakides, S., 2008, Inflation and burst of anisotropic aluminum tubes for hydroforming applications Int’l J Plasticity 24/3, 509–543.
4 Lee, M.G., Korkolis, Y.P., and Kim, J.H., 2014, Recent developments in hydroforming technol- ogy Proc Inst Mech Eng., Part B: J Eng Manuf., 0954405414548463.
5 Arsene, S and Bai, J., 1996 A New Approach to Measuring Transverse Properties of Structural Tubing by a Ring Test, J Test & Eval 24, 386–391.
6 Dick, C.P and Korkolis, Y.P., 2014 Mechanics and full-field deformation study of the ring hoop tension test, Int’l J Solids & Struct, 51, 3042–3057
7 Dick, C.P and Korkolis, Y.P., 2015, Strength and ductility evaluation of cold-welded seams in aluminum tubes extruded through porthole dies, Materials & Design, 67, 631–636
8 Dick, C.P and Korkolis, Y.P Anisotropy of thin-walled tubes by a new method of combined tension and shear loading, (submitted)
9 Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., and Chu, E., 2003, Plane Stress Yield Function for Aluminum Alloy Sheets-Part I: Theory,Int’l J Plasticity, 19, 1297–1319.
An Innovative Procedure for the Experimental Determination of the Forming Limit Curves
Dorel Banabic, Lucian Lazarescu, and Dan-Sorin Comsa
The Forming Limit Curve (FLC) is a crucial tool for quantitatively assessing the formability of sheet metal Numerous methodologies exist for experimentally determining FLCs, which should encompass the full range of deformation relevant to sheet metal forming processes Typically, the strain combinations considered range from uniaxial to biaxial surface loads The subsequent section outlines the experimental methods frequently employed to explore the deformation domain of FLCs.
The uniaxial tension of flat specimens with circular notches enables the exploration of the tension-compression range, as proposed by Brozzo and de Lucca Utilizing wider specimens can achieve the plane strain point, while the positive-positive region of the forming limit curve (FLC) can be replicated using a hydraulic bulging device with circular or elliptical dies By adjusting the eccentricity of the elliptical aperture, various load paths within the tension-tension domain are produced Alternative methods for determining FLCs include punch stretching, as demonstrated by Keeler, who utilized circular specimens and spherical punches of varying radii to alter load paths, although his approach primarily examined the right end of the tension-tension FLC branch Hecker expanded on Keeler's work by enhancing lubrication between the punch and specimen, allowing for a more comprehensive analysis of the tension-tension domain Nakazima further advanced this experimental technique by employing a hemispherical punch with a constant radius alongside rectangular specimens of differing widths.
Dorel Banabic Lucian Lazarescu Dan-Sorin Comsa
Technical University of Cluj-Napoca, Cluj Napoca, Romania e-mail: banabic@tcm.utcluj.ro
A E Tekkaya et al (eds.), 60 Excellent Inventions in Metal Forming,
D Banabic et al investigate both tension-compression and tension-tension domains of the forming limit curve (FLC) Hasek addressed the issue of wrinkling in wide specimens by utilizing circular specimens with lateral notches, improving upon the Nakazima test To minimize friction effects in flat punch drawing tests, Marciniak introduced the double blank method, positioning a specimen atop a carrier blank, and varied the punch cross-section to achieve different load paths Grosnostajski further enhanced Marciniak’s approach by altering the geometries of the specimen and carrier blank However, it is noteworthy that these methods do not fully capture the entire strain domain of the FLC.
An innovative method has been introduced for the experimental determination of the full deformation range of forming limit curves (FLCs) This procedure utilizes the hydraulic bulging technique applied to a double specimen.
The formability test proposed by the authors is based on the hydraulic bulging principle.
Sheet Metal Forming
Frédéric Barlat and Hyuk Jong Bong
Advancements in modern technology have made parameter optimization in forming processes increasingly complex, especially for products requiring high precision, such as bipolar plates made from ultra-thin ferritic stainless steel sheets using servo-press technology The diverse and intricate slide motions necessary for producing such thin materials complicate process optimization, making traditional trial-and-error methods impractical Consequently, finite element (FE) analyses are employed to identify optimal conditions for sheet metal forming, while experimental adjustments complement the model to enhance understanding and narrow the parameter range This article focuses on the plastic behavior aspect, particularly anisotropy, highlighting the need for a multi-scale constitutive description tailored to material microstructure, despite the impractical computation times this may entail Therefore, continuum constitutive models remain a powerful tool in industrial applications.
Frédéric Barlat Hyuk Jong Bong
Pohang University of Science and Technology, Pohang, Korea e-mail: f.barlat@postech.ac.kr
A E Tekkaya et al (eds.), 60 Excellent Inventions in Metal Forming,
In multiaxial loading, stress is represented by a tensor or its deviators for pressure-independent plastic behavior, leading to the yield condition expressed as ˆD N¢ s/¢ r N©/D0 The yield function, ¢ s/N, outlines the yield surface's shape, while N© signifies the effective strain, determining its size via the reference flow curve ¢ r This yield function is a homogeneous first-degree function concerning stress components, simplifying the computation of the loading state The equation encompasses two key elements of plasticity theory: the yield condition and strain hardening Additionally, the flow rule posits that the plastic strain increment arises from the yield function, with the associated flow rule mathematically represented as d©pq DdN© @¢N.
Theoretical insights into crystal plasticity suggest that the yield function is convex and that the associated flow rule is crucial for metals The effective strain serves as a monotonically increasing state variable, representing the accumulated plasticity within the material In isotropic hardening, the yield surface maintains its shape during plastic deformation, with only its size changing Consequently, a consistent effective strain increment is derived from the equivalence of plastic work.
Isotropic materials exhibit plastic behavior that is direction-independent, relying solely on stress tensor invariants, including the three principal stresses, which facilitate the verification of yield function convexity Classical yield conditions for isotropic materials include the Tresca and von Mises criteria, although any yield function expressed in terms of invariants, such as Hershey's, is also valid.
The yield function described by N ¢ D2 1=a fjs 1 s 2 j a C js2s 3 j a C js3s 1 j a g 1=a D¢ r (7.3) simplifies to the von Mises criterion for values of a equal to 2 or 4, and to the Tresca criterion for a equal to 1 or in the limit as a approaches infinity This convex yield function serves as an effective approximation for self-consistent crystal plasticity calculations Research by Hosford demonstrated that using exponents of 6 and 8 yields an almost perfect match for the yield surfaces of isotropic BCC and FCC materials, respectively, as determined through crystal plasticity analysis.
In the context of plastic anisotropy, it is essential to select an appropriate material reference frame for expressing plasticity equations, while also ensuring that stress invariants are consistently modified Among various theories of plastic anisotropy, the linear transformation approach of the stress tensor is highlighted, enabling the extension of any isotropic yield function, defined by principal stresses as invariants, to anisotropic conditions without losing convexity and homogeneity For incompressible materials, a linear transformation is applied to the stress deviator, resulting in the transformed tensor components.
C is a fourth-order tensor that incorporates anisotropy coefficients and represents the macroscopic symmetries of materials This theory allows for the generalization of an isotropic yield function ¢N to anisotropic conditions by substituting the principal deviatoric stress components with the principal values sQ1, sQ2, and sQ3 of the transformed stress deviators A comprehensive extension of Eq 7.3 for orthotropic materials using linear transformations has been proposed It is feasible to implement multiple linear transformations as long as the yield function remains isotropic concerning all principal values of the transformed tensors The principal values sQ0 and sQ00 are computed similarly to the single transformation case In the context of plane stress, the anisotropic yield function Yld2000-2d was introduced by Barlat et al., represented by a specific equation that incorporates the transformation tensor.
The formulation includes eight independent anisotropy coefficients, all set to 1 in the isotropic case, which leads to the recovery of Hershey’s yield condition It is important to note that Yld2000-2d appears in various forms in the literature Additionally, an earlier yield function known as Yld89 was proposed in 1989.
2 ˙ vu ut ¢ xx Nh¢ yy
Fig 1 Normalized yield loci calculated with von Mises and
Yld2000-2d yield functions for ferritic stainless steel sheet sample turns out to be a particular case of Yld2000-2d when the following relationships are used
2hN1.Nc/ 1=a ; ’3 D’4D Nh.Na/ 1=a ; ’5D’6D.Na/ 1=a ;
(7.9) For a general stress state, Eq.7.3was extended as a yield function called Yld2004- 18p [7] in which, out of the 18 available coefficients, only 16 are independent [8] For
Fig 2 FE simulated thickness distribution using different yield functions (I isotropic; A anisotropic) in hemi-spherical punch stretching
In the realm of anisotropic yield functions, the authors primarily reference their own research for brevity; however, numerous other significant contributions are thoroughly discussed in the literature, particularly in Reference [9].
The two-dimensional yield locus for a ferritic stainless steel sheet sample is depicted in Fig 1, showcasing the constitutive relationships utilized in finite element analyses of the Nakazima spherical punch-stretching test Fig 2 illustrates the thickness distribution of the specimen in relation to its distance from the pole, highlighting the significant impact of plastic anisotropy and the yield function exponent on the predicted thickness distribution, which is influenced by the material's crystal structure.
HCP materials demonstrate plastic behavior that is independent of mean stress, yet exhibit differing yield stresses under tension and compression due to the strength-differential effect, which relates to their deformation mechanisms of slip and twinning Cazacu et al proposed an isotropic yield criterion for these materials, defined by a function of principal deviatoric stresses, where the coefficient and a parameter control the tension-compression asymmetry This yield function is characterized as convex and homogeneous, and it has been extended to account for plastic anisotropy through a linear transformation approach.
During plastic deformation, materials often respond differently than expected when the load becomes non-proportional, deviating from isotropic hardening behavior A prominent example of this is the Bauschinger effect, which arises under reverse loading conditions Traditionally, this phenomenon has been modeled using kinematic hardening, which involves translating the yield surface in stress space However, recent research has proposed an alternative approach that captures the Bauschinger effect and other anisotropic hardening behaviors through a distortion of the yield surface, as defined by a specific yield condition.
The microstructure deviator characterizes the deformation history of materials, represented by an isotropic or anisotropic homogeneous yield function of the first degree This constitutive equation incorporates several state variables, such as f1, f2, and h, while maintaining the core principles of plasticity theory for anisotropic materials, ensuring the homogeneity and convexity of the yield function.
This summary outlines the principles behind a plasticity theory for anisotropic metals, integrating lower scale crystal plasticity simulations into the derivation of constitutive equations The approach incorporates microstructural information in an approximate way, resulting in anisotropic yield functions that are user-friendly for optimizing industrial sheet metal forming processes.
Parts of this work were supported by POSCO to whom the authors are very grateful.
1 Hershey, A.V., 1954 The plasticity of an isotropic aggregate of anisotropic face-centered cubic crystals, ASME J Appl Mech 21, 241–249.
2 Hosford, W.F., 1972 A generalized isotropic yield criterion ASME J Appl Mech Trans 39, 607–609.
3 Barlat, F., Lege, D.J., Brem, J.C., 1991 A six-component yield function for anisotropic materi- als Int J Plasticity 7, 693–712.
4 Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003 Plane stress yield function for aluminum alloy sheets–Part I: Theory Int.
5 Barlat, F., Yoon, J.W., Cazacu, O., 2007 On linear transformations of stress tensors for the description of plastic anisotropy Int J Plasticity 23, 876–896.
6 Barlat, F., Lian, J., 1989 Plastic behavior and stretchability of sheet metals Part I: A yield function for orthotropic sheets under plane stress conditions Int J Plasticity 5, 51–66.
7 Barlat, F., Aretz H., Yoon, J.W., Karabin, M.E., Brem J.C., Dick R.E., 2005 Linear transformation-based anisotropic yield functions Int J Plasticity 21, 1009–1039.
8 Van den Boogaard, A.H., Havinga, J., Belin, A., Barlat, F., 2015 Parameter reduction for the Yld2004-18p yield criterion Int J Material Forming, in press, doi:10.1007/s12289-015- 1221-3.
9 Banabic, D., Barlat, F., Cazacu, O., Kuwabara, T., 2010 Advances in Anisotropy and Formabil- ity Int J Material Forming 3, 165–189.
10 Cazacu, O., Plunkett, B., Barlat, F., 2006 Orthotropic yield criterion for hexagonal close packed metals Int J Plasticity 22, 1171–1194.
11 Barlat, F., Vincze, G., Grácio, J.J., Lee, M.G., Rauch, E.F., Tomé, C., 2014 Enhancements of homogenous anisotropic hardening model and application to mild and dual-phase steels Int J.Plasticity 58, 201–218.
BBC2005 Yield Criterion Used in the Numerical Simulation of Sheet Metal Forming Processes
Dorel Banabic and Dan-Sorin Comsa
Over the past four decades, researchers have focused on developing more accurate yield criteria, which rely heavily on the flexibility of their mathematical formulations This flexibility is typically enhanced by incorporating a greater number of material parameters, necessitating extensive experimental data for identification Testing has evolved from uniaxial tension to more complex scenarios like biaxial tension and pure shearing However, yield criteria with numerous parameters can become mathematically intricate, posing challenges for computational efficiency In the case of highly-anisotropic sheet metals, complex models are essential to ensure accurate yield surface descriptions A balanced approach is achieved with models utilizing seven or eight experimental values, as seen in the work of Barlat et al., Cazacu and Barlat, and Vegter The BBC family of yield criteria also falls into this category, offering superior computational efficiency in simulating sheet metal forming processes without relying on linear stress tensor transformations Currently, BBC 2005 is one of the most widely adopted yield criteria in industrial applications, having been integrated as a standard material model in the FE code AutoForm 4.1.1.
Dorel Banabic Dan-Sorin Comsa
Technical University of Cluj-Napoca, Cluj Napoca, Romania e-mail: banabic@tcm.utcluj.ro
A E Tekkaya et al (eds.), 60 Excellent Inventions in Metal Forming,
2 Formulation of the BBC 2005 Yield Criterion
The BBC 2005 model assumes that metallic sheets behave as plastically orthotropic mem- branes under plane-stress conditions This hypothesis allows describing the yield surface by means of the equation
11 ; 22 ; 12 D 21 /DY; (8.1) in whichN is the BBC 2005 equivalent stress (see below),Yis a yield parameter, and˛ˇ